Ref: Synoptic-Dynamic Meteorology in Midlatitudes by Bluestein, p. 57

In rawinsonde (radio wind) observation, the following variables are transmitted from the weather balloon at fixed time intervals:

pressure

temperature

relative humidity

The height of the balloon is than computed using these variables and the hypsometric equation. The hypsometric equation can be derived from the hydrostatic equation.

Hydrostatic Equation:

or

or

Hypsometric Equation:

This equation is obtained first by dividing the above equation by the equation of state:

to get

Integrating from level z1 (pressure p1) to z2 (pressure p2), to yield:

The values of is called the thickness of the pressure layer from to . Where the layer is warm, the thickness is large. Inversely, where the layer is cold, the thickness is small. The thickness map is often used to indicate the layer averaged virtual temperature.

Illustration:

Question: What is virtual temperature?

Question: What is the 1000-500 mb thickness if the average layer virtual temperature is 0°C?

Question: What is the average layer virtual temperature if the 1000-500 mb thickness is 540 dm?

Addition to pressure on constant z surface and geopotential height on constant p surface:

If the constant pressure surface is slanted toward east as illustrated in the following figure, then on the constant z surface. At the same time, on the constant p surfaces. For this reason, an area of high geopotential height on a constant p surfaces is the area of high pressure. When we look at a constant pressure map (such as the 500 mb map), we often point to the area of high geopotential height and say that is the area of high pressure.

Reduction of Pressure to Mean Sea Level:

Starting from the hypsometric equation:

Let subscript 1 be the station (stn) level and 2 be the mean sea level (msl)

Question: How in the above equation is determined?

Altimeter Setting:

The same equation is used but is computed from the standard atmospheric value

Thermal Wind

Definition: The geostrophic wind difference (or vertical wind shear) between two pressure levels.

Considering the wind in the x direction, if is the upper level geostrophic wind and is the lower level geostrophic wind, the thermal wind is defined as:

Similar for the wind in the y direction.

The two pressure levels can be any two levels. The commonly used levels on National Weather Service maps are the 500 and the 1000 mb levels.

Question: What is thermal going to do with it?

Substitute the geostrophic wind equation

into the thermal wind equation, we get:

The relationship between the thickness and the thermal wind is identical to the relationship between the geopotential height and the geostrophic wind.

We now rewrite the hypsometric equation

to

Substitute this equation into the thermal wind equation, we get:

Now, the relationship between the layer averaged temperature and the thermal wind is similar to the relationship between the geopotential height and the geostrophic wind, except that is replaced by . Therefore, many properties of the geostrophic wind will apply to the thermal wind, such as:

1. The thermal wind will blow along the layer averaged isotherms (or the thickness contours) with the low temperature (or low thickness contour) to the left hand side of the thermal wind.

2. The thermal wind speed is proportional to the gradient of the layer averaged isotherms (or the thickness contours) and inversely proportional to the Coriolis parameter.

The combined thermal wind () is the vector sum of the x and y components of the thermal winds. In the following illustration, if the x component of the thermal wind is 10 m/s and the y component of the thermal wind is 5 m/s, the combined thermal wind will be 11.2 m/s from 243.4 °.

If we set the s direction of the Natural coordinate as the direction of the thermal wind, the thermal wind equation in the Natural coordinate can be derived by rotating the axis of the thermal wind equation so the x axis is toward the s direction. The thermal wind equation than can be written as:

or

The relationship between the geostrophic wind and the thermal wind in general is more difficult to be expressed in equation form without resolving vector notation. But we can simply state that:

The upper level geostrophic wind is the vector summation of the lower level geostrophic wind plus the thermal wind between these two levels.