Alexander F. Shchepetkin

Open boundary conditions for long-term integration of regional oceanic models

Regional oceanic models can be developed and used efficiently for the investigation of regional and coastal domains, provided a satisfactory prescription for the open boundary conditions (OBCs) is found. We propose in this paper an adaptive algorithm where inward and outward information fluxes are treated separately. Because of the essentially hyperbolic nature of the incompressible, hydrostatic Primitive Equations, external data are required only for inward boundary fluxes. The outward fluxes are treated with a new algorithm for two-dimensional radiation. Special attention is given to the estimation of the radiation phase speed, essential for detecting the direction of boundary fluxes. The boundary conditions are applied and assessed on a US West Coast (USWC) configuration of the Regional Oceanic Modeling System (ROMS). Our guiding principles are that the numerical solution be stable over multiple years, reach a meaningful statistical equilibrium, and be realistic with respect to the available observational data. A sensitivity analysis suggests that the oblique radiation is robust and sufficiently accurate to detect the direction of information fluxes. The adaptive nudging adequately incorporates the external information minimizing over- and under-specification problems. In addition, a volume constraint based on global correction of normal barotropic velocities improves the overall performances of the open boundary conditions.

Model evaluation experiments in the North Atlantic Basin: simulations in nonlinear terrain-following coordinates

Abstract A primitive equation ocean circulation model in nonlinear terrain-following coordinates is applied to a decadal-length simulation of the circulation in the North Atlantic Ocean. In addition to the stretched sigma coordinate, novel features of the model include the utilization of a weakly dissipative, third-order scheme for tracer advection, and a conservative and constancy-preserving time-stepping algorithm. The objectives of the study are to assess the quality of the new terrain-following model in the limit of realistic basin-scale simulations, and to compare the results obtained with it against those of other North Atlantic models used in recent multi-model comparison studies. The new model is able to reproduce many features of both the wind-driven and thermohaline circulation, and to do so within error bounds comparable with prior model simulations (e.g., CME and DYNAMO). Quantitative comparison with comparable results obtained with the Miami Isopycnic Coordinate Model (MICOM) show our terrain-following solutions are of similar overall quality when viewed against known measures of merit including meridional overturning and heat flux, Florida Straits and Gulf Stream transport, seasonal cycling of temperature and salinity, and upper ocean currents and tracer fields in the eastern North Atlantic Basin. Sensitivity studies confirm that the nonlinear vertical coordinate contributes significantly to model fidelity, and that the global inventories and spatial structure of the tracer fields are affected in important ways by the choice of lateral advection scheme.

A method for computing horizontal pressure‐gradient force in an oceanic model with a nonaligned vertical coordinate

[1] Discretization of the pressure-gradient force is a long-standing problem in terrain-following (or σ) coordinate oceanic modeling. When the isosurfaces of the vertical coordinate are not aligned with either geopotential surfaces or isopycnals, the horizontal pressure gradient consists of two large terms that tend to cancel; the associated pressure-gradient error stems from interference of the discretization errors of these terms. The situation is further complicated by the nonorthogonality of the coordinate system and by the common practice of using highly nonuniform stretching for the vertical grids, which, unless special precautions are taken, causes both a loss of discretization accuracy overall and an increase in interference of the component errors. In the present study, we design a pressure-gradient algorithm that achieves more accurate hydrostatic balance between the two components and does not lose as much accuracy with nonuniform vertical grids at relatively coarse resolution. This algorithm is based on the reconstruction of the density field and the physical z coordinate as continuous functions of transformed coordinates with subsequent analytical integration to compute the pressure-gradient force. This approach allows not only a formally higher order of accuracy, but it also retains and expands several important symmetries of the original second-order scheme to high orders [Mellor et al., 1994; Song, 1998], which is used as a prototype. It also has built-in monotonicity constraining algorithm that prevents appearance of spurious oscillations of polynomial interpolant and, consequently, insures numerical stability and robustness of the model under the conditions of nonsmooth density field and coarse grid resolution. We further incorporate an alternative method of dealing with compressibility of seawater, which escapes pressure-gradient errors associated with interference of the nonlinear nature of equation of state and difficulties to achieve accurate polynomial fits of resultant in situ density profiles. In doing so, we generalized the monotonicity constraint to guarantee nonnegative physical stratification of the reconstructed density profile in the case of compressible equation of state. To verify the new method, we perform traditional idealized (Seamount) and realistic test problems.

Equilibrium structure and dynamics of the California Current, System

This paper addresses the structure and dynamical mechanisms of regional and mesoscale physical variability in the subtropical northeast Pacific Ocean using the Regional Oceanic Modeling System (ROMS). The model is configured with a U.S. West Coast domain that spans the California Current System (CCS) with a mesoscale horizontal resolution up to as fine as 3.5 km. Its mean-seasonal forcing is by momentum, heat, and water fluxes at the surface and adaptive nudging to gyre-scale fields at the open water boundaries. Its equilibrium solutions show realistic mean and seasonal states and vigorous mesoscale eddies, fronts, and filaments. The level of eddy kinetic energy (EKE) in the model is comparable to drifter and altimeter estimates in the solutions with sufficiently fine resolution. Because the model lacks nonseasonal transient forcing, the authors conclude that the dominant mesoscale variability in the CCS is intrinsic rather than transiently forced. The primary eddy generation mechanism is the baroclinic instability of upwelling, alongshore currents. There is progressive movement of meanseasonal currents and eddy energy offshore and downward into the oceanic interior in an annually recurrent cycle. The associated offshore eddy heat fluxes provide the principal balance against nearshore cooling by mean Ekman transport and upwelling. The currents are highly nonuniform along the coast, with important influences by capes and ridges in both maintaining mean standing eddies and launching transient filaments and fronts.

Mesoscale to submesoscale transition in the California Current System Part I: Flow structure, eddy flux, and observational tests

In computational simulations of an idealized subtropical eastern boundary upwelling current system, similar to the California Current, a submesoscale transition occurs in the eddy variability as the horizontal grid scale is reduced to O(1) km. This first paper (in a series of three) describes the transition in terms of the emergent flow structure and the associated time-averaged eddy fluxes. In addition to the mesoscale eddies that arise from a primary instability of the alongshore, wind-driven currents, significant energy is transferred into submesoscale fronts and vortices in the upper ocean. The submesoscale arises through surface frontogenesis growing off upwelled cold filaments that are pulled offshore and strained in between the mesoscale eddy centers. In turn, some submesoscale fronts become unstable and develop submesoscale meanders and fragment into roll-up vortices. Associated with this phenomenon are a large vertical vorticity and Rossby number, a large vertical velocity, relatively flat horizontal spectra (contrary to the prevailing view of mesoscale dynamics), a large vertical buoyancy flux acting to restratify the upper ocean, a submesoscale energy conversion from potential to kinetic, a significant spatial and temporal intermittency in the upper ocean, and material exchanges between the surface boundary layer and pycnocline. Comparison with available observations indicates that submesoscale fronts and instabilities occur widely in the upper ocean, with characteristics similar to the simulations.

Quasi-Monotone Advection Schemes Based on Explicit Locally Adaptive Dissipation

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 twodimensional 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.

Mesoscale to Submesoscale Transition in the California Current System. Part II: Frontal Processes

Abstract This is the second of three papers investigating the regime transition that occurs in numerical simulations for an idealized, equilibrium, subtropical, eastern boundary, upwelling current system similar to the California Current. The emergent upper-ocean submesoscale fronts are analyzed from phenomenological and dynamical perspectives, using a combination of composite averaging and separation of distinctive subregions of the flow. The initiating dynamical process for the transition is near-surface frontogenesis. The frontal behavior is similar to both observed meteorological surface fronts and solutions of the approximate dynamical model called surface dynamics (i.e., uniform interior potential vorticity q and diagnostic force balance) in the intensification of surface density gradients and secondary circulations in response to a mesoscale strain field. However, there are significant behavioral differences compared to the surface-dynamics model. Wind stress acts on fronts through nonlinear Ekman ...

Correction and commentary for Ocean forecasting in terrain-following coordinates: Formulation and skill assessment of the regional ocean modeling system by Haidvogel et al., J. Comp. Phys. 227, pp. 3595-3624

Although our names appear as co-authors in the above article (Haidvogel et al. (2008) [1], hereafter H2008), we were not aware of its existence until after it was published. In reading the article, we discovered that a significant portion of it (~40%, or 10 pages) repeats three large fragments from our own previously published work, Shchepetkin and McWilliams (2005) [2] (hereafter SM2005), but now presented in such a way that the motivation for the specific algorithmic choices made in ROMS and the relations among the different model components are no longer clear. The model equations appearing in H2008, Section 2.1 (taken from an earlier article, Haidvogel et al. (2000) [3]) are not entirely consistent with the actual equations solved in the ROMS code, resulting in contradictions within H2008 itself. In our view the description in H2008 does not constitute a mathematically accurate statement about the hydrodynamic core of ROMS. The purpose of this note is to clarify and correct this, as well as to explain some of the algorithmic differences among ROMS versions now in use.

Mesoscale to Submesoscale Transition in the California Current System. Part III: Energy Balance and Flux

This is the last of a suite of three papers about the transition that occurs in numerical simulations for an idealized equilibrium, subtropical, eastern-boundary upwelling current system similar to the California Current. The transition is mainly explained by the emergence of ubiquitous submesoscale density fronts and ageostrophic circulations about them in the weakly stratified surface boundary layer. Here the high-resolution simulations are further analyzed from the perspective of the kinetic energy (KE) spectrum shape and spectral energy fluxes in the mesoscale-to-submesoscale range in the upper ocean. For wavenumbers greater than the mesoscale energy peak, there is a submesoscale power-law regime in the spectrum with an exponent close to -2. In the KE balance an important conversion from potential to kinetic energy takes place at all wavenumbers in both mesoscale and submesoscale ranges; this conversion is the energetic counterpart of the vertical restratification flux and frontogenesis discussed in the earlier papers. A significant forward cascade of KE occurs in the submesoscale range en route to dissipation at even smaller scales. This is contrary to the inverse energy cascade of geostrophic turbulence and it is, in fact, fundamentally associated with the horizontally divergent (i.e., ageostrophic) velocity component. The submesoscale dynamical processes of frontogenesis, frontal instability, and breakdown of diagnostic force balance are all essential elements of the energy cycle of potential energy conversion and forward KE cascade.

Developments in terrain-following ocean models: intercomparisons of numerical aspects

During the course of developing new numerical algorithms for a terrain-following ocean modeling system (TOMS), different numerical aspects have been evaluated through a comparison between two widely used community ocean models, the Princeton ocean model (POM) and the regional ocean modeling system (ROMS).While both models aim at modeling coastal to basin-scale problems using similar grids, their numerical algorithms, code structure, and parameterization options are very different.Sensitivity studies with an idealized channel flow and a steep seamount configuration demonstrate how different algorithms in the two models may affect numerical errors, the stability of the code and the computational efficiency.For example, new pressure gradient schemes using polynomial fits and new time stepping algorithms may reduce numerical errors and allow using longer time steps than standard schemes do.However, the new schemes may require more careful choices of time steps and the use of higher order advection schemes to maintain numerical stability. 2002 Elsevier Science Ltd.All rights reserved.