11 December 2017 (17:21 - 17:34 hrs), AGU Fall Meeting 2017

NG14A-07, New Orleans Ernest N. Morial Convention Center; 238-239

NG14A-07, New Orleans Ernest N. Morial Convention Center; 238-239

**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)**:*downscale energy cascade*as in**Kolmogorov (1941)**

**Lindborg (2006)**and**Waite & Bartello (2004)**: Stratified turbulence result in thin elongated structures. Vertical length scale $ l_v < l_b = u/N \approx 1 km$

**Callies, Bühler and Ferrari (2016)**: Inertia gravity waves, with frequency $\omega \approx f$. i.e. $l_v \approx$ few metres.

**Vertical resolution?**

**Augier & Lindborg (2013)**: GCMs such as**AFES**and**MPAS**model can simulate mesoscale energy cascade with coarse vertical resolution: 24 levels!

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

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. But in QG limit with strong rotation: $ Q = \frac{f +\zeta }{1+\eta} \rightarrow q = \zeta - f\eta$
- Tendency for waves to develop into shocks giving rise to $k ^ {-2}$ energy spectra

- Kinetic energy (KE) and Available potential energy (APE) must be
**quadratic**and**conserved** - Potential enstrophy conservation
- No shock formation

**Helmholtz decompostion:**

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

- $ {\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.

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

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

- Toy model simulations in
**beta plane** - Large simulation of the toy model over a
**sphere** - Study of
**cyclonic/anticyclonic assymetry**using the toy model - Design of a
**laboratory experiment**which can behave similar to toy-model equations

**Mohanan, A. V., Bonamy C. & Augier, P.***FluidSim: modular, object-oriented Python package for CFD simulations***J. Open Research Software (to be submitted)**| 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