Numerical study of experimentally inspired stratified turbulence forced by waves¶

Jason Reneuve, Pierre Augier and Nicolas Mordant
Université Grenoble Alpes, CNRS, Grenoble INP, LEGI, Grenoble
           
4th July 2022
$$\require{begingroup}\require{newcommand}$$$$\gdef\vec#1{\boldsymbol{#1}}$$$$\gdef\R{\mathcal{R}}$$$$\gdef\FhR{(F_h,\ \R)}$$

Internal gravity wave turbulence in the lab?¶

Oceanic and atmospheric measurements interpreted as IGW turbulence

Reproducible in the lab?

Stable density stratification $\Rightarrow$ Waves (IGW) and "vortices" ($\omega_z$)

Stratified turbulence (forced by vortices)

  • Horizontal Froude number $F_h$ and buoyancy Reynolds number $\R = Re {F_h}^2$

  • LAST regime $F_h<0.02$, $\R>20$: downscale energy cascade and anisotropic spectra

$$b(x,z)$$
Brethouwer, Billant, Chomaz & Lindborg (2007)

  • $\R = Re {F_h}^2 > 20 \Rightarrow Re \gg 1$

    • experiments: very large apparatus
    • simulations: high resolution

Internal gravity wave turbulence in the Coriolis facility!¶

Coriolis platform in Grenoble

  • 13m-diameter, 1m-deep, 130 tons water
  • stratification with salt
  • rotation up to 6 rpm
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Internal gravity wave turbulence in the Coriolis facility!¶

Linearly stratified ($\sim$ 1 ton of salt)

Savaro et al. PRFluids 2020

Internal gravity wave turbulence in the Coriolis facility!¶

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Internal gravity wave turbulence in the Coriolis facility¶

Temporal spectra for different forcing amplitudes¶

Savaro et al. PRFluids 2020

Numerical setup: experimentally inspired stratified turbulence forced by waves¶

  • Pseudo spectral solver ns3d.strat of the open-source CFD framework Fluidsim

  • 3D periodic!

Basically same parameters than for the experiments in the Coriolis platform: size, viscosity, Brunt-Väisälä frequency $N=0.6$ rad/s, forcing frequency $\omega_f = 0.73 N$ and forcing amplitude.

$\Rightarrow$ high resolution (up to $2304\times2304\times384$) and very long time (typically 15 h)

Numerical setup: experimentally inspired stratified turbulence forced by waves¶

$2\pi / N \simeq 10$ s, $2\pi / \omega_f \simeq 15$ s

  • Long simulations at different resolution ($240\times240\times40$, $480\times480\times80$, $1152\times1152\times192$ and $2304\times2304\times384$)

  • Hyper viscosity is decreased when resolution is increased

  • $2304\times2304\times384$ quasi DNS

  • Strong vortical flow!

  • Stationarity?


Response to the forcing: $U_h$ and $\varepsilon_K$

$\Rightarrow F_h \simeq 6.5 \times 10^{{-4}}$ (strongly stratified) and $\R = Re {F_h}^2 \simeq 0.5$ (viscosity affected regime)

Snapshots for $\omega_f = 0.73N$ and $a = 5$ cm¶

  • large scale waves

  • small scale "turbulence"

  • thin vertical layers

  • 2 counter-rotating vortices

Mean flow: comparison simulation / experiment¶

Horizontal cross-section¶

2 counter-rotating vortices at 2 $\neq$ levels

Temporal spectra for $\omega_f = 0.73 N$ and $a = 5$ cm¶

Triads involving the forced frequency and modes of the numerical domain

Vortical flow dominates at very small frequency

Continuum between the peaks

Spatial spectra for $\omega_f = 0.73 N$ and $a = 5$ cm¶

Very strongly anisotropic spectra ("strongly stratified", $F_h \ll 10^{-2}$)

Viscosity affected flow ($\R \simeq 0.5$) but not far from the transition $\R > 1$

Spatio-temporal spectra for $\omega_f = 0.73 N$ and $a = 5$ cm¶

Triad: $\omega_1 + \omega_2 = \omega_f$ and $\vec{k}_1 + \vec{k}_2 = \vec{k}_f$

Spatio-temporal spectra for $\omega_f = 0.73 N$ and $a = 5$ cm¶

Weakly non linear waves only for first vertical modes

Stronger forcing for $\omega_f = 0.73 N$: $a=10$ cm¶

$a = 0.05$ cm ($\R \simeq 0.5$)
$a = 0.10$ cm ($\R \simeq 6.5$)

Stronger forcing for $\omega_f = 0.73 N$: $a=10$ cm¶

Temporal spectra¶

$a = 0.05$ cm ($\R \simeq 0.5$)
$a = 0.10$ cm ($\R \simeq 6.5$)

Stronger forcing for $\omega_f = 0.73 N$: $a=10$ cm¶

Spatial spectra¶

$a = 0.05$ cm ($\R \simeq 0.5$)
$a = 0.10$ cm ($\R \simeq 6.5$)

Slower forcing $\omega_f = 0.4 N$ and $a=10$ cm¶

Temporal spectra¶

$\omega_f = 0.73 N$
$\omega_f = 0.40 N$

Conclusions¶

  • Pseudo spectral 3D periodic simulations with penalization can reproduce some aspects of the dynamics observed in experiments

  • The buoyancy Reynolds number $\R$ is crucial (transition at $\R \gtrsim 1$)

  • Accumulation of energy at large scales (2 counter-rotating vortices and waves)

  • Tank and forcing geometry $\Rightarrow$ normal modes ($\neq$ solutions for exp. and simulations...)

  • "Wave turbulence" between the modes

  • Not far from $\omega^{-2}$ between $\omega_f$ and $N$

Note: Fluidsim (https://fluidsim.readthedocs.io) is now a great tool!¶

conda create -n env-fluidsim ipython fluidsim "fluidfft[build=mpi*]" "h5py[build=mpi*]"