MET 4302 Midterm Study Guide 19FEB18

Transcription

1 The exam will be 4% short answer and the remainder (6%) longer (1- aragrahs) answer roblems and mathematical derivations. The second section will consists of 6 questions worth 15 oints each. Answer Be able to use the erturbation method to obtain linearized versions of nonlinear artial differential equations (see item 14).. Understand the derivation and alication of the linearized Shallow-Water equations: u' u' h' + u = g t v' v' h' + u = g t y h' h' u' v' + u = H + t y 3. Be able to derive the disersion relation for two-dimensional shallow water gravity waves from the linearized governing equations. u v h h + u h' = H + u + = gh + t t y y 4. Understand reresentation of sinusoidal functions in terms of frequency ω and wavenumber k such that ( ) sin πt / T πx / L = sin( ωt kx) where T is the wave eriod and L is the wavelength. 5. Know that the hase seed of a sinusoidal wave is L/T = ω/k. 6. Be able to use Euler s theorem to reresent sinusoidal solutions ψ = Aex{ i( ωt kx)} = Re{( A ia )[cos( ωt kx) + i sin( ωt kx)]} = A cos( ωt kx) + A sin( ωt kx) of linearized artial R I R I iθ differential equations. Remember e = cosθ + isinθ is Euler s theorem. 7. Having made this substitution be able to obtain a Disersion Relation that exresses the frequency as a function of wavenumber(s). 8. Starting with the two-dimensional rotational shallow-water equations: u u u H + u + v fv + g = t y v v v H + u + v + fu + g = t y y H H H u v + u + v + H + = t y y 1

2 where H is the total deth be able to cross differentiate and subtract the momentum equations substitute from the continuity equation and divide by H to obtain the shallow-water otential vorticity equation: 1 D( ζ + f ) ζ + f DH D ζ + f = = H Dt H Dt Dt H 9. Starting with the same shallow-water equations multily the zonal equation by u and the meridional equation by v add them then add g times the continuity equation to obtain the shallow-water energy equation: u v u v u v ( ugh) ( vgh) + + gh + u + + gh + v + + gh = + t y y 1. Know that the first term inside each of the arentheses on the left is the kinetic energy; the second term inside each arentheses is the otential energy. Thus left side of the equation is the individual derivative of the total energy and the right side is the eddy divergence of otential energy. 11. Know the difference between flux form and advective form of the Lagrangian derivative and be able to use the continuity equation to go from one to the other. 1. By way of review from last term know and be able to aly the basic hyotheses that underlie the middle-latitude quasigeostrohic (QG) aroximation: Small Rossby number Ro = U/fL Hydrostatic : φ/ = gα Geostrohic : u = (1/f ) φ/ y v = (1/f ) φ/ Adiabatic: Dθ/Dt = Beta Plane: f = f + βy; βy ζ << f Small divergent comonent 13. Understand the geometric argument that the relative vorticity and the change in lanetary vorticity across a middle latitude domain are less than the Coriolis arameter f at the center of the domain. 14. You should be able to linearize the nonlinear (e.g. advective) terms in a differential equation: v ( v + v ) v v v v v v v u = ( u+ u ) = u + u + u + u u + u + u Where the overbars denote mean quantities rimes denote erturbation s and roducts of erturbation terms are small. 15. Know what Fourier analysis is and be able to describe qualitatively how it is used to reresent a eriodic function as the sum of sinusoidal comonents. 16. Know that in Disersive waves the hase velocity is a function of wavenumber. In a Nondisersive wave the hase velocities do not vary with wavenumber.

3 17. For a wave with frequency ω and wavenumber k the hase velocity is ω/k and the grou velocity is ω/ k. This result is readily generalized to or 3 dimensions. 18. The illustration at the right shows a wavetrain of the form ex ik[x ct]cos δk[x c gt] that results from the suerosition of two wavetrains whose wavenumbers differ by δk and frequencies by δω such that c g = δω/δk. The outer longer wavelength modulation moves with the grou velocity and the shorter carrier moves with the hase velocity. Energy roagates at the grou velocity. 19. Know that in the derivation of one-dimensional sound wave combination of the mass continuity and ideal gas equation casts the former as: 1 1 u u ρ ' u + = + ' = ρ t γ t where ϒ = c /c v. Be able to combine it with the mean state gas law and the momentum equation: 1 ' u + u' = t ρ to get the disersion relation for sound waves ω= ku ± γrt. Are these waves disersive?. Know generally how to derive the disersion relation for twolayer shallow water gravity waves: ( ω ku) g( H + H )( ω ku) k + g σh H k = Recognizing that the first two terms combine to reresent highfrequency solutions and the second two combine to reresent low-frequency solutions. ω = k u± gh + H k u± σgh1h ( ( 1 ) ) 3 ( H1 + H) The former are external gravity waves roagating on the surface density contrast and the latter are internal gravity waves roagating on (small) the density contrast between the layers.. Starting with the shallow-water equations with background rotation (Coriolis force): h + u u fv = g t h + u v + fu = g t y u v + u h= H +. t y

4 be able to derive the disersion relation for rotational shallow-water gravity waves: ω = ku ± gh( k + ) + f be able to sketch and exlain the variation of frequency with wavenumber. Know what the limiting frequency is for very long waves k and. 3. Be able to change the hydrostatic relation and the ressure gradient terms in the momentum equations from density-ressure thermodynamics to otential temerature-exner function thermodynamics. c c R 1 π g g = = = = θ c c z z z c ct c c = = ct = 1 RT R 1 π cθ. ρ c 4. Understand and be able to derive the way that buoyancy emerges from the vertical momentum equation: π π π' g g π' π' cθ cθ' cθ g= cθ cθ' cθ cθ' g z z z cθ cθ z z g π' θ' π' π' =+ g θ' cθ g= g cθ = b cθ θ z θ z z Know that the definition of buoyancy is b= gθ / θ. 5. Know how to recast the linearized adiabatic thermodynamic energy equation in terms of buoyancy and the Brunt-Väisälä frequency by multilying it through by g / θ : g θ' θ' θ + u + w' = θ t z θ' θ' g θ b b g + u g + w = + u + w N = t θ θ θ z t ' '. 6. Understand and be able to exlain the Brunt-Väisälä or buoyancy oscillation. It is a vertical oscillation with a eriod of about ten minutes. Initially uward moving air with the same temerature as its surroundings becomes colder than the air around it through adiabatic exansion. As a result it accelerates downward reversing its uward motion so that is starts sinking back toward its starting oint. By the time it reaches its initial level it has downward momentum so that it asses below its starting oint and warms through adiabatic comression. As a result its 4

5 descending motion decelerates ultimately leading to a return to the initial level with enough uward momentum to carry the air uward into another cycle. 7. Potential-temerature/Exner function thermodynamics lead to the following nonhydrostatic governing equations: u' u' π ' + u = cθ t w' w' π ' + u = b' cθ t z b' b' + u + Nw' = t u' w' + =. z 8. You will not have to derive the disersion relation for internal waves in a stratified fluid but you will need to understand how the derivation roceeds and how to interret the resulting disersion relation: ω = ku ± Nk k +. Also be able to analyze the three-dimensional rotational case as in the homework: ω = ku ± Nk k + f Know what the beta-lane aroximation f= f + ( f/ yy ) = f + βy is and know why the second term is generally significantly smaller than the first. 3. Be able to derive vorticity ζ / t+ v ( ζ + f) + ( ζ + f) v = and otential-vorticity [( ζ + f)/ h]/ t+ v ( ζ + f)/ h= equations from the momentum and mass continuity or thermodynamic energy equations in nondiergent shallow or continuously stratified fluids. 31. Be able to deduce the two-dimensional nondivergent Rossby-wave disersion relation [e.g. ω = ku βk /( k + ) ] from the vorticity equation for atmosheric flows. 3. Be able to deduce the shallow-water Rossby-wave disersion relation [e.g. ω= ku βk /( k + + f / gh) ] from the vorticity/otential vorticity equation for atmosheric flows. 33. Know what the Rossby radius of deformation gh / f is and its role in the dynamics of largescale waves. Recall the role of the inverse square Rossby radius relation for divergent Rossby waves. f / gh in the disersion 5

6 34. Understand and (generally) be able to derive the disersion relation for three-dimensional Rossby waves on a middle-latitude beta lain including hase and grou velocities and the characteristics of the frequency assband. 35. Be able to derive and exlain hase and grou roagation of Rossby waves including the roerties of stationary waves oosite meridional grou and hase velocities and where the zonal grou roagation reverses in still air. 36. Know what the Rossby-wave cutoff frequency ω β ku = / f / gh is and generally how the D nondivergent and D divergent Rossby waves behave as the zonal wavenumber k becomes large or aroaches zero. 37. Understand the derivation for mixed acoustic-buoyancy waves (See lecture B). 38. Given the disersion relation for mixed acoustic-buoyancy waves be able to sketch both branches of their disersion relation. 39. Know and understand the roles of the acoustic cutoff and Brunt-Väisälä frequencies in roagation of mixed acoustic-buoyancy waves. 4. General Advice: Be able to work the homework roblems and to calculate the hase and grou velocities for any given disersion relation. Also be able to describe the above disersion relations. Best of luck with the exam. 6