ALBERT

Description: We study the statistics of a passive scalar Τ(x, t) governed by the advection-diffusion equation with variations in the scalar produced by a steady source. Two important statistical properties of the scalar are the variance, σ2 ≡ 〈 Τ2 〉, and the entropy production, χ ≡ κ 〈 ∇Τ 2〉. Here 〈〉 denotes a space-time average and κ is the molecular diffusivity of χ. Using variational methods we show that the system must lie above a parabola in the (χ, σ2)-plane. The location of the bounding parabola depends on the structure of the velocity and the source. To test the bound, we consider a large-scale source and three two-dimensional model velocities: a uniform steady flow; a statistically homogeneous and isotropic flow characterized by an effective diffusivity; a time-periodic model of oscillating convection cells with chaotic Lagrangian trajectories. Analytic solution of the first example shows that the bound is sharp and realizable. Numerical simulation of the other examples shows that the statistics of Τ(x, t) the parabolic frontier in the (χ, σ2)-plane. Moreover in the homogenization limit, in which the largest scale in the velocity field is much less than the scale of the source, the results of the simulation limit to the bounding parabola. © 2006 Cambridge University Press.

Description: We calculate a rigorous dual bound on the long-time-averaged mechanical energy dissipation rate ε within a channel of an incompressible viscous fluid of constant kinematic viscosity v, depth h and rotation rate f, driven by a constant surface stress τ = ρu*2î, where u* is the friction velocity. It is well known that ε ≤ εStokes = u*4/ν, i.e. the dissipation is bounded above by the dissipation associated with the Stokes flow. Using an approach similar to the variational 'background method' (due to Constantin, Doering & Hopf), we generate a rigorous dual bound, subject to the constraints of total power balance and mean horizontal momentum balance, in the inviscid limit ν → 0 for fixed values of the friction Rossby number Ro* = u*/(fh) = √GE, where G = τh2/(ρν2) is the Grashof number, and E = ν/fh2 is the Ekman number. By assuming that the horizontal dimensions are much larger than the vertical dimension of the channel, and restricting our attention to particular, analytically tractable, classes of Lagrange multipliers imposing mean horizontal momentum balance analogous to the ones used in Tang, Caulfield & Young (2004), we show that ε ≤ εmax = u*4/ν - 2.93u*2f, an improved upper bound from the Stokes dissipation, and ε ≥ εmin = 2.795u*3/h, a lower bound which is independent of the kinematic viscosity ν. © 2005 Cambridge University Press.

Description: New analytic estimates of the rate at which parametric subharmonic instability (PSI) transfers energy to high-vertical-wavenumber near-inertial oscillations are presented. These results are obtained by a heuristic argument which provides insight into the physical mechanism of PSI, and also by a systematic application of the method of multiple time scales to the Boussinesq equations linearized about a 'pump wave' whose frequency is close to twice the inertial frequency. The multiple-scale approach yields an amplitude equation describing how the 2 f0-pump energizes a vertical continuum of near-inertial oscillations. The amplitude equation is solved using two models for the 2 f0-pump: (i) an infinite plane internal wave in a medium with uniform buoyancy frequency; (ii) a vertical mode one internal tidal wavetrain in a realistically stratified and bounded ocean. In case (i) analytic expressions for the growth rate of PSI are obtained and validated by a successful comparison with numerical solutions of the full Boussinesq equations. In case (ii), numerical solutions of the amplitude equation indicate that the near-inertial disturbances generated by PSI are concentrated below the base of the mixed layer where the velocity of the pump wave train is largest. Based on these examples we conclude that the e-folding time of PSI in oceanic conditions is of the order of ten days or less. © 2008 Cambridge University Press.

Description: We consider the dynamics of a hollow cylindrical shell that is filled with viscous fluid and another, nested solid cylinder, and allowed to roll down an inclined plane. A mathematical model is compared to simple experiments. Two types of behaviour are observed experimentally: on steeper slopes, the device accelerates; on shallower inclines, the cylinders rock and roll unsteadily downhill, with a speed that is constant on average. The theory also predicts runaway and unsteady rolling motions. For the rolling solutions, however, the inner cylinder cannot be suspended in the fluid by the motion of the outer cylinder, and instead falls inexorably toward the outer cylinder. Whilst 'contact' only occurs after an infinite time, the system slows progressively as the gap between the cylinders narrows, owing to heightened viscous dissipation. Such a deceleration is not observed in the experiments, suggesting that some mechanism limits the approach to contact. Coating the surface of the inner cylinder with sandpaper of different grades changes the rolling speed, consistent with the notion that surface roughness is responsible for limiting the acceleration. © 2007 Cambridge University Press.

Description: The equilibrium of an idealized flow driven at the surface by wind stress and rapid relaxation to nonuniform buoyancy is analyzed in terms of entropy production, mechanical energy balance, and heat transport. The flow is rapidly rotating, and dissipation is provided by bottom drag. Diabatic forcing is transmitted from the surface by isotropic diffusion of buoyancy. The domain is periodic so that zonal averaging provides a useful decomposition of the flow into mean and eddy components. The statistical equilibrium is characterized by quantities such as the lateral buoyancy flux and the thermocline depth; here, scaling laws are proposed for these quantities in terms of the external parameters. The scaling theory predicts relations between heat transport, thermocline depth, bottom drag, and diapycnal diffusivity, which are confirmed by numerical simulations. The authors find that the depth of the thermocline is independent of the diapycnal mixing to leading order, but depends on the bottom drag. This dependence arises because the mean stratification is due to a balance between the large-scale wind-driven heat transport and the heat transport due to baroclinic eddies. The eddies equilibrate at an amplitude that depends to leading order on the bottom drag. The net poleward heat transport is a residual between the mean and eddy heat transports. The size of this residual is determined by the details of the diapycnal diffusivity. If the diffusivity is uniform (as in laboratory experiments) then the heat transport is linearly proportional to the diffusivity. If a mixed layer is incorporated by greatly increasing the diffusivity in a thin surface layer then the net heat transport is dominated by the model mixed layer.

Description: The radiative flux of internal wave energy (the “tidal conversion”) powered by the oscillating flow of a uniformly stratified fluid over a two-dimensional submarine ridge is computed using an integral-equation method. The problem is characterized by two nondimensional parameters, A and B. The first parameter, A, is the ridge half-width scaled by μh, where h is the uniform depth of the ocean far from the ridge and μ is the inverse slope of internal tidal rays (horizontal run over vertical rise). The second parameter, B, is the ridge height scaled by h. Two topographic profiles are considered: a triangular or tent-shaped ridge and a “polynomial” ridge with continuous topographic slope. For both profiles, complete coverage of the (A, B) parameter space is obtained by reducing the problem to an integral equation, which is then discretized and solved numerically. It is shown that in the supercritical regime (ray slopes steeper than topographic slopes) the radiated power increases monotonically with B and decreases monotonically with A. In the subcritical regime the radiated power has a complicated and nonmonotonic dependence on these parameters. As A → 0 recent results are recovered for the tidal conversion produced by a knife-edge barrier. It is shown analytically that the A → 0 limit is regular: if A ≪ 1 the reduction in tidal conversion below that at A = 0 is proportional to A2. Further, the knife-edge model is shown to be indicative of both conversion rates and the structure of the radiated wave field over a broad region of the supercritical parameter space. As A increases the topographic slopes become gentler, and at a certain value of A the ridge becomes “critical”; that is, there is a single point on the flanks at which the topographic slope is equal to the slope of an internal tidal beam. The conversion decreases continuously as A increases through this transition. Visualization of the disturbed buoyancy field shows prominent singular lines (tidal beams). In the case of a triangular ridge these beams originate at the crest of the triangle. In the case of a supercritical polynomial ridge, the beams originate at the shallowest point on the flank at which the topographic slope equals the ray slope.