Given its importance in parametrizing eddies, we consider the orientation of eddy flux of potential vorticity (PV) in geostrophic turbulence. We take two different points of view, a classical ensemble- or time-average point of view and a second scale decomposition point of view. A net alignment of the eddy flux of PV with the appropriate mean gradient or the large-scale gradient of PV is required. However, we find this alignment to be very weak. A key finding of our study is that in the scale decomposition approach, there is a strong correlation between the eddy flux and a nonlinear combination of resolved gradients. This strong correlation is absent in the classical decomposition. This finding points to a new model to parametrize the effects of eddies in global ocean circulation. CPY 2008 The Royal Society.

Keywords: Fluxes ; Alignment ; Atmospheric turbulence ; Correlation methods ; Flow of fluids ; Gradient methods ; Oceanography ; Turbulence ; Eddy fluxes ; (1 1 1) orientation ; Strong correlations ; Geostrophic turbulence ; new model ; Nonlinear combination ; decomposition approach ; Global ocean circulation ; Potential vorticity (PV)

We demonstrate, in the context of implicit-filtering large eddy simulations (LESs) of geostrophic turbulence, that while the attractor of a well-resolved statistically stationary turbulent flow can be reached in a coarsely resolved LES that is forced by the subgrid scale (SGS) terms diagnosed from the well-resolved computation, the attractor is generically unstable: the coarsely resolved LES system forced by the diagnosed SGS eddy terms has multiple attractors. This points to the importance of interpreting the diagnosed SGS forcing terms in a well-resolved computation or experiment from a combined physical-numerical point of view rather than from a purely physical point of view

Keywords: flow instability ; flow simulation ; geophysical fluid dynamics ; turbulence ; instability ; subgrid model ; implicit-filtering large eddy simulation ; geostrophic turbulence ; attractor ; stationary turbulent flow ; coarsely resolved system ; diagnosed SGS eddy terms ; A4720 ; Hydrodynamic stability and instability ; A4725 ; Turbulent flows, convection, and heat transfer ; A4710 ; General fluid dynamics theory, simulation and other computational methods

Recently, stochastic, as opposed to deterministic, parameterizations are being investigated to model the effects of unresolved subgrid scales ( SGS) in large eddy simulations ( LES) of geophysical flows. We analyse such a stochastic approach in the barotropic vorticity equation to show that ( i) if the stochastic parameterization approximates the actual SGS stresses, then the solution of the stochastic LES approximates the "true" solution at appropriate scale sizes; and that ( ii) when the filter scale size approaches zero, the solution of the stochastic LES approaches the true solution.

Keywords: D PN MATHEMATICS, APPLIED ; D PQ MATHEMATICS ; ISOTROPIC TURBULENCE ; MODEL ; CIRCULATION ; WIND ; EDDIES ; OCEAN

We find that in parameter regimes relevant to the recently observed alternating zonal jets in oceans, the formation of these jets can be explained as due to an arrest of the turbulent inverse-cascade of energy by <i>free</i> Rossby waves (as opposed to Rossby <i>basin</i> modes) and a subsequent redirection of that energy into zonal modes. This mechanism, originally studied in the context of alternating jets in Jovian atmospheres and two dimensional turbulence in zonally-periodic configurations survives in spite of the presence of the meridional boundaries in the oceanic context

Keywords: jets ; oceanography ; turbulence ; zonal jets ; oceans ; jets formation ; turbulent inverse-cascade ; Rossby waves ; Jovian atmospheres ; 2D turbulence ; zonally-periodic configurations ; meridional boundaries ; oceanic context ; A9210F ; Dynamics of the upper ocean ; A9210L ; Turbulence, diffusion, mixing, and convection in the oceans ; A4755C ; Jets in fluid dynamics

On the decadal to centennial time scale, changes in climate are controlled strongly by changes in ocean circulation. However, because of limitations inherent to the time integration schemes used in present-day ocean models,state-of-the-art climate change simulations resolve the oceans only very coarsely. With an aim to enable long-term simulations of ocean circulation at the high resolutions required for a better representation of global ocean dynamics, we have implemented fully-implicit time integration schemes in a version of the popular ocean general circulation model POP (Parallel Ocean Program), employing Jacobian-free Newton-Krylov techniques. Here, we describe the numerical principles underlying iPOP in some detail and present a few computational results. While there are many advantages to this approach, including a consistent and uniform treatment of the terms in the governing equations, the primary advantage lies in the ability to take time steps that are of relevance to the physical phenomenon that is being studied. The time step is not limited (for stability reasons) by the fastest modes of the system. [All rights reserved Elsevier]

Keywords: climatology ; geophysics computing ; integration ; Newton method ; oceanographic techniques ; parallel programming ; ocean modelling ; climate change simulation ; decadal time scale ; centennial time scale ; time integration scheme ; global ocean dynamics ; ocean general circulation model ; parallel ocean program ; Jacobian-free Newton-Krylov iteration ; C7340 ; Geophysics computing ; C4130 ; Interpolation and function approximation (numerical analysis) ; C4160 ; Numerical integration and differentiation ; C6150N ; Distributed systems software

This paper presents analytical and numerical results for a class of turbulence closure models called "alpha models,'' in which Lagrangian averaging and turbulence closure assumptions modify the Eulerian nonlinearity. The alpha models are investigated in the setting of the barotropic, double-gyre circulation in an ocean basin. Two variants of the alpha models for the barotropic vorticity (BV) equation are found to produce the correct four-gyre configuration for the mean barotropic circulation in numerical simulations performed at a resolution 4 times as coarse as that required in a resolved BV model. These are the BV-alpha model and the BV-Leray-alpha model. However, at a resolution 8 times as coarse, only the BV-alpha model produces the proper four-gyre configuration. Thus, the combination of modified nonlinearity and viscous dissipation (the viscosity is the same in all of the runs) in the BV-alpha model is found to provide a promising approach to modeling the mean effects of unresolved mesoscale (subgrid scale) activity in this problem.

Keywords: D SI OCEANOGRAPHY ; STOKES-ALPHA MODEL ; EQUATIONS ; EDDIES ; PARAMETERIZATION ; MOTION ; FLOW

The recently derived Lagrangian-averaged Navier-Stokes equations model the large-scale flow of the Navier-Stokes fluid at spatial scales larger than some <i>a</i> <i>priori</i> fixed α>0, while coarse-graining the behavior of the small scales. In this communication, we numerically study the behavior of the two-dimensional (2D) isotropic version of this model, also known as the α model. The inviscid dynamics of this model exactly coincide with the vortex blob algorithm for a certain choice of smoothing kernel, as well as the equations of an inviscid second-grade non-Newtonian fluid. While previous studies of this system in 3D have noted the suppression of nonlinear interaction between modes smaller than α, we show that the modification of the nonlinear advection term also acts to enhance the inverse-cascade of energy in 2D turbulence and thereby affects scales of motion larger than α as well. This, we note, (a) may preclude a <i>straightforward</i> use of the model as a subgrid model in coarsely resolved 2D computations, (b) is reminiscent of the drag-reduction that occurs in a turbulent flow when a dilute polymer is added, and (c) can be qualitatively understood in terms of known dimensional arguments

Keywords: Navier-Stokes equations ; non-Newtonian flow ; polymer solutions ; turbulence ; vortices ; two-dimensional Lagrangian-averaged Navier-Stokes equations ; energy inverse-cascade ; Lagrangian averaging procedure ; volume-preserving diffeomorphisms ; 2D isotropic version ; vortex blob algorithm ; smoothing kernel ; inviscid second-grade nonNewtonian fluid ; nonlinear interaction ; 2D turbulence ; dilute polymer ; drag-reduction ; A4710 ; General fluid dynamics theory, simulation and other computational methods ; A4725 ; Turbulent flows, convection, and heat transfer ; A4750 ; Non-Newtonian dynamics ; A4730 ; Rotational flow, vortices, buoyancy and other flows involving body forces

The dynamics of an idealized wind-driven double-gyre circulation in an ocean basin are studied from a dynamical systems point of view in an effort to better understand its variability. While previous analyses of this circulation have mostly dealt with local bifurcations of steady states and limit cycles, this study demonstrates the importance of considering global bifurcations as well. In one case, a coherent picture of the global dynamics spanning a range of parameters from where there are only stable steady-state solutions to where there is chaotic eddy shedding is presented. A simple but novel use of power spectra along with dynamical projections of the dynamics suggests that just beyond the regime in which there are only stable steady states, the system exhibits a complicated global bifurcation known as the “Shilnikov phenomenon”

Keywords: chaos ; oceanography ; ocean ; current ; dynamics ; circulation ; global bifurcation ; Shilnikov type ; double gyre ocean model ; wind-driven double-gyre ; ocean basin ; variability ; global dynamics ; chaotic eddy shedding ; Shilnikov phenomenon ; chaos ; A9210F ; Dynamics of the upper ocean

Recently a new class of coarse-grained equations, known as α models, have been proposed for the mean motion of an ideal incompressible fluid. The use of one such model to represent the time-mean component of a turbulent β-plane circulation characterized by potential vorticity mixing is considered. In particular, the focus is on the wind-driven circulation in a shallow ocean basin, a problem well studied as a prototype of more realistic ocean dynamics. The authors demonstrate the ability of an α model to reproduce qualitatively the structure of a four-gyre circulation that forms (in the time mean) when the barotropic vorticity equation is driven by a symmetric, double-gyre wind forcing, and when the dissipation is weak. This is offered as a first step in assessing the utility of the α-model approach to simulating more complex geophysical flows

Keywords: geophysical fluid dynamics ; oceanography ; vortices ; ocean ; turbulence ; dynamics ; dispersive-dissipative eddy parameterization ; eddy ; barotropic model ; dispersion ; dissipation ; coarse grained equations ; α model ; ideal incompressible fluid ; time mean component ; turbulent β-plane circulation ; potential vorticity mixing ; wind driven circulation ; shallow ocean basin ; four gyre circulation ; barotropic vorticity equation ; double gyre wind forcing ; A9210F ; Dynamics of the upper ocean ; A9210D ; Dynamics of the deep ocean ; A4730 ; Rotational flow, vortices, buoyancy and other flows involving body forces ; A9210L ; Turbulence, diffusion, mixing, and convection in the oceans

An inviscid two-dimensional fluid model with nonlinear dispersion that arises simultaneously in coarse-grained descriptions of the dynamics of the Euler equation and in the description of non-Newtonian fluids of second grade is considered. The scaling of the equilibrium states of this model for conserved energy and enstrophy retains the corresponding scaling for the Euler equations on the large scales and at the same time greatly deemphasizes the importance of small scales. This is the first clear demonstration of the beneficial effect of nonlinear dispersion in the model, and should highlight its utility as a subgrid model in more realistic situations

Keywords: non-Newtonian flow ; vortices ; scaling properties ; inviscid mean-motion fluid model ; inviscid two-dimensional fluid model ; nonlinear dispersion ; coarse-grained descriptions ; Euler equation ; nonNewtonian fluids ; equilibrium states ; conserved energy ; enstrophy ; subgrid model ; A4750 ; Non-Newtonian dynamics ; A4730 ; Rotational flow, vortices, buoyancy and other flows involving body forces

Results from a barotropic vorticity equation model driven by symmetric, double-gyre wind forcing are described. The authors work in a regime in which the model reaches a state of turbulent equilibrium. The time-average of the statistically steady state exhibits a four-gyre structure, in contrast to the usual two gyres associated with symmetric double-gyre wind forcing. The four-gyre structure is found in model runs using either free-slip or superslip boundary conditions, and with either Laplacian or biharmonic mixing for the dissipation. It is shown that the vorticity budget of both the inner and outer gyres is dominated by a balance between the wind stress curl and the divergence of the eddy potential vorticity flux, with the explicit dissipation playing a much smaller role. The two inner gyres circulate in the same sense as the wind stress curl and are equilibriated, for the most part, by the eddy flux of potential vorticity. The outer gyres, on the other hand, circulate in the opposite sense to the wind stress curl and are driven by the eddy flux of potential vorticity. It is shown that the gross features of the time-averaged state can be reproduced by a parameterized model in which the divergent part of the potential vorticity flux is represented as a downgradient transfer, and a boundary condition of no normal flux of potential vorticity is applied along the model boundaries. In contrast to the eddy resolving model, the four-gyre structure in the parameterized model depends strongly on the choice of side boundary condition.

Keywords: D SI OCEANOGRAPHY ; QUASI-GEOSTROPHIC OCEAN ; MULTIPLE EQUILIBRIA ; POTENTIAL VORTICITY ; REDUCED-GRAVITY ; INERTIAL GYRES ; DRIVEN ; TURBULENCE ; RECIRCULATION ; EMERGENCE ; FLOWS

We present a new method, based on averaging, to simulate certain systems with multiple time scales efficiently and demonstrate its utility in the context of the shallow-water equations. We first develop the method in a simple linear setting and analytically prove its stability. This is followed by an extension to the full equations and a presentation of a computational model for it. In this preliminary study, we find that the new method produces results that are very close to a fully explicit (spatially and temporally) second-order accurate scheme and much better than a fully explicit (spatially and temporally) first-order accurate scheme, while costing less than the first-order accurate scheme

Keywords: error analysis ; flow simulation ; geophysical fluid dynamics ; numerical stability ; ocean waves ; flow simulation ; multiple time scales ; averages method ; shallow-water equations ; numerical stability ; first-order accurate scheme ; A9210H ; Surface waves, tides, and sea level ; A0620D ; Measurement and error theory ; A0260 ; Numerical approximation and analysis ; A4710 ; General fluid dynamics theory, simulation and other computational methods

We study an interface-capturing two-phase fluid model in which the interfacial tension is modelled as a volumetric stress. Since these stresses are obtainable from a van der Waals-Cahn-Hilliard free energy, the model is, to a certain degree, thermodynamically realistic. Thermal fluctuations are not considered presently for reasons of simplicity. The utility of the model lies in its momentum-conservative representation of surface tension and the simplicity of its numerical implementation resulting from the volumetric modelling of the interfacial dynamics. After validation of the model in two spatial dimensions, two prototypical applications-capillary instability of an initially high-Reynolds-number liquid jet in the gaseous phase and spinodal decomposition in a liquid-gas system-are presented

Keywords: flow instability ; free energy ; jets ; spinodal decomposition ; surface tension ; two-phase flow ; two-phase fluid model ; interface-capturing two-phase fluid ; interfacial tension ; volumetric stress ; van der Waals-Cahn-Hilliard free energy ; thermodynamically realistic model ; thermal fluctuations ; momentum-conservative representation ; numerical implementation ; volumetric modelling ; interfacial dynamics ; prototypical applications ; capillary instability ; high-Reynolds-number liquid jet ; gaseous phase ; spinodal decomposition ; liquid-gas system ; A4755K ; Multiphase flows ; A4755C ; Jets in fluid dynamics ; A6810C ; Fluid surface energy (surface tension, interface tension, angle of contact, etc.) ; A6550 ; Thermodynamic properties and entropy ; A6475 ; Solubility, segregation, and mixing ; A4720 ; Hydrodynamic stability and instability

Three different sets of shallow water equations, representing different levels of approximation are considered. The numerical solutions of these different equations for flow past bottom topography in several different flow regimes are compared. For several cases the full Euler solutions are computed as a reference, allowing the assessment of the relative accuracies of the different approximations. Further, the differences between the dispersive shallow water (DSW) solutions and those of the highly simplified, hyperbolic shallow water (SW) equations is studied as a guide to determining what level of approximation is required for a particular flow. First, the Green-Naghdi (GN) equations are derived as a vertically-integrated rational approximation of the Euler equations, and then the generalized Boussinesq (gB) equations are obtained under the further assumption of weak nonlinearity. A series of calculations, each assuming different values of a set of parameters-undisturbed upstream Froude number, and the height and width of the obstacle, are then presented and discussed. In almost all regions of the parameter space, the SW and DSW theories yield different results; it is only when the flows are entirely subcritical or entirely supercritical and when the obstacles are very wide compared to the depth of the fluid that the SW and DSW theories are in qualitative and quantitative agreement. It is also found that while the gB solutions are accurate only for small bottom topographies (less than 20 the undisturbed fluid depth), the GN solutions are accurate for much larger topographies (up to 65 the undisturbed fluid depth). The limitation of the gB approximation to small topographies is primarily due to the generation of large amplitude upstream propagating solitary waves at transcritical Froude numbers, and is consistent with previous analysis. The GN approximation, which makes no assumptions about the size of the nonlinearity, is thus verified to be a better system to use in cases where the bottom topographies are large or when the bottom topographies are moderate but the flow transcritical

Keywords: external flows ; water waves ; shallow fluid flow ; obstacle ; shallow water equations ; numerical solutions ; flow past bottom topography ; Euler solutions ; dispersive shallow water solutions ; highly simplified hyperbolic shallow water equations ; Green-Naghdi equations ; vertically-integrated rational approximation ; Euler equations ; generalized Boussinesq equations ; undisturbed upstream Froude number ; transcritical flow ; large amplitude upstream propagating solitary waves ; transcritical Froude numbers ; A4735 ; Waves in fluid dynamics

We discuss the parallel implementation of a semi Lagrangian shallow water model on the massively parallel Connection Machine CM-5. The four important issues we address are: (i) two alternative formulations of the elliptic problem and their relative efficiencies; (ii) the performance of two successive orders of a generalized conjugate residual elliptic solver; (iii) the time spent in unstructured communication-an unavoidable feature of semi lagrangian schemes; and (iv) the scalability of the algorithm

Keywords: fluid dynamics ; geophysics computing ; mechanical engineering computing ; oceanography ; parallel machines ; parallel programming ; water ; semi Lagrangian shallow water model ; massively parallel Connection Machine CM-5 ; elliptic problem ; successive orders ; generalized conjugate residual elliptic solver ; unstructured communication ; semi lagrangian schemes ; algorithm scalability ; C7440 ; Civil and mechanical engineering computing ; C6110P ; Parallel programming ; C5440 ; Multiprocessing systems ; C7340 ; Geophysics computing

A new approach to solving the shallow water equations is presented. This involves using discrete velocities of an adaptive nature in a finite volume context. The origin of the discrete-velocity space and the magnitudes of the discrete-velocities are both spatially and temporally variable. The near-equilibrium flow method of Nadiga and Pullin is used to arrive at a robust second-order (in both space and time) scheme - the adaptive discrete velocity (ADV) scheme - which captures hydraulic jumps with no oscillations. The flow over a two-dimensional ridge, over a wide range of undisturbed upstream Frounde numbers prove the robustness and accuracy of the scheme. A comparison of the interaction of two circular vortex patches in the presence of bottom topography as obtained by the ADV scheme and a semi-Lagrangian scheme more than validates the new scheme in two dimensions. 19 refs., 12 figs., 1 tab.

Keywords: 42 ENGINEERING NOT INCLUDED IN OTHER CATEGORIES ; 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS ; WAVE EQUATIONS ; CALCULATION METHODS ; WATER WAVES ; DIFFERENTIAL EQUATIONS ; WAVE FORCES

A new discrete-velocity model is presented to solve the three-dimensional Euler equations. The velocities in the model are of an adaptive nature-both the origin of the discrete-velocity space and the magnitudes of the discrete velocities are dependent on the local flow-and are used in a finite-volume context. The numerical implementation of the model follows the near-equilibrium flow method of Nadiga and Pullin and results in a scheme which is second order in space (in the smooth regions and between first and second order at discontinuities) and second order in time. (The three-dimensional code is included.) For one choice of the scaling between the magnitude of the discrete velocities and the local internal energy of the flow, the method reduces to a flux-splitting scheme based on characteristics. As a preliminary exercise, the result of the Sod shock-tube simulation is compared to the exact solution.

Keywords: 66 PHYSICS ; 42 ENGINEERING NOT INCLUDED IN OTHER CATEGORIES ; GAS FLOW ; FLOW MODELS ; SHOCK WAVES ; COMPUTERIZED SIMULATION ; NUMERICAL SOLUTION ; SHOCK TUBES ; C CODES ; BOLTZMANN STATISTICS

We present a simulation scheme for discrete-velocity gases based on local thermodynamic equilibrium.^Exploting the kinetic nature of discrete-velocity gases, in that context, results in a natural splitting of fluxes, and the resultant scheme strongly resembles the original process.^The kinetic nature of the scheme and the modeling of the infinite collision rate limit, result in a small value of the coefficient of (numerical)-viscosity, the behavior of which is remarkably physical.^A first-order method and two second-order methods using the total variation diminishing principle are developed and an example application is presented.^Given the same computer resources, it is expected that with this approach, a much higher Reynold`s number will be achievable than presently possible with either lattice gas automata or lattice Boltzmann approaches.^The ideas being general, the scheme is applicable to any discrete-velocity model and to lattice gases as well.^17 refs., 4 figs.

Keywords: 420400 - Engineering- Heat Transfer & Fluid Flow ; 990200 - Mathematics & Computers ; BOLTZMANN EQUATION- NUMERICAL SOLUTION ; COUETTE FLOW- COMPUTERIZED SIMULATION ; DIFFERENTIAL EQUATIONS ; EQUATIONS ; FLUID FLOW ; PARTIAL DIFFERENTIAL EQUATIONS ; SIMULATION ; VISCOUS FLOW

We present criteria for monitoring the validity of the NavierendashStokes approximation during the simulation of a rarefied gas. Our approach is based on an underlying kinetic formulation through which one can construct nondimensional non-negative definite matrices from moments of the molecular distribution. We then identify one such 3times3 matrix that can be evaluated intrinsically in the NavierendashStokes approximation. Our criteria are based on deviations of the eigenvalues of this matrix from their equilibrium value of unity. Not being tied to a particular benchmark problem, the resulting criteria are portable and may be applied to any NavierendashStokes simulation. We study its utility here by comparing stationary planar shock profiles computed using the NavierendashStokes equations with those computed using Monte Carlo simulations.copyrightital 1998 American Institute of Physics.

Keywords: 66 PHYSICS ; NAVIER-STOKES EQUATIONS ; GAS-FLOW PROCESSES ; RAREFIED GASES ; SHOCK WAVES ; MATRICES ; MONTE CARLO METHOD ; COMPUTERIZED SIMULATION

The exact structure of a shock is computed in a multiple-speed discrete-velocity gas, the nine-velocity gas, wherein the multiplicity of speeds ensures non-trivial thermodynamics. Obtained as a solution of the model Boltzmann equations, the procedure consists of tracking the shock as a trajectory of a three-dimensional dynamical system connecting an equilibrium upstream state to an equilibrium downstream state. The two equilibria satisfy the jump conditions obtained from the model Euler equations. Comparison of the shock structure to that in a monatomic perfect gas, as given by the Navier-Stokes equation, shows excellent agreement. The shock in the nine-velocity gas has an overshoot in entropy alone, like in a monatomic gas. The near-equilibrium flow technique for discrete-velocity gases [B.T. Nadiga and D.I. Pullin, J. Comp. Phys., submitted], a kinetic flux-splitting method based on the local thermodynamic equilibrium, is also seen to capture the shock structure remarkably well

Keywords: Boltzmann equation ; entropy ; fluid dynamics ; Navier-Stokes equations ; thermodynamics ; nine-velocity gas ; shock structure ; multiple-speed discrete-velocity gas ; speed multiplicity ; nontrivial thermodynamics ; 3D dynamical system ; equilibrium upstream state ; equilibrium downstream state ; model Euler equations ; monatomic perfect gas ; Navier-Stokes equation ; near-equilibrium flow ; kinetic flux-splitting method ; A4710 ; General fluid dynamics theory, simulation and other computational methods ; A0560 ; Transport processes: theory ; A0570C ; Thermodynamic functions and equations of state