| dc.description.abstract | Magnetohydrodynamic (MHD) turbulence remains a challenging physical problem. Different aspects of MHD turbulence are being studied by physicists, engineers, and fluid dynamicists. In this thesis, we concentrate on studies of multiscaling of the velocity and magnetic-field structure functions
Sp=??a(x+r)?a(x)?p?S^p = \langle |a(x + r) - a(x)|^p \rangleSp=??a(x+r)?a(x)?p?
where a is the velocity or magnetic field, for statistically steady MHD turbulence. We also study decaying MHD turbulence.
We use three kinds of models (the last two have been developed by us):
The three-dimensional (3dMHD) equations.
A shell model for MHD turbulence.
A one-dimensional (1dMHD) model whose relation to 3dMHD is the same as that of the Burgers equation to the Navier–Stokes equation for fluid turbulence.
When a magnetized fluid is subjected to sufficiently large external forces or currents, turbulent fluctuations in the velocity and magnetic fields are generated. Viscous losses, magnetic and fluid, lead to a continuous dissipation of energy. If this loss is compensated for by a continuous injection of energy by external forces, then the system eventually achieves a nonequilibrium statistical steady state. Far away from boundaries and at length scales smaller than the energy injection scale, the fluctuations are statistically homogeneous and isotropic (when there is no mean magnetic field). These small scales can be divided into the inertial range, in which the dissipation is insignificant, and the dissipation range, at even smaller length scales where dissipation is predominant.
Before our study, there was some evidence of multiscaling in MHD turbulence from solar-wind data, some earlier shell-model studies, and two-dimensional studies. However, the shell models used earlier were not optimal in one or more of the following ways:
(i) they had some adjustable parameters;
(ii) they did not obey all the conservation laws of 3dMHD in the unforced, inviscid limit; or
(iii) they did not reduce to any well-known shell model for fluid turbulence.
We proposed a new shell model for 3dMHD turbulence which overcame these limitations and used it to calculate the structure functions and the multiscaling exponents. We next showed, by direct numerical simulations of the 3dMHD equations, that our shell-model multiscaling exponents are within error bars of those for 3dMHD. We also calculated the probability distributions of velocity and magnetic-field differences, e.g.,
?v(r)=v(x+r)?v(x),\delta v(r) = v(x + r) - v(x),?v(r)=v(x+r)?v(x),
and found that, as in fluid turbulence, deviations from Gaussian behaviour increase as the separations r decrease and move towards the dissipation range.
We then used our 1dMHD model to find out the influence of a mean magnetic field on the energy spectrum-a long-standing controversy in MHD turbulence. Then we generalized our 1dMHD model to d-dimensions to study decaying MHD turbulence.
Thesis Organization
Chapter 1: A brief account of MHD turbulence along with well-known results from fluid turbulence. We introduce the 3dMHD equations and describe earlier results on homogeneous and isotropic MHD turbulence. We also discuss earlier shell models for MHD turbulence, their results, and limitations. The Burgers equation and its relation to the three-dimensional Navier–Stokes (3dNS) equation are also introduced.
Chapter 2: We propose a new shell model which satisfies all the conservation laws of 3dMHD in the unforced, inviscid limit, obeys the symmetries of the 3dMHD equations, and reduces to the GOY shell model of fluid turbulence in the absence of any magnetic field. We study our shell model numerically and the 3dMHD equations using a pseudospectral code. We calculate the multiscaling exponents for both and show agreement within error bars. We also show that Extended Self Similarity (ESS) holds for 3dMHD and uncover a universal crossover from inertial- to dissipation-range asymptotics using Generalized Extended Self Similarity (GESS). We study the crossover from fluid to MHD turbulence using our shell model, calculate probability distributions for velocity and magnetic-field differences, and examine the influence of a mean magnetic field on multiscaling exponents.
Chapter 3: We derive and discuss various properties and symmetries of the 1dMHD model. We explicitly show that, in this 1d model, the scaling properties of velocity and magnetic-field energy spectra do not depend upon the mean magnetic field. We discuss Galilean invariance leading to Ward identities which ensure non-renormalization of the coefficients of nonlinear terms in the zero-external-wave-vector limit. Using dynamic renormalization group (DRG) and self-consistent calculations, we show that effective fluid and magnetic viscosities are the same in the long-wavelength limit. We also discuss the structure of cross-correlation between noises in the velocity and magnetic-field equations and show that it should be purely imaginary and odd in the wavevector k.
Chapter 4: We study two aspects of decaying MHD turbulence. First, we use our shell model to show that total energy decays as t^(-?) with ? ? 1.3. Next, we generalize our 1d model to d-dimensions (d > 1) and then use it and the 3dMHD equations to study decaying MHD turbulence via one-loop perturbation theory. We define structure functions appropriate for decaying turbulence and show that they decay as ~ t^(-p); the precise value of p depends on the statistics of the initial conditions. We also show that, for a suitable choice of initial cross-correlations between velocity and magnetic fields, even-order structure functions exhibit anomalous scaling in time. We extend our calculation to the case of initial distributions of velocity and magnetic fields which are singular in the long-wavelength limit and discuss our results in the context of initial conditions appropriate for decaying 3dMHD turbulence. | |
