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Wagner, GL, Young WR.  2016.  A three-component model for the coupled evolution of near-inertial waves, quasi-geostrophic flow and the near-inertial second harmonic. Journal of Fluid Mechanics. 802:806-837.   10.1017/jfm.2016.487   AbstractWebsite

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 & Vanneste (J. Fluid Mech., vol 774, 2015, pp. 143-169) to include interactions between near-inertial and waves. Both models possess two conservation laws which together imply that oceanic NIWs forced by winds, tides or flow over bathymetry can extract energy from quasi-geostrophic flows. A second and separate implication of the three-component model is that quasi-geostrophic flow catalyses a loss of NIW energy to freely propagating waves with near-2f(0) frequency that propagate rapidly to depth and transfer energy back to the NIW field at very small vertical scales. The upshot of near-2f(0) generation is a two-step mechanism whereby quasi-geostrophic flow catalyses a nonlinear transfer of near-inertial energy to the small scales of wave breaking and diapycnal mixing. A comparison of numerical solutions with both Boussinesq and three-component models for a two-dimensional initial value problem reveals strengths and weaknesses of the model while demonstrating the extraction of quasi-geostrophic energy and production of small vertical scales.

Rocha, CB, Young WR, Grooms I.  2016.  On Galerkin approximations of the surface active quasigeostrophic equations. Journal of Physical Oceanography. 46:125-139.   10.1175/jpo-d-15-0073.1   AbstractWebsite

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 approximation (C). Approximation A, due to Flierl, represents the streamfunction as a truncated Galerkin series and defines the potential vorticity (PV) that satisfies the inversion problem exactly. Approximation B, due to Tulloch and Smith, represents the PV as a truncated Galerkin series and calculates the streamfunction that satisfies the inversion problem exactly. Approximation C, the true Galerkin approximation for the QG equations, represents both streamfunction and PV as truncated Galerkin series but does not satisfy the inversion equation exactly. The three approximations are fundamentally different unless the boundaries are isopycnal surfaces. The authors discuss the advantages and limitations of approximations A, B, and C in terms of mathematical rigor and conservation laws and illustrate their relative efficiency by solving linear stability problems with nonzero surface buoyancy. With moderate number of modes, B and C have superior accuracy than A at high wavenumbers. Because B lacks the conservation of energy, this study recommends approximation C for constructing solutions to the surface active QG equations using the Galerkin series with standard vertical modes.

Wagner, GL, Young WR.  2015.  Available potential vorticity and wave-averaged quasi-geostrophic flow. Journal of Fluid Mechanics. 785:401-424.   10.1017/jfm.2015.626   AbstractWebsite

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 Buhler & McIntyre (J. Fluid Mech., vol. 354, 1998, pp. 609-646) to non-uniform stratification with buoyancy frequency N(z) and therefore non-uniform background potential vorticity f(0)N(2)(z), and does not require spatial-scale separation between waves and balanced flow. Our interest in non-uniform background potential vorticity motivates the introduction of a new quantity: 'available potential vorticity' (APV). Like Ertel potential vorticity, APV is exactly conserved on fluid particles. But unlike Ertel potential vorticity, linear internal waves have no signature in the Eulerian APV field, and the standard QG potential vorticity is a simple truncation of APV for low Rossby number. The definition of APV exactly eliminates the Ertel potential vorticity signal associated with advection of a non-uniform background state, thereby isolating the part of Ertel potential vorticity available for balanced-flow evolution. The effect of internal waves on QG flow is expressed concisely in a wave-averaged contribution to the materially conserved QG potential vorticity. We apply the theory by computing the wave-induced QG flow for a vertically propagating wave packet and a mode-one wave field, both in vertically bounded domains.

Dewar, WK, Schoonover J, McDougall TJ, Young WR.  2015.  Semicompressible ocean dynamics. Journal of Physical Oceanography. 45:149-156.   10.1175/jpo-d-13-0268.1   AbstractWebsite

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 between internal and gravitational potential fluid energy. To improve upon these equations appears to require the inclusion of acoustics, at which point one should use full Navier-Stokes. This study recommends the new sets for testing in general circulation modeling.

Wagner, GL, Young WR, Lauga E.  2014.  Mixing by microorganisms in stratified fluids. Journal of Marine Research. 72:47-72. AbstractWebsite

We examine the vertical mixing induced by the swimming of microorganisms at low Reynolds and Peclet 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 energy creation to the total rate-of-working on the ocean by microswimmers. The first is based on scaling arguments and estimates eta in terms of the ratio between the typical organism size, a, and an intrinsic length scale for the stratified flow, l = (nu kappa/N-2)(1/4), where. is the kinematic viscosity, kappa the diffusivity, and N the buoyancy frequency. In particular, for small organisms in the relevant oceanic limit, a/l << 1, we predict the scaling eta similar to (a/l)(3). The second estimate of. is formed by solving the full coupled flow-stratification problem by modeling the swimmer as a regularized force dipole, and computing the efficiency numerically. Our computational results, which are examined for all ratios a/l, validate the scaling arguments in the limit a/l << 1 and further predict eta approximate to 1.2 (a/l)(3) for vertical swimming and eta approximate to 0.15 (a/l) 3 for horizontal swimming. These results, relevant for any stratified fluid rich in biological activity, imply that the mixing efficiency of swimming microorganisms in the ocean is at very most 8% and is likely smaller by at least two orders of magnitude.

Gallet, B, Young WR.  2014.  Refraction of swell by surface currents. Journal of Marine Research. 72:105-126. AbstractWebsite

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 +/- 10 degrees on a time scale of one hour. But if the incoming direction is averaged over the duration of an event then, in contrast with the observations by Munk et al. (1963), the sources inferred by great-circle backtracking are most often in good agreement with the location of large storms on weather maps of the Southern Ocean. However there are a few puzzling failures of great-circle backtracking. For example, in one case, the direct great-circle route is blocked by the Tuamoto Islands and the inferred source falls on New Zealand. Mirages like this occur more frequently in the bottom-pressure observations of Munk et al. (1963), where several inferred sources fell on the Antarctic continent. Using spherical ray tracing we investigate the hypothesis that the refraction of waves by surface currents produces the mirages. With reconstructions of surface currents inferred from satellite altimetry, we show that mesoscale vorticity significantly deflects swell away from great-circle propagation so that the source and receiver are connected by a bundle of many rays, none of which precisely follow a great circle. The +/-10 directional fluctuations at the receiver result from the arrival of wave packets that have travelled along the different rays within this multipath. The occasional failure of great-circle backtracking, and the associated mirages, probably results from partial topographic obstruction of the multipath, which biases the directional average at the receiver.

Srinivasan, K, Young WR.  2014.  Reynolds stress and eddy diffusivity of beta-plane shear flows. Journal of the Atmospheric Sciences. 71:2169-2185.   10.1175/jas-d-13-0246.1   AbstractWebsite

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 controlled by a parameter alpha that characterizes whether eddies are elongated along the zonal direction (alpha < 0), are elongated along the meridional direction (alpha > 0), or are isotropic (alpha = 0). The Reynolds stress varies linearly with alpha and nonlinearly and nonmonotonically with gamma, but the Reynolds stress is independent of beta. For positive values of alpha, the Reynolds stress displays an "antifrictional" effect (energy is transferred from the eddies to the mean flow); for negative values of alpha, it displays a frictional effect. When gamma/mu << 1, these transfers can be identified as negative and positive eddy viscosities, respectively. With gamma = beta = 0, the meridional tracer eddy diffusivity is (upsilon'(2)) over bar (2 mu), where upsilon' is the meridional eddy velocity. In general, nonzero beta and gamma suppress the eddy diffusivity below (upsilon'(2)) over bar (2 mu). When the shear is strong, the suppression due to gamma varies as gamma(-1) while the suppression due to beta varies between beta(-1) and beta(-2) depending on whether the shear is strong or weak, respectively.

Young, WR, Wolfe CL.  2014.  Generation of surface waves by shear-flow instability. Journal of Fluid Mechanics. 739:276-307.   10.1017/jfm.2013.617   AbstractWebsite

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-exponential' profile, is solved analytically and studied in detail. We show that there are two types of unstable mode which can, in some cases, co-exist. The first type is the 'Miles mode' resulting from a resonant interaction between a surface gravity wave and a critical level in the air. The second unstable mode is an interaction between surface gravity waves and a critical level in the water, resulting in the growth of ripples. The gravity capillary waves participating in this second resonance have negative intrinsic phase speed, but are Doppler shifted so that their actual phase speed is positive, and matches the speed of the base-state current at the critical level. In both cases, the Reynolds stresses of an exponentially growing wave transfer momentum from the vicinity of the critical level to the zone between the crests and troughs of a surface wave.

Young, WR.  2012.  An exact thickness-weighted average formulation of the Boussinesq equations. Journal of Physical Oceanography. 42:692-707.   10.1175/jpo-d-11-0102.1   AbstractWebsite

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 thickness-weighted average (TWA) formulation is identical to the unaveraged equations, apart from eddy forcing by the divergence of three-dimensional Eliassen-Palm (EP) vectors in the two horizontal momentum equations. These EP vectors are second order in eddy amplitude and, moreover, the EP divergences can be expressed in terms of the eddy flux of the Rossby-Ertel potential vorticity derived from the TWA equations of motion. That is, there is a fully nonlinear and three-dimensional generalization of the one-and two-dimensional identities found by Taylor and Bretherton. The only assumption required to obtain this exact TWA formulation is that the buoyancy field is stacked vertically; that is, that the buoyancy frequency is never zero. Thus, the TWA formulation applies to nonrotating stably stratified turbulent flows, as well as to large-scale rapidly rotating flows. Though the TWA formulation is obtained by working on the equations of motion in buoyancy coordinates, the averaged equations of motion can then be translated into Cartesian coordinates, which is the most useful representation for many purposes.

Srinivasan, K, Young WR.  2012.  Zonostrophic Instability. Journal of the Atmospheric Sciences. 69:1633-1656.   10.1175/jas-d-11-0200.1   AbstractWebsite

Zonostrophic instability leads to the spontaneous emergence of zonal jets on a beta 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. Numerical solution of the QL system shows zonal jets with length scales comparable to jets obtained by solving the nonlinear (NL) system. Starting with the QL system, one can construct a deterministic equation for the evolution of the two-point single-time correlation function of the vorticity, from which one can obtain the Reynolds stress that drives the zonal mean flow. This deterministic system has an exact nonlinear solution, which is an isotropic and homogenous eddy field with no jets. The authors characterize the linear stability of this jetless solution by calculating the critical stability curve in the parameter space and successfully comparing this analytic result with numerical solutions of the QL system. But the critical drag required for the onset of NL zonostrophic instability is sometimes a factor of 6 smaller than that for QL zonostrophic instability. Near the critical stability curve, the jet scale predicted by linear stability theory agrees with that obtained via QL numerics. But on reducing the drag, the emerging QL jets agree with the linear stability prediction at only short times. Subsequently jets merge with their neighbors until the flow matures into a state with jets that are significantly broader than the linear prediction but have spacing similar to NL jets.

Hazewinkel, J, Paparella F, Young WR.  2012.  Stressed horizontal convection. Journal of Fluid Mechanics. 692:317-331.   10.1017/jfm.2011.514   AbstractWebsite

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. We obtain two-dimensional numerical solutions showing a regime in which there is an upper cell with thermally indirect circulation (buoyant fluid is pushed downwards by the applied stress and heavy fluid is elevated), and a second deep cell with thermally direct circulation. In this two-cell regime the driving mechanisms are competitive in the sense that neither dominates the flow. A scaling argument shows that this balance requires that surface stress vary as the horizontal Rayleigh number to the three-fifths power.

Sukhatme, J, Young WR.  2011.  The advection-condensation model and water-vapour probability density functions. Quarterly Journal of the Royal Meteorological Society. 137:1561-1572.   10.1002/qj.869   AbstractWebsite

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 supplies moisture at the deep-tropical southern boundary of the domain and the saturation humidity is specified as a monotonically decreasing function of distance from the boundary. The boundary source balances the interior condensation sink, so that a stationary spatially inhomogeneous humidity distribution emerges. An exact solution of the Fokker-Planck equation delivers a simple expression for the resulting probability density function (PDF) of the wate-rvapour field and also the relative humidity. This solution agrees completely with a numerical simulation of the process, and the humidity PDF exhibits several features of interest, such as bimodality close to the source and unimodality further from the source. The PDFs of specific and relative humidity are broad and non-Gaussian. The domain-averaged relative humidity PDF is bimodal with distinct moist and dry peaks, a feature which we show agrees with middleworld isentropic PDFs derived from the ERA interim dataset. Copyright (C) 2011 Royal Meteorological Society

Balmforth, NJ, Young WR.  2011.  An interacting particle model with compact hierarchical structure. Physica D-Nonlinear Phenomena. 240:101-113.   10.1016/j.physd.2010.09.016   AbstractWebsite

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 other (the killer), thus maintaining the total number. The special feature of the model is that it possesses a closed hierarchical structure in which the statistical moments of the governing master equation lead to closed equations for the reduced distribution functions (the concentration, pair correlation function, and so on). In one spatial dimension, we show that persistent spatial correlations (clusters) arise in this model and we characterize the dynamics in terms of analytical properties of the pair correlation function. As the range of the reaction kernel is increased, the dynamics varies from an ensemble of largely independent random walkers at small range to tightly bound clusters with longer-range reaction kernels. (C) 2010 Elsevier B.V. All rights reserved.

Young, WR.  2010.  Dynamic enthalpy, conservative temperature, and the seawater Boussinesq approximation. Journal of Physical Oceanography. 40:394-400.   10.1175/2009jpo4294.1   AbstractWebsite

A new 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 del . u = 0, employs the nonlinear equation of state of seawater to obtain the buoyancy force, and uses the conservative temperature introduced by McDougall as a thermal variable. The conserved energy consists of the kinetic energy plus the Boussinesq dynamic enthalpy h(double dagger), which is the integral of the buoyancy with respect to geopotential height Z at a fixed conservative temperature and salinity. In the Boussinesq approximation, the full specific enthalpy h is the sum of four terms: McDougall's potential enthalpy, minus the geopotential g(0)Z, plus the Boussinesq dynamic enthalpy h(double dagger), and plus the dynamic pressure. The seawater Boussinesq approximation removes the large and dynamically inert contributions to h, and it reveals the important conversions between kinetic energy and h(double dagger).

Vanneste, J, Young WR.  2010.  On the energy of elliptical vortices. Physics of Fluids. 22   10.1063/1.3474703   AbstractWebsite

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 Delta E =-Gamma(2) ln[q+q(-1))/2]/(4 pi), where q(2) is the ratio of the major to the minor axis of any particular elliptical vorticity contour. It is notable that Delta E is independent of the details of vorticity profile of the axisymmetric vortex and, in particular, independent of its average radius. The implications of this result for the two-dimensional inverse cascade are briefly discussed. (c) 2010 American Institute of Physics. [doi:10.1063/1.3474703]

Winters, KB, Young WR.  2009.  Available potential energy and buoyancy variance in horizontal convection. Journal of Fluid Mechanics. 629:221-230.   10.1017/s0022112009006685   AbstractWebsite

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 transfer rates around each cycle, we first show that the volume-averaged dissipation rate of buoyancy variance chi equivalent to kappa , where b is the buoyancy, is bounded from above by 4.57h(-1) kappa(2/3) nu(-1/3) b(max)(7/3). Here h is the depth of the container, kappa the molecular diffusion, nu the kinematic viscosity and b(max) the maximum buoyancy difference that exists on the surface. The bound on chi is used to estimate the generation rate of available potential energy E-a and the rate at which E-a is irreversibly converted to background potential energy via diapycnal fluxes, both of which are shown to vanish at least as fast as kappa(1/3) in the limit kappa -> 0 at fixed Prandtl number Pr = nu/kappa As a thought experiment, consider a hypothetical ocean insulated at all boundaries except at the upper surface, where the buoyancy is prescribed. The bounds on the energy transfer rates in the mechanical energy budget imply that buoyancy forcing alone is insufficient by at least three orders of magnitude to maintain observed oceanic dissipation rates and that additional energy sources such as winds, tides and perhaps bioturbation are necessary to sustain observed levels of turbulent dissipation in the world's oceans.

Birch, DA, Young WR, Franks PJS.  2009.  Plankton layer profiles as determined by shearing, sinking, and swimming. Limnology and Oceanography. 54:397-399.   10.4319/lo.2009.54.1.0397   AbstractWebsite
Tsang, YK, Young WR.  2009.  Forced-dissipative two-dimensional turbulence: A scaling regime controlled by drag. Physical Review E. 79   10.1103/PhysRevE.79.045308   AbstractWebsite

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 mu. Simulations at statistical equilibrium reveal a scaling regime in which epsilon proportional to mu(1/3), with no significant dependence of epsilon on hyperviscosity, domain size, or numerical resolution. This power-law scaling is explained by a crude closure argument that identifies advection by the energetic large-scale eddies as the crucial process that limits epsilon by disrupting the phase relation between the body force and fluid velocity. The average input epsilon is due mainly to spatial regions in which the large-scale velocity is much less than the root-mean-square velocity. We argue that epsilon proportional to mu(1/3) characterizes energy injection by a steady or slowly changing spectrally confined body force.

Birch, DA, Young WR, Franks PJS.  2008.  Thin layers of plankton: Formation by shear and death by diffusion. Deep-Sea Research Part I-Oceanographic Research Papers. 55:277-295.   10.1016/j.dsr.2007.11.009   AbstractWebsite

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 H-0 and L-0, respectively. Our model treats the plankton as an inert passive tracer with vertical diffusivity k(v) and subject to a vertically-sheared horizontal current with shear alpha. After a transient of duration L-0/alpha H-0 the vertical thickness H of the patch decreases with H(t) approximate to L-0/alpha t. This shear-driven thinning is halted by diffusion at a time of order alpha(-2/3)k(v)(-1/3)L(0)(2/3), and at this time the layer achieves a minimum layer thickness of order alpha(1/3)k(v)(1/3)L(0)(1/3). For typical oceanic parameters, such as k(v)similar to 10(-5)m(2)s(-1), alpha similar to 10(-2)s(-1), and L-0 similar to 1000 m the initial transient is about 3 h and the layer achieves a minimum thickness of order 1 m in a time of order 1 day. During the shear thinning the intensity of the layer decreases by a factor of 3(-1/2) approximate to 0.58, which means that the intensity of the thin layer is comparable to the intensity of the patch from which it was formed. Subsequently the layer thickens and its intensity decreases; the coup de grace is delivered by shear dispersion at a time of order H-0(2)/k(v). The lifetime of the thin layer, defined by the condition that the maximum concentration is comparable to the initial maximum concentration, is the same order as the time to achieve minimum thickness. Additionally, analysis of a nutrient phytoplankton model shows that phytoplankton growing in a sheared patch of nutrients can result in a layer of phytoplankton that develops as an initially thin feature. (c) 2007 Elsevier Ltd. All rights reserved.

Young, WR, Tsang YK, Balmforth NJ.  2008.  Near-inertial parametric subharmonic instability. Journal of Fluid Mechanics. 607:25-49.   10.1017/s0022112008001742   AbstractWebsite

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 f(0)-pump energizes a vertical continuum of near-inertial oscillations. The amplitude equation is solved using two models for the 2 f(0)-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.

Tsang, YK, Young WR.  2008.  Energy-enstrophy stability of beta-plane Kolmogorov flow with drag. Physics of Fluids. 20   10.1063/1.2958321   AbstractWebsite

We develop a nonlinear stability method, the energy-enstrophy (EZ) method, that is specialized to two-dimensional hydrodynamics and basic state flows consisting of a single Helmholtz eigenmode. 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 [H. Fukuta and Y. Murakami, J. Phys. Soc. Jpn. 64, 3725 (1995)] shows that the laminar solution is monotonically and globally stable in a certain portion of the (mu,beta)-parameter space. The EZ method proves nonlinear stability in a larger portion of the (g,,6)-parameter space than does the energy method. Moreover, by penalizing high wavenumbers, the EZ method identifies a most strongly amplifying disturbance that is more physically realistic than that delivered by the energy method. Linear instability calculations are used to determine the region of the (mu,beta)-parameter space where the flow is unstable to infinitesimal perturbations. There is only a small gap between the linearly unstable region and the nonlinearly stable region, and full numerical solutions show only small transient amplification in that gap. (C) 2008 American Institute of Physics.

Thompson, AF, Young WR.  2007.  Two-layer baroclinic eddy heat fluxes: Zonal flows and energy balance. Journal of the Atmospheric Sciences. 64:3214-3231.   10.1175/jas4000.1   AbstractWebsite

The eddy heat flux generated by statistically equilibrated baroclinic turbulence supported on a uniform, horizontal temperature gradient is examined using a two-layer beta-plane quasigeostrophic model. The dependence of the eddy diffusivity of temperature, D-tau, on external parameters such as beta, bottom friction kappa, the deformation radius lambda, and the velocity jump 2U, is provided by numerical simulations at 110 different points in the parameter space beta(* =) beta lambda(2)/U and kappa(*) = kappa lambda/U. There is a special "pivot" value of beta(*), B-*(piv approximate to) 11/16, at which D-tau depends weakly on K-*. But otherwise D-tau has a complicated dependence on both beta(*) and kappa(*), highlighted by the fact that reducing kappa(*) leads to increases (decreases) in D-tau if beta is less than (greater than) beta(piv)(*). Existing heat-flux pararneterizations, based on Kolmogorov cascade theories, predict that D-tau is non-zero and independent of kappa(*) in the limit kappa(*) -> 0. Simulations show indications of this regime provided that kappa(*) <= 0.04 and 0.25 <= beta(*) <= 0.5. All important length scales in this problem, namely the mixing length, the scale of the energy containing eddies, the Rhines scale, and the spacing of the zonal jets, converge to a common value as bottom friction is reduced. The mixing length and jet spacing do not decouple in the parameter regime considered here, as predicted by cascade theories. The convergence of these length scales is due to the formation of jet-scale eddies that align along the eastward jets. The baroclinic component of these eddies helps force the zonal mean flow, which occurs through nonzero Reynolds stress correlations in the upper layer, as opposed to the barotropic mode. This behavior suggests that the dynamics of the inverse barotropic cascade are insufficient to fully describe baroclinic turbulence.

Balmforth, NJ, Bush JWM, Vener D, Young WR.  2007.  Dissipative descent: rocking and rolling down an incline. Journal of Fluid Mechanics. 590:295-318.   10.1017/s0022112007008051   AbstractWebsite

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.

Birch, DA, Tsang YK, Young WR.  2007.  Bounding biomass in the Fisher equation. Physical Review E. 75   10.1103/PhysRevE.75.066304   AbstractWebsite

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 constrained using variational arguments and delimited by simulations. The statistical steady state reached when the system is in the survival region of parameter space is characterized by integral constraints and upper and lower bounds on the biomass and productivity that follow from variational arguments and direct inequalities. In the limit of zero-decorrelation time the velocity field is shown to act as Fickian diffusion with an eddy diffusivity much larger than the molecular diffusivity: this allows a one-dimensional model to predict the biomass, productivity, and extinction transitions. All results are illustrated with a simple growth and stirring model.

Cessi, P, Young WR, Polton JA.  2006.  Control of large-scale heat transport by small-scale mixing. Journal of Physical Oceanography. 36:1877-1894.   10.1175/jpo2947.1   AbstractWebsite

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 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.