Stochastic Chaos of Exponential Oscillons and Pulsons

An exact three-dimensional solution for stochastic chaos of I wave groups of M random internal waves governed by the Navier-Stokes equations is developed. The Helmholtz decomposition is used to expand the Dirichlet problem for the Navier-Stokes equations into the Archimedean, Stokes, and Navier problems. The exact solution is obtained with the help of the method of decomposition in invariant structures. Differential algebra is constructed for six families of random invariant structures: random scalar kinematic structures, time-complementary random scalar kinematic structures, random vector kinematic structures, time-complementary random vector kinematic structures, random scalar dynamic structures, and random vector dynamic structures. Tedious computations are performed using the experimental and theoretical programming in Maple. The random scalar and vector kinematic structures and the time-complementary random scalar and vector kinematic structures are applied to solve the Stokes problem. The random scalar and vector dynamic structures are employed to expand scalar and vector variables of the Navier problem. Potentialization of the Navier field becomes available since vortex forces, which are expressed via the vector potentials of the Helmholtz decomposition, counterbalance each other. On the contrary, potential forces, which are described by the scalar potentials of the Helmholtz decomposition, superimpose to generate the gradient of a dynamic random pressure. Various constituents of the kinetic energy are ascribed to diverse interactions of random, three-dimensional, nonlinear, internal waves with a two-fold topology, which are termed random exponential oscillons and pulsons. Quantization of the kinetic energy of stochastic chaos is developed in terms of wave structures of random elementary oscillons, random elementary pulsons, random internal, diagonal, and external elementary oscillons, random wave pulsons, random internal, diagonal, and external wave oscillons, random group pulsons, random internal, diagonal, and external group oscillons, a random energy pulson, random internal, diagonal, and external energy oscillons, and a random cumulative energy pulson.