# 7 Geostrophy thermal and gradient winds

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```					Chapter 7: Geostrophic, Thermal and Gradient Winds

The Navier-Stokes equation, Conservation of Mass and the equation of state make
up five equations to solve for the five unknowns of u, p and  . These equations along
with the additional information contained in the boundary conditions gives us all the
mathematical tools we need to find the motion of the atmosphere or ocean in a general
physical situation. The non-linear character of the Navier-Stokes equation, prevent us
from developing simple analytic solutions for a given scenario in the ocean or
atmosphere. Even more, numerically solving the above non-linear equations in the
atmosphere or ocean is a challenge for even the most sophisticated supercomputers. It is
due to this sensitivity to the conditions of the problem that we still are unable to predict
weather or the trajectory of a hurricane with 100% accuracy.
To circumvent these difficulties, it is up to the oceanography or meteorologist to
use their physical understanding of the problem in order to simplify the equations of
motion to a manageable computational problem. In this chapter, we will use scaling
arguments to simplify the equations of motion to the most fundamental atmospheric
problems and examine the characteristics of the associated solutions.

Rossby Number:

The inviscid equations of motion in a rotating frame with small density variations
are:

u    u     u     u           1 p
u     v      w  fv              2 u
t    x     y      z          o x
v    v     v     v          1 p
 u  v  w  fu                    2 v              (1)
t    x     y     z           o y
w    w      w      w    1 p 
u     v      w                   g   2 w
t     x     y      z     o z  o
u v w
         0
x y z

We will first assume that the ocean or atmospheric medium we are examining is
Z
shallow such that, for vertical scale Z and horizontal scale L,      1 . Given this
L
assumption we can see using dimensional analysis of the continuity equation, that the
vertical flow, W, field is significantly smaller than the horizontal flow, U

 u v w  U W               W   Z
 x  y  z   L  Z  0  U  L  1
              
We can therefore neglect the vertical velocity field in all advective terms and in the
vertical momentum equation. Vertical conservation of motion then yields the equation of
hydrostatic balance in the vertical direction:

p
  g
z
We will now assume that we are considering flows where the Coriolis force
Du h
dominates over horizontal inertial effects (horizontal material derivative terms:         ). If
Dt
the local derivative is of the same order as the advective term in the material derivative
then we can consider either in our scaling analysis. It is often easier to consider length
scales over time scales in our problems so we will express our inertial dimensional scales
U2
in terms of the horizontal advective term - u  u h      . Now introduce the Rossby
L
number; which relates the comparative scales of the inertial terms with the Coriolis
force1.
U2
Inertial contributions              U
RL                            L       
Coriolis Force             fU     fL

Reynolds number:

We are also interested in flows where we can neglect the effects of friction. Using a
similar analysis as above, we can compare the inertial effects with the frictional terms in
the equations of motion. The resulting dimensionless number is called the Reynolds
number:
U2
Inertial contributions             UL
Re                          L       
Friction Force           U    
L2

We can see for low Reynolds flow ( Re  1) frictional effects dominate and for large
Reynolds number flows ( Re  1) we may neglect friction. For Geostrophic flows, we
are interested in large Reynolds number flow.

Geostrophic Wind:

1
The L subscript in the Rossby number is to indicate that we considered the advective contribution for the
inertial effects. An additional Rossby number can be developed if we consider time scales and thus the
local time derivative (This form of the Rossby number would be more appropriate in a spectral analysis of
a normal mode solution for example) :
U
1
RT  T          
fU       fT
We are interested in problems where the Rossby number is small and Reynolds
number is large so we can neglect the inertial terms of the equation of motion. For a
m
Coriolis parameter of ~ 10 4 , an atmospheric flow of U ~ 30 with a length scale of ~
s
1000km (the size of a high or low pressure system) and laminar kinematic viscosity of
m
 ~ 0.01 2 , the Rossby number is ~ 0.3 and the Reynolds number is 3 109 . This
s
shows that for standard atmospheric parameters, neglecting friction is obvious and the
assumption of neglecting the inertial terms with respect to rotation is fairly valid. If this
assumption holds then we obtain the equations for Geostrophic flow which is indicated
by a balance between the gradient and Coriolis force:

1 p
fv 
 o x
1 p
fu                                                                           (2)
 o y
p
  g
z

For the mid-latitudes, equations (2) holds surprisingly well for many local
1
barotropic weather systems. When expressed in vector form f  u h   p , we can
o
immediately see that the flow field is parallel along the isobars , u h  p  0 .
Alternatively, this also means that pressure is constant along the streamlines of the flow
field. Figure 1 shows the Geostrophic flow field around an axi-symmetric low pressure
system. The flow is counterclockwise around low pressure systems and clockwise around
high pressure systems in the northern hemisphere as expected by our physical understand
between the relationship of the pressure gradient to the Coriolis force.
1000

800

600

400

200

distance (km)
0

-200

-400

-600

-800

-1000
-1000   -800   -600   -400   -200         0         200   400   600   800   1000
distance (km)

Figure 1 – Geostrophic flow around an axi-symmeteric low pressure system.
Taylor-Proudman Theorem:

The results of the Geostrophic and thermal wind equations lead to the amazing
physical result that when these equations hold, the entire flow field has no dependence in
the vertical direction. This leads to very strange phenomena such as when we move a
cylinder slowly in a rotating frame the entire fluid column will move and not just the
water surrounding the cylinder. This result is called the Taylor –Proudman theorem
which states that an unbound, rotating, frictionless and non-inertial fluid has no
dependence on the direction of rotation. We will now prove this theorem quantitatively.
geostrophic balance assumes an unbound, rotating, frictionless and non-inertial fluid:
1 p
fv 
 o x
1 p
fu  
 o y
Taking the derivative of the first equation with respect to y and adding it to the derivative
of the second equation with respect to x we obtain:

 u v 
f  0
 x y 
       

Application of the continuity equation,   u  0 , shows us that the vertical velocity has
no dependence in the z-direction:

w
0
z
^    ^ 
Now, using the horizontal gradient operator,  h  i        j , take the curl of the
x    y
original geostrophic equations:

                1      
 h  f  u   h    h p   0  0,0,0
       
 o      
Notice that the right hand side is equal to the vector 0 and not scalar 0 so all three
components of the vector is equal to 0.
Expanding the left hand side, and use of the continuity equation,   u  0 leads to the
result:

u v
f  u  0           0
z z

In other words, no component of the velocity vector depends on z:

u
0
z

This proves the Taylor-Proudman Theorem. The results of this theorem have been
confirmed in the laboratory. A cylinder was placed in a rotating tank and moved slowly,
perpendicular to the direction of rotation. A colored dye was placed ahead of the motion
of the cylinder and at a depth higher then the top of the cylinder. It was observed that as
the cylinder passed below the colored dye, the dye separated. This result showed the
independence of the velocity field in the direction of rotation, requires any horizontal
motion of the fluid to include the entire vertical column of the fluid.

Thermal wind:

We know from experience that the density and temperature change with altitude
in the atmosphere. With the use of the ideal gas law we can see that these density and the
associated temperatures changes will relate to change in horizontal flow with attitude.
First note that the ideal gas law is baroclinic and we will want to consider horizontal
density variations driven by temperature changes. We can therefore no longer assume the
Boussinesq approximation. Consider representing the density field as follows

    p , T      o  z    '  x, y 

Substitution of this density field in the geostrophic equations and operating by
d
, we obtain:
dz
u    1  o     g  '     ln  o     g '  ln  '
        u                      u
z     o z    f o y        z         f o y
(3)
v    1  o     g  '     ln  o     g '  ln  '
        v                      v
z     o z    f o x        z         f o x

Approximating the equation of state in the atmosphere with the use of the ideal gas law
p o  p   o   RT , we can relate the above density gradients with temperature
gradients (see appendix 1 at the end of the chapter). The resulting equations are called
the thermal wind equations since they show the vertical variations in the horizontal wind
field as a function of the temperature gradients.

u 1 T     g T
     u
z T z    fT y
(4)
v 1 T     g T
     u
z T z    fT x

It can be shown empirically or by scaling analysis that the first term in the above two
equations are negligible compared to horizontal gradients of the temperature field (notice
g
the second term is multiplied by the factor ) so we can approximate the thermal wind
f
equations as:

u    g T

z    fT y
(5)
v   g T

z fT x

u ^ v ^
If we express the vertical gradients in equation (4) as a thermal wind, VT        i   j,
z z
then the thermal wind equations are expressed in vector form as

g
f  VT       T                                                              (6)
T

Notice the above equation shows us that the direction of VT is parallel to the
isotherms with the colder air to the left of the vector just as the gradient wind is parallel
to isobars with lower pressure to the left. It is common to refer to the wind shear in the
thermal wind equations as a thermal wind, indicated by equation (6), but we must make
note that we are actually referring to the vertical shear of the horizontal wind field.
Recall that upper air observations provide various physical parameters of the
atmosphere with respect to altitude. Equation (6) is often used in reverse with these
observations to indicate the horizontal transport of temperature. If you have a
circumstance where the Geostrophic wind is veering with height, then by simple vector
addition in figure 2, you can see the Geostrophic wind must be transporting warmer air to
colder regions. The opposite holds in the case of Geostrophic wind backing with height.

Colder air
T  21o C
VT
22 o C
T
V g  z1        V g z 2 
23o C
24 o C
Warmer air

Figure 2 – Diagram showing the orientation of thermal wind to the temperature field and
a veering Geostrophic wind with height. In the above figure, z 2  z1 to indicate that the
gradient wind is veering with height.

Looking back at Figure 1, we notice that curvature is significant in the flow field.
The continually changing direction of the fluid parcel as it travels around the low
pressure system will lead to a centrifugal acceleration and associated force to the
equations of motion. By converting the original equations of motion, given by equations
(1), to cylindrical polar coordinates we can derive a simple relationship for the axial flow
around a low or high pressure system that includes the effects of rotation.
The details of converting from rectangular coordinates to cylindrical polar
coordinates were given in SO 355 and only the results are provided here. The spatial
coordinates in our new CS are x, y, z   r , , z  with associated velocity components
u, v, w  u r , u , w . Equation (1) then takes the following form in cylindrical polar
cords:

u r       u    u u r     u                1 p
2
u
 ur r           w r    fu  
t         r    r        z     r          o r
u        u    u u       u    u u            1 p
 ur            w    r  fu r  
t         r    r         z      r          o r 
w        w u w        w     1 p 
 ur           w                  g
t        r    r       z      o z  o
u v w
         0
x y z
We now make the following assumptions:

W
1) The medium is shallow and therefore       1 . We can then neglect the vertical
U
velocity component in all advection terms. Also, the vertical equation of motion is simply
the hydrostatic equation.

2) The flow is steady – therefore    0
t

3) The flow is axi-symmetric – therefore     0

4) Another consequence of assumptions 2 and 3 is u r  0

Given the above assumptions, the radial component of the conservation of momentum is

1 p
2
u
 fu                                                                    (7)
r          o r

All terms of the axial component of conservation of momentum are eliminated and the
vertical component of momentum conservation is the equation of hydrostatic balance.
The axial flow in equation (7) is called the gradient wind and accounts for both
the Coriolis and centrifugal force balancing with the pressure gradient. Since equation
(7) is simply a quadratic equation for u , we can obtain a simple analytic solution for the

r p
2
fr      fr 
u                                                            (8)
2       2      o r
There are three physical situations for equation (8) depending on the sign of the square
root and the sign of the pressure gradient. They are:

r p
2
fr     fr                     p
1) Normal Cyclonic flow - u                             0 and      0:
2      2       o r           r
Since the radial gradient of the pressure field is greater than 0, we are talking about a
pressure field that increases as one travels away from the origin which is the case in any
low pressure system. Both terms in the radical are of the same sign and the sum is
fr
greater than      implying that the axial flow is counter-clockwise or cyclonic. From a
2
force relationship perspective, consider first a geostrophic balance. The straight flow is a
balance between an inward directed pressure gradient and an outward directed Coriolis
force. In the case of the gradient wind, an inward directed centripetal force then causes
the flow to curve in the cyclonic direction.
r p
2
fr     fr                       p
2) Normal anti-cyclonic flow - u                           0 and       0:
2      2        o r           r
The decreasing radial pressure gradient indicates a high pressure system. The root term
fr
is positive but less than    indicating the axial flow is clockwise. The Coriolis and
2
Centripetal force is directed inward and balanced by the outward directed pressure
gradient. Notice that, since the radial gradient of pressure is less than 0, there is a limit
p
on how large        can be to ensure the radical is real. The restriction is
r
p  o f 2 r

r       4
r p
2
fr    fr 
3): Anomalous anti-cyclonic flow about a high: u                        0 and
2     2    o r
p
0
r

This third type of flow can exist theoretically but is not really seen empirically. It has the
same force balance as example 2 but utilizes the negative root of the quadratic equation.
Simple calculation shows that the required pressure gradients for the above flow to exist
as a gradient wind are extremely small.

Ocean applications to Geostrophic flow (Geostrophic technique):

In the above sections, we evaluated the geostrophic flow and thermal wind in the
atmosphere. In most circumstances in the ocean, the Rossby number should be small and
so equation (2) can be used in the ocean medium as well. Given density and surface
variations along an ocean front, as shown in figure 3, we can find the current shear across
the front.

Colder water               Warmer water
1                                                Ocean
 isopycnals                     surface
u1


2
u2
 isobars

Figure 3 – Density (dashed lines) and Pressure surfaces (solid lines) in the ocean.
Let us consider the effects of density variations with depth on the north-south
flow (into the page) in figure 3. We are interested in the north-south velocity component
in the geostrophic equation:

1 p
fv                                                                     (9)
 x

px, z  p  z    
Applying the chain rule to the pressure field,                                and use of the
x      z  x


P

p
hydrostatic equation,       g , where  is an integrated average density value in the
z
fluid column, the above geostrophic equation takes the form
 g  z 
v                                                       (10)
 f  x P 
      

 z 
We see from the geometry of figure 3 that    x  is simply the slope of the constant

 P
 z 
pressure surfaces and that  x   tan  .

 P
Now, we are interested in finding the shear in the northerly flow of the ocean
frontal current. We can identify the current by various lines of constant density. Larger
density values will be closer to the surface in the colder water whereas as smaller density
values will be at surface in areas where the water is warm. These isopycnals howvere will
slope downaward at the frontal interface and we can use this information to identify
velocity values at specific depths. Using equation (9) for two different density values,
and subtracting the two velocities, we obtain,
1 1      1  p 1   2  1  p
v1  v2                                                               (11)

f  1  2  x f  1  2  x
                 

Now using hydrostatic balance like in equation (10) we obtain,

g    2  1  z           g  1   2  z
v1  v2                                                                           (12)
f  1  2  x
                P       f    x
                P
2
The second approximation in equation (12) is 1  2 ~  which is reasonable because, at

most, the variations of density in the ocean give us     0.07 .


Equation (12) shows us that if we are given the general slope of the pressure
surfaces across an ocean front such as the Gulf-stream as well as the density values at
different depths, we can evaluate the change in current speeds at those depths. We might
also use equation (12) to find density variations provided we know the ocean currents
across the front.

Example

At a latitude of 35 o North, f  8.37  10 5 s 1 , the density variations across the gulf
stream are
kg
 2  1.0275  103 3
m
and
kg
1  1.0265  103 3
m
 z   700m
 x   28km  0.025 .
and the slope of the pressure surface is        
 P

Find the difference in the north-south current with respect to depth:

kg
  1  2  1.0270  103        and
m3

  10     
v1  v2            
9.8
5
m
 0.025   2.85 m
 1.0270    8.37  10 s                     s

m
Thus the vertical shear (Ocean equivalent to the atmospheric thermal wind) is 2.85
s
across the gulf stream with the faster flow at the surface.

Appendix 1 – Derivation of temperature gradients in the thermal wind equation
from the ideal gas law.

Starting from the thermal wind equation as gradients in the density field given by
equation (3): (the primes have been dropped)
u       ln  o      g  ln 
               u
z          z         f o y
v      ln  o      g  ln  
               v
z         z          f o x
Using the same assumptions that the pressure and density are functions of a
vertical stratified background density and pressure field superimposed on a local pressure
and density field, Po   o RT and P  RT , we can obtain the following logarithms:
ln Po   ln  o   ln T   ln R 
and
ln P   ln    ln T   ln R 

Focusing on the u component of the thermal wind equations, substitution of the above
logarithms yields:

u     ln Po      ln T     g  ln P g  ln T
            u           u                    .
z       z            z        f o y    f o y

The first term of the above equation can be simplified using the hydrostatic equation of
and the ideal gas law:

 ln Po      1 Po    g o     g
              u       u      u    u
z          Po z     Po     RT

The third term can also be simplified by the use of equations (2) and the ideal gas law:

g  ln P  g 1 P        g
               u
f o y       f o P y    RT

Notice that the last two equations are the same but of opposite sign and therefore cancel
out in the thermal wind equation. The resulting equation after applying the above
analysis is given in equation (4)

u 1 To    g 1 T
      u
z To z    f o T y
v
A similar analysis can be carried out for      to obtain:
z
v 1 T     g T
     u
z T z    fT  o x

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