Abstract
The paper describes a technology for the automated compilation of equations for shell models of turbulence in the computer algebra system Maple. A general form of equations for the coefficients of nonlinear interactions is given, which will ensure that the required combination of quadratic invariants and power-law solutions is fulfilled in the model. Described the codes for the Maple system allowing to generate and solve systems of equations for the coefficients. The proposed technology allows you to quickly and accurately generate classes of shell models with the desired properties.
Highlights
The main idea of shell models of turbulence is to construct a chain of ordinary differential equations describing the energy distribution over the scale spectrum in developed turbulence
We describe the developed technology for the automated construction of cascade models, in which the calculation of the coefficients of nonlinear interactions is implemented using one of the computer algebra systems (CAS) - - the Maple system
Various cascade models differ in the coefficients of nonlinear interactions S ni j, Hni j, S ni j, Lni j, Wni j, which ensure the existence of different quadratic invariants
Summary
The main idea of shell models of turbulence is to construct a chain of ordinary differential equations describing the energy distribution over the scale spectrum in developed turbulence. The absolute values of Xn(t) are interpreted as measures of pulsations of the field X in the ranges of scales with wave numbers from [kn; kn+1) For these variables, a system of equations is drawn up that is qualitatively similar to the initial equations of the turbulent system. The main mathematical problem in deriving the equations of cascade models is to calculate the coefficients of nonlinear terms that would ensure the existence of certain quadratic invariants. We describe the developed technology for the automated construction of cascade models, in which the calculation of the coefficients of nonlinear interactions is implemented using one of the computer algebra systems (CAS) - - the Maple system. The authors of this paper considered the use of CAS for the automated derivation of shell models of MHD turbulence [4, 5]. All the details of the mathematical calculations can be found in the work of the authors [6]
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