The problem of two-dimensional turbulence with a linear friction can be considered as the active version of the transport of a passive scalar field with finite lifetime.
Indeed, there is a formal analogy between
two-dimensional Navier-Stokes equation for the scalar vorticity
, supplemented by the linear friction term
It is clear that the analogy is just apparent: while Eq. (2.7) is a linear equation for the field , Eq. (2.6) is strongly non-linear, because vorticity is the curl of velocity. Moreover while the equation for the passive scalar needs to be supplemented by an equation for the evolution of the advecting velocity field , the equation for vorticity determines also the evolution of velocity. Indeed the incompressible velocity field can be expressed in terms of the stream-function as , and the stream-function can be obtained from the vorticity solving the Poisson equation .
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It is interesting to notice that the forcings and
used in our simulations are chosen as independent stochastic
processes with the same statistics.
Were the two forcings identical
they would cancel each other in the equation for the difference field
which consequently
would decay to in a finite time.
This means that if the passive field is forced exactly
in the same way of the vorticity field, after some time it becomes
identical to the active one, although it has no feedback on the fluid.
This trivial observation is revealing of the crucial
role played by the correlation between forcing
and fluid trajectories in the passage from the
passive to the active problem.