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Analogies with the passive scalar problem

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 $\omega = \nabla \times {\mbox{\boldmath$u$}}$, supplemented by the linear friction term

\begin{displaymath}
{\partial \omega \over \partial t} + {\mbox{\boldmath$u$}} \...
... \omega =
\nu \nabla^2 \omega - \alpha \omega + f_{\omega} \;.
\end{displaymath} (2.6)

and the dynamics of a concentration field $\theta ({\mbox{\boldmath$x$}},t)$ with a finite lifetime $\tau$, transported by the velocity field ${\mbox{\boldmath$u$}}({\mbox{\boldmath$x$}},t)$, which is described by the reaction-diffusion-advection equation
\begin{displaymath}
{\partial \theta \over \partial t} + {\mbox{\boldmath$u$}} \...
...kappa \nabla^2 \theta - {1 \over \tau} \theta + f_{\theta} \;.
\end{displaymath} (2.7)

where $\kappa$ is the molecular diffusivity and $f_{\theta}$ is the source of scalar fluctuations.

It is clear that the analogy is just apparent: while Eq. (2.7) is a linear equation for the field $\theta ({\mbox{\boldmath$x$}},t)$, 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 ${\mbox{\boldmath$u$}}({\mbox{\boldmath$x$}},t)$, the equation for vorticity determines also the evolution of velocity. Indeed the incompressible velocity field can be expressed in terms of the stream-function $\psi$ as ${\mbox{\boldmath$u$}} = (\partial_y \psi, -\partial_x \psi)$, and the stream-function can be obtained from the vorticity solving the Poisson equation $\omega= - \Delta \psi$.

Figure 2.1: Snapshots of the vorticity field (left) and passive scalar field with finite lifetime (right) resulting from the simultaneous integration of Eq. (2.7) and Eq. (2.6). Detail of the simulation are reported in Sec. (2.5)
\includegraphics[draft=false,scale=0.7, clip=true]{F_frameactive.eps} \includegraphics[draft=false,scale=0.7, clip=true]{F_framepassive.eps}
Nevertheless comparing a snapshot of the vorticity field obtained from numerical integration of Eq. (2.6), with a simultaneous snapshot of the passive scalar (see Fig. 2.1) obtained by the parallel integration of Eq. (2.7), using identical parameters $\kappa= \nu$ and $\alpha = 1/\tau$, it is evident that many similarities exist between the passive and the active field. Both vorticity and passive scalar fields are characterized by filamental structures, whose thickness can be as small as the smallest active length-scales. These ``active'' regions, where the vorticity experiences relatively strong excursions, are alternated to ``quiescent'' areas, the patches, where vorticity changes smoothly. This is the visual counterpart of the intermittency phenomenon, which is originated by the identical mechanism for vorticity and passive scalar.

It is interesting to notice that the forcings $f_{\omega}$ and $f_{\theta}$ used in our simulations are chosen as independent stochastic processes with the same statistics. Were the two forcings identical $f_{\omega} \equiv f_{\theta}$ they would cancel each other in the equation for the difference field $\theta({\mbox{\boldmath$x$}},t) - \omega({\mbox{\boldmath$x$}},t)$ which consequently would decay to $0$ 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.


next up previous contents
Next: Passive scalar with finite Up: Effects of friction in Previous: Steepening of the energy   Contents
Stefano Musacchio 2004-01-09