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.