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

**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)**

**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

- Velocities and the scalar can be decomposed as a sum of

**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?**

$$\frac{\partial {\bf u}} {\partial t} + {\bf u}\cdot \nabla{\bf u} + f {\bf e}_z \times {\bf u} = -c^2 \nabla \eta $$ $$\frac{\partial \eta}{\partial t}+ {\bf u} \cdot \nabla \eta = - (1+\eta) \nabla \cdot {\bf u}$$

Explain many geophysical phenomena, including waves

Conserves potential vorticity and enstrophy.

- 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

- 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

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

$$\frac{\partial {\bf u}} {\partial t} + {{\bf u}\cdot \nabla} {\bf u} + f {\bf e}_z \times {\bf u} = -c^2 \nabla \eta $$ $$\frac{\partial \eta}{\partial t}+ {{\bf u} \cdot \nabla} \eta = \red{- (1+\eta) \nabla \cdot {\bf u}}$$

**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}} $

$$\frac{\partial {\bf u}} {\partial t} + \red{{\bf u}\cdot \nabla} {\bf u} + f {\bf e}_z \times {\bf u} = -c^2 \nabla \eta $$ $$\frac{\partial \eta}{\partial t}+ \red{{\bf u} \cdot \nabla} \eta = \green{-\nabla \cdot {\bf u}}$$

**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 $|\zeta| >> |d|$

**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.

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

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

- 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!

- Toy model simulations in
**beta plane** - Large simulation of the toy model over a
**sphere** - Study of
**cyclonic/anticyclonic assymetry**using the toy model

Bitbucket fluiddyn/fluidsim | Github fluiddyn/fluidsim

- 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