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 Breeze
using Oceananigans: Oceananigans
using Oceananigans.Units
using Printf
using CairoMakie

Problem parameters

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

p₀ = 100000  # Pa - surface pressure
θ₀ = 300     # K - reference potential temperature
U  = 20      # m s⁻¹ - mean wind
N² = 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, 10
Lx, Lz = 300kilometers, 10kilometers

grid = 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 amplitude
a  = 5000    # m - perturbation half-width parameter
x₀ = Lx / 3  # m - perturbation center in x

constants = ThermodynamicConstants()
g = constants.gravitational_acceleration
pˢᵗ = 1e5

θᵇᵍ(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()
surface_pressure = p₀
potential_temperature = θ₀

300

Case 1: Anelastic

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

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

Case 2: Boussinesq (constant reference density)

constant_density_reference_state = ReferenceState(grid, constants; surface_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_anomaly: not materialized

Case 3: Compressible (fully explicit, no substepping)

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

CompressibleDynamics{ExplicitTimeStepping}
├── density: not materialized
├── pressure: not materialized
├── terrain_metrics: Nothing
├── 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();
                                           surface_pressure,
                                           reference_potential_temperature = θᵇᵍ)

CompressibleDynamics{SplitExplicitTimeDiscretization}
├── density: not materialized
├── pressure: not materialized
├── terrain_metrics: Nothing
├── 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)
end

for 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 / Nz
Rᵈ = dry_air_gas_constant(constants)
cᵖᵈ = constants.dry_air.heat_capacity
ℂᵃᶜ = sqrt(cᵖᵈ / (cᵖᵈ - Rᵈ) * Rᵈ * θ₀)

cfl = 0.5
Δt_advective    = cfl * min(Δx, Δz) / U
Δt_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 # seconds

case_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_existing = true)
    simulations[key] = sim
end

for (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′): 3.9610e-03, max|w|: 0.0062
[ Info: Compressible (explicit) - Iter: 1000, t: 22.529 minutes, max(θ′): 3.3875e-03, max|w|: 0.0046
[ Info: Compressible (explicit) - Iter: 1500, t: 33.787 minutes, max(θ′): 3.0844e-03, max|w|: 0.0036
[ Info: Compressible (explicit) - Iter: 2000, t: 45.045 minutes, max(θ′): 2.8742e-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′): 2.9077e-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′): 2.9630e-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].times
Nt = 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 => [1.30669e-5 4.00307e-5 6.541e-5 8.60533e-5 9.94584e-5 0.000103677 9.76217e-5 8.11158e-5 5.49759e-5 2.02637e-5; 1.28774e-5 3.94319e-5 6.43655e-5 8.45329e-5 9.74674e-5 0.0001013 9.5063e-5 7.87236e-5 5.32048e-5 1.95764e-5; 1.2734e-5 3.89694e-5 6.35265e-5 8.32465e-5 9.56903e-5 9.90746e-5 9.25757e-5 7.63321e-5 5.14003e-5 1.88686e-5; 1.26289e-5 3.862e-5 6.28587e-5 8.21564e-5 9.40963e-5 9.69855e-5 9.01619e-5 7.39574e-5 4.95821e-5 1.81496e-5; 1.25545e-5 3.83622e-5 6.23319e-5 8.12352e-5 9.26731e-5 9.50448e-5 8.7859e-5 7.16515e-5 4.77977e-5 1.74397e-5; 1.25042e-5 3.81776e-5 6.19236e-5 8.04686e-5 9.14294e-5 9.32932e-5 8.57378e-5 6.95002e-5 4.61203e-5 1.67693e-5; 1.24726e-5 3.8052e-5 6.16191e-5 7.98553e-5 9.0391e-5 9.1792e-5 8.38921e-5 6.76107e-5 4.46389e-5 1.61753e-5; 1.24553e-5 3.79745e-5 6.14096e-5 7.94016e-5 8.95918e-5 9.06101e-5 8.24209e-5 6.60933e-5 4.34442e-5 1.5695e-5; 1.24488e-5 3.79373e-5 6.12898e-5 7.91163e-5 8.90647e-5 8.98099e-5 8.14108e-5 6.5043e-5 4.26141e-5 1.5361e-5; 1.24505e-5 3.79353e-5 6.1257e-5 7.90067e-5 8.88344e-5 8.94376e-5 8.0925e-5 6.45288e-5 4.22046e-5 1.51962e-5; 1.24595e-5 3.79665e-5 6.13109e-5 7.90785e-5 8.89157e-5 8.95186e-5 8.09974e-5 6.45868e-5 4.22449e-5 1.52125e-5; 1.24764e-5 3.80329e-5 6.14546e-5 7.93351e-5 8.93115e-5 9.00549e-5 8.16285e-5 6.5217e-5 4.27339e-5 1.54092e-5; 1.25033e-5 3.81395e-5 6.16933e-5 7.97765e-5 9.00116e-5 9.10229e-5 8.27841e-5 6.638e-5 4.36388e-5 1.57725e-5; 1.25435e-5 3.82941e-5 6.2033e-5 8.03975e-5 9.09914e-5 9.23742e-5 8.43956e-5 6.80001e-5 4.48973e-5 1.62768e-5; 1.26009e-5 3.85056e-5 6.24793e-5 8.11884e-5 9.22149e-5 9.40403e-5 8.63674e-5 6.99719e-5 4.64239e-5 1.68871e-5; 1.26795e-5 3.87828e-5 6.30376e-5 8.21375e-5 9.36404e-5 9.59422e-5 8.85891e-5 7.21747e-5 4.8121e-5 1.75633e-5; 1.2783e-5 3.9135e-5 6.37144e-5 8.32352e-5 9.52285e-5 9.80031e-5 9.0952e-5 7.44893e-5 4.98916e-5 1.82659e-5; 1.29156e-5 3.95722e-5 6.45187e-5 8.44777e-5 9.69497e-5 0.000100161 9.33664e-5 7.68157e-5 5.16534e-5 1.8961e-5; 1.30813e-5 4.01059e-5 6.54635e-5 8.58699e-5 9.87909e-5 0.000102379 9.5773e-5 7.9085e-5 5.33481e-5 1.96246e-5; 1.32845e-5 4.07486e-5 6.65654e-5 8.7426e-5 0.000100757 0.000104648 9.81487e-5 8.12648e-5 5.49461e-5 2.02438e-5; 1.35292e-5 4.1512e-5 6.78425e-5 8.91679e-5 0.000102871 0.000106988 0.000100507 8.33614e-5 5.64483e-5 2.08184e-5; 1.38178e-5 4.24048e-5 6.93119e-5 9.11227e-5 0.00010517 0.000109446 0.000102898 8.54212e-5 5.78887e-5 2.13616e-5; 1.41514e-5 4.34325e-5 7.09887e-5 9.3322e-5 0.000107707 0.000112094 0.000105408 8.75293e-5 5.93332e-5 2.18998e-5; 1.45301e-5 4.45983e-5 7.28875e-5 9.58014e-5 0.000110546 0.000115028 0.000108154 8.98039e-5 6.08741e-5 2.24701e-5; 1.49542e-5 4.59059e-5 7.50238e-5 9.86002e-5 0.00011376 0.000118356 0.000111271 9.23868e-5 6.2623e-5 2.31172e-5; 1.54248e-5 4.73613e-5 7.74163e-5 0.000101761 0.000117428 0.000122197 0.00011491 9.54311e-5 6.4701e-5 2.38897e-5; 1.59442e-5 4.89738e-5 8.00884e-5 0.000105333 0.000121632 0.00012667 0.000119215 9.90877e-5 6.72268e-5 2.48352e-5; 1.65165e-5 5.07586e-5 8.30714e-5 0.00010937 0.000126456 0.000131891 0.000124325 0.000103492 7.03047e-5 2.59949e-5; 1.71489e-5 5.2739e-5 8.64077e-5 0.000113937 0.000131991 0.000137969 0.000130359 0.000108759 7.402e-5 2.74021e-5; 1.78523e-5 5.49486e-5 9.01532e-5 0.000119111 0.000138327 0.000145007 0.000137423 0.000114982 7.84396e-5 2.90822e-5; 1.86414e-5 5.74316e-5 9.43778e-5 0.000124979 0.000145563 0.000153102 0.000145603 0.000122231 8.361e-5 3.10525e-5; 1.95341e-5 6.02413e-5 9.9164e-5 0.000131641 0.000153799 0.000162343 0.000154966 0.000130548 8.95541e-5 3.33202e-5; 2.05506e-5 6.34383e-5 0.000104603 0.000139201 0.000163135 0.000172804 0.000165557 0.000139949 9.62685e-5 3.58811e-5; 2.17129e-5 6.70873e-5 0.000110793 0.000147769 0.000173665 0.000184547 0.000177393 0.000150416 0.000103725 3.87208e-5; 2.30429e-5 7.12535e-5 0.000117828 0.000157448 0.000185474 0.000197619 0.000190473 0.000161914 0.000111877 4.18175e-5; 2.45604e-5 7.59948e-5 0.000125793 0.000168328 0.000198635 0.000212052 0.000204788 0.000174397 0.000120674 4.5148e-5; 2.62806e-5 8.1356e-5 0.000134754 0.000180479 0.000213201 0.00022787 0.000220321 0.000187824 0.000130072 4.86916e-5; 2.82139e-5 8.73669e-5 0.000144754 0.000193947 0.00022921 0.00024509 0.000237069 0.00020217 0.000140039 5.24338e-5; 3.03663e-5 9.40461e-5 0.000155822 0.00020877 0.000246704 0.000263753 0.000255058 0.000217447 0.000150578 5.63744e-5; 3.27412e-5 0.000101405 0.000167983 0.000224986 0.000265738 0.000283927 0.000274364 0.000233726 0.000161739 6.05328e-5; 3.53414e-5 0.000109455 0.000181261 0.000242643 0.000286388 0.000305716 0.000295111 0.000251128 0.000173618 6.49466e-5; 3.81731e-5 0.000118218 0.000195698 0.000261814 0.000308767 0.000329274 0.000317477 0.000269831 0.000186348 6.96682e-5; 4.12523e-5 0.000127743 0.000211389 0.000282644 0.000333074 0.000354847 0.000341737 0.000290093 0.000200121 7.47715e-5; 4.46093e-5 0.00013813 0.000228503 0.000305375 0.000359615 0.000382791 0.000368264 0.000312262 0.000215196 8.03575e-5; 4.82857e-5 0.000149508 0.000247265 0.000330315 0.000388763 0.000413511 0.000397468 0.000336713 0.000231853 8.65379e-5; 5.23269e-5 0.000162021 0.000267911 0.000357778 0.000420873 0.000447372 0.000429695 0.000363751 0.000250323 9.34042e-5; 5.67769e-5 0.000175805 0.000290665 0.000388054 0.000456275 0.000484703 0.000465247 0.000393628 0.000270779 0.000101023; 6.16754e-5 0.000190982 0.000315732 0.000421427 0.000495312 0.000525886 0.0005045 0.000426663 0.000293435 0.00010947; 6.70476e-5 0.000207631 0.000343254 0.000458119 0.000538307 0.000571329 0.000547905 0.000463266 0.00031857 0.000118845; 7.28846e-5 0.000225727 0.000373207 0.000498147 0.000585348 0.000621216 0.000595718 0.000503701 0.000346376 0.00012922; 7.91365e-5 0.000245121 0.000405353 0.000541209 0.000636118 0.000675253 0.000647702 0.000547793 0.000376747 0.00014056; 8.57437e-5 0.000265628 0.000439383 0.000586868 0.000690066 0.000732825 0.000703234 0.000594995 0.000409312 0.00015273; 9.26971e-5 0.000287217 0.000475227 0.000634982 0.000746953 0.000793591 0.000761904 0.000644902 0.000443782 0.000165624; 0.000100074 0.000310123 0.000513258 0.000686016 0.000807263 0.000857995 0.000824058 0.000697758 0.000480304 0.000179294; 0.000108003 0.000334744 0.000554132 0.000740837 0.000872001 0.000927084 0.00089067 0.000754372 0.000519416 0.000193932; 0.000116578 0.000361364 0.000598318 0.000800079 0.000941927 0.00100167 0.000962526 0.000815408 0.00056156 0.000209697; 0.000125785 0.000389943 0.000645742 0.000863641 0.00101692 0.0010816 0.00103949 0.000880743 0.000606639 0.00022655; 0.000135523 0.000420166 0.000695874 0.000930789 0.00109609 0.00116592 0.0011206 0.000949539 0.000654067 0.000244271; 0.000145691 0.000451715 0.000748183 0.0010008 0.00117856 0.00125366 0.00120492 0.00102098 0.000703278 0.000262651; 0.000156263 0.000484513 0.000802538 0.0010735 0.00126415 0.00134461 0.00129226 0.00109491 0.000754171 0.000281655; 0.000167284 0.0005187 0.00085918 0.00114923 0.00135326 0.00143926 0.00138311 0.00117176 0.000807048 0.000301393; 0.000178793 0.000554396 0.000918315 0.00122828 0.00144626 0.00153801 0.00147785 0.00125189 0.00086216 0.000321959; 0.000190745 0.000591465 0.000979718 0.00131035 0.00154278 0.00164049 0.00157614 0.001335 0.00091931 0.00034328; 0.000203003 0.000629481 0.00104269 0.00139451 0.00164174 0.00174554 0.00167686 0.00142016 0.000977863 0.000365125; 0.00021539 0.000667905 0.00110634 0.00147959 0.00174177 0.00185174 0.00177869 0.00150627 0.0010371 0.000387231; 0.000227765 0.000706297 0.00116996 0.00156466 0.00184185 0.00195805 0.0018807 0.00159257 0.0010965 0.000409415; 0.000240043 0.000744397 0.00123312 0.00164918 0.00194137 0.00206389 0.00198237 0.00167868 0.00115581 0.00043157; 0.000252171 0.000782033 0.00129553 0.00173276 0.00203988 0.00216877 0.00208323 0.00176418 0.00121472 0.000453576; 0.000264069 0.000818957 0.00135677 0.00181481 0.00213665 0.00227183 0.00218242 0.00184829 0.00127267 0.00047522; 0.000275593 0.000854721 0.00141609 0.00189429 0.00223037 0.00237163 0.00227844 0.00192971 0.00132876 0.000496167; 0.000286532 0.000888667 0.00147238 0.00196968 0.00231924 0.00246619 0.00236936 0.00200679 0.00138185 0.000515992; 0.000296629 0.000919995 0.00152431 0.00203921 0.00240114 0.00255328 0.00245305 0.0020777 0.0014307 0.000534238; 0.000305612 0.000947863 0.0015705 0.00210102 0.00247392 0.00263065 0.00252736 0.00214068 0.00147409 0.000550449; 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0.000234815 0.000691597 0.00103671 0.00119044 0.00113327 0.000906082 0.000595775 0.000303205 0.000104629 1.70037e-5; 0.000210033 0.000613881 0.000906047 0.00101295 0.000921436 0.000678282 0.00037489 0.000114619 -2.61623e-5 -3.21147e-5; 0.000176757 0.000510385 0.00073463 0.000784788 0.000655472 0.000399349 0.000110891 -0.000106078 -0.000176842 -8.8218e-5; 0.000136896 0.000387121 0.000532699 0.000520167 0.000352909 8.88162e-5 -0.000176604 -0.000341649 -0.000335208 -0.000146687; 9.29383e-5 0.000251862 0.0003133 0.00023688 3.51757e-5 -0.000229942 -0.000464579 -0.000572181 -0.00048733 -0.000202276; 4.78333e-5 0.00011376 9.16056e-5 -4.47417e-5 -0.000273702 -0.000531252 -0.000728231 -0.000776553 -0.000618611 -0.000249525; 4.81362e-6 -1.72248e-5 -0.000116073 -0.000303212 -0.000548856 -0.000789136 -0.00094302 -0.000934288 -0.000715115 -0.000283253; -3.28377e-5 -0.000131041 -0.000293506 -0.000517592 -0.000766676 -0.000979652 -0.00108702 -0.0010276 -0.000765013 -0.000299125; -6.2058e-5 -0.000218397 -0.000425969 -0.00066942 -0.000907115 -0.00108338 -0.00114336 -0.00104354 -0.000760076 -0.000294206; 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0.000138255 0.000391262 0.000577729 0.000709782 0.000815636 0.000884035 0.000865111 0.000720646 0.000468814 0.000163334; 0.000139768 0.000389891 0.000561777 0.000670775 0.000750551 0.000792705 0.000750776 0.00059665 0.000364619 0.000119567; 0.000124449 0.000337076 0.000463671 0.000527951 0.000571293 0.000588077 0.000534266 0.000390517 0.00020579 5.56851e-5; 9.42281e-5 0.000239276 0.000295632 0.00030019 0.000303183 0.000300629 0.000248763 0.000134 1.6592e-5 -1.86714e-5; 5.39667e-5 0.000111826 8.37098e-5 2.30801e-5 -1.1395e-5 -2.39977e-5 -6.03978e-5 -0.000132197 -0.000173173 -9.19502e-5; 1.09411e-5 -2.2722e-5 -0.000135033 -0.000255048 -0.000318034 -0.000330807 -0.000341979 -0.00036462 -0.000332956 -0.000152517; -2.61477e-5 -0.000137768 -0.000317922 -0.000480346 -0.000558743 -0.000564311 -0.000547353 -0.000524221 -0.000436226 -0.000190471; -4.90879e-5 -0.000208337 -0.000425523 -0.000604446 -0.000683444 -0.000679095 -0.000639385 -0.000583401 -0.000465578 -0.000199554; -5.2201e-5 -0.000217581 -0.000431996 -0.000596948 -0.000662574 -0.000650803 -0.000601126 -0.000532354 -0.000416547 -0.000178566; 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3.69551e-5 3.99279e-5 6.67071e-5 7.8222e-5 8.12567e-5 7.88816e-5 6.3715e-5 4.44753e-5 3.00486e-5 -1.6147e-5; 2.436e-5 1.03186e-6 -1.42207e-6 -9.7605e-6 -2.68682e-5 -3.62803e-5 -4.94575e-5 -5.02205e-5 -3.73123e-5 -4.11164e-5; 8.67028e-6 -4.74493e-5 -8.7065e-5 -0.000119997 -0.000158285 -0.000174926 -0.000180984 -0.000158444 -0.00011445 -6.99467e-5; -3.5047e-6 -8.4643e-5 -0.000154531 -0.000204546 -0.000255503 -0.000276157 -0.000273164 -0.000232305 -0.000168501 -9.02672e-5; -3.45134e-6 -8.35112e-5 -0.000158547 -0.000203869 -0.000249665 -0.000268609 -0.000259798 -0.000217008 -0.00016192 -8.79361e-5; 1.38836e-5 -2.86976e-5 -7.35081e-5 -8.49913e-5 -0.00010525 -0.000118528 -0.000113088 -9.07541e-5 -8.03536e-5 -5.77785e-5; 4.49133e-5 6.83726e-5 8.02043e-5 0.000121152 0.000137362 0.000129243 0.00012122 0.000107374 5.10585e-5 -8.78741e-6; 7.79236e-5 0.000171879 0.000244004 0.000336025 0.000385795 0.000379017 0.00035198 0.000300274 0.000180609 4.0191e-5; 9.90542e-5 0.00023944 0.000349403 0.000470739 0.000539676 0.000531218 0.00048883 0.000412926 0.000257337 7.03945e-5; 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5.74385e-5 0.000177892 0.000294001 0.000392027 0.000459887 0.000487085 0.000466063 0.000393214 0.000269873 0.00010057; 6.22571e-5 0.000192847 0.000318793 0.00042522 0.000499005 0.000528708 0.00050607 0.000427112 0.000293227 0.000109298; 6.75705e-5 0.00020933 0.000346102 0.000461748 0.000541982 0.000574343 0.000549854 0.000464172 0.000318739 0.000118826; 7.33701e-5 0.000227318 0.000375899 0.000501599 0.00058884 0.000624052 0.00059752 0.000504504 0.000346484 0.000129178; 7.96057e-5 0.000246657 0.00040794 0.00054447 0.000639268 0.000677569 0.000648866 0.000547963 0.000376371 0.000140325; 8.62166e-5 0.00026716 0.000441916 0.000589951 0.000692806 0.000734442 0.000703486 0.000594214 0.000408186 0.000152196; 9.31844e-5 0.000288769 0.000477727 0.000637893 0.000749263 0.000794467 0.000761169 0.000643067 0.000441813 0.000164755; 0.000100565 0.000311657 0.000515655 0.000688655 0.000809037 0.000858043 0.000822267 0.000694807 0.000477449 0.000178073; 0.000108465 0.000336155 0.000556251 0.000742972 0.000872978 0.000926054 0.000887602 0.00075013 0.000515564 0.000192319; 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4.68552e-5 0.000141909 0.000226993 0.000289296 0.000320403 0.000317517 0.000283175 0.000223255 0.00014476 5.1873e-5; 1.39118e-5 3.92792e-5 5.58038e-5 5.82842e-5 4.56765e-5 2.23577e-5 -3.19246e-6 -2.1387e-5 -2.48193e-5 -1.17191e-5; -1.80254e-5 -6.02721e-5 -0.00011042 -0.000166335 -0.000221868 -0.000265552 -0.000282953 -0.000260697 -0.000190864 -7.40194e-5; -4.81256e-5 -0.000154164 -0.000267398 -0.000378838 -0.000475509 -0.000539093 -0.0005493 -0.000488935 -0.000349435 -0.000133558; -7.5526e-5 -0.000239717 -0.000410686 -0.000573285 -0.000708272 -0.000790885 -0.000795181 -0.000700158 -0.000496456 -0.000188817; -9.93574e-5 -0.000314233 -0.000535818 -0.000743717 -0.000913174 -0.00101355 -0.00101355 -0.000888441 -0.000627865 -0.00023828; -0.000118776 -0.000375099 -0.00063848 -0.000884393 -0.00108351 -0.00120001 -0.00119769 -0.00104812 -0.000739784 -0.000280505; -0.000133 -0.000419899 -0.000714698 -0.000990048 -0.00121314 -0.00134384 -0.00134147 -0.00117408 -0.000828713 -0.000314188; -0.000141346 -0.000446529 -0.000761034 -0.00105616 -0.00129684 -0.00143956 -0.00143974 -0.00126201 -0.000891712 -0.000338237; 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6.64769e-5 0.000185809 0.000247671 0.000220224 0.000102972 -6.83685e-5 -0.000231418 -0.000318558 -0.000282942 -0.000119219; 0.000108856 0.00031512 0.00045503 0.000484622 0.000395864 0.000222083 2.8117e-5 -0.000113062 -0.000148762 -7.05918e-5; 0.000149737 0.000439675 0.000654174 0.000737368 0.000674041 0.000495679 0.000270261 7.67706e-5 -2.58784e-5 -2.6292e-5; 0.00018717 0.000553478 0.000835317 0.000965645 0.00092278 0.000737157 0.000480696 0.000239051 7.76392e-5 1.06883e-5; 0.000219273 0.000650737 0.000989042 0.00115718 0.00112808 0.000932137 0.00064606 0.000362791 0.000154391 3.76178e-5; 0.000244331 0.000726196 0.00110684 0.00130097 0.00127754 0.00106804 0.000754848 0.000438673 0.000198182 5.22265e-5; 0.000260903 0.000775443 0.00118162 0.00138799 0.00136116 0.001135 0.000798303 0.000459816 0.000204554 5.29048e-5; 0.000267906 0.000795197 0.00120821 0.00141184 0.00137219 0.00112676 0.000771272 0.00042251 0.000171292 3.88934e-5; 0.000264704 0.000783569 0.00118379 0.00136937 0.00130787 0.00104144 0.000673014 0.000326897 9.88858e-5 1.04535e-5; 0.000251175 0.000740299 0.00110831 0.00126128 0.00117008 0.000882338 0.000507848 0.0001775 -9.10953e-6 -3.09992e-5; 0.000227773 0.000666939 0.000984819 0.00109247 0.000965836 0.000658291 0.000285559 -1.64671e-5 -0.000145887 -8.28299e-5; 0.000195563 0.000566962 0.000819578 0.000872296 0.000707424 0.000383795 2.13731e-5 -0.000241178 -0.000301463 -0.000141217; 0.000156221 0.00044574 0.000622025 0.000614305 0.000412054 7.85108e-5 -0.000264607 -0.000478755 -0.000463106 -0.00020132; 0.00011198 0.000310361 0.000404384 0.000335667 0.000100983 -0.000233849 -0.000548626 -0.000708403 -0.000616163 -0.000257599; 6.55311e-5 0.000169278 0.000180986 5.61012e-5 -0.000201914 -0.000527237 -0.000805094 -0.000908062 -0.000745258 -0.000304272; 1.98635e-5 3.17965e-5 -3.26711e-5 -0.000203551 -0.000472023 -0.000775472 -0.00100889 -0.00105651 -0.000835791 -0.00033589; -2.19259e-5 -9.25573e-5 -0.000221058 -0.000422996 -0.00068616 -0.000954749 -0.00113794 -0.00113559 -0.000875569 -0.000347957; -5.68572e-5 -0.000194745 -0.000369853 -0.000584265 -0.000824799 -0.00104614 -0.00117577 -0.0011326 -0.000856441 -0.000337549; 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0.000103405 0.000314827 0.000529983 0.000730661 0.000878959 0.000941923 0.000895492 0.000733359 0.000479662 0.000169743; 0.000113106 0.000337127 0.000545874 0.000716514 0.000816744 0.000829345 0.000748211 0.000581018 0.000360181 0.000121823; 0.000106447 0.000309187 0.000480731 0.000598394 0.000638512 0.00060114 0.000497379 0.000346288 0.000186311 5.39609e-5; 8.38145e-5 0.000233156 0.000341211 0.000390756 0.000368526 0.000290861 0.000182126 6.71561e-5 -1.34357e-5 -2.27434e-5; 4.86957e-5 0.000120417 0.000148155 0.000125076 4.85005e-5 -5.22617e-5 -0.00014603 -0.000209861 -0.000205664 -9.55798e-5; 7.44331e-6 -9.27169e-6 -6.53305e-5 -0.000153457 -0.000267442 -0.000370171 -0.000431076 -0.000437266 -0.000357749 -0.000152398; -3.14596e-5 -0.000130289 -0.000258361 -0.00039248 -0.000521085 -0.000605388 -0.000622141 -0.000575123 -0.000443766 -0.000183831; -5.91184e-5 -0.000216055 -0.000389818 -0.000542204 -0.000661477 -0.000713033 -0.000684648 -0.000599659 -0.000449747 -0.000185115; -6.83689e-5 -0.000245669 -0.000428683 -0.000567691 -0.000657224 -0.000671099 -0.000607552 -0.000507907 -0.000376327 -0.000156999; 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2.6838e-5 2.07672e-5 3.05046e-5 2.48251e-5 2.7416e-5 1.39993e-5 1.83328e-6 -5.96811e-6 -1.00014e-5 -3.20878e-5; 1.5997e-5 -1.0157e-5 -2.75022e-5 -5.45062e-5 -6.97075e-5 -9.09253e-5 -0.000102199 -9.18895e-5 -7.23289e-5 -5.63535e-5; 5.80644e-6 -4.02709e-5 -8.51858e-5 -0.000131316 -0.00015967 -0.000183861 -0.000190242 -0.000161536 -0.000123371 -7.62375e-5; 1.0178e-6 -5.47402e-5 -0.000117406 -0.000172294 -0.000203518 -0.000225634 -0.000226701 -0.000187191 -0.000143803 -8.39985e-5; 6.85421e-6 -3.72166e-5 -9.68133e-5 -0.000141683 -0.000161028 -0.000177159 -0.000177512 -0.000141956 -0.000114865 -7.26125e-5; 2.521e-5 1.85251e-5 -1.28031e-5 -2.65847e-5 -2.06103e-5 -3.02702e-5 -3.81088e-5 -2.21489e-5 -3.30642e-5 -4.08809e-5; 5.18749e-5 9.98925e-5 0.000114032 0.000144227 0.000181894 0.000176488 0.000154078 0.00014135 8.09886e-5 3.53773e-6; 7.7435e-5 0.000177924 0.000235906 0.00030687 0.000372924 0.000369658 0.000331703 0.000292347 0.000188353 4.61437e-5; 9.16682e-5 0.000220811 0.000300118 0.000391877 0.000474932 0.000474684 0.000428543 0.000374978 0.000249001 7.16493e-5; 8.84737e-5 0.00020918 0.000273648 0.000355582 0.000440565 0.000447833 0.000406904 0.000356408 0.000237593 6.97937e-5; 6.84067e-5 0.000144406 0.000157782 0.000199123 0.000271413 0.000291439 0.000269129 0.000236578 0.00015098 3.82835e-5; 3.80777e-5 4.69248e-5 -1.43952e-5 -3.51458e-5 1.32082e-5 4.8335e-5 5.19732e-5 4.42364e-5 7.00385e-6 -1.7023e-5; 7.31973e-6 -5.25187e-5 -0.000192153 -0.000281655 -0.00026342 -0.00021576 -0.000187962 -0.00017417 -0.000162082 -8.40356e-5; -1.43266e-5 -0.00012379 -0.000324894 -0.000474057 -0.000486245 -0.000432732 -0.000390539 -0.000366986 -0.000317953 -0.000147538; -2.0456e-5 -0.000146404 -0.000377322 -0.000564682 -0.000601761 -0.000550096 -0.000506767 -0.000489197 -0.000425024 -0.000192803; -9.21345e-6 -0.000114013 -0.000337195 -0.000534809 -0.000586329 -0.000541562 -0.000508432 -0.000511461 -0.000458101 -0.000208992; 1.67953e-5 -3.43265e-5 -0.000215686 -0.000395786 -0.000448217 -0.000410306 -0.000392678 -0.00042559 -0.000407541 -0.000191499; 5.17774e-5 7.48848e-5 -4.09595e-5 -0.000181572 -0.000221113 -0.000184858 -0.000180704 -0.000245442 -0.000280872 -0.000142708; 8.8604e-5 0.000191501 0.000151014 6.25518e-5 4.70968e-5 8.99866e-5 8.86152e-5 -2.6699e-6 -0.000100168 -7.10512e-5; 0.000120583 0.000294575 0.000325468 0.000291288 0.00030604 0.000363816 0.000367701 0.000260598 0.000103318 1.11527e-5; 0.000142703 0.000368241 0.000455419 0.000468054 0.000512739 0.000590561 0.000609789 0.000500388 0.00029574 9.0357e-5; 0.000152174 0.000403443 0.000524694 0.000569421 0.000637411 0.00073545 0.000776689 0.000678677 0.000446856 0.000154291; 0.000148379 0.000397826 0.000527995 0.000585804 0.000665334 0.000778398 0.000843595 0.000768825 0.000534601 0.000193842; 0.000132463 0.000354532 0.000469155 0.000519784 0.000595986 0.000714533 0.000801015 0.000758337 0.000547584 0.000204121; 0.000106823 0.000280653 0.000358779 0.00038345 0.000440782 0.000552845 0.000654495 0.0006493 0.000485703 0.000184753; 7.46688e-5 0.000185896 0.00021213 0.000195807 0.000220407 0.000313841 0.000422684 0.000456884 0.000359134 0.000139492; 3.96825e-5 8.15465e-5 4.74568e-5 -1.9442e-5 -3.77872e-5 2.66644e-5 0.000134314 0.000206421 0.00018606 7.53192e-5; 5.73243e-6 -2.04438e-5 -0.000115597 -0.000236155 -0.000302716 -0.000274471 -0.000175819 -7.0574e-5 -1.04647e-5 1.16841e-6; -2.34444e-5 -0.000108468 -0.000257763 -0.000428342 -0.000542999 -0.000554525 -0.000471643 -0.000341122 -0.000206168 -7.35286e-5; -4.46088e-5 -0.00017243 -0.000362171 -0.000573061 -0.000730707 -0.000782117 -0.000720723 -0.000575485 -0.000379135 -0.000140252; -5.53906e-5 -0.000204912 -0.000416332 -0.000653164 -0.000844789 -0.000933454 -0.000898361 -0.000750953 -0.0005126 -0.000192482; -5.45721e-5 -0.000202052 -0.000413582 -0.000659234 -0.000873373 -0.000994826 -0.000990056 -0.000853984 -0.000596466 -0.00022627; -4.22e-5 -0.000163873 -0.000353604 -0.000590232 -0.000814443 -0.000963177 -0.000991914 -0.000880442 -0.000627381 -0.000240253; -1.94974e-5 -9.40034e-5 -0.000241927 -0.000452775 -0.00067488 -0.000844918 -0.000909272 -0.000834213 -0.00060768 -0.000235228; 1.13849e-5 1.08733e-6 -8.86761e-5 -0.000259471 -0.000468444 -0.000653632 -0.000754309 -0.000725019 -0.00054375 -0.000213516; 4.76953e-5 0.000113081 9.29966e-5 -2.69086e-5 -0.000213469 -0.000407624 -0.00054362 -0.000566241 -0.000444473 -0.000178371; 8.64079e-5 0.000232783 0.000288423 0.000226225 6.92009e-5 -0.000127864 -0.000296319 -0.000373352 -0.000320147 -0.000133565; 0.000124498 0.000350945 0.000482693 0.000480689 0.000357833 0.000163539 -3.27439e-5 -0.000162921 -0.000181844 -8.31649e-5; 0.000159164 0.000458933 0.000661697 0.000717977 0.00063116 0.000444574 0.000226543 4.80886e-5 -4.1004e-5 -3.13942e-5; 0.000188 0.000549278 0.000813043 0.000921511 0.000869698 0.000694606 0.000461843 0.00024312 9.1054e-5 1.75374e-5; 0.000209146 0.000616141 0.00092687 0.00107778 0.00105712 0.000895823 0.000655669 0.000407123 0.000203858 5.96981e-5; 0.000221385 0.000655662 0.000996496 0.0011773 0.00118148 0.00103466 0.000794188 0.000527806 0.000288658 9.17632e-5; 0.000224194 0.000666118 0.00101874 0.00121518 0.00123608 0.00110286 0.000868374 0.000596719 0.000339225 0.000111324; 0.000217706 0.000647842 0.000993855 0.00119109 0.00121955 0.00109777 0.000874531 0.000609843 0.000352327 0.000117078; 0.000202616 0.000602908 0.000925004 0.00110874 0.00113536 0.00102201 0.000814084 0.000567551 0.000327781 0.000108869; 0.000180028 0.000534697 0.000817587 0.000974891 0.000990818 0.000882475 0.000692795 0.00047405 0.000268124 8.75713e-5; 0.00015132 0.000447439 0.00067847 0.000798444 0.000795909 0.000689202 0.000519714 0.000336545 0.000178074 5.48964e-5; 0.000118022 0.000345855 0.000515395 0.000589577 0.000562354 0.000454314 0.000306165 0.000164379 6.39359e-5 1.3174e-5; 8.17373e-5 0.000234918 0.00033656 0.000359181 0.000302859 0.000191194 6.48874e-5 -3.17353e-5 -6.69488e-5 -3.48604e-5; 4.41016e-5 0.000119695 0.000150354 0.000118457 3.059e-5 -8.61712e-5 -0.000190672 -0.000240388 -0.000206702 -8.62583e-5; 6.73453e-6 5.21459e-6 -3.4893e-5 -0.000121453 -0.000241325 -0.000363789 -0.000447022 -0.000450097 -0.000347379 -0.000138042; -2.88095e-5 -0.0001037 -0.000211195 -0.000349878 -0.000500326 -0.000628295 -0.000691296 -0.000649931 -0.000481425 -0.000187382; -6.11089e-5 -0.000202644 -0.00037127 -0.000557107 -0.000735017 -0.000867607 -0.000911916 -0.000830097 -0.000602098 -0.000231759; 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0.000172334 0.000532243 0.000876675 0.00116508 0.00136294 0.00144032 0.00137583 0.0011594 0.000795045 0.00029599; 0.000199499 0.000616758 0.00101729 0.00135413 0.00158671 0.0016795 0.00160674 0.00135583 0.000930777 0.000346803; 0.000224279 0.000693854 0.00114556 0.00152657 0.00179081 0.00189764 0.00181731 0.00153494 0.00105453 0.00039313; 0.000246412 0.000762717 0.00126013 0.00168061 0.00197313 0.00209251 0.00200543 0.00169495 0.0011651 0.000434522; 0.000265719 0.000822793 0.0013601 0.00181503 0.00213225 0.00226261 0.00216966 0.00183467 0.00126165 0.000470669; 0.000282099 0.000873771 0.00144494 0.00192913 0.00226737 0.00240708 0.0023092 0.00195341 0.00134371 0.000501398; 0.000295524 0.000915559 0.00151451 0.00202274 0.00237825 0.00252569 0.00242379 0.00205096 0.00141116 0.000526656; 0.000306025 0.000948259 0.00156898 0.00209607 0.00246517 0.00261873 0.00251373 0.00212756 0.00146414 0.000546503; 0.000313688 0.000972136 0.00160879 0.00214972 0.00252882 0.00268693 0.00257973 0.00218381 0.00150307 0.000561094; 0.000318639 0.000987584 0.0016346 0.00218457 0.00257027 0.00273143 0.00262286 0.00222063 0.00152858 0.000570662; 0.000321039 0.000995105 0.00164723 0.00220175 0.00259083 0.00275363 0.00264449 0.00223917 0.00154147 0.000575508; 0.000321073 0.000995276 0.00164766 0.00220255 0.00259204 0.00275518 0.0026462 0.00224079 0.00154267 0.000575981; 0.000318947 0.00098873 0.00163694 0.00218839 0.00257556 0.00273786 0.00262974 0.00222697 0.00153322 0.000572469; 0.000314874 0.000976138 0.00161617 0.00216075 0.00254319 0.00270359 0.00259695 0.00219929 0.00151421 0.000565379; 0.000309079 0.000958194 0.00158652 0.0021212 0.00249675 0.00265434 0.00254972 0.00215936 0.00148674 0.000555131; 0.000301788 0.000935602 0.00154915 0.0020713 0.0024381 0.00259206 0.00248997 0.0021088 0.00145195 0.000542144; 0.000293226 0.000909058 0.00150522 0.0020126 0.00236906 0.00251872 0.00241957 0.0020492 0.00141093 0.000526826; 0.000283609 0.000879239 0.00145585 0.00194661 0.00229141 0.00243619 0.00234032 0.0019821 0.00136474 0.000509574; 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6.17358e-5 0.000141221 0.000148576 0.000133022 0.00013961 0.000153657 0.000126583 4.26851e-5 -4.15242e-5 -4.049e-5; 9.79601e-5 0.000255043 0.000336554 0.000377573 0.000416302 0.000437318 0.000392542 0.000266521 0.000114889 1.94433e-5; 0.000123341 0.000336864 0.00047638 0.000565362 0.000634503 0.000666911 0.000614913 0.000461195 0.00025597 7.46484e-5; 0.000133371 0.00037257 0.000544287 0.000664482 0.000757281 0.00080407 0.000756975 0.000594758 0.000358782 0.000116294; 0.000126541 0.000357065 0.000530632 0.000660355 0.000765829 0.000827172 0.000795969 0.000645755 0.000407048 0.000137937; 0.000104238 0.000294065 0.000439851 0.000556133 0.000660537 0.000733129 0.000725417 0.000605681 0.000393119 0.000136326; 7.01653e-5 0.000194408 0.000287999 0.000370067 0.000458807 0.000536075 0.000554781 0.000479308 0.000318499 0.00011165; 2.95032e-5 7.35462e-5 9.89161e-5 0.00013099 0.000190624 0.000263656 0.000306756 0.000282983 0.000192946 6.72573e-5; -1.20089e-5 -5.11967e-5 -0.000100042 -0.000126887 -0.000106863 -4.79722e-5 1.3107e-5 4.15017e-5 3.25023e-5 8.94996e-6; 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6.90171e-5 0.000182413 0.00021745 0.000147905 -1.15857e-5 -0.000211774 -0.000383 -0.000455737 -0.000383117 -0.000158848; 0.00011539 0.000323312 0.000442362 0.000432744 0.00030113 9.51933e-5 -0.000111244 -0.000241795 -0.000243514 -0.000108039; 0.000158373 0.000454428 0.000653364 0.000703284 0.000603051 0.000397459 0.000162144 -2.21004e-5 -9.77976e-5 -5.45269e-5; 0.000195043 0.000566779 0.000835697 0.00093994 0.000871279 0.000670822 0.000414036 0.000183839 4.06379e-5 -3.31266e-6; 0.000223112 0.000653275 0.000977573 0.00112681 0.00108686 0.000894834 0.000624518 0.000358961 0.00015994 4.11494e-5; 0.000241016 0.000709035 0.00107072 0.00125246 0.0012358 0.00105403 0.000778166 0.000489798 0.000250628 7.52726e-5; 0.00024795 0.000731485 0.00111063 0.00131034 0.00130965 0.00113858 0.000864796 0.000567182 0.000306117 9.65395e-5; 0.000243842 0.000720305 0.00109642 0.00129869 0.00130552 0.00114447 0.000879695 0.000586512 0.000322934 0.00010359; 0.000229282 0.000677216 0.00103062 0.00122023 0.00122573 0.00107322 0.000823427 0.000547637 0.000300659 9.62071e-5; 0.000205425 0.000605673 0.0009186 0.00108154 0.00107717 0.000931188 0.00070127 0.000454423 0.000241657 7.52155e-5; 0.000173858 0.000510484 0.000767997 0.000892263 0.000870397 0.000728752 0.000522405 0.00031409 0.000150628 4.23212e-5; 0.000136474 0.000397401 0.000588052 0.000664229 0.000618664 0.00047923 0.000298942 0.000136416 3.40707e-5 -9.1172e-8; 9.53296e-5 0.000272708 0.000388918 0.000410587 0.000336853 0.000197808 4.48944e-5 -6.71369e-5 -0.000100317 -4.91792e-5; 5.25278e-5 0.000142833 0.000181032 0.000144935 4.04954e-5 -9.9524e-5 -0.000224839 -0.000284274 -0.000244222 -0.000101864; 1.00993e-5 1.39993e-5 -2.54642e-5 -0.000119446 -0.000255144 -0.000396936 -0.000495407 -0.000502665 -0.00038927 -0.000155034; -3.00879e-5 -0.000108061 -0.000221204 -0.000370237 -0.000535834 -0.000679591 -0.000752816 -0.000710633 -0.000527503 -0.000205727; -6.64022e-5 -0.000218334 -0.000397972 -0.000596592 -0.000788993 -0.000934306 -0.000984571 -0.000897715 -0.000651765 -0.000251279; -9.74987e-5 -0.000312682 -0.000548972 -0.000789509 -0.00100413 -0.00115004 -0.00118017 -0.00105508 -0.000756001 -0.000289426; -0.00012235 -0.000387938 -0.000668987 -0.000942049 -0.00117312 -0.00131821 -0.00133138 -0.00117578 -0.000835441 -0.000318388; -0.000140257 -0.000441943 -0.000754444 -0.00104943 -0.00129034 -0.0014328 -0.00143245 -0.00125493 -0.000886703 -0.000336896; -0.000150841 -0.000473525 -0.000803388 -0.00110902 -0.00135262 -0.00149043 -0.00148005 -0.00128967 -0.000907806 -0.000344205; -0.000154029 -0.000482445 -0.000815392 -0.00112017 -0.00135915 -0.00149012 -0.00147319 -0.00127911 -0.000898109 -0.00034007; -0.000150016 -0.000469295 -0.000791399 -0.00108409 -0.00131124 -0.00143318 -0.00141302 -0.00122417 -0.000858202 -0.000324705; -0.000139235 -0.000435393 -0.000733539 -0.00100353 -0.00121203 -0.00132282 -0.00130253 -0.00112729 -0.000789747 -0.000298725; -0.000122306 -0.00038264 -0.000644912 -0.00088256 -0.00106618 -0.00116387 -0.00114624 -0.000992237 -0.000695294 -0.000263076; -9.99972e-5 -0.000313394 -0.000529361 -0.000726218 -0.000879517 -0.000962409 -0.000949879 -0.000823779 -0.000578091 -0.000218967; -7.31798e-5 -0.000230325 -0.000391254 -0.000540254 -0.000658687 -0.00072537 -0.000720005 -0.000627399 -0.000441878 -0.000167791; -4.27837e-5 -0.000136289 -0.000235264 -0.000330833 -0.000410833 -0.000460231 -0.000463695 -0.000409018 -0.000290702 -0.000111054; -9.7606e-6 -3.42104e-5 -6.61806e-5 -0.000104273 -0.000143288 -0.000174678 -0.000188234 -0.000174742 -0.000128737 -5.03123e-5; 2.49504e-5 7.30258e-5 0.000111267 0.000133177 0.00013669 0.000123677 9.91511e-5 6.93709e-5 3.98708e-5 1.28879e-5; 6.04509e-5 0.000182658 0.000292552 0.000375535 0.000422147 0.000427532 0.000391527 0.000317501 0.00021114 7.70618e-5; 9.5907e-5 0.000292123 0.00047347 0.000617238 0.000706614 0.000730091 0.000682438 0.000564228 0.000381358 0.000140825; 0.000130566 0.000399107 0.000650224 0.000853267 0.000984249 0.00102522 0.000966047 0.000804651 0.00054717 0.000202926; 0.000163769 0.000501583 0.000819491 0.00107922 0.00124994 0.00130753 0.00123724 0.00103447 0.000705632 0.000262266; 0.000194959 0.000597839 0.000978458 0.00129139 0.00149934 0.00157247 0.00149169 0.00125006 0.000854253 0.000317916; 0.000223683 0.00068648 0.00112484 0.00148673 0.00172894 0.00181634 0.00172586 0.00144845 0.000991008 0.000369122; 0.000249589 0.000766429 0.00125686 0.00166291 0.00193602 0.00203629 0.00193706 0.00162736 0.00111434 0.0004153; 0.000272428 0.000836916 0.00137328 0.00181828 0.00211865 0.00223027 0.00212334 0.00178519 0.00122313 0.000456038; 0.000292043 0.000897459 0.00147328 0.00195177 0.0022756 0.00239703 0.00228351 0.00192091 0.00131671 0.00049108; 0.00030836 0.000947836 0.00155652 0.00206292 0.00240634 0.00253599 0.00241704 0.00203409 0.00139476 0.000520314; 0.000321384 0.000988058 0.00162301 0.00215177 0.00251091 0.00264721 0.00252398 0.00212479 0.00145734 0.000543758; 0.000331184 0.00101834 0.00167311 0.00221878 0.00258988 0.0027313 0.00260492 0.0021935 0.00150478 0.000561541; 0.000337883 0.00103906 0.00170745 0.00226481 0.00264423 0.00278932 0.00266088 0.0022411 0.00153768 0.000573886; 0.00034165 0.00105075 0.0017269 0.002291 0.00267533 0.00282268 0.00269323 0.00226873 0.00155685 0.000581092; 0.00034269 0.00105403 0.00173248 0.00229873 0.00268478 0.00283312 0.00270361 0.00227779 0.00156324 0.000583518; 0.000341231 0.0010496 0.00172536 0.00228955 0.00267441 0.00282255 0.00269387 0.00226984 0.00155792 0.000581565; 0.00033752 0.00103823 0.0017068 0.00226513 0.00264617 0.00279305 0.002666 0.00224656 0.00154205 0.000575665; 0.000331814 0.00102071 0.0016781 0.0022272 0.00260208 0.00274677 0.00262204 0.00220968 0.00151682 0.000566265; 0.000324369 0.000997836 0.00164056 0.00217751 0.0025442 0.00268586 0.00256408 0.00216096 0.00148344 0.000553816; 0.000315442 0.000970388 0.00159548 0.00211778 0.00247455 0.00261248 0.00249416 0.00210213 0.0014431 0.000538764; 0.00030528 0.000939134 0.00154413 0.00204968 0.00239508 0.0025287 0.00241427 0.00203486 0.00139695 0.00052154; 0.000294121 0.000904802 0.0014877 0.00197482 0.00230768 0.00243649 0.00232631 0.00196077 0.00134611 0.000502557; 0.000282185 0.000868075 0.00142732 0.0018947 0.00221409 0.00233773 0.00223205 0.00188135 0.00129159 0.000482201; 0.000269678 0.000829587 0.00136403 0.00181069 0.00211594 0.00223412 0.00213315 0.001798 0.00123437 0.00046083; 0.000256788 0.000789916 0.00129878 0.00172407 0.00201472 0.00212725 0.00203112 0.00171199 0.00117531 0.000438773; 0.000243683 0.000749582 0.00123244 0.00163599 0.00191176 0.00201853 0.00192729 0.00162446 0.0011152 0.000416322; 0.000230514 0.000709048 0.00116576 0.00154744 0.00180826 0.00190922 0.00182289 0.00153644 0.00105474 0.00039374; 0.000217411 0.000668716 0.00109941 0.00145933 0.00170525 0.00180041 0.00171896 0.0014488 0.000994548 0.000371256; 0.000204488 0.000628935 0.00103397 0.0013724 0.00160362 0.00169305 0.0016164 0.00136231 0.000935145 0.000349066; 0.00019184 0.00059 0.000969907 0.00128732 0.00150413 0.00158795 0.001516 0.00127764 0.000876979 0.000327338; 0.000179546 0.000552154 0.000907639 0.0012046 0.00140742 0.00148577 0.00141837 0.0011953 0.000820421 0.00030621; 0.00016767 0.000515596 0.000847486 0.0011247 0.00131398 0.00138705 0.00132405 0.00111575 0.00076577 0.000285794; 0.000156262 0.000480478 0.000789703 0.00104794 0.00122422 0.0012922 0.00123343 0.00103931 0.000713263 0.000266179; 0.000145361 0.000446918 0.000734482 0.000974582 0.00113843 0.00120155 0.00114682 0.000966256 0.000663075 0.00024743; 0.000134991 0.000414996 0.000681957 0.000904805 0.00105682 0.00111533 0.00106442 0.00089676 0.000615333 0.000229595; 0.000125171 0.000384765 0.000632213 0.000838721 0.000979539 0.00103366 0.00098639 0.000830939 0.000570115 0.000212703; 0.000115908 0.000356249 0.000585291 0.000776386 0.000906638 0.000956628 0.000912781 0.00076885 0.00052746 0.000196769; 0.000107202 0.00032945 0.000541196 0.000717806 0.000838128 0.000884233 0.000843604 0.0007105 0.000487375 0.000181794; 9.90491e-5 0.000304352 0.000499899 0.000662944 0.000773966 0.000816434 0.000778819 0.000655854 0.000449835 0.000167771; 9.14375e-5 0.000280922 0.000461347 0.000611729 0.000714071 0.000753144 0.000718343 0.000604843 0.000414792 0.000154681; 8.43529e-5 0.000259115 0.000425466 0.000564063 0.000658326 0.00069424 0.000662059 0.000557369 0.000382179 0.000142498; 7.77772e-5 0.000238874 0.000392163 0.000519824 0.00060659 0.000639573 0.000609825 0.000513312 0.000351915 0.000131194; 7.16897e-5 0.000220137 0.000361335 0.000478873 0.000558702 0.000588973 0.000561478 0.000472534 0.000323904 0.000120731; 6.6068e-5 0.000202834 0.000332868 0.00044106 0.000514483 0.000542252 0.000516838 0.000434885 0.000298042 0.000111071; 6.08884e-5 0.000186892 0.000306642 0.000406224 0.000473749 0.000499214 0.000475719 0.000400206 0.000274222 0.000102175; 5.61265e-5 0.000172237 0.000282533 0.000374202 0.000436306 0.000459656 0.000437926 0.000368333 0.000252331 9.39985e-5; 5.17573e-5 0.000158791 0.000260415 0.000344826 0.000401958 0.00042337 0.000403261 0.0003391 0.000232253 8.65003e-5; 4.77563e-5 0.000146479 0.000240163 0.00031793 0.000370512 0.00039015 0.000371528 0.000312341 0.000213876 7.96372e-5; 4.40989e-5 0.000135225 0.000221652 0.000293348 0.000341775 0.000359794 0.000342531 0.000287891 0.000197085 7.33672e-5; 4.07613e-5 0.000124955 0.000204763 0.000270921 0.000315557 0.000332102 0.000316082 0.000265591 0.000181772 6.76491e-5; 3.77204e-5 0.000115599 0.000189377 0.000250492 0.000291678 0.000306882 0.000291995 0.000245284 0.000167828 6.2443e-5; 3.49538e-5 0.000107089 0.000175382 0.000231911 0.000269961 0.000283948 0.000270094 0.000226821 0.000155152 5.77103e-5; 3.24404e-5 9.9357e-5 0.000162669 0.000215036 0.000250239 0.000263123 0.000250208 0.000210058 0.000143644 5.34143e-5; 3.01599e-5 9.23426e-5 0.000151137 0.000199728 0.000232352 0.000244237 0.000232177 0.000194861 0.000133212 4.95201e-5; 2.8093e-5 8.59862e-5 0.000140688 0.000185861 0.000216149 0.000227132 0.000215847 0.000181099 0.000123766 4.59945e-5; 2.62219e-5 8.02323e-5 0.000131231 0.00017331 0.000201488 0.000211656 0.000201075 0.000168652 0.000115223 4.28064e-5; 2.45294e-5 7.50286e-5 0.000122679 0.000161964 0.000188234 0.000197669 0.000187726 0.000157405 0.000107505 3.99264e-5; 2.29997e-5 7.03261e-5 0.000114952 0.000151713 0.000176263 0.000185036 0.000175671 0.000147251 0.000100539 3.73269e-5; 2.1618e-5 6.60791e-5 0.000107975 0.000142458 0.000165456 0.000173636 0.000164794 0.000138089 9.4254e-5 3.49825e-5; 2.03703e-5 6.22449e-5 0.000101677 0.000134106 0.000155706 0.000163351 0.000154984 0.000129828 8.85877e-5 3.2869e-5; 1.9244e-5 5.8784e-5 9.59935e-5 0.000126571 0.000146911 0.000154075 0.000146138 0.000122381 8.34804e-5 3.09643e-5; 1.82269e-5 5.56596e-5 9.08636e-5 0.000119771 0.000138976 0.000145709 0.000138161 0.000115666 7.88767e-5 2.92477e-5; 1.73081e-5 5.28378e-5 8.62317e-5 0.000113632 0.000131815 0.000138161 0.000130965 0.00010961 7.47258e-5 2.77003e-5; 1.64775e-5 5.02874e-5 8.20461e-5 0.000108087 0.000125347 0.000131345 0.00012447 0.000104145 7.09805e-5 2.63044e-5; 1.57257e-5 4.79795e-5 7.82596e-5 0.000103071 0.000119499 0.000125184 0.0001186 9.92081e-5 6.75977e-5 2.50439e-5; 1.50442e-5 4.58879e-5 7.48289e-5 9.85285e-5 0.000114204 0.000119607 0.000113288 9.47412e-5 6.4538e-5 2.3904e-5; 1.44252e-5 4.39887e-5 7.17147e-5 9.4406e-5 0.0001094 0.000114548 0.000108472 9.06924e-5 6.17654e-5 2.28714e-5; 1.38617e-5 4.22604e-5 6.88816e-5 9.06566e-5 0.000105032 0.000109951 0.000104096 8.70147e-5 5.92476e-5 2.19339e-5; 1.33475e-5 4.06837e-5 6.62977e-5 8.72383e-5 0.000101051 0.000105762 0.00010011 8.36659e-5 5.69557e-5 2.10807e-5; 1.28771e-5 3.92416e-5 6.39353e-5 8.41139e-5 9.74137e-5 0.000101936 9.647e-5 8.0609e-5 5.48641e-5 2.03024e-5; 1.24457e-5 3.79196e-5 6.17701e-5 8.12512e-5 9.40821e-5 9.84331e-5 9.31385e-5 7.78117e-5 5.29508e-5 1.95905e-5; 1.20492e-5 3.67049e-5 5.97815e-5 7.86229e-5 9.10243e-5 9.52187e-5 9.00826e-5 7.52466e-5 5.11968e-5 1.89382e-5; 1.16844e-5 3.55876e-5 5.79527e-5 7.62065e-5 8.82138e-5 9.22653e-5 8.72757e-5 7.28913e-5 4.95867e-5 1.83395e-5; 1.13486e-5 3.45594e-5 5.62704e-5 7.39844e-5 8.56301e-5 8.9551e-5 8.46969e-5 7.07279e-5 4.81082e-5 1.779e-5]

Contour comparison

fig = Figure(size=(1200, 700))

# 2×2 layout
axes_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; end
for ax in axes; if ax.ylabel[] == ""; hideydecorations!(ax, grid=false); end; end

cfs = [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; end
for ax in anim_axes; if ax.ylabel[] == ""; hideydecorations!(ax, grid=false); end; end

hm_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[] = nn
end

Cross-section comparison

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

z_mid = Nz ÷ 2
linestyles = [: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])
end

axislegend(ax, position=:rt)

fig_cross

Julia version and environment information

This example was executed with the following version of Julia:

using InteractiveUtils: versioninfo
versioninfo()

Julia Version 1.12.6
Commit 15346901f00 (2026-04-09 19:20 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-18.1.7 (ORCJIT, znver3)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 8 virtual cores)
Environment:
  JULIA_GPG = 3673DF529D9049477F76B37566E3C7DC03D6E495
  JULIA_LOAD_PATH = :@breeze
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
  JULIA_VERSION = 1.12.6
  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 Pkg
Pkg.status()

Status `/__w/Breeze.jl/Breeze.jl/docs/Project.toml`
  [86bc3604] AtmosphericProfilesLibrary v0.1.7
  [660aa2fb] Breeze v0.5.3 `.`
⌅ [052768ef] CUDA v6.0.0
  [13f3f980] CairoMakie v0.15.11
⌅ [6a9e3e04] CloudMicrophysics v0.35.0
  [e30172f5] Documenter v1.17.0
  [daee34ce] DocumenterCitations v1.4.1
  [7da242da] Enzyme v0.13.154
  [46192b85] GPUArraysCore v0.2.0
  [63c18a36] KernelAbstractions v0.9.41
  [98b081ad] Literate v2.21.0
  [85f8d34a] NCDatasets v0.14.15
  [9e8cae18] Oceananigans v0.110.1
  [a01a1ee8] RRTMGP v0.21.9
  [3c362404] Reactant v0.2.264
  [b77e0a4c] InteractiveUtils v1.11.0
  [44cfe95a] Pkg v1.12.1
  [9a3f8284] Random v1.11.0
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated`

This page was generated using Literate.jl.