AOS611, Ch.5, Z.Liu, 03/09/16
1
Chapter 5. Stratified Quasi-Geostrophic Rossby Waves
Sec. 5.1: Quasi-Geostrophic Equation in Stratified Fluid
1. Nondimensional Equations
We’ll use the oceanic equations (4.1.15)
 t u  u   u  fv  
 t v  u   v  fu 
1
 z p  
1
0
1
0
 x p 
 y p 
1
0
1
0
Fx
Fy
g '
(5.1.1)
0
0
 xu   y v   z w  0
 t    u      S 0
We choose the scales as
u, v ~ U,
x,y ~ L,
w~U
D
,
L
z ~ D, t ~
L
U
and denote the Rossby number as:
=
U
fo L
The density and pressure can be written as
’(x,y,z,t) (= Ocean-0) = S(z)+(x,y,z,t),
(5.1.2a)
P(x,y,z,t) (=POcean +0gz ) =P(z)+p(x,y,z,t),
(5.1.2b)
where Ocean and POcean are the total density and pressure, 0 is the average density of the
ocean, and
dP
  g S (z )
dz
represents the static part associated with the mean stratification and
dp
  g
dz
AOS611, Ch.5, Z.Liu, 03/09/16
2
is the dynamic pressure associated with the horizontal density variations. For large scale
flows with small Rossby number, similar to the shallow water case in section 2.1, the
dynamic pressure is also scaled as
p ~ foLU0
(5.1.3)
such that the pressure gradient force is comparable with the Coriolis force; the
source/sink is assumed weak, with
-plane approximation
L
fo
1
F  G , and G  O(1); we also use the local
f oU 0
  , where r~ 1. Then, we can write the two momentum
equations in dimensionless variables (subscripted with “*”) as:
u
p
 { *  (u *  * )u*  y*v* }  v*   *  Gx
t*
x*
v
p
 { *  (u *  * )v*  y*u* }  u*   *  G y
t*
y*
The continuity equation is
u* v* w*


0
 x*  y *  z *
The scale of  can be derived from the hydrostatic equation
p
  g .
z
Notice (5.1.3), we have
~
f LU
L
p
0 ~ 0 ( )2   
~ o
gD
LD
gD
2
where LD 
(5.1.4)
gD
is the external deformation radius. We can therefore define  as
fo 2
 ~ *
where * ~ O(1). The hydrostatic equation can be written as
p*
  *
z*
With all these scalings, the thermodynamic equation becomes
(5.1.4a)
AOS611, Ch.5, Z.Liu, 03/09/16
3
U *
UD d S
{
 (u *   * ) * } 
w*
 S0 .
L t*
L
dz
Thus,
d
S

UD
 { *  (u *   * ) * } 
w* S  o
t*
f o L
dz
fo
d
S

D
 { *  (u *   * ) * } 
w* S  o
t*

dz
fo
The scale of
(5.1.5)
d S
can be derived at the first order from the adiabatic condition. We can
dz
show from (5.1.5) that
d S 1  '
 '


dz  z
z
Indeed, for adiabatic flows, the solution is always along isopycnals
d '
 0. Therefore,
dt
u x  '  w z  ' . But, we know that QG equations require at the first order non-divergent:
w
D

u
L
Therefore, the slope of the isopycnal surface must be  
D
L
 '  ' w
D
 
x z u
L
Since
 '   D
 S

~
, and therefore
 0 , we have
x x z L
x
  (  S   )

z
z

z
d S

dz
(5.1.6)
This implies that any horizontal variation of the static stability (since ’ is the horizontal
variation part) must be small. This is a weak assumption in many cases. (especially in
marine eddies .. )
AOS611, Ch.5, Z.Liu, 03/09/16
4
z  zS
ρ=const
z <<zS
Using (5.1.4) and (5.1.6), we have the scale
d S


* ( z )
dz
D
where * ( z )  O(1) .
The thermodynamic equation becomes

 { *  (u ** ) * }  w*  S
t*
where
S 
S0
and S  O (1) such that at the leading order the flow is adiabatic.
fo
Note 1: If we define a buoyancy frequency N(z)
N2  
g d S
 0 dz
we have
d S
2
N 2 0 / g
 LI 
O (1)  * ( z )  dz 

 
 / D  0 L2f 02 / gD / D  L 
where LI2=(ND/f)2 is the interior deformation radius.
Therefore,
d

 S
z
dz

 L

 LI
2

  1

This requires the scale to be not too large, similar to the homogeneous case.
AOS611, Ch.5, Z.Liu, 03/09/16
5
The complete set of dimensionless equations are therefore (drop the subscript “*”):
p
u
  {  (u  )u  ryv  G x }
x
t
p
v
u 
  {  (u  )v  ryu  G y }
y
t

 w*   {  (u  )   Q}
t
p
 
z
u v w
 
0
x y z
v
(5.1.7)
Similar to the shallow water case in Section 2.1, we solve this set of equations by
expanding variables as powers of ε:
u  uo  u1  O ( 2 )
v  vo  v1  O ( 2 )
p  po  p1  O ( 2 ) .
(5.1.8)
   o  1  O ( 2 )
w  wo  w1  O ( 2 )
2. O(1) Equation and Dynamics
Substitute (5.1.8) into (5.1.7), at the leading order, we have
po
x
p
uo   o
y
vo 
wo  0
(5.1.9a-e)
po
  o
z
uo vo wo


0
x y
z
As in the shallow water case, (a), (b), (c) can be used to derive (e). Thus, there are only 4
independent equations, but with 5 unknowns. This is the “Geostrophy degeneracy”.
To better understand the O(1) dynamics, we write (5.1.9a,b) in dimensional form as the
geostrophic balance:
AOS611, Ch.5, Z.Liu, 03/09/16
6
p
  x
f o  o x
1 p
ug  
  y
f o  o y
vg 
where  
p
o fo
1
(5.1.10)
is the geostrophic stream function. The hydrostatic balance (5.1.9c)
can be written as
f 

 o
0
g z
(5.1.11)
Differentiate (5.1.10) with respect to z and use (5.1.11), we have
v g  2


z xz
u g
 2


z
yz

f o  0 x
g
g

(5.1.12)
f o  0 y
This is the thermal wind relation, a direct result of geostrophy and hydrostatic balance.
This relation has been used frequently to infer ocean currents from the density field, i.e.
the so called “dynamic method”.
Note 2: It also represents a balance between the baroclinic vorticity generation and the
title of planetary vorticity in y- and x- directions. Indeed, the general vorticity equation is:
dξ a
  p
 (ξ a  )u  ξ a   u 
dt
2
For large scale ε<<1, we have ξ a  2Ω  O( ) . In addition, zyP<<yzP, which is
equivalent to the Boussinesq approximation in the ocean. The x and y component of the
vorticity equation can therefore be written as
 y  z p   z  y p
 y  z p
d ax
 2 z u 

2


u

z
dt
2
2
day
dt
 2z v 
z x p  x z p
  p
 2z v  x 2 z
2


AOS611, Ch.5, Z.Liu, 03/09/16
7
Notice hydrostatic balance: z p   g , in the steady state
d ax d ay

 0 , we have the
dt
dt
thermal wind relationship. On the RHS in the two equations above, the first term is the
tilting term while the second term is the baroclinic term. The balance can be seen
schematically as follows:

zu>0
z
x>0 due to tilting
y
x
denser
lighter
A westerly wind shear z u  0 generates positive vorticity in the x direction
 z u  0   x  0 . This vorticity is balanced by the opposite rotation that is forced by
the northward density gradient  y   0,  z p  0   y  z p  0 .
Note 3: Taylor - Proudman Theorem
For large scale low frequency processes, we have ξ a  2Ω, and  t   . The vorticity
equation, assuming incompressiblity, is
(2Ω  )u 
  p
2
If furthermore, the fluid is barotropic   p  0 , we have
(Ω  )u  0
or assuming Ω  k , this is
 z u  0
AOS611, Ch.5, Z.Liu, 03/09/16
8
or there is no shear of velocity in the vertical direction.
 zu   zv   z w  0
The water column therefore behaves like a column of solid body.
Water column
Taylor Column
Solid body
Even in stratified case, this still shows the tendency of rotation to constraint fluid motion
variation along . In other words, rotation tends to couple the flow in the direction of .
Thus, rotation produces a “stiffening” effect that tends to align the vortex tube in the
direction of rotation. This is also why in a layered model we can assume no shear for
large scales GFD processes.
3. O() Equation and QGPV Equation
At the next order, we have the equations
v1 
p1 D*g

u o  ryv o  G x
x
Dt
 u1 
p1 D*g

vo  ryu o  G y
y
Dt
 w1* 
D*g
Dt
o  Q
(5.1.13)
 p1
 1  0
z
u1 v1 w1


0
x y z
where
D*g
Dt




. (This becomes a horizontal total derivative because
 uo
 vo
t
x
y
wo  0 ). The vorticity equation is therefore:
AOS611, Ch.5, Z.Liu, 03/09/16
9
D*g vo uo
G y Gx
u1 v1


(

)  rvo 

x y
dt x y
x
y
Here, we used:
v u
u v v u
 D*g
 D*g
(
vo )  (
uo )  D*gt ( o  o )  ( o  o )( o  o ) .
x y
x y x y
x dt
y dt
Thus,
D*gt (
vo u o
w

 ry )  1  curlG
x y
z
(5.1.14)
As in the shallow water case, the evolution of the O(1) variables uo , vo are determined
by the O(ε) variables. Since
w1  

1
Q
Q
D*gt  o 
  D*gt ( o ) 
*
*
*
*
 z w1   Dgt [
(5.1.15)
 0

Q
( )]   ( )
z *
z *
Thus, we have the nondimensional QG P.V. equation:
D*gt {ry 
vo u o   o
Q

 ( )}  curlG   z ( )
x y z *
*
(5.1.16)
or in the dimensional form
Dg q  Sq
(5.1.17a)
where
q  f o  y     z (
Sq 
1

fo 
d s
k F   z (
)
(5.1.17b)
dz
f oQ
d s
)
(5.1.17c)
dz
   x v g   y u g   2 H
(5.1.17d)
D g   t  u g  x  v g  y   t  J ( , )
(5.1.17e)
 0 f o 
,
g z
(5.1.18)
Since now
 
AOS611, Ch.5, Z.Liu, 03/09/16
N2  
10
g d s
,
 0 dz
(5.1.19)
we have

f 

 o2
d s
N z
dz
The dimensional QGPV equation can be written in terms of ψ as:
 t q  J ( , q)  S q
(5.1.20a)
f 
q  f o   y   H   z ( o 2
)
N z
2
2
(5.1.20b)
For unforced, adiabatic flow, q is conserved along the geostrophic flow (which is
different from the original 3-D flow!)
AOS611, Ch.5, Z.Liu, 03/09/16
11
4. The Atmosphere Case
The QGPV equation in the atmosphere can be derived parallel to the oceanic equation.
Typical scales are chosen as
u, v ~ U,
x, y ~ L, w ~ U
the Rossby number is =
D
,
L
z ~ D, t ~
L
,
U
U
, the potential temperature is written as
fo L
(x,y,z,t)=(z)+'(x,y,z,t)
and the geostrophic potential high as
(x,y,z,t)=(z)+'(x,y,z,t)
where
d
(z)
g
dz
s
d '
'
g
dz
s
For large scale flows, the geopotential height anomaly is scaled as
 ~ foLU
such that the pressure gradient is comparable to the Coriolis force at the first order.
The source/sink is also weak such that
1
F   ,
f oU
where |G|O(1).
The local -plane is adopted as
L
fo
  , O(r) ~ 1.
The two momentum equations are represented in dimensionless variables as:
u

 { *  (u *  * )u*  y*v* }  v*   *  Gx
t*
x*
v

 { *  (u *  * )v*  y*u* }  u*   *  G y
t*
y*
The mass equation is
AOS611, Ch.5, Z.Liu, 03/09/16
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u* v* 1 


(pw* )  0
x* y* p z*
The scale of ‘ is estimated from the hydrostatic equation
'
'
g
z
s
This gives
‘ ~

gD
where LD 2 
s ~
fo L U
L
 s ~  s ( )2   
gD
LD
gD
is the external deformation radius Thus,
fo 2
‘ ~ *
where * ~ O(1),
and the hydrostatic balance is
*
 *
z*
With these scaling, the thermodynamic equation becomes
U  *
UD d
{
 (u *   * ) * } 
w*
 Qa
L t*
L
dz
 {
 *
UD
d Q a
 (u *   * ) * } 
w*

t*
f o L
dz
fo
The scale of
d 1  '
'
d


can be anticipate to satisfy
. For adiabatic flows,
dz
dz  z
z
the solution is always along isentropics,
d
 0 . Since QG equations require
dt
w
D
D

, the slope of the isentropic surface must be  
u
L
L

x
Since
D
     L
z

 0 , we have
x
  '  ' D

~
x x
z L
and therefore
AOS611, Ch.5, Z.Liu, 03/09/16
13
 '  (   ' )

z
z
 '  ()

z
z
Thus, we are led to the scale of
d


 (z)
dz
D *
where * ( z )  O(1) .
If we define a buoyancy frequency N(z), then
N2 
g d
 s dz
The thermodynamic equation becomes

 { *  (u ** ) * }  w*  Q
t*
where
Q 
Qa
and Q  O(1) is consistent with leading order adiabatic.
f o
The complete set of dimensionless equations are now (drop *):

u
  {  (u  )u  ryv  G x }
x
t

v
u 
  {  (u  )v  ryu  G y }
y
t

 w*   {  (u  )  Q}
t

  0
z
u v 1 


( pw)  0
x y p z
v
The variables will be expanded as follows:
u  uo   u1  O( )
2
v  v o  v1  O( 2 )
  o  1  O( 2 )
   o   1  O( 2 )
w  wo   w1  O( 2 )
AOS611, Ch.5, Z.Liu, 03/09/16
14
The O(1) equations are
 o
x

uo   o
y
vo 
wo  0

 o
z
 uo  v o 1 


(pwo )  0
x y p z
In dimensional form, the geostrophic balance is:
vg 
1  '
  x
f o x
ug  
where  
or
'
1  '
  y
f o y
is the geostrophy stream function. The hydrostatic equation is
fo
  f o 
. This leads to the thermal wind relationship

s
g z
v g  2
u g
g  '
 2
g  '


,


z xz f o s x
z
yz
f o s y
At the next order, we have
v1 
1 D*g

u o  ryv o  G x
x
Dt
 u1 
1 D*g

vo  ryu o  G y
y
Dt
 w1* 
D*g
Dt
o  Q

  0
z
u1 v1 1  ( pw1 )


0
x y p z
 '
'
z
AOS611, Ch.5, Z.Liu, 03/09/16
where
D*g
Dt
D*gt (

15



. The vorticity equation can be derived as:
 uo
 vo
t
x
y

vo uo
1 

 ry ) 
( pw1 )  curlG
x y
p z
Since
w1  

1
Q
Q
D*gt o 
  D*gt ( o ) 
*
*
*
*

1
1 
1 
Q
 z ( pw1 )   Dgt [
( p 0 )] 
 (p )
p
p z *
p z
*
we have the nondimensional QG P.V. equation:
D* gt {ry 
vo uo 1  p o
1
pQ


(
)}  curlG   z (
)
x y p z *
p
*
In the dimensional form
Dg q  Sq
where
q  f o  y   
pf 
pf Q
1
1
1
 z ( o ) , Sq  k    F   z ( o a )
d
d
p

p
dz
dz
   x v g   y u g   2 H ,
Dg  t  ugx  vgy
Now, with
'
 s f o 
g d
, N2 
g z
s dz
and therefore
'
d
dz

f o 
, we have the QGPV equation for the atmosphere as:
N 2 z
 t q  J ( , q)  S q
(5.1.21a)
pf 
1
q  f o  y   H   z ( o2
).
p
N z
2
2
(5.1.21b)
AOS611, Ch.5, Z.Liu, 03/09/16
16
Sec. 5.2: Rossby Waves in Stratified Flows
1. Dispersion Relationship
As in the shallow water case, we study small perturbations linearized on a mean zonal
flow. The linearized QGPV equation is:
( t  U x )q 'v' Q y  0
where the mean and perturbation potential vorticity are

1  pf 2 o
Qy    yyU  z  2 zU  ,
p  N

1  pf o2  
q'   xx   yy   z  2

p  N z 
We have used the atmospheric equation (5.1.21) and the oceanic equation can be
recovered simply by setting p=p0. The QGPV equation can be written in the perturbation
streamfuction  as
( t  U x ){ xx   yy 

1  pf 02
 z  2  z }   xQ y  0
p N

(5.2.1)
We will assume the basic state is slowly varying, that is the wave length in y, z directions
are short relative to the scales at which U, N2 and Qy vary. (But, we don't need to assume
p(z) slowly varying!). Assuming the solution of the form
z
i 


2H
 ( x, y, z , t )  Re  ( y, z )e



where
y
z
  k ( x  ct )   l ( y ' )dy '  m( z ' )dz '
We have approximately
 xx  k 2
and
,  yy  l 2
AOS611, Ch.5, Z.Liu, 03/09/16
17
pf o2
f o2 1
1
 z ( 2  z )  2  z ( p z )
p
N
N p
z

f o2 Hz
H
e

(
e
 z )
z
N2
z
f o2
1
2
i H
  2 (m 
) e e
N
4H 2
f o2
1
  2 (m 2 
)
N
4H 2

Therefore, (5.2.1) gives the dispersion relationship as
(U  c)[k 2  l 2 
f o2
1
(m 2 
)] Qy  0 .
2
N
4H 2
That is
Qy
c  U
k2  l2 
fo2
1
(m2 
)
2
N
4H 2
(5.2.2)
for the atmosphere. For the ocean, we can set H infinitely large (incompressible), so that
c  U
Qy
(5.2.3)
f2
k  l  o2 m 2
N
2
2
The relation between the shallow water Rossby waves and the baroclinic Rossby waves
here are readily seen if we make
2
L Dm 
N2
f 2 (m 2 
(5.2.4)
1
)
4H2
The dispersion relationship can be put exactly the same form as the shallow water Rossby
wave
cU 
Qy
k 2  l 2  L2Dm
The LDm is the deformation radius for baroclinic flows with a vertical wave number m:
Since LDm ~ 1 / m , the deformation radius increases and the wave speed faster for smaller
m (or larger vertical scale), and vise versa. For typical atmospheric stratification, we have
LD1 ~ 1000 km, while for typical oceanic stratification, we have LD1 ~ 50 km.
AOS611, Ch.5, Z.Liu, 03/09/16
18
2. Group Velocity and Vertical Propagation
For a given k, l, the dispersion relationship gives the vertical wave number
N 2  Qy

1
m  2 
 k 2  l 2 
f o U  c
 4H 2
2
(5.2.5)
When the RHS > 0, m is real, and the Rossby wave propagate vertically; when the
RHS < 0, m becomes imaginary, and the waves are trapped vertically.
For propagation waves (real m), the group velocity can be calculated as
f2

1
 U  [k 2  l 2  o2 (m 2 
)]
k
N
4H 2


 2kl
l
f2


 2 o2 km
m
N
C gx 
C gy
C gz
(5.2.6)
where
  Q y [k 2  l 2 
f o2
1
(U  c) 2
2
2
(
m

)]

Qy
N2
4H 2
Note 1: If l is replaced by
(5.2.7)
f o2
m, we have C gy the same as C gz Therefore,
N2
mathematically, the y direction and z direction are very similar for Rossby waves.
However, later, we will see that the physical meaning of the group velocity in these two
directions differ dramatically.
In the case of U=0, we have Qy =β>0, we have C px 

k
 0 , so the wave always
propagates westward. Define the phase velocity as
  
C p  (C px , C py , C pz )  ( , , )
k l m
We see from (5.2.2) and (5.2.6) that
sign(C py )  sign(l) ,but sign (C gy )   sign (l )
sign(C pz)  sign(m) ,but sign (C gz )   sign (m)
Thus, the phase and group velocity are in the opposite directions.
AOS611, Ch.5, Z.Liu, 03/09/16
19
It should be pointed out that in the atmosphere, even if m is real (when the basic state
allows),  does not vary with height simply as eimz; instead, its amplitude increases with
height as
 ~e e
imz
z
2H
or the energy increases with height as
||2 ~ ez/H ~ 1/p
The amplification of the streamfunction with height is caused by the reduction of
atmospheric density.
z
ez/H

Finally, for stationary forcing (topography or large scale heating/cooling) c=0.
Eqn.(5.2.5) shows that only those largest scale waves (smaller k, l) can propagate
vertically in the westerly wind (U>0) (real m). This has been used to explain the
observed stratosphere. Stratosphere disturbances are believed to originate from the
troposphere. Observations show that the mid- and high latitude stratosphere is dominated
by disturbances at planetary scales (wave number 1, 2 3), although the most energetic
disturbances in the troposphere are at higher wave numbers (6 and 7). This is because
only those very long waves can propagate into the stratosphere according to (5.2.2).
AOS611, Ch.5, Z.Liu, 03/09/16
Fig.5.1: Vertical propagation of
atmospheric waves
20
AOS611, Ch.5, Z.Liu, 03/09/16
Fig.5.2: Geopotential height
anomaly in the stratosphere
21
AOS611, Ch.5, Z.Liu, 03/09/16
22
Sec. 5.3 Vertical Normal Modes
1: Vertical Modes in the Ocean
Consider an ocean with U=0 and N uniform. The mean PV gradient is therefore
Qy   . The linearized QGPV equation is:
 t [ xx   yy
z
2
f
 o 2  zz ]   x  0
N
D
On the bottom (assume flat)
w(x,y,0,t) = 0
D
(5.3.1)
(5.3.2)
At the top z=D+(x,y,t)
0
We have a rigid lid
w(x,y,D,t) = 0.
(5.3.3)
To write the vertical boundary condition in terms of , we resort to the thermodynamic
equation:
 t ' w
d s
0
dz
The hydrostatic balance gives:

g '

z
fo s
(5.3.4)
We then have
w
 t '
f   2
f
 o s
 o2  tz
d s
d s tz N
dz
g
(5.3.5)
dz
Thus, the vertical boundary conditions (5.3.2) and (5.3.3) become
 tz  0
at z=0, D
(5.3.6)
Returning to the QGPV equation. We look for separable solutions of the form
   ( z )( x, y)e it
Substitute this into the QGPV equation, we have the equation for the vertical structure as
2
d  f o d 
2

   
2
dz  N ( z ) dz 
and the equation for the horizontal structure as
(5.3.7)
AOS611, Ch.5, Z.Liu, 03/09/16

23

 i  xx    yy   2   

0
x
(5.3.8)
The vertical boundary conditions (5.3.6) becomes
d
 0 on z=0,D
dz
(5.3.9)
Therefore, (5.3.7) and (5.3.9) form an eigenvalue problem. The eigenvalues are real.

Indeed, notice (5.3.9),

2


D
0
d
dz
N2
fo
2
D
0
 dz  
2

D
0
N2
fo
2
 (5.3.7) gives
D  d  d 
d 2
 d 
 2 dz        
0
dz
 dz  dz   dz 

dz

D  d 
D  d 
    dz      dz.
0
0
 dz 
 dz 
2
D
0
2
2
Therefore, the eigenvalue is
 d 
  dz

0
dz
2
   2 
0
D N
2
0 f 2  dz
o
2
D
In the case of a uniform N, the eigenfunctions and eigenvalues can be easily solved as
 N2
 m  z   cos
 fo
N2
fo
2
m 
2

 m z 
(5.3.10)

m
,
D
m  0,1,2, 
(5.3.11)
Substitute them into (5.3.8), we have the dispersion relationship

 k
f 2 m 2
k 2  l 2  02 (
)
D
N
Thus, for each vertical mode, m, the dispersion relationship is exactly the same as that for
the shallow water Rossby wave, provided that we replace the effective deformation
radius for each mode as:
L2Dm 
ND2
 f o m 2
.
(5.3.12)
AOS611, Ch.5, Z.Liu, 03/09/16
24
The deformation radius vanishes, with an increasing m.
The correspondence of the deformation radius in (5.3.12) with that in the 1.5-layer model
(1.5.4) can be readily seen below. Since
N2 
 1
g ds
1
~g
 g ,
o dz
o D
D
we have
2
2
LDm 
g'
D
g' D 1
g' Dm
2 
2 
2
2
D fo m 
fo m 
f o2
where
Dm 
D
m 2
is the equivalent depth. Thus, each baroclinic mode propagate exactly as a 1.5 layer
model Rossby wave with an equivalent depth of Dm. (Some people also use the
LDm 
expression of
gDˆ m

, such that the equivalent depth is Dˆ m 
D )
2
 m
fo
The LDm is called the internal (baroclinic) deformation radius. This gives a close analogy
between shallow water dynamics and the stratified dynamics. In the ocean,
LD1 

 1 .
LDo

The vertical structure of the normal modes are further discussed below. The m=0 mode is
the barotropic mode or external mode (deformation radius infinitely large in the absence
of free surface elevation here). The velocity does not have shear in the vertical direction
and there is no density perturbation for this mode.
The m  1 mode is the mth baroclinic mode or internal mode. These modes have m node
points in the velocity field and are all accompanied with density perturbations. For all the
baroclinic modes, the vertically integrated net transport vanishes.
This can be shown directly from (5.3.7)

and (5.3.9). Integrate (5.3.7) from
z=0 to D gives 
2
m
and therefore

D
0

D
0
m z'dz'  0
m z 'dz '  0, if m  0 .
m=1,2,3
m=0
z
AOS611, Ch.5, Z.Liu, 03/09/16
25
Note 1: Equivalent particle examples of external and internal modes. Consider two balls
connected by a spring. There are two possible normal modes. The first has both balls
moving in the same direction, as if there is only one ball. This is the”barotropic mode”.
The second has the two balls always moving in the opposite directions. This is the
“baroclinic mode”.
“barotropic mode”
“baroclinic mode”
When more balls are added, there are more freedoms and more “modes”. ||
In reality, N(z) is not uniform at all (Fig.5.3). Analytical solution becomes usually
impossible. Nevertheless, for slowly varying N(z), we can still use the WKB method such
that, with U=0, (5.2.5) becomes :
m2  
N 2 (z) 
(  k 2  l2 )
2
f
C
for the oceanic case. If m is real at any height, it is real at all height, although N(z) may
change. So there is no internal reflection. The normal mode is caused by reflection at the
top and bottom boundaries.
z
AOS611, Ch.5, Z.Liu, 03/09/16
26
After a couple of reflections, normal mode is established in the z direction. The
establishment of the normal mode is similar to the normal mode in the case of horizontal
boundaries. The key is that the wave energy is trapped within a finite region.
The most dramatic change of N is in the oceanic thermocline. The WKB solution shows
that m(z) changes, small for small N, but large for large N. (Fig.5.4).
Indeed, for a general N(z), the veritical eigenvalue problem (5.3.7) and (5.3.9) can be
solved numerically to give m. Then, the horizontal structure satisfies (5.3.8), which is the
same as the shallow water Rossby wave, except for replacing the deformation radius by
LD2=m-2. The Rossby wave of the mth mode has the dispersion relationship

 k
k  l 2  m
2
2
.
2. Atmospheric Case
The normal mode in the atmosphere is much more complicated, because of the lack of a
upper boundary. It turns out that the normal mode usually doesn't exist in the atmosphere.
This is not hard to imagine, because, in the absence of a reflective top boundary, a normal
mode can't be established.
z
Mathematically, one can see this crudely here. If the vertical mode equation allows the
solution
 ~ Ae imz  Be imz
AOS611, Ch.5, Z.Liu, 03/09/16
The absence of an energy source from above requires that the wave energy radiates
upward only. This selects B=0. Furthermore, at the bottom,  z  0 determines that
A  0 . Therefore, there is no normal mode. However, normal modes may exist when a
strong shear of U(z) produces internal reflections.
27
AOS611, Ch.5, Z.Liu, 03/09/16
Fig.5.3: Vertical profiles of N in the
atmosphere and ocean
28
AOS611, Ch.5, Z.Liu, 03/09/16
Fig.5.4: Normal modes in the presence of
realistic oceanic stratification.
29
AOS611, Ch.5, Z.Liu, 03/09/16
30
Sec 5.4: The Eliassem-Palm Theorem
1. E-P theorem:
The shallow water E-P theorem in section 2.6 can be generalized to the stratified fluid.
The QGPV equation.of the atmosphere is
Dg q  (t  ux  v y )q  Sq
Consider a basic state
U  U ( y, z) ,   ( z ) ,   ( y , z) ,   f o ( y , z )   ( z )
where

d g( z ) g p k
 U ( y , z ) ,

 ( ) 
y
dz
Ts
Ts p*
and the mean temperature field
T  T ( y, z )  ( z )  T ( y, z )  (
p k
) ( z )
p*
where the basic state satisfies the thermal wind relationship

g T
U
 fo
Ts y
z
The mean QGPV is therefore:
2
 2 1  pf o 
Q( y , z )  f o  y 

(
)
y
p z N 2 z
and the mean PV gradient is
 2U 1  pf o 2 U
Qy ( y ,z )    2 
(
)
y
p z N 2 z
Write
   ( y , z )   ' ( x, y , z , t )
where  '   , is a small perturbation. The QGPV equation can be linearized as
(t  Ux )q 'v' Qy  S ' q
(5.4.1)
Multiplying the equation by pq'/Qy, we have
(  t  U x )
Since
pq' 2
 pv' q'  pq' S' q / Q y
2Q y
(5.4.2)
AOS611, Ch.5, Z.Liu, 03/09/16
31
v'   x ' , q'   xx ' yy '
v'  xx ' 
f
1
 z ( p o2  z ' )
p
N
1
 x [( x ' ) 2 ]
2
1
v'  yy '   y [ x '  y ' ]   x [( y ' ) 2 ]
2
f
f
pf

[ p o 2  z ' ]   z [ p o 2  x '  z ' ]   xz2  ' o2  z '
z N
N
N
2
v'
2
  z[ p
2
2
2
fo
pf
1
 x '  z ' ]   x [ o2 ( z ' ) 2 ]
2
2
N
N
pv’q’ can be written in the form of flux divergence
pv' q '    F
(5.4.3)
Here, the flux F is the generalized E-P flux
2
p
  p [v' 2 u ' 2 ( N  ' ) 2 ]
fo
' 2
' 2
2

[(

)

(

)

(


'
)
]
d
 2
x
y
z
 Fx   2
2


N
dz


 (5.4.4)
F   Fy   
p x '  y '
 pu ' v'



2
pf
v
'

'


o
 Fz 
pf o




 x '  z '
d

2


N

 
dz
and
g d
' N 2
N 
,  z ' 
 s dz
f o d
dz
2
The perturbation PV equation (5.4.2) can be written in the wave activity equation:
( t  U x ) A    F  S
(5.4.5)
where
pq  2
A
2Q y
is the wave activity, and
(5.4.6)
S  S q pq' / Q y . The conventional E-P equation is the zonal
mean of the generalized E-P equation.
t A F  S
(5.4.7)
AOS611, Ch.5, Z.Liu, 03/09/16
32
where
1
  pu ' v ' 
p q '2


F


A 2
, F   y    pf v' ' 
Qy
 Fz   o d 
dz 

(5.4.8)
This gives the Elliassn - Palm Theorem:
For (i) steady amplitude t A = 0, and (ii) conservative S = 0, the E-P flux F is nondivergent. (Therefore, F can’t originate from nowhere and end in nowhere, like the mass
flux of an incompressible fluid)
Note 1: For the ocean:



2
 ' 2
 u ' 2 v' 2 ( N
)2 
fo
' 2
2
d S
 Fx  ( x )  ( y )  N 2 ( z ' )  
dz 

 


 |||
F   Fy   
 x '  y '
 u ' v'


f o v' 
2
 
 Fz  
fo


 
 x '  z '
d s
2

N

 
dz


2. Wave Activity Flux and Group Velocity
For almost-plane waves, under the WKB assumption, the solution can be assumed of the
form

 ' ( x, y, z , t )  Re  ( y, z )e
i 

y
z
2H



z
where   k ( x  ct )   l ( y ' )dy '  m( z ' )dz ' . We can derive the wave activity as
A
p 2
p
q' / Q y 
|  |2
2
4
where we have used

f2
q   k 2  l 2  2
N

1 
 2
m 
 ' .
4 H 2 

and (5.2.7) with
  Q y [k 2  l 2 
f o2
1
(U  c) 2
2
2
(
m

)]

Qy
N2
4H 2
AOS611, Ch.5, Z.Liu, 03/09/16
33
Similarly, the flux is
Fy  p x  y  
1
2
kl 
2
(5.4.9)
*
2
2
2

f0

1 f0
1    1 f0
2
Fz  p 2  x  z  
Re
ik

im



km 






2
2
2N
2 H    2 N
N


Notice the group velocity of the baroclinic Rossby waves in (5.2.6), we therefore have
2


f
F  Fy , Fz   2kl, 2 0 2 km  A  (C gy , C gz ) A .
N


3 Vertical Propagation and Meridianal Heat Transport
The vertical component of the E-P flux is directly related to the meridional heat flux
f0
' N 2







v' '  v' '  km |  | 2
x
z
2
d

fo
N
dz
2
Fz 
(5.4.10)
An upward E-P flux ( Fz  0 ) corresponds to a northward heat transport, and vise versa.
In the atmosphere, Rossby waves are usually forced from the surface to propagate
upward. ( Fz  0 ). This corresponds to a westward tilt (km>0 ) and should transport heat
poleward. In addition, waves are also caused by baroclinic instability (Chapter 6). The
unstable waves also tilt westward and transport heat poleward.
max
min
max
z
=0
=0
=0
=0
MfN, z
(k,,m)
Cg
k, x
AOS611, Ch.5, Z.Liu, 03/09/16
34
The northward heat transport by Rossby waves contributes to the major part of the
atmospheric poleward heat transport in the midlatitude region. Interestingly, in the ocean,
the wind forces Rossby waves at the surface and therefore downward. These waves will
transport heat equatorward, against the mean gradient.
4. Applications
Case 1: Wave-mean flow interaction in vertically sheared flow. Assuming a westerly
wind with a maximum speed in the middle level. In the lower half, the westward tilting
trough produces an upward E-P flux. The accompanied northward heat transport is down
the mean temperature gradient (Ty<0 for Uz>0) and therefore tends to reduce the mean
temperature gradient. The perturbation grows by extracting APE from the mean APE.
Similar discussions show that the perturbation in the upper half is also unstable.
Alternatively, the E-P flux converges, increasing the wave activity at the expense of the
mean flow strength. (tA increases and tU decreases).
z
U(z)
Fz<0
x
Fz >0
Case 2:
Vertical propagation of the atmospheric Rossby waves (see the end of last section).
AOS611, Ch.5, Z.Liu, 03/09/16
35
The amplitude increases for a vertically propagation wave with height inversely
proportional to pressure. For a wave packet originate at the surface (1000mb) propagating
into the stratosphere (10mb), its amplitude increases by 10 times. This can be seen using
the E-P theorem. For steady, conservation waves,
.F =0.
For plane waves, Fy is independent of y, so that
 z F   y F  0 ,
or
 pf o2
z
N
2

 x'  z'   0

If N is constant,


 z p x'  z'  0
(5.4.11)
(5.4.12)
Now, if
 '  Re[ ( y, z )e i ]
z
where   e 2 H . We have  x '  Re( ike i ) ,  z '  Re[( im 
 x '  z ' 
1
)e i ] . Therefore,
2H
1
km |  | 2 . Thus, (5.4.12) gives
2
 z (p |  | 2 )  0
In reality, the amplitude can be changed by dissipation, nonlinearity, wave refraction , etc
AOS611, Ch.5, Z.Liu, 03/09/16
36
Questions for Chapter 5
Exercises for Chapter 5
E5.1. (Vertical Rossby wave propagation diagram) The basic state is motionless and has
a constant Brunt-Vasara frequency N. Baroclinic Rossby waves have the form of
e i(kx+ly+mz)
a) Discuss the mathematical similarity between the vertical propagation of stratified
baroclinic Rossby waves and the meridional propagation of shallow water Rossby waves.
Plot the wave vector and direction of the group velocity in the wave number (k,m) plane
(or the dispersion diagram circle).
b) In light of (a), consider an upward/westward propagating baroclinic Rossby wave that
is incident on a vertical wall (or tall mountain). What will be the direction of the reflected
Rossby wave ? What will be the wave phase pattern ?
Z
X
Cg
Aei(kx+ly+mz)
(k,m)
(c) If a baroclinic system tilts westward with height, what is the direction of Rossby wave
energy propagation?
AOS611, Ch.5, Z.Liu, 03/09/16
=0
=0
max
H
warm
Low
37
=0
min
L
cold
high
=0
max
H
min
L
warm
cold
Low
high
What kind of weather synoptic system does this correspond to? Which direction does
this system transport heat flux? What is the direction of the E-P flux in the vertical
direction? (For simplicity, you can assume an infinite scale height, or incompressibility).
E5.2. (Wind forced stratified ocean) A stratified linear ocean is forced by a spatially
uniform Ekman pumping on the surface with a frequency . The ocean basin has a zonal
scale L (such that the wave number is k=2/L) that is much longer than the internal
deformation radius:
(a)
Find the direction of the downward group velocity.
(b)
What is the direction of the group velocity when the forcing frequency approaches zero
(the limit of steady forcing)?
(c)
In light of (b), is it possible to have subsurface motion under a steady wind forcing?
(d)
Is Sverdrup relation valid in the limit of a steady wind?
(e)
What is the implication of (c) and (d)?
Z
Cg
L
X
AOS611, Ch.5, Z.Liu, 03/09/16
38
E5.3 (Non-Doppler shift effect) In a stratified ocean, we will consider planetary scale
perturbations that are governed by the potential vorticity equation
2
2


f o  
f o  
 t  z ( 2
)   x  J  ,  z ( 2
)  0
N z 
N z 


.

We project the streamfunction on vertical modes:    m ( x, y, t ) m ( z ) where the mth
m 0
vertical mode m is determined by the eigenvalue equation (5.3.7) as
d  f o d m ( z ) 
2

   m  m ( z ) .
2
dz  N ( z )
dz 
2
(a) If the flow is projected only on a single vertical mode m=M, what is the advection
term in the potential vorticity equation.
(b) In light of (a), how do you interpret the perfect non-Doppler-shift effect of the
planetary Rossby wave in the shallow water or the 1.5-layer model?
(c) If the flow is projected on more than one vertical modes, show that the mth mode of
the flow only advects the part of the stretching voriticty that excludes mode m.
(d) Based on (c), under what condition, Rossby waves will be advected by mean flow (or
Doppler-shift occurs) in a general continuously stratified ocean?
E5.4: (Wave-mean flow interaction of baroclinic waves). Based on the wave activity
equation and E-P flux (5.4.7) and (5.4.8), discuss the wave-mean flow interaction of the
following disturbances in a westerly shear flow.
U(z)
z
x
(a) Will the disturbance grow or decay? Will the mean flow intensify or weaken?
AOS611, Ch.5, Z.Liu, 03/09/16
(b) Discuss the difference and similarity from the corresponding barotropic case (in
section 2.6) of negative viscosity.
39