Balanced EquationsDM Ch 15

Shallow water model configuration

In pure inertial wave motion, horizontal pressure gradients are zero.

Consider now waves in a layer of rotating fluid with a free surface where horizontal pressure gradients are associated with free surface displacements.

f

H(1 + η) H

pa

z = 0

p = pa + ρg(H(1 + η) – z)

undisturbed depth

Reminder: Inertia-gravity waves

Consider hydrostatic motions - then

ap(x , y, z, t ) p g[H{1 (x , y, t )} z ]= + ρ + η −

1ρ

η∇ = ∇h hp gH

independent of z

The fluid acceleration is independent of z.

If the velocities are initially independent of z, then they will remain so.

Linearized equations - no basic flow

u fv gHt xv fu gHt yu vx y t

∂ ∂η− = −

∂ ∂∂ ∂η

+ = −∂ ∂∂ ∂ ∂η

+ = −∂ ∂ ∂

Consider wave motions which are independent of y.

A solution exists of the form u u kx tv v kx t

kx t

= −

= −

= −

cos ( ),sin ( ),cos ( ),

ω

ω

η η ω, tanu v and cons tsη

if

ω η

ω

ωη

,,,

u fv gHkfu vku

− − =

− =

− + =

000

These algebraic equations for have solutions only if

,u v and η

ωω

ωω ω

− −−

−= − + + =

f gHkfk

f gHk00

03 2 2( )

ω ω= = +0 2 2 2or f gHk

The solution with ω = 0 corresponds with the steady solution (∂/∂t = 0) of the equations and represents a steady current in strict geostrophic balance in which

The other two solutions correspond with so-called inertia-gravity waves, with the dispersion relation

gHvf x

∂η=

∂

The phase speed of these is

ω2 2 2= +f gHk

c k gH f kp = = ± +ω / [ / ]2 2

The waves are dispersive

Replace the v-momentum equation by the vorticity equation =>

0

0

u f v gHt x

v ft t

u v 0t x y

v ux y

∂ ∂η− = −

∂ ∂∂ζ ∂η

+ β =∂ ∂∂η ∂ ∂

+ + =∂ ∂ ∂

∂ ∂ζ = −

∂ ∂where

Assume that f can be approximated by its value f0 at a particular latitude, except when differentiated with respect to y in the vorticity equation. This is justified provided that meridional particle displacements are small.

Inclusion of the beta effect

Again assume that ∂/∂y ≡ 0 and consider travelling-wave solutions of the form:

u u kx tv v kx t

kx t

= −

= −

= −

cos ( ),sin ( ),cos ( ),

ω

ω

η η ω

ω η

ω β ωη

ωη

,

( ) ,

,

u f v gHk

k v f

ku

− − =

+ − =

− + =

0

0

0

0

0

These are consistent only if the determinant is zero

ω

ω β ω

ω

ω ω β ω

− −

+ + −

−

= − + − =

f gHkk f

kgHk k f k

0

02 2

020

00( ) ( )( )

Now expanding by the second row

A cubic equation for ω with three real roots.

When ω >> β/k, the two non-zero roots are given approximately by the formula

ω2 202= +gHk f

This is precisely the dispersion relation for inertia-gravity waves.

When ω2 << gHk2, there is one root given approximately by2 2

0k /(k f / gH)ω = −β +

This has the solution , where

Equations:

0

gHvf x

∂η=

∂

If ∂/∂y ≡ 0 as before, the vorticity equation reduces to an equation for η, and since ζ = ∂v/∂x =>

2

020 0

gH gHf 0t f x f x

⎛ ⎞∂ ∂ η ∂η− η + β =⎜ ⎟∂ ∂ ∂⎝ ⎠

η η ω= −cos ( )kx t

Filtering

0

0

u f v gHt x

v ft t

u v 0t x y

∂ ∂η− = −

∂ ∂∂ζ ∂η

+ β =∂ ∂∂η ∂ ∂

+ + =∂ ∂ ∂

ω β= − +k k f gH/ ( / )202

There is no other solution for ω as there was before.

In other words, making the geostrophic approximation when calculating v has filtered out in the inertia-gravity wave modes from the equation set, leaving only the low frequency planetary wave mode.

This is not too surprising since the inertia-gravity waves, by their very essence, are not geostrophically-balanced motions.

Dispersion relation for a divergent planetary wave

The idea of filtering sets of equations is an important one in geophysical applications.

The quasi-geostrophic equations are often referred to as 'filtered equations' since, as in the above analysis, the consequence of computing the horizontal velocity geostrophically from the pressure or stream-function suppresses the high frequency inertia-gravity waves which would otherwise be supported by the Boussinesq equations.

Furthermore, the Boussinesq equations themselves form a filtered system in the sense that the approximations which lead to them filter out compressible, or acoustic waves.

Filtered equations more generally

The full nonlinear form of the shallow-water equations in a layer of fluid of variable depth h(x,y,t) is

u u u hu v fv gt x y xv v v hu v fu gt x y y

h h h u vu v h 0t x y x y

∂ ∂ ∂ ∂+ + − = −

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

+ + + = −∂ ∂ ∂ ∂

⎛ ⎞∂ ∂ ∂ ∂ ∂+ + + + =⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

ζ ∂ ∂ ∂ ∂= − = +x y x yv u D u vrelative vorticity horizontal divergence

The u- and v-equations can be replaced by the vorticity and divergence equations: =>

The Balance Equations

vorticity equation

u v v ( f ) D 0t x y

∂ζ ∂ζ ∂ζ+ + + β + ζ + =

∂ ∂ ∂

divergence equation

2

2

D D Du v D f 2J(u, v)t x y

u g h 0,

∂ ∂ ∂+ + + − ζ −

∂ ∂ ∂

+ β + ∇ =β = df/dy

Combine the vorticity equation and the continuity equation to form the potential vorticity equation:

q q qu v 0 ,t x y

∂ ∂ ∂+ + =

∂ ∂ ∂potential vorticity is q = (ζ + f)/h

2 2

u v v ( f ) D 0t x yD D Du v D f 2J(u, v) u g h 0t x yq q qu v 0t x y

∂ζ ∂ζ ∂ζ+ + + β + ζ + =

∂ ∂ ∂∂ ∂ ∂

+ + + − ζ − + β + ∇ =∂ ∂ ∂∂ ∂ ∂

+ + =∂ ∂ ∂

Equivalent to: u u u hu v fv gt x y xv v v hu v fu gt x y y

h h h u vu v h 0t x y x y

∂ ∂ ∂ ∂+ + − = −

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

+ + + = −∂ ∂ ∂ ∂

⎛ ⎞∂ ∂ ∂ ∂ ∂+ + + + =⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

Choose representative scales:

U for the horizontal velocity components u, v,

L for the horizontal length scale of the motion,

H for the undisturbed fluid depth,

δH for depth departures h − H,

f0 for the Coriolis parameter, and

T = L/U - an advective time scale.

Define two nondimensional parameters:

the Rossby number, Ro = U/(f0L), and

the Froude number, Fr = U2/(gH).

Scale analysis

f f Ro y= + ′0 1( ) ,β

where y is nondimensional and .′ =β βL U2 /

Let

For middle latitude systems

U m s m s L m~ , ~ , ~10 10 101 11 1 1 6− − − −β

β' is of order unity.

Treatment of f

The nondimensional forms of the u-momentum and continuity equations are:

0

u g H hRo yv v ,t f UL x

∂ δ ∂⎛ ⎞′+ − β − = −⎜ ⎟∂ ∂⎝ ⎠…

H h D 0 ,H t

δ ∂⎛ ⎞+ + =⎜ ⎟∂⎝ ⎠…

Let us examine first the quasi-geostrophic scaling:

Ro u yv v g H f UL ht x[ ] ( / ) ,∂ β δ ∂+ − ′ − = −… 0

( / )[ ] ,δ ∂H H h Dt + + =… 0

In quasi-geostrophic motion (Ro << 1), the term proportional to Ro can be neglected.

δH H Ro Fr/ .= −1

Then gδH/f0UL ≈ O(1).

We could choose the scale H so that this quantity is exactly unity, i.e. δH = f0UL/g. Then the term δH/H may be written

⇒ to lowest order in Ro

The momentum equations are

0

0

u g H hRo yv vt f UL xv g H hRo yv ut f UL y

∂ δ ∂⎛ ⎞′+ − β − = −⎜ ⎟∂ ∂⎝ ⎠∂ δ ∂⎛ ⎞′+ − β + = −⎜ ⎟∂ ∂⎝ ⎠

…

…

a consistent scaling for quasi-geostrophic motion is that δH/H = Ro-1 Fr = O(Ro) so that the term proportional to δH/H → 0 as Ro → 0.

D = 0 (or, more generally, D ≤ O(Ro) ).

H h D 0 ,H t

δ ∂⎛ ⎞+ + =⎜ ⎟∂⎝ ⎠…

h hv ux y

∂ ∂= = −

∂ ∂

Ro v D Ro( y ) D 0t

∂ζ⎛ ⎞′ ′+ + β + + β + ζ =⎜ ⎟∂⎝ ⎠…

and

2 2DRo D 2J(u, v)] (1 Ro y) Ro u h 0,t

∂⎛ ⎞ ′ ′+ + − − ζ + β + β + ∇ =⎜ ⎟∂⎝ ⎠…

where J(u,v) is the Jacobianu v v ux y x y

∂ ∂ ∂ ∂−

∂ ∂ ∂ ∂

In nondimensional form, the vorticity and divergence equations are

The vorticity and divergence equations

2 2

R o v D R o ( y ) D 0tDR o D 2 J (u , v ) (1 R o y ) R o u h 0t

∂ζ⎛ ⎞′ ′+ + β + + β + ζ =⎜ ⎟∂⎝ ⎠∂⎛ ⎞ ′ ′+ + − − ζ + β + β + ∇ =⎜ ⎟∂⎝ ⎠

…

…

At lowest order in Rossby number

The quasi-geostrophic approximation

1R o v D 0t

∂ζ⎛ ⎞′+ + β + =⎜ ⎟∂⎝ ⎠…

∇ =2h ζ

where D = RoD1

When δH/H = Ro−1Fr, the continuity equation ⇒

1h 1 D 0t

∂+ =

∂ μ… where μ= Ro−2Fr

and D = Ro D1.

These are the quasi-geostrophic forms of the vorticity, divergence and continuity equations.

Note that the divergence equation has reduced to a diagnostic one relating the fluid depth to the vorticity which is consistent with ζ obtained from

u v v ( f ) D 0t x y

∂ζ ∂ζ ∂ζ+ + + β + ζ + =

∂ ∂ ∂

1Ro v D 0t

∂ζ⎛ ⎞′+ + β + =⎜ ⎟∂⎝ ⎠…

∇ =2h ζ

1h 1 D 0t

∂+ =

∂ μ…

D = 0 at O(Ro0) => there exists a streamfunction ψsuch that

ψ = h +constant (e.g. h − H).

0

u g H hRo yv vt f UL x

∂ δ ∂⎛ ⎞′+ − β − = −⎜ ⎟∂ ∂⎝ ⎠…

and from

u vy x

∂ψ ∂ψ= − =

∂ ∂

The potential vorticity equation in nondimensional form can be obtained by eliminating D1 between

1Ro v D 0t

∂ζ⎛ ⎞′+ + β + =⎜ ⎟∂⎝ ⎠… and

1h 1 D 0t

∂+ =

∂ μ…

a single equation for ψ (or h) =>

2u v ( y ) 0t x y

⎛ ⎞∂ ∂ ∂ ′+ + ∇ ψ + β − μψ =⎜ ⎟∂ ∂ ∂⎝ ⎠

analogous to the form for a stably-stratified fluid.

Note: in the case with a continuous vertical stratification, the ψ term would be replaced by a second-order vertical derivative of ψ.

The potential vorticity equation

The quasi-geostrophic approximation leads to an elegant mathematical theory, but calculations based upon it tend to be inaccurate in many atmospheric situations such as when the isobars are strongly curved.

In the latter case, we know that centrifugal forces are important and gradient wind balance gives a more accurate approximation.

We try to improve the quasi-geostrophic theory by including terms of higher order in Ro in the equations.

u u u= ψ χ+uψ = ∧ ∇ψk

ψ is the streamfunction.

uχ χ= ∇

χ the velocity potential

We decompose the horizontal velocity into rotational and divergent components

ζ ψ= ∇2 and D = ∇2χ

It follows that

This decomposition is general (see Holton, 1972, Appendix), but is not unique : ⇒

One can add equal and opposite flows with zero vorticity and divergence to the two components without affecting the total velocity u.

Consistent with the quasi-geostrophic scaling, where χ is zero to O(Ro0), we scale χ with ULRo so that in nondimensional form

u u u= ψ χ+ u u u= +ψ χRo

Equations Ro v D Ro( y Dt[ ] ) ,∂ ζ β β ζ+ + ′ + + ′ + =… 0

Ro D D J u v Ro y

Ro u ht[ ( , )] ( )

,

∂ ζ β

β

+ + − − + ′

+ ′ + ∇ =

… 2

2

2 1

0

[ ( . ) ( ) ]

( ) ,

∂ ψ ψ β ∂ ψ χ

β ψ χ

ψ χt xRo

y

∇ + + ⋅∇ ∇ + ′ + ∇

+ ′ + ∇ ∇ =

2 2 2

2 2 0

u u

and

Ro J Ro

Ro J u v Ro J u v J u v Ro J u v

Ro y Ro u Ro u h

t2 2 2 3 2 2 2

2 3

2 2 2

2 2 2

1 0

[ ( , )] [ ( ) ( ) ]

( , ) [ ( , ) ( , )] ( , )

( ) .

∂ χ ψ χ χ χ

ψ β β β

χ

ψ ψ χ ψ ψ χ χ χ

ψ χ

∇ + ∇ + ⋅∇ ∇ + ∇ −

− + − −

∇ + ′ + ′ + ′ + ∇ =

u

The idea is to neglect terms of order Ro2 and Ro3 in these equations, together with the equivalent approximation in the continuity equation

In dimensional form they may be written

[ ]( ) ( )∂ ψ ψ χt f f+ ⋅ ∇ ∇ + + ∇ + ∇ =u 2 2 2 0

2 02 2[( ) ( ) ( ) ] ( )∂ ψ ∂ ψ ∂ ψxx yy xy f h− + ∇ ⋅ ∇ψ − ∇ =

∂ χth h h+ ⋅ ∇ + ∇ =u 2 0

∂ ∂ ∂ ∂ ∂t x y x yh u h v h h u v+ + + + =( ) ,0

u = uψ + uχ.

Note: the divergence equation has been reduced to a diagnostic one relating ψ to h.

∇ +2ψ f

is by the total wind u and not just the nondivergent component of u as in the quasi-geostrophic approximation.

Moreover the advection of in

[ ]( ) ( )∂ ψ ψ χt f f+ ⋅ ∇ ∇ + + ∇ + ∇ =u 2 2 2 0

∂ χth h h+ ⋅ ∇ + ∇ =u 2 0and of h in

2 2 2

22 2 22

2 2

2

( f ) ( f ) 0t

2 (f ) g h 0x y x y

h h h 0t

∂⎛ ⎞+ ⋅∇ ∇ ψ + + ∇ ψ + ∇ χ =⎜ ⎟∂⎝ ⎠⎛ ⎞⎛ ⎞∂ ψ ∂ ψ ∂ ψ⎜ ⎟− + ∇ ⋅ ∇ψ − ∇ =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

∂+ ⋅∇ + ∇ χ =

∂

u

u

These equations are called the balance equations.

They were first discussed by Charney (1955) and Bolin (1955).

The Balance Equations

It can be shown that for a steady axisymmetric flow on an f-plane, the equation:

The Balance Equations

22 2 22

2 22 (f ) h 0x y x y

⎛ ⎞⎛ ⎞∂ ψ ∂ ψ ∂ ψ⎜ ⎟− + ∇ ⋅ ∇ψ − ∇ =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

reduces to the gradient wind equation.

=> we may expect the equations to be a better approximation than the quasi-geostrophic system for strongly curved flows.

Unfortunately it is not possible to combine the balance equations into a single equation for y as in the quasi-geostrophic case and they are rather difficult to solve.

Methods of solution are discussed by Gent and McWilliams (1983).

Note: although the balanced equations were derived by truncating terms of O(Ro2) and higher, the only equation where approximation is made is the divergence equation.

=> the equations represent an approximate system valid essentially for sufficiently small horizontal divergence.

As long as this is the case, the Rossby number is of no importance.

Methods of solution

Elimination of between∇2χ

gives the potential vorticity equation:

q q qu v 0t x y

∂ ∂ ∂+ + =

∂ ∂ ∂

2h h h 0t

∂+ ⋅∇ + ∇ χ =

∂u

and

Thus an alternative form of the balance system is

fqh

ζ +=

Given q, the first two equations can be regarded as a pair of simultaneous equations for diagnosing ψ and h, subject to appropriate boundary conditions.

Equation enables the prediction of q,

while may be used to diagnose χ.2h h h 0t

∂+ ⋅∇ + ∇ χ =

∂u

q q qu v 0t x y

∂ ∂ ∂+ + =

∂ ∂ ∂

2h h h 0t

∂+ ⋅∇ + ∇ χ =

∂u

22 2 22

2 22 (f ) g h 0x y x y

⎛ ⎞⎛ ⎞∂ ψ ∂ ψ ∂ ψ⎜ ⎟− + ∇ ⋅ ∇ψ − ∇ =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

ψand h

Under certain circumstances [e.g. Ro << 1, β' > O(1)], the nonlinear terms in Eq.(11.47) may be neglected in which case the equation becomes

∇⋅ ∇ψ − ∇ =( )f g h2 0

With this approximation, the system (11.46), (11.48) and (11.49) or (11.30), (11.31), (11.48), (11.49) constitute the linear balance equations.These systems are considerably easier to solve than the balance equations.

The Linear Balance Equations