William R. Young

University of California, San Diego | UCSD · Scripps Institution of Oceanography (SIO)

The Nusselt numbers of horizontal convection - Volume 894 - Cesar B. Rocha, Navid C. Constantinou, Stefan G. Llewellyn Smith, William R. Young

We use a multiple-scale expansion to average the wave action balance equation over an ensemble of sea-surface velocity fields characteristic of the ocean mesoscale and submesoscale. Assuming that the statistical properties of the flow are stationary and homogeneous, we derive an expression for a diffusivity tensor of surface-wave action density. Th...

For the problem of horizontal convection the Nusselt number based on entropy production is bounded from above by C Ra 1/3 as the horizontal convective Rayleigh number Ra → ∞ for some constant C (Siggers et al., J. Fluid Mech., vol. 517, 2004, pp. 55-70). We reexamine the variational arguments leading to this 'ultimate regime' by using the Wentzel-K...

We consider the problem of horizontal convection in which non-uniform buoyancy, b_s(x,y), is imposed on the top surface of a container and all other surfaces are insulating. Horizontal convection produces a net horizontal flux of buoyancy, J, defined by vertically and temporally averaging the interior horizontal flux of buoyancy. We show that \over...

We study stimulated generation – the transfer of energy from balanced flows to existing internal waves – using an asymptotic model that couples barotropic quasi-geostrophic flow and near-inertial waves with $\text{e}^{\text{i}mz}$ vertical structure, where $m$ is the vertical wavenumber and $z$ is the vertical coordinate. A detailed description of...

A heat exchanger can be modeled as a closed domain containing an incompressible fluid. The moving fluid has a temperature distribution obeying the advection-diffusion equation, with zero temperature boundary conditions at the walls. Starting from a positive initial temperature distribution in the interior, the goal is to flux the heat through the w...

Using a one-layer QG model, we study the effect of random monoscale topography on forced beta-plane turbulence. The forcing is a uniform steady wind stress that produces both a uniform large-scale zonal flow U(t) and smaller-scale macroturbulence (both standing and transient eddies). The flow U(t) is retarded by Ekman drag and by the domain-average...

Starting from the hydrostatic Boussinesq equations, we derive a time-averaged `hydrostatic wave equation' that describes the propagation of inertia-gravity internal waves through quasi-geostrophic flow. The derivation uses a multiple-time-scale asymptotic method to isolate wave field evolution over intervals much longer than a wave period, assumes...

We derive an asymptotic model that describes the nonlinear coupled evolution of (i) near-inertial waves (NIWs), (ii) balanced quasi-geostrophic flow and (iii) near-inertial second harmonic waves with frequency near $2f_{0}$ , where $f_{0}$ is the local inertial frequency. This ‘three-component’ model extends the two-component model derived by Xie &...

The interaction of the barotropic tide with a tall, two-dimensional ridge is examined analytically and numerically at latitudes where the tide is subinertial, and contrasted to when the tide is superinertial. When the tide is subinertial, the energy density associated with the response grows with latitude as both the oscillatory along-ridge flow an...

We derive a wave-averaged potential vorticity equation describing the evolution of strongly stratified, rapidly rotating quasi-geostrophic (QG) flow in a field of inertia-gravity internal waves. The derivation relies on a multiple-time-scale asymptotic expansion of the Eulerian Boussinesq equations. Our result confirms and extends the theory of Büh...

This study investigates the representation of solutions of the three-dimensional quasigeostrophic (QG) equations using Galerkin series with standard vertical modes, with particular attention to the incorporation of active surface buoyancy dynamics. This study extends two existing Galerkin approaches (A and B) and develops a new Galerkin approximati...

We study the representation of approximate solutions of the three-dimensional quasigeostrophic (QG) equations using Galerkin series with standard vertical modes. In particular, we show that standard modes are compatible with nonzero buoyancy at the surfaces and can be used to solve the Eady problem. We extend two existing Galerkin approaches (A and...

The equations of motion are reexamined with the objective of improving upon the Boussinesq approximation. The authors derive new equations that conserve energy, filter out sound waves, are more accurate than the Boussinesq set, and are computationally competitive with them. The new equations are partly enabled by exploiting a reversible exchange be...

Using recordings of swell from pitch-and-roll buoys, we have reproduced the classic observations of long-range surface wave propagation originally made by Munk et al. (1963) using a triangular array of bottom pressure measurements. In the modern data, the direction of the incoming swell fluctuates by about $\pm 10^\circ$ on a time scale of one hour...

We examine the vertical mixing induced by the swimming of microorganisms at low Reynolds and P\'eclet numbers in a stably stratified ocean, and show that the global contribution of oceanic microswimmers to vertical mixing is negligible. We propose two approaches to estimating the mixing efficiency, $\eta$, or the ratio of the rate of potential ener...

The Reynolds stress induced by anisotropically forcing an unbounded Couette flow, with uniform shear gamma, on a beta plane, is calculated in conjunction with the eddy diffusivity of a coevolving passive tracer. The flow is damped by linear drag on a time scale mu(-1). The stochastic forcing is white noise in time and its spatial anisotropy is cont...

We consider the linear stability of an inviscid parallel shear flow of air over water with gravity and capillarity. The velocity profile in the air is monotonically increasing upwards from the sea surface and is convex, while the velocity in the water is monotonically decreasing from the surface and is concave. An archetypical example, the 'double-...

We investigate solutions of the two-dimensional Navier–Stokes equation in a square box with stress-free boundary conditions. The flow is steadily forced by the addition of a source to the vorticity equation; attention is restricted to even so that the forcing has zero integral. Numerical solutions with and show that at high Reynolds numbers the sol...

The author shows that a systematic application of thickness-weighted averaging to the Boussinesq equations of motion results in averaged equations of motion written entirely in terms of the thickness-weighted velocity; that is, the unweighted average velocity and the eddy-induced velocity do not appear in the averaged equations of motion. This thic...

Zonostrophic instability leads to the spontaneous emergence of zonal jets on a β plane from a jetless basic-state flow that is damped by bottom drag and driven by a random body force. Decomposing the barotropic vorticity equation into the zonal mean and eddy equations, and neglecting the eddy–eddy interactions, defines the quasilinear (QL) system....

We consider the problem of a Boussinesq fluid forced by applying both non-uniform temperature and stress at the top surface. On the other boundaries the conditions are thermally insulating and either no-slip or stress-free. The interesting case is when the direction of the steady applied surface stress opposes the sense of the buoyancy driven flow....

The statistically steady humidity distribution resulting from an interaction of advection, modelled as an uncorrelated random walk of moist parcels on an isentropic surface, and a vapour sink, modelled as immediate condensation whenever the specific humidity exceeds a specified saturation humidity, is explored with theory and simulation. A source s...

The statistically steady humidity distribution resulting from an interaction of advection, modeled as an uncorrelated random walk of moist parcels on an isentropic surface, and a vapour sink, modeled as immediate condensation whenever the specific humidity exceeds a specified saturation humidity, is explored with theory and simulation. A source sup...

a b s t r a c t We present a model interacting particle system with a population of fixed size in which particles wander randomly in space, and pairs interact at a rate determined by a reaction kernel with finite range. The pairwise interaction randomly selects one of the particles (the victim) and instantly transfers it to the position of the othe...

The dynamics of a forced, low-mode oceanic internal tide propagating poleward on a beta plane are investigated numerically. We focus on the instability that transfers energy from the forced wave to waves at subharmonic frequency near the critical latitude where the subharmonic and local inertial frequencies match. Through parametric subharmonic ins...

Consider a two-dimensional axisymmetric vortex with circulation Gamma. Suppose that this vortex is isovortically deformed into an elliptical vortex. We show that the reduction in energy is DeltaE=-Gamma2 ln[(q+q-1)/2]/(4pi), where q2 is the ratio of the major to the minor axis of any particular elliptical vorticity contour. It is notable that Delta...

Anew seawater Boussinesq system is introduced, and it is shown that this approximation to the equations of motion of a compressible binary solution has an energy conservation law that is a consistent approximation to the Bernoulli equation of the full system. The seawater Boussinesq approximation simplifies the mass conservation equation to Δ. u =...

We consider the mechanical energy budget for horizontal Boussinesq convection and show that there are two distinct energy pathways connecting the mechanical energy (i.e. kinetic, available potential and background potential energies) to the internal energy reservoir and the external energy source. To obtain bounds on the magnitudes of the energy tr...

We consider two-dimensional turbulence driven by a steady prescribed sinusoidal body force working at an average rate epsilon. Energy dissipation is due mainly to drag, which damps all wave number at a rate micro. Simulations at statistical equilibrium reveal a scaling regime in which epsilon proportional, variant micro;{1/3}, with no significant d...

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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...

We develop a new nonlinear stability method, the Energy-Enstrophy (EZ) method, that is specialized to two-dimensional hydrodynamics; the method is applied to a beta-plane flow driven by a sinusoidal body force, and retarded by drag with damping time-scale mu^{-1}. The standard energy method (Fukuta and Murakami, J. Phys. Soc. Japan, 64, 1995, pp 37...

We show that a steady vertically-sheared current can produce a thin layer of plankton by differentially advecting an initial patch whose vertical and horizontal dimensions are H0 and L0, respectively. Our model treats the plankton as an inert passive tracer with vertical diffusivity κv and subject to a vertically-sheared horizontal current with she...

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 cyl...

The eddy heat flux generated by statistically equilibrated baroclinic turbulence supported on a uniform, horizontal temperature gradient is examined using a two-layer β-plane quasigeostrophic model. The dependence of the eddy diffusivity of temperature, D_τ, on external parameters such as β, bottom friction κ, the deformation radius λ, and the velo...

The Fisher-Kolmogorov-Petrovskii-Piskunov equation with a variable growth rate and advection by an incompressible velocity field is considered as a model for plankton dispersed by ocean currents. If the average growth rate is negative then the model has a survival-extinction transition; the location of this transition in the parameter space is cons...

In the study of accretion disks around central objects with specified gravitational potentials, it is commonly assumed that the distribution of the mean azimuthal velocity is Keplerian. A similar assumption of centrifugal balance underlies the usual determinations of galactic mass distributions from rotation curves. The authors offer a justificatio...

The equilibrium of an idealized flow driven at the surface by wind stress and rapid relaxation to non-uniform 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 diff...

We derive a closed master equation for an individual-based population model in continuous space and time. The model and master equation include Brownian motion, reproduction via binary fission, and an interaction-dependent death rate moderated by a competition kernel. Using simulations we compare this individual-based model with the simplest approx...

Numerical calculations of the rate at which energy is converted from the external to internal tides at steep oceanic ridges are compared with estimates from analytic theories. The numerical calculations are per- formed using a hydrostatic primitive equation ocean model that uses a generalized s-coordinate system as the vertical coordinate. The mode...

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 �h...

We study the statistics of a passive scalar $T({\bm{x},t)$ 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, $\sigma^2\equiv \langle T^2 \rangle$, and the entropy production, $\chi \equiv \kappa \langle|\bm{\nabla} T|^2\rangle$...

The eddy heat flux generated by the statistically equilibrated baroclinic instability of a uniform, horizontal temperature gradient is studied using a two-mode f-plane quasigeostrophic model. An overview of the dependence of the eddy diffusivity D on the bottom friction , the deformation radius , the vertical variation of the large-scale flow U, an...

We calculate a rigorous dual bound on the long-time-averaged mechanical energy dissipation rate $\varepsilon$ within a channel of an incompressible viscous fluid of constant kinematic viscosity $\nu$, depth $h$ and rotation rate $f$, driven by a constant surface stress ${\bm\tau}\,{=}\,\rho u^2_\star\xvec$, where $u_\star$ is the friction velocity....

We calculate the optimal upper and lower bounds, subject to the assumption of streamwise invariance, on the long-time-averaged mechanical energy dissipation rate $\varepsilon$ within the flow of an incompressible viscous fluid of constant kinematic viscosity $\nu$ and depth $h$ driven by a constant surface stress $\tau=\rho u^2_\star$, where $u_\st...

Using linear wave theory, the rate at which energy is converted into internal gravity waves by the interaction of the barotropic tide with topography in an ocean is calculated. Bell's formula for the conversion rate is extended to the case of an ocean of finite depth H with weak two-dimensional topography h(x, y) and arbitrary buoyancy frequency N(...

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.T he radiated power (watts per metre of barrier) is 1 4 πρ0b 2U 2N 1 − (f/ω)2M(b/ h), where ρ0 is the mean density of seawater, U cos(ωt )t hetidal veloci...

We calculate the optimal upper and lower bounds, subject to the assumption of streamwise invariance, on the long-time-averaged mechanical energy dissipation rate ɛ within the flow of an incompressible viscous fluid of constant kinematic viscosity ν and depth h that is driven by a constant stress τ, defining an appropriate Grashof number G=τ h^2/ν^2...

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, $\kappa$, is essential in ‘lifting’ $...

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. If b/h is small and N is constant, the radiated power (watts per metre of barrier) is (pi/4) rho_0 b^2 U^2 N sqrt{1-(f/omega)^2} where rho_0 is the mean...

Analytical estimates of the rate at which energy is extracted from the barotropic tide at topography and converted into internal gravity waves are given. The ocean is idealized as an inviscid, vertically unbounded fluid on the f plane. The gravity waves are treated by linear theory and freely escape to z 5' . Several topographies are investigated:...

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, [nu], and thermal diffusivity, [kappa], are lowered to zero, with [sigma] [identical with] [nu]/[kappa] fixed, then the energy dissipation per un...

Using linear wave theory, we calculate the rate at which energy is converted into internal gravity waves by the interaction of the barotropic tide with topography in an ocean. We extend Bell's (1975a) formula for the conversion rate to the case of an ocean of finite depth H with two-dimensional topography h(x,y) and arbitrary buoyancy frequency N(z...

We show that the geophysical β-effect strongly affects the linear stability of a sinusoidal Kolmogorov flow. If α denotes the angle between the flow direction and the planetary vorticity gradient then the critical Reynolds’ number, Rc(α,β), is zero for β≠0, provided that . In particular, the small β limit is discontinuous: , rather than the classic...

We show that the geophysical β-effect strongly affects the linear stability of a sinusoidal Kolmogorov flow. If α denotes the angle between the flow direction and the planetary vorticity gradient then the critical Reynolds' number, R c (α, β), is zero for β = 0, provided that sin 2α = 0. In particular, the small β limit is discontinuous: lim β→0 R...

The surface mixed layer of the ocean is often characterized by density compensation between the horizontal temperature and salinity gradients. In this contribution we present a combination of theoretical arguments and numerical simulations to investigate how compensation might emerge as a result of processes at work within the mixed layer. The dyna...

Many dispersive processes have moments of displacements with large-t behavior 〈∣x∣p〉∼tγp. The study of γp as a function of p provides a more complete characterization of the process than does the single number γ2. Also at long times, the core of the concentration relaxes to a self-similar profile, while the large-x tails, consisting of particles wh...

A# ection-di#usion-reaction (adr) models link physical oceanography and biological oceanography. These models,which describe biology using continuous concentration fields,usually neglect individual-scale fluctuations. I describe a stochastic individual-based model,called the Brownian bug process,which illustrates some of the surprising issues assoc...

Clustering of organisms can be a consequence of social behaviour, or of the response of individuals to chemical and physical cues. Environmental variability can also cause clustering: for example, marine turbulence transports plankton and produces chlorophyll concentration patterns in the upper ocean. Even in a homogeneous environment, nonlinear in...

Inviscid spatially compact vortices (such as the Rankine vortex) have discrete Kelvin modes. For these modes, the critical radius, at which the rotation frequency of the wave matches the angular velocity of the fluid, lies outside the vortex core. When such a vortex is not perfectly compact, but has a weak vorticity distribution beyond the core...

We present analytical estimates of the rate at which energy is extracted from the barotropic tide at topography. We use two model topographies: a sinusoidal ripple and a family of profiles which approximates a periodically spaced set of Gaussian bumps. The conversion rate is expressed as a function of a parameter, ε, which is the ratio of the slope...

In this study we investigate stratified Kolmogorov shear flow. We derive the amplitude equations for this system and solve them numerically to explore the effect of a weak stabilizing stratification. We then explore the non-diffusive limit of this system, and solve amplitude equations for this system to study the weakly nonlinear evolution of the i...

An idealized model of the transmission of near-inertial waves from the mixed layer into the deeper ocean is studied in order to assess the combined effects of background geostrophic vorticity and the planetary vorticity gradient. The model geostrophic flow is steady and barotropic with a streamfunction =- cos (2y); the planetary vorticity gradient...

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 re...

The authors study the stability of a barotropic sinusoidal meridional flow on a plane. Because of bottom drag and lateral viscosity, the system is dissipative and forcing maintains a basic-state velocity that carries fluid across the planetary vorticity contours; this is a simple model of forced potential vorticity mixing. When the Reynolds number...

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;...