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2.4 The non-linear energy cascade

The basic properties of turbulence, as derived both from the Navier-Stokes equation and from phenomenological considerations, is the legacy of A.N. Kolmogorov3 (Frisch, 1995Jump To The Next Citation Point). Phenomenology is based on the old picture by Richardson who realized that turbulence is made by a collection of eddies at all scales. Energy, injected at a length scale L, is transferred by non-linear interactions to small scales where it is dissipated at a characteristic scale l D, the length scale where dissipation takes place. The main idea is that at very large Reynolds numbers, the injection scale L and the dissipative scale lD are completely separated. In a stationary situation, the energy injection rate must be balanced by the energy dissipation rate and must also be the same as the energy transfer rate e measured at any scale l within the inertial range l « l « L D. Fully developed turbulence involves a hierarchical process, in which many scales of motion are involved. To look at this phenomenon it is often useful to investigate the behavior of the Fourier coefficients of the fields. Assuming periodic boundary conditions the a-th component of velocity field can be Fourier decomposed as
sum ua(r,t) = ua(k, t)exp(ik .r), k

where k = 2pn/L and n is a vector of integers. When used in the Navier-Stokes equation, it is a simple matter to show that the non-linear term becomes the convolution sum

sum @ua(k,-t) = Mabg(k) ug(k - q, t)ub(q,t), (13) @t q
where Mabg(k) = - ikb(dag- kakb/k2) (for the moment we disregard the linear dissipative term).

MHD equations can be written in the same way, say by introducing the Fourier decomposition for Elsässer variables

sum z±a(r,t) = z±a (k, t)exp(ik .r), k

and using this expression in the MHD equations we obtain an equation which describes the time evolution of each Fourier mode. However, the divergence-less condition means that not all Fourier modes are independent, rather ± k .z (k,t) = 0 means that we can project the Fourier coefficients on two directions which are mutually orthogonal and orthogonal to the direction of k, that is,

sum 2 z±(k,t) = za±(k,t)e(a)(k), (14) a=1
with the constraint that (a) k .e (k) = 0. In presence of a background magnetic field we can use the well defined direction B0, so that
ik × B0 ik e(1)(k) = --------- ; e(2)(k) = ---× e(1)(k). |k × B0 | |k|

Note that in the linear approximation where the Elsässer variables represent the usual MHD modes, z±1 (k,t) represent the amplitude of the Alfvén mode while z±2 (k,t) represent the amplitude of the incompressible limit of the magnetosonic mode. From MHD Equations (12View Equation) we obtain the following set of equations:

[ ] @-- ± ( L--)3 sum d sum 2 ± ± @t ± i(k .cA) za (k,t) = 2p Aabc(- k,p, q)zb (p, t)zc (q,t). (15) p+q=k b,c=1
The coupling coefficients, which satisfy the symmetry condition Aabc(k,p, q) = - Abac(p,k,q), are defined as
[ ][ ] A (-k, p,q) = (ik)*.e(c)(q) e(a)*(k) .e(b)(p) , abc

and the sum in Equation (15View Equation) is defined as

d ( )3 sum =_ 2p- sum sum dk,p+q, p+q=k L p q

where d k,p+q is the Kronecher’s symbol. Quadratic non-linearities of the original equations correspond to a convolution term involving wave vectors k, p and q related by the triangular relation p = k - q. Fourier coefficients locally couple to generate an energy transfer from any pair of modes p and q to a mode k = p + q.

The pseudo-energies E ±(t) are defined as

1 1 integral 1 sum sum 2 E ± (t) = ----3 3|z± (r, t)|2d3r = -- |z±a (k, t)|2 2 L L 2 k a=1

and, after some algebra, it can be shown that the non-linear term of Equation (15View Equation) conserves separately E ±(t). This means that both the total energy E(t) = E+ + E - and the cross-helicity EC(t) = E+ - E-, say the correlation between velocity and magnetic field, are conserved in absence of dissipation and external forcing terms.

In the idealized homogeneous and isotropic situation we can define the pseudo-energy tensor, which using the incompressibility condition can be written as

( L )3 < > ( kakb ) Ua±b(k,t) =_ --- z±a (k, t)z±b (k,t) = dab - --2-- q ±(k), 2p k

brackets being ensemble averages, where ± q (k) is an arbitrary odd function of the wave vector k and represents the pseudo-energies spectral density. When integrated over all wave vectors under the assumption of isotropy

[ integral ] integral T r d3k U± (k,t) = 2 oo E ±(k,t)dk, ab 0

where we introduce the spectral pseudo-energy E± (k,t) = 4pk2q± (k, t). This last quantity can be measured, and it is shown that satisfies the equations

@E-±(k,t)- ± 2 ± ± @t = T (k,t) - 2nk E (k, t) + F (k,t). (16)
We use n = j in order not to worry about coupling between + and - modes in the dissipative range. Since the non-linear term conserves total pseudo-energies we have
integral oo ± 0 dk T (k,t) = 0,

so that, when integrated over all wave vectors, we obtain the energy balance equation for the total pseudo-energies

dE ±(t) integral oo integral oo ------- = dk F ±(k,t) - 2n dk k2E ±(k,t). (17) dt 0 0
This last equation simply means that the time variations of pseudo-energies are due to the difference between the injected power and the dissipated power, so that in a stationary state
integral oo integral oo dk F ±(k,t) = 2n dk k2E ±(k,t) = e±. 0 0

Looking at Equation (16View Equation), we see that the role played by the non-linear term is that of a redistribution of energy among the various wave vectors. This is the physical meaning of the non-linear energy cascade of turbulence.

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