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## Abstract

The one-dimensional bulk mixed-layer model of Niiler (1975) is extended to two (or three) dimensions to take account of horizontal variation in wind stress on mixed-layer dynamics. Both surface stirring (Kraus and Turner, 1967) and bulk shear (Pollard *et al*., 1973) entrainment mechanisms are included. The development of horizontal structure in the upper ocean an the subseasonal to seasonal time scale is the focus of interest. An asymptotic two-timing technique is employed to simplify the dynamical equations. Wind-driven advection can be important in establishing and concentrating horizontal gradients of the sea surface temperature. Wind stress curl-driven vertical velocity can be as important as entrainment velocity in determining the horizontal distribution of mixed-layer depth. Several illustrative calculations are discussed. A case with initially horizontally uniform temperature, 0.05°C m^{−1} initial vertical gradient, and wind stress of 1 dyn cm^{−2} and scale of 1000 km, shows horizontal temperature gradients of ∼0.01°C km^{−1} in the vicinity of the wind-driven convergence zone after 100 days integration. A similar case, except with initial horizontal gradient of 0.006°C km^{−1}, shows temperature gradients of 0.02°C km^{−1} after 100 days.

In wind-driven convergence zones, mixed-layer depths of 120 m can be achieved after 100 days, by a combination of entrainment and downwelling, mostly the latter, especially after long times. In divergent zones, steady mixed-layer depths can be achieved in less than 100 days through the competition between the effects of upwelling and entrainment. These steady depths range from 20 to 90 m, depending on location.

## Abstract

The one-dimensional bulk mixed-layer model of Niiler (1975) is extended to two (or three) dimensions to take account of horizontal variation in wind stress on mixed-layer dynamics. Both surface stirring (Kraus and Turner, 1967) and bulk shear (Pollard *et al*., 1973) entrainment mechanisms are included. The development of horizontal structure in the upper ocean an the subseasonal to seasonal time scale is the focus of interest. An asymptotic two-timing technique is employed to simplify the dynamical equations. Wind-driven advection can be important in establishing and concentrating horizontal gradients of the sea surface temperature. Wind stress curl-driven vertical velocity can be as important as entrainment velocity in determining the horizontal distribution of mixed-layer depth. Several illustrative calculations are discussed. A case with initially horizontally uniform temperature, 0.05°C m^{−1} initial vertical gradient, and wind stress of 1 dyn cm^{−2} and scale of 1000 km, shows horizontal temperature gradients of ∼0.01°C km^{−1} in the vicinity of the wind-driven convergence zone after 100 days integration. A similar case, except with initial horizontal gradient of 0.006°C km^{−1}, shows temperature gradients of 0.02°C km^{−1} after 100 days.

In wind-driven convergence zones, mixed-layer depths of 120 m can be achieved after 100 days, by a combination of entrainment and downwelling, mostly the latter, especially after long times. In divergent zones, steady mixed-layer depths can be achieved in less than 100 days through the competition between the effects of upwelling and entrainment. These steady depths range from 20 to 90 m, depending on location.

## Abstract

An improved bound is obtained for the radius of the semicircle in the complex plane containing the complex phase speed of baroclinically unstable plane wave disturbances. In the limit of long waves, this bound contains a term increasing with *β* and decreasing with the mean stratification (i.e., decreasing with the baroclinic Rossby radius of deformation). An extension of the bound, valid for finite wavelengths longer than order (*δ*
*u*/*β*)^{1/2}, where *δ*
*u* is half the range of velocities in the mean shear flow, is also obtained.

## Abstract

An improved bound is obtained for the radius of the semicircle in the complex plane containing the complex phase speed of baroclinically unstable plane wave disturbances. In the limit of long waves, this bound contains a term increasing with *β* and decreasing with the mean stratification (i.e., decreasing with the baroclinic Rossby radius of deformation). An extension of the bound, valid for finite wavelengths longer than order (*δ*
*u*/*β*)^{1/2}, where *δ*
*u* is half the range of velocities in the mean shear flow, is also obtained.

## Abstract

An alongshore wind stress is applied at the eastern boundary of an ocean model, and its effects on the structure of the circulation in the ventilated pycnocline are studied. The offshore surface Ekman flow associated with such a wind is taken to be balanced by onshore geostrophic transport distributed over the surface and subsurface layers. Hence, the layer interfaces at the boundary slope downward and equatorward, giving nonzero thicknesses. Because of this, some geostrophic contours (lines of constant potential vorticity) emerge from the eastern wall and strike northward, while some strike southward, giving rise to northern and southern limbs of the eastern shadow zones described by Luyten et al. The position of the layer-interface outcrop that forms the northern boundary of the northern limb of a shadow zone may not be freely chosen, but is determined so as to be consistent with the eastern boundary conditions, whose constraints are transmitted along the geostrophic contours. The westward extent of the northern shadow zone is usually quite modest, O(100 km) for typical alongshore winds. For stronger alongshore winds, or weaker interior forcing, greater westward penetration is possible.

The shadow zones furnish a means for sharply turning midocean currents and density outcrops to the south as they approach the eastern boundary, and forming alongshore boundary currents, which partly feed the wind-driven boundary upwelling and partly recirculate into the ocean interior.

When the subsurface layer is allowed to carry part of the onshore transport at the eastern boundary, motion is permitted in what would otherwise be a stagnant shadow zone. The potential vorticity of this layer is determined at the eastern boundary and conserved along geostrophic contours even against the sense of the circulation—a form of upstream influence.

## Abstract

An alongshore wind stress is applied at the eastern boundary of an ocean model, and its effects on the structure of the circulation in the ventilated pycnocline are studied. The offshore surface Ekman flow associated with such a wind is taken to be balanced by onshore geostrophic transport distributed over the surface and subsurface layers. Hence, the layer interfaces at the boundary slope downward and equatorward, giving nonzero thicknesses. Because of this, some geostrophic contours (lines of constant potential vorticity) emerge from the eastern wall and strike northward, while some strike southward, giving rise to northern and southern limbs of the eastern shadow zones described by Luyten et al. The position of the layer-interface outcrop that forms the northern boundary of the northern limb of a shadow zone may not be freely chosen, but is determined so as to be consistent with the eastern boundary conditions, whose constraints are transmitted along the geostrophic contours. The westward extent of the northern shadow zone is usually quite modest, O(100 km) for typical alongshore winds. For stronger alongshore winds, or weaker interior forcing, greater westward penetration is possible.

The shadow zones furnish a means for sharply turning midocean currents and density outcrops to the south as they approach the eastern boundary, and forming alongshore boundary currents, which partly feed the wind-driven boundary upwelling and partly recirculate into the ocean interior.

When the subsurface layer is allowed to carry part of the onshore transport at the eastern boundary, motion is permitted in what would otherwise be a stagnant shadow zone. The potential vorticity of this layer is determined at the eastern boundary and conserved along geostrophic contours even against the sense of the circulation—a form of upstream influence.

## Abstract

A layered model of steady geostrophic ocean circulation driven by wind stress and buoyancy flux at the surface is derived. Potential vorticity, or thickness, of the two near-surface layers is driven by Ekman pumping and buoyancy pumping. The latter is represented as a flow of mass proportional to the modified buoyancy flux, across the first submerged layer interface. This mass flux is modified by the advection of buoyancy in the wind-driven Ekman layer. Though diffusive diapycnal buoyancy flux across deeper layers is neglected at lowest order, it is essential for the global balance of the buoyancy budget. The global buoyancy balance requirement determines such parameters as the midocean outcrop latitudes of layers that outcrop in the subtropical gyre, and the depths of interfaces at the eastern boundary of layers that do not. These parameters control the mean thicknesses of the layers and, with the diapycnal diffusivity, the mean diffusive flux of buoyancy through each active layer. In this way the area-mean stratification is determined by the wind-driven circulation and the surface buoyancy flux.

Model solutions were computed for two idealized runs differing only by the amplitude of buoyancy forcing In run A, the surface buoyancy flux was chown to give a meridional buoyancy transport equivalent to 0.15 PW (1 PW = 1 petawatt) across the subtropical-subarctic gyre boundary. In run B, the buoyancy forcing was adjusted to give an intergyre meridional buoyancy transport equivalent to 0.51 PW. In both runs diapycnal diffusivities in the layers were held at O(10^{−4} m^{2} s^{−1}). These two runs gave density contrasts over the active layers of 8 kg m^{−3} (run A) and 18 kg m^{−3} (run B). The latter is an extremely large figure compared to the maximum density contrast across the ocean pycnocline observed in nature. The author concludes that the ocean cannot accomplish meridional buoyancy transport equivalent to O(1 PW), while diapycnal diffusivities are O(10^{−4} m^{2} s^{−1}) and density gradients across the pycnocline are ≲O(4 kg m^{−3}/1000 m). It is necessary for global buoyancy and heat balance that there are regions in the oceans with far larger diapycnal diffusivities than O(10^{−4} m^{2} s^{−1}). Likely candidates for such regions are the upper layers of the ocean, where extremely powerful mixing can be driven by surface wind stirring and convection, and the high-energy zones of the western boundary currents.

## Abstract

A layered model of steady geostrophic ocean circulation driven by wind stress and buoyancy flux at the surface is derived. Potential vorticity, or thickness, of the two near-surface layers is driven by Ekman pumping and buoyancy pumping. The latter is represented as a flow of mass proportional to the modified buoyancy flux, across the first submerged layer interface. This mass flux is modified by the advection of buoyancy in the wind-driven Ekman layer. Though diffusive diapycnal buoyancy flux across deeper layers is neglected at lowest order, it is essential for the global balance of the buoyancy budget. The global buoyancy balance requirement determines such parameters as the midocean outcrop latitudes of layers that outcrop in the subtropical gyre, and the depths of interfaces at the eastern boundary of layers that do not. These parameters control the mean thicknesses of the layers and, with the diapycnal diffusivity, the mean diffusive flux of buoyancy through each active layer. In this way the area-mean stratification is determined by the wind-driven circulation and the surface buoyancy flux.

Model solutions were computed for two idealized runs differing only by the amplitude of buoyancy forcing In run A, the surface buoyancy flux was chown to give a meridional buoyancy transport equivalent to 0.15 PW (1 PW = 1 petawatt) across the subtropical-subarctic gyre boundary. In run B, the buoyancy forcing was adjusted to give an intergyre meridional buoyancy transport equivalent to 0.51 PW. In both runs diapycnal diffusivities in the layers were held at O(10^{−4} m^{2} s^{−1}). These two runs gave density contrasts over the active layers of 8 kg m^{−3} (run A) and 18 kg m^{−3} (run B). The latter is an extremely large figure compared to the maximum density contrast across the ocean pycnocline observed in nature. The author concludes that the ocean cannot accomplish meridional buoyancy transport equivalent to O(1 PW), while diapycnal diffusivities are O(10^{−4} m^{2} s^{−1}) and density gradients across the pycnocline are ≲O(4 kg m^{−3}/1000 m). It is necessary for global buoyancy and heat balance that there are regions in the oceans with far larger diapycnal diffusivities than O(10^{−4} m^{2} s^{−1}). Likely candidates for such regions are the upper layers of the ocean, where extremely powerful mixing can be driven by surface wind stirring and convection, and the high-energy zones of the western boundary currents.

## Abstract

The evolution of the covariance of two tracers involves a quantity called codissipation, proportional to the covariance of the gradients of the two tracers, and analogous to the dissipation of tracer variance. The evolution of the variance of a composite tracer—a linear combination of two simple, primary tracers—depends on the“composite dissipation,” a combination of the individual simple tracer dissipations and the codissipation. The composite dissipation can be negative (implying growth of the variance of the composite tracer) for structures in which the correlation of the simple tracer gradients are large enough (i.e., large codissipation). This situation occurs in the phenomena of double diffusion and salt fingering. A particular composite tracer called watermass variation, a measure of water-type scatter about the mean tracer versus tracer relationship, lacks production terms of the conventional form—tracer flux multiplying tracer gradient—in its variance evolution balance. Only codissipation can produce variance of watermass variation. The requirements that watermass variance production and dissipation be in equilibrium, and that no other composite tracer variance be tending to grow due to codissipation, lead to a particular relation among codissipation and the simple dissipations and between the simple dissipations themselves. The latter are proportional to one another, the proportionality factor being the square of the slope of the mean tracer versus tracer relation. The same results can be obtained by modifying Batchelor’s argument to give the equilibrium cospectrum of two tracer gradients at high wavenumbers in a well-developed field of isotropic turbulence. As a consequence of these arguments, the turbulent eddy tracer fluxes are also proportional, with the mean tracer–tracer slope as proportionality factor. Further, the ratio of turbulent diffusivities of two tracers is unity. The dissipation of buoyancy, a composite tracer constructed from temperature and salinity, is proportional at equilibrium to thermal dissipation multiplied by a factor that depends on the stability ratio. This previously established result is obtained here under less restrictive conditions.

## Abstract

The evolution of the covariance of two tracers involves a quantity called codissipation, proportional to the covariance of the gradients of the two tracers, and analogous to the dissipation of tracer variance. The evolution of the variance of a composite tracer—a linear combination of two simple, primary tracers—depends on the“composite dissipation,” a combination of the individual simple tracer dissipations and the codissipation. The composite dissipation can be negative (implying growth of the variance of the composite tracer) for structures in which the correlation of the simple tracer gradients are large enough (i.e., large codissipation). This situation occurs in the phenomena of double diffusion and salt fingering. A particular composite tracer called watermass variation, a measure of water-type scatter about the mean tracer versus tracer relationship, lacks production terms of the conventional form—tracer flux multiplying tracer gradient—in its variance evolution balance. Only codissipation can produce variance of watermass variation. The requirements that watermass variance production and dissipation be in equilibrium, and that no other composite tracer variance be tending to grow due to codissipation, lead to a particular relation among codissipation and the simple dissipations and between the simple dissipations themselves. The latter are proportional to one another, the proportionality factor being the square of the slope of the mean tracer versus tracer relation. The same results can be obtained by modifying Batchelor’s argument to give the equilibrium cospectrum of two tracer gradients at high wavenumbers in a well-developed field of isotropic turbulence. As a consequence of these arguments, the turbulent eddy tracer fluxes are also proportional, with the mean tracer–tracer slope as proportionality factor. Further, the ratio of turbulent diffusivities of two tracers is unity. The dissipation of buoyancy, a composite tracer constructed from temperature and salinity, is proportional at equilibrium to thermal dissipation multiplied by a factor that depends on the stability ratio. This previously established result is obtained here under less restrictive conditions.

## Abstract

The thermobaric nonlinearity in the equation of state for seawater density—namely, the dependence of thermal expansibility on pressure—coupled with spatial variation of the oceanic temperature–salinity (*θ*–*s*) relation generates a nonlinear behavior in the buoyant force that can counter the linear dispersion of baroclinic Rossby waves and produce solitary waves. A Korteweg–deVries equation is derived in which the coefficient of the nonlinear term depends on the thermobaric parameter and the spatial gradient of the anomaly of the *θ*–*s* relation. Quantitative estimates can be made of the magnitude of the effect in terms of these parameters. For example, given first-baroclinic-mode spatial variations of order 0.1 psu (1000 km)^{−1} or 0.7°C (1000 km)^{−1}, from a *θ*–*s* relation with a density ratio of 2, a solitary Rossby wave of maximum vertical displacement of approximately 100 m and horizontal scale of approximately 30 baroclinic Rossby radii of deformation can be generated.

## Abstract

The thermobaric nonlinearity in the equation of state for seawater density—namely, the dependence of thermal expansibility on pressure—coupled with spatial variation of the oceanic temperature–salinity (*θ*–*s*) relation generates a nonlinear behavior in the buoyant force that can counter the linear dispersion of baroclinic Rossby waves and produce solitary waves. A Korteweg–deVries equation is derived in which the coefficient of the nonlinear term depends on the thermobaric parameter and the spatial gradient of the anomaly of the *θ*–*s* relation. Quantitative estimates can be made of the magnitude of the effect in terms of these parameters. For example, given first-baroclinic-mode spatial variations of order 0.1 psu (1000 km)^{−1} or 0.7°C (1000 km)^{−1}, from a *θ*–*s* relation with a density ratio of 2, a solitary Rossby wave of maximum vertical displacement of approximately 100 m and horizontal scale of approximately 30 baroclinic Rossby radii of deformation can be generated.

## Abstract

The forms of the primitive equations of motion and continuity are obtained when an arbitrary thermodynamic state variable=mrestricted only to be vertically monotonic=mis used as the vertical coordinate. Natural generalizations of the Montgomery and Exner functions suggest themselves. For a multicomponent fluid like seawater the dependence of the coordinate on salinity, coupled with the thermobaric effect, generates contributions to the momentum balance from the salinity gradient, multiplied by a thermodynamic coefficient that can be completely described given the coordinate variable and the equation of state. In the vorticity balance this term produces a contribution identified with the baroclinicity vector. Only when the coordinate variable is a function only of pressure and in situ specific volume does the coefficient of salinity gradient vanish and the baroclinicity vector disappear.

This coefficient is explicitly calculated and displayed for potential specific volume as thermodynamic coordinate, and for patched potential specific volume, where different reference pressures are used in various pressure subranges. Except within a few hundred decibars of the reference pressures, the salinity-gradient coefficient is not negligible and ought to be taken into account in ocean circulation models.

## Abstract

The forms of the primitive equations of motion and continuity are obtained when an arbitrary thermodynamic state variable=mrestricted only to be vertically monotonic=mis used as the vertical coordinate. Natural generalizations of the Montgomery and Exner functions suggest themselves. For a multicomponent fluid like seawater the dependence of the coordinate on salinity, coupled with the thermobaric effect, generates contributions to the momentum balance from the salinity gradient, multiplied by a thermodynamic coefficient that can be completely described given the coordinate variable and the equation of state. In the vorticity balance this term produces a contribution identified with the baroclinicity vector. Only when the coordinate variable is a function only of pressure and in situ specific volume does the coefficient of salinity gradient vanish and the baroclinicity vector disappear.

This coefficient is explicitly calculated and displayed for potential specific volume as thermodynamic coordinate, and for patched potential specific volume, where different reference pressures are used in various pressure subranges. Except within a few hundred decibars of the reference pressures, the salinity-gradient coefficient is not negligible and ought to be taken into account in ocean circulation models.

## Abstract

We formulate analytically and solve numerically a semigeostrophic model for wind-driven thermocline upwelling at a coastal boundary. The model has a variable-density entraining mixed layer and two homogeneous interior layers. All variables are uniform alongshore. The wind stress and surface heating are constant. The system is started from rest, with constant layer depths and mixed layer density. A modified Ekman balance is prescribed far offshore, and the normal-to-shore velocity field responds on the scales of the effective local internal deformation radii, which themselves adjust in response to changes in layer depths, interior geostrophic vorticity, and mixed layer density. Sustained upwelling results in a steplike horizontal profile of mixed layer density, as the layer interfaces “surface” and are advected offshore. The upwelled fronts have width O(*u*
_{*}/*f*), as in the two-layer model of de Szoeke and Richman (1984). For fixed initial layer depths, the interior response and the horizontal separation of the upwelled fronts scale with the initial internal deformation radii. Around the fronts, surface layer divergence occurs that is equal in magnitude to the divergence in the upwelling zone adjacent to the coast, but its depth penetration is inhibited by the stratification.

## Abstract

We formulate analytically and solve numerically a semigeostrophic model for wind-driven thermocline upwelling at a coastal boundary. The model has a variable-density entraining mixed layer and two homogeneous interior layers. All variables are uniform alongshore. The wind stress and surface heating are constant. The system is started from rest, with constant layer depths and mixed layer density. A modified Ekman balance is prescribed far offshore, and the normal-to-shore velocity field responds on the scales of the effective local internal deformation radii, which themselves adjust in response to changes in layer depths, interior geostrophic vorticity, and mixed layer density. Sustained upwelling results in a steplike horizontal profile of mixed layer density, as the layer interfaces “surface” and are advected offshore. The upwelled fronts have width O(*u*
_{*}/*f*), as in the two-layer model of de Szoeke and Richman (1984). For fixed initial layer depths, the interior response and the horizontal separation of the upwelled fronts scale with the initial internal deformation radii. Around the fronts, surface layer divergence occurs that is equal in magnitude to the divergence in the upwelling zone adjacent to the coast, but its depth penetration is inhibited by the stratification.

## Abstract

The purpose of this paper is to understand how long planetary waves evolve when propagating in a subtropical gyre. The steady flow of a wind-driven vertically sheared model subtropical gyre is perturbed by Ekman pumping that is localized within a region of finite lateral extent and oscillates periodically at about the annual frequency after sudden initiation. Both the background flow and the infinitesimal perturbations are solutions of a 2½-layer model. The region of forcing is located in the eastern part of the gyre where the steady flow is confined to the uppermost layer (shadow zone). The lateral scales of the forcing and of the response are supposed to be small enough with respect to the overall gyre scale that the background flow may be idealized as horizontally uniform, yet large enough (greater than the baroclinic Rossby radii) that the long-wave approximation may be made. The latter approximation limits the length of time over which the solutions remain valid. The solutions consist of (i) a forced response oscillating at the forcing frequency in which both stable (real) and zonally growing (complex) meridional wavenumbers are excited plus (ii) a localized transient structure that grows as it propagates away from the region of forcing. Application of the method of stationary phase provides analytical solutions that permit clear separation of the directly forced part of the solution and the transient as well as estimation of the temporal growth rate of the transient, which proves to be convectively unstable. The solutions presented here are relevant to understanding the instability of periodic (including annual period) perturbations of oceanic subtropical gyres on scales larger than the baroclinic Rossby radii of deformation.

## Abstract

The purpose of this paper is to understand how long planetary waves evolve when propagating in a subtropical gyre. The steady flow of a wind-driven vertically sheared model subtropical gyre is perturbed by Ekman pumping that is localized within a region of finite lateral extent and oscillates periodically at about the annual frequency after sudden initiation. Both the background flow and the infinitesimal perturbations are solutions of a 2½-layer model. The region of forcing is located in the eastern part of the gyre where the steady flow is confined to the uppermost layer (shadow zone). The lateral scales of the forcing and of the response are supposed to be small enough with respect to the overall gyre scale that the background flow may be idealized as horizontally uniform, yet large enough (greater than the baroclinic Rossby radii) that the long-wave approximation may be made. The latter approximation limits the length of time over which the solutions remain valid. The solutions consist of (i) a forced response oscillating at the forcing frequency in which both stable (real) and zonally growing (complex) meridional wavenumbers are excited plus (ii) a localized transient structure that grows as it propagates away from the region of forcing. Application of the method of stationary phase provides analytical solutions that permit clear separation of the directly forced part of the solution and the transient as well as estimation of the temporal growth rate of the transient, which proves to be convectively unstable. The solutions presented here are relevant to understanding the instability of periodic (including annual period) perturbations of oceanic subtropical gyres on scales larger than the baroclinic Rossby radii of deformation.

## Abstract

The hydrostatic equations of motion for ocean circulation, written in terms of pressure as the vertical coordinate, and without making the Boussinesq approximation in the continuity equation, correspond very closely with the hydrostatic Boussinesq equations written in terms of depth as the vertical coordinate. Two mathematical equivalences between these non-Boussinesq and Boussinesq equation sets are demonstrated: first, for motions over a level bottom; second, for general motions with a rigid lid. A third non-Boussinesq equation set, for general motions with a free surface, is derived and is shown to possess a similar duality with the Boussinesq set after making due allowance for exchange of the roles of bottom pressure and sea surface height in the boundary conditions, a reversal of the direction of integration of the hydrostatic equation, and substitution of specific volume for density in the hydrostatic equation. The crucial simplification in these equations of motion comes from the hydrostatic approximation, not the Boussinesq approximation. A practical consequence is that numerical ocean circulation models that are based on the Boussinesq equations can, with very minimal rearrangement and reinterpretation, be made free of the strictures of the Boussinesq approximation, especially the ones that follow from its neglect of density dilatation in the conservation of mass.

## Abstract

The hydrostatic equations of motion for ocean circulation, written in terms of pressure as the vertical coordinate, and without making the Boussinesq approximation in the continuity equation, correspond very closely with the hydrostatic Boussinesq equations written in terms of depth as the vertical coordinate. Two mathematical equivalences between these non-Boussinesq and Boussinesq equation sets are demonstrated: first, for motions over a level bottom; second, for general motions with a rigid lid. A third non-Boussinesq equation set, for general motions with a free surface, is derived and is shown to possess a similar duality with the Boussinesq set after making due allowance for exchange of the roles of bottom pressure and sea surface height in the boundary conditions, a reversal of the direction of integration of the hydrostatic equation, and substitution of specific volume for density in the hydrostatic equation. The crucial simplification in these equations of motion comes from the hydrostatic approximation, not the Boussinesq approximation. A practical consequence is that numerical ocean circulation models that are based on the Boussinesq equations can, with very minimal rearrangement and reinterpretation, be made free of the strictures of the Boussinesq approximation, especially the ones that follow from its neglect of density dilatation in the conservation of mass.