Studies in geophysical fluid mechanics

Ashwin Vishnu Mohanan*, Pierre Augier^, Erik Lindborg*

* Linne Flow Center, Department of Mechanics, KTH, Stockholm, Sweden
^ LEGI / CNRS, Université Grenoble Alpes, Grenoble, France

   
15 Feb 2018 (13:00 - 13:15 hrs), MISU
Stockholm University, Stockholm, Sweden
  1. Augier, P., Mohanan A.V. & Lindborg, E. Shallow water wave turbulence J. Fluid Mech. (under review).
  2. Lindborg, E. & Mohanan, A. V. A two-dimensional toy model for geophysical turbulence. Phys. Fluids (2017).
  3. Augier, P., Mohanan A.V. & Bonamy, C. FluidDyn: a Python open-source framework for research and teaching in fluid dynamics J. Open Res. Softw. (under review).
  4. Mohanan A.V., Bonamy, C. & Augier, P. FluidFFT: common API (C++ and Python) for Fast Fourier Transform HPC libraries J. Open Res. Softw. (under review).
  5. Mohanan A.V., Bonamy, C., Linares, M. C. & Augier, P. FluidSim: modular, object-oriented Python package for high-performance CFD simulations J. Open Res. Softw. (under review).

Article [2] selected as featured research by AIP (Nov 22, 2017)

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1. Background

1.1. Atmospheric energy spectra from aircraft data: Nastrom and Gage (1985)

Nastrom and Gage
  • Synoptic scale spectra ($\lambda > 1000$ km) ~ $k^{-3}$
  • Mesoscale spectra ($\lambda = 1$ to $500$ km) ~ $k^{-5/3}$

1. Background

1.2. Possible explanations for the mesoscale energy $k^{-5/3}$ spectra

  • Gage (1979) & Lilly (1983): inverse energy cascade as in Kraichnan (1967)

  • Dewan (1979):forward energy cascade as in Kolmogorov (1941)

Theoretical predictions for turbulent structures involved
  • Lindborg (2006) and Waite & Bartello (2004): Stratified turbulence result in thin elongated structures. Vertical length scale $ l_v \sim u/N \approx 1 km$
  • Callies, Bühler and Ferrari (2016): Inertia gravity waves, with frequency $\omega \approx f$. i.e. $l_v \approx$ 100 metres.

1. Background

1.3. Stratified Turbulence

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3D Boussinesq equation simulations in Lindborg (2006) demonstrated that

  • energy spectra scales as $k^{-5/3}$
  • energy flux indicates a forward energy cascade

1. Background

1.4. Results from General Circulation Models

  • Augier & Lindborg (2013): A GCM called AFES can simulate mesoscale energy cascade with coarse vertical resolution: 24 levels!
  • Other GCMs (ECMWF) cannot!
  • Energy spectra and fluxes computed from spherical harmonics . Spherical harmonic indices $l$ & $m$ correspond to latitude and longitude angles.
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1. Background

1.4. Motivation for the present study

Questions and contradictions

  • Stratified turbulence predicts a fine vertical resolution to be needed for obtaining $k^{-5/3}$ spectra
  • GCM simulations required only 24 pressure levels


  • Minimum number of levels required to reproduce $k^{-5/3}$ spectra?
  • Is it possible with a single level model? 1-layer Shallow-water equation?

2. Quest for the simplest model

Why shallow-water equations?
$$\frac{\partial {\bf u}} {\partial t} + {\bf u}\cdot \nabla{\bf u} + f {\bf e}_z \times {\bf u} = -c^2 \nabla \eta $$$$\frac{\partial \eta}{\partial t}+ {\bf u} \cdot \nabla \eta = - (1+\eta) \nabla \cdot {\bf u}$$
  • Explain many geophysical phenomena, including waves

  • Conserves potential vorticity and enstrophy.

Why not shallow-water equations?
  • Kinetic energy is not quadratic, but cubic: $E_K = (H + \eta) \frac{\mathbf{u}.\mathbf{u}}{2}$
  • Potential enstrophy is not quadratic in general.
  • Tendency for waves to develop into shocks giving rise to $k ^ {-2}$ energy spectra

2. Quest for the simplest model

Divergence fields ($\nabla\cdot\bf{u}$) of shallow-water equations and toy model: forcing at $k_f = 6$

Wave-vortex fields

2. Quest for the simplest model

Quasi-Geostrophy (QG): Charney (1971)

  • In QG limit, potential vorticity can be approximated as $ Q = \frac{f +\zeta }{1+\eta} \rightarrow q = \zeta + \beta y - f_0\eta$. Thus QG potential enstrophy, $\Omega = \frac{1}{2} q^2$ is quadratic.
  • Inverse energy cascade and forward enstrophy cascade: just like 2D turbulence in Kraichnan (1971).

Shallow water equation is often studied as QG equations:

$$\frac{D}{Dt}\left(\nabla^2 \psi + \beta y - \frac{1}{L_d^2} \psi \right)= \frac{D}{Dt}\left(\zeta + \beta y - f_0 \eta \right)=0$$

Important assumptions:

  1. Rossby number, $Ro < 1 \implies$ strong rotation
  2. Burger number, $1 / Bu = L_d / L < 1 \implies$ planetary scales
  3. Variations in coriolis term ($\beta$) is small $\implies$ mid-latitudes and above
$$\require{color} \newcommand{\red}[1]{\mathbin{\textcolor{red}{#1}}} \newcommand{\green}[1]{\mathbin{\textcolor{green}{#1}}}$$

2. Quest for the simplest model

Motivations for the Modifications: (Lindborg and Mohanan 2017)

$$\frac{\partial {\bf u}} {\partial t} + {{\bf u}\cdot \nabla} {\bf u} + f {\bf e}_z \times {\bf u} = -c^2 \nabla \eta $$$$\frac{\partial \eta}{\partial t}+ {{\bf u} \cdot \nabla} \eta = \red{- (1+\eta) \nabla \cdot {\bf u}}$$
  • Assumption #1: Surface displacement much smaller compared to the mean fluid layer height, $\eta << 1$.

     Replace  $ \red{-(1+\eta) \nabla \cdot {\bf u}}$   with   $ \green{-\nabla \cdot {\bf u}} $

2. Quest for the simplest model

Motivations for the Modifications: (Lindborg and Mohanan 2017)

$$\frac{\partial {\bf u}} {\partial t} + \red{{\bf u}\cdot \nabla} {\bf u} + f {\bf e}_z \times {\bf u} = -c^2 \nabla \eta $$$$\frac{\partial \eta}{\partial t}+ \red{{\bf u} \cdot \nabla} \eta = \green{-\nabla \cdot {\bf u}}$$
  • Assumption #2: Velocities in the large scale are dominated by rotational part, $ |\bf u_r| >> |\bf u_d| $. Use Helmoltz decomposition to make this distiction.



 Replace   $\red{{\bf u} \cdot \nabla}$  with   $\green{{\bf u_r} \cdot \nabla}$

While allowing, $|\zeta| \sim |d|$ in contrast with QG where $|\zeta| >> |d|$

2. Quest for the simplest model

The toy model equations

$$\frac{\partial {\bf u}} {\partial t} + \green{{\bf u}_r\cdot \nabla} {\bf u} + f {\bf e}_z\times {\bf u} = -c \nabla \theta $$$$\frac{\partial \theta}{\partial t}+ \green{{\bf u}_r \cdot \nabla} \theta = - c\green{\nabla \cdot {\bf u}} $$

where, $\theta = c\eta$

  • Pros: No shocks, KE and APE are quadratic and conserved, linearised potential vorticity conserved in the limit $Ro \rightarrow 0$: $q = \zeta - f\eta$

  • Cons: Full potential vorticity $Q$ is not exactly conserved

3. Results

Energy spectra and spectral energy fluxes

Toy model forced at $k_f = 6$

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3. Results

Spectral energy fluxes

Toy-model forced at $k_f = 30$ compared to GCM results from Augier & Lindborg(2013)

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3. Results

Energy spectra

Toy model forced at $k_f = 30$

spectra

3. Results

Potential vorticity ($q$) and wave ($a_+$) fields

Toy model forced at $k_f = 30$

3. Results

Potential vorticity ($q$) and wave ($a_+$) fields: Anticyclone formation

Toy model forced at $k_f = 30$

Wave-vortex fields

4. FluidDyn project

fluidsim: modular and reproducible simulation framework

Python framework to run sequential and parallel (MPI) CFD simulations for a variety of problems (Navier-Stokes, Shallow Water, Föppl von Kármán equations, ...).

  • highly modular, object-oriented structure, on-the-fly postprocessing

  • specialized in pseudo-spectral methods (based on fluidfft),

  • user friendly, documented

  • efficient (much faster than Dedalus, faster than SpectralDNS).

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4. FluidDyn project

fluidfft: API for FFT libraries in sequential and in parallel, with CPU and GPU.

Hierarchy of C++ and Cython classes to use different FFT libraries: FFTW, P3DFFT, PFFT, CuFFT .. possibly more?

  • Python operators classes (2d and 3d) to write code independently of the library used for the computation of the FFT.

  • Pythran to speedup critical code. Performance $\simeq$ Fortran.

  • Command line utilities (fluidfft-bench and fluidfft-bench-analysis).

  • Unit tests!

5. MILESTONE project on Coriolis platform, Grenoble

  • The Coriolis platform 13 m diameter)
  • Used by international researchers through European projects (EUHIT, Hydralab).

5. MILESTONE project on Coriolis platform, Grenoble

Top view of setup used for 2 sets of experiments

  • Summer 2016 (a collaboration between KTH, Stockholm, Sweden and LEGI).
  • Summer 2017: focused on mixing without rotation

5. MILESTONE project on Coriolis platform, Grenoble

Carriage:

  • 3 m $\times$ 1 m
  • runs on tracks (13 m long)
  • good control in position ($\Delta x<$ 5 mm) and in speed ($U< 25$ cm/s)


Thank you for your attention!

Outlook

  • Study of wave-vortex interactions using shallow-water and toy-model equations

  • Large simulation of the toy model over a sphere

  • Study of cyclonic/anticyclonic assymetry using the toy model

Summary
  • Toy model reproduces $k^{-5/3}$ energy spectra similar to atmospheric mesoscale spectra.

  • FluidDyn open-source project

  • MILESTONE project

CC-BY-SA: Ashwin Vishnu Mohanan