Compute advection in Natural coordinate

Recall that the expansion of the total derivative in dry adiabatic process can be written as:

or

In Natural coordinate, these equations are written as:

and

Considering only the horizontal advection, the equations become

Working on the previous example, we need to compute only one advection term if we use the natural coordinate and make the s direction along the wind.

The finite difference form to compute the advection can be written as:

using the forward time and centered space difference, or

using the forward time and upstream space difference. In using the second equation, we may find it advantageous to choose as the distance across two isotherms. That will predetermine the value of and all we have to do is to measure the distance of .

Estimate the advection and change of stability using a hodograph

Looking back at the equation for temperature advection

On a constant p surface, it can be written as

Remember the thermal wind equation?

In reference to the above figure, let us use the Natural coordinate and choose the s direction along the thermal wind (and along the isotherms with cold air to the left). Rotating the x axis to the s direction, we get the advection equation as

is the average wind speed perpendicular to the thermal wind.

The sign of

If the wind veers with height, is positive and there is warm advection. If the wind is back with height, is negative and there is cold advection.

Ignoring the difference between temperature and virtual temperature, the temperature change due to advection can be written as:

or

Procedure to compute the advection using this equation.

1. Plot a hodograph showing the wind and the thermal wind.

2. For the layer of interest, measure , the wind speed perpendicular to the thermal wind. Be sure that is positive if the wind is veer with height (warm advection) and negative is the wind is back with height (cold advection)

3. Compute advection.

This procedure can be used to determine the temperature change due to advection in any layer. If there is warm advection in the lower layer, or cold advection in the upper layer, or both, the sounding will become more unstable.

Temperature advection using 1000-500 mb thickness

The hypsometric equation derive in Lecture 6

can be rewritten as

where is the thickness. Using the conclusion that, for a fixed layer where is a constant, the thickness is proportional to the layer averaged temperature. Take derivative with respect to n, we get

Again ignore the difference between temperature and virtual temperature, we can eliminate from this equation and the equation

to get

Look at this figure, we find that , , and all have the same , the wind component perpendicular to the thermal wind (or in the n direction). The component along the thermal wind (or in the s direction) does not contribute the temperature advection. Therefore, we will get the same temperature advection no matter we use the 1000 mb geostrophic wind, 500 mb geostrophic wind, or the mean geostrophic wind.

Using the surface pressure and 1000-500 mb thickness chart to compute advection.

The surface pressure pattern should be similar to the 1000 mb geopotential height pattern and the geostrophic wind on these two maps should be similar. The following explanation is quite straightforward in vector calculus. It is a little messy without it. Using the geostrophic equation in x, y, z coordinate

and rotate the x axis to the direction of the thermal wind, we get

We have two components here because now the s direction in not along the wind.

Substitute the second equation into the temperature change equation

we get

In finite difference form, it is

If we consider the rhomboid formed by the pressure and thickness lines in the above figure, = 400 Pa, and = 60 m. is the distance between points a and b, and is the distance between the points b and b'. is therefore the area of the rhomboid. We can conclude that the temperature advection is inversely proportional to the area formed by the intersection of the isobars and thickness lines.

We can similarly superimpose the 500 mb height contour map to the 1000-500 mb thickness map to look for the temperature advection. However, these two set of lines are often parallel and it will be hard to determine the magnitude of the advection.

The effect of temperature advection on vertical velocity

Horizontal warm advection is one of the key factors in creating upward motion. In an area of warm advection, cold air is replaced by the warm air and the surface pressure will have to fall. That will create a low pressure in this area, which in turn will create convergence and upward motion. The magnitude of the upward motion is proportional to the intensity of the warm advection and inversely proportional to the area of warm advection. The first point is obvious. The second point can be illustrated below:

Suppose the pressure is constant at the beginning. The two circles are areas of warm advection and the pressure drop due to the warm advection. Even though the pressure drops are the same, the pressure gradient in area B is 2/3 of that in area A. The magnitude of the convergence and vertical velocity in area B will be proportionally smaller. The wind created by pressure change is called the isallobaric wind.

Cold advection will increase the surface pressure and produce divergence and downward motion.

The effect of vertical advection on temperature

In area of warm advection, the associated upward motion will cool the atmosphere and partially offset the effect of the warm advection. The temperature change due to vertical velocity can be estimated first using the potential temperature change equation and convert to temperature.

or

The effect of radiation on temperature

Without considering the cloud, radiation has a net cooling effect in the free atmosphere of about 1°C per day. Radiation will also have a net warming effect near the surface. Radiation is the cause of the diurnal surface temperature variation. If the atmospheric lapse rate is large (more unstable), the radiational effect on temperature is small. If the lapse rate is small or it has an inversion (more stable), the radiational effect of temperature is large. It is illustrated below:

If we know the value of radiation (short wave and long wave), we can compute the exact temperature change due to radiation. The other way is to check the climatic diurnal variation of temperature and adjust it according to the stability, soil moisture, vegetation, and cloud cover.