Margules Equation for Frontal Slope
¶ug
g ¶T
=¶z
f 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.
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With these constraints, in Fig.1 , there would be no southerly/northerly
winds and the west wind, uw , would be uniform at all elevations in the warm
air and the west wind, uc , 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:
¶ ug u2 - u1 uw - uc
=
=
¶z
Dz
Dz
¶T
T2 - T1 Tc - Tw
¶ y = Dy = Dy
¶ ug u2 - u1 uc - uw
=
=
¶y
Dy
Dz
Put (2a,b) into (1) and solve for the frontal slope
Dz
f T æ uw - uc ö
=
Dy g çè T - T ÷ø
c
w
Dz
f T æ uc - uw ö
=ç
÷
Dy
g è Tc - Tw ø
3
(3a,b)
(2a,b, c)
Put (2a) into (2c)
æ uc - uw ö
=
Dy çè Dy ÷ø
(4)
Du = - Du
Dz
Dy
Du
Put (4) and (2b) into far right term in brackets in (3b) and convert to differential
calculus form.
æ
ö
æ uw - uc ö ç -¶u ¶y ÷
ç
÷ » ç ¶T ÷
è Tc - Tw ø
è ¶y ø
z = ¶v¶ x - ¶u ¶ y
(5)
(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
æ -¶u ö
æ
ö
¶y÷ fT ç z ÷
Dz » f T ç
Dy g ç - ¶ T ÷ = g ç - ¶ T ÷ (7)
è
¶ yø
è
¶ yø
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.
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