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Invert geofluid problems based on xarray, using SOR iteration

Project description

xinvert

DOI GitHub Documentation Status PyPI version Workflow

animate plot

1. Introduction

Researches on meteorology and oceanography usually encounter inversion problems that need to be solved numerically. One of the classical inversion problem is to solve Poisson equation for a streamfunction $\psi$ given the vertical component of vorticity $\zeta$ and proper boundary conditions.

$$\nabla^2\psi=\zeta$$

Nowadays xarray becomes a popular data structure commonly used in Big Data Geoscience. Since the whole 4D data, as well as the coordinate information, are all combined into xarray, solving the inversion problem become quite straightforward and the only input would be just one xarray.DataArray of vorticity. Inversion on the spherical earth, like some meteorological problems, could utilize the spherical harmonics like windspharm, which would be more efficient using FFT than SOR used here. However, in the case of ocean, SOR method is definitely a better choice in the presence of irregular land/sea mask.

More importantly, this could be generalized into a numerical solver for elliptical equation using SOR method, with spatially-varying coefficients. Various popular inversion problems in geofluid dynamics will be illustrated as examples.

One problem with SOR is that the speed of iteration using explicit loops in Python will be e-x-t-r-e-m-e-l-y ... s-l-o-w! A very suitable solution here is to use numba. We may try our best to speed things up using more hardwares (possibly GPU).

Classical problems include Gill-Matsuno model, Stommel-Munk model, QG omega model, PV inversion model, Swayer-Eliassen balance model... A complete list of the classical inversion problems can be found at this notebook.

Why xinvert?

  • Thinking and coding in equations: User APIs are very close to the equations: unknowns are on the LHS of =, whereas the known forcings are on its RHS;
  • Genearlize all the steady-state problems: All the known steady-state problems in geophysical fluid dynamics can be easily adapted to fit the solvers;
  • Very short parameter list: Passing a single xarray forcing is enough for the inversion. Coordinates information is already encapsulated.
  • Flexible model parameters: Model paramters can be either a constant, or varying with a specific dimension (like Coriolis $f$), or fully varying with space and time, due to the use of xarray's broadcasting capability;
  • Parallel inverting: The use of xarray, and thus dask allow parallel inverting, which is almost transparent to the user;
  • Pure Python code for C-code speed: The use of numba allow pure python code in this package but native speed;

2. How to install

Requirements xinvert is developed under the environment with xarray (=version 0.15.0), dask (=version 2.11.0), numpy (=version 1.15.4), and numba (=version 0.51.2). Older versions of these packages are not well tested.

Install via pip (not yet)

pip install xinvert

Install from github

git clone https://github.com/miniufo/xinvert.git
cd xinvert
python setup.py install

3. Example: Helmholtz decomposition

This is a classical problem in both meteorology and oceanography that a vector flow field can be deomposed into rotational and divergent parts, where rotational and divergent parts are represented by the streamfunction and velocity potential. Given vorticity (vor) and divergence (div) as the forcing functions, one can invert the streamfunction and velocity potential directly.

3.1 Atmospheric demonstration

Here is an atmospheric demonstration with no lateral boundaries:

import xarray as xr
from xinvert import invert_Poisson

dset = xr.open_dataset('data.nc')

vor = dset.vor

# specify boundary conditions in invert parameters
# 'extend' for lat, 'periodic' for lon
iParams = {'BCs': ['extend', 'periodic']}

# Invert within lat/lon plane, with extend and periodic boundary
# conditions in lat and lon respectively
psi = invert_Poisson(vor, dims=['lat','lon'], iParams=iParams)

atmospheric plot

3.2 Oceanic demonstration

Here is a oceanic demonstration with complex lateral boundaries of land/sea:

import xarray as xr
from xinvert import invert_Poisson

dset = xr.open_dataset('mitgcm.nc')

vor = dset.vor

# specify boundary conditions in invert parameters
# 'fixed' for lat, 'periodic' for lon; undefined value is 0
iParams = {'BCs':['fixed', 'periodic'], 'undef':0}

# Invert within YG/XGplane, with fixed and periodic boundary respectively.
# Kwarg undef is used as a mask for land value.
psi = invert_Poisson(vor, dims=['YG','XG'], iParams=iParams)

oceanic plot

3.3 Animate the convergence of iteration

One can see the whole convergence process of SOR iteration as:

from xinvert import invert_Poisson_animated

# input of vor need to be two dimensional only;
# psi has one more dimension than vor as iteration, which could be animated over.
# Here psi has 40 frames and loop 1 per frame (final state is after 40 iterations)
psi = animate_iteration(invert_Poisson, vor, iParams=iParams,
                              loop_per_frame=1, max_frames=40)

animate plot

More examples can be found in these notebooks:

  1. Poisson equation for streamfunction/velocity potential;
  2. Poisson equation for meridional overturning and zonal Walker circulations;
  3. Geopotential model for balanced mass field;
  4. Eliassen model for the meridional overturning circulation;
  5. PV inversion for 2D reference state (TODO);
  6. PV inversion for 2D QGPV (TODO);
  7. Matsuno-Gill model for heat-induced tropical circulation
  8. Stommel-Munk model for wind-driven ocean circulation
  9. Omega equation for quasi-geostrophic vertical motion;
  10. 3D oceanic flow;

more to be added...

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