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## Abstract

Three-dimensional numerical simulations of rotating, statically and inertially stable, mesoscale flows show that wave packets, with vertical velocity comparable to that of the balanced flow, can be spontaneously generated and amplified in the frontal part of translating vortical structures. These frontal wave packets remain stationary relative to the vortical structure (e.g., in the baroclinic dipole, tripole, and quadrupole) and are due to inertia–gravity oscillations, near the inertial frequency, experienced by the fluid particles as they decelerate when leaving the large speed regions. The ratio between the horizontal and vertical wavenumbers depends on the ratio between the horizontal and vertical shears of the background velocity. Theoretical solutions of plane waves with varying wavenumbers in background flow confirm these results. Using the material description of the fields it is shown that, among the particles simultaneously located in the vertical column in the dipole’s center, the first ones to experience the inertia–gravity oscillations are those in the upper layer, in the region of the maximum vertical shear. The wave packet propagates afterward to the fluid particles located below.

## Abstract

Three-dimensional numerical simulations of rotating, statically and inertially stable, mesoscale flows show that wave packets, with vertical velocity comparable to that of the balanced flow, can be spontaneously generated and amplified in the frontal part of translating vortical structures. These frontal wave packets remain stationary relative to the vortical structure (e.g., in the baroclinic dipole, tripole, and quadrupole) and are due to inertia–gravity oscillations, near the inertial frequency, experienced by the fluid particles as they decelerate when leaving the large speed regions. The ratio between the horizontal and vertical wavenumbers depends on the ratio between the horizontal and vertical shears of the background velocity. Theoretical solutions of plane waves with varying wavenumbers in background flow confirm these results. Using the material description of the fields it is shown that, among the particles simultaneously located in the vertical column in the dipole’s center, the first ones to experience the inertia–gravity oscillations are those in the upper layer, in the region of the maximum vertical shear. The wave packet propagates afterward to the fluid particles located below.

## Abstract

The processes involved in the vertical splitting of vortices in geophysical dipoles, rotating and stably stratified, are investigated using a three-dimensional numerical model under the *f*-plane and Boussinesq approximations. Vertical splitting in asymmetric dipoles is possible when the vortices have a similar amount of potential vorticity but significantly differ in vertical extent. One representative case of vertical splitting is analyzed, and it is found that prior to the splitting there is a shearing period characterized by vertical unalignment and loss of horizontal axisymmetrization. The splitting starts when the upper and lower parts of the deep vortex independently experience vertical alignment and horizontal axisymmetrization. Vertical splitting also involves vortex horizontal splitting in the intermediate layers, which might explain the vertical asymmetry found in some isolated subsurface vortices in the ocean interior.

## Abstract

The processes involved in the vertical splitting of vortices in geophysical dipoles, rotating and stably stratified, are investigated using a three-dimensional numerical model under the *f*-plane and Boussinesq approximations. Vertical splitting in asymmetric dipoles is possible when the vortices have a similar amount of potential vorticity but significantly differ in vertical extent. One representative case of vertical splitting is analyzed, and it is found that prior to the splitting there is a shearing period characterized by vertical unalignment and loss of horizontal axisymmetrization. The splitting starts when the upper and lower parts of the deep vortex independently experience vertical alignment and horizontal axisymmetrization. Vertical splitting also involves vortex horizontal splitting in the intermediate layers, which might explain the vertical asymmetry found in some isolated subsurface vortices in the ocean interior.

## Abstract

Potential vorticity (PV) is usually defined as *α*
** ω** · grad

*ϕ,*where

*α*is the specific volume,

**is vorticity, and**

*ω**ϕ*is any quantity, usually a conserved one. The most common derivation of the PV theorem therefore uses the component of the vorticity equation normal to the

*ϕ*surfaces. Since PV can also be expressed as

*α*div(

**u**× grad

*ϕ*) and

*α*div(

*ω**ϕ*), alternative derivations of the PV conservation law are introduced. In these derivations the PV conservation theorem is considered as the divergence of the projection (weighted by |grad

*ϕ*|) of the equation of motion onto the direction of grad

*ϕ,*or, alternately, as the divergence of a

*ϕ*-weighted vorticity equation. The first of these interpretations is closely related to the procedure of considering every

*ϕ*surface as a surface of constraint for the infinitesimal virtual displacements used in variational methods, and therefore it is closely related to a Hamiltonian derivation of the PV theorem. The different expressions are presented using the spatial as well as the material description of the fields. The kinematical foundations of the PV theorem in the material description are especially simple because they only involve derivative commutations with respect to the material variables.

It is also provided a precise mathematical expression for the so-called impermeability theorem, clarifying the sense in which such a theorem can be understood. In order to do so it is necessary to introduce a suitable transformation of the fluid velocity. An immediate consequence of such a transformation is that the quantity *ϕ* and the quantity ** ω** · grad

*ϕ*(also called potential vorticity substance per unit volume) behave as a label of the particles and as the “density,” respectively, of the transformed fluid. The impermeability theorem is then an expression of the conservation of the “mass” and of the conservation of the identity of the particles in the transformed fluid.

## Abstract

Potential vorticity (PV) is usually defined as *α*
** ω** · grad

*ϕ,*where

*α*is the specific volume,

**is vorticity, and**

*ω**ϕ*is any quantity, usually a conserved one. The most common derivation of the PV theorem therefore uses the component of the vorticity equation normal to the

*ϕ*surfaces. Since PV can also be expressed as

*α*div(

**u**× grad

*ϕ*) and

*α*div(

*ω**ϕ*), alternative derivations of the PV conservation law are introduced. In these derivations the PV conservation theorem is considered as the divergence of the projection (weighted by |grad

*ϕ*|) of the equation of motion onto the direction of grad

*ϕ,*or, alternately, as the divergence of a

*ϕ*-weighted vorticity equation. The first of these interpretations is closely related to the procedure of considering every

*ϕ*surface as a surface of constraint for the infinitesimal virtual displacements used in variational methods, and therefore it is closely related to a Hamiltonian derivation of the PV theorem. The different expressions are presented using the spatial as well as the material description of the fields. The kinematical foundations of the PV theorem in the material description are especially simple because they only involve derivative commutations with respect to the material variables.

It is also provided a precise mathematical expression for the so-called impermeability theorem, clarifying the sense in which such a theorem can be understood. In order to do so it is necessary to introduce a suitable transformation of the fluid velocity. An immediate consequence of such a transformation is that the quantity *ϕ* and the quantity ** ω** · grad

*ϕ*(also called potential vorticity substance per unit volume) behave as a label of the particles and as the “density,” respectively, of the transformed fluid. The impermeability theorem is then an expression of the conservation of the “mass” and of the conservation of the identity of the particles in the transformed fluid.

## Abstract

A new derivation and interpretation of the semigeostrophic (SG) material invariant in the theory of geophysical flows is introduced. First, a generalized three-dimensional equation of the SG dynamics is established and the generalized equations for the rate of change of vorticity and for the rate of change of the velocity gradient cofactor tensor are obtained. Next, a conservation equation for the vorticity–velocity gradient cofactor tensor (denoted **Ξ̃**) is derived. The specific potential **Ξ̃**, that is, **Ξ̃** in the reference configuration per unit of mass, is defined and an expression for its rate of change is obtained. The SG invariant is interpreted as the vertical component of the specific potential **Ξ̃**. Under the SG assumptions (advection of the geostrophic velocity, hydrostatic, and *f*-plane approximations) this vertical component is materially conserved in the SG flow. The generalized SG invariant (i.e., the specific potential **Ξ̃**) differs conceptually from the Beltrami–Rossby–Ertel specific potential vorticity. Its conservation in the SG flow seems to be highly dependent on the SG assumptions, especially on the *f*-plane approximation and on the horizontal nature of the geostrophic velocity.

## Abstract

A new derivation and interpretation of the semigeostrophic (SG) material invariant in the theory of geophysical flows is introduced. First, a generalized three-dimensional equation of the SG dynamics is established and the generalized equations for the rate of change of vorticity and for the rate of change of the velocity gradient cofactor tensor are obtained. Next, a conservation equation for the vorticity–velocity gradient cofactor tensor (denoted **Ξ̃**) is derived. The specific potential **Ξ̃**, that is, **Ξ̃** in the reference configuration per unit of mass, is defined and an expression for its rate of change is obtained. The SG invariant is interpreted as the vertical component of the specific potential **Ξ̃**. Under the SG assumptions (advection of the geostrophic velocity, hydrostatic, and *f*-plane approximations) this vertical component is materially conserved in the SG flow. The generalized SG invariant (i.e., the specific potential **Ξ̃**) differs conceptually from the Beltrami–Rossby–Ertel specific potential vorticity. Its conservation in the SG flow seems to be highly dependent on the SG assumptions, especially on the *f*-plane approximation and on the horizontal nature of the geostrophic velocity.

## ABSTRACT

A 2019 comment by Hochet and Tailleux and the corresponding reply by Holmes et al. discuss the volume and mass balance on a control volume bounded by a given isotherm and the ocean free surface. This note partly reconciles the terms of discrepancy on volume or mass transport appearing in these publications by proving that the integral expressions in the comment of Hochet and Tailleux, using a particular parameterization of the moving surfaces in Cartesian coordinates, correspond to the mass transport across the moving surfaces, as long as the mass density is included, as given in direct vector notation by Holmes et al. in their reply.

## ABSTRACT

A 2019 comment by Hochet and Tailleux and the corresponding reply by Holmes et al. discuss the volume and mass balance on a control volume bounded by a given isotherm and the ocean free surface. This note partly reconciles the terms of discrepancy on volume or mass transport appearing in these publications by proving that the integral expressions in the comment of Hochet and Tailleux, using a particular parameterization of the moving surfaces in Cartesian coordinates, correspond to the mass transport across the moving surfaces, as long as the mass density is included, as given in direct vector notation by Holmes et al. in their reply.

## Abstract

The *β* term, usually defined as the northward gradient of the vertical planetary vorticity, is here defined and interpreted as the northward planetary vorticity. The *β* term in the vorticity equation, relative to a rotating reference frame, is put into an exact correspondence with the planetary vorticity tilting term, which has a meaning independent of the coordinate system. For barotropic, isochoric, horizontal flow there is an exact balance between the rate of change of vertical vorticity and the planetary vorticity tilting term due to the northward components of velocity and planetary vorticity. This interpretation of the *β* term in the vorticity equation seems to be simpler than the interpretation based on the northward gradient of the vertical planetary vorticity because it only involves the components, not the derivatives, of the planetary vorticity, and because it is independent of the coordinate system. It is also shown, as a consequence of this new interpretation, that the Sverdrup relation is practically equivalent to the vertical component of the Taylor–Proudman constraint.

## Abstract

The *β* term, usually defined as the northward gradient of the vertical planetary vorticity, is here defined and interpreted as the northward planetary vorticity. The *β* term in the vorticity equation, relative to a rotating reference frame, is put into an exact correspondence with the planetary vorticity tilting term, which has a meaning independent of the coordinate system. For barotropic, isochoric, horizontal flow there is an exact balance between the rate of change of vertical vorticity and the planetary vorticity tilting term due to the northward components of velocity and planetary vorticity. This interpretation of the *β* term in the vorticity equation seems to be simpler than the interpretation based on the northward gradient of the vertical planetary vorticity because it only involves the components, not the derivatives, of the planetary vorticity, and because it is independent of the coordinate system. It is also shown, as a consequence of this new interpretation, that the Sverdrup relation is practically equivalent to the vertical component of the Taylor–Proudman constraint.

## Abstract

A consistent explanation for the anticyclonic curvature of the Atlantic jet as it passes through the Strait of Gibraltar and flows into the Mediterranean Sea (eastern side of the strait) is provided. The anticyclonic curvature of the Atlantic jet, which is the key feature to understand the upper-layer circulation in the western Alboran Sea, is simply related to the positive net evaporation over the Mediterranean. The result of this positive net evaporation, that mainly occurs in areas of the Mediterranean far from the Strait of Gibraltar, is a net inflow transport through the strait. It is proposed that the positive net evaporation is able to produce such a net inflow in the strait because of an anomalous large-scale pressure gradient. This anomalous pressure gradient is found to be approximately collinear to the strait orientation. The time-averaged inflow of Atlantic water at the eastern side of the Strait of Gibraltar must therefore be supergeostrophic, and hence it must have anticyclonic curvature.

## Abstract

A consistent explanation for the anticyclonic curvature of the Atlantic jet as it passes through the Strait of Gibraltar and flows into the Mediterranean Sea (eastern side of the strait) is provided. The anticyclonic curvature of the Atlantic jet, which is the key feature to understand the upper-layer circulation in the western Alboran Sea, is simply related to the positive net evaporation over the Mediterranean. The result of this positive net evaporation, that mainly occurs in areas of the Mediterranean far from the Strait of Gibraltar, is a net inflow transport through the strait. It is proposed that the positive net evaporation is able to produce such a net inflow in the strait because of an anomalous large-scale pressure gradient. This anomalous pressure gradient is found to be approximately collinear to the strait orientation. The time-averaged inflow of Atlantic water at the eastern side of the Strait of Gibraltar must therefore be supergeostrophic, and hence it must have anticyclonic curvature.

## Abstract

It is shown that there is an exact correspondence between the scalar Rossby–Ertel's potential vorticity (PV) for a field ε, and the component of Beltrami's material vorticity along the ε-coordinate line (first equivalence theorem). Thus, Rossby–Ertel's PV can be interpreted as a particular case (scalar) of the vectorial Beltrami's material vorticity. The rate of change of Beltrami's vorticity only depends on the curl of the acceleration (or baroclinic–diffusive terms in the rate of change of vorticity) and not on the convective rate of change (involving advection, stretching, and divergence terms). When the motion is circulation preserving (the acceleration is irrotational) Cauchy's vorticity formula states that Beltrami's material vorticity is conserved. However, when the curl of the acceleration is zero only along the direction normal to certain ε surfaces, only the ε component of Beltrami's material vorticity is conserved. Thus, a second equivalence theorem states that Ertel's PV conservation theorem is equivalent to Cauchy's vorticity formula along the ε-coordinate line.

Beltrami's material vorticity is first introduced via Piola's transformations from the spatial vorticity field, but it is shown that a direct definition of Beltrami's material vorticity using the referential velocity and in terms of material variables is also possible. In this approach the current *definition* of specific PV in terms of the spatial vorticity and the gradient of ε becomes instead a *relation* between both vorticities that can be derived from their respective definitions.

## Abstract

It is shown that there is an exact correspondence between the scalar Rossby–Ertel's potential vorticity (PV) for a field ε, and the component of Beltrami's material vorticity along the ε-coordinate line (first equivalence theorem). Thus, Rossby–Ertel's PV can be interpreted as a particular case (scalar) of the vectorial Beltrami's material vorticity. The rate of change of Beltrami's vorticity only depends on the curl of the acceleration (or baroclinic–diffusive terms in the rate of change of vorticity) and not on the convective rate of change (involving advection, stretching, and divergence terms). When the motion is circulation preserving (the acceleration is irrotational) Cauchy's vorticity formula states that Beltrami's material vorticity is conserved. However, when the curl of the acceleration is zero only along the direction normal to certain ε surfaces, only the ε component of Beltrami's material vorticity is conserved. Thus, a second equivalence theorem states that Ertel's PV conservation theorem is equivalent to Cauchy's vorticity formula along the ε-coordinate line.

Beltrami's material vorticity is first introduced via Piola's transformations from the spatial vorticity field, but it is shown that a direct definition of Beltrami's material vorticity using the referential velocity and in terms of material variables is also possible. In this approach the current *definition* of specific PV in terms of the spatial vorticity and the gradient of ε becomes instead a *relation* between both vorticities that can be derived from their respective definitions.

## Abstract

This work provides the general theory of the volume and mass conservation in terms of its transport across surfaces (open or closed) defined by a constant value of an oceanographic property obeying a balance law. This fluid property can be any spatial density *ϕ* (amount of quantity per unit volume), or its specific value *ϕ̂**αϕ* (amount of quantity per unit mass, where *α* is the specific volume). The main expressions obtained relate the volume transport across a *ϕ*-surface to the flux of the quantity (**h**
_{ϕ}) across the *ϕ*-surface boundary, and the mass transport across a *ϕ̂***h**
_{ϕ} across the *ϕ̂**ϕ*-surface, being however equivalent for closed (unlimited) *ϕ*-surfaces. The main expressions are generalized to the three-dimensional case, and the relation to previous results is discussed.

## Abstract

This work provides the general theory of the volume and mass conservation in terms of its transport across surfaces (open or closed) defined by a constant value of an oceanographic property obeying a balance law. This fluid property can be any spatial density *ϕ* (amount of quantity per unit volume), or its specific value *ϕ̂**αϕ* (amount of quantity per unit mass, where *α* is the specific volume). The main expressions obtained relate the volume transport across a *ϕ*-surface to the flux of the quantity (**h**
_{ϕ}) across the *ϕ*-surface boundary, and the mass transport across a *ϕ̂***h**
_{ϕ} across the *ϕ̂**ϕ*-surface, being however equivalent for closed (unlimited) *ϕ*-surfaces. The main expressions are generalized to the three-dimensional case, and the relation to previous results is discussed.

## Abstract

Frontal collisions of mesoscale baroclinic dipoles are numerically investigated using a three-dimensional, Boussinesq, and *f*-plane numerical model that explicitly conserves potential vorticity on isopycnals. The initial conditions, obtained using the potential vorticity initialization approach, consist of twin baroclinic dipoles, balanced (void of waves) and static and inertially stable, moving in opposite directions. The dipoles may collide in a close-to-axial way (cyclone–anticyclone collisions) or nonaxially (cyclone–cyclone or anticyclone–anticyclone collisions). The results show that the interacting vortices may bounce back and interchange partners, may merge reaching a tripole state, or may squeeze between the outer vortices. The formation of a stable tripole from two colliding dipoles is possible but is dependent on diffusion effects. It is found that the nonaxial dipole collisions can be characterized by the interchange between the domain-averaged potential and kinetic energy. Dipole collisions in two-dimensional flow display also a variety of vortex interactions, qualitatively similar to the three-dimensional cases.

## Abstract

Frontal collisions of mesoscale baroclinic dipoles are numerically investigated using a three-dimensional, Boussinesq, and *f*-plane numerical model that explicitly conserves potential vorticity on isopycnals. The initial conditions, obtained using the potential vorticity initialization approach, consist of twin baroclinic dipoles, balanced (void of waves) and static and inertially stable, moving in opposite directions. The dipoles may collide in a close-to-axial way (cyclone–anticyclone collisions) or nonaxially (cyclone–cyclone or anticyclone–anticyclone collisions). The results show that the interacting vortices may bounce back and interchange partners, may merge reaching a tripole state, or may squeeze between the outer vortices. The formation of a stable tripole from two colliding dipoles is possible but is dependent on diffusion effects. It is found that the nonaxial dipole collisions can be characterized by the interchange between the domain-averaged potential and kinetic energy. Dipole collisions in two-dimensional flow display also a variety of vortex interactions, qualitatively similar to the three-dimensional cases.