Quasi-geostrophic motion

Chapter 8

Quasi-geostrophic motion

¾ Simplification of the basic equations can be obtained for synoptic scale motions.

¾ Consider the Boussinesq system ⇒ ρis assumed to be constant in as much as it affects the fluid inertia and continuity.

Scale analysis for synoptic-scale motions

¾ 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'; b = Σb '; and f = f₀f', f₀ is a typical middle latitude value of f.

The horizontal component of the momentum equation takes the nondimensional form

Ro t

W U

L H w

z f P

ULf p

h

∂

∂

∂

∂ ρ

' '

' ' '

+ ⋅∇ +

L

NM O

L QP

NM O

QP

u^'_h ′ u^'_h k u^'_{h h}^'

0

¾ Definition of scales => all ⁽´⁾-quantities have magnitude ~ O(1).

¾ Typical values of the scales for middle latitude synoptic systems are: L = l0⁶m, H = l0⁴m, U = 10 ms⁻¹, P = 10³Pa(10 mb), b = gδT/T = 10*3/300 = 10 ms⁻², ρ= 1 kg m⁻³and f₀~ 10⁻⁴s.

¾ Clearly, we can take P = ULf₀.

¾ Then, assuming that (WL/UH) ~ O(1), the key parameter is the Rossby number.

where V'_hdenotes the operator (∂/∂x', ∂/∂y', 0) and Rois the nondimensional parameter U/(f₀L), the Rossby number.

¾ For synoptic scale motions at middle latitudes, Ro ~ 0.lso that, to a first approximation, the D'u'_h/Dt'can be neglected and the equation reduces to one of geostrophic balance.

¾ In dimensionalform it becomes fk∧u_h = − ∇1 _hp

ρ

We solve it by takingk^of both sides.

u_g k _h

f p

= + 1 ∧ ∇ ρ

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

Ro t

W U

L H w

z f P

ULf p

h

∂

∂

∂

∂ ρ

' '

' ' '

+ ⋅∇ +

L

NM O

L QP

NM O

QP

u^'_h ′ u^'_h k u^'_h ^'_h

0

¾ As noted earlier, it is a diagnostic equationfrom which the wind can be inferred at a particular time when the pressure gradient is known.

¾ In other words, the limit of Ro t

W U

L H w

z f P

ULf p

h

∂

∂

∂

∂ ρ

' '

' ' '

+ ⋅∇ +

L

NM O

L QP

NM O

QP

u^'_h ′ u^'_h k u^'_{h h}^'

0

as Ro →0is degeneratein the sense that time derivatives drop out.

¾ We cannot use the geostrophic equation to predictthe evolution of the wind field.

¾ If f is constant the geostrophic wind is horizontally nondivergent; i.e., V_h⋅u_g= 0.

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

∂

∂

∂

∂ ρ

' '

' ' '

+ ⋅∇ +

L

NM O

L QP

NM O

QP

u^'_h ′ u^'_h k u^'_{h h}^'

0

for Ro << 1, u_h~u_gwhile u_ais of order Ro.

A suitable scale for |u_a|is URo.

Because V_h⋅u_g= 0, the continuity equation reduces to the nondimensional form (assuming that fis constant).

Ro W H

L U

w

h a z

∇ ⋅ +

L

NM O

QP

' ' '

u ∂ '

∂ 0

is important if W

H L

U =Ro≈0 1. The second term of

a typical scale forw isU(H/L)Ro = 10⁻²ms⁻¹. the operator u_h' ⋅V_h'in

Ro t

W U

L H w

z f P

ULf p

h

∂

∂

∂

∂ ρ

' '

' ' '

+ ⋅∇ +

L

NM O

L QP

NM O

QP

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.

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' ⋅V_h' is u'_g⋅V_h'

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

Two important results

In nondimensionalform, the vertical momentum equation is

0

W L D ' w ' p ' H

Ro b '

H U Dt ' z ' ULf

∂ Σ

⎡ ⎤ = − +

⎢ ⎥ ∂

⎣ ⎦

It is easy to check that ΣH/(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

∂

∂t+ _g⋅ ∇_{h g} f _a

L NM O

QP

u u k u 0

1 p b z

∂ = ρ ∂

∇ ⋅_{h a}+ w = u ∂z

∂ 0

2

g h b N w 0

t

⎡∂ + ⋅∇ ⎤ + =

⎢∂ ⎥

⎣ u ⎦

u_g k _h

f p

= + 1 ∧ ∇ ρ

where at present fis assumed to be a constant.

an approximated form of the thermodynamic equation

The governing equations for quasi-geostrophic motion in dimensional form

First derive the vorticity equation:

u_g⋅ ∇u_g = ∇(¹₂u_g²)−u_g ∧ ∇ ∧( u_g) Use:

∂

∂

u^g u_g u_g u_g k u_a t + ∇(¹₂ ²)− ∧ ∇ ∧( )+f ∧ =0

∂

∂ ζ

t + _g⋅ ∇_{h g} f f _{h a}

L NM O

QP

u ( + )= - u Taking k ·∇^ gives

where ζ_g= k ^ u_gis the vertical component of relative vorticitycomputed using the geostrophic wind.

If fis constant, ∂ .

∂t + _g⋅ ∇_h f

L NM O

QP

u =0

A prediction equation for the flow at small Ro

AssumeN²is constant

2

g h b N w 0

t

⎡∂ + ⋅∇ ⎤ + =

⎢∂ ⎥

⎣ u ⎦

∂

∂z of Take

g 2

g h h

b w

+ b + N 0

t z z z

∂ ∂ ∂ ∂

⎡ + ⋅∇ ⎤ ⋅∇

⎢∂ ⎥∂ ∂ ∂

⎣ ⎦

u u =

1 p b z

∂ = ρ ∂

∂

∂ ρ

∂

∂ u

z^g k _h

f

p

= + 1 ∧ ∇ z

u_g k _h

f p

= + 1 ∧ ∇ ρ

Using ^{g 1} _hb

f

∂ = + ∧ ∇

∂

u k

z

∂

∂ w

z = −∇ ⋅^h u^a

g 2

g h h

b w

+ b + N 0

t z z z

∂ ∂ ∂ ∂

⎡ + ⋅∇ ⎤ ⋅∇

⎢∂ ⎥∂ ∂ ∂

⎣ ⎦

u u =

∂

∂z of

g h g 2

f b

f 0

t N z

∂ ∂

⎡ + ⋅∇ ⎤ ⎡ζ + + ⎤

⎢∂ ⎥ ⎢ ∂ ⎥

⎣ u ⎦ ⎣ ⎦=

2

g h h a

b = N

t z

∂ ∂

⎡ + ⋅∇ ⎤ − ∇ ⋅

⎢∂ ⎥∂

⎣ u ⎦ u

∂

∂ ζ

t + _g⋅ ∇_{h g} f f _{h a}

L NM O

QP

u ( + )= - u

¾ Assumes that f is a constant (then we can omit the single f in the middle bracket).

¾ If the meridional displacement of air parcels is not too large, we can allow for meridional variations in fwithin the small Rossby number approximation - see exercise 8.1.

∂

∂t + ^g⋅ ∇^h

L NM O

QP

u q=0

0

g 2

f b

q f

N z

= ζ + + ∂ where ∂

¾ Suppose that f = f₀+ βy, then

¾ This is an equation of fundamental importance in dynamical meteorology; it is the quasi-geostrophic potential vorticity equation

¾ It states that the quasi-geostrophic potential vorticityqis conserved along geostrophically computed trajectories.

¾ It is the prognosticequation which enables us to calculate the time evolution of the geostrophic wind and pressure fields.

The quasi-geostrophic potential vorticity equation

ζ^g = ⋅ ∇ ∧k u^g = ρ1f ∇^h p

0 2

q f p f f p

h z

= 1 ∇ + +

0

2 0

2

ρ ρ 2

∂

∂ N²

0

g 2

f b

q f

N z

= ζ + + ∂

∂

b 1 p z

= ∂ ρ ∂ Expression of q in terms of pressure

Write in the form

∂

∂ q

t = − ⋅ ∇u_{g h}q

u_g k _h

f p

= 1 ∧ ∇ ρ 0

∂

∂t + ^g⋅ ∇^h

L NM O

QP

u q=0

Solution procedure

¾ Suppose that we make an initial measurement of the pressure field p(x,y,z,0)at time t = 0.

¾ Calculate q(x,y,z,0)using

¾ Predict the distribution of q(x,y,z, Δt)using

q f p f f p

h z

= 1 ∇ + +

0

2 0

2

ρ ρ 2

∂

∂ N²

¾ Calculated u_g(x,y,z,0)using

u_g k _h

f p

= 1 ∧ ∇ ρ 0

∂

∂ q

t = − ⋅ ∇u_{g h}q

¾ Diagnose p(x,y,z,Δt)by solving the elliptic partial differential equationfor p:

¾ Diagnose u_g(x,y,z,Δt)using

∇_h p+ f p = −

z f q f

2 0

2 2

2 0

N²

∂

∂ ρ ( )

u_g k _h

f p

= 1 ∧ ∇ ρ 0

¾ Repeat the process ...

∂

∂

∂

∂ t

p

g h z + ⋅ ∇

L NM O

QP

u 0 at z = 0

¾ In order to carry out the integrations, appropriate boundary conditions must be prescribed.

¾ For example, for flow over level terrain, w = 0at z = 0.

2

g h b N w 0

t

⎡∂ + ⋅∇ ⎤ + =

⎢∂ ⎥

⎣ u ⎦

Use and 1 p

b z

= ∂ ρ ∂ Boundary conditions

¾ When Nis a constant, the nondimensional form of this equation is

D b 1

w 0

Dt B

′ ′+ ′ =

′

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 radius of deformation

¾ 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

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 b 1}w 0 Dt B

′ ′+ ′ =

′

¾ Since in quasi-geostrophic theory the total derivative D/Dtis approximated by ∂/∂t+ u_g⋅V_h, the rate-of-change of buoyancy is computed following the (horizontal) geostrophic velocity u_g.

¾ The vertical advection of buoyancy w∂/∂zis negligible.

¾ Thus quasi-geostrophic flows "see" only the stratification of the basic state characterized by N²= (g/θ)dθ₀/dz ---- this is independent of time; such flows cannot change the ‘effective static stability’ characterized locally by N²+ ∂b/∂z.

¾ The derivation of the potential vorticity equation for a compressible atmosphere is similar to that for a Boussinesq fluid.

¾ The equation for the conservation of entropy, or equivalently, for potential temperature θ, replaces the equation for the conservation of density:

s= c_plnθ

D Dt

ρ =0 D

Dt θ =0

¾ Other changes are b g o

* ρ − ρ

= − ρ b g ^o

* θ − θ

= θ N g d

dz

2 = o

θ θ

*

The quasi-geostrophic equation for a compressible atmosphere

For a deep atmospheric layer, the continuity equation must include the verticaldensity variation ρ₀(z):

∇ ⋅_{h a}+

z w

u 1 =

0

0

ρ 0

∂

∂ (ρ ) The vorticity equation is

D

Dt f f

z w

(ζg ) ( )

ρ

∂

∂ ρ

+ = ⁰ ₀

0

The theory applies to small departures from an adiabatic

atmospherein whichθ₀(z) is approximately constant, equal toθ*.

The potential vorticity equation is

∂

∂ ζ

ρ

∂

∂

ρ ∂ψ

∂

t f f

z z z

N z

g h g

+ ⋅ ∇

L NM O

QP

L

NM O

L QP

NM O

QP

u ⁰

2

0

0

2 0

( )

( )

¾ For steady flow (∂/∂t≡ 0) the quasi-geostrophic potential vorticity equation takes the form u_g⋅V_hq = 0.

¾ Assume that fis a constant,

¾ u_g⋅V_hq = 0is satisfied (e.g.) by solutions of the form q = f.

¾ For these solutions, psatisfies

∇_h p+ f p = − =

z f q f

2 0

2 2

2 0 0

N²

∂

∂ ρ ( ) or ∂ ψ

∂

∂ ψ

∂

∂ ψ

∂

2 2

2 2

2 2

2

2 0

x y

f

N z

+ + =

ψis the geostrophic streamfunction (= p/ρf).

These solutions have zero perturbation potential vorticity Quasi-geostrophic flow over a bell-shaped

mountain

Omit the zero subscript on f, and assumethat Nis a constant.

∂ ψ

∂

∂ ψ

∂

∂ ψ

∂

2 2

2 2

2 2

2

2 0

x y

f

N z

+ + =

Put ^{z=(N f z/ )} Laplace’s equation

∂ ψ

∂

∂ ψ

∂

∂ ψ

∂

2 2

2 2

2

2 0

x + y + z = Two particular solutions are:

ψ = −Uy ψ = −S/ 4πr

u = −ψ_y = U

a uniform flow

2 2 2

r = x +y +( z +z )*

a source solution, strengthS at ^z= −^z*

ψ = −U y− S/ 4πr Streamfunction equation is linear

is a solution.

In qg-flow 1 p

b f

z z

∂ ∂ψ

= =

ρ ∂ ∂ because ψ = p/ρf

The vertical displacement of a fluid parcel, ηis related to σby

2

b η = −N

Since bis a constant on isentropic surfaces, the displacement of the isentropic surface fromz = constant for the flow defined by ψ= −Uy−S/4πris given (in dimensional form) by

η= − π

L

NM O

QP

∗

−

∗

S

f x y N

f z z z z

4

2 2

2 2

2 3 2

( ) ( )

/

η= − π

L

NM O

QP

∗

−

∗

S

f x y N

f z z z z

4

2 2

2 2

2 3 2

( ) ( )

/

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

h x y Sz

f x y N

f z h

R R

m

( , )

( / )

/

/

= −

L

NM O

QP

= +

∗ ∗

−

∗

4

1

2 2

2 2

2 3 2

2 3 2

π

h_m= −S/(4πNR_*²) R = (x² +y²)

R_*= Nz_*/f

is an isentropic surface of the quasi-geostrophic flow defined byψ= −Uy−S/4πr.

h x y h

R R ( , ) m

( / ) ^/

= _∗ ² +1^{3 2}

When S = 4πNR_*²h_m and z = f R_*/N, ψ= −Uy−S/4πr represents the flow in the semi-infinite regionz >= h of a uniform currentU past the bell-shaped mountainwith circular contours given byh(x,y).

The mountain height ish_mand its characteristic width isR_*. In terms ofh_metc., the displacement of an isentropic surface in this flow is

η( , , ) ( / )

( / ) ( / ) ^/

x y z h z z

R R z z

= m +

+ +

∗

∗ ∗

1

2 12 3 2

A B O R_* x U

The vorticity changes in stratified quasi- geostrophic flow over an isolated mountain

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

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.

η₂-η₁

A B 0 R_* x 5

−5

End of

Chapter 8