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## Abstract

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## Abstract

The problem of geostrophic adjustment, originally considered by C.G. Rossby, is solved in an axisymmetric geometry for a continuously stratified fluid, where the adjusted final state is in hydrostatic, gradient-wind balance. This problem is relevant to the generation of submesoscale coherent vortices in the ocean: diapycnal mixing events can create a local anomaly of less strongly stratified fluid, which then develops a balancing circulation through adjustment. An analytical solution is obtained for a few uniform-density layers, and this is compared with numerical solutions for continuous stratification. In both representations, two-dimensional solutions are compared with axisymmetric ones.

## Abstract

The problem of geostrophic adjustment, originally considered by C.G. Rossby, is solved in an axisymmetric geometry for a continuously stratified fluid, where the adjusted final state is in hydrostatic, gradient-wind balance. This problem is relevant to the generation of submesoscale coherent vortices in the ocean: diapycnal mixing events can create a local anomaly of less strongly stratified fluid, which then develops a balancing circulation through adjustment. An analytical solution is obtained for a few uniform-density layers, and this is compared with numerical solutions for continuous stratification. In both representations, two-dimensional solutions are compared with axisymmetric ones.

## Abstract

From measurements made during the 1973 Mid-Ocean Dynamics Experiment in the western North Atlantic, horizontal maps of the total dynamic pressure (or, in the geostrophic approximation, streamfunction) have been constructed for different vertical levels and time periods by the interpolation technique of objective analysis. The space and time sampling of the observations—several tens of kilometers horizontally, hundreds of meters vertically and several days in time—were adequate for resolving mesoscale eddies. The data consisted of velocities (displacement rates) at 1500 m depth from neutrally buoyant floats and vertical density profiles throughout the water column. The resulting maps have been considered from several, essentially phenomenological, points of view. These include descriptions of the synoptic eddy structure, the time evolution and propagation of the eddies, the adequacy of linear modal vertical structure, and the correspondences and energy partition between motions in the two vertical modes.

## Abstract

From measurements made during the 1973 Mid-Ocean Dynamics Experiment in the western North Atlantic, horizontal maps of the total dynamic pressure (or, in the geostrophic approximation, streamfunction) have been constructed for different vertical levels and time periods by the interpolation technique of objective analysis. The space and time sampling of the observations—several tens of kilometers horizontally, hundreds of meters vertically and several days in time—were adequate for resolving mesoscale eddies. The data consisted of velocities (displacement rates) at 1500 m depth from neutrally buoyant floats and vertical density profiles throughout the water column. The resulting maps have been considered from several, essentially phenomenological, points of view. These include descriptions of the synoptic eddy structure, the time evolution and propagation of the eddies, the adequacy of linear modal vertical structure, and the correspondences and energy partition between motions in the two vertical modes.

## Abstract

Observations of float trajectories and vertical density profiles from the Mid-Ocean Dynamics Experiment are analyzed in terms of a likely equation of motion for mesoscale eddies involving the conservation of quasi-geostrophic potential vorticity along horizontal particle paths. From maps of the potential vorticity a careful scale analysis is made to estimate both local values for the Rossby number and the relative dynamical contributions of the planetary vorticity gradient, the relative vorticity and the stretching of vortex lines in the vertical. The proposed conservation is verified, to within estimates of the likely error, at several depths where this error is sufficiently small. Furthermore, two regimes in time are found, one in which the dynamical balances are highly nonlinear and another, for longer time scales, in which they are marginally linear.

## Abstract

Observations of float trajectories and vertical density profiles from the Mid-Ocean Dynamics Experiment are analyzed in terms of a likely equation of motion for mesoscale eddies involving the conservation of quasi-geostrophic potential vorticity along horizontal particle paths. From maps of the potential vorticity a careful scale analysis is made to estimate both local values for the Rossby number and the relative dynamical contributions of the planetary vorticity gradient, the relative vorticity and the stretching of vortex lines in the vertical. The proposed conservation is verified, to within estimates of the likely error, at several depths where this error is sufficiently small. Furthermore, two regimes in time are found, one in which the dynamical balances are highly nonlinear and another, for longer time scales, in which they are marginally linear.

## Abstract

The quasi-geostrophic, small-amplitude free modes of oscillation are examined for a midlatitude ocean basin with mean currents. Attention is restricted to a particular class of mean currents which are solutions of nonlinear, inviscid and unforced equations and whose free modes are all stable ones. Among the free modes are ones confined to the narrow regions where the mean jets are strongest. These modes, dubbed “jet modes”, have the following properties: 1) their phase speed is in the direction of and of the order of magnitude of the mean jet maximum velocity; 2) they are vertically in phase and upper-layer intensified when the mean jet is upper-layer intensified in phase and the thermocline is shallow; 3) they have a broader horizontal scale in the deep water than in the thermocline; 4) they have horizontal critical layers whose local balance is a nonlinear rather than a frictional one; 5) their Doppler-shifted frequencies are proportional to a mean potential vorticity gradient dominated by the horizontal curvature of the, mean jet; 6) and their mean energy and potential vorticity flux divergences are small or—in the particular geometry of a channel—zero. It is argued that many of these features should characterize the transience of narrow jets in general, especially those features relating to the spatial structure of the modes. (The stability and dispersion relation characteristics should be more peculiar to the type of jet present.)

## Abstract

The quasi-geostrophic, small-amplitude free modes of oscillation are examined for a midlatitude ocean basin with mean currents. Attention is restricted to a particular class of mean currents which are solutions of nonlinear, inviscid and unforced equations and whose free modes are all stable ones. Among the free modes are ones confined to the narrow regions where the mean jets are strongest. These modes, dubbed “jet modes”, have the following properties: 1) their phase speed is in the direction of and of the order of magnitude of the mean jet maximum velocity; 2) they are vertically in phase and upper-layer intensified when the mean jet is upper-layer intensified in phase and the thermocline is shallow; 3) they have a broader horizontal scale in the deep water than in the thermocline; 4) they have horizontal critical layers whose local balance is a nonlinear rather than a frictional one; 5) their Doppler-shifted frequencies are proportional to a mean potential vorticity gradient dominated by the horizontal curvature of the, mean jet; 6) and their mean energy and potential vorticity flux divergences are small or—in the particular geometry of a channel—zero. It is argued that many of these features should characterize the transience of narrow jets in general, especially those features relating to the spatial structure of the modes. (The stability and dispersion relation characteristics should be more peculiar to the type of jet present.)

## 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.

## 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.

## Abstract

The low-order, nine-component, primitive equation model of Lorenz (1980) is used as the basis for a comparative study of the quality of several intermediate models. All the models are intermediate between the primitive equations and quasi-geostrophy and will not support gravity-wave oscillations; this reduces to three the number of independent components in each. Strange attractors, stable limit cycles, and stable and unstable fixed points are found in the models. They are used to make a quantitative intercomparison of model performance as the forcing strength, or equivalently the Rossby number, is varied. The models can be ranked from best to worst at small Rossby number as follows: the primitive equations, the balance equations, hypogeostrophy, geostrophic momentum approximation, the linear balance equations, and quasi-geostrophy. At intermediate Rossby number the only change in this ranking is the demotion of hypogeostrophy to the position of worst. Caveats about the low-order model, and hence the generality of the conclusions, are also discussed.

## Abstract

The low-order, nine-component, primitive equation model of Lorenz (1980) is used as the basis for a comparative study of the quality of several intermediate models. All the models are intermediate between the primitive equations and quasi-geostrophy and will not support gravity-wave oscillations; this reduces to three the number of independent components in each. Strange attractors, stable limit cycles, and stable and unstable fixed points are found in the models. They are used to make a quantitative intercomparison of model performance as the forcing strength, or equivalently the Rossby number, is varied. The models can be ranked from best to worst at small Rossby number as follows: the primitive equations, the balance equations, hypogeostrophy, geostrophic momentum approximation, the linear balance equations, and quasi-geostrophy. At intermediate Rossby number the only change in this ranking is the demotion of hypogeostrophy to the position of worst. Caveats about the low-order model, and hence the generality of the conclusions, are also discussed.

## Abstract

Large-scale extratropical motions (with dimensions comparable to, or somewhat smaller than, the planetary radius) in the atmosphere and ocean exhibit a more restricted range of phenomena than are admissible in the primitive equations for fluid motions, and there have been many previous proposals for simpler, more phenomenologically limited models of these motions. The oldest and most successful of these is the quasi-geostrophic model. An extensive discussion is made of models intermediate between the quasi-geostrophic and primitive ones, some of which have been previously proposed [e.g., the balance equations (BE), where tendencies in the equation for the divergent component of velocity are neglected, or the geostrophic momentum approximation (GM), where ageostrophic accelerations are neglected relative to geostrophic ones] and some of which are derived here. Virtues of these models are assessed in the dual measure of nearly geostrophic momentum balance (i.e., small Rossby number) and approximate frontal structure (i.e., larger along-axis velocities and length scales than their cross-axis counterparts), since one or both of these circumstances is usually characteristic of planetary motions. Consideration is also given to various coordinate transformations, since they can yield simpler expressions for the governing differential equations of the intermediate models. In particular, a new set of coordinates is proposed, isentropic geostrophic coordinates,(IGC), which has the advantage of making implicit the advections due to ageostrophic horizontal and vertical velocities under various approximations. A generalization of quasi-geostrophy is made. named hypo-geostrophy (HG), which is an asymptotic approximation of one higher order accuracy in Rossby number. The governing equations are simplest in IGC for both HG and GM; we name the latter in these coordinates isentropic semi-geostrophy (ISG), in analogy to Hoskins’ (1975) semi-geostrophy (SG). HG, GM and BE are, in our opinion, the three most valuable intermediate models for future consideration. HG and BE are superior to GM asymptotically in small Rossby number, but HG in IGC and GM are superior to HG in other coordinates and BE in frontal asymptotics. GM has global (not asymptotic) integral invariants of energy and enstrophy, which HG lacks, and this may assure physically better solutions in weakly asymptotic situations. BE has one global (energy) and one asymptotic (enstrophy) invariant. BE has difficulties of solution existence and uniqueness. Further progress in the search for intermediate models requires obtaining an extensive set of solutions for these models for comparison with quasi-geostrophic and primitive equation solutions.

## Abstract

Large-scale extratropical motions (with dimensions comparable to, or somewhat smaller than, the planetary radius) in the atmosphere and ocean exhibit a more restricted range of phenomena than are admissible in the primitive equations for fluid motions, and there have been many previous proposals for simpler, more phenomenologically limited models of these motions. The oldest and most successful of these is the quasi-geostrophic model. An extensive discussion is made of models intermediate between the quasi-geostrophic and primitive ones, some of which have been previously proposed [e.g., the balance equations (BE), where tendencies in the equation for the divergent component of velocity are neglected, or the geostrophic momentum approximation (GM), where ageostrophic accelerations are neglected relative to geostrophic ones] and some of which are derived here. Virtues of these models are assessed in the dual measure of nearly geostrophic momentum balance (i.e., small Rossby number) and approximate frontal structure (i.e., larger along-axis velocities and length scales than their cross-axis counterparts), since one or both of these circumstances is usually characteristic of planetary motions. Consideration is also given to various coordinate transformations, since they can yield simpler expressions for the governing differential equations of the intermediate models. In particular, a new set of coordinates is proposed, isentropic geostrophic coordinates,(IGC), which has the advantage of making implicit the advections due to ageostrophic horizontal and vertical velocities under various approximations. A generalization of quasi-geostrophy is made. named hypo-geostrophy (HG), which is an asymptotic approximation of one higher order accuracy in Rossby number. The governing equations are simplest in IGC for both HG and GM; we name the latter in these coordinates isentropic semi-geostrophy (ISG), in analogy to Hoskins’ (1975) semi-geostrophy (SG). HG, GM and BE are, in our opinion, the three most valuable intermediate models for future consideration. HG and BE are superior to GM asymptotically in small Rossby number, but HG in IGC and GM are superior to HG in other coordinates and BE in frontal asymptotics. GM has global (not asymptotic) integral invariants of energy and enstrophy, which HG lacks, and this may assure physically better solutions in weakly asymptotic situations. BE has one global (energy) and one asymptotic (enstrophy) invariant. BE has difficulties of solution existence and uniqueness. Further progress in the search for intermediate models requires obtaining an extensive set of solutions for these models for comparison with quasi-geostrophic and primitive equation solutions.

## Abstract

Imagery and numerical modeling show an abundance of submesoscale oceanic eddies in the upper ocean. Large-eddy simulation (LES) is used to elucidate eddy impacts on the atmospheric boundary layer (ABL) forced by winds, convection, and an eddy with varying radius; the maximum azimuthal eddy speed is 1 m s^{−1}. Simulations span the unstable regime −1/*L* = [0, ∞], where *L* is the Monin–Obukhov (M–O) stability parameter. A linearized Ekman model and the LES couple ABL winds to an eddy through rough-wall M–O boundary conditions. The eddy currents cause a surface stress anomaly that induces Ekman pumping in a dipole horizontal pattern. The dipole is understood as a consequence of surface winds aligned or opposing surface currents. In free convection a vigorous updraft is found above the eddy center and persists over the ABL depth. Heterogeneity in surface temperature flux is responsible for the full ABL impact. With winds and convection, current stress coupling generates a dipole in surface temperature flux even with constant sea surface temperature. Wind, pressure, and temperature anomalies are sensitive to an eddy under light winds. The eddy impact on ABL secondary circulations is on the order of the convective velocity scale

## Abstract

Imagery and numerical modeling show an abundance of submesoscale oceanic eddies in the upper ocean. Large-eddy simulation (LES) is used to elucidate eddy impacts on the atmospheric boundary layer (ABL) forced by winds, convection, and an eddy with varying radius; the maximum azimuthal eddy speed is 1 m s^{−1}. Simulations span the unstable regime −1/*L* = [0, ∞], where *L* is the Monin–Obukhov (M–O) stability parameter. A linearized Ekman model and the LES couple ABL winds to an eddy through rough-wall M–O boundary conditions. The eddy currents cause a surface stress anomaly that induces Ekman pumping in a dipole horizontal pattern. The dipole is understood as a consequence of surface winds aligned or opposing surface currents. In free convection a vigorous updraft is found above the eddy center and persists over the ABL depth. Heterogeneity in surface temperature flux is responsible for the full ABL impact. With winds and convection, current stress coupling generates a dipole in surface temperature flux even with constant sea surface temperature. Wind, pressure, and temperature anomalies are sensitive to an eddy under light winds. The eddy impact on ABL secondary circulations is on the order of the convective velocity scale

## Abstract

Atmospheric variability on timescales of a month or longer is dominated by a small number of large-scale spatial patterns (“teleconnections”), whose time evolution has a significant stochastic component because of weather excitation. One may expect these patterns to play an important role in ocean–atmosphere interaction. On interannual and longer timescales, horizontal advection in the ocean can also play an important role in such interaction. The authors develop a simple one-dimensional stochastic model of the interaction between spatially coherent atmospheric forcing patterns and an advective ocean. The model may be considered a generalization of the zero-dimensional stochastic climate model proposed by Hasselmann. The model equations are simple enough that they can be solved analytically, allowing one to fully explore the parameter space. The authors find that the solutions fall into two regimes: (i) a *slow*–*shallow* regime where local damping effects dominate and (ii) a *fast*–*deep* regime where nonlocal advective effects dominate. An interesting feature of the fast–deep regime is that the ocean–atmosphere system shows preferred timescales, although there is no underlying oscillatory mechanism in the uncoupled ocean or in the uncoupled atmosphere. Furthermore, the existence of the preferred timescale in the ocean does not depend upon a strong atmospheric response to SST anomalies. The timescale is determined by the advective velocity scale associated with the upper ocean and the length scale associated with low-frequency atmospheric variability. For the extratropical North Atlantic basin, this timescale would be of the order of a decade, indicating that advective ocean–atmosphere interaction could play an important role in decadal climate variability. The solutions also highlight the differences between local thermodynamic feedbacks associated with changes in the air–sea temperature difference and nonlocal dynamic feedbacks associated with horizontal ocean advection.

## Abstract

Atmospheric variability on timescales of a month or longer is dominated by a small number of large-scale spatial patterns (“teleconnections”), whose time evolution has a significant stochastic component because of weather excitation. One may expect these patterns to play an important role in ocean–atmosphere interaction. On interannual and longer timescales, horizontal advection in the ocean can also play an important role in such interaction. The authors develop a simple one-dimensional stochastic model of the interaction between spatially coherent atmospheric forcing patterns and an advective ocean. The model may be considered a generalization of the zero-dimensional stochastic climate model proposed by Hasselmann. The model equations are simple enough that they can be solved analytically, allowing one to fully explore the parameter space. The authors find that the solutions fall into two regimes: (i) a *slow*–*shallow* regime where local damping effects dominate and (ii) a *fast*–*deep* regime where nonlocal advective effects dominate. An interesting feature of the fast–deep regime is that the ocean–atmosphere system shows preferred timescales, although there is no underlying oscillatory mechanism in the uncoupled ocean or in the uncoupled atmosphere. Furthermore, the existence of the preferred timescale in the ocean does not depend upon a strong atmospheric response to SST anomalies. The timescale is determined by the advective velocity scale associated with the upper ocean and the length scale associated with low-frequency atmospheric variability. For the extratropical North Atlantic basin, this timescale would be of the order of a decade, indicating that advective ocean–atmosphere interaction could play an important role in decadal climate variability. The solutions also highlight the differences between local thermodynamic feedbacks associated with changes in the air–sea temperature difference and nonlocal dynamic feedbacks associated with horizontal ocean advection.