A Physical Interpretation for the Stability Property of a Localized Disturbance in a Deformation Flow

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  • 1 Cooperative Institute for Climate Studies, Department of Meteorology, University of Maryland, College Park, Maryland
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Abstract

The relationship between the local shape of an unstable disturbance and the basic deformation field has been put forward by Mak and Cai as a general condition for barotropic instability of a zonally varying nondivergent basic flow. The general condition states that an unstable disturbance has to be elongated locally at an angle of less than 45° along the axis of contraction of the basic deformation field. The conventional condition for barotropic instability of a zonally uniform basic flow (“an unstable disturbance necessarily leans against the basic shear”) is a special case of the general condition.

To physically interpret the general condition, we have analyzed the immediate subsequent evolution of a localized elliptic-shaped disturbance (defined in terms of streamfunction) embedded in a purely deformation flow. The localized disturbance has the minimum kinetic energy and enstrophy when its shape is circular. Under the influence of the basic deformation, the disturbance tends to shrink along the axis of contraction and to expand along the axis of dilatation. Hence, the disturbance with the major axis along the axis of contraction would deform toward a circle shape. The change in eccentricity of such a disturbance alone acts to reduce its total energy and enstrophy. Because of the conservation constraint of the total perturbation enstrophy, the amplitude of the disturbance has to increase as its eccentricity decreases. The energy change due to the change in amplitude overwhelms that resulting from the change in eccentricity. Therefore, the overall kinetic energy of the localized disturbance tends to increase with time during the course of its evolution. The same arguments also explain why the disturbance with major axis along the axis of dilatation is decaying.

Abstract

The relationship between the local shape of an unstable disturbance and the basic deformation field has been put forward by Mak and Cai as a general condition for barotropic instability of a zonally varying nondivergent basic flow. The general condition states that an unstable disturbance has to be elongated locally at an angle of less than 45° along the axis of contraction of the basic deformation field. The conventional condition for barotropic instability of a zonally uniform basic flow (“an unstable disturbance necessarily leans against the basic shear”) is a special case of the general condition.

To physically interpret the general condition, we have analyzed the immediate subsequent evolution of a localized elliptic-shaped disturbance (defined in terms of streamfunction) embedded in a purely deformation flow. The localized disturbance has the minimum kinetic energy and enstrophy when its shape is circular. Under the influence of the basic deformation, the disturbance tends to shrink along the axis of contraction and to expand along the axis of dilatation. Hence, the disturbance with the major axis along the axis of contraction would deform toward a circle shape. The change in eccentricity of such a disturbance alone acts to reduce its total energy and enstrophy. Because of the conservation constraint of the total perturbation enstrophy, the amplitude of the disturbance has to increase as its eccentricity decreases. The energy change due to the change in amplitude overwhelms that resulting from the change in eccentricity. Therefore, the overall kinetic energy of the localized disturbance tends to increase with time during the course of its evolution. The same arguments also explain why the disturbance with major axis along the axis of dilatation is decaying.

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