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Abstract
Incompressible, homogeneous magnetohydrodynamic (MHD) turbulence
consists of
fluctuating vorticity and magnetic fields, which are represented
in terms of their Fourier coefficients. Here, a set of five Fourier spectral
transform method numerical simulations of two-dimensional (2-D) MHD turbulence
on a $512^2$ grid is described. Each simulation is a numerically realized
dynamical system consisting of Fourier modes associated with wave vectors $\mathbf{k}$,
with integer components, such that $k = |\mathbf{k}| \le k_{max}$. The simulation set consists
of one ideal (non-dissipative) case and four real (dissipative) cases. All
five runs had equivalent initial conditions. The dimensions of the dynamical
systems associated with these cases are the numbers of independent real and
imaginary parts of the Fourier modes. The ideal simulation has a dimension
of $366104$, while each real simulation has a dimension of $411712$. The real
runs vary in magnetic Prandtl number $P_M$, with $P_M \in {0.1, 0.25, 1, 4}$. In the
results presented here, all runs have been taken to a simulation time of $t = 25$.
Although ideal and real Fourier spectra are quite different at high $k$, they are
similar at low values of $k$. Their low $k$ behavior indicates the existence of broken
symmetry and coherent structure in real MHD turbulence, similar to what
exists in ideal MHD turbulence. The value of $P_M$ strongly affects the ratio
of kinetic to magnetic energy and energy dissipation (which is mostly ohmic).
The relevance of these results to 3-D Navier-Stokes and MHD turbulence is
discussed.
Mathematics Subject Classification: 76F05, 76F20, 76W05.
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