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John K. Dukowicz

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John K. Dukowicz

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Various approximations of the governing equations of compressible fluid dynamics are commonly used in both atmospheric and ocean modeling. Their main purpose is to eliminate the acoustic waves that are potentially responsible for inefficiency in the numerical solution, leaving behind gravity waves. The author carries out a detailed study of gravity wave dispersion for seven such approximations, individually and in combination, to exactly evaluate some of the often subtle errors. The atmospheric and oceanic cases are qualitatively and quantitatively different because, although they solve the same equations, their boundary conditions are entirely different and they operate in distinctly different parameter regimes. The atmospheric case is much more sensitive to approximation. The recent “unified” approximation of Arakawa and Konor is one of the most accurate. Remarkably, a simpler approximation, the combined Boussinesq–dynamically rigid approximation turns out to be exactly equivalent to the unified approximation with respect to gravity waves. The oceanic case is insensitive to the effects of any of the approximations, except for the hydrostatic approximation. The hydrostatic approximation is inaccurate at large wavenumbers in both the atmospheric and oceanic cases because it eliminates the entire buoyancy oscillation flow regime and is therefore to be restricted to low aspect ratio flows. For oceanic applications, certain approximations, such as the unified, dynamically rigid, and dynamically stiff approximations, are particularly interesting because they are accurate and approximately conserve mass, which is important for the treatment of sea level rise.

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John K. Dukowicz

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Existing ocean models often contain errors associated with the computation of the density and the associated pressure gradient. Boussinesq models approximate the pressure gradient force, ρ −1p, by ρ−10p, where ρ 0 is a constant reference density. The error associated with this approximation can be as large as 5%. In addition, Cartesian and sigma-coordinate models usually compute density from an equation of state where its pressure dependence is replaced by a depth dependence through an approximate conversion of depth to pressure to avoid the solution of a nonlinear hydrostatic equation. The dynamic consequences of this approximation and the associated errors can be significant. Here it is shown that it is possible to derive an equivalent but “stiffer” equation of state by the use of modified density and pressure, ρ* and p*, obtained by eliminating the contribution of the pressure-dependent part of the adiabatic compressibility (about 90% of the total). By doing this, the errors associated with both approximations are reduced by an order of magnitude, while changes to the code or to the code structure are minimal.

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John K. Dukowicz and Richard J. Greatbatch

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Numerical experiments are performed over a wide range of parameters to show that mean flows in the form of Fofonoff gyres, characterized by a linear relationship between streamfunction and potential vorticity, are universally produced in the statistically steady state of inviscid unforced barotropic quasigeostrophic turbulence, provided that the initial state is sufficiently well resolved. Further, as the resolution is increased, the mean-flow energy approaches the total energy, and the mean-flow potential enstrophy reaches a minimum value, which is lower than the value with no flow. This is in agreement with the predictions of the theory of equilibrium statistical mechanics. The timescale for the appearance of these flows is on the order of 5–10τ ϵ, where τ ϵ is a mean eddy turnover time. When viscosity is turned on, the mean-flow Fofonoff gyres become internally homogenized and eventually disappear entirely as the flow decays to zero. This evolution of the gyres can be universally scaled with a timescale τ ν = δ 2/ν, where δ is the Rhines scale and ν is the viscosity coefficient. There is an initial period of very rapid adjustment on a timescale of ∼0.005τ ν at the enstrophy accumulated at very high wavenumbers is dissipated, followed by an intermediate period with a timescale of ∼0.04τ ν during which the gyres are homogenized, and finally a period of gyre decay on a timescale of ∼0.3τ ν. In general, there is a competition between the statistical tendency to organize the mean flow into Fofonoff gyres and the tendency for homogenization, with the tendency to form Fofonoff gyres being always overwhelmed given a sufficiently long time. Thus, the issue of whether statistical mean flows, such as Fofonoff gyres, emerge and play a role depends on the relative magnitude of the two timescales, τ ϵ and τ ν.

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John K. Dukowicz and Richard J. Greatbatch

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A stochastic theory of tracer transport in compressible turbulence has recently been developed and then applied to the ocean case because stratified flow in isopycnal coordinates is analogous to compressible flow with the isopycnal layer thickness playing the role of density. The results generalize the parameterization of Gent and McWilliams in the sense that the eddy-induced transport velocity (i.e., the bolus velocity, which is directly related to the thickness–velocity correlation) is given by downgradient Fickian diffusion of thickness with a general mixing tensor K. This result is however dependent on an imprecise postulate relating the Lagrangian to the Eulerian mean velocities. In this paper the authors show that this postulate is unnecessary in a certain flow regime. This regime exists whenever the effect of thickness–velocity correlations dominates over the effect of relative vorticity–velocity correlations. The primary example is planetary geostrophic turbulence, but it may also exist in homogeneous quasigeostrophic turbulence. The bolus velocity is modified and becomes equivalent to upgradient diffusion of potential vorticity along isopycnals with the same mixing tensor K. This changes the conceptual basis of the bolus velocity parameterization of Gent and McWilliams at least in this regime, but practically, the effective change is of rather small magnitude.

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Elizabeth C. Hunke and John K. Dukowicz

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A new discretization for the elastic–viscous–plastic (EVP) sea ice dynamics model incorporates metric terms to account for grid curvature effects in curvilinear coordinate systems. A fundamental property of the viscous–plastic ice rheology that is invariant under changes of coordinate system is utilized; namely, the work done by internal forces, to derive an energy dissipative discretization of the divergence of the stress tensor that includes metric terms. Comparisons of simulations using an older EVP numerical model with the new formulation highlight the effect of the metric terms, which can be significant when ice deformation is allowed to affect the ice strength.

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John K. Dukowicz, Richard D. Smith, and Robert C. Malone

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Certain aspects of the Semtner-Chervin version of the Bryan-Cox-Semtner global ocean model are reformulated for improved efficiency on parallel computer architectures and on the Connection Machine CM-2 in particular. These changes involve (a) the switch from a streamfunction to a surface pressure formulation in the barotropic equations, (b) the splitting off of the Coriolis terms from the barotropic equations to produce a symmetric surface pressure equation, which then permits (c) the use of a preconditioned conjugate-gradient method for the solution of this equation. The switch to a surface pressure formulation (a) eliminates global equations associated with island boundary conditions and therefore improves performance as well as allows an unlimited number of islands, (b) reduces sensitivity to rapidly varying bottom topography and therefore obviates the need for smoothing the topography, and (c) makes the surface pressure a prognostic variable, thus potentially making it easier to assimilate surface altimetry data. Care is taken to retain the energetic consistency built into the original model. The nine-point operator for the pressure equation is found necessary to maintain energy consistency, in contrast to the streamfunction formulation where both nine-point and five-point operators are usable. Computational results closely resemble those of the original model, but with significantly improved performance. The barotropic part of the calculations in the surface pressure formulation is newly three times faster than in the streamfunction formulation, and the full calculation involving both barotropic and baroclinic parts is nearly two times faster. These changes are described and details of the performance of the new formulation on the CM-2 are given.

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Kirk Bryan, John K. Dukowicz, and Richard D. Smith

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Mesoscale eddies in the ocean play an important role in the ocean circulation. In order to simulate the ocean circulation, mesoscale eddies must be included explicitly or parameterized. The eddy permitting calculations of the Los Alamos ocean circulation model offer a special opportunity to test aspects of parameterizations that have recently been proposed. Although the calculations are for a model in level coordinates, averages over a five-year period have been carried out by interpolating to instantaneous isopycnal surfaces. The magnitude of “thickness mixing” or bolus velocity is found to coincide with areas of intense mesoscale activity in the western boundary currents of the Northern Hemisphere and the Antarctic Circumpolar Current. The model also predicts relatively large bolus fluxes in the equatorial region. The analysis does show that the rotational component of the bolus velocity is significant. Predictions of the magnitude of the bolus velocity, assuming downgradient mixing of thickness with various mixing coefficients, have been compared directly with the model. The coefficient proposed by Held and Larichev provides a rather poor fit to the model results because it predicts large bolus velocity magnitudes at high latitudes and in other areas in which there is only a small amount of mesoscale activity. A much better fit is obtained using a constant mixing coefficient or a mixing coefficient originally proposed by Stone in a somewhat different context. The best fit to the model is obtained with a coefficient proportional to λ 2/T, where λ is the radius of deformation, and T is the Eady timescale for the growth of unstable baroclinic waves.

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Shan Sun, Rainer Bleck, Claes Rooth, John Dukowicz, Eric Chassignet, and Peter Killworth

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Buoyancy anomalies caused by thermobaricity, that is, the modulation of seawater compressibility by potential temperature anomalies, underlie a long-standing argument against the use of potential-density-framed numerical models for realistic circulation studies. The authors show that this problem can be overcome by relaxing the strict correspondence between buoyancy and potential density in isopycnic-coordinate models. A parametric representation of the difference between the two variables is introduced in the form of a “virtual potential density,” which can be viewed as the potential density that would be computed from the in situ conditions using the compressibility coefficient for seawater of a fixed (but representative) salinity and potential temperature. This variable is used as a basis for effective dynamic height computations in the dynamic equations, while the traditionally defined potential density may be retained as model coordinate. The conservation properties of the latter assure that adiabatic transport processes in a compressibility-compliant model can still be represented as exactly two-dimensional. Consistent with its dynamic significance, the distribution of virtual potential density is found to determine both the local static stability and, to a lesser degree, the orientation of neutrally buoyant mixing surfaces. The paper closes with a brief discussion of the pros and cons of replacing potential density by virtual potential density as vertical model coordinate.

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Stephen M. Griffies, Anand Gnanadesikan, Ronald C. Pacanowski, Vitaly D. Larichev, John K. Dukowicz, and Richard D. Smith

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This paper considers the requirements that must be satisfied in order to provide a stable and physically based isoneutral tracer diffusion scheme in a z-coordinate ocean model. Two properties are emphasized: 1) downgradient orientation of the diffusive fluxes along the neutral directions and 2) zero isoneutral diffusive flux of locally referenced potential density. It is shown that the Cox diffusion scheme does not respect either of these properties, which provides an explanation for the necessity to add a nontrivial background horizontal diffusion to that scheme. A new isoneutral diffusion scheme is proposed that aims to satisfy the stated properties and is found to require no horizontal background diffusion.

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