Semi-geostrophic frontogenesis

Semi geostrophic Quasi geostrophic fvz = bx

Cross-front thermal-wind balance  fvz = bx

is the total Brunt-Väisälä frequency, rather than that based on the basic state potential temperature distribution. To maintain thermal-wind balance ( fvz = bx )

This is an equation for the vertical circulation in the semi-geostrophic case. It is elliptic provided that the so-called Ertel potential vorticity, This condition which ensures that the flow is stable to symmetric baroclinic disturbances. Compare with the QG-circulation equation

Frontogenesis in a field of geostrophic confluence adiabatic warming adiabatic cooling C D cold warm z . A The role of the ageostrophic circulation may be seen by reference to the figure, which depicts a frontogenetic situation in which there is large-scale geostrophic confluence in the x-direction, represented by Ql > 0 , tending to increase the temperature gradient sx. The circulation in the (x, z) plane given by (14.4.11) is as shown. The circulation is direct, in the sense that rising motion occurs in the warm air and subsidence in the cold air. Thus, according to (14.3.5), there is adiabatic cooling on the warm side and adiabatic warming on the cold side of the convergence line (x = 0). As described by the second term on the right-hand-side of (14.4.8), this effect opposes the increase in horizontal temperature gradient represented by the term Ql in this equation, except near the horizontal boundaries where w is small and frontogenesis is unimpeded. The ageostrophic circulation has also an effect on the shear; the Coriolis torque implied by the sign of ua in Eq. (14.4.3) serves to increase the shear vz , which, according to (14.4.6) would otherwise be reduced at the rate Ql/f. B .  y x x = 0 (northern hemisphere case)

C D cold warm z . A B . x x = 0 The ageostrophic velocity ua is clearly convergent (uax < 0) in the vicinity of A on the warm side of the maximum Tx (bx). If included in the advection of b it would lead to a larger gradient bx.

C D cold warm z . B . A x x = 0 At A, the generation of cyclonic relative vorticity z is underestimated because of the exclusion of the stretching term zwz in the vertical vorticity equation,

C D cold warm z . B . A x x = 0 Similar arguments apply to the neighbourhood of C on the cold side of the maximum temperature gradient at upper levels. In the vicinity of B and D, the ageostrophic divergence would imply weaker gradients in z and the neglect of zwz would imply smaller negative vorticity.

Transformation to geostrophic coordinates x1 x2 z z X1 X2 X x (a) (b) X = x + vg(x,z)/f (a) The circulation in the (X, Z) plane in a region of active frontogenesis (Ql > 0) . (b) The corresponding circulation in (x,z)-space. The dashed lines are lines of constant X which are close together near the surface, where there is large cyclonic vorticity.

Frontogenesis in a deformation field y x ug = -ax v = ay Dq = 12oC

q(x,z) v(x,z)

A 1000-500 mb thickness chart over Australia

Semi-geostrophic frontogenesis

Frontogenesis in a deformation field y x ug = -ax vg = ay Dq very small

Semi-geostrophic equations

Define deviation flow (′) basic deformation flow

Equations for deviation flow

A third conservation property:

Geostrophic coordinates

PV

This is a Poisson equation for 

: DM Sign slip?

x1 x2 z z X1 X2 X x (a) (b) X = x + vg(x,z)/f

Figure 7.2 Figure 7.2

See next

warm cold x Figure 7.3

-1

η ξ Streamlines

Term often small

Solution by coordinate transform

q = …

Next figure

Figure 7.10

The End