Pseudo-spectral solver for 2D-HIT

This project contains a pseudo-spectral solver for forced HIT (homogeneous isotropic turbulence).

We consider the NS equations in vorticity formulation:

$$\frac{\partial \omega}{\partial t} + \mathcal{J}(\omega, \psi) = \nu \nabla^2 \omega + \mu(f - \omega),$$

$$\nabla^2 \psi= \omega,$$

where $J$ is the advection operator

$$ \mathcal{J}(\omega, \psi) = \frac{\partial \psi}{\partial x} \frac{\partial \omega}{\partial y} - \frac{\partial \psi}{\partial y} \frac{\partial \omega}{\partial x}. $$

Let $\hat{u}_{\boldsymbol{k}}$ denote the Fourier coefficient of wave number vector $\boldsymbol{k}$ for the Fourier transform of the scalar function $u(x,y)$. So

$$u(x,y) = \sum_{\boldsymbol{k} \in \mathbb{Z}^2} \hat{u}_{\boldsymbol{k}} e^{i(k_1 x + k_2 y)} .$$

Requirments

This project requires PyTorch. It is developed to use GPU acceleration, but it should also work on "CPU only" devices.

Workflow for setting up a subgrid parametrization

High fidelity reference simulation

Both the CNN base-parametrization and the tau-orthogonal method require training data. For the CNN this training data consists of {input: low fidelity fields; output: SGS field}. The tau orthogonal method needs the trajectory of the quanities of interest in a high fidelity simulation (its targets to track).

The training datasets can be created using "HF_solver.py". Which implements a pseudo-spectral AB/BDI2 scheme. For details, please see "input_explained.md".

The CNNs are typically trained on a data set with 2000 input -> output fields. These can be obtained from a 2000-day simulation. Such a simulation takes up to 3 hours with GPU acceleration. After which it has created a file with training data for the CNNs and a file with reference data for the tau-orthogonal method.

200.day.high.fidelity.simulation.mp4