shallowwater 0.1.1

Shallow-water model on a C-grid over a beta-plane (educational)

shallowwater

A minimal shallow-water equations solver on an Arakawa C-grid over a (f, β)-plane, with Rayleigh friction and SSP-RK3 time stepping.

• Flat bottom, square basin, closed walls (no-normal-flow).
• Linear or Momentum advection (vector-invariant form).
• Staggering: η[j,i] → (Ny, Nx); u[j,i] → (Ny, Nx+1); v[j,i] → (Ny+1, Nx).
• Forcing at the surface via wind stress (τx on u, τy on v), mass source Q (at η), and optional tidal geopotential φ (at η) such that momentum sees η_total = η + φ/g.

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Install (editable)

cd shallowwater
python -m venv .venv && source .venv/bin/activate  # or use conda/uv/mamba
pip install -e .

Run the example notebook

Launch Jupyter from the same environment:

jupyter notebook notebooks/01_wind_gyre.ipynb

Batch-running notebooks

The repo includes a helper script run_notebooks.sh to execute notebooks in place (outputs are written back into the .ipynb files using jupyter nbconvert).

Make it executable:

chmod +x run_notebooks.sh

Run a single notebook:

./run_notebooks.sh -n notebooks/01_wind_gyre.ipynb

Run all notebooks in the current directory:

./run_notebooks.sh -a

Common options:

• -t SECONDS — cell timeout (default: 3600)
• -k NAME — kernel name (e.g. python3)
• -x GLOB — exclude pattern (repeatable), e.g. -x "*WIP*.ipynb"
• -c — continue on errors (maps to --allow-errors)

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What’s inside (model at a glance)

Equations (linearized, flat bottom) At η-points (centers):

• Continuity: (\eta_t = -\nabla\cdot \mathbf{F} + Q), with (\mathbf{F} = (H u, H v)).

At u/v points (faces/edges):

• Zonal momentum: (u_t = +f v - g,\partial_x(\eta+\phi/g) + \tau_x/(\rho H) - r,u).
• Meridional momentum: (v_t = -f u - g,\partial_y(\eta+\phi/g) + \tau_y/(\rho H) - r,v).

Optionally, momentum advection and laplacian viscosity can be added.

Here (f=f_0+\beta (y-y_0)). φ is the surface geopotential perturbation (e.g., for tides or pressure loading). When φ is omitted, the code assumes φ=0.

Time stepping

• Strong-stability-preserving RK3; use compute_dt_cfl(...) for a safe CFL step based on (c=\sqrt{gH}).

Boundary conditions

• Rectangular closed basin with no-normal-flow on walls (C-grid-friendly masking/ghosting).

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Source layout

src/
  shallowwater/
    __init__.py            # exports core API
    initial.py             # initial states (e.g., gaussian bump); 
                           # also: geostrophic_velocities_from_eta(η, grid, params, ...)
    forcing.py             # wind/pressure/mass forcings (wind gyre, tides via φ, storms, etc.)
    visualize.py           # animate_eta, coast_hovmoller, plot_forcings, animate_eta_spectrum
    ...                    # (operators/dynamics/integrator are used internally)

Top-level:

pyproject.toml
README.md
run_notebooks.sh
notebooks/   # study cases (see list below)
tests/

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Public API (Python)

Shapes use the C-grid staggering noted above.

• ModelParams(H, g, rho, f0, beta, y0, r, linear=True)

• make_grid(Nx, Ny, Lx, Ly) Returns a grid object with dx, dy, Lx, Ly and C-grid coordinates (x_c, y_c, x_u, y_u, x_v, y_v).

• compute_dt_cfl(grid, params, cfl=0.5)

• setup_initial_state(grid, params, mode="rest" | "gaussian_bump", **kwargs) -> (eta, u, v)

• geostrophic_velocities_from_eta(eta, grid, params, *, degree_of_balance=1.0, alpha=1.0, sponge=0, fmin=1e-6) -> (u, v) Compute (u, v) that are geostrophic w.r.t. a given η. Useful to start from a balanced (or partially balanced) state.

• Forcings (return conventions):

  • 3-tuple (taux_u, tauy_v, Q_eta) — wind stress on u/v, mass source on η.
  • 4-tuple (taux_u, tauy_v, Q_eta, phi_eta) — same, plus geopotential φ at η (for tides/pressure). Provided helpers include:
  • zero_forcing(...)
  • wind_gyre_forcing(t, grid, params, tau0=...)
  • tidal_potential_forcing(t, grid, params, amp_eta_eq=..., omega=..., kx=..., ky=...) → via φ
  • stommel_arons_forcing(t, grid, params, Q0=..., R=..., time_ramp=...)
  • storm_surge_forcing(t, grid, params, Vmax=..., delta_p=..., ...) → wind + inverse barometer (φ)
  • coastal_alongshore_wind_forcing(t, grid, params, coast=..., direction=..., ...)

• tendencies(state, t, grid, params, forcing_fn, hooks=None) -> (deta_dt, du_dt, dv_dt) (internally uses either 3- or 4-tuple forcing; φ is optional).

• run_model(tmax, dt, grid, params, forcing_fn, ic_fn, save_every=10, out_vars=('eta','u','v'), hooks=None)

• Visualization (shallowwater.visualize):

  • animate_eta(out, grid, remove_mean=True, cmap="RdBu_r", contours=False, frames=None, ...)
  • coast_hovmoller(out, grid, units_x="km")
  • plot_forcings(forcing_fn, t, grid, params, what="all"|"wind"|["taux","|tau|","Q"], ...)
  • animate_eta_spectrum(out, grid, quadrant="full"|"ur", log10=True, ...)

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Typical workflow

from shallowwater import (ModelParams, make_grid, setup_initial_state,
                          compute_dt_cfl, run_model, zero_forcing)

Nx, Ny = 128, 128
Lx, Ly = 2.0e6, 2.0e6
grid = make_grid(Nx, Ny, Lx, Ly)
params = ModelParams(H=1000.0, g=9.81, rho=1025.0, f0=1e-4, beta=2e-11, y0=Ly/2, r=1/(10*86400), linear=True)

dt = compute_dt_cfl(grid, params, cfl=0.5)
ic_fn = lambda g, p: setup_initial_state(g, p, mode="gaussian_bump", amp=0.1, R=2e5)
forcing_fn = lambda t, g, p: zero_forcing(t, g, p)

out = run_model(tmax=5*86400, dt=dt, grid=grid, params=params,
                forcing_fn=forcing_fn, ic_fn=ic_fn,
                save_every=24, out_vars=("eta",))

To start near geostrophic balance:

from shallowwater.initial import geostrophic_velocities_from_eta
def ic_balanced(g, p):
    eta, _, _ = setup_initial_state(g, p, mode="gaussian_bump", amp=0.5, R=2e5)
    u, v = geostrophic_velocities_from_eta(eta, g, p, degree_of_balance=0.9, alpha=0.95, sponge=6)
    return eta, u, v

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Study cases (notebooks)

All study cases live in notebooks/ (animated previews eta_*.gif are included):

1. 01_wind_gyre.ipynb — classic β-plane wind-driven gyre (Sverdrup interior + western boundary current).
2. 02_gravity_waves.ipynb — linear gravity/Poincaré waves from a perturbed surface.
3. 03_tsunami.ipynb — unforced propagation from a localized uplift (deep water; no rotation/friction).
4. 04_tides.ipynb — equilibrium-tide forcing via variable geopotential φ (M2-like).
5. 05_abyssal_flow.ipynb — Stommel–Arons-like source (NE) + uniform sink (zero net Q) on a β-plane.
6. 06_seiche.ipynb — standing basin modes (m,n) with no forcing (clean f-plane seiche).
7. 07_equatorial_waves.ipynb — equatorial Kelvin/Rossby packets (β-plane, equator at mid-domain).
8. 08_storm_surge.ipynb — moving cyclone: wind + inverse barometer (φ) over a shallow shelf.
9. 09_wind_driven_kelvin_wave.ipynb — alongshore wind band builds coastal setup; after shutoff a coastal Kelvin wave propagates.
10. 10_geostrophic_adjustment.ipynb — Gaussian dome adjusts on an f-plane; start from rest or from a partially geostrophic state using geostrophic_velocities_from_eta.
11. 11_Rossby_wave_propagation.ipynb — Gaussian dome adjusts on an beta-plane; start from geostrophic state using geostrophic_velocities_from_eta.

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Notes & conventions

• Units: SI throughout. Typical H in meters, τ in N m⁻², Q in m s⁻¹, φ in m² s⁻², space in meters, time in seconds.

• C-grid shapes:

  • eta: (Ny, Nx) at cell centers (x_c, y_c)
  • u: (Ny, Nx+1) staggered in x at (x_u, y_u)
  • v: (Ny+1, Nx) staggered in y at (x_v, y_v)

• Forcing return: 3-tuple (τx_u, τy_v, Q) or 4-tuple (τx_u, τy_v, Q, φ). If you don’t need φ, omit it.

• CFL: external celerity (c=\sqrt{gH}). compute_dt_cfl picks a safe step using min(dx,dy).

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Testing

A tests/ folder is provided as a placeholder; feel free to contribute simple regression checks (e.g., energy decay with Rayleigh friction, tide period checks, or Kelvin phase speed comparisons).

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License

MIT.