Numerical solutions of the three-dimensional magnetohydrodynamic alpha model.

We present direct numerical simulations and alpha -model simulations of four familiar three-dimensional magnetohydrodynamic (MHD) turbulence effects: selective decay, dynamic alignment, inverse cascade of magnetic helicity, and the helical dynamo effect. The MHD alpha model is shown to capture the long-wavelength spectra in all these problems, allowing for a significant reduction of computer time and memory at the same kinetic and magnetic Reynolds numbers. In the helical dynamo, not only does the alpha model correctly reproduce the growth rate of magnetic energy during the kinematic regime, it also captures the nonlinear saturation level and the late generation of a large scale magnetic field by the helical turbulence.

[1]  Darryl D. Holm,et al.  Camassa-Holm Equations as a Closure Model for Turbulent Channel and Pipe Flow , 1998, chao-dyn/9804026.

[2]  P. Sulem,et al.  Linear and non-linear dynamos associated with ABC flows , 1992 .

[3]  Darryl D. Holm,et al.  The Camassa-Holm equations and turbulence , 1999 .

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

[5]  P. Mininni,et al.  Understanding turbulence through numerical simulations , 2004 .

[6]  Annick Pouquet,et al.  On the non-linear stability of the 1:1:1 ABC flow , 1994 .

[7]  J. McWilliams,et al.  Coherent structures and turbulent cascades in two‐dimensional incompressible magnetohydrodynamic turbulence , 1995 .

[8]  Darryl D. Holm,et al.  The Three Dimensional Viscous Camassa–Holm Equations, and Their Relation to the Navier–Stokes Equations and Turbulence Theory , 2001, nlin/0103039.

[9]  U. Frisch,et al.  Helicity cascades in fully developed isotropic turbulence , 1973 .

[10]  Dynamo action in turbulent flows , 2003, astro-ph/0306069.

[11]  Darryl D. Holm,et al.  Direct numerical simulations of the Navier–Stokes alpha model , 1999, Physica D: Nonlinear Phenomena.

[12]  W. Matthaeus,et al.  Long-time states of inverse cascades in the presence of a maximum length scale , 1983, Journal of Plasma Physics.

[13]  P. D. Mininni,et al.  A numerical study of the alpha model for two-dimensional magnetohydrodynamic turbulent flows , 2005 .

[14]  M. Lesieur,et al.  Influence of helicity on the evolution of isotropic turbulence at high Reynolds number , 1977, Journal of Fluid Mechanics.

[15]  Enhancement of the inverse-cascade of energy in the two-dimensional Lagrangian-averaged Navier-Stokes equations , 2001 .

[16]  David Montgomery,et al.  Turbulent relaxation processes in magnetohydrodynamics , 1986 .

[17]  Douglas K. Lilly,et al.  Numerical Simulation of Two‐Dimensional Turbulence , 1969 .

[18]  Darryl D. Holm,et al.  The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories , 1998, chao-dyn/9801015.

[19]  Darryl D. Holm.  Averaged Lagrangians and the mean effects of fluctuations in ideal fluid dynamics , 2001 .

[20]  Y. Ponty,et al.  Numerical study of dynamo action at low magnetic Prandtl numbers. , 2004, Physical review letters.

[21]  K. Subramanian,et al.  Astrophysical magnetic field and nonlinear dynamo theory , 2004, astro-ph/0405052.

[22]  Jerrold E. Marsden,et al.  EULER-POINCARE MODELS OF IDEAL FLUIDS WITH NONLINEAR DISPERSION , 1998 .

[23]  The inverse cascade and nonlinear alpha-effect in simulations of isotropic helical hydromagnetic turbulence , 2000, astro-ph/0006186.

[24]  U. Frisch,et al.  Helical and Nonhelical Turbulent Dynamos , 1981 .

[25]  Hantaek Bae.  Navier-Stokes equations , 1992 .

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

[27]  An alternative interpretation for the Holm alpha model , 2002, physics/0207056.

[28]  Edriss S. Titi,et al.  Attractors for the Two-Dimensional Navier–Stokes-α Model: An α-Dependence Study , 2003 .

[29]  Jerrold E. Marsden,et al.  Numerical simulations of the Lagrangian averaged Navier-Stokes equations for homogeneous isotropic turbulence , 2003 .

[30]  Darryl D. Holm,et al.  The Navier–Stokes-alpha model of fluid turbulence , 2001, nlin/0103037.

[31]  W. Matthaeus,et al.  SELECTIVE DECAY HYPOTHESIS AT HIGH MECHANICAL AND MAGNETIC REYNOLDS NUMBERS * , 1980 .

[32]  W. Matthaeus,et al.  The evolution of cross helicity in driven/dissipative two‐dimensional magnetohydrodynamics , 1988 .

[33]  A. P. Kazantsev.  Enhancement of a magnetic field by a conducting fluid , 1968 .

[34]  Los Alamos National Laboratory,et al.  The joint cascade of energy and helicity in three-dimensional turbulence , 2002, nlin/0206030.

[35]  Darryl D. Holm.  Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics. , 2002, Chaos.

[36]  Darryl D. Holm,et al.  A connection between the Camassa–Holm equations and turbulent flows in channels and pipes , 1999, chao-dyn/9903033.