NONLINEAR EFFECTS
IN THE
BETWEEN WAVES AND
INTERACTION
TOPOGRAPHY
by
RICHARD CRAIG
B.S.,
DEININGER
State University of New
(1978)
York at Albany
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSPHY
at the
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
December
© Massachusetts
1982
Institute of Technology
1982
Signature of Author
Department of Meteorology and Physical
Oceanography
Certified by
Thesis
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983
NONLINEAR EFFECTS
IN THE INTERACTION
BETWEEN WAVES AND TOPOGRAPHY
by
RICHARD CRAIG DEININGER
Submitted to the Department of Meteorology and Physical
Oceanography, Massachusetts Institute of Technology, on
December 3, 1981, in partial fulfillment of the requirements
for the degree of Doctor of Philosophy.
ABSTRACT
This thesis consists of three separate problems unified
by two basic features.
First, in all the problems a
nonlinear effect either in the form of a wave-wave
interaction or a wave-zonal flow interaction plays a key
role.
Second, all the problems involve the instability of a
topographically forced wave for which resonance plays a key
role.
The method of multiple time scales is used to analyze
each problem which exploits the assumptions of weak
nonlinearity in each case as well as the assumption that the
flow is nearly neutral or near resonance or both depending
on the problem under consideration.
In Chapter 2 the weakly nonlinear effects on the
instability of a basic state consisting of a topographically
forced wave in an inviscid, barotropic beta-plane model are
studied.
The results obtained differ substantially from
those obtained when the basic state is a free Rossby wave.
Here the basic-state wave is fixed in phase with respect to
the mountain, while the amplitude of the topographic wave
and perturbation evolve.
The nonlinear feedback between the
topographic wave and perturbation gives rise to an
oscillation for a topographically subresonant zonal flow and
an explosive nonlinear instability for a topographically
superresonant zonal flow.
In the subresonant case, the
effect of the perturbation on the forced wave is a
dissipative one, when averaged over the course of the
nonlinear oscillation. The standing topographic wave
interacts with the traveling instability on the topographic
wave through the convergence of Reynolds' stresses which is
suggestive of the way in which standing and traveling eddies
interact in the atmosphere.
In Chapter 3, under the assumption of quasi-resonant
flow, the topographic instability of a normal mode of the
Charney problem is examined and found to draw its energy
from both the available potential energy of the mean flow
and from the zonal momentum via form drag.
No apriori
truncation is necessary.
The instability occurs for either
superresonant or subresonant flow depending on the zonal and
meridional scale of the topography.
In addition it is
limited to narrow bands in meridional wavenumber with
relatively little dependence on zonal wavenumber.
It is
suggested that this may explain the more well defined
meridional structure of blocking.
In Chapter 4, the evolution equations are obtained for
the instability of a small but finite-amplitude
topographically forced nearly neutral Charney mode embedded
in a mean shear flow.
The topography is found to have a
stabilizing effect on the baroclinic instability, but a new
topographically induced stationary instability arises even
for baroclinically subcritical flow.
It is most easily
excited in superresonant flow near resonance for narrow
bands in meridional wavenumber.
This topographic
instability of the Charney mode draws its energy from the
available potential energy of the mean flow and survives
even in the presence of 0(1) Ekman friction if the Ekman
friction is not too strong.
Thus, it is suggested that the
topographic instability of the Charney mode may be just as
dynamically important as the baroclinic instability of the
Charney mode.
In addition, it is suggested that the normal
modes of the Charney model, due to their topographic
instability, are just as dynamically important as the
baroclinically unstable modes of the Charney model in the
presence of topography.
Thesis
Title:
Supervisor:
Dr. G. R. Flierl
Associate Professor of Oceanography
Massachusetts Institute of Technology
AKNOWLEDGEMENTS
First,
thesis
I would
advisor,
like to to express my gratitude to my
Glenn
Flierl.
His
scientific
insight,
open
accessibility and patience has been a great help to me
throughout my graduate studies.
During
benefit
the early part of my graduate studies,
of many
Jule Charney.
discussions
with my advisor
Through these discussions,
I had the
at that
time,
I developed an
interest in topographically forced waves about which this
thesis
is concerned.
inspiration
My memory of him is a great
to me.
I have also greatly benefited from discussions with
Joe Pedlosky,
Ray
Pierrehumbert
and Peter
Stone.
These
people have helped me see through some particularly
troubling scientific problems.
appreciated.
Their help
is much
Thanks are also due Poala Malanotte-Rizzoli
and K. K. Tung who served on my thesis committee and made
helpful comments.
I would also like to take this opportunity to thank
Arthur Loesch who introduced me to and sparked my interest
in
geophysical
scientific
fluid dynamics.
discussions
My stay
at M.I.T.
friendship of my
I continue
with him and
his
has been made
fellow students.
questions.
friendship.
enjoyable
Their
greatly appreciated as well as their
to enjoy
by the
friendship is
help with many
4
Most
for
importantly
I would
like
to thank
Elizabeth
Cary
her encouragement and support when it was genuinely
needed.
I'd like to thank
Isabel Kole for her careful and
speedy preparation of the figures and Francis Doughty and
Rhea
Epstein
Finally,
for
their
careful
the support
in
typing
the
of
the
form of a
thesis.
UCAR Fellowship
and NSF Grant ATM-76-20070 is greatly appreciated.
5
TABLE OF CONTENTS
Page
ABSTRACT
ACKNOWLEDGEMENTS
TABLE OF CONTENTS
CHAPTER 1
Overall introduction
CHAPTER 2
Topographically forced wave
instability at finite amplitude
Section 1
Introduction
Section 2
The model
Section 3
Formulation of the nonlinear
analysis
Section 4
The nonlinear analysis
Section 5
Concluding remarks
APPENDIX A:
Results for the Three-Term Truncation
APPENDIX B:
Forced Solution Coefficients
CHAPTER 3
Topographic instability of the normal
modes in the Charney problem
Section 1
Introduction
40
Section 2
The model and basic equations
44
Section 3
Brief review of the Charney model
47
Section 4
Derivation of the nonlinear
evolution equations for the
topographic instability of the
normal modes
52
Section 5
Topographic instability and its
energetics
Page
Section 6
Multiple equilibria
Section 7
Conclusions
CHAPTER 4
95
The Charney mode under the influence of
topography
Section 1
Introduction
Section 2
Effect of topography on an unforced
growing Charney mode
100
Instability of a topographically
forced Charney mode with very weak
zonal flow at the ground
120
Influence of topography on the
baroclinic instability of the
Charney mode and its topographic
instability
134
The topographically forced Charney
mode and the influence of 0(1)
Ekman friction
157
Conclusions
191
Section 3
Section 4
Section 5
Section 6
CHAPTER 5
REFERENCES
0'verall
Conclusions
97
196
208
Chapter
1
Introduction
Historically speaking,
most stability problems with
atmospheric applications have
states
treated zonally uniform basic
which, in general, were allowed to vary in both the
vertical and meridional directions. The
assumption of
zonality in these unforced and usually inviscid stability
problems was considered
justified as a rough approximation
to the quasi-steady Hadley circulation and by its
explaining
various aspects of the observed global
circulation.
been
success in
In particular,
such stability problems have
successful in explaining the existence of
the
midlatitude cyclones by the structure of the most unstable
modes arising
from the assumed basic state.
Of course as these wave disturbances grow the
assumption of linearity breaks down which
forces one to
consider the effects of nonlinearity on the evolution of
disturbance.
The
first systematic
given in a paper by Pedlosky
study of this
sort was
(1970) in which a small but
finite-amplitude disturbance was shown
the
the
to be able
to modify
linear instability such that the disturbance grew
exponentially only in its
nascent stage.
In particular he
demonstrated that as the baroclinic disturbance grows,
decreases the vertical shear, on which it depends for
growth.
its
This slows the growth of the disturbance and
eventually causes
decays
it
it to decay.
Then as the disturbance
the zonal available potential energy is replenished
8
until
the disturbance again becomes unstable.
In this
manner an oscillation ensues due to the nonlinear feedback
between
the
shear of
finite-amplitude
the zonal
disturbance
and
the vertical
flow.
However, having established
that the baroclinic
disturbance can grow to finite amplitude, one is
a
further question.
What are
faced with
the stability properties of
the resulting wave disturbances?
This question forces the
consideration of more general basic states.
As a prototype model of wave instability in the
atmosphere Lorenz
(1972) considered
stationary one-dimensional
the instability of a
Rossby wave.
Rossby wave is indeed unstable and
He found that the
suggested that
such
instability explained a portion of the unpredictability of
the atmosphere.
Gill
(1974) considered a basic state Rossby
wave which varied both cross-stream and downstream and
showed
is
that,
for small basic wave amplitude, the instability
confined to a small portion of wavenumber space, which
corresponds to the locus of wavenumbers satisfying the
kinematic resonance conditions
conditions) for
three waves.
(or near resonance
For
small amplitudes,
instability is thus limited in wavenumber space, while
instability, for larger basic wave amplitudes,
Lorenz
(1972),
is much less limited
studied by
in wavenumber
space.
Gill refers to the instability as resonant instability for
small basic wave amplitude and Rayleigh instability for
large basic wave amplitude.
The weakly nonlinear behavior
of a resonant triad of waves has been examined by
Longuet-Higgins and Gill
(1967).
In
that work they
described the nonlinear oscillation between the three
resonant waves which ensues after the
linear
instability.
Deininger,
1982)
have
the weakly nonlinear evolution of the Rayleigh
considered
type wave
Recently I (i.e.
initial stage of the
instability.
The result obtained was analogous to
that obtained by Pedlosky
fundamental
(1970) but differed in a
way due to the wavy basic state
with as opposed
I was dealing
to the vertically sheared zonal
Pedlosky's model.
flow in
Namely, there was a feedback between the
growing perturbation and Rossby wave amplitude analogous to
the feedback between the growing baroclinic wave and zonal
shear in Pedlosky's case but,
in addition, there was a
feedback between the disturbance and Rossby wave phase. This
phase
feedback has no analog
Pedlosky
(1970) and was
in the case considered by
suggested as a possible mechanism
for the nonconstant speed of progression of midlatitude high
and low pressure systems.
Lin (1980a,b) has studied the stability of the more
realistic basic state consisting of a one-dimensional free
neutral baroclinc wave embedded in a vertically sheared
zonal
flow.
He demonstrates as one might expect that for
strong wave amplitude as compared
instability
to zonal shear the
is fundamentally wavelike, that is,
the
instability excites a spectrum of waves which is generally
not centered at
the scale of
the most unstable mode of the
10
vertically sheared zonal
For relatively weak
flow alone.
state wave amplitude the instability behaves like
basic
instability of
flow but
the vertically sheared zonal
slightly modified by the presence of the wave.
he considered a basic
as well,
the
However, had
state wave with meridional structure
he would have found that a locus of other
wavenumbers might also be excited corresponding to those
neutral modes which can form a resonant
unstable baroclinic mode.
triad with most
In fact Loesch
(1974) considered
the weakly nonlinear evolution of a weakly unstable
baroclinic wave which formed a resonant triad with two
neutral baroclinic waves.
He demonstrated
that as the
unstable baroclinic wave draws available potential energy
out of the mean flow it can be leaked
to the neutral
baroclinic modes via the resonant triad wave-wave
interaction.
Until now only free wave instabilities have been
discussed.
Another mechanism for distorting
the
zonal
uniformity of the Hadley circulation involves the diversion
of the zonal flow by topography.
The
linear stability of
such a topographically forced wave has received relatively
little
treatment until recently.
analyzed the linear
Charney
instability of a barotropic wave forced
by the diversion of a uniform zonal
topography.
and Flierl (1981)
In general
flow by sinusoidal
the instability of the
topographically forced wave is due both to the shear of the
forced wave and
the way in which the disturbance
interacts
11ii
Thus,
with the topography.
the instability of the
forced
wave is shown to be unstable to Rayleigh or resonant
triad
In Chapter 2,
instabilities as is the free Rossby wave.
I
examine the weakly nonlinear evolution of a topographically
forced
it
wave and
the traveling
in an effort to understand
instability which develops on
in a simple internally
consistent manner, the interaction of their atmospheric
relatives.
The basic
technique employed
in this regard is
the method of multiple time scales and is the same as that
used in the weakly nonlinear analysis of the free Rossby
wave instability problem (i.e.,
Deininger, 1982).
stated, we proceed as follows:
The perturbation
nonlinearity is required to be weak.
That is to
Briefly
say, while
the interaction between the basic state wave and
perturbation is 0(1),
with
the interaction of the perturbation
itself and with corrections to
the basic
state wave is
required to be weak. Having made this assumption we choose
the small amplitude parameter such that
the weak
nonlinearity balances the slow growth of the slightly
unstable perturbation.
The slow growth is attained by
hinging the nonlinear stability analysis about the point of
neutral
stability.
However, the results of the nonlinear
forced wave problem in Chapter 2 differ markedly from those
of the free wave problem.
Therefore a comparison of the two
problems will be particularly interesting.
In the linear stability analysis of Charney and Flierl
(1981),
they also show that there is a limit for which the
12
triad type.
instability is not of the Rayleigh or resonant
In
this limit,
if the zonal
superresonant, the
flow is
instability becomes stationary and
to as a
is referred
instability.
form-drag instability or topographic
first examined in a paper by Charney and
Devore
It was
(1979) in
which they sought to explain the quasi-steady existence of
anomalously high pressure
(blocking).
in certain geographical regions
They suggested that
stationary topographic
the
instability is responsible for the
transition from the
normal relatively zonal state of the atmosphere
blocking situation as the zonal
The mechanism for the
as follows:
An
to the
flow became superresonant.
instability can be most
simply stated
increase in the topographically forced wave
amplitude increases the
form drag.
This increase in form
drag then decreases the zonal flow on which the forced wave
depends.
If the
flow is superresonant
flow increases the amplitude of the
turn
further decreases the
the decrease in zonal
forced wave which in
zonal flow via the
Clearly this cycle leads to instability.
removes momentum from the zonal
form drag.
The form drag
flow which increases the
forced wave amplitude.
The
study of Charney and Devore
(1979) was later
extended to a two-layer baroclinic model by Charney and
Strauss
(1980) with no zonal flow at
topographic
instability is
found but
the ground.
Again
in this case the
perturbation grows at the expense of the zonal available
potential energy.
The topography and
therefore the form
for this energy transfer.
drag merely act as the catalyst
Had there been nonzero flow at
have been a conversion
of
there would also
the ground
zonal momentum by the
In both the studies of Charney and
Charney and Strauss (1980) an ad hoc
Devore
form drag.
(1979) and
severely truncated
spectral representation of the motion is employed.
led
This has
subsequent authors to consider ways of isolating the
topographic
(1979) did
topographic
instability in a more systematic manner.
so in a barotropic
model by restricting the
features he considered
infinite meridional extent.
Hart
to those of
nearly
In this manner he neglected the
asymptotically small Reynolds stresses which were neglected
in an ad hoc manner in the truncated models.
taken in the forced wave instability
Flierl
(1981)
The limit
study of Charney and
for which the wave instability
reduced to
topographic instability, corresponds to the assumptions made
by Hart
(1979)
for the validity of his asymptotic
theory.
However, Hart's analysis remains unsatisfactory due to
unrealistic
the
assumption of topography with nearly infinite
meridional extent --
to begin with the very assumption which
allowed the simplified analysis.
An alternative approach has been taken by Pedlosky
(1981),
Plumb (1981a,b)
and Jacobs
(1979a,b)
within the
context of various models based on the analysis of the
forced Duffing equation (Stoker, 1950).
The asymptotic
technique employed by these authors exploits the importance
of the linear
resonance in the topographic
instability and
the quasi-linear nature of the
latter
topographic
instability.
The
point, based on the previous studies of topographic
instability, says that the important
sake of the topographic
instability
field topography and the zonal
flow.
interaction
for the
is between the wave
As a result wave-wave
interactions can be neglected, although this should be done
in a systematic manner.
To accomplish this,
the flow was
restricted to near linear resonance and weak nonlinearity.
Both Pedlosky (1981) and Jacobs
the barotropic model.
topographic
(1979a) worked within
Pedlosky pointed out
that
instability may also occur for subresonant
flow if the zonal wavelength is long enough.
extended
this
study to a two-layer model
at the ground as was the case
Strauss (1980).
however he did not consider a real
boundary but rather he imposed a moving
flow and wave
field
at
zonal
flow
in the work of Charney and
In an independent study, Plumb
boundary which eliminated
zonal
He then
with non
employed a similar approach in the context of
model,
the
(1981a)
the
two-layer
topographic lower
forcing on the lower
the feedback between the zonal
the ground.
The topic of Chapter 3 is a study of the topographic
instability in the continuously stratified Charney (1947)
model, employing most closely, the asymptotic
by Pedlosky
(1981)
baroclinic models.
in
the
This
barotropic
technique used
and two-layer
study is undertaken in order to
15
the nature of the
understand,
in a deductive way,
topographic
instability in a simple
continuously stratified
model.
Specifically the topographic instability of a normal
mode of
the Charney model
is examined.
The normal modes
which are found on the long wave or weak vertical shear
of the unstable Charney mode neutral
stationary and
curve, become
therefore resonant only for a given eastward
zonal velocity at
the ground.
the mode in question.
Thus,
The exact value depends on
it is to be expected that the
disturbance will draw its energy
from both the available
potential energy of the vertically sheared
the topography acting as a catalyst
momentum via
side
the form drag.
and
zonal flow with
from the
zonal
As in the barotropic model this
topographic instability might be thought of as a special
case of a more general wave instability of the Rayleigh
type,
in which
the topography acts
as a catalyst for the
instability of a normal mode of the Charney problem.
In
this limit the normal modes would not be unstable in the
absence of
topography.
Plumb (1981b) and Jacobs
have also considered the question of topographic
in a continuously stratified model.
those studies and mine will
(1979b)
instability
The difference between
also be discussed in Chapter
In Chapter 4 the main topic of study is the
of a small but finite-amplitude topographically
3.
stability
forced
Charney mode which, of course, is embedded in a uniform
vertically sheared flow.
Note we distinguish between the
neutral modes of the Charney problem and
the normal modes of
16
the
Charney problem, the latter of which can only become
We will
on a line
or nearly neutral,
neutral,
in parameter space.
be restricted to the nearly neutral
Thus,
parameter space.
portion of
a mode which
Chapter 4 deals with
can be weakly unstable even in the absence of topography in
contrast to
mind
the subject of Chapter 3.
attempt
two questions we shall
the baroclinic
presence of
flow and
(2)
in parameter regimes for
flow is baroclinically stable due to the
topography?
One might view this
(1980a,b)
to
to answer in Chapter 4.
instability of the zonal
does any new instability arise
which the zonal
fact brings
sinusoidal topography
(1) How does the presence of the
affect
This
study as
study of a neutral
vertical shear flow in the
the analog
of Lin's
wave embedded
baroclinic
in
a
limit of small but finite wave
amplitude except, in the present work, the wave is forced by
topography and
not limited to strict neutrality because of
its small amplitude.
to that
The method of analysis
to study
used by Pedlosky (1979)
the
amplitude dynamics of a growing Charney mode.
is very similar
finite
In particular
the analysis pivots about the point of neutral stability for
the Charney mode.
The method
of multiple time scales is
used to obtain the evolution of a weakly growing Charney
mode under the assumption of weak nonlinearity and,
present case,
weak sinusoidal topography.
For the purpose of summary
I will
reiterate the
subjects to be discussed in Chapters 2, 3 and 4.
In
in the
17
Chapter 2 the finite-amplitude dynamics of the
topographically forced wave instability in a barotropic
model are discussed.
The linear disturbance draws its
energy from the shear of the
this
forced wave.
linear instability is the topographic
Chapter 3 the topographic
the Charney model
A special case of
instability.
In
instability of a normal mode of
is discussed.
be unstable in the absence of
This normal mode would not
topography.
The subject of
Chapter 4 is the instability of a topography forced Charney
mode which
can be unstable in the absence of topography.
will be shown, all
the
these topics involve nonlinear effects
interaction between waves and
topography.
As
in
Chapter 2 Topographically Forced Wave Instability
at Finite
Amplitude
1.
Introduction
The interaction between standing and transient eddies
in the atmosphere
meteorologists.
importance to
is of fundamental
In an observational
interaction, Holopainen (1978) has
study of this
shown the vertically
averaged horizontal fluxes of relative vorticity by smalland large-scale transient eddies
maintaining the vertically
eddies' vorticity balance.
to be important
averaged, annual mean,
in
standing
In this chapter, as a first step
in understanding the interaction between
transient eddies in the atmosphere,
the standing and
a barotropic model will
be studied in which the nonlinear evolutions of a stationary
topographically forced wave and the travelling
which develops on it are considered.
instability
The topographic wave
is forced by the diversion of a uniform zonal current by
sinusoidal topography.
The tacit assumptions here are
that
the barotropic topographically forced wave will represent
the vertically averaged standing wave in the atmosphere and
the
traveling, linear, barotropic disturbance which develops
on the topographic wave will represent the large-scale
vertically averaged transient
atmospheric disturbances.
Obviously, the cyclone-scale transient eddies
properly represented
in a barotropic model.
representations of atmospheric
cannot be
Although these
topographic stationary and
large-scale transient eddies are crude,
the interactions
between them can be studied in a simply internally
consistent manner in order to provide some insight into the
interaction of their atmospheric counterparts.
To study the problem of the interaction between a
forced standing eddy and traveling disturbances which
develop on it as instabilities, the finite-amplitude
characteristics of linearly unstable perturbations to a
topographically forced wave will be analyzed.
The linear
stability problem has been discussed by Charney and Flierl
(1981).
In order to study the nonlinear effects on the
evolution of the perturbation and the topographic wave, the
non-linearity is required to be weak.
Then, as in Deininger
(1982, hereafter referred to as DE), the method of multiple
time scales can be used to close the problem for a slowly
growing perturbation.
In performing this analysis, it will
be found that the nonlinear evolution of this
topographically forced wave stability problem is markedly
different from that of the free Rossby wave stability
problem described in DE.
This contrast will make the
comparison of the nonlinear aspects of the topographically
forced wave stability problem and the free Rossby wave
stability problem particularly interesting.
Clearly, the atmosphere is baroclinic; however, based
on some preliminary calculations, the behavior to be
described in this paper using the barotropic model to be
discussed in the next section, does have a counterpart in
baroclinic models.
Therefore, the results of this chapter
20
should be regarded as providing a foundation for the
understanding of stationary and
transient wave interactions
in more realistic baroclinic models.
2.
The model
The nondimensional vorticity equation governing the
barotropic motion of a quasigeostrophic,
homogeneous
fluid on an infinite beta-plane, bounded in the
vertical direction by an upper flat
a lower corrugated plate
a
--
2
V
as topography,
is
2
+ J(,
-
horizontal plate and by
+ n)
V2
= 0
(2.1)
ax
n=hB/ED
variation.
to act
a
+
at
where
inviscid,
is
0(1)
and represents
hB is the height of the
the topographic
topography, E=U/f
the Rossby number, and D is the mean depth of
0L
is
the fluid.
All other variables have their conventional meanings and are
the same as those in DE.
sinusoidal,
The topography is assumed to be
i.e.,
n = nO sine,
(2.2a)
6 = kx + ly.
(2.2b)
where
3.
Formulation of the nonlinear problem
An exact solution of (2.1) for sinusoidal topography
(2.2)
whose
stability
Charney
and
the topographically forced wave
Flierl
solution
(1981)
studied,
is
+ F sine,
' = -Uy
(3.1)
where
nOU
K 2 (U -
and
c
(3.2)
c)
-K2
This solution exhibits a singularity when
U = c,
in which case
(3.3)
the Doppler-shifted wave field becomes
stationary relative to the topography.
analysis it will be assumed
U-c=0(1),
In the subsequent
so the forced wave is
topographically nonresonant.
If * is
the disturbance streamfunction which is to be
superimposed on topographic wave solution, i.e.,
Y = -Uy
p,
+ F sine +
(3.4)
substitution of (3.4) into (2.1) using
(3.2) yields the
problem for the perturbation streamfunction, which can be
written
a
a
2
(-- + U--) V,2
at
ax
a
+
-
* is
c)
a
-
1 -- )(UV
ay
If
K 2 (U -
ax
a
x cosO(k
o
-
+
= -
J(,
2
V).
(3.5)
ax
infinitesimally small,
justified and yields
)
linearization of
(3.5) is
a
a
(--
+ U-)
at
2
a
no
+
-
+
V2
ax
K2 (U -
ax
x cos(k
a
a
-- ay
1 -- )(UV
ax
(3.6) is a more general
Eq.
2
+ a)
= 0.
to (3.6) is of
= e i Xt
(3.6)
form of the linear stability
(1981).
solved by Charney and Flierl
problem
c)
The
solution
the form
I P neien + *,
n
(3.7)
with
ln) = (kO,
(kn,
10)
+ n(k,l).
Substitution of (3.7) into (3.6) results in a recursion
for the Pn quite similar
relation
in
form to Gill's
(1974)
recursion relation for the stability of a free Rossby wave.
It is
nodan+iQn+l
+ (X + 6 n)Qn + nodan-1Qn-1
= 0,
(3.8)
where
b
d =
K 2 (U -
,
(3.9a)
c)
1
b = -
(kl 0 -
ik 0 ),
(3.9b)
2
an = U - cn,
Cn =
Kn
2
B/Kn 2 ,
= kn 2 + 1n 2 ,
(3.9c)
(3.9d)
(3.9e)
23
6n = knan,
(3.9f)
Qn = Kn2 Pn.
(3.9g)
Solution of (3.8) will require the truncation of the
infinite set of homogeneous equations
(3.8).
As in
for Pn generated
the free Rossby wave analysis of DE,
by
the
qualitative nature of the weakly nonlinear dynamics can be
reasonably represented by the two mode approximation, i.e.,
* = e iXt (P
0
ei6
0
+ Plei61)
+ *.
The analysis to follow can be carried out
(3.10)
for any finite
trnucation and yield the same qualitative results, so the
truncation of (3.10) is not as restrictive as it first may
seem.
The results of a three mode truncation of (3.7) are
given in Appendix A.
After truncating the sum (3.7) to (3.10) the infinite
set of equations reduces to
(X + 61)Q
nOdalQ
0
+ nOda
0
0 Qo = 0,
1
+ (X + 60)Q
1
0
= 0.
(3.11)
For the nontrivial solution of (3.11) it is required that
X= -
(
61 + 6 0
1
) + --
2
[(61 - 60)2 + 4nO
2
d 2 a1 a 0 ]1 / 2
2
(3.12)
Then for instability to occur,
ala 0 < 0
(3.13a)
24
is
the mountain height must exceed a critical
required and
value which
is
-
S(61
60)2
4d 2 ala
Since
it
the perturbation was chosen
8/K 1 2
follows that
along
(3.13b)
0
< 8/K 2
such
< a/K 0 2 .
that K 0 2
Then,
< K2
<K 1 2 ,
from (3.13a)
with (3.9c,d) it can be seen that both subresonant and
superresonant
flow is possible within the parameter range
corresponding
to linear
If
instability.
the topography is just slightly higher than nc by a
small amount A(A << nc)
such that
no = nc + A,
(3.14)
the growth rate of the perturbation is proportional to
A11 / 2 , i.e.,
Xi2 = -2d
As in the
2
nc
2
alaoA.
(3.15)
free Rossby wave problem, (3.15) suggests the long
time scale
A1 1 /2 t
T =
as the one over which the nonlinear evolution will
when the perturbation is allowed to be finite but
Now the
time operator
a
-- +
at
IAl1 /2
a
.
aT
in
(3.5)
is
replaced
by
occur
small.
25
so (3.5) can be rewritten as
a
(-
A11/
+
2
at
a
fnc +A
+
K 2 (U - c)
-
where
a
J(0,
2a
+ U -)
aT
ax
V2
B-ax
a
cos6(k--
+
-
1
ay
a
2
-)(UV
+ B)9
ax
V 2).
(3.16)
(3.14) was also used.
appropriate expansion for
It
to choose
now remains
the
, so the weak effects of
nonlinearity and instability can be balanced.
As in DE, the
appropriate choice is
S=
A1I/
2
4(1)
+
2
IAI(
) +
IA1
3
/ 2 4(
3
) + .
..
(3.17)
Substitution of (3.17) into (3.16) results in a series of
problems
for the
successive powers of
problem leads to the
=
1
1
n=O
2
.
The O(IA11/
2
)
specification of the neutral
perturbation already calculated.
*(1)
A11 /
The neutral solution is
A
LnPoeiOn + *
(3.18)
where
61 + 60
On =
L1
On -
(
)t,
2
(61 - 60)Ko
=
2dnalK2
2dncalKl 2
LO = 1.
(3.19a)
2
(3.19b)
(3.19b)
(3.19c)
26
So far the analysis is essentially the
the time
stretching
necessary
in
the
analysis has not been done here.
unnecessary.
This is due
wave is forced, causing
same as
in DE except
free Rossby
Doing
so would
wave
later prove
to the fact that the basic-state
the basic
state wave structure,
produced by the self-interaction of the perturbation at the
next order,
to be nonresonant.
As a result, the
self-interaction of the perturbation produces a forced
solution
instead of producing a phase change of the basic
state wave which
coordinate.
the
in DE required
the use
of a stretched
time
This foreshadows the major difference between
free Rossby and topographically
forced wave analyses.
This difference and the effects of nonlinearity will begin
to be apparent at the next order.
4.
The nonlinear analysis
0(IAI)
The nonlinearity at
is due to the
self-interaction of the perturbation field.
Using (3.18) to
calculate the inhomogeneous terms at O(IAI),
and retaining
only those terms consistent with the original
truncation,
results in the equation
(--+
at
U
-) -)
vV
22)
+
+
ax
x cos6(k -- ay
ax
a
a
1 -- )(UV
ax
2
+ B)9
K2 (U -
c)
= K
+ 2b(K 1 2 -
^
o
2
eie
+ K 1 L 1 --aT
Po
ei60
-~
3T
2
2b(K 1
2
K 0 2 )L 1
-
K0
2
2
0
ei6
-
1
)LlP01o2 ei6 + *,
(4.1)
where
61 = klx + flY -
in
-
three inhomogeneous terms
free Rossby wave analysis
the
60)
t ,
= (2k 0 + nk, 210 + nl).
(kn,ln)
The first
(61
and
in
(4.1) are the same as
therefore
give rise
to
the particular solutions
(2)
1
(2)
e
Pn
=
p
n=O
+
(4.2)
*,
where
(2)
P1
2iL 1
aP
61 -
(2)
PO
= 0
(2)
*f
=
aT
60
and
1
(2)
ei
Fn
A
n + *
(4.3)
n=0
where
(2)
Fn
ifn(PO)2
the fn are given
term is analogous
in Appendix
B.
The remaining inhomogeneous
to the one which in the
free Rossby wave
28
analysis was resonant and therefore produced a phase
correction of the Rossby wave.
nonresonant for
basic
However, here
it is
U*c and produces a forced solution in the
state structure which represents an amplitude
correction to the topographically
forced basic
state wave.
This solution is
(2)
= FBeiO
*B
FB = ifB P0
+ *,
(4.4a)
2
(4.4b)
,
and
(K 1 2 fB
K 0 2 )(6
-
60)K02
2
2kncalKl
This difference
is due
structure using
2
(4.4c)
to the basic state being
forced as
The solution to O(IAI)
in the basic
free.
opposed to being
state
-
1
(3.1),
(3.2) and
(4.4), is
(nc + A)
K 2 (U c)
+
A
(K 1
2
-
K0
2
)(6
kncalKl
1
-
6 0 )K 0
2
and has an interesting property.
2
IPl
2 ]
sine,
Note the O(IAI)
(4.5)
correction
to the basic state wave is always a positive contribution to
the effective basic wave amplitude which is defined as the
coefficient of sine in (4.5).
conditions al
occur.
> 0 and a0
This is so
< 0 must be true
For subresonant flow (U < c)
since the
for instability to
the absolute value of
the effective basic state wave amplitude decreases as the
29
perturbation grows, but
flow (U > c) the
increases.
wave amplitude
basic-state
effective
for superresonant
Thus,
for
flow there seems to be a further
superresonant
destabilization which should be apparent when
the behavior
of PO is determined at the next order.
The O(1A1
a
/
2
) problem can be written
a
+
(--
3
at
2
#p
v
)
3
v2
+
A
1
(2)
AI K2 (U
aT
a
x cose(k -ay
J(4(1),
v
(3.18),
2
,(
a
1 -- )(UV
ax
2
(4.2),
))
terms only
3
c
)
+
c
K 2 (U -
to ei6 0
produce
3
6)(
-
c)
)
c)
2
+ 6),(1)
2
_ j(,(
(4.3) and
V2 ,
),
1
(4.4),
)).
(4.6)
the inhomogeneities of
At this order only terms
(4.6) can be evaluated.
proportional
(
--
a
1 -- )(UV
ax
a
Using
8
ax
a
x cosO(k -- ay
-
+
)
ax
-
3
a
--
U
and eie
forced
1
need be retained,
solutions
determine the behavior of PO on the
which
long time scale T.
terms,
becomes
at
+
U --
ax
nc
a
a
a
)
v2
(3
)
+
B -
ax
(
)
the other
are not needed
After evaluating the relevant inhomogeneous
--
as
+
K 2 (U - c)
(4.6)
to
2
wv
x cos6(k -ay
=
aPl(
1 -- )(UV
ax
2
A
)
ida 0 K 0 2 P 0
+
[K12
+
[---
Al1
+
2ib(KO2
-
-
K2 )fBPOIP0
IAI
aT
+ 2ib(K 0 2 -
)
a)*(3
+
K')flPo0P01 2 ]
ida 1K 1 2 L 1 P
-
ei
6
1
2ib(K 1 2 - K2 )LlflP1pO 2 ] e i 60
(4.7)
*
the secularities from (4.7) results in the
Removal of
evolution equation for PO which
d2P0
A
dT2
ai2po + NPOIP0I
IAI
2
= 0,
(4.8)
where the growth rate
Xi
2
aiA
2
-
agrees with
the
2d 2f n ca
linear
la
o,
result
(3.15).
The nonlinear
coefficient
N = NB
+ NH,
is comprised of a part due to the feedback with
state,
i.e.,
b 2 K0 2
NB =
kK2 K 1 2
(K
1
2
-
K02)(61 -
al(U - c)
6o)
the basic
2
31
K2
KO2 S[(
2
K02
and a part due
)a
+
to the
(
K12 - K2
K1
)a]
KI2
,
feedback with higher
(K 0 2 - K 1 2 ) (K12 - K2)
L1
(K12 - K2)
The latter feedback is essentially the
Rossby wave analysis.
mechanism is unique to this
(4.10)
same as
However,
chapter.
it can be seen for subresonant
NB
harmonics, i.e.,
bfl(61 - 60)
NH
the free
(4.9)
in
the former feedback
Upon examination of NB
(U < c)
flow
discussed
that
> 0,
which says the nonlinear feedback is stabilizing and
therefore gives rise to an oscillatory exchange of energy
between
the effective basic
perturbation.
state wave amplitude and
When this oscillation
is averaged over one
cycle, the effect of the perturbation reinforces the
topographic wave.
topographic
wave is
observational
Sasamori
This dissipative effect on the
in
qualitative
agreement
with the
and numerical model results of Youngblut and
(1980).
But
for superresonant
flow NB < 0,
nonlinear
feedback is destabilizing and
results in an
explosive
instability
up in
in
This was to be expected
showed
which Po blows
from the
so the
a finite
solution at 0(IAI),
time.
which
the effective amplitude of the basic state to be
increased as the perturbation grew, causing it to be more
unstable.
This explosive instability is somewhat
reminiscent of
the form-drag
Charney and Devore (1979) in
flow.
superresonant
instability discussed by
that it occurs
However, the explosive
is actually a distinct phenomenon.
interaction of the topographically
for
instability here
It arises by the
forced wave with its
perturbation via the convergence of Reynolds' stresses.
Thus,
this instability
is independent
due to the
flow via the
of form drag
the form-drag
(note NB
instability is
topographically forced wave
form drag.
However, whether or
the explosive instability occurs actually depends on the
sign of the
the
whereas
interaction of the
with a zonal
not
of nc),
is independent
full nonlinear coefficient N, not just NB.
sign and magnitude of
important.
In fact,
NH relative
to NB is
Thus
also
if NH is positive and large as compared
to NB, the explosive instability could occur only near
resonance where the analysis is not valid.
specific
Therefore, a
numerical calculation of N is desirable.
Unfortunately,
I would not expect the
calculation of N using
the two-mode truncation to be representative of
the N
calculated using less restrictive truncations.
Therefore,
this calculation would not be meaningful until
the
convergence properties of N can be established, which is
beyond the scope of the present work.
As previously mentioned,
the
flow can be barotropically
resonant only for unrealistically large values
velocity for planetary-scale waves
explosive instability unlikely.
of zonal
in this model, making the
However, in a baroclinic
system, resonances corresponding
may be more easily realized
zonal
velocity
required,
to free baroclinic modes
from the
which may
point of view of
the
allow the explosive
instability to be more easily excited provided the
topographically forced wave and
vertically trapped.
Also,
linear instability are
in a baroclinic model the
baroclinic nature of the topogrpahically forced basic state
(e.g.,
transient
(e.g.,
Holopainen, 1970) as well as short wavelength
eddies and
Bottger
and
some
large-scale
Fraedrich,
1980)
Therefore, extending this analyses
worth further consideration.
transient
eddies
can be accounted
for.
to a baroclinic model is
The neglect of baroclinic and
frictional effects in this study precludes any comparison of
the present results with the atmosphere.
A few remarks concerning truncation which apply to this
analysis as well as that given in DE are in order.
course of this analysis it has been convenient
the perturbation (3.7) to
(3.18);
In the
to truncate
however, it is possible to
carry out this analysis for any finite truncation.
restrictive truncations the growth rate,
nonlinear coefficient N of
ai
2
, and
For less
the
(4.8) are, of course, different
from those given here and depend on parameters
such as the
phase speed and the critical amplitude, which are determined
in the course of the linear analysis.
It
is known
that
successive estimates of the eigenvalue do converge to a
value
(although somewhat slowly in some parameter regimes)
as the truncation is made less
severe
(Tung, 1976).
Thus,
ai
we expect
clear,
2
will converge to a finite value.
however, that the
is not
nonlinear ocefficient N will
converge to a finite value.
nonlinear coefficient imply?
What
would a diverging
It would
nonlinearity is felt immediately and
the weakly
It
suggest
would then
nonlinear analysis as well
that the
imply that
as the linear
stability analysis is meaningless in the region of parameter
space
in which N diverged. This
is because the time scale of
the nonlinearity is much faster than that of the linear
instability when N diverges, which implies that
separation of the linear
instability and
the
nonlinear time
scales used to close the problem is no longer valid.
this case we might expect
In
turbulent behavior to be present
immediately as opposed to the case of finite N where
turbulent behavior may develop more
slowly as the wavenumber
spectrum fills out and stronger nonlinearity develops.
Presumably,
time
1AI-1/
this
2
longer valid.
occurs
over times
scales
after which the theory in
longer than
the
this study becomes no
Hence, the question of truncation is a
serious one and whould
be borne in mind
in interpreting any
study which employs truncation, such as the present work, or
numerical
5.
simulations where truncation
is implied.
Concluding remarks
In this chapter the nonlinear evolution equations were
obtained which govern the interaction between a
topographically forced wave and
its weakly unstable
perturbation.
This analysis, valid away from topographic
resonance, showed that the topographically forced wave
remained in a fixed position relative to the mountain while
its amplitude was altered as the perturbation evolved.
explosive instability occurs
for superresonant
which the analysis is not valid after a short
smooth oscillatory exchange of energy
the perturbation and
topographic
An
flow for
time and a
takes place between
wave for subresonant flow.
When this oscillation is averaged over one cycle, the effect
of the perturbation
dissipative one.
that of
on the
topographically
This behavior
forced
wave is
a
is markedly different from
the free-wave analysis by Deininger (1982) in which
the phase of the basic-state Rossby wave was altered as the
oscillatory exchange took place between the Rossby wave
amplitude and perturbation.
The mechanism behind
study
the nonlinear feedback in this
is the convergence of vorticity flux,
essentially the tilted
trough mechanism.
been shown by Holopainen
(1978)
i.e.,
This mechanism has
to be important
in the
maintenance of the standing eddies' vorticity balance in the
atmosphere.
The preceeding analysis, then,
is suggestive of
a type of interaction which may occur between the
topographically forced standing eddies and
in the atmosphere.
transient eddies
36
Appendix A
Results
for the Three-Term Truncation
When
(3.7) is truncated to
*(1)
= L 1 P 0 ei6l + p 0 ei0
+ L- 1 POeie-1
(Al)
where
On
= en +
(A2a)
Xrt,
ncda0K02
Ln =
, for n = +1
(A2b)
6n)Kn 2
(Xr +
and
Xr =
(A2c)
Xr + 60-
the coefficients of (4.8) become
2
ai 2
al
a-1
Hnc
),
+
(
-
Xr + 61
Xr +
N =
6-1
K0 2 -
8b 2 (KO 2 - K 2 )
K1 2
[-HkK 2 (U
- c)
[
al
+
Xr + 61
-
K-1
2
x
4
a- 1
K-12(Ar +
]
Xr + 6-1
(r
+
61)
K0 2
+
8nc 2 b 2 d 2 (K0 2 -K 2 )2 (K- 1 2 -K 1 2 )
HK1 4 K-1
-
A
K1 2 (Xr + 61)
x
(A3a)
2
(Xr
+ 6_ 1 )2 kK 2 (4U -
c)
[(K12-K2)(4K2-K_12)+(K_12-K2)(4K2-K12)]
6.-1)
1
+ H
K 2 )(K 1 2 -
2b ( 12 -
dnc
K1
4b(Kl 2 -
K1
2
2
(K 0
K-1
2
K-1
2
K2 )(Xr
-
-
2
) (K
2
K0
2
)fl
+ 61)
0o2)fo
( Xr + 61)(
+
r
6-1)
2b (K_ 1 2 - K 2 )(K_12 - K02)f_
dnc K- 1 2
1
(A3b)
(K 0
2
2
-
+ 6-1i)
K )(Xr
where
al
a-1
(Xr + 61)
fl
3
6_-1)3
[-ncd(K0 2
=
Z12(61 +
+ b(K
+
(Xr
2
1
f- 1 =
-
-
K0
2
2
-
K2)f0
Xr)
(A4b)
)],
-2
"
K_12(6_
(A4a)
[ncd(K O 2
1
2
+
-
K 2 )f 0
Xr)
+ b(K- 1 2 - K 0 2 )],
(A4c)
1
2
fo =
[4b(K
-
4bdncal
(K 1 2 -
-
4bdnca-l
(K 0 2
1
-
K02)(6_
-
K-
+
K-12)LL-1(61
I
2
)(6
+ 2Xr)L
1
1
+
2
2
Xr)(
6
-1
+
2
Xr)
1
X,)L-l]
(A4d)
D = -4d 2 nc 2 a0[al
+ (61 +
2
Xr)(60 +
(6_ 1 +
2
2
Xr) + a-1(61 + 2Xr)]
Xr)(6-1 +
and nc and Xr can be determined
3Xr 2 + 2(61 +
,nc
2
=
Xr),
+ 61)Xr2
(A4e)
from the two equations
-1)Xr + 616-1
,
d 2 a 0 (al + a- 1 )
Xr3 + (61
-
6
2
+ [616-1 -
nc
2
(A5a)
d 2 a0(al + a-1)r
nc 2 d 2 a0(al6_1 + a- 1 61 ) = 0.
2
(A5b)
singularity in N when U = c/4 corresponds to a
The
resonance with the second harmonic of the basic-state wave
and the analysis is not valid there.
For the special
in which 1 = kO = 0, one must go to the 0(1A1
closure as done in Deininger and Loesch
2
case
) to obtain
(1982).
Appendix B
Forced Solution Coefficients
fn are
The
fl
=
2b(2Xr + 60 )(K 1 2 - K02)L 1
..
_..
..
S=
K2D_ ...
-4ncbdal(Kl 2 fO02D
K02)L
(Bla)
1
(Blb)
where
61 + 60
Xr
=
)
- (
(B2a)
and
D = (2 Xr + 60)
(2 Xr + 61)
-
4nc
2
d 2 alao.
(B2b)
The barred quantities are
K,2 = -k2
en =
+ 1,
2
8/K,
(B3)
an = U -
6,
cn
= kn(U -
Cn)
Chapter
3
Topographic instability of the normal modes in
the Charney problem
1.
Introduction
The
first theories of topographic
and Devore, 1979;
an ad hoc
Charney and Strauss,
simplification
baroclinic models.
was
instability (Charney
1980)
of barotropic
Specifically,
and
were based on
two-layer
the ad hoc
simplification
the severely truncated spectral representation of the
already simplified equations
of motion. The primary effect
of the particular truncation employed was
wave-wave interactions but
flow interaction.
to neglect the
ingeniously retain
This wave-zonal
the wave-zonal
flow interaction was
fundamental
to the topographic instability.
above cited
studies remain unsatisfying due
However, the
to the
nondeductive nature of the simplifications required to
isolate the topographic instability.
Hart (1979) was
successful
in eliminating the necessity of spectral
truncation
in a barotropic model
with nearly
this
by considering topography
infinite horizontal extent.
study is the unrealistic
The drawback of
restriction placed on the
topography in order to isolate the instability.
Independently Jacobs
(1981)
studied the
(1979a),
Plumb
topographic
(1981a) and Pedlosky
instability in barotropic and
two-layer baroclinic models using another asymptotically
valid technique which neglected
This deductive method exploits
in the topographic
the wave-wave
interactions.
the role of linear resonance
instability and
its quasi-linear nature
41
discovered in the earlier studies.
Specifically, the flow
is assumed to be near linear resonance and weakly nonlinear.
Thus the lowest order solution consists of a free,
stationary, linear wave.
Its amplitude is determined by the
wave-mean flow nonlinearity which arises in the presence of
topography as the flow is slightly detuned from resonance.
As mentioned in Chapter 1, Plumb's (1981b) analysis is
somewhat different since he imposes a propagating forcing.
Of course a wave resonant with this forcing must be moving
with it.
Furthermore, as noted by Pedlosky (1981) there is
no feedback affecting the topographic forcing.
The purpose of this chapter is to remove one of the
simplifications of the above cited models paralleling, most
closely, the technique employed by Pedlosky (1981) to
isolate the topographic instability in the layered models.
Namely, the limiting assumption of a barotropic or two-layer
baroclinic model will be removed.
Here, topographic
instability will be examined in a continuously stratified
baroclinic model.
The particular model to be used in this
chapter and in Chapter 4 is the nonlinear version of the
Charney (1947) model with sinusoidal topography, Ekman
friction and Newtonian cooling.
It is chosen for its
relative simplicity which is due to the assumptions of
constant vertical shear and constant Brunt-Vaisala
frequency.
The Charney model admits as its vertically
trapped solutions both normal modes and unstable modes. The
Charney mode is the most unstable mode.
However, the
unstable modes can become neutral on a curve in parameter
space
in which case they
are referred to as neutral modes.
The neutral modes should not be confused with the normal
modes.
The
normal modes which owe their existence to the
shear remain baroclinically
presence of vertical
neutral
throughout parameter space as opposed to the unstable modes.
near neutrality is
The unstable modes
Chapter
In this chapter we examine the topographically
4.
induced instability
analogous
the subject of
to Pedlosky
to a free,
stationary
determined by the
Thus in a manner
of the normal modes.
(1981) the 0(1)
solution corresponds
Its amplitude is
normal mode.
topographically induced wave-mean flow
interaction with a zonal flow slightly detuned from
resonance.
Weak
Ekman
also allowed to affect
friction and Newtonian cooling are
the dynamics.
The principal differences
between the
layered
baroclinic model and the continuous stratified baroclinic
model are due
to the way
topography are
model, Ekman
in each
included in each.
friction enters
layer.
budget of
in which Ekman friction and
In the
layered baroclinic
the potential vorticity budget
Topography enters
the entire lower layer.
the potential vorticity
Thus potential vorticity
is not conserved and multiple equilibria can
balance between Ekman
friction and topographic forcing.
However, in the continuous model
dissipation potential vorticity
presence of
result as a
in the absence of internal
is conserved even in the
topography and Ekman friction.
Therefore
internal dissipation in the form of Newtonian cooling is
included which destroys the conservation of potential
vorticity in the interior.
This along with the inclusion of
nonlinearity makes possible the existence of multiple
equilibria.
Ekman friction alone is not sufficient to
produce multiple equilibria in a continuous model.
During the course of this work two other studies
(Jacobs, 1979b; Plumb, 1981b) of topographic instability in
a continuous model became known to me.
Both authors
consider a more general shear flow which makes their results
less analytically explicit.
complexity, Jacobs
In fact, due to this additional
(1979b) found it necessary to limit his
consideration to the steady problem.
The relative
simplicity of the present model allows the derivation of the
fully transient amplitude equations and an explicit
expression for the growth rate of the topographic
instability.
Plumb's work also differed from the present
work in the type of topography specified.
He considered a
propagating topography for the purpose of emulating previous
studies of stratospheric warming.
However this does not
allow a feedback between the zonal flow and wave field at
the ground.
As will be seen, this feedback is a feature of
the present study.
2.
The Model
The model to be used in this and the next chapter,
consists of a quasigeostrophic continuously stratified fluid
on a B-plane bounded horizontally by two vertical walls
separated by a distance L, and bounded vertically by the
ground with its sinusoidal topographic undulations and
infinity.
The basic density field is assumed to have a
constant scale height H and constant Brunt-Vaisala frequency
N associated with it.
The basic state velocity field
consists of a latitudinally uniform zonal flow with constant
vertical shear.
The flow at the ground is not zero.
Thus
the full nondimensional stream function * can be written
= -
(x,y,z,t)
Uy
-
zy +
We(x,y,z,t)
(2.1)
where x,y,z,t are the zonal, meridional, vertical and time
coordinates, respectively.
and z.
In (2.1) U is independent of y
The first two terms of (2.1) represent the basic
state zonal flow while the third term represents the
disturbance field.
The variables have been
nondimensionalized as follows:
horizontal lengths by L,
vertical distance by
fL
D = -N
(2.2a)
and horizontal velocities by
3U,
U o = D ---
3z
(2.2b)
where 3U*/az is the constant shear of the basic zonal flow
45
in dimensional
form. The parameter c measures
magnitude of the basic state zonal velocity
disturbance
field.
Using the above scaling, the
for the disturbance under the
the
field to the
version of the potential vorticity equation
nondimensional
takes
the relative
influence of Newtonian cooling
form
+ z)
[-+ (U
at
= - n(--
az
h-
I + q -
-- ]
ax
1
+ eJ(4,II)
ax
-),
az
2
(2.3)
where
H=
+
V2
-
h-1
az 2
az
(2.4)
is the potential vorticity of the disturbance,
q = 8 + h-
and
1
(2.5)
the operators J and V2 are defined by
3a ab
J(a,b) E -.
ax ay
2
V
32
-+ -ax 2
3y 2
a2
aa ab
-.
ax ay
,
.
The nondimensional parameters
8 = 8*L
h = H/D
2
8, h and n are
/Uo
(2.6a)
,
(2.6b)
46
n = L/(UoTN),
where
TN is
(2.6c)
the Newtonian cooling
time and
B* is the
f.
northward gradient of the Coriolis parameter
At the bottom boundary just above the
Ekman layer the
vertical velocity must match the Ekman pumping velocity plus
the vertical velocity due to topographyu.
condition the nondimensional
After using
this
lower boundary condition for
€
at z=0 can be written
a
-)--
+ U
at
a
a3
_a
(--
ax az
-
-
ax
+
a
sJ(4,--)
-
J(,n)
1
-
az
-
rV
-
E
U nx
-
-
az
(2.7)
where
fL
hB
n = -(-)
D
Uo
r
1
-
SE
2 D
In
6
(2.8)
hB(x,y)
is
fL
(--)
Uo
,
(2.8a)
,
(2.8b)
E = (2v'/f) 1 / 2 is the Ekman layer
the elevation
of
the
thickness and
lower boundary
above z=0.
At the channel sidewalls the meridional velocity must
vanish,
i.e.
= 0 at y = 0,1,
3a/x
(2.9a)
and as a result
lim
x+0
1
-2x
x
-x
32
---- dx' = 0
ay3t
must also be true.
(2.9b)
47
The model as stated above is essentially the same model
as formulated by Pedlosky (1979) except for the inclusion of
topographic effects.
It also could be thought of as the
nonlinear version of the Charney (1947) model for baroclinic
instability with the addition of Newtonian cooling,
Ekman
pumping and topography.
3.
Brief review of the Charney model
The analysis in this chapter and the next chapter
exploits the simplifications that occur when the Charney
mode or the normal modes of the Charney model are nearly
resonant.
This linear resonance occurs when these modes
become stationary.
Thus a brief review of the stationary
neutral Charney mode and stationary normal modes of the
Charney problem is necessary.
In doing so we consider the linear, undamped version of
the basic equations (2.3,2.7,2.9a) for the disturbance 4O in
the absence of topography.
a
a
[--
+
at
a2
2
+ z)--][V
(U
They are
o +
at
-
az2
aoo
-
h-1
-]
ax
+ q -
= 0,
ax
(3.1a)
a
(--+
at
_
U
a
ap 0
-)-
ax az
a 0
-
= 0,
ax
(3.1b)
Pox = 0 on y = 0,1.
(3.1c)
and
48
(3.1)
A propagating wave solution of
can be sought in the
form
90 = A FO(z)
(3.2)
sin ly + *,
eik(x-ct)
where 1 is an integral multiple of
i.
into (3.1) results in the following
Substitution of (3.2)
vertical
structure
equations
(z
+ U -
(U
-
d 2 Fo
c)(dz
dF o
c) -dz
The equation
Fo
dF o
dz
1
h
= 0
on z
K 2 Fo)
+ qF o = 0,
(3.3a)
=
0.
(3.3b)
for F o can be reduced
hypergeometric equation with the
1
Fo =
(z + U -
c)e
S=
(z + U -
c),
(h
- 1
-
to a standard confluent
transformation
)z
2
(3.4a)
F(E),
(3.4b)
where
=
(h- 2 + 4K 2 )1/ 2 ,
(3.4c)
K 2 = k 2 + 12
(3.4d)
The standard equation and associated
condition takes the
lower boundary
form
E FEE + (2-E)Fg h-
1
p(U-c)2[FE + (
(1-ro)F = 0,
-
p
)F]
2p
(3.5a)
= 0 on
E =
Eo-
U-c)
(3.5b)
49
where
B + h- 1
q
ro -
-
1/2
,
(h- 2 + 4K 2 )
1
(3.6)
modes can be
Both the neutral Charney mode and normal
discussed relatively simply by considering only integral
For integral ro,
values of ro .
solution to
the
as
which Fo decays as z+o can be written
Stegun,
(3.5a) for
(Abramowitz and
1964)
F(S) = M(1-r
o ,
2,
5),
(3.7a)
where
(1-ro)(2-ro)E 2
(l-ro)E
M(1-ro,2,)
+
E 1 +
2
2
0
(1-ro)nEn
+...+
+...
...
=
2
3
x
2!
(1-ro)m Em
,P
I
m=O
( )nn!
x
2
( )m
(3.7b)
m!
and if a is any number
(a)n =
(a)(a+l)(a+2)
... (a+n-1);
For a given integral ro
(a) o =
1.
there are two ways by which the
lower boundary condition (3.5b) may be satisfied.
E = Eo
-
On
(U-c) either
U = c,
(3.8a)
or
(h-1_
Fg +
)
F = 0
order
2In
to
distinguish between
the two roots
In order to distinguish between the two roots
(8b)
(3.8b)
the
(3.8a,b) the
the
convention in terminology adopted here is as follows:
wave corresponding to the double root (3.8a) is referred to
as the neutral mode and the wave(s) corresponding to the
single root(s) of (3.8b) is(are) referred to as the normal
The neutral mode becomes unstable for nonintegral
mode(s).
ro .
The normal modes also exist for nonintegral ro and are
vertically trapped waves propagating zonally through the
shear flow.
As will be seen later, the distinction between
the modes is
important since the two different
modes require different treatments.
types of
Next we discuss the
structures of the different types of modes when they are
stationary.
When ro=1, the solution (3.7) is simply F=1 for which
it is only possible to satisfy
(3.8a) not (3.8b).
This
single root for ro=l is known as the neutral Charney mode
and is stationary when U=O.
In this case the vertical
structure is
-Bo
Fo(z)
where qgo =
Bo
Miles
=
z/2
(3.9)
= ze
or equivalently
-h-1 + (h- 2 + 4K2 )1 / 2
(3.10)
(1964) has shown that near the neutral curve for the
Charney mode, i.e.
6 = Bo - 6 with
161<<1,
the growth rate is 0(61/2) for 6>0 and 0(63/2)
(3.11)
for 6<0.
The
51
interest here is in the stronger
(1979) points out that
Pedlosky
instability for which 6>0.
a small
amount of Ekman
damping is enough to squelch the weak instability.
that FO is zero at z=0.
of
The analysis of Chapter 4 makes use
these facts concerning the Charney mode as
in Pedlosky
But in Chapter 4 the purpose is to study the effect
(1979).
of
Notice
topography on the baroclinic
instability and
its
equilibration.
then (3.7) implies
If r 0 =2,
F (5)
and
= 1 -
1
- 5,
2
(3.12)
(3.6) becomes
2p =
8 + h-1
(3.13)
In this case there is both a neutral mode which becomes
unstable for nonintegral rO and a normal mode.
and
Using (3.12)
(3.13) in (3.5b) it can be seen that the normal mode
becomes
stationary when
28
(3.14)
The vertical
structure function for this simplest stationary
normal mode is
1
-(h
Fo(z) = (z+U)[1 -
where
- (z+U)]
2
p and U are given by
This normal mode has
In addition,
e
-
1)z
(3.15)
(3.13) and
an analytic
1 -
2
(3.14),
respectively.
continuation for all
for higher values of ro more
normal modes
ro>l.
52
exist.
are a-i normal modes
In fact for any rO>1 there
where a is the next integer larger than ro.
However, to
keep the calculations relatively simple the example in
section 5 is restricted to the r0=2 normal mode.
I do not
expect the restriction to integral ro to alter the
fundamental physics of the problem.
In fact, as we will see
in section 4, the derivation of the evolution equations
which govern the destabilization of the normal modes by
topography,
is independent of ro
.
Of course, the numerical
results would certainly depend on the structure of the
normal modes and therefore on ro .
This dependence should
perhaps be investigated in the future.
2.4
Derivation of the evolution equations for the normal
modes of the Charney problem in the presence of
topography
The method of analysis used to describe the behavior of
the normal modes of this continuously stratified baroclinic
model is analogous to that used in the barotropic and two
layer baroclinic model by Pedlosky
Pedlosky's work,
(1981).
Here as in
the analysis pivots about the resonance of
a normal mode described in the last section.
This occurs
for some value of the zonal current, say ur, which depends
on the mode considered and the various parameters in the
problem such as B, k,
h,
and r O .
However, it is not
necessary at this point to specify which normal mode is to
be dealt with.
It suffices to say that the concern here is
53
the stationary normal modes of
with one of
problem which exist
for either
the Charney
integral or nonintegral
rO .
In the absence of nonlinearity and damping, a normal
resonant with topography grows secularly in time.
mode
In
this occurrence, weak nonlinearity and
order to prevent
included and the flow is slightly detuned from
damping are
resonance, i.e. with s<<1 we choose
2
r =
=
n
(4. la)
r',
e2n',
(4.lb)
U = ur + e 2 A'.
In addition
(4.1c)
the sinusoidal topography
is required to be
The appropriate choice for topography
weak.
is
n = E3 b = E3 Meikx sin ly + *,
where
0(E
3
)
* represents
(4.1d)
the complex conjugate.
The choice of
topography in this case differs from the choice made
in the baroclinic case discussed by Pedlosky (1981) but
the same choice made in his barotropic analysis.
for this
is
The reason
reverts to the properties of the normal modes in
each case.
Here, as in Pedlosky's barotropic case, the
preexisting zonal flow directly forces
wave in which case only O(e3)
significant
0(e 3
)
the 0(1)
stationary
topography is required for
wave-topographic
interaction.
In his
baroclinic case he chooses a prescribed zonal flow which
does not directly force the stationary wave.
In this case
an 0(E 3
on the
)
wave-topographic interaction depends
54
zonal momentum from the upper to lower
mean
transfer of 0(s)
This, in the presence of 0(e) topography, forces an
layer.
wave which interacts with the topography at 0(3).
O(e2)
As in Pedlosky (1981), given the above choices, the
otherwise stationary normal mode evolves on a time scale
T = E2 t.
(4.2)
If we substitute (4.2) along with (4.1) into the governing
(2.7) and (2.9) they become
equations (2.3),
(Ur+Z)-
E2(_
= -
+ q-
ax
+ A'--)l-sJ(p,)
at
ax
a2
-
2 n'(-
az
a
_a24
-
ur
-
a
=
-
at
-
E2r' 2
2
,
3a
2
a
- h
a
A'-)---
at
ax az
e2 Urbx -
n .
-)
az
2(-. +
3x
(4.3a)
ax
3¢
eJ(,--)
E3 J(4,b)
az
(4.3b)
ax
on
z = 0 and
(4.3c)
Ox = 0,
lim
x+W
x
1
f
--
320
-
C2
2x -x
dx' = 0
ayaT
(4.3d)
on y= 0,1.
Substitution of the asymptotic series
=
€0
+
E
1i +
62
2 +
**
(4.4)
55
into
(4.3) results in a sequence of
upon removal of
determine
linear problems, which
the secularities at
the behavior of
each order, will
the stationary normal mode and
topographically induced zonal
flow.
The 0(1)
the
problem for
0
is
2
(Z+Ur)-- (V
ax
a2
0
ur azax
-
a2 0
4o +
o
-
h
az 2
a3o
-=
ax
o
----) + q-ax
ax
= 0
(4.5a)
(4.5b)
on z = 0,
0
and leads to the specification of what
stationary normal mode of
is to 0(1),
the Charney problem.
a free
As discussed
in section 3, it can be written
€0 = AFO(z) eikx sin ly + *
(4.6)
where FO satisfies
d 2 Fo
(z + Ur)(---dz 2
dFo
ur dz
-
h
- 1
Fo = 0
dF o
--dz
- K 2 Fo) + qF o = 0,
(4.7a)
on z =0,
(4.7b)
and the value of the zonal current required
depends on which mode is chosen.
It
the purpose of the derivation in this
for resonance ur
is not necessary for
section to make
specific the normal mode under consideration, but it
necessary
to limit the
is
initial spectrum to just one mode
thereby neglecting resonant triad
interactions.
56
Nevertheless,
the restriction in the
is self consistent.
single mode
initial spectrum to a
is a consequence of
This
the weak nonlinearity.
The O(e) problem can be written
(z
- J(
+ ur)--+ q --ax
ax
ax
Ur ---3z3x
Upon integration of
, Ho)
(4.8a)
)
-az
J( 0 o,
-
0o
(4.5a,b) once
on z=0.
in x the
(4.8b)
relations
q
Ho
=
"
€O
,
Z + Ur
3 o
1
az
ur
= --
are obtained.
(4.9a)
z = 0,
on
(4.9b)
Using (4.9a,b) in the Jacobian nonlinearity
of (4.8a,b) it is easily
seen that the right hand side of
(4.8a,b) vanishes since
40) = 0.
J(4 0 ,
As a result
(4.8a,b) reduce to
32€1
-
ur
azax
which are
av1
ax
a31
32 1
2
-- (V2
ax
(z+ur)
1 +
-
-
az
2
h
-)
az
3a1
= 0,
+ q ax
= O on z = 0,
(4.10b)
identical in form to the 0(1)
problem
Therefore a solution with the same structure as
be added at
(4.10a)
this order.
(4.5a,b).
(4.6) might
However, this solution can be
57
eliminated by renormalizing
40.
Alternatively, if 40 is
fixed as specified in (4.6) we need
solution at
not
this order.
Based on the experience of Pedlosky
a new nontrivial solution
(1981a,b)
must be included which corresponds
correction
include the similar
to the zonal current.
(1981) and Plumb
satisfying
to a wave
(4.10a,b)
induced
This solution which can be
written
€1 =
1j
(4.11)
(y,z,T),
has the potential vorticity
a 2 z1
H1
a 2 ¢1
¢i1
-
+--
-
h-1
az 2
ay 2
(4.12)
az
Its evolution and structure will be determined by the
removal of
zonal secularities at 0(E3).
might have been
zonal
by
secularities at 0(s
the wave
Ekman
included at 0(1).
field
friction.
and
Thus,
is
2
its
However, removal of the
) would show that
damped
by Newtonian
if the 0(1)
zero initially, it would remain
obviates
Such a solution
0 is not
forced
cooling and
zonal correction were
so forever which of course
introduciton.
Having specified the free stationary normal mode and
the zonal current, we proceed to higher order where the
effects of
topographic
forcing, damping and wave-mean flow
nonlinearity appear and determine the evolution of
flow.
the
58
The problem at
0(E2)
a
aH2
(Z+ur)--ax
+ qax
is
a
a
2
-
(-
+ A'--)
aT
ax
a2 o
-
J(1PIo) -n
'
- h-
(-
32
a
2
axaz
-
Using
J(
1
a
2
(--
=-
----
ax
aT
, --- ) az
(4.1d),
a
+
r'V
0o,
1)
(4.13a)
ao
a*1
A'--) ---ax
az
J(
(4.9) the
o,-)
az
- n'--az
(4.6) and
J(
ao
--- )
az
1
az 2
Ur
Ho -
on z = 0.
(4.13b)
inhomogeneous terms of
(4.13) can be evaluated and rewritten as
a12
(z+Ur)--ax
- n'K
-
+ q
2
ax
a
Fo
= [q----(-- + ikA'
z+ur aT
AFo(sin ly) -
Fo
ikqA ---z+ur
a2
ur
2
a
a
2
2
[ax
+ r'K 2 AFo (sin ly)
Fo
a
+ ikA'
(ur
9T
-
ikAF o
(4.14a)
+ n')A(sin
(4.14b)
The removal of the secular terms
the amplitude equation for A.
accomplished
operator which
ly)
(sin ly)]eikx +
Olyz
on z = 0.
can be
ly)
ly)
Oly(sin ly)]eikx + *,
axaz
ikFoAJly(sin
+ n')A(sin
by first
from
(4.14) results in
Removal of the secularities
operating on
corresponds to multiplying
adjoint solution of the homogeneous
(4.14a)
with
the
(4.14a) by the
side of (4.14) and
integrating over the domain, i.e. the operator is
-z/h
f
Fo
0
where
1
-
dze
f
z+ur
(
ly) e-l
dy2(sin
kx
(
),
0
) denotes a zonal average.
This will ensure
the
validity of the expansion (4.4) by removing those
inhomogeneous
homogeneous
operator
terms which have a projection on the
solutions of
to (4.14a),
and using
(4.7a,b),
(4.14).
results
in the equation
a
Fo(O)f
the
integrating by parts a number of times
1
-
Application of
dy2(sin
ly)
e - ikx
0
- -
ax
a3 2
["--2]
az
1
ur
z = 0
W -z/h
aA
Fo 2
dz
= (-- + ikA'A + n 'A) q f e
aT
0
2
(z+ur)
W
-
n'K
2
f
A
-z
CO
Fo 2
--z+ur
-z/h
0
Fo
-z/h
(--+ur-
x f dze
ikA f
(z+ur)
f dze
0
1
dz -
0
x
Fo 2
/h
e
dy 2 (sin
2
ly)
0
1
ly -
ikqA
f
dy2(sin 2ly)
0
2
)2
Oly•
(4.15)
0
Substitution of the boundary condition
(4.14b) into
(4.15)
with the definitions
1
Pl1
f 0 2 sin
2
1
ly dy,
1y 'H
(4.16a)
-fU1
u
f
ly dy =
sin21ly
(4.16b)
2 sin2ly ul dy
2
u1
(4.16c)
yields
Fo(O)
Ur 2
+
2
r'K
aT
Ur
ik
A(ullz Ur
1
- ull)
Ur
=
z=0
Fo 2
-z/h
x qf
dz - n'K
(z+u)
2
A f
2
aA
(-- + ikA'A + n'A)
aT
Fo2
dz
-_z+ur
-z/h
O
e
0
-
-
ikA'A + n 'A)
Fo2 (0)
-
2
Fo (0)
A + ikFo(O)M
--
aA
(--
-2
e
0
Fo 2
® -z/h
ik f e
0
- qull]
[(z+ur)pll
dz,
(4.17)
(z+ur) 2
or equivalently
O
aA
+ ikA'A + n'A)[q
(-
F o 2 (0)
2
[r'
+ K A
0
e
ikAf
(z+u
r
A(ullz -
Ur
z+ur
- [(z+ur)pll
)2
Fo 2 (0)
ik
dz]
-
2
e
0
Ur 2
Fo2
-z/h
0
Fo
F,2(0)
dz -
-
0
-z/h
2
(z+ur) 2
-n'
ur
+
e
0
aT
-
f
F0
-z/h
-
qull]
1
_-- ull)
Ur
dz
= ikFo(O)M.
z=0
(4.18)
This is the equation governing the evolution of the wave
amplitude A which is
forced by sinusoidal
topography of
amplitude M, damped by Ekman friction and Newtonian cooling
and which interacts with the correction to the zonal flow.
61
flow is as yet
However, the correction to the zonal
undetermined;
thus, we seek an equation for
correction driven by the wave
be obtained
from the 0(E
3
the zonal
flow
Such an equation can
field.
) zonally averaged problem which
is
written
aIl
32(1
--+ n'(--- haT
az 2
+ J(0
2
= 0
,fo)
3201
1
3D1
-- ) + J(
az
o,112 )
(4.19a)
,
a2~1
D1
+ r'
+ n'
3y 2
aTaz
+ J(0 2
az
a¢2
-)
az
+ J(co,
a o
, --- ) + J(o 0 , b) = 0.
3z
(4.19b)
the time change of
In (4.19a,b) the first terms represent
x-independent potential vorticity and
respectively.
thermal
terms
temperature
Terms proportionaln' and r' damp the O(s)
and momentum fields respectively.
of
(4.19a,b) can be rewritten as
J(fo,n2)
+ J
+
J(fo,,2z)
(02,
J(2,
Ho)
oz)
= -ay
(Vo
=
(voe2
ay
where
Vi =
The Jacobian
ix,
2
+ V2 fo),
+
(4.20a,b)
V260),
(4.20a,b)
8i =
Thus,
iz-
it is clear the nonlinear terms
convergence of
produced by
the meridional
wave
the
fluxes of potential vorticity
the wave field in
the meridional
field in
(4.19a) and
flux of heat at
(4.19b).
the phase shift
topography.
the convergence of
the ground produced by
The final term of
form drag caused by
relative to the
represent the
Note Ekman
(4.19b) represents
in the wave field
friction and form
drag do not act directly on the interior of
described by
(4.19a),
discussed by Pedlosky
boundary condition
It
zonal
in
contrast
(1981).
to
the
the flow
the layered cases
Rather they only
enter in
the
(4.19b).
is possible to evaluate the nonlinear terms of
flow equations
(4.19a,b) or,
the
equivalently according to
(4.20a,b), the convergence of potential vorticity
and heat
fluxes without explicitly calculating
To do this
we first use
(4.5a,b) to rewrite
2 and H 2 .
the potential vorticity and
heat fluxes as
a
-ay
a
(vo2
+ v2Ho)
= -
a
_
_2a2
-- (vo2 + v20) = ay
ay
aH 2
o ax
-
o (-
-
ay
q
z+ur
a 2
-),
ax
(4.21a)
1
3 2
- -)
axaz
ur 3x
(4.21b)
+
+
Then by substituting the O(E2 ) equations (4.14a,b) into
(4.21a,b) in order to eliminate the 0(e2 ) variables in terms
63
of the 0(1) wave field, the convergence of the fluxes can be
written
Fo 2 (z)
a
(voHI2
ay
+ 2n'K
+ v 2 Ho)
2
a
IA1 2 -
(z)
]
-
(sin
F 0 2 (0)
2
+ v26o)
Fo 2 (0)
2
r'K
ur
2
=
z=0
Finally using
1A1
]
Ur 2
a
- - (sin
ay
2
(sin
A* M)-
2
ly),
2
ly)
- ikFo(0)
(4.22b)
(4.22a,b) to evaluate the
in the equations
for
the zonal current
in terms of the 0(1) wave field,
(4.19a,b),
a2 ¢ 1
az 2
a3
32 1
--. (----+
3y 2
aT
aD 1
h- 1 -)
az
[-
r
+ 2 n')
(-aT
)2
(sin 2 ly)
x--
3¢ 1
-
az 2
h
-
)
az
K 2 Fn 2 (z)
q
(z+u
yields
3201
+ n'(--
Fo 2 (z)
+
(4.22a)
ly)
(4.20a,b) and
nonlinear terms
2
+ 2n')
a
(-+ 2n')
aT
[
a
-
x (AM*
2
ay
Z+Ur
a
(vo,
-ay
a
(--aT
[-q
(Z+Ur)
Fo
2
=
+ 2n'
] A1 2
z+ur
= 0
(4.23a)
a2 t
a32
1
a
1
+
+ r'
aTaz
'
az
2Fo 2 (0)
-
r'K
+ [.
-
3y 2
2
]
1A1
2
F 0 2 (0)
1
Ur 2
a
(--- + 2 n')
aT
a
-
(sin21y)
= 0
.
(4.23b)
64
In obtaining
(4.23b) note the form drag term present in
(4.19b) has been cancelled by a term of the same form but
opposite sign arising from the evaluation of the nonlinear
As a result, both the convergence of the
term in (4.19b).
flux of potential vorticity in (4.23a) and the
meridional
sum of the form drag and the convergence of the meridional
flux of heat at the ground in (4.23b) produced by the wave
field are written solely in terms of wave transience and
There is no explicit topographic contribution to
damping.
the wave forcing of the zonal flow which consists of the
convergence of the meridional potential vorticity flux and
the sum of the form drag and convergence of meridional heat
flux.
Therefore, as in Pedlosky (1979, 1981) the wave
forcing of the zonal flow vanishes, as it must according to
the noninteraction theorems, for steady waves in the absence
of friction.
However, just because there is no explicit
contribution from topography in the wave forcing of the
zonal flow does not mean the wave forcing is completely
In fact, from the energy
independent of the topography.
form of the wave field evolution equation (4.17), i.e.
0
-z/h
e
[qf
0
Fo 2
(Z+ur) 2
r'K
2
n')
|A
2
=
[
z+ur
2
Fo 2 (0)
]
Ur
1A1 2
2n'K 2
-
a
Fo 2 (0)
dz]
e
0
-
a
(-+
aT
F0 2
-z/h
x
dz
-
ikFo(O)(AM*
.-
Ur 2
-
(-
+ 2n')
aT
A M),
(4.24)
65
it
is
easily
zonal
seen that
the
wave transience
forcing
of the
flow in the absence of damping is due to the
topographic
form drag.
Thus, the
form drag drives the wave
transience which then drives the zonal
according to
flow correction
(4.23a,b).
In general, at every height z the meridional
flux of
potential vorticity has contributions due to the convergence
Reynolds stress and meridional heat
of both
-z/h
a
--
-z/h
(vo112 + V2 1o)
-z/h
a
+ -e
az
(e.g.
i.e.
Pedlosky, 1979),
e
flux
(Voe
2
= -e
(U o V 2
ay
+ u 2 Vo)
+ V28 0 )
(4.25)
Thus both terms contribute to the wave forcing in (4.22a)
and in the interior zonal flow equation (4.23a) which is the
convergence of
this potential vorticity
density weighted vertical
integral of
c
x
(z-ur)
dz]
2
1A1 2
(sin
2
ly)
-z/h
e
0
+ 2n'K
z+ur
®
f
2
+ 2n')
(-aT
Fo 2
S-z/h
f e
0
= -
then integrate
a
Fo 2
dz
o
(4.25) and
to obtain
e
qf
[-
-z/h
To show this we use the
(4.22a) in
over the vertical domain
However, the
integral of the convergence of the
Reynolds stress vanishes.
meridional
flux.
(uo2
+ u 2Vo)dz
-
(vo
2
+ v2
0o
)
z=
0
(4.26)
Then using
the meridional
integral of the evaluation of the
convergence of the heat flux at the ground
(4.26),
eliminate the heat flux in
(4.22b) to
in
we obtain
-z/h
[-
-z/h
Fo2 (Z)
oe
qf
a
dz
(z+ur) 2
0
+ 2 n'K 2 f
(+ 2n')
aT
0
x
A
dz]
x
2
(sin
2
ly) = -
f
-
[
-
ur
2
a
+ 2 n')
(-aT
z+ur
a
-z/h
-
e
0
Fo 2 (0)
Fo 2 (z)
c e
(uov 2 + u 2 vo)dz
ay
2Fo 2 (0)
r'K
-
2
] A1
2
(sin21y)
ur
+ ikFo(O)(AM* - A*M) sin 2 ly.
(4.27)
Now, if the modular wave field equation
(4.27),
the desired result obtains,
0
f
-z/h
e
o
(4.28)
integral of the convergence of
meridional momentum flux is zero.
redistributes
i.e.
a
(uov 2 + u 2 Vo)dz = 0.
ay
Therefore the vertical
convergence of
(4.24) is used in
As a result the
the momentum flux at each height z merely
the x independent potential vorticity
1
in
the vertical but does not alter
f
e-z/h(vo
f
e-z/h
E2 + v 2
1O)
dz
and
Eq.
H1 dz.
(4.28) also has a consequence on the vertically
integrated x-independent momentum balance.
In general,
the
67
vertically integrated x-independent momentum can be changed
only by friction such as Ekman friction, the vertically
integrated Coriolis torque due to the wave induced
meridional circulation and the vertically integrated
convergence of the Reynolds stress.
By (4.28) the last of
these is zero leaving only friction and the Coriolis torque
to alter the vertically integrated zonal momentum.
The
Coriolis torque can be related to the form drag using the
zonally averaged heat and mass conservation equations for
the wave induced meridional circulation.
However, we can
more directly obtain an equation for the changes in the
vertically integrated x-independent zonal momentum in terms
of friction and form drag, by first multiplying e-z/h times
(4.23a) or equivalently times
a2 (1
a
-
a2 1
(----
-h-1
az2
ay2
aT
a
+
a2,1
1
+ n'
-)
a 1
-
(-
3z2
az
a
h-
-)
az
a
- -- (uov 2 + u 2 v o ) + ez/h ,
e-z/h (vo6 2 + V28o),
az
ay
and integrating in z using (4.28) to obtain
a
--
aT
a2 ¢ 1
(-
+
ay2
1
a2c
-
az 2
a( 1
1
-
h--.
)
a
= _- -
az
ay
(Uo
2
+ u2 v o
)
a
+ ----e-z/h (Voe 2 + v26o),
e-z/h az
(4.29)
Then using the evaluation of the heat flux at the ground
(4.22b) and the bottom boundary zonal flow equation (4.23b)
in (4.29) and integrating once in y using the sidewall
68
boundary condition (4.3d),
the desired relation can be
written
aul
f
0
r'
=-
dze-z/h
U1
(z=0)
aT
+ ikFo(0)(AM* - A*M)(sin
2
ly),
(4.30)
where
1 /3ay.
ul = -a
Thus the vertically integrated momentum at a given latitude
is damped by Ekman
topographic
friction at the ground and
form drag at the ground.
The
forced by the
;sign of the
topographic effect depends upon whether or not the wave
is
leading or lagging behind the topography.
Note the form drag here is proportional to AM*-A*M as
it
is in the barotropic case discussed by Pedlosky (1981).
However,
this
is in contrast to his barocliic case where the
form drag is proportional to
this
(dA/dT)M* - (dA*/dT)M.
Again
is a reflection of the different properties previously
alluded to of
the 0(1)
stationary waves in each case.
Pedlosky's two-layer baroclinic case
the 0(1)
wave is nonzero in the upper layer only and
directly forced by the topography.
development of
transient
form drag
(in the absence of
wave field forces an 0(s)
proportional to dlAI 2 /dT.
friction)
zonal flow
stationary
therefore is not
As a result,
relies on the
In
the
following essentially
process:
the 0(1)
in the lower layer
This interacts with the O(e)
topography to produce an 0(2) wave
field in the lower layer
69
The 0(E
proportional to dA/dT.
2
to produce an 0(E
the topography
) wave then
3
)
interacts with
form drag proportional to
This is in contrast to the present
(dA/dT)M* - (dA*/dT)M.
case and Pedlosky's barotropic analysis where the normal
mode of amplitude A is directly
hence,
forced by the topography and
it directly interacts with the 0(E 3 ) topography
produce an 0(E
3 )
In closing
to
to AM*-A*M.
form drag proportional
this section we restate the evolution
equations and compare them to those derived by Plumb
the
nonlinear equations describing
The closed set of
(1981b).
flow interaction
topographically induced wave-mean
consists of
(4.18),
(4.23a,b).
With ul = -
a
1
/3y
they can
be rewritten
aA
0
-z/h
(--- + ikA'A + n'A)[q f
e
0
aT
Fo 2 (0)
+ K A[r' -n'
ur
x
O -z/h
f e
0
f
F0
)2
dz
2
(0)
Ur 2
]
Fo 2
dz]
e
-
ikA
2
(z+ur)
0
Fo2
[(z+ur)pll -
qull] dz
(z+ur) 2
aull
Fo 2 (0)
A (--+ ik --az
ur
aul
a
a2 u 1
(----I
+--2
2
az
ay
aT
Fo 2 (z)
[-
2
(z+
-z/h
C
2
F0
h-1 .)
az
a
(z+ur)
2
aT
+ n'(
aul
-
az
(4.31a)
ikFoM,
=
a2 u 1
ul
(-
q
1
- ull)
z=0
ur
h-
2
-
)
az
Fo2(z)
+ 2n')lA12
+ 2n'K
2
A 12
z+ur
70
32
x --ay2
(sin)
2
(ly),
(4.31b)
a2 u 1
au I
=
+ r' + n'--az
3y 2
aTaz
32u1
F0
2
(0)
[
ur
2
a
(--+
aT
2n')IA1 2
a2
Fo 2 (0)
- 2
ur
r'K21AI
2 ]
(sin21y),
(4.31c)
ay2
where
a2 u
1
f-
P11 =
2 (sin21y)
(-
0
1
ay2
a 2 uI
+ az 2
- h
1
aul
-)dy,
az
(4.32a)
1
Ull
2 (sin 2 ly) ul dy.
= f
0
(4.32b)
The zonal velocity ul must satisfy the boundary condition
(4.33)
aul/3T = 0,
on y = 0,1.
The evolution equations
in the
(4.31a,b,c,d), even
absence of Newtonian cooling and
Ekman friction differ
fundamentally from those derived by Plumb (1981b).
in contrast
to Plumb's model the present model does not
a top which here eliminates
reflection.
have
the possibility of any spurious
However, since the normal modes of
the Charney
problem are vertically trapped, the constraint of
would
First,
a top
not alter the basic physics of the wave mean flow
interaction.
parallel other
More importantly, Plumb,
imposed forcing
lead by his desire
treatments of
stratospheric
to
71
warmings, is not considering real topographic forcing which
in effect eliminates the feedback between the zonal flow and
wave field at the ground.
This is evidenced by the absence
of the entire right hand side of (4.31c) and the last term
on the left side of (4.31a) in his comparable equations
(2.26) and
(2.39).
This is not important for the existence
of the initial topographic instability.
Such a feedback is
only necessary when describing the later stages of real
topographic instability.
However, the presence of the
feedback does alter the growth rate obtained.
72
5.
The topographic instability and
its energetics
In this section we shall demonstrate the existence of
the instability
analysis, only
(4.31) become
aA
(-- + ikA'A)[q
aT
-z/h
-
e
ikA f
0
Fo
2
(0)
u-----r
ur
32u
1
3y 2
3T
aT
-z/h
f
3T3z
F0
e
(Z+ur)
0
2
dz -
2
(0)
------ ]
Ur
2
Fo2
-- 2 [(z+ur)pll - qull] dz + ikA
(e+r--
aull
(---az
-
Ull)
=
(5.1a)
ikFo(O)M,
z=0
ur
32U1
+ ---- - h-1
a3 2
Fo
aul
-___ =
2
diAl
(z)
2
_
3 z
(Z+Ur)
2
dT
ly),
Fo 2 (0) diAl
2
dT
---Ur-----dT
Ur 2
1
Fo2
(z+ur )2
a2
(sin 2
x -2
ay
32u
in the absence of damping
Without damping the evolution
will be demonstrated.
equations
In order to simplify the
instability.
the topographic
(5.1b)
2
32
-3y2
(sin
2
ly),
(5.1c)
on z = 0, and
aul/
3T
on y = 0,1.
= 0,
(5.1d)
73
The steady
examine
(5.1) whose stability we will
solution of
is
Fo(O)M
As =
Fo 2 (O)
A'(qI1
Ur 2
Uls
(5.21)
O,
0
(5.2b)
where
Il
=
Fo 2
-z/h
f
e
dz
0
(z+Ur)
2
For simplicity M and therefore As is
taken to be real.
The
steady solution corresponds to a topographically forced
wave.
Since the wave is steady and
inviscid, by
know there is no form drag associated with it.
resonance
(A' = 0)
(4.27) we
Note exact
can't be considered because the steady
wave amplitude would be infinite
in the absence of damping.
The instability of the topographically forced wave
(5.2) can be studied by linearizing
solution.
A = As
ul =
Therefore if we write
+ A'
'
,
the linearized equations are
(5.1)
about the steady
74
aA'
(-- + ikA'A')[qI
aT
S--- U' l l )
=
1
-
Fo 2
-z/h
-
ur 2
qu'll]
-
x [(Z+ur)p'll
Fo 2 (0)
f
ikA s
e
(z+ur) 2
0
dz + ikA
Fo 2
--ur
s
au'll
(--az
O,
Ur
z=0
(5.3a)
32 u'
32 u' 1
S(---- +
2
3z
ay 2
aT
a
dAr' a2
x ---ay2
dT
(sin
- 2
Ur
aTaz
au'1
- h 1 ---- ) = - 2q
az
As
(Z+ur) 2
ly),
2
dAr' 32
-As dT
ay2
2
(sin
is the real part of A'.
where A',
Fo 2 (z)
(5.3b)
F0 2 (0)
a3' 1
2
1
ly)
on
Z=0
,
(5.3c)
A'i will be the imaginary
It is particularly easy to calculate pll by
part.
integrating over the
(5.3b) by -2 sin21ly and
multiplying
meridional domain using the definition
of this
(4.32a).
The result
is
Fo 2 (z)
ap 1 1
-
2ql
2
(z+Ur) 2
3T
dAr'
As ---dT
,
which when integrated in time yields
P11 =
-
2q1 2 F 2 (z)A2
)
(z+r
(5.4)
75
the purpose of
plus a constant which for
to zero.
problem can be set equal
forced to solve
In order to obtain ul we are
sought
u'
(5.3b-d)
the solution to ul is
(1979),
Following Pedlosky
directly.
the stability
in the form
1
uj
=
sin jy,
(z,T)
j
(5.5)
(5.5) satisfies
Note
where j = Jn, J = 1,2,3,....
sidewall boundary condition (5.1d).
the zonal
flow equation
(5.3b,c),
the
Substituting (5.5) into
multiplying each by sinjy,
integrating over the meridional domain and time yields
-
az 2
h-
1
2
uj
=
-
8q1 2
j2-412
az
Fo 2 (z)
x
AsAr' ,
(z+ur)
auj
-
2
(5.6a)
81 2 j
(1-(-1)
Fo2(0)
AsA'r
j2
3z
J )
Again, in obtaining
-
on
z=0.
Ur 2
412
(5.6b)
(5.6a,b) since the concern
stability problem and
is the
not the initial value problem, the
constants arising from the initial value problem are taken
to be zero.
The solution to
(5.6a) is of the
form
j (1-(-1)J)
uj = -8ql
2
AsAr'
j2 -
412
(Bj(z)
+ Cje-aJz),
(5.7)
76
where
1
aj =
1
+ (4h 2
2h
-
(5.8)
forced solution of the equation
and Bj(z) is the
32Bj
1/2
+ j2 )
Fo 2 (z)
aBj
-
-
h1
@z 2
j2Bj =
az
(Z+ur)
2
.
(5.9)
The constant Cj can be obtained by demanding
(5.7) satisfy the lower boundary condition (5.6b).
solution
The result of
this is
aBj(0)
F 0 2 (0)
)
----+
--az
qur 2
1
=
Cj
that the
cj
Then by (5.5),
,
(5.10)
(5.7) and (5.10) the correction to the zonal
current can be written
J
j(l-(-l)
u'
1
1
+ aj
which
= C j
Fo
2
8q42
j2
_ 412
aBj(O)
(O)
(-
+
) e-ajz]
qur 2
1
+ aj
32q1 4
= C j
F 0 2 (0)
(
(5.11)
2
(4.16b) yields
(1-(-1)J
j
(j2 _ 412)
AsAr'
[Bj(z)
aBj(0)
+
qu
sin jy,
az
in combination with
u'll
[Bj(z)
AsAr'
)
az
e-ajz]
(5.12)
77
Finally, using
(5.4) and
(5.12) in the
linearized wave field
equation
(5.3a) to write the wave-zonal flow interaction
terms of
wave amplitude results in the
following
in
equation
for the wave amplitude
A'/aT + ikA'A' - ik N O As 2 Ar'
= 0
(5.13)
where
Fo
No = -
2
-
(qIj
0
2
{ql f
)-1
Ur
2
j
(j
F 0 2 (0)
2
-
qur 2
(1-(-1)
Fo2(0)
j
(j 2 -
F 0 2 (0)
1
+--
(
aj
+
qur
2
dz
2
[Bj(z)
Fo2(0)
32q1 4
ur
J )
j
412)2
2
(1-(-l)J)
.
(j2 - 412)2
3Bj(O)
-)}.
az
Separating (5.13)
(z+Ur )2
(Z+Ur)
dz -
+
2
Fo 4
Fo2
-z/h
x [
qur
e
ff e
412)2 0
aBj
+ ---. )e-ajz]
z
(
-z/h
0
(1-(-1)J)2
+ 32q 2 14
1
+ --aj
(0)
[Bj(O)
(5.14)
into real and
imaginary parts and
solving for the imaginary part yields
d2 A
dT
-= k 2
(A'NoAs
2
-
A'
2
) Ai.
2
If a solution
(5.15)
is sought
Ai = Ai(O)eOT
in the form
(5.16)
78
the growth
rate a is given by
= (A'NOAs 2 =2/k2
From (5.17) we see
A' 2 ).
(5.17)
the topographically
forced wave is
unstable whenever
IA'NOI
> 0,
(5.18a)
1 > A'.
(5.18b)
and
IN O As
2
Thus in this continuously stratified case as in Pedlosky's
(1981)
and Plumb's
(1981a,b) analyses, instability exists
for either super- or sub-resonant
flow, depending on the
sign of NO .
if NO>O
According to
occurs on the
(5.18),
superresonant side of resonance whereas for
N 0 <0 the opposite is true.
NO is
(1981)
the instability
analogous
equation
It is
E2 (3k
to the quantity
(3.7).
In his
interesting to note that
2
-1
case the
2
)/8
in
Pedlosky's
side of resonance on
which the topographic instability occurs depends on the
relative meridional and zonal scales of the topography.
the present
case it also depends
of the Charney mode excited.
In
on the vertical structure
However, the exact parameter
dependence of N O on zonal and meridional wave numbers and
vertical
structure is extremely complicated.
As a result,
an explicit calculation of N O is done only for
the normal
Charney mode with the simplest vertical structure (i.e.
r0 =2)
Figure
under
the Boussinesq-like assumption
>>h
-1
.
In
(3.5.2a) the regions for which N O is positive and
M
14
1312
II
10
9
8
U NSTABLE
7
6
NEUTRAL
WAVES
5
NEUTRAL
4
WAVES
3
2
I
-I
0
Fig. (3.5.1)
I
2
Stability regime in
3
4
,i,"
space for the topographic
instability of the ro=2 normal mode of the Charney problem assuming
and taking
z1I.
I0
9
8
/
/ -)
7
6
5
I
-I-
*
4 -I
3
2
-II
O
(+)
*
/
-
I
2
3
4
5
6
7
I
I
8
9
10
k
Fig. (3.5.2a) Regions of wavenumber space for which
N>O and N<O. In the region below the dot-dashed line,
the quantity & S3k-t/8 is greater than zero. This
is the counterpart in Pedlosky's (1981)
barotropic
analysis to the more complicated N of the present
analysis.
81
negative are shown.
NO is positive in a band centered at
1 = 4.75 and in a wedge centered at
above the dot-dashed line
region
space
for which
topotropic
1 = 1.5
for k<6.5.
The
is the region of wavenumber
the nonlinear coefficient in the barotropic
instability
analysis of Pedlosky
(1981) is
greater than zero, i.e.
e 2 (3k 2 -1 2 )
>0
8
Thus it is clear the regions for which the topographic
instability
occurs for subresonant or superresonant flow
differ markedly from the barotropic model to the baroclinic
model considered here.
to the fact
This is due
contains information on the vertical
that now N O
structure as well as
information on the horizontal structure of the mode.
Figure
k=1=1.0
(3.5.1)
regimes
the stability
of the
normal
mode
(for which NO>O) are plotted against the topography
height M and the detuned zonal
flow A'.
feature of this diagram is its
demonstration that less
topographic height is required
for instability
is approached.
of topographic
instability
is
In
This
The principal
is reminiscent of the earlier studies
instability
which showed that topographic
However, as M
is a quasi-resonant phenomenon.
increased, the topographic
further from resonance.
instability can be excited
In Figure
(3.5.2b) the stability
regimes in wavenumber space are shown.
of the topography,
quite closely
as resonance
Due to the strength
the regions of instability
to where N O plotted
in Figure
correspond
(3.5.2a) is
I
UNSTABLE
I
NEUTRAL
WAVES
I
IUNSTABLE
I
0
:)
I
2
Fig. (3.5.2b)
3
4
5
I
I
I
I
6
7
8
9
10
Stability regime in wavenumber space for
the topographic instability
with A=LO
and
=1M.O
0
I
Fig.
(3.5.2c)
2
3
4
5
k
Same as (b)
6
except
7
8
)'=--.0
9
10
and M =S.O .
0
I
2
Fig. (3.5.2d)
3
4
5
k
6
Same as (c) except
7
8
'= -/.0
9
10
and
/
A = 7,5.
greater than zero.
These are, as we have already mentioned,
the only regions in which the instability can occur for
superresonant flow when NO>O.
However, note that not all
waves with larger zonal wavenumber which could be unstable
when No>O, are unstable.
Thus, the instability is limited
to the larger scales for which No>O.
In Figures (3.5.2c,d)
the stability diagrams in wavenumber space are plotted for
the subresonant flow A' = -1.0 and increasing topographic
amplitude M = 5.0 and M = 7.5, respectively.
As the size of
topography is increased, the regions of instability, which
now occupy regions in Figure (3.5.2a) for which NO<0, grow
toward larger values of k.
Figures (3.5.2c,d).
This can be seen by comparing
These regions of instability avoid, as
they must, regimes where NO>O which can be seen by comparing
Figures (3.5.2c,d) to Figure (3.5.2a).
Above the existence of the topographic instability was
demonstrated for the normal modes of the Charney problem.
In order to better understand this study of topographic
instability we now investigate its energetics.
The equation
for the wave field energetics of the inviscid topographic
instability is
a3 c
4 _
I
3T
e
0
2
2
oy + coz]dz
2
-z/h
6
= s6
-z/h 1
2
- [ox +
e
[Hox P2y + c2x coy]
uly dz
+
E4
f
-z/h
4
e
[#ox
4
c2z +
2x loz]
0
E 4 Ur
+
where
of
[
a
--az
(Ur + z) dz
[ obx] + 0(e 6 )
] represents
(5.19)
the meridional
average.
The left side
(5.19) is the time change of total wave energy, kinetic
The terms on the right side of
plus available.
(5.19) are
the rate of conversion of mean
in their order of appearance
flow kinetic energy to perturbation kinetic energy by the
Reynolds stress,
the rate of conversion of mean available
potential energy to perturbation available potential energy
by
the eddy heat
flux, and
the rate of conversion of mean
flow kinetic energy to perturbation energy by the form drag.
Note the Reynolds stress conversion of energy from the
horizontally sheared part of the
zonal
flow is O(s6) and
therefore negligible compared to the 0(e4) terms.
This
due to the absence of horizontal
zonal
flow.
The 0(4)
energy balance, i.e.
a
C -z/h
-- f
e
aT
0
x [Cox
€2z
shear in the 0(1)
is
2
1
2
- [Lox +
2
+ €2x
Poy
+
2
oz]dz =
0
-z/h
e
0
oz] -
(Ur + z)- dz + ur
az
(5.20)
is somewhat different than that of the
case considered by Pedlosky (1981).
[Obx],
layered baroclinic
Here both the heat flux
conversion of energy from the vertically sheared part of the
zonal
flow and
the form drag conversion of energy from the
zonal
flow contribute to the topographic
instability.
later
conversion was O(E 6
(1981)
case since the 0(1)
)
in Pedlosky's
zonal flow was zero
Thus, only the heat flux conversion
instability in his case.
topographic
and Devore
in the lower layer.
contributed to the
In barotropic models of
(1979) and Pedlosky (1981)
both the
topographic
the heat
baroclinic
instability such as those discussed in Charney
conversion contributes
Thus,
The
to the
only the
topographic
form drag conversion,
form drag
instability.
singly responsible for
instability in previous barotropic models, and
flux conversion,
topographic
singly responsible for
the
instability in previous baroclinic models, are
responsible for
the instability
in the present baroclinic
model.
It is a simple matter to calculate
the contributions
to
the time change of the total wave energy in the wave energy
equation
(5.20).
The left side of
using the solution (4.6) for (0,
(5.20) can be calculated
integrating by parts
employing the equations for the vertical eigenfunction
(4.7a,b).
The result for the time change of the total wave
energy is
a
-z/h
I
-aT
e
0
1
-
2
C
(q
f
0
1
2
- [ox
2
-z/h
e
2
+ Coy +
F0 2 (0)
F0 2
--
z+ur
2
oz]dz
dz -
---
diAl
2
) --
ur
dT
(5.21)
88
The two terms on the right side of (5.20) must
the y-averaged version
first integrate by parts in z and use
between
relation
of the
and heat flux
flux,
the vorticity
Reynolds
[Vo6
+ V26o]
2
= e-z/h [voH2 + V2
vorticity flux and the heat flux at the ground
the wave amplitude yields the heat
-z/h
+ €2x 4oz]
[Cox €2z
e
0
-z/h Fo 2
f
e
0
z+Ur
C
1
+ - ikur Fo(O)
2
Using
the 0(1)
form drag
dz
(AM* -
ol].
(5.22)
(4.22a,b) to write the
Then using the inviscid form of
1
- (q
2
stress
(4.25), i.e.
-e -z/h
az
f
the
conversion, we
flux
To calculate the heat
(5.21).
result
add up to
in terms of
flux conversion
a
3z
(Ur + z)dz
F 0 2 (0)
diAl 2
ur
dT
-
A*M).
(5.23)
solution (4.6) and
(4.1d) to calculate the
term yields
1
ur[#obx]
Thus,
the
field
- ikur Fo(O)
2
sum of the heat
drag conversion
wave
-
(AM* -
(5.24)
flux conversion
(5.24) is equal to the
energy and
is
A M).
proportional
from the inviscid form of (4.24),
i.e.
(5.23) and form
time change of total
to
31A1 2 /3T.
We note
-z/h
[qj
F0
2
F0
2
(0)
diAl
]
dz-
e
(z+ur ) 2
0
2
r
2
dT
= -ikFo(O)(AM *-A* M)
is entirely due to the presence of
the wave transience
Thus,
topography.
(5.25)
the topography has
acted as a catalyst
causing a normal mode of Charney problem, which is stable to
the usual baroclinic instabiltiy, to be unstable.
6.
Multiple equilibria
In the
interest of completeness, we now briefly discuss
the evolution of the
lead to the
initial
linear instability which will
consideration of the special
requirements
for
multiple equilibria in this continuously stratified model.
The nonlinear complex amplitude equation corresponding to
the
fully nonlinear set of
(5.1a-d)
inviscid evolution equations
is simply
dA
+ ikA'A dT
1
- ikNoA (1A1 2 2
JA(0)1
) = ikA'As,
(6.1)
where A s is given by (5.2a) and A(O)
the amplitude.
2
This describes the
oscillations which occur after the
is the initial value of
inviscid nonlinear
initial instability.
We
note however, that these oscillations never lose their
memory
of the
initial
of viscosity thus
condition.
far.
This
is
due to the neglect
Therefore, we now consider the
90
damped equations (4.31a-c) of this continuously stratified
baroclinic model.
This is done to show the existence of the
multiple equilibria to which the instability may tend hence
losing sight of the initial value.
Such multiple equilibria
have been shown to exist in previous barotropic models,
layered baroclinic models, and in the continuously
stratified model considered by Jacobs
(1980).
All the
multiple equilibria previously calculated were the result of
a balance between the form drag and the damping.
In
barotropic models Ekman friction is sufficient to produce
However, this is not necessarily the
multiple equilibria.
case in baroclinic models.
In a layered baroclinic model
Ekman friction only enters into the energy budget of the
layer on which it
acts.
As a result,
a two layer model only determines
the wave amplitude
remains
in
a lower Ekman layer in
the zonal flow in
the lower layer.
infinitely degenerate.
terms of
The upper layer
In a continuously stratified
model the situation is even more severe where the Ekman
friction only determines the zonal flow at the ground.
interior is left infinitely degenerate.
The
Therefore in order
to completely determine the steady problem, an internal
dissipation is necesary.
In the present model, the internal dissipation is the
Newtonian cooling.
(4.31a,b,c), i.e.
We now consider the steady form of
Fo 2 (0)
Fo 2 (0)
ikA'(q
2
-
1
2
F0 2 (0)
+ r'k
2
)A + [n'(qI 1 - k 12 -
]A - ikA
U
I
F0
qull]dz
e
[(z+ur)Pll
(z+ur) 2
0
Ur
-
F0 2
-z/h
_
+ ik
)
Ur 2
2
(0)_
aull
1
A (--- ull)
ur
az
ur
= ikFo(O)M,
z=0
(6.2a)
41 2 Fo 2 (z)
u) 2
(z+ur) 2
aul
Sh 1
az
32-U
U
3z 2
2
[(Z+Ur)k
2
-
q]
I1I
cos
21y
(6.2b)
a2TUl
aul
r' ---az2
41 2 Fo 2 (0)
(n'
n'az
ur
-
r'urK 2 ) A
2
cos
21y
2
(6.2c)
on z=0,
where
f
I1 = I
-z/h
e
0
Fo2
(z+Ur ) 2
-z/h Fo 2
---f e
0
z+ur
dz
(6.3a)
00
12 =
dz
(6.3b)
The steady problem (6.2a,b,c) is similar to
the one
considered by Pedlosky (1979) where Newtonian cooling was
also introduced to break the conservation of potential
vorticity
study,
in the interior of
the steady limit of
the fluid.
Here, as in that
(4.31b) is singular
in the y
92
layers at
boundary
according
steady
narrow
the sidewalls whose widths
to T 1 / 2 as
the solution approaches the steady
state is approached, the
Therefore, as a steady
state.
to narrow transient
This singularity gives rise
interval.
zonal flow equation (6.2b) is valid over a larger
(6.2b) can
As a result
domain.
portion of the meridional
be
steady zonal
used to calculate the frictionally induced
flow.
for the zonal flow and substituting the result
Solving
in the steady
(6.2a) results
into the steady wave equation
complex amplitude equation for
Separating
A.
that into real
and imaginary parts yields
Fo 2 (0)
-kA'Ai + (qI1
+ r'K
2
-
)-1
r 22
Fo 2 (0)
-------- ]Ar ur
Fo2(0)
kPAi
-
[n'
(qI1
2
+ Ai2)
(Ar
(qIl
= 0,
(6.4a)
F0 2 (0)
Fo 2 (0)
kA'Ai +
-
)-1
[n'
(qI
K212)
1
Ur 2
2
+ r'K
2
Fo2(0)
- K 2 12 )
- 2
Ur
2
]Ai + kPAr (Ar
Ai 2 ) = kA'As,
+
(6.4b)
ur
where As is given by
(2.5.2a) and P is
Fo 2 (0)
P =
(qI1
=
)-1
{-
2
Ur 2
0
Fo
x (u(z)
-z/h
e
f
Fo2
-
(z+ur) 2
2
+ De-z/h)dz z+ur
((z+ur)K 2 -q)]
[412
(z+ur)
93
C
e
f
+ 2q
Fo 2
-z/h
Fo
+ 2
(u(z) + De-z/h)dz
-
0
2
(z+ur)
2
(0)
.
ur 2
h-1 D -
(0) -
z
^
(u(0) + D)]},
1
ur
A
[u
(6.5)
where u is the forced solution of
Fo 2 (z)
A
uzz -
h-1 U z =
[(z+ur)K
2
-
q]
2
(z+ur,)
and
D
-41
n'Uz(0)
=
2
Fo
r'u(0)
2
n'
(0)
2
2
)
(
...
41
- r'urK
ur
+ h-ln,
r'
It is very difficult to solve
However, in the limit of small
41 2 r'
2
+ h-inr
(5.33a,b) in general.
friction it is relatively
simple to show the existence of multiple equilibria.
'+0 keeping
can be found
the ratio
in the
Ar = Ar(O)
Ai
Note
=
+
r'/
fixed
a
solution to
r'
(6.4a,b)
form
n' Ar
( 1
),
(6.6a)
1
)
(6.6b)
+ n' Ai(
from (6.4a) Ai
(0
PAr( 0 )3 + A'A(
0
The cubic
'
If
) vanishes.
) - A'A s = 0
(6.7) has multiple real
multiple equilibria exist when
Ar( 0 ) satisfies
(6.7)
roots or equivalently,
94
A'/P < - 27/4 A 2 s .
(6.8)
The essence of this result
only for
found
a sufficiently detuned zonal
flow.
This was also
to be the case in the earlier studies of topographic
instability.
that
is that multiple equilibria exist
The result
(6.8) is particularly analogous to
obtained in the barotropic analysis of Pedlosky (1981)
where it is also demonstrated that
for sufficiently detuned flow.
upper bound
multiple equilibria exist
In addition, he finds an
on the detuning parameter
for multiple
equilibria to exist which is inversely proportional to the
Ekman friction.
Therefore as the Ekman friction is
decreased the range of the detuning parameter for which
multiple equilibria occur becomes larger and
in fact becomes
infinite as the Ekman friction approaches zero.
present study, the condition
Ekman
friction approached
(1981),
it
is
not possible
the limit of very small
upper limit, the
larger.
(6.8) was obtained assuming
zero.
As is
to obtain
friction.
the case
an upper
in Pedlosky
limit on
A'
in
In order to obtain this
Ekman friction must be allowed
In contrast to the case
calculation
In the
to be
studied by Pedlosky, this
is technically difficult and therefore is not
carried out here.
Previous studies
(e.g,. Jacobs
that we can expect one of
(1979))
the equilibria to be unstable.
However, to describe the evolution of the
the unstable
have demonstrated
equilibrium requires
the
solution away from
solution of
the fully
95
transient and damped evolution equations
(4.31a-c) which is
beyond the scope of the present work.
7.
Conclusions
In this chapter, the fully transient evolution
equations were derived governing the topographic instability
of a general normal mode of the Charney problem.
These
evolution equations include the wave-zonal flow interaction
at the ground not found by Plumb (1981b) due to the nature
of the forcing used in his problem.
The growth rate of the
topographic instability was obtained in analytic but
unclosed form for the normal mode of the Charney model with
the simplest vertical structure under a Boussinesq-like
assumption.
Thus, this calculation is not quite as explicit
as that in the two-layer model
(Pedlosky, 1981), but it is
easier to follow than the analogous calculation in the study
of topographic
instability in a continuously stratified
model by Plumb (1981b).
It was found that while the
wave-zonal flow interaction at the ground
(not present in
Plumb's (1981b) analysis) is not required for the existence
of the topographic instability, this interaction does modify
the growth rate of the disturbance.
The topographic
instability is excited most easily near resonance as in
earlier studies and exists for both subresonant and
superresonant flow depending on the sign of the nonlinear
interaction coefficient N O .
Multiple equilibria also exist
in this model for a sufficiently detuned flow as in the
96
earlier studies.
superresonant
The regions of subresonant and
instability differ markedly from those in the
barotropic model.
In the present model the superresonant
instability is very limited in meridional wavenumber.
This
strong dependence of the instability on meridional
wavenumber is due to the strong meridional dependence of the
nonlinear wave-mean flow interaction.
The fact that
meridional dependence is not a feature of the barotropic
analysis of Pedlosky (1981) suggests that this is due to the
more realistic vertical structure in the present model.
However, the role of the channel walls in this regard should
be investigated in the future.
The subresonant instability
is not so limited in meridional wavenumber but for both
subresonant and superresonant flow the larger zonal scales
are most easily excited by the topographic instability.
Overall this suggests that the topographic instability is
quite sensitive to the vertical structure of the mean flow
and normal mode under study.
The strong dependence on
meridional wavenumber suggests that given the zonal flow in
the atmosphere, one might be able to locate a stationary
instability with a fairly well defined meridional structure.
However, we must be cautious about the applicability of the
instability found at very large space scales in this model.
At these scales, the quasi-Boussinesq approximation breaks
down and moreover, the quasi-geostrophic theory
begins to
break down. Thus more realistic models of the large-scale
atmospere should be considered in the future.
97
Chapter 4.
1.
The Charney mode under the influence of
topography.
Introduction
In this chapter we examine the behavior, in the
presence of topography, of the Charney mode which is either
weakly stable or unstable and embedded in a uniform shear
flow.
The two questions to be answered are:
(1) what
effect does topography have on the unstable Charney mode and
(2) does the presence of topography lead to instability in
parameter regimes which in teh absence of topography would
be stable?
The studies in the next four sections follow the
work of Pedlosky
(1979) quite closely with the important
exception that weak topography is allowed as well as a weak
uniform zonal flow in some cases.
In each section either
the size of topography or the size of the weak zonal flow
are altered in order to examine the above stated questions
in different parameter regimes.
Pedlosky
(1979) considered
the finite-amplitude dynamics of a weakly unstable Charney
mode.
To examine this oproblem he employs the method of
multiple time scales and pivots the analysis about the point
of neutral stability to obtain the necessary time scale
separation.
The expansion parameter is then chosen such
that the amplitude of the weak instability can be determined
by the wave-mean flow nonlinearity.
We shall see that this
wave-mean flow nonlinearity will be important in modifying
the instability.
In Section 2 the effect of sinusoidal topography on an
infinitesimal weakly-growing Charney mode, of the same
98
the topography, will
structure as
(1981)
studied this question in a two-layer baroclinic model
and found the effect
There is a
to be stabilizing.
in section 2
difference between his analysis and the one
which is a consequence of structure of
neutral wave amplitude does not vanish
it is necessary to require that
topography
only
the neutral
two-layer model the
In the
baroclinic wave in each case.
Thus
Pedlosky
be studied.
in the lower layer.
it be in phase with the
so as not to produce a form drag.
produced by the transient
growing wave.
Form drag is
phase shifted part of the
In the continuous model the neutral Charney
mode vanishes at the ground making such a requirement
unnecessary
but the form drag is still produced only
transient phase-shifted part of the wave.
The effect of
of the Charney
topography on the instability
2 is a result of the topographically
by the
mode in section
induced wave-zonal flow
interaction at the ground.
Section 3 is quite similar to section 2 but a very weak
uniform zonal flow is allowed.
flow by
the topography
The diversion of this zonal
forces a nearly
neutral Charney mode.
The presence of this forced wave futher alters the stability
properties of the previous section by
of
the wave-mean flow
further modification
interaction at all depths of the
fluid.
In section 4 the
size of
increased again. As a result
the uniform zonal
flow is
the topography must be assumed
to be weaker in order that the Charney
mode is not too
99
strongly
forced.
The consequence of
this
the portion of the wave-zonal feedback
at
was solely responsible for modifying the
section 2 and partially responsible
is to eliminate
the ground which
instability
in section
3.
in
However,
the uniform zonal velocity is now so large that the
advective time scale associated with it is of the order of
the long time scale for the wave-mean flow interaction.
Therefore,
the weakly
unstable Charney mode is now
propagating at the speed of the zonal flow in the absence of
topography.
Plumb (1979) considered an analogous problem in
the two layer model with propagating topography.
analysis is restricted to weak
However, he was
baraclinic stability.
able to show even in baroclinically stable
parameter regimes that there was
topography were strong enough.
an instability
the neutral modes
stationary and therefore coherent with the forced
wave which allowed a strong wave-zonal
leading to the instability.
energy
if the
He found taht as the
topographic height increased, one of
became
His
for the instability
potential energy
In addition, it was
shown the
is drawn from the available
of the mean flow.
examine this effect
flow interaction
In
for baroclinically
section 4 we will
subcritical
flow and
also the effect of increasing topography on teh propagating
baroclinic
instability
supercritical flow).
problem to that
Ekman
(i.e.,
also for baroclinically
Section 5 contains an analogous
considered in section 4 but with
friction and small Newtonian cooling.
strong
100
2.
Effect
of topography on a stationary weakly growing
Charney mode.
In the previous chapter
modes of
the nearly stationary normal
the Charney problem were shown to be unstable due
to the wave-mean flow interaction for which the topography
acted as a catalyst.
in altering
In this chapter the role of topography
the baroclinic
instability of the stationary,
nearly-neutral Charney mode is examined.
this alteration
is due to
We will
show that
the topographically induced
modification of the wave mean flow interaction which,
absence of topography, arises
in the
from the finite-amplitude
equilibration of the stationary, weakly unstable Charney
mode.
Pedlosky
(1979) studied the equilibration of the
weakly growing Charney mode in the absence of topography.
The present analysis is identical
to that in his section 3
with the important exception that weak O(s)
topography
sinusoidal
is included here.
The analysis pivots about the
stationary neutral
Charney mode for which, according to
1/2
Bo + h-1 = (h- 2 + 4K 2 )
(3.3.10),
(2.1a)
,
and
U
=
(2.1b)
0.
As described in seciton
(3.3),
if
is perturbed from the
value defined by (2.1a) by an amount of 0(62),
6 = Bo - E2
6
,
i.e.,
(2.2)
101
the growth rate
choice of the
is
for
0(e)
6 > 0.
This
fact motivates
long time scale
T = et.
If
(2.1a,b),
(2.3)
(2.2)
and
(2.3) are used
in the potential
vorticity equation (3.2.3) and
the Newtonian cooling is
assumed
0(-
to be very small
z ---
the
+ qo
-m
ax
ax
qo =
So + h-1
= -
(n
<<
2
)),
(3.2.3)
e-- -
becomes
),
ax
aT
(2.4)
where
(2.5)
Upon inspection of (2.4) it becomes apparent that there is a
singularity near the ground.
near
the
the
0(e)
ground
"O(E))
slow time scale
for z ~ 0(1)
shear of
(z
This
is due
the advective
of the
to the
fact that
time scale matches
temporal
evolution,
whereas
the advective derivative due to the vertical
the basic state flow is 0(1).
In order to deal
with this critical layer singularity, the
inner
independent
variable
z
e
is defined.
(2.6)
The inner stream function 4(x,y,C,T)
satisfies the potential vorticity equation
then
102
A
a
T
x
Sa
= -J
2
-
3a 2
ch-1
2
(e,
a
)
+
E(q
-
E26)
ax
a
-
62V2
+
-
3~
E h-
2
2V 2 ~ ),
+
a C2
subject
A
32a
a
-- )(-
+
(--
A
ac
(2.7)
to a boundary condition at the ground.
If we choose
the topography according to
n = eb = eMeikx
(sinly)
(2.8)
+ *
and if the Ekman spinup time is assumed to be of the order
of
the
long time
scale, i.e.,
r = er'
(2.9)
then the lower boundary condition for
on
=0 can be
written
A
-
A
-
3T
A
+ J (,
ax
-- ) = ac
r'
-
J(,b).
(2.10)
Therefore, the potential vorticity equation
defines the outer problem for
whereas
(2.4)
the potential
vorticity equation (2.7) and lower boundary condition (2.10)
A
define the inner problem for
in powers of
solved
e
A
.
By expanding both
both the outer and
4
inner problems can be
independently yielding the equations
amplitude and the associated zonal
and
for the wave
flow correction.
Then
the problem can be closed by the asymptotic matching of the
inner solution to the outer solution which passes
103
information about
boundary
the effect of
topography and other lower
information in € to the
fluid away
from the
boundary described by €.
the outer problem.
We begin by considering
The
appropriate outer expansion is
€ =
Using
po
€1i
+
(2.11)
+
2~ 2
+
(2.11)
....
in the outer potential
results in a sequence of problems
C.
The 0(1)
vorticity equation
(2.4)
for consecutive powers of
problem is
affo
a¢0
z --+ qo =
ax
ax
0,
(2.12)
and leads to the specification of a single stationary
neutral
Charney mode which can be written
o = AFo(z)eikx
(sinly) + *
,
(2.13a)
where
-aoz/2
Fo(z)
and
= ze
,
(2.13b)
Bo is given by (2.1a).
It is
interesting to note that
the vertical eigenfunction given in
(2.13b) vanishes at z=0.
Therefore the stationary neutral Charney mode does not
interact directly with the
In
addition
note that
topography to produce
€o near
the ground
is
form drag.
only O(s),
i.e.,
so
~ €eAeikx
(sinly) + *.
(2.14)
104
As a result the inner solution which will match the neutral
Charney mode will be O(e).
It will be helpful to keep this
in mind when we get to the inner sequence of problems.
The O(e)
outer problem can be written
anHI
31
ax
-8oz/2
dA
qo -
+ qo -
z -
e
eikx(sin
ly)
+ *
dT
ax
(2.15)
where the 0(1) solution has already been used to evaluate
the inhomogeneous terms.
1
1 = --
dA
--
The full solution to (2.15) is
-aoz/2
eikx(sin ly) + *
e
+ 01(y,z,T),
ik dT
(2.16)
where Dl is the O(e) correction of the zonal flow.
It will
1 is
be apparent at the next order that the introduction of
necessary due to the forcing provided by the self
The solution (2.16) has only
interaction of the wave field.
x-independent potential vorticity, i.e.,
2 1
a2 1
HI
+ -----
=-
ay2
The O(e)
az 2
h-1 ...
1
(2.17)
az
solution (2.16) remains O(e near the ground,
thus requiring the matchable inner solution to be O(e).
Since the transient wave part of
1 does not vanish at the
ground we can expect form drag to be produced by its
interaction with topography which in turn alters the zonal
flow.
In
addition,
determined that
if
from the inner problem at O(e)
it
is
1 does not vanish at the ground, the zonal
105
l1 (y,O,T),
the ground
at
flow correction
will
with
interact
Thus we
the topography thereby altering the wave amplitude.
the presence of a wave-zonal
anticipate
due
flow feedback loop
to the presence of topography which is in addition to
flow feedback loop
the topography independent wave-zonal
in his section 3.
discussed by Pedlosky (1979)
clearly
However, to
the consideration of
see this requires
the inner
We postpone this until after
sequence of problems.
2
consideration of the final outer problem at 0(e
The O(e
an2
z --
2
+
) outer equation
3 2
q
ax
is
TI1
--
o
+
-
ax
).
---
-
J(o 0
ax
3T
,fl
1 ) - J(1o)*
(2.18)
The
zonally average version of (2.18) can be written
2
2
-- (-aT
3y 2
+
a_
a
_
=
h-1 -)
3z
2
-
J(
1)
-
1o)
(2.19)
The nonlinear terms in
(2.19) are easily evaluated using the
solutions at previous orders
(2.13) and
(2.16).
After doing
so (2.19) becomes
32
(----
3
--
1
3y 2
3T
diAl
2
a2
321
+
1
-
-o
-
h-
1
az 2
-)
= qo
z
e
3a
a
(sin 2 ly),
xdT
(2.20)
Dy
which describes the evolution of the zonal
of the
fluid
in terms of the
transient wave
still
required.
field.
flow in the body
forcing provided by the
A lower boundary condition on 01 is
106
for
We now seek the equation
and 0(E)
the 0(1)
to evaluate
solutions
wave-like
the
terms of (2.18) yields
inhomogeneous
1a2
2
= ikSFoAeikx
+ qo
z -ax
ax
a n1
ikF o -ay
-
Using
amplitude.
the wave
(sin ly) + *
a 1
Aeikx
(sin ly) + * -
ik
ay
Fo
Sqo-- Aeikx (sin ly) + *.
z
Removal of the secularities from
desired amplitude equation.
(2.21)
(2.21) results in the
This can be accomplished by
(2.21) by
multiplying
Fo
e-z/h (--)
z
e - ik
x
2
(sin
ly)
and integrating over the domain.
The resulting solvability
condition is
1
f
6
dy 2 (sin ly) e-1 i
x ¢2 (x,y,O,T) =-
0
qo 2
A
aull
-
dze-qoz (zpl
- Af
0
--- ) + Aull (O,T),
az
(2.22)
where
1
P11
2(sin
=
0
2
an 1
1y) ay
dy,
(2.23a)
and
ull = -
f
as 1
1
2(sin21y)
0
ay
dy.
(2.23b)
107
In order
to obtain the amplitude equation, 42 (x,y,0,T) must
first be
found.
Thus,
the sequence of
inner problems is
considered next for the purpose of determining ¢2 (x,y,0,T)
a lower boundary condition on the zonal
and obtaining
flow
correction ¢1.
The appropriate inner expansion takes
A
A
A
A
2
S= €o + E~1 + C
However,
no 0(1)
which
¢2 +
*..
(2.24)
*
recall from the outer problem
function near
the form
the ground
is
0(e)
or
that
the outer stream
higher order.
inner solution is matchable to the outer solution
requires
o = 0.
The first
of
(2.25)
nonzero inner solution appears at O(c).
In view
inner equations can be written
(2.25) the O(s)
a
a
15C = 0.
(-+ 5--)
aT
ax
(2.26a)
=
aT3C
The
Therefore
0
on
= 0.
(2.26b)
ax
solution to (2.26a,b) which can
be matched to
the outer
solution is
€1
Matching
to O(s)
= (D
1
C + C 1 ) eikx(sin
ly)
+ *
+ fl
(y,t)
(2.27)
the inner solution (2.27) to the outer solution up
it is found
108
1
dA
ik dT
D
= A(T),
1
fl = DI(Y,
As a result of
final
(2.28a)
(2.28b)
O
T).
(2.28c)
(2.28a,b,c) the O(s)
inner
solution takes the
form
1 dA
-)
= (AC + ik dT
1
From the
(sin
eikx
consequence of this
(2.29)
-- )(--aT
ax
a
(--- +
a
2
2
) inner problem.
are
aTa
oAe i k x
+
(sin ly)
+ *)
= 0
2
(2.30a)
v2
2
r'v2
-
The
in the presence of bottom topography
The 0(e2) inner equations
A
-
A
J(l 1 ,b)
-
J(jl,
-)
ac
ax
where the inhomogeneous terms in
been evaluated
(2.30b)
the vorticity equation have
using the O(s)
the definition of qo (2.5).
equation
(y,O,T).
zonal flow at the ground.
will be apparent in the O(E
already
1
inner solution (2.29) it is now clear there is both
nonvanishing wave and
a2
+ * +
ly)
solution
(2.29)
and
The solution of the vorticity
(2.30a) is
Bo
^
2
2
-
2
+ * + CX2
Aeikx
(y,T).
(sin ly)
+ * +
(C 2 +D 2
)eikx(sin
ly)
(2.31)
109
If
is substituted
(2.31)
into the lower boundary condition
(2.30b) and a zonal average
condition on the zonal
aX2
aT
In
a 2 01
+ r' --ay 2
the lower boundary
is taken,
flow is obtained, i.e.,
^
(v1 ,6 1 ).
a
(y,O,T)
-
(2.32)
ay
(2.32) X2 is as yet undetermined,
,
v1 =
81 =
ax
and the Jacobians have been rewritten to make
contribution to the wave forcing
due to the
interaction
of the
This
portion of the additional wave-zonal
(2.32) would
The
nonvanishing
at the ground with the topography.
the presence of topography.
form drag and
from the
convergence of heat flux at the ground.
clear the
form drag
0(s)
wave
is
field
forms an essential
feedback loop due to
The heat
flux contribution to
Using
exist even in the absence of topography.
(2.29) to evaluate the right side of (2.32) yields
a2 t 1
ax2
- + r'
aT
ay 2
d
(y,O,T) = -
[1A1
zonal
+ AM* + A*M]
dT
a
x -- (sin 2ly),
ay
from which it
2
(2.33)
becomes clear that the wave forcing of the
flow at the ground is due to
the wave transience.
110
When the
and O(E
O(E)
2
respectively, are substituted into
condition
Note
3a 1
(M + A) -ay
next to last
from the
term
2 )
lower boundary
(y,O,T).
(2.34)
the zonal
in (2.34) that
the other necessary ingredient
is
for the topographically
induced portion of the wave-zonal
flow feedback loop.
(2.34) D 2 is as yet undetermined as is X2 in (2.33)
In
To determine these quantities
¢2 (x,y,,T) in (2.22).
the outer solution is Taylor expanded about z=0,
So
-F2_2 +
2
-
4 = A[e
1 dA
(1 + E --ik dT
+
This
with the topography.
flow at the ground interacts
and
the O(e
(2.31)
(2.30b), the wave-like portion yields
r'K 2 dA
1
3D 2
+
= ----k2
dT
ik aT
C2
and
(2.29)
) solutions
P25C-
a3 1
... ]
Bo
-2
+
ly)
(sin
eikx
+ *
3
-
+ e5 1 (y,O,T)
... )eikx (sin ly)
(y,O,T) + C2 2 (x,y,O,T) + 0(e
i.e.,
),
(2.35)
az
and matched to the
inner solution,
i.e.,
1
S=
+
-
E2
+-
(AC
{_-
dA
--- )eikx
ik dT
ao
_
2
(A + M)
2
a3
ay
(sin
+ *
ly)
3D 2
1
A + D 2 C + -+
ik aT
r'K
+ c(1
2
(y,O,T)
dA
+
k2
dT
1
(y,O,T)}
+ E2 CX2 + O(E3)
eikx (sin ly) + *
(2.36)
11i
The results are
a3
1
X2 =
(y,O,T)
(2.37a)
az
8
dA
2ik dT
D2 =
2(x,y,O,T)
x eikx
(2.37b)
Oo d 2 A
= [--2k 2 dT 2
r'k 2 dA
+ ----k2
dT
3I
(A+M) -ay
(sin ly) + *.
(y,O,T)]
(2.37c)
Substitution of (2.37c) into the solvability condition
(2.22) and substitution of (2.37a) into
condition for
equation for
the zonal flow
the zonal flow
the lower boundary
(2.33) yields, along with
(2.20) already obtained, the
final closed set of nonlinear equations.
d2 A
dT 2
-
2r'K
--8o
2
dA
2k 2 6A
dT
8oq
2
2k 2
aull
--- )dz + --- Mull
az
ao
a2 u 1
a2 u
+
(---az 2
ay2
a
-aT
diAl 2
x -
2k 2 A
+-f
8o
0
They are
e-qoZ
(zp11
(O,T) = 0,
(2.38a)
aul
h-1
.-
) =-
qoe-S
0
z
az
a2
-
dT
1
0
the
ay2
(sin21y),
(2.38b)
112
32 u
1
(y,O,T)
aTaz
+
|A 1
2
a2
) -
+ r'
a2 u 1
--ay2
(y,O,T)
d
= dT
+ A*M
(AM*
(sin 2 1y),
2
(2.38c)
ay
where
ul =
(2.39)
ay
and
aul
= 0,
(2.40)
aT
on y = 0,1.
The set of equations
(2.38a,b,c) governs
the
equilibration of the weakly unstable stationary Charney mode
in the presence of topography and the associated wave mean
flow interaction.
Pedlosky
except
(1979)
to those derived by
They are identical
(his equations
for the presence of the
(3.47),
(4.4a))
form drag term in (2.38c) and
the presence of the last term in
(2.38a) which represents
the interaction of the topography with
correction at the ground.
(4.3) and
the zonal
These terms are
flow
responsible for
the topographically induced portion of the wave mean flow
interaction.
We now investigate their effect on the wave mean
interaction.
This requires the solution of
which entails the elimination of pll
amplitude equation (2.38a).
flow
(2.38a,b,c)
and ull from the
It is a simple matter to solve
113
for P 1 1 in
keeping
terms of wave amplitude
in
mind
its
definition
1
P11 =
f
2(sin
2
0
32 u
x
32 u
1
(--
h - 1 --
az 2
Multiplying
2
2
1y)
) dy
(2.38b) by 2(sin
P11 = - q 0 1
2(sin
0
az
(1A1 2 -
2
1y).
integrating over the
in time yields
domain and once
meridional
1
f
-
aul
1
+ --
ay 2
in
aH 1
dy
1y) -ay
IA( 0
) 12
)e- Ooz
(2.41)
,
to obtain
order
1
ull = f
2(sin
2
1y) ul dy,
0
first the solution of ul must be found.
is sought in the
ul =
ulj
If a solution to ul
form
(z,T)
sin jy,
j
(2.42)
where
Using
J = 1i, 2, 3,
= Jfr,
j
to solve
(2.38b,c)
-8qo1
uij
=
(Boqo-j
x
(e-Soz
-
2
2
j
)(j
2
j2
aj qo
-41
for
2
...
.
ulj(z,T)
[( A1
yields
2
-
IA(0)12)
)
e-ajz + Bje-ajz]
(2.43)
114
where
1
1
+ (j
--
=
aj
2
+
1/2
-)
4h
2h
2
(2.44)
and
r'j
dBj
--- + ---dT
aj
Bj
j2
aj
= -
j2 a
- (AM*
aT
Boqo +
qoaj
From (2.43),
2
j2
)( AI 2
(1 - qoaj
-
IA(0) 12)
+ A*M).
(2.45)
If
ull can be calculated.
the result is
substituted in (2.38a), the amplitude equation becomes
d2 A
2r'k
2
+-
dT
+
Z
NjBj]
2
dA
2k
2
6
A + A[No(1A1 2 -
dT
Boqo
IA(0)1
2
)
2
+ M[Qo(IA1 2 -
IA(0)1
2
) +
QjBj]
=
0,
(2.46)
where
16qo14(1-(-1)J) 2
2k 2
No
Bo
j
2
-
(Boqo
so
j2)(j2-412)2
( 0 +qo
q(aj + qo)
Bo+qo
qo(aj + qo)
qo1 2
(qo +
so)2
(2.47a)
2k 2
Nj = ---
so
(B oqoj2)(j2-412)2(qo+a j
)
(2.47b)
115
2k 2
Qo = --8
o
J
16qol 4 (1-(-1)J)2
2
( oqo-j
0
)(j
2
-41
2
j2
)2
),
(1 - -j
qo
0
(2.47c)
16qol4(1-(-l)J) 2
2k 2
Qj -
2
(Boqo-j2)(j2-412)
Bo
(2.47d)
The solution of the coupled amplitude equations (2.45)
and
(2.46) determines the evolution of the wave amplitude
given the inital conditions.
In general they are very
difficult to solve.
in the absence of friction the
effect of
the
instability
However,
topography on the linear baroclinic
is easy to see.
linearized version of
Consider
(2.45) and
the inviscid
(2.46),
i.e.,
2k 2 6
d2 A
-
QjBj = 0,
A + M
2
oqo
0
dT 2
8oqo -
j
(2.48a)
j2
Bj =
(AM* + A*M).
(2.48b)
qoaj
If,
for
simplicity, A and M are assumed
elimination of Bj
d2A
dT 2
2
to be real,
the
from (2.48a) yields
A = 0,
(2.49)
where
2
-2
Boqo-
(M 2
k6
j2
Qj
J
goaj
Boqo 2
2.50)
(2.50)
116
From the growth rate (2.50) it
is easily seen that the
instability towards
topography modifies the baroclinic
stability provided
Boqo - j2
SQj
qoaj
j
= 64M
Indeed this is the case.
k 2 14
---Bo j
2
In
(1-(-1)J)2
>
aj(j
2
-41
the stability
0.
)2
(2.51)
fact each term of the sum is
zero and the sum converges.
greater than
2
In Fig.
diagram plotted in topographic
(4.2.1)
amplitude M and
supercriticality 6 space when the Boussinesq-like
approximation
ao
the topographic
>>
h-1
is made.
It clearly shows that as
amplitude increases
the baroclinic
instability is stabilized and that as the supercriticality
is increased the topographic amplitude required for
stabilization
is greater.
The physical mechanism
stabilization is as follows:
for this topographic
As the baroclinically unstable
wave grows, a wave, phase shifted
from the initial
baroclinic disturbance, develops, which interacts with the
topography to produce the
form drag which according to
(2.38c) modifies the zonal shear at the ground in such a way
as to stabilize the baroclinic wave.
Closer examination of the growth rate rewritten in the
form
2k 2
2
a
[32 M 2 14
So
6
(1-(-l)J)2
- M]
I
q0 2
j aj(j 2 - 412)2
(2.52)
117
reveals that the topographic stabilization is most effective
for waves with short meridional
Furthermore, for those scales
(1-(-1)J)
3214
j
aj(j
2
-
>> 1).
for which
main effect of topography
limit the baroclinic
This
2
(2.53)
412)2
the baroclinic instability
stability
14
6/qo 2
M2 >
scales.
scale (i.e.,
is rendered stable.
on baroclinic instability
instability
are plotted
in
is to
to the larger meridional
is apparent in Fig.
regimes
Thus the
(4.2.2) in which the
k,l
space
under the
Boussinesq-like assumption 8o >> h-1.
This stabilizing
effect of topography was also noted by
Pedlosky
the context of a two layer model.
(1981) in
He also pointed out
the presence of topography caused the
that
inviscid nonlinear
oscillation to be asymmetric about zero amplitude.
Inspection of the present amplitude equations
(2.46) in their inviscid limit reveals
(2.45) and
this to be the case
here also.
The stabilization of the baroclinic instability by
topography
is of special interest in
Namely, while
the previous chapter.
light of the result
topography destabilizes
the normal modes of the Charney model via topographic
it
instability,
Charney
mode.
stabilizes
Thus,
the
in
baroclinically
unstable
in the presence of topography the
118
NEUTRAL
WAVE S
UNSTABLE
-6
-4
-2
Fig.(4.2.1)
0
2
4
6
8
10
12
Stability regime diagram vs. M and
14
S
16
18
20
for k=1=1.0.
22
24
119
10
1
1
I
NEUTRAL
I
5j '
WAVES
41
I- I
I
0
0
UNSTABLE
I
I
I
I
I
I
I
I
I
I
2
3
4
5
6
7
8
9
10
k
Fig. (4.2.2)
Stability regime diagram in k,l space for fixed
topography M =.5 and supercriticality
E=12.0.
In the absence
of topography the entire range of wavenumbers would be unstable.
120
normal modes are at least as dynamically important as the
baroclinically unstable modes in the Charney model.
3.
Instability of a topographically forced Charney mode
with very weak zonal flow at the ground.
In this section we will see how the stability problem
of section 2 is altered as the zonal flow at the ground is
increased enough to force a small but finite-amplitude
Charney mode near neutrality.
In section 2 we studied the
effect of topography on a weakly unstable infinitesimal free
Here and in sections 4 and 5 the stability of
Charney mode.
a topographically forced nearly neutral Charney mode
embedded in a shear flow will be examined.
The wave is
forced by the diversion of the zonal flow by the sinusoidal
topography.
The difference between this and the next
section is due to the choice of the size of the zonal
velocity at the ground and the topography.
Here the choice
of topography is the same as in the last section but now a
very weak zonal flow will be allowed at the ground, i.e.,
U = 62 A'.
(3.1)
Except for the presence of this weak zonal flow all aspects
of this analysis are identical to those of the last seciton.
Then if
(2.3),
(2.1b) is
(2.6),
replaced with (3.1)
(2.8)
and
(2.9)
but the choices
are made as
and inner problems, respectively, become
before
(2.2),
the outer
121
z + qo
ax
ax
2
-
A'
-+
ax
aT
J(
2
),
ax
(3.2a)
and
a
aa
cA'
+
aT
(qo
+ e
-- )(ax
a~2
- +
ax
-
6)-- + J( 4 , --ax
aC2
-
h-
a
1
2
2V)
+
-
sh- 1
+
E2V
2
) = 0
a
(3.2b)
subject
to
A
A
A
SA' -
E2 A'bx -
on
-
axac
aT a
A
+
-
J(¢,
ax
--- )
3C2
= -
Er
2
eJ(¢,b),
(3.2c)
z=0.
Again outer and inner expansions of the
(2.24) will be used.
that
Before proceeding it
(3.2a,b,c) differ
form (2.11) and
is useful to note
from their counterparts
in section 2
only by the presence of terms proportional to A'.
Examination of these
terms in (3.2a,b,c) reveals that they
all have a zero zonal average.
(2.38b,c)
for
the zonal flow obtained in the previous
section also obtain here,
a
-
aT
32 u 1
---
3y2
Therefore, the equations
32 u
+
1
-
az2
h
i.e.,
1
aul
-)
az
diAl 2
= -
qoe-
0o
z
dT
122
a2
(sin21ly),
x 3y2
(3.3a)
a2 u 1
32u1
(y,O,T) + r'-aTaz
ay 2
+
A
2
a2
)-(sin
Dy2
2
d
(y,O,T) = dT
ly).
(3.3b)
Thus it is only possible for
the wave equation.
the additional
field problems are different
the outer and
terms to alter
In fact a further examination of
(3.2a,b,c) reveals that only the O(e
result
(AM* + A*M
2
) outer and
from those
inner wave
field
inner wave
in section 2.
solutions
As a
up to O(c)
are the same as in section 2 which are
*
= Aze-aoZ/ 2 eikx
x
e-
*
= ,
0 oz/
2
eikx
E dA
(sin ly) + * + ik dT
(sin
ly)
+ *
,
(3.41)
and
(A
1 dA
+ -- )
ik dT
eikx (sin
ly)
+ *
(3.4b)
respectively.
Now we turn to the consideration of the O(c2)
problem.
Using
(3.4a) to evaluate the
the 0(E2)
outer
equation
outer
inhomogeneous terms
for the wave field takes the form
123
a
2
z-- (V22 +
ax
32
-
-
ik Fo
-
h
Aeikx (sin
a
32 1
(ay
ay 2
x (sin
ly) + *
3 1
--- )Aekx(sin ly)
az
Aeikx
(3.5)
ly) e-lK
x
0
- Af
h'-
.
1
2(sin
ax
'8D
Fo
1
qo ly) - ik -ay
z
(sin
The solvability condition for
f
32
+ qo
ly) + ikSFoAeikx(sin ly)
a2 1
+ -3z 2
30 1
--- ) Aeikx
az
1
h-
3z2
Fo
ikA' qo z
=
3a2
-)
az
2
6
¢2 (x,y,O,T)dy = A'A + --- A
qo 2
-
dze-qoz (zplI
0
(3.5) is
aull
-) + Aull
az
(O,T).
(3.6)
In (3.6) it appears as if the term A'A will play a role in
the amplitude equation.
contribution
However, shortly we will see that a
from the lefthand side of (3.6) will cancel
with A'A on the right
Now the O(e
2
side of
(3.6).
) inner problem will
be considered whose
solution when matched to the outer solution will determine
€2(x,y,O,T).
If
(3.4b) is used
inhomogeneous terms,
can be written
the O(e
2
to evaluate the
) inner wave field equations
124
a2¢2
--= a 2
a8
---
2
-
(sin
(3.7a)
eikx (sin ly) -
ik(A + M) Oly
The wave
ly) + *
a 2
--= -ikA'Aeikx
ax
-
x
BoAeikx
solution
to
r'K
(sin
(2
ly)
(3.7a)
dA
ik
ikA'Meikx
(y,O,T)
2
+
eikx
(sin ly)
(sin
which
dT
ly)
+ *
(3.7b)
can be matched
to the
outer solution is
8o
8Bo
¢2
2 Aeikx
=-
2
x eikx
(sin
ly)
(sin ly) + (-
o
d2A
2k 2
dT 2
C2 =
[
+ A'
(A
Matching
+ M)]
the lower boundary condition
It is
r'k 2 dA
a4 1
------ + (A + M) --- (y,O,T)
+
k2
dT
ay
eikx
(sin
ly)
+ *.
(3.9)
¢2 to ¢2 yields
¢2 (x,y,O,T)
= C 2 eikx
where C 2 is given by (3.9).
condition
+ C2 )
(3.8)
C 2 can be determined.
8
-2ik dT
+ *
When (3.5) is substituted into
(3.7b),
dA
(sin
ly)
Using
+ *.
(3.10)
(3.10) in the solvability
(3.6) we obtain the wave amplitude equation
125
d2 A
2r'k 2
-+
dT 2
ao
-
dA
dT
2k
2
6
2
oqo
0
aull
2k 2
) dz + --M ull
az
Bo
2k 2
A + -A f
e-qoZ
ao
0
2k 2 A,
(O,T) = -
M.
ao
Equation (3.11) differs from its counterpart
only by the presence of the
(zP11
(3.11)
in section 2
inhomogeneous term.
No other
terms proportional to A' arise because the advective time
scale associated
with the weak zonal
which is an order slower
than the
flow s 2 A' is
2
)
long time scale.
The coupled set of equations (3.11) and
a steady solution of the
O(C
(3.3a,b) allow
form
qo 2 A 'M
As
=
6
(3.12a)
Uls = 0
(3.12b)
The
study of
the stability
point of this section.
forced wave.
This is
the
of
this
steady solution
the
It corresponds to a topographically
Notice the wave has a resonance at
point at which the neutral
become stationary relative to
happens to correspond to
is
6 = 0.
propagating
the topography.
It
the stability threshold
waves
also
for an
infinitesmal baroclinic wave in the absence of topography.
Thus we expect the instability of the
on whether the flow is superresonant
(6<0).
forced wave to depend
(6>0) or subresonant
In any case, in order to avoid the resonance, the
126
following stability analysis must
be restricted
to values of
6 which are 0(1).
If
the Ekman friction is assumed to be negligible, the
evolution equations, when
linearized about
topographically forced wave solution
d 2 A'
2k 2 6
8oqo 2
2
dT
2k 2
+ --So
M u'll
a 2 u'
(2A
2
1
-
+
3y
x
e-qoz
(zp'll
0
-
au'Ill
----) dz
az
(3.13a)
(
aT
f
(3.12a,b) take the form
(O,T) = 0,
a 2 u' 1
a
-
2k 2
A' + -As
Bo
the
az
-
2
(sin
1
qoe-aoz
= -
2
ly),
(3.13b)
82u1
(2A s
=
au'
-)
az
a2
dAr'
) --ay 2
dT
(y,O,T)
h-1
aTaz
dAr'
dAr'
a2
-+ 2M --) ay 2
dT
dT
(sin
2
ly),
(3.13c)
au'
au'
1
1
---(O,z,T) aT
(1,z,T) = 0,
aT
where Ar' and and Ai'
(3.13d)
are
the real and
imaginary parts of
the perturbation wave amplitude A' and M is assumed to be
real
for simplicity.
The small amplitude solution of
(3.11) consists of a
baroclinically unstable mode plus a topographically forced
wave.
Thus, the equation for the perturbation wave
amplitude
(3.13a) now involves the
interaction of the
perturbation zonal flow with both the
forced wave and the
127
topography which modifies the baroclinic
is
in contrast to
flow was only
Also in section 2,
case,
only interacts with
the perturbation zonal
forced by the interaction of
distubance with
However,
the topography.
the perturbation zonal flow is also
interaction of
forced wave.
the wave
in
the wave disturbance with the
Thus
in addition
section there
the present
forced by the
topographically
to the topographically
induced modifications of the wave-mean flow
the last
This
the previous section where the
perturbation zonal flow at the ground
the topography.
instability.
interaction in
is a further modification due to the
presence of the forced wave.
Thus,
the stability problem is
altered fundamentally.
The linearized equations for the perturbation zonal
flow (3.13b,c,d) can be solved just
as the nonlinear zonal
flow equations were solved in the last
resultant perturbation zonal
section.
flow is substituted
When the
into
(3.13a) the resulting perturbation wave equation is
d 2 A'
2k 2
6
0o
qo 2
4k 2 M 2
A'
dT 2
qo24.
+ --6
2
ao
[
ao
62
(X 1 + X 2 ) + X 3 ] Ar'
(3.14)
rate a = kc,
The growth
c2
+-
qo 4 A' 2 X o
corresponding
qo4A' 2 X o
6
I
qo 2
-
2M
2
[
+
62
to (3.14) is
qo2A'
"
6
(X 1 + X 2 ) + X 3 ]}
(3.15)
128
where
Bo
- 2k
X0
2
No
(3.16a)
ao
X1 = 2k- 2 Qo
(3.16b)
oqo -
8o
Nj
--j 2k 2
X2
=
X3
8o
Qj
= I
j 2k 2
where No,
Nj,
J2
)
(
qoaj
Boqo -
(3.16c)
J2
(
)
qoaj
(3.16d)
Qo and Qj are defomed by
respectively.
The term responsible
the baroclinic
instability
brackets of
in
(2.47a,b,c,d)
for the stabilization of
section
2 is
X3
in
(3.15) which is always greater than
the square
zero.
The
remaining terms in the square brackets are due to the
presence of
the forced wave.
Their effect on the
instability depends on their sign and relative magnitudes.
In order
to evaluate the nonlinear coefficients Xo, X 1 , X
2
and X 3 more easily, the Boussinesq-like approximation
Bo >> h -
will be made.
1
c
2
=
K
l
(3.17)
Using (3.17) the dispersion relation becomes
16K 4 A, 2
6
{-4K 2
-
2M
2
[
62
Xo
8K 2 A,
+
6
X1
+ X3]}
,
(3.18)
129
where
1
(m + 214
xo
=-
1))
0
r5
m= 1
(m
K
+ - -
1
-)2((m
-
2
7T
i
K
(- -
1
_)2
12
.)2
2
72
5
16K1
2
(3.19a)
214
X 1 = X2
=
_
it
5
m=1
(m
K
1
+ - - -)
((m -
2
i
1
-)2
12
-
)2
72
2
(3.19b)
X3
From
214
_
=
i5
m=1
1
(m - -)
2
(3.19b,c) it is easy
than zero.
1
((m
and
-
12
-- )2
iT2
2
(3.19c)
to see X 1 , X 2 and X 3 are greater
show that X0
to Xo
is
stabilizing and
stabilizing
for
Thus the relative signs
6 determine whether or not the
all
relation and therefore the
The term
the terms
but the
largest
and magnitudes of A'
term proportional to X 1
in (3.18) is stabilizing or destabilizing.
the
is greater
for extremely large scale waves.
proportional to X 3 is always
scale waves.
.)2
Numerical calcuations
than zero except
proportional
-
The dispersion
instability depends on whether
flow is eastward or westward.
When the
flow is
130
M
I
I
I
2
I
I
I
I
I
I
I
I
I
I
I
I
NEUTRAL
WAVES
NEUTRAL
WAVES
II
I
I
I
I
UNSTABLE
I
-6
-4
-2
Fig. (4.3.1a)
and
Z =1.0.
0
2
4
6
8
Stability regimes in 1 ,
10
12
14
16
18
space with k=1=1.0
20
22
24
131
M
NEUTRAL
NEUTRAL
WAVES
I
I
II
UNSTABLE
I[
-6
-4
-2
0
2
4
6
8
10
8
Fig.(4.3.1b)
Same as (a) but
A
=-1.O.
12
14
16
18
20
22
24
132
baroclinically supercritical (6>0) the term proportional to
X1 is stabilizing for eastward flow (A'>O) and destabilizing
for westward flow (A'<0).
Just the opposite is true for
baroclinically subcritical flow (6<0).
However, it seems
clear that overall, we should expect an increase in
topography to be stabilizing.
The numerical computations
displayed in Figs. (4..3.1a,b) and (4.3.2a,b) bear this out.
In Figs.
(4.3.1a,b) we have plotted the stability regimes in
M,6 space for A' = 1.0 and A' = -1.0 respectively.
The
eastward flow case shows stronger stabilization than in the
westward flow case.
The eastward flow case is also more
stable than the A' = 0 case of Fig. (4.2.1).
This is due to
the interaction of the forced wave with the perturbation
zonal flow.
The westward flow case in Fig. (4.3.1b) is less
stabilized than the A' = 0 case of Fig. (4.2.1).
This is a
result of the term in (3.18) proportional to X1 which can be
destabilizing for westward flow (A'<0).
However, on the
whole topography is stabilizing except for unlikely
parameter regimes.
We will close this section by considering the
wave-field energetics.
In general the total wave energy can
be modified by the conversion of mean flow kinetic energy to
perturbation kinetic energy by Reynold's stress, conversion
of mean flow kinetic energy to perturbation energy by the
form drag and
conversion of mean available potential energy
to perturbation available potential energy by the eddy heat
flux.
In the present case, the energy budget can be written
133
a
C3
C
f
aT
+
2+
oz2dz
2
I
ly + S1x 4oy]
e-z/h [ ox
[#ox
1z + ~Ix Ooz]
----
0
4
+
(A'
~lyy dz
a(z)
3 c
e-z/h
f
e3
+
2
[o
0
:5
=
1
e-z/h_
dz
3z
Oly)[obx]
-
z=0,
[
where
Eq.
(3.20)
] is a meridional avrage and
(3.20) shows the total
(
)
is a zonal average.
energy is modified primarily by
the conversion of available zonal potential energy.
The
other contributions are one or two orders of magnitude
smaller.
(3.20) but A' = 0.
same as
zonal
The energetics of the previous section are the
flow at the ground
Thus it is clear that
has altered
the O(e
2
the instability problem
but leaves the leading order energy budget unaltered.
In
the
next
be O(E)
but
the
section
the zonal
topography will
(3.20) but
-
Oly
flow at
the ground will
be O(e 2 ) instead of O(s).
The leading order energetics will
remain unaltered from
the term
[ obx]
z=0
will fall to 0(e
modified
5
further.
).
)
However, the instability will be
134
4.
Influence of topography on the baroclinic instability of
the Charney mode and its
topographic instability.
The topic of the present section is,
of
the last
topographically forced Charney mode near
baroclinic neutrality.
In the
zonal
the Charney mode
flow which
the absence of
the order of
forced
last section the uniform
topography it has
interest.
is allowed
baroclinic
the topic
section, the linear instability of a small but
finite-amplitude
flow
as was
is so weak that in
no effect on the problem to
In this
section,
the uniform zonal
to be stronger such that the stationary weak
instability
of
the last
slowly propagating weak baroclinic
section
will
become
a
instability in this
section.
Nevertheless, the present analysis is very similar to
that of the last two sections and section 3 of
(1979).
This will allow the discussion of its development
to be considerably more brief.
about
Pedlosky
Again
the analysis pivots
the stationary neutral Charney mode which,
absence of forcing,
in the
will grow or decay on the long
time
scale
T = et,
(4.1)
if
6 =
Bo -
(4.2)
26
and
r = e2 r'
As in section 3 this weak instability is
against
the forcing and
(4.3)
then balanced
the usually stabilizing
135
nonlinearity, but now the weak advection of the Charney mode
by the
uniform zonal flow also comes
into play.
The wave is
forced by a stronger weak uniform zonal current
flowing over
the topography than in the
last section.
choose the magnitude of this current such
time
scale associated
with
it
matches
the disturbance which is also
flow interaction.
We
that the advection
the e-folding
time of
the time scale of wave-zonal
Therefore we write U as
U = e'
Then the appropriate choice
e2 b = e 2 Meikx
n =
With
(4.5) the
(4.4)
for
ly) + *
(sin
,
(4.5)
This approach was also used
(1979) in a two-layer model
topography is an order of magnitude
last section.
is
forcing will be balanced against the growth,
friction and nonlinear terms.
by Plumb
the topography
Note that the
smaller here than in the
As a result the topographically induced
portion of the wave mean
flow interaction of the ground
the last two sections will
be an order weaker.
in
Thus it will
be negligible to the order of interest here.
Substituting
outer,
(4.1-5) into (3.2.3) and
(3.2.7) the
inner and lower boundary equations can be written
+
z -ax
-
ax
+
(
aT
'
)T
-ax
+
26
ax
-
J(,
H),
(4.6a)
136
a
a
+
(--
+
--
A'
aT
32
a
--
ax
)(-
ax
-
£26) -
-
J
-
3a 2
2
E
(qo
A
+ S2V2
h- -
3a 2
ax
+
)
,2V
a
-
(4,
2
+
ac
22
S
-
a
- 1
h
),
sc
(4.6b)
and
a2
-aTa
-
2
+ A'
-axac
- -- + J(
ax
3a
a
, -)
2
= er'V2¢
-
E2A'bx
ac
E2 J( 4 , b),
respectively.
(4.6c)
In (4.6a,6b,6c),
4 and 6 are the inner
stream function and inner vertical coordinate, respectively
as
in the previous sections.
Expanding 4 and 4 as before,
i.e.,
=
o + CEl + E2
€2 +
**.,
(4.7a)
*..,
(4.7b)
AA
€ =
+ C€i + E2 d2 +
and substituting into
(4.6a,6b,6c) results in a sequence of
equations for both the inner and outer problems.
The outer
problem as defined by
solved in
section 2 except
the 0(1)
outer
(4.6a) is identical
for the presence of
to that
the -£A'Hx term.
Thus
equation is the same as (2.12) whose solution
is the stationary neutral Charney mode,
o = Aze-aoz/
2
eikx (sin
ly)
+ *
,
i.e.
(4.8)
137
At
O(e)
the
the effect
0(1)
of the
solution
(4.8)
additional
the
O(s)
term
outer
is
felt.
Using
problem can be
written as
an 1
ap
ax
x
It
ax
(sin
has
the
ly)
1
(ik
+ *
(4.9)
2
(sin
eikx
+ *
(4.10)
1
=
a2 01
+
,
- haz 2
(4.11)
az
the solution (4.10) from the
term -EA'Ix
is
not phase shifted relative to the
neutral Charney mode solution.
interaction with the 0(1)
solution
As a result its
in the O(E2)
problem does not produce additional wave
flow.
to
a¢1
that the contribution to
additional
of (4.10) contributes
vorticity
ay 2
zonal
ly)
(y,z,T).
a2 s
0(1)
eikx
.
dA
+ A'A)e-aoZ/
dT
the potential
Note
2
dT
Again only the x-independent part
1
e-8oZ/
solution
1 =
+ €1
dA
+ ikA'A)
qo (-
-
+ qo
z --
Because
of this
fact and
outer
forcing of the
because
the 0(s)
potential vorticity remains x-independent, the result of the
O(c2)
outer problem
is
the same
describes the evolution of the
transient wave
forcing and
as
in
section
2 where
(2.20)
zonal flow in terms of the
(2.22) is the solvability
138
condition on the wave-like inhomogenieties.
(2.22) is
condition
1
6
f
dy2(sin ly) e- 1kx
2
(x,y,O,T)
=-
qo 2
0
-
-aull
dze-qoz (zpI
Af
-
--
0
)+
Aull(O,T),
az
(4.12)
where pll and ull are defined by (2.23a,b),
Again
p2 (x,y,,T)
forces us
for
to turn to the consideration of the inner problem
its determination as well as for the purpose of
Based on the experience of
the O(s)
near
respectively.
is undetermined by the outer problem which
obtaining a lower boundary condition on the
limit
The solvabiliy
zonal flow.
section 2 the solution to
inner problem can easily be obtained by taking the
of the 0(1)
the
ground.
and O(s)
outer
Thus the O(s)
solutions
inner
(4.8)
solution
is
and
(4.10)
found to
be
1i =(AC
1
+ -
dA
-
+ A'A)
eikx(sin
ik dT
ly) + * + 4 1 (y,O,T)
(4.13)
Since
the
O(s2)
inner
problem differsn
substantially from that of section 2,
careful consideration.
If we use
evaluate the inhomogeneous terms,
can be written
a
(-+ A'
aT
more
it will be given more
(4.13) as well as
the O(e2)
(4.5) to
inner equations
as
a
32
a
a
+
ax
-)(-----a 2
ax
2
+
oAeikx(sin
ly)
+ *)
= 0,
(4.14a)
139
A
A
32¢2
32¢
---+
'
aT3
A
a
2
axa
+ * -
ikAeikx
x (sin
ikA'Me i k x
ly(y,O,T) (sin ly) + * -
+ * -
ly)
1
dA
-= r'K 2 (- - + A'A) eikx(sin ly)
ax
ik dT
2
r'
32 1
3y2
diAl
(y,O,T)
2
a
(sin 2 ly)
dT
ay
(4.14b)
As opposed
2,
2
to the O(e
) lower boundary
condition in
section
(4.14b) contains additional terms proportional to A'A and
the forcing
A'M but does not contain a form drag term and
interaction term between the topography
an
and the zonal flow
correction at the ground which formed the topographically
induced portion of the wave mean
2.
As before,
the solution
flow interaction in section
(4.14a,b) is of the form
^o
2
2 Aeikx (sin ly)
-
+ * + (C
2
x
(sin
ly)
+ * + 5 -az
2
+ D2 4)
eikx
(y,O,T)
(4.15)
Matching this solution to the otuer solution yields
Bo
D2
2
1 dA
(-+ A'A)
ik dT
¢2 (x,y,O,T) = C 2 eikx (sin
Substituting the O(e
2
(4.16a)
ly) + *
) inner solution
(4.16b)
(4.15) along with
140
(4.16a) into the
(4.14b) then
lower boundary condition
yields
Bo d 2 A
+
2k 2 dT 2
C2
2
ir'K
dA
iBoA'
dT
k
3¢i
A'A + A'M - A -ay
+
k
o80A2 A
r'K
2
k2
2
dA
dT
(y,O,T),
(4.17a)
and
(y,O,T)
a321
aT3z
3a2 1 (y,O,T)
+ r' ay 2
diAl
2
a
(sin21ly)
-
dT
ay
(4.17b)
The later result
is the required lower boundary condition on
the zonal
Use of
flow.
(4.16b) along with (4.17a) in the
solvability condition (4.12) results in the wave amplitude
equation which along with (2.20) and
(4.17b) form the closed
set of equations governing this damped and
This set of equations is
flow interaction.
d2 A
+ 2(ikA'
--2
dT
26k 2
-
-
)
2
forced wave mean
dA
r'K 2
) + (2ir'
+ Bo
dT
2k 2
A + -A
8o
f
kK 2
- A'
Bo
W
e-qoz (zpll 0oqo
O
2
- k2A,
aull
--- ) = az
2k 2
A'M,
So
(4.18a)
3
aT
32u,
32u1
(--+ az 2
3y 2
-
h-1
aul
.)
az
dJAl
_
q=e-o-
2
32
z
(sin21y)
dT
ay 2
(4.18b)
141
2
3a
a2 u 1
1
---(y,O,T) + r'
3aTz
diAl
2
a2
(y,O,T) 2
ay
(sin
dT
2
ay
2
ly).
2
(4.18c)
The zonal velocity is also subject
to the sidewall boundary
condition
aul
0,
-
aT
(4.19)
on y = 0,1.
The equations
the equations
Pedlosky
(4.18a,b,c) and
(3.47),
(1979) except
(4.3) and
(4.19) are
identical
to
(4.4a,b) obtained by
for the presence of terms
proportional to the weak zonal velocity A' and an
inhomogeneous
M.
term proportional to the topogrpahic amplitude
We now consider the linear instability of
topographically forced wave
the
in the absence of Ekman
friction
whose steady amplitude is
2A'M
As =
Bo('2 +
By
26
oqo
2 2
(4.20)
(4.20) is the topographic
real,
then As also is.
subcritical
A'2
This
=
amplitude is assumed to be
Note that
for baroclinically
(6<0) flow As becomes infinite when
26
-.
.
2
This
corresponds
to a resonance which occurs when the
corresponds
to a resonance
which
occurs when
the
142
subcritical propagating neutral mode becomes stationary
relative to the topography.
In contrast to section 3, since
the advective time scale of the zonal
flow EA'
is the same
as that of the wave-mean flow interaction, the resonance
to the stability transition
point no longer corresponds
for
an infinitesimal baroclinic wave.
Nevertheless, based on
the experience of the two previous
chapters we can expect
the resonance to play an important
role in the instability.
Our inviscid
However,
stability
analysis
the stability analysis
must
avoid
this
point.
in section 5 will
include
friction which will allow this restriction to be removed.
Linearizing
the evolution equations
the steady wave solution
d 2 A'
dA'
---+ 2ikA' dT
dT 2
2k 2
+
-- As
o
a
3 2 u'
(
-.
2
aT
ay
x
dAr'
---dT
a 2 u'
e-qoz
26
k2
(A'
2
1
a2 u
+
(zp'll -
A'
-)
2
au'll
--)
az
dz = 0,
(4.21a)
au' 1
1
-az
+
80qo
0
h-1
2
) = -
qoe- 0o
2A s
(4.21b)
dAr'
1
z
az
32
--- (sin21y),
3y 2
(y,0,T)
3T3z
(4.20) yields
m
f
(4.18a,b,c) about
a2
= 2A s
(sin
dT
3y
2
2
ly),
(4.21c)
143
au'1
-
0
3T
(4.21d)
on y = 0,1.
The definitions of P'll
a 2 u'
1
P'll
=
-
2(sin
2
1y)
0
(
1
ay 2
and u'll are
a 2 u' 1
+
az 2
-
h
-1
au' 1
) dy,
az
(4.22a)
1
U'll
= -
f
2(sin 2 1y)
u' 1
dy
0
The last
(4.22b)
term in
(4.21a) represents
the interaction of
forced wave with the perturbation zonal
(4.21b,c) the disturbance zonal flow
interaction of
disturbance.
the topographically
According to
is forced by the
forced wave and
the wave
Thus there is a feedback loop between the
forced wave and
which
flow.
the
the disturbance wave
field and
is key in the instability of the
zonal flow
forced Charney mode.
Again the zonal velocity can be obtained by solving
(4.21b,c,d) as in previous sections.
used
in
(4.22a,b)
to write P'll and
wave amplitude alone.
into (4.21a) the
The
u'll
When P'll and
result can then be
in
u'll
terms
of the
are substituted
folowing linear equation
for the
perturbation wave amplitude is obtained:
d 2 A'
dA'
---- + 2ikA' -- k2
2
dT
dT
26
(A' 2 +
)A'
Soqo
2
+ 2NoAs 2 Ar' = 0,
(4.23)
where Ar'
is the real part of the perturbation amplitude and
144
2k
= k
2
No
2
o
J
j2
(ooqo-j
2
q0
qo(aj + qo)
2
)(j)
-41
2
5o
(------
)2
o+qo
12
(qo +
(4.23) is split into
result
(1-(-l)J)2
N' 8
If
16qo1 4
80)
2
the real and
imaginary parts the
is
d 2 Ar'
----2 kA'
dT 2
dAi'
- dT
26
k2
(A' 2 +
-)Ar'
aoqo 2
+ 2NoAs 2 Ar
'
= 0,
(4.25a)
d 2 Ar'
dT
2
where Ai'
dAr'
+ 2 kA' -- k2
dT
26
(A' 2 + ---- )Ai'
$oqo 2
is the imaginary part of A'.
associated with
a2
-
= 0,
The growth
(4.25b)
rate a
(4.25a,b) is
(N'As2 + A' 2
26
{(N'As 2 + A' 2 -
t
-)
Ooqo 2
k2
26
+ (A' 2 + --)[2N'As2
2
$oqo
-
(A,
2
+
26
.)2
aoqo 2
26
.
)]1/2
$oqo 2
(4.26)
The
first thing to note about (4.26) is that where A'
appears it does as A'
ground
order
(A'>O)
2
so remarks
approximation
flow at the
also apply to easterly flow at the ground.
to simplify the calculations
approximation
for westerly
the Boussinesq-like
ao >> h-1 will be made.
Under this
In
145
qo =
aj = j
Bo
(4.27a)
ao
=
= Jr
(4.27b)
2ao n
(4.27c)
where
a0 =
(4.27d)
As a consequence of
(4.27) the nonlinear coefficient
(4.24)
reduces to
1
m + 2
2k 2 L 4
No
=
(ao-1)
m=
m=1
z2a o
(m
+ ao
-
1
-)2
2
((m
-
1
_)2
2
-
16aoL2
and
(4.28)
the growth rate
(2.7.26) reduces
to
a2
6
(N'As
k2
+
(A'
2
+
2
+ A
2
-)(2N'As
4K 3
4K 3
2
2
+ A' 2
)4K3_ {(N'As
6
-
4K 3
where
J
1
m = - + 1i, L= i
2
No
and N'
k2
)2
-
4K 3
1/2
(4.29)
146
No>O for all but
implies
that
the smallest meridional
wavenumber.
the nonlinearity would stabilize a
baroclinically growing wave as in Pedlosky
the contribution to the growth rate
(1979).
However,
(4.29) from the terms
proportional to N' is not necessarily stabilizing.
this
This
To show
the parameter regime of (4.29) will be considered
corresponding
to baroclinic stability of an infinitesimal
disturbance, i.e.
6 < 0.
Plumb
(4.30)
(1979) discussed this parameter regime in the context
If
of his two-layer model.
V is
defined
such
that
4K 3
(4.31)
and the steady wave amplitude in
(4.29) is rewritten as
A'2M2
2K2(2
=
the growth
k
(4.32)
rate takes the form
a2
-
u2)2
-
2
[A'
+
,2 +
2
N'A' 2 M2
_]
K2(A,2 - U2)2
_
{[A,2
+
12
N' A' 2 M 2
+ K2(
K2(,2
+ (A'
2
+ p,2)
[
(A'
2
-
2)
2)2
-
(4.33)
2N'A'
k2
k2(A'2
2
-
2
M
1/2
2)2
2)2
147
If
(4.33) are
the operations within the curly brackets in
carried out
When a2 /k 2 <0 the modes are propagating, but
real.
a 2 /k
2
is easy to see that a 2 /k 2 must always be
it
>0 one mode becomes unstable.
are assumed
to be zero for
If
the moment,
the
when
nonlinear effects
(4.33) reduces to
a2
= -
-
k
; 1)2
(A'
2
(4.34)
whose roots correspond to a slowly propagating mode (+ sign)
and
to a faster propagating mode
increasing
the
(-sign).
forcing a small amount,
which mode's phase speed
we can determine
is decreased.
which will become unstable.
Now, by
This
is the mode
For small forcing a 2 /k 2 can be
approximated by
a2
.
k2
N'A'
=
-
(A'
T
2)
2
M2
K2(12-
When the zonal
A' 2
-
A'
U2 ) 2
flow is subresonant,
U
nonlinear effect.
i.e.
(4.36)
Thus,
increase due
Indeed this is the case.
see that
there is no instability possible
However,
-
if the
12 > 0
to the
we expect both of them to remain
stable.
A, 2
(4.35)
12 < 0
the phase speeds of both the modes
flow.
-
+
From
(4.33) it is easy to
for subresonant
flow is superresonant,
i.e.
(4.37)
148
(4.35) shows that the nonlinearity decreases the phase speed
of the
slower mode while increasing
mode.
Thus the slower mode is expected to be unstable.
This decrease
increases
the speed of the
faster
in phase speed continues as the forcing
until
2N'A' 2 M 2 = K2
(A, 2
-
V2)3
(4.38)
a2 /k 2 = 0 for the slower mode.
at which point
From this
point on the disturbance is in phase with the forced wave
which allows a coherent interaction between the
perturbation,
forcing
forced wave and mean flow.
Therefore, as the
increases further such that
2N'A, 2 M 2 > K2 (A, 2
instability occurs.
destabilizing
-
p2)3
(4.39)
Thus the nonlinear effect
is
here.
Recall that as
4K 3
increases the flow becomes more baroclinically subcritical.
As U is increased toward A',
a resonance is approached
which the slowest baroclinically subcritical
becomes
for
neutral mode
stationary relative to the topography.
Thus, since
the neutral modes are slower as resonance is approached,
less topographic height should be necessary to excite the
stationary instability.
Figure
As is shown by (4.39) and
(4.41), less topography is required to make the flow
149
unstable as resonance is approached.
This behavior is
similar to that of the topographic instability discussed in
Chapter 3 which is also more easily excited near resonance
for superresonant flow only.
This stationary instability,
then, might be thought of as the topographic instability of
the Charney mode as opposed to the topographic instability
of a normal mode of the Charney model.
We now consider the parameter regime of (4.29)
corresponding to the baroclinic instability of an
infinitesimal disturbance, i.e.
6 > 0.
(4.40)
In this case U is defined as
1~=-
4K 3
(4.41)
and the growth rate is written as
a2
k2
= -
+ K
2
N'A'2M2
[A,'2
-
k2(A,2
N'A'
(
2
2N'A'
K2(A
'2
2 M2
2
+
+ (A'
2 M2
1/2
+
]
2 +
2
±_
{[
2,
2
+ U2)2
+ U2)[
_ (A'
2
+ p2)
U2)2
(4.42)
Here, when the topographic forcing is zero,
the instability
corresponds to an unstable baroclinic wave propagating as
the speed of the zonal velocity, i.e.
modes
for the unstable
150
K
(4.43)
forcing M is increased by a small amount, the growth
As the
rate of the unstable modes is decreased.
This is easily
seen by an examination of te approximation to
(4.42) for
M, i.e.
small
N' A'
a
+ iA'
-
2
M2
2
+
-
2UK 2 (A'
k
(4.44)
p2)
The phase speed is also modified but this effect is of
higher
The numerical
order.
calculations shown in Figure
(4.41) demonstrate that this tendency continues until the
quantity in curly brackets in
N'A'
K2(2
2
M2
N'A'
4A' 2
2
+
(4.42) is equal to zero, i.e.
2
M2
[
+
)2 + 4A2
_
.2]
K2 (A, 2 + U2)2
= 0
(4.45)
at which point the growth rate vanishes leaving neutral
propagating waves of equal phase speed.
As M is increased
further, this quantity continues to increase.
The result is
to cause the phase speeds of the two modes to diverge.
The
mode whose phase speed is decreased with increasing M
becomes stationary when M is increased enough
2N'A' 2 M2
=
....
K2(A,2 + p2 ) 2
and becomes
2N' A'
A'2
+
such that
2
2
(4.46)
unstable when
2
M2
S>
K2(A,2 + U )2
2
A' 2 +
U2
(4.47)
151
Therefore, the role of topography depends on its size.
Smaller topography tends to stabilize the propagating
baroclinic wave which grows at the expense of the mean
vertical shear.
the presence of
However,
as the topography becomes larger,
the topographically
forced wave modifies the
nonlinear wave-mean flow interaction enough
an unstable
disturbance.
to give rise
to
This instability, which still
draws its energy from the available potential energy of the
mean
flow, occurs because the neutral propagating wave
disturbance becomes stationary relative to
as M is increased and
wave.
This is the
for which
transition
6<0.
therefore coherent with the
forced
same instability mechanism as in the case
In fact,
line
the forced wave
in Figure
(4.41) note how the
from neutral waves to the
stationary
instability continues smoothly from negative to positive
super-criticalities.
Thus,
the Charney mode occurs
supercritical baroclinic
the topographic
instability of
for both subcritical and
flow.
Also of interest is the dependence of the instability
on wavenumber space, all else being
(4.4.2a,b,c),
for three
different
fixed.
values
In Figures
of the
shear
supercriticality, the regions of stationary instability,
propagating instability and neutral propagating waves have
been plotted vs.
both k and 1.
The most striking feature of
these figures is the relatively weak dependence of the
instability on k and the very strong dependence
negative shear supercriticality
(6<0)
on 1.
For
the only instability
152
STATIONARY
INSTABILITY
NEUTRAL
PROPAGATING
WAVES
INSTABILITY
WAVES
-12
-10
Fig. (4.4.1)
-8
-6
-4
-2
2
4
The regions of the topographic amplitude
and shear supercriticality
S
6
A
,
, space for which there are
neutral waves(N), stationary(S) or propagating(P) instabilities.
The calculation was done with
A'=1, k=l1,
and 1=1.
153
10
I
9.
I
I
I
7
N
I
6-
5-
II
4-
I
0
I
2
3
5
4
6
7
8
9
k
Fig. (4.4.2a)
For negative shear supercriticality the regions
of zonal and meridional wavenumber space for which there are
neutral waves, stationary or propagating instabilities.
calculation
A'=1,
A'1 =1
and
a =-1.
In the
154
10
9
8
I
N
7
6
IS
23-
N
L
P
0
I
2
3
4
5
6
7
8
9
10
k
Fig. (4.4.2b)
( 8 =1).
Same as (a) except for positive shear supercriticality
155
N
)
N
I
S
p
IIIP
4 I
6.
I
L
I
P
N
3
O
0
N
rI
I
I
I
I
I
I
II
I
2
3
4-
5
6
7
Fig. (4.4.2c)
Same as (b) except
S =3.
I
8
9
10
156
possible is of
very
k<8
the stationary type and
narrow bands centered
at
shown in Figure (4.4.2a).
1=1.75
for k<4 and 1=4.75
for
This instability which relies
nonlinear wave-mean flow interaction remains
on the
virtually unchanged for positive
shown
is limited to two
in Figures (4.4.2b,c).
of propagating
unstable
However, when 6>0 wider bands
instability begin to appear which tend to get
wider as k gets larger.
For 6=1
these bands are centered at
As the flow becomes more baroclinically
1=3.25.
1=.75 and
shear supercriticalities as
for larger
6=3 two additional
instability centered at 1=6.5 and
bonds of propagating
1=9.5 develop.
Thus,
for
larger supercriticality, the propagating instability is not
limited to
the larger meridional
supercriticality.
stationary
To summarize,
scales as it is
it can be said
instability or topographic
its existence to the topographically
for smaller
that the
instability which owes
induced wave-mean flow
interaction is limited to larger meridional scales while the
propagating instability which is essentially a baroclinic
wave modified by the topographically
induced wave-mean flow
interaction is limited to larger meridional scales for only
weak supercriticalities.
Furthermore,
instability tends to occupy the larger
the baroclinic
instability, under the
the topographic
zonal
scales while
influence of
topography tends to be more effective at shorter zonal
scales.
Finally, an increase in topographic
amplitude tends
to stabilize the barolcinic instability of the zonal
but gives rise to a new topographic
In
the next
section
instability.
we will examine
friction on the instability of the
flow
the
role to strong
forced wave.
157
5.
The topographically
influence of 0(1)
forced Charney mode and the
Ekman
friction
The subject of this section
stronger
is a study of the effect of
friction on the instability of a topographically
The basic approach is
forced Charney mode near neutrality.
the same as in section 4 in that
the analysis pivots about
the neutral Charney mode which is weakly forced
diversion of weak zonal
by the
flow by topography and made weakly
unstable by letting
B = Bo-
,
2
6
(5.1)
The nonlinearity due to the wave mean flow interaction
is
then balanced against the linear growth or decay and forcing
terms in order to obtain an equation for the wave amplitude.
However, in the present analysis both the Ekman friction and
Newtonian cooling are assumed to be stronger than iin
section 4.
is assumed to be 0(1).
In fact Ekman friction
This modifies
the properties of the
instability.
of these modifications is the purpose of this
Pedlosky (1979),
The study
sction.
in section 5, studied the nonlinear
evolution and eventual equilibration of the weakly unstable
Charney mode in the presence of
the absence
of
forcing.
identical
to that except
forcing.
Here
The present
F2n,
Ekamn friction but
problem
is
for the inclusion of
as in Pedlosky
parameter is assumed to be 0(e
n =
0(1)
in
essentially
topographic
(1979) the Newtonian cooling
2
),
i.e.
(5.2)
158
As a result,
is O(e
2
the growth rate associated with
) as opposed to O(E)
in
appropriate long time scale
is
T
=
section 4.
the instability
Therefore,
.
Et
the
(5.3)
This slower long time scale necessitates a change in the
definition of the inner independent variable C,
which now
becomes
S= z/
2
.
(5.4)
As previously mentioned, the wave forcing is provided
by the diversion of a weak zonal
topography.
that
flow by sinusoidal
Again the uniform zonal velocity is chosen such
the advection time scale associated with it is of the
same order as
choice for
the e-folding time scale.
the uniform zonal velocity is
U = C2 A'
.
(5.5)
The topography is 0(e)
as in section 4,
n = eb = eMeikx(sin
ly)
z ax
(5.6)
inner and
lower
equations can be written
+ qo -ax
-
2 (aT
+
A'
E J(,
-
E2n
C) '
(
;z 2
-)I
ax
+ e2 6ax
3
32
-
i.e.
+ *.
Under the above assumptions the outer,
boundary
In this case, the
- h-
1
az
(5.7a)
159
a
a
+ (A' +
[aT
aa4
2
aC
S
+
e
2
(qo
-
+ E,4V 2 )
s2 h-1
-
)--](
ax
a
a2
1
J(,
26)-
ax
a
+ c2h-1
-
E
asc
3
+
4
V 2 p)
a2
-
n'(
E2
-
h-
1
),
2
3a
(5.7b)
ar
and
A
a32
+ A'
aTaC
-
1
32
rV 2
-
1
+ -J(4,
- axac
n'..
ac
ax
a¢
-- ) = ac
s
e J(
,b)
e2 A'bx
-
(5.7c)
respectively.
Again, by expanding the outer and
inner stream
functions € and p as
=
o + C#1 + E2
+
**.
,
(5.8a)
42 +
*.
,
(5.8b)
A
A
S=
'2
+
fo
01 + ,2
we begin consideration of
the resulting sequence of outer
As is now the usual the 0(1)
and inner problems.
outer
problem leads to the specification of the neutral Charney
mode which is
written as
-8oz/2
=
€o = Aze
Aze
e±KX~Kx
e!
(sny-+-(59
(sin
ly)
+ *
.
(5.9)
160
Keeping
(5.9) in mind,
the 0(c) outer equation takes the
form
n1
z -ax
a€1
+ qo ax
= 0,
(5.10)
to which the only new solution
€1 =
D1
is
(5.11)
(y,z,T).
It will become apparent that this solution represents the
0(s)
zonal
flow correction forced by the self
interaction of
the wave fields.
Using
the solutions at previous orders
to evaluate the
(5.9) and
(5.11)
inhomogeneous terms of the 0(c2) equation
becomes
an 2
z -ax
a¢ 2
dA
F
+ qo = [Eqo (-- + ikA'A + n'A)-- + (ik6
ax
dT
z
n'K 2 )Fo A -
-
Fo
ikFoIIlyA + ikqo z
ulA]eikx(sin ly) + *
(5.12)
where
-Boz/2
Fo(z) = ze
(5.13)
Attempting to remove the secular
results
terms from
in the solvability condition
1
f
dA
6
+ A'A + -A
qo 2
ik dT
1
2 (sin ly)e -lK x
0
(5.12) as before
¢2 (x,y,O,T) dy
-
161
-
K2
--qo 2
n'
ik
+ Aull
"
-
1
aull
--)
-qoz
)A-Af
dze
(zp 1 1
-
az
0
(O,T) .
(5.14)
Evaluation of the term on the
left hand side of (5.14) will
lead to the determination of the amplitude equation for the
wave.
However,
€ 2 (x,y,O,t)
to do so requires
the knowledge of
which can only be determined through the
asymptotic matching of € and € after consideration of the
0(e
2
) inner problem.
Thus,
we are forced to postpone the
final evaluation of (5.14) until then.
We now consider the 0(e3) zonally averaged problem.
(5.11)
is used,
it takes the
a
32 (b1
a321
-(-+ 3y 2
az 2
aT
= -
J (0o,
R2)
- J(0
2
form
3 1
-)
az
h 1
If
+
3201
,1
n' (--- h-1
az
1
--
)
az
, Ho),
(5.15)
or equivalently
a
321
a2 ¢ 1
3a
-(-+ --aT
ay 2
az 2
a
ay
where
o (R2x
qo
+ -z
321
1
h-1
-)
az
+ n'(
3z
2
3
h-1
1
- )
az
€2x),
(5.16)
the relation
1o =
-
qo
-z
o
(5.17)
162
was
used as was
integration by parts in x.
Jacobian terms of
(5.15) are put
(5.16) it is easy to see that
When the
into the convenient form in
(5.12) can be used to
eliminate the 0(e2) variables in
(5.16), which have not been
0(1)
explicitly calculated, in terms of the known
field.
The
result
of doing
a
a2
3I~2l
(+
3T
ay 2
az 2
-Bo
= qoe
z
diA1 2
[----dT
so is
2
aa
-)
3z
h
wave
+ n'
1
1
(-
-)
- h
3z 2
zK2
+ 2n'(1 - -)
q0 2
az
a
(sin 2 1y).
A1 2 ]
ay
(5.18)
This is the desired equation for the
forced by the transient and
zonal flow correction
internally damped wave field.
The next step is to solve the sequence of inner problems
obtain
Since
2
(x,y,O,T) and a lower boundary condition on tl.
the only wave-like solution up to 0(e
(5.9) which also becomes 0(E2)
O(e2).
2
(y,z,T) must have a counterpart in the O(s)
0(s) zonal
inner
it is easy to
inner solutions
However, the outer O(e) zonal
since tl need not
) is given by
near the ground
see that there are no matchable wave-like,
until
to
solution tl
inner problem
necessarily vanish at the ground.
solution
is
written
fl(y,T),
If
the
the 0(e)
zonally averaged lower boundary condition yields
a2f
r' ay2
0,
(5.19)
163
which implies
fl
(5.20)
= 0
Then matching fl to
1
which
the
l1(y,O,T) yields
(5.21)
= 0
(y,O,T)
is the desired lower boundary condition
inner solution is at most 0(s
We now push on-to the O(e2)
2
Thus,
on el.
).
inner problem which
A
[-- + (5 + a')-- +n,]
- =
,
3a 2
ax
3T
(5.22a)
A
32
a2 2
axa
3T3
The solution of
solution
¢2
3
2
-
+ A'
-
2
2^
+ rV 02 +
3a2
n'
-
ax
(5.22b)
3a
(5.22a) which will match to the outer
is
=
+ C2) eikx (sin ly)
(A
+ *
(5.23)
,
where C2 can be determined by requiring
(5.23)
A'bx
that the
satisfy the lower boundary condition
1
C2 =
ik
+ rK2
+ rK 2
dA
(-+ ikA'A +
dT
solution
(5.22b).
It
is
ikA'
n'A)
+
M.
ik
+ rK
2
(5.24)
Matching the inner solution (5.23) to the outer solution
yields
42 (x,y,O,T) = C2 eikx (sin ly) + *
164
which becomes
1
¢2 (x,y,O,T)
=
[
dA
(-- + ikA'A + n'A)
dT
ik + rK 2
ikA'
+ rK2IMI eikx (sin ly) + *
ik + rK 2
S
(5.25)
when (5.24) is used.
Now that ¢ 2 (x,y,O,T) is determined, it is a simple
matter to obtain the equation for the wave amplitude.
Substituting
(5.25) into the solvability condition (5.14)
and performing the indicated operations yields the following
equation for the wave amplitude, i.e.
A
dA
+ ikA'A +
(--
k2
+ ---
n'A)
K2 )
-
[n'(qo2
-
qo 2
dT
n'
ikA
+ -
(6 -
go
2
-- )
rK2
1
+ Ak
2
i
(
C
)f
rK 2
r
6]
-
k
-qoz
e --
aull
(zp
-
11
---
)dz
3z
0
k 2 A'
+
-
M = 0
rk 2
(5.26a)
which along with the following set of equations for the
zonal
flow, i.e.
a3
--
3T
a3
a2,1
a21
(----
ay
+
2
-
3z 2
32(1
1
-
h-1
-)
az
+ n'(
3s 1
-
az
h -1
-)
az
2
165
-so
z
= qoe
dIA
[--dT
2
zK2
+ 2n'(1 - -)
qo 2
tAt
2
a
] (sin
ay
2
ly),
(5.26b)
1
(Y,O,T) = 0 ,
(5.26c)
and
-0
(5.26d)
aTay
on y=0,1 form the closed set of equations describing the
evolution of the wave-zonal
flow interaction.
Notice that
the wave equation (5.26a) is first order in time as opposed
to second order in time which was the case in previous
The strong friction present in this case is
sections.
responsible
for this decrease in order.
decrease
order was
in
first
Such a frictional
noted by Pedlosky (1970)
in
the
context of the two layer model.
The
the zonal
(1979).
zonal
flow equations
(5.6b,c,d)
flow equations (5.23),
are identical
to
(5.24a,b) of Pedlosky
The equation for the wave amplitude (2.8.26a) is
the same as Pedlosky's wave amplitude
equation
(5.27)
except
for the presence of a term proportional to the weak zonal
velocity A' and an inhomogeneous term proportional to the
topographic amplitude M.
This topographic inhomogeneous
term forces a wave whose stability we will now study in the
absence of Newtonian cooling.
equations
Assuming n'=O, the amplitude
(5.26a-d) take the form
166
0
dA
-
(A'
+ ik
+ s)A - ksfA + Ak(f-i)
f
dT
-qoz
e
0
aull
x
(zp
-
11
----
) dz + kfA'M = 0,
ax
a2
a
3a2
1
-(---aT
ay 2
(5.27a)
1
1
+ --az 2
-
h-1
-oz
)
= qo
e
diA1
2
a
.
.- - (sin21y)
dT
ay
(5.27b)
az
¢ 1 (Y,O,T) = 0
;2t
(5.27c)
a2
1
(O,z,T) -
aTa
2
1
aTa 2
(l,z,T)
= 0
(5.27d)
where the inverse friction parameter,
f
is defined
2
= k/rK
(5.28a)
and the supercriticality parameter,
s
f,
s,
is defined
6/qo 2
=
(5.28b)
The steady topographically forced wave solution of
(5..27a-d) is
[fs
As =
fA'
f
If
M is
real,
2
+ i(A'
s 2 + (A'
+ s)]M
+ s)
2
(5.29)
the real part of As reaches a finite maximum
when A' + s = 0.
Since the maximum is finite it is possible
to explore the nature of the instability near this
resonance.
This is in contrast to the case in section 4
167
where
the
analysis
stability
was
resonance because the lack of
valid
away
only
from
to an infinite
friction lead
response at resonance.
For
simplicity, M will
amplitude equations
same
Fourier
sections,
+ Asi
(A'
and A'r and
in
the disturbance
+ s)A'
-
ksfA'
+ 2k(f-i)NoAs
wave
A'r
(5.30)
Asi are
A'i are
previous
(Asr
= 0,
A'i)
Asr and
y used
solved using
form
takes the
dA'
-+ ik
dT
for
they are
in
technique
the linear equation
amplitude A'
where
expansion
the
linearized about the
(5.27a-d) are
topographically forced wave (5.29) and
the
When
assigned to be real.
the real
the
real
and
and
imaginary parts
imaginary
of As,
parts of A'.
Under the Boussinesq-like approximation
h-1
the
<<
B0,
nonlinear coefficient,
K1
No
-
4
-
(I
N6
m=l1
the
NO, takes
form
1
k
(m + i
1
-
-)[(m
2
-
1
12
-)2
2
"2
12
]2)
8K 2
(5.31)
which is
nearly always greater than
zero and
the
supercriticality parameter becomes
s = 6/4K
2
.
(5.32)
168
Note in
(5.30) that as the
increased
the
steady wave
makes the
last
term of
topographic
amplitude
problem
the
forced wave and the perturbation
forced by the
the
forced
If
interaction of
the
zonal
in previous
linear stability
in the
interaction between
increased which
As
(5.30) only when the topography
represents
is
hence altering the
(5.30).
term which arises
sections this
As is
(5.30) larger,
stability characteristics of
amplitude
is nonzero,
topographically
flow which in turn
the perturbation wave
field and
wave.
A'. is
assumed
to be proportional
to eaT in
after much algebra, the dispersion relation
for
(5.30),
a can be
written
NoM2
a
= sf f2s2 +
k
+
2NoM
2
f 2 A, 2 (A,
f2 s 2 +
(A'
In the absence of
Thus,
zero.
disturbance
case
the
zonal
A ,
2
f
(A'
+ {[
+ s)
+ s)
-
Recall
= sf
2
(A'
f2s2 +
+ s)
2
2
f
(A'
3
]2
+ s)
2
1/2
}
+ S)2
(5.33)
topography, the forced wave amplitude is
interaction
between
flow is nonexistent
the dispersion relation
k
NoM2 A,
3
± i(A' + s)
forced
wave and
when M=0.
In this
the
(5.33) reduces
to
.
(5.34)
s = 6/4K
relation
for
2
f = k/rK 2 .
This
is the dispersion
an infinitesimally small
Charney mode near
and
is
169
neutrality under the influence of strong Ekman friction.
For s>O the propagating Charney mode is unstable while for
s<O the propagating Charney mode decays.
However, in the
presence of Newtonian cooling, absent here, Pedlosky
(1979)showed that there is a minimum supercriticality
K2
6c
=
k2
rn'
(3K 2
+ h-
2
),
(5.35)
which must be exceeded for instability to occur.
We will now examine how the baroclinic instability of
the Charney mode is altered as the topographic amplitude is
increased.
If the topographic amplitude is small, the
dispersion relation will reflect at least a relatively weak
interaction between the perturbation mean flow and the
forced wave which will help us understand the full
dispersion relation (5.33).
NoM2 A'
a
-
f
3
sf -
±
f2 s 2
k
2
For M<<1, (5.33) reduces to
+ (A' + s)
NoM2 f 2 A' 2
...
(A'
+ s)[f
2
s
2
+
i(A'
{1
+ s)
2
' + O(M4
) .
(A' + s)2]
(5.36)
We can see immediately from (5.36) that the small increase
of topographic amplitude stabilizes the baroclinic
instability for s>O and further stabilizes the baroclinic
stability for s<O.
The phase speed is decreased for A'+s>O
and increased for A'+s<O.
From Figures (4.5..la,b,c) which
are plots, for A'=1.0 and various values of the Ekman
parameter r, of the regions in topographic amplitude,
170
supercriticality space
instability
(PI),
it is
is propagating
propagating decaying
stationary decaying waves
(SI),
which there
for
(SD) and
waves
stationary instability
clear that this stabilizing
topographic
amplitude is
increased
influence of
further.
The next
transition as M is increased depends on the
6 and
value of r.
instability
two
persists
terms in
sf
(2)
-
2
until
either
f
-
2
s
(A'
+ s) 2
(5.37)
2
+
2
(A'
+ s) 2
first
f
3
2NoM2 f
+ s)2
f
2
s
in (5.33) becomes
in Figures
(4.5.1a)
the
(A'
+ s)
+ s)
2
that occurs in all
(4.5.1a,b,c) except
where r=.2.
As
to a transition from
to a propagating decaying
is the situation
for
In
the case
(1)
next transition is from
6>3.0
cases
in
friction is
the Ekman
transition occurs at larger
supercriticality.
satisfied,
2
(A'
possibility corresponds
This
this
+
A'
(5.38)
wave regime.
increased
2
2
> 0.
instability regime
given
first
A 2f3
a propagating
plotted
the
i.e.
(A'
The
sum of
i.e.
negative,
the quantity in curly brackets
NoM 2 A,
Figure
the
(1)
< 0,
f2 s 2 +
positive,
supercriticality
6>0 the regime of propagating
(5.33) becomes
NoM
or
For
(PD),
values of M for a
where
(5.37) is
the propagating
decaying wave regime to the stationary decaying wave regime
171
The next transition is
(5.38) is satisfied.
and occurs when
to the regime of stationary instability which occurs for
I[
NoM 2 A'
f
2
s
-2
2
+
2
(A'
f
2NoM2 f
3
+ S)
+
2
f2 s 2 +
2
A'
2
+ s)
(A'
(A'
+ s)2
NoM 2 A' 2 f3
-
(A'
+ s) 2 1>1
sf -
f2 s2
(5.39)
+
(A
+
)2
At this point there is an interesting difference between the
transition to the stationary instability here where r is
0(1)
and the transition to the stationary instability in the
inviscid case discussed in the last section.
In the last
section, before the transition was to occur, the phase speed
of one of the neutral modes was slowing down as the
topographic amplitude increased.
As the topographic
amplitude increased further, the neutral mode became
stationary relative to the forced wave.
Therefore, the
neutral mode became coherent with it and could easily draw
energy from it leading to the instability.
However, here in
the presence of friction the mode becomes stationary and yet
i-t still decays until the topography is increased (thereby
enhancing the interaction between the perturbation zonal
flow and
forced wave)
enough to overcome
the
friction.
In
summary, the inviscid case becomes unstable as soon as the
mode becomes coherent
with the forced
wave but in
the
viscous case the topography must increase past the point
where the waves are stationary to enhance the wave mean flow
interaction enough to become unstable.
vir
10
19
8
SD
St
7
6
5
4
PD
PD
-9
-8
-7
-6
Fig. (4.5.1a)
amplitude
M
-5
-4
-3
-2
-I
4
5
Regions of SI, PI, SD, PD (see text for definitions) plotted vs. topographic
and supercriticality
S
with
S
=1.0, r=.2, k-1.0, and
1=1.0.
6
7
8
173
When the
(5.37)
is not,
sf
sf
the
second possibility
-
i.e.
NoM 2 A 2 f 3
2
+ (A' + s)2
f2
> 0
> 0
(5.40)
first and only transition is
instability
regime to the
This occurs
in Figure
also
(5.38) is satisfied but
from the propagating
stationary
instability regime.
(4.5.1a) where r=.2
for
6>3.0.
This
is a uniquely viscous result.
When
the
flow is
regime of propagating
transitions
decaying
subcritical
instability.
as M increases if
waves
(6 and
The
s<O)
there
is
no
succession of
from a regime of propagating
to a regime of stationary decaying waves as
(5.38) is satisfied, to a regime of
stationary instability
when
the succession of
(5.39) is
transitions
satisfied.
That
as M is increased
is,
if
A' + s > 0
(5.41a)
A'
(5.41b)
When
+ s < 0
the only transition is from a regime
waves
to one
of propagating decaying
of stationary decaying waves.
stationary instability in the region where
satisfied.
A'
+ s
There is no
(5.41b) is
If
= 0
the dispersion relation (5.33) reduces to
(5.41c)
174
SD
PD
PD
-9
-8
Fig. (4.5.1b)
-7
-6
-5
-4
Same as (a) but r=1.0.
-2
-1
1
2
175
M
14
-'
13
-
12
-
II
I0
9
8
PD
PO
7
6
5
4
3
2
-9
-8
Fig. (4.5.1c)
-7
-6
-5
-4
Same as (a) but r=5.0.
I
II
I
-3
-2
-I
0
I
2
176
= -
c
which
A'f -
NoM
2
f ± NoM
2
f,
(5.42)
says there are only stationary decaying modes present
for any topographic amplitude M.
instability
is again limited
superresonant
A'
+ s
Thus,
the stationary
to what might be called a
flow for which
> 0.
This is suggestive of its topographic origins.
A further comparison of Figures (4.5.1a,b,c) reveals
several interesting features.
First note that the
stationary instability seems to transcend the
supercriticality boundary 6=0 while the propagating
instability does not.
The latter requires 6>0.
As was
pointed out in the last section, this is due to the fact
that
the propagating
instability is really just a
topographically modified baroclinic
supercriticality is required while
instability
the
instability
the stationary
relies on the presence of
zonal available potential energy.
Figures
(4.5.1a,b,c)
the
stationary
for which
the
forced wave to tap
However,
instability
in all
under
the
the
influence of large friction requires a larger topographic
amplitude as resonance is approached.
previously mentioned
In
fact,
there is no stationary
as
instability
A' + ( 0.
This
discussed
is
in direct contrast
in section 4 where
to the
inviscid case
the topographic
amplitude
for
177
required
for
approached,
A
instability decreases
including the point
of interest
second point
the transition
in
Figure
distinguish
(4.5.1c)
is that as
not on M.
where
friction is
This
some
and
from the vertical
is
at which resonance occurs.
lines become more vertical,
highly dependent on 6 but
evident
as resonance
increased,
that is to say,
is particularly
lines
may
be hard
the stationary instability
is limited
to a very narrow region in 6.
stationary
instability is squeezed more closely near the
side of resonance as
superresonant
In
addition, the
friction is
increased.
This we should keep in mind when we examine the
instability
and decay for
waves plotted vs. k,1
The
third
the
propagating
decaying waves, especially
This
discuss the
This
is
also
a useful
fact
stability regimes in k and
ground is eastward
(A'>O)
dispersion relation (5.33) A'
also as A'.
whether the
the ground
(4.5.2) the stability
when we
feature not
the
(A'<O).
There,
flow at
the
Here in the
appears not only as A'
Thus, the viscous
flow at
mind
in section 4.
of whether
or westward
in
1 space.
strongly damped flow has another
instability is independent
figure
for larger values of
to keep
present in the inviscid case discussed
the
point of
small area in the diagrams of
interest is
friction.
relatively
regions of
waves and stationary
propagating
for r=5.0.
to
2
but
instability does depend on
is eastward of westward.
regimes
in M,6 space are
westward flow at the ground A'=-1.0 with
plotted
for the
r=1.0.
The progression of stability
transitions as M
In
178
M
14
SD
13
12
II
P
PO
10
9
SI
8-
7
6
5
4-
pl
32
1
-2
-I
Fig. (4.5.2)
0
I
2
3
4
Stability regimes plotted vs.
westward flow at the ground,
5
6
7
i and
&
for the
8
A =-I.0 and Ekman friction r=1.0.
9
10
179
increases is essentially the same as that just discussed for
A'=1.0.
In addition,
Figure (4.5.2)
shows that there are
only propagating decaying waves for
6<0
and that the stationary instability occurs only near 6=8.0.
Furthermore, at 6=8.0 the stationary instability occurs for
every value of M.
This is due to the shifting of the line
for which
A'
+ S = 0,
from 6=-8.0 to the supercritical value 6=8.0 when A'=1.0 is
changed to A'=-1.0.
When A'+s=O the dispersion relationn
(5.33) reduces to (5.42) which shows that there is always at
least one growing mode for which the growth rate is c=f if
A'=-1.0 which is in agreement with the numerical results.
The instability occuring about A'=1.0 is baroclinic and not
topographic.
Finally we note that since the region of SI is
near the resonance defined by A'+s=O the region of SI in k,l
space will be near the circle defined by
k2+1 2 = -6/(4A')
= 6/4.
Thus, when A'<O
the stationary
instability occurs around the resonance A'+s=O but for A'>O
the stationary instability occurs only for superresonant
flow, i.e.
A'+s>O.
We will now examine the regions in k and 1 space of
propagating instability (PI), propagating decaying waves
(PD) and stationary instability (SI) for A'=1.0.
Figures
180
(4.5.3a,b,c) are
6=-2.0,
the
1.0,
and 6.0 respectively.
stability regime diagrams for
1.0,
respectively and
of plot
figure
for
r=5.0 and
for r=5.0
wavenumber
regime.
and
There are
6<0 is
r=1.0 and
6=.05,
not shown
regimes of
r=.2 and
(4.5.4a,b,c) are
6=-2.0,
(4.5.5a,b) are
6=.01 and
from PD to SI
friction but
0.5 and
the same type
respectively.
since
A
the entire
decay
stationary decay near the
especially
they are too small
for smaller
to be drawn.
values of
Recall
from M
6 stability diagrams that regions of PD were small
also.
In
the present diagrams
because of the
Let us
friction.
subcritical
occurs
first compare diagrams
In Figures
(4.5.3a) and
flow (6<0),
in very
wavenumber
it is particularly
As
the only
grow to the left
for
instability present is the
in the inviscid case of section 4,
narrow bands
space
(4.5.4a) which are
in 1.
The rest of
is one of the propagating
to include
it
the
decaying waves.
to positive values,
propagating instability appear
increased.
true
for a given value of
the supercriticality is increased
band of
there
choise of parameters made.
stationary type.
As
Figures
Figures
for
space is occupied by the propagating
transition
vs.
the stability regime diagrams
first for
larger k values as
These wedges develop between
small k and
6 is
the bands of
stationary instability which stay essentially the same as
is
increased.
6 is increased
The wedges of PI
for
6
develop more rapidly in k as
larger values of
location in 1 tends to be
a
friction but
independent of
r.
the
This can be
181
ISI
I
IPD
6
ISI
IPD
2
S
I
2
3
4
5
6
7
8
k
Fig. (4.5.3a)
Stability regimes vs. k and 1 for fixed
topographic amplitude
M
=1.0 and r=.2,
S =.2.
See text for the definition of SI, PD, and PI.
9
182
10
IPI
S I
8I
7-
PD
6
PI
5.
0
4-
SPI I
PD
I
2-
I
0
Pt
1
3
4
6
5
k
Fig. (4.5.3b)
Same as (a)
but
g
=1.0.
7
8
9
10
183
0
I
2
3
4
5
6
k
Fig. (4.5.3c)
Same as (a) but
8 =6.0.
7
8
9
10
184
I
8
ISI
I
H-
5
I
PP
I
SI
I
21
I
I
S
I
I
2
3
4
5
6
7
8
9
k
Fig. (4.5.4a)
Stability regimes vs.
r=1.0 and
9 =-2.0.
k and 1 for
V=1. 0,
10
185
OL
O
I
2
3
4
5
k
Fig.(4.5.4b)
Same as (a) but
6
7
8
9
10
186
10
PI
I
9
I
PD
I
IPI
PD
7I
4-
I
I
0
I
Pi
I
I
2
3
I
4
5
6
k
Fig. (4.5.4c)
Same as (b) but
1.0.
7
8
9
10
187
SI
SP1
PD
5.
IP
0
I
I
I
II
I
2
3
4
5
I
i
I
I
6
7
8
9
k
Fig.
r=5.0
(4.5.5a)
and
Stability regimes vs. k and 1 for
=.01.
44 =1.0,
10
188
I
7P1
6.
0
5-
4.
3
3PI
I
6
0
I
2
3
4
5
6
k
Fig. (4.5.5b)
Same as (a) but
S =.05.
7
8
9
10
189
seen by comparing
easily
values of r and 6.
the figures
keeping
mind the
in
As r gets larger the regions of SI
become smaller and in fact disappear for r=5.0.
also to be expected
from the M vs.
6 stability
This is
diagrams wich
showed a decreasing region of SI as r increased.
Nevertheless, when regions of SI do appear they appear in
approximately the same bands in 1 independent of the
friction.
As already mentioned, the center of the wedges of
PI in 1 also tend to be independent of 1.
there are no bands of SI such as in
Figures
Thus, even when
(4.5.5a,b)
for
r=5.0, the wedges of PI still develop around those bands in
1 which contain the stationary instability for lesser values
of r.
This discussion can be summarized by making three
points.
First, the bands of stationary instability
transcend the supercriticality boundary 6=0 while the wedges
of propagating instability are limited to 6>0.
This is due
to the different nature of the two types of instability.
The propagating instability is essentially a topographically
modified baroclinic instability which, of course, requires
supercritical flow (6>0) while the stationary instability
which also feeds on the zonal available potential energy
requires sufficient topographic amplitude which increases
the interaction between the forced wave and perturbation
zonal
flow thereby instigating the instability.
The second
point to be made is that the 0(1) Ekman friction stabilizes
the stationary instability and the propagating
instability.
190
The final point is
that the introduction of large viscosity
has made the instability depend on the direction of the
zonal flow at the ground.
For eastward flow at the ground
the stationary topographic instability is confined to
superresonant flow which exists closer to resonance as the
friction is increased.
ground,
If the flow is westward at the
the stationary instability is
confined to one side.
near resonance but not
However, when the flow is westward,
the nature of the stationary instability is baroclinic
instead of topographic.
191
6.
Conclusions
In this chapter we have
concerning
considered two questions
a Charney mode embedded in a shear
either weakly baroclinically
subcritical
flow which
is
or supercritical
using a weakly nonlinear theory.
First, what
of
growing Charney mode and
topography on a baroclinically
second does
where
section 2 we found that
stabilizes
the
topography strongly
infinitesimal baroclinic
stabilizing mechanism is due
to the
zonal
flow and wave at the ground.
phase
shift develops relative
produces
a form drag.
vertical
shear near the ground
stabilizing
instability except
As
to the topography which
The form drag modifies
towards
stability, hence
be
flow at the
forced.
section 3 allows a very weak zonal
finite-amplitude
considered.
section 2 but
Thus
forced
The results differ somewhat
again topography is found
additional
for
wave-zonal
flow which is
the stability of a small
topographically
baroclinic instability except
The
the diversion of the
a Charney mode via
flow by the topography.
The
the mean
the disturbance.
enough to force
scales.
The
the wave grows a
ground by which a topographic wave might
analysis of
scale.
feedback between the
In section 2 there is no uniform zonal
but
to instability
those waves with a very large meridional
zonal
the effect
there would otherwise be none?
In
for
the presence of topography lead
is
Charney
mode was
from these of
to stabilize the
the very large meridional
flow
feedback due
to the
192
presence of the
responsible
for the differences
feedback depends on the
This
If
flow.
the uniform
this additional
than
essence
stabilizes
is
ground
flow
if the
at
of sections
the
ground
is
3 is
in
section 2.
that
topography
largest meridional scales
all but the
flow at the ground
is allowed to be relatively
larger;
time scale associated with
e-folding
time
scale of
the earlier
section 4 the propagating baroclinic
stabilized by
scale.
In
in fact
the
the same as
found which
sections.
In
This is
small meridional
The
scale.
zonal
induced stationary
is most easily excited near
to the
The resonance corresponds
point at which
neutral mode becomes stationary relative
the topography.
the
instability is
addition, a topographically
subcritical
flow at the
is less effective for waves with small
instability is
resonance.
for waves of
of the
The results differ
the presence of topography.
especially true
stabilization
it is
the disturbance.
from those of
Thus
is very weak.
both sections 4 and 5 the uniform zonal
substantially
eastward,
than
advection
a
the uniform zonal
the ground is westward
flow at
2 and
3.
topography more stabilizing
less stabilizing
Charney mode when the
In
zonal
between section 2 and
direction of
feedback makes
in section 2 but
the topography
the
topographically forced Charney mode is
The instability whose
available potential energy occurs as
source is
to
the zonal
the disturbance becomes
stationary relative to the forced wave which allows a strong
coherent
interaction between
the
perturbation,
forced wave
193
superresonant
flow and occurs
baroclinically
values of meridional
instability
limited to very narrow
centered at moderate and
(1979)
low
from large
the topographic
in a two-layer baroclinic
by a strong wave-mean
have a counterpart
The
zonal scales.
found by Plumb
model, marked
In wavenumber
wavenumber which extend
scales to moderate
to
is either
supercritical.
in meridional wavenumber
zonal
flow which
topographic instability is
space the
bands
for
subcritical or
is limited
instability
The topographic
and mean flow.
flow interaction, does
in the continuous
baroclinic Charney
model.
In
section 5 the effect of 0(1)
topographically forced Charney mode
consequently
instability
the effect of
Ekman
found in section
the analysis differ according
or westward
baroclinic
zonal flow at
Ekman
friction on the
is studied
friction on
4 is examined.
to the presence of an eastward
the ground.
The stabilization is least effective
topographic
scales.
the topographic
The results of
For
eastward flow
instability is again stabilized by
large meridional
and
topography.
for
large zonal or
The stationary
instability is
in nature and is most easily excited
resonant
flow.
In addition, the
in order
for
is large
to begin with and
flow must be
this instability to develop.
it
is
If
in near
superresonant
the friction
increased the topographic
instability
is harder to excite and confined closer to
resonance.
This
baroclinic
instability
is in contrast
which
fills
the
to the propagating
more
of wavenumber
space
194
friction
as
is
Charney mode under
is due
This
increased.
the influence of
which changes
the vertical phase
from the mean
flow.
Ekman
0(1)
Ekman
friction
the
friction
to draw energy
so as
shift
is increased this
shift becomes more destabilizing.
phase
In
wavenumber
space
instability is confined to very narrow bands
topographic
the
As
nature of
to the
in meridional wavenumber
but not
limited
in zonal
wavenumber.
The story
at
the
ground.
baroclinically
instability
occurs
In
subcritical
serves
the
The stationary
to make
it a stationary
growing
large meridional
instability as opposed
this
scales where the
otherwise be a
the Charney mode
easily
near
resonance
zonal
clear
the topographic
scales.
or
is less
for superresonant flow most
and
to that
similar
narrow bands in meridional wavenumber
for larger
small zonal
stabilization
of the Charney problem.
found
for
is a new topographic
there
instability of
the normal modes
chapter shows that
instability except
Furthermore,
excited
The topography
one.
the inviscid work of
baroclinically
instability
flow and is not
in nature but rather baroclinic.
topography stabilizes what would
effective.
instability for
The propagating baroclinic
flow.
is again stabilized.
to a propagating
Overall
the flow is westward
is no
this case there
for both sub- and superresonant
topographic
only
is quite different when
Thus,
It
is
found
limited
for
to
space and most easily
on
this
basis,
instability and baroclinic
it
is
instability
195
of
the
Charney mode are
However with
be observable.
topographic
instability.
observability of
in
0(1)
Ekman
instability becomes more
the baroclinic
mode
separate phenomena
friction
the
scarce as compared to
This makes less clear, the
the topographic
the atmosphere.
and therefore may
instability of the Charney
196
Chapter 5
Overall Conclusions
In this thesis I have considered three separate
problems concerning the instability of a topographically
forced wave.
The topographically forced wave considered in
each problem is of a different type.
In Chapter 2 the amplitude of the stationary,
barotropic, forced, plane wave is 0(1).
As a result, a
traveling instability which develops upon this wave can
produce a spectrum of waves which interact with the forced
wave and amongst themselves as the instability grows to
finite amplitude.
The assumptions of weak nonlinearity and
weak growth have been made to make the problem analytically
tractable.
This study provides a self-consistent model for
the interaction of the barotropic components of the
planetary scale stationary and transient waves in the
atmosphere.
The study suggests in the atmosphere the
possibility of an explosive instability when the zonal flow
is superresonant and an oscillation between the planetary
scale stationary and transient eddies due to the convergence
of Reynolds stresses for subresonant flow.
However, there
are a number of assumptions made which make difficult the
direct comparison of the model with the atmosphere.
on the long time scale required by the theory,
effects become important in the atmosphere.
First,
frictional
As in other
weakly nonlinear stability studies the introduction of small
friction may lead to chaotic behavior in time associated
197
with multiple
friction
0(1)
quasi-steady
may lead
equilibration
(e.g.
states
Second,
limiting
baroclinicity is an important
some planetary scale transient
(e.g.
1980)
state
factor
in the
A baroclinic
the baroclinic nature
eddies
(e.g.
can model
Bottger and
the existence of
the cyclone scale eddies and the placement
scale eddies relative to the
(e.g. Neihaus,
to expect
1980;
of the cyclone
planetary scale stationary
Fredriksen,
1979).
It
is
the relative location of the cyclones
to the stationary wave to be important
in modeling the
interaction between the stationary and
transient waves in
the atmosphere.
Based on some preliminary
seems a two-layer baroclinic
calculations, it
analysis might be carried
through in much the same way as the barotropic one
Chapter 2.
of
Perhaps more
1970) in the atmosphere.
importantly, a baroclinic model
reasonable
and
and the planetary scale stationary eddies
Holopainen,
eddies
1971)
the neglect of
the model to the atmosphere.
model would be capable of modeling
Fraedrich,
steady
to finite-amplitude
(Pedlosky, 1970).
application of
Pedlosky,
The details of
in
this calculation are left
for
future study.
Yet another important restriction has been placed on
the problem, that of weak nonlinearity.
to expect that the results of removing
would depend on
strong
the
the degree of
It
seems reasonable
this restriction
truncation employed.
nonlinear behavior of the
model within
The
the context
severe truncations employed in Chapter 2 could be
of
198
examined by numerically solving an ad hoc truncated version
of the nonlinear stability problem.
In this way the
amplitude of each wave in the perturbation is no longer
linearly related to each other and it might be determined if
the explosive nonlinear instability remains when the weakly
nonlinear assumption is removed but the truncation remains
severe.
In addition, if the severity of the truncation were
lessened in this ad hoc truncated model, the cascading of
energy in wavenumber space also becomes possible.
This
effect might also kill the explosive nonlinear instability.
The resolution of the survivability of the explosive
nonlinear instability is important since such a strong
instability would likely dominate other instabilities in the
atmosphere.
To determine whether or not the nonlinear
instability occurs when the flow is superresonant in the
barotropic model, requires the calculation of the feedback
between the instability and its harmonics.
It is possible
that such a feedback is large and stabilizing, hence
eliminating the nonlinear instability except near resonance
for which the analysis is invalid.
However, the calculation
depends on the truncation and hence is postponed until the
convergence of the entire nonlinear coefficient can be
examined.
We might also ask if the instability can still play a
role in a baroclinic model.
I would expect this to be the
case at least in a two-layer baroclinic model.
However, it
is clear that although the initial tendency of the explosive
199
its
instability may be meaningful,
not.
it
Thus,
process,
seems
term behavior is
long
likely that an important
physical
Above,
as yet undetermined, has been neglected.
weak nonlinearity or
that the assumption of
have suggested
However, there are
truncation may be responsible.
possibilities which might be examined within
a weakly nonlinear model.
two other
the context of
first possibility
The
I
is the
inclusion of a small amplitude propagating wave with the
spatial
structure of
This may produce the
the topography.
feedback necessary to stabilize the nonlinear instability.
Another possibility (perhaps in combination with the
previous one)
is to confine the fluid
would produce additional nonlinear
force a correction of
this zonal
flow with
the zonal
This
to a channel.
interaction which would
flow.
The
interaction of
could
the topographically forced wave
be stabilizing.
In
the remainder of
this thesis
I investigated
various types
influence of topography on the
possible in a continuously stratified,
specifically the Charney model.
studied are
neutral
to the
zonal
flow.
westerly zonal
the
A
constant shear model,
baroclinically
given normal mode becomes
topography
for a value of the
current which depends on which normal mode is
The stationary normal mode then
topography which
interaction.
of modes
first type of modes
the normal modes which are
stationary relative to the
chosen.
The
the
can resonate with
leads to a significant wave mean
The simplifications which occur when the
flow
200
normal mode is
wave-mean
is
nearly stationary are exploited
flow interaction.
The second
type of mode studied
the Charney mode which is unstable to the
However, it
becomes neutral on a line
simultaneously
zero.
stationary if
In this case,
topography
is assumed
exploited
zonal
flow.
in parameter space and
the zonal
flow at the ground
the neutral mode can
is
interact with the
to produce a significant wave-mean
interaction.
to study the
flow
The simplifications which occur when the
flow
to be weakly unstable and nearly stationary are
to study this wave-mean flow
interaction.
In Chapter 3 I have extended the study of topographic
instability
examine
to a continuously stratified baroclinic model
the topographic instability of
Charney problem.
and near
scales
The flow is assumed
resonance, which allows
to be
instability,
As
the instability is more easily
potential
the mean flow and
energy of
momentum via
form drag.
barotropic study of
excited near
from both the available
In a manner
from the
similar
zonal
to the
topographic instability by Pedlosky
instability can occur
superresonant
the
to be weakly nonlinear
in earlier studies of topographic
its energy
over
the
the method of multiple
resonance but draws
the
mode of
used to obtain the evolution equations without
the use of truncation.
(1981),
a normal
to
for either subresonant or
flow and would develop with a high centered
topography which would move upstream as
quasi-steady state was approached.
wavenumber space
for which the
the
However, the regions of
topographic
instability of
201
the
normal
mode of
very weakly on
the zonal scale.
Therefore,
the
instability
earlier models,
the simpler
narrow bands
that the stationary topographic
a
defined meridional
upstream as
to
This suggests
instability may be
structure but
less well
instability causes
state
initially
the topography which would move
the quasi-steady state
Charney and Devore
zonal
scales.
feature of the development of highs
centered over the peaks of
as
is limited
instability would have the
The
zonal structure.
additional
to
in contrast
in the atmosphere at planetary zonal scales with
fairly well
defined
scale and only
in meridional wavenumber and is more
for planetary zonal
easily excited
observable
superresonant or
depend strongly on the meridional
subresonant
somewhat
problem is
the Charney
(1979)
is approached.
first suggested,
Then,
if
topographic
the transition in the atmosphere
from the
to the blocking state, blocking would propagate
upstream initially, involve planetary zonal scales and the
meridional structure of blocking would
definition than
be the case.
its zonal
Blocks
tend
and have planetary zonal
structure.
have better
Indeed this seems to
to propagate upstream
initially
scales and a meridional
which consists of a split jet.
structure
However, I remain somewhat
baffled as to why the instability is so selective
meridionally.
It is possible the meridional walls of the
channel are artifically playing a role in the meridional
selectivity.
This seems somewhat unlikely due to the
absence of such selectivity in the barotropic channel model
202
study of
topographic instability by Pedlosky (1981).
Nevertheless
the question of the
geometry requires
role of
the channel
further investigation.
To
this end
a
similar study might be done with spherical geometry.
would also be helpful
instability
in another respect.
effects can be
important.
scales quasigeostrophic theory begins
the
stratification
be
The topographic
is more easily excited at planetary scales for
which spherical
Therefore,
This
inclusion of at
least
Also at planetary
to break down.
the additional
term in the thermodynamic
equation which can
important at planetary scales as discussed by White
(1977) would be another worthwhile extension
of the present
theory.
There are
two other questions concerning
the
applicability of the analysis to the atmosphere.
analysis
the
flow was assumed
to be
In
the
near global resonance
with a single component of the topography.
Such resonance
is
likely at the planetary scales in the atmosphere as long
as
the wave can't propagate meridionally to any significant
extent.
However, a resonant response
localized topography is less
presence of
likely especially in the
friction which would damp
moved away from the topography.
only applicable
to small scale
the response as it
Therefore, the analysis is
to the large atmospheric
is confined by a meridional waveguide.
scales if the wave
The response of
atmosphere to localized topography involving a
resonance" remains an open question.
"local
the
203
Another assumption made in the course of
was
that of weak nonlinearity.
instability
As
is concerned, this
However, the assumption
far as
form a resonant
topographic
is not a serious restriction.
limits the interaction of
topographically unstable wave with other
which
the
the analysis
triad with the
waves.
Those
interaction would be an interesting
which the removal of
make
future
study of
study after
the weakly nonlinear assumption would
4,
the instability of a small
finite-amplitude stationary topographically
mode embedded
in a mean shear
flow
but
forced Charney
is examined.
The
analysis hinges on the neutral
stability curve
nonlinearity is assumed.
these assumptions
With
evolution equations for the zonal
In
problem,
the
we
topography
seek to
find out whether
stabilizes or destabilizes
instability and
and weak
the
flow are obtained using
the method of multiple time scales.
if the presence of
studying
this
presence
of
the baroclinic
topography leads
to any
instability.
In
the second section, the problem treated
exceptional
to
A
sense.
In Chapter
new
waves
topographically
unstable wave will interact most easily with it.
this
the
since the
force a wave.
zonal flow at
The effect of
the ground
is somewhat
is too weak
topography on the
infinitesimal growing baroclinic wave
is studied and
to be strongly stabilizing except
large meridional
scales.
for
This stabilization is due to
the
found
form drag which
204
develops as the wave grows and which alters the mean shear
near the ground thus stabilizing the instability.
In the subsequent sections, a uniform zonal flow is
present which by its diversion by the topography produces a
small but finite-amplitude topographally forced Charney mode
near neutrality.
Its instability in the presence of the
uniform vertically sheared flow is studied.
The instability
feeds on the available potential energy of the shear flow.
Again the effect of topography is stabilizing on the
baroclinic instability.
This makes less likely the
possibility that baroclinic instability leading to a growing
Charney mode is the source of the planetary scale transient
waves as was suggested by Hartman (1979).
Furthermore, it
seems likely that this would also be the effect of
topography on the Green modes which has also been proposed
as the source of the planetary scale transient waves.
However, the calculation concerning the effect of topography
on the Green modes still remains to be carried out.
A new stationary instability develops even in parameter
ranges which are baroclinically stable characterized by a
strong wave-mean flow interaction.
This strong wave-mean
flow interaction is due to the disturbance becoming
stationary relative to the
strong interaction with it.
forced wave, which allows a
It is most easily excited near
resonance and the flow must be superresonant for it to
occur.
Thus the instability is reminiscent of the
topographic instability of the normal mode.
As a result, we
205
refer to it as the topographic instability of the Charney
mode.
This instability is also limited to narrow bands in
meridional wavenumber and is more easily excited at larger
zonal scales as is the topographic instability of the normal
modes of the Charney problem.
Therefore, the topographic
instability of the Charney mode may be responsible for the
existence of the baroclinic component of the planetary scale
stationary eddies in the atmosphere.
This might be tested
by examining if the planetary scale stationary eddies have a
better defined meridional structure than zonal structure.
Knowledge of the position of the finite-amplitude stationary
wave relative to the topography would also be useful
information for an atmospheric test of the theory.
Unfortunately, this involves the technically difficult
search for the finite-amplitude quasi-steady states of the
model.
Due to its difficulty, this is left for future
consideration.
The presence of 0(1) Ekman friction reduces the growth
rate of the topographic instability and if Ekman friction
becomes strong enough the topographic instability
disappears.
This casts some doubt on its realizability in
the atmosphere.
In addition, there are several features of
the analysis which make the applicability of the analysis to
the atmosphere questionable.
The analysis pivots about the
neutral point of the Charney mode.
Also in the absence of a
zonal flow at the ground this neutral Charney mode is
stationary which allows it to resonate with the topography.
206
Therefore the interaction between the topography and the
wave is strong.
The effect of topography on the barolcinic
instability and the existence of topographic instability of
the Charney mode away from the neutral curve is unclear.
Compounding this problem is that for more realistic vertical
profiles of the zonal wind there may be no neutral points
(e.g. Geisler and Garcia, 1977) and even if there are
neutral points, the waves may not be stationary.
Therefore
this problem requires further investigation before the
actual effect of planetary scale topography on the transient
and stationary planetary scale eddies in the atmosphere can
be ascertained.
Nevertheless, if in the atmosphere there
are no neutral modes, it seems physically reasonable to
expect the topography to most readily influence those
baroclinic waves which travel the slowest relative to the
topography.
Based on the various problems considered in this thesis
it is clear topography has a profound influence on
atmospheric flow.
The conclusions are as follows:
First,
in the atmosphere the interaction between the planetary
scale topographically forced stationary eddies and transient
eddies which may develop as an instability on the stationary
waves is fundamentally different than the interaction
between a free stationary wave and the traveling instability
which develops on it.
In fact the interaction between the
topographically forced wave and its instability may lead to
an explosive growth for superresonant flow.
207
Second, planetary
scale baroclinically stable modes
in
the atmosphere become unstable in the presence of topography
for selective meridional scales.
The selectivity in
meridional scale may explain why blocking has a better
defined meridional structure than zonal structure.
Finally, propagating planetary scale waves which are
baroclinically unstable are stabilized by the presence of
topography.
Thus, it seems somewhat unlikely that the
source of the planetary scale transient eddies is the
baroclinic instability of the zonal flow.
However, the
presence of topography leads to a planetary scale stationary
instability which feeds on available potential energy of the
flow.
This instability may provide a source for the
planetary scale transient eddies in the atmosphere.
On the whole the effect of the planetary scale
topography on the atmosphere is to act as a catalyst for the
removal of energy from the zonal available potential energy
and/or the zonal momentum.
208
References
and K.
H.,
Bottger,
wavenumber-frequency
Phys.
Atmos.,
Charney, J.
G.,
1980.
Fraedrich,
53,
G.
J. Meteor.,
and J. G.
1979.
Devore,
equilibria in the atmosphere and
36,
Sci.,
Charney, J.
G.
and G.
Flierl,
R.
blocking.
Oceanic
1981.
G.
M.
and D.
Deininger, R. C.
wave
39,
J. Atmos.
analogues of
Wunsch, eds.,
MIT
504-548.
planetary wave
1980.
Strauss,
sytems. J.
1982.
amplitude.
Deininger,
135-162.
Evolution of Physical
and C.
A. Warren
instability, multiple equilibria
finite
4,
1205-1216.
Oceanography, B.
Charney, J.
in a
Multiple flow
large-scale atmosphereic motions.
Press,
Beitr.
90-105.
baroclinic westerly current.
Charney, J.
50 0 N.
domain observed along
The dynamics of long waves
1947.
in the
Disturbances
R.
C.
and
A.
Free Rossby
J.
Atmos.
Z.
and propagating
Sci.,
37,
1157-1176.
wave instability at
Sci.,
Loesch,
instability at finite
688-690.
Atmos.
Form drag
1982.
39,
563-572.
A note
amplitude.
J.
on Rossby
Atmos. Sci.,
209
Fredriksen,
J.
S.
1979.
The effects of
on the regions of cyclogenesis:
Atmos.
Geisler,
J.
Sci.,
36,
E. and
R.
long planetary wavs
Linear
Gill,
A.
Sci.,
E.,
R. Garcia,
1977.
Baroclinic
J.
1979.
Geophys.
D. L.,
zonal-mean
36,
Fluid
Dyn.,
6,
Barotropic quasi-geostrophic
anisotropic mountains.
Hartman,
1979.
J.
Atmos.
Sci.,
29-47.
flow over
36,
1736-1746.
Baroclinic instability of realistic
states to planetary waves.
J.
Atmos.
0.,
1970.
An observational
study of
energy balance of the
stationary disturbances
atmosphere. Quart. J.
Roy.
Holopainen,
E.
long-term
Jacobs,
Sci.,
2336-2349.
Holopainen, E.
in
J.
stability of planetary waves on an
beta-plane.
E.,
a 8-plane.
311-321.
1974. The
infinite
Hart,
34,
J.
195-204.
instability at long wavelengths on
Atmos.
theory.
the
0.,
mean
1978.
flow by
atmosphere.
S. J.,
Preprint.
On
1979a.
J.
Meteor.
the dynamic
the
Atmos.
Soc.,
forcing of
large-scale
Sci.,
96,
the
in the
626-644.
the
Reynolds'
35,
stress
1596-1604.
Nonlinear resonant mountain waves.
210
Jacobs.
S.
J.,
1979b.
A baroclinic
resonant mountain waves.
Lin,
C.
A.,
1980a.
Eddy heat
planetary waves. Part
model
for nonlinear
Preprint.
fluxes
I. J.
and
stability of
Atmos. Sci.,
37,
2353-2372.
Lin,
A.,
C.
1980b.
Eddy
heat
planetary waves. Part
fluxes and
II.
J.
stability of
Atmos. Sci.,
37,
2373-2380.
Loesch,
A.
Z.,
1974. Resonant interactions between unstable
and
neutral
31,
1177-1201.
barolcinic
S.
Longuet-Higgins, M.
waves:
and A.
E.
Part
Gill,
interactions between planetary
London,
A299,
E. N.,
Lorenz,
motion.
Miles,
J.
W.,
Atmos.
1964.
M.
flows.
1967.
Atmos.
Sci.,
Resonant
waves. Proc.
Roy.
Soc.
Sci.,
instability of Rossby wave
29,
258-264.
A note on Charney's
instability. J.
Niehaus,
J.
120-140.
1972. Barotropic
J.
I.
Atmos.
C.
W.,
J.
Atmos. Sci.,
1980.
Sci.,
21,
model
of
zonal-wind
451-452.
Instability of non-zonal baroclinic
37,
1447-1463.
211
Pedlosky,
J.,
Atmos.
Pedlosky,
Sci.,
J.,
small
1970.
Finite
27,
baroclinic
waves.
J.
15-30.
Finite-amplitude baroclinic waves with
1971.
dissipation.
Pedlosky, J.,
amplitude
1979.
J.
Atmos.
Sci.,
28,
587-597.
Finite amplitude baroclinic waves in a
continuous model of
the atmosphere. J.
Atmos.
Sci.,
36,
1908-1924.
Pedlosky, J.,
flows.
and baroclinic
R.
A.,
1979.
Part
1:
Undamped
Plumb,
A.,
2:
near-resonant
R.
A.,
evolution near
Stoker, J.
Sci.,
J.,
J.
Damped and
forcing.
1981b.
night vortex:
Atmos.
Atmos. Sci.,
38,
2626-2641.
shear
Atmos.
J.
flow.
the baroclinic
Sci.,
36, 205-216.
1981a. Forced waves in a baroclinic
flow. Part
Plumb,
J.
in barotropic
Forced waves in a baroclinic
instability threshold.
Plumb, R.
topographic waves
Resonant
1981.
shear
undamped response to weak
Atmos.
Instability
Sci.,
38,
1856-1869.
of the distorted polar
A theory of stratospheric warmings. J.
38,
1950.
2514-2531.
Nonlinear Vibrations.
Publishers, New York.
Interscience
212
Tung,
K.
K.,
1976.
On the
A reexamination of
distorted
convergence
the
background
of spectral
theory of wave
flows.
J.
Atmos.
series
-
propagation in
Sci.,
33,
1816-1820.
White,
A.
using
J.
1977. Modified quasi-geostrophic equations
geometric height as vertical coordinate.
R. Met.
Youngblut,
of
A.,
Soc.,
C. and
Sci.,
37,
383-396.
T. Sasamori,
transient and
circulation.
103,
Part
1980.
The
nonlinear effects
stationary eddies on
I:
1944-1957.
Quart.
Diagnostic
the winter mean
analysis.
J.
Atmos.