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## Abstract

Geostrophic adjustment and frontogenesis are examined by means of a two-dimensional, inviscid, rotating and nonlinear fluid model that satisfies the condition of zero potential vorticity. The fluid is bounded top and bottom by level, rigid lids. The initial state is one of no motion, but an unbalanced horizontal temperature gradient is prescribed. The subsequent motion is represented as the sum of an inertial oscillation, with the frequency of the local Coriolis frequency *f,* and an evolving geostrophic flow. When a nondimensional parameter *a,* a Rossby number, satisfies *a* < 1, the gradient of the evolving geostrophic flow increases (frontogenesis) during the period 0 < *t* â©½ *Ï€*/*f*; the gradient decreases during the period *Ï€*/*f* < *t* â©½ 2*Ï€*/*f* (frontolysis). When *a* â‰¥ 1, the relative vorticity of the evolving geostrophic flow becomes infinite: a discontinuity forms at the top and bottom boundaries during the period 0 < *t* â©½ *Ï€*/*f.* There is an equipartition of energy between the inertial oscillation and the geostrophic flow, and nonlinear interactions occur between them. An exact (Fourier) spectral representation of the solution on the bottom boundary is used to display the kinetic energy spectrum and the transfer of energy through the spectrum at the time that the discontinuity forms. Applications of the model to oceanic and to atmospheric frontogenesis and to restratification of the surface mixed layer, following a storm, are noted.

## Abstract

Geostrophic adjustment and frontogenesis are examined by means of a two-dimensional, inviscid, rotating and nonlinear fluid model that satisfies the condition of zero potential vorticity. The fluid is bounded top and bottom by level, rigid lids. The initial state is one of no motion, but an unbalanced horizontal temperature gradient is prescribed. The subsequent motion is represented as the sum of an inertial oscillation, with the frequency of the local Coriolis frequency *f,* and an evolving geostrophic flow. When a nondimensional parameter *a,* a Rossby number, satisfies *a* < 1, the gradient of the evolving geostrophic flow increases (frontogenesis) during the period 0 < *t* â©½ *Ï€*/*f*; the gradient decreases during the period *Ï€*/*f* < *t* â©½ 2*Ï€*/*f* (frontolysis). When *a* â‰¥ 1, the relative vorticity of the evolving geostrophic flow becomes infinite: a discontinuity forms at the top and bottom boundaries during the period 0 < *t* â©½ *Ï€*/*f.* There is an equipartition of energy between the inertial oscillation and the geostrophic flow, and nonlinear interactions occur between them. An exact (Fourier) spectral representation of the solution on the bottom boundary is used to display the kinetic energy spectrum and the transfer of energy through the spectrum at the time that the discontinuity forms. Applications of the model to oceanic and to atmospheric frontogenesis and to restratification of the surface mixed layer, following a storm, are noted.

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## Abstract

Nonlinear geostrophic adjustment is examined with a Boussinesq model. The motion is restricted to a two-dimensional channel in the horizontal and vertical (*x*, *z*) plane; the fluid is in uniform rotation, is stably stratified, inviscid, and incompressible. The flows considered fall under two classes: zero and uniform potential vorticity flows. Steady geostrophic flow fields are determined from initial man imbalances, represented by both symmetric and antisymmetric density anomalies that vary along the *x* axis. The distinguishing characteristic of these solutions is the development of a front, defined as a zero-order discontinuity in both density and geostrophic velocity at one or both vertical boundaries. Frontal formation occurs, as previously discovered by Ou for zero potential vorticity flow, when the initial horizontal density gradient is sufficiently large. The critical values are displayed for different cases in terms of the initial amplitude and initial scale of the density anomaly.

The conversion of initial potential energy into geostrophic kinetic Î”KE and potential Î”PE energies during adjustment is also derived. Ou's result that Î³ = Î”KE/Î”PE = 1/2, independent of the initial scale is confirmed. It is shown, however, that Î³â‰¤1/2 for uniform potential vorticity flow. Large initial scales *a*
^{âˆ’1}, large compared to the deformation radius, have the largest values of Î³, approaching Î³ = 1/2 as *a*â†’0. This limit approaches the solution and energy ratio for zero potential vorticity flow. The energy ratio associated with an antisymmetric density anomaly is characterized by Î³â†’1/3 and *a* â†’âˆž: that is, the initial mass imbalance becomes a step function. In the other case, when the initial disturbance is symmetric and vanishes with *a*â†’âˆž, Î³ also vanishes. These results unify previous studies that have not provided the distinction between zero and uniform potential vorticity flows in examinations of the energy conversion process. Yet the reason for this distinction has not been delineated.

## Abstract

Nonlinear geostrophic adjustment is examined with a Boussinesq model. The motion is restricted to a two-dimensional channel in the horizontal and vertical (*x*, *z*) plane; the fluid is in uniform rotation, is stably stratified, inviscid, and incompressible. The flows considered fall under two classes: zero and uniform potential vorticity flows. Steady geostrophic flow fields are determined from initial man imbalances, represented by both symmetric and antisymmetric density anomalies that vary along the *x* axis. The distinguishing characteristic of these solutions is the development of a front, defined as a zero-order discontinuity in both density and geostrophic velocity at one or both vertical boundaries. Frontal formation occurs, as previously discovered by Ou for zero potential vorticity flow, when the initial horizontal density gradient is sufficiently large. The critical values are displayed for different cases in terms of the initial amplitude and initial scale of the density anomaly.

The conversion of initial potential energy into geostrophic kinetic Î”KE and potential Î”PE energies during adjustment is also derived. Ou's result that Î³ = Î”KE/Î”PE = 1/2, independent of the initial scale is confirmed. It is shown, however, that Î³â‰¤1/2 for uniform potential vorticity flow. Large initial scales *a*
^{âˆ’1}, large compared to the deformation radius, have the largest values of Î³, approaching Î³ = 1/2 as *a*â†’0. This limit approaches the solution and energy ratio for zero potential vorticity flow. The energy ratio associated with an antisymmetric density anomaly is characterized by Î³â†’1/3 and *a* â†’âˆž: that is, the initial mass imbalance becomes a step function. In the other case, when the initial disturbance is symmetric and vanishes with *a*â†’âˆž, Î³ also vanishes. These results unify previous studies that have not provided the distinction between zero and uniform potential vorticity flows in examinations of the energy conversion process. Yet the reason for this distinction has not been delineated.

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## Abstract

Geostrophic adjustment of fluid in constant rotation is considered. The initial state is characterized by a mass imbalance, the initial velocity field is geostrophic, and zero potential vorticity flow is imposed. The final balanced state is determined from conservation of potential vorticity (zero), linear momentum, and mass. The potential energy released during the adjustment process is partitioned into the kinetic and potential energies of the balanced state in the ratio 1:2, independent of the scale of the initial state. The development of a front (infinite relative vorticity and a zero-order discontinuity in density) does, however, depend on the spatial scales of the initial fields. Moreover, the initial position of the maximum vorticity relative to the initial position of the maximum density gradient significantly affects the balanced flow. Physical interpretations of the principal features are provided.

## Abstract

Geostrophic adjustment of fluid in constant rotation is considered. The initial state is characterized by a mass imbalance, the initial velocity field is geostrophic, and zero potential vorticity flow is imposed. The final balanced state is determined from conservation of potential vorticity (zero), linear momentum, and mass. The potential energy released during the adjustment process is partitioned into the kinetic and potential energies of the balanced state in the ratio 1:2, independent of the scale of the initial state. The development of a front (infinite relative vorticity and a zero-order discontinuity in density) does, however, depend on the spatial scales of the initial fields. Moreover, the initial position of the maximum vorticity relative to the initial position of the maximum density gradient significantly affects the balanced flow. Physical interpretations of the principal features are provided.