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J. S. Allen and P. A. Newberger

Abstract

Intermediate models contain physics between that in the primitive equations and that in the quasigeostrophic equations. The specific objective here is to investigate the absolute and relative accuracy of several intermediate models for stratified flow by a comparison of numerical finite-difference solutions with those of the primitive equations (PE) and with those of the quasigeostrophic (QG) equations. The numerical experiments involve initial-value problems for the time-dependent development of an unstable baroclinic jet on an f plane in a doubly periodic domain with flat bottom. Although the geometry is idealized, the problem is set up so that the dynamics should be similar to that of the baroclinic jet observed off the northern California coast in the Coastal Transition Zone (CTZ) field program. Three numerical experiments are conducted where the flow fields are characterized by local Rossby numbers that range from moderately small to O(1). The unstable jet develops finite amplitude meanders that grow in amplitude until they pinch off to form detached eddies on either side of the jet. The instability process is characterized by the transfer of potential to kinetic energy accompanied by a large increase in the barotropic component of the flow. Although the initial jet velocity profiles are symmetric about the jet centerline, as the Rossby number of the jet increases the meander growth and eddy detachment process becomes more asymmetrical about the jet axis. A meander on the positive vorticity side of the jet pinches off first to form a relatively large anticyclonic eddy followed in time by the detachment of a smaller cyclonic eddy on the negative vorticity side. The intermediate models that we consider are the balance equations (BE), the balance equations based on momentum equations (BEM), the iterated geostrophic models (IG2 and IG3), the linear balance equations (LBE), the linear BEM (LBEM), and the geostrophic momentum approximation (GM). We also include a second-order quasigeostrophic approximation (QG2) and a primitive equation model with semi-implicit time differencing (PESI). The results of the numerical experiments for moderate Rossby number flow show that the QG, QG2, LBE, and GM models give large errors and produce flow fields that have substantial qualitative differences from the PE. The LBEM model is somewhat better, while IG2 gives considerably smaller errors. The BE, BEM, IG3, and PESI models give highly accurate approximate solutions to PE, and that result holds also for those models applied to the O(1) Rossby number flow.

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J. S. Allen and P. A. Newberger

Abstract

Model studies of two-dimensional, time-dependent, wind-forced, stratified downwelling circulation on the continental shelf have shown that the near-bottom offshore flow can develop time- and space-dependent fluctuations involving spatially periodic separation and reattachment of the bottom boundary layer and accompanying recirculation cells. Based primarily on the observation that the potential vorticity Π, initially less than zero everywhere, is positive in the region of the fluctuations, this behavior was identified as finite amplitude slantwise convection resulting from a symmetric instability. To further support that identification, a direct stability analysis of the forced, time-dependent, downwelling circulation would be useful, but is difficult because the instabilities develop as an integral part of the evolving flow field. The objectives of the present study are 1) to examine the linear stability of a near-bottom oceanic flow over sloping topography with conditions dynamically similar to those in the downwelling circulation and 2) to establish a link between the instabilities observed in the wind-forced downwelling problem and the results of recent theoretical studies of bottom boundary layer behavior in stratified oceanic flows over sloping topography. These objectives are addressed by investigating the two-dimensional linear stability and the nonlinear behavior of the steady, inviscid, “arrested Ekman layer” solution produced by transient downwelling in one-dimensional models of stratified flow adjustment over a sloping bottom. A linear stability analysis shows that this solution is unstable to symmetric instabilities and confirms that a necessary condition for instability is Π > 0 in the bottom layer. Numerical experiments show that the unstable, time-dependent, nonlinear behavior in the boundary layer involves the formation of slantwise circulation cells with characteristics similar to those found in the wind-forced downwelling circulation and the development of weak stable stratifiction close to that corresponding to marginally stable conditions with Π = 0.

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J. S. Allen and P. A. Newberger

Abstract

Time-dependent downwelling on the Oregon continental shelf is studied with a two-dimensional approximation, that is, spatial variations across shelf and with depth, using the Blumberg-Mellor, finite-difference, stratified, hydrostatic, primitive equation model. The time-dependent response of a coastal ocean at rest to constant, downwelling-favorable wind stress is examined. Topography and stratification representative of the Oregon continental shelf are used for the basic case experiment. The wind stress forces onshore flow in a turbulent surface boundary layer. The compensating flow below the surface layer advects the density field downward and offshore and accelerates an alongshore current in the form of a vertically and horizontally sheared coastal jet. The dominant feature of the response flow field is a downwelling front that moves slowly offshore, leaving behind an inshore region where the density is well mixed. The downwelling front in the density field is concentrated near the bottom, while the front in alongshore velocity extends over the full depth and is nearly vertical, separating weak alongshore velocities inshore from the coastal jet offshore. The front contains strong vertical motion from the surface to the bottom and some recirculation. Much of the offshore flow from the base of the front is characterized by time- and space-dependent fluctuations involving spatially periodic separation and reattachment of the bottom boundary layer and accompanying recirculation cells. This flow has positive potential vorticity and appears to be finite-amplitude slantwise convection resulting from a hydrostatic symmetric instability. Additional experiments show the dependence of the response flow field an the magnitude of the wind stress, the initial stratification, and the shelf topography. Experiments with the vertical turbulent kinematic viscosity and diffusivity parameterized by a different turbulence closure scheme, as a function of a local Rich-ardson number, or as constants show dependence of the response flow field on the choice of turbulence submodel. The occurrence of a well-mixed region inshore and the existence of time- and space-dependent fluctuations associated with slantwise convection in the near-bottom offshore flow appear to be robust features of the two-dimensional downwelling response.

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J. A. Barth, J. S. Allen, and P. A. Newberger

Abstract

As part of a program to improve understanding of the dynamics of the complicated, vigorous eddy and jet flow fields recently observed over the continental shelf and slope, we investigate the potential of intermediate models for use in both process and data assimilation studies of these flows. Intermediate models incorporate physics simpler than that contained in the full primitive equations yet more complete than in the quasi-geostrophic equations, and are capable of representing subinertial flows over O(1) bottom topographic variations and/or with O(1) isopycnal slopes. In addition, intermediate models dynamically filter out high-frequency gravity- inertial motions leading, potentially, to higher computational efficiency and well-posed limited area forecast/hindcast models. Initial studies focus on single layer flows on an f-plane with a free surface, governed by the shallow-water equations. In Part I, various intermediate models are formulated and their accuracy assessed by comparing some exact nonlinear analytical solutions that exist for the shallow-water equations with corresponding analytical solutions of the intermediate models. Here in Part II, an extensive set of numerical finite-difference solutions to initial-value problems in doubly periodic domains (to isolate model differences from the influence of boundary condition implementation on solid walls) is used to determine the accuracy of various intermediate models by comparing their predictions with those of a shallow-water equation model that uses a potential enstrophy and energy conserving numerical scheme (SWE). Intermediate model results are also contrasted with those from a quasi-geostrophic (QG) model. The intermediate models considered are based on the geostrophic momentum (GM) approximation, the derivation of Salmon utilizing Hamilton's principle (HP), a geostrophic vorticity (GV) approximation, a combination of the quasi-geostrophic momentum and full continuity equations (IM), the linear balance equations (LBE), the balance equations (BE), the related balance-type (HBE, BEM, NBE) and modified linear balance equations (LQBE), and on Lynch's slow equations in their original form (SE) and in a modified form (MSE). In addition, a semi-implicit version (SEMI) of the shallow-water equations, which numerically filters high-frequency motions, is included in the study. The basic initial-value problem used to test the various intermediate models involves sinusoidal flow over a symmetric Gaussian-shaped bottom topographic feature. Comparisons are made for a range of the relevant dimensionless model parameters including the strength of the flow (as measured by the Rossby number), the square of the ratio of a characteristic horizontal length scale to the Rossby radius of deformation, and the height of the topographic feature. A second initial-value problem involves the evolution of a rotating elliptical vortex over both flat and variable bottom topography. Results show that in cases with low local Rossby number flow, but with large topographic height, most of the intermediate models are substantially more accurate than QG. Even for flows with O(1) local Rossby numbers, some of the intermediate models continue to give excellent results. Specifically, BE, BEM and SEMI consistently do the best in the comparisons. Although the relative ordering of the remaining models is somewhat parameter-dependent, in general the next most accurate models are MSE and LQBE followed by HP and NBE. These are followed in quality by GV and GM while the remaining models, HBE, IM and LBE, perform least accurately for the range of parameters studied. Generally, intermediate models that have an integral invariant corresponding to conservation of potential enstrophy do better than those without, with the best results coming from models which have this property and a nonlinear balance equation.

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J. S. Allen, J. A. Barth, and P. A. Newberger

Abstract

The study of intermediate models for barotropic continental shelf and slope flow fields initiated in Parts I and II is continued. The objective is to investigate the possible use of intermediate models for process and data assimilation studies of nonlinear mesoscale eddy and jet current fields over the continental shelf and slope. Intermediate models contain physics between that in the primitive equations and that in the quasi-geostrophic equations and are capable of representing subinertial frequency motion over the O(1) topographic variations typical of the continental slope while filtering out high-frequency gravity–inertial waves. We concentrate on the application of intermediate models to the f-plane shallow-water equations. The accuracy of several intermediate models is evaluated here by a comparison of numerical finite-difference solutions with those of the primitive shallow-water equations (SWE) and with those of the quasi-geostrophic equations (QG) for flow in a periodic channel. The intermediate models that we consider are based on the balance equations (BE), the balance equations derived from momentum equations (BEM), the potential vorticity conserving linear balance equations (LQBE), the hybrid balance equations (HBE), the near balance equation (NBE), a geostrophic vorticity (GV) approximation, the geostrophic momentum (GM) approximation, and the quasi-geostrophic momentum and full continuity equations (IM). The periodic channel provides a basic geometry for the study of physical flow processes over the continental shelf and slope. Wall boundary conditions are formulated for the intermediate models and implemented in the numerical finite-difference approximations. The ability of intermediate models to represent linear ageostrophic coastally trapped waves, i.e., Kelvin and continental shelf waves, is verified by numerical experiments. The results of numerical solution intercomparisons for initial-value problems involving O(1) topographic variations are as follows. For flow at small local Rossby number |εL| < 0.2, where εL is given by the magnitude of the vorticity divided by f, all of the intermediate models do well, while the QG model does poorly. For flows with larger values of |εL|, e.g., |εL| ≈ 0.5, the performance of the different intermediate models varies. BEM and BE consistently give extremely accurate solutions while the solutions from LQBE are almost as good. The other models are substantially less accurate with errors generally increasing in the order NBE, HBE, GV, GM, IM. The QG solution always has the largest errors. Consistent with the results from the studies in Part II in a doubly periodic domain, the balance equations BE and BEM, followed closely by LQBE, appear to be the most accurate intermediate models.

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J. S. Allen, J. A. Barth, and P. A. Newberger

Abstract

Motivated by the general objective of pursuing oceanographic process and data assimilation studies of the complex, nonlinear eddy and jet current fields observed over the continental shelf and slope off the west coast of the United States, we investigate the use of intermediate models for that purpose. Intermediate models contain physics between that in the primitive equations and that in the quasigeostrophic equations and are capable of representing subinertial frequency motion over the O(1) topographic variations typical of the continental slope, while filtering out high-frequency gravity-inertial waves. As an initial step, we compare and evaluate several intermediate models applied to the f-plane shallow-water equations for flows over topography. The accuracy and utility of the intermediate models are assessed by a comparison of exact analytical and numerical solutions with those of the primitive shallow-water equations (SWE) and with those of the quasi-geostrophic equations (QG). The intermediate models that we consider are based on the geostrophic momentum (GM) approximation, the derivation of Salmon (1983) utilizing Hamilton's principle (HP), a geostrophic vorticity (GV) approximation, the quasi-geostrophic momentum and full continuity equations (IM), the linear balance equations (LBE), the balance equations (BE), the related balance-type (HBE, BEM, NBE) and modified linear balance equations (LQBE), the slow equations (SE) of Lynch (1989), and the modified slow equations (MSE). In Part I, we discuss the intermediate models and develop formulations that are suitable for numerical solution in physical coordinates for use in Parts II and III. We investigate the capability of the intermediate models to represent linear ageostrophic coastally trapped waves, i.e., Kelvin and continental shelf waves, and demonstrate that they do so with accuracy consistent with standard linear low-frequency approximations. We also assess the accuracy of the models by a comparison of exact nonlinear analytical solutions to the SWE for steady flow in an elliptic paraboloid and for unsteady motion of elliptical vortices in a circular paraboloid with corresponding analytical solutions to the intermediate models and to QG. General results from the exact solution comparisons include the following. Many of the intermediate models are capable of producing more accurate solutions than QG over a range of Rossby numbers 0 < ε < 1. In some cases, the intermediate models provide accurate approximate solutions where QG is not applicable and fails to give a relevant solution. Considerable parameter-dependent variation in quality exists, however, among the different intermediate models. For the particular problems considered here, BE, HBE, BEM, NBE, and MSE reproduce the exact results of the SWE while LBE and LQBE give the same approximation as QG. The accuracy of the models is typically in the order GV, GM, IM, HP, and QG, with GV most accurate and IM and HP sometimes less accurate than QG.

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J. S. Allen, P. A. Newberger, and J. Federiuk

Abstract

Time-dependent upwelling on the Oregon continental shelf is studied with a two-dimensional approximation, that is, spatial variations across-shelf and with depth, using the Blumberg–Mellor, finite-difference, stratified, primitive equation model. The time-dependent response of a coastal ocean at rest to constant, upwelling favorable, wind stress is examined. Topography and stratification representative of the Oregon continental shelf are used for the basic case experiment. The wind stress forces offshore flow in a turbulent surface boundary layer. The compensating onshore flow below the surface layer accelerates an alongshore current in the form of a vertically and horizontally sheared coastal jet. Dense water is advected onshore and upward into the surface layer. An upwelling front characterized by relatively large horizontal gradients in density and alongshore velocity v is formed near the surface at the inshore edge of the coastal jet. Large vertical gradients in v and large values of turbulent kinetic energy are also found in the frontal region. Additional experiments show the dependence of the response flow field on the initial stratification and the shelf topography. Experiments with the vertical turbulent kinematic viscosity and diffusivity parameterized as constants or as functions of a local Richardson number show substantial dependence of the response flow field on the choice of turbulence submodel.

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