Margules Equation for Frontal Slope

(1)

The Thermal Wind Relation

Cross-section showing frontal zone. The assumption is that the atmosphere is barotropic (or equivalent barotropic) except in the frontal zone. If that is the case, the geostrophic wind is independent of height north and south of the frontal zone (thermal wind is zero) and the thermal wind relation can be used to deduce the characteristics of the atmosphere around the frontal zone.

(2a,b)

Put
(2a,b) into (1) and solve for the frontal slope

(3a,b)

But,
from fig. 1,

(4)

Put
(4) into (3b) to obtain

(5)

Margules Equation for Frontal
Slope

Where
the numerator is positive if the
west wind decreases northward (which means that the relative vorticity is
positive), and the denominator is positive if temperatures decrease
northward. The numerator is just
the relative vorticity and the denominator is the zonal temperature gradient. If the frontal slope is positive, then
the height of the front above the ground increases northward (towards the cold
air)

Equation
(5) states that the front slopes upwards toward lower temperature. The slope is inversely proportional to
the strength of the temperature gradient (shallower slope for stronger fronts)
and directly related to the relative geostrophic vorticity. For zonal flow, the relative geostrophic
vorticity is directly related to the wind shear, with strongest positive values
north of the jet and greatest negative values south of the yet. Thus, the front slopes steeply upwards
on the poleward side of the jet in the upper troposphere.

Incidentally,
since the zonal temperature gradient changes sign in the stratosphere during
the winter, it is common to note that the frontal zone slopes back towards the
equator over the tropopause.

The
impact of friction modifies the frontal slopes as determined from MargulesŐ
Equation. It acts to create steeper
slopes for cold fronts and shallower slopes for warm fronts, for a given zonal
temperature gradient as shown below.