Inertia-gravity waves

This example simulates the propagation of inertia-gravity waves in a stably stratified atmosphere, following the classical benchmark test case described by Skamarock and Klemp (1994). This test evaluates the accuracy of numerical pressure solvers by introducing a small-amplitude temperature perturbation into a stratified environment with constant Brunt-Väisälä frequency, triggering propagating inertia-gravity waves.

The test case is particularly useful for validating anelastic and compressible solvers, as discussed at the CM1 inertia-gravity wave test page.

Physical setup

The background state is a stably stratified atmosphere with constant Brunt-Väisälä frequency $N$, which gives a potential temperature profile

\[θ^{\rm bg}(z) = θ_0 \exp\left( \frac{N^2 z}{g} \right)\]

where $θ_0 = 300 \, {\rm K}$ is the surface potential temperature and $g$ is the gravitational acceleration.

The initial perturbation is a localized temperature anomaly centered at $x = x_0$:

\[θ'(x, z) = Δθ \frac{\sin(π z / L_z)}{1 + (x - x_0)^2 / a^2}\]

with amplitude $Δθ = 0.01 \, {\rm K}$, half-width parameter $a = 5000 \, {\rm m}$, and perturbation center $x_0 = L_x / 3$. A uniform mean wind $U = 20 \, {\rm m \, s^{-1}}$ advects the waves.

Comparison of dynamical formulations

We compare four dynamical formulations:

1. Anelastic: Filters acoustic waves via the anelastic approximation
2. Boussinesq: Anelastic with constant reference density
3. Compressible (explicit): Fully compressible with explicit time stepping (small Δt)
4. Compressible (split-explicit / substepper): Wicker–Skamarock RK3 with acoustic substepping — uses the advective $Δt$ for the outer step and absorbs acoustic modes in an inner loop with an auto-computed substep count.

using Breezeusing Oceananigans: Oceananigansusing Oceananigans.Unitsusing Printfusing CairoMakie

Problem parameters

We define the thermodynamic base state and mean wind following Skamarock and Klemp (1994):

p₀ = 100000  # Pa - surface pressureθ₀ = 300     # K - reference potential temperatureU  = 20      # m s⁻¹ - mean windN² = 0.01^2  # s⁻² - Brunt-Väisälä frequency squared

0.0001

Grid configuration

The domain is 300 km × 10 km with 300 × 10 grid points, matching the nonhydrostatic case configuration in the paper by Skamarock and Klemp (1994).

Nx, Nz = 300, 10Lx, Lz = 300kilometers, 10kilometersgrid = RectilinearGrid(CPU(), size=(Nx, Nz), halo=(5, 5),                       x=(0, Lx), z=(0, Lz),                       topology=(Periodic, Flat, Bounded))

300×1×10 RectilinearGrid{Float64, Periodic, Flat, Bounded} on CPU with 5×0×5 halo
├── Periodic x ∈ [0.0, 300000.0) regularly spaced with Δx=1000.0
├── Flat y                       
└── Bounded  z ∈ [0.0, 10000.0]  regularly spaced with Δz=1000.0

Initial conditions

The perturbation parameters from the paper by Skamarock and Klemp (1994):

Δθ = 0.01    # K - perturbation amplitudea  = 5000    # m - perturbation half-width parameterx₀ = Lx / 3  # m - perturbation center in xconstants = ThermodynamicConstants()g = constants.gravitational_accelerationpˢᵗ = 1e5θᵇᵍ(z) = θ₀ * exp(N² * z / g)θᵢ(x, z) = θᵇᵍ(z) + Δθ * sin(π * z / Lz) / (1 + (x - x₀)^2 / a^2)ρᵢ(x, z) = adiabatic_hydrostatic_density(z, p₀, θ₀, pˢᵗ, constants)

ρᵢ (generic function with 1 method)

Build all four models

advection = WENO()base_pressure = p₀potential_temperature = θ₀

300

Case 1: Anelastic

reference_state = ReferenceState(grid, constants; base_pressure, potential_temperature)anelastic_dynamics = AnelasticDynamics(reference_state)

AnelasticDynamics(p₀=100000.0, θ₀=300.0)
├── pressure_solver_float_type: Float64
└── pressure_anomaly: not materialized

Case 2: Boussinesq (constant reference density)

constant_density_reference_state = ReferenceState(grid, constants; base_pressure, potential_temperature)ρ₀ = adiabatic_hydrostatic_density(0, p₀, θ₀, pˢᵗ, constants)set!(constant_density_reference_state.density, ρ₀)boussinesq_dynamics = AnelasticDynamics(constant_density_reference_state)

AnelasticDynamics(p₀=100000.0, θ₀=300.0)
├── pressure_solver_float_type: Float64
└── pressure_anomaly: not materialized

Case 3: Compressible (fully explicit, no substepping)

compressible_dynamics = CompressibleDynamics(ExplicitTimeStepping();                                             base_pressure,                                             reference_potential_temperature = θᵇᵍ)

CompressibleDynamics{ExplicitTimeStepping}
├── dry_density: not materialized
├── pressure: not materialized
├── terrain_metrics: Breeze.TerrainFollowingDiscretization.SlopeOutsideInterpolation
├── time_discretization: ExplicitTimeStepping
└── reference_state: θᵇᵍ (generic function with 1 method)

Case 4: Compressible split-explicit (substepper)

Wicker–Skamarock RK3 acoustic substepping with the default ThermalDivergenceDamping (coefficient $0.1$) and an auto-computed substep count chosen from the horizontal acoustic CFL each step.

substepper_dynamics = CompressibleDynamics(SplitExplicitTimeDiscretization();                                           base_pressure,                                           reference_potential_temperature = θᵇᵍ)

CompressibleDynamics{SplitExplicitTimeDiscretization}
├── dry_density: not materialized
├── pressure: not materialized
├── terrain_metrics: Breeze.TerrainFollowingDiscretization.SlopeOutsideInterpolation
├── time_discretization: SplitExplicitTimeDiscretization
└── reference_state: θᵇᵍ (generic function with 1 method)

Build all models:

models = Dict(    :anelastic    => AtmosphereModel(grid; advection, dynamics=anelastic_dynamics),    :boussinesq   => AtmosphereModel(grid; advection, dynamics=boussinesq_dynamics),    :compressible => AtmosphereModel(grid; advection, dynamics=compressible_dynamics),    :substepper   => AtmosphereModel(grid; advection, dynamics=substepper_dynamics),)

Set initial conditions:

for name in (:anelastic, :boussinesq)    set!(models[name]; θ=θᵢ, u=U)endfor name in (:compressible, :substepper)    ref = models[name].dynamics.reference_state    set!(models[name]; θ=θᵢ, u=U, qᵗ=0, ρ=ref.density)end

Time stepping constraints

The anelastic and split-explicit models use the advective CFL, while the fully explicit compressible model is limited by the sound speed.

Δx, Δz = Lx / Nx, Lz / NzRᵈ = dry_air_gas_constant(constants)cᵖᵈ = constants.dry_air.heat_capacitycᵃᶜ = sqrt(cᵖᵈ / (cᵖᵈ - Rᵈ) * Rᵈ * θ₀)cfl = 0.5Δt_advective    = cfl * min(Δx, Δz) / UΔt_compressible = cfl * min(Δx, Δz) / (cᵃᶜ + U)time_steps = Dict(    :anelastic    => Δt_advective,    :boussinesq   => Δt_advective,    :compressible => Δt_compressible,    :substepper   => Δt_advective,)@info "Time steps" Δt_advective Δt_compressible

┌ Info: Time steps
│   Δt_advective = 25.0
└   Δt_compressible = 1.3618178767657314

Run all simulations

stop_time = 3000 # secondscase_names = Dict(    :anelastic    => "Anelastic",    :boussinesq   => "Boussinesq",    :compressible => "Compressible (explicit)",    :substepper   => "Compressible (substepper)",)

Dict{Symbol, String} with 4 entries:
  :compressible => "Compressible (explicit)"
  :boussinesq => "Boussinesq"
  :substepper => "Compressible (substepper)"
  :anelastic => "Anelastic"

Background θ field for computing perturbation

θᵇᵍ_field = CenterField(grid)set!(θᵇᵍ_field, (x, z) -> θᵇᵍ(z))simulations = Dict{Symbol, Simulation}()for (key, model) in models    Δt = time_steps[key]    sim = Simulation(model; Δt, stop_time, verbose=false)    Oceananigans.Diagnostics.erroring_NaNChecker!(sim)    # Progress callback    θ′ = PotentialTemperature(model) - θᵇᵍ_field    name = case_names[key]    function progress(sim)        w = sim.model.velocities.w        @info @sprintf("%s - Iter: %4d, t: %s, max(θ′): %.4e, max|w|: %.4f",                       name, iteration(sim), prettytime(sim), maximum(θ′), maximum(abs, w))        return nothing    end    callback_interval = key === :compressible ? IterationInterval(500) : IterationInterval(100)    add_callback!(sim, progress, callback_interval)    # Output    outputs = merge(model.velocities, (; θ′))    sim.output_writers[:jld2] = JLD2Writer(model, outputs;                                           filename = "igw_$(key).jld2",                                           schedule = TimeInterval(100),                                           overwrite_files = true)    simulations[key] = simendfor (key, sim) in simulations    @info "Running $(case_names[key])..."    run!(sim)end

[ Info: Running Compressible (explicit)...
[ Info: Compressible (explicit) - Iter:    0, t: 0 seconds, max(θ′): 9.7791e-03, max|w|: 0.0000
[ Info: Compressible (explicit) - Iter:  500, t: 11.271 minutes, max(θ′): 3.9608e-03, max|w|: 0.0062
[ Info: Compressible (explicit) - Iter: 1000, t: 22.529 minutes, max(θ′): 3.3873e-03, max|w|: 0.0046
[ Info: Compressible (explicit) - Iter: 1500, t: 33.787 minutes, max(θ′): 3.0845e-03, max|w|: 0.0036
[ Info: Compressible (explicit) - Iter: 2000, t: 45.045 minutes, max(θ′): 2.8737e-03, max|w|: 0.0028
[ Info: Running Boussinesq...
[ Info: Boussinesq - Iter:    0, t: 0 seconds, max(θ′): 9.7791e-03, max|w|: 0.0000
[ Info: Boussinesq - Iter:  100, t: 41.667 minutes, max(θ′): 2.8429e-03, max|w|: 0.0028
[ Info: Running Compressible (substepper)...
[ Info: Compressible (substepper) - Iter:    0, t: 0 seconds, max(θ′): 9.7791e-03, max|w|: 0.0000
[ Info: Compressible (substepper) - Iter:  100, t: 41.667 minutes, max(θ′): 2.9126e-03, max|w|: 0.0029
[ Info: Running Anelastic...
[ Info: Anelastic - Iter:    0, t: 0 seconds, max(θ′): 9.7791e-03, max|w|: 0.0000
[ Info: Anelastic - Iter:  100, t: 41.667 minutes, max(θ′): 2.9627e-03, max|w|: 0.0028

Results

We compare the potential temperature perturbation at the final time for all five formulations. This comparison reveals how well each formulation captures inertia-gravity wave propagation.

cases = [:anelastic, :boussinesq, :compressible, :substepper]titles = [case_names[k] for k in cases]θ′ts = Dict(k => FieldTimeSeries("igw_$(k).jld2", "θ′") for k in cases)times = θ′ts[:anelastic].timesNt = length(times)x_km = range(0, Lx / 1000, length=Nx)z_km = range(0, Lz / 1000, length=Nz)levels = range(-Δθ / 2, stop=Δθ / 2, length=21)θ′_final = Dict(k => interior(θ′ts[k][Nt], :, 1, :) for k in cases)

Dict{Symbol, SubArray{Float64, 2, Array{Float64, 4}, Tuple{UnitRange{Int64}, Int64, UnitRange{Int64}, Int64}, false}} with 4 entries:
  :compressible => [1.30866e-5 4.00924e-5 6.55113e-5 8.61838e-5 9.96009e-5 0.000103811 9.77279e-5 8.11857e-5 5.50126e-5 2.02753e-5; 1.29036e-5 3.95093e-5 6.4479e-5 8.46525e-5 9.75562e-5 0.000101327 9.50179e-5 7.86258e-5 5.31033e-5 1.95311e-5; 1.27639e-5 3.90545e-5 6.36401e-5 8.33416e-5 9.57142e-5 9.89893e-5 9.23795e-5 7.6072e-5 5.11676e-5 1.87698e-5; 1.26599e-5 3.87057e-5 6.29631e-5 8.22204e-5 9.40567e-5 9.68012e-5 8.98395e-5 7.3566e-5 4.92451e-5 1.80086e-5; 1.25844e-5 3.84424e-5 6.24213e-5 8.12683e-5 9.25829e-5 9.47907e-5 8.74547e-5 7.118e-5 4.73992e-5 1.72741e-5; 1.25313e-5 3.82477e-5 6.19952e-5 8.04769e-5 9.13095e-5 9.30081e-5 8.53064e-5 6.90091e-5 4.57105e-5 1.66002e-5; 1.24952e-5 3.81088e-5 6.16729e-5 7.98489e-5 9.02673e-5 9.15197e-5 8.34912e-5 6.71614e-5 4.42684e-5 1.60234e-5; 1.2472e-5 3.80159e-5 6.14471e-5 7.93918e-5 8.94907e-5 9.03937e-5 8.2106e-5 6.57434e-5 4.31584e-5 1.55788e-5; 1.24586e-5 3.79619e-5 6.13125e-5 7.91119e-5 8.90073e-5 8.96851e-5 8.12284e-5 6.48405e-5 4.24497e-5 1.52944e-5; 1.24529e-5 3.79419e-5 6.12652e-5 7.90122e-5 8.88324e-5 8.94252e-5 8.09034e-5 6.45032e-5 4.21839e-5 1.51876e-5; 1.24543e-5 3.79545e-5 6.13036e-5 7.90931e-5 8.8969e-5 8.96197e-5 8.11388e-5 6.47415e-5 4.23704e-5 1.5263e-5; 1.24637e-5 3.80018e-5 6.14298e-5 7.93532e-5 8.94087e-5 9.0251e-5 8.19093e-5 6.55276e-5 4.2987e-5 1.55118e-5; 1.24835e-5 3.80898e-5 6.16482e-5 7.97883e-5 9.01321e-5 9.12807e-5 8.31607e-5 6.68008e-5 4.39838e-5 1.59132e-5; 1.25175e-5 3.82267e-5 6.19648e-5 8.0391e-5 9.11092e-5 9.26513e-5 8.48128e-5 6.84727e-5 4.52882e-5 1.64372e-5; 1.25698e-5 3.84225e-5 6.23864e-5 8.1152e-5 9.23027e-5 9.42916e-5 8.6766e-5 7.04339e-5 4.68113e-5 1.70471e-5; 1.2645e-5 3.86881e-5 6.29216e-5 8.20636e-5 9.36751e-5 9.61268e-5 8.89142e-5 7.25678e-5 4.84581e-5 1.7704e-5; 1.27479e-5 3.90354e-5 6.35817e-5 8.31235e-5 9.51966e-5 9.80915e-5 9.11612e-5 7.47668e-5 5.014e-5 1.83715e-5; 1.28832e-5 3.94774e-5 6.43815e-5 8.43371e-5 9.68514e-5 0.000100141 9.34358e-5 7.69474e-5 5.17863e-5 1.902e-5; 1.30554e-5 4.0027e-5 6.53386e-5 8.57183e-5 9.86403e-5 0.000102257 9.56995e-5 7.90595e-5 5.33522e-5 1.96308e-5; 1.32686e-5 4.06964e-5 6.64708e-5 8.72866e-5 0.000100579 0.000104447 9.79482e-5 8.10903e-5 5.48235e-5 2.01973e-5; 1.35256e-5 4.14948e-5 6.77937e-5 8.90641e-5 0.000102695 0.000106741 0.000100212 8.30658e-5 5.62188e-5 2.07269e-5; 1.38277e-5 4.24269e-5 6.93188e-5 9.10735e-5 0.000105027 0.000109192 0.000102554 8.50493e-5 5.75876e-5 2.12393e-5; 1.41744e-5 4.34932e-5 7.10539e-5 9.33385e-5 0.00010762 0.000111873 0.000105066 8.71362e-5 5.90048e-5 2.17647e-5; 1.45641e-5 4.46922e-5 7.30059e-5 9.5885e-5 0.00011053 0.000114871 0.000107862 8.94443e-5 6.05634e-5 2.23406e-5; 1.49954e-5 4.60232e-5 7.51832e-5 9.87427e-5 0.000113819 0.000118286 0.000111067 9.21046e-5 6.23676e-5 2.30087e-5; 1.54687e-5 4.7489e-5 7.75989e-5 0.000101946 0.000117554 0.000122217 0.000114811 9.52526e-5 6.4525e-5 2.38124e-5; 1.59856e-5 4.9097e-5 8.02722e-5 0.000105536 0.000121804 0.00012677 0.000119223 9.90215e-5 6.71406e-5 2.4794e-5; 1.65503e-5 5.08615e-5 8.3232e-5 0.000109563 0.000126648 0.000132048 0.000124425 0.000103535 7.03099e-5 2.59913e-5; 1.71703e-5 5.28063e-5 8.65206e-5 0.000114089 0.000132169 0.000138154 0.000130532 0.0001089 7.4113e-5 2.74357e-5; 1.78573e-5 5.4968e-5 9.0198e-5 0.000119195 0.000138462 0.000145193 0.000137644 0.000115204 7.86112e-5 2.91504e-5; 1.86282e-5 5.73969e-5 9.43437e-5 0.00012498 0.000145635 0.000153267 0.000145852 0.000122515 8.38464e-5 3.11499e-5; 1.95037e-5 6.01557e-5 9.9054e-5 0.000131559 0.000153804 0.000162474 0.000155223 0.000130872 8.98365e-5 3.3439e-5; 2.05073e-5 6.33143e-5 0.000104435 0.000139054 0.000163082 0.000172899 0.000165803 0.000140285 9.65713e-5 3.60102e-5; 2.16635e-5 6.69451e-5 0.000110596 0.000147587 0.000173576 0.00018461 0.000177615 0.000150736 0.000104019 3.88469e-5; 2.29951e-5 7.11157e-5 0.000117636 0.000157268 0.000185379 0.000197664 0.000190665 0.000162196 0.000112138 4.19298e-5; 2.4521e-5 7.58819e-5 0.000125637 0.000168184 0.000198564 0.000212097 0.000204953 0.000174633 0.000120891 4.52404e-5; 2.62548e-5 8.12829e-5 0.000134657 0.000180396 0.000213175 0.000227929 0.000220465 0.000188013 0.000130238 4.87612e-5; 2.82044e-5 8.73418e-5 0.000144727 0.000193939 0.000229241 0.000245171 0.000237191 0.000202306 0.000140149 5.24785e-5; 3.03739e-5 9.40712e-5 0.000155868 0.000208838 0.00024679 0.000263849 0.00025515 0.000217522 0.000150626 5.63919e-5; 3.27653e-5 0.000101479 0.000168097 0.000225122 0.000265867 0.000284022 0.00027441 0.000233726 0.000161717 6.05198e-5; 3.53801e-5 0.00010957 0.000181432 0.000242829 0.000286539 0.000305788 0.000295088 0.000251035 0.000173512 6.48981e-5; 3.82228e-5 0.000118363 0.000195907 0.000262027 0.000308914 0.000329299 0.000317366 0.000269627 0.000186146 6.95794e-5; 4.13073e-5 0.000127902 0.000211611 0.000282855 0.000333188 0.000354804 0.000341526 0.000289776 0.000199824 7.46437e-5; 4.46624e-5 0.000138281 0.000228706 0.00030555 0.000359672 0.000382668 0.000367959 0.000311852 0.000214828 8.02015e-5; 4.83293e-5 0.000149629 0.000247418 0.000330424 0.000388745 0.000413312 0.000397095 0.000336252 0.000231455 8.63715e-5; 5.23549e-5 0.000162096 0.000267992 0.000357804 0.000420781 0.000447121 0.000429302 0.000363298 0.000249944 9.32482e-5; 5.67863e-5 0.000175825 0.000290669 0.000388001 0.000456127 0.000484443 0.000464895 0.000393249 0.000270472 0.000100899; 6.16675e-5 0.000190953 0.000315669 0.000421316 0.000495144 0.000525665 0.000504247 0.000426415 0.000293244 0.000109394; 6.70275e-5 0.000207568 0.000343152 0.000457988 0.000538162 0.000571192 0.000547792 0.000463184 0.000318519 0.000118827; 7.28604e-5 0.000225656 0.000373105 0.000498041 0.000585274 0.000621201 0.000595771 0.000503798 0.000346471 0.000129261; 7.91167e-5 0.000245067 0.00040529 0.000541176 0.000636157 0.000675393 0.000647938 0.000548073 0.000376985 0.000140658; 8.57335e-5 0.000265606 0.000439381 0.000586931 0.00069024 0.000733131 0.000703652 0.000595448 0.000409681 0.00015288; 9.2695e-5 0.000287222 0.000475275 0.000635125 0.000747235 0.00079403 0.000762464 0.000645487 0.000444248 0.000165813; 0.000100073 0.000310135 0.000513321 0.000686184 0.000807584 0.000858484 0.000824672 0.000698393 0.000480807 0.000179497; 0.000107998 0.00033474 0.000554168 0.000740967 0.000872272 0.000927512 0.000891219 0.000754946 0.000519873 0.000194117; 0.000116566 0.000361337 0.000598307 0.00080013 0.00094208 0.00100194 0.000962901 0.000815814 0.00056189 0.000209832; 0.000125768 0.0003899 0.000645689 0.000863606 0.00101693 0.00108168 0.00103962 0.000880909 0.000606782 0.000226609; 0.000135507 0.000420118 0.000695796 0.000930685 0.00109597 0.00116579 0.00112048 0.00094944 0.000653997 0.000244244; 0.000145678 0.000451671 0.000748092 0.00100064 0.00117833 0.00125335 0.00120456 0.00102062 0.000703002 0.000262541; 0.000156252 0.000484469 0.000802433 0.00107329 0.00126381 0.00134414 0.00129169 0.00109433 0.000753717 0.000281473; 0.000167272 0.00051865 0.000859056 0.00114898 0.00135285 0.00143867 0.00138239 0.00117103 0.000806475 0.000301163; 0.000178778 0.000554336 0.000918172 0.001228 0.0014458 0.00153737 0.00147708 0.00125111 0.000861554 0.000321716; 0.000190728 0.0005914 0.000979572 0.00131008 0.00154236 0.0016399 0.00157544 0.00133431 0.000918771 0.000343066; 0.000202986 0.000629421 0.00104256 0.00139429 0.00164141 0.00174509 0.00167635 0.00141965 0.000977481 0.000364974; 0.000215377 0.000667858 0.00110625 0.00147945 0.00174158 0.0018515 0.00177843 0.00150602 0.00103692 0.000387162; 0.000227755 0.000706266 0.00116991 0.0015646 0.00184179 0.001958 0.00188067 0.00159256 0.00109651 0.000409421; 0.000240034 0.000744374 0.00123309 0.00164917 0.00194139 0.00206396 0.00198249 0.00167882 0.00115593 0.000431623; 0.000252158 0.000782001 0.0012955 0.00173275 0.00203992 0.00216887 0.00208339 0.00176436 0.00121487 0.000453641; 0.000264045 0.000818893 0.00135669 0.00181474 0.00213662 0.00227186 0.0021825 0.0018484 0.00127277 0.000475266; 0.000275553 0.000854605 0.00141592 0.0018941 0.0022302 0.00237151 0.00227837 0.00192969 0.00132876 0.000496174; 0.000286472 0.000888485 0.00147209 0.00196934 0.00231888 0.00246586 0.00236909 0.00200659 0.00138173 0.000515953; 0.000296547 0.000919744 0.00152391 0.0020387 0.00240058 0.00255274 0.00245257 0.00207734 0.00143046 0.000534154; 0.000305512 0.000947556 0.00157 0.00210038 0.0024732 0.00262993 0.00252671 0.00214016 0.00147375 0.000550326; 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3.72145e-5 4.07287e-5 6.79205e-5 7.99689e-5 8.31129e-5 8.07731e-5 6.56068e-5 4.57637e-5 3.08895e-5 -1.59118e-5; 2.47407e-5 2.38203e-6 7.04113e-7 -6.82663e-6 -2.34574e-5 -3.26467e-5 -4.57457e-5 -4.71961e-5 -3.52245e-5 -4.04499e-5; 9.06749e-6 -4.59261e-5 -8.46208e-5 -0.000116711 -0.000154267 -0.000170621 -0.000176644 -0.000154623 -0.000111748 -6.90626e-5; -3.35214e-6 -8.38583e-5 -0.000153219 -0.000202802 -0.000253167 -0.000273663 -0.000270677 -0.000229873 -0.000166733 -8.97188e-5; -3.73046e-6 -8.41328e-5 -0.000159523 -0.000205268 -0.000251088 -0.00027019 -0.000261461 -0.000218083 -0.000162554 -8.82621e-5; 1.31933e-5 -3.07028e-5 -7.68115e-5 -8.96074e-5 -0.000110583 -0.00012434 -0.000119042 -9.56333e-5 -8.36496e-5 -5.9074e-5; 4.4024e-5 6.56737e-5 7.57413e-5 0.000114974 0.000130105 0.000121379 0.000113225 0.000100544 4.63266e-5 -1.05862e-5; 7.7174e-5 0.000169551 0.000240146 0.000330728 0.000379536 0.000372307 0.000345265 0.00029441 0.000176461 3.86385e-5; 9.87477e-5 0.00023842 0.000347666 0.000468334 0.00053681 0.000528226 0.000485912 0.000410297 0.000255409 6.96827e-5; 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5.73527e-5 0.000177599 0.000293454 0.000391209 0.000458824 0.000485839 0.000464757 0.000392049 0.000269067 0.000100275; 6.21776e-5 0.00019257 0.000318261 0.000424398 0.000497883 0.00052733 0.000504586 0.000425781 0.000292304 0.000108957; 6.74819e-5 0.000209023 0.000345517 0.000460853 0.000540763 0.000572843 0.00054825 0.00046274 0.000317736 0.000118448; 7.32565e-5 0.000226931 0.000375187 0.000500557 0.000587484 0.000622449 0.000595867 0.000503053 0.000345457 0.000128786; 7.94565e-5 0.00024616 0.000407058 0.00054324 0.000637756 0.000675888 0.000647222 0.000546559 0.000375379 0.000139946; 8.603e-5 0.000266552 0.000440871 0.000588546 0.000691155 0.000732707 0.000701874 0.00059288 0.000407262 0.000151848; 9.29666e-5 0.000288072 0.000476561 0.000636363 0.000747513 0.000792694 0.000759571 0.000641775 0.000440942 0.000164432; 0.000100326 0.000310905 0.000514425 0.000687063 0.000807232 0.000856236 0.000820638 0.000693496 0.000476581 0.000177753; 0.000108217 0.00033538 0.000555003 0.000741364 0.00087115 0.000924206 0.000885902 0.000748742 0.000514641 0.000191974; 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4.78236e-5 0.00014461 0.00023057 0.000292578 0.000322383 0.000317712 0.000281762 0.000220975 0.000142668 5.09891e-5; 1.45734e-5 4.10785e-5 5.80515e-5 6.00918e-5 4.63602e-5 2.1693e-5 -4.93643e-6 -2.35782e-5 -2.66617e-5 -1.24711e-5; -1.77515e-5 -5.95878e-5 -0.000109747 -0.000166161 -0.000222477 -0.0002669 -0.000284728 -0.000262466 -0.000192193 -7.45342e-5; -4.8266e-5 -0.00015466 -0.000268356 -0.000380305 -0.000477365 -0.000541057 -0.000551052 -0.00049026 -0.000350254 -0.000133842; -7.60401e-5 -0.000241274 -0.000413111 -0.000576235 -0.000711308 -0.000793552 -0.000797165 -0.000701395 -0.000497086 -0.000189006; -0.000100134 -0.000316546 -0.000539317 -0.000747826 -0.000917293 -0.00101715 -0.00101631 -0.000890288 -0.000628913 -0.000238628; -0.000119642 -0.000377692 -0.000642452 -0.000889177 -0.00108856 -0.00120485 -0.00120193 -0.00105149 -0.000742037 -0.000281338; -0.00013374 -0.000422178 -0.000718387 -0.000994883 -0.00121888 -0.00135017 -0.00134791 -0.00117991 -0.000832975 -0.000315838; -0.000141734 -0.000447864 -0.000763621 -0.00106033 -0.0013029 -0.0014475 -0.00144891 -0.00127103 -0.000898635 -0.000340975; 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6.77246e-5 0.000189007 0.00025102 0.000221445 0.000100354 -7.51309e-5 -0.000241071 -0.000328779 -0.000290954 -0.000122399; 0.000109498 0.000316685 0.000456446 0.000484636 0.000393662 0.000217705 2.24126e-5 -0.00011882 -0.000153183 -7.23327e-5; 0.000150029 0.000440363 0.000654753 0.000737312 0.000673078 0.000493914 0.000268081 7.46234e-5 -2.75485e-5 -2.69586e-5; 0.000187544 0.000554513 0.000836736 0.000967089 0.000923941 0.000737841 0.000480853 0.000238764 7.71492e-5 1.04362e-5; 0.000220243 0.00065356 0.000993187 0.00116175 0.00113204 0.00093463 0.000646735 0.00036198 0.000153034 3.69361e-5; 0.000246382 0.000732126 0.00111535 0.00130992 0.00128455 0.00107133 0.000753974 0.000434854 0.000193906 5.02823e-5; 0.000264356 0.000785323 0.00119544 0.00140184 0.00137095 0.00113796 0.000794149 0.000451127 0.000195903 4.91225e-5; 0.000272812 0.000809099 0.00122723 0.00143016 0.00138403 0.00112852 0.000763075 0.000408428 0.000157992 3.31903e-5; 0.000270772 0.000800624 0.0012067 0.00139072 0.00132068 0.00104174 0.000661439 0.000308727 8.22285e-5 3.394e-6; 0.000257768 0.000758686 0.0011326 0.00128328 0.00118257 0.000881627 0.000495102 0.000158447 -2.62319e-5 -3.81947e-5; 0.000233966 0.000684054 0.00100702 0.00111206 0.000976595 0.000657618 0.00027506 -3.17387e-5 -0.000159362 -8.84419e-5; 0.000200268 0.000579759 0.000835688 0.000886 0.000714949 0.000384457 1.70381e-5 -0.00024742 -0.000306699 -0.000143332; 0.00015836 0.000451211 0.000628117 0.000618753 0.000414854 8.16288e-5 -0.000259253 -0.000471234 -0.000455966 -0.000198216; 0.000110696 0.000306106 0.000397323 0.000328149 9.77616e-5 -0.000227675 -0.000531275 -0.000683963 -0.000593884 -0.000248126; 6.03849e-5 0.000154075 0.000159174 3.52156e-5 -0.000211943 -0.000518061 -0.000775218 -0.000865909 -0.000707149 -0.000288149; 1.09988e-5 6.05172e-6 -6.8682e-5 -0.000237325 -0.00048877 -0.000763928 -0.000967942 -0.000998568 -0.000783586 -0.000313855; -3.37106e-5 -0.00012659 -0.000268255 -0.000466977 -0.000708395 -0.000941853 -0.00108921 -0.00106646 -0.0008134 -0.000321757; -7.01518e-5 -0.000233084 -0.000422913 -0.000633679 -0.000850149 -0.00103312 -0.00112405 -0.00105907 -0.000790427 -0.000309771; 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1.89799e-5 3.89039e-6 7.78106e-6 7.97218e-7 1.36277e-5 1.27256e-5 1.55e-5 1.63652e-5 1.04488e-5 -2.23011e-5; 8.09087e-6 -2.80513e-5 -5.12295e-5 -8.05647e-5 -8.61951e-5 -9.38476e-5 -9.21906e-5 -7.3048e-5 -5.37846e-5 -4.71975e-5; -9.1402e-7 -5.53584e-5 -0.000103927 -0.00015324 -0.000174797 -0.000186832 -0.00018513 -0.000149649 -0.000108935 -6.85023e-5; -3.57265e-6 -6.40545e-5 -0.000126696 -0.000185098 -0.000213771 -0.000227083 -0.000226092 -0.000183388 -0.000134473 -7.80858e-5; 5.00711e-6 -3.89835e-5 -9.42815e-5 -0.000142182 -0.000163834 -0.000175832 -0.00017992 -0.000145815 -0.000111623 -6.91093e-5; 2.64117e-5 2.49319e-5 1.76967e-6 -1.42012e-5 -1.53007e-5 -2.61355e-5 -4.23417e-5 -3.29518e-5 -3.77287e-5 -4.09827e-5; 5.60743e-5 0.000113965 0.000139099 0.000168028 0.000194377 0.000182088 0.000147764 0.000123723 6.64472e-5 -1.55778e-6; 8.41057e-5 0.000197916 0.000268455 0.000339091 0.000390446 0.000374418 0.000321755 0.000267824 0.000163448 3.54953e-5; 9.96608e-5 0.000243601 0.000335806 0.000428289 0.000494675 0.000476433 0.000413709 0.000344909 0.000216493 5.67283e-5; 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8.48319e-5 0.000178068 0.000129547 4.24032e-5 3.51291e-5 8.43613e-5 8.41211e-5 -8.6726e-6 -0.000107095 -7.46308e-5; 0.000124522 0.000303524 0.000335521 0.00030278 0.00031752 0.000367837 0.000356536 0.000234636 7.39707e-5 -2.72323e-6; 0.000153609 0.000397651 0.000494937 0.000510079 0.000548076 0.000606591 0.00059684 0.000461515 0.000250785 6.91875e-5; 0.000167846 0.000447008 0.000584945 0.000633411 0.000690688 0.000762145 0.000766146 0.000635481 0.000395301 0.000129881; 0.000165762 0.000446623 0.000596112 0.000658312 0.000726124 0.000811184 0.000837945 0.000729377 0.000485527 0.00017036; 0.000148448 0.000399411 0.000531739 0.000586446 0.000652666 0.000747457 0.000801168 0.000728982 0.000508594 0.000185115; 0.000118972 0.000314424 0.000405246 0.000432803 0.000483882 0.000580734 0.000660444 0.000634127 0.000461961 0.000172666; 8.16415e-5 0.000204606 0.000236659 0.000221511 0.000244613 0.000333585 0.000433929 0.00045747 0.000352852 0.000135397; 4.12794e-5 8.46017e-5 4.91649e-5 -1.84011e-5 -3.3514e-5 3.71743e-5 0.000149683 0.000221617 0.000196334 7.88507e-5; 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4.05841e-5 9.15665e-5 5.98827e-5 -6.49394e-5 -0.00024748 -0.000429527 -0.000549947 -0.000560152 -0.000434284 -0.000173363; 8.18255e-5 0.000218921 0.000267284 0.00020276 5.01946e-5 -0.00013634 -0.000292154 -0.000360139 -0.000305931 -0.000127224; 0.000122692 0.000345396 0.000474338 0.000472469 0.000354177 0.000168424 -1.84643e-5 -0.00014307 -0.000164004 -7.56527e-5; 0.000160153 0.000461689 0.000665927 0.000724414 0.000641713 0.000461089 0.000248798 7.24238e-5 -2.11966e-5 -2.33661e-5; 0.000191614 0.000559804 0.000828943 0.000941271 0.000892645 0.00072046 0.00048947 0.000269392 0.00011088 2.53118e-5; 0.000215071 0.000633523 0.000953064 0.00110918 0.00109057 0.000929002 0.000686614 0.000433425 0.000222272 6.66575e-5; 0.000229197 0.000678678 0.00103122 0.00121838 0.00122359 0.00107361 0.000827241 0.000553241 0.000305075 9.76918e-5; 0.000233366 0.00069324 0.00105978 0.00126352 0.00128471 0.0011461 0.000902791 0.0006211 0.000353721 0.000116288; 0.00022762 0.000677235 0.00103847 0.00124362 0.00127195 0.00114336 0.000909388 0.000633074 0.000365181 0.00012125; 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0.00012298 0.000335754 0.000474563 0.000562926 0.000631593 0.000663794 0.000611915 0.000458625 0.000254161 7.39631e-5; 0.000133231 0.000372118 0.000543498 0.000663334 0.000755808 0.000802436 0.000755404 0.000593423 0.000357847 0.000115942; 0.000126628 0.000357299 0.000530933 0.000660605 0.000765952 0.00082721 0.000796008 0.000645808 0.000407089 0.000137956; 0.000104524 0.000294907 0.000441132 0.000557665 0.000662154 0.000734753 0.00072699 0.000607051 0.000394084 0.000136694; 7.05978e-5 0.000195702 0.000290023 0.000372592 0.000461607 0.000538973 0.000557592 0.000481744 0.000320208 0.000112299; 3.00164e-5 7.50931e-5 0.000101369 0.000134112 0.000194162 0.000267368 0.000310365 0.000286107 0.000195138 6.80882e-5; -1.14833e-5 -4.9604e-5 -9.74949e-5 -0.000123601 -0.000103089 -4.39744e-5 1.70078e-5 4.48822e-5 3.48775e-5 9.84996e-6; -4.84302e-5 -0.000161719 -0.000279751 -0.000366105 -0.000391481 -0.000355915 -0.000286552 -0.000212604 -0.000140868 -5.51292e-5; -7.62774e-5 -0.000247253 -0.000422425 -0.000563059 -0.000635792 -0.000631463 -0.00056506 -0.000456569 -0.000311541 -0.000119135; 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0.000157994 0.000453271 0.000651484 0.000700812 0.000600164 0.000394388 0.000159173 -2.46472e-5 -9.95725e-5 -5.51955e-5; 0.000194719 0.00056579 0.000834094 0.000937833 0.000868814 0.000668189 0.000411475 0.00018163 3.90879e-5 -3.89838e-6; 0.000222863 0.000652515 0.000976345 0.0011252 0.00108497 0.000892808 0.000622533 0.000357232 0.000158715 4.06846e-5; 0.000240854 0.00070854 0.00106993 0.00125143 0.00123459 0.00105271 0.000776861 0.000488641 0.000249794 7.49535e-5; 0.000247879 0.000731271 0.0011103 0.00130991 0.00130915 0.00113802 0.000864214 0.000566636 0.000305704 9.63776e-5; 0.00024386 0.000720366 0.00109654 0.00129886 0.00130572 0.00114465 0.000879824 0.00058657 0.000322937 0.000103585; 0.000229383 0.000677529 0.00103115 0.00122095 0.00122656 0.00107407 0.000824208 0.000548251 0.000301049 9.63465e-5; 0.000205595 0.000606201 0.00091948 0.00108272 0.00107854 0.00093262 0.000702609 0.000455513 0.000242379 7.54801e-5; 0.000174083 0.000511179 0.000769151 0.000893802 0.000872193 0.000730634 0.000524179 0.000315557 0.000151614 4.26854e-5; 0.000136737 0.000398213 0.000589394 0.000666016 0.000620749 0.000481421 0.000301017 0.000138145 3.52429e-5 3.43835e-7; 9.5614e-5 0.000273585 0.000390362 0.000412506 0.000339093 0.000200165 4.71343e-5 -6.52617e-5 -9.90379e-5 -4.87033e-5; 5.2817e-5 0.000143723 0.000182496 0.000146875 4.276e-5 -9.71379e-5 -0.000222566 -0.000282364 -0.000242914 -0.000101376; 1.03784e-5 1.48574e-5 -2.40569e-5 -0.000117583 -0.00025297 -0.000394645 -0.000493221 -0.000500821 -0.000388003 -0.00015456; -2.98317e-5 -0.000107274 -0.000219917 -0.000368535 -0.000533851 -0.000677498 -0.000750817 -0.000708942 -0.000526336 -0.000205291; -6.61788e-5 -0.000217649 -0.000396855 -0.000595119 -0.000787277 -0.000932495 -0.000982839 -0.000896245 -0.000650747 -0.000250897; -9.73156e-5 -0.000312121 -0.000548061 -0.000788311 -0.00100274 -0.00114857 -0.00117876 -0.00105388 -0.000755165 -0.000289112; -0.000122212 -0.000387515 -0.000668304 -0.000941156 -0.00117209 -0.00131711 -0.00133033 -0.00117488 -0.000834809 -0.000318149; -0.000140165 -0.000441664 -0.000753998 -0.00104886 -0.00128967 -0.0014321 -0.00143178 -0.00125434 -0.000886286 -0.000336737; -0.000150796 -0.000473389 -0.000803177 -0.00110875 -0.00135232 -0.00149011 -0.00147974 -0.00128939 -0.0009076 -0.000344125; -0.000154027 -0.000482443 -0.000815402 -0.0011202 -0.00135919 -0.00149016 -0.00147323 -0.00127913 -0.000898105 -0.000340066; -0.000150055 -0.000469416 -0.000791609 -0.00108438 -0.00131159 -0.00143355 -0.00141337 -0.00122444 -0.000858381 -0.000324769; -0.000139308 -0.000435619 -0.000733923 -0.00100406 -0.00121265 -0.00132347 -0.00130314 -0.0011278 -0.000790084 -0.000298848; -0.000122407 -0.000382955 -0.000645439 -0.000883271 -0.00106702 -0.00116476 -0.00114708 -0.000992933 -0.000695763 -0.000263249; -0.000100121 -0.000313778 -0.000529999 -0.000727073 -0.000880522 -0.00096347 -0.000950889 -0.000824621 -0.000578661 -0.000219179; -7.33195e-5 -0.000230757 -0.00039197 -0.000541212 -0.000659809 -0.000726555 -0.000721134 -0.000628343 -0.00044252 -0.000168029; -4.29332e-5 -0.000136751 -0.000236027 -0.00033185 -0.000412024 -0.000461489 -0.000464893 -0.000410023 -0.000291388 -0.000111309; -9.91409e-6 -3.46835e-5 -6.69611e-5 -0.000105312 -0.000144503 -0.00017596 -0.000189457 -0.000175769 -0.000129441 -5.05747e-5; 2.4798e-5 7.25566e-5 0.000110495 0.000132151 0.000135492 0.000122413 9.79452e-5 6.83553e-5 3.9174e-5 1.26277e-5; 6.03042e-5 0.000182207 0.000291811 0.000374551 0.000420999 0.000426322 0.000390373 0.000316527 0.00021047 7.68114e-5; 9.57694e-5 0.0002917 0.000472777 0.000616322 0.000705545 0.000728965 0.000681363 0.00056332 0.000380732 0.000140591; 0.000130441 0.000398722 0.000649594 0.000852436 0.000983281 0.0010242 0.000965074 0.000803827 0.000546601 0.000202713; 0.000163658 0.000501242 0.000818934 0.00107849 0.00124908 0.00130663 0.00123638 0.00103375 0.00070513 0.000262078; 0.000194864 0.000597546 0.000977981 0.00129076 0.00149862 0.00157171 0.00149096 0.00124944 0.000853824 0.000317755; 0.000223603 0.000686237 0.00112444 0.00148621 0.00172835 0.00181572 0.00172527 0.00144794 0.000990654 0.000368988; 0.000249526 0.000766237 0.00125655 0.00166251 0.00193556 0.0020358 0.0019366 0.00162697 0.00111406 0.000415195; 0.000272381 0.000836774 0.00137305 0.00181798 0.00211831 0.00222992 0.00212301 0.0017849 0.00122293 0.00045596; 0.000292011 0.000897364 0.00147313 0.00195158 0.00227538 0.0023968 0.0022833 0.00192072 0.00131657 0.000491028; 0.000308343 0.000947784 0.00155644 0.00206282 0.00240623 0.00253588 0.00241694 0.002034 0.00139469 0.000520287; 0.00032138 0.000988045 0.001623 0.00215176 0.0025109 0.00264721 0.00252398 0.00212478 0.00145733 0.000543753; 0.00033119 0.00101836 0.00167315 0.00221884 0.00258996 0.00273139 0.00260501 0.00219357 0.00150482 0.000561556; 0.000337899 0.00103911 0.00170754 0.00226494 0.00264439 0.00278948 0.00266104 0.00224123 0.00153777 0.000573917; 0.000341674 0.00105082 0.00172703 0.00229118 0.00267555 0.00282292 0.00269345 0.00226891 0.00155697 0.000581137; 0.000342721 0.00105412 0.00173264 0.00229895 0.00268505 0.0028334 0.00270388 0.00227801 0.00156339 0.000583573; 0.000341267 0.00104971 0.00172555 0.0022898 0.00267471 0.00282288 0.00269418 0.0022701 0.00155809 0.000581629; 0.00033756 0.00103836 0.00170701 0.00226541 0.0026465 0.00279341 0.00266634 0.00224684 0.00154224 0.000575736; 0.000331855 0.00102084 0.00167831 0.00222749 0.00260243 0.00274714 0.0026224 0.00220997 0.00151702 0.00056634; 0.000324412 0.000997969 0.00164078 0.00217781 0.00254456 0.00268624 0.00256444 0.00216126 0.00148364 0.000553892; 0.000315485 0.000970522 0.0015957 0.00211808 0.0024749 0.00261285 0.00249452 0.00210243 0.0014433 0.00053884; 0.000305323 0.000939266 0.00154435 0.00204998 0.00239543 0.00252906 0.00241462 0.00203516 0.00139715 0.000521615; 0.000294162 0.000904929 0.00148791 0.00197511 0.00230801 0.00243684 0.00232664 0.00196105 0.0013463 0.000502629; 0.000282224 0.000868196 0.00142752 0.00189496 0.0022144 0.00233806 0.00223237 0.00188162 0.00129178 0.000482269; 0.000269715 0.000829701 0.00136421 0.00181094 0.00211623 0.00223444 0.00213345 0.00179826 0.00123454 0.000460895; 0.000256822 0.000790022 0.00129896 0.00172431 0.00201499 0.00212754 0.00203139 0.00171222 0.00117547 0.000438832; 0.000243714 0.000749679 0.0012326 0.0016362 0.00191201 0.0020188 0.00192754 0.00162468 0.00111534 0.000416377; 0.000230542 0.000709135 0.00116591 0.00154763 0.00180848 0.00190945 0.00182311 0.00153663 0.00105487 0.000393789; 0.000217436 0.000668794 0.00109954 0.00145949 0.00170545 0.00180062 0.00171916 0.00144897 0.000994664 0.000371299; 0.00020451 0.000629004 0.00103408 0.00137255 0.00160379 0.00169324 0.00161658 0.00136246 0.000935247 0.000349104; 0.000191859 0.000590059 0.000970004 0.00128744 0.00150428 0.00158811 0.00151615 0.00127776 0.000877067 0.000327371; 0.000179562 0.000552205 0.000907721 0.00120471 0.00140754 0.0014859 0.0014185 0.00119541 0.000820496 0.000306238; 0.000167684 0.000515638 0.000847554 0.00112479 0.00131408 0.00138716 0.00132416 0.00111584 0.000765833 0.000285818; 0.000156274 0.000480513 0.000789759 0.00104801 0.0012243 0.00129229 0.00123351 0.00103939 0.000713314 0.000266198; 0.00014537 0.000446945 0.000734526 0.00097464 0.00113849 0.00120162 0.00114688 0.000966314 0.000663116 0.000247446; 0.000134998 0.000415018 0.000681991 0.000904849 0.00105687 0.00111538 0.00106447 0.000896804 0.000615364 0.000229607; 0.000125176 0.000384781 0.000632238 0.000838753 0.000979575 0.0010337 0.000986426 0.00083097 0.000570137 0.000212712; 0.000115911 0.00035626 0.000585308 0.000776407 0.000906661 0.000956652 0.000912803 0.00076887 0.000527475 0.000196774; 0.000107204 0.000329457 0.000541205 0.000717817 0.000838139 0.000884245 0.000843616 0.00071051 0.000487383 0.000181797; 9.90499e-5 0.000304355 0.000499902 0.000662947 0.000773968 0.000816436 0.000778821 0.000655856 0.000449837 0.000167772; 9.14373e-5 0.000280921 0.000461345 0.000611725 0.000714065 0.000753137 0.000718336 0.000604838 0.000414789 0.00015468; 8.43519e-5 0.000259111 0.000425459 0.000564053 0.000658313 0.000694226 0.000662046 0.000557358 0.000382173 0.000142496; 7.77754e-5 0.000238868 0.000392153 0.000519809 0.000606572 0.000639553 0.000609806 0.000513296 0.000351905 0.00013119; 7.16873e-5 0.000220129 0.000361322 0.000478854 0.000558679 0.000588948 0.000561454 0.000472514 0.000323891 0.000120726; 6.60651e-5 0.000202825 0.000332853 0.000441038 0.000514457 0.000542224 0.000516811 0.000434862 0.000298028 0.000111066; 6.08852e-5 0.000186882 0.000306625 0.0004062 0.00047372 0.000499183 0.000475689 0.000400181 0.000274206 0.000102169; 5.61229e-5 0.000172226 0.000282514 0.000374176 0.000436275 0.000459622 0.000437894 0.000368307 0.000252313 9.39921e-5; 5.17537e-5 0.000158779 0.000260395 0.000344799 0.000401926 0.000423335 0.000403228 0.000339073 0.000232235 8.64936e-5; 4.77525e-5 0.000146467 0.000240143 0.000317902 0.000370479 0.000390115 0.000371494 0.000312313 0.000213857 7.96304e-5; 4.40951e-5 0.000135213 0.000221632 0.000293321 0.000341742 0.000359759 0.000342498 0.000287863 0.000197067 7.33604e-5; 4.07575e-5 0.000124943 0.000204743 0.000270894 0.000315525 0.000332068 0.000316049 0.000265563 0.000181753 6.76423e-5; 3.77166e-5 0.000115588 0.000189357 0.000250465 0.000291646 0.000306848 0.000291963 0.000245257 0.00016781 6.24362e-5; 3.49502e-5 0.000107077 0.000175362 0.000231885 0.00026993 0.000283915 0.000270062 0.000226794 0.000155134 5.77037e-5; 3.24368e-5 9.93459e-5 0.000162651 0.00021501 0.000250209 0.000263091 0.000250178 0.000210033 0.000143627 5.34079e-5; 3.01564e-5 9.23319e-5 0.000151119 0.000199704 0.000232323 0.000244207 0.000232148 0.000194837 0.000133195 4.9514e-5; 2.80898e-5 8.5976e-5 0.000140671 0.000185837 0.000216121 0.000227103 0.000215819 0.000181076 0.00012375 4.59887e-5; 2.62188e-5 8.02226e-5 0.000131215 0.000173288 0.000201461 0.000211629 0.000201049 0.00016863 0.000115208 4.28008e-5; 2.45264e-5 7.50194e-5 0.000122664 0.000161943 0.000188209 0.000197642 0.0001877 0.000157384 0.000107491 3.99211e-5; 2.29969e-5 7.03174e-5 0.000114938 0.000151693 0.000176239 0.000185012 0.000175648 0.000147231 0.000100525 3.7322e-5; 2.16154e-5 6.60709e-5 0.000107961 0.00014244 0.000165435 0.000173612 0.000164772 0.000138071 9.42414e-5 3.49778e-5; 2.03679e-5 6.22372e-5 0.000101664 0.000134089 0.000155686 0.000163329 0.000154963 0.000129811 8.8576e-5 3.28646e-5; 1.92417e-5 5.87769e-5 9.59816e-5 0.000126555 0.000146892 0.000154055 0.000146119 0.000122364 8.34695e-5 3.09602e-5; 1.82248e-5 5.5653e-5 9.08526e-5 0.000119756 0.000138958 0.00014569 0.000138143 0.000115651 7.88666e-5 2.9244e-5; 1.73062e-5 5.28318e-5 8.62215e-5 0.000113619 0.000131798 0.000138143 0.000130949 0.000109597 7.47164e-5 2.76968e-5; 1.64757e-5 5.02818e-5 8.20368e-5 0.000108074 0.000125332 0.000131329 0.000124455 0.000104133 7.09719e-5 2.63012e-5; 1.57241e-5 4.79744e-5 7.82511e-5 0.00010306 0.000119485 0.000125169 0.000118586 9.91966e-5 6.75899e-5 2.5041e-5; 1.50427e-5 4.58833e-5 7.48212e-5 9.85181e-5 0.000114191 0.000119593 0.000113275 9.47308e-5 6.45309e-5 2.39014e-5; 1.44239e-5 4.39846e-5 7.17078e-5 9.43967e-5 0.000109389 0.000114537 0.000108461 9.06831e-5 6.17591e-5 2.2869e-5; 1.38605e-5 4.22567e-5 6.88755e-5 9.06484e-5 0.000105022 0.000109941 0.000104086 8.70064e-5 5.9242e-5 2.19318e-5; 1.33465e-5 4.06805e-5 6.62924e-5 8.72311e-5 0.000101043 0.000105753 0.000100101 8.36587e-5 5.69508e-5 2.10789e-5; 1.28762e-5 3.92389e-5 6.39307e-5 8.41077e-5 9.74064e-5 0.000101929 9.64626e-5 8.06027e-5 5.48599e-5 2.03008e-5; 1.2445e-5 3.79173e-5 6.17663e-5 8.1246e-5 9.40759e-5 9.84265e-5 9.31322e-5 7.78064e-5 5.29472e-5 1.95892e-5; 1.20486e-5 3.67031e-5 5.97784e-5 7.86186e-5 9.10192e-5 9.52133e-5 9.00775e-5 7.52423e-5 5.11939e-5 1.89371e-5; 1.16839e-5 3.55861e-5 5.79503e-5 7.62032e-5 8.82099e-5 9.22611e-5 8.72717e-5 7.28879e-5 4.95844e-5 1.83387e-5; 1.13482e-5 3.45583e-5 5.62686e-5 7.39819e-5 8.56272e-5 8.95479e-5 8.46939e-5 7.07254e-5 4.81066e-5 1.77894e-5]

Contour comparison

fig = Figure(size=(1200, 700))# 2×2 layoutaxes_layout = [(1, 1), (1, 2), (2, 1), (2, 2)]axes = [Axis(fig[r, c]; title=titles[i],             ylabel = c == 1 ? "z (km)" : "",             xlabel = r == 2 ? "x (km)" : "")        for (i, (r, c)) in enumerate(axes_layout)]for ax in axes; if ax.xlabel[] == ""; hidexdecorations!(ax, grid=false); end; endfor ax in axes; if ax.ylabel[] == ""; hideydecorations!(ax, grid=false); end; endcfs = [contourf!(axes[i], x_km, z_km, θ′_final[k]; colormap=:balance, levels)       for (i, k) in enumerate(cases)]Colorbar(fig[1:2, 3], first(cfs); label="θ′ (K)")fig[0, :] = Label(fig, "Inertia-gravity waves: θ′ at t = $(prettytime(times[Nt]))", fontsize=20)n = Observable(1)fig_anim = Figure(size=(1200, 700))anim_axes = [Axis(fig_anim[r, c]; title=titles[i],                   ylabel = c == 1 ? "z (km)" : "",                   xlabel = r == 2 ? "x (km)" : "")             for (i, (r, c)) in enumerate(axes_layout)]for ax in anim_axes; if ax.xlabel[] == ""; hidexdecorations!(ax, grid=false); end; endfor ax in anim_axes; if ax.ylabel[] == ""; hideydecorations!(ax, grid=false); end; endhm_anims = [contourf!(anim_axes[i], @lift(θ′ts[k][$n]);                      colormap=:balance, levels, extendhigh=:auto, extendlow=:auto)            for (i, k) in enumerate(cases)]Colorbar(fig_anim[1:2, 3], last(hm_anims); label="θ′ (K)")anim_title = @lift "Inertia-gravity waves: θ′ at t = $(prettytime(times[$n]))"fig_anim[0, :] = Label(fig_anim, anim_title, fontsize=20, tellwidth=false)record(fig_anim, "inertia_gravity_wave.mp4", 1:Nt, framerate=8) do nn    n[] = nnend

Cross-section comparison

A vertical cross-section at mid-height shows the wave phase and amplitude differences:

z_mid = Nz ÷ 2linestyles = [:solid, :dash, :dot, :dashdot]colors = Makie.wong_colors()fig_cross = Figure(size=(900, 400))ax = Axis(fig_cross[1, 1], xlabel="x (km)", ylabel="θ′ (K)",          title="θ′ at z = $(round(z_km[z_mid], digits=1)) km, t = $(prettytime(times[Nt]))")for (i, k) in enumerate(cases)    lines!(ax, x_km, θ′_final[k][:, z_mid];           label=titles[i], linewidth=2, linestyle=linestyles[i], color=colors[i])endaxislegend(ax, position=:rt)fig_cross

Julia version and environment information

This example was executed with the following version of Julia:

using InteractiveUtils: versioninfoversioninfo()

Julia Version 1.13.2
Commit 45b6d15e34b (2026-10-14 20:33 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 8 × AMD EPYC 7R13 Processor
  WORD_SIZE: 64
  LLVM: libLLVM-20.1.8 (ORCJIT, znver3)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 5 GC (on 8 virtual cores)
Environment:
  JULIA_NUM_GC_THREADS = 4,1
  JULIA_GPG = 64B779A570972FFF7BFC2B54EAD471E1A1F2C10A
  JULIA_LOAD_PATH = :@breeze
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
  JULIA_VERSION = 1.13.2
  JULIA_DEPOT_PATH = /usr/local/share/julia:
  JULIA_PATH = /usr/local/julia
  JULIA_PROJECT = @breeze

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

import PkgPkg.status()

Status `/__w/Breeze.jl/Breeze.jl/docs/Project.toml`
⌃ [86bc3604] AtmosphericProfilesLibrary v0.1.8
  [660aa2fb] Breeze v0.15.1 `.`
⌃ [052768ef] CUDA v6.4.1
  [13f3f980] CairoMakie v0.15.17
  [6a9e3e04] CloudMicrophysics v0.49.1
  [e30172f5] Documenter v1.20.0
  [daee34ce] DocumenterCitations v1.5.0
  [b6400b83] DocumenterCodeBlocks v1.5.1
  [1c52b33b] DocumenterLandingPage v0.2.2
  [7da242da] Enzyme v0.13.219
⌅ [46192b85] GPUArraysCore v0.2.1
⌅ [63c18a36] KernelAbstractions v0.9.47
  [98b081ad] Literate v2.21.1
  [85f8d34a] NCDatasets v0.14.15
  [cd8119b0] NumericalRadiation v0.2.1
  [9e8cae18] Oceananigans v0.118.1
  [a01a1ee8] RRTMGP v1.1.1
  [3c362404] Reactant v0.2.308
  [276daf66] SpecialFunctions v2.9.2
  [10745b16] Statistics v1.11.5
  [b77e0a4c] InteractiveUtils v1.11.0
  [44cfe95a] Pkg v1.13.0
  [9a3f8284] Random 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`

This page was generated using Literate.jl.