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  • 2015-2019  (6)
  • 2010-2014  (5)
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  • 1
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    On the energy of elliptical vortices (2010)
    American Institute of Physics
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    Publication Date: 2010-08-01
    Print ISSN: 1070-6631
    Electronic ISSN: 1089-7666
    Topics: Physics
    Published by American Institute of Physics
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  • 2
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    Available potential vorticity and wave-averaged quasi-geostrophic flow (2015)
    Cambridge University Press
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    Publication Date: 2015-11-23
    Description: 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 Bühler & McIntyre (J. Fluid Mech., vol. 354, 1998, pp. 609-646) to non-uniform stratification with buoyancy frequency and therefore non-uniform background potential vorticity , 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. © 2015 Cambridge University Press.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 3
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    A three-component model for the coupled evolution of near-inertial waves, quasi-geostrophic flow and the near-inertial second harmonic (2016)
    Cambridge University Press
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    Publication Date: 2016-08-10
    Description: 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, where 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-frequency that propagate rapidly to depth and transfer energy back to the NIW field at very small vertical scales. The upshot of near-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. © 2016 Cambridge University Press.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 4
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    Stratified tidal flow over a tall ridge above and below the turning latitude (2016)
    Cambridge University Press
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    Publication Date: 2016-03-29
    Description: 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 and near-ridge isopycnal displacement become large. Where f=0, nonlinear processes lead to the formation of along-ridge jets, which become faster at high latitudes. Dissipation and mixing is larger, and peaks later in the tidal cycle when the tide is subinertial compared with when the tide is superinertial. Mixing occurs mainly on the flanks of the topography in both cases, though a superinertial tide may additionally generate mixing above topography arising from convective breaking of radiating waves. © 2016 Cambridge University Press.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 5
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    An asymptotic model for the propagation of oceanic internal tides through quasi-geostrophic flow (2017)
    Cambridge University Press
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    Publication Date: 2017-09-12
    Description: We derive a time-averaged 'hydrostatic wave equation' from the hydrostatic Boussinesq equations that describes the propagation of inertia-gravity internal waves through quasi-geostrophic flow. The derivation uses a multiple-scale asymptotic method to isolate wave field evolution over intervals much longer than a wave period, assumes the wave field has a well-defined non-inertial frequency such as that of the mid-latitude semi-diurnal lunar tide, assumes that the wave field and quasi-geostrophic flow have comparable spatial scales and neglects nonlinear wave-wave dynamics. As a result the hydrostatic wave equation is a reduced model applicable to the propagation of large-scale internal tides through the inhomogeneous and moving ocean. A numerical comparison with the linearized and hydrostatic Boussinesq equations demonstrates the validity of the hydrostatic wave equation model and illustrates how the model fails when the quasi-geostrophic flow is too strong and the wave frequency is too close to inertial. The hydrostatic wave equation provides a first step toward a coupled model for energy transfer between oceanic internal tides and quasi-geostrophic eddies and currents. © 2017 Cambridge University Press.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 6
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    Stressed horizontal convection (2012)
    Cambridge University Press
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    Publication Date: 2012-01-05
    Description: Abstract 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. © 2012 Cambridge University Press.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 7
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    Generation of surface waves by shear-flow instability (2013)
    Cambridge University Press
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    Publication Date: 2013-12-18
    Description: 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.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 8
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    Radiation of internal waves from groups of surface gravity waves (2017)
    Cambridge University Press
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    Publication Date: 2017-09-15
    Description: Groups of surface gravity waves induce horizontally varying Stokes drift that drives convergence of water ahead of the group and divergence behind. The mass flux divergence associated with spatially variable Stokes drift pumps water downwards in front of the group and upwards in the rear. This 'Stokes pumping' creates a deep Eulerian return flow that sets the isopycnals below the wave group in motion and generates a trailing wake of internal gravity waves. We compute the energy flux from surface to internal waves by finding solutions of the wave-averaged Boussinesq equations in two and three dimensions forced by Stokes pumping at the surface. The two-dimensional (2-D) case is distinct from the 3-D case in that the stratification must be very strong, or the surface waves very slow for any internal wave (IW) radiation at all. On the other hand, in three dimensions, IW radiation always occurs, but with a larger energy flux as the stratification and surface wave (SW) amplitude increase or as the SW period is shorter. Specifically, the energy flux from SWs to IWs varies as the fourth power of the SW amplitude and of the buoyancy frequency, and is inversely proportional to the fifth power of the SW period. Using parameters typical of short period swell (e.g. 8Â s SW period with 1Â m amplitude) we find that the energy flux is small compared to both the total energy in a typical SW group and compared to the total IW energy. Therefore this coupling between SWs and IWs is not a significant sink of energy for the SWs nor a source for IWs. In an extreme case (e.g. 4 m amplitude 20s period SWs) this coupling is a significant source of energy for IWs with frequency close to the buoyancy frequency. © 2017 Cambridge University Press.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 9
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    Reynolds Stress and Eddy Diffusivity of β-Plane Shear Flows (2014)
    American Meteorological Society
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    Publication Date: 2014-05-30
    Description: The Reynolds stress induced by anisotropically forcing an unbounded Couette flow, with uniform shear γ, on a β 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 μ−1. The stochastic forcing is white noise in time and its spatial anisotropy is controlled by a parameter α that characterizes whether eddies are elongated along the zonal direction (α 〈 0), are elongated along the meridional direction (α 〉 0), or are isotropic (α = 0). The Reynolds stress varies linearly with α and nonlinearly and nonmonotonically with γ, but the Reynolds stress is independent of β. For positive values of α, the Reynolds stress displays an “antifrictional” effect (energy is transferred from the eddies to the mean flow); for negative values of α, it displays a frictional effect. When γ/μ ≪ 1, these transfers can be identified as negative and positive eddy viscosities, respectively. With γ = β = 0, the meridional tracer eddy diffusivity is , where υ′ is the meridional eddy velocity. In general, nonzero β and γ suppress the eddy diffusivity below . When the shear is strong, the suppression due to γ varies as γ−1 while the suppression due to β varies between β−1 and β−2 depending on whether the shear is strong or weak, respectively.
    Print ISSN: 0022-4928
    Electronic ISSN: 1520-0469
    Topics: Geography , Geosciences , Physics
    Published by American Meteorological Society
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  • 10
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    Zonostrophic Instability (2012)
    American Meteorological Society
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    Publication Date: 2012-05-01
    Description: 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. 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.
    Print ISSN: 0022-4928
    Electronic ISSN: 1520-0469
    Topics: Geography , Geosciences , Physics
    Published by American Meteorological Society
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