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
Stockholm University, Stockholm, Sweden
Related research¶
Augier, P., Mohanan A.V. & Lindborg, E. Wave energy cascade in forced-dissipative one-layer shallow-water flows. J. Fluid Mech. (to be submitted).
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)
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.
- 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
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: $\Pi > 0$
- Length scale:
3D Boussinesq equation simulations in Lindborg (2006) demonstrated that
- energy spectra scales as $k^{-5/3}$
- energy flux indicates a forward energy cascade
- 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.
- 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:
- Rossby number, $Ro < 1 \implies$ strong rotation
- Burger number, $1 / Bu = L_d / L < 1 \implies$ planetary scales
- Variations in coriolis term ($\beta$) is small $\implies$ mid-latitudes and above
- 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
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!
4. Outlook¶
- Toy model simulations in beta plane
- Large simulation of the toy model over a sphere
- 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
Bitbucket fluiddyn/fluidsim | Github fluiddyn/fluidsim
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