The thermal wind
1
The thermal wind
Introduction
The geostrophic wind is determined by the gradient of the isobars (on a horizontal
surface) or isohypses (on a pressure surface). On a pressure surface the gradient of the
isohypses reflects the tilt of the pressure surface. If this tilt changes with pressure then
also the geostrophic wind will change with pressure in magnitude and/or direction.
Generally speaking the thermal wind is the change of the geostrophic wind with
pressure (or height): it is the vector difference of the geostrophic wind at two different
levels and as such it is not a real wind. In this note we will treat the causes of the thermal
wind, and we will also see how this quantity can be used in practice.
The geostrophic wind
The expressions for the geostrophic wind are:
ug  
1 p
g y
and
vg 
1 p
g x
,
(1)
where the minus sign in the expression of the left is only important for the direction of the
u-component of the geostrophic wind. On a pressure surface with isohypses these
expressions may be rewritten assuming hydrostatic equilibrium:
p
  g ,
z
(2)
This leads to:
ug  
g z
f y
and
vg 
g z
f x
(3)
From formula (3) it appears that the magnitude of the geostrophic wind depends on the
tilt of the pressure surface (z/y and z/x). For a given distance (y or x) of the
isohypses the magnitude of the geostrophic wind can be determined with the aid of
Equation 3. In the example in Figure 1 it is obvious that z/x < 0 (height z of the
pressure surface decreases in the positive x-direction) and consequently for the vcomponent of the geostrophic wind we have: vg < 0. The larger the tilt of the pressure
surface the smaller the distance (x) between the isohypses and hence the larger the
geostrophic wind speed. A similar conclusion applies to the u-component of the
geostrophic wind. It appears that the magnitude (and the direction) of the geostrophic
wind may change if pressure surfaces are NOT parallel to one another. Just when such
a situation may occur is explained in the next section.
The thermal wind
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Figure 1. The magnitude of the geostrophic wind is determined by the tilt of the pressure surfaces.
For a small tilt (left) the isohypses are far apart and the magnitude of the geostrophic wind is
small; for a large tilt (right) the isohypses are close together and the magnitude of the geostrophic
wind is large.
The relation with temperature
According to the equation of state (i.e. the gas law) the temperature determines the air
density () for a given value of the pressure (p):
p  Rd T
hence

p
Rd T
(4)
Thus in an area with high temperatures the density at a given pressure is lower than in
an area with low temperatures (where the density is higher). Because of the high
density the decrease of pressure with height in the cold area is larger than the decrease
of pressure in the warm area. In Figure 2 (left) this is clearly visible. We have assumed
that the pressure at the surface (z = 0) is the same everywhere (p0). In the cold area less
vertical distance is needed to have the same decrease in pressure (dp).
Figure 2. The effect of a horizontal temperature difference (left) and a
horizontal temperature gradient (right) on the height and tilt of a pressure
surface (assuming equal surface pressure (p0).
The thermal wind
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If temperature changes in the horizontal, then the pressure surface aloft is no longer
horizontal but will have a tilt (Figure 2 right). A horizontal temperature gradient influences
the tilt of all pressure surfaces. From Figure 3 it appears that the tilt of the pressure
surfaces increases higher up in the atmosphere. This will have consequences on the
magnitude of the geostrophic wind: in the case sketched in Figure 3 it will increase with
height. From Figures 1 and 3 the following conclusion can be drawn:
A horizontal temperature gradient causes a change of the geostrophic wind with
height (or with pressure).
The difference between the geostrophic wind at the two pressure surfaces is called the
thermal wind.
Figure 3. The effect of a horizontal temperature
gradient on the tilt of subsequent pressure surfaces.
The magnitude of the thermal wind thus depends on the value of the horizontal
temperature gradient. An expression for the magnitude of the thermal wind is determined
by differentiating the expression of the geostrophic wind with relation to pressure (p):

  g z 
g   z 
g   z 
  
   
 
u g   
p
p  f y 
f p  y 
f y  p 
(5)
For the factor z p we use the assumption of hydrostatic equilibrium:
z
1
1 Rd T


.
p
g
g p
Here we have used the gas law. Substituting this in Equation (5) and multiplying by p
leads to:
The thermal wind
p
ug
p

ug
 ln p

Rd T
.
f y
4
(6)
We integrate this expression from p0 to p1, with p0>p1:
p1
 u g 
p0
Rd
f
p1
T
 y  ln p .
p0
This leads to an expression for the components of the thermal wind (uT, vT):
uT  ug ( p1 )  ug ( p0 )  
Rd T
ln( p0 p1 )
f y
(7)
vT  vg ( p1 )  vg ( p0 )  
Rd T
ln( p0 p1 ) ,
f x
(8)
and
In the above integration we have used an average value ( T ) for the temperature in the
layer p0-p1 in order to simplify the integration. From these expressions it is obvious that
the thermal wind depends on the horizontal temperature gradient. The following rule also
follows from these equations:
The thermal wind blows parallel to the isotherms with the cold air on the left hand
side. The closer the isotherms the stronger the thermal wind.
Advection of warm and cold air
From Equations (7) and (8) it appears that the direction of the thermal wind is parallel to
the isotherms. This leads to the following two possible situations.
Figure 4. Wind backing with height is an
indication of cold air advection.
Figure 5. Wind veering with height is an
indication of warm air advection.
The thermal wind
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In Figure 4 the wind is backing going from p0 (lower level) to p1<p0 (higher level), in this
case from west to southwest. The thermal wind is the wind vector on the high level
minus the wind vector on the low level. The thermal wind blows parallel to the isotherms
with the cold air on the left hand side. The geostrophic wind on both levels is blowing
from the cold area: this is a case of cold air advection. In Figure 5 the wind is veering
from west northwest. The thermal wind again has cold air on its left hand side, and in
this case it appears that the wind on both pressure levels is blowing from the warm area:
this is a case of warm air advection.
The value of the temperature advection is determined by the value of the thermal wind
(i.e. by the distance of the isotherms, i.e. by the horizontal temperature gradient) and by
the component of the geostrophic wind which is at right angles to the isotherms. From
Figure 6 it appears the area of the triangle with sides vg(p0), vg(p1) and vT must be
proportional to the value of the advection: the larger the area of the triangle the larger
the temperature advection.
Figure 6. The area of the triangle is
proportional to the value of the advection.
Qualitative application: hodograph and stability
Advection of air from a different average temperature in a number layers in the vertical
influences the stability of the atmosphere. It is relatively easy to gain an overview of the
advection in different layers by performing what is called a hodograph analysis. This
involves the construction of a radial diagram with the station in the centre. From this
centre the wind speed at several or all pressure levels is plotted as a vector (Figure 7).
The vector is plotted in the direction of the wind. This means that a westerly wind is
plotted as an arrow to the east (Figure 7). Going to the next higher pressure level the
next arrow is plotted etc. The end points of all vectors are connected by straight lines (or
arrows). These straight lines are the vectors of the thermal wind (ref. Figure 6). The end
result is called a hodograph.
The next step is to consider the veering or backing of the wind, starting from the lowest
level to ever higher pressure levels. In this way the warm or cold air advection in each
layer can be determined. By comparing the areas of the triangles in relation to warm or
cold air advection it is possible to determine the changes in atmospheric stability. As an
example in Figure 7 between 70 and 50 kPa there is more cold air advection than
The thermal wind
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between 90 and 70 kPa. This means that in this case the vertical temperature gradient
must increase and instability is increasing.
Figure 7. Example of a hodograph. Wind vectors on all pressure
levels (black, pressure in kPa) are all plotted from the centre, thermal
wind vectors (red) connect the black arrows.
In Figure 8 a number of examples of parts of hodographs are presented and the
following conclusions about the increase of decrease of stability can be drawn:
Figure 8. Examples of the application of hodograph analysis for determining the change
in stability of a number of layers (see text). The layers are indicated by increasing
numbers from bottom to top.
A: This is the case where the thermal wind is equal in each layer. The areas of the three
triangles are therefore equal. It means that: at every level the same amount of cold air
The thermal wind
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advection is taking place. The temperature will drop everywhere, but the vertical
temperature gradient and hence the stability will not change.
B: In this case the thermal wind, the areas of the triangles and hence the cold air
advection increase from bottom to top. As a result this part of the atmosphere will
become more unstable (or less stable) because temperature decrease near the top will
be stronger than near the bottom.
C: Each layer shows cold air advection, but in the top layer this cold air advection is
larger than in the middle layer: above level 1 the instability will increase. In the bottom
layer the cold air advection is larger than in the middle layer: below level 2 the instability
will decrease.
D: This is a case of warm air advection. Above level 1 the instability will decrease
because temperatures in the top layer will rise more than temperatures in the middle
layer. Below level 2 the instability will increase because the temperatures in the middle
layer will rise less than temperatures in the bottom layer.
In practice hodograph analysis will indicate if, at that moment, the stability on a certain
location will increase or decrease. Changes in the environmental lapse rate have direct
consequences for the stability and e.g. may be important for the development of
convective clouds and showers. Using hodograph analysis it may be possible to
determine if a warm front is approaching or if a cold front has passed.
Quantitative application: determination of change in temperature
From the thermal wind equations (7) and (8) the value of the thermal wind components
can be calculated for a given horizontal temperature gradient. If we rewrite this equation,
then the value of the horizontal temperature gradient can be calculated for given values
of the thermal wind components. If in an air layer the value of the component of the wind
which is at right angles to the isotherms (VN) has been determined (Figure 6), then the
change in temperature in that layer can be calculated with the following expression:
T
T
f
1
  VN
  VNVT
t
n
Rd ln( p0 p1 )
(9)
This calculation assumes that no other processes such as vertical motions and diabatic
processes are acting at the same time, as these may also effect the change in
temperature.