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Internal Tide Test Case

The internal tide test case simulates tidal flow over a Gaussian seamount in a stratified ocean. This configuration isolates the role of time discretization, as spatial advection plays a secondary role in this mostly linear regime.

Configuration

The domain spans 2000 km horizontally (periodic) and 2 km vertically (bounded), with a Gaussian seamount of height 250 m and width 20 km. The domain is initially stratified with N² = 10⁻⁴ s⁻² and forced by an oscillatory tidal forcing.

Parameters

using TimestepperTestCases

# Get default parameters
params = internal_tide_parameters()

Key parameters:

  • Domain: 2000 km × 2 km
  • Grid: 256 × 128 points
  • Seamount: 250 m height, 20 km width
  • Tidal period: 12.421 hours (M₂ tide)
  • Initial stratification: N² = 10⁻⁴ s⁻²
  • Time step: 5 minutes (AB2) or 10 minutes (RK3)
  • Simulation duration: 40 days

Running the Simulation

using TimestepperTestCases

# Run with RK3 and split-explicit free surface
sim = internal_tide(:SplitRungeKutta3)

# Or with implicit free surface
sim = internal_tide(:SplitRungeKutta3; 
                    free_surface=ImplicitFreeSurface(internal_tide_grid()))

# Run with AB2 for comparison
sim = internal_tide(:QuasiAdamsBashforth2)

Output Fields

The simulation outputs:

  • u, w: Horizontal and vertical velocities
  • b, c: Buoyancy and passive tracer
  • η: Free surface elevation
  • Δtc², Δtb²: Tracer variance dissipation rates
  • Gbx, Gbz, Gcx, Gcz: Gradient squared fields

Expected Results

As shown in the paper, RK3 schemes retain more energy than AB2:

  • Sharper velocity structures (u′, w)
  • Better preserved stratification (N²)
  • Lower numerical diffusivity throughout the water column
  • Higher kinetic energy and available potential energy

Loading and Analyzing Results

# Load results
case = load_internal_tide("output_folder/", "SplitRungeKutta3", "split_free_surface_DFB")

# Access time series
case[:KE]   # Kinetic energy
case[:RPE]  # Reference potential energy (irreversible mixing)
case[:APE]  # Available potential energy
case[:abx]  # Volume-averaged x-direction dissipation
case[:abz]  # Volume-averaged z-direction dissipation
case[:κb]   # Effective numerical diffusivity

Physical Interpretation

This test case demonstrates that temporal discretization alone can introduce significant numerical mixing, with AB2 showing diffusivity comparable to low-order spatial schemes. The RK3 formulation substantially reduces this mixing, preserving wave energy and stratification over the 40-day integration.