Dynamics of Distributed Sources

Dynamics of distributed sources is described by nonlinear partial differential equations. Lagrangian analytical solutions of these (and associated) equations are obtained and discussed in the context of the Lagrangian mod-eling-from the Lagrangian Invariants to dynamics. Possible applications of distributed sources and sinks to geophysical fluid dynamics and to gravitation are indicated. Singular vortices, sources (sinks) and vortex-sinks are well known models in fluid dynamics (including geophysical fluid dynamics), magnetized plasma, superfluidity and superconductivity (see, for example, Refs. 1-3 and references therein). Dynamics of distributed vortices is well developed classical area of research. However, to our knowledge, dynamics of distributed sources has not been considered before. Structure and intensity of distributed sources can change in time and, in this sense, they are more rich and flexible models for various media than singular sources with fixed intensity. In this letter we show that distributed sources have a nontrivial dynamics, which in some cases can be described analytically in Lagrangian variables. Local intensity of distributed sources (sinks) in moving continuous media is characterized by the divergency of the velocity field α(t, x) = ∂v i (t, x)/∂x i , where summation over the repeated subscripts is assumed from 1 to the dimension of the space s = 1, 2, 3. To get an equation for α , we consider superposition of localized sources, which move each other with induced velocity field [2,3]. Thus, the equation can be written in form: dα dt ≡ ∂α ∂t + v k ∂α ∂x k = f (1) Here f may include diffusion and some other effects (see below). We start with consideration of " free " sources with f = 0. In this case α is the La-grangian Invariant (LI), which is conserved along trajectories. For the po