More on Fronts and Frontogenesis

Chapter 14

More on Fronts and Frontogenesis

Semi-geostrophic frontogenesis

 The so-called semi-geostrophic theory of frontogenesis is obtained from the unapproximated forms of the frontal equations:

 in other words, we do not approximate D/Dt by D _g /Dt and

therefore advection by the total wind is included.

0    _z P  

D

Dt   N w ₀ ²  0

   fv  _x P

Dv

Dt  fu   _y P

 _x u   _y v   _z w  0

   fv  _x P

D v

Dt ^g  fu _a  0

0    _z P  

D

Dt ^g  N w

 ₀ ²  0

 _{x a} u   _z w  0

 _{x g} u   _{y g} v  0

   fv  _x P

0    _z P   ^{fv z =  x}

y

f of Dv fu P

z Dt

    



As before, cross-front thermal-wind balance  fv _z =  _x

D

Dt ( fv _z )   Q ₁  F u ² _az  S w ² _z

F ²  f f v (  _x ) S ²  fv _z   _x

D

Dt  _x  Q ₁  u _{ax x}   w N _x ²

N ²  N ₀ ²   _z

Now

2 o

of D N w 0

x Dt

    

also 

D

Dt ( fv _z )   Q ₁  F u ² _az  S w ² _z

2

x 1 ax x x

D Q u w N

Dt     

is the total Brunt-Väisälä frequency, rather than that based on the basic state potential temperature distribution.

N ²  N ₀ ²   _z

To maintain thermal-wind balance ( fv _z =  _x )

N ²  _xx  2 S ²  _xz  F ²  _zz   2 Q ₁

2 2 2

1 ax x x 1 az z

Q  u   w N   Q  F u  S w

N ²  _xx  2 S ²  _xz  F ²  _zz   2 Q ₁

This is the 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 as discussed in a later course (Advanced Lectures on Dynamical Meteorology).

1 2 2 4

q f (F N  ^  S ) 0 

N ₀ ²  _xx  f ²  _zz   2 Q ₁

Compare with the QG-circulation equation

z

X

z

x X ₁ X ₂

x ₁ x ₂

(a) The circulation in the (X, Z) plane in a region of active frontogenesis (Q

> 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.

X = x + v _g (x,z)/f

(a) (b)

y

x

u _g = x v = y _i ¹

o

2 x

(x) tan

L

  

       

Frontogenesis in a deformation field

 = 12 ^o C

H H

H

A 1000-500 mb thickness chart over Australia

The End