Let us consider a wavefield
in a periodic cube of with
side
and let the Fourier transform of this field be
where
index
marks the mode with wavenumber
on the grid in the
-dimensional Fourier space. For
simplicity let us assume that there is a maximum wavenumber
(fixed e.g. by dissipation) so that no modes with wavenumbers greater
than this maximum value can be excited. In this case, the total
number of modes is
. Correspondingly, index
will only take values in a finite box,
which is centred at 0 and all sides of which are equal to
. To consider homogeneous turbulence, the
large box limit
will have to be taken.
1
Let us write the complex as
where
is a
real positive amplitude and
is a phase factor which takes
values on
, a unit circle centred at zero in the
complex plane. Let us define the
-mode joint PDF
as the probability for the wave intensities
to be in the
range
and for the phase factors
to be on
the unit-circle segment between
and
for all
. In terms of this PDF, taking the averages will
involve integration over all the real positive
's and along all
the complex unit circles of all
's,
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