Free Solutions of the Barotropic Vorticity Equation

Grant Branstator National Center for Atmospheric Research, Boulder, Colorado

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J. D. Opsteegh IMOU, University of Utrecht, Utrecht, the Netherlands

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Abstract

Using a variational procedure, we numerically search for steady solutions to the unforced, inviscid barotropic vorticity equation on the sphere. The algorithm produces many states that have extremely small tendencies within the triangular 15 spherical harmonic truncation employed in the calculation and which can thus be considered to be free modes. Often these states are similar to the planetary scale structure of the first guess fields; when observed 500 mb flow patterns are used as first guesses, the resulting free solutions can have structures reminiscent of time-mean atmospheric states. The functional relationship between streamfunction and absolute vorticity in the solutions is usually nonlinear, and thus the solutions are unlike any previously known free solutions of the barotropic vorticity equation.

Each first guess considered in the study leads to a distinct, free, steady stale, but the collection of such states is not dense in phase space. The distribution of these states is nonuniform. There seems to be a concentration of such states in the part of phase space in which the atmosphere resides. There, neighboring free states appear to be separated from each other by at most the distance that typically separates independent observed flows. Some evidence suggests that in certain cases free modes may be tightly clustered or even connected to each other.

Experiments with a forced–dissipative time–dependent model indicate that free modes like the ones we have found can influence model behavior. For sufficiently strong dissipation and forcing, the existence of such states leads to a resonant response when the forcing is chosen such that a particular free mode is close to being a solution of the forced dissipative system. Furthermore, at lower levels of dissipation, for which a free state is linearly unstable, model trajectories can still be periodically attracted to the free state. An example is given of a lime integration where the forced—dissipative system vacillates between two steady states. One of these states is a free state of the unforced inviscid barotropic vorticity equation.

We conclude that the existence of these free modes may redden the spectrum of the, atmosphere and enhance the prospects for long-range prediction.

Abstract

Using a variational procedure, we numerically search for steady solutions to the unforced, inviscid barotropic vorticity equation on the sphere. The algorithm produces many states that have extremely small tendencies within the triangular 15 spherical harmonic truncation employed in the calculation and which can thus be considered to be free modes. Often these states are similar to the planetary scale structure of the first guess fields; when observed 500 mb flow patterns are used as first guesses, the resulting free solutions can have structures reminiscent of time-mean atmospheric states. The functional relationship between streamfunction and absolute vorticity in the solutions is usually nonlinear, and thus the solutions are unlike any previously known free solutions of the barotropic vorticity equation.

Each first guess considered in the study leads to a distinct, free, steady stale, but the collection of such states is not dense in phase space. The distribution of these states is nonuniform. There seems to be a concentration of such states in the part of phase space in which the atmosphere resides. There, neighboring free states appear to be separated from each other by at most the distance that typically separates independent observed flows. Some evidence suggests that in certain cases free modes may be tightly clustered or even connected to each other.

Experiments with a forced–dissipative time–dependent model indicate that free modes like the ones we have found can influence model behavior. For sufficiently strong dissipation and forcing, the existence of such states leads to a resonant response when the forcing is chosen such that a particular free mode is close to being a solution of the forced dissipative system. Furthermore, at lower levels of dissipation, for which a free state is linearly unstable, model trajectories can still be periodically attracted to the free state. An example is given of a lime integration where the forced—dissipative system vacillates between two steady states. One of these states is a free state of the unforced inviscid barotropic vorticity equation.

We conclude that the existence of these free modes may redden the spectrum of the, atmosphere and enhance the prospects for long-range prediction.

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