Hydrodynamics. — Application of a model system to illustrate some points of the statistical theory of free turbulence

Vortex Motion State of the Dry Atmosphere with Nonzero Velocity Divergence

“…The functional form of the zonal velocity components differs from the corresponding expressions in the previous studies [7][8][9][10][11][14][15][16] because of the fact that the velocity divergence is nonzero. Thus, we seek the expression of u in the form:…”

“…A typical three-dimensional basic state is cyclonic vorticity with rising motion due to surface convergence and anticyclonic vorticity as well as sinking motion due to surface divergence. Exact solutions of the Navier-Stokes equations for a three-dimensional vortex have been discovered [7][8][9][10][11]; of particular interest is Sullivan's two-cell vortex solution, because the flow not only spirals in toward the axis and out along it, but it also has a region of reverse flow near the axis.…”

A New Efficient Method for Solving Two-Dimensional Burgers' Equation

“…where p ∈ [0, 1] is an embedding parameter and u 0 (x, y, t) and v 0 (x, y, t) are initial approximation of solution of (1). By applying Laplace transform on both sides of (3), we have…”

“…The Burgers model of turbulence is a very important fluid dynamic model, and the study of this model and the theory of shock waves have been considered by many authors, both to obtain a conceptual understanding of a class of physical flows and for testing various numerical methods. The mathematical properties of Burgers' equation have been studied by Burgers [1]. Nonlinear phenomena play a crucial role in applied mathematics and physics.…”

Hopf-Cole Transformation

“…For the study of shock waves, the Burgers Green's function is used in the construction of the Green's function, [28]. The main motivation of J. M. Burgers for the study of the equation u t + uu x = µu xx was to understand certain key elements of turbulence for compressible flows, [3,4,5]. It is the simplest model with strong nonlinearity and dissipation.…”