README.md

Shallow Water Equations Dataset

Numerical solutions to the viscous shallow water equations on a sphere using spectral methods.

Equations

The dataset simulates the viscous shallow water equations on a rotating sphere:

Momentum equation:

∂u/∂t + ν∇⁴u + g∇h + 2Ω × u = -u·∇u

Continuity equation:

∂h/∂t + ν∇⁴h + H∇·u = -∇·(hu)

Where:

• u is the velocity field (2-component vector)
• h is the height perturbation field
• ν is kinematic viscosity
• g is gravitational acceleration
• Ω is the Earth's rotation vector
• H is the mean fluid height

Variables

The dataset returns a dictionary with the following fields:

Coordinates

• spatial_coordinates: (Nphi×Ntheta, 2) - Flattened (phi, theta) coordinate pairs
• phi_coords: (Nphi, Ntheta) - Longitude grid
• theta_coords: (Nphi, Ntheta) - Colatitude grid
• lat_coords: (Nphi, Ntheta) - Latitude grid
• time_coordinates: (time_steps,) - Time points in hours

Initial Conditions

• u_initial: (2, Nphi, Ntheta) - Initial velocity field
• h_initial: (Nphi, Ntheta) - Initial height perturbation
• vorticity_initial: (Nphi, Ntheta) - Initial vorticity field

Solution Trajectories

• u_trajectory: (time_steps, 2, Nphi, Ntheta) - Velocity field evolution
• h_trajectory: (time_steps, Nphi, Ntheta) - Height field evolution
• vorticity_trajectory: (time_steps, Nphi, Ntheta) - Vorticity field evolution

Physical Parameters

• alpha: Perturbation width parameter
• beta: Perturbation latitude scale parameter
• R: Earth radius (6.37122×10⁶ m)
• Omega: Earth rotation rate (7.292×10⁻⁵ rad/s)
• nu: Kinematic viscosity
• g: Gravitational acceleration (9.80616 m/s²)
• H: Mean fluid height (10⁴ m)

Grid Parameters

• Nphi: Number of longitude points
• Ntheta: Number of latitude points

Dataset Parameters

• Domain: Spherical surface (φ ∈ [0, 2π], θ ∈ [0, π])
• Default grid: 256×128 (longitude × latitude)
• Time range: 0 to 360 hours (default)
• Spatial resolution: Spectral (dealiased at 3/2 factor)
• Temporal resolution: 600 seconds (10 minutes)
• Save interval: Every 5 hours (default)

Physical Parameters

• Earth radius: R = 6.37122×10⁶ m
• Rotation rate: Ω = 7.292×10⁻⁵ rad/s
• Viscosity: ν = 1×10⁵ m²/s / 32² (Reynolds number dependent)
• Gravity: g = 9.80616 m/s²
• Mean height: H = 10⁴ m
• Jet maximum velocity: 80 m/s
• Perturbation amplitude: 120 m

Random Parameters (per sample)

• Alpha range: [0.05, 0.6] - Controls perturbation width in longitude
• Beta range: [0.02, 0.3] - Controls perturbation width in latitude

Physical Context

This dataset simulates atmospheric or oceanic flows on a rotating sphere governed by the shallow water equations. The system models large-scale geophysical fluid dynamics, including:

• Zonal jet streams with realistic velocity profiles
• Geostrophic balance between pressure gradients and Coriolis forces
• Rossby waves and other planetary-scale motions
• Vorticity dynamics in rotating reference frames

Each sample begins with a balanced zonal jet (resembling atmospheric jet streams) perturbed by a localized height anomaly. The evolution showcases:

• Jet instability and wave breaking
• Vortex formation and interaction
• Energy cascades in geophysical turbulence
• Conservation of potential vorticity

This dataset is relevant for:

• Atmospheric and oceanic modeling
• Climate dynamics
• Geophysical fluid dynamics research
• Weather prediction algorithms
• Machine learning for Earth system modeling

Usage

from dataset import ShallowWaterDataset

# Create dataset
dataset = ShallowWaterDataset(
    Nphi=256,           # Longitude points
    Ntheta=128,         # Latitude points
    stop_sim_time=360,  # Hours
    save_interval=5     # Save every 5 hours
)

# Generate a sample
sample = next(iter(dataset))

# Access solution data
u_evolution = sample["u_trajectory"]      # Velocity field
h_evolution = sample["h_trajectory"]      # Height field
vorticity = sample["vorticity_trajectory"] # Vorticity field
times = sample["time_coordinates"]        # Time points
alpha, beta = sample["alpha"], sample["beta"] # Parameters

Visualization

Run the plotting scripts to visualize samples:

python plot_sample.py      # 4-panel static visualization on sphere
python plot_animation.py   # Animated vorticity evolution

The plotting functions create 3D spherical visualizations showing the evolution of vorticity fields over time, with proper Earth-like pole orientation.

Data Generation

Generate the full dataset:

python generate_data.py

This creates train/test splits saved as chunked parquet files in the data/ directory. The generation uses smaller default parameters (128×64 grid, 100-hour simulations) for efficiency while maintaining the same physics.

File Structure

datasets-shallow-water-dedalus/
├── dataset.py              # Main dataset class
├── plot_sample.py           # 4-panel spherical visualization
├── plot_animation.py        # Animated GIF generation
├── generate_data.py         # Dataset generation script
├── requirements.txt         # Python dependencies
├── README.md               # This file
└── sample_plot.png         # Example visualization