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• 1. [Article] On Intermediate Models for Barotropic Continental Shelf and Slope Flow Fields. Part II: Comparison of Numerical Model Solutions in Doubly Periodic Domains

  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 ...

  Citation × Citation Citation Abstract Copy Title: On Intermediate Models for Barotropic Continental Shelf and Slope Flow Fields. Part II: Comparison of Numerical Model Solutions in Doubly Periodic Domains Author: Newberger, P. A., Allen, J. S., Barth, J. A. 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. Title: On Intermediate Models for Barotropic Continental Shelf and Slope Flow Fields. Part II: Comparison of Numerical Model Solutions in Doubly Periodic Domains Author: Newberger, P. A., Allen, J. S., Barth, J. A. 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. Close

• 2. [Article] Upwelling Circulation on the Oregon Continental Shelf. Part II: Simulations and Comparisons with Observations

  Sixty-day simulations of flow on the Oregon continental shelf are performed using the Blumberg and Mellor sigma coordinate, primitive equation model. The model is two-dimensional (an across-shelf section) ...

  Citation × Citation Citation Abstract Copy Title: Upwelling Circulation on the Oregon Continental Shelf. Part II: Simulations and Comparisons with Observations Author: Federiuk, J., Allen, J. S. Sixty-day simulations of flow on the Oregon continental shelf are performed using the Blumberg and Mellor sigma coordinate, primitive equation model. The model is two-dimensional (an across-shelf section) with high spatial resolution and realistic shelf topography. Forcing consists of surface heat flux, either hourly or low-pass filtered wind stress, and in one case, a constant alongshore pressure gradient. Model results are compared with current and hydrographic measurements from the CUE-2 program. The horizontal scale of the alongshore coastal jet is significantly influenced by the structure of the initial density and velocity fields. The model successfully reproduces the vertical shear in the alongshore velocity field v, but the model's mean v field is too strongly southward, and the variance in both the vvu and v fields is underpredicted. Inclusion of the alongshore pressure gradient, while improving prediction of the mean alongshore velocities, does not improve the model-data correlation. The time-mean model density agrees with observations at middepths over the shelf, but shows larger values than observed near the surface. The results demonstrate the importance of including a surface heat flux and of specifying realistic initial density and alongshore velocity fields. Title: Upwelling Circulation on the Oregon Continental Shelf. Part II: Simulations and Comparisons with Observations Author: Federiuk, J., Allen, J. S. Sixty-day simulations of flow on the Oregon continental shelf are performed using the Blumberg and Mellor sigma coordinate, primitive equation model. The model is two-dimensional (an across-shelf section) with high spatial resolution and realistic shelf topography. Forcing consists of surface heat flux, either hourly or low-pass filtered wind stress, and in one case, a constant alongshore pressure gradient. Model results are compared with current and hydrographic measurements from the CUE-2 program. The horizontal scale of the alongshore coastal jet is significantly influenced by the structure of the initial density and velocity fields. The model successfully reproduces the vertical shear in the alongshore velocity field v, but the model's mean v field is too strongly southward, and the variance in both the vvu and v fields is underpredicted. Inclusion of the alongshore pressure gradient, while improving prediction of the mean alongshore velocities, does not improve the model-data correlation. The time-mean model density agrees with observations at middepths over the shelf, but shows larger values than observed near the surface. The results demonstrate the importance of including a surface heat flux and of specifying realistic initial density and alongshore velocity fields. Close

• 3. [Article] Iterated Geostrophic Intermediate Models

  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 O(1) topographic ...

  Citation × Citation Citation Abstract Copy Title: Iterated Geostrophic Intermediate Models Author: Allen, J. S. 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 O(1) topographic variations typical of the continental slope while filtering out high-frequency gravity-inertial waves. We present here a formulation for stratified flow of a set of new intermediate models, termed iterated geostrophic (IG) models, derived under the assumption that the Rossby number ϵ is small. We consider the emotion of a rotating, continuously stratified fluid governed by the hydrostatic, Boussinesq, adiabatic primitive equations (PE) with a spatially variable Coriolis parameter and with weak biharmonic momentum diffusion. The IG models utilize the pressure field as the basic variable [as in the quasigeostrophic (QG) approximation], are capable of providing solutions of formally increasing accuracy in powers of ϵ in a systematic manner, and are straightforward to solve numerically. The 1G models are obtained by iteration, at a fixed time t = t₀, of the momentum and thermodynamic equations using the known pressure field ϕ(x, t). The iteration procedure products a sequence of estimates of increasing accuracy for the velocity components and for the time derivative of the pressure field ϕt(x, t₀). The formulation is asymptotic in the sense that, given the pressure field, at each iteration the velocity components and ϕ₁(x, t₀) are formally determined to a higher order of accuracy in powers of ϵ. The order of the IG model is specified by the predetermined fixed number of iterations N. Thus, a set of models is produced depending on the choice for N, and the different models are denoted by IGN. The value of ϕ₁(x, t₀) obtained from iteration N is used with a time difference scheme to advance the pressure field in time, and the process may be repeated. Energy and potential enstrophy conservation in the IG models are asymptotic. In the following companion paper (Allen and Newberger), the accuracies of several intermediate models, including IG2 and IG3, are investigated by a comparison of numerical finite-difference solutions to those of the primitive equations. For moderate Rossby number flows, it is found that IG2 gives approximate solutions of reasonable accuracy, with errors substantially smaller than those obtained from QG and several other intermediate models. The IG3 model is found to give extremely accurate approximate solutions for flows with Rossby numbers that range from moderately small to O(1). Title: Iterated Geostrophic Intermediate Models Author: Allen, J. S. 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 O(1) topographic variations typical of the continental slope while filtering out high-frequency gravity-inertial waves. We present here a formulation for stratified flow of a set of new intermediate models, termed iterated geostrophic (IG) models, derived under the assumption that the Rossby number ϵ is small. We consider the emotion of a rotating, continuously stratified fluid governed by the hydrostatic, Boussinesq, adiabatic primitive equations (PE) with a spatially variable Coriolis parameter and with weak biharmonic momentum diffusion. The IG models utilize the pressure field as the basic variable [as in the quasigeostrophic (QG) approximation], are capable of providing solutions of formally increasing accuracy in powers of ϵ in a systematic manner, and are straightforward to solve numerically. The 1G models are obtained by iteration, at a fixed time t = t₀, of the momentum and thermodynamic equations using the known pressure field ϕ(x, t). The iteration procedure products a sequence of estimates of increasing accuracy for the velocity components and for the time derivative of the pressure field ϕt(x, t₀). The formulation is asymptotic in the sense that, given the pressure field, at each iteration the velocity components and ϕ₁(x, t₀) are formally determined to a higher order of accuracy in powers of ϵ. The order of the IG model is specified by the predetermined fixed number of iterations N. Thus, a set of models is produced depending on the choice for N, and the different models are denoted by IGN. The value of ϕ₁(x, t₀) obtained from iteration N is used with a time difference scheme to advance the pressure field in time, and the process may be repeated. Energy and potential enstrophy conservation in the IG models are asymptotic. In the following companion paper (Allen and Newberger), the accuracies of several intermediate models, including IG2 and IG3, are investigated by a comparison of numerical finite-difference solutions to those of the primitive equations. For moderate Rossby number flows, it is found that IG2 gives approximate solutions of reasonable accuracy, with errors substantially smaller than those obtained from QG and several other intermediate models. The IG3 model is found to give extremely accurate approximate solutions for flows with Rossby numbers that range from moderately small to O(1). Close

• 4. [Article] On Intermediate Models for Stratified Flow

  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 ...

  Citation × Citation Citation Abstract Copy Title: On Intermediate Models for Stratified Flow Author: Newberger, P. A., Allen, J. S. 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. Title: On Intermediate Models for Stratified Flow Author: Newberger, P. A., Allen, J. S. 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. Close

• 5. [Article] On Intermediate Models for Barotropic Continental Shelf and Slope Flow Fields. Part III: Comparison of Numerical Model Solutions in Periodic Channels

  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 ...

  Citation × Citation Citation Abstract Copy Title: On Intermediate Models for Barotropic Continental Shelf and Slope Flow Fields. Part III: Comparison of Numerical Model Solutions in Periodic Channels Author: Newberger, P. A., Barth, J. A., Allen, J. S. 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 |ϵ[subscript]L| < 0.2, where ϵ[subscript]L 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 |ϵ[subscript]L|, e.g., |ϵ[subscript]L| ≈ 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 model Title: On Intermediate Models for Barotropic Continental Shelf and Slope Flow Fields. Part III: Comparison of Numerical Model Solutions in Periodic Channels Author: Newberger, P. A., Barth, J. A., Allen, J. S. 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 |ϵ[subscript]L| < 0.2, where ϵ[subscript]L 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 |ϵ[subscript]L|, e.g., |ϵ[subscript]L| ≈ 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 model Close