Optimal Growth Rates in the Quasigeostrophic Initial Value Problem

Enda O'Brien University of Miami, Rosenstiel School of Marine and Atmospheric Science, Miami, Florida

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Abstract

A large class of wave structures in quasigeostrophic flow have instantaneous growth rates significantly larger than normal-mode growth rates. Since energy and potential enstrophy growth rates can be defined as functions of the perturbation structure, this structure can be varied in order to maximize the growth rates. Green's model is used as an illustration of the method. The only essential constraint in the inviscid case is the conservation of wave action. When friction is included, an appropriate modification of this constraint can be found. The problem reduces to a fourth-order nonlinear ordinary differential equation for the perturbation structure function. An analytic solution is obtained for the continuous inviscid problem with no interior potential vorticity gradient (i.e., β = 0) and yields structures that include the classical Eady wave as a special case. For the more general problem, the equation is discretized and solved numerically.A general feature of the fastest-growing baroclinic waves is that their phase tilts are more uniform with height than normal modes. As the waves evolve toward (unstable) normal-mode structures, the phase tilts concentrate at lower levels and the potential vorticity fluxes become shallower. There is no long-wave or short-wave cutoff in the initial value problem, even though the corresponding normal-mode instability may have long-wave or short-wave cutoffs or both. Indeed, the largest instantaneous growth rates in both the energy and potential enstrophy norms are obtained for the longest waves.While the variational principle requires no information about normal modes in order to find the fastest-growing structures, the principle may also be used to obtain normal modes as particular cases.

Abstract

A large class of wave structures in quasigeostrophic flow have instantaneous growth rates significantly larger than normal-mode growth rates. Since energy and potential enstrophy growth rates can be defined as functions of the perturbation structure, this structure can be varied in order to maximize the growth rates. Green's model is used as an illustration of the method. The only essential constraint in the inviscid case is the conservation of wave action. When friction is included, an appropriate modification of this constraint can be found. The problem reduces to a fourth-order nonlinear ordinary differential equation for the perturbation structure function. An analytic solution is obtained for the continuous inviscid problem with no interior potential vorticity gradient (i.e., β = 0) and yields structures that include the classical Eady wave as a special case. For the more general problem, the equation is discretized and solved numerically.A general feature of the fastest-growing baroclinic waves is that their phase tilts are more uniform with height than normal modes. As the waves evolve toward (unstable) normal-mode structures, the phase tilts concentrate at lower levels and the potential vorticity fluxes become shallower. There is no long-wave or short-wave cutoff in the initial value problem, even though the corresponding normal-mode instability may have long-wave or short-wave cutoffs or both. Indeed, the largest instantaneous growth rates in both the energy and potential enstrophy norms are obtained for the longest waves.While the variational principle requires no information about normal modes in order to find the fastest-growing structures, the principle may also be used to obtain normal modes as particular cases.

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