Front

Properties of fronts: A front is characterized by a large temperature gradient and a strong cyclonic wind shear.

Surface map showing cold and warm fronts.

Vertical cross-section of the cold and warm fronts.

Margules formula of frontal structure

Let us look at the pressure, which is continuous across a front. (NOTE: temperature and wind are discontinuous along a front).

Now using the geostrophic wind approximation

and the hydrostatic approximation

we can get

Now use the equation of state

to get

Implication:

1. Since and , it implies that

and there must be cyclonic wind shear across a front. This is the reason that the isobars across the front must be kinked.

2. If the temperature contrast across the front is a constant, large horizontal wind shear across the front implies steep frontal slope.

3. If the wind shear across the front is a constant, small temperature contrast across the front implies steep frontal slope.

4. If the frontal slope is a constant, large temperature contrast across the front implies large horizontal wind shear across the front (large vorticity).

Frontogenesis equation in two-dimension

Let us consider a weak front along the east-west direction and be the indicator of the frontal strength.

Let us start with

Taking partial derivatives of the above equation with y, we can get the frontogenesis equation:

is frontogenesis

is frontolysis

Term (1) is the confluence term. Confluence across the front produces frontogenesis.

Confluence is used here to indicate the horizontal convergence in one direction and divergence in another direction. It is not necessary for the confluence to produce a vertical velocity. Confluence can be associated with deformation and mathematically, deformation can be written as:

Term (2) is the tilting term. Upward motion in the cold air area and downward motion in the warm air area implies frontogenesis.

Term (3) is the diabatic term. Cooling in the cold area and warming in the warm area implies frontogenesis.

The relative strength of fronts on the surface and on the upper levels

If a weak front in strengthened by a confluence as shown below. If there is no vertical velocity, the confluence term will intensify the front at the same rate in all the levels. However, in the north, the cold advection in the north will produce the sinking motion and the warm advection in the south will produce a rising motion. A frontolysis effect will be produced due to the tilting term and weaken the front. On the surface, the vertical velocity is zero because of the surface boundary condition and tilting term is zero. The frontogenesis effect will continue through the confluence term. The next results is that the surface front is much stronger than the front in other levels.

Another way to think of the frontal structure

Let us start will a vertical wall separating the cold air to the west and the warm air to the east. Below the cold air, there is a relatively high pressure and below the warm air, there is a relatively low pressure. The pressure gradient is the largest on the surface but the wall prevents the pressure gradient to produce a motion.

If the wall is taken away, what will happen?

1. The cold air will gush toward the warm air in the low levels and tilt the front. The complete circulation in a vertical cross-section is shown below:

2. The Coriolis force will turn the wind to the right hand side and produce a strong north wind on the surface and a strong south wind in the upper levels. In either case, there is cyclonic circulation.

3. If the temperature contrast is large, the pressure gradient is also large and that will produce strong wind and small frontal slope. If the temperature contrast is small, the frontal slope will be large. In the extreme case of infinitesimally small temperature contrast, the front will stay vertical.

4. The vertical motion produced by the frontal circulation will almost always weaken the front in the upper levels.

Precipitation over the fronts

Occlusion

Occlusion of the fronts will lift the warm air above the cold air, making the atmosphere more stable.

Sounding across a front

One way to detect the height of the front in to look for the temperature inversion across the front. The moisture above the frontal inversion is likely to be high. This is different from the subsidence inversion, above which the moisture is very low.

Jet stream and fronts

The thermal wind equation tells us that below the upper level jet stream, there is a strong temperature contrast or a front. This front is usually called the polar front, which may be unstable and produce cyclogenesis.