Instabilities in the shallow-water system with a semi-Lagrangian, time-centered discretization

** Figure generating scripts **

Christopher Subich, July 2021

This repository contains the scripts necessary to regenerate the figures and data used in the above-titled paper, submitted to Monthly Weather Review in March 2021.

Basic execution:

To generate all data and figures, run:

# Generate data for growth-rate figures (about 72h on a desktop system)
generate_data # produces saved data files

# Generate figures
mkdir paper_figs
plot_paper

The ancillary files are:

• cubic_instab.m, fourier_instab.m, linear_instab.m -- calculates maximal growth rates of the shallow-water system over a wide range of Courant numbers and topographic wavelengths; parameters are modifiable in their respective preambles.
• fd_ops.m -- Helper function to generate finite-difference operators on the set of staggered grids
• gen_ic.m -- Helper function to generate a background steady state corresponding to a specific (and provided) $u_0$, $\Delta t$, $\phi_0$, and hill profile, via iteration of the linearized steady-state system
• get_linearized_op.m -- Helper function to provide the linearized shallow-water operators for a given background state
• interesting_spectra.m -- Collection of "interesting" points in Courant number / topographic wavenumber space, selected by the CASE variable
• interp_cubic.m, interp_fourier.m, interp_linear.m -- helper functions to compute the semi-Lagrangian operators for a given set of trajectories, using cubic, Fourier (trigonometric), or linear interpolation respectively.
• label_eigvecs.m -- Helper script to associate the eigenvectors of the advection or timestepping operator with a dominant physical mode
• plot_growthrate.m -- Helper script to give a contour plot of maximum growth rate over varying Courant number and hill wavenumbers, using the data produced by *_instab.m
• plot_offcentering_graph.m -- Separate script (called by paper_figs.m) to calculate and graph growth rates over varying hill amplitudes and off-centering parameters
• plot_paper.m -- Main plotting script, which generates all figures in the submitted paper
• semilag_traj.m -- Helper function to compute (iteratively) semi-Lagrangain departure points corresponding to a given velocity field
• show_spectrum.m -- Helper script to calculate and plot unstable modes in a particular realization of the shallow water operator
• test_with_offcentering.m -- Helper script to define an off-centered shallow water operator and compute its modes
• time_convergence.m -- Script to generate a sample stable/unstable comparison, with reference solution (ode45 and Fourier spectral method), from identical initial conditions
• timestep.m -- script to evaluate a shallow water case over time, to observe any instability growing