2.7 Shell models for turbulence cascade

Since numerical simulations, in some cases, cannot be used, simple dynamical systems can be introduced to investigate, for example, statistical properties of turbulent flows which can be compared with observations.

The shell model mimics the gross features of the time evolution of spectral Navier-Stokes or MHD equations. The 3D hydrodynamic shell model is usually quoted in literature as the GOY model, and has been introduced some time ago by Gledzer (1973 ) and by Ohkitani and Yamada (1989 ). The MHD shell model, which coincide with the GOY model when the magnetic variables are set to zero, has been introduced independently by Frick and Sokoloff (1998 ) and Giuliani and Carbone (1998 ). In the following, we will refer to the MHD shell model as the FSGC model. The shell model can be built up through four different steps:

a) Introduce discrete wave vectors:
As a first step we divide the wave vector space in a discrete number of shells whose radii grow according to a power $kn = k0cn$ , where $c > 1$ is the inter-shell ratio, $k0$ is the fundamental wave vector related to the largest available length scale $L$ , and $n = 1,2,...,N$ .

b) Assign to each shell discrete scalar variables:
Each shell is assigned two or more complex scalar variables $un(t)$ and $bn(t)$ , or Elsässer variables $Zn±(t) = un ± bn(t)$ . These variables describe the chaotic dynamics of modes in the shell of wave vectors between $kn$ and $kn+1$ . It is worth noting that the discrete variable, mimicking the average behavior of Fourier modes within each shell, represents characteristic fluctuations across eddies at the scale $-1 ln ~ kn$ . That is, the fields have the same scalings as field differences, for example $± ± ± h Z n ~ |Z (x + ln)- Z (x)|~ ln$ in fully developed turbulence. In this way, the possibility to describe spatial behavior within the model is ruled out. We can only get, from a dynamical shell model, time series for shell variables at a given $kn$ , and we loose the fact that turbulence is a typical temporal and spatial complex phenomenon.

c) Introduce a dynamical model which describes non-linear evolution:
Looking at Equation (15 ) a model must have quadratic non-linearities among opposite variables $Zn±(t)$ and $Z± (t) n$ , and must couple different shells with free coupling coefficients.

d) Fix as much as possible the coupling coefficients:
This last step is not standard. A numerical investigation of the model might require the scanning of the properties of the system when all coefficients are varied. Coupling coefficients can be fixed by imposing the conservation laws of the original equations, namely the total pseudo-energies

sum | | E± (t) = 1- ||Z ±||2, 2 n n

that means the conservation of both the total energy and the cross-helicity:

E(t) = 1- sum |u |2 + |b |2 ; H (t) = sum 2 R e (u b*), 2 n n n c n n n

where $R e$ indicates the real part of the product $unb* n$ . As we said before, shell models cannot describe spatial geometry of non-linear interactions in turbulence, so that we loose the possibility of distinguishing between two-dimensional and three-dimensional turbulent behavior. The distinction is, however, of primary importance, for example as far as the dynamo effect is concerned in MHD. However, there is a third invariant which we can impose, namely

which can be dimensionally identified as the magnetic helicity when $a = 1$ , so that the shell model so obtained is able to mimic a kind of 3D MHD turbulence (Giuliani and Carbone, 1998

After some algebra, taking into account both the dissipative and forcing terms, FSGC model can be written as

where

( ) ( ) Pn± = 2---a---c Z±n+2Zn±+1 + a-+-c Zn±+1Zn+±2 + ( 2) ( 2) c--a- ± ± a-+-c ± ± + 2c Zn-1Z n+1- 2c Z n-1Zn+1 + (c - a ) (2 - a - c) - ---2- Z±n-2Zn±-1- ------2-- Zn±-1Zn±- 2, (21) 2c 2c

where⁵ $c = 2$ , $a = 1/2$ , and $c = 1/3$ . In the following, we will consider only the case where the dissipative coefficients are the same, i.e., $n = m$ .