spectral_gradients.jl

const DEFAULT_RADIUS = 1

"""
$(TYPEDSIGNATURES)
Curl of a vector `u, v` written into `curl`, `curl = ∇×(u, v)`.
`u, v` are expected to have a 1/coslat-scaling included, otherwise `curl` is scaled.
Acts on the unit sphere, i.e. it omits 1/radius scaling as all inplace gradient operators.
`flipsign` option calculates -∇×(u, v) instead. `add` option calculates `curl += ∇×(u, v)` instead.
`flipsign` and `add` can be combined. This functions only creates the kernel and calls the generic
divergence function _divergence! subsequently with flipped u, v -> v, u for the curl."""
function curl!( curl::LowerTriangularMatrix,
                u::LowerTriangularMatrix,
                v::LowerTriangularMatrix,
                S::SpectralTransform;
                flipsign::Bool=false,
                add::Bool=false,
                )

    # = -(∂λ - ∂θ) or (∂λ - ∂θ), adding or overwriting the output curl
    kernel(o, a, b, c) = flipsign ? (add ? o-(a+b-c) : -(a+b-c)) :
                                    (add ? o+(a+b-c) :   a+b-c )    
    _divergence!(kernel, curl, v, u, S)             # flip u, v -> v, u
end

"""
$(TYPEDSIGNATURES)
Divergence of a vector `u, v` written into `div`, `div = ∇⋅(u, v)`. 
`u, v` are expected to have a 1/coslat-scaling included, otherwise `div` is scaled.
Acts on the unit sphere, i.e. it omits 1/radius scaling as all inplace gradient operators.
`flipsign` option calculates -∇⋅(u, v) instead. `add` option calculates `div += ∇⋅(u, v)` instead.
`flipsign` and `add` can be combined. This functions only creates the kernel and calls
the generic divergence function _divergence! subsequently."""
function divergence!(   div::LowerTriangularMatrix,
                        u::LowerTriangularMatrix,
                        v::LowerTriangularMatrix,
                        S::SpectralTransform;
                        flipsign::Bool=false,
                        add::Bool=false,
                        )

    # = -(∂λ + ∂θ) or (∂λ + ∂θ), adding or overwriting the output div
    kernel(o, a, b, c) = flipsign ? (add ? o-(a-b+c) : -(a-b+c)) :
                                    (add ? o+(a-b+c) :   a-b+c )                
    _divergence!(kernel, div, u, v, S)
end

"""
$(TYPEDSIGNATURES)
Generic divergence function of vector `u`, `v` that writes into the output into `div`.
Generic as it uses the kernel `kernel` such that curl, div, add or flipsign
options are provided through `kernel`, but otherwise a single function is used.
Acts on the unit sphere, i.e. it omits 1/radius scaling as all inplace gradient operators.
"""
function _divergence!(  kernel,
                        div::LowerTriangularMatrix{Complex{NF}},
                        u::LowerTriangularMatrix{Complex{NF}},
                        v::LowerTriangularMatrix{Complex{NF}},
                        S::SpectralTransform{NF}
                        ) where {NF<:AbstractFloat}

    @boundscheck size(u) == size(div) || throw(BoundsError)
    @boundscheck size(v) == size(div) || throw(BoundsError)

    (; grad_y_vordiv1, grad_y_vordiv2 ) = S
    @boundscheck size(grad_y_vordiv1) == size(div) || throw(BoundsError)
    @boundscheck size(grad_y_vordiv2) == size(div) || throw(BoundsError)
    lmax, mmax = size(div) .- (2, 1)              # 0-based lmax, mmax 

    lm = 0
    @inbounds for m in 1:mmax+1                 # 1-based l, m
        
        # DIAGONAL (separate to avoid access to v[l-1, m])
        lm += 1                                 
        ∂u∂λ  = ((m-1)*im)*u[lm]
        ∂v∂θ1 = zero(Complex{NF})               # always above the diagonal
        ∂v∂θ2 = grad_y_vordiv2[lm]*v[lm+1]
        div[lm] = kernel(div[lm], ∂u∂λ, ∂v∂θ1, ∂v∂θ2)

        # BELOW DIAGONAL (but skip last row)
        for l in m+1:lmax+1
            lm += 1
            ∂u∂λ  = ((m-1)*im)*u[lm]
            ∂v∂θ1 = grad_y_vordiv1[lm]*v[lm-1]
            ∂v∂θ2 = grad_y_vordiv2[lm]*v[lm+1]  # this pulls in data from the last row though
            div[lm] = kernel(div[lm], ∂u∂λ, ∂v∂θ1, ∂v∂θ2)
        end

        # Last row, only vectors make use of the lmax+1 row, set to zero for scalars div, curl
        lm += 1
        div[lm] = zero(Complex{NF})
    end

    return nothing
end

"""
$(TYPEDSIGNATURES)
Divergence (∇⋅) of two vector components `u, v` which need to have size (n+1)xn,
the last row will be set to zero in the returned `LowerTriangularMatrix`.
This function requires both `u, v` to be transforms of fields that are scaled with
`1/cos(lat)`. Acts on the unit sphere, i.e. it omits 1/radius scaling unless
`radius` keyword argument is provided.
An example usage is therefore

    RingGrids.scale_coslat⁻¹!(u_grid)
    RingGrids.scale_coslat⁻¹!(v_grid)
    u = spectral(u_grid, one_more_degree=true)
    v = spectral(v_grid, one_more_degree=true)
    div = divergence(u, v, radius = 6.371e6)
    div_grid = gridded(div)
"""
function divergence(u::LowerTriangularMatrix,
                    v::LowerTriangularMatrix;
                    radius = DEFAULT_RADIUS)

    @assert size(u) == size(v) "Size $(size(u)) and $(size(v)) incompatible."

    S = SpectralTransform(u)
    div = similar(u)
    divergence!(div, u, v, S, add=false, flipsign=false)

    if radius != 1
        div .*= inv(radius)
    end

    return div
end

# called by divergence or curl
function _div_or_curl(  
    kernel!,
    u::Grid,
    v::Grid;
    radius = DEFAULT_RADIUS
) where {Grid<:AbstractGrid}

    @assert size(u) == size(v) "Size $(size(u)) and $(size(v)) incompatible."

    u_grid = copy(u)
    v_grid = copy(v)

    RingGrids.scale_coslat⁻¹!(u_grid)
    RingGrids.scale_coslat⁻¹!(v_grid)

    S = SpectralTransform(u_grid, one_more_degree=true)
    us = spectral(u_grid, S)
    vs = spectral(v_grid, S)

    div_or_vor = similar(us)
    kernel!(div_or_vor, us, vs, S, add=false, flipsign=false)

    if radius != 1
        div_or_vor .*= inv(radius)
    end

    return div_or_vor
end

"""
$(TYPEDSIGNATURES)
Divergence (∇⋅) of two vector components `u, v` on a grid.
Applies 1/coslat scaling, transforms to spectral space and returns
the spectral divergence. Acts on the unit sphere, i.e. it omits 1/radius scaling unless
`radius` keyword argument is provided.
"""
divergence(u::Grid, v::Grid; kwargs...) where {Grid<:AbstractGrid} = _div_or_curl(divergence!, u, v; kwargs...)

"""
$(TYPEDSIGNATURES)
Curl (∇×) of two vector components `u, v` on a grid.
Applies 1/coslat scaling, transforms to spectral space and returns
the spectral curl. Acts on the unit sphere, i.e. it omits 1/radius scaling unless
`radius` keyword argument is provided."""
curl(u::Grid, v::Grid; kwargs...) where {Grid<:AbstractGrid} = _div_or_curl(curl!, u, v; kwargs...)

"""
$(TYPEDSIGNATURES)
Curl (∇×) of two vector components `u, v` of size (n+1)xn, the last row
will be set to zero in the returned `LowerTriangularMatrix`. This function
requires both `u, v` to be transforms of fields that are scaled with
`1/cos(lat)`. Acts on the unit sphere, i.e. it omits 1/radius scaling unless
`radius` keyword argument is provided. An example usage is therefore

    RingGrids.scale_coslat⁻¹!(u_grid)
    RingGrids.scale_coslat⁻¹!(v_grid)
    u = spectral(u_grid)
    v = spectral(v_grid)
    vor = curl(u, v, radius=6.371e6)
    vor_grid = gridded(div)
"""
function curl(  u::LowerTriangularMatrix,
                v::LowerTriangularMatrix;
                radius = DEFAULT_RADIUS)

    @assert size(u) == size(v) "Size $(size(u)) and $(size(v)) incompatible."

    S = SpectralTransform(u)
    vor = similar(u)
    curl!(vor, u, v, S, add=false, flipsign=false)

    if radius != 1
        vor .*= inv(radius)
    end

    return vor
end

"""
$(TYPEDSIGNATURES)
Get U, V (=(u, v)*coslat) from vorticity ζ spectral space (divergence D=0)
Two operations are combined into a single linear operation. First, invert the
spherical Laplace ∇² operator to get stream function from vorticity. Then
compute zonal and meridional gradients to get U, V.
Acts on the unit sphere, i.e. it omits any radius scaling as all inplace gradient operators.
"""
function UV_from_vor!(  U::LowerTriangularMatrix{Complex{NF}},
                        V::LowerTriangularMatrix{Complex{NF}},
                        vor::LowerTriangularMatrix{Complex{NF}},
                        S::SpectralTransform{NF}
                        ) where {NF<:AbstractFloat}

    (; vordiv_to_uv_x, vordiv_to_uv1, vordiv_to_uv2 ) = S
    lmax, mmax = size(vor) .- (2, 1)                  # 0-based lmax, mmax
    
    @boundscheck lmax == mmax || throw(BoundsError)
    @boundscheck size(U) == size(vor) || throw(BoundsError)
    @boundscheck size(V) == size(vor) || throw(BoundsError)
    @boundscheck size(vordiv_to_uv_x) == size(vor) || throw(BoundsError)
    @boundscheck size(vordiv_to_uv1) == size(vor) || throw(BoundsError)
    @boundscheck size(vordiv_to_uv2) == size(vor) || throw(BoundsError)

    lm = 0
    @inbounds for m in 1:mmax                       # 1-based l, m, exclude last column

        # DIAGONAL (separated to avoid access to l-1, m which is above the diagonal)
        lm += 1

        # U = -∂/∂lat(Ψ) and V = V = ∂/∂λ(Ψ) combined with Laplace inversion ∇⁻², omit radius R scaling
        U[lm] = vordiv_to_uv2[lm]*vor[lm+1]         # - vordiv_to_uv1[lm]*vor[l-1, m] <- is zero
        V[lm] = im*vordiv_to_uv_x[lm]*vor[lm]

        # BELOW DIAGONAL
        for l in m+1:lmax                           # skip last two rows
            lm += 1

            # U = -∂/∂lat(Ψ) and V = V = ∂/∂λ(Ψ) combined with Laplace inversion ∇⁻², omit radius R scaling
            # U[lm] = vordiv_to_uv2[lm]*vor[lm+1] - vordiv_to_uv1[lm]*vor[lm-1]
            U[lm] = muladd(vordiv_to_uv2[lm], vor[lm+1], -vordiv_to_uv1[lm]*vor[lm-1])
            V[lm] = im*vordiv_to_uv_x[lm]*vor[lm]
        end

        # SECOND LAST ROW
        lm += 1
        U[lm] = -vordiv_to_uv1[lm]*vor[lm-1]        # meridional gradient again (but only 2nd term from above)
        V[lm] = im*vordiv_to_uv_x[lm]*vor[lm]       # zonal gradient again (as above)

        # LAST ROW (separated to avoid out-of-bounds access to l+2, m)
        lm += 1
        U[lm] = -vordiv_to_uv1[lm]*vor[lm-1]        # meridional gradient again (but only 2nd term from above)
        V[lm] = zero(Complex{NF})                   # set explicitly to 0 as Ψ does not contribute to last row of V
    end

    # LAST COLUMN
    @inbounds begin
        lm += 1                     # second last row
        U[lm] = zero(Complex{NF})
        V[lm] = im*vordiv_to_uv_x[lm]*vor[lm]

        lm += 1                     # last row
        U[lm] = -vordiv_to_uv1[lm]*vor[lm-1]
        V[lm] = zero(Complex{NF})
    end

    return nothing
end

"""
$(TYPEDSIGNATURES)
Get U, V (=(u, v)*coslat) from vorticity ζ and divergence D in spectral space.
Two operations are combined into a single linear operation. First, invert the
spherical Laplace ∇² operator to get stream function from vorticity and
velocity potential from divergence. Then compute zonal and meridional gradients
to get U, V.
Acts on the unit sphere, i.e. it omits any radius scaling as all inplace gradient operators.
"""
function UV_from_vordiv!(   U::LowerTriangularMatrix{Complex{NF}},
                            V::LowerTriangularMatrix{Complex{NF}},
                            vor::LowerTriangularMatrix{Complex{NF}},
                            div::LowerTriangularMatrix{Complex{NF}},
                            S::SpectralTransform{NF}
                            ) where {NF<:AbstractFloat}

    (; vordiv_to_uv_x, vordiv_to_uv1, vordiv_to_uv2 ) = S
    lmax, mmax = size(vor) .- (2, 1)                  # 0-based lmax, mmax
    @boundscheck lmax == mmax || throw(BoundsError)
    @boundscheck size(div) == size(vor) || throw(BoundsError)
    @boundscheck size(U) == size(vor) || throw(BoundsError)
    @boundscheck size(V) == size(vor) || throw(BoundsError)
    @boundscheck size(vordiv_to_uv_x) == size(vor) || throw(BoundsError)
    @boundscheck size(vordiv_to_uv1) == size(vor) || throw(BoundsError)
    @boundscheck size(vordiv_to_uv1) == size(vor) || throw(BoundsError)

    lm = 0
    @inbounds for m in 1:mmax                       # 1-based l, m, skip last column

        # DIAGONAL (separated to avoid access to l-1, m which is above the diagonal)
        lm += 1
        
        # div, vor contribution to meridional gradient
        ∂ζθ =  vordiv_to_uv2[lm]*vor[lm+1]          # lm-1 term is zero
        ∂Dθ = -vordiv_to_uv2[lm]*div[lm+1]          # lm-1 term is zero
        
        # the following is moved into the muladd        
        # ∂Dλ = im*vordiv_to_uv_x[lm]*div[lm]       # divergence contribution to zonal gradient
        # ∂ζλ = im*vordiv_to_uv_x[lm]*vor[lm]       # vorticity contribution to zonal gradient

        z = im*vordiv_to_uv_x[lm]
        U[lm] = muladd(z, div[lm], ∂ζθ)             # = ∂Dλ + ∂ζθ
        V[lm] = muladd(z, vor[lm], ∂Dθ)             # = ∂ζλ + ∂Dθ

        # BELOW DIAGONAL (all terms)
        for l in m+1:lmax                               # skip last row (lmax+2)
            lm += 1
            
            # div, vor contribution to meridional gradient
            # ∂ζθ = vordiv_to_uv2[lm]*vor[lm+1] - vordiv_to_uv1[lm]*vor[lm-1]
            # ∂Dθ = vordiv_to_uv1[lm]*div[lm-1] - vordiv_to_uv2[lm]*div[lm+1]
            ∂ζθ = muladd(vordiv_to_uv2[lm], vor[lm+1], -vordiv_to_uv1[lm]*vor[lm-1])
            ∂Dθ = muladd(vordiv_to_uv1[lm], div[lm-1], -vordiv_to_uv2[lm]*div[lm+1])

            # The following is moved into the muladd
            # ∂Dλ = im*vordiv_to_uv_x[lm]*div[lm]   # divergence contribution to zonal gradient
            # ∂ζλ = im*vordiv_to_uv_x[lm]*vor[lm]   # vorticity contribution to zonal gradient

            z = im*vordiv_to_uv_x[lm]
            U[lm] = muladd(z, div[lm], ∂ζθ)         # = ∂Dλ + ∂ζθ
            V[lm] = muladd(z, vor[lm], ∂Dθ)         # = ∂ζλ + ∂Dθ            
        end

        # SECOND LAST ROW (separated to imply that vor, div are zero in last row)
        lm += 1
        U[lm] = im*vordiv_to_uv_x[lm]*div[lm] - vordiv_to_uv1[lm]*vor[lm-1]
        V[lm] = im*vordiv_to_uv_x[lm]*vor[lm] + vordiv_to_uv1[lm]*div[lm-1]

        # LAST ROW (separated to avoid out-of-bounds access to lmax+3
        lm += 1
        U[lm] = -vordiv_to_uv1[lm]*vor[lm-1]        # only last term from 2nd last row
        V[lm] =  vordiv_to_uv1[lm]*div[lm-1]        # only last term from 2nd last row
    end

    # LAST COLUMN 
    @inbounds begin
        lm += 1                                     # second last row
        U[lm] = im*vordiv_to_uv_x[lm]*div[lm]       # other terms are zero
        V[lm] = im*vordiv_to_uv_x[lm]*vor[lm]       # other terms are zero

        lm += 1                                     # last row
        U[lm] = -vordiv_to_uv1[lm]*vor[lm-1]        # other terms are zero
        V[lm] =  vordiv_to_uv1[lm]*div[lm-1]        # other terms are zero
    end
end

"""
$(TYPEDSIGNATURES)
Laplace operator ∇² applied to the spectral coefficients `alms` in spherical
coordinates. The eigenvalues which are precomputed in `S`.
∇²! is the in-place version which directly stores the output in the first argument `∇²alms`.
Acts on the unit sphere, i.e. it omits any radius scaling as all inplace gradient operators.

Keyword arguments
=================

  - `add=true` adds the ∇²(alms) to the output
  - `flipsign=true` computes -∇²(alms) instead
  - `inverse=true` computes ∇⁻²(alms) instead

Default is `add=false`, `flipsign=false`, `inverse=false`. These options can be combined."""
function ∇²!(   ∇²alms::LowerTriangularMatrix{Complex{NF}}, # Output: (inverse) Laplacian of alms
                alms::LowerTriangularMatrix{Complex{NF}},   # Input: spectral coefficients
                S::SpectralTransform{NF};                   # precomputed eigenvalues
                add::Bool=false,                            # add to output array or overwrite
                flipsign::Bool=false,                       # -∇² or ∇²
                inverse::Bool=false,                        # ∇⁻² or ∇²
                ) where {NF<:AbstractFloat}

    @boundscheck size(alms) == size(∇²alms) || throw(BoundsError)
    lmax, mmax = size(alms) .- (1, 1)     # 0-based degree l, order m of the Legendre polynomials
    
    # use eigenvalues⁻¹/eigenvalues for ∇⁻²/∇² based but name both eigenvalues
    eigenvalues = inverse ? S.eigenvalues⁻¹ : S.eigenvalues
    @boundscheck length(eigenvalues) >= lmax+1 || throw(BoundsError)

    @inline kernel(o, a) = flipsign ? (add ? (o-a) : -a) :
                                      (add ? (o+a) :  a)

    lm = 0
    @inbounds for m in 1:mmax+1     # order m = 0:mmax but 1-based
        for l in m:lmax+1           # degree l = m:lmax but 1-based
            lm += 1
            ∇²alms[lm] = kernel(∇²alms[lm], alms[lm]*eigenvalues[l])
        end
    end

    return ∇²alms
end

"""
$(TYPEDSIGNATURES)
Laplace operator ∇² applied to input `alms`, using precomputed eigenvalues from `S`.
Acts on the unit sphere, i.e. it omits 1/radius^2 scaling unless
`radius` keyword argument is provided."""
function ∇²(
    alms::LowerTriangularMatrix,    # Input: spectral coefficients
    S::SpectralTransform;           # precomputed eigenvalues
    radius = DEFAULT_RADIUS,
)
    ∇²alms = similar(alms)
    ∇²!(∇²alms, alms, S, add=false, flipsign=false, inverse=false)

    if radius != 1
        ∇²alms .*= inv(radius^2)
    end

    return ∇²alms
end

"""
$(TYPEDSIGNATURES)
Returns the Laplace operator ∇² applied to input `alms`.
Acts on the unit sphere, i.e. it omits 1/radius^2 scaling unless
`radius` keyword argument is provided."""
∇²(alms::LowerTriangularMatrix; kwargs...) = ∇²(alms, SpectralTransform(alms); kwargs...)

"""
$(TYPEDSIGNATURES)
InverseLaplace operator ∇⁻² applied to input `alms`, using precomputed
eigenvalues from `S`. Acts on the unit sphere, i.e. it omits radius^2 scaling unless
`radius` keyword argument is provided."""
function ∇⁻²(
    ∇²alms::LowerTriangularMatrix,  # Input: spectral coefficients
    S::SpectralTransform;           # precomputed eigenvalues
    radius = DEFAULT_RADIUS,
)

    alms = similar(∇²alms)
    ∇⁻²!(alms, ∇²alms, S, add=false, flipsign=false)

    if radius != 1
        ∇²alms .*= radius^2
    end

    return alms
end

"""
$(TYPEDSIGNATURES)
Returns the inverse Laplace operator ∇⁻² applied to input `alms`.
Acts on the unit sphere, i.e. it omits radius^2 scaling unless
`radius` keyword argument is provided."""
∇⁻²(∇²alms::LowerTriangularMatrix; kwargs...) = ∇⁻²(∇²alms, SpectralTransform(∇²alms); kwargs...)

"""$(TYPEDSIGNATURES) Calls `∇²!(∇⁻²alms, alms, S; add, flipsign, inverse=true)`."""
function ∇⁻²!(  ∇⁻²alms::LowerTriangularMatrix{Complex{NF}}, # Output: inverse Laplacian of alms
                alms::LowerTriangularMatrix{Complex{NF}},   # Input: spectral coefficients
                S::SpectralTransform{NF};                   # precomputed eigenvalues
                add::Bool=false,                            # add to output array or overwrite
                flipsign::Bool=false,                       # -∇⁻² or ∇⁻²
                ) where {NF<:AbstractFloat}

    inverse = true
    return ∇²!(∇⁻²alms, alms, S; add, flipsign, inverse)
end

"""$(TYPEDSIGNATURES) Applies the gradient operator ∇ applied to input `p` and stores the result
in `dpdx` (zonal derivative) and `dpdy` (meridional derivative). The gradient operator acts
on the unit sphere and therefore omits the 1/radius scaling"""
function ∇!(dpdx::LowerTriangularMatrix{Complex{NF}},       # Output: zonal gradient
            dpdy::LowerTriangularMatrix{Complex{NF}},       # Output: meridional gradient
            p::LowerTriangularMatrix{Complex{NF}},          # Input: spectral coefficients
            S::SpectralTransform{NF}                        # includes precomputed arrays
            ) where {NF<:AbstractFloat}

    lmax, mmax = size(p) .- (1, 1)                            # 0-based, include last row
    @boundscheck size(p) == size(dpdx) || throw(BoundsError)
    @boundscheck size(p) == size(dpdy) || throw(BoundsError)

    (; grad_y1, grad_y2 ) = S

    lm = 0
    @inbounds for m in 1:mmax           # 1-based l, m, skip last column

        # DIAGONAL (separated to avoid access to l-1, m which is above the diagonal)
        lm += 1

        dpdx[lm] = (m-1)*im*p[lm]       # zonal gradient: d/dlon = *i*m
        dpdy[lm] = grad_y2[lm]*p[lm+1]  # meridional gradient: p[lm-1]=0 on diagonal
       
        # BELOW DIAGONAL (all terms)
        for l in m+1:lmax               # skip last row
            lm += 1
            
            dpdx[lm] = (m-1)*im*p[lm]
            dpdy[lm] = grad_y1[lm]*p[lm-1] + grad_y2[lm]*p[lm+1]
        end

        # LAST ROW (separated to avoid out-of-bounds access to lmax+2
        lm += 1
        dpdx[lm] = (m-1)*im*p[lm]
        dpdy[lm] = grad_y1[lm]*p[lm-1]  # only first term from 2nd last row
    end

    # LAST COLUMN
    @inbounds begin
        lm += 1 
        dpdx[lm] = mmax*im*p[lm]
        dpdy[lm] = grad_y2[lm]*p[lm+1]  # only 2nd term

        lm += 1
        dpdx[lm] = mmax*im*p[lm]
        dpdy[lm] = grad_y1[lm]*p[lm-1]  # only 1st term
    end

    return dpdx, dpdy
end

"""$(TYPEDSIGNATURES) The zonal and meridional gradient of `p`
using an existing `SpectralTransform` `S`. Acts on the unit sphere,
i.e. it omits 1/radius scaling unless `radius` keyword argument is provided."""
function ∇(p::LowerTriangularMatrix, S::SpectralTransform; radius = DEFAULT_RADIUS)
    dpdx = similar(p)
    dpdy = similar(p)
    ∇!(dpdx, dpdy, p, S)

    if radius != 1
        dpdx .*= inv(radius)
        dpdy .*= inv(radius)
    end

    return dpdx, dpdy
end

"""$(TYPEDSIGNATURES) The zonal and meridional gradient of `p`.
Precomputes a `SpectralTransform` `S`. Acts on the unit-sphere,
i.e. it omits 1/radius scaling unless `radius` keyword argument is provided."""
function ∇(p::LowerTriangularMatrix; kwargs...)
    S = SpectralTransform(p, one_more_degree=true)
    return ∇(p, S; kwargs...)
end

"""$(TYPEDSIGNATURES) The zonal and meridional gradient of `grid`.
Transform to spectral space, takes the gradient and unscales the 1/coslat
scaling in the gradient. Acts on the unit-sphere, i.e. it omits 1/radius scaling unless
`radius` keyword argument is provided. Makes use of an existing spectral transform `S`."""
function ∇(grid::AbstractGrid, S::SpectralTransform; kwargs...)
    p = spectral(grid, S)
    dpdx, dpdy = ∇(p, S; kwargs...)
    dpdx_grid = gridded(dpdx, S, unscale_coslat=true)
    dpdy_grid = gridded(dpdy, S, unscale_coslat=true)
    return dpdx_grid, dpdy_grid
end

"""$(TYPEDSIGNATURES) The zonal and meridional gradient of `grid`.
Transform to spectral space, takes the gradient and unscales the 1/coslat
scaling in the gradient. Acts on the unit-sphere, i.e. it omits 1/radius scaling unless
`radius` keyword argument is provided."""
function ∇(grid::AbstractGrid; kwargs...)
    S = SpectralTransform(grid, one_more_degree=true)
    return ∇(grid, S; kwargs...)
end