Green's function for a generalized two-dimensional fluid

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  • 一般化された2次元流体系のグリーン関数(渦・回転(1),一般講演)

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A Green's function for a generalized two-dimensional (2D) fluid in an unbounded domain (the so-called α turbulence system) is discussed. The generalized 2D fluid is characterized by a relationship between an advected quantity q and the stream function ψ: namely, q=-(-Δ)^<α/2>ψ. Here, α is a real number and q is referred to as the vorticity. In this study, the Green's function refers to the stream function produced by a delta-functional distribution of q, i.e., a point vortex with unit strength. The Green's function is given by the Riesz potential, except when α is an even number. When α is a positive even number, the logarithmic correction to the Riesz potential appears. The transition of the small-scale behavior of q at α=2, a well-known property of forced and dissipative α turbulence, is explained in terms of the Green's function. Moreover, the azimuthal velocity around the point vortex is derived from the Green's function. The functional form of the azimuthal velocity indicates that physically realizable systems for the generalized 2D fluid exist only when α≤3.

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