**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

Stockholm University, Stockholm, Sweden

**Augier, P., Mohanan A.V. & Lindborg, E.***Shallow water wave turbulence***J. Fluid Mech. (under review)**.**Lindborg, E. & Mohanan, A. V.***A two-dimensional toy model for geophysical turbulence.***Phys. Fluids (2017)**.**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)**.**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)**.**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)

**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.

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?**

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

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

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

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

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

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).

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!

- The Coriolis platform 13 m diameter)
- Used by international researchers through European projects (EUHIT, Hydralab).

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

**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)

Study of

**wave-vortex**interactions using shallow-water and toy-model equationsLarge simulation of the toy model over a

**sphere**Study of

**cyclonic/anticyclonic assymetry**using the toy model

Toy model reproduces $k^{-5/3}$ energy spectra similar to atmospheric mesoscale spectra.

FluidDyn open-source project

Bitbucket fluiddyn/fluidsim | Github fluiddyn/fluidsimMILESTONE project

CC-BY-SA: Ashwin Vishnu Mohanan