Realizability Condition and Inverse Cascade

In this work the 3D-MHD turbulence is described by statistical averages of phys-ical quantities. Fully periodic boundary conditions ensure that the viscous boundary layers are not present in the system, so approximate statistical homogeneity of the turbulent system is preserved [17]. Isotropy of the system is also assumed. Next the three quantities of equations (1.20 - 1.22) are represented in their statistically averaged forms. With the above assumptions in place and without assuming invariance under planar reexions, following the arguments given in [5] for a non-helical turbulence,the realizability condition for magnetic helicity is obtained, which is reproduced here:

|H^M(k)| ≤E^M(k)/k ≤E(k)/k. (1.46) Here E^M and E are magnetic and total energies respectively. Suppose that an initial state of maximal helicity is conned to two wavenumbersp andqwith (p < q)and let this excitation be entirely transferred to the wave numberk. From the conservation of total energy and magnetic helicity it is seen that

E(k) = E(p) +E(q) (1.47)

H^M(k) =H^M(p) +H^M(q) = E(p)/p+E(q)/q. (1.48) Using the realizability condition, and performing few simple manipulations, the above equations are written as

k ≤ p|H^M(p)|+q|H^M(q)|

|H^M(p)|+|H^M(q)| . (1.49)

The expression on the r.h.s. of the above equation is a weighted mean of p and q and thus

min(p,q)≤ p|H^M(p)|+q|H^M(q)|

|H^M(p)|+|H^M(q)| ≤max(p,q) (1.50)

1.3 Inverse Cascade of Magnetic Helicity 17 Thereforek ≤max(p,q). Thus simultaneous up-transfer of total energy and magnetic helicity is not possible. Also the invariance of magnetic helicity holds only under the assumption that b(r,t) vanishes at innity, in the statistically homogeneous case or that the mean magnetic eld vanishes. Hence the transfer of magnetic helicity takes place from large wavenumbers to small wavenumbers and this is known as `inverse cascade'. A more detailed version of this process and the physics involved will be discussed in chapter 3.

Importance of Inverse Cascade of Magnetic Helicity

Inverse cascade of magnetic helicity in 3D-MHD turbulence, is believed to be one of the processes responsible for the formation of large-scale magnetic structures in the universe, as the movement of this quantity is towards smaller wavenumbers or large scales. In the celestial bodies with rotation, it is believed that the dierence of kinetic helicity (twists in the velocity eld) and magnetic helicity (twists in the magnetic eld) results in the so called α-dynamo, where kinetic helicity injection results in enhance-ment of the magnetic eld [1], but not necessarily lead to stable large-scale magnetic structure formation. The relation deduced from equation (1.48) i.e. equation (1.50), suggests that the magnetic helicity always moves to large scales. Thus in all probability,

`inverse cascade' of magnetic helicity might be an important process for the formation of the stable large-scale magnetic structures seen in the celestial atmospheres and their vicinities. Currently no clear evidence of magnetic helicity transfer from comparatively very small scales to very large scales, has been put forward. In this work an attempt to gather such an evidence is being made using DNS. Two cases: forced turbulence and decaying turbulence are reported. In the following chapters the numerical method is described rst and then data analysis of the simulations is presented followed by a discussion on the ndings from this work.

Chapter 2

Direct Numerical Simulations of 3D-MHD Turbulence

In this chapter the numerical methods employed for the simulation of 3D-MHD turbu-lence are described. First the spectral scheme used for this purpose is discussed along with the aliasing error problem and its solution. Next the integration scheme followed by initial conditions is discussed. The forcing mechanism used for the simulations is ex-plained next. The concept of hyperviscosity is mentioned along with a short discussion on Reynolds number. Finally the software and hardware that make these simulations work are mentioned, as well as the diagnostics from these simulations.

2.1 Motivation for Direct Numerical Simulations (DNS) and Equation Set

The inverse cascade of magnetic helicity is best understood in the spectral do-main. It is noteworthy that not only this property but many other properties of MHD turbulence demonstrate interesting characteristics in the spectral domain. Of these characteristics, the most important one is the so called `inertial range' of wavenumbers (discussed in detail in chapter 3) exhibited by the spectra of certain quantities of MHD turbulence like total energy. In the inertial range the spectra show self-similar power law behavior, which is a predictable property of a randomly uctuating system. The investigation of inertial ranges and the universality of the power laws forms one of the important aspects of turbulence studies. Numerical simulations of turbulence in the spectral domain are performed using several methods like large eddy simulations (LES), shell models or direct numerical simulations (DNS) [15]. LES methods and

shell models approximate the nonlinear terms of equations (1.39 - 1.41) in one or other form (see for e.g. [32, 22]), whereas DNS methods do not use any such approxima-tions and deal with the equaapproxima-tions in their true form. Thus methods other than DNS usually involve additional assumptions. If the equation set is studied without any ad-ditional physical approximations, a better understanding of the turbulent ows could be obtained. The DNS methods stay closest to the underlying dierential equations describing the turbulent systems although they are computationally expensive. With appropriate choice of numerical methods however the computational overhead can be reduced.

In studying the equation set (1.39 - 1.41) in the Fourier domain, the spatial deriva-tives are transformed into simple multiplications with wave vectors. Here the time evolution of the equations directly yields the spectra of the physical quantities. Al-though the Fourier methods have several advantages, they also have a major drawback, namely, the Gibbs phenomenon. The Gibbs phenomenon manifests itself as character-istic oscillations of Fourier series near steep gradients. By assuming incompressibility, such discontinuities of physical quantities are excluded [17]. For incompressible ows, spectral methods are more accurate than nite dierence schemes as they require less discretization points for achieving the same accuracy (see [33] for a detailed description of dierent numerical schemes).

This study aims at a better understanding of nonlinear inertial range dynamics of two types of MHD turbulent ows: forced MHD turbulence and decaying MHD turbulence in three dimensions. For this purpose the set of equations (1.39 - 1.41) is actually written in the following manner:

∂tω˜ =ik×[v^×ω−b×^(∇ ×b)]−µkˆ ²ω˜ +Fv (2.1)

∂_tb˜ =ik×v^×b−ˆηk²˜b+F_b (2.2)

ik·˜v=ik·b˜=0 (2.3)

whereFvandFb are the forcing terms for the velocity and magnetic elds respectively.

Thus, if F_v and F_b are set to zero, the equation set represents a decaying turbulence case and if they are non-zero, it is a forced turbulent system. In this set of equations, the nature of forcing very much inuences how the turbulent ow and its characteris-tics evolve. In the forced system studied here a random forcing is employed, which is discussed in the section 2.4.2.

2.2 Pseudospectral Scheme 21 Fully periodic boundary conditions are chosen so that inuence of the boundary on the system remains minimum. This also ensures that the turbulence remains approx-imately statistically homogeneous. Therefore inherent properties of MHD turbulence in the system can be studied, with considerable detachment from the inuence of the boundary.

2.2 Pseudospectral Scheme

The equation set (2.1 - 2.3) is solved in the Fourier space in a regular cubic box of linear size 2π, discretized with N points in each direction. This corresponds to the Fourier wavenumber range ^−N₂ +1 ≤ k ≤ ^N₂ −1. All physical quantities are approximated by truncated Fourier series, e.g. for the Fourier counterpart of the real quantity ω(x_j,t), ωˆ_k(t) as The modek= (0,0,0)of all physical quantities, i.e. their spatial average, is set to zero.

As already mentioned in section 1.3, the physical quantities are real valued and satisfy symmetry (ωˆ−k(t) =ωˆ^∗_k(t)) in Fourier space, hence it is enough to only store one half of Fourier modes. This symmetry property helps in reducing the memory requirement and also speeds up the calculations. The convolution terms in the equations (2.1) and (2.2) may in general be represented as

^[a b]_k = X

k=p+q

fa_p be_q where |k|,|p|,|q| ≤ N

2 −1. (2.5)

A simple calculation shows that numerically evaluating such an expression in three dimensions requires O(N⁶) operations. This fact limits the application of spectral methods to small Fourier data sets [34]. In order to overcome this limitation, the vari-ables in the relation are rst transformed into real space. A multiplication is performed here and the value retransformed into the Fourier space. This mathematical operation is facilitated by the fact that a convolution in Fourier domain is a multiplication in real space. The method explained here is the `pseudospectral scheme' [34]. This method reduces the complexity of the order of operations performed to O(N³log2N), which is only possible with FFT (Fast Fourier Transform). But this method suers from

`aliasing error' caused by the nite discretization, shown in equation (2.4).