NumbaCS

Documentation: https://numbacs.readthedocs.io/

Source Code (MPL-2.0): https://github.com/alb3rtjarvis/numbacs

NumbaCS (Numba Coherent Structures) is a Python package for performing coherent structure analysis in a user-friendly and efficient manner. It implements methods referred to as "coherent structure methods", an umbrella term for any method that can be used to infer or extract Lagrangian and Eulerian coherent structures. These are tools for performing analysis of time-dependent dynamical systems, mainly with a focus on material transport in fluid flows. While this package can be used with any (incompressible) flow, methods are provided which make it simple for the user to take advantage of these tools for large-scale geophysical flows.

Features

NumbaCS currently implements the following features:

Diagnostics:

• FTLE -- finite time Lyapunov exponent
• iLE -- instantaneous Lyapunov exponent
• LAVD -- Lagrangian averaged vorticity deviation
• IVD -- instantaneous vorticity deviation

Feature extraction:

• LCS -- hyperbolic and elliptic Lagrangian coherent structures
• OECS -- hyperbolic and elliptic objective Eulerian coherent structures
• Ridge extraction -- FTLE and iLE ridge extraction

Installation

Conda: conda install -c conda-forge numbacs Pip: pip install numbacs

NOTE: It is strongly recommended to use conda for installation due to some reported issues when using pip for one of the dependencies (see issues #20 and #29 of the numbalsoda package -- these seem to primarily affect Windows installations).

Basic usage

Below we show basic demonstrations for computing FTLE. See the Examples Gallery for more examples showcasing functionality.

Predefined flow

```python import numpy as np from math import copysign from numbacs.flows import getpredefinedflow from numbacs.integration import flowmapgrid2D from numbacs.diagnostics import ftlegrid2D import matplotlib.pyplot as plt

set initial time, integration span, and integration direction

t0 = 0. T = -10. int_direction = copysign(1, T)

get ode to be used by 'flowmapgrid2D'

funcptr, params, domain = getpredefinedflow('doublegyre', intdirection=int_direction)

set up domain

nx,ny = 401,201 x = np.linspace(domain[0][0], domain[0][1], nx) y = np.linspace(domain[1][0], domain[1][1], ny) dx = abs(x[1] - x[0]) dy = abs(y[1] - y[0])

computes final position of particle trajectories over grid

flowmap = flowmapgrid2D(funcptr, t0, T, x, y, params)

compute FTLE over grid

ftle = ftlegrid2D(flowmap, T, dx, dy)

plot FTLE

fig, ax = plt.subplots(dpi=200) ax.contourf(x, y, ftle.T, levels=50) ax.set_aspect('equal') plt.show() ```

Numerical velocity data

```python import numpy as np from math import copysign from numbacs.flows import getinterparrays2D, getflow2D from numbacs.integration import flowmapgrid2D from numbacs.diagnostics import ftlegrid_2D import matplotlib.pyplot as plt

load in atmospheric data

dates = np.load('../data/merrajune2020/dates.npy') dt = (dates[1] - dates[0]).astype('timedelta64[h]').astype(int) t = np.arange(0, len(dates)*dt, dt, np.float64) lon = np.load('../data/merrajune2020/lon.npy') lat = np.load('../data/merra_june2020/lat.npy')

NumbaCS uses 'ij' indexing, most geophysical data uses 'xy'

indexing for the spatial coordintes. We need to switch axes and

scale by 3.6 since velocity data is in m/s and we want km/hr.

u = np.moveaxis(np.load('../data/merrajune2020/u500800hPa.npy'),1, 2)*3.6 v = np.moveaxis(np.load('../data/merrajune2020/v500800hPa.npy'),1, 2)*3.6 nt,nx,ny = u.shape

set domain on which ftle will be computed

dx = 0.2 dy = 0.2 lonf = np.arange(-100, 35+dx, dx) latf = np.arange(-5, 45+dy, dy)

set integration span and integration direction

day = 16 t0_date = np.datetime64("2020-06-{:02d}".format(day)) t0 = t[np.nonzero(dates == t0_date)[0][0]] T = -72.0 params = np.array([copysign(1, T)])

get interpolant arrays of velocity field

gridvel, Cevalu, Cevalv = getinterparrays2D(t, lon, lat, u, v)

retrieve flow and set spherical = 1 since flow is on spherical domain

and lon is from [-180, 180)

funcptr = getflow2D(gridvel, Cevalu, Ceval_v, spherical=1)

compute final position of particle trajectories over grid

flowmap = flowmapgrid2D(funcptr, t0, T, lonf, latf, params)

compute FTLE over grid

ftle = ftlegrid2D(flowmap, T, dx, dy)

plot coastlines and FTLE

coastlines = np.load('../data/merrajune2020/coastlines.npy') fig,ax = plt.subplots(dpi=200) ax.scatter( coastlines[:,0], coastlines[:,1], 1, 'k', marker='.', edgecolors=None, linewidths=0 ) ax.contourf(lonf, latf, ftle.T, levels=80, zorder=0) ax.setxlim([lonf[0], lonf[-1]]) ax.setylim([latf[0], latf[-1]]) ax.setaspect('equal') plt.show() ```

Key dependencies

NumbaCS is built on top of three main packages: Numba, numbalsoda, and interpolation. Numba is a JIT-compiler for Python array and numerical functions that generates optimized machine code just-in-time (using the LLVM compiler library) to significantly speed up numerical operations in Python. Numbalsoda is a Python wrapper to ODE solvers in both C++ (LSODA) and FORTRAN (DOP853) that is compatible with Numba (standard Python ODE solvers cannot be used within Numba functions) and bypasses the Python interpreter, speeding up the most computationally expensive piece of Lagrangian coherent structure methods, particle integration. The interpolation package is a Numba-compatible package which optimizes interpolation in Python. It is used in NumbaCS to generate JIT-compiled interpolant functions of numerical velocity fields which can be fed into solvers from the numbalsoda package, resulting in efficient particle integration for flows defined by numerical velocity data. All of this interfacing, which is done behind the scenes through modules the user can call in a straightforward manner, is how NumbaCS achieves impressive speeds while maintaining a relatively simple user experience. We are grateful to the creators, maintainers, and contributors of each of these packages, as well as the other dependencies which NumbaCS relies on (NumPy, SciPy, and ContourPy).

Roadmap

Future releases aim to extend certain methods to higher dimensions, implement new features that should be straightforward within this framework (shape coherent sets, lobe dynamics, etc.), and further streamline and optimize the process for large-scale geophysical applications.

Contributing

Pull requests are welcome. For major changes, please open an issue first to discuss what you would like to change.

Please make sure to update tests as appropriate. See the Contributing guide for more details.

Similar software

This section lists similar packages, their functionality, what ODE solvers are available (and what language they are implemented in), and the available interpolation routines. For performance comparisons of some packages on core functionality, see the Benchmarks repository.

Lagrangian -- Python package for computing FSLE, FTLE, and eigenvectors of Cauchy-Green tensor with a focus on geophysical flows. Only works with NetCDF files for velocity data. Largely written in C++ with pybind11 used for binding, resulting in fast runtimes. - Features: FTLE, FSLE, Cauchy Green eigenvectors - Integration: RK4 (C++) - Interpolation: Linear

Dynlab -- Object oriented Python package which computes Lagrangian and Eulerian diagnostics along with ridge extraction. Provides a large collection of predefined flows and is very user friendly. - Features: FTLE, iLE, Trajectory repulsion rate, FTLE ridge extraction, iLE ridge extraction - Integration: LSODA (Python) - Interpolation: N/A

TBarrier -- Collection of Jupyter notebooks accompanying the book Transport Barriers and Coherent Structures in Flow Data -- Advective, Diffusive, Stochastic, and Active methods by George Haller. Python code which implements a wide variety of Lagrangian and Eulerian diagnostics and extraction methods for a variety of different transport settings, in both 2 and 3 dimensions (NumbaCS currently only implements purely advective methods in 2D). - Features (both 2D and 3D): FTLE, iLE, variational hyperbolic LCS and OECS, variational elliptic LCS and OECS, variational parabolic LCS, active hyperbolic LCS and OECS, active elliptic LCS and OECS, DBS, diffusive and stochastic elliptic LCS and OECS - Integration: RK4 (Python) - Interpolation: Linear in time, cubic in space

Newman -- Fast C++ code for computing FTLE which works with geophysical flows and storm tracking. Velocity data must be raw binary or ASCII format. No longer maintained. - Features (both 2D and 3D): FTLE, Cauchy Green eigenvectors - Integration: RK45, RK4, Euler (C++) - Interpolation: Linear

Aquila-LCS -- Python code designed to compute FTLE for high-speed turbulent boundary layers in 3D. Utilizes Numba to implement GPU and CPU versions of the code for fast runtimes. - Features (both 2D and 3D): FTLE, FSLE - Integration: Euler (Python/Numba) - Interpolation: Linear

CoherentStructures.jl -- Julia toolbox for computing LCS/FTCS in aperiodic flows. Implements elliptic LCS methods, FEM-based methods (FEM approximation of dynamic Laplacian for FTCS extraction), and Graph Laplacian-based methods (spectral clustering and diffusion maps for coherent sets). - Features: FTLE, finite time coherent sets (via both FEM approximation of dynamic Laplacian and graph Laplacian-based methods), variational elliptic LCS and OECS, diffusive and stochastic elliptic LCS and OECS - Integration: DifferentialEquations.jl, a very advanced and efficient suite of DE solvers (Julia) - Interpolation: Linear, cubic, B-Spline

LCS Tool -- MATLAB code used to compute elliptic LCS, hyperbolic LCS, and FTLE. - Features: FTLE, variational hyperbolic LCS, variational elliptic LCS - Integration: ode45 - based off of RK5(4) due to Dormand and Prince (MATLAB) - Interpolation: Linear, cubic

LCS MATLAB Kit -- MATLAB GUI for computing FTLE from a time series of 2D velocity data. Has FTLE implementation for intertial particles as well (iFTLE). - Features: FTLE, iFTLE - Integration: Version 1.0 -- RK4 (MATLAB), Version 2.3 -- Euler (MATLAB) - Interpolation Version 1.0 -- Cubic, Version 2.3 -- Linear

NumbaCS -- Numba accelerated Python package which efficiently computes a variety of coherent structure methods. - Features: FTLE, iLE, FTLE ridge extraction, variational hyperbolic LCS and OECS, LAVD-based elliptic LCS, IVD-based elliptic OECS, flow map composition - Integration: DOP853 (FORTRAN), LSODA (C++) - Interpolation: Linear, cubic

Acknowledgments

This work was partially supported by the National Science Foundation (NSF) under grant number 1821145 and the National Aeronautics and Space Administration (NASA) under grant number 80NSSC20K1532 issued through the Interdisciplinary Research in Earth Science (IDS) and Biological Diversity & Ecological Conservation programs.