Soil heat conduction at global scale

Here we extend the single column soil heat conduction example to do global scale simulations, accelerated by GPU (if available).

using Terrarium

using CUDA
using Dates
using Rasters, NCDatasets
using NumericalEarth.DataWrangling
using NumericalEarth.SoilGrids
using Statistics

using CairoMakie, GeoMakie

import RingGrids

[ Info: Current working directory: /home/runner/work/Terrarium.jl/Terrarium.jl/docs/src/examples

First we check if GPU is available and choose the architecture correspondingly.

arch = CUDA.functional() ? GPU() : CPU()

[ Info: Setting up simulation on CPU()

Next, we load a land-sea mask at ~1° resolution. The mask is a full Gaussian grid, as defined by the FullGaussianGrid from RingGrids.jl. Irrespective of the architecture used for simulation, the land-sea mask is kept on the CPU for easy scalar indexing, which is by default not allowed for GPU arrays (see here).

NF = Float32
land_sea_frac_native = RingGrids.Field(arch, ERA5LandInvariants(), "lsm"; NF)
ring_grid = on_architecture(arch, RingGrids.FullGaussianGrid(72))
land_sea_frac_N72 = RingGrids.interpolate(ring_grid, land_sea_frac_native)
fig = heatmap(land_sea_frac_N72)

Then we set up a masked ColumnRingGrid, selecting only grid points with >50% land cover:

land_mask = land_sea_frac_N72 .> 0.5
land_mask_cpu = on_architecture(CPU(), land_mask)
grid = ColumnRingGrid(arch, NF, ExponentialSpacing(N = 30), land_mask.grid, land_mask)
grid_lon, grid_lat = RingGrids.get_lonlats(grid.rings) # in radians, on CPU
grid_latd = rad2deg.(grid_lat) # latitude in degrees, as expected by `LatitudinalClimatology`

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Remember from the documentation section on grids, that the x-axis of the Oceananigans RectilinearGrid corresponds to a single index following the ring order (for more details, see the corresponding section in the RingGrids.jl documentation).

To make the simulation a bit more interesting, we will use spatially periodic initial and boundary conditions. The climatology will be determined by latitude with a maximum of 20 °C at the equator and minimum of -20°C at the poles, as provided by LatitudinalClimatology:

climatology = LatitudinalClimatology(NF)

fig = heatmap(RingGrids.Field(climatology.(grid_latd), grid.rings))

The initial temperature profiles are linear in depth, as given by QuasiThermalSteadyState. Its surface temperature T₀ accepts spatially varying values, so we pass the climatology directly; with a geothermal heat flux of 0.05 W/m² and a bulk thermal conductivity of 1 W/m/K, this yields a geothermal gradient of 0.05 K/m. We leave the (unused) hydrology at its default (dry) state. Now we create our SoilModel with this initializer:

energy_initializer = QuasiThermalSteadyState(NF; T₀ = climatology, Qgeo = NF(0.05))
initializer = SoilInitializer(NF; energy = energy_initializer, hydrology = DefaultInitializer(NF))
soil = SoilEnergyWaterCarbon(NF; hydrology = SoilHydrology(NF, NoFlow()))
model = SoilModel(grid; soil, initializer)

SoilModel{Float32} on CPU()
├── grid:  ColumnRingGrid with dimensions (14018, 1, 30)
├── soil:  SoilEnergyWaterCarbon{Float32, SoilStratigraphy{Float32, 1, Tuple{ConstantSoilHorizon{Float32, :soil, ConstantSoilPorosity{Float32}}}}, SoilThermodynamics{Float32, SoilEnergyTemperatureClosure, SoilThermalProperties{Float32, FreeWater, InverseQuadratic, SoilThermalConductivities{Float32}}}, SoilHydrology{Float32, NoFlow, SoilSaturationPressureClosure, SoilHydraulicsSURFEX{Float32, BrooksCorey{FreezeCurves.SoilWaterVolume{Float32, Float32, Float32}, Float32, Float32}, UnsatKBrooksCorey{Float32}}, Nothing}, ConstantSoilCarbonDensity{Float32}}
├── constants:  PhysicalConstants{Float32}
├── initializer:  SoilInitializer{Float32, QuasiThermalSteadyState{Float32, LatitudinalClimatology{Float32}, Float32}, DefaultInitializer{Float32}, DefaultInitializer{Float32}}
├── timestepper:  ForwardEuler{Float32}

We will impose a periodic temperature boundary condition at the surface to represent the daily cycle. We can specify it directly as a continuous function thanks to the power of Oceananigans Fields. However, we will need to use an enclosing function here to i) copy the vector of latitudes onto the device specified by arch, and ii) ensure that the compiler is able to infer the correct type of the coordinate values in the boundary condition function periodic_bc, which returns a temperature value based on the coordinate x of the RectiLinearGrid and the time t (s). The climatology struct is captured by the closure as well; it is a plain isbits value, so this is safe on the GPU.

function get_temperature_bc(lon::AbstractVector, latd::AbstractVector, amplitude = 10.0)
    # make sure coordinate arrays are on the same device
    lon_device = on_architecture(arch, NF.(lon))
    lat_device = on_architecture(arch, NF.(latd))
    # function matching the expected signature for boundary conditions on a column-based grid
    function periodic_bc(x::NF, t::NF) where {NF}
        # x coordinate is just the grid cell index
        lonₓ = lon_device[round(Int, x)]
        latₓ = lat_device[round(Int, x)]
        # use climatology at latₓ (degrees) as the mean of BC
        T₀ = climatology(latₓ)
        seconds_per_day = NF(24 * 3600)
        # shift BC by longitude in radians to (roughly) mimic the global daily cycle
        T = T₀ + NF(amplitude) * sin(2π * t / seconds_per_day - lonₓ)
        return T
    end
    return periodic_bc
end


lon_masked = grid_lon[land_mask_cpu]
latd_masked = grid_latd[land_mask_cpu] # mask out non-land points
bc = PrescribedSurfaceTemperature(:T_ub, get_temperature_bc(lon_masked, latd_masked))

(temperature = (top = ValueBoundaryCondition: ContinuousBoundaryFunction periodic_bc at (Nothing, Nothing, Nothing),),)

We are finally ready to initialize our model with the above boundary conditions:

integrator = initialize(model, boundary_conditions = bc)

Integrator of SoilModel{Float32, ColumnRingGrid{Float32, CPU, RingGrids.FullGaussianGrid{SpeedyWeatherInternals.Architectures.CPU{KernelAbstractions.CPU}, Vector{UnitRange{Int64}}, Vector{Int64}, Int64}, Oceananigans.Grids.RectilinearGrid{Float32, Oceananigans.Grids.Periodic, Oceananigans.Grids.Flat, Oceananigans.Grids.Bounded, Oceananigans.Grids.StaticVerticalDiscretization{Oceananigans.Utils.StaticStridesOffsetArray{Float32, 1, (37,), (-3,), Vector{Float32}}, Oceananigans.Utils.StaticStridesOffsetArray{Float32, 1, (36,), (-3,), Vector{Float32}}, Oceananigans.Utils.StaticStridesOffsetArray{Float32, 1, (37,), (-4,), Vector{Float32}}, Oceananigans.Utils.StaticStridesOffsetArray{Float32, 1, (36,), (-3,), Vector{Float32}}}, Float32, Float32, Oceananigans.Utils.StaticStridesOffsetArray{Float32, 1, (14024,), (-3,), StepRangeLen{Float32, Float64, Float64, Int64}}, Oceananigans.Utils.StaticStridesOffsetArray{Float32, 1, (14024,), (-3,), StepRangeLen{Float32, Float64, Float64, Int64}}, Float32, Float32, Nothing, Nothing, CPU, Oceananigans.Grids.GridSize{14018, 1, 30, 3, 0, 3}}, RingGrids.Field{Bool, 1, Vector{Bool}, RingGrids.FullGaussianGrid{SpeedyWeatherInternals.Architectures.CPU{KernelAbstractions.CPU}, Vector{UnitRange{Int64}}, Vector{Int64}, Int64}, SpeedyWeatherInternals.ArrayDimensions.XY}}, SoilEnergyWaterCarbon{Float32, SoilStratigraphy{Float32, 1, Tuple{ConstantSoilHorizon{Float32, :soil, ConstantSoilPorosity{Float32}}}}, SoilThermodynamics{Float32, SoilEnergyTemperatureClosure, SoilThermalProperties{Float32, FreeWater, InverseQuadratic, SoilThermalConductivities{Float32}}}, SoilHydrology{Float32, NoFlow, SoilSaturationPressureClosure, SoilHydraulicsSURFEX{Float32, BrooksCorey{FreezeCurves.SoilWaterVolume{Float32, Float32, Float32}, Float32, Float32}, UnsatKBrooksCorey{Float32}}, Nothing}, ConstantSoilCarbonDensity{Float32}}, SoilInitializer{Float32, QuasiThermalSteadyState{Float32, LatitudinalClimatology{Float32}, Float32}, DefaultInitializer{Float32}, DefaultInitializer{Float32}}, ForwardEuler{Float32}} with timestepper ForwardEuler{Float32}(300.0f0)
├── Current time: 0.0
├── StateVariables{Float32}(clock = Clock{Float32, Float64}(time=0 seconds, iteration=0, last_Δt=Inf days), prognostic = (:internal_energy,), auxiliary = (:temperature, :liquid_water_fraction, :ground_temperature, :saturation_water_ice, :water_table, :hydraulic_conductivity), inputs = (:initial_surface_temperature, :geothermal_heat_flux), namespaces = (:soil,), timestepper_cache = EmptyCache)

Let's already plot the initial surface temperature state to see what it looks like:

T_surface_initial = RingGrids.Field(arch, interior(integrator.state.ground_temperature), grid)
fig = heatmap(T_surface_initial[:, 1, 1], title = "Temperature of uppermost soil layer", colorrange = (-20, 20))

We will do a quick check, advancing the simulation by one timestep with Δt = 15 minutes (900 seconds)

timestep!(integrator)
@time timestep!(integrator)

  0.181149 seconds (182.24 k allocations: 2.781 MiB)

...then run the simulation for 12 hours to see the temperature change!

@time run!(integrator, period = Hour(12), Δt = 600.0)

Integrator of SoilModel{Float32, ColumnRingGrid{Float32, CPU, RingGrids.FullGaussianGrid{SpeedyWeatherInternals.Architectures.CPU{KernelAbstractions.CPU}, Vector{UnitRange{Int64}}, Vector{Int64}, Int64}, Oceananigans.Grids.RectilinearGrid{Float32, Oceananigans.Grids.Periodic, Oceananigans.Grids.Flat, Oceananigans.Grids.Bounded, Oceananigans.Grids.StaticVerticalDiscretization{Oceananigans.Utils.StaticStridesOffsetArray{Float32, 1, (37,), (-3,), Vector{Float32}}, Oceananigans.Utils.StaticStridesOffsetArray{Float32, 1, (36,), (-3,), Vector{Float32}}, Oceananigans.Utils.StaticStridesOffsetArray{Float32, 1, (37,), (-4,), Vector{Float32}}, Oceananigans.Utils.StaticStridesOffsetArray{Float32, 1, (36,), (-3,), Vector{Float32}}}, Float32, Float32, Oceananigans.Utils.StaticStridesOffsetArray{Float32, 1, (14024,), (-3,), StepRangeLen{Float32, Float64, Float64, Int64}}, Oceananigans.Utils.StaticStridesOffsetArray{Float32, 1, (14024,), (-3,), StepRangeLen{Float32, Float64, Float64, Int64}}, Float32, Float32, Nothing, Nothing, CPU, Oceananigans.Grids.GridSize{14018, 1, 30, 3, 0, 3}}, RingGrids.Field{Bool, 1, Vector{Bool}, RingGrids.FullGaussianGrid{SpeedyWeatherInternals.Architectures.CPU{KernelAbstractions.CPU}, Vector{UnitRange{Int64}}, Vector{Int64}, Int64}, SpeedyWeatherInternals.ArrayDimensions.XY}}, SoilEnergyWaterCarbon{Float32, SoilStratigraphy{Float32, 1, Tuple{ConstantSoilHorizon{Float32, :soil, ConstantSoilPorosity{Float32}}}}, SoilThermodynamics{Float32, SoilEnergyTemperatureClosure, SoilThermalProperties{Float32, FreeWater, InverseQuadratic, SoilThermalConductivities{Float32}}}, SoilHydrology{Float32, NoFlow, SoilSaturationPressureClosure, SoilHydraulicsSURFEX{Float32, BrooksCorey{FreezeCurves.SoilWaterVolume{Float32, Float32, Float32}, Float32, Float32}, UnsatKBrooksCorey{Float32}}, Nothing}, ConstantSoilCarbonDensity{Float32}}, SoilInitializer{Float32, QuasiThermalSteadyState{Float32, LatitudinalClimatology{Float32}, Float32}, DefaultInitializer{Float32}, DefaultInitializer{Float32}}, ForwardEuler{Float32}} with timestepper ForwardEuler{Float32}(300.0f0)
├── Current time: 43800.0
├── StateVariables{Float32}(clock = Clock{Float32, Float64}(time=12.167 hours, iteration=74, last_Δt=10 minutes), prognostic = (:internal_energy,), auxiliary = (:temperature, :liquid_water_fraction, :ground_temperature, :saturation_water_ice, :water_table, :hydraulic_conductivity), inputs = (:initial_surface_temperature, :geothermal_heat_flux), namespaces = (:soil,), timestepper_cache = EmptyCache)

Now we can plot the resulting soil (surface) temperature map using RingGrids + GeoMakie:

T_surface = RingGrids.Field(arch, interior(integrator.state.ground_temperature), grid)
fig = heatmap(T_surface[:, 1, 1], title = "Temperature of uppermost soil layer", colorrange = (-20, 20))

We can also wrap the integrator in an Oceananigans Simulation which can be used to add callbacks and save outputs.

using Oceananigans.OutputWriters: JLD2Writer, AveragedTimeInterval
using Oceananigans.Units: days

sim = Simulation(integrator; Δt = 600.0, stop_time = 2days)
run!(sim)

[ Info: Initializing simulation...
[ Info:     ... simulation initialization complete (217.593 ms)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (295.701 ms).
[ Info: Simulation is stopping after running for 39.709 seconds.
[ Info: Simulation time 2 days equals or exceeds stop time 2 days.

Julia version and environment information

This example was executed with the following version of Julia:

using InteractiveUtils: versioninfo
versioninfo()

Julia Version 1.13.4
Commit c1ac32ab733 (2026-11-13 17:30 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-20.1.8 (ORCJIT, znver3)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_DEBUG = Documenter

These were the top-level packages installed in the environment:

import Pkg
Pkg.status()

Status `~/work/Terrarium.jl/Terrarium.jl/docs/Project.toml`
  [c7e460c6] ArgParse v1.2.0
⌃ [052768ef] CUDA v6.4.2
  [13f3f980] CairoMakie v0.15.17
  [0b91fe84] DisplayAs v0.1.6
  [e30172f5] Documenter v1.20.0
  [daee34ce] DocumenterCitations v1.5.0
  [d12716ef] DocumenterInterLinks v1.1.0
  [db073c08] GeoMakie v0.7.18
  [033835bb] JLD2 v0.6.7
  [0f8b85d8] JSON3 v1.14.4 [deprecated]
⌅ [63c18a36] KernelAbstractions v0.9.47
  [98b081ad] Literate v2.21.1
  [ee78f7c6] Makie v0.24.17
  [85f8d34a] NCDatasets v0.14.15
  [904d977b] NumericalEarth v0.9.3
⌅ [9e8cae18] Oceananigans v0.117.2
  [a3a2b9e3] Rasters v0.15.0
  [d1845624] RingGrids v1.0.2
  [80418d68] Terrarium v0.1.11-DEV `~/work/Terrarium.jl/Terrarium.jl`
  [b77e0a4c] InteractiveUtils v1.11.0
  [d6f4376e] Markdown v1.11.0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`
Info Packages marked with [deprecated] are no longer maintained. Use `status --deprecated` to see more information.

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