====== randomfield ======

Create a velocity field with 
   * random spectral coefficients that decay exponentially
   * zero divergence
   * Dirichlet boundary conditions at the walls

===== options =====

<code>
  -Nx       --Nx             <int>                     # x gridpoints
  -Ny       --Ny             <int>                     # y gridpoints
  -Nz       --Nz             <int>                     # z gridpoints
  -a        --alpha          <real>   default == 0     Lx = 2 pi/alpha
  -g        --gamma          <real>   default == 0     Lz = 2 pi/gamma
  -lx       --lx             <real>                    Lx = 2 pi lx
  -lz       --lz             <real>                    Lz = 2 pi lz
  -sd       --seed           <int>    default == 1     seed for random number generator
  -s        --smoothnes      <real>   default == 0.4   smoothness of field, 0 < s < 1
  -m        --magnitude      <real>   default == 0.2   magnitude  of field, 0 < m < 1
  -mf       --meanflow                                 perturb the mean
  -s1       --s1-symmetry                              satisfy s1 symmetry
  -s2       --s2-symmetry                              satisfy s2 symmetry
  -s3       --s3-symmetry                              satisfy s3 symmetry
  <fieldname>  (trailing arg 1)                        output file
</code>
===== mathematics =====

The field takes the form 

<latex>
{\bf u}(x,y,z)  = \sum_{jklm} u_{jklm} T_l(y) e^{2 \pi \, i (jx/Lx + kz/Lz)} {\bf e}_m
</latex>

where the spectral coefficients are assigned according to 

<latex>
u_{jklm} = (\text{random # in }[-1, 1]) * \text{(1-smoothness)}^{|l| + |j| + |k|}
</latex>

with corrections to meet boundary and divergence conditions and rescaling so that L2Norm(u) = magnitude.

The form of spectral decay chosen is crude, but normally what is needed in a random field is that 
it meets the BCs and zero-div, is controllably smooth, and excites modes with all symmetries. It would probably be better to work the length scales Lx,Ly,Lz into the exponent of ''(1-smoothness)'', so that the variations in the random field are roughly spatially isotropic.