SELECTIVE DECAY HYPOTHESIS AT HIGH MECHANICAL AND MAGNETIC REYNOLDS NUMBERS *

Implications of certain applications of turbulence theory to two-dimensional turbulence and magnetohydrodynamic flow are discussed. It is shown that the use of the Navier-Stokes equation (NSE) for measurements of turbulent fluctuations has been effective only for three-dimensional flows. For two-dimensional flows, used for the study of large-scale motions in the atmosphere or ocean, enstrophy is cascaded to high wave numbers and dissipated at a finite rate even at infinite Re. MHD flows are numerically calculated for the two-dimensional case and analytically for the three-dimensional case, for which discrepancies in the relative rates of energy and cross helicity decay lead to a recommendation that numerical calculations for the three-dimensional case be carried out to determine the precise decayed states.

[1]  O. Reynolds.  III. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels , 1883, Proceedings of the Royal Society of London.

[2]  U. Frisch,et al.  Strong MHD helical turbulence and the nonlinear dynamo effect , 1976, Journal of Fluid Mechanics.

[3]  F. Bretherton,et al.  Two-dimensional turbulence above topography , 1976, Journal of Fluid Mechanics.

[4]  W. Wolibner.  Un theorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long , 1933 .

[5]  Dissipative, forced turbulence in two-dimensional magnetohydrodynamics , 1977 .

[6]  D. Montgomery.  Two-dimensional vortex motion and 'negative temperatures.' , 1972 .

[7]  Y. Salu,et al.  Two-dimensional turbulence in inviscid fluids or guiding center plasmas , 1975 .

[8]  O. Reynolds.  IV. On the theory of lubrication and its application to Mr. Beauchamp tower’s experiments, including an experimental determination of the viscosity of olive oil , 1886, Philosophical Transactions of the Royal Society of London.

[9]  S. Orszag.  Numerical Simulation of Incompressible Flows Within Simple Boundaries. I. Galerkin (Spectral) Representations , 1971 .

[10]  A. Pouquet.  On two-dimensional magnetohydrodynamic turbulence , 1978, Journal of Fluid Mechanics.

[11]  C. Leith,et al.  Developments in the theory of turbulence , 1973 .

[12]  S. Orszag,et al.  Small-scale structure of two-dimensional magnetohydrodynamic turbulence , 1979, Journal of Fluid Mechanics.

[13]  Magnetic dynamo action in two-dimensional turbulent magneto-hydrodynamics , 1977 .

[14]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[15]  David Montgomery,et al.  Two-Dimensional Turbulence , 2012 .

[16]  Y. Salu,et al.  Use of the spectral method for two- and three-dimensional guiding center plasmas , 1975 .

[17]  R. Sudan.  Stability of Field-Reversed, Force-Free, Plasma Equilibria with Mass Flow , 1979 .

[18]  George Vahala,et al.  Three‐dimensional magnetohydrodynamic turbulence in cylindrical geometry , 1978 .

[19]  T. Tatsumi.  Theory of Homogeneous Turbulence , 1980 .

[20]  Uriel Frisch,et al.  Possibility of an inverse cascade of magnetic helicity in magnetohydrodynamic turbulence , 1975, Journal of Fluid Mechanics.

[21]  J. Taylor.  Negative temperatures in two dimensional vortex motion , 1972 .

[22]  C. Leith.  Diffusion Approximation for Two‐Dimensional Turbulence , 1968 .

[23]  Remarks on turbulence theory , 1975 .

[24]  D. Fyfe,et al.  High-beta turbulence in two-dimensional magnetohydrodynamics , 1975, Journal of Plasma Physics.

[25]  H. L. Grant,et al.  Turbulence spectra from a tidal channel , 1962, Journal of Fluid Mechanics.

[26]  M. Rosenbluth,et al.  MHD stability of Spheromak , 1979 .

[27]  R. Kraichnan.  Inertial Ranges in Two‐Dimensional Turbulence , 1967 .