Semigeostrophic frontogenesis Semi geostrophic Quasi geostrophic fvz bx

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 y B x (northern hemisphere case) . . x=0 A

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

C D cold warm z B x . . A x=0 Ø 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 x . . A x=0 Ø Similar arguments apply to the neighbourhood of C on the cold side of the maximum temperature gradient at upper levels. Ø 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 z (a) z X 1 X 2 X (b) x 1 x 2 x 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 = 12 o. C

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

A 1000 -500 mb thickness chart over Australia H H H

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?

z (a) z X 1 X 2 X X = x + vg(x, z)/f (b) x 1 x 2 x

Figure 7. 2

See next

warm cold x Figure 7. 3

-1

η ξ Streamlines

Term often small

Solution by coordinate transform

Next figure

Figure 7. 10

The End