TY - JOUR

T1 - Stability properties of exact nonzonal solutions for flow over topography

AU - Frederiksen, J. S.

AU - Carnevale, G. F.

PY - 1986

Y1 - 1986

N2 - The nonlinear stability properties of stationary exact nonzonal solutions for inviscid ilow over topography are examined within a barotropic model in spherical geometry. For stationary solutions, such that the potential vorticity is proportional to the streamfunction, necessary and sufficient conditions for nonlinear stability are established. For a truncated system with rhomboidal truncation wave number J these are that the solid body rotation component of the zonal wind u1 be negative, corresponding to westward flow, as J→∞. These results are established by using the methods of statistical mechanics. The sufficient condition for stability is also established by applying Arnol'd's method. The results are illustrated by numerical calculations. The stationary solutions are perturbed by disturbances in the streamfunction fields or by small changes in the topographic height; the climatic states for the system are obtained directly using statistical mechanics methods. The nonlinear stability properties of the stationary solutions are obtained by comparing the stationary solution with the climate, which for inviscid flow is shown to be unique. Stationary flows for which u is eastward, are found to be unstable even in the limit as the streamfunction perturbation or change in the topographic height vanish. Large amplitude transient waves are generated which break the time invariance symmetry of the initial stationary flows. In contrast, for stationary flows with westward ui, the climate is identical to the initial flow in the limit as the initial streamfuncton perturbation or the change in the topographic height vanishes. The linear instability characteristics of the stationary solutions are also obtained by solving a linear eigenvalue problem. The difficulties in establishing the stability properties of more general exact solutions, where the streamfunction is a general differentiable function of the potential vorticity, within numerical spectral models are discussed.

AB - The nonlinear stability properties of stationary exact nonzonal solutions for inviscid ilow over topography are examined within a barotropic model in spherical geometry. For stationary solutions, such that the potential vorticity is proportional to the streamfunction, necessary and sufficient conditions for nonlinear stability are established. For a truncated system with rhomboidal truncation wave number J these are that the solid body rotation component of the zonal wind u1 be negative, corresponding to westward flow, as J→∞. These results are established by using the methods of statistical mechanics. The sufficient condition for stability is also established by applying Arnol'd's method. The results are illustrated by numerical calculations. The stationary solutions are perturbed by disturbances in the streamfunction fields or by small changes in the topographic height; the climatic states for the system are obtained directly using statistical mechanics methods. The nonlinear stability properties of the stationary solutions are obtained by comparing the stationary solution with the climate, which for inviscid flow is shown to be unique. Stationary flows for which u is eastward, are found to be unstable even in the limit as the streamfunction perturbation or change in the topographic height vanish. Large amplitude transient waves are generated which break the time invariance symmetry of the initial stationary flows. In contrast, for stationary flows with westward ui, the climate is identical to the initial flow in the limit as the initial streamfuncton perturbation or the change in the topographic height vanishes. The linear instability characteristics of the stationary solutions are also obtained by solving a linear eigenvalue problem. The difficulties in establishing the stability properties of more general exact solutions, where the streamfunction is a general differentiable function of the potential vorticity, within numerical spectral models are discussed.

UR - http://www.scopus.com/inward/record.url?scp=84963159838&partnerID=8YFLogxK

U2 - 10.1080/03091928608245892

DO - 10.1080/03091928608245892

M3 - Article

AN - SCOPUS:84963159838

VL - 35

SP - 173

EP - 207

JO - Geophysical and Astrophysical Fluid Dynamics

JF - Geophysical and Astrophysical Fluid Dynamics

SN - 0309-1929

IS - 1-4

ER -