Fluid Dynamics in Earth and Planetary Sciences
Glenn Flierl, EAPS, MIT

Lecture 3: Jets – stability and nonlinear stability- Gulf Stream, Kuroshio, North Brazil Current, Agulhas Current, ...
- Meandering
Quasigeostrophic: Now, we will go back to the QG two-layer system
Stability
Appendix A. Stability Program- Contour dynamics
Numerical results
Conclusions: numerical
PulsesContinuum modes
Zero lower layer flow
- Ring formation- Diagnostics of baroclinic amplification
1 —Gulf Stream, Kuroshio, North Brazil Current, Agulhas Current, ...
The wind-driven circulations result in intensified currents on the western boundariesof the ocean, related with the asymmetries produced by the beta-e↵ect (the sphericity ofthe earth and the westward propagation of Rossby waves) as realized by Stommel (1948).
In essence, the wind puts in negative vorticity, and, in the mid-ocean where the large-scale
relative vorticity is negligible, the fluid responds by decreasing f : moving equatorward.
The return poleward flow has to be in a boundary layer which can inject cyclonic vorticity
back into the flow. This can happen by side-wall or bottom friction on the western sidesof the oceans.
These major currents are several orders of magnitude larger than the mean flows (butnot than the mesoscale eddies) and detach from the boundaries in various ways to form
free jets. These characteristically meander and form detached eddies (like the cut-o↵ lowsin the atmosphere). We shall discuss the dynamics of meandering and eddy formation.

1

2 —Meandering
There are several models which have been applied to analyze the meandering of oceanic
inertial jets:• instability theory (almost always linear)• thin-jet models which assume the length scale of the meanders is large compared to
the width of the jet• contour-dynamic models which follow PV contours
We will concentrate on the last one (which uses QG) , but consider first the long wave
instability in the full two-layer model (Flierl, 1999). The upper and lower layer velocity
perturbations have the form uj exp(ıkx ıkct), ıkvj exp(ıkx ıkct) with the perturbation upper layer thickness and deep pressure also proportional to exp(ıkx ıkct). The
perturbations satisfy(Uc)u1 + (Uyf )v1 + g 0 h +f u1 + g 0 h y +u1 h + (v1 h)y + (Ucu2h) + [v2 (H2y = k (Uc)v1c)h = 0f v2 +
f u2 +

u2 (H

=0

=0
y =

k 2 cv2h)]y + ch = 0(with the lower-layer pressure). The lowest order solution in the interior region (wherec ⇠ U ) is found by dropping the k 2 terms and recognizing that long waves look like
displacements of the jet plus the flow necessary to shift the axis: u1 = Uy , v1 = U c,
g 0 h = g 0 hy = f U , u2 = 0, v2 = c, and = f c.
Operate by multiplying the y-momentum equation by h, the mass equation by f ,subtracting, and integrating over y. We do this in each layer to find
Z
Z
Z
1
2
c[hu1 ] 1
hy + f c h = kh(U c)v1
Z
2
⇡k
hU 2
Z
Z
Z
1
2
c[(H h)u2 ] 1 +hy f c h = k(H h)( c)v2⇡0where the approximate versions of the right-hand sides are obtained by substituting thelowest order solution and assuming that c is small.
These interior solutions do not decay away from the jet; therefore, we must look fordecaying exterior solutions and match them to the interior forms. We have two cases toconsider: the two-layer model and the equivalent barotropic model.

2

Two-layer case: The interior v field becomes barotropic at the wings of the jet, so
we must seek a barotropic, trapped wave in the exterior. We find the required matchingcondition by summing the two equationsZ1
1
2
c[hu1 ] 1 + c[(H h)u2 ] 1 ⇡ k
hU 2
(2.1)
The barotropic external field to the north looks like:u1 h + u2 (Hh) =ckH exp( ky)v1 h + v2 (Hh) =cH exp( ky)H(g 0 h + p) =ckH(c + f /k) exp( ky)where the amplitude is set by matching to the interior v field. Similar solutions apply on
the southern side. If we substitute these forms in the matching condition (2.xx), we find
Z
k
2
c =
hU 2
2H
Thus, c ⇠ ık 1/2 and the jet is unstable. Furthermore, the result (2.1) that the net momentum transport is related to the external barotropic wave field applies on the -plane, since
the jet width is small compared to f / . The implication is that the external BT field will
be central in determining the instability properties even on the plane.
Equivalent barotropic case: When the lower layer is motionless, the exterior,trapped solutions are baroclinic withu1 =c` exp( `y)v1 =c exp( `y)0cf exp( `y)gh=having a meridional decay rate

`2 = f 2 /g 0 hwhich is di↵erent on the northern and southern sides of the jet. The upper layer integrated
equation (with dropped, of course)ZZ12c[hu1 ] 1 + f c h ⇡ k
hU 2
simplifies because the boundary term is order c2 and c is now order k 2 . Thus, the boundary
terms are negligible, and
R
R
2
hU
hU 2
c = k2 R= k2
f hf [h( 1) h(1)]as in Cushman-Roisin, et al. (1993).

3

The di↵erences show up in QG theory (as we shall see) and can be traced to the kstructure of the Green’s function: for the barotropic modeG(yy0 ) =

1exp( k|y2k

y 0 |)

is order 1/k, whereas the baroclinic modes areG(yy0 ) =

1exp( K|y2K

y 0 |)

,

K=

p

k2 +

2

is regular and leads to c order k 2 (the westward propagation nearly balances the eastward
mean advection).
2.1 — Quasigeostrophic:
Now, we will go back to the QG two-layer system with both layers active. Defining
the Green’s function matrix by
✓

r2

h2 2

h1 2

h2 2
r 2 h1 2

◆✓

G11
G21

G12
G22

◆

=

✓

x0 )

(x
0

0
(x

x0 )

◆

(2.2)

p
with hj the fractional thickness of the j th layer and= f0 (H1 + H2 )/g 0 H1 H2 theinverse of the deformation radius. This has the property that h2 G21 = h1 G12 (required
for showing that H
i .)
qi =
The Hamiltonian is
ZZ
1 X
H=
dx dx0 hi qi (x)Gij (x x0 )qj (x0 )2 i,j=1,2
and the bracket is


X 1 Z
F G
{F, G} =
dx qi
,
h
q
qi
i
i
i=1,2

4

2.2 — Stability
If we split the PV into a zonal basic state and perturbations,@ 0
q = [q,@t0] + [q 0 , ] + [q 0 ,0]with the last term dropped. If we define the displacements of PV contours by
q0 =
then

@
⌘=
@t

or

@
@
U
⌘+
@x
@x
@
⌘=
@t

@
U
⌘
@x

0

qy ⌘

,

0

Z

dx0 G(x=Z

dx0 G(xx0 )q y (y 0 )We can write this as a symmetric kernel usingZ⇣@qy ⌘ =
dx0 U (y 0 )q y (y 0 ) (x x0 ) + q y (y)G(x@tx0 )q y (y 0 )⌘(x0 )@⌘(x0 )
0@xx0 )q y (y 0 )

(2.3)

⌘ @

@x0

⌘(x0 )For a Fourier mode ⌘ = ⌘(y, t) exp(ıkx), we end up with@qy ⌘ =@tıkZ

dy 0 G(y y 0 )⌘(y 0 ) , G(y y 0 ) = U (y 0 )q y (y 0 ) (y y 0 )+q y (y)Gk (y y 0 )q y (y 0 )2@and the 1-D Green’s function Gk (y y 0 ) is found by replacing r2 in (2.2) by @y
k2 .
2
R
RR
@
@
The symmetry implies that @t
q y ⌘ ⇤ ⌘ = 0 (Rayleigh’s theorem). Also @t⌘ ⇤ (y)G(yy 0 )⌘(y 0 ) = 0 (Arnold’s thm). Note that we have not explicitly included the sum overlayers, but it is implicit.
Since the problem of interest does not satisfy either of the sufficient conditions forstability, we now want to examine the stability more directly. If we look for disturbances@@of the form @t⌘ = c @x
⌘ in (2.3), we have

U (y)⌘(y) +Z

dy 0 Gk (yy 0 )q y (y 0 )⌘(y 0 ) = c⌘(y)(2.4)

Discretizing the integral in y leads to a standard matrix eigenvalue equation. The simplest
such comes from replacing q y by delta-functions at the discretization points yi : q y =qi (y yi ). Then
Mij ⌘j = c⌘i , Mij = Ui + Gk (yi yj )qj(2.5)
delta=1
delta=0.2
delta=0.2,w=1,v=0bt
delta=0.001,w=1
delta=0.001,w=3
delta=0.2,w=1
delta=0.2,w=2
delta=0.2,w=3
delta=0.2,deep flow zero5

delta=0.001,w=2
delta=0.2,w=3

2.3 — Appendix A. Stability Program% u,y,l: jet velocities, front locations, layer indicesw=3;v=0.5;
u=[0,1,0,0,v,0];y=[-w,0,w,-w,0,w];l=[1,1,1,2,2,2];% gamma,f: 1/R d, modal amp. each mode[columns] & each layer[rows]
del=0.2;
gamma=[0,1];
f=[1,1/sqrt(del);1,-sqrt(del)];% end of required input
yv=abs(ones(length(y),1)*y-y’*ones(1,length(y)));% calculate the delta’s (PV jumps) from u. In some cases, g is
% singular and you need to specify delta and perhaps then calculate u
g=jetg(gamma,f,l,yv,0);delta=-diag(g\u’)
ud=diag(u);% calculate the dispersion relationship
os=[];
k=0.02:0.02:2;
for kk=kg=jetg(gamma,f,l,yv,kk);
mm=ud+g*delta;
om=eig(mm); os=[os;om’];endplot(k,real(os),’+’,k,imag(os),’o’)------------- jetg.m -----------function gs=jetg(gamma,f,l,yv,k)% function gs=jetg(gamma,f,l,yv,k)%% Greens function for jets
% gamma: row vector of inv. def. rad.;f: vertical eigenmode% amplitudes in each layer.
l: layer indices% yv: |y i-y j|; k: wavenumber
gs=zeros(size(yv)*[1;0],size(yv)*[0;1]);for m=1:size(gamma)*[0;1];
ff=f(l,m)*f(l,m)’;ak=sqrt(gamma(m)^2+k^2);
if ak==0
gs=gs+ff.*yv/2.0;
else
gs=gs-ff.*exp(-ak*yv)/2/ak;
end
end
6

3 —Contour dynamics
Point vortices provide one useful “toy model” for understanding the dynamics. At
the next level up, we can consider the PV field as consist of patches of uniform PV. This
gives a good method for examining nonlinear e↵ects on the growth of perturbations. (Fora smooth T field, it may not be so sensible.)Q = qi H y

Yi (x, t),Yi = yi + ⌘i (x, t)with the mean of ⌘i zero.
The flow can be calculated by integrating around the boundaries of the various patches,as can the streamfunction and the Hamiltonian.
Z 1 Z
1
0
0
H=
hi qi qj Ii Ij Gij (x x ) , Ii =
dy dx
2
Yi
The notation has been simplified here: hi is for the layer in which interface i is embedded
and the indices on G correspond to 1 or 2 depending on the layers holding i and j.
We now need the form for the Poisson bracket. We use
Z
Z
Z
F
F
F = dx dy
qi (Yi y)r⌘i = dx
⌘i
q
⌘i
)

F
=
q

1 F
qi ⌘i

The Poisson bracket becomesZZ
1
1 A 1 B
{A, B} =dx dy,qi H(yhi
q i ⌘i q i ⌘i

ZZ
1
A B
=
dx dy
, H(y Yi )q i hi
⌘i ⌘i
Z
1
A @ B
=
dx
q i hi⌘i @x ⌘iYi )For reference, the bracket for a single curve parameterized by X( ), Y ( ) is1
{A, B} =
q0

I

d

✓

Y AX X AY
X2 + Y 2

◆

@
@

✓

Y BX X BY
X2 + Y 2◆(subscripts for functional derivatives also). For single-valued contours, using
have AX = ⌘x A⌘ , AY = A⌘ , X = 1, Y = ⌘x , and this turns into
1
{A, B} =q0

Z

as above.
7

dx A⌘

@
B⌘
@x

= x, weWe can find the evolution for the Fourier amplitudes
Z
1⌘i,k =
dx e ıkx ⌘i
2⇡
Z
Z
@
1
H
ık
H
ıkx @
⌘i,k =
dx e=
dx e ıkx
@t2⇡hi qi
@x ⌘i
⇡qi
⌘i
If we use Y i = y + ⌘k exp(ıkx), (suppressing the index for the particular contour) then
H=

Z

H
@H
dx
⌘k eıkx =
⌘n
⌘i
@⌘n

)

Z

dx

H ıkx
@H
e
=
⌘i
@⌘k⇤

and we arrive at the dynamical equation for the Fourier amplitudes@
ık@H⌘i,k =⇤@t2⇡hi qi @⌘i,k(H

f ourier)

If we can write an expansion of H in the modal amplitudes ⌘k , the equation above
will give us information about nonlinear e↵ects on growth. If mode k (denoted by ⌘) isunstable, we expect to find mode 2k (denoted by ⇠) will also be excited. We will notdevelop amplitude in mode 0 because of area conservation. To approximate, we expand H
around a straight state using
H ' H2 [⌘i , ⌘j ] + H3 [⌘i , ⌘j , ⌘m ] + H4 [⌘i , ⌘j , ⌘m , ⌘n ] . . .
– a sum of quadratic, cubic, quartic ... functionals of the interface displacements. We
truncate to this order. If we keep the k (⌘) and 2k (⇠) terms in the Fourier expansion, wehave
H2 = Aij ⌘i⇤ ⌘j + Âij ⇠i⇤ ⇠j
H3 =Bijk (⌘i ⌘j ⇠k⇤ + ⌘i⇤ ⌘j⇤ ⇠k )H4 =Cijkl ⌘i⇤ ⌘j⇤ ⌘k ⌘l + . . .

k
The coefficients are related to the values of the Greens function matrix Gij=
yj , zi , zj ).

8R

dx0 cos(kx0 )G(x0 , yi

The lowest order (H2 , ⌘) gives the same linear equation for the stability problem. We
take⌘i = A(t)ei@
As conventional, we use the quasi-equilibrium form in the @t⇠ equation to find ⇠ ⇠ ⌘⌘ ⇤ .
In the fourth order, we end up with

@2
A=
@t2

2

A + N A|A|2

We build a jet using 6 contours, three in the upper layer and three in the lower. The
jumps are chosen so that the zonal velocity is zero outside the outer contours. We canadjust the deep centerline velocity and the width
1. The long sinuous waves are unstable with c ⇠ ık 1/2 , ! ⇠ ık 3/2 , even for very thicklower layers (e.g., = 0.1).
2. When the deep velocity is prograde and the jet is sufficiently narrow, there is a second,
unstable, long wave mode with ci ! const as k ! 0. In the barotropic case, v =1, Squires’ theorem shows that a baroclinic
p mode of wave number k will be like abarotropic mode with wavenumber µ = 1 + k 2 . Thus ci will be non-zero at k = 0
for baroclinic perturbations if the barotropic mode with k = 1 is unstable. Therefore,
for W < 1.8327, the very long waves k ! 0 will be unstable to a baroclinic mode
disturbance, and this mode will have a larger growth rate than the “meandering”
mode discussed in 1.
3. We can separate out the varicose modes explicitly by imposing antisymmetry upon
the perturbations and working with a reduced 2 ⇥ 2 matrix Although the geometryof the varicose mode perturbations is similar to Pedlosky’s “heton cloud” case, thevaricose modes here are stable since the zonal flow vanishes at the y = L interfaces.
4. A narrow jet has a barotropic type of instability even in the limit of an infinitely
deep lower layer, but the growth rates decrease as the jet gets wider. In the case offinite lower layer depth, the growth rates level out and correspond to the baroclinic
instability of the center fronts.
5. Asymmetric jets have rather similar dispersion characteristics to the symmetric profiles.
6 Generally, we find that nonlinearity enhances the instability for widths less than 2.3(in units of the deformation radius; recall that the “width” W is the distance to thezero in the velocity). For wider jets, the nonlinearity leads to equilibration as occurs
for the baroclinic front Flierl (1988). For fairly barotropic jets, the nonlinearity isagain destabilizing; in this case, numerical studies of the barotropic jet suggests thatvortex street formation will occur.
The case of the “heton cloud” with just one front in each layer is especially interesting.
The linear matrix when h1 = h2 = 0.5 gives Green’s functions
G11 = G22 =1 k|y y0 |
e
4k

0
1
e K|y y | , G12 = G21 =4K

9

0
1 k|y y0 |1e+e K|y y |4k
4K

with K 2 = k 2 +2. We takeu = ±U e|y|corresponding to qi = ±2U . The resulting matrix
✓
◆
✓
◆✓
1
1 0
k K k K
2 U
M=
+
0
1
k
K
k
K
0
4kK
leading to a dispersion relationc = ±U

r

(k

Z

Z

dx0 G(⇢)

0
2 U

◆

)(K
)
kK
which becomes unstable when k < . Short waves are trapped near the front in either theupper or lower layer, depending on the branch of the dispersion relation, and propagate
non-dispersively with the speed at the front in the appropriate layer. Long waves are
unstable with a growth rate = =(!) / k 3/2 , in agreement with the prediction of thin jet
theory (Flierl and Robinson, 1984, Howard and Drazin, 19xx). We would expect the thin
jet theory to be applicable in this case since the momentum transport of the basic jet isfinite.
The nonlinear results indicate this is a supercritical bifurcation, although the nonlinear
term changes sign when k/ becomes smaller. When the perturbation is no longer bounded
N > 0, the symmetry of the system results in a wave which grows to very large amplitudeuntil the resulting N-S jets interact and break up the feature.
To investigate, this, we used a standard QG model with k = 1 as the longest wave inthe domain. We also calculated steady states using simulated annealing.
4 —Nonlinear coefficient
H=

1
qi qj
2

Let

H=
=

1
qi qj
2
1
qi qj
2

Z

dx
Ai

dx
A

Z i
A

Z i

dx

Z

Aj

,

⇢ = |xr2 (⇢) = G(⇢)
A

dx0 r02

A

ds0 r0 · n̂0

I j
I j

1
qi qj
dx
ds0 ⇢ r0 ⇢ · n̂02AA
Z i I j1
= qi qjdx
ds0 ⇢ r · (⇢n̂0 ) since
2
A
A
Z i
I j
1
= qi qjdx
ds0 r · ( n̂0 )2AA
I i I j1
= qi qjds
ds0 n̂ · n̂0
2
Ai
Aj
=

x0 |

10

r0 ⇢ =r⇢

For contours Yi = yi + ⌘i (x)n̂ =⇢2 = (x1⌘xp,p
1 + ⌘x2
1 + ⌘x2

x0 )2 + yi + ⌘i (x)yj!
⌘j (x0 )2Taylor expand in ⌘ to some order (⌘ 4 ) and Fourier expand (k, 2k).
dTurn into equations for dt⌘i (k) and ⌘i (2k) and make the usual amplitude expansion
neat the critical point. Or use⌘ˆi (k) = a(t)ei +

1 @
afi
ık @t

,

⌘ˆi (2k) = a2 gi@
in A = H + cr P and set @t
A = 0. Here ei is the eigenvector at the critical point k0 , fi isthe correction when k is slightly less than k0 , and gi is the forced 2k vector.

@2
a=
@t2

2

a + N a|a|2

4.1 — Numerical results
For the heton cloud, max(Y1 ) is:
heton k=1/1.05
k=1/1.1
sa
0.98
1.02
1.04
1.05
1.06
1.08
1.1

0.025
0.097266
0.73157
1.27101.59901.99953.4278–0.05
1.87380.75829
1.2812
1.60492.00673.4361–

4.2 — Conclusions: numerical
1) Nonlinearly stable regions exist but seem to be very narrow.
2) Most disturbances will break via eddy formation for the more baroclinic jet or street
formation for the more barotropic cases.

11

4.3 — Pulses
Although almost all stability computations are for zonal flow and normal modes or atbest wavenumber k, most atmospheric and oceanic disturbances are much more localizedin space and time.
If we consider the evolution of a localized initial disturbance in 1D, using the methodof Farrell (19xx), who shows that the perturbation behaves likeZ⌘(x, t) = dkA(k)eıkx ı⌦(k)t⌘(x, t) = lim

t!1

Z

dkA(k)eı(kU⌦(k))tThe steps are:• Deform the integration contour in the complex k plane to pass through the saddle
point ks such that U = d⌦/dk. Then
✓
◆
Z
d2 ⌦
ı(ks U ⌦(ks ))t002⌘(x, t) ! A(ks )e
dk expıt 2 kdk ks
The envelope will be growing at a rate
= = ⌦(ks )U ks• The saddle point wavenumber is determined by
d
(!
dk

x
k )=0
t

in the complex k plane. The required derivative
d!
dc
=c+k
dk
dk
can be found using the matrix eigenvalue problem (4.5). If we di↵erentiate this withrespect to k, we find




d
d
d
d
Mij ⌘j + Mij⌘j =c ⌘i + c
⌘i
dk
dk
dk
dk
Multiplying by the left eigenvector ⌘i+ of Mij givesdd+[ c ] = ⌘i
Mij ⌘j /⌘i+ ⌘jdkdk
We then search out the zeros of d!i /dk in the complex k space beginning with theone known to be on the real k axis, The envelope shape as a function of x/t = d!r /dk
12

for the two values we’ve been using. For both values of , absolute instability canoccur, though the most rapidly growing part of the envelope moves downstream at a
speed of about 12% of the maximum upper layer jet velocity.
examplesThe pulse asymptotics is a bit more complex for the six front jets, since there may be
several unstable modes. However, the picture which emerges is fairly simple: the jets with
prograde lower layer flow are convectively unstable, with the peak of the envelope growing(as it must) at the maximum temporal growth rate and moving downstream at about 23%of the maximum jet speed. When the lower layer flows are retrograde at the center, thejet has an absolute instability, though the peak of the envelope still moves (more slowly)
downstream.
4.4 — Continuum modes
One criticism of contour dynamics models for jets is that they lack the continuous
spectrum. To compensate for this lack, however, they have more neutral normal modes.
As the number of contours becomes large, the set of the frequencies of these modes becomes dense. If we consider an initial condition with many of these modes excited andappropriately out of phase, they can very gradually come into phase and then shift out
again. The recurrence (or near recurrence) time can be very long. Therefore transient
growth and decay of an initial disturbance is actually similar: there is a long period withthe amplitude proportional to t 2 as obtained for the continuous jet by Brown and Stewartson (19xx). Notice that a large number of contours (xx) is required to give the longrecurrence time.
4.5 — Zero lower layer flow
The basic state we assume in this section also has a continuous PV distribution, witha single PV front in the upper layer and no basic state flow whatsoever in the lower layer.
This basic state can be approximated by a large number of small amplitude fronts in thelower layer to mimic the no-flow condition there. We have also derived the dispersion
relation from a Frobenius expansion of the integral eigenvalue problem (4.4). The results
are identical.
The growth rates show a single unstable mode corresponding to a baroclinic instability.
The maximum growth rate occurs at a scale x.x and decreases as the lower layer becomesdeeper, being proportional to . These results are the same as the standard two–layer
baroclinic instability problem for a y–independent zonal flow. The waves on the upper
layer PV contour with a positive northward jump reinforce the perturbations in the lower
layer negative PV gradient regions.

13

5 —Ring formation
The heton model and the upper-layer front both have the essential elements for ringformation. In essence, the upper layer positive PV jump interacts with the negative deep
PV jump or gradient so that the perturbations amplify each other. The propagation speeds
need to match so that the phase o↵set is maintained. As the bump elongates, the anomaly
begins to move the south-eastern side backwards until it pinches o↵. The meander needs
to be long enough to allow this to occur while retaining the water properties within thering.
sketch
0-20
30-50
60-80
90-100
6 —Diagnostics of baroclinic amplification
The equations for the upper layer@
q1 = [q1 ,
@t

1] =



q1 ,

Z

G11 (x00

x )q1 (x ) + q1 ,Z

G12 (xx0 )q2 (x0 )show that we can calculate separately the motion tendency from vortex interactions within
the same layer and from the other layer. This “piecewise inversion” allows us to diagnose
baroclinic amplification: the velocities induced on the central upper layer PV contour by
the lower layer flows tend to elongate it, and likewise the upper layer influence on the lower
layer is amplifying the displacement.
from thin jetfrom thin jetfrom westrex

14

References[1] Stommel, H., 1948: The westward intensification of wind-driven ocean currents, Transactions, American Geophysical Union., 29, 202-206. (p1)[2] Flierl, G.R., 1999: Thin jet and contour dynamics models of Gulf Stream meandering, Dyn. Atmos.
Oceans., 29, 189-215. (p2)[3] Cushman-Roisin, B., Pratt, L.J. and Ralph, E., 1993: A general theory for equivalent barotropic
thin jets, J. Phys. Oceanogr., 23, 91-103. (p3)[4] Flierl, G.R., 1988: On the instability of geostrophic vortices, J. Fluid Mech., 197, 349388. (p9)[5] Flierl, G.R., and Robinson, A.R., 1984: On the time dependent meandering of a thin jet, J. Phys.
Oceanogr., 14, 412423. (p10)[6] Howard, L.N., and Drazin, P.G., 1964: On instability of parallel flow of inviscid fluid in a rotating
system with variable Coriolis parameter, J. Math. Phys., 43, 83-99. (p10)[7] Farrell, B.F., 1982: Pulse asymptotics of the Charney baroclinic instability problem, J. Atmos. Sci.,
39, 507-517 (p12)[8] Brown, S.N., and Stewartson, K., 1980: On the algebraic decay of disturbances in a stratified shear
flow, J. Fluid Mech., 100, 811-816. (p13)