chapter_01_0_intro.md

\chapter{The MILESTONE experiment}\label{ch:exp}

This chapter describes the experiment, Mixing and Length Scales in Stratified Turbulence [MILESTONE, @campagne2016] which aimed at testing the theory of stratified turbulence [@Billant2001;@Lindborg2006]. The development of this theory is briefly reviewed in @sec:strat. In particular this experiment intended to:

1. verify that layered structures emerge with a vertical length scale of $u/N$,

2. verify that there is a forward energy cascade with spectrum scaling as $E(k) \sim \epsilon^{2/3} k^{-5/3}$, or alternatively the second order structure function scaling as $\braket{\delta \mathbf{u}. \delta \mathbf{u}} \sim \epsilon^{2/3} r^{2/3}$, and

3. measure the mixing efficiency in the strongly stratified regime.

Stratified turbulence theory can potentially lead to improved ocean models which has profound implications in our ability to forecast weather and climate. It is well known that

a global system of ocean currents are responsible for redistribution of heat from the tropics to higher latitudes, replenishing nutrients to the surface which are critical for sustenance of marine life, and melting of ice in high latitudes.

{#fig:thermohaline width=80%}

This phenomenon, interchangeably known as thermohaline circulation or the meridional overturning circulation (MOC) shown in @fig:thermohaline, is driven by a combination of wind forcing, fluxes of temperature and salinity at several isolated regions near the surface and turbulent mixing in the interior of the ocean [see chapter 21 in @vallis_atmospheric_2017]. Through mixing, the kinetic energy of the ocean currents get converted into potential energy, resulting in a net upward flux of dense water. In ocean models this flux of buoyancy is often represented using eddy diffusivity models parametrized based on mixing: $$

• \nabla \cdot (\uu b) \simeq \kappa_\tau \nabla^2 b $$ In the literature, there are different but closely related quantities which characterize mixing [@Gregg]:

• mixing efficiency, a ratio of dissipation of potential energy and total energy $\eta = \epsP / (\epsK + \epsP)$,

• mixing coefficient, $\Gamma = \epsP / \epsK$,

• flux Richardson number, a ratio of buoyancy flux to turbulence production, $Ri_f = B / (\epsK + B)$, where $B = -\braket{bu_z}$

Conventionally ocean models rely on a nominal value of $\Gamma = 0.2$ due to @Osborn, to set the eddy diffusivity parameter $\kappa_\tau = \epsP / N^2 = \Gamma \epsK / N^2$ [@OsbornCox;@Lindborg-vertical-2008]. However numerical studies [@BrethouwerLindborg2009;@Salehipour-diapycnal-2015;@maffioli_mixing_2016] have shown that mixing efficiency is not a constant and depends on the strength of stratification. In the limit of strong stratification it approaches a constant value, and approaches zero in the limit of weakly stratified turbulence, scaling with horizontal Froude number as $\Gamma \propto F_h ^{-2}$ [@maffioli_mixing_2016]. At intermediate levels of stratification, it is found to vary in the interval $\Gamma \in [0.26, 0.51]$ and peaking at $F_h \approx 0.33$.

The strongly stratified turbulence regime is characterized by two non-dimensional numbers [@Brethouwer2007], $$ F_h = \frac{\epsK}{NU^2} \ll 1,\text{ and } \R = \frac{\epsK}{\nu N^2} > 10. $$ a Froude number based on horizontal velocity and buoyancy Reynolds number respectively. The regime of strongly stratified turbulence, in which $F_h$ is small and $\mathcal{R}$ is large, is highly relevant for applications in the ocean [@gargett_composite_1981;@RileyDeBruynKops2003;@Lindborg2006]. However, due to the fact that the buoyancy Reynolds number relates to the conventional Reynolds number ($Re_h$) as $\R = Re_hF_h^2$, reaching this regime is a challenging task, both numerically and experimentally. In trying to reach the strongly stratified regime experimentally by increasing the degree of stratification, the buoyancy Reynolds number often becomes so low that turbulence is totally suppressed. The fluid has to be under a strong background stratification and high Reynolds number forcing simultaneously and this state is difficult to maintain as mixing tends to reduce stratification over time. In simulations, a high resolution DNS is required to reach the stratified turbulence regime. Despite these difficulties numerical simulations have pushed the limits to be as close to the predicted values of $F_h$ and $\R$ as possible [@Brethouwer2007;@BrethouwerLindborg2009;@Maffioli2016;@maffioli_mixing_2016] of which some are depicted on @fig:stratified-regime along with estimates of the values attained by the MILESTONE experiment.

{#fig:stratified-regime}

Thus by testing the validity of the stratified turbulence theory, one can have a good picture of the vertical length scales, the direction of energy cascade, and of what parametrizations of mixing are appropriate, which can greatly benefit modelling of ocean turbulence. In the following sections, the experimental setup and the open-source software stack which was developed to perform the experiment and post-process the data are briefly described, whereafter we highlight some of the results.