Atmospheric Waves Perturbation Theory Based on Chapter 7

Atmospheric Waves: Perturbation Theory Based on Chapter 7 of Holton’s An Introduction to Dynamic Meteorology Chapter 7 - Waves MT 454

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 Chapter 7 - Waves MT 454

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. Chapter 7 - Waves MT 454

Perturbation Method: Assumptions Assume: 1. One can view much important atmospheric behavior as perturbations about a basic state, e. g. , 2. Basic state is given (known), but it must be a solution to the governing equations 3. Perturbations much smaller than basic state, e. g. , or Applying 1 - 3 gives linearized equations 7. 1 Perturbation Method MT 454

Class Slide 7. 1 Perturbation Method MT 454

Class Slide 7. 1 Perturbation Method MT 454

Class Slide 7. 1 Perturbation Method MT 454

Class Slide 7. 1 Perturbation Method MT 454

Class Slide 7. 1 Perturbation Method MT 454

Class Slide 7. 1 Perturbation Method MT 454

• 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. Class Slide 7. 1 Perturbation Method MT 454

Class Slide 7. 1 Perturbation Method MT 454

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) 7. 2 Wave Properties MT 454

Class Slide (Holton gives another example. ) 7. 2 Wave Properties MT 454

Class Slide 7. 2 Wave Properties MT 454

Possible solution: Class Slide Can test: substitute into equation. Note that 2 ond derivatives of trig functions return -(original function). E. g. : d 2 cos( t)/dt 2 = - 2 cos( t) 7. 2 Wave Properties MT 454

Class Slide 7. 2 Wave Properties MT 454

Class Slide 7. 2 Wave Properties MT 454

1. = frequency of oscillations 2. One wave period or cycle = 2 / 3. is independent of Xo (amplitude) 4. Phase of oscillation is = t - o 7. 2 Wave Properties MT 454

Propagating Waves Example is stationary oscillator. Propagating oscillations? 1. Similarity: characterization by amplitude & phase 2. Phase now function of time and space: e. g. , in 1 -D: = kx - t + o 3. k = 2 /Lx (wavenumber) 4. 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 MT 454

Class Slide 7. 2 Wave Properties MT 454

Class Slide 7. 2 Wave Properties MT 454

If observer is moving with the wave, then phase is constant. Thus: Class Slide 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 MT 454

In 2 or More Dimensions Lines of constant k = (phase-change in x-direction)/(unitlength) l = (phase-change in y-direction)/(unitlength) 7. 2 Wave Properties |K| =(phase-change)/(unitlength) MT 454

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) MT 454

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 7. 2 Wave Properties MT 454

Phase Speed in Coordinate Directions Move with point of constant phase - e. g. , crest Move only in xdirection: Similarly, looking at phase change only in y direction (e. g. , crest movement in y) 7. 2 Wave Properties MT 454

C Is Not A Vector! - 1 Cx is rate of phase advance in xdirection (e. g. , rate of advance of point P on crest) Cx increases with decreasing projection of K vector onto x axis: P 7. 2 Wave Properties P Cx MT 454

C Is Not A Vector! - 2 Cx increases with decreasing projection of K vector onto x axis. Thus: As angle 90˚, Cx ! P Cx thus > speed of light => not a physical velocity Rather, this is location change of a geometric point Thus, phase “speed”, not “velocity” 7. 2 Wave Properties MT 454

A Physical Vector: Group Velocity Class Slide The group velocity describes energy propagation. 7. 2 Wave Properties MT 454

Class Slide 7. 2 Wave Properties MT 454

Class Slide (See also figures shown in class) 7. 2 Wave Properties MT 454

Class Slide 7. 2 Wave Properties MT 454

Class Slide 7. 2 Wave Properties MT 454

Class Slide 7. 2 Wave Properties MT 454

Class Slide 7. 2 Wave Properties MT 454

Class Slide 7. 2 Wave Properties MT 454

Simple Wave Types 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Conservation Laws Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Review: Real and Imaginary Parts of Complex Numbers Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Shallow-Water Gravity Waves Restoring Force: Gravity Need density discontinuity Force is transverse to direction of propagation 7. 3 Wave Types MT 454

Horizontal Propagation Water flows in from sides at rest, creating new depressions 7. 3 Wave Types MT 454

Example Configuration x 2 z 2 po z 1 po + p. A A z 1 po + p. B B h Two incompressible fluids, lower-density fluid on top. 7. 3 Wave Types MT 454

Assumptions - 1 Assumptions: 1. Incompressible => no sound waves (simplifies equations) 2. Hydrostatic Density I is constant in layer I, so p/ x is independent of depth: ( p/ x)/ z = ( p/ z)/ x = - ( g)/ x = 0 3. p/ x = 0 in upper layer (simplicity) 4. Motion in x-z plane => v=0 and ( )/ y = 0 7. 3 Wave Types MT 454

Assumptions - 2 Assumptions: 5. Time scale << 1 day (ignore Coriolis force) 6. Flat bottom => w(z=0) = 0 7. 3 Wave Types MT 454

Pressure at Two Points: A & B 2 po 1 po + p. A po + p. B What is increase in pressure going from po down to levels A & B? h at A: pressure = po + p. A = po + 2 g z at B: pressure = po + p. B = po + 2 g z 2+ 1 g z 1 but z 1 = x (∂h/∂x) z 2 = z - x (∂h/∂x) 7. 3 Wave Types (Relative to point A: go farther from A, z 1 increases according to slope of h) MT 454

Pressure Difference between A & B Or, as x 0 and recognizing canceling terms: 1. po same at A & B because assumed p = 0 in upper layer 2. Implicitly assume x 0 already when using z 1 = ( h/ x) x 7. 3 Wave Types MT 454

Conservation Laws Conservation of u momentum: Need only lower layer and only u on basis of assumptions. Conservation of mass: Incompressible (which also => no need for a thermodynamic equation) 7. 3 Wave Types MT 454

Solving the Equations 1. Assume u u(z). (since ( h/ x) not function of z) 3. But … 7. 3 Wave Types MT 454

Re-written Continuity Equation With momentum equation: 2 equations & 2 unknowns 7. 3 Wave Types MT 454

Perturbation Procedure - Step 1 Assume separation into basic state and perturbations: 7. 3 Wave Types MT 454

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 => ANSWER: YES 7. 3 Wave Types MT 454

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( )' ] : 7. 3 Wave Types MT 454

Eliminate u' - Substitute Eliminate u' by applying { ( )/ t + u ( )/ x} to second eq. and substitute into first eq. Note: Linear in h' 7. 3 Wave Types MT 454

Wave Form for Solution Assume h' = A exp{ik(x-ct)} (Only real part is physical) Recall d exp{ik(x-ct)} /dx = ik exp{ik(x-ct)} d exp{ik(x-ct)} /dt = -ikc exp{ik(x-ct)} 7. 3 Wave Types MT 454

Solutions Trivial: h' = 0 Non-Trivial: Require other terms add to zero: 7. 3 Wave Types MT 454

Notes on Solutions 1. Shallow water - Require Lx >> H. Otherwise too deep for hydrostatic assumption. 2. Function of SQRT(H). 3. Ocean: H ~ 4 km => (for u = 0) c = 200 m/s = 720 km/hr. 4. Ocean thermocline: / 1 ~ 0. 01 => c = 20 m/s. 5. (cg)x = c, since / k = (ck)/ k = c. 7. 3 Wave Types MT 454

7. 7 Class Slide 7. 7 Rossby Waves MT 454

Class Slide 7. 7 Rossby Waves MT 454

Class Slide 7. 7 Rossby Waves MT 454

Class Slide 7. 7 Rossby Waves MT 454

Class Slide 7. 7 Rossby Waves MT 454

Class Slide 7. 7 Rossby Waves MT 454

Class Slide 7. 7 Rossby Waves MT 454

Class Slide 7. 7 Rossby Waves MT 454

Class Slide 7. 7 Rossby Waves MT 454

Internal Gravity Waves 7. 4 Internal Gravity Waves MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Class Slide 7. 3 Wave Types MT 454

Geostrophic Adjustment Motivation • 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 MT 454

Analysis: Use Simplest Q-G System 1. f = constant (f-plane) 2. Shallow-water equations (no stratification, no consideration of T, but include “f” terms) 3. Basic state flow = 0 (perturbations about a rest state) 4. Air over water => lower ≈ 1 7. 6 Geostrophic Adjustment MT 454

Perturbation Equations Note: momentum equations for u’ and v’ continuity includes u’/ x and v’/ y 7. 6 Geostrophic Adjustment MT 454

h Equation - 1 Take (last equation of previous slide)/ t: Substitute from u’ and v’ momentum equations: ? See next slide. 7. 6 Geostrophic Adjustment MT 454

h Equation - 2 Where … … and we recognize c 2 = g. H , the 1 -D shallow-water gravity-wave phase speed 7. 6 Geostrophic Adjustment MT 454

Extremes - Case 1 Suppose f=0: • 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 exp{i(kx+ly- t)} , and 2 = g. H(k 2+l 2) • Earlier, had simply 2 = g. Hk 2 , so this is a generalization 7. 6 Geostrophic Adjustment MT 454

Extremes - Case 2 Suppose f≠ 0: • There is coupling of h’ and ’ • Estimate sizes of terms: t ~ 1/fo ; x, y ~ L This is << 1 if n ≥ 4 (i. e. , H >> 1 km, or deep) Then have balanced state: 7. 6 Geostrophic Adjustment MT 454

Steady State Geostrophic balance gives Steady state is in geostrophic balance 7. 6 Geostrophic Adjustment MT 454

Two Extremes 1. f = 0 (or H very small): gravity waves 2. f ≠ 0, H very big: geostrophic steady state What happens in between? Considered by Rossby (1930 s) 7. 6 Geostrophic Adjustment MT 454

Assume Unbalanced What is evolution given by this equation? Need in terms of h , i. e. , have 2 unknowns, need 2 equations 7. 6 Geostrophic Adjustment MT 454

Second Equation: Vorticity Take (v equation)/ x - (u equation)/ y Tie to h ? 7. 6 Geostrophic Adjustment MT 454

Eliminate u' and v' Tie to h ? Recall continuity equation Then substituting for perturbation divergence 7. 6 Geostrophic Adjustment MT 454

Perturbation Potential Vorticity Q' Define perturbation potential vorticity: Then Q / t = 0, or Q = constant (is conserved) Q (t=0) Q (t>0) Do not have to solve a time-dependent differential equation 7. 6 Geostrophic Adjustment MT 454

Example: Initial State u’, v’ = 0 h’ = -ho sgn(x) Step function for initial h'. This is not balanced! (What is PGF at x=0? ) 7. 6 Geostrophic Adjustment MT 454

Once and Future Potential Vorticity Solve for Substitute into 2 h / t 2 7. 6 Geostrophic Adjustment MT 454

Eliminating Then: 7. 6 Geostrophic Adjustment MT 454

Intertio-Gravity Waves If ho = 0, then have homogeneous equation (almost like the shallow-water gravity waves equation): Gives inertio-gravity waves: 2 = f 2 + g. H(k 2 + l 2) If waves have long enough wavelength (k & l small enough), then f is important and coriolis force affects waves. 7. 6 Geostrophic Adjustment MT 454

ho ≠ 0 Since h (t=0) 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) 7. 6 Geostrophic Adjustment MT 454

Assume: Steady State Obtained 7. 6 Geostrophic Adjustment MT 454

Adjusted State - 1 SOLUTION: Rossby Radius of Deformation 7. 6 Geostrophic Adjustment MT 454

Adjusted State - 2 7. 6 Geostrophic Adjustment MT 454

Adjusted State - 3 1. For |x| >> R, h' unchanged 2. u' = - (g/f) h / y = 0 3. v' = + (g/f) h / x = -(gho/ R) exp{-|x|/ R} 7. 6 Geostrophic Adjustment MT 454

Importance of Q' 1. Using h'/ t = 0 alone does not yield unique steady state (two unknowns, 1 equation) 2. Combined with Q' conservation: only 1 final state 7. 6 Geostrophic Adjustment MT 454

Time Evolution 1. Complex 2. Involves energy transfer by gravity waves 3. How much by gravity waves? 7. 6 Geostrophic Adjustment MT 454

Energy Partitioning Determine how much energy goes into gravity waves. Initially: Use this with care! If integrate over all x, get infinite energy. 7. 6 Geostrophic Adjustment MT 454

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) 7. 6 Geostrophic Adjustment MT 454

… result Compute instead change in potential energy: 7. 6 Geostrophic Adjustment MT 454

Energy Change: f = 0 For f = 0: • Rossby radius ∞ and h' 0 • So potential energy change ∞ • All P K • This is gravity-wave solution (energy into gravity waves) • h' = 0 for |x| ∞ as t ∞ : No energy left at finite x. Gravity waves carry it away. 7. 6 Geostrophic Adjustment MT 454

Energy Change: f ≠ 0 For f ≠ 0: • Change in P is finite • For steady part of flow: 7. 6 Geostrophic Adjustment MT 454

Missing Energy? Thus we have: These are not equal. Where is the rest of P? Radiating away in interio-gravity waves 7. 6 Geostrophic Adjustment MT 454

Final Notes - 1 1. Only a small part of P K, since P(t ∞) also ∞ 2. Potential vorticity conservation is a very useful tool: time integration done "automatically" 3. Rossby radius of deformation is a fundamental scale. P occurs primarily for |x| < R Steady-flow K occurs primarily for |x| < R Inertio-gravity wave K radiates to |x| >> R 7. 6 Geostrophic Adjustment MT 454

Final Notes - 2 4. This scale appears in other flow problems. It is a natural scale for large-scale, quasi-geostrophic flow. 5. For H ~ 10 km, f ~ 10 -4 s-1, R ~ SQRT(10. 104)/10 -4 ~ 3. 103 km, 4. 5. or, the synoptic scale. The Rossby radius of deformation is thus fundamental to the appearance of this numeric scale in synoptic flow. 7. 6 Geostrophic Adjustment MT 454

Atmospheric Waves END Chapter 7 - Waves MT 454