MULTIPLE EQUILIBRIA AND THEIR STABILITY
IN A
BAROTROPIC AND BAROCLINIC
ATMOSPHERE
by
SANDRO RAMBALDI
Laurea in Fisica, University of Bologna
(1974)
SUBMITTED TO THE DEPARTMENT-OF
METEOROLOGY AND PHYSICAL OCEANOGRAPHY
IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
August 1982
Massachusetts Institute of Technology 1982
Signature
of
Author
Department of Meteorology and Physical Oceanography
August
1982
Certified by
Glenn R. Flier1
Thesis Supervisor
Accepted bp:
Ronald G. Prinn
Chairman, Departmental Graduate Committee
MIT
PAGE 2
MULTIPLE EQUILIBRIA AND THEIR STABILITY
IN A
BAROTROPIC AND BAROCLINIC
ATMOSPHERE
by
SANDRO RAMBALDI
Submitted to the Department of Meteorology
in August 1982
Physical Oceanography
and
in partial fulfillment of the requirements
Philosophy
Degree of Doctor of
for the
ABSTRACT
This work is divided into three independent sections; the
of
in the forcing
topography
common
subject is the role of
stationary waves.
first section we study the multiple equilibria
In the
problem
in
a barotropic
atmosphere and
we
examine
the
considering
in
in
a simple model,
form-drag
instability
how the
study
We
processes
involved.
some
detail
the
or
departs
equilibria (stable case)
approaches the
system
Then we consider the non-linear
them (unstable case).
from
of the
solution using
the
properties
discussing
problem
and
of
energy
and
potential
enstrophy
the
conservation
The
zonal flow.
function for the
introducing a potential
dissipation and forcing is then discussed
small
limit
for
and compared with numerical solutions.
highly
examine a two-layer
section we
second
In the
multiple
study
to
with topography
channel model
truncated
that
We find
stability.
their
equilibria and to discuss
require meridional
all the Charney-Straus multiple equilibria
temperature gradients which are highly baroclinically unstable.
After a detailed study we demonstrate that the flux of angular
produce one
is able to
latitudes
tropical
from
momentum
(in the two layer
baroclinically stable
equilibrium which is
model).
For
low values of the external radiation
heating
the stationary topographically forced wave alone satisfies all
PAGE 3
the
balances,
while
for
larger values of the
external
topographically
forced
forcing
the
stationary
radiation
wave collaborates with shorter, more baroclinically unstable,
moving, synoptic scale waves.
last section we have presented some
In
the third and
transport of
upward
theorems on the
simple
but general
stationary quasi-geostrophic
energy by
momentum and
zonal
induced by
circulation
meridional
the mean
and by
waves
these waves.
Thesis Supervisor
Title :
Dr. Glenn R. Flierl
-Associate Professor of Oceanography
PAGE 4
Acknowledgments
This work began and was almost completed under the guidance
of the late Prof. Jule G. Charney.
His constructive criticisms
and helpful comments deepened my understanding of my particular
problem and
its relation to more general ones.
Throughout all my studies
and
Ray
Pierrehumbert
were
Profs. Glenn Flierl, Peter Stone
ready
sources
for
advice
and
consultation.
During
1982
Prof.
Glenn
Flierl
became
my advisor and
available for
supported my efforts. He was always
discussion,
providing useful alternative viewpoints.
The manuscript was
carefully
corrected by Joel Sloman and
the figures were skillfully and promptly
Kole.
The
research
was
supported
by
Foundation under grant NSF 76-20070ATM.
drawn by Ms. Isabelle
the
National Science
PAGE 5
Table of contents
Page
Abstract
...........................................
Acknowledgments ....................................
4
Table of contents....................................
5
SECTION 1
FORM-DRAG INSTABILITY AND MULTIPLE EQUILIBRIA
IN THE BAROTROPIC CASE
1.
Introduction ...................................
7
2.
Equations of motion ............................
10
3.
Form-drag instability ..........................
14
4.
Non-linear behaviour for the inviscid case .....
21
5.
Approach to the equilibria in the viscous case
27
6.
Stability of the equilibria ....................
33
7.
Numerical
integration . ..........................
33
References . .........................................
35
Figure captions . ....................................
36
F igur es...............................................
37
SECTION 2
MULTIPLE EQUILIBRIA IN A BAROCLINIC MODEL
1.
Introduction . ...................................
48
PAGE 6
Page
2.
Equations of motion
3.
Low-order model ....
55
4.
Multiple equilibria
58
5.
Search for more stab
6.
Conclusion .........
76
References .............
78
Figure captions ........
79
Table captions .........
82
Figures ................
81
Tables .................
102
71
equilibr
SECTION 3
STATIONARY ADIABATIC FRICTIONLESS FLOW OVER TOPOGRAPHY
.......
1.
Introduction
2.
Equations of motion . . .. ..... ..
3.
Simple
4.
A simple non-linear analytical
theorems ....
103
...........
104
107
. ..........
References ............ . -.- -.- -.- -.- -.-
solution
115
123
PAGE 7
SECTION 1
FORM-DRAG INSTABILITY AND MULTIPLE EQUILIBRIA
IN THE BAROTROPIC CASE
1.
Introduction
In 1979, Charney and his coauthor-s wrote a series of
which
papers
developed the idea that the atmosphere
first
may have several possible equilibria
forcing.
external
in a
beta
In the simplest problem, barotropic flow
channel
plane
same
the
for
states
with
topography,
forced
by
a
momentum source, Charney and Devore (1979) demonstrated that
truncation
forcing,
did
have
within
multiple
certain
Given the
states.
stable
parameter
spectral
order
the non linear equations arising from a low
ranges,
possible
two
equilibria were found, one with a weak zonal flow and strong
waves and one with strong
higher
resolution
zonal
numerical
flow
code
and
was
weak
waves.
A
used to assess the
degree to which their truncated model rapresented
solutions
of the original problem.
In further work, Charney and Straus (1980),
examined
the effects of baroclinicity, Charney, Shukla and Mo (1981),
PAGE B
of
suggests possible directions in which
(1981),
Reinhold
work
the
finally
applied the model to earth's topography;
the principles of multiple equilibria might be modified when
scale
synoptic
occur
instabilities
in
addition
is
to
to weak
topographic forcing of the long waves.
model
simplest
with
form-drag
to
a model similar
equilibria:
and Flierl
Charney
however,
here,
purpose
Our
(1980).
(1979)
Hart
of
multiple
and
instability
that
the
examine
or
We shall consider the processes
involved in form drag instability in some detail in order to
the
explicate
in
mechanism
the
straightforward
most
barotropic case.
how
As new physical contributions we point out
form drag
is felt by the zonal flow through the ageostrophic
component of the flow and in paragraph 3 and 5
the
properties
the
and
mechanism
of
describe
we
the
form-drag
emphasizing the study of the quantities that can
instability
be
the
paragraph
In
measured.
6
we
instability as a
simple
consequence
of
energy
and
conservation
discuss
of
potential
the
the form-drag
simultaneous
In the
enstrophy.
viscous case some numerical calculations have been
done
to
show how the system approaches the stable equilibria.
As technical contribution we have introduced
potential
function;
in
this
a
new
way we have three degree of
freedom and three conserved quantities and we
are
able
to
PAGE 9
give
a
simple
but
complete qualitative discussion of the
properties of our system in the inviscid case.
PAGE 10
2.
Equations of motion
The simplest set of equations which
form
drag
Hart
(1979)
illustrate
the
instability and multiple equilibria are those of
describing
the
coupled
evolution
of
the
x-independent flow and the topographically forced waves.
derive
similar
expansion.
equations
This
new
without
derivation
using
any
clearly
asymptotic
points
importance of the ageostrophic components in the
between
the
waves
and
the
vorticity
equation
for
wave
topography
is
+~
zonal
u
flow.
flows
The
over
out the
interaction
potential
y-independent
,+
~
We
C~±2.1I)
is the geostrophic streamfunction
where
U
is the eastward
component
of
the
geostrophic
component
of
the
geostrophic
wind
V
is
the northward
wind
f, is the Coriolis parameter at mid-latitude
is
derivative
(df /dy)
of
the
Coriolis
parameter
=
PAGE 11
h is the height of the topography
H is the depht of the fluid
t'
and
upper
is the dissipation time scale (due
and
lower Ekman layers ).
zonal wind U cames from
the
to
friction
in
The evolution of the mean
x-averaged
momentum
equation
where periodicity has been assumed
T~--(V/
(2.2)
-
When the topography is y-independent the Reynolds'
disappear
and
the
changes
in
dissipation, external forcing, U
part
of
the
y-component
of
the
zonal flow are due to
and to
the
stresses
the
velocity
ageostrophic
V' .
This
ageostrophic part of the velocity can be directly related to
the
form drag as follows:
we calculate the mean meridional
flux of mass here, denoted by M
M
(H-
The form drag
LX
H
-
V-
(2.3)
is given by
L2
LaL
rDx
X
rax
-
PAGE 12
where P is the atmospheric ground
angle
from
pressure
and
G
the
is
horizontal to the inward normal to the surface.
To lower order the meridional mass flux simplifies to
M
+
+(2.5)
where the first term represent the geostrophic
contribution
to the form drag.
In a y-independent problem, the mean meridional mass
flux
could
still
on time but th'e simplest case is
depend
that of no net flux
M= o
(which can be
confined
2h
.=-_.
in
justified
if
the
a very wide channel).
flow
is
(2.6)
assumed
to
be
With these assumptions,
our system of equations becames
X'D
-j
(2.8)D
(2.8)
The solution of the system (2.7) and
a
with
found
Fourier
decomposition
PAGE
13
can
be
topography
of
and
streamfunction
t4)=
Equations
sin(
h
h =
.4
-
Uy
+
A Al
(2.7) and
H
k x+ T)
AA
sin(
.M~%ILJ4J~A
k x+
+ B cos(
14
x+
-4
f)
(2.8) became
~A4A4
(QU)-
k
t)i
A4
(2.9)
U*)
(2.10)
t
AAN
U-
_
-BM
(2.11)
PAGE 14
3.
Form-drag
The
instability
simplest
system
containing
form
the
drag
interaction can be obtained with a single mode topography
naL
)(
(3.1)
Under this assumption our system of equations can be divided
into a set
of sub-systems
for k #h k we get
(3.2)
-
A
-
Even if U is time dependent
(3.3)
we
can
see
from
the
energy equation
--
(.
(3. 4)
that these modes simply dissipate.
(3.3) can be solved exactly;
the solution is
given by:
The
system
(3.2)
and
introducing a complex notation
PAGE 15
(3.5)
' A
This is just a decaying travelling Rossby wave.
To
determine the time dependence of the zonal flow and to study
the role of the topography we must consider the case k = k
we then have the following system
(jil~U*)
-~
(U= A)(U-
=A.cU-/3Id)A
a
To clarify
involved
let
the
us
role
introduce
(3. 6)
--
(3.7)
+AiU--LB
'-'
H
of
the
(3.8)
different
non-dimensional
parameters
variables
as
follows:
(~IA) LI
Z)-U
(/3J V
(3. 9)
r6
H
"i')
PAGE 16
With these substitutions we get two
non-dimensional
parameters
-^f-
(3.10)
that describes the importance of the mountain and
-
_
which describes the dissipative
zonal
number
wave
the earth radius and
to
h A/
I.-4
f
.2,
Y
k = n*( 27V/2 rr a cos
,
f
is
the
Introducing
effects.
a middle latitude,
),where a is
chosen
here
and the following typical atmospheric numbers,
45*,
be
n
11)
__(3.
I
--
sec
10
t 4. 10 days
4S
.6*10
5
sec,
we get the following values:
. -
'
.
0
With these non-dimensional parameters our system
of
(3.6)-(3.7)-(3.8),dropping the primes, becomes
equations,
-
~~*
--
(3. 12)
(3. 13)
PAGE 17
The mechanism of form-drag instability can
be
seen
quite clearly from these equations and figures 1 and 2.
Let
us neglect forcing and dissipation and
the
situation
consider
first
UJ4C
when U is less than the resonant speed
U (
,
or
in dimensional quantities.
The wave generated by the flow has
vorticity
which
is in phase with the topography since the flow is dominantly
constrained
to follow the
geostrophic
south
contours,
on
and north on downslopes, corresponding to cyclonic
upslopes
curvature over the peaks.
situation
there
wave-mean flow
is
non
no
In
form
interaction
this
inviscid
drag,
in
an
the
steady-state
illustration
absence
of
of
time
dependence or friction (Pedlosky, private comunication).
If we perturb this
which
by
introducing
vorticity
is 9e out of phase with the topography, the flow will
evolve in time.
flow:
flow
There are two different tendencies for
the
an acceleration or deceleration of the mean flow and
a change in the vorticity distribution.
If
topography
the
initial
vorticity
by 90 (figure 1.b),
corresponding to B
This discussion of form-drag instability was
by Prof. Glenn R. Flierl
lags
perturbation
the
O, the
0?
contributed
PAGE 18
pressure disturbance will
form
drag
lead the topography
decelerate
will
the
Changes in vorticity are induced
propagation
and
subresonant
mean
tendency
eastward
flow,
mean
by
so
flow
that
the
(figure 1.c).
westward
Rossby
wave
advection by the mean flow.
the
westward
For
propagation
wave
larger than the eastward mean flow so that the
is
vorticity begins to increase over the crest (figure 1.d).
From the first derivative patterns, we
the
second
whether
generating
field,
to see
There
three
are
different
mechanisms for
wave propagation and mean advection of the
,:
by
generation
topographic
the
(4
additional advection of the basic state vorticity
the
L4
continues to lie
propagation
wave
The
flow.
opposite
or
reinforced
is
perturbation
initial
counteracted.
explore
derivative of the vorticity,
time
the
can
the
toward
field
by
plus mean advection
leading
west,
and
field
to
)
a
of
from the original perturbation (figure 1.e).
sign
The decrease in the flow over topography leads to a decrease
in
in
vorticity
Finally,
topography by 90*.
the
a
which
advection
(the term which would correspond to
vorticity
differentiated
since
troughs,
the
correlated
negative,
with
contributions
to
h.
W
For
are
and
U < I
the
- LQ
form of Eq. (3.14)) also leads the
is
U
of
the
leads
basic
in the
topography
the
basic
vorticity
,
then,
all
of
is
the
of opposite sign to the initial
PAGE 19
that
perturbation so
is
wave
topographic
the
neutrally
stable.
U >
When
different
{
the
however,
(figure 2).
The basic equilibrium distribution of
since
vorticity is anti-correlated with the topography
in
vorticity
tend to balance the
we still get a form drag retarding the
propagation
tendency
now
has
but
the
a reversed sign.
The
we
As
westward.
a
than
propagates
it
cha nges in the
term,
flow leading the topogra phy (figure 2.c).
The
over
the
the
decrease
contribution now
in
flow
mean
the
But the
has
the
since
sign
opposite
(figure 2.g).
possibility
Thus there is the
when the equilibrium state vorticity is sufficiently
which occurs near resonance) that this
other
two
and
I
will
term
basic mechanism for form drag
as
follows:
large (
will
dominate
the
original
reinforce
perturbation, leading to an instability of
summarized
the
equilibrium vorticity is now of opposite sign from
the topography
the
still
propagation
wave
is also the same as before (figure 2.f).
topography
original
the
doppler-shifted
the
tendency from
third
at
look
flow
mean
perturbation advects downstream faster
generates
the
If we introduce the same form of perturbation,
topography.
however,
over
stretching
vortex
the
planetary
than
rather
vorticity
relative
changes
wave
quite
is
situation
the
flow.
The
instability, therefore, can be
when
the
zonal
flow
is
PAGE 20
supercritical,
topography.
the
vorticity
Decreases in the
anti-correlated
vorticity
is
anticorrelated
zonal
shifting
flow
result
westward
with the
in
leading
pressure patterns which have highs on the upslopes and
on
the downslopes.
this
to
lows
This in turn leads to a form drag which
decelerates the zonal flow further.
PAGE 21
4.
Non-linear behaviour for the inviscid case
_I
As we have seen
can
our
solve
parameter
and
we
by developing all our unknowns in a
system
:
power series of
(F-B)
is a small
)
,
=
(iA
+
+
(
We shall study the non-linear behaviour only of
the
zero-order equations which are:
(4. 1)
0
.:
-.
(4.2)
.(
0 - B
B
- (U,0-- ) A ,
U0
-
This inviscid problem has two independent
(4.3)
conserved
quantities
1
.4-I
40
0
2-
(0- -)
At~
~
+-BI-
(4. 4)
(%U+
.4
xno
(4.5)
(.5
PAGE 22
pointed
DeVore
and
Charney
As
first
the
out,
quantity represents the total energy and the second quantity
a
allows
solution
analytical
complete
a
potential
the
with
energy
total
existence of these two integrals of motion
The
enstrophy.
the
of
combination
system
the
of
(4.1)-(4.2)-(4.3) in terms of elliptic functions.
As we shall see a full discussion of the
of
properties
the solutions of this system can be done without solving
in
only
equation
order
As first step we derive a simple second
explicitly.
we substitute the definition of Q in
U;
Eq. (4.3) getting
---- 4Q +o(
o~
do---
After
of
derivative
time
a
4.6 )
and
(4.1)
the
substitution of B,,from (4.6) we get:
---(L)--iMU-')-- q ---
It is useful at
expression
explicit
for
this
Q
point
as
to
U
(4.7)
introduce
a
more
function of the stationary
solutions.
If
U
has
any
chosen
value
U1
the
system
PAGE 23
(4.i)-(4.2)-(4.3)
gives
a
stationary
forced
wave
with
amplitude
0~o
(4.8)
(4.9)
U-i
On figure 3, the
constant
3,,
Eq.
(4.5),
light
lines
while the heavy
stationary solutions, from here
curve,
Eq.
(4.9).
on
lines
of
line represents the
called
the
equilibria
At least one intersection on the branch
number one always exists;
0 can
represent
therefore the conserved
quantity
always be written as a function of the coordinates of
this
intersection,
here
denoted
with
( U , A , B
),as
follows
Q -.
(0--)
-+.
X
(4.10)
Using this definition and defining
u = U, -
A
=
-
U
"distance" from the
U
"distance of the equilibrium from the
resonance"
equilibrium
PAGE 24
(4.7) can be rewritten as
Eq.
LL--~AU
+
QL
(4. 11)
L
or multiplying both sides by u.
(4. 12)
e +
LL)
where
LL
+
We have a
evolution
of
defined
is a potential function
the
L.
-
analogy
mathematical
4-
by
LL
(4.13)
the
between
time
wind and the motion of a particle
zonal
moving in a one-dimensional space with
potential
a
energy
given by
With this
behaviour
the
of
potential
our solutions:
we
can
discuss
the
general
in particular we can see that
linear stability properties of the points of equilibrium
are governed by the sign of
I+i
, in agreement with
the linear analysis of Charney and DeVore;
behaviour
is
dominated
the
non
linear
by the positive sign of the fourth
PAGE 25
order term and we have a non linear stability
for
all
the
points of equilibrium.
Depending on the number of points of equilibrium our
system
A
behaves
,the
in
points
=0
,
different
of
ways;
equilibrium
for
are
a
given
given
Q,,or
u = 0
by
or
i. e.
d tA
6ALt
the
u
u = 0
root
A,
B),
- S
represents
previously
the
discussed
.3=0
<4.14)
equilibrium
point,
while
the
other two
roots
AZ,3
represent the other intersections of
with
the equilibria curve.
the
(4.15)
constant
The point P =U ,A
B)
G.
line
will be
the only equilibrium if
<(4.16)
In this case the potential function, Eq. (4. 13),
has only one
PAGE 26
minimum
( U,
A_,
oscillates
the
for
potential functions
evident that the second equilibrium
one
on the branch 2.
if
periodic;
we
I
is
The
motion
equilibria
the
basin to the other.
system
is
start close to the equilibria it
consists of a simple oscillation while if we start far
the
always
In figure 4 we have
drawn the potential function for this case.
always
unstable
an
Some straight-forward
analysis shows that the minimum on the branch
the
(4.13), it is
is
point
point while the other two are stable.
than
from
three intersections are present;
>
expression
lower
around
periodically
B ).
For
the
system
the
and
oscillates
from
from one attractor
PAGE 27
5.
Approach
case;
to the equilibria
in the viscous case
Until now we
have
in
of a small dissipation and forcing the
presence
considered
the
non-dissipative
evolution of our system is essentially similar
now
we
except
that
no longer have an equilibrium curve but only one or
three equilibria and from any points of our phase-space
system
goes
conserving
better
towards
the stable points while approximately
To get a
the potential enstrophy and the energy.
understanding
of
the
way
in
which
the
approaches the stable equilibria we can study the
system
behaviour
of small perturbations around these stable equilibria.
Eqs.
the
From
(3.12)-(3.13)-(3.14) we find that these equilibria are
defined by:
8)
-U
-
-- U(5.1)
B
(5.2)
.
C
(5.3)
U-A
It
coordinates
for
'Xft *. -
is
of
convenient
the
0
to
get
equilibrium
we have:
an
expansion
of
point in powers of
the
2?
PAGE 28
(5.4)
Pt;
U
+(5.5)
+
)(5.6)
where A and U are related by:
(5.7)
This is a zero order relation and
equilibria
implies
that
the
point must be on the inviscid equilibria curve.
Secondly,
P ~
we have
~
-
(5.8)
which comes from the first order equations and
implies
the
balance between the generation and the dissipation.
A graphical solution of
this
system
is
shown
in
figure I for different values of the external forcing
U*
The heavy line represents the equilibria curve (5.7),
while
the light curve rapresents points where the generation is in
balance with
the
(U,A).
we
As
dissipation,
can
Eq.
(5.8),
in
the
plane
see for small forcing we have only one
PAGE 29
stable equilibrium and for larger values of the
forcing
can
the
get
three
equilibria
with
the
one
in
we
middle
unstable.
Denoting, as before, with (u,a, b) the
from (U ,A ,B
perturbations
) we get the following equations:
(5.9)
S-
S=
--
(5.10)
i
-
(5.11)
U)
Having a system with constant coefficients,
look
for
a
the form (u,ab)*e
solution in
we
can
and we get
the following eigenvalue problem:
(t+
+
Ci'+2)
-
)-
-
(5.12)
-o
where
expanding the eigenvalues
-.
in power of
)(5.
13)
PAGE 30
-
C3
+
>
(I
0
+.
-3
(.
)
=
we get
Q
-
-o
Z.
.
+-_-_
-i
(5. 15)
one real
We have three solutions:
g = -
(5. 14)
(I-
)
+
(
(5. 16)
0+
o
(5. 17)
and two complex coniugates
The first eigenvalue has for its eigenvector
(5. 18)
curve
in
the
plane
B = 0
of
figure
eigenvalues have an oscillatory part
time
scale
and
therefore,
on
the
the
to
tangent
corresponding to the direction
with
equilibria
The other two
3.
a
much
shorter
average, the solution
PAGE 31
approaches
the
point
of
equilibrium
moving
the
along
"equilibria curve" with a damped motion.
These
two
complex
eigenvalues
with
their
eigenvectors
b
C
LL
(5.
give rise to a damped oscillation
with
damped
standing
and
19)
in the zonal flow together
propagating waves.
In fact the
stream-function for the wavy part of these perturbations can
be written as:
FWZCx,~)
e~~L/
0
(5.
20)
where
Z(~,t)
=
0-
z.
1~
(~)
-
We have two cases:
L> d.
Z (xQ)
,
we have
=
.4.
c
{
-IZ.
for
-l
.- c
and
)
+-x
z-.
the
super-resonant
case,
PAGE 32
an
where the first term rapresents
eastward
moving
"slow
Rossbu wave" and the second term a smaller standing wave out
of phase with the topography;
U
we have
where the first term
Rossby
for
the
sub
resonant-case,
and
represents
a
westward
moving
"fast
wave" and the second term a smaller standing wave in
phase with the topography.
PAGE 33
6.
Stability of the equilibria
Using figures 6. (a) and 6. (b)
properties
of
the
equilibria;
we
for
a
parabola in figure 6.(a), we have three
the
equilibria curve, heavy line;
can
discuss
chosen 0
we
can
Eq.
For
the
ellipsoid intersects
with
fields
can
(4.4).
see, for the two stable equilibria C and E, any
reached.
points
with
for each intersection we
nearby point, on the QGcurve, has larger energy
be
dotted
,
intersections
have drawn the energy ellipsoid, dashed lines,
As
the
unstable
the
0
and
cannot
equilibrium D, the energy
surface
so
that
neighboring
finite B's and smaller energies in the U and A
be
enstrophy;
this
reached
conserving
trajectory
is
energy
the
and
potential
heavy dashed line of
figure 6. (b).
Numerical integration
7.
To illustrate part of
dynamics
with
a
of
fourth
= .8
considered,
linear
and
non
linear
the preceeding discussions we have integrated,
order
(3.12)-(3.13)-(3.14).
x
the
and
for
Runge
method,
the
system
We
have chosen the following values
.05
.
different
with three equilibria
Kutta
In figures 7a,7b,7c
initial
(CD,E);
as
we
have
conditions, the system
we
can
see,
the
two
PAGE 34
stable
equilibria
almost
on
the
displacement
(C,E)
isolines
are
of
approached with oscillations
G,
together
with
a
net
along the inviscid equilibria curve toward the
stable equilibrium.
In figure 8 we have considered parameters such
the
system has only one equilibrium.
that
We no longer have any
forced and damped equilibria on the branch number two of the
equilibrium curve, but the equilibrium curve is still a zero
order
equilibrium
oscillations,
solution
so
that
with 0 nearly conserved,
we
can
see
together with a mean
motion along the equilibrium curve toward the saddle
After
crossing
damped
point.
this point, the oscillations grow again and
the system evolves toward the only equilibrium left.
PAGE 35
REFERENCES
J.G.
Charney, J.G.,
DeVore, 1979:
Multiple flow equilibria
in the atmosphere and blocking.
J. Atmos. Sci.,
36,
1205-1216.
Charney, J.G.
and D.M. Strauss, 1980:
multiple
in
equilibria
and
propagating
baroclinic, orographically
systems.
J. Atmos. Sci.,
Charney, J.G.,
Form-drag instability,
forced, planetary
37,
j. Shukla and K.C. Mo,
barotropic
blocking
planetary
waves
wave
1157-1176
1981:
Comparison of a
theory with observation. J. Atmos.
Sci. 38,
Charney, J.G.
and
G.R. Flier1, 1980:
large-scale
atmospheric motions.
oceanography.
Oceanic
analogues
of
Evolution of physical
Edited by B.A. Warren and C.Wunsch M.I.T.
Press. 504-548
Hart, J.E.,
1979:
anisotropic
Reinhold, B.B.,
stationary
Barotropic
mountains.
1981:
quasi-geostrophic
J. Atmos. Sci.,
Dynamics of weather
waves and
Cambridge, Mass. 02139.
blocking.
flow
over
36, 1736-1746
regimes:
quasi
Ph.D. Thesis, M.I.T.,
PAGE 36
FIGURE CAPTIONS
Figure 1 : see text
Figure 2 : see text
Figure 3 : The
heavy
line
represents
the equilibrium
curve, and the light lines are contours of constant
Q in the plane (UA).
Figure 4 : Shape of the potential function for different
value of
Figure 5 : Graphical solution of Eq. (1.48),
for different values of the forcings U
Figure 6 : The equilibria
the
which
,
line,
Eq. (1.49).
points (C,DE) are points in
the
and
energy
heavy
potential
enstrophy
surfaces-are tangent and the stability is determined
by whether nearby points can be reached.
Figure 7 : Examples
evolutions, starting
of time
from
different regions of the phase space in presence of
three equilibria
(CD,E). The heavy line represents
the trajectory of the system, the medium heavy line
represents the equilibria curve and the light lines
are the intersections
of the
with the (UA) plane.
For this
As
Figure 8
one
9= .05 and
U = 4.8,
ch osen:
in
figure
equilibrium, C.
chosen:
U = 4.8,
constant G
:?
7
but in
For
.05 and
this
evolution
surfaces
we have
.8
presence
of only
evolution we have
.8.
PAGE 37
(U <#8/k'
CASE)
(0)
+
basic state
/~ Th
+
perturbation
-1
-
(b)
7%>
/
(c)
Ut
/
/
(d)
+
+
/
-I-
(e)
+
-
-
from wave
propagation
(f)
+
from
ftt
Ut * Vh
ttt from
Ut 'V
FIG.
1
-
/
(9)
PAGE 38
(U>
CASE)
/ /k 2
(a)
+
basic state
/
/
-I-
(b)
+
-
perturbation
/
(c)
Ut
//
/
(d)
St
+
00,
/0
(e)
tt from wave
propagation
-
Ctt
(f)
from
Ut.
Vh
(g)
Ctt
f rom
Ut 0 Vt
FIC.
2
PAGE 39
branch *2
bran
PAGE 40
A = 3.0
v (U)
A = 2.6
A = 1.6
30-
20-
-5
-4
PAGE 41
-4
-3
-2
-I
10
-5
T r
i
U'= 6
=2
PAGE 42
N.
V'
S-I
/
-
PAGE 43
PAGE 44
PAGE 45
PAGE 46
PAGE 47
CTfl
0
PAGE 48
SECTION 2
MULTIPLE EQUILIBRIA IN A BAROCLINIC MODEL
1.
Introduction
In
equilibria
the
barotropic
is
the
multiplicity
Because the
form-drag
oresents
a
maximum
the speed of the zonal flow goes close to a resonance,
the mean
for
a
constant
zonal flow
external
has
forcing,
on
the
resonance.
of
view,
From
this
point
the
of
statistical equilibrium values
airplane
possibility,
reaching different
two
sides
we
analogy with the two different equilibrium
an
of
due essentially to the resonant structure of
the form-drag.
when
case
of
have a strong
velocities
a
supersonic
considerations continue to
case,
but
here
we
speed.
apply
also
to
the
baroclinic
to
(Reinhold, 1981).
We have organized our study in a
to
the
have some evidence that the equilibria
synoptic scale baroclinic instability
us
a
These general qualitative
found could be unreachable if the system is too unstable
allows
that
can reach, for a constant propelling force:
subsonic speed or, after having overcome the maximum in
form-drag,
the
compact
way
that
discuss the multiple equilibria problem in a
PAGE 49
more general and simple way
We
forcing.
thermal
function
as
of
the
external
demonstrated that the multiple
have
Straus
and
equilibria suggested by Charney
for
realistic
parameter values are baroclinically unstable to short waves.
We have found that the balance implied by one of
equations,
the
barotropic
zonal
flow
model
equation, is quite
different from the real atmospheric balance.
shown
the
We
have
then
that a simple correction to this equation, simulating
a constant flux of zonal angular momentum from the
leads
to
baroclinically
more
stable
tropics,
equilibria.
New
equilibria are then hypothesized in which the solutions have
a
large
scale
stationary
dependent
part.
stability
boundary
this
but
part
large
the
and a synoptic scale time
scale
state
is
near
the
synoptic motions also play an
important role in the heat and momentum balances.
PAGE 50
Equations of motion
2.
In the barotropic case it was decided to
that
arguing
directly,
flow
zonal
forcing,
this
baroclinic atmosphere, would correspond to the thermal
the
driven
by
direct
thermal
field.
radiation
forcing
and
in
a
wind
To take into account the
effects
the
study
to
the
force
of
baroclinicity of the flow, Charney and Straus (1980) decided
for
to look
two-layer
equilibria
multiple
in
a
highly
truncated
Following their notation, we consider the
model.
potential vorticity equations for the two layers
Lb
(4.
C,
(2. 1)
+1 ( qz.) .)19
_j
7-
= _F
V1 V
2-
q7 t
where the subindex 1 indicates the upper layer
the subindex 2 indicates the lower layer
1represents internal dissipation
V<
represents bottom Ekman dissipation
represents the Rossby deformation radius
The
starred
quantities
represent
the
external
PAGE 51
forcing
of
the
temperature field, here parameterized as a
Newtonian relaxation with an inverse time scale of
The potential vorticities
in
the
two
layers
are
defined by
+
[
-
where h is the height of the topography and H
is
the
mean
depth of each layer.
We consider this flow in a channel
walls
at y = 0 and y =
by
,
u and v by
L.
T
with
meridional
Let x and y be scaled by L, t
L
by
/9
L
by
,
h
by H and define the non dimensional parameters
XL
respectively.
variables,
functions,
that
Vol
y~K~'~V
To get a spectral model we
now
here
F-
on
dimensionless,
and
complete
F
by
denoted
F F -T.
U
a
develop
;
LV
we let
all
set
our
of
such
PAGE 52
f3
=
(cZ.o)
=
F'
F
and we define
horizontal
crii
average
(c:-
CI
on
the
cj-)f
where
the
channel.
bar
implies
Substituting
a
these
definitions in (2. 1) we get
L-
L
AL
(2. 2)
LL
P
j9All
L9Q
~
~~
~(~
WL
P I
[ i
±~\
97IA"4kzVIL
~
Y J &:j WJ
i
wh ere
Qi
=L4
1L 1--
.
(W
and
An
introd uced
their
alternative
set
to
the
discuss
interactions:
the
of
variables
is
usually
dynamics of the two layers and
barotropic
compon ents of the stream-function
and
baroclinic
PAGE 53
eat)
\~'iL'
8*-.
L
±
-
4 ~
(
.Q)
and the barotropic and baroclinic component of the potential
vorticity
(9)=
(9
93)
=
-(
the
With these new variables
)
+
tK
-
of
equations
motion
(2. 1) bec ome
kwz
p) +i (91~ fX
zj
'~Q
(2.3)
=
&.
~A()~-
~
(v~~±
(Q~O9
-.-).A
- aCX~Q
In our
equation
obtained
multiplied by
'(-and
we
discussion
1I
summing
from
to
shall
consider
the
energy
adding the first equation of (2.2)
second
the
over
\('L
i.
After
equation
changing
multiplied
to
the
by
new
PAGE 54
variables, the energy equation takes the form
ole.
-.- 2:
We
equations
shall
for
. o9+ A1
4.//
also
both
consider
layers
-v.Y
2.
-
the
A~3~q
dCL
9.
potential
(2.4)
enstrophy
obtained by summing over i the
.
first equation of (2.2) multiplied by
equation multiplied by
(e)9L9
;
for the
and
the
second
upper layer we get
zU
-C1
(2
LiL
%L
5
and for the lower layer
a4
L -L.
9. .IL
L
9
..
R( J- (2.6)
PAGE 55
Low-order model
3.
Following the
to
model
procedure
for
the
barotropic
equilibria we consider only three
multiple
find
used
now
on
system
to
from
modes, the same as Charney and Straus (1980;
referred as C-B):
a zonal mode
Fa
tw =
and
=
c ose(
y)
I
two wavy modes
F =
2 sin( y )*cos( m*x
F =
.3
2 sin(
x=
1+ .A7
=
y )*sin( m*x
.3
This choice, in particular, enables
I
-M
our
have baroclinic instability and to balance the generation of
heat with the divergence of
The
heat.
interaction
the
eddy
fluxes
of
sensible
coefficient between these modes is
given by
The
and
we choose
term has projection only on F ,
/
to match the C-S value;
PAGE 56
we consider a single mode topography with projection only on
F,
and
The evolution equations for the zonal flow become
3
A.
+
_j+
-
In a long time
(3.
-(
.2K
mean
we
can
+
(-
see
(3.2)
from
equation
that,
the zonal
flow to balance the dissipation due to
bottom
the
can
boundary
see
first
in the lower layer, the form drag must push
layer.
the
that
the
Ekman
In this simple model the Reynolds'
stresses are identically zero, and from the second
we
1)
eddy
fluxes
equation
of sensible heat, the
internal dissipation and the Newtonian cooling must
balance
the external generation of heat.
If complex variables are
part of the flow:
/141+
L
introduced
for
the
wavy
PAGE 57
Eqs. (2. 3) can be rewritten as:
OX1
+ IZ. - 6
(3.3)
C 2.1
+-
6
(3.4)
+09
where aI
aI
a#
and b
a
and
zonal quantit ies,
-L
Z.
19i
=
G
are
+
functions
only
defined by
x x , LC N - (-X?-- 1)
?
I
)
--
QVZ
Q
aZ
G '7
of
the
PAGE 58
4.
Multiple equilibria
Without the topography
behavior.
For
a
this
small forcing
system
has
a
it has a stable stationary
solution, with no waves and with the heat balance
by
the
mean
increased
meridional
this
instability
circulation;
solution
and
the
becomes
system
if
unstable
reaches
a
the
satisfied
the forcing is
to
baroclinic
different
equilibration with the meridional temperature
to
simple
stable
gradient
set
two-layer critical value for baroclinic instability
and with a
translating
baroclinic
wave
large
enough
to
transfer
all
forcing.
This second state corresponds to a stable cycle in
the
heat
necessary
to balance the external
the phase space.
The presence of topography enriches the system
several
new
behaviors
and,
as C-S discussed, it can show
stationary, periodic or aperiodic
the
range
of
parameters
with
solutions,
chosen
and
depending
on
the
on
initial
conditions.
In
this
paper
we
shall
mainly
consider
the
stationary solutions and their stability.
To
compute
Eqs. (3.1)-(3.2)-(3.3)-(3.4)
dependence.
is
each
are
solved
A graphical solution in the
presented
point
stationary
(
in
kV
,
Fig.1.
GI
solutions
without
plane
the
(
time
)
This is obtained by computing at
) the wavy part,
from
Eqs. (3.3)-(3.4)
PAGE 59
looking
of
curves
represent
9
forcing
computing the external
lines
zonal flow is specified, then
the
once
linear
are
which
balance
in
are
straight heavy line at forty
solutions
Hadley
no
with
and
form-drag
finally
Ekman
the
(heavy lines), Eq. (3.1).
The
represents
the
five
degrees
wave present and with zero wind
the closed heavy
speed in the lower layer;
and
)
constant
for the points where the
dissipation
from Eq. (3.2) (the light
line
represents
non trivial balances between form drag and dissipation.
The intersections between
line
light
selected
the
heavy
lines
and
a
define the stationary solutions for a
selected strengh of forcing
In the case here
studied
the
external
parameters
have been set to the following values:
channel width
7TL = 5000 Km
H =
5
mean depth of each layer
Km
Rossby deformation radius
= 1000 Km
2L
A
=
5.
The internal and Ekman dissipation
well
as
the
time
scales
as
Newtonian heating time scale have been set to
ten days corresponding
=
h
to:
=.0114
PAGE 60
For the topographically forced wave
wave
number
three,
n = 3 and
we
m =
have
chosen
( n/2.83 ),
zonal
and
the
amplitude of the sinusoidal topography has been set to 1 Km.
With these choices we get the
same
equations
with
the same parameter values as C-S.
Before discussing the stationary
full
system
it
-solutions
of
the
is useful to consider the stationary waves
that are forced by the topography for a
fixed
(
In Figs.2 and 3 the amplitude and the phase, relative to the
topography, of the barotropic and
are
drawn.
baroclinic
forced
waves
Because the topography interacts only with the
lower-layer flow we also present the amplitude and the phase
of the forced wave on this layer
divides the plane in two
part
it
the
has
zonal
form
The straight line
regions;
in
the
upper
flow has easterlies while in the lower part
westerlies
positive
(Fig.4).
in
drag,
flow eastward (i.e.,
have been shaded
the
lower-layer.
The
region
of
where the topography pushes the zonal
the phase is
between
- Tr and
0 ),
(Fig.4b).
As we can see the amplitude of the forced
solutions
is dominated by the intensity of the zonal flow in the lower
layer,
In
the
( 'j-91),
inviscid
and by
case
the presence of
this
a
resonant
curve.
resonant curve (heavy line of
Fig.5) is the two-layer baroclinic extension of the resonant
barotropic point,
0 and
4-
,
and in particular
PAGE 61
interior
it can be shown that, for westerly flows in the
(shaded
semicircle
this
baroclinic
forced waves are in phase
cyclonic
with
curvature
topography,
the
with
peaks
the
on
barotropic and the
the
region),
anticyjclonic
and
subresonant
barotropic
curvature in the valleys) as in the
of
case.
It is interesting
point
to
system
the
that
out
(3.3)-(3.4) has a true resonance, infinite response, even in
P
that
lies
fact
In
the presence of dissipation.
on the closed heavy
point
a
is
there
line of Fig.1, where the
dissipation is balanced by the generation and it is possible
to
have
a stationary wave also without the topography;
course, this stationary
finite
value
of
the
solution cannot
external
resonance when the determinant of
,
Eqs. (3.3)-(3. 4),
QO
-Q
be
coefficients
for
a
We find a
.
forcing
4
of
and
vanishes.
Q
=
The real part vanishes on the heavy
while
reached
of
line
of
the immaginary part vanishes on the dashed line;
Fig.5
the
intersection between this two curves defines the position of
the
infinite
resonance.
Thus
unlike the barotropic case
where the amplitude of the wave is limited by dissipation as
PAGE 62
the
forcing
is
increased,
here grow as
the stationary wave amplitudes
when the forcing
is increased.
Due to the presence of this true
anticipate
that
in
resonance
we
can
its neighborhood the forced stationary
wave are large enough to tranfer meridionally all
the
heat
necessary to balance the external heat forcing and therefore
we can expect multiple equilibria.
having
solved
forcing
(
As
we
can
see,
after
the problem, (Fig.1) for small values of the
K. 140)
we
have
only
one
intersection
corresponding to a stable direct Hadley circulation, without
any forced wave.
For values of the forcing between .140 and
.175 two new equilibria are present;
wind
(
> PI)
one has easterly zonal
in the lower layer and the forced waves exert
a negative form drag while the other has westerly zonal wind
).in the
positive
lower layer and the forced
form drag.
westerlies
ground.
exert
a
For values of the forcing between .175
and .200 two other equilibria
have
waves
and for
When the forcing
appear;
for
C.195
both
,>.195 one has easterlies at the
is
increased
further,
the
two
solutions
with
equilibria.
One is the Hadley solution that has only a mean
meridional
efficient
gives
easterlies
circulation
in transfering
meridional
forced ones
&
.
and,
heat
temperature
disappear
as
we
can
leaving
see, is not very
meridionally
gradients
The other two equilibria
three
and
9i
therefore
close
are
to the
locked
to
PAGE 63
the true resonance and, in analogy with the barotropic case,
other
one can be called subresonant and the
whether
on
depending
superresonant,
it is inside or outside the resonant
curve of Fig.5.
Before continuing our study of these equilibria, let
us
try to state why these equilibria are important and from
which characteristics they get their importance.
more
or,
points
equilibria
of
presence
The
generally, the existence of regions in the phase space where
some
are
components
predictability
In our case these are
these components.
of
the
increases
varying,
slowly
the mean zonal flow and those modes directly
the
by
forced
Following Speranza (private comunication) these
topography.
regions can be seen as examples of islands of predictability
in
a stochastic, unpredictable space.
we
are
stationary
considering
predictability
In our case, because
high
the
solutions.
is a consequence of the high persistence.
islands
There are four classes of persistence
that
we can find:
*
a)
simple
and
fortunate
stable solution,
inside its
b)
solution *
stationary
Stable
for
most
case we could find a full stationary
which
attractor basin,
* Unstable
the
In
system
the
once it
stationary
confined
remains
gets there.
solution *
In
a
s t ill
PAGE 64
case
simple
full non-linear stationary
a
find
could
we
is close enough to the equilibrium
system
the
case,
once
point,
it continues to go closer in some directions, on
linear
the
of
system
their
of
values
the
rates.
growth
system
Lorenz
stationary points of the
such
point becomes directly
this
eigenvectors
related to the amplitudes of the most unstable
and
of
the amount of time that
i.e.,
to
close
spends
around
theory
stability
equilibrium, its persistence ,
the
unstable
the
on
the
Once the system enters the region
manifold, it departs.
validity
others,
along
while
manifold,
stable
this
In
solution with some linearly unstable eigenvectors.
in
unstable
The
the
aperiodic
reqime are examples of this case.
c)
*
corresponds
to
stationary
solution *
quasi-stationary
Stable
This
a stable solution with only some components
nearly
or
this
where
stationary,
quasi-stationarity is due to the non-linear interaction with
the full non-stationary
some constant components
d)
* Unstable
A
components.
stable
cycle
with
is a simple example of this case.
quasi-stationary
solution*
In
the
more general case we can find an unstable solution with only
some components nearly stationary.
time
behavior
of
the
system
point, we have to consider the
Here, to understand
the
in the neighborhood of this
instability
of
this
state
PAGE 65
role
the
account
into
taking
the
of
non-stationary
components.
full
that
evidence
We do not have any
stationary
solutions do exist, but we have some experimental evidence blocking situations - and
until
found
highly truncated models
spectral
seen
been
have
have
stationary
these
of
now
therefore,
only
are
solutions
must
be
to real persistent
question
open
on
the
truncated solutions, but we
realize that these solutions can be very
they
-
have been obtained with
and,
an
The
part.
approximations
Here we still
solutions.
validity
possible
as
only
suggestions
of the existence of solutions
quasi-stationary
with a non trivial
that
theory -
equilibria
multiple
theoretical
some
relevant
even
if
approximations of unstable quasi-stationary
solutions.
Once the system is supposed to be in the presence of
one
these islands we still have to determine its impact
of
on the time behavior of our
importance
the
from
system.
island
An
gets
its
amount of time spent by the system in
its neighborhood and this is a function of the extent of the
attractor
its
in
basin
stable manifold and on the growth
rates present in the unstable manifold.
Here we
believing
conclude
our
that
approximations
of
our
truncated
real
general
considerations
and
solutions
are
solutions
we
stationary
quasi-stationary
PAGE 66
begin
To
study.
our
continue
unstable manifold around the
in the
instabilities
subspace including only the forced waves and the zonal
and
we
solutions
stationary
"internal"
linear
the
study
shall
C-S
understanding of the
our
flow
then we shall consider some "external" instabilities in
the subspace containing the most unstable baroclinic wave.
Let us first consider the
of
zonal flow and
meridional
perturbations in the
solutions, i.e.,
stationary
these
instabilities
"internal"
in the wave field with
the
and
zonal
same
number of the topography, described by the
wave
the
system of Eqs. (3.1)-(3.2)-(3.3)-(3.4) linearized around
chosen
As
solution.
stationary
we
as
see,
shall
a
consequence of the introduction of another external forcing,
all
the
states;
unstable
(
points
therefore
modes
on
I,
91 )
can
the
compute
we
correspond
the whole plane.
growth
roots
is
plotted
unstable non propagating
in
rates
of
the
We have a sixth-order
Fig.6a
number
The
eigenvalue problem with real coefficients.
unstable
stationary
to
of
the number of
and
(real) roots is plotted in
Fig.6b.
Without the topography we do not find any finite area region
of growing stationary perturbations;
(dashed
line
in
Fig.6b)
that
there is only
the
divides
propagating from the westward propagating ones.
a
line
eastward
As we
see,
while this line now has been broadened to a region of finite
area new regions of orographic
instabilities
(here
defined
PAGE 67
as
instabilities
that
grow
in
place) have appeared.
In
Fig.7 we have plotted isolines of the largest growth rate in
presence
of the
of
topography.
largest
instability
growth
problem
In Fig.8 we have plotted isolines
rate
without
obtained,
solving
topography;
basic state that we perturb has no wave
baroclinic
instability.
the
same
in this case the
and
we
have
pure
As we can see the barotropic part
of the flow influences the growth rate as well as the
speed
of
the
perturbation;
this
is
because
the
describing the zonal flow always vanishes at the
and
a
change in the barotropic
Galileian transformation.
the
presence
of
a
phase
mode
two
walls
part is not equivalent to a
Comparing Fig.7 with Fig.B we see
strong
orographic
that
instability
dominates the picture for low values of the baroclinic zonal
flow
near
true
the
resonance.
The
indentation
isolines in the upper part of the figure shows the
influence
of
the
of orography
of instabilities:
pure
baroclinic
stronger
topography on those unstable modes that,
also without topography, were growing almost in
presence
on the
place.
In
(Fig.7) we have essentially two kinds
for high values
of
and
instability
61
the
we
have
source
almost
of
the
perturbation energy is the available potential energy of the
zonal
flow;
for
low
value
of
&
and close to the true
resonance we have almost a pure orographic
the
instability
and
forced stationary wave is the main source of energy for
PAGE 68
super-resonant
the instability as in the
The
barotropic
case.
line of Fig.7 indicates the stationary solutions
heavy
of the C--S model for different values of the forcing and
see
that
some
of
these
unstable, to "internal"
some
instabilities,
are stable, or weakly
solutions
perturbations
suggesting
topography),
of
we
wavenumber
(same
as
are the dominant
that,
if
these
these
equilibria
can
play
an
important role in the time behavior of the system.
of
At this first level
find
a
stable stationary
truncation
we
could
Another subspace that
solutions.
has been studied by C-S is the one including modes with
same
theu
number;
wave
stationary
solutions
The effects of this
found
the
but with a higher
zonal wave number of the topography,
meridional
still
all
that
their
were unstable to these perturbations.
y-mode
higher
instability
have
been
collaborators
(private
comunication) who ran a two-layer highly truncated
spectral
by
studied
Arakawa
model
including
ones;
they
found
and
his
one zonal wave number and three meridional
that,
instead
of
multiple
stationary
solutions,1 multiple statistical steady states can be reached
resemblina
the C-S multiple equilibria.
In
this
way
they
have shown that these instabilities are not able to take and
to keep the system completely away from the
they
weakly modify the equilibration.
equilibria
but
In particular, these
unstable modes participate in the meridional
heat
transfer
PAGE 69
meridional
value
equilibrium
temperature
lowering
the
gradient.
At this second level of truncation they show that
there may be stable quasi-stationary solutions.
baroclinic
unstable
containing the most
the
C-S
consider the subspace
we
when
arise
equilibria
multiple
of
importance
Serious concerns about the
the
With
wave.
parameters chosen in this model the most unstable baroclinic
wave has the largest meridional wave lenght and
direct topographic forcing on the most
wave
baroclinic
unstable
zonal number five and we study a new instability
with
probleem,
wave
we neglect the
influence
its
consider
To
five.
number
zonal
to
adding
in
problem can be factored
the
describing
of
number
five.
This
the
first
sub-problems:
two
zonal flow and the topographic forced waves
and
that we have just studied
instability
wave
zonal
for
equations
analogous
the
(3.1)-(3.2)-(3.3)-(3.4)
Eq.s
wave
the
second
the
describing
number five in a constant zonal flow.
In Fig. 9 we have plotted isolines of the largest growth rate
obtained
in
this
wave number five).
stationary
states
Fig.9 (heavy lines).
problem (no topography and
To understand the stability of
zonal
the
C-S
for the most unstable baroclinic wave
solutions
we have superimposed
stationary
second
the
for
curve
different
that
delineates
the
values of the forcing
C-S
in
As we can see only the Hadley solution
for small value of the forcing is stable.
PAGE 70
Stone
real
(1978) has shown that at middle latitudes
atmosphere
baroclinic
result
is
very
close
to
the neutral value for
instability in the two-layer model) and the
solutions
require
The
fact
that
the
C-S
indeed
does
equilibria.
it
not
To
has
been
stay
prove
shown
close
two
zonal
wave
importance,
(Reinhold, 1981)
to
any
of
these
that the
multiple
the preceding point Reinhold used a
two-layer highly truncated model
and
stationary
a meridional temperature gradient almost
twice as big raises some suspicions about their
system
same
has been confirmed with a numerical time integration
of a two-layer model.
and
the
numbers,
including
one
the
zonal
directly forced by the
topography and another more baroclinicly unstable, with
meridional
interactions.
wave
numbers,
to
flow
allow
some
two
non-linear
PAGE 71
5.
Search for more stable equilibria
From Fig.9
equilibria
we
are
have
seen
strongly
how
all
the
baroclinically
C-S
wavy
unstable.
To
understand if this negative result is a consequence of
neglected
important
physical
assumptions of the C-S model.
parameters
involved
and,
verified that this model
in
phenomena,
we
some
studied
the
We started by considering the
as
pointed
out by Reinhold, we
is sensitive to reasonable
changes
the parameters, but is also very consistent in producing
multiple
equilibria
which
are
strongly
baroclinically
unstable.
To understand the reason for
must
understand
why
their
Because
.
we
the non-trivial balances between form
drag and dissipation are obtained only for
9B
instability
high
values
of
this is a complex balance we did not find a
simple explanation and we decided to compare this balance in
our model with
the one in the real atmosphere.
The model
or
equation (3.1)
states that, in equilibrium
in a long time average, the mountain acts to balance the
dissipation of zonal
boundary
layer.
This
observational studies
torque
and
surface
Bowman
(1974)
found
torque
and
the
momentum
due
statement
on
the
friction.
that,
on
to
does
relative
Newton
the
the
Ekman
not agree with the
role
(1971)
average,
surface friction work
surface
of
mountain
and Oort and
the
mountain
together against the
PAGE 72
convergence of angular momentum flux,
neglected
in
the Reynolds' stresses
our model.
It is interesting to point out that this problem
not
is
in the barotropic studies where the zonal flow
present
directly
forced,
and
that
also
if
we
compute
used by Charney and DeVore (1979),
we get a somewhat
higher
still realistic value of the atmospheric convergence of
momentum due to Reynolds'
the
computed
In
stresses.
Table
1
we
the
60* N for
30* to
we
calculation
this
for
the data of Oort and
using
seasons
extreme
Rasmusson (1971);
divided
the
in two layers, one from the ground to 500 mb and
atmosphere
As we can see
the other from 500 mb to 50 mb.
source
a
have
barotropic and baroclinic convergence of eddy
flux of angular momentum in the latitudinal band from
are
the
of momentum necessary to give the zonal forcing
convergence
but
is
of
both
barotropic
the
tropics
and baroclinic angular
momentum for the mid-latitudes, and these momentum exchanges
are
(that
neglected
condition,
boundary
in channel models when the meridional
v = 0,
is
applied)
can
play
an
important role on our stationary solutions.
To take this momentum source for the zonal flow into
we
account
(
,
F
equation:
decided
to
include
a
constant
forcing,
), in the barotropic and baroclinic zonal momentum
PAGE 73
C- 3
k (Z
±y
The
balance
realistic
- 9
for
and
3t
the
-+(5.1)
(&''(ti~)± +
3
barotropic
part
flux convergence.
balance
P
The balance for the baroclinic
drastically influenced;
in
fact
momentum
the
the
91
,
(
zonal flow is not so
flux
of
baroclinic
vertical
can simply continue our discussion by
we
discussion
of
function
.
I"
however
,
the form
the
drag
data
in
the
(
have
to
as
becomes less
positive;
become
that this is not the parameter
interest, since the momentum forcing is
lower layer.
between a selected heavy line
light line
parameterized
Presumably as
will
suggests
range of atmospheric
positive
However
the baroclinic flux of momentum will no
of the wave field.
Nw
shear
introducing a
given by
longer be so simple once this quantity is
than
the
in
in trying to set a larger
new effective forcing
the
more
momentum simply adds its contribution to the external
radiation forcing
and
now
To discuss the effects of this forcing we
looked for the equilibria solving for
zonal
becames
we can have stationary states where mountain
torque and friction work together to
plane.
(5.2)
(
In Fig. 10 the intersections
const) and
a
selected
const) define the stationary solutions for
PAGE 74
e. and
the particular values of the forcings
As
Irt.
we
can see the equilibria move in the right direction, becoming
values
for realistic
more baroclinicly stable and,
the
of
momentum forcing, we start to have, in the lower left
zonal
which
waves
long
stationary
with
equilibria
Fig.10,
side of
energy
considerable
are
in
baroclinicly stable for
shorter, synoptic-scale waves.
into
taken
been
have
Along with the other factors that
account, we want to mention that changes in the ground
ground
the
to
(extrapolation
parameterization
friction
instead of lower layer value being used to compute the Ekman
dissipation,
the Ekman term
i.e.,
V V4z..
Eq. (2.1)
has
but not to the point
improve,
been replaced b y34)
in
of stability, the stability of the wavy equilibria (Fig.11).
neglected,
being
until
term
physical
interesting
Another
to parameterize, but which can
difficult
play a role in the stabilization of our equilibria,
latitudinally
varying
moist convection.
equilibria
to
simple Newtonian
term
*
equilibria
forcing.
forcing
heat
due
is
heat
non-symmetric
perturbation
for
a
our
forcing we considered a
the
zonal
same
wave
we introduced a wavy forcing
Fig.12a, b, c, d
In
in Eq. (2.3).
obtained
i.e.
with
the
to radiation and
To get an idea of the sensitivity of
number as the topography,
now
strong
present
we
zonally
varying
The amplitude of the forced wave is 40
K and
the
heat
the
PAGE 75
time
Newtonian
scale
is
ten
days,
with different phase
relations between the topography and the heat forcing.
Fig.12b
we
From
have the suggestion that when the heating is in
phase with the
topography
multiple
baroclinicly stable can be obtained.
equilibria
which
are
PAGE 76
6.
Conclusion
We have seen how, taking into account
way
the
fluxes
of
angular
regions, our highly
stationary
This
truncated
forced
the baroclinic
momentum
in
from
two-layer
a
the
simple
tropical
system
can
large scale waves stable with respect to
instability of shorter synoptic scale
state requires a barotropic
baroclinic zonal
have
waves.
zonal flow about twice the
flow in reasonable agreement with
what
we
measure in the real atmosphere.
To discuss the properties of this equilibrium state,
varying
V
~v,
and
we
consider a low order model with
only one meridional wave number
numbers:
a
planetary
wave number three,
wave,
and
with
topographically
the
most
two
zonal
wave
forced wave, zonal
baroclinically
unstable
zonal wave number five in our model.
For small value
equilibria
with
and
no
the
tropical
forcing
1g
the
obtained with a stationary wave number three and
wave
number
>
I
unstable,
9
baroclinic
instability,
therefore
the
close
reaching
of
to
its
five
are
strongly
9
(where
dashed
is the neutral value for
line
of
shorter wave develops and
9
preventing
baroclinically
the
Fig.10),
and
keeps the value of
forced
long
wave
from
equilibria.
When we increase
becomes tangent to
0 =
and the
isoline
tp
constant
(dashed line of Fig.10), we have
PAGE 77
external
the
two situations depending upon the strength of
fC
radiation
forcing.
0,
value of
For
values of
less than
91 =
on the intersection between
(the
and
the
) we can have a stationary solution in which
isoline
of
the long
stationary wave is the only- wave
and
forcing
the external radiation
and
it
the heat necessary to balance
all
meridionally
transports
present
meridional
the
keeps
temperature gradient below the critical value for baroclinic
For values of
instability.
G
while
the
isoline
the
between
intersection
9,
than
greater
F
difference between
constant
and
9
to
balance
zonal
and
states
the
atmospheric
in which long stationary waves are
present and we have found that a determinant factor for
realization
the
.
We have explored the possibility of the
reaching
at
long
shorter baroclinic wave reaches an amplitude
large enough to transfer the heat required
system
the
state
equilibrium
the
reaches
wave
forced
9 e.
the
of such a state is the presence of a barotropic
momentum
mid-latitudes.
flux
coming
This
angular
from
the
momentum
tropics
the
flux is especially
strong in winter when examples of atmospheric
more frequent.
into
blocking
are
PAGE 78
REFERENCES
Multiple flow equilibria
DeVore, 1979:
J.G.
Charney, J.G.,
J. Atmos. Sci.,
in the atmosphere and blocking.
36,
1205-1216.
Charney, J.G.
equilibria
multiple
in
Form-drag
and D.M. Straus, 1980:
and
propagating
momentum balance.
Qort, A.H.,
and its
28,
1974:
interannual
hemisphere. J. Atmos. Sci.
Oort, A.H.,
wave
1157-1176.
Sci.
Atmos.
J.
and Bowman II H. D.,
torque
waves
Mountain torques in the global angular
1971:
Newton, C. W.,
37,
J. Atmos. Sci.,
planetary
forced, planetary
baroclinic, orographically
systems.
instability,
31,
623-628.
A study of the mountain
variations in the
northern
1974-1982.
and E.M. Rasmusson, 1971:
Atmospheric circulation
NOAA Prof. Paper 5, Govt. Printing Office,
statistics.
323 pp.
Reinhold, B. B.,
stationary
1981:
Dynamics of weather
waves and
blocking.
regimes:
quasi
Ph. D. Thesis, M. I.T.,
Cambridge, Mass. 02139.
Stone, P. H.,
561-571.
1978:
Baroclinic adjustment. J. Atmos. Sci. 35,
PAGE 79
FIGURE CAPTIONS
Figure 1 : Graphical
(
.
&
91 ,
),
solution of the equilibria in the plane
represent lines of constant
the light lines
indicates balances between the
the heavy line
form drag and the Ekman dissipation.
Figure 2a:
Amplitude
dimensional
Figure 2b:
of
barotropic forced wave in non-
the
units.
Phase, relative to topography, of the
barotropic
forced wave.
Figure 3a:
Amplitude
baroclinic forced wave in non-
of the
dimensional units.
Figure 3b:
Phase, relative to topography, of the
baroclinic
forced wave.
Figure 4a:
Amplitude
in non-dimensional
Figure 4b:
forced wave in the lower layer
of the
units.
Phase, relative to topography, of the forced wave
lower layer.
in the
Figure 5 : Resonant
without
(heavy
dissipation
dissipation
there
is
channel model
our two layer
curve for
still
line).
a
line
of
presence
In
on
the
which
dissipative effects balance and in P we therefore have
a true infinite response.
Figure 6a:
areas
Number of "Internal
we have
instabilities". In the white
no unstable
roots, while in the
other
regions we find 1,2 or 3 unstable roots as written.
PAGE 80
Figure 6b:
Number
of
"Internal
instabilities"
with zero
phase velocity. In the white areas we have no stationary
unstable roots, while in the
other regions we find 1,2
or 3 unstable stationary roots as written . The dashed
line
represents the
only
unstable
stationary root in
absence of topography.
Figure 7 : Largest growth rate for the "internal
The heavy
instability".
line, representing balances between form drag
and Ekman dissipation,
indicates possible equilibria.
Figure 8 : Largest growth rate for the "internal instability"
without topography;
i.e. growth rate for the baroclinic
instability of wave number three.
Figure 9 :Largest
baroclinic
growth
mode
rate
for
present in our
the
most
unstable
channel model:
zonal.
wave number five.
Figure 10:
(
Graphical solution of the equilibria in the plane
,9)
in presence of a zonal momentum
the light
heavy
Figure 11:
forcing
lines represent lines of constant
line
represent
lines of
constant
,the
,
where
As in Fig.10 but with ground extrapolation to
compute the Ekman dissipation.
Figure 12a:
(
,01:)
Graphical solution of the equilibria in the plane
in presence of a zonal momentum
and a latitudinal Newtonian heat forcing
of 400 and a
heating
Newtonian time
scale of
is in phase with respect to the
forcing
(with amplitude
ten
days).
The
topography. The
PAGE 81
light lines represent lines of constant
lines represent lines of constant
Figure 12b:
is out of phase
topography.
As in Fig.12a but here the heating is leading the
topography
Figure 12d:
the heavy
where
As in Fig.12a but here the heating
with respect to the
Figure 12c:
e
by 904 .
As in Fig.12a but here the
topography
by 90".
heating is behind the
PAGE 82
TABLE CAPTION
Table 1 :Barotropic and
baroclinic
eddy
flux of angular
momentum and its convergence between 300
in
m/sec
(or
in
and 60* north.
our non--dimensional units).
PAGE 83
20-
.15-
.10
5-
5
0
PT(0
1
,10
.15
.20
f1
.25
PAGE 84
.20
.15
.10
5
CrrT
7n
T
.10
.15
.20
.25
PAGE 85
.2
5
al
.20-
.15-
10
.5
0.5
,10
.15
-
.20
.,-25
PAGE 86-
.20
.04
.02
.00
.02
5
FIG,
3a
10
.15
-20
#,
.25
PAGE 87
.25
0i
.20-
.15
-
,10-
5.-
O0
CT
5
''Th
.10
.15
.20
*4s .25
PAGE 88
.25
0,
.20-
.15-
.10-
.5-
.5
r-Tr
/
A/
.10
.15
.20
qll .25
PAGE 89
.251
.20.
.15-
10
5
.10
.15
.20
*,
.25
PAGE 90
.25
20
0
5
P
T(
Ci
lo
5
5
e
20
2
.25
PAGE 91
.20
.15.\0-
.10-
O5
5
yr- r r%
1-
.10
.15
.20
.25
PAGE 92
e,
.20
o
15
.10
.15
.20
PAGE 93
5
PT0
7
.10
.15
.20
.25
PAGE 94
25
e,
20
15
10
.5
0.5
.10
.15
.20
PAGE 95
25
.18
.16
)0
.14
.12
15
.to
10
.06
00
.02
5
0
C-T f\
..
O
10
.15
.20
'I,
.25
PAGE 96
.20
.15
.10
.5-
0
.10
.15
20
*pg
.25
PAGE 97
.25
al
20.
15-
10-
5
.10
.15
.20
-25
.p
PAGE 98
25
ei
20-
I5
.10
.5
5
r-
Ir flX
-d 111 -
.10
.15
.20
#i
.25
PAGE 99
.20
#1I
,25
PAGE 100
25
200
.1=.28
.24
.22
.0
.10 -
.
.N
-14
.12
9
10
.6--
.5IN
.l O
.15
.20
.25
PAGE 101
.25
e,
20
26
.15-
.22
.20
.18
.10-
.16
.12
.10o
0
'1
.5
>vas
.10
.1
PAGE 102
Summer
Winter
Eddy fluxes of barotropic
angular momentum
(m /s ) :
BR
Eddy convergence of barotropic
angular momentum between 30*
(m/s)
and 60 N.
at 300
16.4
7.4
at 60"
-3. 6
-1. 2
8. 5*10
3. 6*10
o
*
-3
.53*10
(non-dimensional units)
at 30*
Eddy fluxes of baroclinic
angular momentum
(m'/s )
at 600
-
-1.9
(non-dimensional units)
TABLE 1
-
-
-3
-0.7
-
-.
angular momentum between 300
(m/sE
23*10
4.2
7.0
Eddy convergence of baroclinic
and 60* N.
.
3. 8*10
2. 1*10
.24*10
.13*10
a C
PAGE 103
SECTION 3
STATIONARY ADIABATIC FRICTIONLESS FLOW OVER TOPOGRAPHY
1.
Introduction
In this last section we shall examine
quasi-geostrophic
vertical
analyze
inviscid
over
stationary
topography
propagation of zonal momentum and energy;
and the
we
then
in some detail how these transports are accomplished
in a simple fully non-linear
shall
flow
a
consider
solution.
In
particular
we
the implication of a top radiation boundary
condition on the mean meridional circulation.
Our study clarifies the role of the stationary waves
in
forcing
a mean meridional circulation and in particular
suggests that there will be a strong meridional mass flux in
the layer containing the topography.
It also points out the
necessity of studying the time dependent problem to increase
our
understanding of the mean meridional circulation in the
upper atmosphere.
PAGE 104
2.
Equations of motion
We study a quasi-geostrophic adiabatic
flow
beta channel.
periodic
zonally
a
in
frictionless
We assume the
validity of the hydrostatic balance and we choose to work in
pressure coordinates.
the
consider
We
following
quasi-geostrophic
equations:
zonal momentum equation
-
4
L-
-
-
_(
-
(2. 1)
+o
(2.2)
+
(2.3)
vorticity equation
avk'
thermodunamic
--
-
.J (
q+7
equation
+ J
- --
mass conservation equation
PAGE 105
I
-
--
0
(2. 4)
and the following energy equation
I (
-
es
,I±f.I
4-
((
S
-)
-
(
.W adding
>
where
T
/
(2. 5)
(2.2) by
,
(2.3) by
IVq1Z
J3t-
which can be obtained by multiplying
S ')P
TT"D
41
tP
them together and averaging over a p-surface.
LAL
eastward
V
northward component of the geostrophic wind
component of the geostrophic wind
eastward component of the ageostrophic wind
V
northward
component of the ageostrophic wind
Coriolis parameter =
t
2.bvMMf
Coriolis parameter evaluated
.j
at mid-latitude
derivative of the Coriolis parameter
geopotential
y
V
+
stream-function for the geostrophic wind
horizontal divergence or gradient
d4-
PAGE 106
C2D
individual pressure change
d1
dry
static
stability =
-
zonal average at constant p and y
C horizontal
average at constant pressure
PAGE 107
3.
Simple theorems
We now proceed to prove three statements.
Theorem 1.
For a stationary channel flow the
flux of energy,
Proof.
This
thermodynamic
kV
total
upward
must vanish.
is
a
straightforwar-d
equation.
In
fact,
consequence
multiplying
of
the
(2.3)
by
and horizontally averaging, we have:
CV
--
-
A4I~y /?3(V
=-
'
I
Dq
(3. 1)
where we have used the zonal periodicity and the presence of
lateral meridional walls.
Lemma 1.
energy
If we
upward
have
we
a
must
stationary
wave
that
propagates
have a mean meridional circulation
that transports back the same amount of energy.
Lemma 2.
No level
Theorem 2.
to
can be a net source of energy.
For a stationary channel flow the form drag
due
the topography must be balanced by the Coriolis force in
the layer containing the topography.
Proof.
We start from the
zonal
momentum
equation
(2. 1),
PAGE 108
and, to avoid the problem arising where p-surfaces intersect
the
ground,
(quantities
we
integrate
evaluated
at
vertically
from
ground
the ground are hereafter denoted
p = P (xy), to a
with the sub-index g),
the
generic
level
p,
then we take the spatial derivatives outside the integral to
get
P
PF
P
IDX
/ZX
p
P
where the term
in
the
(u
L-) )
(3.2)
PCP
has been neglected been
Rossby number.
Taking a p-level,
order
second
p = P
just above
the topography we can perform a horizontal average.
All the
terms having in front a horizzontal derivative vanish due to
our zonal
periodicity and our meridional boundary conditions
and we are left with
- V+/cP
P
Lemma 3.
=
(3.3)
If the net meridional flux of mass is zero at
latitudes
valid
e-3
once
particular
the
p
form
Is
drag must van i sh.
lower
we can choose
Because
we
a Te
than
any
all
In fact Eq. (3.2) is
ground
pressure;
in
p = 0.
working
with
quasi
geostrophic
PAGE 109
equation
we
conclude
that
the form-drag must be a second
order term in the Rossby number.
this
result
To point out more
clearly
we can do the quasi-geostrophic scaling
on the
zonal momentum equation
D X
/-k
/~,
Dp
/D
where now (U,V,cO) are the full components of
field.
x
the
velocity
We scale all our variables in the quasi geostrophic
framework
L/
((L
))
/
L.
where Ro is the
typical
scale
value
the
number
Rossby
of
vertical
and
P
the
the ground pressure, has been choosen to
variations
of
the
pressure
field.
Substituting we get
-,
4
V
4-
-,
r-
PAGE 110
Integrating from P
to P=0
and
horizontally
averaging
we
find:
D
4)
Q~J'
P
and assuming no meridional mass flux
Z
1
_x
>.
;IAw~
Transforming the x-derivative following
to
the
ground
x-derivative at constant height, using the definition of
geostrophic
wind
in
z-coordinates
and
neglecting
horizontal variations of the ground density,
1
4)
the
we can
write:
"R7(''
where
now
H =(g
h/R
h/H
/P0;
2 Ro,
/10J
VyI'J7
A/-
H
is
the
Assuming
we-find that
internal
a
-the
height
realistic
part
of
the
1
as
scale
defined
small
topography,
meridional
wind
PAGE 111
correlated
number.
waves
with
the
In the
topography
linear
theory
must
of
be
of order Rossby
topographically
forced
this does not hold when we .go close to a resonance so
that the linear theory must break down.
Our third
drag
and
the
statement is a relation between
vertical
the
form
component of the Palm and Eliassen
flux.
We start by considering the zonally
averaged
zonal
momentum equation
.- x&
f/'X
4-
We write the ageostrophic
wind
(3. 4)
using
the
zonally
averaged continuity and thermodynamic equations
V=
We
boundary
CDPl'f77
---
integrate
condition
V
once
in
V
0
y,
using
the
meridional
on the lateral boundary, to
get
jX
.--
I
--a P
5
(3. 5)
/
PAGE 112
X
We then substitute
(3.4) getting
.-0/'O
7-VX
VX
where now on the right hand side we have the
the Palm and Eliassen flux
divergence
(Andrew and McIntyre, 1976).
second term on the R. H. S can be interpreted
as
convergence
flux
of
(3. 6)
"geostrophic
vertical
a
of
The
vertical
of
zonal
momentum."
Theorem 3.
vertical
and it
For a stationary
flux
flow
the
total
"geostrophic
of zonal momentum" is constant with pressure
is related to the topography by:
-
LA
Proo f.
We
consider
-p e-D
10 X
the
zonal
X k4-
averaged
(3.7)
thermodynamic
equation
on a
pressure
vertically
surface
integrate
P above
the
topography,
then
we
the conservation of mass from P (xy)
PAGE 113
to P0 and we zonally average to get:
=
S
Equating these two
-
V+
V')
--
expressions
for
-Z3 X
and
integrating
once in the meridional direction we get:
-D
(3. 8)
P
Using
neglecting
rewrite
Eq. (3. 3),
second--order
substituting
geostrophic
__
-~
proving the theorem.
streamfunction,
x-derivative outside the integral sign
integrating once by part we find:
-V
and
~4~
substituting the definition of
partial
-Py
X14P
the
f"
as
x eA
taking
by
terms in the Rossby number, we can
the R.H.S of Eq. (3.8)
1
and
f
PAGE 114
Lemma 4.
For small topography
_____x
~L v~V
.5
P
R[D
(3. 9)
comes from (3.8)
This result, without any y-integration
we
neglect
the
the topography;
if
coupling between the ageostrophic wind and
i. e.
we neglect
p
PAGE 115
4.
A simple non-linear analytical solution
We now solve a particular
case
and
we
study
how
these requirements are satisfied.
We use
equation,
the
quasi-geostrophic
obtained
by
eliminating
potential
e-)
vorticity
between
(2.2) and
(2. 3),
-
(4.1)
To define the bottom boundary condition we consider that the
ground is a material surface:
Where in the stationary case the ground pressure
only, a
function
of
x and y.
Transforming the horizontal
derivatives following the ground to
at
constant
height,
is
horizontal
derivatives
and using the hydrostatic balance, we
can write:
(V~v)
-~
~~S
4 3~
O~.Vh
(4.2)
PAGE 116
In the last expression we have neglected
the ageostrophic pressure, which
topography
and
ground density
level P0
,
instead
that
of
we
assuming
assume
a
a constant
(4.2) on a constant pressure.
on p = P
1
Eliminating
(2.3) and
ch
between the
thermodynamic
equation
(4.2) we get
J~n4}
9)
the
f(4.3)
3O
An analytical solution of (4. 1) and
assuming
condition
consistently
and we apply
advection
is a second order term.
We approximate this boundary
small
the
(4.3)
can
be
obtained
quasi-geostrophic potential vorticity to be a
linear function of the geostrophic stream-function
4
-
--
;a
(4.4)
and assuminQ
-P
Putting
-~
in explicity the barotropic zonal flow
(4.5)
PAGE 117
.
and
setting
C
-
--A/U
= -
4
(4.6)
we can rewrite
(4. 5)
and
(4.6)
as:
I
~
-s ~~Z:~p
U
(4. 7)
These equations are linear and can be easily solved for
any
assigned topography.
To
solve
we
approximate,
the dry static
stability as:
Wiin-Nielsen (1959),
_S
where
H
problem
the
following
V
V4
z
VO
4-,
at mid-latitude is
1-4 .
approximatly
AAA 6
1.07 *
We define the following non-dimensional quantities:
T0P
L
4z
L
>'
H h
'
(4.8)
PAGE 118
where
7i-L
reference
is the meridional width of the channel,
pressure, P =
ground
and H is an internal
With
these
scale
substitutions
coordinate
1000 mb,
height,
and
H
k
is
a
To
Po
J
introducing
.
8 km.
a new vertical
Eq. (4.7) becomes:
(4.9)
Assuming no topography on the
meridional
walls
of
Because
of
the channel we can use a Fourier expansion
m'
2-M I
Aim
X4
)
rL
=A. 1-
Where
the
and L
is the channel
lenght.
linearity of the problem we can consider each component
independenly.
-/3
L(
We must consider two cases, for
L
(pp)Lz~
the forced wave decays exponentially
given by:
L7
and
the
solution
is
PAGE 119
X+
+ C
(4. 10)
and applying the bottom boundary condition we get
'Y;o
MI
(4. 11)
topography
We see that the forced wave is in phase with the
and theref~ore the form drag vanishes.
Uii
L-
the second case,
In
L -
ZC-Z
the solution has upward, A and A'
and downward, B
and
B',
energy propagating waves and it is given by:
Ada X +Am
4-
4-?
Cr+
cl
(4. 12)
and applying the bottom boundary condition we get
A(A-A
=
(
B
+B)
± (4--B)
+
PAGE 120
Applying
a
top
radiation
boundary
condition
to
determine the solution we have:
B
0
(4. 13)
I
A
4 -L4
=7, - -3'LU R
Because the solution is not in phase with
now
have
a topographic form drag.
the topography
we
To get no form drag and
to satisfy the bottom boundary condition we must have
A
i.e.,
we
FI= B=
B'= O
0=
have
two
waves
with
the
same
--
Al
amplitude,
one
propagating upward and the other downward.
The most interesting solution is that obtained
the
top
radiation
boundary
condition because it does not
require a source of wave eneTgy at infinity;
a single-mode topography
with
in presence of
this is given by
L(4.14
(4. 14)
S-Z 'U4 c.'O"
(
J4X, --
AU)
PAGE 121
As we have said,
hence,
for
this solution transfers wave energy upward;
Theorem
1,
we
circulation that transfer
energy
must
energy
have
a
mean meridional
downward.
The
vertical
flux due to the mean meridional circulation is given
TL
by
The
---x
vertical
motion
can
be
computed
from
the
thermodynamic equation (2.3)
(4. 15)
and the zonal average
--X
c.
reduces to
(4.16)
independent
meridional
of
the
motion
pressure.
we
have
a
Hence,
downward
due
flux
to
the
of
mean
energy,
independent of height, that balances exactly the upward flux
of
wave energy.
To understand the source of this energy at
infinity we should consider the time dependent
problem
and
PAGE 122
study
the
upward
propagation of the wave front;
meridional circulation should be
front
set
up
behind
the mean
the
wave
that should be also the source of the energy which is
advected downward.
From Eg. (4.16) we also see that at some latitudes we
get
a
constant
upward mass flux starting from the surface
p =P and leaving the atmosphere while at other latitudes
get
a
down
mass
to
flux
the
we
entering the upper atmosphere and coming
p =P
surface
.
From
Theorem
2
we
can
anticipate that in the layer containing the topography there
will be a net meridional
lower
part
of
the
transfer of mass able to close
meridional
upper part of the mean meridional
we
should,
as
the wave front.
circulation.
the
To close the
circulation we think
that
we said before, study the time evolution of
PAGE 123
REFERENCES
Andrews, D.G.,
and M.E. McIntyre, 1976:
horizzontal
and
vertical
shear:
Planetary
the
waves
in
generalized
Eliassen-Palm relation and the mean zonal acceleration.
J.
Atmos.
Sci.,
Wiin-Nielsen, A.,
33,
1959:
2031-2048
On
barotropic
and baroclinic models,
with special emphasis on ultra-long waves. Mon. Wea. Rev.
87,
171-183.