Sea Ice Timestepping

ClimaSeaIce.jl supports two timestepping schemes for evolving the sea ice state: Forward Euler and Split Runge-Kutta 3rd order. The default is Split Runge-Kutta 3rd order.

Overview

Sea ice timestepping in ClimaSeaIce separates the physics into three distinct steps that are performed sequentially within each substep:

1. Dynamic step: Advection of ice thickness and concentration
2. Thermodynamic step: Column physics (melting and freezing)
3. Momentum step: Evolution of ice velocities

This operator-splitting approach allows each physical process to be treated with an appropriate numerical method.

Forward Euler Timestepper

The ForwardEuler is the simplest timestepping scheme advances the state via:

\[U^{n+1} = U^n + Δt \, G^n\]

where $U^n$ is the state at timestep $n$ and $G^n$ is the tendency.

To use ForwardEuler, use the timestepper keyword argument when constructing the SeaIceModel:

using ClimaSeaIce
using Oceananigans

grid = RectilinearGrid(size=(16, 16, 1), extent=(1, 1, 1))
model = SeaIceModel(grid; timestepper = :ForwardEuler)

# output
SeaIceModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 16×16×1 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×1 halo
├── timestepper: ForwardEulerTimeStepper
├── ice_thermodynamics: SlabThermodynamics
├── advection: Nothing
└── external_heat_fluxes:
    ├── top: Int64
    └── bottom: Int64

Forward Euler is first-order accurate and may require smaller time steps for stability.

Split Runge-Kutta Timestepper (Default)

The Split Runge-Kutta 3rd order (SplitRungeKutta3) scheme is a 3rd-order accurate method that performs three substeps per full time step. This is the default timestepper and is recommended for most applications.

model = SeaIceModel(grid; timestepper = :SplitRungeKutta3)  # default

# output
SeaIceModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 16×16×1 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×1 halo
├── timestepper: SplitRungeKuttaTimeStepper(3)
├── ice_thermodynamics: SlabThermodynamics
├── advection: Nothing
└── external_heat_fluxes:
    ├── top: Int64
    └── bottom: Int64

Algorithm

The SplitRungeKutta3 scheme follows Oceananigans' implementation. At the beginning of each full time step, the current state is cached via cache_current_fields!. Then, three substeps are performed via rk_substep!, each with a different time increment and weighting.

Each substep performs the following operations in sequence:

1. Compute advective tendencies for ice thickness $h$ and concentration $ℵ$
2. Update $h$ and $ℵ$ via dynamic_time_step!
3. Apply thermodynamic fluxes (melting/freezing)
4. Update ice momentum (implicit or split-explicit)

Advantages

• Third-order temporal accuracy
• Larger stable time steps compared to Forward Euler
• Consistent with Oceananigans' ocean model timestepping

Choosing a Timestepper

For coupled ocean-sea ice simulations with ClimaOcean.jl, the SplitRungeKutta3 timestepper is recommended as it matches the ocean model's default timestepping scheme.