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Equilibrated Channel Flow

The channel simulation is an idealized re-entrant periodic channel forced by wind stress and surface heat flux. Unlike the transient configurations, this case relaxes toward statistical equilibrium, allowing assessment of how time-stepping schemes affect the equilibrated solution and how numerical mixing interacts with parameterized physical diffusivity.

Configuration

The domain is 1000 km × 2000 km × 3 km with periodic zonal boundaries and bounded meridional boundaries. The channel is forced by sinusoidal wind stress and variable surface heat flux, with buoyancy restored at the northern boundary.

Parameters

using TimestepperTestCases

# Run with default settings
sim = channel_simulation(timestepper=:SplitRungeKutta3)

# Customize closure
sim = channel_simulation(
    timestepper=:SplitRungeKutta3,
    closure=default_closure(),
    zstar=true
)

# Use custom grid
sim = channel_simulation(
    timestepper=:SplitRungeKutta3,
    grid=custom_grid
)

Key parameters:

  • Domain: 1000 km × 2000 km × 3 km
  • Grid: 200 × 400 × 90 points (non-uniform vertical spacing)
  • Time step: 5 minutes (AB2), 10 minutes (RK3), or 20 minutes (RK6)
  • Simulation duration: 40 years (last 5 years averaged)
  • Closure: CATKE with background diffusivity κ = 10⁻⁵ m²/s, ν = 10⁻⁴ m²/s
  • Wind stress: Sinusoidal profile τ = -0.1/ρ * sin(π * y / Ly)
  • Heat flux: Variable profile Qᵇ * cos(3π * y / Ly) for y < 5/6 * Ly

Running the Simulation

using TimestepperTestCases

# Standard run
sim = channel_simulation(timestepper=:SplitRungeKutta3)

# Restart from checkpoint
sim = channel_simulation(
    timestepper=:SplitRungeKutta3,
    restart_file="channel_checkpoint_0.jld2"
)

Output Fields

The simulation outputs:

  • u, v, w: Velocity components
  • b: Buoyancy
  • η: Free surface elevation
  • Abx, Aby, Abz: Advective buoyancy dissipation
  • Gbx, Gby, Gbz: Buoyancy gradient squared
  • Snapshots: Every 360 days
  • Averages: 5-year averages

Expected Results

As shown in the paper:

  • RK3-SE achieves κ_phys / κ_num > 2 throughout most of the water column
  • AB2-SE hovers around κ_phys / κ_num ≈ 1
  • RK3-UP (with low-order advection) shows κ_phys / κ_num < 1
  • RK3-SE produces cooler equilibrated solutions (less spurious mixing)
  • Horizontal numerical diffusivity dominates over vertical

Loading and Analyzing Results

# Load channel results
case = load_channel("output_folder/", "0"; arch=CPU())

# Access diagnostics
case[:KE]   # Kinetic energy time series
case[:MKE]  # Mean kinetic energy
case[:RPE]  # Reference potential energy
case[:APE]  # Available potential energy
case[:η2]   # Free surface variance

Physical Interpretation

This test case demonstrates the critical importance of maintaining κ_phys / κ_num > 1 for long-term climate integrations. When numerical mixing overwhelms physical mixing (as in RK3-UP or near-surface AB2-SE), the equilibrated solution is significantly warmer and less stratified. RK3-SE maintains physical processes as dominant throughout most of the water column, which is essential for preserving stratification and mesoscale variability in climate applications.