Geostrophic Adjustment: Frontogenesis and Energy Conversion

William Blumen Astrophysical, Planetary, and Atmospheric Sciences Department, University of Colorado, Boulder, Colorado

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Rongsheng Wu Department of Atmospheric Sciences, Nanjing University, People's Republic of China

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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 ΔKE and potential ΔPE energies during adjustment is also derived. Ou's result that γ = ΔKE/ΔPE = 1/2, independent of the initial scale is confirmed. It is shown, however, that γ≤1/2 for uniform potential vorticity flow. Large initial scales a−1, large compared to the deformation radius, have the largest values of γ, approaching γ = 1/2 as a→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 γ→1/3 and a →∞: that is, the initial mass imbalance becomes a step function. In the other case, when the initial disturbance is symmetric and vanishes with a→∞, γ 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 ΔKE and potential ΔPE energies during adjustment is also derived. Ou's result that γ = ΔKE/ΔPE = 1/2, independent of the initial scale is confirmed. It is shown, however, that γ≤1/2 for uniform potential vorticity flow. Large initial scales a−1, large compared to the deformation radius, have the largest values of γ, approaching γ = 1/2 as a→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 γ→1/3 and a →∞: that is, the initial mass imbalance becomes a step function. In the other case, when the initial disturbance is symmetric and vanishes with a→∞, γ 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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