3.2 EDQNM
3.2.2 Summary of Important Results of 3D-MHD Turbulence Obtained
The unique way in which the MHD equations have been approximated in EDQNM, in terms of physical quantities that are not explicitly seen at rst glance from the
3.2 EDQNM 47 original equations, help in the understanding of some important physical phenomenon like the Alfvén eect and inverse cascade of magnetic helicity, to name a few. Some of them are briey explained here.
Alfvén Eect
Kraichnan [55] noticed that in the presence of large-scale magnetic energy, Alfvén waves can bring small-scale velocity and magnetic energies to equipartition and relax triple correlations (due to triadic interactions) in a time which may be shorter than the local eddy turnover time [7]. This can be illustrated using the EDQNM equations.
For this, EDQNM equations are expanded to represent local and nonlocal eects and all the nonlocal eects are ignored. In the next step, only terms that represent large-scale interactions are retained. With these two operations, the eect of random Alfvén waves on the kinetic and magnetic energy spectra can be analyzed. When the Alfvén contribution to the eddy-damping rate dominates the self-distortion and dissipation terms, it will essentially represent the r.m.s. magnetic eld (b0), which is also the typical group velocity of the Alfvén waves. Under the action of random Alfvén waves, it is observed that magnetic and kinetic energy spectra relax to equipartition, in a time of order of(kb0)−1, as predicted by Kraichnan [55]. Also it was observed that the helicity spectra relax to equipartition given by the relation
HkV =k2HkM. (3.35)
When the energy spectra and helicity spectra deviate from equipartition, the dif-ference between them gives rise to residual energy and residual helicity, given by EkR = EkV −EkM and HkR = HkV −k2HkM respectively. When these two dierences relax to zero, the eect is called `Alfvén' eect.
Helicity or α Eect
The above discussion was centered around the large-scale eects alone. Since EDQNM equations allow the study of eects at various scales separately, now only the small-scale eects are considered.
The same treatment of expanding the equations in terms of local and nonlocal eects, retaining only the local terms and nally ignoring the large-scale eects results in a
small-scale phenomenon known as the `torsality' αRk, which is given by:
Thus when αkR is known, it is easy to integrate the equations representing the small scale eects (see equations 3.16 and 17 of [7]). This results in the exponential growth (decay) of magnetic energy and magnetic helicity, at a rate given by k|αRk|. The im-portant conclusion from this study is that the small-scale residual helicity destabilizes the large-scale magnetic energy and magnetic helicity. This is similar to the `helicity or α eect'. Both the kinetic and magnetic helicities produce a destabilizing eect and it is the dierence, as measured by the residual helicity, that acts as a true driver of instability in the ow. This instability is responsible for the `inverse cascade', of magnetic helicity and eventually the large-scale magnetic structure. The helicity eect could also give a rate of growth of mean magnetic eld, when such a eld is present.
Such an eect is very important for the production of a α dynamo [7].
Inverse Cascade of Magnetic Helicity and Inertial Range
The physics of inverse cascade of magnetic helicity can be explained from the above discussed two eects, in a MHD turbulent system forced at small-scale. It is explained in steps below:
• 1) The helicity injection at a wavenumber say k ∼ kE, produces a growth of both magnetic energy and magnetic helicity in a small wavenumber k∼(1/2)kE, through the `helicity eect'.
• 2) The growing magnetic energy at this wavenumber, reduces the residual helicity near kE by the Alfvén eect, while the growing magnetic helicity at (1/2)kE, destabilizes the small wavenumbers.
• 3) It is easily noticeable, that the steps 1 and 2 could go on and on to drive the magnetic helicity spectra into ever smaller wave numbers, resulting in what is called the `inverse cascade of magnetic helicity'.
From the numerical simulations of EDQNM equations for forced 3D-MHD turbu-lence, it was observed that the magnetic helicity shows ak−2 power law behavior in its spectra while magnetic energy showed ak−1 power law behavior. The ver-ication or conformation of these power law behaviors, in high resolution DNS is
3.2 EDQNM 49 one of the key motivations to perform this study.
Other aspects
Importance of EDQNM
The EDQNM equations are a exible set having equations and parameters that allow the study of interactions on local and nonlocal scales separately, without disturbing each other. This property also indicates the robustness of the equation set. A dimensional study of these equations together with the results from the direct numerical simulations, can be used to gain further insights into the prop-erties of MHD turbulence. For this purpose stationarity is assumed. Further the analysis is restricted to inertial range thus eliminating the inuence of the driving or dissipation scales. Finally a dynamical equilibrium between the local and non-local eects (discussed above), which tends to result in an energy equipartition between the velocity and magnetic elds, is also assumed [18]. In [14] using such assumptions, phenomenological description of residual energy has been obtained.
Further, numerical simulations of the EDQNM equations could be conducted.
In these simulations, very high values for Reynolds numbers could be achieved, larger than what are possible in direct numerical simulations [31]. Using the EDQNM equations, systems like nonlinear turbulent dynamo have been studied [7]. The correlations between velocity and magnetic elds have also been studied using a modied set of EDQNM equations [56, 57]. The set of equations have been used to verify, prove or disprove several aspects of 3D-MHD turbulence for example: a) verication of power law behavior for total energy spectra in 3D-MHD turbulence (through the numerical simulations of EDQNM), b) existence of a power law behavior for quantities like the magnetic helicity and magnetic energy (through numerical simulations and phenomenological arguments) and c) a comment on the scales of the magnetic structures (based on pure theoretical arguments backed by numerical simulations). With all these positives, there do exist few negative aspects for the EDQNM equations.
Shortcomings of EDQNM
The major shortcoming of EDQNM is that the eddy-damping parameter, so cru-cial to the equations is set from outside the system. As it is also a closure theory it suers from general weaknesses of closure theories and also suers from their constraints. Some of these are a) the value of the Kolmogorov constant is not
been determined using this model. b) real space structures and intermittency as-pects of the turbulent ow are not explained because of the Gaussian assumption of the elds's fourth and higher order moments. c) Overemphasis is given to the strength of nonlinear interactions, ignoring the fact that the local rearrangements of the elds, often give rise to a depletion of nonlinearity. d) The random charac-ter of the turbulence is over emphasized in closure theories. Anisotropic nature of the MHD turbulence is hard to be explained using closure theories, though not impossible [1]. Mean square of the random variable can have a negative value, violating the realizability.
Despite these short comings, the already mentioned exibility and robustness oered by the EDQNM equations is what is most attractive to work with. The focus of this work will be on applying dimensional analysis to EDQNM equa-tions together with some of the results from DNS studies of MHD equaequa-tions (see section 4.5) to gain further insights into the properties of MHD turbulence.