A two-dimensional toy model for geophysical turbulence

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

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

   
6 March 2018 (11:15 - 12:15 hrs), MISU
Stockholm University, Stockholm, Sweden
  1. Augier, P., Mohanan A.V. & Lindborg, E. Wave energy cascade in forced-dissipative one-layer shallow-water flows. J. Fluid Mech. (to be submitted).

  2. Lindborg, E. & Mohanan, A. V. A two-dimensional toy model for geophysical turbulence. Phys. Fluids (2017).

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. Quick recap of turbulence fundamentals

  • Kinetic ($E_K$) and Available Potential Energy($E_A$)
  • Cumulative sum of transfer terms $T(k_h)$ = Spectral energy fluxes $\Pi(k_h)$
  • Normal mode decomposition (for Bousinessq and shallow-water equations)
    • Velocities and the scalar can be decomposed as a sum of one vortex mode and two wave modes
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1. Background

1.3. Stratified Turbulence

  • Lindborg (2006): Postulated scaling laws for non-rotating stratified turbulence

    • Length scale:
      • $l_v \sim u/N$
      • $\frac{l_v}{l_h} \sim F_h$, the horizontal Froude number. Typically $F_h << 1$ for strong stratification.
    • Energy spectrum:
      • $E_K(k_h) = C_1 \epsilon_K^{2/3}k_h^{-5/3}$;
      • $E_A(k_h) = C_2 \epsilon_P \epsilon_K^{-1/3}k_h^{-5/3}$
    • Forward energy cascade: $\Pi > 0$

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

2.1 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

2. Quest for the simplest model

2.3 Desirable properties for turbulence studies

  • Kinetic energy (KE) and Available potential energy (APE) should be quadratic and conserved
  • Potential enstrophy conservation in the QG limit
  • No shock formation

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

2. Quest for the simplest model

2.4. 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

2.4. 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| $.

     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

2.4. Motivations for the Modifications

Helmholtz decomposition:

${\bf u} = \bf{u}_r + \bf{u}_d$

  • $ {\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.

2. Quest for the simplest model

2.5. 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

2. Quest for the simplest model

2.6. Normal modes of toy model equations

  1. Linearize the toy-model equations
  2. Convert the equations in the spectral plane
  3. Solve the eigenvalue problem

 Normal modes: $\hat a_1, \hat a_2, \hat b$

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

3. Results

3.0 Numerical Simulations

  • Pseudospectral method solved in a bi-periodic domain.
  • Grid-size: 1920x1920
  • Time-stepping: $4^\text{th}$ order Runge-Kutta, with a time step to resolve the fast gravity waves.
  • Toy-model equations transformed and solved in terms of the normal modes.
  • Forcing: Narrow band forcing in moderately large wavenumbers, random in time, controlled by calculating the forcing power apriori.
  • Perfect energy conservation!

3. Results

3.1 Divergence fields ($\nabla\cdot\bf{u}$)

Shallow-water equations and toy model: forcing at $k_f = 6$

Wave-vortex fields

3. Results

3.2 Energy spectra and spectral energy fluxes

Toy model forced at $k_f = 6$

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3. Results

3.3 Energy spectra

Toy model forced at $k_f = 6$ using different forcing methods

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3. Results

3.4 Spectral energy fluxes

Toy model forced at $k_f = 6$ using different forcing methods

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3. Results

3.5 Spectral energy fluxes

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

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3. Results

3.6 Energy spectra

Toy model forced at $k_f = 30$

spectra

3. Results

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

Toy model forced at $k_f = 30$

3. Results

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

Toy model forced at $k_f = 30$

Wave-vortex fields

4. Outlook


  1. Toy model simulations in beta plane
  2. Large simulation of the toy model over a sphere
  3. Study of cyclonic/anticyclonic assymetry using the toy model


Thank you for your attention!

Open source and reproducible
Mohanan, A. V., Bonamy C. & Augier, P. FluidSim: modular, object-oriented Python package for CFD simulations (to be submitted)
Bitbucket fluiddyn/fluidsim | Github fluiddyn/fluidsim
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Summary
  • Toy model developed by adding two modifications to the 1-layer shallow water equations.
  • Able to reproduce $k^{-5/3}$ energy spectra similar to atmospheric mesoscale spectra.
  • Conserves K.E., A.P.E. and linear potential enstrophy in the quadratic form: useful in turbulence studies.
  • Further reading: Lindborg, E. & Mohanan, A. V. A two-dimensional toy model for geophysical turbulence. Phys. Fluids (2017)
CC-BY-SA: Ashwin Vishnu Mohanan