Fluid trajectories and exit-times

We consider the two-dimensional Navier-Stokes equation for the scalar vorticity $\omega= \nabla \times {\mbox{\boldmath$v$}}$:

$${\partial \omega \over \partial t} + {\mbox{\boldmath$v$}} \... ... \omega = \nu \nabla^2 \omega - \alpha \omega + f_{\omega} \;,$$ (2.24)

with the additional friction term $- \alpha \omega$.

As already told in Sec. 2.2 the presence of friction regularizes the flow, removing scale by scale part of the enstrophy which is transported to small scales by the cascade process. As a consequence, at variance with the frictionless case where the the flux of enstrophy towards small scale is constant in the scaling range, in presence of friction it decays as $k^{-\xi}$ (see Figure 2.2). At viscous wave-numbers $k_d \sim \nu^{-1}$ the enstrophy flux is stopped by viscous dissipation, with a dissipation rate $\epsilon_{\omega} \sim \nu^{\xi}$ which vanishes in the inviscid limit $\nu \to 0$, since $\xi > 0$. In other words, in the limit of vanishingly small viscosity, there is no dissipative anomaly for enstrophy[19].

Figure 2.2: Enstrophy flux $\Pi _{\omega }(k) \sim k^{-\xi }$ for $\nu =5 \cdot 10^{-5}$ ($+$) and $\nu = 1.5 \cdot 10^{-5}$ ($\times$). Here $\alpha = 0.15$. Reducing $\nu$ the remnant enstrophy flux at small scales tends to zero as $\nu ^{\xi }$, allowing to disregard viscous dissipation.

The absence of dissipative anomaly for any $\alpha$ strictly positive allows to disregard the viscous term in Eq. (2.24) as far as we are interested in the statistical properties in the scaling range. In the limit $\nu \to 0$ it is possible to solve Eq. (2.24) by the method of characteristics yielding the expression:

$$\omega({\mbox{\boldmath$x$}},t)= \int_{-\infty}^{t} f_\omega({\mbox{\boldmath$X$}}(s),s)\,\exp[-\alpha(t-s)]\,ds$$ (2.25)

where ${\mbox{\boldmath$X$}}(s)$ denotes the trajectory of a particle transported by the flow:

$$\dot{\mbox{\boldmath$X$}}(s)={\mbox{\boldmath$v$}}({\mbox{\boldmath$X$}}(s),s)$$ (2.26)

ending at ${\mbox{\boldmath$X$}}(t)={\mbox{\boldmath$x$}}$. The uniqueness of the trajectory ${\mbox{\boldmath$X$}}(s)$ in the limit $\nu \to 0$ is ensured by the fact that the velocity field is Lipschitz-continuous ^2.1as it can be seen from the velocity spectrum $E(k) \sim k^{-3-\xi}$, always steeper than $k^{-3}$ (see Fig. 2.4).

The physical meaning of Eq. (2.26) is clear: the value of the vorticity field in a certain position ${\mbox{\boldmath$x$}}$ at time $t$ is just the sum of all the contributions given by the forcing along the fluid trajectory which ends in ${\mbox{\boldmath$x$}}$ at time $t$, depleted by an exponential factor because of the linear damping.

Vorticity differences $\delta_r \omega$ at scale $r$ smaller than the forcing correlation length $L$ are then associated to couples of fluid trajectories which at time $t$ are at distance $\vert{\mbox{\boldmath$X$}}'(t) - {\mbox{\boldmath$X$}}(t)\vert = r$

$$\omega({\mbox{\boldmath$x$}}',t)-\omega({\mbox{\boldmath$x$}... ...s)-f_\omega({\mbox{\boldmath$X$}}(s),s)] e^{-\alpha(t-s)} \;ds$$ (2.27)

Inside the time integral, the difference between the value of $f_\omega$ at ${\mbox{\boldmath$X$}}'$ and that at ${\mbox{\boldmath$X$}}$ is negligibly small as long as the two fluid particles lie at a distance smaller than $L$, the correlation length of the forcing; conversely, when the pair is at a distance larger than $L$, it approximates a Gaussian random variable $\Omega$. One then obtains:

$$\delta_r \omega \sim \Omega \int_{-\infty}^{t-T_L(r)} e^{-\alpha(t-s)}\;ds \sim \Omega e^{-\alpha T_L(r)}$$ (2.28)

where $T_L(r)$ is the time that a couple of particles at distance $r$ at time $t$ takes to reach a separation $L$ backward in time.

Equation (2.28) simply tells that a fluctuation of vorticity is originated by the forcing when the two trajectories are at distance larger than the forcing correlation length, and then it decays exponentially for all the time $T_L(r)$ required to transport by chaotic advection the fluctuation down to the small scale $r$ in the direct cascade process. At a fixed scale $r$, large vorticity fluctuations arise from couples of particles with relatively short exit-times $T_L(r) \ll \langle T_L(r) \rangle$, whereas small vorticity fluctuations are associated to large exit-times.