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- Author or Editor: Richard J. Greatbatch x

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## Abstract

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## Abstract

A novel and efficient numerical method is used to investigate the nonlinear equations of motion for the upper layer of a two-layer ocean in which the lower layer is infinitely deep and at rest. The efficiency is achieved by seeking solutions that are in a steady state, translating in equilibrium with the storm. Oscillations are found in the wake of the storm. Two features of the response are attributed to the nonlinear terms in the equation of motion: 1) a rapid transition from a maximum in the downwelling phase, to a maximum in the upwelling phase of each oscillation, followed by a gradual relaxation to the next downwelling maximum; and 2) a displacement of the maximum response, usually to the right of the storm track, by ∼40 km. It is shown that the horizontal pressure gradient terms can be neglected from the momentum equations for “fast”, “large” storms, in which case a Lagrangian integration can be performed, following fluid particles. This enables feature 1) to be attributed to the along-track advection terms and 2) to be associated with the cross-track advection terms. When the horizontal pressure gradient terms are more important, feature 1) remain but the maximum response is displaced, in the wake, to the left of the track from the right. It is shown that even a symmetric storm can produce a strongly asymmetric response. Finally, results are compared with observations of the response of the ocean to hurricanes.

## Abstract

A novel and efficient numerical method is used to investigate the nonlinear equations of motion for the upper layer of a two-layer ocean in which the lower layer is infinitely deep and at rest. The efficiency is achieved by seeking solutions that are in a steady state, translating in equilibrium with the storm. Oscillations are found in the wake of the storm. Two features of the response are attributed to the nonlinear terms in the equation of motion: 1) a rapid transition from a maximum in the downwelling phase, to a maximum in the upwelling phase of each oscillation, followed by a gradual relaxation to the next downwelling maximum; and 2) a displacement of the maximum response, usually to the right of the storm track, by ∼40 km. It is shown that the horizontal pressure gradient terms can be neglected from the momentum equations for “fast”, “large” storms, in which case a Lagrangian integration can be performed, following fluid particles. This enables feature 1) to be attributed to the along-track advection terms and 2) to be associated with the cross-track advection terms. When the horizontal pressure gradient terms are more important, feature 1) remain but the maximum response is displaced, in the wake, to the left of the track from the right. It is shown that even a symmetric storm can produce a strongly asymmetric response. Finally, results are compared with observations of the response of the ocean to hurricanes.

## Abstract

A framework for mesoscale eddy parameterization based on density-weighted averaging at fixed height is developed. The method uses the fully non-Boussinesq equations of motion and is connected to the equations carried by Boussinesq ocean models only after the averaged equations have been developed. The framework applies to the continuity, tracer, and momentum equations within a single formalism. Two methods for applying parameterizations in ocean models are obtained. The first, based on the tracer equation, corresponds to the approach commonly taken when including eddy effects in ocean models. The second puts the forcing for the eddy-induced transport into the averaged momentum equation where it appears as the divergence of a generalized Eliassen–Palm flux.

It is then shown how to solve for the tracer transport velocity. The solutions form a family closely related to the temporal residual mean (TRM) velocity of McDougall and McIntosh, valid to *O*(*α*
^{3}), where *α* is perturbation amplitude. The analysis is extended to obtain a family of exact solutions for the eddy-induced mass transport, valid at any order in perturbation amplitude. It is also shown how to obtain a generalization of the TRM to take account of diffusion and time dependence in the instantaneous equations. The solution suggests that the tracer transport velocity could be different for different tracers, depending primarily on the structure of the mean field. This conclusion also applies in the case of isopycnal averaging; it is not a result that is peculiar to averaging at fixed height.

Finally, it is shown how the non-Boussinesq analysis presented in the paper can be modified to analyze output from eddy-resolving, Boussinesq ocean models.

## Abstract

A framework for mesoscale eddy parameterization based on density-weighted averaging at fixed height is developed. The method uses the fully non-Boussinesq equations of motion and is connected to the equations carried by Boussinesq ocean models only after the averaged equations have been developed. The framework applies to the continuity, tracer, and momentum equations within a single formalism. Two methods for applying parameterizations in ocean models are obtained. The first, based on the tracer equation, corresponds to the approach commonly taken when including eddy effects in ocean models. The second puts the forcing for the eddy-induced transport into the averaged momentum equation where it appears as the divergence of a generalized Eliassen–Palm flux.

It is then shown how to solve for the tracer transport velocity. The solutions form a family closely related to the temporal residual mean (TRM) velocity of McDougall and McIntosh, valid to *O*(*α*
^{3}), where *α* is perturbation amplitude. The analysis is extended to obtain a family of exact solutions for the eddy-induced mass transport, valid at any order in perturbation amplitude. It is also shown how to obtain a generalization of the TRM to take account of diffusion and time dependence in the instantaneous equations. The solution suggests that the tracer transport velocity could be different for different tracers, depending primarily on the structure of the mean field. This conclusion also applies in the case of isopycnal averaging; it is not a result that is peculiar to averaging at fixed height.

Finally, it is shown how the non-Boussinesq analysis presented in the paper can be modified to analyze output from eddy-resolving, Boussinesq ocean models.

## Abstract

Gent et al. have emphasized the role of the eddy-induced transport (or bolus) velocity as a mechanism for redistributing tracers in the ocean. By writing the momentum equations in terms of the isopycnal flux of potential vorticity, the author shows that any parameterization of the eddy-induced transport velocity must be consistent with the conservation equation for potential vorticity. This places a constraint on possible parameterizations, a constraint that is satisfied by the Gent and McWilliams parameterization only if restrictions are placed on the diffusivity coefficient. A new parameterization is suggested that is the simplest extension of Gent and McWilliams based on the potential vorticity formulation. The new parameterization parameterizes part of the time-mean flow driven by the Reynolds stress terms in addition to the eddy-induced transport velocity. It is also shown that the eddy-induced transport velocity can always be written as the Ekman velocity associated with the vertical derivative of a horizontally directed eddy stress. The author shows how the eddy stress is related to the “inviscid pressure drag” or “form drag” associated with the eddies, although the correspondence is not exact.

## Abstract

Gent et al. have emphasized the role of the eddy-induced transport (or bolus) velocity as a mechanism for redistributing tracers in the ocean. By writing the momentum equations in terms of the isopycnal flux of potential vorticity, the author shows that any parameterization of the eddy-induced transport velocity must be consistent with the conservation equation for potential vorticity. This places a constraint on possible parameterizations, a constraint that is satisfied by the Gent and McWilliams parameterization only if restrictions are placed on the diffusivity coefficient. A new parameterization is suggested that is the simplest extension of Gent and McWilliams based on the potential vorticity formulation. The new parameterization parameterizes part of the time-mean flow driven by the Reynolds stress terms in addition to the eddy-induced transport velocity. It is also shown that the eddy-induced transport velocity can always be written as the Ekman velocity associated with the vertical derivative of a horizontally directed eddy stress. The author shows how the eddy stress is related to the “inviscid pressure drag” or “form drag” associated with the eddies, although the correspondence is not exact.

## Abstract

This paper has two purposes: One is to present a new and efficient multilevel numerical model for calculating the response of the ocean to a moving storm; the second is to show how, on a time scale of a few inertial periods following the arrival of the storm, the maximum horizontal and vertical velocities found in the wake can be calculated using a linear Ekman model and a knowledge of that part of the change in the depth of the wind mixed layer due to entrainment. This is demonstrated over a range of experiments with the multilevel numerical model. These integrate the full nonlinear equations of motion with realistic ocean stratification and involve substantial entrainment of water into the wind mixed layer.

It is also shown that on this time scale, the horizontal currents are confined near the surface but that the vertical velocity field extends throughout the depth of the ocean. It is shown in Appendix B that the wind forcing need only be “large” or “fast” for the forced response not to feel the effect of the ocean stratification and to extend through the depth of the ocean in this way.

The parameter which determines the horizontal structure of the response, in coordinates scaled with respect to the scale *L* of the storm, is *k* = *U*/*Lf*. Here *U* is the storm translation speed and *f* the Coriolis parameter. This parameter also determines the magnitude of the response, after suitable nondimensionalization.

Finally, it is shown how to apply these results to an interpretation of observations and other model results.

## Abstract

This paper has two purposes: One is to present a new and efficient multilevel numerical model for calculating the response of the ocean to a moving storm; the second is to show how, on a time scale of a few inertial periods following the arrival of the storm, the maximum horizontal and vertical velocities found in the wake can be calculated using a linear Ekman model and a knowledge of that part of the change in the depth of the wind mixed layer due to entrainment. This is demonstrated over a range of experiments with the multilevel numerical model. These integrate the full nonlinear equations of motion with realistic ocean stratification and involve substantial entrainment of water into the wind mixed layer.

It is also shown that on this time scale, the horizontal currents are confined near the surface but that the vertical velocity field extends throughout the depth of the ocean. It is shown in Appendix B that the wind forcing need only be “large” or “fast” for the forced response not to feel the effect of the ocean stratification and to extend through the depth of the ocean in this way.

The parameter which determines the horizontal structure of the response, in coordinates scaled with respect to the scale *L* of the storm, is *k* = *U*/*Lf*. Here *U* is the storm translation speed and *f* the Coriolis parameter. This parameter also determines the magnitude of the response, after suitable nondimensionalization.

Finally, it is shown how to apply these results to an interpretation of observations and other model results.

## Abstract

In the Stommel box model, the strength of the overturning circulation is parameterized in terms of the density (and hence the pressure) difference between the two boxes. Straub has pointed out that this parameterization is not consistent with the Stommel–Arons model for the abyssal circulation. In particular, the zonally averaged density field implied by the Stommel–Arons model is unrelated to the strength or the direction of the meridional overturning circulation. Here, the inconsistency is examined using the abyssal circulation model of Kawase and a variant to include the effect of Southern Hemisphere wind forcing. The important parameter is *R,* the ratio of two timescales: the timescale for a perturbation to the density field to propagate, by either wave or advective processes, from a high-latitude source to the equator and the timescale for the dissipation of a perturbation to the density field by diapycnal mixing. If the model is forced only by a deep water source in the northern basin, it is found that the model behaves like the Stommel–Arons model when *R* ≪ 1 (the “weak” damping regime) and like the Stommel box model when *R* ≫ 1 (the “strong” damping regine). Estimates of *R* suggest that coarse-resolution models generally reside in or near the Stommel box model regime (*R* ≫ 1), which is probably why these models generally support the Stommel box model hypothesis and corroborate the momentum-based closure used in zonally averaged models. On the other hand, it is not clear that the real world is also in the strong damping regime. Indeed, it is easy to obtain estimates for *R,* using realistic parameter values, that sit in the weak damping regime. It is shown that, even in the weak damping regime (*R* ≪ 1), adding forcing by the Southern Hemisphere circumpolar westerlies generally moves the model into the Stommel box model regime. It therefore is concluded that, at least in the context of the Kawase model, the inconsistency noted by Straub can be removed by including the effect of Southern Hemisphere wind forcing and that the Stommel box model approach probably has wider applicability than is suggested by estimates of *R* alone.

## Abstract

In the Stommel box model, the strength of the overturning circulation is parameterized in terms of the density (and hence the pressure) difference between the two boxes. Straub has pointed out that this parameterization is not consistent with the Stommel–Arons model for the abyssal circulation. In particular, the zonally averaged density field implied by the Stommel–Arons model is unrelated to the strength or the direction of the meridional overturning circulation. Here, the inconsistency is examined using the abyssal circulation model of Kawase and a variant to include the effect of Southern Hemisphere wind forcing. The important parameter is *R,* the ratio of two timescales: the timescale for a perturbation to the density field to propagate, by either wave or advective processes, from a high-latitude source to the equator and the timescale for the dissipation of a perturbation to the density field by diapycnal mixing. If the model is forced only by a deep water source in the northern basin, it is found that the model behaves like the Stommel–Arons model when *R* ≪ 1 (the “weak” damping regime) and like the Stommel box model when *R* ≫ 1 (the “strong” damping regine). Estimates of *R* suggest that coarse-resolution models generally reside in or near the Stommel box model regime (*R* ≫ 1), which is probably why these models generally support the Stommel box model hypothesis and corroborate the momentum-based closure used in zonally averaged models. On the other hand, it is not clear that the real world is also in the strong damping regime. Indeed, it is easy to obtain estimates for *R,* using realistic parameter values, that sit in the weak damping regime. It is shown that, even in the weak damping regime (*R* ≪ 1), adding forcing by the Southern Hemisphere circumpolar westerlies generally moves the model into the Stommel box model regime. It therefore is concluded that, at least in the context of the Kawase model, the inconsistency noted by Straub can be removed by including the effect of Southern Hemisphere wind forcing and that the Stommel box model approach probably has wider applicability than is suggested by estimates of *R* alone.

## Abstract

The residual effect of surface gravity waves on mean flows in the upper ocean is investigated using thickness-weighted mean (TWM) theory applied in a vertically Lagrangian and horizontally Eulerian coordinate system. Depth-dependent equations for the conservation of volume, momentum, and energy are derived. These equations allow for (i) finite amplitude fluid motions, (ii) the horizontal divergence of currents, and (iii) a concise treatment of both kinematic and viscous boundary conditions at the sea surface. Under the assumptions of steady and monochromatic waves and a uniform turbulent viscosity, the TWM momentum equations are used to illustrate the pressure- and viscosity-induced momentum fluxes through the surface, which are implicit in previous studies of the wave-induced modification of the classical Ekman spiral problem. The TWM approach clarifies, in particular, the surface momentum flux associated with the so-called virtual wave stress of Longuet-Higgins. Overall, the TWM framework can be regarded as an alternative to the three-dimensional Lagrangian mean framework of Pierson. Moreover, the TWM framework can be used to include the residual effect of surface waves in large-scale circulation models. In specific models that carry the TWM velocity appropriate for advecting tracers as their velocity variable, the turbulent viscosity term should be modified so that the viscosity acts only on the Eulerian mean velocity.

## Abstract

The residual effect of surface gravity waves on mean flows in the upper ocean is investigated using thickness-weighted mean (TWM) theory applied in a vertically Lagrangian and horizontally Eulerian coordinate system. Depth-dependent equations for the conservation of volume, momentum, and energy are derived. These equations allow for (i) finite amplitude fluid motions, (ii) the horizontal divergence of currents, and (iii) a concise treatment of both kinematic and viscous boundary conditions at the sea surface. Under the assumptions of steady and monochromatic waves and a uniform turbulent viscosity, the TWM momentum equations are used to illustrate the pressure- and viscosity-induced momentum fluxes through the surface, which are implicit in previous studies of the wave-induced modification of the classical Ekman spiral problem. The TWM approach clarifies, in particular, the surface momentum flux associated with the so-called virtual wave stress of Longuet-Higgins. Overall, the TWM framework can be regarded as an alternative to the three-dimensional Lagrangian mean framework of Pierson. Moreover, the TWM framework can be used to include the residual effect of surface waves in large-scale circulation models. In specific models that carry the TWM velocity appropriate for advecting tracers as their velocity variable, the turbulent viscosity term should be modified so that the viscosity acts only on the Eulerian mean velocity.

## Abstract

There is an ongoing discussion in the community concerning the wave-averaged momentum equations in the hybrid vertically Lagrangian and horizontally Eulerian (VL) framework and, in particular, the form stress term (representing the residual effect of pressure perturbations) that is thought to restrict the handling of higher-order waves in terms of a perturbation expansion. The present study shows that the traditional pressure-based form stress term can be transformed into a set of terms that do not contain any pressure quantities but do contain the time derivative of a wave-induced velocity. This wave-induced velocity is referred to as the pseudomomentum in the VL framework, as it is analogous to the generalized pseudomomentum in Andrews and McIntyre. This enables the second expression for the wave-averaged momentum equations in the VL framework (this time for the development of the total transport velocity minus the VL pseudomomentum) to be derived together with the vortex force. The velocity-based expression of the form stress term also contains the residual effect of the turbulent viscosity, which is useful for understanding the dissipation of wave energy leading to a transfer of momentum from waves to circulation. It is found that the concept of the virtual wave stress of Longuet-Higgins is applicable to quite general situations: it does not matter whether there is wind forcing or not, the waves can have slow variations, and the viscosity coefficient can vary in the vertical. These results provide a basis for revisiting the surface boundary condition used in numerical circulation models.

## Abstract

There is an ongoing discussion in the community concerning the wave-averaged momentum equations in the hybrid vertically Lagrangian and horizontally Eulerian (VL) framework and, in particular, the form stress term (representing the residual effect of pressure perturbations) that is thought to restrict the handling of higher-order waves in terms of a perturbation expansion. The present study shows that the traditional pressure-based form stress term can be transformed into a set of terms that do not contain any pressure quantities but do contain the time derivative of a wave-induced velocity. This wave-induced velocity is referred to as the pseudomomentum in the VL framework, as it is analogous to the generalized pseudomomentum in Andrews and McIntyre. This enables the second expression for the wave-averaged momentum equations in the VL framework (this time for the development of the total transport velocity minus the VL pseudomomentum) to be derived together with the vortex force. The velocity-based expression of the form stress term also contains the residual effect of the turbulent viscosity, which is useful for understanding the dissipation of wave energy leading to a transfer of momentum from waves to circulation. It is found that the concept of the virtual wave stress of Longuet-Higgins is applicable to quite general situations: it does not matter whether there is wind forcing or not, the waves can have slow variations, and the viscosity coefficient can vary in the vertical. These results provide a basis for revisiting the surface boundary condition used in numerical circulation models.

## Abstract

The dynamical origin of the spectral and autocorrelation structure of annular variability in the troposphere is investigated by a deductive approach. Specifically, the structure of the power spectrum and autocorrelation function of the zonal-mean geopotential is analyzed for the case of a quasigeostrophic spherical atmosphere subject to a white noise mechanical forcing applied in a single Hough mode and concentrated at a particular level in the vertical, with vertically uniform Newtonian cooling and Rayleigh drag concentrated at a rigid lower boundary. Analytic expressions for the power spectrum are presented together with expressions for an approximate red noise (i.e., a Lorentzian-shaped) power spectrum. It is found that for an infinitely deep atmosphere the power spectrum can be well approximated by a red noise process for the first few Hough modes (associated with large Rossby heights), provided the distance from the forcing is not larger than about one Rossby height. When a frictional rigid lower boundary is included, however, the approximation is generally bad. The high-frequency part of the power spectrum exhibits near-exponential behavior and the autocorrelation function shows a transition from a rapid decay at short lags to a much slower decay at longer lags, if the thermal and mechanical damping time scales are sufficiently well separated. Since observed annular variability exhibits the same characteristics, the above results lead to the hypothesis that these characteristics may, to some extent, be intrinsic to the linear zonal-mean response problem—although the need for an additional contribution from eddy feedbacks is also implied by the results.

## Abstract

The dynamical origin of the spectral and autocorrelation structure of annular variability in the troposphere is investigated by a deductive approach. Specifically, the structure of the power spectrum and autocorrelation function of the zonal-mean geopotential is analyzed for the case of a quasigeostrophic spherical atmosphere subject to a white noise mechanical forcing applied in a single Hough mode and concentrated at a particular level in the vertical, with vertically uniform Newtonian cooling and Rayleigh drag concentrated at a rigid lower boundary. Analytic expressions for the power spectrum are presented together with expressions for an approximate red noise (i.e., a Lorentzian-shaped) power spectrum. It is found that for an infinitely deep atmosphere the power spectrum can be well approximated by a red noise process for the first few Hough modes (associated with large Rossby heights), provided the distance from the forcing is not larger than about one Rossby height. When a frictional rigid lower boundary is included, however, the approximation is generally bad. The high-frequency part of the power spectrum exhibits near-exponential behavior and the autocorrelation function shows a transition from a rapid decay at short lags to a much slower decay at longer lags, if the thermal and mechanical damping time scales are sufficiently well separated. Since observed annular variability exhibits the same characteristics, the above results lead to the hypothesis that these characteristics may, to some extent, be intrinsic to the linear zonal-mean response problem—although the need for an additional contribution from eddy feedbacks is also implied by the results.

## Abstract

The wintertime northern annular mode (NAM) at the surface is known to undergo slow intraseasonal variations in association with stratospheric variability, which leads the surface signal by up to several weeks. The relative contributions, however, of potentially relevant stratosphere–troposphere coupling mechanisms are not yet fully understood.

In this study the relative roles of (i) the downward effect of the zonal-mean secondary circulation induced by quasigeostrophic (QG) adjustment to stratospheric wave drag and radiative damping and (ii) wave drag local to the troposphere are estimated. For this purpose, a spectral tendency equation of the QG zonal-mean zonal wind is derived and used, in a first step, to obtain the external mechanical forcing that, in the QG framework, drives exactly the observed stratospheric and tropospheric daily NAM. In a second step, the equation is then integrated in time to reconstruct the daily NAM, but with the forcing restricted to either stratospheric or tropospheric levels, each case leaving a characteristic NAM surface signal.

The relative roles of the above-mentioned mechanisms are found to be of similar quantitative importance, but to differ in a qualitative sense. The downward effect of stratospheric QG adjustment is responsible for the initiation of the NAM surface signal, whereas subsequently local tropospheric wave drag actively maintains and persists the signal over several weeks. Furthermore, the downward effect of QG adjustment to stratospheric radiative damping is shown to have only a minor impact, compared to that from stratospheric wave drag. The robustness of these conclusions is demonstrated by a sensitivity study with respect to various model parameters.

## Abstract

The wintertime northern annular mode (NAM) at the surface is known to undergo slow intraseasonal variations in association with stratospheric variability, which leads the surface signal by up to several weeks. The relative contributions, however, of potentially relevant stratosphere–troposphere coupling mechanisms are not yet fully understood.

In this study the relative roles of (i) the downward effect of the zonal-mean secondary circulation induced by quasigeostrophic (QG) adjustment to stratospheric wave drag and radiative damping and (ii) wave drag local to the troposphere are estimated. For this purpose, a spectral tendency equation of the QG zonal-mean zonal wind is derived and used, in a first step, to obtain the external mechanical forcing that, in the QG framework, drives exactly the observed stratospheric and tropospheric daily NAM. In a second step, the equation is then integrated in time to reconstruct the daily NAM, but with the forcing restricted to either stratospheric or tropospheric levels, each case leaving a characteristic NAM surface signal.

The relative roles of the above-mentioned mechanisms are found to be of similar quantitative importance, but to differ in a qualitative sense. The downward effect of stratospheric QG adjustment is responsible for the initiation of the NAM surface signal, whereas subsequently local tropospheric wave drag actively maintains and persists the signal over several weeks. Furthermore, the downward effect of QG adjustment to stratospheric radiative damping is shown to have only a minor impact, compared to that from stratospheric wave drag. The robustness of these conclusions is demonstrated by a sensitivity study with respect to various model parameters.