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- Author or Editor: James C. McWilliams x

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## Abstract

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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.

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## 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.)

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## 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.

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## 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.

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## 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.

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## Abstract

Ocean convection often occurs in regions of mesoscale eddy activity, where convective mixing and geostrophic eddy dynamics interact. The authors examine the interactions between a group of geostrophic eddies and convective mixing induced by surface buoyancy loss through a series of numerical simulations using a nonhydrostatic Boussinesq model. The eddies are initially baroclinic, with a surface-intensified density anomaly and sheared flow, but they are stable to baroclinic instability because of their small size. In the absence of buoyancy loss, eddy mergers occur much as in previous studies of geostrophic turbulence. With the addition of surface buoyancy loss, the surface stratification is eroded by small-scale convection. The convective mixing is highly heterogeneous, being deeper in regions of weaker initial stratification and shallower in more strongly stratified regions. The deformation radius is reduced in mixing regions and the weakly stratified eddies become baroclinically unstable. The barotropic component of kinetic energy increases as convection proceeds, largely due to the conversion of the available potential energy of the eddies in the baroclinic instability process. The convective forcing therefore provides a means of increasing the barotropic component of the eddy kinetic energy, by enabling the baroclinic instability. The fluid is efficiently homogenized by the energetic eddy field, leading to a few isolated eddies separated by a well-mixed fluid. These simulations provide a possible explanation for energetic eddy fields observed during convective periods in the Labrador Sea.

## Abstract

Ocean convection often occurs in regions of mesoscale eddy activity, where convective mixing and geostrophic eddy dynamics interact. The authors examine the interactions between a group of geostrophic eddies and convective mixing induced by surface buoyancy loss through a series of numerical simulations using a nonhydrostatic Boussinesq model. The eddies are initially baroclinic, with a surface-intensified density anomaly and sheared flow, but they are stable to baroclinic instability because of their small size. In the absence of buoyancy loss, eddy mergers occur much as in previous studies of geostrophic turbulence. With the addition of surface buoyancy loss, the surface stratification is eroded by small-scale convection. The convective mixing is highly heterogeneous, being deeper in regions of weaker initial stratification and shallower in more strongly stratified regions. The deformation radius is reduced in mixing regions and the weakly stratified eddies become baroclinically unstable. The barotropic component of kinetic energy increases as convection proceeds, largely due to the conversion of the available potential energy of the eddies in the baroclinic instability process. The convective forcing therefore provides a means of increasing the barotropic component of the eddy kinetic energy, by enabling the baroclinic instability. The fluid is efficiently homogenized by the energetic eddy field, leading to a few isolated eddies separated by a well-mixed fluid. These simulations provide a possible explanation for energetic eddy fields observed during convective periods in the Labrador Sea.

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## Abstract

In layered models of the ocean, the assumption of a deep resting layer is often made, motivated by the surface intensification of many phenomena. The propagation speed of first-mode, baroclinic Rossby waves in such models is always faster than in models with all the layers active. The assumption of a deep-resting layer is not crucial for the phase-speed enhancement since the same result holds if the bottom pressure fluctuations are uncorrelated from the overlying wave dynamics.

*C*

_{fast}is the enhanced phase speed,

*C*

_{standard}the standard phase speed,

^{′}

_{1}(

*z*)

*H*

_{0}is the reference depth serving to define it. In the case WKB theory is applicable in the vertical direction, the above formula reduces to

*N*is the deep Brunt–Väisälä frequency and

_{b}*N*

The amplification factor is computed from a global hydrographic climatology. The comparison with observational estimates shows a reasonable degree of consistency, although with appreciable scatter. The theory appears to do as well as the previously published mean-flow theories of Killworth et al. and others. The link between the faster mode and the surface-intensified modes occurring over steep topography previously discussed in the literature is also established.

## Abstract

In layered models of the ocean, the assumption of a deep resting layer is often made, motivated by the surface intensification of many phenomena. The propagation speed of first-mode, baroclinic Rossby waves in such models is always faster than in models with all the layers active. The assumption of a deep-resting layer is not crucial for the phase-speed enhancement since the same result holds if the bottom pressure fluctuations are uncorrelated from the overlying wave dynamics.

*C*

_{fast}is the enhanced phase speed,

*C*

_{standard}the standard phase speed,

^{′}

_{1}(

*z*)

*H*

_{0}is the reference depth serving to define it. In the case WKB theory is applicable in the vertical direction, the above formula reduces to

*N*is the deep Brunt–Väisälä frequency and

_{b}*N*

The amplification factor is computed from a global hydrographic climatology. The comparison with observational estimates shows a reasonable degree of consistency, although with appreciable scatter. The theory appears to do as well as the previously published mean-flow theories of Killworth et al. and others. The link between the faster mode and the surface-intensified modes occurring over steep topography previously discussed in the literature is also established.

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## Abstract

In a large-scale deformation flow, lateral and vertical buoyancy gradients sharpen through baroclinic frontogenesis near the surface boundary. A “thermally direct” ageostrophic secondary circulation cell arises during frontogenesis to maintain geostrophic, hydrostatic (thermal wind) momentum balance for the alongfront flow. Unstable three-dimensional fluctuations can grow during frontogenesis by baroclinic instability of the alongfront shear flow that converts frontal potential energy to fluctuation energy. At finite amplitude, the fluctuations provide alongfront-averaged eddy momentum and buoyancy fluxes that arrest the frontal sharpening even while the deformation flow persists. The frontal ageostrophic secondary circulation reverses to become a “thermally indirect” cell in the center of the front. This allows an approximate opposition between ageostrophic advection and eddy-flux divergence in the frontal buoyancy gradient variance (i.e., frontal strength) balance equation, implying frontal equilibration. During the approximately equilibrated phase, the energy exchange rates among the deformation flow, front, and fluctuations are all reduced in comparison with a solution without eddy-flux feedback on the frontal evolution. The mean stratification is enhanced by both frontogenesis and eddy vertical buoyancy flux. The thermally indirect secondary circulation arises from eddy fluxes acting to force a departure in thermal-wind balance for the alongfront flow, overwhelming the single-cell thermally direct circulation induced by the deformation flow. The equilibrated thermal-wind imbalance in the frontal flow is appreciable, and its magnitude is set by the cross-front eddy flux of alongfront vorticity. This demonstrates an essentially inviscid, baroclinic, dynamical process for frontogenetic arrest through frontal instability.

## Abstract

In a large-scale deformation flow, lateral and vertical buoyancy gradients sharpen through baroclinic frontogenesis near the surface boundary. A “thermally direct” ageostrophic secondary circulation cell arises during frontogenesis to maintain geostrophic, hydrostatic (thermal wind) momentum balance for the alongfront flow. Unstable three-dimensional fluctuations can grow during frontogenesis by baroclinic instability of the alongfront shear flow that converts frontal potential energy to fluctuation energy. At finite amplitude, the fluctuations provide alongfront-averaged eddy momentum and buoyancy fluxes that arrest the frontal sharpening even while the deformation flow persists. The frontal ageostrophic secondary circulation reverses to become a “thermally indirect” cell in the center of the front. This allows an approximate opposition between ageostrophic advection and eddy-flux divergence in the frontal buoyancy gradient variance (i.e., frontal strength) balance equation, implying frontal equilibration. During the approximately equilibrated phase, the energy exchange rates among the deformation flow, front, and fluctuations are all reduced in comparison with a solution without eddy-flux feedback on the frontal evolution. The mean stratification is enhanced by both frontogenesis and eddy vertical buoyancy flux. The thermally indirect secondary circulation arises from eddy fluxes acting to force a departure in thermal-wind balance for the alongfront flow, overwhelming the single-cell thermally direct circulation induced by the deformation flow. The equilibrated thermal-wind imbalance in the frontal flow is appreciable, and its magnitude is set by the cross-front eddy flux of alongfront vorticity. This demonstrates an essentially inviscid, baroclinic, dynamical process for frontogenetic arrest through frontal instability.

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## Abstract

The phenomenon of oceanic Ekman layer rectification refers to how the time-mean, Ekman layer velocity profile with depth differs as a consequence of variability in the surface wind in addition to the time-mean wind. This study investigates rectification using the *K*-Profile Parameterization (KPP) model for the turbulent surface boundary layer under simple conditions of uniform density and no surface buoyancy flux or surface wave influences. The rectification magnitude is found to be significant under typical conditions. Its primary effects are to extend the depth profile deeper into the interior, reduce the mean shear, increase the effective eddy viscosity due to turbulent momentum mixing, and rotate slightly the surface velocity farther away from the mean wind direction. These effects are partly due to the increase in mean stress because of its quadratic dependence on wind speed but also are due to the nonlinearity of the turbulent mixing efficiency. The strongest influence on the rectification magnitude is the ratio of transient wind amplitude to mean wind speed. It is found that an accurate estimate of the mean current usually can be obtained by using a quasi-stationary approximation that is a weighted integral of the steady Ekman layer response over the probability density function for the wind, independent of the detailed wind history. Rectification occurs even for very high frequency wind fluctuations, though the accuracy of the quasi-steady approximation degrades in this limit (as does the validity of the KPP model). This theory is extended to include the effects of the horizontal component of the Coriolis frequency, *f ^{ y}
*. Based on published computational turbulence solutions, a simple parameterization is proposed that amplifies the turbulent eddy diffusivity in KPP by a factor that decreases with latitude and depends on the wind orientation. The effect of

*f*≠ 0 is to increase both the shear and the surface speed in the time-mean Ekman current for winds directed to the northeast and decrease both quantities for winds to the southwest, with weaker influences on these properties for the orthogonal directions of southeast and northwest. Furthermore, with transient winds there is significant coupling between

^{ y}*f*≠ 0 and the rectification effect; for example, the mean surface current direction, relative to the mean wind, is significantly changed for these orthogonal directions.

^{ y}## Abstract

The phenomenon of oceanic Ekman layer rectification refers to how the time-mean, Ekman layer velocity profile with depth differs as a consequence of variability in the surface wind in addition to the time-mean wind. This study investigates rectification using the *K*-Profile Parameterization (KPP) model for the turbulent surface boundary layer under simple conditions of uniform density and no surface buoyancy flux or surface wave influences. The rectification magnitude is found to be significant under typical conditions. Its primary effects are to extend the depth profile deeper into the interior, reduce the mean shear, increase the effective eddy viscosity due to turbulent momentum mixing, and rotate slightly the surface velocity farther away from the mean wind direction. These effects are partly due to the increase in mean stress because of its quadratic dependence on wind speed but also are due to the nonlinearity of the turbulent mixing efficiency. The strongest influence on the rectification magnitude is the ratio of transient wind amplitude to mean wind speed. It is found that an accurate estimate of the mean current usually can be obtained by using a quasi-stationary approximation that is a weighted integral of the steady Ekman layer response over the probability density function for the wind, independent of the detailed wind history. Rectification occurs even for very high frequency wind fluctuations, though the accuracy of the quasi-steady approximation degrades in this limit (as does the validity of the KPP model). This theory is extended to include the effects of the horizontal component of the Coriolis frequency, *f ^{ y}
*. Based on published computational turbulence solutions, a simple parameterization is proposed that amplifies the turbulent eddy diffusivity in KPP by a factor that decreases with latitude and depends on the wind orientation. The effect of

*f*≠ 0 is to increase both the shear and the surface speed in the time-mean Ekman current for winds directed to the northeast and decrease both quantities for winds to the southwest, with weaker influences on these properties for the orthogonal directions of southeast and northwest. Furthermore, with transient winds there is significant coupling between

^{ y}*f*≠ 0 and the rectification effect; for example, the mean surface current direction, relative to the mean wind, is significantly changed for these orthogonal directions.

^{ y}