Abstract
Mixing and dispersion in stratified flows is a topic of utmost importance for environmental and climate processes. In a recent paper we have analysed and derived relations for the vertical dispersion of fluid particles in stably stratified flows governed by the Boussinesq equations \cite{lindbret2008}. Assuming that the flow is homogeneous and statistically stationary, we derived the relation % \begin{equation} \la \delta z^2 \ra = \frac{2}{N^2} \left [ \ep t \left ( 1 - \mathcal{O} (\mathcal{R}^{-1/2} ) \right ) + 2 E_P \right ], \label{verdis} \end{equation} % for the mean square of the vertical particle displacement $\delta z$ at large times $t$. Here, $N$ is the Brunt-V{\"a}is{\"a}l{\"a} frequency, $E_P$ and $\varepsilon_P$ are the potential kinetic energy and its dissipation respectively, and $\mathcal{R} = \varepsilon_K/\nu N^2$ where $\varepsilon_K$ is the kinetic energy dissipation rate and $\nu$ the viscosity. % The last term on the right-hand-side of (\ref{verdis}) is the adiabatic dispersion term whereby the density of the fluid particle does not change. In stably stratified, vertical adiabatic particle displacements are constrained by the available energy and consequently the adiabatic contribution to vertical dispersion is finite. The first term is the diabatic contribution due to density changes of the fluid particles and grows linearly in time. Since $\mathcal{R} \gg 1$ in most geophysical flows the term involving $\mathcal{R}$ is often negligible, but in lower Reynolds number laboratory experiments and numerical simulations it can give a significant contribution.Vertical dispersion has mainly been studied in decaying stratified turbulence. In that case diabatic dispersion does not give a long-time contribution to particle dispersion and we predict a finite value for $\la \delta z^2 \ra$ as $t \rightarrow \infty$ \cite{lindbret2008}.
So far, only van Aartrijk {\em et al.} \cite{aart} have studied particle dispersion in statistically stationary stratified turbulence through direct numerical simulations (DNS). At large times they observed $\la \delta z^2 \ra \sim t$, in accordance with (\ref{verdis}). % The aim of our work is to study adiabatic and diabatic dispersion in stratified flows, and validate (\ref{verdis}) and other relations presented in \cite{lindbret2008}. We have carried out a series of DNS of stratified turbulence with a range of Reynolds numbers and stratification strengths. Figure \ref{vis} shows snapshots of the buoyancy field of DNS with strong and weak stratification respectively. We can see the typical layered structure versus the isotropy in the strongly and weakly stratified case. Forcing was used to obtain statistically stationary flows. A large number of fluid particles were tracked in the DNS for many eddy turnover times $T$ \cite{bretlind2008b}. Similar simulations without particles have been carried out by \cite{bret2007} who studied the features of strongly stratified turbulence. % % Figures \ref{dns128} and \ref{dns} show DNS results of the time development of $\langle \delta z^2 \rangle$ and the prediction $\langle \delta z^2 \rangle = (4 E_P + 2\varepsilon_P t)/N^2$, i.e. relation (\ref{verdis}) without the term involving $\mathcal{R}$. % In the initial period dispersion is dominated by adiabatic processes but around $t \sim T$ this contribution levels off and for longer times diabatic dispersion gives the only contribution. In all simulations the limit $\langle \delta z^2 \rangle \sim t$ due to diabatic dispersion is clearly observed. Furthermore, we can see that for $t \rightarrow \infty$ the results of the DNS with $\mathcal{R} \gg 1$ closely agree with (\ref{verdis}). For decreasing values of $\mathcal{R}$ the deviation from $\langle \delta z^2 \rangle = (4 E_P + 2\varepsilon_P t)/N^2$ becomes larger, in accordance with (\ref{verdis}). Our simulations are the first to reveal the asymptotic diabatic dispersion $\langle \delta z^2 \rangle \rightarrow 2\varepsilon_P t/N^2$ since this is only observed in stationary flows at long times.
One simulation with hyperviscosity is carried out to test the relation $ \langle \delta z ^{2} \rangle = (1+\pi C_{PL}) 2 \ep t/N^{2} $, which should be valid for shorter time scales in the range $ N^{-1} \ll t \ll T $, where $ T $ is the turbulent eddy turnover time \cite{bretlind2008}. The result of the hyperviscosity simulation is in reasonable agreement with this relation, with $ C_{PL}$ about $3$. Based on the simulation results it is argued that the time scale determining the evolution of $ \langle \delta z ^{2} \rangle $ is the eddy turnover time, $ T $, rather than the buoyancy time scale $ N^{-1} $, as suggested in previous studies. The simulation results are also consistent with the prediction of Lindborg \& Brethouwer that the nearly flat plateau of $ \langle \delta z ^{2} \rangle $ observed at $ t \sim T $ should scale as $ 4 E_{P} /N^{2} $, where $ E_{P} $ is the mean turbulent potential energy.
Another subject of much interest is the relative dispersion of two fluid particles. Richardson (1926) suggested % \begin{equation} \langle \Delta^2 \rangle = g \varepsilon_K t^3, \label{rich} \end{equation} % in high Reynolds number Kolmogorov turbulence for $\tau \ll t \ll T$ ($\tau$ is the Kolmogrov time scale and $T$ is the eddy turnover time). Here, $\Delta = | \vetx^1(t)-\vetx^2(t) |$ is the separation distance between the particle pair, $g$ is the Richardson constant and $\varepsilon_K$ is the mean kinetic energy dissipation rate. For similar reasons as in isotropic turbulence, we can expect Richardson dispersion in stratified turbulence in the horizontal direction. In order to test this hypothesis, we have carried out numerical simulations of strongly stratified turbulence. The simulations are with hyperviscosity and have the same numerical setup as in Brethouwer \& Lindborg (2008). Figure \ref{hyp} shows $\langle \Delta^2 \rangle$ in the horizontal direction in four different simulations. We see, indeed, a powerlaw-scaling, but it appears that $\langle \Delta^2 \rangle \sim t^4$. This is a surprising finding and further simulations are necessary to verify if this result is robust.