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

• Good morning all!
• I am here to talk about how we modified the shallow water equations, into a toy-model for turbulence studies.
• This work has been a culmination of the efforts from Erik, Pierre and myself Ashwin.

Related research

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)

• Here, I have listed the articles related to this talk
• Pleased to annouce that - distinction of being featured among several AIP publications once a week.

1. Background

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

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

• Picture in front of you is the result of aircraft measurements in the atmosphere accumulated over several years
• Planetery scale and mesoscale
• Why does this happen? There are several answers...

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.

• Kraichnan: Inverse ener. E(k) ~ k^{-5/3} & Forward ens.: E(k)~k^{-3}
• Emerging consensus that it could be explained by downscale or forward energy cascade.
• Theories tried to estimate the type of vertical structures and quantify the length scale
• Lindborg and Waite & Bartello in seperate studies attributed this to elongated structures (zig-zag instability) estimated to be bands of 1 km
• Callies et al. pointed out this could be because of gravity waves few metres
• That raises the question is there is a minimum Vertical resolution? to get the mesoscale spectra?

1. Background

1.3. Stratified Turbulence

 

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

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

• Confirmed with DNS results.
• Compenstated spectra on the left -> k-5/3
• Flux on the right -> Forward 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.

 

• Fact: GCM such as AFES able to reproduce mesoscale spectra with very few pressure levels
• l < 3 ( planetary scales) APE is forced; l
• l = [8, 30] baroclinic instability: conversion from APE -> KE. reflected also in green $C_{cum}$ curve
• l > 50 reasonably flat energy flux -> inertial range and some energy goes upscale
• l < 10 inverse energy cascade: source of large scale circulations

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?

• How many levels do we need in the vertical?
• Most extreme case: the shallow water equations?

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

• For those who are unfamiliar, SWE solves for horizontal velocity vector $\bf{u}$ (momentum conservation) and non-dimensional surface displacement ${\eta}$ (mass conservation)
• Many good properties
• Not suitable for turbulence studies... why?
• Reason: inability to represent K.E. and Pot. Enstrophy in quadratic form
• More importantly: Shocks result in $k^{-2}$, not $k^{-5/3}$ spectra

2. Quest for the simplest model

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

• Shocks appear as sharp thin fronts with negative divergence
• Divergence field (also related to vertical velocity)
• In toy model, alternating bands -> waves!

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

• Energies must be conserved in the quadratic form
• Potential enstrophy conservation in the QG limit
• No shocks!

$$\require{color} \newcommand{\red}[1]{\mathbin{\textcolor{red}{#1}}} \newcommand{\green}[1]{\mathbin{\textcolor{green}{#1}}}$$

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 $|\zeta| >> |d|$

Apply two modifications on the classical shallow water equations

• replace the right hand side of the scalar equation to make it linear. Reason: \eta << 1 when Fr -> 0
• in the advection term we use rotational velocity. Reason: large scale motions dominated by rotation
• use Helmholtz decomposition to calculate u_r

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

• Putting in the assumptions mentioned earlier, we get the toy-model
• Replaced \eta with a proportional \theta variable. Takes the form of potential temperature.

• Advantages and disadvantages

  ${\bf u} = \bf{u}_r + \bf{u}_d$
  
  
  • $ {\bf u}_r = -\nabla \times ( {\bf e_z} \Psi)$ is the rotational component
  • $\bf {u}_d = \nabla \chi$ is the divergent component

with $\Psi$ and $\chi$ being the stream function and the velocity potential respectively.

$$\begin{align} \widehat{a}_{1} &= \frac{k}{\sqrt{2} \omega} ( c \widehat{\theta} + f \widehat{\Psi} - {\mbox{i}} \omega \widehat{\chi} ) \\ \widehat{a}_{2} &= \frac{k}{\sqrt{2} \omega} ( c \widehat{\theta} - f \widehat{\Psi} + {\mbox{i}} \omega \widehat{\chi} ) \\ \widehat{b} &= \frac{1}{\omega} (c k^2 \widehat{\Psi}-f \widehat{\theta} ) \\ \end{align} $$

where, $\omega = \sqrt{f^2+k^2c^2}$

3. Results

Energy spectra and spectral energy fluxes

Toy model forced at $k_f = 6$

 

• Wave spectra scales as k^{-5/3}, forward energy cascade, dominates
• Vortex spectra scales as k^{-3}, forward enstrophy cascade
• Tendency for inverse energy cascade

3. Results

Spectral energy fluxes

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

 

• Similar as before, only difference is the forcing wavenumber
• Toy model demonstrates similar energetics compared to GCM
• Conversion from APE to KE, forward cascade and inverse cascade of K.E.
• Difference: equipartition of K.E. and A.P.E. in smaller scales signifying waves. Not present in GCM.

3. Results

Energy spectra

Toy model forced at $k_f = 30$

• Stronger inverse cascade, with vortical energy dominating at larger scales
• Smaller scales, wave and vortex spectra scale (approx) as before.
• Ripples appear when coherent vortices begin to form

3. Results

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

Toy model forced at $k_f = 30$

• Notice the coherent vortices form
• Waves form around vortices, strong interactions

3. Results

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

Toy model forced at $k_f = 30$

• Presence of a cyclonic core, surrounded by a strong anti cyclonic filaments around it
• Also note, the waves gathering around the vortex

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).

 

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

  Bitbucket fluiddyn/fluidsim | Github fluiddyn/fluidsim

• MILESTONE project

CC-BY-SA: Ashwin Vishnu Mohanan

• The work here has been made using fully open source code
• I have summarized the talk in this slide
• Will be happy to take any questions
• Thank you all for your attention