Quasi-geostrophic motion

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Chapter 8 Quasi-geostrophic motion

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 Simplification of the basic equations can be obtained for synoptic scale motions.

 Consider the Boussinesq system   is assumed to be constant in as much as it affects the fluid inertia and continuity.

Scale analysis for synoptic-scale motions

 Introduce nondimensional variables, (  ), and typical scales (in capitals) as follows:

(x, y) = L(x', y') z = Hz' t = (L/U)t'

(u,v) = U(u', v') w = Ww' p = Pp';  =  '; and f = f₀ f', f₀ is a typical middle latitude value of f.

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The horizontal component of the momentum equation takes the nondimensional form

Ro t

W U

L H w

z f P

ULf p

h



 

' '

' ' '

  

L

N M O Q P L N

M O

Q P

^ ^ ^{ } ^

u^'_h  u^'_h k u^'_h ^'_h

0

 Definition of scales => all ⁽´⁾-quantities have magnitude ~ O(1).

 Typical values of the scales for middle latitude synoptic systems are: L = l0⁶ m, H = l0⁴ m, U = 10 ms^1 , P = 10³ Pa (10 mb),

 = gT/T = 10*3/300 = 10 ms^2,  = 1 kg m^3 and f₀ ~ 10^4 s.

 Clearly, we can take P = ULf₀.

 Then, assuming that (WL/UH) ~ O(1), the key parameter is the Rossby number.

where '_h denotes the operator (/x', /y', 0) and Ro is the nondimensional parameter U/(f₀L), the Rossby number.

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 For synoptic scale motions at middle latitudes, Ro ~ 0.l so that, to a first approximation, the D'u'_h/Dt' can be neglected and the equation reduces to one of geostrophic balance.

 In dimensional form it becomes

fk u _h   1 _hp



We solve it by taking k _^ of both sides.

u_g k _h

f p

  1  



This equation defines the geostrophic wind. Our scaling shows to be a good approximation to the total horizontal wind u_h.

Ro t

W U

L H w

z f P

ULf p

h



 

' '

' ' '

  

L

N M O Q P L N

M O

Q P

^ ^ ^{ } ^

u^'_h  u^'_h k u^'_h ^'_h

0

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 As noted earlier, it is a diagnostic equation from which the wind can be inferred at a particular time when the pressure gradient is known.

 In other words, the limit of Ro t

W U

L H w

z f P

ULf p

h



 

' '

' ' '

  

L

N M O Q P L N

M O

Q P

^ ^ ^{ } ^

u^'_h  u^'_h k u^'_h ^'_h

0

as Ro  0 is degenerate in the sense that time derivatives drop out.

 We cannot use the geostrophic equation to predict the evolution of the wind field.

 If f is constant the geostrophic wind is horizontally nondivergent; i.e., _hu_g .

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The difference between the horizontal wind and the geostrophic wind is called the ageostrophic wind:

u_a  u_h  u_g Now Ro

t

W U

L H w

z f P

ULf p

h



 

' '

' ' '

  

L

N M O Q P L N

M O

Q P

^ ^ ^{ } ^

u^'_h  u^'_h k u^'_h ^'_h

0

for Ro << 1, u_h ~ u_g while u_a is of order Ro.

A suitable scale for |u_a| is URo.

Because _hu_g , the continuity equation reduces to the nondimensional form (assuming that f is constant).

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Ro W H

L U

w

h a z

  

L

N M O Q P

^

' ' '

u  '

 0

is important if W

H L

U  Ro 0 1. The second term of

a typical scale for w is U(H/L)Ro = 10^2 ms^1. the operator u_h'_h' in

Ro t

W U

L H w

z f P

ULf p

h



 

' '

' ' '

  

L

N M O Q P L N

M O

Q P

^ ^ ^{ } ^

u^'_h  u^'_h k u^'_h ^'_h

0

is much larger than w'/z‘ .

To a first approximation, advection by the vertical velocity can be neglected, both in the momentum and thermodynamic equations.

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To a first approximation, advection by the vertical velocity can be neglected, both in the momentum and thermodynamic equations.

Also the dominant contribution to u_h'_h' is u'_g_h'

in quasi-geostrophic motion, advection is by the geostrophic wind.

Two important results

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In nondimensional form, the vertical momentum equation is

Ro W H

L U

D w Dt

p z

H

L

ULf

N M O Q P

^{' '}_' ^{ } ^ ^'_' ^  ^'

0

It is easy to check that H/(ULf₀) = 1

Ro(WH/UL) = Ro ²(H/L)² = 10^6.

Synoptic scale perturbations are in a very close state of hydrostatic balance.

Vertical momentum equation

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

t  _g _h _g f _a

L N

M O

Q P

^ ^ ^

u u k u 0

1





p  z 

 _h _a  w  u z

 0



 

t  _g _h N w

L N

M O

Q P

^ ^

u ² 0

u_g k _h

f p

  1  



where at present f is assumed to be a constant.

an approximated form of the thermodynamic equation

The governing equations for quasi-geostrophic

motion in dimensional form

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First derive the vorticity equation:

u_g u_g  (¹₂ u_g²)  u_g   ( u_g)

Use:



u^g u_g u_g u_g k u_a

t  (¹₂ ²)    ( )  f   0



 

t  _g  _h _g f f _h _a

L N

M O

Q P

^{ }

u ( + ) = - u Taking k ·  _^gives

where _g = k _^u_g is the vertical component of relative vorticity computed using the geostrophic wind.

If f is constant, .

t  _g  _h f

L N

M O

Q P

u = 0

A prediction equation for the flow at small Ro

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Assume N² is constant



 

t  _g _h N w

L N

M O

Q P

^ ^

u ² 0



z of Take







  



t z z

w

g h g z

  h

L N

M O

Q P

^

u u

=

+ + N² 0

1





p  z 



 



 u

z^g k _h

f

p

  1   z u_g k _h

f p

  1  



Using ^

u 

z^g k _h

  1f  



 w

z   ^h u^a







  

t z z 

w

g h g z

  h

L N

M O

Q P

^

u u

=

+ + N² 0



z of

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

  



t f f

g h g z

 

L N

M O

Q P L

^{ }

N M O Q P

u =

N² 0







t  ^g ^h z ^h ^a

L N

M O

Q P

^{  }

u = N² u



 

t  _g  _h _g f f _h _a

L N

M O

Q P

^{ }

u ( + ) = - u

 Assumes that f is a constant (then we can omit the single f in the middle bracket).

 If the meridional displacement of air parcels is not too large, we can allow for meridional variations in f within the small Rossby number approximation - see exercise 8.1.

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

t  ^g ^h

L N

M O Q P

u q 0=

q f f

g

z

    



0

N

²

where

 Suppose that f = f₀ + y, then

 This is an equation of fundamental importance in dynamical meteorology; it is the quasi-geostrophic potential vorticity equation

 It states that the quasi-geostrophic potential vorticity q is conserved along geostrophically computed trajectories.

 It is the prognostic equation which enables us to calculate the time evolution of the geostrophic wind and pressure fields.

The quasi-geostrophic potential vorticity equation

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^g    k u^g  1f ^h p

0

2

q f p f f p

h z

 1   

0

2 0

2

  2



 N²

q f f

g

z

    



0

N

²

 



 1 p z

Expression of q in terms of pressure

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Write in the form



 q

t    u

_g _h

q

u

_g

k

_h

f p

 1  



0



t  ^g ^h

L N

M O Q P

u q 0=

Solution procedure

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 Suppose that we make an initial measurement of the pressure field p(x,y,z,0) at time t = 0.

 Calculate q(x,y,z,0) using

 Predict the distribution of q(x,y,z, t) using

q f p f f p

h z

 1   

0

2 0

2

  2



 N²

 Calculated u_g(x,y,z,0) using

u_g k _h

f p

 1  

 0



 q

t    u

_g _h

q

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 Diagnose p(x,y,z,t) by solving the elliptic partial differential equation for p:

 Diagnose u_g(x,y,z,t) using

_h p  f p  

z f q f

2 0

2 2

2 0

N²



  ( )

u_g k _h

f p

 1  

 0

 Repeat the process ...

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

 t 

p

g h

z

 

L N

M O Q P ^

u 0

_{at z = 0}

 In order to carry out the integrations, appropriate boundary conditions must be prescribed.

 For example, for flow over level terrain, w = 0 at z = 0.



 

t  _g _h N w

L N

M O

Q P

^ ^

u ² 0

Use and 





 1 p z

Boundary conditions

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 When N is a constant, the nondimensional form of this equation is

D

Dt  B w

   

 1

0

where B f L

N H

L L_R

 ⁰₂^{2 2}₂  ₂² and ^L ^NH

R  f

0

the Burger

number the Rossby length

 An important feature of quasi-geostrophic motion is the assumption that L ~ L_R, or equivalently that B ~ 1.

More on the approximated thermodynamic

equation

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The rate-of-change of buoyancy (and temperature) experienced by fluid parcels is associated with vertical motion in the presence of a stable stratification.

When B ~ 1, ^D

Dt  B w

   

 1

0

 Since in quasi-geostrophic theory the total derivative D/Dt is approximated by /t+ u_g_h , the rate-of-change of

buoyancy is computed following the (horizontal) geostrophic velocity u_g.

 The vertical advection of buoyancy w/zis negligible.

 Thus quasi-geostrophic flows "see" only the stratification of the basic state characterized by N² = (g/)d₀/dz ---- this is independent of time; such flows cannot change the ‘effective static stability’ characterized locally by N² + /z.

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 The derivation of the potential vorticity equation for a

compressible atmosphere is similar to that for a Boussinesq fluid.

 The equation for the conservation of entropy, or equivalently, for potential temperature , replaces the equation for the

conservation of density:

s c _p ln

D Dt

  0 D

Dt

  0

 Other changes are

  

  

g ^o

* ^

 

 

g ^o

*

N g d

dz

2



o



*

The quasi-geostrophic equation for a

compressible atmosphere

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For a deep atmospheric layer, the continuity equation must include the vertical density variation ₀(z):

 _h _a 

z w

u 1 =

0

 0



 ( ) The vorticity equation is

D

Dt f f

z w

(g ) ( )





 

  ⁰ ₀

0

The theory applies to small departures from an adiabatic

atmosphere in which ₀(z) is approximately constant, equal to *.

The potential vorticity equation is



 





 



t f f

z z

z

N z

g h g

 

L N

M O

Q P

^{ }

L

N M O Q P L N

M O

Q P

^

u ⁰

2 0

0

2 0

( )

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 For steady flow (/t0) the quasi-geostrophic potential vorticity equation takes the form u_g _hq = 0.

 Assume that f is a constant,

 u_g _hq = 0 is satisfied (e.g.) by solutions of the form q = f.

 For these solutions, p satisfies

_h p  f p    z f q f

2 0

2 2

2 0 0

N²



  ( ) _or ^{ }



 



 



2 2

2

2 0

x y

f

N z

  

 is the geostrophic streamfunction (= p/f).

These solutions have zero perturbation potential vorticity

Quasi-geostrophic flow over a bell-shaped

mountain

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Omit the zero subscript on f, and assume that N is a constant.

 



 



 



2 2

2

2 0

x y

f

N z

  

Put ^z ^ ^{( / )}^{N f z} Laplace’s equation

 



 



 



2 2

2

2 0

x  y  z 

Two particular solutions are:

  Uy

  S / 4r

u   

_y

 U

a uniform flow

2 2 2

r  x  y  (z  z )*

a source solution, strength S at ^z ^{ }^z ^*

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  Uy  S / 4r Streamfunction equation is linear

is a solution.

In qg-flow 







 1 p   z f

z because  = p/f

The vertical displacement of a fluid parcel,  is related to  by

 

  N²

Since  is a constant on isentropic surfaces, the displacement of the isentropic surface from z = constant for the flow defined by  = Uy  S/4r is given (in dimensional form) by

   

L

  

N M O

Q P

^







S

f x y N

f z z z z

4

2 2

2

3 2

( ) ( )

/

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   

L

  

N M O

Q P

^







S

f x y N

f z z z z

4

2 2

2

3 2

( ) ( )

/

The displacement of fluid parcels which, in the absence of motion would occupy the plane at z = 0 is

h x y Sz

f x y N

f z h

R R

m

( , )

( / )

/

 

L

 

N M O Q P

 

 





4

1

2 2 2

2

3 2

2 3 2



h_m = S/(4NR_*²)

R  (x²  y²) R_* = Nz_*/f

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is an isentropic surface of the quasi-geostrophic flow defined by  = Uy  S/4r.

h x y h

R R ( , ) m

( / ) ^/

 _ ² 1 ^{3 2}

When S = 4NR_*²h_m and z = f R_*/N,  = Uy  S/4r represents the flow in the semi-infinite region z >= h of a uniform current U past the bell-shaped mountain with circular contours given by h(x,y).

The mountain height is h_m and its characteristic width is R_*. In terms of h_m etc., the displacement of an isentropic surface in this flow is

( , , ) ( / )

( / ) ( / ) ^/

x y z h z z

R R z z

 m 

 



 

1

2 1 2 3 2

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A B O R_* x U

The vorticity changes in stratified quasi-

geostrophic flow over an isolated mountain

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The incident flow is distorted by the mountain anticyclone, but the perturbation velocity and pressure field decay away from the mountain (after R. B. Smith, 1979).

NH case

The streamline pattern in quasi-geostrophic

stratified flow over an isolated mountain

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Height of the lowest isentrope above the topography in as a function of x.

Unit scale equals the length of the four vertical lines in Fig. 8.1.

₂ - ₁

A B

0

R_*

x 5

5

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