Margules Equation for Frontal Slope u g z = g f T T y (1) Equation (1) is the thermal wind relation for the west wind geostrophic component of the flow. For the purposes of this derivation, we assume that the real winds are geostrophic when we obtain estimates of the left hand side derivative. The thermal wind relation provides us with a powerful tool in understanding the relationship between flow at upper and lower levels of the troposphere. As you will see, it also provides us with a conceptual tool for understanding frontal slopes. Consider Fig. 1, which shows a schematic frontal zone in the northern hemisphere. Fig. 1: Cross-section showing frontal zone. with a slope z/ y
We make the assumption that the atmosphere is barotropic (or equivalent barotropic) except in the frontal zone that has a slope z/ y. If that is the case, the height surfaces have constant slope in the warm air mass (but with a higher mean temperature) and a constant slope in the cold air mass (but with a lower mean temperature). Thus, the geostrophic wind is independent of height in the cold air north and south of the frontal zone (thermal wind is zero in the cold and warm air masses). However, the thermal wind relation can be used to deduce the characteristics of the atmosphere around the frontal zone. That is because the frontal zone itself is greatly baroclinic with changing height gradients for each of the isobaric surfaces across the frontal zone. Recall this figure from Metr 201. Our refinement of it involves the SLOPE of the frontal zone between the surface cold front (the X) and the northern bound of the baroclinic zone. In Metr 201 we just assumed this was a vertical wall. 2
With these constraints, in Fig.1, there would be no southerly/northerly winds and the west wind, u w, would be uniform at all elevations in the warm air and the west wind, u c, would be uniform at all elevations in the cold air. From Fig. 1, we can compute the vertical shear of the west wind through the baroclinic zone by constructing simple finite difference increments, as shown in the figure. Then, the derivatives in Equation (1) can be estimated as follows: u g z = u u 2 1 Δz = u w u c Δz T y = T 2 T 1 u g y = u u 2 1 = T c T w = u c u w Δz (2a,b, c) Put (2a,b) into (1) and solve for the frontal slope Δz Δz = f T # u w u c & % ( g $ T c T w ' Δz = f T # u c u w % g $ T c T w & ( ' (3a,b) 3
Put (2a) into (2c) # Δu = u u & c w % ( $ ' Δu (4) = Δu Δz Put (4) and (2b) into far right term in brackets in (3b) and convert to differential calculus form. # % $ u w u c T c T w # & u & y ( % ( ' % T $ y ' ( (5) ζ = v x u y (6) Remembering that there is no v component of the wind for zonal flow and so v/ x=0, put (6) into (5). Then put the result into (3b) to obtain 4
Δz f T g u y T = f T g y ζ T y (7) Margules Equation for Frontal Slope 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) The numerator is positive if the west wind decreases northward (which means that the relative vorticity is positive). The denominator is positive if temperatures decrease northward. Equation (7) states that the front slopes upwards towards 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 Margules 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. 5
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