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A modulated point‐vortex model for geostrophic, β‐plane dynamics
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7.H. Aref and N. Pomphrey, Proc. R. Soc. London Ser. A 380, 359 (1982). The latter paper contains a comprehensive discussion of the ‐vortex problem in the plane with new results for the four vortex problem. The point vortex evolution equations are transformed into a canonical Hamiltonian form and it is easy to see that the system is “nonintegrable” for Additional results will be presented in H. Aref, Ann. Rev. Fluid Mech. (1983).
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16.The finite‐difference model is that used in McWilliams and Zabusky, Geophys. Astrophys. Fluid Dyn. 19, 207 (1982). Numerical parameters are a grid spacing of 0.1, a periodicity length of 20, and a time step of 0.01. A damping term is added to the right side of (1), viz.,
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20.A technical detail is that the spatial average of must vanish on a periodic domain. Hence, a small constant is added to (66) to ensure this for the finite‐difference solution. Because the periodicity length is much larger than A. this correction is small (of the order of their ratio squared) and can be disregarded for our purposes.
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