Fundamentals Of Geophysical Fluid Dynamics Part 9 pot

6.2 Oceanic Wind Gyre and Western Boundary Layer 261 H L 0 0 H H z z 0 0 y y H H H L Fig. 6.19. A meridional cross-section at mid-longitude in the basin in an 8- layer model of a double wind-gyre. (Top) Mean zonal velocity, u (y, z). The contour interval is 0.05 m s −1 , showing a surface jet maximum of 0.55 m s −1 , a deep eastward flow of 0.08 m s −1 , and a deep westward recirculation-gyre flow 0.06 m s −1 . (Bottom) Eddy kinetic energy, 1 2 (u  ) 2 (y, z). The contour interval is 10 −2 m 2 s −2 , showing a surface maximum of 0.3 m 2 s −2 and a deep maximum of 0.02 m 2 s −2 . (Holland, 1986.) 262 Boundary-Layer and Wind-Gyre Dynamics eddies because of the averaging. The Sverdrup gyre circulation is evi- dent (cf., Fig. 6.14), primarily in the upper ocean and away from the strong boundary and separated jet currents. At greater depth and in the neighborhood of these strong currents are recirculation gyres whose peak transport is several times larger than the Sverdrup transport. These re- circulation gyres arise in response to downward eddy momentum flux 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 in- stantaneous structure is vigorously meandering and has a eddy kinetic energy 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). Overall, the mean currents and eddy fluxes and their mean dynami- cal balances have a much more complex spatial structure in turbulent wind gyres than in zonal jets. While simple analytic models of steady linear 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 a mathematically 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 and enstrophy 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 (and important topographic form stress if B = 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 synthesis remains illusive. Rather than pursue this problem further, this seems an appropriate point to end this introduction to GFD — contemplating the relationship between simple, idealized analyses and the actual complexity of geophys- ical flows evident in measurements and computational simulations. Afterword This book, of necessity, gives only a taster’s sampling of the body of posed and partially solved problems that comprises GFD. A comparison of the material here with other survey books (e.g., those cited in the Bibliography) indicates the great breadth of the subject, and, of course, the bulk of the scientific record for GFD is to be found in journal articles, only lightly cited here. 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 are many. Much of the GFD literature is an exploration of different com- binations of the basic ingredients, always with the goal of discovering better paradigms for understanding the outcome of experiments, obser- vations, and computational simulations. Mastery of this literature is a necessary part of a research career in GFD, 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 in the process. My hope is that the material covered in this book will pro- vide novice researchers with enough of an introduction, orientation, and motivation to go forth and multiply. 263 Exercises Fundamental Dynamics 1. Consider a two-dimensional (2D) velocity field that is purely rota- tional (i.e., it has a vertical component of vorticity, ζζζ = ∇∇∇×u, but its divergence, δ = ∇∇∇·u, is zero in both 2D and 3D): 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 u everywhere 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 spatial curve traced over time by a parcel moving with the velocity) that passes through 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 vor- ticity. [Secs. 2.1.1 and 2.1.5] 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 Exercises 265 by the net of absorbed and emitted radiation at a rate of 0.5 m 2 s −3 . What is the local rate of change of temperature? (Assume the motion is along an isobar, so that there is no change of pressure following the flow.) [Sec. 2.1.2] 3. Derive the vertical profiles of pressure and density for a resting, conservative ocean with temperature and salinity profiles, T (z) = T 0 + T ∗ e z/H , S(z) = S 0 + S ∗ e z/H , for z ≤ 0, assuming a Boussinesq equation of state as an approxima- tion to the true equation of state for seawater (i.e., completely neglect compressibility, including the small difference between temperature and potential temperature). Estimate approximate values for T 0 , S 0 , T ∗ , S ∗ , and H in Earth’s ocean. [Secs. 2.2.1 and 2.3.2] 4. If an atmospheric pressure fluctuation of −10 3 Pa passes slowly over the ocean, how much will the local sea-level change? If a geostrophic surface current in the ocean is 0.1 m s −1 in magnitude and 100 km in width, how big is the sea-level change across it? Explain how each of these behaviors is or is not consistent with the rigid-lid approximation for oceanic dynamics. [Secs. 2.2.3 and 2.4.2] 5. Starting with the equations of state, internal energy, and entropy for an ideal gas, derive the evolution equation for potential temperature (2.52). Estimate the temperature change that occurs if an air parcel is lifted over a mountain 2 km high. [Secs. 2.3.1-2.3.2] 6. Describe the propagation of sound in the ˆ x direction for an initial wave form that is sinusoidal with wavenumber, k, and pressure ampli- tude, p ∗ . Derive the relations among p, ρ, u, ζζζ, and δ. (Hint: Linearize the compressible, conservative fluid equations for an ideal gas around a thermodynamically uniform state of rest, neglecting the gravitational force.) [Sec. 2.3.1] 7. Derive a solution for an oceanic surface gravity wave with a surface elevation of the form h(x, y, t) = h 0 sin[kx − ωt], h 0 , k, ω > 0 . Assume conservative fluid dynamics, constant density, constant pressure at the free surface, incompressibility, small h 0 (such that the free surface condition can be linearized about z = 0 in a Taylor series expansion and that the dynamical equations can be linearized about a state of rest), irrotational motion (i.e., ζζζ = 0), and infinite water depth (i.e., with all motion vanishing as z → −∞). Explain what happens to this solution 266 Exercises when the rigid-lid approximation is made. [Sec. 2.2.3 and Stern (Chap. 1, 1975)] 8. Derive the vertical profiles of pressure and density for a resting at- mosphere with a temperature profile, T (z) = T 0 + T ∗ e −z/H , for z ≥ 0, assuming an ideal gas law. For positive T 0 , 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 disad- vantages for these alternative transformations. (Hint: Make sure that you use the alternative forms for ω ≡ D t ˜z, the flow past an isobaric surface.) [Sec. 2.3.5] 10. Demonstrate the rotational transform relations for D t , ∇, ∇ · u, and D t u, 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 a geostrophic velocity for f = f 0 , and f = f 0 + β 0 (y − y 0 )? Explain why f = f (y) is geophysically relevant. In combination with the hy- drostatic relation, derive the thermal-wind balance (i.e., eliminate the geopotential 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 de- rive the condition under which the hydrostatic approximation would be valid for approximately geostrophic motions with Ro  1 (i.e., the re- placement for (2.72)). Comment on its relevance to large-scale oceanic and atmospheric motions. (Hint: The geopotential, density, and ver- tical velocity scaling estimates — expressed in terms of the horizontal velocity and length scales and the environmental parameters — should be consistent with geostrophic and hydrostatic balances.) [Sec. 2.4.2] 13. Which way do inertial-wave velocity vectors rotate in the southern hemisphere? 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 267 Barotropic and Vortex Dynamics 1. Show that the area inside a closed material curve — like the one in Fig. 3.2 — is conserved with time in 2D flow. (Hint: Mimic in 2D the 3D relations in Sec. 2.1.5 among volume conservation, surface normal flow, and interior divergence.) Does this result depend upon whether or not the non-conservative force, F, is zero? [Sec. 3.1] 2. Show that the following quantities are integral invariants (i.e., they are conserved with time) for conservative 2D flow and integration over the whole plane. Assume that ψ, u, ζ → 0 as |x| → ∞, F = F = 0, and f(y) = f 0 + β 0 (y − y 0 ). [Sec. 3.1] (a) 1 2  (u 2 + v 2 ) dx dy [kinetic energy] (Hint: Multiply the 2D vorticity equation by -ψ and integrate over the area of the whole plane; consider integrations by parts.) (b)  ζ dx dy [circulation] (c)  ζ 2 dx dy [enstrophy] (d)  q 2 dx dy [potential enstrophy] (e)  xζ dx dy,  yζ dx dy (for β 0 = 0) [spatial centroid] 3. For a stationary, axisymmetric vortex with a monotonic pressure anomaly, how do the magnitude and radial scale of the associated cir- culation in gradient-wind balance differ depending upon the sign of the pressure anomaly when Ro is small but finite? [Sec. 3.1.4] 4. Calculate the trajectories of three equal-circulation point vortices initially located at the vertices of an equilateral triangle. [Sec. 3.2.1] 5. Calculate using point vortices the evolution of a tripole vortex with a total circulation of zero. Its initial configuration consists of one vortex in between two others that are on opposite sides, each with half the strength and the opposite parity of the central vortex. [Sec. 3.2.1] 6. Is the weather more or less predictable than the trajectories of N point 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 shear layer 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 growth rates, and what are the eigenfunction profiles in y? In particular, what are the discontinuities, if any, in ψ  , u  , v  , and φ  across y = 0? [Sec. 268 Exercises 3.3.3 and Drazin & Reid (Sec. 4, 1981) (The latter solves the prob- lem of a vortex-sheet instability for a non-rotating fluid; interestingly, the growth rate formula is the same, but their method, which assumes pressure continuity across the sheet, is not valid when f = 0.)] 9. Derive eddy–mean interaction equations for enstrophy and potential enstrophy, analogous to the energy equations in Sec. 3.4. Derive eddy- viscosity relations for these balances and interpret them in relation to the jet flows depicted in Fig. 3.13. Explain the relationship between eddy Reynolds stress and vorticity flux profiles. [Secs. 3.4-3.5] 10. Explain how the emergence of coherent vortices in 2D flow is or is not consistent with the general proposition that entropy and/or disorder can 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 in Fig. 3.2 — is not conserved with time in a shallow-water flow. Does this 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 be- tween two inert layers with different densities (lighter above and denser below), assuming that the layer interfaces are free surfaces. What is the appropriate formula for a potential vorticity conserved on parcels when F = 0? [Sec. 4.1] 3. Show that the total energy E defined by E ≡   dx dy 1 2  hu 2 + gη 2  is an integral invariant of the conservative Shallow-Water Equations, given favorable lateral boundary conditions. [Sec. 4.1.1] 4. In conservative Shallow-Water Equations with mean depth H, assume there is a uniform zonal geostrophic flow, u = u 0 ˆ x, coming from x = −∞ towards a mountain whose height is ∆H. Calculate the vorticity of fluid parcels that (a) move from x = −∞ on to the top of the mountain, (b) move from on top of the mountain away towards x = ∞, and (c) do both movements in sequence. Assume that the surface height variations are smaller than the topographic height variations. (Hint: Consider potential vorticity conservation.) [Sec. 4.1.1] Exercises 269 5. In the conservative Shallow-Water Equations, (a) show that station- 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 = ˆ z ×∇Ψ. (b) Show that this implies flow along contours of f /h = (f + β 0 y)/(H −B), when the spatial scale of the flow is large enough and/or the velocity is weak enough so that ζ and η are negligible compared to δf and δh. (c) Under these conditions sketch the trajectories for an incident uniform eastward flow across a mid-oceanic ridge with B = B(x) > 0 or, alterna- tively, with B = B(y) > 0. What about with steady uniform meridional flow? [Sec. 4.1.1] 6. (a) Derive the inertia-gravity wave dispersion relation for small- amplitude fluctuations in the conservative 3D Boussinesq equations with a simple equation of state, ρ = ρ 0 (1 − αT ) (i.e., so that density is con- served on parcels), for a basic state of rest with uniform rotation and stratification, f = f 0 and ρ(z) = ρ 0  1 − N 2 0 z/g  , in an unbounded do- main. Demonstrate that (b) ω depends only on the direction of k, not its magnitude K = |k|, (c) N 0 and f 0 are the largest and smallest fre- quencies allowed for the inertia-gravity modes, and (d) the phase and group velocities, c p ≡ ω k ≡ ωk/K 2 , c g ≡ ∂ω ∂k ≡  ∂ ω ∂k (x) , ∂ ω ∂k (y) , ∂ ω ∂k (z)  , are orthogonal and have opposite-signed vertical components for the inertia-gravity modes. [Sec. 4.2 and Holton (Secs. 7.4 and 7.5, 2004)] 7. Describe qualitatively the end states of geostrophic adjustment for the following initial sea-level and velocity configurations in the Shallow- Water Equations with β = 0: (a) an axisymmetric mound with no motion and a flat bottom; (b) an axisymmetric depression with no mo- tion and a flat bottom; (c) a velocity patch (i.e., a horizontal square with uniform horizontal velocity inside and zero velocity outside) with a flat surface and bottom; and (d) an initial state with no motion, flat upper surface, and a topographic bump in the bottom. (Hint: Con- sider gradient-wind balance to compare (a) vs. (b), as well as the two sides 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 aspect ratio, H  L, to approach a singularity in the surface shape? What happens in a 3D fluid afterward? [Sec. 4.4] 9. For a deep-water surface gravity wave propagating in the ˆ x direction 270 Exercises with dispersion relation ω 2 = gk and orbital velocity components, u(x, z, t) = gkη o ω e kz sin[kx−ωt] & w(x, z, t) = − gkη o ω e kz cos[kx−ωt] , derive the Stokes drift velocity profile starting from the Lagrangian tra- jectory equation (2.1). (Hint: Make sure you obtain (4.96) en route.) How does it compare with the shallow-water Stokes drift (4.98)? [Sec. 4.5] 10. Derive the non-dimensional quasigeostrophic potential vorticity equation (4.113)-(4.114) directly from the non-dimensional Shallow-Water Equations (4.109)-(4.110) as  → 0. (Note: This is an alternative path to the derivation in the text, where the Shallow-Water potential vortic- ity equation (4.24) was non-dimensionalized and approximated in the limit  → 0.) [Sec. 4.6] 11. Derive the quasigeostrophic potential vorticity equation for a 3D Boussinesq fluid that is uniformly rotating and stratified (i.e., f and N are constant). Assume a simple equation of state where density is ad- vectively conserved as an expression of internal energy conservation, and assume approximate geostrophic and hydrostatic momentum balances. Assume that Rossby, Ro = V/fL, and Froude, F r = V/NH, numbers are comparably small and that density fluctuations, ρ  , are small com- pared to the mean stratification, ρ; i.e., the dimensional density field is ρ(x, y, z, t) = ρ 0  1 − N 2 H g  z H  + Ro B b(x, y, z, t)  , where the Burger number, B = (NH/fL) 2 , is assumed to be O(1), and b = −(gH/ρ 0 fV L)ρ  is the non-dimensional, fluctuation buoyancy field. Where does the deformation radius appear in this derivation, and what does it indicate about the relative importance of the component terms in q qg ? (Hints: Use geostrophic scaling to non-dimensionalize the momentum, continuity, and internal energy equations; then form the vertical vorticity equation and use continuity to replace the horizontal divergence (related to the higher-order, ageostrophic velocity field) with the the vertical velocity; combine the result with the internal energy equation 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. 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