Atmospheric Waves: Perturbation Theory

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Atmospheric Waves Perturbation Theory
Based on Chapter 7 of Holtons An Introduction
to Dynamic Meteorology
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Waves in Atmosphere

• Wavelike behavior commonly observed
• Wave solutions to conservation laws help us
  understand physical interactions and energy
  propagation
• As first approximation, one can superimpose wave
  solutions of different scales to depict
  atmospheric flow

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Simplification Needed

• Full equations too complicated for physical
  insight - need simplified models
• Chapter 6 Primitive equations simplified to
  quasi-geostrophic system
• Chapter 7 Q-G equations simplified to
  linearized equations.

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Perturbation Method Assumptions

• Assume
• One can view much important atmospheric behavior
  as perturbations about a basic state, e.g.,
• Basic state is given (known), but it must be a
  solution to the governing equations
• Perturbations much smaller than basic state,
  e.g., or
• Applying 1- 3 gives linearized equations

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7.1 Perturbation Method
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• Resulting equation is linear in ( )' variables
• Since basic state is given, applying same method
  to all of our conservation laws gives a set of
  linearized equations in ( )' variables.
• Linear equations are much easier to solve than
  nonlinear equations.
• Linear equations often give wave solutions.
• Typically assume ( ) sinusoidal waves.

7.1 Perturbation Method
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Solving for Waves

• Look to find specific properties
• Phase speed
• Energy propagation
• Vertical structure
• Conditions for existence, growth and decay of
  waves (when where we might expect to see
  physical interactions represented by the waves)

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(Holton gives another example.)
7.2 Wave Properties
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7.2 Wave Properties
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Possible solution
Can test substitute into equation. Note that
2ond derivatives of trig functions return
-(original function). E.g. d2cos(?t)/dt2
-?2cos(?t)
7.2 Wave Properties
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7.2 Wave Properties
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7.2 Wave Properties
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• ? frequency of oscillations
• One wave period or cycle 2?/?
• ? is independent of Xo (amplitude)
• Phase of oscillation is ? ?t - ?o

7.2 Wave Properties
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Propagating Waves
Example is stationary oscillator. Propagating
oscillations?

• Similarity characterization by amplitude
  phase
• Phase now function of time and space e.g., in
  1-D ? kx - ?t ?o
• k 2?/Lx (wavenumber)
• Phase speed c ?/k
• Speed observer must move for phase of wave to be
  constant (e.g., speed of trough/crest movement)

7.2 Wave Properties
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7.2 Wave Properties
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If observer is moving with the wave, then phase
is constant. Thus
This gives the change in position x in time t,
hence speed, for point maintaining constant phase
with respect to wave.
7.2 Wave Properties
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In 2 or More Dimensions
Lines of constant ??
k (phase-change in x-direction)/(unit-length)
K (phase-change)/(unit-length)
l (phase-change in y-direction)/(unit-length)
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Wavelength in 2 or More Dimensions
K (? phase) (unit-length)
Lines of constant ??
Then (? phase) (length-moved) x (?
phase)/(unit-length) If (? phase) 2?, then
wavelength ? 2?/K Wavelength distance
for wave form to repeat (e.g., crest-to-crest
distance)
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Phase Speed in 2 or More Dimensions
Move with point of constant phase - e.g., crest
By analogy with 1-D, for phase speed C,
perpendicular to lines of constant ??
Lines of constant ??
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Phase Speed in Coordinate Directions
Move with point of constant phase - e.g., crest
Move only in x-direction
Similarly, looking at phase change only in y
direction (e.g., crest movement in y)
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C Is Not A Vector! - 1
Cx is rate of phase advance in x-direction (e.g.,
rate of advance of point P on crest)
Cx increases with decreasing projection of K
vector onto x axis


P
P
Cx
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C Is Not A Vector! - 2
Cx increases with decreasing projection of K
vector onto x axis. Thus

As angle ? ? 90, Cx ? ?! Cx thus gt speed of
light gt not a physical velocity Rather, this is
location change of a geometric point Thus, phase
speed, not velocity
?????
P
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A Physical Vector Group Velocity
The group velocity describes energy propagation.
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(See also figures shown in class)
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7.2 Wave Properties
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Simple Wave Types
7.3 Wave Types
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Conservation Laws
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Review Real and Imaginary Parts of Complex
Numbers
7.3 Wave Types
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Restoring Force Gravity
Shallow-Water Gravity Waves
Need density discontinuity
Force is transverse to direction of propagation
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Horizontal Propagation
Water flows in from sides at rest, creating new
depressions
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Example Configuration
dx
r2
dz2
po
dz
dz1
r1
po dpB
po dpA
A
B
h
Two incompressible fluids, lower-density fluid on
top.
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Assumptions - 1

• Assumptions
• Incompressible gt no sound waves (simplifies
  equations)
• Hydrostatic
• Density ?I is constant in layer I, so ?p/?x is
  independent of depth ?(?p/?x)/?z ?(?p/?z)/?x
  - ?(?g)/?x 0
• ?p/?x 0 in upper layer (simplicity)
• Motion in x-z plane gt v0 and ?( )/?y 0

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Assumptions - 2

• Assumptions
• Time scale ltlt 1 day (ignore Coriolis force)
• Flat bottom gt w(z0) 0

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Pressure at Two Points A B
r2
What is increase in pressure going from po down
to levels A B?
po
r1
po dpB
po dpA
h
at A pressure po dpA po r2g dz
at B pressure po dpB po r2g dz2 r1g dz1
but dz1 dx (?h/?x)
(Relative to point A go farther from A, dz1
increases according to slope of h)
dz2 dz - dx (?h/?x)
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Pressure Difference between A B
Or, as ?x ? 0 and recognizing canceling terms

• po same at A B because assumed ?p 0 in upper
  layer
• Implicitly assume ?x ? 0 already when using ?z1
  (?h/?x)?x

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Conservation Laws
Conservation of u momentum
Need only lower layer and only u on basis of
assumptions.
Conservation of mass
Incompressible (which also gt no need for a
thermodynamic equation)
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Solving the Equations
1. Assume u ? u(z). (since (?h/?x) not function
of z)
3. But
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Re-written Continuity Equation
With momentum equation 2 equations 2 unknowns
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Perturbation Procedure - Step 1
Assume separation into basic state and
perturbations
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Perturbation Procedure - Step 2
Specify basic state (All variables are constants)
Is this a solution? Substitute into our 2
equations Momentum Get 0 0 Modified
Continuity Get 0 0 Both equations satisfied
gt ANSWER YES
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Perturbation Procedure - Step 3
Subsitute pertubation basic state into both
equations, Drop derivatives of constants e.g.,
basic state and products of perturbations e.g.,
( )'x( )'
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Eliminate u' - Substitute
Eliminate u' by applying ?( )/?t u?( )/?x to
second eq. and substitute into first eq.
Note Linear in h'
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Wave Form for Solution
Assume h' A expik(x-ct) (Only real part is
physical)
Recall d expik(x-ct) /dx ik
expik(x-ct) d expik(x-ct) /dt -ikc
expik(x-ct)
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Solutions
Trivial h' 0
Non-Trivial Require other terms add to zero
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Notes on Solutions

• Shallow water - Require Lx gtgt H. Otherwise too
  deep for hydrostatic assumption.
• Function of SQRT(H).
• Ocean H 4 km gt (for u 0) c 200 m/s 720
  km/hr.
• Ocean thermocline ??/?1 0.01 gt c 20 m/s.
• (cg)x c, since ??/?k ?(ck)/?k c.

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7.7
7.7 Rossby Waves
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Internal Gravity Waves
7.4 Internal Gravity Waves
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Motivation
Geostrophic Adjustment

• Synoptic midlatitude motions geostrophic
  balance
• What if knocked out of balance?
• (e.g., sudden impulse - convective heating,
  downdraft)
• How does atmosphere return to geostrophic
  balance?
• (It must or we would not observe it.)

Answer will show role for gravity waves!
7.6 Geostrophic Adjustment
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Analysis Use Simplest Q-G System

• f constant (f-plane)
• Shallow-water equations (no stratification, no
  consideration of T, but include f terms)
• Basic state flow 0 (perturbations about a rest
  state)
• Air over water gt ????lower 1

7.6 Geostrophic Adjustment
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Perturbation Equations
Note momentum equations for u and v
continuity includes ?u/?x and ?v/?y
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h? Equation - 1
Take ?(last equation of previous slide)/?t
Substitute from u and v momentum equations
??? See next slide.
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h? Equation - 2
Where
and we recognize c2 gH , the 1-D
shallow-water gravity-wave phase speed
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Extremes - Case 1

• Suppose f0
• Height and vorticity equations decouple (no h in
  vorticity eq., no vorticity in height eq.)
• Equation is then 2-D shallow-water gravity wave
  eq., where h A expi(kxly-?t) , and  ?2
  gH(k2l2)
• Earlier, had simply ?2 gHk2 , so this is a
  generalization

7.6 Geostrophic Adjustment
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Extremes - Case 2

• Suppose f?0
• There is coupling of h and ?
• Estimate sizes of terms t 1/fo x,y L

This is ltlt 1 if n 4 (i.e., H gtgt 1 km, or deep)
Then have balanced state
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Steady State
Geostrophic balance gives
Steady state is in geostrophic balance
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Two Extremes

• f 0 (or H very small) gravity waves
• f ? 0, H very big geostrophic steady state

What happens in between? Considered by Rossby
(1930s)
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Assume Unbalanced
What is evolution given by this equation?
Need ?? in terms of h? , i.e., have 2 unknowns,
need 2 equations
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Second Equation Vorticity
Take ?(v? equation)/?x - ?(u? equation)/?y
Tie to h??
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Eliminate u' and v'
Tie to h?? Recall continuity equation
Then substituting for perturbation divergence
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Perturbation Potential Vorticity Q'
Define perturbation potential vorticity
Then ?Q?/?t 0, or Q? constant (is conserved)
Q?(t0) ? Q?(tgt0) Do not have to solve a
time-dependent differential equation
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Example Initial State
u, v 0 h -ho sgn(x)
Step function for initial h'. This is not
balanced! (What is PGF at x0?)
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Once and Future Potential Vorticity
Solve for ?? Substitute into ?2h?/?t2
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Eliminating ??
Then
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Intertio-Gravity Waves
If ho 0, then have homogeneous equation (almost
like the shallow-water gravity waves equation)
Gives inertio-gravity waves ?2 f2 gH(k2
l2) If waves have long enough wavelength (k l
small enough), then f is important and coriolis
force affects waves.
7.6 Geostrophic Adjustment
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ho ? 0

• Since h?(t0) is independent of y, then
• There is no PGF acting in y direction
• Thus nothing can change in y direction
• So h' remains independent of y
• (Due to symmetry in y)

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Assume Steady State Obtained
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Adjusted State - 1
SOLUTION
Rossby Radius of Deformation
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Adjusted State - 2
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Adjusted State - 3

• For x gtgt ?R, h' unchanged
• u' - (g/f)?h?/?y 0
• v' (g/f)?h?/?x -(gho/?R) exp-x/?R

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Importance of Q'

• Using ?h'/?t 0 alone does not yield unique
  steady state (two unknowns, 1 equation)
• Combined with Q' conservation only 1 final
  state

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Time Evolution

• Complex
• Involves energy transfer by gravity waves
• How much by gravity waves?

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Energy Partitioning
Determine how much energy goes into gravity
waves. Initially
Use this with care! If integrate over all x, get
infinite energy.
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Energy Change
Compute instead change in potential energy
Here P with is the energy per unit y (i.e,
integrate one unit of length in y)
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result
Compute instead change in potential energy
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Energy Change f 0

• For f 0
• Rossby radius ? 8 and h' ? 0
• So potential energy change ? 8
• All P ? K
• This is gravity-wave solution (energy into
  gravity waves)
• h' 0 for x ? 8 as t ? 8 No energy left at
  finite x. Gravity waves carry it away.

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Energy Change f ? 0

• For f ? 0
• Change in P is finite
• For steady part of flow

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Missing Energy?
Thus we have These are not equal. Where is
the rest of P? Radiating away in interio-gravity
waves
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Final Notes - 1

• Only a small part of P ? K, since P(t?8) also ? 8
• Potential vorticity conservation is a very useful
  tool time integration done "automatically"
• Rossby radius of deformation is a fundamental
  scale.

?P occurs primarily for x lt ?R Steady-flow K
occurs primarily for x lt ?R Inertio-gravity
wave K radiates to x gtgt ?R
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Final Notes - 2

• This scale appears in other flow problems. It is
  a natural scale for large-scale,
  quasi-geostrophic flow.
• For H 10 km, f 10-4 s-1,
• ?R SQRT(10.104)/10-4 3.103 km,
• or, the synoptic scale.
• The Rossby radius of deformation is thus
  fundamental to the appearance of this numeric
  scale in synoptic flow.

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Atmospheric Waves
END