Lesson 10 – Barotropic and Baroclinic instability
The common approach for analyzing the instability of linear geophysical flow
requires examining the long term time dependence of the solution. In particular, for a
stream function  x, y, t  of a linear system, the normal mode solution can be
represented as
    y e i kx t     y e ik  x ct 
^
^
where we have the used phase speed relationship c 

in the second equality. We can
k
assess the stability of the solution based on form phase speed. If we consider a complex
phase speed and break up c into real and imaginary components, c  cr  ic i , then we can
deduce the time dependent nature of the solution based on the magnitude and sign of ci .
The three possibilities are
ci  0 Periodic solution
  0 as t  
ci  0 Unstab le solution.    as t  
ci  0 Stable solution.
Let us use the above analysis for some basic barotropic and baroclinic geophysical
systems.
1. Barotropic instability: Rayleigh’s inflection point criteria
Recall the general equation for conservation of vorticity from the previous chapter
D
1
   f u   v  2   p   2
Dt



(1)
Let us simplify equation (1) by assuming the system is 2-D, inviscid, barotropic,
incompressible and a apply a background shear flow of the form
^
u  U  yi
Background flows of this form are representative of those found in a boundary layer. We
will consider our flow to be in a horizontal medium with boundaries at y1 and y 2 .
Equation 1 then simplifies to the form:
D
  f   0
Dt
Where   
(2)
d
U  y .
dy
Now let us consider a perturbation to this background current:
u  U  y   u ' i  v ' j
^
^
(3)
Since the system is incompressible and 2-D, we can represent the perturbed velocity field
as a stream function so the total vertical vorticity is then of the form:
     ' 
d
U  y    2
dy
The linear form of equation (1) then simplifies to

 2

d 2U  
   U  2     2 
0
t
x
dy  x

(4)
Substitute a normal mode solution into equation (4):
    y e ik  x ct 
^
^
Where in this case,  may be complex. The resulting equation is

2


U^
  0
dy 2 
U  c  d 2  k 2      d
 dy

^

2
Dividing through by U  c , we obtain
 d2
^
1 
d 2U  ^
 2  k 2  
   2   0
U c
dy 
 dy

^
Multiplying by the complex conjugate of the stream function amplitude,  * , and
integrating over the boundaries y1 and y 2 , we obtain

y2
y1
^
 ^

^ ^ 
y2
 d d *
1 
d 2U  ^ ^ * 
2
*
 dy dy  k    dy  y1 U  c    dy 2   dy  0








(5)
Integration by parts has been used for the first term with the no slip condition,
  y1     y 2   0 . The first term in equation (5) is always real but the second term can
^
^
be either real or complex based on c  cr  ic i . If there is a complex component of the
phase speed, then we know the flow is unstable. The imaginary portion of equation (5)
gives the equation
ci 
y2
y1
1
U  c  U  c *



d 2U  ^ ^ * 
   2   dy  0
dy 


(6)
There are two possibilities of equation (6) being equal to 0. The first is that the flow is
periodic with ci  0 . The other possibility is that the flow is unstable with

y2
y1

d 2U  ^ ^ * 
   2   dy  0 . The only way this integral can equal zero is
dy 


2
d U
 2  changes sign somewhere within the fluid column or boundary
dy 
1
U  c  U  c *


if the term  





d 2U  d
f
layer. Thus a necessary requirement for instability is that    2  
dy  dy

changes sign somewhere within the fluid column. This result is called the Rayleigh’s
inflection point criterion for instability.
Recall that one major assumption for the above analysis is that the flow is
barotropic. For atmospheric flows, this result is applicable for the tropic regions as well
as fully mature frontal systems. To examine the instability of evolving frontal systems,
we must consider Baroclinic instability.
2. Baroclinic instability
Weather maps for the midlatitudes frequently show horizontal wavy variations of
temperature and pressure superimposed on westerly mean flows. Similar variations are
found in the ocean on westerly currents, such as the northern portion of the Gulf Stream
as it moves eastward across the north Atlantic. Unlike under barotropic conditions,
where gradients of density and pressure are parallel, these waves occur juxtaposed with
different gradients of density and pressure.
For geophysical flows, temperature varies as a function of latitude due to the
effects of solar heating. As temperature decrease towards the poles, density will increase
and lead to the formation of a baroclinic system. Density also decreases upward.
According to the thermal wind relation,
f  VT  
g
T ,
T
a westerly flow (such as the jet stream or northern Gulf Stream) superimposed on this
temperature gradient must have a velocity that increases with height.
A system where density surfaces slope with height, such as a front, has more
potential energy than a system with horizontal density surfaces. If a flow (such as the jet
stream) can convert this potential energy to kinetic energy by flattening out the sloping
density surfaces, it can acquire “baroclinic instability.” Baroclinic instability theory was
developed by Bjerknes et al. in the 1940s and is still considered one of the “major
triumphs” of fluid mechanics.
i. Basic setup
Consider a base state where density is stably stratified in the vertical (  decreases with
g 
, be uniform and
 o z


increasing northward at a constant rate of
. From thermal wind, since
=constant,
y
y
the vertical shear of the base flow U  z  must also be constant. We neglect variations in
height). Let the buoyancy frequency N, where N 2  
Coriolis (  is neglected).
The resulting governing equations are
u
u
u
u
1 p
u
 v  w  fv  
t
x
y
z
0 x
v
v
v
v
1 p
 u  v  w  fu  
t
x
y
z
 0 y
p
 g
z
u v w
 
0
x y z
D  




u
v
w
0
Dt
t
x
y
z
0
(7)
The total flow is composed of a westerly jet U  z  that is in geostrophic equilibrium with
base density   y, z  plus perturbations:
u  U  z   u   x, y , z 
v  v  x, y , z 
w  w  x, y, z 
(8)
    y , z      x, y , z 
p  p  y , z   p   x, y , z 
The base flow is in geostrophic and hydrostatic balance, fU  
p
1 p
and 0     g
z
0 y
U0 z
. By cross-differentiating equation (7) and
H
subtracting the y-derivative of the first from the x-derivative of the second, we get a
vorticity equation (similar to the last chapter but without the beta-effect):
. If U=0 at the surface, then U 




w
u
v
w
   f 
0
t
x
y
z
z
Decomposing using equation (8) and neglecting the nonlinear terms gives a perturbation
vorticity equation:
 
 
w
U
f
0
t
x
z
(9)
Perturbation vorticity can be expressed in terms of perturbation pressure by assuming the
1 p
perturbation velocity components are in geostrophic balance, u   
and
0 f y
1 p
v  
. Thus perturbation vorticity (after cross-differentiating) becomes
0 f x
1

 2H p . Now all that remains is to express w in equation (9) in terms of
0 f
perturbation pressure. The steps are skipped (left as an exercise for the reader?) and the
resulting equation that governs (nearly) geostrophic perturbations of a westerly flow is:
  2
f 2  2 p 



U

p

 0.

 H
x  
N 2 z 2 
 t
ii. Baroclinic interpretation and instability criterion
The underlying point for this analysis is that, since the density field is now a function of
1
pressure and temperature, the term 2   p  in equation (1) cannot be neglected.

Further, due to the more complex stratification of the density field, we must consider
vertical variations and thus requiring analysis of all three dimension of the system as
opposed to the simple 2-D problem in the previous section. We will still consider the
flow to be inviscid and examine perturbations of a uniform background shear flow. We
will neglect the analysis of the equations of motion and look directly at the solution.
The linear solution is usually represented as waves in the pressure field of the
form
^
p  p  z  ei kx ly t 
(10)
 
 
H 
H 
N2 2 2
2
Where pz   A cosh   z     B sinh    z    and   2 k  l with
2 
2 
f
 
 
g 
.
N2  
 o z

^

The pressure field is related to the density through the hydrostatic approximation

1 p
g z
d 2
 0.
With some work, we can verify that this solution is barotropic provided that
dz 2
Application of the above solution to the underlying equations and boundary condition (no
normal flow to the system) leads to the following requirement on the phase speed
c
U0 U0

2 H
H
 H    H
 H  
 tanh 
 coth 
 


 2   2
 2 
 2
(11)
Unstable solutions will occur if the radical is imaginary. Examining the nature of
the phase speed as a function of H , we observe that the radical will be imaginary when
H
 H 
 coth 
 . The critical point of instability occurs when the radical is zero and
2
 2 
cH
 H 
 coth  c  which leads to the result that solution are unstable for
2
 2 
H  2.4
(12)


N2 2 2
k  l so we can relate the point of stability to the length scales
f2
HN 2.4
of the system.
where K is the magnitude of the horizontal wavenumber.

f
K
Using the definition of the wavenumber, the inequality of equation (11) can be related to
the wavelength of the system leading to the result
Recall that  2 
  2.6
HN
f
(13)
Equation (12) shows that instability will occur only for relatively long waves in the
atmosphere or ocean. Standard values for H,N and f indicate that   2700km in the
atmosphere and   133km in the ocean with maximum instability occurring around
  4000km in the atmosphere and   200km in the ocean.
The above criteria for instability means that longwave westerly jets in the
atmosphere such as Rossby waves are prone to the spontaneous formation of
disturbances. From the initial supposition of that the system is baroclinic implies that the
intensification of small perturbations is due to extracting potential energy from the
sloping mean density surfaces of the atmosphere in an attempt to make these density
surfaces more horizontal. This also leads to the transport of heat from lower latitudes to
higher latitudes as the lower density air is transported northward.