THANS TION FROM ONE WAVE REGIME TO ANOTHER'
I-1 A TWO-LAYER GEOSTROPHIC MODEL
U
by
MAN-KIN MA
B.A.8$,University of Toronto
(1903)
SUBMITTSD IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF
SCIENCE
at the
MASSACRUSETTS INSTITUTE OF TECHNOLOGY
January 1986
Signature of Author.
.
.
.
.
.
Department of Meteorology, January 1966
Certified by.
Thesis Suverzvisor
(
Accepted by .
Na
Cirm,,
bepartmental dwittee on
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1'
' -- -
I
- -
Iq
TRANSITOA FRhM ONE WAVE REGIME TO PNTHER
IN A TWO-LAYER GEOSTROPHIC MODEL
by
Man-Kin Mak
Submitted to the Department of Meteorology on December
24, 1965 in partial fulfillment of the requirements for
the degree of Master of Science
ABSTRACT
A truncated spectral representation of a to-layer geostrophic
model with 'parameterised effects of friction and heating is used
for investigating the steady states as well an the transient behaviour of an idealized rotating fluid when it undergoes a transition from one wave number to another in the Rossby regime. Wave
number 2 and 3 are studied in this paper.
For the equilibrium states, it i found that the total energy
of the system is independent of the thermal forcing, &A , as
is within the range where both waves coexist stably.
long as o
A given increase of $* leads to a certain increase of energy
associated with wave 2 and a decrease In equal amount of energy
associated with wave 3. but the zonal component remains unchanged.
appears
The transient behaviour following a change of /
to have throe distinct stages of development. In the case of
increasing 8,f from the lower to the upper bound of the transition range, the initial development is a growth of all components;
for the smaller rate of increase of
I a the initial developmat
lasts longer but has a weaker intensity. The second stage of
development is characterised by an exponential growth of one wave
and decay of the other wave, the growth rate being relatively
independent of how &A is actually varied. The third stage is a
gradual readjustment to a now state of equilibrium.
No transition can take place if we hold the static stability
fixed even though its initial and final values are known to be
equal.
A dynamical account for the outlined major characteristics is
proposed. The transition phenomenon is viewed as a chain of
processes with baroclinic instability of the waves with respect to
the zonal flow as the basic mechanism. Variability of the static
stability serves the role of intensifying the sonal flow until
instability begins0 and thus is an indispensable factor.
It is hoped that this study sheds some light on an analogous,
but much more complicated phenomenon of wave transition observed
in the dishpan eperiment.
Thesis Supervisor:
Title:
Edward N. Lorenz
Professor of Meteorology
TABLE OF CONTEfT6
INTRODUCTION
TWO-LAYER GEOSTROPHIC MDDEL
3
EQUILIMIUM STEADY STATES
7
TRANSIENT BEHAVIOUR DURING TRANSITION FROM
ONE WAVE REGIE T ANOTHER
17
EFFECT OF VARIABILITY OF TE STATIC STABILIT
27
CONCLUSION
34
APPENDIX
A
37
APPENDIX
B
42
ACKNOWLEDGEMENTS
I wish to express my thanks to my thesis advisor,
Professor Edward N. Lorenz, for taking active interest in.
Inspiring much optimism of and giving some valuable advice
throughout this research.
I also greatly appreciate the enthusiastic spirit of
my follow student, Mr. John Young, in many of our unending
discussions.
My thanks are also due to Mles Pauline Look Bong for
her able assistance in helping me to prepare the manuscript.
NRODUCTION
The dish-pan apparatus was first found indepeandenty by Fultz
a'd Hide (see Fultz (1953) Hide (1953))
to be a powerful tool for
stdying atmospheric motions, particularly those of planetary scale.o
Such experimental flows and the observed large-scale atmospheric
flows have been shown in many studio
but als
to be not only visually similar
dynamically and energetically similar.
Specifically, Fults
and Hide demonstrated that axially symmetric flow, asymmetric flows
and essentially turbulent flow can be reproduced by controlling the
combination of the heating rate and the rotation rate of the apparatue.
The symmetric and asymmetric flows have since then been named the
Hadley regime and the Rossby regime respectively as analogs of the
atmospheric flows observed in the tropics and the mid-latitudes.
Fults and Hide also obtained quantitative criteria for the transition
between the Hadley and Rossby regimes
and the transition between
different wave numbers within the Rosuby regime.
Since then several
theoretical hydrodynaists have attempted to explain such observations
withmathematical models.
(1956) and Lorenz (1962).
that of Lorenz.
Notable are those of Kuo (1954)0, Davies
Of particular relevance to this study is
With the use of a two-layer geostrophic "numerical-
predicton*' model 0 and with the technique of spectral representation
of each dynamic and thermodynamic quantity of the system, Lorenz was
able to obtain stability criteria for the symetric and asymmetric
flows.
His theoretical stability criteria were found to agree
qualitatively with the experimental resulti.
However, his analysis is only applicable toi the steady equili-
brium situation.
It Is incapable of describing how the transition
from one wave number to another actually occura.
Thus we cannot
deduce what the mechanisms governing the rate of evolution are.
This
study is primarily aimed at answering these unsolved questions.
Before
we investigate the transient behaviour, we will first verify the theoretical predictions of the steady states, with particular interest in
the variation with
A of the distribution f
the energy among the
various components of the system.
In this study we adopt Lorens's basic approach,) but usenumerh-.a IntegratAon' to investigate the following aspectn of the problem of wave
transition:
(1)
Structure and energy of the steady states as functions of the
externally imposed heatingr
(2)
and
9A
Transient behaviour of the eyatem under four different types
of time varying thermal forcing
1;
, namely:
(1)
sudden increase of 6 from an initial value to a
final value
(1i)
e is increased hyperbolically towards a final value
(111) exponential increase of 6A*
(iv) sinusoidal fluctuation of Q*
(3)
The effect of the variation of static stability on the wave
transition within the range of
in finite amplitude.
A
where both waves coexist
i-.-
c~m~
~.
Th~ 2'w~ y~
The physical
Bystem
that we wish to investigate is a rotating
cylinder with a controllable temperature distribution at the bottom.
The numerical model that is used for representing this system can
best be described pietorially as follows:
c~ o
=o
-
*
4+
-------> q
e
3
-
*
W=O
4
.7
-,ez
rr
-P -------------
0
~sII,
jill
t-T
-aC -2X
cz k
The mathematical formulation of this system wa first done by Lorenz
(1962).
For the sake of completeness the detailed derivation is given
in Appendix A. Only the subsequent set of equations that govern the
dynamical and thermodynamical quantities are presented in this chapter.
Lorenz*@ approach is based on the idea that each physical quantity be
adequately approximated by a truncated series of orthogonal functions.
II-
4'
The Choice V9 the orthognalu
the system to be studied.
dicatd
by the geonetry of
In the cae of a cylindrical domainu
Besel expansion for / o T
0 and
a
V
series after the five terms.
Is appropriate.
We truncate the
Such a truncation correeponds to suppressing
all components except the sonal . the wave number 2 and the wave number 3
components.
The expansions are as follows:
0-
t
'433
2 2
21
-1
Where
C-=
radius of outer cylinder
= depth of the liquid
= coefficient of thermal expansion
= gravity
f Coriolis
parameter
As shown in the Appendix A
9A
,
,- 'r
9
1
for 71
and also the quantities
=
2.3 are governed by the following
set of 12 ordinary differential equations.
For the special choice:
(2)
0
(i)
6,=-ff
Udi
-act-19
,')
4
£>4,- 9
40
=Z
-
L
Lv)
&3d-
,
,174
A')
2,1]
00
2
N-414
1')
0-44i7+2-rX
)
tJn:
r2~~o~
(] X) v
-
,-c,.F
-
4-
./~IAhJ~~PAl
-Q]
(X;)
(Y;;)
~
4
=
r.<.-
A
= &( 0 -4i * ' 0)
+
+-
,-p
4
C,
(te
,0
o
F,by
-4]
40
+&[
iA3
9,,z
r.3.
7
,'4
(4
-j +2
)
,,)+
+0
2'
0 '#
-i*)
,,j Ilipiil
bU
il lloilillillkil, , 11,
6.
It is very difficult to study this set o2 equations with an
arbitrary forcing $ . We therefore resort to the technique of numeri.4
cal integration. Although this is aesthetically less appealing and loss
conclusive than analytic analysis, it nevertheless is a powerful tool
for bringing to us some insight about the behaviour of this system when
two finite amplitude Rossby waves simultaneously interact with the
thermally forced sonal component.
It is a relatively staple matter to numerically integrate such a
set of ordinary differential equations.
However, in order to avoid
computational instability and excessive truncation error, care should
be given to the choice of the integration scheme.
procedure is used in this study.
A double-approximation
Such a scheme has been showin (Lorenz,
1963) to be appropriate for both periodic and non-periodic flows.
The
validity of this computational procedure is checked by applying Lorenz s
analytic expressions to the quantities of the steady state obtained
with numerical integration.
If those expressions agree with the com-
putations we can then say with some confidence that our computational
procedure is reliable.
CHAIPTER
Equ1A-Ai liM
S§ta&
tesa
hbat of the features related to the various equilibrium ster icy
states of such a system have been investigated analytically by
Lorenz (1962),
Of particular relevance to this study are those that
are concerned with the coexistence of two Rossby waves.
these two waves by wave numbers/4 and )) *
We designate
For mathematical cot-
venience we only permit interaction between each wave and the son1
component and also we choose the frictional heating coefficient to
Z
2
Lorenz found that for a system in which all wave
allowed to exist
e
numbers are
if two waves are to coexist stably the wave numbe s
must be consecutive integers, i.e.
and P =7
f /
.
Two stabIl4 ty
criteria have to be satisfied simultaneously.
(3)
7 A
(a)
(b)
&
A
P
Furthermore, the other quantities of the system are found to satisfy
the following relationship:
(c)
(e)
+
A
(2)
,k;
4
I
44
(k)
(J)
*-
A
+ AJ
p
=os
/
(n)
(0)
(n)
(p)
(ci)
&<;4
These equations would degenerate to those that describe the system
consisting of one wave, say wave number/
. if we put A
and remve all quantities with subscript P
An shown by Lorens, the stability criteria
when represented
graphically, do have a strong resemblance to the corresponding experimental curves.
But in view of the omlicated algebraic relation, it
opet
4S difficult to estimate the relatIve Magnitude-v oft02rou
ln facto the thaoretical
of the flow for dlfferent thermal forcings.
analyals does not yield a functional dependence of
on
and
and therefo
/
With the aid of a computer, we cano however, read1.ly
*
obtain that information from the various steady states corresponding
This is done by numerically integratIng
to different thermal forcing.
the original set of equations with an arbitrary initial condition.
This is indeed the starting point of this study.
Our objective is t
find the quantitative dependence of the steady state on
9
within
The theoretical results described above can
the transition range.
thereby be checked directly.
The friction and
Wave numbers 2 and 3 are used in this study.
heating parameters have the following values;
=0.1
#
=
Ge
=1.
I
0.
is varied over the range from 0.8 to 1.8.
An arbitrary set of
initial conditions is used.
It is found that the two waves coxist
stably for the range of
from 1.00 to 1.28.
Lorenx's analysiso
range of 0'
temperature
,
A
As predicted by
all do not vary within that
and
a whereas the wave amplitudes of the stream function and
J
dependence on
.,
A
and
.
70
=,
3)
do have a dofinite
7
These
results are shown in Figure 1.
Before we
discuss their physical implications, we should first ascertain the
reliability of these results.
(3c)
To do that we compute
with equation
based on the quantities of the steady state obtained by numerical
integrations and then find out by how much this value differs from the
actual value of 0 used.
A
This was done and it was found that
iMiUi*iinin.~
W1111iiiia~m
I
to.
~
"4
.5A
Fig la
Streamfunction amplitude of the waves,
vs thermal forcing, g
,
A
Pig lb
Static stabilit',
zonal temperature,
O
in equilibrium states.
zonal streanfunction,4
,
,
A
vs O
A
,
and
in equilibrium states.
*9987 While
P=
1.00.
Thum we have a Uab-
stantial evidence concerning the validity of the double appromination
procedure.
We now examine the physical implications of the results displayed
in Figure 1.
That
;
,, 7
remain constant implies that the
strength of the zonal flow, the zonal temperature field, the static
stability and the overall average of the temperature field are uneffected after a partial or complete transition.
It is difficult to
tell how the waves* amplitude varies with &A.
However, if we plot
the square of the wave amplitude from Fig. la instead of the amplit;ude
itself, against
, we would get a simple linear relationship. This
&
is shown in Figure 2.
This finding immediately suggests that there
might be some simple energetic relationship between the two waves in
the transition region.
With this expectation we will first examine the
analytic expression for the energy of each wave and then seek a simple
link between them, which hopefully will be consistent with the finding
mentioned above.
The derivation of the energy components of the system
is quite straightforward.
It is presented in Appendix B.
According to equation (B-4b) and (B-40), total kinetic energy of the
waves (EAT)
-
2
Substituting equation (4g),
(4h) we get
Fig 2
Square of the streamfunction of the maves,
('+
W,(,)
vs
and
stateso
'30-
2')
'2-0
( NK
)
'1
.,.I
/.'
i equilibrium
13.
k3t
A
Since according to (3-p) and (3-q)o
are constant for the range of
,
slop* is equal to ... IA)(/f*
)
thersal forcing.
Figure 3 Is such a
Indeed we find a straight line whose slope is -. 73.
(3-p),
(3-q) we find that
)
the
we can then say that
plot.
(/
2(/--4
is a linear plot and if
A+z - e
u
remains constant for
(
where both waves coexist stably.
42 (
Thus If
are functions
and
alone, the two coefficients
of i
EKE
A
Using
0/7
This is the physical implication, implicit in Fig. 20 that we set out
to seek.
Now we are in a position to describe how the various energy components
vary with
0
in steady states.
For sufficiently low value of 19*
i.e. weak heating, we should expect only wave number 3.
The sonal ft.
(A)-D
the .E of wave 3 ( K ) and the total potential energy (TP ) are expected
to increase with
tion region i.e. -
6
until
1.0.
#
reaches the lower limit of the transi-
Beyond this point E , 7P( and
Cr
remain
constant, and wave 2 progressively intensifies at the expense of wave 3.
When
reaches the upper limit of the transition zone,
3 virtually vanishes.
From thereon the system absorbs energy again
through the bottom; A ,Pe
bigger.
~ 1.25, wave
as well as K2 all become progressively
In view of other studies, we should expect that at a certain
Fig
( (~
&+
3 W3
FWg
). . vs
S.W *
ot
)j,)within the range of
C4
'IiIi
Ii,1
4
1O-1.25
1o.
otageo wava 2 also otops growing and avenitually
regime will dominate.
i
e
da Hadly
Since our study is merely one coneemring the
phenomenon of transition between two Roasby waves, our investigation
stops at the point where wave 3 clearly vanishes and wave 2 Is clearly
well-established.
Knowing the various steady states of the system for
a large range of
f (see Table 1)0 we can readily verify the description
of the variations of the various energy coWonents.
Figure 4.
The previous description is thus proved to be accurate.
TABLE 1.
(kA
4
klPg
42,
A
ex
Steady States and Energy Variations with
.103
.0000
.2004
.0
.0000
.0
.2018
-. 2474
.183
-. 0000
-. 1513
-. 3400
.2003
.0
.2158
.096
.2158
.2683
.2203
.2405
-. 0220
-. 4434
-. 1883
-. 0089
.2763
.2818
-. 0622
.4601
*2167
-. 2093
.2157
.1160
.1053
.1411
.2157
.2092
.0
.0
.2203
-. 2569
.0
.0
.2404
.1034
-. 0492
.1535
.0225
.1696
.3109
.0844
.0
.0
-. 0000
4
,
04
6.
-. 0224
-. 1741
-. 1781
-. 0743
1.3379
.3379
1.3830
.3830
A'
.0435
.0
.0507
.0276
.0
.188
*1858
.0517
.0
.0409
.0400
.0
.2414
.2414
Ag K3
1.2703
.2703
.0
.2158
-. 3578
-. 1299
.2158
.0460
1'1
A,
They are shown in
-. 1737
.0119
.0881
-. 1472
1.3830
.3630
1.3630
.3630
1.3781
.3781
1.4419
.4419
.05061
.0055
.0420
.0466
.0125
.2489
.2614
.0547
.0435
.0256
.0465
.1022
.1560
.2562
.0534
.0837
.0085
.0465
.2014
.0832
.2546
.0531
.1037
.0
.0485
.2650
.0
.2680
.0554
.1030
.0
.0578
.3175
.0
.3175
.0178
.0
.0
4'
Fig 4
Kinetic energy of variouB components
equilibrium states*
.25-
2K
-2I'
1.4
vs
Ain
C.APTER I I
Transient BehaAour of the System durinM Transition frM
Oe..PredJminatt ae_Numb2r-to A thLr
The reason that we want to examine the transient behaviour of the
system is twofold.
First, the transient behaviour is interesting and
important in its own right.
Second, it is hoped that studies of the
major features during the transition evolution of the system might lead
to a deeper understanding of the transition phenomenon.
To fulfill the
first objective Is essentially to give a synoptic description of the
evolution of the system as revealed in the numerically integration.
But the second objective is a mauch more difficult one to be fulfilled.
The synoptic picture is most probably the net effects of several unknown
interwoven processes.
We should therefore follow the approach of a
chemist who tries to
aalyse the structure of a complicated molecule.
In other words, we should change the controllable factors one at a time
and examine the corresponding effects in the hope of being able to isolate
the crucial factors,
In this problem, there are two factors that we can
control and hope for interesting results.
thermal forcingo Of
One is the wave that the
, approaches its final value.
or not the static stability is allowed to vary.
The other is whether
Although a lot of other
parameters such an the frictional and heating coefficients can also be
changed arbitrarily, doing so is unlikely to yield qualitatively difforent results.
Figure 5 shows the evolution of the amplitude of the wave components
of the vertical mean streamfunction for three different types of variation of
9A
from 1.01 V;to 1.20.
Curve (1) corresponds to PA being
increased stepwisely; curve (1I)
l.-IIly.
hyperbo'lically; curve (MI)
egonn
Figure 6 shows the evolution of the static stabilit1 and the
zonal component as gall as the
variation.
The main features of the evolution of the three components can be
summarized as follows:
1.
The faster 0 is increased to the final value, the
will the transition be completed.
2.
A is
Except for case (I) where the rate of increase of
infinite, the initial growth rate of wave 2 is very much
smaller than the growth rate during the major portion of
the transition. Curves (I), (II) and (III) of wave 2 are
to a first approximation parallel to one another. The
slope of the main portion gives the major growth rate of
an e-folding time equal to 200 non-dimensional units, or
equivalent to 16 revolutions (days).
ooner
3. Wave 3 also has an initial growth although it subsequently
decays exponentially to a smaller steady value. The portions of the curves that correspond to this exponential
decay for the three cases are also parallel to one another.
4.
The zonal component and the static stability also increase
initially for about the same period as wave 3 does before
they return to their original value.
5.
The initial growth of all components have the following
comn characteristics: the faster 6A is increased,
the smaller is the period of the initial growth, and also
the larger is the initial growth rate. Furthermore, the
period of the initial growth of all components is roughly
the same.
6. The phase speed of both waves changes in a similar manner
as the zonal flow does.
Feature 1 is what we would intuitively expect, but it yields little
information about the underlying processes.
Feature 6 indicates that
the waves are advected along by the monal flow, thus their phase speed
is proportional to the strength of the zonal flow.
-ir-
"4
4oo
a
0
4e'o
Fig 5
Evelution of 4
(1)
6
goo
40/000
= 1.01
= 1.20
+)f
and
e t=0
ot>0
/00
zoo
4eo/zoo
with time for three cases
(II)
-= i.01 + (.19)tanh(.02t)
(III)tfA*;- 1.01 exni(.oo14t) 4/-ZO
therea
1.20
Fig :A
Varition of the
t-
therrnal forcing
with time
3o
Fig it
Evolution of the
6Ob static stabilit/
T0
'3o'
4
.25-
/. ,,
;00
Pig & i; olution of the
,e. zonal streamfunction
i
2R0
Featurer 2 to
v
tholwn
Umplications.
Ao -a002ew
waves are concern&d. the whole transitaon can be divided into threO
stages: an initial development, a major development and a final
adjustment to a new equilibrium.
The initial development i qua7-
tatively different from the major development.
For wave 2 the Itajor
development is much more intense and lasts much longer than tk e '
initial development.
For wave 3 apart from the large difference In
magnitude, the initial development is even in opposite senso of the
major development.
Above all, the initial development has a distinct
dependence on how the thermal forcingo
I
e is varied, Wile the
major development is relatively independent of it.
Such difference
suggests that the initial development is of a forced neture and the
major development is of an inertial nature.
Since the major develop-
ment of the waves begins at about the time when the rxrnal flow and
its vertical shear reach their maximum values, it it suggestive to
associate this development with a baroclinic instability process of
the waves with respect to the sufficiently intensified xonal flow.
This suggestion is consistent with the fact that there is not a counterpart major development in the evolution of the zanal flow because during
the baroclinic instability process the growing wave gets its energy
mainly from the decaying wave.
So it appears that the transition is a
consequence of baroclinic instability of the waves with respect to an
intensified zonal flow which is caused by the initial development.
In
order to complete the reasoning, we need to know the crucial factor
responsible for the intensification of the zonal flow.
That piece of
Ii
___________________________________
______________________________________
missing inte~rnation0
Is
founMd by the following attempt
Two sets of runs were made with the objective of exploring the
effect of the variation of static stability.
e Is
In each run,
A
stepwisely increased from 1.01 to a final value with the equilibrium
A= 1.01 as the initial state.
state for
free to vary while in
stability is
The result is
In one set (A) the static
the other set (B)
it
is
held fixed.
very remarkable and indeed is against what one would
Figure 7 shows the evolution of the kinetic
intuitively expect.
energy of each wave in differenO cases.
where
is that for set (B).
The most surprising feature
= 0s no transition from wave 3 to
wave 2 can occur.
Even though the initial and final values of
are the same for
0A
within the range from 1.0 to 1.25,
the whole
transition process is suppressed merely by suppressing the variability
of the static stability.
If we examine closely the effects displayed
in Fig. 100 we find that the initial growth of the zonal flow Is also
suppressed although that of wave 2 and 3 still exists.
So we now come
to preliminary conclusions - variability of the static stability is a
crucial factor responsible for the intensification of the zonal flow
that subsequently leads to the baroolinic instability of the waves.
As
to how this factor works from the energy point of view is yet to be
explored.
This aspect is the subject matter of Chapter IV.
It is of interest to check whether anything qualitatively different
might occur if
1.20 to 1.01.
PA
is decreased in a similar manner as above from
The results are given in Fig. 8. The basic features
are again apparent.
So the proposed dynamical account should also be
-------------
I-
h' ,
-
c/A
-
I
-.
1
/
l.~o
/
I
Z%0~
tA =/-&o
/0o
/0o
/90
and / , due to a stepwise increase
Fig 7 variation with time of the waves kinetic energy,
of $4* from 1.01 to 1.20 in two caseso
<r is free to vary (broken line)
( )
(ii ) Cr- is held fixed at the equilibrium value (solid line)
C4-i
2-2
A
Ar
2'ooo
Pig 8 1:volution of'
(IV)
6,*= 1.
__ -Pl-
j/++1
20
= 1.01
(VI)
and
with time for three cases.
^ t =0
R t > 0
(V)A = 1.20 - (.19)tanh(.02t)
1.2 exp(-.oo0114t)
t*=
L01
> 1.01
0,
03o
4oo
Fig 9 Evolution of
due to thermal forcing
.+
(g
t
i
Lastlyo w e xmine the behwAvour of the system if
9A
is varied
sinusoidally between 1.01 and 1.20 over a wide range of frequency. in
the light of the above findings
we should expect a simple forced
oscillation of the system for high frequency variation of
because
the system would be always in the stage of initial dvelopmanto i.e.
before the major development,, namely baroolinic instability of the
waves 0 can set in
A would be decreased and thus a reverse "initial"
development would begin.
period of the
9
shown in Fig. 9.
Indeed, this is what we observe when the
A oscillation it
112 non-dimensional units.
This is
Presumably if a nwh lower frequency is used0 part
of the expected major development may set in and thus a distorted
sinusoidal response should be found.
Perhaps it
is worth pointing out
two other features in Fig. 9.
(I)
that of
The amplitude of
4
A fluctuation is larger than that of
Is larger than that of (O* -I.
) , although they fluctuate
about their steady state value as empeoted.
(11)
is in phase with
whereas all other quantities have
definite but different phase shift with respect to ,
I cannot offer convincing reasons to account for these features.
I.
___________________
CHAPTER IV
Eff-
o:1Xf at
.
t
t
_
A-
One fruitful approach to examining the dynamical effects of the
variation of the static stability is to study the energy transformaGates (1961) took this approach.
tion processes.
Hie analysis reveals
two additional terms, which. would aot exist if the static stability Is
held constant, relating the eddy available potential energy and the
But without direct verification he
zonal available potential energy.
had to contend with just a qualitative discussion of the effects of
the variation of static stability.
In this study0 with the benefit
of using spectral representation, we can examine its effects on each
wave through numerical integration.
As was shown In Chapter III the
transition is actually suppressed when
* we use.
of what value of
=e=mination, of
Cr is held constant regardless
Further insight can be gained only from
the analytic expressions of the energy transformation.
Those expressions can be obtained by simply taking the time derivative
of equations (B-da)
:
(B-4f).
(Note:
Begause of the difference in the
definitions of the temperature field, our expressions are different
from Gates*.)
m..
(7)
{4.K
-2
284
wh{re
14
=
--
-
CA,
3' ) 3K
3
g
%3
t
6 43
C
1[4
and .jare frictional dissipation.
4+c
Since -- ej is positive for
warm air rising and cold air sinking, the other terms on the right
sides of equations (6),, (7) and (8) are then energy transformation
terus from XPr to kinetic energy
[..
(9)
where
....- .
2 Oet&A
Similarly
(10)
vdiere expressions for
3'A
3
f 3 3 fA 3
]3
are similar to
the corresponding ones in (9) except interchanging subscripts 2 and 3.
203
Equations (6) to (11) can be sensarised schematically In the folloudag
energy diagram.
2U E
S/N/K
/
-
To F r
f/NAL
,2?,DISS1 P4 7/ OIV
We should notice that
f
'
Vs allow %o to vary with time.
and
arise from the fact that
Wenever they differ from zero it would
mean that these are two additional channels for energy flow which would
30.
have been wuppreosd hac
baen held constant,
Their signS have
explicit phyeical interpretation,
~L%~Lb
~-
(ocq~ e~j
(&~+
L~tA~
A-
A3
Each side of the inequality stands for the rate of conversion of
per unit amount of
zonal combined.
-
where ;
stands for either 2 or 3 and
It means that static stability would vary in such
a way as to channel further amount
A
to
Each side can be called the "specific" conversion rate
from A.P.E. to K.E.
loss rate of
.
and A3per unit (
of 4 2 to
4A 3 )
A
when the combined
exceeds that associated
with wave 2.
Similarly
jA3
A3
A
.
The not effect of the variation of the static stability in then to
replenish the A.P.E. of a component
- iben its "specific" conversion
rate is larger than the specific conversion rate of the other components
combined.
This mechanism should therefore be very effective soon after
one component begins to grow and acts as a boost for further growth.
Hence it is highly suggestive to infer that this mechanism is a major
process to precipitate the exponential growth of wave 2 soon after its
growth rate becomes perceptible.
In order to see the detailed effects of the variation of
, we examine
1.
3A3
04.
/
3/
/
/A
9
'
K
a- 4o
e
Fig 10
-
too
&,o
--
V)
Ks
/
/000"o
;j
ee$
0(
Variation of the kinetic energy and the available potential energy of various
components due to a stepuise increase of 9* from 101 'to 1.20 in two cases,
(
(i
)
(broken lines)
, is free to vary
1 is held fixed at the equilibrium value
(solid lines)
32.
Fig. 10 which is a
mmary of the evolution of each energy component
is stepwisely increased from 1.01 to 1.2
when
for two different
cases: (1) C, Is held constant and (2) <q is free to adjust itself.
Besides the obvious fact that no transition takes place in case (1)
we also notice two distinct features. First, grows noticeably slower in
Secondly, when
case (2) than case (1).
,
=
0
K is
esentially suppressed,
is free to vary.
whereas it grows significantly when TO
Now we are in a position to recapitulate and complete our reasoning
behind the dynamical account for the transition phenomenon.
Z is
When
first increased, there is a period of initial growth of all components.
This arises from the fact that there is a not Inflow of energy into the
fluid system through the bottom interface.
Of the four processes that
convert potential available energy between the wave and the zonal components0 two arise from the variability of the static stability.
is positive.
tA
the initial condition and the initial growth,
Although part of the increase of
A
is simultaneously transferred to
the transfer can be so large that there is a net increase of
fore substantial growth of K
is possible.
A
A
2
and there-
Now as soon as K is large
enough, instability for wave 2 is triggered on.
time
For
We notice that by this
CQ increases at a such lower rate and soon begins to decrease.
As long as A- remains sufficiently strong wave 2 continues to be unstable.
While K and
A
are continually drained by the growing wave 2.
they are simultaneously replenished by decaying wave 3.
stained .growth of wave 2 is possible.
Thus a sub-
The montonic decrease of CJ, has
334
two effects.
At firato the baroolinic growth of wave 2 io zurtr
boosted.
This Is largely due to the existence of
on as k
Is reduced to a sufficiently low value,
-
.
ter
a. decreases at
a very low rate, wave 3 Is stabilized and the growth of wave 2 olows
down.
Eventually a new equilibrium state Is reached.
However, there is a serious weakness in our approach.
some external processes
Even if by
C( is maintained approximately the same,
our previous results does not necessarily predict correctly that
transition would not occur.
The reason lies In the fact that we use
one rather restrictive assumption in our model.
That is that each
Rossby wave is assumed only to Interact with the sonal component and
is not able to interact with one another directly.
When one deals
with finite amplitude waves like those encountered in this study,
such an assumption is a very drastic one.
Therefore, even for
non-linear interaction between the waves could conceivably bring
about transition through direct transfer of energy without relying
on mechanIsms represented
fA,-
and
-A,
.
4.
CHAPTER V
It is found in this study that with the technique of numerical
integration a truncated spectral representation of each dynamical and
thermodynamical quantity offers an acceptable approach for investigating the transient behaviour of a fluid system like the dish pan
experiment.
We have only considered the simplest type of transition
between Rossby waves, namely between 2 waves of consecutive wave nupbers.
In addition to verifying Lorenz's
theoretical results concerning
the relationships among the marious quantities in steady state, we also
find that the total energy of the system remains constant within the
range of
ever, as
0A
where two Rossby waves coexist in finite amplitude.
How-
is increased, wave number 2 in the new state has more
energy than initially by an amount exactly equal to that lost by wave
number 3.
The reverse is true when
A is decreased.
This finding
was shown to be implicit in Lorenas theoretical results.
In examining the transient behaviour of the system during the
transition, we find that the system evolves in a rather surprising
manner for a short period initially before the actual transition takes
place.
It is found that both the zonal component and the wave component
that subsequently decays actually intensity for a while.
that the other growing wave dominates the developments
flow gradually returns to the original value.
Soon after
while the zonal
Although the rate of
transition is found to be proportional to the rate of increase of
the difference is reflected more by the delay of the growth of wave
35.
number 2 than by the dependence of the growth rates on the rate of
increase of
0 4.
The variation of the static stability is found to play a central
role in the transition phenomenon.
By holding the static stability
constant, we find that transition cannot occur.
Instead the system
quickly adjusts itself to a new equilibrium state very different from
the laboratory and Lorenz"e theoretical results.
It is proposed that transition from one wave number to another in
primarily a consequence of baroolinic instability of the waves with
respect to the sufficiently intensified zonal flow when
A
is changed.
Energy is transferred from one wave to another via the sonal component.
Thus the whole phenomenon is hinged upon vhether or not the zonal flow
can be sufficiently intensified before it returns to its steady value.
This crucial factor is found to be determined by the variability of
the static stability.
the rate with which
bility.
The rate of increase of
K
only determines
is increased to the critical value for insta-
Once instability begins the growing wave derives its energy
from that stored in the other wave and therefore should be relatively
unaffected by the actual manner whereby
is increased.
As more
and more energy is transferred from one wave to another by way of the
zonal flow, the original unstable configuration is eased up and the
variability of
03
in turn acts to oppose further development.
Thus
the system eventually approaches a new steady state.
The objectives of this study as outlined in the introduction are
essentially fulfilled.
However, one aspect of the transient behaviour
36.
of the system is still
development.
not fully understood, and that Ia the Initial
Although the initial intensification of the mona1
flows is found to depend critically on the variability of the static
stability
we still do not know about the processes responilble for
the initial growth of the waves.
Those unexplored processes however
do have interesting properties.
In particular, we find some definite
dependence of the amplitude and phase of each quantity upon the sinusoidal variation of
.
It seems quite difficult to identify the
crucial factors behind those processes and I cannot offer suggestions
to do so.
Fortunately the transition phenomenon does not critically
depend upon those processes.
37.
APPNDIX A
Tmrncated Functional Representation of a TwoLayer Quai-Gostrophic Model
A frictionless, hydrostatic and adiabatic fluid system in (XzyapOt)
coordinate is governed by the following met of equations.
Vortieity equation:
+ lk-cu XI
(A-2)
v~
0
-0V
x -C)
Divergence equation:
=
v~
.vv~
First Law of Themodynaics:
Equation of state:
C< k(= T )
+ Q.)
(per fect gas)
(nc
ompressible liquid)
The hydrostatic equation and the continuity equation are embodied in
these equations.
The next step is to introduce mmxiinm simplification
which are based on the property of quasi-geostrophic flows, but they
must not violate two fundamental invariants of the system, namely
(1)
total vorticity is conserved; (2) total energy is conserved.
those problems of geophysical interests,
are both small.
Wx
and
l'
For
W
If both of then are neglected, the two mentioned con-
straints will still not be violated.
In addition, quasi-geostrophic
38,
and (ii)
(1)
Elows have two distinctive properties:
V'
* a
It follows that advoction Is essentially duo to the non-divergent
part of the horizontal wilnd
in (2).
Thus if w write V
=. /
' V,
.
(
) can be safely ignored
4- V,7(
(A-(2)0 (A-(3) for constant
forms of (A-() 0
(A-4)
and that (
then the simplified
f
becomes
)_-
(5a) can further be simplified by using the equation of state and
hydrostatic equation.
(perfect gas)
2
0(liquid)
1
= gas constant
where
coefficient of thermal expansion
2!)
depth of fluid
Now it we substitute the quantities of the nodel described an page 3
into (A-4),
(A-5b),
(A-6) ve would obtain the consistent and maximum
simplified set of equations as follows:
39.
(A-7)
(4)4
(c)
.D
gCt)
D
4'e)
+
+
(/cr)
...
D
V
)-
-
+
=
=0
VX
+ Jc(-(,)-6-
jV
X)
-ZO
0
Frictional and heating effects are accomdated into (A-7) as
follows.
They are parameterised with the use of the following physi-
cal approximations.
We introduce a coefficient of friction,
at the underlying surface and
at the separating surface.
2
The
drag then is assumed to be proportional to the relative velocity at
each surface.
Similarly we introduce a coefficient of heatng, 2
at the underlying surface, and
at the separating surface.
The
heat flux is assumed to be thettemperature difference across each
surface.
With such assumptions, the following terms
U .?0
+
/T
(
-~~
-
~
4-2
L2~
If~
19
i
40#
ahould be added to (7a)O (7b),
(70,) (7d) and (7a) respectiveiy.
In this study we want to study the simplest transition phenomenonthe one between two Rossby waves, wave number 2 and 3.
Thus a trun-
cated functional representation for each quantity is appropriate.
For mathematical convenience we only allow interaction between each
wave and the zonal component.
But we permit the static stability -9.
vary with time, although not with space.
Because of the cylindrisal
geometry of the systemu, a logical choice of the orthogonal funct c-n
is Fourier-Bessel function.
The actual expansion of each quant'ty is
given on page 4 and is therefore not repeated here.
them in equation (A-7)
It we subtitute
and make use of the following properti
i
of
the Fourier-Bessel functions
of
are conwitants
we can equate the coefficient of each orthogonal function.
Tat end
result then is the following set of ordinary differential eqations:
(A-8)
Complete set of the governing equations:
(IV
13
3 3A
L
A3
41.
(v)
'4' --
(V) (0
(var)
,
fd
-0(
4 A e)
-
A
= -d
iK
+
- 4"
-(
ft+4
-Q
%ki-de24-
+A,0+46
. 0-
c;.
it.
--- pu
19
- Ct+2E3
-+
(va)
-) #A
-
%+Ry-
0ff3( %* O9'PN3) -0-
by
(Ex)
) -0
649A4
C
P
619
+ bd(-k{t k+*)3
3
-.
(1i)
60
l'iii)
9,
(xiii)
-C7
- -,(%,
PI A
(xv)
%
(yv;)
b
(xv)i i
'G)
K-
0,(
=-
k3
--'fee's
Ak
t T
~-G~#4)
01
'
9
-
-AW
S-4,
kJ440
LL.
Ke
T. 6UL3
)
-+
OALA kCu-'L
*A
-ion+
+ T
0+ ,)
9K ...3
Since thetas *s in (A-8v)
- 0
K(k
4~.L3
Q9 'k
'
r;-
=~~
-*
+ '
3 k3
e~
"
c
A
3L
'-A-
- (A-Sxvii) Can easily be eliminatedo we will
eventually obtain 12 ordinary differential equations (21) - (2xii) for
12 variables depending on Z
alone.
They are given on page 8.
L
42.
APPENDIX B
Derivation of the Various Energy Components
of the System
Total kinetic energy (TR) of the horizontal flow.
(B-1)
TCE
-
>l
OJ40
//
4?y v3v
+ t7 , V'k-0- 7t -7 -)
ar 4veraoe
Are4/
at'Pra11e
~1
vo -
vib
.01
F
4 Vekit -711
4F
4-
Lh 211
A 0/
12.
4;tvlFit/
11+
C4[o
F~i- -(~F~ I
LA.?~S
k,~ 4
F 24
V1A:
c--
I4ZL
L7?V2
z
4-
k
q-1 4'~..
/
c
(k-A
(f
1t2~
C
2
4..
~1
9
r
L3
430
Total poiential and
itrne.1 energy
(E)
WIVERC
4
f 0 -I
0
)-2)
7r
4 z
-
(
c-
Available potential energy
[6w3' ,
+
A4
-kdo)
.APE)
A.,Jlad
0
'7-
E
.4-
2 ,
8,
+ 01:n"F
'41
c'
+.
2.
L
Writing
g-.
=-L,
+
9
t0
I.
7-
9 +
+
T-, -+-
&
T
3
lj;
We can associate a part of each type with one of the three components
of the system, namely the zonal part, wave : imber 2, and wave number 3.
"4 e)
(B-4)
14CT
Kj
f-
14A1z
2)
Bibi omaAh
Davis, T. V.0 1956: The forced flow due to heating of a rotating
liquid. Royal Trans. Roy. Soc. London (A) 249g p. 27-64.
Fults, D., 1953: A survey of certain thermally and mechanically
driven fluid systems of meteorological interests. Fluid
models in geophysics Proc. lot Symp. on the use of models
in geophysical fluid dynamics, Baltimore, p. 27-63.
Gates, W. L. 1961: The stability properties and energy transformations of the two-layer model on variable static stability.
Dynamical Weather Prediction Project sei. Rqport No. 6
UCLA.
Hide, R., 1983: Some experiments on thermal convection in a rotating
liquid. Quart. g. R. Meteor. 8oo. 79, p. 161.
Kuo, H. L., 19531 On convective instability of a rotating liquid.
Fluid Models in Geonhsi.. Proc. 1st symposium on the use
of models in geophysical fluid dynamics, Baltimore, p. 101-116.
Lorens, g. N., 1962: Simplified dynamic equations applied to the
rotating basin experiments. J. Atm. Sci., Vol* 19, No. l
p. 39-51.
Lorens, B. N.0 1963: Deterministic Nonperiodie Flow.
Vol. 20, No. 20 p. 130-141.
i. Atm. Sci.