More on Fronts and Frontogenesis

Chapter 14

More on Fronts and Frontogenesis

 The foregoing theory is concerned solely with the

kinematics of frontogenesis and shows how particular flow patterns can lead to the intensification of horizontal

temperature gradients.

 We consider now the dynamical consequences of increased horizontal temperature gradients

 We know that if the flow is quasi-geostrophic, these increased gradients must be associated with increased vertical shear through the thermal wind equation.

 We show now by scale analysis that the quasi-geostrophic approximation is not wholly valid when frontal gradients become large, but the equations can still be simplified.

Dynamics of frontogenesis

 The following theory is based on the review article by Hoskins (1982).

 It is observed, inter alia, that atmospheric fronts are marked by large cross-front gradients of velocity and temperature.

 Assume that the curvature of the front is locally unimportant and choose axes with x in the cross-front direction, y in the along-front direction and z upwards:

z

y x

front

L

l

cold warm

V

U

u_h v

u y

x Frontal scales and coordinates

Observations show that typically, U ~ 2 ms ^-1

V ~ 20 ms ^-1 L 1000 km

 ~ 200 km

=> V >> U and L >> .

The Rossby number for the front, defined as

The relative vorticity (~V/) is comparable with f and the motion is not quasi-geostrophic.

Ro  V f/  ~ 20  (10^4  2 10⁵) is typically of order unity.

Du

Dt fv U fV

U V

V / ~ ² / ² f

 1



F

H GIKJ

and Dv

Dt fu UV fU

V / ~ / f

 ~

  1

A more detailed scale analysis is presented by Hoskins and Bretherton (1972, p15), starting with the equations in

orthogonal curvilinear coordinates orientated along and normal to the surface front.

The motion is quasi -geostrophic across the front, but not along it.

The ratio of inertial to Coriolis accelerations in the x and y directions =>

 The scale analysis, the result of Exercise (14.3), and making the Boussinesq approximation, the equations of motion for a front are

0   

P  

D

Dt  N w₀²  0

   fv 

P

Dv

Dt  fu  _yP

_xu  _yv  _zw  0

P  p / 

N₀ = the Brunt-Väisälä frequency of the basic state

N₀²  ( /g ₀)(d₀ / dz)

buoyancy force per unit mass

 I assume that f and N₀ are constants.

 While the scale analysis shows that frontal motions are not quasi-geostrophic overall, much insight into frontal

dynamics may be acquired from a study of frontogenesis within quasi-geostrophic theory.

 Such a study provides also a framework in which later modifications, relaxing the quasi-geostrophic assumption, may be better appreciated.

Quasi-geostrophic frontogenesis

D

Dt t u

x v

y

g    g  g











where v_g = v is computed from as it stands and u_g  ( / )1 f _yP

Set u  u_g  u_a

Dv

Dt  fu_a  0

_{x a}u  _zw  0 and

The quasi-geostrophic approximation involves replacing D/Dt by

fv  

P

0   

P  

D

Dt  N w₀²  0

   fv 

P

Dv

Dt  fu  _yP

_xu  _yv  _zw  0

   fv 

P

D v

Dt^g  fu_a  0

0   

P  

D

Dt^g N w

 ₀²  0

_{x a}u  _zw  0

_{x g}u  _{y g}v  0

   fv 

P

0   

P  

Let us consider the maintenance of cross-front thermal wind balance expressed by fv_z = _x .

D

Dt^g (fv_z)  Q₁  f u² _az

D

Dt^g  _x Q₁  N w₀² _x

Note that u_gx + v_y = 0

These equations describe how the geostrophic velocity field

acting through Q_l attempts to destroy thermal wind balance by changing fv_z and _x by equal and opposite amounts and how ageostrophic motions (u_a, w) come to the rescue!

Q u v v

gx x x y x y

1        



( , ) ( , )

N w₀² _x  f u² _az  2Q₁

Also from u_gx + v_y = 0, there exists a streamfunctionfor the cross-frontal circulation satisfying

( , ) (u w_a  _z, _x)

N₀²_xx  f²_zz  2Q₁

This is a Poisson-type elliptic partial differential equation for the cross-frontal circulation, a circulation which is forced by Q_l.

Q

 u



 v



y

x

u_g = x

v = y ^{Q u v}

v

gx x x y x y

1        



( , )

( , ) = _x

Frontogenesis in a deformation field

cold warm

. .

cold warm

z

x

B A

C D

adiabatic warming adiabatic cooling

(northern hemisphere case)

x = 0

Frontogenesis in a field of geostrophic confluence

 If w = 0, Q_l is simply the rate at which the buoyancy (or

temperature) gradient increases in the cross-front direction following a fluid parcel, due to advective rearrangement of the buoyancy field by the horizontal motion.

Q u v v

gx x x y

x y

1

       



( , )

( , )

 _x increases due to confluence (u_x < 0) acting on this

component of buoyancy gradient and due to along-front horizontal shear v_x acting on any along-front buoyancy gradient _y.

 D_x/Dt is an alternative measure of frontogenesis to the Boussinesq form of the frontogenesis function D|_h|/Dt analogous to the left hand side of this, i.e., T₁ + T₂ + T₃ + T₄.

D

Dt^g  _x Q₁  N w₀² _x

1 gx x x y

Q  u    v

 The quasi-geostrophic theory of frontogenesis in a field of pure geostrophic deformation was developred by Stone (1966), Williams and Plotkin (1968), and Williams (1968).

 The solutions obtained demonstrate the formation of large horizontal gradients near boundaries, but away from

boundaries, the induced ageostrophic circulation prevents the contraction of the horizontal length scale of the

temperature field below the Rossby radius of deformation, L_R = NoH/f; where H is the depth of the fluid.

 Because the ageostrophic circulation does not contribute to advection in quasi-geostrophic theory, the largest

horizontal temperature gradient at each height remains coincident with the line of horizontal convergence (x = 0).

Limitations of quasi-geostrophic theory

 Many unrealistic features of the quasi-geostrophic theory result from the omission of certain feedback mechanisms.

 The qualitative effect of some of these feedbacks can be deduced from the quasi-geostrophic results.

. .

cold warm

z

x

B A

C D

x = 0

 The ageostrophic velocity u_a is clearly convergent (u_ax < 0) in the vicinity of A on the warm side of the maximum T_x (_x).

 If included in the advection ofit would lead to a larger gradient _x.

. .

cold warm

z

x

B A

C D

x = 0

 At A, the generation of cyclonic relative vorticityis

underestimated because of the exclusion of the stretching term _wz in the vertical vorticity equation,

D

Dt  f w



  ( )

 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and the neglect of _wz would imply smaller negative vorticity.

. .

cold warm

z

x

B A

C D

x = 0

D

Dt  (f )w_z

 In summary, QG-theory points to the formation of sharp

surface fronts with cyclonic vorticity on the warm side of the temperature contrast, and with the maximum horizontal

temperature gradient sloping in the vertical from A to C, even though these effects are excluded in the QG-solutions.

 The theory highlights the role of horizontal boundaries in

frontogenesis and shows that the ageostrophic circulation acts to inhibit the formation of large gradients in the free

atmosphere.

 Hoskins (1982) pointed out that unless the ageostrophic convergence at A and C increase as the local gradients

increase, the vorticity and the gradients incan only increase exponentially with time.

 Quasi-geostrophic theory does not even suggest the formation of frontal discontinuities in a finite time.

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   

P  

D

Dt  N w₀²  0

   fv 

P

Dv

Dt  fu  _yP

_xu  _yv  _zw  0

   fv 

P

D v

Dt^g  fu_a  0

0   

P  

D

Dt^g N w

 ₀²  0

_{x a}u  _zw  0

_{x g}u  _{y g}v  0

   fv 

P

0   

P  

y

f of Dv fu P

z Dt

    



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

D

Dt (fv_z)  Q₁  F u² _az  S w² _z

F

 f f v ( 

)

D

Dt _x  Q₁  u_{ax x}  w N_x ²

N²  N₀²  _z

Now

2 o

of D N w 0

x Dt

    

also 

D

Dt (fv_z)  Q₁  F u² _az  S w² _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.

N²  N₀²  _z

To maintain thermal-wind balance ( fv_z = _x )

N²_xx  2S²_xz  F²_zz  2Q₁

2 2 2

1 ax x x 1 az z

Q  u   w N  Q  F u S w

N²_xx  2S²_xz  F²_zz  2Q₁

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

N₀²_xx  f²_zz  2Q₁ Compare with the QG-circulation equation

z

X

z

x X₁ X₂

x₁ x₂

(a) The circulation in the (X, Z) plane in a region of active frontogenesis (Q_l > 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^oC

H H

H

A 1000-500 mb thickness chart over Australia

The End