More on wave motions, filtering

Chapter 11

More on wave motions, filtering

v

» Consider a homogeneous layer of inviscid fluid on an f-plane confined between rigid horizontal boundaries.

z

y

x

f Inertial waves

» Suppose that the entire layer is set impulsivelyin motion in the y-direction with the constant velocity v =v at t = 0.

u fv 0 t

v fu 0 t

∂ − =

∂

∂ + =

∂ Equations

2 2 2

v f v 0 t

∂ + =

∂

The same equation is satisfied also by u.

2 2 2

v f v 0

t

∂ + =

∂ ^{v= v cos ωt}

constant ω = ±f

The full solution is: ( , )u v = v(sin ft, cos ft)

( , )u v = v(sin ft, cos ft)

The vector velocity has magnitudeVwhere

V² = u² +v² = =v v² = constant

The directionchanges periodically with time with period2π/f.

called the inertial period f is sometimes referred to as the inertial frequency.

» The perturbation velocity is independent of spatial position.

» All fluid parcels move with the same velocityVat any instant ---assumesthat the flow domain is infinite and unconstrained by lateral boundaries.

» => at each instant, the layer moves as would a solid block.

( , )

[ cos , sin ]

x x y y v t

dt v

f ft ft

− ⁰ − ⁰ =

z

Some notes

» Consider a fluid parcel initially at the point (x₀, y₀) and suppose that it is at the point (x,y) at time t.

» Integrating the velocity =>

The parcel executes a circular path, an inertia circle, with centre at (x₀ +v f y / , ₀) and radius .v f /

v

v

ft

parcel trajectory

(x y_o, _o) (x_o +v f y / , _o) The motion is anticyclonic in sense.

The period of motion2π/ω= 2π/f = half a pendulum day, =>

the time for a Focault pendulumto turn through 180°.

v

v R fv

2 = centrifugal

force

fv Coriolis

force

R v

= f

There is no pressure gradientin the flow - the only forces are the centrifugal and Coriolis forces. In circular motion these must balance. This is possible only in anticyclonic motion.

Force balance on a fluid parcel undergoing pure inertial oscillations (northern hemisphere).

» Pure inertial oscillationsdo not seem to be important in the earth's atmosphere, but

» Time spectra of ocean currentsoften exhibit significant amounts of energy at the inertial frequency (see Holton, p.60).

» Nevertheless, inertial effectsare observed in the atmosphere.

» Examples are:

- sea breeze circulations ( => next pictures) - the nocturnal low level jet

Inertial effects

The Morning Glory

A Gulf cloud line

A Gulf cloud line

Coral Sea

Cape York Peninsula

Gulf of Carpentaria

800 km

800 km

Sea breeze circulations over Cape York Peninsula

1200 h

1600 h

1800 h

2000 h

2200 h

0000 h

0200 h

Movie by Gerald Thomsen

The profile of wind component in the direction of the geostrophic wind (u_g) showing a nocturnal jet, compared with the profile the previous afternoon.

1430 0814

0001

1548

θ^oC 1000

500

z m

1500

day

night night day

5 0

0

u_g

u m s⁻¹ 10

1000

10 15 20

1500

500

z m

Shallow water model configuration

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

» Consider now wavesin a layer of rotating fluid with a free surfacewhere horizontal pressure gradients are associated with free surface displacements.

f

H(1 + η) H

p_a

z = 0

p = p_a+ ρg(H(1 + η) – z)

undisturbed depth

Inertia-gravity waves

Consider hydrostatic motions - then

p( x , y, z , t ) = pa + ρg[ H {1+ η( x , y, t )}− z ] 1

ρ ∇_hp = gH∇_hη

independent ofz

The fluid acceleration is independent ofz.

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

Linearized equations - no basic flow

u fv gH

t x

v fu gH

t y

u v

x y t

∂ − = − ∂η

∂ ∂

∂ + = − ∂η

∂ ∂

∂ + ∂ = −∂η

∂ ∂ ∂

Consider wave motions which are independent of y.

A solution exists of the form u u kx t

v v kx t

kx t

= −

= −

= −

cos ( ),

sin ( ),

cos ( ),

ω ω

η η ω

, tan

u v and η cons ts

if

ω η ω ωη

,

,

,

u fv gHk fu v ku

− − =

− =

− + =

0 0 0

These algebraic equations for have solutions only if

,

u v and η

ω

ω

ω

ω ω

− −

−

−

= − + + =

f gHk

f k

f gHk

0 0

3 2 2 0

( )

ω = 0 or ω² = f² +gHk²

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

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

v gH

f x

= ∂η

∂

The phase speed of these is

ω² = f² + gHk²

c_p =ω /k = ± [gH+f² /k²] The waves are dispersive

» The existence of a positive and negative root for ω(or c_p) shows that these waves can propagate in eitherx-direction.

» In the limit as k→0, ω →fand the x-dependence of the motion drops out.

» Then the motion corresponds with the pure inertial wave discussed earlier.

» In the absence of rotation (i.e., f = 0) the theory reduces to that for small amplitude surface gravity waves on shallow water.

» Such waves are non-dispersive:

ω² = f² + gHk²

2

p 2

c gH f

k k

= ω = ± +

c_p = ± gH

Note that the hydrostatic assumption restricts the validity of the whole analysis to long waves on shallow water; strictly for waves with wavelength λ >> H.

Rotation is important if => f² / k² ~ gH λ =2π(gH)^{1 2/} / f =2πL_R

the Rossby radius of deformation for a homogeneous fluid

ω² = f² + gHk²

In the deep ocean, H ~ 4 km= ×4 10 m³

At 45°latitude, f = 10⁻⁴ s⁻¹

3 1 / 2 4 7 4

2 (10 4 10 ) / 10⁻ 1.3 10 m ~ 10 km

λ = π × × × = ×

Rotation effects are important only on the planetary scale.

gH ÷200 ms⁻¹ and .

» Replace the v-momentum equation by the vorticity equation

=>

0

0

u f v gH

t x

v f

t t

u v

t x y 0

v u

x y

∂ − = − ∂η

∂ ∂

∂ζ + β = ∂η

∂ ∂

∂η ∂+ + ∂ =

∂ ∂ ∂

∂ ∂

ζ = −

∂ ∂

where

Assume thatf can be approximated by its value f₀at a particular latitude, exceptwhen differentiated with respect to yin the vorticity equation. This is justified provided that meridional particle displacements are small.

Inclusion of the beta effect

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

u u kx t

v 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 gHk

k f

k

gHk k f k

0

0 2 2

02

0

0

0

( ) ( )( )

Now expanding by the second row

A cubic equation for ωwith three real roots.

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

ω² =gHk² +f₀²

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

When ω²<< gHk², there is one rootgiven approximately by

2 2

k /(k f / gH)0

ω = −β +

» For wavelengths smallcompared with 2πtimes the Rossby length 2π gH / f₀

reduces to the dispersion relation for nondivergent Rossby waves.

k ≤ 0(f₀ gH)

the effects of divergence due to variations in the free surface elevation become important, or even dominant.

2

2 0

k k f

gH ω = − β

+

For longer wavelengths,

2 0

~ kgH f ω β For extremely long waves

These are nondispersive.

» The importance of divergence effects on the ultra-long waves explains why the calculated phase speeds for planetary waves of global wave-numbers-1and -2were unrealistically large.

» The latter were obtained from the phase speed formula for nondivergent waves.

» In the atmosphere, divergence effects are not associated with the displacement of a free surface!

» To understandthe effect of divergence on planetary waves, we introduce the meridional displacement ξof a fluid parcel.

» This is related to the meridional velocity component by v = Dξ/Dt, or to a first approximation by

t v

∂ξ =

∂

» Substituting for vin and integrating with respect to time, assuming that all perturbation quantities vanish at t = 0, gives

∂ ζ β_t + v = f₀∂ η_t

ζ= −βξ+f₀η

This formula is equivalent (within a linear analysis) to the conservation of potential vorticity (see exercise 11.2).

f + =ζ f₀ +βξ ζ+ =f₀

holds for a fluid parcel which hasξ andζinitially zero so that ζ = −βξ.

In the absence of divergence,

The term f₀ηrepresents the increase in relative vorticity due to the stretching of planetary vorticity associated with horizontal convergence (positive η).

1

−1 0

ξ ξ η η/|| , /|| ζ ζ/||

v v/||

π 2π

HI LO θ= kx− ωt

The relative phases of the quantities v, ξ, ηand ζ over one wavelength

northern hemisphere case

The relative vorticity tendency due to stretchingopposes that due to meridional motion and thereby reduces the restoring tendencyof the induced velocity field.

Phase diagram for Rossby waves

This has the solution , where

Equations:

0

v gH

f x

= ∂η

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

2 2 0

0 0

gH gH

f 0

t f x f x

⎛ ⎞

∂ ⎜ ∂ η− η + β⎟ ∂η =

∂ ⎝ ∂ ⎠ ∂

η η= cos (kx−ωt) Filtering

0

0

u f v gH

t x

v f

t t

u v

t x y 0

∂ − = − ∂η

∂ ∂

∂ζ + β = ∂η

∂ ∂

∂η ∂+ + ∂ =

∂ ∂ ∂

ω = −βk / (k² +f₀² /gH)

» There is no other solution for ωas there was before.

» In other words, making the geostrophic approximation when calculating vhas filteredout 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 equationsare 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 equationsthemselves form a filtered systemin the sense that the approximations which lead to them filter out compressible, or acoustic waves.

Filtered equations