Internal tide by a seamount

In this example, we show how internal tide is generated from a barotropic tidal flow sloshing back and forth over a sea mount.

Install dependencies

First let's make sure we have all required packages installed.

using Pkg
pkg"add Oceananigans, CairoMakie"

using Oceananigans
using Oceananigans.Units

Grid

We create an ImmersedBoundaryGrid wrapped around an underlying two-dimensional RectilinearGrid that is periodic in $x$ and bounded in $z$.

Nx, Nz = 250, 125

H = 2kilometers

underlying_grid = RectilinearGrid(size = (Nx, Nz),
                                  x = (-1000kilometers, 1000kilometers),
                                  z = (-H, 0),
                                  halo = (4, 4),
                                  topology = (Periodic, Flat, Bounded))

250×1×125 RectilinearGrid{Float64, Periodic, Flat, Bounded} on CPU with 4×0×4 halo
├── Periodic x ∈ [-1.0e6, 1.0e6) regularly spaced with Δx=8000.0
├── Flat y                       
└── Bounded  z ∈ [-2000.0, 0.0]  regularly spaced with Δz=16.0

Now we can create the non-trivial bathymetry. We use GridFittedBottom that gets as input either (i) a two-dimensional function whose arguments are the grid's native horizontal coordinates and it returns the $z$ of the bottom, or (ii) a two-dimensional array with the values of $z$ at the bottom cell centers.

In this example we'd like to have a Gaussian hill at the center of the domain.

\[h(x) = -H + h_0 \exp(-x^2 / 2σ^2)\]

h₀ = 250meters
width = 20kilometers
hill(x) = h₀ * exp(-x^2 / 2width^2)
bottom(x) = - H + hill(x)

grid = ImmersedBoundaryGrid(underlying_grid, PartialCellBottom(bottom))

250×1×125 ImmersedBoundaryGrid{Float64, Periodic, Flat, Bounded} on CPU with 4×0×4 halo:
├── immersed_boundary: PartialCellBottom(mean(zb)=-1993.93, min(zb)=-2000.0, max(zb)=-1754.95, ϵ=0.2)
├── underlying_grid: 250×1×125 RectilinearGrid{Float64, Periodic, Flat, Bounded} on CPU with 4×0×4 halo
├── Periodic x ∈ [-1.0e6, 1.0e6) regularly spaced with Δx=8000.0
├── Flat y                       
└── Bounded  z ∈ [-2000.0, 0.0]  regularly spaced with Δz=16.0

Let's see how the domain with the bathymetry is.

x = xnodes(grid, Center())
bottom_boundary = interior(grid.immersed_boundary.bottom_height, :, 1, 1)
top_boundary = 0 * x

using CairoMakie

fig = Figure(size = (700, 200))
ax = Axis(fig[1, 1],
          xlabel="x [km]",
          ylabel="z [m]",
          limits=((-grid.Lx/2e3, grid.Lx/2e3), (-grid.Lz, 0)))

band!(ax, x/1e3, bottom_boundary, top_boundary, color = :mediumblue)

fig

Now we want to add a barotropic tide forcing. For example, to add the lunar semi-diurnal $M_2$ tide we need to add forcing in the $u$-momentum equation of the form:

\[F_0 \sin(\omega_2 t)\]

where $\omega_2 = 2π / T_2$, with $T_2 = 12.421 \,\mathrm{hours}$ the period of the $M_2$ tide.

The excursion parameter is a nondimensional number that expresses the ratio of the flow movement due to the tide compared to the size of the width of the hill.

\[\epsilon = \frac{U_{\mathrm{tidal}} / \omega_2}{\sigma}\]

We prescribe the excursion parameter which, in turn, implies a tidal velocity $U_{\mathrm{tidal}}$ which then allows us to determing the tidal forcing amplitude $F_0$. For the last step, we use Fourier decomposition on the inviscid, linearized momentum equations to determine the flow response for a given tidal forcing. Doing so we get that for the sinusoidal forcing above, the tidal velocity and tidal forcing amplitudes are related via:

\[U_{\mathrm{tidal}} = \frac{\omega_2}{\omega_2^2 - f^2} F_0\]

Now we have the way to find the value of the tidal forcing amplitude that would correspond to a given excursion parameter. The Coriolis frequency is needed, so we start by constructing a Coriolis on an $f$-plane at the mid-latitudes.

coriolis = FPlane(latitude = -45)

FPlane{Float64}(f=-0.000103126)

Now we have everything we require to construct the tidal forcing given a value of the excursion parameter.

T₂ = 12.421hours
ω₂ = 2π / T₂ # radians/sec

ϵ = 0.1 # excursion parameter

U_tidal = ϵ * ω₂ * width

tidal_forcing_amplitude = U_tidal * (ω₂^2 - coriolis.f^2) / ω₂

@inline tidal_forcing(x, z, t, p) = p.tidal_forcing_amplitude * sin(p.ω₂ * t)

u_forcing = Forcing(tidal_forcing, parameters=(; tidal_forcing_amplitude, ω₂))

ContinuousForcing{@NamedTuple{tidal_forcing_amplitude::Float64, ω₂::Float64}}
├── func: tidal_forcing (generic function with 1 method)
├── parameters: (tidal_forcing_amplitude = 1.8218611749508242e-5, ω₂ = 0.00014051439111137024)
└── field dependencies: ()

Model

We built a HydrostaticFreeSurfaceModel:

model = HydrostaticFreeSurfaceModel(; grid, coriolis,
                                      buoyancy = BuoyancyTracer(),
                                      tracers = :b,
                                      momentum_advection = WENO(),
                                      tracer_advection = WENO(),
                                      forcing = (; u = u_forcing))

HydrostaticFreeSurfaceModel{CPU, ImmersedBoundaryGrid}(time = 0 seconds, iteration = 0)
├── grid: 250×1×125 ImmersedBoundaryGrid{Float64, Periodic, Flat, Bounded} on CPU with 4×0×4 halo
├── timestepper: QuasiAdamsBashforth2TimeStepper
├── tracers: b
├── closure: Nothing
├── buoyancy: BuoyancyTracer with ĝ = NegativeZDirection()
├── free surface: SplitExplicitFreeSurface with gravitational acceleration 9.80665 m s⁻²
│   └── substepping: FixedTimeStepSize(39.986 seconds)
├── advection scheme: 
│   ├── momentum: WENO(order=5)
│   └── b: WENO(order=5)
└── coriolis: FPlane{Float64}

We initialize the model with the tidal flow and a linear stratification.

uᵢ(x, z) = U_tidal

Nᵢ² = 1e-4  # [s⁻²] initial buoyancy frequency / stratification
bᵢ(x, z) = Nᵢ² * z

set!(model, u=uᵢ, b=bᵢ)

Now let's build a Simulation.

Δt = 5minutes
stop_time = 4days

simulation = Simulation(model; Δt, stop_time)

Simulation of HydrostaticFreeSurfaceModel{CPU, ImmersedBoundaryGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 5 minutes
├── Elapsed wall time: 0 seconds
├── Wall time per iteration: NaN days
├── Stop time: 4 days
├── Stop iteration: Inf
├── Wall time limit: Inf
├── Minimum relative step: 0.0
├── Callbacks: OrderedDict with 4 entries:
│   ├── stop_time_exceeded => Callback of stop_time_exceeded on IterationInterval(1)
│   ├── stop_iteration_exceeded => Callback of stop_iteration_exceeded on IterationInterval(1)
│   ├── wall_time_limit_exceeded => Callback of wall_time_limit_exceeded on IterationInterval(1)
│   └── nan_checker => Callback of NaNChecker for u on IterationInterval(100)
├── Output writers: OrderedDict with no entries
└── Diagnostics: OrderedDict with no entries

We add a callback to print a message about how the simulation is going,

using Printf

wall_clock = Ref(time_ns())

function progress(sim)
    elapsed = 1e-9 * (time_ns() - wall_clock[])

    msg = @sprintf("iteration: %d, time: %s, wall time: %s, max|w|: %6.3e, m s⁻¹\n",
                   iteration(sim), prettytime(sim), prettytime(elapsed),
                   maximum(abs, sim.model.velocities.w))

    wall_clock[] = time_ns()

    @info msg

    return nothing
end

add_callback!(simulation, progress, name=:progress, IterationInterval(200))

Diagnostics/Output

Add some diagnostics. Instead of $u$ we save the deviation of $u$ from its instantaneous domain average, $u' = u - (L_x H)^{-1} \int u \, \mathrm{d}x \mathrm{d}z$. We also save the stratification $N^2 = \partial_z b$.

b = model.tracers.b
u, v, w = model.velocities

U = Field(Average(u))

u′ = u - U

N² = ∂z(b)

filename = "internal_tide"
save_fields_interval = 30minutes

simulation.output_writers[:fields] = JLD2OutputWriter(model, (; u, u′, w, b, N²);
                                                      filename,
                                                      schedule = TimeInterval(save_fields_interval),
                                                      overwrite_existing = true)

JLD2OutputWriter scheduled on TimeInterval(30 minutes):
├── filepath: internal_tide.jld2
├── 5 outputs: (u, u′, w, b, N²)
├── array type: Array{Float64}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 64.4 KiB

We are ready – let's run!

run!(simulation)

[ Info: Initializing simulation...
[ Info: iteration: 0, time: 0 seconds, wall time: 3.008 minutes, max|w|: 2.248e-03, m s⁻¹
[ Info:     ... simulation initialization complete (25.032 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (24.724 seconds).
[ Info: iteration: 200, time: 16.667 hours, wall time: 51.816 seconds, max|w|: 5.237e-03, m s⁻¹
[ Info: iteration: 400, time: 1.389 days, wall time: 16.607 seconds, max|w|: 4.014e-03, m s⁻¹
[ Info: iteration: 600, time: 2.083 days, wall time: 16.653 seconds, max|w|: 1.928e-03, m s⁻¹
[ Info: iteration: 800, time: 2.778 days, wall time: 16.385 seconds, max|w|: 4.494e-03, m s⁻¹
[ Info: iteration: 1000, time: 3.472 days, wall time: 15.588 seconds, max|w|: 2.978e-03, m s⁻¹
[ Info: Simulation is stopping after running for 2.404 minutes.
[ Info: Simulation time 4 days equals or exceeds stop time 4 days.

Load output

First, we load the saved velocities and stratification output as FieldTimeSerieses.

saved_output_filename = filename * ".jld2"

u′_t = FieldTimeSeries(saved_output_filename, "u′")
 w_t = FieldTimeSeries(saved_output_filename, "w")
N²_t = FieldTimeSeries(saved_output_filename, "N²")

umax = maximum(abs, u′_t[end])
wmax = maximum(abs, w_t[end])

times = u′_t.times

Visualize

Now we can visualize our resutls! We use CairoMakie here. On a system with OpenGL using GLMakie is more convenient as figures will be displayed on the screen.

We use Makie's Observable to animate the data. To dive into how Observables work we refer to Makie.jl's Documentation.

using CairoMakie

n = Observable(1)

title = @lift @sprintf("t = %1.2f days = %1.2f T₂",
                       round(times[$n] / day, digits=2) , round(times[$n] / T₂, digits=2))

u′ₙ = @lift u′_t[$n]
 wₙ = @lift  w_t[$n]
N²ₙ = @lift N²_t[$n]

axis_kwargs = (xlabel = "x [m]",
               ylabel = "z [m]",
               limits = ((-grid.Lx/2, grid.Lx/2), (-grid.Lz, 0)),
               titlesize = 20)

fig = Figure(size = (700, 900))

fig[1, :] = Label(fig, title, fontsize=24, tellwidth=false)

ax_u = Axis(fig[2, 1]; title = "u'-velocity", axis_kwargs...)
hm_u = heatmap!(ax_u, u′ₙ; nan_color=:gray, colorrange=(-umax, umax), colormap=:balance)
Colorbar(fig[2, 2], hm_u, label = "m s⁻¹")

ax_w = Axis(fig[3, 1]; title = "w-velocity", axis_kwargs...)
hm_w = heatmap!(ax_w, wₙ; nan_color=:gray, colorrange=(-wmax, wmax), colormap=:balance)
Colorbar(fig[3, 2], hm_w, label = "m s⁻¹")

ax_N² = Axis(fig[4, 1]; title = "stratification N²", axis_kwargs...)
hm_N² = heatmap!(ax_N², N²ₙ; nan_color=:gray, colorrange=(0.9Nᵢ², 1.1Nᵢ²), colormap=:magma)
Colorbar(fig[4, 2], hm_N², label = "s⁻²")

fig

Finally, we can record a movie.

@info "Making an animation from saved data..."

frames = 1:length(times)

record(fig, filename * ".mp4", frames, framerate=16) do i
    @info string("Plotting frame ", i, " of ", frames[end])
    n[] = i
end

[ Info: Making an animation from saved data...
[ Info: Plotting frame 1 of 193
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