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Home , Seiche, Shallow water equations

ASSIGNEMENT 1 Shallow water This problem surrounds the , linearized about a motionless equilibrium with depth H(x): ut = −gηx, ηt + (Hu)x = 0. (1) The equilibrium is a uniform tank of length L and depth H = constant. The perturbations satisfy u(0, t) = u(L, t) = 0. The practical part: Find a suitable container. Pour in a layer of water. Tilt the tank to excite a . Measure the length and the depth of the water , and observe the period of the seiche. The theoretical part: Calculate what the period should be theoretically by searching for separable solutions (normal modes) of the linearized shallow water equations, [u(x, t), η(x, t)] = [uˇ(x), ηˇ(x)]e−iωt, where ω is the frequency (giving a period of 2π/ω). Comparison: Compare theory and experiment, and offer explanations for why the two may not match.

2 2 (2) The equilibrium is a lake with H(x) = H0(1 − x /L ). Look for separable solutions of the shallow water equations once again, and set up a differential equation for ηˇ(x). The solutions should be regular, even at the lake’s edges, x = L. Find the normal-mode frequencies, ω, and sketch a few of the modes with lowest frequencies; comment on how the solutions differ from the modes of (1). (You might want to recall/research Legendre’s equation.)

1 ASSIGNEMENT 2 in a rotating channel Consider shallow-water waves in a rotating channel of depth H and width L on the f−plane:

ut − fv = −gηx

vt + fu = −gηy

ηt + (Hu)x + (Hv)y = 0. At y = 0 and L, v = 0. We are interested in -like (normal-mode) solutions with (u, v, η) ∝ exp i(kx − ωt) (and a more complicated dependence on y). (a) Take H to be constant. Find and classify all the non-trivial normal modes. Be careful not to omit the “,” and determine the spatial structure of that special normal mode. (b) Now assume that H = H0(1 − sy/L), with s  1, and look for solutions with ω = O(s). Interpret the results, highlighting how variations in topography can mimic the beta-effect. Summarize all your results by drawing a “ diagram” showing k versus ω.

2 ASSIGNEMENT 3 Internal waves Consider a rotating fluid layer in which there is a (constant) background density gradient, β. Linear perturbations satisfy the equations:

0 px pz ρ g ut = − , wt = − − , ρ0 ρ0 ρ0

2 0 ρ0N u + w = 0, ρ − w = 0, x z t g 2 where N (the square of the buoyancy frequency) is proportional to |β|, ρ0 is a constant reference density, and ρ0(x, z, t) is the density perturbation, which is only taken into account in the buoyancy term (the “Boussinesq approximation”). Look for solutions with the dependence, exp i(k · x − ωt), where

k ≡ (k, m) ≡ K(cos θ, sin θ) is the wavenumber vector, and θ is the angle of phase propagation with respect to the horizontal. Derive the dispersion relation, ω = ω(k, m). Hence show that phase of internal gravity waves always propagates with the same angle with respect to the vertical given the frequency and background fluid properties, whatever the wavenumber K. What is that angle? Show that the group velocity of internal gravity waves, cg = ∂ω/∂k, is orthogonal to the phase velocity (i.e. k). What distinctive features of a wave packet of internal gravity waves does this imply? Search the internet for the “St Andrew’s cross” gravity-wave pattern, and explain this beast using your results. Comment on what the results signify for wave reflection from an inclined wall. After searching the internet for “ attractors”, comment also on what they also mean for internal waves generated inside a closed body of fluid.

3 ASSIGNEMENT 4 Thermohaline convection Consider a two-dimensional layer of fluid between two plates at z = 0 and z = d. The Boussinesq equations of motion are 1 u + uu + wu = − p + ν(u + u ), t x z ρ x xx zz 1 w + uw + ww = − p + ν(w + w ) + g(α T − α S), t x z ρ z xx zz T S

Tt + uTx + wTz = κT (Txx + Tzz),

St + uSx + wSz = κS(Sxx + Szz), where T and S represent temperature and salinity, αT and αS are coefficients of thermal expansion, and κT and κS are diffusivities. If the plates are held at fixed temperature and salinity, so that T = T1 and S = S1 and z = 0, and T = T2 and S = S2 and z = d, show that there is a motionless state of pure diffusion with z z T = T1 − (T1 − T2) , S = S1 − (S1 − S2) d d and a given pressure distribution. When T1 > T2 and S1 > S2, the fluid is hotter and saltier at the bottom; since salt increases the density, but heat lowers it, the two contributors to the buoyancy force compete with one another (heat is destabilizing and salt is stabilizing). Let z z T = T1 − (T1 − T2) + (T1 − T2)Θ(xˆ, zˆ, tˆ), S = S1 − (S1 − S2) + (S1 − S2)Σ(xˆ, zˆ, tˆ), d d κT (u, w) = (ψˆ, −ψˆ), ψ = ψ(xˆ, zˆ, tˆ) d z x d2 (x, z) = d(xˆ, zˆ), t = t.ˆ κT Show that, if the perturbation amplitudes, ψ, Θ and Σ are all small, and after dropping the hat decoration, 1 ∇2ψ = ∇4ψ − R Θ + R Σ , σ t T x s x 2 Θt + ψx = ∇ Θ, 2 Σt + ψx = τ∇ Σ, 2 2 2 where τ, σ, RT and RS are dimensionless groups that you should determine, and ∇ ≡ ∂ /∂x + 2 2 ∂ /∂z . Assume that the boundary conditions are stress free, so that (ψ, Θ, Σ) ∝ eλt+ikx sin nπz is an admissable solution. Determine the cubic equation for λ. Show that (a) for thermal convection with RS = 0, the layer is unstable to the mode with wavenumbers k and n when RT > RC (n, k). By minimizing RC over n and k, establish the onset of convection. (b) when RS 6= 0, steady convection (λ real and passing through zero) still occurs provided RT exceeds a threshold that you should calculate. (c) instability can arise through oscillatory convection (λ = iω + λr, with λr passing through zero) when RT surpasses a different threshold (what is it?). (d) given that κS  κT for heat and salt in water, what are the implications for oceanic convection? (e) the roles of heat and salt can be effectively interchanged if we put κS  κT and interpret S as heat and T as salt (so that salt is destabilizing and heat is now stabilizing). What are the implications now?

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