ALBERT

Description: Using a matched asymptotic expansion we analyse the two-dimensional, near-critical reflection of a weakly nonlinear internal gravity wave from a sloping boundary in a uniformly stratified fluid. Taking a distinguished limit in which the amplitude of the incident wave, the dissipation, and the departure from criticality are all small, we obtain a reduced description of the dynamics. This simplification shows how either dissipation or transience heals the singularity which is presented by the solution of Phillips (1966) in the precisely critical case. In the inviscid critical case, an explicit solution of the initial value problem shows that the buoyancy perturbation and the alongslope velocity both grow linearly with time, while the scale of the reflected disturbance is reduced as 1/t. During the course of this scale reduction, the stratification is Overturned' and the Miles-Howard condition for stratified shear flow stability is violated. However, for all slope angles, the Overturning' occurs before the Miles-Howard stability condition is violated and so we argue that the first instability is convective. Solutions of the simplified dynamics resemble certain experimental visualizations of the reflection process. In particular, the buoyancy field computed from the analytic solution is in good agreement with visualizations reported by Thorpe & Haines (1987). One curious aspect of the weakly nonlinear theory is that the final reduced description is a linear equation (at the solvability order in the expansion all of the apparently resonant nonlinear contributions cancel amongst themselves). However, the reconstructed fields do contain nonlinearly driven second harmonics which are responsible for an important symmetry breaking in which alternate vortices differ in strength and size from their immediate neighbours.

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: This article considers the instabilities of rotating, shallow-water, shear flows on an equatorial βplane. Because of the free surface, the motion is horizontally divergent and the energy density is cubic in the field variables (i.e. in standard notation the kinetic energy density is 1/2h(u2+v2)). Marinone & Ripa (1984) observed that as a consequence of this the wave energy is no longer positive definite (there is a cross-term Uh'ú). A wave with negative wave energy can grow by transferring energy to the mean flow. Of course total (mean plus wave) energy is conserved in this process. Further, when the basic state has constant potential vorticity, we show that there are no exchanges of energy and momentum between a growing wave and the mean flow. Consequently when the basic state has no potential vorticity gradients an unstable wave has zero wave energy and the mean flow is modified so that its energy is unchanged. This result strikingly shows that energy and momentum exchanges between a growing wave and the mean flow are not generally characteristic of, or essential to, instability. A useful conceptual tool in understanding these counterintuitive results is that of disturbance energy (or pseudoenergy) of a shear mode. This is the amount of energy in the fluid when the mode is excited minus the amount in the unperturbed medium. Equivalently, the disturbance energy is the sum of the wave energy and that in the modified mean flow. The disturbance momentum (or pseudomomentum) is defined analogously. For an unstable mode, which grows without external sources, the disturbance energy must be zero. On the other hand the wave energy may increase to plus infinity, remain zero, or decrease to minus infinity. Thus there is a tripartite classification of instabilities. We suggest that one common feature in all three cases is that the unstable shear mode is roughly a linear combination of resonating shear modes each of which would be stable if the other were somehow suppressed. The two resonating constituents must have opposite-signed disturbance energies in order that the unstable alliance has zero disturbance energy. The instability is a transfer of disturbance energy from the member with negative disturbance energy to the one with positive disturbance energy. © 1987, Cambridge University Press. All rights reserved.

Description: We study fixed-flux convection in a long, narrow slot which is inclined to the horizontal. (Gravity is in the vertical direction, and horizontal is perpendicular to this.) Because of the fixed-flux boundary conditions the convective modes have much larger lengthscales in the along-slot direction than in the transverse direction. In the case of a horizontal slot this disparity in scales has been previously exploited to obtain an amplitude equation for the single mode which first becomes unstable a8 the Rayleigh number is increased above critical. When the slot is tilted we show that there is a distinguished limit in which there are two active modes in the slightly supercritical regime. This new limit is when the horizontal wavenumber, the supercriticality, and the tilt of the slot away from vertical, are all small. A modification of the well-known expansion for fixed flux convection in a horizontal slot leads to a coupled system of partial differential equations for the amplitudes of the two modes. Numerical solution of this system suggests that all initial conditions eventually evolve into one of the two states, both of which consist of a single, steady roll in the cavity. The states are distinguished by the direction of circulation of the roll, and by the buoyancy fields, which are quite different in the two cases. © 1992, Cambridge University Press. All rights reserved.

Description: This paper formulates a model of mixing in a stratified and turbulent fluid. The model uses the horizontally averaged vertical buoyancy gradient and the density of turbulent kinetic energy as variables. Heuristic 'mixing-length' arguments lead to a coupled set of parabolic differential equations. A particular form of mechanical forcing is proposed; for certain parameter values the relationship between the buoyancy flux and the buoyancy gradient is non-monotonic and this leads to an instability of equilibria with linear stratification. The instability results in the formation of steps and interfaces in the buoyancy profile. In contrast to previous ones, the model is mathematically well posed and the interfaces have an equilibrium thickness that is much larger than that expected from molecular diffusion. The turbulent mixing process can take one of three forms depending on the strength of the initial stratification. When the stratification is weak, instability is not present and mixing smoothly homogenizes the buoyancy. At intermediate strengths of stratification, layers and interfaces form rapidly over a substantial interior region bounded by edge layers associated with the fluxless condition of the boundaries. The interior pattern subsequently develops more slowly as interfaces drift together and merge; simultaneously, the edge layers advance inexorably into the interior. Eventually the edge layers meet in the middle and the interior pattern of layers is erased. Any remaining structure subsequently decays smoothly to the homogeneous state. Both the weak and intermediate stratified cases are in agreement with experimental phenomenology. The model predicts a third case, with strong stratification, not yet found experimentally, where the central region is linearly stable and no steps form there. However, the edge layers are unstable; mixing fronts form and then erode into the interior.

Description: We obtain an analytic solution for the generation of internal gravity waves by tidal flow past a vertical barrier of height b in a uniformly stratified ocean of depth h 〉 b and buoyancy frequency N. The radiated power (watts per metre of barrier) is 1/4πρ0b2U2N√1 - (f/ω 2M(b/h), where ρ0 is the mean density of seawater, U cos(ωt) the tidal velocity, and f the Coriolis frequency. The function M(b/h) increases monotonically with M(0) = 1, M(0.92) = 2 and M(1) = ∞. As b/h → 1, M diverges logarithmically and consequently the radiated power grows as In[(h - b /b]. We also calculate the conversion in a realistically stratified ocean with strongly non-uniform buoyancy frequency, N(z). A rough approximation to the radiated power in this case is 1/4πρ0b2U2N(b)√1 - (f/ω)2M(b/π), where N(b) is the buoyancy frequency at the tip of the ridge and B is the height of the ridge in WKB coordinates. (The WKB coordinate is normalized so that the total depth of the ocean is π.) The approximation above is an over-estimate of the actual radiation by as much as 20% when B/π ≈ 0.8. But the formula correctly indicates the strong dependence of conversion on stratification through the factor N(b).

Description: Consider the problem of horizontal convection: a Boussinesq fluid, forced by applying a non-uniform temperature at its top surface, with all other boundaries insulating. We prove that if the viscosity, v, and thermal diffusivity, K, are lowered to zero, with σ Ξ v/k fixed, then the energy dissipation per unit mass, ε, also vanishes in this limit. Numerical solutions of the two-dimensional case show that despite this anti-turbulence theorem, horizontal convection exhibits a transition to eddying flow, provided that the Rayleigh number is sufficiently high, or the Prandtl number σ sufficiently small. We speculate that horizontal convection is an example of a flow with a large number of active modes which is nonetheless not 'truly turbulent' because ε → 0 in the inviscid limit.

Description: We study advection-diffusion of a passive scalar, T, by an incompressible fluid in a closed vessel bounded by walls impermeable to the fluid. Variations in T are produced by prescribing a steady non-uniform distribution of T at the boundary. Because there is no flow through the walls, molecular diffusion, κ, is essential in 'lifting' Τ off the boundary and into the interior where the velocity field acts to intensify ▽Τ. We prove that as κ → O (with the fluid velocity fixed) this diffusive lifting is a feeble source of scalar variance. Consequently the scalar dissipation rate χ - the volume integral of κ ▽Τ2 - vanishes in the limit κ → O. Thus, in this particular closed-flow configuration, it is not possible to maintain a constant supply of scalar variance as κ → O and the fundamental premise of scaling theories for passive scalar cascades is violated. We also obtain a weaker bound on χ when the transported field is a dynamically active scalar, such as temperature. This bound applies to the Rayleigh-Bénard configuration in which Τ = ±1 on two parallel plates at Ζ = ±η/2. In this case we show that χ → 3.252 × (κε/νh2 1/3 where ν is the viscosity and ε is the mechanical energy dissipation per unit mass. Thus, provided that ε and ν/κ are non-zero in the limit κ → O, χ might remain non-zero.

Description: We calculate the optimal upper and lower bounds, subject to the assumption of streamwise invariance, on the long-time-averaged mechanical energy dissipation rate ε within the flow of an incompressible viscous fluid of constant kinematic viscosity ν and depth h driven by a constant surface stress τ = ρu*2, where u* is the friction velocity. We show that ε ≤ εmax = τ2/(ρ2ν), i.e. the dissipation is bounded above by the dissipation associated with the laminar solution u = τ(z+h)/(ρν)î, where î is the unit vector in the streamwise x-direction. By using the variational 'background method' (due to Constantin, Doering and Hopf) and numerical continuation, we also generate a rigorous lower bound on the dissipation for arbitra Grashof numbers G, where G = τh2/(ρν2). Under the assumption of streamwise invariance as G → ∞, for flows where horizontal mean momentum balance and total power balance are imposed as constraints, we show numerically that the best possible lower bound for the dissipation is ε ≥ εmin = 7.531u*3/h, a bound that is independent of the flow viscosity. This scaling (though not the best possible numerical coefficient) can also be obtained directly by applying the same imposed constraints and restricting attention to a particular, analytically tractable, class of mean flows. © 2004 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.