∂t(˜bkeˆηk2t) = eηkˆ 2tik×v^×b (2.8) Here the dissipation term is included implicitly and stability and accuracy properties do not depend anymore on the dissipation term, but only on the non-linear term. With this modication the leapfrog scheme for the equations is:
ω˜n+1 = ˜ωn−1e−ˆµk2∆t +2∆te−ˆµk2∆t[v^×ω−b×^(∇ ×b)]n (2.9) b˜n+1 = ˜bn−1e−ˆηk2∆t+2∆te−ˆηk2∆t[v^×b]n (2.10) where n is a time step index and ∆t denotes the time interval of one time step. The solution obtained with this scheme is often modied by temporal oscillations with the period 2∆t. These oscillations arise due to the inaccurate approximation of time derivatives. They can be avoided by temporal averaging of the obtained solution over every two subsequent time steps (see [20, 17, 35]). For nonlinear partial dierential equations like the ones under consideration there are no clear rules to guarantee the numerical stability of a simulation, and therefore no recipes to indicate how small∆t ought to be. The Courant-Friedrichs-Lewy (CFL) condition, an estimate originally developed for advection, provides the upper bound as:
∆t ≤ ∆x
vmax ∼ π
kmaxvmax (2.11)
where vmax is the maximal speed of propagation in the system. As incompressibility was assumed magneto-acoustic waves are excluded. A good estimate for the maximum speed of propagation isvmax =√
Etot. Although equation (2.11), forms a good estimate for stability, the time step additionally can be adjusted in particular simulations for maximum stability [20, 17, 35, 36].
2.4 Initial Conditions and Forcing
2.4.1 Initial Conditions
The simulation is initiated by providing denite amounts of kinetic and magnetic energies, following [20, 13].
Step1:The initial velocity and magnetic elds are symmetrical Gaussian uctuations centered around a particular wave number km with a functional form ae
−(k−km)2 (2k 2
0) , with a being the amplitude, k0 as its width and k the wavenumber.
Step2 : An orthonormal basis of the form n
em := |kkm
m|,e1,e2
o is generated with em⊥km and e2 :=em ×e1, where e1 and e2 are random vectors that are orthogonal tokand are normalized. This orthonormal basis is generated to preserve solenoidality of the initial magnetic and velocity elds.
Step3: A random vector potential is now generated using the functions from step-1 and step-2 for a grid point m whosejth component is dened as:
Aˆmj =ae rotational operators. Local magnetic helicity is then generated with the help of these eigenvectors, using the relationHm = 12A∗m ·(ikm ×Am) =ke−
k 2m
k 20 [q+2(φ)−q−2(φ)].
Note that the parameterq±(φ) sets the amount of magnetic helicity(HM) introduced into the initial condition, and can vary between ±HmaxM while HmaxM ∼= EM/k0, EM
In the helicity expression above the relation ikm ×Am is the magnetic eld bm, gen-erated from the already obtained magnetic vector potential. The factor φ determines the amount of magnetic helicity in the system; φ = 0 is the state with no magnetic helicity andφ =±1 the state of maximum helicity.
For generating the initial velocity eld, the magnetic eld is rotated using a set of transformation matrices in thee1,e2- plane:
vm =D−1kmDϕDkmbm. (2.14) This also sets the cross helicity HC of the initial state. The transformation matrices are dened using the orthonormal vectors em,e1,e2 as:
2.4 Initial Conditions and Forcing 25 cross helicity is zero and the case ϕ∈ {0, π} correspond to the case of maximum cross helicity. The magnetic eld and velocity eld are normalized in such a way that the magnitudes of initial kinetic and magnetic energiesEV,EM, can be set to any required value by appropriate normalization factors. This procedure also ensures that the initial magnetic and velocity elds are divergence free.
An initial condition generated at a low resolution can be used for performing high resolution simulations with the help of some simple numerical operations. In this strategy, the initial setup is padded with the necessary number of additional zeros at time t = 0, without disturbing the initial energy budget. This procedure allows the switch from any lower resolution to any wanted higher resolution, without eecting the initial values of the physical quantities.
The table containing the exact initial conditions, is shown in chapter 4, table 4.1. The initial kinetic and magnetic energies for the forced turbulence case are chosen to be small in magnitude, while for decaying case they are chosen to be high. In the forced case, the forcing term contributes to the energy budget and hence to avoid unwanted increases in energies which could violate the CFL criterion or would evolve the system relatively slowly while satisfying the CFL; the initial energies are deliberately kept low.
In the decaying case, no such external energy contribution is present, but the initial energy is decreasing with time. Hence, to study the system for a considerable time, in this decaying phase, the initial energy is kept high. The position of the initial velocity and magnetic elds are in the highk to intermediate k regions, to facilitate the study of inverse cascade. In the system, initial cross helicity is zero, while the fraction of magnetic helicity present is set to 0.5.
2.4.2 Forcing
In the forced case i.e. when ((Fv and Fb)6= 0) they are generated in a manner very similar to the way the initial conditions were obtained. First as in step-2 above
two orthonormal basis setsn
ek1 := |kk1
1|,e11/21,e12/22
oare generated. Using these unit vectors the forcing terms are obtained as in equations (2.17 - 2.24). The parameters φ1 and φ2 essentially determine the amount of magnetic helicity and kinetic helicity generated by these forcing terms. ψ1 andψ2 are the amplitudes of these forcing terms respectively. It is important to note that the terms Fv and Fb are only limited to a certain wave number region, i.e. the forcing is limited to a band of wave numbers [kst ,kend]. The random numbers α1/2 and β1/2 are as above in the range [0,1] and are generated every time step. The threshold valuesψ1 and ψ2 have been decided in such a way that CFL is not violated, after some trial and error.
Fb = It is possible, in principal, with this forcing setup, to have varying levels of magnetic and kinetic helicities generated and added to the system, over a small band of wavenum-bers. But, here the studies concentrates on two cases: 1) maximum amount of magnetic
2.5 Hyperviscosity and Reynolds Number 27