Vertical motions of heavy inertial particles smaller than the smallest scale of the turbulence in strongly stratified turbulence

Abstract

We study the statistics of the vertical motion of inertial particles in strongly stratified turbulence. We use Kinematic Simulation (KS) and Rapid Distortion Theory (RDT) to study the mean position and the root mean square (rms) of the position fluctuation in the vertical direction. We vary the strength of the stratification and the particle inertial characteristic time. The stratification is modelled using the Boussinesq equation and solved in the limit of RDT. The validity of the approximations used here requires that$\sqrt{{L}/{g}}<{2\pi}/{\mathcal{N}}<\tau\_{\eta}$,where $\tau\_{\eta}$ is the Kolmogorov time scale, $g$ the gravitational acceleration, $L$ the turbulence integral length scale and $\mathcal{N}$ the Brunt-V\"ais\"al\"a frequency. We introduce a drift Froude number$Fr\_{d} = \tau\_p g / \mathcal{N} L$. When $Fr\_{d} < 1$, the rms of the inertial particle displacement fluctuation is the same as for fluid elements, i.e.$\langle(\zeta\_3 - \langle \zeta\_3 \rangle)^2\rangle^{1/2} = 1.22\, u'/\mathcal{N} + \mbox{oscillations}$. However, when $Fr\_{d} > 1$,$\langle(\zeta\_3 - \langle \zeta\_3 \rangle)^2\rangle^{1/2} = 267 \, u' \tau\_p$. That is the level of the fluctuation is controlled by the particle inertia $\tau\_p$ andnot by the buoyancy frequency $\mathcal{N}$. In other words it seems possible for inertial particles to retain the vertical capping while loosing the memory of theBrunt-V\"ais\"al\"a frequency.