Chris Hill

Hydrostatic, quasi‐hydrostatic, and nonhydrostatic ocean modeling

Ocean models based on consistent hydrostatic, quasi-hydrostatic, and nonhydrostatic equation sets are formulated and discussed. The quasi-hydrostatic and nonhydrostatic sets are more accurate than the widely used hydrostatic primitive equations. Quasi-hydrostatic models relax the precise balance between gravity and pressure gradient forces by including in a consistent manner cosine-of-latitude Coriolis terms which are neglected in primitive equation models. Nonhydrostatic models employ the full incompressible Navier Stokes equations; they are required in the study of small-scale phenomena in the ocean which are not in hydrostatic balance. We outline a solution strategy for the Navier Stokes model on the sphere that performs efficiently across the whole range of scales in the ocean, from the convective scale to the global scale, and so leads to a model of great versatility. In the hydrostatic limit the Navier Stokes model involves no more computational effort than those models which assume strict hydrostatic balance on all scales. The strategy is illustrated in simulations of laboratory experiments in rotating convection on scales of a few centimeters, simulations of convective and baroclinic instability of the mixed layer on the 1- to 10-km scale, and simulations of the global circulation of the ocean.

Implementation of an Atmosphere–Ocean General Circulation Model on the Expanded Spherical Cube

A hydrodynamical kernel that drives both an atmospheric and oceanic general circulation model is implemented in general orthogonal curvilinear coordinates using the finite-volume method on the sphere. The finite-volume method naturally describes arbitrary grids, and use of the vector-invariant form of the momentum equations simplifies the generalization to arbitrary coordinates. Grids based on the expanded spherical cube of Rancic et al., which contain eight singular points, are used. At these singularities the grid is nonorthogonal. The combined use of vector-invariant equations and the finite-volume method is shown to avoid degeneracy at these singular points. The model is tested using experiments proposed by Williamson et al. and Held and Saurez. The atmospheric solutions are examined seeking evidence of the underlying grid in solutions and eddy statistics. A global ocean simulation is also conducted using the same code. The solutions prove to be accurate and free of artifacts arising from the cubic grid.

Estimates and Implications of Surface Eddy Diffusivity in the Southern Ocean Derived from Tracer Transport

Abstract Near-surface “effective diffusivities” associated with geostrophic eddies in the Southern Ocean are estimated by numerically monitoring the lengthening of idealized tracer contours as they are strained by surface geostrophic flow observed by satellite altimetry. The resulting surface diffusivities show considerable spatial variability and are large (2000 m2 s−1) on the equatorward flank of the Antarctic Circumpolar Current and are small (500 m2 s−1) at the jet axis. Regions of high and low effective diffusivity are shown to be collocated with regions of, respectively, weak and strong isentropic potential vorticity gradients. The maps of diffusivity are used, along with climatological estimates of surface wind stress and air–sea buoyancy flux, to estimate surface meridional residual flows and the relative importance of Eulerian and eddy-induced circulation in the streamwise-averaged dynamics of the Antarctic Circumpolar Current.

Scales, Growth Rates, and Spectral Fluxes of Baroclinic Instability in the Ocean

AbstractAn observational, modeling, and theoretical study of the scales, growth rates, and spectral fluxes of baroclinic instability in the ocean is presented, permitting a discussion of the relation between the local instability scale; the first baroclinic deformation scale Rdef; and the equilibrated, observed eddy scale. The geography of the large-scale, meridional quasigeostrophic potential vorticity (QGPV) gradient is mapped out using a climatological atlas, and attention is drawn to asymmetries between midlatitude eastward currents and subtropical return flows, the latter of which has westward and eastward zonal velocity shears. A linear stability analysis of the climatology, under the “local approximation,” yields the growth rates and scales of the fastest-growing modes. Fastest-growing modes on eastward-flowing currents, such as the Kuroshio and the Antarctic Circumpolar Current, have a scale somewhat larger (by a factor of about 2) than Rdef. They are rapidly growing (e folding in 1–3 weeks) and d...

NASA supercomputer improves prospects for ocean climate research

Estimates of ocean circulation constrained by in situ and remotely sensed observations have become routinely available during the past five years, and they are being applied to myriad scientific and operational problems [Stammer et al., 2002]. Under the Global Ocean Data Assimilation Experiment (GODAE), several regional and global estimates have evolved for applications in climate research, seasonal forecasting, naval operations, marine safety, fisheries, the offshore oil industry coastal management, and other areas. This article reports on recent progress by one effort, the consortium for Estimating the Circulation and Climate of the Ocean (ECCO), toward a next-generation synthesis of ocean and sea-ice data that is global, that covers the full ocean depth, and that permits eddies.

OpenAD/F: A Modular Open-Source Tool for Automatic Differentiation of Fortran Codes

The Open/ADF tool allows the evaluation of derivatives of functions defined by a Fortran program. The derivative evaluation is performed by a Fortran code resulting from the analysis and transformation of the original program that defines the function of interest. Open/ADF has been designed with a particular emphasis on modularity, flexibility, and the use of open source components. While the code transformation follows the basic principles of automatic differentiation, the tool implements new algorithmic approaches at various levels, for example, for basic block preaccumulation and call graph reversal. Unlike most other automatic differentiation tools, Open/ADF uses components provided by the Open/AD framework, which supports a comparatively easy extension of the code transformations in a language-independent fashion. It uses code analysis results implemented in the OpenAnalysis component. The interface to the language-independent transformation engine is an XML-based format, specified through an XML schema. The implemented transformation algorithms allow efficient derivative computations using locally optimized cross-country sequences of vertex, edge, and face elimination steps. Specifically, for the generation of adjoint codes, Open/ADF supports various code reversal schemes with hierarchical checkpointing at the subroutine level. As an example from geophysical fluid dynamics, a nonlinear time-dependent scalable, yet simple, barotropic ocean model is considered. OpenAD/Fs reverse mode is applied to compute sensitivities of some of the models transport properties with respect to gridded fields such as bottom topography as independent (control) variables.

Design and Implementation of Components in the Earth System Modeling Framework

The Earth System Modeling Framework is a component-based architecture for developing and assembling climate and related models. A virtual machine underlies the component-level constructs in ESMF, providing both a foundation for performance portability and mechanisms for resource allocation and component sequencing.

A New Treatment of the Coriolis Terms in C-Grid Models at Both High and Low Resolutions

Abstract Numerical models of the ocean typically employ gridpoint techniques in which the dynamical variables defining the state of the ocean are held on a staggered grid. One common arrangement of the variables, known as the Arakawa C-grid, is particularly prone to gridscale noise that is due to spatial averaging of Coriolis terms and that is manifest when the grid resolution is coarse with respect to the deformation radius. Here, the authors analyze the problem in the context of linear inertia–gravity waves and discuss the reason for the prevalence of noise. They suggest a solution to the problem in which the C-grid model variables are augmented with D-grid velocity variables. An analysis of the resulting C–D grid indicates favorable behavior and numerical results are presented to demonstrate this. Finally, they discuss the similarity in nature between the C–D grid and the Z-grid, to explain why the C–D grid works well at both high and low resolution.

Conservation of properties in a free-surface model

In height coordinate ocean models, natural conservation of tracers (temperature, salinity or any passive tracer) requires that the thickness of the surface cell varies with the free-surface displacement, leading to a non-linear free-surface formulation (NLFS). However, NLFS does not guarantee exact conservation unless special care is taken in the implementation, and in particular the time stepping scheme, as pointed out by Griffies et al. (Monthly Weather Rev. 129 (2001) 1081). This paper presents a general method to implement a NLFS in a conservative way, using an implicit free surface formulation. Details are provided for two tracer time stepping schemes, both second order in time and space: a two time-level scheme, such as Lax–Wendroff scheme, guarantees exact tracer conservation;a three time-level scheme such as the Adams–Bashforth II requires further adaptations to achieve exact local conservation and accurate global conservation preventing long term drift of the model tracer content. No compromise is required between local and global conservation since the method accurately conserves any tracer. In addition to the commonly used backward time stepping, the implicit free surface formulation also offers the option of a Crank–Nickelson time stepping which conserves the energy. The methods are tested in idealized experiments designed to emphasize problems of tracer and energy conservation. The tests show the ability of the NLFS method to conserve tracers, in contrast to the linear free-surface formulation. At test of energy conservation reveals that free-surface backward time-stepping strongly damps the solution. In contrast, Crank–Nickelson time stepping exactly conserves energy in the pure linear case and confirms the NLFS improvement relative to the linear free-surface when momentum advection is included. � 2003 Elsevier Ltd. All rights reserved.

Atmosphere–Ocean Modeling Exploiting Fluid Isomorphisms

Mathematical isomorphisms between the hydrostatic equations that govern the evolution of a compressible atmosphere and an incompressible ocean are described and exploited to guide the design of a hydrodynamical kernel for simulation of either fluid.