Fundamentals Of Geophysical Fluid Dynamics Part 9 pot

Starting with the equations of state, internal energy, and entropyfor an ideal gas, derive the evolution equation for potential temperature2.52.. Hint: Linearizethe compressible, conserv

6.2 Oceanic Wind Gyre and Western Boundary Layer 261

H

L0

H

L

Fig 6.19 A meridional cross-section at mid-longitude in the basin in an layer model of a double wind-gyre (Top) Mean zonal velocity, u (y, z) Thecontour interval is 0.05 m s−1, showing a surface jet maximum of 0.55 m s−1,

8-a deep e8-astw8-ard flow of 0.08 m s−1, and a deep westward recirculation-gyreflow 0.06 m s−1 (Bottom) Eddy kinetic energy, 1

2(u0)2(y, z) The contourinterval is 10−2m2s−2, showing a surface maximum of 0.3 m2s−2and a deepmaximum of 0.02 m2s−2 (Holland, 1986.)

262 Boundary-Layer and Wind-Gyre Dynamics

eddies because of the averaging The Sverdrup gyre circulation is dent (cf., Fig 6.14), primarily in the upper ocean and away from thestrong boundary and separated jet currents At greater depth and in theneighborhood of these strong currents are recirculation gyres whose peaktransport is several times larger than the Sverdrup transport These re-circulation gyres arise in response to downward eddy momentum flux

evi-by isopycnal form stress (cf., Sec 5.3.3) The separated time-mean jet

is a strong, narrow, surface-intensified current (Fig 6.19, top) Its stantaneous structure is vigorously meandering and has a eddy kineticenergy envelope that extends widely in both the horizontal and verti-cal directions away from the mean current that generates the variability(Fig 6.19, bottom)

in-Overall, the mean currents and eddy fluxes and their mean cal balances have a much more complex spatial structure in turbulentwind gyres than in zonal jets While simple analytic models of steadylinear gyre circulations (Sec 6.2.2) and their normal-mode instabilities(e.g., as in Secs 3.3 and 6.2) provide a partial framework for inter-preting the turbulent equilibrium dynamics, obviously they do so in amathematically and physically incomplete way The eddy–mean interac-tion includes some familiar features — e.g., Rossby waves and vortices;barotropic and baroclinic instabilities; turbulent cascades of energy andenstrophy and dissipation (Sec 3.7); turbulent parcel dispersion (Sec.3.5); lateral Reynolds stress and eddy heat flux (Secs 3.4 and 5.2.3);downward momentum and vorticity flux by isopycnal form stress (andimportant topographic form stress if B 6= 0) (Sec 5.3.3); top and bot-tom planetary boundary layers (Sec 6.1); and regions of potential vor-ticity homogenization (Sec 5.3.4) — but their comprehensive synthesisremains illusive

dynami-Rather than pursue this problem further, this seems an appropriatepoint to end this introduction to GFD — contemplating the relationshipbetween simple, idealized analyses and the actual complexity of geophys-ical flows evident in measurements and computational simulations

Most of the important GFD problems have been revisited frequently.The relevant physical ingredients — fluid dynamics, material properties,gravity, planetary rotation, and radiation — are few and easily stated,but the phenomena that can result from their various combinations aremany Much of the GFD literature is an exploration of different com-binations of the basic ingredients, always with the goal of discoveringbetter paradigms for understanding the outcome of experiments, obser-vations, and computational simulations.

Mastery of this literature is a necessary part of a research career inGFD, but few practitioners choose to read the literature systematically.Instead the more common approach is to address a succession of spe-cific research problems, learning the specifically relevant literature inthe process My hope is that the material covered in this book will pro-vide novice researchers with enough of an introduction, orientation, andmotivation to go forth and multiply

263

Fundamental Dynamics

1 Consider a two-dimensional (2D) velocity field that is purely tional (i.e., it has a vertical component of vorticity, ζζζ = ∇ × u, but its∇divergence, δ = ∇ · u, is zero in both 2D and 3D):∇

rota-u = −∂ψ∂y, v = +∂ψ

∂x, w = 0 ,where the streamfunction, ψ, is a scalar function of (x, y, t)

(a) Show that an isoline of ψ is a streamline (i.e., a line tangent to ueverywhere at a fixed time)

(b) For the Eulerian expression,

ψ = −Uy + A sin[k(x − ct)] ,sketch the streamlines at t = 0 Without loss of generality assume that

U, A, k, and c are positive constants (Note: If the second component

of ψ represents a wave, then A is its amplitude; k is its wavenumber;and c is its phase speed.)

(c) For ψ in (b) find the equation for the trajectory (i.e., the spatialcurve traced over time by a parcel moving with the velocity) that passesthrough the origin at t = 0

(d) For ψ in (b) sketch the trajectories for c = −U, 0, U, and 3U;summarize in words the dependence on c (Hint: Pay special attention

to the case c = U and derive it as the limit c→ U.)

(e) For ψ in (b) show that the divergence is zero and evaluate the ticity [Secs 2.1.1 and 2.1.5]

vor-2 T is 3 K cooler 25 km to the north of a particular point, and the wind

is northerly (i.e., from the north) at 10 m s−1 The air is being heated

264

approxima-4 If an atmospheric pressure fluctuation of − 103Pa passes slowly overthe ocean, how much will the local sea-level change? If a geostrophicsurface current in the ocean is 0.1 m s−1 in magnitude and 100 km inwidth, how big is the sea-level change across it? Explain how each ofthese behaviors is or is not consistent with the rigid-lid approximationfor oceanic dynamics [Secs 2.2.3 and 2.4.2]

5 Starting with the equations of state, internal energy, and entropyfor an ideal gas, derive the evolution equation for potential temperature(2.52) Estimate the temperature change that occurs if an air parcel islifted over a mountain 2 km high [Secs 2.3.1-2.3.2]

6 Describe the propagation of sound in the ˆx direction for an initialwave form that is sinusoidal with wavenumber, k, and pressure ampli-tude, p∗ Derive the relations among p, ρ, u, ζζζ, and δ (Hint: Linearizethe compressible, conservative fluid equations for an ideal gas around

a thermodynamically uniform state of rest, neglecting the gravitationalforce.) [Sec 2.3.1]

7 Derive a solution for an oceanic surface gravity wave with a surfaceelevation of the form

h(x, y, t) = h0 sin[kx− ωt], h0, k, ω > 0

Assume conservative fluid dynamics, constant density, constant pressure

at the free surface, incompressibility, small h0(such that the free surfacecondition can be linearized about z = 0 in a Taylor series expansion andthat the dynamical equations can be linearized about a state of rest),irrotational motion (i.e., ζζζ = 0), and infinite water depth (i.e., with allmotion vanishing as z → −∞) Explain what happens to this solution

for z≥ 0, assuming an ideal gas law For positive T0, T∗, and H, what

is a necessary condition for this to be a gravitationally stable profile?(Hint: What is required for θ to have a positive gradient?) [Sec 2.3.2]

9 Derive the equations for hydrostatic balance and mass conservation

in pressure coordinates using alternatively the two different ˜z definitions

in (2.74) and (2.75) Discuss comparatively the advantages and vantages for these alternative transformations (Hint: Make sure thatyou use the alternative forms for ω ≡ Dtz, the flow past an isobaric˜surface.) [Sec 2.3.5]

disad-10 Demonstrate the rotational transform relations for Dt, ∇, ∇ · u,and Dtu, starting from the relations defining the rotating coordinates,unit vectors, and velocities [Sec 2.4.1]

11 What are the vertical vorticity and horizontal divergence for ageostrophic velocity for f = f0, and f = f0 + β0(y − y0)? Explainwhy f = f (y) is geophysically relevant In combination with the hy-drostatic relation, derive the thermal-wind balance (i.e., eliminate thegeopotential between the approximated vertical and horizontal momen-tum equations) [Sec 2.4.2]

12 Redo the scaling analysis in Sec 2.3.4 for a rotating flow to rive the condition under which the hydrostatic approximation would bevalid for approximately geostrophic motions with Ro 1 (i.e., the re-placement for (2.72)) Comment on its relevance to large-scale oceanicand atmospheric motions (Hint: The geopotential, density, and ver-tical velocity scaling estimates — expressed in terms of the horizontalvelocity and length scales and the environmental parameters — should

de-be consistent with geostrophic and hydrostatic balances.) [Sec 2.4.2]

13 Which way do inertial-wave velocity vectors rotate in the southernhemisphere? Which way does the air flow geostrophically around a low-pressure center in the southern hemisphere (e.g., a cyclone in Sydney,Australia)? [Sec 2.4.3]

Exercises 267Barotropic and Vortex Dynamics

1 Show that the area inside a closed material curve — like the one inFig 3.2 — is conserved with time in 2D flow (Hint: Mimic in 2D the3D relations in Sec 2.1.5 among volume conservation, surface normalflow, and interior divergence.) Does this result depend upon whether ornot the non-conservative force, F, is zero? [Sec 3.1]

2 Show that the following quantities are integral invariants (i.e., theyare conserved with time) for conservative 2D flow and integration overthe whole plane Assume that ψ, u, ζ → 0 as |x| → ∞, F = F = 0,and f (y) = f0+ β0(y− y0) [Sec 3.1]

yζ dx dy (for β0= 0) [spatial centroid]

3 For a stationary, axisymmetric vortex with a monotonic pressureanomaly, how do the magnitude and radial scale of the associated cir-culation in gradient-wind balance differ depending upon the sign of thepressure anomaly when Ro is small but finite? [Sec 3.1.4]

4 Calculate the trajectories of three equal-circulation point vorticesinitially located at the vertices of an equilateral triangle [Sec 3.2.1]

5 Calculate using point vortices the evolution of a tripole vortex with atotal circulation of zero Its initial configuration consists of one vortex

in between two others that are on opposite sides, each with half thestrength and the opposite parity of the central vortex [Sec 3.2.1]

6 Is the weather more or less predictable than the trajectories of Npoint vortices for large N ? Explain your answer [Sec 3.2.2]

7 State and prove Rayleigh’s inflection point theorem for an inviscid,steady, barotropic zonal flow on the β-plane [Sec 3.3.1]

8 Derive the limit for the Kelvin-Helmholtz instability of a free shearlayer with a piecewise linear profile, U (y), as the width of the layer be-comes vanishingly thin and approaches a vortex sheet (i.e., take the limit

kD→ 0 for the eigenmodes of (3.87)) What are the unstable growthrates, and what are the eigenfunction profiles in y? In particular, whatare the discontinuities, if any, in ψ0, u0, v0, and φ0 across y = 0? [Sec

268 Exercises

3.3.3 and Drazin & Reid (Sec 4, 1981) (The latter solves the lem of a vortex-sheet instability for a non-rotating fluid; interestingly,the growth rate formula is the same, but their method, which assumespressure continuity across the sheet, is not valid when f6= 0.)]

prob-9 Derive eddy–mean interaction equations for enstrophy and potentialenstrophy, analogous to the energy equations in Sec 3.4 Derive eddy-viscosity relations for these balances and interpret them in relation tothe jet flows depicted in Fig 3.13 Explain the relationship betweeneddy Reynolds stress and vorticity flux profiles [Secs 3.4-3.5]

10 Explain how the emergence of coherent vortices in 2D flow is or isnot consistent with the general proposition that entropy and/or disordercan only increase with time in isolated dynamical systems [Secs 3.6-3.7]

Rotating Shallow-Water and Wave Dynamics

1 Show that the area inside a closed material curve — like the one inFig 3.2 — is not conserved with time in a shallow-water flow Doesthis result depend upon whether the non-conservative force, F, is zero

or not? [Sec 4.1]

2 Derive the shallow-water equations appropriate to an active layer tween two inert layers with different densities (lighter above and denserbelow), assuming that the layer interfaces are free surfaces What is theappropriate formula for a potential vorticity conserved on parcels when

Exercises 269

5 In the conservative Shallow-Water Equations, (a) show that ary solutions are ones in which q =G[Ψ], where G is any functional op-erator and Ψ is a transport streamfunction, such that hu = ˆz× ∇Ψ (b)Show that this implies flow along contours of f /h = (f + β0y)/(H− B),when the spatial scale of the flow is large enough and/or the velocity isweak enough so that ζ and η are negligible compared to δf and δh (c)Under these conditions sketch the trajectories for an incident uniformeastward flow across a mid-oceanic ridge with B = B(x) > 0 or, alterna-tively, with B = B(y) > 0 What about with steady uniform meridionalflow? [Sec 4.1.1]

station-6 (a) Derive the inertia-gravity wave dispersion relation for amplitude fluctuations in the conservative 3D Boussinesq equations with

small-a simple equsmall-ation of stsmall-ate, ρ = ρ0(1− αT ) (i.e., so that density is served on parcels), for a basic state of rest with uniform rotation andstratification, f = f0 and ρ(z) = ρ0 1− N2z/g

con-, in an unbounded main Demonstrate that (b) ω depends only on the direction of k, notits magnitude K =|k|, (c) N0 and f0 are the largest and smallest fre-quencies allowed for the inertia-gravity modes, and (d) the phase andgroup velocities,

7 Describe qualitatively the end states of geostrophic adjustment forthe following initial sea-level and velocity configurations in the Shallow-Water Equations with β = 0: (a) an axisymmetric mound with nomotion and a flat bottom; (b) an axisymmetric depression with no mo-tion and a flat bottom; (c) a velocity patch (i.e., a horizontal squarewith uniform horizontal velocity inside and zero velocity outside) with

a flat surface and bottom; and (d) an initial state with no motion, flatupper surface, and a topographic bump in the bottom (Hint: Con-sider gradient-wind balance to compare (a) vs (b), as well as the twosides across the initial flow direction in (c) for either Ro = o(1) or

Ro = O(1).) [Sec 4.3]

8 What controls the time it takes for a gravity wave with small aspectratio, H  L, to approach a singularity in the surface shape? Whathappens in a 3D fluid afterward? [Sec 4.4]

9 For a deep-water surface gravity wave propagating in the ˆx direction

10 Derive the non-dimensional quasigeostrophic potential vorticityequation (4.113)-(4.114) directly from the non-dimensional Shallow-WaterEquations (4.109)-(4.110) as → 0 (Note: This is an alternative path

to the derivation in the text, where the Shallow-Water potential ity equation (4.24) was non-dimensionalized and approximated in thelimit → 0.) [Sec 4.6]

vortic-11 Derive the quasigeostrophic potential vorticity equation for a 3DBoussinesq fluid that is uniformly rotating and stratified (i.e., f and Nare constant) Assume a simple equation of state where density is ad-vectively conserved as an expression of internal energy conservation, andassume approximate geostrophic and hydrostatic momentum balances.Assume that Rossby, Ro = V /f L, and Froude, F r = V /N H, numbersare comparably small and that density fluctuations, ρ0, are small com-pared to the mean stratification, ρ; i.e., the dimensional density fieldis

ρ(x, y, z, t) = ρ0



1 − N

2Hg

 zH

+ Ro

B b(x, y, z, t)

,

where the Burger number, B = (NH/fL)2, is assumed to be O(1),and b =−(gH/ρ0f V L)ρ0 is the non-dimensional, fluctuation buoyancyfield Where does the deformation radius appear in this derivation, andwhat does it indicate about the relative importance of the componentterms in qqg? (Hints: Use geostrophic scaling to non-dimensionalize themomentum, continuity, and internal energy equations; then form thevertical vorticity equation and use continuity to replace the horizontaldivergence (related to the higher-order, ageostrophic velocity field) withthe the vertical velocity; combine the result with the internal energyequation to eliminate the vertical velocity; then use the geostrophic re-lations to evaluate the remaining horizontal velocity, vertical vorticity,and buoyancy terms.) [Sec 4.6 and Eq (5.28) in Sec 5.1.2]

12 From the Shallow-Water Equations, derive the dispersion relationfor a quasigeostrophic topographic wave, assuming a uniform bottom

Exercises 271slope upward to the south Compare it to the Rossby wave dispersionrelation (4.120) [Sec 4.7]

13 Solve for the steady-state wave field in the conservative reflection

of an incident shallow-water wave impinging upon a western boundary

at x = 0 for two separate types of waves: (a) an inertia-gravity wave onthe f -plane, and (b) a Rossby wave on the β-plane (Hint: Representthe wave field as a sum of incident and reflected components that add

up to satisfy the boundary condition of no normal flow.) [Sec 4.7]

Baroclinic and Jet Dynamics

1 Derive the energy conservation law for a conservative 2-layer fluid.Also derive it for a N -layer fluid and compare with the 2-layer energyconservation law [Secs 5.1.1-5.1.2]

2 Derive the quasigeostrophic approximation to the N -layer energyconservation law in problem #1 Then decompose the flow into time-mean and time-variable components, and derive the eddy–mean energyconversion terms (referred to as barotropic and baroclinic conversion,respectively) [Secs 3.4 and 5.1]

3 Calculate the vertical modes for N = 3 and comment on the relations

of the deformation radii and vertical structures, both among the modesand relative to the modes for a N = 2 model (Hint: Pose the modalproblem for general g0

n+.5 and Hn, but then decide whether assumingequal density jumps and layer depths make the solution so much easierthat it suffices to illustrate the nature of the answer.) [Sec 5.1.3]

4 Evaluate the vertical modes for a 2-layer fluid with H1 = H2.Then evaluate the vertical modes for a continuously stratified fluid with

N (z) = No, a positive constant Compare the two sets of modes [Sec.5.1.3]

5 Derive a Rayleigh theorem for the necessary condition for an inviscid,2-layer, baroclinic instability of a mean zonal flow Start from

∂q0 n

∂t + un

∂q0 n

∂x + v

0 n

∂qn

∂y = 0with n = 1, 2 Then assume a normal-mode form as in (5.56); multiplyeach layer equation by −ψ0∗

n; sum over layers with weights Hn/H (tomimic a volume integral, the horizontal part of which is trivial given thisparticular normal-mode form); and determine the necessary conditionfor ω and C to have a nonzero imaginary part [Sec 5.2]