Topical Collection "Geophysical Fluid Dynamics"

Editor

Collection Editor
Prof. Dr. Pavel S. Berloff

Department of Mathematics, Imperial College London, London, UK
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Phone: +44-(0)20-7594-9662
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Interests: geophysical fluid dynamics; ocean circulation and modelling; nonlinear processes; geophysical turbulence and waves

Topical Collection Information

Dear Colleagues,

Geophysical Fluid Dynamics (GFD) is a relatively young, but rapidly growing, branch of fluid mechanics that deals with a great variety of complex multiscale flow patterns and distributions of material properties arising in planetary atmospheres and oceans. These flow patterns are typically controlled by planetary rotation, various boundary conditions, and ubiquitous fluid density gradients. They interact with each other and combine on large scales to establish the climate. GFD employs mathematical analysis and computational modeling to deal with fundamental aspects, analyses and, ultimately, interpretations of the observed phenomena. To a large degree, the observed complexity of geophysical motions is due to the nonlinearity of the fluid dynamics, which connects GFD research with other branches of fluid mechanics. The Special Issue, “Geophysical Fluid Dynamics”, of the journal welcomes your new research contributions to the field.

Prof. Dr. Pavel S. Berloff
Collection Editor

Manuscript Submission Information

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Keywords

  • nonlinear dynamics
  • general circulation of atmospheres and oceans
  • geophysical turbulence, vortices and waves
  • parameterizations of small-scale processes
  • material transport and mixing
  • hydrodynamic instabilities
  • buoyancy driven processes
  • boundary layer processes

Published Papers (31 papers)

2018

Jump to: 2017, 2016

Open AccessArticle Optimal Perturbations of an Oceanic Vortex Lens
Received: 27 June 2018 / Revised: 8 August 2018 / Accepted: 15 August 2018 / Published: 31 August 2018
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Abstract
The stability properties of a vortex lens are studied in the quasi geostrophic (QG) framework using the generalized stability theory. Optimal perturbations are obtained using a tangent linear QG model and its adjoint. Their fine-scale spatial structures are studied in details. Growth rates
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The stability properties of a vortex lens are studied in the quasi geostrophic (QG) framework using the generalized stability theory. Optimal perturbations are obtained using a tangent linear QG model and its adjoint. Their fine-scale spatial structures are studied in details. Growth rates of optimal perturbations are shown to be extremely sensitive to the time interval of optimization: The most unstable perturbations are found for time intervals of about 3 days, while the growth rates continuously decrease towards the most unstable normal mode, which is reached after about 170 days. The horizontal structure of the optimal perturbations consists of an intense counter-shear spiralling. It is also extremely sensitive to time interval: for short time intervals, the optimal perturbations are made of a broad spectrum of high azimuthal wave numbers. As the time interval increases, only low azimuthal wave numbers are found. The vertical structures of optimal perturbations exhibit strong layering associated with high vertical wave numbers whatever the time interval. However, the latter parameter plays an important role in the width of the vertical spectrum of the perturbation: short time interval perturbations have a narrow vertical spectrum while long time interval perturbations show a broad range of vertical scales. Optimal perturbations were set as initial perturbations of the vortex lens in a fully non linear QG model. It appears that for short time intervals, the perturbations decay after an initial transient growth, while for longer time intervals, the optimal perturbation keeps on growing, quickly leading to a non-linear regime or exciting lower azimuthal modes, consistent with normal mode instability. Very long time intervals simply behave like the most unstable normal mode. The possible impact of optimal perturbations on layering is also discussed. Full article
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Open AccessArticle Submesoscale Turbulence over a Topographic Slope
Received: 27 December 2017 / Revised: 29 April 2018 / Accepted: 2 May 2018 / Published: 7 May 2018
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Abstract
Regions of the ocean near continental slopes are linked to significant vertical velocities caused by advection over a sloping bottom, frictional processes and diffusion. Oceanic motions at submesoscales are also characterized by enhanced vertical velocities, as compared to mesoscale motions, due to greater
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Regions of the ocean near continental slopes are linked to significant vertical velocities caused by advection over a sloping bottom, frictional processes and diffusion. Oceanic motions at submesoscales are also characterized by enhanced vertical velocities, as compared to mesoscale motions, due to greater contributions from ageostrophic flows. These enhanced vertical velocities can make an important contribution to turbulent fluxes. Sloping topography may also induce large-scale potential vorticity gradients by modifying the slope of interior isopycnal surfaces. Potential vorticity gradients, in turn, may feed back on mesoscale stirring and the generation of submesoscale features. In this study, we explore the impact of sloping topography on the characteristics of submesoscale motions. We conduct high-resolution (1 km × 1 km) simulations of a wind-driven frontal current over an idealized continental shelf and slope. We explore changes in the magnitude, skewness and spectra of surface vorticity and vertical velocity across different configurations of the topographic slope and wind-forcing orientations. All of these properties are strongly modulated by the background topography. Furthermore, submesoscale characteristics exhibit spatial variability across the continental shelf and slope. We find that changes in the statistical properties of submesoscale motions are linked to mesoscale stirring responding to differences in the interior potential vorticity distributions, which are set by frictional processes at the ocean surface and over the sloping bottom. Improved parameterizations of submesoscale motions over topography may be needed to simulate the spatial variability of these features in coarser-resolution models, and are likely to be important to represent vertical nutrient fluxes in coastal waters. Full article
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Open AccessArticle Kinematics of a Fluid Ellipse in a Linear Flow
Received: 31 December 2017 / Revised: 5 February 2018 / Accepted: 7 February 2018 / Published: 12 February 2018
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Abstract
A four-parameter kinematic model for the position of a fluid parcel in a time-varying ellipse is introduced. For any ellipse advected by an arbitrary linear two-dimensional flow, the rates of change of the ellipse parameters are uniquely determined by the four parameters of
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A four-parameter kinematic model for the position of a fluid parcel in a time-varying ellipse is introduced. For any ellipse advected by an arbitrary linear two-dimensional flow, the rates of change of the ellipse parameters are uniquely determined by the four parameters of the velocity gradient matrix, and vice versa. This result, termed ellipse/flow equivalence, provides a stronger version of the well-known result that a linear velocity field maps an ellipse into another ellipse. Moreover, ellipse/flow equivalence is shown to be a manifestation of Stokes’ theorem. This is done by deriving a matrix-valued extension of the classical Stokes’ theorem that involves a spatial integral over the velocity gradient tensor, thus accounting for the two strain terms in addition to the divergence and vorticity. General expressions for various physical properties of an elliptical ring of fluid are also derived. The ellipse kinetic energy is found to be composed of three portions, associated respectively with the circulation, the rate of change of the moment of inertia, and the variance of parcel angular velocity around the ellipse. A particular innovation is the use of four matrices, termed the I J K L basis, that greatly facilitate the required calculations. Full article
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2017

Jump to: 2018, 2016

Open AccessArticle On the Applicability of Linear, Axisymmetric Dynamics in Intensifying and Mature Tropical Cyclones
Received: 31 May 2017 / Revised: 20 November 2017 / Accepted: 30 November 2017 / Published: 13 December 2017
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Abstract
The applicability of linearized axisymmetric dynamics to the intensification and structure change of tropical cyclones is investigated. The study is motivated by recent work that presented axisymmetric solutions to the linearized, non-hydrostatic, vortex-anelastic equations of motion (the so-called 3DVPAS model). The work called
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The applicability of linearized axisymmetric dynamics to the intensification and structure change of tropical cyclones is investigated. The study is motivated by recent work that presented axisymmetric solutions to the linearized, non-hydrostatic, vortex-anelastic equations of motion (the so-called 3DVPAS model). The work called into question the importance of a recently proposed nonlinear, system-scale boundary-layer spinup mechanism both in intensifying storms and in mature storms undergoing secondary eyewall formation. The issue is examined using a three-dimensional mesoscale simulation of an intensifying tropical cyclone, alongside the linear 3DVPAS model. Solutions to the linear model, for imposed eddy forcing terms derived from the mesoscale simulation, are shown to be valid only for short times ( t < 1 h) in the inner-core region of the vortex. At later times, the neglected nonlinear terms become significant and the linear results invalid. It follows that the linear results cannot be used to describe all aspects of the tropical cyclone dynamics at later times. In particular, they cannot be used (a) to dismiss the importance of the nonlinear boundary-layer spinup mechanism, nor (b) to isolate the separate effects of diabatic heating from those of friction, within the nonlinear boundary layer at least. Such separation depends on the linear superposition principle, which fails whenever nonlinearity is important. Similar caveats apply to the use of another linear model, the traditional Sawyer–Eliassen balance model. Its applicability is limited not only by linearity, but also by its assumption of strictly balanced motion. Both are incompatible with nonlinear spinup. Full article
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Open AccessArticle The Interaction of Two Surface Vortices Near a Topographic Slope in a Stratified Ocean
Received: 21 July 2017 / Revised: 10 October 2017 / Accepted: 17 October 2017 / Published: 25 October 2017
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Abstract
We study the influence of bottom topography on the interaction of two identical vortices in a two-layer, quasi-geostrophic model. The two vortices have piecewise-uniform potential vorticity and are lying in the upper layer of the model. The topography is a smooth bottom slope.
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We study the influence of bottom topography on the interaction of two identical vortices in a two-layer, quasi-geostrophic model. The two vortices have piecewise-uniform potential vorticity and are lying in the upper layer of the model. The topography is a smooth bottom slope. For two cyclones, topography modifies the merger critical distance and the merger efficiency: the topographic wave and vortices can advect the two cyclones along the shelf when they are initially far from it or towards the shelf when they are initially closer to it. They can also advect the two cyclones towards each other and thus favour merger. The cyclones deform, and the potential vorticity field undergoes filamentation. Regimes of partial vortex merger or of vortex splitting are then observed. The interaction of the vorticity poles in the two layers are analysed to explain the evolution of the two upper layer cyclones. For taller topography, two new regimes appear: vortex drift and splitting; and filamentation and asymmetric merger. They are due to the hetonic coupling of lower layer vorticity with the upper layer vortices (a heton is a baroclinic vortex dipole, carrying heat and momentum and propagating horizontally in the fluid), or to the strong shear that the former exerts on the latter. The interaction of two anticyclones shows regimes of co-rotation or merger, but specifically, it leads to the drift of the two vortices away from the slope, via a hetonic coupling with oppositely-signed vorticity in the lower layer. This vorticity originates in the breaking of the topographic wave. The analysis of passive tracer evolution confirms the inshore or offshore drift of the fluid, the formation of tracer fronts along filaments and its stirring in regions of vortex merger. The trajectories of particles indicate how the fluid initially in the vortices is finally partitioned. Full article
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Open AccessArticle The Impact of Topographic Steepness on Tidal Dissipation at Bumpy Topography
Received: 31 July 2017 / Revised: 29 September 2017 / Accepted: 11 October 2017 / Published: 14 October 2017
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Abstract
Breaking internal waves are an important contributor to mixing in the stratified ocean interior. We use two-dimensional, nonhydrostatic numerical simulations to examine the breaking of internal waves generated by tidal flow over sinusoidal bottom topography. We explore the sensitivity of the internal wave
[...] Read more.
Breaking internal waves are an important contributor to mixing in the stratified ocean interior. We use two-dimensional, nonhydrostatic numerical simulations to examine the breaking of internal waves generated by tidal flow over sinusoidal bottom topography. We explore the sensitivity of the internal wave breaking to the topographic steepness and Coriolis frequency, focusing on the vertical structure of kinetic energy dissipation and the ratio of local dissipation to the barotropic-to-baroclinic energy conversion. When the tidal frequency is twice the local Coriolis frequency, wave breaking above the topography is driven by wave–wave interactions which transfer wave energy from the tidal forcing frequency to the inertial frequency. The greater shear associated with the inertial frequency waves leads to enhanced dissipation in a thick layer above the topography. The topographic steepness strongly modulates this dependence of dissipation on Coriolis frequency; for some steep sinusoidal topographies, most wave energy propagates downward into the topographic troughs, eliminating the possibility for significant breaking above the topographic peaks. Current parameterizations of tidal dissipation in use in global ocean models need to be adapted to include the dependence of the local dissipation on both the Coriolis frequency and the topographic steepness. Full article
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Open AccessArticle The Eddy Diffusivity in Barotropic β-Plane Turbulence
Received: 1 September 2017 / Revised: 4 October 2017 / Accepted: 6 October 2017 / Published: 12 October 2017
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Abstract
Geostrophic turbulent eddies play a crucial role in the oceans, mixing properties such as heat, salt, and geochemical tracers. A useful reduced model for geostrophic turbulence is barotropic (2D) turbulence. The focus of this study is on 2D β-plane turbulence with quadratic
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Geostrophic turbulent eddies play a crucial role in the oceans, mixing properties such as heat, salt, and geochemical tracers. A useful reduced model for geostrophic turbulence is barotropic (2D) turbulence. The focus of this study is on 2D β -plane turbulence with quadratic drag, which, although arguably the most realistic barotropic model of ocean turbulence, has remained unexplored thus far. We first review and test classical scaling arguments for the eddy diffusivity in three regimes: the strong friction limit, the weak friction/strong β limit, and a transition regime. We then develop a generalized theory by parameterizing the nonlinear eddy–eddy interactions as a stochastic process, which leads to an analytical solution for the eddy diffusivity spectrum, whose integral yields a “bulk” diffusivity. The theory successfully predicts the smooth transition of diffusivity across the three regimes, and echoes with the recent argument that eddy phase propagation relative to the mean flow suppresses the eddy diffusivity. Moreover, the generalized theory reduces to the classical scaling arguments in both the strong friction and strong β limits, which has not been clear from the previous work on diffusivity suppression by flow-relative phase propagation. Full article
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Open AccessArticle Hyperbolic Covariant Coherent Structures in Two Dimensional Flows
Received: 22 August 2017 / Revised: 20 September 2017 / Accepted: 21 September 2017 / Published: 27 September 2017
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Abstract
A new method to describe hyperbolic patterns in two-dimensional flows is proposed. The method is based on the Covariant Lyapunov Vectors (CLVs), which have the properties of being covariant with the dynamics, and thus, being mapped by the tangent linear operator into another
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A new method to describe hyperbolic patterns in two-dimensional flows is proposed. The method is based on the Covariant Lyapunov Vectors (CLVs), which have the properties of being covariant with the dynamics, and thus, being mapped by the tangent linear operator into another CLVs basis, they are norm independent, invariant under time reversal and cannot be orthonormal. CLVs can thus give more detailed information about the expansion and contraction directions of the flow than the Lyapunov vector bases, which are instead always orthogonal. We suggest a definition of Hyperbolic Covariant Coherent Structures (HCCSs), which can be defined on the scalar field representing the angle between the CLVs. HCCSs can be defined for every time instant and could be useful to understand the long-term behavior of particle tracers. We consider three examples: a simple autonomous Hamiltonian system, as well as the non-autonomous “double gyre” and Bickley jet, to see how well the angle is able to describe particular patterns and barriers. We compare the results from the HCCSs with other coherent patterns defined on finite time by the Finite Time Lyapunov Exponents (FTLEs), to see how the behaviors of these structures change asymptotically. Full article
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Open AccessArticle The Impact of Horizontal Resolution on Energy Transfers in Global Ocean Models
Received: 15 July 2017 / Revised: 15 August 2017 / Accepted: 24 August 2017 / Published: 28 August 2017
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Abstract
The ocean is a turbulent fluid with processes acting on a variety of spatio-temporal scales. The estimates of energy fluxes between length scales allows us to understand how the mean flow is maintained as well as how mesoscale eddies are formed and dissipated.
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The ocean is a turbulent fluid with processes acting on a variety of spatio-temporal scales. The estimates of energy fluxes between length scales allows us to understand how the mean flow is maintained as well as how mesoscale eddies are formed and dissipated. Here, we quantify the kinetic energy budget in a suite of realistic global ocean models, with varying horizontal resolution and horizontal viscosity. We show that eddy-permitting ocean models have weaker kinetic energy cascades than eddy-resolving models due to discrepancies in the effect of wind forcing, horizontal viscosity, potential to kinetic energy conversion, and nonlinear interactions on the kinetic energy (KE) budget. However, the change in eddy kinetic energy between the eddy-permitting and the eddy-resolving model is not enough to noticeably change the scale where the inverse cascade arrests or the Rhines scale. In addition, we show that the mechanism by which baroclinic flows organise into barotropic flows is weaker at lower resolution, resulting in a more baroclinic flow. Hence, the horizontal resolution impacts the vertical structure of the simulated flow. Our results suggest that the effect of mesoscale eddies can be parameterised by enhancing the potential to kinetic energy conversion, i.e., the horizontal pressure gradients, or enhancing the inverse cascade of kinetic energy. Full article
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Open AccessArticle Stable Vortices in a Continuously Stratified Ocean with Thin Active Layer
Received: 27 June 2017 / Revised: 9 July 2017 / Accepted: 20 July 2017 / Published: 25 July 2017
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Abstract
This paper presents a model which yields examples of stable vortices in a continuously stratified rotating fluid, thus providing a possible explanation of the observed longevity of oceanic eddies. The model is based on two assumptions. Firstly, the ocean comprises a thin upper
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This paper presents a model which yields examples of stable vortices in a continuously stratified rotating fluid, thus providing a possible explanation of the observed longevity of oceanic eddies. The model is based on two assumptions. Firstly, the ocean comprises a thin upper (active) layer and a thick lower (passive) one, with large and small vertical gradients of density, respectively. Secondly, the Rossby number is small, justifying the use of the geostrophic and quasi-geostrophic approximations for the active and passive layers (the two are treated differently because the vortex-induced displacement of the isopycnal surfaces is comparable to the depth of the active layer, but is much smaller than that of the passive one). Using the asymptotic equations derived on the basis of the above assumptions, we prove a stability criterion and thus identify a class of stable vortex profiles. This class is much wider than the one following from the standard requirement that the potential vorticity be monotonic in the whole bulk of the fluid. Full article
Open AccessArticle Regimes of Axisymmetric Flow and Scaling Laws in a Rotating Annulus with Local Convective Forcing
Received: 8 June 2017 / Revised: 14 July 2017 / Accepted: 16 July 2017 / Published: 19 July 2017
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Abstract
We present a numerical study of axisymmetric flow in a rotating annulus in which local thermal forcing, via a heated annular ring on the outside of the base and a cooled circular disk in the centre of the top surface, drives convection. This
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We present a numerical study of axisymmetric flow in a rotating annulus in which local thermal forcing, via a heated annular ring on the outside of the base and a cooled circular disk in the centre of the top surface, drives convection. This new configuration is a variant of the classical thermally-driven annulus, where uniform heating and cooling are applied through the outer and inner sidewalls respectively. The annulus provides an analogue to a planetary circulation and the new configuration, with its more relaxed vertical thermal boundary conditions, is expected to better emulate vigorous convection in the tropics and polar regions as well as baroclinic instability in the mid-latitude baroclinic zone. Using the Met Office/Oxford Rotating Annulus Laboratory (MORALS) code, we have investigated a series of equilibrated, two dimensional axisymmetric flows across a large region of parameter space. These are characterized in terms of their velocity and temperature fields. When rotation is applied several distinct flow regimes may be identified for different rotation rates and strengths of differential heating. These regimes are defined as a function of the ratio of the horizontal Ekman layer thickness to the non-rotating thermal boundary layer thickness and are found to be similar to those identified in previous annulus experiments. Convection without rotation is also considered and the scaling of the heat transport with Rayleigh number is calculated. This is then compared with existing work on the classical annulus as well as horizontal and Rayleigh-Bénard convection. As with previous studies on both rotating and non-rotating convection the system’s behaviour is found to be aspect ratio dependent. This dependence is seen in the scaling of the non-rotating Nusselt number and in transitions between regimes in the rotating case although further investigation is required to fully explain these observations. Full article
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Open AccessArticle Interaction between a Quasi-Geostrophic Buoyancy Filament and a Heton
Received: 15 June 2017 / Revised: 27 June 2017 / Accepted: 28 June 2017 / Published: 3 July 2017
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Abstract
We investigate the interaction between a heton and a current generated by a filament of buoyancy anomaly at the surface. Hetons are baroclinic dipoles consisting of a pair of vortices of opposite signs lying at different depths. Such structures have self-induced motion whenever
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We investigate the interaction between a heton and a current generated by a filament of buoyancy anomaly at the surface. Hetons are baroclinic dipoles consisting of a pair of vortices of opposite signs lying at different depths. Such structures have self-induced motion whenever the pair of vortices are offset horizontally. A surface buoyancy filament generates a shear flow since the density perturbation locally modifies the pressure field. The vertical shear induced by the filament offsets the vortices of the heton if vertically aligned initially. Moreover, if the vortex nearer the surface is in adverse horizontal shear with the buoyancy filament, the heton tends to move towards the filament. Conversely, if the upper vortex is in cooperative horizontal shear with the buoyancy filament, the heton moves away from it. The filament is also naturally unstable and eventually breaks into a series of billows as it is perturbed by the heton. Moderate to large intensity surface buoyancy distributions separate the vortices of the heton, limiting its advection as a baroclinic dipole. Instead, the vortices of the heton start to interact strongly with surface billows. Additionally, the vortices of the heton can be partially destroyed by the filament if the shear it induces is sufficiently large. Full article
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Open AccessArticle An Oceanic Ultra-Violet Catastrophe, Wave-Particle Duality and a Strongly Nonlinear Concept for Geophysical Turbulence
Received: 12 March 2017 / Revised: 14 June 2017 / Accepted: 16 June 2017 / Published: 29 June 2017
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Abstract
There is no theoretical underpinning that successfully explains how turbulent mixing is fed by wave breaking associated with nonlinear wave-wave interactions in the background oceanic internal wavefield. We address this conundrum using one-dimensional ray tracing simulations to investigate interactions between high frequency internal
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There is no theoretical underpinning that successfully explains how turbulent mixing is fed by wave breaking associated with nonlinear wave-wave interactions in the background oceanic internal wavefield. We address this conundrum using one-dimensional ray tracing simulations to investigate interactions between high frequency internal waves and inertial oscillations in the extreme scale separated limit known as “Induced Diffusion”. Here, estimates of phase locking are used to define a resonant process (a resonant well) and a non-resonant process that results in stochastic jumps. The small amplitude limit consists of jumps that are small compared to the scale of the resonant well. The ray tracing simulations are used to estimate the first and second moments of a wave packet’s vertical wavenumber as it evolves from an initial condition. These moments are compared with predictions obtained from the diffusive approximation to a self-consistent kinetic equation derived in the ‘Direct Interaction Approximation’. Results indicate that the first and second moments of the two systems evolve in a nearly identical manner when the inertial field has amplitudes an order of magnitude smaller than oceanic values. At realistic (oceanic) amplitudes, though, the second moment estimated from the ray tracing simulations is inhibited. The transition is explained by the stochastic jumps obtaining the characteristic size of the resonant well. We interpret this transition as an adiabatic ‘saturation’ process which changes the nominal background wavefield from supporting no mixing to the point where that background wavefield defines the normalization for oceanic mixing models. Full article
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Open AccessArticle Construction of Hamiltonian and Nambu Forms for the Shallow Water Equations
Received: 10 March 2017 / Revised: 28 April 2017 / Accepted: 11 May 2017 / Published: 13 May 2017
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Abstract
A systematic method to derive the Hamiltonian and Nambu form for the shallow water equations using the conservation for energy and potential enstrophy is presented. Different mechanisms, such as vortical flows and emission of gravity waves, emerge from different conservation laws for total
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A systematic method to derive the Hamiltonian and Nambu form for the shallow water equations using the conservation for energy and potential enstrophy is presented. Different mechanisms, such as vortical flows and emission of gravity waves, emerge from different conservation laws for total energy and potential enstrophy. The equations are constructed using exterior differential forms and self-adjoint operators, and result in the sum of two Nambu brackets—one for the vortical flow and one for the wave-mean flow interaction—and a Poisson bracket representing the interaction between divergence and geostrophic imbalance. The advantage of this approach is that the Hamiltonian and Nambu forms can here be written in a coordinate-independent form. Full article
Open AccessArticle Reply to “Comment on Tailleux, R. Neutrality Versus Materiality: A Thermodynamic Theory of Neutral Surfaces. Fluids 2016, 1, 32.”
Received: 13 April 2017 / Revised: 19 April 2017 / Accepted: 20 April 2017 / Published: 26 April 2017
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Abstract
McDougall, Groeskamp and Griffies (MGG) strongly criticise all aspects of Tailleux (2016) that challenge the current conventional wisdom about the use of neutral density concepts for studying and parameterising lateral ocean stirring and mixing. However, their claim that most of Tailleux (2016)’s results
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McDougall, Groeskamp and Griffies (MGG) strongly criticise all aspects of Tailleux (2016) that challenge the current conventional wisdom about the use of neutral density concepts for studying and parameterising lateral ocean stirring and mixing. However, their claim that most of Tailleux (2016)’s results or conclusions are incorrect is easily shown to originate: (1) from mistakingly confusing Tailleux’s Eulerian arguments for Lagrangian ones; (2) from their irrational belief that only one particular kind of quasi-material surface is somehow endorsed by Nature and hence relevant to the description of stirring and mixing—namely the locally-defined neutral tangent planes—stating at one point: “why should the ocean care about human constructed density variables”? MGG appear to overlook the simple fact that solutions of the Navier–Stokes equations—which synthesise our ideas about how Nature works—never require the introduction of any form of quasi-material or quasi-neutral density variable. This implies that the empirical isopycnal/isentropic stirring property is necessarily an emergent property of the Navier–Stokes equations, and hence that all forms of isopycnal surfaces—both neutral and not—are necessarily all human constructs. To establish the relevance of any particular construct to the actual ocean, an explicit model of stirring is needed to elucidate the nature of the dynamical/energetics constraints on lateral stirring. Even in the simplest model of stirring, neutral stirring represents only one possible mode out of a continuum of stirring modes responsible for lateral stirring in the ocean, without any evidence that it should dominate over the other ones. To help clarify the issues involved, it is proposed to regard the rigorous study of ocean stirring and mixing as relying on at least five distinct stages, from defining a model of stirring to constructing physically-based mixing parameterisations in numerical ocean models. Full article
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Open AccessComment Comment on Tailleux, R. Neutrality versus Materiality: A Thermodynamic Theory of Neutral Surfaces. Fluids 2016, 1, 32
Received: 27 March 2017 / Revised: 27 March 2017 / Accepted: 20 April 2017 / Published: 26 April 2017
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Abstract
Tailleux has written about the concept of epineutral mixing and has attempted to justify it from an energetic viewpoint. However, Tailleux’s approach is incorrect because it ignores the unsteady nature of the density field during baroclinic motions, which in turn leads to incorrect
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Tailleux has written about the concept of epineutral mixing and has attempted to justify it from an energetic viewpoint. However, Tailleux’s approach is incorrect because it ignores the unsteady nature of the density field during baroclinic motions, which in turn leads to incorrect conclusions. Tailleux also asserts that “adiabatic and isohaline parcel exchanges can only be meaningfully defined on material surfaces” that are functions of only Absolute Salinity and Conservative Temperature and are not separately a function of pressure. We disagree with this assertion because there is no physical reason why the ocean should care about a globally-defined function of Absolute Salinity and Conservative Temperature that we construct. Rather, in order to understand and justify the concept of epineutral mixing, we consider the known physical processes that occur at the in situ pressure of the mixing. The Tailleux paper begins with two incorrect equations that ignore the transience of the ocean. These errors echo throughout Tailleux, leading to sixteen conclusions, most of which we show are incorrect. (Comment on Tailleux, R. Neutrality Versus Materiality: A Thermodynamic Theory of Neutral Surfaces. Fluids 2016, 1, 32, doi:10.3390/fluids1040032.) Full article
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Open AccessReview Surface Quasi-Geostrophy
Received: 29 November 2016 / Revised: 1 February 2017 / Accepted: 3 February 2017 / Published: 16 February 2017
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Abstract
Oceanic and atmospheric dynamics are often interpreted through potential vorticity, as this quantity is conserved along the geostrophic flow. However, in addition to potential vorticity, surface buoyancy is a conserved quantity, and this also affects the dynamics. Buoyancy at the ocean surface or
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Oceanic and atmospheric dynamics are often interpreted through potential vorticity, as this quantity is conserved along the geostrophic flow. However, in addition to potential vorticity, surface buoyancy is a conserved quantity, and this also affects the dynamics. Buoyancy at the ocean surface or at the atmospheric tropopause plays the same role of an active tracer as potential vorticity does since the velocity field can be deduced from these quantities. The surface quasi-geostrophic model has been proposed to explain the dynamics associated with surface buoyancy conservation and seems appealing for both the ocean and the atmosphere. In this review, we present its main characteristics in terms of coherent structures, instabilities and turbulent cascades. Furthermore, this model is mathematically studied for the possible formation of singularities, as it presents some analogies with three-dimensional Euler equations. Finally, we discuss its relevance for the ocean and the atmosphere. Full article
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Open AccessArticle A Reduced Model for Salt-Finger Convection in the Small Diffusivity Ratio Limit
Received: 27 December 2016 / Accepted: 23 January 2017 / Published: 30 January 2017
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Abstract
A simple model of nonlinear salt-finger convection in two dimensions is derived and studied. The model is valid in the limit of a small solute to heat diffusivity ratio and a large density ratio, which is relevant to both oceanographic and astrophysical applications.
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A simple model of nonlinear salt-finger convection in two dimensions is derived and studied. The model is valid in the limit of a small solute to heat diffusivity ratio and a large density ratio, which is relevant to both oceanographic and astrophysical applications. Two limits distinguished by the magnitude of the Schmidt number are found. For order one Schmidt numbers, appropriate for astrophysical applications, a modified Rayleigh–Bénard system with large-scale damping due to a stabilizing temperature is obtained. For large Schmidt numbers, appropriate for the oceanic setting, the model combines a prognostic equation for the solute field and a diagnostic equation for inertia-free momentum dynamics. Two distinct saturation regimes are identified for the second model: the weakly driven regime is characterized by a large-scale flow associated with a balance between advection and linear instability, while the strongly-driven regime produces multiscale structures, resulting in a balance between energy input through linear instability and energy transfer between scales. For both regimes, we analytically predict and numerically confirm the dependence of the kinetic energy and salinity fluxes on the ratio between solutal and thermal Rayleigh numbers. The spectra and probability density functions are also computed. Full article
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2016

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Open AccessArticle The Effects of Mesoscale Ocean–Atmosphere Coupling on the Quasigeostrophic Double Gyre
Received: 30 August 2016 / Revised: 3 October 2016 / Accepted: 8 October 2016 / Published: 21 October 2016
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Abstract
We investigate the effects of sea surface temperature (SST)-dependent wind stress on the wind-driven quasigeostrophic (QG) double gyre. The main effects are to reduce the strength of the circulation and to shift the inter-gyre jet to the south. The SST front across the
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We investigate the effects of sea surface temperature (SST)-dependent wind stress on the wind-driven quasigeostrophic (QG) double gyre. The main effects are to reduce the strength of the circulation and to shift the inter-gyre jet to the south. The SST front across the inter-gyre jet induces a zonal wind stress anomaly over the jet that accelerates the southern flank of the jet and decelerates the northern flank. This local wind stress anomaly causes the jet to shift southwards. Shifting the jet south, away from the peak wind stress, reduces the net power input to the ocean circulation. Allowing the wind stress to depend on the difference between the atmospheric and oceanic velocity also reduces the net wind power input, and has a larger impact than SST dependence. When wind stress depends only on SST, the impact on the circulation is stronger than when wind stress depends on both SST and ocean surface velocity. Ocean surface velocity dependence leads to direct extraction of mesoscale energy by the winds. In contrast, SST dependence leads to injection (extraction) of mesoscale energy in the subtropical (subpolar) gyres, with almost complete cancellation because of the symmetric wind field. Full article
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Open AccessArticle Estimating Eulerian Energy Spectra from Drifters
Received: 22 July 2016 / Revised: 4 October 2016 / Accepted: 8 October 2016 / Published: 15 October 2016
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Abstract
The relations between the kinetic energy spectrum and the second-order longitudinal structure function for 2D non-divergent flow are derived, and several examples are considered. The transform from spectrum to structure function is illustrated using idealized power-law spectra of turbulent inertial ranges. The results
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The relations between the kinetic energy spectrum and the second-order longitudinal structure function for 2D non-divergent flow are derived, and several examples are considered. The transform from spectrum to structure function is illustrated using idealized power-law spectra of turbulent inertial ranges. The results illustrate how the structure function integrates contributions across wavenumber, which can obscure the dependencies when the inertial ranges are of finite extent. The transform is also applied to the kinetic energy spectrum of Nastrom and Gage (1985), derived from aircraft data in the upper troposphere; the resulting structure function agrees well with that of Lindborg (1999), calculated with the same data. The transform from structure function to spectrum is then tested with data from 2D turbulence simulations. When applied to the (Eulerian) structure function obtained from the transform of the spectrum, the result closely resembles the original spectrum, except at the largest wavenumbers. The deviation at large wavenumbers occurs because the transform involves a filter function which magnifies contributions from large separations. The results are noticeably worse when applied to the structure function obtained from pairs of particles in the flow, as this is usually noisy at large separations. Fitting the structure function to a polynomial improves the resulting spectrum, but not sufficiently to distinguish the correct inertial range dependencies. Furthermore, the transform of steep (non-local) spectra is largely unsuccessful. Thus, it appears that with Lagrangian data, it is probably preferable to focus on structure functions, despite their shortcomings. Full article
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Open AccessArticle Neutrality Versus Materiality: A Thermodynamic Theory of Neutral Surfaces
Received: 16 February 2016 / Revised: 12 September 2016 / Accepted: 14 September 2016 / Published: 28 September 2016
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Abstract
In this paper, a theory for constructing quasi-neutral density variables γ directly in thermodynamic space is formulated, which is based on minimising the absolute value of a purely thermodynamic quantity Jn. Physically, Jn has a dual dynamic/thermodynamic interpretation as the
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In this paper, a theory for constructing quasi-neutral density variables γ directly in thermodynamic space is formulated, which is based on minimising the absolute value of a purely thermodynamic quantity J n . Physically, J n has a dual dynamic/thermodynamic interpretation as the quantity controlling the energy cost of adiabatic and isohaline parcel exchanges on material surfaces, as well as the dependence of in-situ density on spiciness, in a description of water masses based on γ, spiciness and pressure. Mathematically, minimising | J n | in thermodynamic space is showed to be equivalent to maximising neutrality in physical space. The physics of epineutral dispersion is also reviewed and discussed. It is argued, in particular, that epineutral dispersion is best understood as the aggregate effect of many individual non-neutral stirring events (being understood here as adiabatic and isohaline events with non-zero buoyancy), so that it is only the net displacement aggregated over many events that is approximately neutral. This new view resolves an apparent paradox between the focus in neutral density theory on zero-buoyancy motions and the overwhelming evidence that lateral dispersion in the ocean is primarily caused by non-zero buoyancy processes such as tides, residual currents and sheared internal waves. The efficiency by which a physical process contributes to lateral dispersion can be characterised by its energy signature, with those processes releasing available potential energy (negative energy cost) being more efficient than purely neutral processes with zero energy cost. The latter mechanism occurs in the wedge of instability, and its source of energy is the coupling between baroclinicity, thermobaricity, and density compensated temperature/salinity anomalies. Such a mechanism, which can only exist in a salty ocean, is speculated to be important for dissipating spiciness anomalies and neutral helicity. The paper also discusses potential conceptual difficulties with the use of neutral rotated diffusion tensors in numerical ocean models, as well as with the construction of neutral density variables in physical space. It also emphasises the irreducible character of thermobaric forces in the ocean. These are argued to be the cause for adiabatic thermobaric dianeutral dispersion, and to forbid the existence of density surfaces along which fluid parcels can be exchanged without experiencing buoyancy forces, in contrast to what is assumed in the theory of neutral surfaces. Full article
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Open AccessArticle Meridional and Zonal Wavenumber Dependence in Tracer Flux in Rossby Waves
Received: 22 July 2016 / Revised: 26 August 2016 / Accepted: 29 August 2016 / Published: 6 September 2016
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Abstract
Eddy-driven jets are of importance in the ocean and atmosphere, and to a first approximation are governed by Rossby wave dynamics. This study addresses the time-dependent flux of fluid and a passive tracer between such a jet and an adjacent eddy, with specific
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Eddy-driven jets are of importance in the ocean and atmosphere, and to a first approximation are governed by Rossby wave dynamics. This study addresses the time-dependent flux of fluid and a passive tracer between such a jet and an adjacent eddy, with specific regard to determining zonal and meridional wavenumber dependence. The flux amplitude in wavenumber space is obtained, which is easily computable for a given jet geometry, speed and latitude, and which provides instant information on the wavenumbers of the Rossby waves which maximize the flux. This new tool enables the quick determination of which modes are most influential in imparting fluid exchange, which in the long term will homogenize the tracer concentration between the eddy and the jet. The results are validated by computing backward- and forward-time finite-time Lyapunov exponent fields, and also stable and unstable manifolds; the intermingling of these entities defines the region of chaotic transport between the eddy and the jet. The relationship of all of these to the time-varying transport flux between the eddy and the jet is carefully elucidated. The flux quantification presented here works for general time-dependence, whether or not lobes (intersection regions between stable and unstable manifolds) are present in the mixing region, and is therefore also easily computable for wave packets consisting of infinitely many wavenumbers. Full article
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Open AccessArticle Eddy Backscatter and Counter-Rotating Gyre Anomalies of Midlatitude Ocean Dynamics
Received: 14 May 2016 / Revised: 18 August 2016 / Accepted: 25 August 2016 / Published: 2 September 2016
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Abstract
This work concerns how two competing mechanisms—eddy backscatter and counter-rotating gyre anomalies—influence the midlatitude ocean dynamics, as described by the eddy-resolving quasi-geostrophic (QG) model of wind-driven gyres. We analyzed dynamical balances and effects of different eddy forcing components, as well as their dependencies
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This work concerns how two competing mechanisms—eddy backscatter and counter-rotating gyre anomalies—influence the midlatitude ocean dynamics, as described by the eddy-resolving quasi-geostrophic (QG) model of wind-driven gyres. We analyzed dynamical balances and effects of different eddy forcing components, as well as their dependencies on increasing vertical resolution and decreasing eddy viscosity and found that the eastward jet and its adjacent recirculation zones are maintained mostly by the eddy forcing via the eddy backscatter mechanism, whereas the time-mean eddy-forcing component plays not only direct jet-supporting but also indirect jet-inhibiting role. The latter is achieved by inducing zonally elongated anticyclonic/cyclonic Counter-rotating Gyre Anomaly (CGA) in the subpolar/subtropical gyre. The indirect effect of CGAs on the eastward jet is found to be moderate relative to the dominant eddy backscatter mechanism. We also found that the higher the vertical baroclinic mode, the weaker its backscatter role and the stronger its CGA-driving role. Although the barotropic and first baroclinic modes are the most efficient ones in maintaining the backscatter, the higher, up to the fifth baroclinic modes also have significant but reverse impact that reduces the backscatter. Full article
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Open AccessArticle Scalar Flux Kinematics
Received: 16 May 2016 / Revised: 15 July 2016 / Accepted: 18 July 2016 / Published: 25 August 2016
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Abstract
The first portion of this paper contains an overview of recent progress in the development of dynamical-systems-based methods for the computation of Lagrangian transport processes in physical oceanography. We review the considerable progress made in the computation and interpretation of key material features
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The first portion of this paper contains an overview of recent progress in the development of dynamical-systems-based methods for the computation of Lagrangian transport processes in physical oceanography. We review the considerable progress made in the computation and interpretation of key material features such as eddy boundaries, and stable and unstable manifolds (or their finite-time approximations). Modern challenges to the Lagrangian approach include the need to deal with the complexity of the ocean submesoscale and the difficulty in computing fluxes of properties other than volume. We suggest a new approach that reduces complexity through time filtering and that directly addresses non-material, residual scalar fluxes. The approach is “semi-Lagrangian” insofar as it contemplates trajectories of a velocity field related to a residual scalar flux, usually not the fluid velocity. Two examples are explored, the first coming from a canonical example of viscous adjustment along a flat plate and the second from a numerical simulation of a turbulent Antarctic Circumpolar Current in an idealized geometry. Each example concentrates on the transport of dynamically relevant scalars, and the second illustrates how substantial material exchange across a baroclinically unstable jet coexists with zero residual buoyancy flux. Full article
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Open AccessArticle Diapycnal Velocity in the Double-Diffusive Thermocline
Received: 22 May 2016 / Revised: 30 July 2016 / Accepted: 12 August 2016 / Published: 25 August 2016
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Abstract
A series of large-scale numerical simulations is presented, which incorporate parameterizations of vertical mixing of temperature and salinity by double-diffusion and by small-scale turbulence. These simulations reveal the tendency of double-diffusion to constrain diapycnal volume transport, both upward and downward. For comparable values
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A series of large-scale numerical simulations is presented, which incorporate parameterizations of vertical mixing of temperature and salinity by double-diffusion and by small-scale turbulence. These simulations reveal the tendency of double-diffusion to constrain diapycnal volume transport, both upward and downward. For comparable values of mixing coefficients, the average diapycnal velocity in the double-diffusive thermocline is much less than in the corresponding turbulent regime. The insulating effect of double-diffusion is rationalized using two theoretical models. The first argument is based on the assumed vertical advective-diffusive balance. The second theory uses the Rhines and Young technique to evaluate the net diapycnal transport across regions bounded by closed streamlines at a given density surface. The numerical simulations and associated analytical arguments in this study underscore fundamental differences between double-diffusive mixing and mechanically generated small-scale turbulence. When both double-diffusion and turbulence are taken into account, we find that the constraints on diapycnal velocity loosen (tighten) with the increase (decrease) of the fraction of the overall mixing attributed to turbulence. The range of diapycnal velocities that could be realized in doubly-diffusive fluids is determined by the variation in the heat/salt flux ratio. We hypothesize that the unique ability of double-diffusive mixing to actively control diapycnal volume transport may have significant ramifications for the structure and dynamics of thermohaline circulation in the ocean. Full article
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Open AccessArticle Stabilization of Isolated Vortices in a Rotating Stratified Fluid
Received: 20 June 2016 / Revised: 28 July 2016 / Accepted: 12 August 2016 / Published: 24 August 2016
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Abstract
The key element of Geophysical Fluid Dynamics—reorganization of potential vorticity (PV) by nonlinear processes—is studied numerically for isolated vortices in a uniform environment. Many theoretical studies and laboratory experiments suggest that axisymmetric vortices with a Gaussian shape are not able to remain circular
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The key element of Geophysical Fluid Dynamics—reorganization of potential vorticity (PV) by nonlinear processes—is studied numerically for isolated vortices in a uniform environment. Many theoretical studies and laboratory experiments suggest that axisymmetric vortices with a Gaussian shape are not able to remain circular owing to the growth of small perturbations in the typical parameter range of abundant long-lived vortices. An example of vortex destabilization and the eventual formation of more intense self-propagating structures is presented using a 3D rotating stratified Boussinesq numerical model. The peak vorticity growth found during the stages of strong elongation and fragmentation is related to the transfer of available potential energy into kinetic energy of vortices. In order to develop a theoretical model of a stable circular vortex with a small Burger number compatible with observations, we suggest a simple stabilizing procedure involving the modification of peripheral PV gradients. The results have important implications for better understanding of real-ocean eddies. Full article
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Open AccessArticle The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time
Received: 11 April 2016 / Revised: 4 July 2016 / Accepted: 11 July 2016 / Published: 19 July 2016
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Abstract
The nonlinear Schrödinger equation (NLS) is a canonical evolution equation, which describes the dynamics of weakly nonlinear wave packets in time and space in a wide range of physical media, such as nonlinear optics, cold gases, plasmas and hydrodynamics. Due to its integrability,
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The nonlinear Schrödinger equation (NLS) is a canonical evolution equation, which describes the dynamics of weakly nonlinear wave packets in time and space in a wide range of physical media, such as nonlinear optics, cold gases, plasmas and hydrodynamics. Due to its integrability, the NLS provides families of exact solutions describing the dynamics of localised structures which can be observed experimentally in applicable nonlinear and dispersive media of interest. Depending on the co-ordinate of wave propagation, it is known that the NLS can be either expressed as a space- or time-evolution equation. Here, we discuss and examine in detail the limitation of the first-order asymptotic equivalence between these forms of the water wave NLS. In particular, we show that the the equivalence fails for specific periodic solutions. We will also emphasise the impact of the studies on application in geophysics and ocean engineering. We expect the results to stimulate similar studies for higher-order weakly nonlinear evolution equations and motivate numerical as well as experimental studies in nonlinear dispersive media. Full article
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Open AccessArticle Dynamically Consistent Parameterization of Mesoscale Eddies—Part II: Eddy Fluxes and Diffusivity from Transient Impulses
Received: 4 May 2016 / Revised: 18 June 2016 / Accepted: 21 June 2016 / Published: 30 June 2016
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Abstract
This work continues development of the framework for dynamically consistent parameterization of mesoscale eddy effects in non-eddy-resolving ocean circulation models. Here, we refine and extend the previous results obtained for the double gyres and aim to account for the eddy backscatter mechanism that
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This work continues development of the framework for dynamically consistent parameterization of mesoscale eddy effects in non-eddy-resolving ocean circulation models. Here, we refine and extend the previous results obtained for the double gyres and aim to account for the eddy backscatter mechanism that maintains eastward jet extension of the western boundary currents. We start by overcoming the local homogeneity assumption and by taking into account full large-scale circulation. We achieve this by considering linearized-dynamic responses to finite-time transient impulses. Feedback from these impulses on the large-scale circulation are referred to as footprints. We find that the local homogeneity assumption yields only quantitative errors in most of the gyres but breaks down in the eastward jet region, which is characterized by the most significant eddy effects. The approach taken provides new insights into the eddy/mean interactions and framework for parameterization of unresolved eddy effects. Footprints provide us with maps of potential vorticity anomalies expected to be induced by transient eddy forcing. This information is used to calculate the equivalent eddy potential vorticity fluxes and their divergences that partition the double-gyre circulation into distinct geographical regions with specific eddy effects. In particular, this allows approximation of the real eddy effects that maintain the eastward jet extension of the western boundary currents and its adjacent recirculation zones. Next, from footprints and their equivalent eddy fluxes and from underlying large-scale flow gradients, we calculate spatially inhomogeneous and anisotropic eddy diffusivity tensor. Its properties suggest that imposing parameterized source terms, that is, equivalent eddy flux divergences, is a better parameterization strategy than implementation of the eddy diffusion. Full article
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Open AccessArticle A Theoretical Model of Long Rossby Waves in the Southern Ocean and Their Interaction with Bottom Topography
Received: 25 April 2016 / Revised: 19 May 2016 / Accepted: 24 May 2016 / Published: 27 May 2016
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Abstract
An analytical model of long Rossby waves is developed for a continuously-stratified, planetary geostrophic ocean in the presence of arbitrary bottom topography under the assumption that the potential vorticity is a linear function of buoyancy. The remaining dynamics are controlled by equations for
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An analytical model of long Rossby waves is developed for a continuously-stratified, planetary geostrophic ocean in the presence of arbitrary bottom topography under the assumption that the potential vorticity is a linear function of buoyancy. The remaining dynamics are controlled by equations for material conservation of buoyancy along the sea surface and the sea floor. The mean, steady-state surface circulation follows characteristics that are intermediate to f and f / H contours, where f is the Coriolis parameter and H is the ocean depth; for realistic stratification and weak bottom currents, these characteristics are mostly zonal with weak deflections over the major topographic features. Equations are derived for linear long Rossby waves about this mean state. These are qualitatively similar to the long Rossby wave equations for a two-layer ocean, linearised about a state of rest, except that the surface characteristics in the wave equation, which dominate the propagation, follow precisely the same path as the mean surface flow. In addition to this topographic steering, it is shown that a weighted integral of the Rossby propagation term vanishes over any area enclosed by an f / H contour, which has been shown in the two-layer model to lead to Rossby waves “jumping” across the f / H contour. Finally, a nonlinear Rossby wave equation is derived as a specialisation of the result previously obtained by Rick Salmon. This consists of intrinsic westward propagation at the classical long Rossby speed, modified to account for the finite ocean depth, and a Doppler shift by the depth-mean flow. The latter dominates within the Antarctic Circumpolar Current, consistent with observed eastward propagation of sea surface height anomalies. Full article
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Open AccessArticle Fofonoff Negative Modes
Received: 15 February 2016 / Revised: 6 April 2016 / Accepted: 11 April 2016 / Published: 15 April 2016
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Abstract
The classic solution of Fofonoff to the problem of free inertial flow in a closed basin is extended to the case where the potential vorticity, q, is linearly proportional to the streamfunction, with a negative definite constant, K2. Such
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The classic solution of Fofonoff to the problem of free inertial flow in a closed basin is extended to the case where the potential vorticity, q, is linearly proportional to the streamfunction, with a negative definite constant, K 2 . Such a relation arises naturally in the presence of an eastward flow, instead of Fofonoff’s westward zonal flow on the β plane. The resulting solutions can be wavelike if K 2 = β L 2 / U π 2 exceeds the critical value of 1 where U is the magnitude of the eastward flow and L is the characteristic meridional scale of the motion. Solutions are presented with various boundary conditions on the basin boundaries, and conditions for which the solutions suffer a resonance are also obtained. It is suggested that oceanic circulations with eastward flows naturally excite these Fofonoff negative modes. The possibility of resonance and instability adds additional physical complexity to the modes. Full article
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