where and 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
MHD equations can be written in the same way, say by introducing the Fourier decomposition for Elsässer variables
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 means that we can project the Fourier coefficients on two directions which are mutually orthogonal and orthogonal to the direction of , that is,
Note that in the linear approximation where the Elsässer variables represent the usual MHD modes, represent the amplitude of the Alfvén mode while represent the amplitude of the incompressible limit of the magnetosonic mode. From MHD Equations (12) we obtain the following set of equations:
and the sum in Equation (15) is defined as
where is the Kronecher’s symbol. Quadratic non-linearities of the original equations correspond to a convolution term involving wave vectors , and related by the triangular relation . Fourier coefficients locally couple to generate an energy transfer from any pair of modes and to a mode .
The pseudo-energies are defined as
and, after some algebra, it can be shown that the non-linear term of Equation (15) conserves separately . This means that both the total energy and the cross-helicity , 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
brackets being ensemble averages, where is an arbitrary odd function of the wave vector and represents the pseudo-energies spectral density. When integrated over all wave vectors under the assumption of isotropy
where we introduce the spectral pseudo-energy . This last quantity can be measured, and it is shown that satisfies the equations
so that, when integrated over all wave vectors, we obtain the energy balance equation for the total pseudo-energies
Looking at Equation (16), 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.
© Max Planck Society and the author(s)