Atmospheric Waves Chapter 6climate.ucdavis.edu/...Chapter06-Part01-WaveMotion.pdf · x,*East...

Atmospheric Waves Chapter 6

Paul A. Ullrich

paullrich@ucdavis.edu

Part 1: Wave-like Motion

Atmospheric Waves

Paul Ullrich Atmospheric Waves March 2014

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.

Atmospheric Waves

Paul Ullrich Atmospheric Waves March 2014

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.  

x,  East  

y,  North   Crest

Trough Trough

Atmospheric Waves

k ⌘ 2⇡

Lx

Definition: The wavelength Lx of a wave is the distance between neighboring troughs (or crests).

Paul Ullrich Atmospheric Waves March 2014

Lx

Definition: The wavenumber k of a wave is defined as

Atmospheric Wave Example

Paul Ullrich Atmospheric Waves March 2014

Wavenumber vector

Wavenumbers: crest

trough

= ki+ `j

k =2⇡

Lx

` =2⇡

Ly

Ly

Lx

L =2⇡

=

pk2 + `2

x,  East  

y,  North  

Wavelength

Paul Ullrich Atmospheric Waves March 2014

Phase line

Wavelengths:

From geometric considerations:

Wavenumber vector:

Constant phase lines:

Vector position:

Waves in 2D

Lx

=2⇡

kLy

=2⇡

`1

L2=

1

L2x

+1

L2y

=k2 + `2

4⇡2

ji

L =2⇡

=

pk2 + `2

= ki+ `j

kx+ `y = k · r

r = xi+ yj

L

Paul Ullrich Atmospheric Waves March 2014

with T wave period

Geometric considerations provide:

Phase speed:

Wave frequency:

crest at t2 crest at t1

at fixed position (x,y)

with wavenumber

2D Traveling Waves

⌫ =2⇡

T

1

�s

2=

1

�x

2+

1

�y

2

�s =⌫�t

c =�s

�t=

⌫

=p

k2 + `2

⌫t2`

⌫t1`

⌫t1/k �x

⌫t2/k

�s

�y

Paul Ullrich Atmospheric Waves March 2014

The crests are translated over a distance:

Propagation speed of the wave:

in x-direction

in y-direction

Propagation speed of the crest line: phase speed

direction is parallel to wavenumber vector

2D Traveling Waves

�x =⌫t2

k

� ⌫t1

k

=⌫�t

k

c

x

=�x

�t

=⌫

k

c

y

=�y

�t

=⌫

`

c =�s

�t=

⌫

Phase  speed  c  is  less  than  either  cx  or  cy

Paul Ullrich Atmospheric Waves March 2014

If the frequency ν is a function of the wavenumber components, so is the phase speed:

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.

Dispersion Relationship

c(k, `) =⌫(k, `)pk2 + `2

Definition: The dispersion relationship is then defined by ⌫(k, `)

Paul Ullrich Atmospheric Waves March 2014

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

Wave Envelope

Paul Ullrich Atmospheric Waves March 2014

Source: Wikipedia

A wave pattern is a succession of wave packets. Within each packet (here 1D), the wave propagates at the phase speed While the packet or envelope (and therefore the energy) travels at the group velocity (here in 1D)

Components of group velocity vector:

Wave Envelope

c =⌫

k

cg =@⌫

@k

cgx

=@⌫

@kcgy

=@⌫

@`

Paul Ullrich Atmospheric Waves March 2014

Definition: The group velocity of a system is defined as The group velocity vector (in 2D) is defined as cg ⌘ rk⌫

Group Velocity

Source: Wikipedia

Source: Wikipedia

Paul Ullrich Atmospheric Waves March 2014

This is a very general approach, and can always be done.

Linear Perturbation Theory

u = u+ u0

Variable Mean Difference (perturbation)

Assume that each variable is equal to the mean state plus a perturbation from that mean:

Paul Ullrich Atmospheric Waves March 2014

Impose assumptions and constraints on the mean and perturbations Assumptions need to be physically sensible and justified For example:

€

u = u + " u

Linear Perturbation Theory

Variable Mean (independent

of time and longitude)

Perturbation: Perturbations are “small”, and so products of perturbations are very small.

Paul Ullrich Atmospheric Waves March 2014

u

@u

@x

= (u+ u

0)@

@x

(u+ u

0)

= u

@u

@x

+ u

0 @u

@x

+ u

@u

0

@x

+ u

0 @u0

@x

With these assumptions non-linear terms (like the one below) become linear:

Linear Perturbation Theory

These terms are zero since the mean is independent of x.

Terms with products of perturbations are very small

and will be ignored.

u

@u

@x

⇡ u

@u

0

@x

Paul Ullrich Atmospheric Waves March 2014

Assume that the mean state satisfies the equations of motion For example: u-momentum equation becomes (here with constant density, frictionless):

Plug in:

Linearize:

Linearization

Du

Dt

= �1

⇢

@p

@x

+ fv

@u

@t

+ u

@u

@x

+ v

@u

@y

= �1

⇢

@p

@x

+ fv

u = u+ u0 v = v0 p = p+ p0

@u

0

@t

+ u

@u

0

@x

+ v

0 @u

@y

= �1

⇢

@p

0

@x

+ fv

0

Paul Ullrich Atmospheric Waves March 2014

Assume we have derived a set of linearized equations with constant coefficients. Look for simple wave-like solutions:

However, this form of the wave fixes the “phase” over the wave. That is at x=0 and t=0 all solutions must have . More general solution: Use complex numbers

1D Wave Solutions

u

0(x, t) = u0 cos(kx� ⌫t) k =

2⇡

Lx

⌫ =2⇡

Twith

u0 = u0

Paul Ullrich Atmospheric Waves March 2014

i =p�1

u0 = Re(u0) + iIm(u0)u

0(x, t) = Re [u0 exp(ikx� i⌫t)]

2D waves that propagate horizontally in the x and y direction with constant wave amplitude u0

Note: Only the real part Re[ ] has physical meaning!

2D Wave Solutions

u

0(x, y, t) = Re [u0 exp(i(kx+ `y � ⌫t))]

Real part of Wave amplitude, may be complex

exp(i�) = cos�+ i sin�Recall:

Re [u0 exp(i�)] = Re(u0) cos�� Im(u0) sin�

Paul Ullrich Atmospheric Waves March 2014

In 3D search for general wave solutions of the form:

3D Wave Solutions

u

0(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:

u

0(x, y, z, t) = Re [u0(z) exp(i(kx+ `y � ⌫t))]

Paul Ullrich Atmospheric Waves March 2014