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 , where is the inter-shell ratio, is the fundamental wave vector related to
the largest available length scale , and .

b) Assign to each shell discrete scalar variables:

Each shell is assigned two or more complex scalar variables and , or Elsässer variables
. These variables describe the chaotic dynamics of modes in the shell of wave vectors
between and . 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 . That is, the fields have the same scalings as field differences, for example
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 , 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
and , 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

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

where indicates the real part of the product . 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 , 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 wherehttp://www.livingreviews.org/lrsp-2005-4 |
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