# Grids

The grids currently supported are:

• RectilinearGrids with either constant or variable grid spacings and
• LatitudeLongitudeGrid on the sphere.

## RectilinearGrid

A RectilinearGrid is constructed by specifying the size of the grid (a Tuple specifying the number of grid points in each direction) and either the extent (a Tuple specifying the physical extent of the grid in each direction), or by prescribing x, y, and z. Keyword arguments x, y, and z could be either (i) 2-Tuples that define the the end points in each direction, or (ii) arrays or functions of the corresponding indices i, j, or k that specify the locations of cell faces in the x-, y-, or z-direction, respectively.

A regular rectilinear grid with $N_x \times N_y \times N_z = 32 \times 64 \times 256$ grid points and an extent of $L_x = 128$ meters, $L_y = 256$ meters, and $L_z = 512$ meters is constructed by

julia> grid = RectilinearGrid(size = (32, 64, 256), extent = (128, 256, 512))
32×64×256 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 128.0)  regularly spaced with Δx=4.0
├── Periodic y ∈ [0.0, 256.0)  regularly spaced with Δy=4.0
└── Bounded  z ∈ [-512.0, 0.0] regularly spaced with Δz=2.0
Default domain

When using the extent keyword, e.g., extent = (Lx, Ly, Lz), then the $x \in [0, L_x]$, $y \in [0, L_y]$, and $z \in [-L_z, 0]$ – a sensible choice for oceanographic applications.

### Specifying the grid's architecture

The first positional argument in either RectilinearGrid or LatitudeLongitudeGrid is the grid's architecture. By default architecture = CPU(). By providing GPU() as the architecture argument we can construct the grid on GPU:

julia> grid = RectilinearGrid(GPU(), size = (32, 64, 256), extent = (128, 256, 512))
32×64×256 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on GPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 128.0)  regularly spaced with Δx=4.0
├── Periodic y ∈ [0.0, 256.0)  regularly spaced with Δy=4.0
└── Bounded  z ∈ [-512.0, 0.0] regularly spaced with Δz=2.0

### Specifying the grid's topology

Another crucial keyword is a 3-Tuple that specifies the grid's topology. In each direction the grid may be Periodic, Bounded or Flat. By default, both the RectilinearGrid and the RectilinearGrid constructors assume the grid topology is horizontally-periodic and bounded in the vertical, such that topology = (Periodic, Periodic, Bounded).

A "channel" model that is periodic in the $x$-direction and wall-bounded in the $y$- and $z$-dimensions is build with,

julia> grid = RectilinearGrid(topology = (Periodic, Bounded, Bounded), size = (64, 64, 32), extent = (1e4, 1e4, 1e3))
64×64×32 RectilinearGrid{Float64, Periodic, Bounded, Bounded} on CPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 10000.0) regularly spaced with Δx=156.25
├── Bounded  y ∈ [0.0, 10000.0] regularly spaced with Δy=156.25
└── Bounded  z ∈ [-1000.0, 0.0] regularly spaced with Δz=31.25

The Flat topology comes in handy when running problems with fewer than 3 dimensions. As an example, to use a two-dimensional horizontal, doubly periodic domain the topology is (Periodic, Periodic, Flat). In that case, the size and extent are 2-tuples, e.g.,

julia> grid = RectilinearGrid(topology = (Periodic, Periodic, Flat), size = (32, 32), extent = (10, 20))
32×32×1 RectilinearGrid{Float64, Periodic, Periodic, Flat} on CPU with 3×3×0 halo
├── Periodic x ∈ [0.0, 10.0)      regularly spaced with Δx=0.3125
├── Periodic y ∈ [0.0, 20.0)      regularly spaced with Δy=0.625
└── Flat z

### Specifying domain end points

To specify a domain with a different origin than the default, the x, y, and z keyword arguments must be used. For example, a grid with $x \in [-100, 100]$ meters, $y \in [0, 12.5]$ meters, and $z \in [-\pi, \pi]$ meters is constructed via

julia> grid = RectilinearGrid(size = (32, 16, 256), x = (-100, 100), y = (0, 12.5), z = (-π, π))
32×16×256 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── Periodic x ∈ [-100.0, 100.0)     regularly spaced with Δx=6.25
├── Periodic y ∈ [0.0, 12.5)         regularly spaced with Δy=0.78125
└── Bounded  z ∈ [-3.14159, 3.14159] regularly spaced with Δz=0.0245437

### Grids with non-regular spacing in some of the directions

For a "channel" model, as the one we constructed above, one would probably like to have finer resolution near the channel walls. We construct a grid that has non-regular spacing in the bounded dimensions, here $y$ and $z$ by prescribing functions for y and z keyword arguments.

For example, we can use the Chebychev nodes, which are more closely stacked near boundaries, to prescribe the $y$- and $z$-faces.

julia> Nx, Ny, Nz = 64, 64, 32;

julia> Lx, Ly, Lz = 1e4, 1e4, 1e3;

julia> chebychev_spaced_y_faces(j) = - Ly/2 * cos(π * (j - 1) / Ny);

julia> chebychev_spaced_z_faces(k) = - Lz/2 - Lz/2 * cos(π * (k - 1) / Nz);

julia> grid = RectilinearGrid(size = (Nx, Ny, Nz),
topology = (Periodic, Bounded, Bounded),
x = (0, Lx),
y = chebychev_spaced_y_faces,
z = chebychev_spaced_z_faces)
64×64×32 RectilinearGrid{Float64, Periodic, Bounded, Bounded} on CPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 10000.0)    regularly spaced with Δx=156.25
├── Bounded  y ∈ [-5000.0, 5000.0] variably spaced with min(Δy)=6.02272, max(Δy)=245.338
└── Bounded  z ∈ [-1000.0, 0.0]    variably spaced with min(Δz)=2.40764, max(Δz)=49.0086

We can easily visualize the spacing of $y$ and $z$ directions.

using CairoMakie

fig = Figure(resolution=(800, 900))

ax1 = Axis(fig[1, 1]; xlabel = "y (m)", ylabel = "y-spacing (m)", limits = (nothing, (0, 250)))
lines!(ax1, grid.yᵃᶜᵃ[1:Ny], grid.Δyᵃᶜᵃ[1:Ny])
scatter!(ax1, grid.yᵃᶜᵃ[1:Ny], grid.Δyᵃᶜᵃ[1:Ny])

ax2 = Axis(fig[2, 1]; xlabel = "z-spacing (m)", ylabel = "z (m)", limits = ((0, 50), nothing))
lines!(ax2, grid.Δzᵃᵃᶜ[1:Nz], grid.zᵃᵃᶜ[1:Nz])
scatter!(ax2, grid.Δzᵃᵃᶜ[1:Nz], grid.zᵃᵃᶜ[1:Nz]) ## LatitudeLongitudeGrid

A simple latitude-longitude grid with Float64 type can be constructed by

julia> grid = LatitudeLongitudeGrid(size = (36, 34, 25),
longitude = (-180, 180),
latitude = (-85, 85),
z = (-1000, 0))
36×34×25 LatitudeLongitudeGrid{Float64, Periodic, Bounded, Bounded} on CPU with 3×3×3 halo and with precomputed metrics
├── longitude: Periodic λ ∈ [-180.0, 180.0) regularly spaced with Δλ=10.0
├── latitude:  Bounded  φ ∈ [-85.0, 85.0]   regularly spaced with Δφ=5.0
└── z:         Bounded  z ∈ [-1000.0, 0.0]  regularly spaced with Δz=40.0

For more examples see RectilinearGrid and LatitudeLongitudeGrid.