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  • Author or Editor: Alexander F. Shchepetkin x
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Alexander F. Shchepetkin and James C. McWilliams

Abstract

The authors develop and test computational methods for advection of a scalar field that also include a minimal dissipation of its variance in order to preclude the formation of false extrema. Both of these properties are desirable for advectively dominated geophysical flows, where the relevant scalars are both potential vorticity and material concentrations. These methods are based upon the sequential application of two types of operators: 1) a conservative and nondissipative (i.e., preserving first and second spatial moments of the scalar field), directionally symmetric advection operator with a relatively high order of spatial accuracy; and 2) a locally adaptive correction operator of lower spatial accuracy that eliminates false extrema and causes dissipation. During this correction phase the provisional distribution of the advected quantity is checked against the previous distribution, in order to detect places where the previous values were overshot, and thus to compute the excess. Then an iterative diffusion procedure is applied to the excess field in order to achieve approximate monotone behavior of the solution.

In addition to the traditional simple flow tests, we have made long-term simulations of freely evolving two-dimensional turbulent flow in order to compare the performance of the proposed technique with that of previously known algorithms, such as UTOPIA and FCT. This is done for both advection of vorticity and passive scalar. Unlike the simple test flows, the turbulent flow provides nonlinear cascades of quadratic moments of the advected quantities toward small scales, which eventually cannot be resolved on the fixed grid and therefore must be dissipated. Thus, not only the ability of the schemes to produce accurate shape-preserving advection, but also their ability to simulate subgrid-scale dissipation are being compared. It is demonstrated that locally adaptive algorithms designed to avoid oscillatory behavior in the vicinity of steep gradients of the advected scalars may result in overall less dissipation, yet give a locally accurate and physically meaningful solution, whereas algorithms with built-in hyperdiffusion (i.e., those traditionally used for direct simulation of turbulent flows) tend to produce a locally unsufficient and, at the same time, globally excessive amount of dissipation. Finally, the authors assess the practial trade-offs required for large models among the competing attributes of accuracy, extrema preservation, minimal dissipation (e.g., appropriate to large Reynolds numbers), and computational cost.

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Alexander F. Shchepetkin and James J. O'Brien

Abstract

Dissipation in numerical ocean models has two purposes: to simulate processes in which the friction is physically relevant and to prevent numerical instability by suppressing accumulation of energy in the smallest resolved scales. This study shows that even for the latter case the form of the friction term should be chosen in a physically consistent way. Violation of fundamental physical principles reduces the fidelity of the numerical solution, even if the friction is small. Several forms of the lateral friction, commonly used in numerical ocean models, are discussed in the context of shallow-water equations with nonuniform layer thickness. It is shown that in a numerical model tuned for the minimal dissipation, the improper form of the friction term creates finite artificial vorticity sources that do not vanish with increased resolution, even if the viscous coefficient is reduced consistently with resolution. An alternative numerical implementation of the no-slip boundary conditions for an arbitrary coast line is considered. It was found that the quality of the numerical solution may be considerably improved by discretization of the viscous stress tensor in such a way that the numerical boundary scheme approximates not only the stress tensor to a certain order of accuracy but also simulates the truncation error of the numerical scheme used in the interior of the domain. This ensures error cancellation during subsequent use of the elements of the tensor in the discrete version of the momentum equations, allowing for approximation of them without decrease in the order of accuracy near the boundary.

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Changming Dong, James C. McWilliams, and Alexander F. Shchepetkin

Abstract

Density stratification and planetary rotation distinguish three-dimensional island wakes significantly from a classical fluid dynamical flow around an obstacle. A numerical model is used to study the formation and evolution of flow around an idealized island in deep water (i.e., with vertical island sides and surface-intensified stratification and upstream flow), focusing on wake instability, coherent vortex formation, and mesoscale and submesoscale eddy activity. In a baseline experiment with strong vorticity generation at the island, three types of instability are evident: centrifugal, barotropic, and baroclinic. Sensitivities are shown to three nondimensional parameters: the Reynolds number (Re), Rossby number (Ro), and Burger number (Bu). The dependence on Re is similar to the classical wake in its transition to turbulence, but in contrast the island wake contains coherent eddies no matter how large the Re value. When Re is large enough, the shear layer at the island is so narrow that the vertical component of vorticity is larger than the Coriolis frequency in the near wake, leading to centrifugal instability on the anticyclonic side. As Bu decreases the eddy size shrinks from the island breadth to the baroclinic deformation radius, and the eddy generation process shifts from barotropic to baroclinic instability. For small Ro values, the wake dynamics is symmetric with respect to cyclonic and anticyclonic eddies. At intermediate Ro and Bu values, the anticyclonic eddies are increasingly more robust than cyclonic ones as Ro/Bu increases, but for large Re and Ro values, centrifugal instability weakens the anticyclonic eddies while cyclonic eddies remain coherent.

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James C. McWilliams, Edward Huckle, and Alexander F. Shchepetkin

Abstract

The K-profile parameterization scheme is used to investigate the stratified Ekman layer in a “fair weather” regime of weak mean surface heating, persistently stable density stratification, diurnal solar cycle, and broadband fluctuations in the surface stress and buoyancy flux. In the case of steady forcing, the boundary layer depth typically scales as hu */Nf, where u * is the friction velocity, f is the Coriolis frequency, and N is the interior buoyancy frequency that confirms empirical fits. The diurnal cycle of solar forcing acts to deepen the boundary layer because of net interior absorption and compensating surface cooling. Parameterized mesoscale and submesoscale eddy-induced restratification flux compresses the boundary layer. With transient forcing, the mean boundary layer profiles are altered; that is, rectification occurs with a variety of causes and manifestations, including changes in h and in the Ekman profile u(z). Overall, stress fluctuations tend to deepen the mean boundary layer, especially near the inertial frequency. Low- and high-frequency surface buoyancy-flux fluctuations have net shallowing and deepening effects, respectively. Eddy-induced interior profile fluctuations are relatively ineffective as a source of boundary layer rectification. Rectification effects in their various combinations lead to a range of mean velocity and buoyancy profiles. In particular, they lead to a “rotated” effective eddy-viscosity profile with misalignment between the mean turbulent stress and mean shear and to a “flattening” of the velocity profile with a larger vertical scale for the current veering than the speed decay; both of these effects from rectification are consistent with previous measurements.

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James C. McWilliams, Jonathan Gula, M. Jeroen Molemaker, Lionel Renault, and Alexander F. Shchepetkin

Abstract

A submesoscale filament of dense water in the oceanic surface layer can undergo frontogenesis with a secondary circulation that has a surface horizontal convergence and downwelling in its center. This occurs either because of the mesoscale straining deformation or because of the surface boundary layer turbulence that causes vertical eddy momentum flux divergence or, more briefly, vertical momentum mixing. In the latter case the circulation approximately has a linear horizontal momentum balance among the baroclinic pressure gradient, Coriolis force, and vertical momentum mixing, that is, a turbulent thermal wind. The frontogenetic evolution induced by the turbulent mixing sharpens the transverse gradient of the longitudinal velocity (i.e., it increases the vertical vorticity) through convergent advection by the secondary circulation. In an approximate model based on the turbulent thermal wind, the central vorticity approaches a finite-time singularity, and in a more general hydrostatic model, the central vorticity and horizontal convergence are amplified by shrinking the transverse scale to near the model’s resolution limit within a short advective period on the order of a day.

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Jun Choi, Annalisa Bracco, Roy Barkan, Alexander F. Shchepetkin, James C. McWilliams, and Jeroen M. Molemaker

Abstract

Four numerical simulations are used to characterize the impact of submesoscale circulations on surface Lagrangian statistics in the northern Gulf of Mexico over 2 months, February and August, representative of winter and summer. The role of resolution and riverine forcing is explored focusing on surface waters in regions where the water column is deeper than 50 m. Whenever submesoscale circulations are present, the probability density functions (PDFs) of dynamical quantities such as vorticity and horizontal velocity divergence for Eulerian and Lagrangian fields differ, with particles preferentially mapping areas of elevated negative divergence and positive vorticity. The stronger the submesoscale circulations are, the more skewed the Lagrangian distributions become, with greater differences between Eulerian and Lagrangian PDFs. In winter, Lagrangian distributions are modestly impacted by the presence of the riverine outflow, while increasing the model resolution from submesoscale permitting to submesoscale resolving has a more profound impact. In summer, the presence of riverine-induced buoyancy gradients is the key to the development of submesoscale circulations and different Eulerian and Lagrangian PDFs. Finite-size Lyapunov exponents (FSLEs) are used to characterize lateral mixing rates. Whenever submesoscale circulations are resolved and riverine outflow is included, FSLEs slopes are broadly consistent with local stirring. Simulated slopes are close to −0.5 and support a velocity field where the ageostrophic and frontogenetic components contribute stirring at scales between about 5 and 7 times the model resolution and 100 km. The robustness of Lagrangian statistics is further discussed in terms of their spatial and temporal variability and of the number of particles available.

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Roy Barkan, James C. McWilliams, Alexander F. Shchepetkin, M. Jeroen Molemaker, Lionel Renault, Annalisa Bracco, and Jun Choi

Abstract

Realistic, submesoscale-resolving numerical simulations are used to characterize the flow’s statistics and the geography of surface submesoscale currents in the northern Gulf of Mexico. This study examines the role of the Mississippi–Atchafalaya River system in driving submesoscale currents during winter and summer, on and off the shelf, by investigating two sets of statistically equilibrated solutions, with and without river forcing. In this paper, the first of three, the authors analyze vorticity ζ, horizontal divergence δ, and available potential energy to eddy kinetic energy conversion and show that river forcing has an important effect on the spatial distribution and magnitudes of submesoscale currents in both seasons. During winter, solutions without river forcing display an increase in seasonal-mean values of ζ, δ and compared to solutions with river forcing, particularly east of the Mississippi River delta and offshore. On the contrary, during summer, seasonal-mean values are larger in solutions with river forcing throughout the entire region. The river effects can be rationalized in terms of scaling arguments that relate submesoscale current magnitudes to the surface boundary layer depth and lateral buoyancy gradients. River outflow enhances submesoscale currents by increasing lateral buoyancy gradients but suppresses them by decreasing the boundary layer depth. A discussion of the submesoscale-generating mechanisms that in each season may determine whether the enhancement effect overcomes the suppression effect or vice versa is presented. Regional comparisons of horizontal velocity spectra, root-mean-square ζ, root-mean-square δ, and root‐mean‐square across different resolutions show no sign of convergence even at 150-m horizontal resolution. This demonstrates the numerical challenge of modeling the full range of submesoscale currents.

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Lionel Renault, M. Jeroen Molemaker, James C. McWilliams, Alexander F. Shchepetkin, Florian Lemarié, Dudley Chelton, Serena Illig, and Alex Hall

Abstract

In this study, uncoupled and coupled ocean–atmosphere simulations are carried out for the California Upwelling System to assess the dynamic ocean–atmosphere interactions, namely, the ocean surface current feedback to the atmosphere. The authors show the current feedback, by modulating the energy transfer from the atmosphere to the ocean, controls the oceanic eddy kinetic energy (EKE). For the first time, it is demonstrated that the current feedback has an effect on the surface stress and a counteracting effect on the wind itself. The current feedback acts as an oceanic eddy killer, reducing by half the surface EKE, and by 27% the depth-integrated EKE. On one hand, it reduces the coastal generation of eddies by weakening the surface stress and hence the nearshore supply of positive wind work (i.e., the work done by the wind on the ocean). On the other hand, by inducing a surface stress curl opposite to the current vorticity, it deflects energy from the geostrophic current into the atmosphere and dampens eddies. The wind response counteracts the surface stress response. It partly reenergizes the ocean in the coastal region and decreases the offshore return of energy to the atmosphere. Eddy statistics confirm the current feedback dampens the eddies and reduces their lifetime, improving the realism of the simulation. Finally, the authors propose an additional energy element in the Lorenz diagram of energy conversion: namely, the current-induced transfer of energy from the ocean to the atmosphere at the eddy scale.

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Roy Barkan, James C. McWilliams, M. Jeroen Molemaker, Jun Choi, Kaushik Srinivasan, Alexander F. Shchepetkin, and Annalisa Bracco

Abstract

This paper, the second of three, investigates submesoscale dynamics in the northern Gulf of Mexico under the influence of the Mississippi–Atchafalaya River system, using numerical simulations at 500-m horizontal resolution with climatological atmospheric forcing. The Turner angle Tu, a measure of the relative effect of temperature and salinity on density, is examined with respect to submesoscale current generation in runs with and without riverine forcing. Surface Tu probability density functions in solutions including rivers show a temperature-dominated signal offshore, associated with Loop Current water, and a nearshore salinity-dominated signal, associated with fresh river water, without a clear compensating signal, as often found instead in the ocean’s mixed layer. The corresponding probability distribution functions in the absence of rivers differ, illustrating the key role played by the freshwater output in determining temperature–salinity distributions in the northern Gulf of Mexico during both winter and summer. A quantity referred to as temperature–salinity covariance is proposed to determine what fraction of the available potential energy that is released during the generation of submesoscale circulations leads to the destruction of density gradients while leaving spice gradients untouched, thereby leading to compensation. It is shown that the fresh river fronts to the east of the Bird’s Foot can evolve toward compensation in concert with a gradual release of available potential energy. It is further demonstrated that, during winter, the cross-shelf freshwater transport mechanism to the west of the Bird’s Foot is well approximated by a diffusive process, whereas to the east is better represented by a ballistic process associated with the Mississippi water that converges in a jetlike pattern.

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