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:
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:
This leads to:
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 Μz/Μx < 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 relation with temperature
According to the equation of state (i.e. the gas law) the temperature determines the air density (ρ) for a given value of the pressure (p):
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).
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.
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):
For the factor ∂z/∂p we use the assumption of hydrostatic equilibrium:
Here we have used the gas law. Substituting this in Equation (5) and multiplying by p leads to:
We integrate this expression from p0 to p1 with p0>p1:
This leads to an expression for the components of the thermal wind (uT, vT):
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.
In Figure 4 the wind is backing going from p0 (lower level) to p1< p0 (higher level),
inthis 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.
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 between 90 and 70 kPa. This means that in this case the vertical
temperature gradient must increase and instability is increasing.
