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Stephen M. Griffies

Abstract

This paper formulates tracer stirring arising from the Gent–McWilliams (GM) eddy-induced transport in terms of a skew-diffusive flux. A skew-diffusive tracer flux is directed normal to the tracer gradient, which is in contrast to a diffusive tracer flux directed down the tracer gradient. Analysis of the GM skew flux provides an understanding of the physical mechanisms prescribed by GM stirring, which is complementary to the more familiar advective flux perspective. Additionally, it unifies the tracer mixing operators arising from Redi isoneutral diffusion and GM stirring. This perspective allows for a computationally efficient and simple manner in which to implement the GM closure in z-coordinate models. With this approach, no more computation is necessary than when using isoneutral diffusion alone. Additionally, the numerical realization of the skew flux is significantly smoother than the advective flux. The reason is that to compute the skew flux, no gradient of the diffusivity or isoneutral slope is taken, whereas such a gradient is needed for computing the advective flux. The skew-flux formulation also exposes a striking cancellation of terms that results when the GM diffusion coefficient is identical to the Redi isoneutral diffusion coefficient. For this case, the horizontal components to the tracer flux are aligned down the horizontal tracer gradient, and the resulting computational cost of Redi diffusion plus GM skew diffusion is roughly half that needed for Redi diffusion alone.

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Houssam Yassin and Stephen M. Griffies

Abstract

The discrete baroclinic modes of quasigeostrophic theory are incomplete, and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal-mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradients activates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence of multiple modes with no internal zeros, a finite number of modes with negative norms, and the fact that their vertical structures form a basis capable of representing any quasigeostrophic state with a differentiable series expansion. These properties are a consequence of the Pontryagin-space setting of the Rossby wave problem in the presence of boundary buoyancy gradients (as opposed to the usual Hilbert-space setting). We also examine the quasigeostrophic vertical velocity modes and derive a complete basis for such modes as well. A natural application of these modes is the development of a weakly nonlinear wave-interaction theory of geostrophic turbulence that takes topography into account.

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Tapio Schneider and Stephen M. Griffies

Abstract

A conceptual framework is presented for a unified treatment of issues arising in a variety of predictability studies. The predictive power (PP), a predictability measure based on information–theoretical principles, lies at the center of this framework. The PP is invariant under linear coordinate transformations and applies to multivariate predictions irrespective of assumptions about the probability distribution of prediction errors. For univariate Gaussian predictions, the PP reduces to conventional predictability measures that are based upon the ratio of the rms error of a model prediction over the rms error of the climatological mean prediction.

Since climatic variability on intraseasonal to interdecadal timescales follows an approximately Gaussian distribution, the emphasis of this paper is on multivariate Gaussian random variables. Predictable and unpredictable components of multivariate Gaussian systems can be distinguished by predictable component analysis, a procedure derived from discriminant analysis: seeking components with large PP leads to an eigenvalue problem, whose solution yields uncorrelated components that are ordered by PP from largest to smallest.

In a discussion of the application of the PP and the predictable component analysis in different types of predictability studies, studies are considered that use either ensemble integrations of numerical models or autoregressive models fitted to observed or simulated data.

An investigation of simulated multidecadal variability of the North Atlantic illustrates the proposed methodology. Reanalyzing an ensemble of integrations of the Geophysical Fluid Dynamics Laboratory coupled general circulation model confirms and refines earlier findings. With an autoregressive model fitted to a single integration of the same model, it is demonstrated that similar conclusions can be reached without resorting to computationally costly ensemble integrations.

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Stephen M. Griffies and Eli Tziperman

Abstract

The interdecadal variability of a stochastically forced four-box model of the oceanic meridional thermohaline circulation (THC) is described and compared to the THC variability in the coupled ocean–atmosphere GCM of Delworth, Manabe, and Stouffer. The box model is placed in a linearly stable thermally dominant mean state under mixed boundary conditions. A linear stability analysis of this state reveals one damped oscillatory THC mode in addition to purely damped modes. The variability of the model under a moderate amount of stochastic forcing, meant to emulate the random variability of the atmosphere affecting the coupled model's interdecadal THC variability, is studied. A linear interpretation, in which the damped oscillatory mode is of primary importance, is sufficient for understanding the mechanism accounting for the stochastically forced variability. Direct comparison of the variability in the box model and coupled GCM reveals common qualitative aspects. Such a comparison supports, although does not verify, the hypothesis that the coupled model's THC variability can be interpreted as the result of atmospheric weather exciting a linear damped oscillatory THC mode.

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Yalin Fan and Stephen M. Griffies

Abstract

The impacts of parameterized upper-ocean wave mixing on global climate simulations are assessed through modification to Large et al.’s K-profile ocean boundary layer parameterization (KPP) in a coupled atmosphere–ocean–wave global climate model. The authors consider three parameterizations and focus on impacts to high-latitude ocean mixed layer depths and related ocean diagnostics. The McWilliams and Sullivan parameterization (MS2000) adds a Langmuir turbulence enhancement to the nonlocal component of KPP. It is found that the Langmuir turbulence–induced mixing provided by this parameterization is too strong in winter, producing overly deep mixed layers, and of minimal impact in summer. The later Smyth et al. parameterization modifies MS2000 by adding a stratification effect to restrain the turbulence enhancement under weak stratification conditions (e.g., winter) and to magnify the enhancement under strong stratification conditions. The Smyth et al. scheme improves the simulated winter mixed layer depth in the simulations herein, with mixed layer deepening in the Labrador Sea and shoaling in the Weddell and Ross Seas. Enhanced vertical mixing through parameterized Langmuir turbulence, coupled with enhanced lateral transport associated with parameterized mesoscale and submesoscale eddies, is found to be a key element for improving mixed layer simulations. Secondary impacts include strengthening the Atlantic meridional overturning circulation and reducing the Antarctic Circumpolar Current. The Qiao et al. nonbreaking wave parameterization is the third scheme assessed here. It adds a wave orbital velocity to the Reynolds stress calculation and provides the strongest summer mixed layer deepening in the Southern Ocean among the three experiments, but with weak impacts during winter.

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A. J. George Nurser and Stephen M. Griffies

Abstract

We detail the physical means whereby boundary transfers of freshwater and salt induce diffusive fluxes of salinity. Our considerations focus on the kinematic balance between the diffusive fluxes of salt and freshwater, with this balance imposed by mass conservation for an element of seawater. The flux balance leads to a specific balanced form for the diffusive salt flux immediately below the ocean surface and, in the Boussinesq approximation, to a specific form for the salinity flux. This balanced form should be used in specifying the surface boundary condition for the salinity equation and the contribution of freshwater to the buoyancy budget.

Open access
Stephen M. Griffies and Robert W. Hallberg

Abstract

This paper discusses a numerical closure, motivated from the ideas of Smagorinsky, for use with a biharmonic operator. The result is a highly scale-selective, state-dependent friction operator for use in eddy-permitting geophysical fluid models. This friction should prove most useful for large-scale ocean models in which there are multiple regimes of geostrophic turbulence. Examples are provided from primitive equation geopotential and isopycnal-coordinate ocean models.

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Trevor J. McDougall, Sjoerd Groeskamp, and Stephen M. Griffies

Abstract

The small-slope approximation to the full three-dimensional diffusion tensor of epineutral diffusion gives exactly the same tracer flux as the commonly used projected nonorthogonal diffusive flux of layered ocean models and of theoretical studies. The epineutral diffusion achieved by this small-slope approximation is not exactly in the direction of the correct epineutral tracer gradient. That is, the use of the small-slope approximation leads to a very small flux of tracer in a direction in which there is no epineutral gradient of tracer. For (the tracer) temperature or salinity, the difference between the correct epineutral gradient and the small-slope approximation to it is proportional to neutral helicity. The authors also make the point that small-scale turbulent mixing processes act to diffuse tracers isotropically (i.e., the same in each spatial direction) and hence it is strictly a misnomer to call this process “dianeutral diffusion” or “vertical diffusion.” This realization also has implications for the diffusion tensor.

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Stephen M. Griffies, Ronald C. Pacanowski, Martin Schmidt, and V. Balaji

Abstract

This paper details a free surface method using an explicit time stepping scheme for use in z-coordinate ocean models. One key property that makes the method especially suitable for climate simulations is its very stable numerical time stepping scheme, which allows for the use of a long density time step, as commonly employed with coarse-resolution rigid-lid models. Additionally, the effects of the undulating free surface height are directly incorporated into the baroclinic momentum and tracer equations. The novel issues related to local and global tracer conservation when allowing for the top cell to undulate are the focus of this work. The method presented here is quasi-conservative locally and globally of tracer when the baroclinic and tracer time steps are equal. Important issues relevant for using this method in regional as well as large-scale climate models are discussed and illustrated, and examples of scaling achieved on parallel computers provided.

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Sulagna Ray, Andrew T. Wittenberg, Stephen M. Griffies, and Fanrong Zeng

Abstract

The heat budget of the Pacific equatorial cold tongue (ECT) is explored using the GFDL-FLOR coupled GCM (the forecast-oriented low ocean resolution version of CM2.5) and ocean reanalyses, leveraging the two-layer framework developed in Part I. Despite FLOR’s relatively weak meridional stirring by tropical instability waves (TIWs), the model maintains a reasonable SST and thermocline depth in the ECT via two compensating biases: 1) enhanced monthly-scale vertical advective cooling below the surface mixed layer (SML), due to overly cyclonic off-equatorial wind stress that acts to cool the equatorial source waters; and 2) an excessive SST contrast between the ECT and off-equator areas, which boosts the equatorward heat transport by TIWs. FLOR’s strong advective cooling at the SML base is compensated by strong downward diffusion of heat out of the SML, which then allows FLOR’s ECT to take up a realistic heat flux from the atmosphere. Correcting FLOR’s climatological SST and wind stress biases via flux adjustment (FA) leads to weaker deep advective cooling of the ECT, which then erodes the upper-ocean thermal stratification, enhances vertical mixing, and excessively deepens the thermocline. FA does strengthen FLOR’s meridional shear of the zonal currents in the east Pacific, but this does not amplify either the simulated TIWs or their equatorward heat transport, likely due to FLOR’s coarse zonal ocean resolution. The analysis suggests that to advance coupled simulations of the ECT, improved winds and surface heat fluxes must go hand in hand with improved subseasonal and parameterized ocean processes. Implications for model development and the tropical Pacific observing system are discussed.

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