Atmospheric Waves Chapter 6
Paul A. Ullrich [email protected]
Part 1: Wave-like Motion Atmospheric Waves
The equations of motion contain many forms of wave- like solutions, true for the atmosphere and ocean. Waves are important since they transport energy and mix the air (especially when breaking).
Some are of interest depending on the problem: Rossby waves, internal gravity (buoyancy) waves, inertial waves, inertial-gravity waves, topographic waves, shallow water gravity waves.
Some are not of interest to meteorologists, for instance sound waves.
Paul Ullrich Atmospheric Waves March 2014 Atmospheric Waves
Large-scale mid-la tudinal waves (Rossby waves) are cri cal for weather forecas ng and transport (see Atmospheric Waves).
Large-scale waves in the tropics (Kelvin waves, mixed Rossby-gravity waves) are also important, but of very different character.
Waves can be unstable. That is they start to grow, rather than just bounce back and forth.
Paul Ullrich Atmospheric Waves March 2014 Atmospheric Waves Crest y, North
x, East
Trough Trough
Lx
Defnition: The wavelength Lx Defnition: The wavenumber k of a wave is the distance of a wave is defned as between neighboring troughs 2⇡ (or crests). k ⌘ Lx
Paul Ullrich Atmospheric Waves March 2014 Atmospheric Wave Example
Paul Ullrich Atmospheric Waves March 2014 Phase line Wavenumber vector = ki + `j
crest Wavenumbers: 2⇡ 2⇡ k = ` = Ly Lx Ly
2⇡ y, North L = = k2 + `2 p
x, East
trough Lx Wavelength
Paul Ullrich Atmospheric Waves March 2014 Waves in 2D 2⇡ 2⇡ Wavelengths: L = L = x k y ` 1 1 1 k2 + `2 From geometric considerations: 2 = 2 + 2 = 2 L Lx Ly 4⇡ 2⇡ L = = k2 + `2 p
Wavenumber vector: = ki + `j
Constant phase lines: kx + `y = k r L · Vector position: r = xi + yj j i
Paul Ullrich Atmospheric Waves March 2014 2D Traveling Waves 2⇡ Wave frequency: ⌫ = with T wave period T
Geometric considerations provide: 1 1 1 = + s2 x2 y2 at fxed position (x,y) y ⌫ t s = ⌫t2 s ` ⌫t1 Phase speed: crest crest at t2 ` at t s ⌫ 1 c = = t ⌫t1/k x with wavenumber = k2 + `2 ⌫t2/k
Paul Ullrich p Atmospheric Waves March 2014 2D Traveling Waves The crests are translated over a distance: ⌫t ⌫t ⌫ t x = 2 1 = k k k Propagation speed of the wave: x ⌫ c = = x t k in x-direction y ⌫ c = = in y-direction y t ` Propagation speed of the crest line: phase speed
s ⌫ direction is parallel to c = = wavenumber vector t
Phase speed c is less than either cx or cy
Paul Ullrich Atmospheric Waves March 2014 Dispersion Relationship
If the frequency ν is a function of the wavenumber components, so is the phase speed: ⌫(k, `) c(k, `)= pk2 + `2
Physically, this implies that various waves of a composite signal will all travel at different speeds.
As a result, there is distortion of the signal over time.
This phenomenon is called wave dispersion.
Defnition: The dispersion relationship is then defned by ⌫(k, `)
Paul Ullrich Atmospheric Waves March 2014 Wave Envelope In general, a wave pattern consists of a series of superimposed waves, leading to destructive and constructive interference.
Therefore: Energy distribution is a property of a set of waves rather than a single wave. wave packet
Source: Wikipedia
Paul Ullrich Atmospheric Waves March 2014 Wave Envelope A wave pattern is a succession of wave packets.
Within each packet (here 1D), the wave propagates at the phase speed ⌫ c = k
While the packet or envelope (and therefore the energy) travels at the group velocity (here in 1D)
@⌫ Defnition: The group velocity of a system is defned as cg = @k The group velocity vector (in 2D) is defned as c ⌫ g ⌘rk @⌫ @⌫ Components of group velocity vector: cgx = cgy = @k @`
Paul Ullrich Atmospheric Waves March 2014 Group Velocity
Source: Wikipedia
Source: Wikipedia
Paul Ullrich Atmospheric Waves March 2014 Linear Perturbation Theory
Assume that each variable is equal to the mean state plus a perturbation from that mean:
u = u + u0
Variable Mean Difference (perturbation)
This is a very general approach, and can always be done.
Paul Ullrich Atmospheric Waves March 2014 Linear Perturbation Theory
Impose assumptions and constraints on the mean and perturbations
Assumptions need to be physically sensible and justifed
For example: u = u + u"
Variable Mean Perturbation: Perturbations (independent are “small”, and so products of of time and perturbations are very small. longitude) € Paul Ullrich Atmospheric Waves March 2014 Linear Perturbation Theory
With these assumptions non-linear terms (like the one below) become linear:
@u @ u =(u + u0) (u + u0) @x @x @u @u @u0 @u0 = u + u0 + u + u0 @x @x @x @x
These terms are zero since the Terms with products of mean is independent of x. perturbations are very small and will be ignored. @u @u u u 0 @x ⇡ @x
Paul Ullrich Atmospheric Waves March 2014 Linearization Assume that the mean state satisfes the equations of motion
For example: u-momentum equation becomes (here with constant density, frictionless): Du 1 @p = + fv Dt ⇢ @x @u @u @u 1 @p + u + v = + fv @t @x @y ⇢ @x
Plug in: u = u + u0 v = v0 p = p + p0
@u0 @u0 @u 1 @p0 Linearize: + u + v0 = + fv0 @t @x @y ⇢ @x
Paul Ullrich Atmospheric Waves March 2014 1D Wave Solutions
Assume we have derived a set of linearized equations with constant coefficients.
Look for simple wave-like solutions: 2⇡ 2⇡ u0(x, t)=u0 cos(kx ⌫t) with k = ⌫ = Lx T
However, this form of the wave fxes the “phase” over the wave. That is at x=0 and t=0 all solutions must have u 0 = u 0 .
More general solution: Use complex numbers i = p 1
u0(x, t)=Re[u exp(ikx i⌫t)] u =Re(u )+iIm(u ) 0 0 0 0
Paul Ullrich Atmospheric Waves March 2014 2D Wave Solutions
2D waves that propagate horizontally in the x and y
direction with constant wave amplitude u0
u0(x, y, t)=Re[u exp(i(kx + `y ⌫t))] 0
Real part of Wave amplitude, may be complex
Recall: exp(i ) = cos + i sin
Re [u exp(i )] = Re(u ) cos Im(u )sin 0 0 0 Note: Only the real part Re[ ] has physical meaning!
Paul Ullrich Atmospheric Waves March 2014 3D Wave Solutions
In 3D search for general wave solutions of the form:
u0(x, y, z, t)=Re[u0 exp(i(kx + `y + mz ⌫t))]
Wave amplitude, may be complex
Such a wave propagates in all three dimensions.
Alternatively: Horizontally propagating waves with varying amplitude u0(z) in the vertical direction:
u0(x, y, z, t)=Re[u (z)exp(i(kx + `y ⌫t))] 0
Paul Ullrich Atmospheric Waves March 2014