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Difference Operators

Finite-difference operators for computing spatial derivatives on staggered Arakawa C-grids.

finitevolx.Difference1D

Bases: Module

Finite-difference operators on a 1-D Arakawa C-grid.

Parameters:

Name Type Description Default
grid ArakawaCGrid1D

The underlying 1-D grid.

 required
Source code in finitevolx/_src/operators/difference.py 
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class Difference1D(eqx.Module):
    """Finite-difference operators on a 1-D Arakawa C-grid.

    Parameters
    ----------
    grid : ArakawaCGrid1D
        The underlying 1-D grid.
    """

    grid: ArakawaCGrid1D

    def diff_x_T_to_U(self, h: Float[Array, "Nx"]) -> Float[Array, "Nx"]:
        """Forward difference in x: T-point -> U-point.

        dh_dx[i+1/2] = (h[i+1] - h[i]) / dx

        Parameters
        ----------
        h : Float[Array, "Nx"]
            Scalar field at T-points.

        Returns
        -------
        Float[Array, "Nx"]
            Forward x-difference at U-points, same shape as input.
        """
        out = interior(diff_x_fwd_1d(h) / self.grid.dx, h)
        return out

    def diff_x_U_to_T(self, u: Float[Array, "Nx"]) -> Float[Array, "Nx"]:
        """Backward difference in x: U-point -> T-point.

        du_dx[i] = (u[i+1/2] - u[i-1/2]) / dx

        Parameters
        ----------
        u : Float[Array, "Nx"]
            Velocity field at U-points.

        Returns
        -------
        Float[Array, "Nx"]
            Backward x-difference at T-points, same shape as input.
        """
        out = interior(diff_x_bwd_1d(u) / self.grid.dx, u)
        return out

    def laplacian(self, h: Float[Array, "Nx"]) -> Float[Array, "Nx"]:
        """Laplacian at T-points.

        nabla2_h[i] = (h[i+1] - 2*h[i] + h[i-1]) / dx^2

        Parameters
        ----------
        h : Float[Array, "Nx"]
            Scalar field at T-points.

        Returns
        -------
        Float[Array, "Nx"]
            Laplacian at T-points, same shape as input.
        """
        out = interior((diff_x_fwd_1d(h) - diff_x_bwd_1d(h)) / self.grid.dx**2, h)
        return out

diff_x_T_to_U(h)

Forward difference in x: T-point -> U-point.

dh_dx[i+1/2] = (h[i+1] - h[i]) / dx

Parameters:

Name Type Description Default
h Float[Array, Nx]

Scalar field at T-points.

 required

Returns:

Type Description
Float[Array, Nx]

Forward x-difference at U-points, same shape as input.
Source code in finitevolx/_src/operators/difference.py 
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def diff_x_T_to_U(self, h: Float[Array, "Nx"]) -> Float[Array, "Nx"]:
    """Forward difference in x: T-point -> U-point.

    dh_dx[i+1/2] = (h[i+1] - h[i]) / dx

    Parameters
    ----------
    h : Float[Array, "Nx"]
        Scalar field at T-points.

    Returns
    -------
    Float[Array, "Nx"]
        Forward x-difference at U-points, same shape as input.
    """
    out = interior(diff_x_fwd_1d(h) / self.grid.dx, h)
    return out

diff_x_U_to_T(u)

Backward difference in x: U-point -> T-point.

du_dx[i] = (u[i+1/2] - u[i-1/2]) / dx

Parameters:

Name Type Description Default
u Float[Array, Nx]

Velocity field at U-points.

 required

Returns:

Type Description
Float[Array, Nx]

Backward x-difference at T-points, same shape as input.
Source code in finitevolx/_src/operators/difference.py 
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def diff_x_U_to_T(self, u: Float[Array, "Nx"]) -> Float[Array, "Nx"]:
    """Backward difference in x: U-point -> T-point.

    du_dx[i] = (u[i+1/2] - u[i-1/2]) / dx

    Parameters
    ----------
    u : Float[Array, "Nx"]
        Velocity field at U-points.

    Returns
    -------
    Float[Array, "Nx"]
        Backward x-difference at T-points, same shape as input.
    """
    out = interior(diff_x_bwd_1d(u) / self.grid.dx, u)
    return out

laplacian(h)

Laplacian at T-points.

nabla2_h[i] = (h[i+1] - 2*h[i] + h[i-1]) / dx^2

Parameters:

Name Type Description Default
h Float[Array, Nx]

Scalar field at T-points.

 required

Returns:

Type Description
Float[Array, Nx]

Laplacian at T-points, same shape as input.
Source code in finitevolx/_src/operators/difference.py 
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def laplacian(self, h: Float[Array, "Nx"]) -> Float[Array, "Nx"]:
    """Laplacian at T-points.

    nabla2_h[i] = (h[i+1] - 2*h[i] + h[i-1]) / dx^2

    Parameters
    ----------
    h : Float[Array, "Nx"]
        Scalar field at T-points.

    Returns
    -------
    Float[Array, "Nx"]
        Laplacian at T-points, same shape as input.
    """
    out = interior((diff_x_fwd_1d(h) - diff_x_bwd_1d(h)) / self.grid.dx**2, h)
    return out

finitevolx.Difference2D

Bases: Module

Finite-difference operators on a 2-D Arakawa C-grid.

Parameters:

Name Type Description Default
grid ArakawaCGrid2D

The underlying 2-D grid.

 required
Source code in finitevolx/_src/operators/difference.py 
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class Difference2D(eqx.Module):
    """Finite-difference operators on a 2-D Arakawa C-grid.

    Parameters
    ----------
    grid : ArakawaCGrid2D
        The underlying 2-D grid.
    """

    grid: ArakawaCGrid2D

    # ------------------------------------------------------------------
    # Forward differences
    # ------------------------------------------------------------------

    def diff_x_T_to_U(self, h: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
        """Forward x-difference: T-point -> U-point.

        dh_dx[j, i+1/2] = (h[j, i+1] - h[j, i]) / dx

        Parameters
        ----------
        h : Float[Array, "Ny Nx"]
            Scalar field at T-points.

        Returns
        -------
        Float[Array, "Ny Nx"]
            Forward x-difference at U-points.
        """
        out = interior(diff_x_fwd(h) / self.grid.dx, h)
        return out

    def diff_y_T_to_V(self, h: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
        """Forward y-difference: T-point -> V-point.

        dh_dy[j+1/2, i] = (h[j+1, i] - h[j, i]) / dy

        Parameters
        ----------
        h : Float[Array, "Ny Nx"]
            Scalar field at T-points.

        Returns
        -------
        Float[Array, "Ny Nx"]
            Forward y-difference at V-points.
        """
        out = interior(diff_y_fwd(h) / self.grid.dy, h)
        return out

    def diff_y_U_to_X(self, u: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
        """Forward y-difference: U-point -> X-point (corner).

        du_dy[j+1/2, i+1/2] = (u[j+1, i+1/2] - u[j, i+1/2]) / dy

        Parameters
        ----------
        u : Float[Array, "Ny Nx"]
            Velocity field at U-points.

        Returns
        -------
        Float[Array, "Ny Nx"]
            Forward y-difference at X-points.
        """
        out = interior(diff_y_fwd(u) / self.grid.dy, u)
        return out

    def diff_x_V_to_X(self, v: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
        """Forward x-difference: V-point -> X-point (corner).

        dv_dx[j+1/2, i+1/2] = (v[j+1/2, i+1] - v[j+1/2, i]) / dx

        Parameters
        ----------
        v : Float[Array, "Ny Nx"]
            Velocity field at V-points.

        Returns
        -------
        Float[Array, "Ny Nx"]
            Forward x-difference at X-points.
        """
        out = interior(diff_x_fwd(v) / self.grid.dx, v)
        return out

    def diff_y_X_to_U(self, q: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
        """Backward y-difference: X-point -> U-point.

        dq_dy[j, i] = (q[j+1/2, i+1/2] - q[j-1/2, i+1/2]) / dy

        Parameters
        ----------
        q : Float[Array, "Ny Nx"]
            Scalar field at X-points.

        Returns
        -------
        Float[Array, "Ny Nx"]
            Backward y-difference at U-points.
        """
        out = interior(diff_y_bwd(q) / self.grid.dy, q)
        return out

    def diff_x_X_to_V(self, q: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
        """Backward x-difference: X-point -> V-point.

        dq_dx[j, i] = (q[j+1/2, i+1/2] - q[j+1/2, i-1/2]) / dx

        Parameters
        ----------
        q : Float[Array, "Ny Nx"]
            Scalar field at X-points.

        Returns
        -------
        Float[Array, "Ny Nx"]
            Backward x-difference at V-points.
        """
        out = interior(diff_x_bwd(q) / self.grid.dx, q)
        return out

    # ------------------------------------------------------------------
    # Backward differences
    # ------------------------------------------------------------------

    def diff_x_U_to_T(self, u: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
        """Backward x-difference: U-point -> T-point.

        du_dx[j, i] = (u[j, i+1/2] - u[j, i-1/2]) / dx

        Parameters
        ----------
        u : Float[Array, "Ny Nx"]
            Velocity field at U-points.

        Returns
        -------
        Float[Array, "Ny Nx"]
            Backward x-difference at T-points.
        """
        out = interior(diff_x_bwd(u) / self.grid.dx, u)
        return out

    def diff_y_V_to_T(self, v: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
        """Backward y-difference: V-point -> T-point.

        dv_dy[j, i] = (v[j+1/2, i] - v[j-1/2, i]) / dy

        Parameters
        ----------
        v : Float[Array, "Ny Nx"]
            Velocity field at V-points.

        Returns
        -------
        Float[Array, "Ny Nx"]
            Backward y-difference at T-points.
        """
        out = interior(diff_y_bwd(v) / self.grid.dy, v)
        return out

    # ------------------------------------------------------------------
    # Compound operators
    # ------------------------------------------------------------------

    def divergence(
        self,
        u: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """Divergence of (u, v) at T-points.

        delta[j, i] = du_dx[j, i] + dv_dy[j, i]
                    = (u[j, i+1/2] - u[j, i-1/2]) / dx
                    + (v[j+1/2, i] - v[j-1/2, i]) / dy

        Parameters
        ----------
        u : Float[Array, "Ny Nx"]
            x-velocity at U-points.
        v : Float[Array, "Ny Nx"]
            y-velocity at V-points.

        Returns
        -------
        Float[Array, "Ny Nx"]
            Divergence at T-points.
        """
        return _divergence_2d(u, v, self.grid.dx, self.grid.dy)

    def curl(
        self,
        u: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """Curl (relative vorticity) of (u, v) at X-points (corners).

        zeta[j+1/2, i+1/2] = dv_dx[j+1/2, i+1/2] - du_dy[j+1/2, i+1/2]
                            = (v[j+1/2, i+1] - v[j+1/2, i]) / dx
                            - (u[j+1, i+1/2] - u[j, i+1/2]) / dy

        Parameters
        ----------
        u : Float[Array, "Ny Nx"]
            x-velocity at U-points.
        v : Float[Array, "Ny Nx"]
            y-velocity at V-points.

        Returns
        -------
        Float[Array, "Ny Nx"]
            Relative vorticity at X-points (corners).
        """
        return _curl_2d(u, v, self.grid.dx, self.grid.dy)

    def laplacian(self, h: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
        """Laplacian at T-points.

        nabla2_h[j, i] = (h[j, i+1] - 2*h[j, i] + h[j, i-1]) / dx^2
                       + (h[j+1, i] - 2*h[j, i] + h[j-1, i]) / dy^2

        Parameters
        ----------
        h : Float[Array, "Ny Nx"]
            Scalar field at T-points.

        Returns
        -------
        Float[Array, "Ny Nx"]
            Laplacian at T-points.
        """
        # nabla2_h[j, i] = d^2h/dx^2 + d^2h/dy^2
        d2x = (diff_x_fwd(h) - diff_x_bwd(h)) / self.grid.dx**2
        d2y = (diff_y_fwd(h) - diff_y_bwd(h)) / self.grid.dy**2
        out = interior(d2x + d2y, h)
        return out

    def grad_perp(
        self,
        psi: Float[Array, "Ny Nx"],
        mask_u: Float[Array, "Ny Nx"] | None = None,
        mask_v: Float[Array, "Ny Nx"] | None = None,
    ) -> tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]:
        """Perpendicular gradient: T-point streamfunction to geostrophic velocity.

        Maps a streamfunction ψ at T-points to face-centred geostrophic
        velocities (u, v) = (-∂ψ/∂y, ∂ψ/∂x) on an Arakawa C-grid.

        Expanding the T→X bilinear interpolation into the backward X→U / X→V
        differences gives a compact stencil that reads the T-point ghost cells
        of ψ directly:

        u[j, i+1/2] = -(ψ[j+1,i] + ψ[j+1,i+1] - ψ[j-1,i] - ψ[j-1,i+1]) / (4·dy)
        v[j+1/2, i] =  (ψ[j,i+1] + ψ[j+1,i+1] - ψ[j,i-1] - ψ[j+1,i-1]) / (4·dx)

        The resulting (unmasked) velocity field is discretely non-divergent:
        div(u, v) = 0 at T-points. When masks are applied, the divergence-free
        property no longer holds in general.

        Parameters
        ----------
        psi : Float[Array, "Ny Nx"]
            Streamfunction at T-points.
        mask_u : Float[Array, "Ny Nx"] | None, optional
            Binary mask at U-points. If provided, u is zeroed where mask is 0.
        mask_v : Float[Array, "Ny Nx"] | None, optional
            Binary mask at V-points. If provided, v is zeroed where mask is 0.

        Returns
        -------
        tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]
            (u, v) — zonal velocity at U-points and meridional velocity at
            V-points.

        References
        ----------
        .. [1] louity/MQGeometry ``fd.py`` — ``grad_perp`` function.
           https://github.com/louity/MQGeometry/blob/main/fd.py
        .. [2] louity/qgsw-pytorch ``finite_diff.py`` — ``grad_perp`` function.
           https://github.com/louity/qgsw-pytorch/blob/main/src/finite_diff.py

        Examples
        --------
        >>> from finitevolx import ArakawaCGrid2D, Difference2D
        >>> grid = ArakawaCGrid2D.from_interior(8, 8, 1e3, 1e3)
        >>> diff = Difference2D(grid=grid)
        >>> psi = jnp.ones((grid.Ny, grid.Nx))
        >>> u, v = diff.grad_perp(psi)
        """
        # u[j, i+1/2] = -(ψ[j+1,i] + ψ[j+1,i+1] - ψ[j-1,i] - ψ[j-1,i+1]) / (4·dy)
        u = interior(
            -(psi[2:, 1:-1] + psi[2:, 2:] - psi[:-2, 1:-1] - psi[:-2, 2:])
            / (4.0 * self.grid.dy),
            psi,
        )

        # v[j+1/2, i] = (ψ[j,i+1] + ψ[j+1,i+1] - ψ[j,i-1] - ψ[j+1,i-1]) / (4·dx)
        v = interior(
            (psi[1:-1, 2:] + psi[2:, 2:] - psi[1:-1, :-2] - psi[2:, :-2])
            / (4.0 * self.grid.dx),
            psi,
        )

        if mask_u is not None:
            u = u * mask_u
        if mask_v is not None:
            v = v * mask_v

        return u, v

curl(u, v)

Curl (relative vorticity) of (u, v) at X-points (corners).

zeta[j+1/2, i+1/2] = dv_dx[j+1/2, i+1/2] - du_dy[j+1/2, i+1/2] = (v[j+1/2, i+1] - v[j+1/2, i]) / dx - (u[j+1, i+1/2] - u[j, i+1/2]) / dy

Parameters:

Name Type Description Default
u Float[Array, 'Ny Nx']

x-velocity at U-points.

 required
v Float[Array, 'Ny Nx']

y-velocity at V-points.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Relative vorticity at X-points (corners).
Source code in finitevolx/_src/operators/difference.py 
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def curl(
    self,
    u: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """Curl (relative vorticity) of (u, v) at X-points (corners).

    zeta[j+1/2, i+1/2] = dv_dx[j+1/2, i+1/2] - du_dy[j+1/2, i+1/2]
                        = (v[j+1/2, i+1] - v[j+1/2, i]) / dx
                        - (u[j+1, i+1/2] - u[j, i+1/2]) / dy

    Parameters
    ----------
    u : Float[Array, "Ny Nx"]
        x-velocity at U-points.
    v : Float[Array, "Ny Nx"]
        y-velocity at V-points.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Relative vorticity at X-points (corners).
    """
    return _curl_2d(u, v, self.grid.dx, self.grid.dy)

diff_x_T_to_U(h)

Forward x-difference: T-point -> U-point.

dh_dx[j, i+1/2] = (h[j, i+1] - h[j, i]) / dx

Parameters:

Name Type Description Default
h Float[Array, 'Ny Nx']

Scalar field at T-points.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Forward x-difference at U-points.
Source code in finitevolx/_src/operators/difference.py 
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def diff_x_T_to_U(self, h: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
    """Forward x-difference: T-point -> U-point.

    dh_dx[j, i+1/2] = (h[j, i+1] - h[j, i]) / dx

    Parameters
    ----------
    h : Float[Array, "Ny Nx"]
        Scalar field at T-points.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Forward x-difference at U-points.
    """
    out = interior(diff_x_fwd(h) / self.grid.dx, h)
    return out

diff_x_U_to_T(u)

Backward x-difference: U-point -> T-point.

du_dx[j, i] = (u[j, i+1/2] - u[j, i-1/2]) / dx

Parameters:

Name Type Description Default
u Float[Array, 'Ny Nx']

Velocity field at U-points.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Backward x-difference at T-points.
Source code in finitevolx/_src/operators/difference.py 
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def diff_x_U_to_T(self, u: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
    """Backward x-difference: U-point -> T-point.

    du_dx[j, i] = (u[j, i+1/2] - u[j, i-1/2]) / dx

    Parameters
    ----------
    u : Float[Array, "Ny Nx"]
        Velocity field at U-points.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Backward x-difference at T-points.
    """
    out = interior(diff_x_bwd(u) / self.grid.dx, u)
    return out

diff_x_V_to_X(v)

Forward x-difference: V-point -> X-point (corner).

dv_dx[j+1/2, i+1/2] = (v[j+1/2, i+1] - v[j+1/2, i]) / dx

Parameters:

Name Type Description Default
v Float[Array, 'Ny Nx']

Velocity field at V-points.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Forward x-difference at X-points.
Source code in finitevolx/_src/operators/difference.py 
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def diff_x_V_to_X(self, v: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
    """Forward x-difference: V-point -> X-point (corner).

    dv_dx[j+1/2, i+1/2] = (v[j+1/2, i+1] - v[j+1/2, i]) / dx

    Parameters
    ----------
    v : Float[Array, "Ny Nx"]
        Velocity field at V-points.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Forward x-difference at X-points.
    """
    out = interior(diff_x_fwd(v) / self.grid.dx, v)
    return out

diff_x_X_to_V(q)

Backward x-difference: X-point -> V-point.

dq_dx[j, i] = (q[j+1/2, i+1/2] - q[j+1/2, i-1/2]) / dx

Parameters:

Name Type Description Default
q Float[Array, 'Ny Nx']

Scalar field at X-points.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Backward x-difference at V-points.
Source code in finitevolx/_src/operators/difference.py 
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def diff_x_X_to_V(self, q: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
    """Backward x-difference: X-point -> V-point.

    dq_dx[j, i] = (q[j+1/2, i+1/2] - q[j+1/2, i-1/2]) / dx

    Parameters
    ----------
    q : Float[Array, "Ny Nx"]
        Scalar field at X-points.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Backward x-difference at V-points.
    """
    out = interior(diff_x_bwd(q) / self.grid.dx, q)
    return out

diff_y_T_to_V(h)

Forward y-difference: T-point -> V-point.

dh_dy[j+1/2, i] = (h[j+1, i] - h[j, i]) / dy

Parameters:

Name Type Description Default
h Float[Array, 'Ny Nx']

Scalar field at T-points.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Forward y-difference at V-points.
Source code in finitevolx/_src/operators/difference.py 
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def diff_y_T_to_V(self, h: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
    """Forward y-difference: T-point -> V-point.

    dh_dy[j+1/2, i] = (h[j+1, i] - h[j, i]) / dy

    Parameters
    ----------
    h : Float[Array, "Ny Nx"]
        Scalar field at T-points.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Forward y-difference at V-points.
    """
    out = interior(diff_y_fwd(h) / self.grid.dy, h)
    return out

diff_y_U_to_X(u)

Forward y-difference: U-point -> X-point (corner).

du_dy[j+1/2, i+1/2] = (u[j+1, i+1/2] - u[j, i+1/2]) / dy

Parameters:

Name Type Description Default
u Float[Array, 'Ny Nx']

Velocity field at U-points.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Forward y-difference at X-points.
Source code in finitevolx/_src/operators/difference.py 
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def diff_y_U_to_X(self, u: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
    """Forward y-difference: U-point -> X-point (corner).

    du_dy[j+1/2, i+1/2] = (u[j+1, i+1/2] - u[j, i+1/2]) / dy

    Parameters
    ----------
    u : Float[Array, "Ny Nx"]
        Velocity field at U-points.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Forward y-difference at X-points.
    """
    out = interior(diff_y_fwd(u) / self.grid.dy, u)
    return out

diff_y_V_to_T(v)

Backward y-difference: V-point -> T-point.

dv_dy[j, i] = (v[j+1/2, i] - v[j-1/2, i]) / dy

Parameters:

Name Type Description Default
v Float[Array, 'Ny Nx']

Velocity field at V-points.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Backward y-difference at T-points.
Source code in finitevolx/_src/operators/difference.py 
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def diff_y_V_to_T(self, v: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
    """Backward y-difference: V-point -> T-point.

    dv_dy[j, i] = (v[j+1/2, i] - v[j-1/2, i]) / dy

    Parameters
    ----------
    v : Float[Array, "Ny Nx"]
        Velocity field at V-points.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Backward y-difference at T-points.
    """
    out = interior(diff_y_bwd(v) / self.grid.dy, v)
    return out

diff_y_X_to_U(q)

Backward y-difference: X-point -> U-point.

dq_dy[j, i] = (q[j+1/2, i+1/2] - q[j-1/2, i+1/2]) / dy

Parameters:

Name Type Description Default
q Float[Array, 'Ny Nx']

Scalar field at X-points.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Backward y-difference at U-points.
Source code in finitevolx/_src/operators/difference.py 
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def diff_y_X_to_U(self, q: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
    """Backward y-difference: X-point -> U-point.

    dq_dy[j, i] = (q[j+1/2, i+1/2] - q[j-1/2, i+1/2]) / dy

    Parameters
    ----------
    q : Float[Array, "Ny Nx"]
        Scalar field at X-points.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Backward y-difference at U-points.
    """
    out = interior(diff_y_bwd(q) / self.grid.dy, q)
    return out

divergence(u, v)

Divergence of (u, v) at T-points.

delta[j, i] = du_dx[j, i] + dv_dy[j, i] = (u[j, i+1/2] - u[j, i-1/2]) / dx + (v[j+1/2, i] - v[j-1/2, i]) / dy

Parameters:

Name Type Description Default
u Float[Array, 'Ny Nx']

x-velocity at U-points.

 required
v Float[Array, 'Ny Nx']

y-velocity at V-points.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Divergence at T-points.
Source code in finitevolx/_src/operators/difference.py 
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def divergence(
    self,
    u: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """Divergence of (u, v) at T-points.

    delta[j, i] = du_dx[j, i] + dv_dy[j, i]
                = (u[j, i+1/2] - u[j, i-1/2]) / dx
                + (v[j+1/2, i] - v[j-1/2, i]) / dy

    Parameters
    ----------
    u : Float[Array, "Ny Nx"]
        x-velocity at U-points.
    v : Float[Array, "Ny Nx"]
        y-velocity at V-points.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Divergence at T-points.
    """
    return _divergence_2d(u, v, self.grid.dx, self.grid.dy)

grad_perp(psi, mask_u=None, mask_v=None)

Perpendicular gradient: T-point streamfunction to geostrophic velocity.

Maps a streamfunction ψ at T-points to face-centred geostrophic velocities (u, v) = (-∂ψ/∂y, ∂ψ/∂x) on an Arakawa C-grid.

Expanding the T→X bilinear interpolation into the backward X→U / X→V differences gives a compact stencil that reads the T-point ghost cells of ψ directly:

u[j, i+1/2] = -(ψ[j+1,i] + ψ[j+1,i+1] - ψ[j-1,i] - ψ[j-1,i+1]) / (4·dy) v[j+1/2, i] = (ψ[j,i+1] + ψ[j+1,i+1] - ψ[j,i-1] - ψ[j+1,i-1]) / (4·dx)

The resulting (unmasked) velocity field is discretely non-divergent: div(u, v) = 0 at T-points. When masks are applied, the divergence-free property no longer holds in general.

Parameters:

Name Type Description Default
psi Float[Array, 'Ny Nx']

Streamfunction at T-points.

 required
mask_u Float[Array, 'Ny Nx'] | None

Binary mask at U-points. If provided, u is zeroed where mask is 0.

 None
mask_v Float[Array, 'Ny Nx'] | None

Binary mask at V-points. If provided, v is zeroed where mask is 0.

 None

Returns:

Type Description
tuple[Float[Array, 'Ny Nx'], Float[Array, 'Ny Nx']]

(u, v) — zonal velocity at U-points and meridional velocity at V-points.
References

.. [1] louity/MQGeometry fd.py — grad_perp function. https://github.com/louity/MQGeometry/blob/main/fd.py .. [2] louity/qgsw-pytorch finite_diff.py — grad_perp function. https://github.com/louity/qgsw-pytorch/blob/main/src/finite_diff.py

Examples:

>>> from finitevolx import ArakawaCGrid2D, Difference2D
>>> grid = ArakawaCGrid2D.from_interior(8, 8, 1e3, 1e3)
>>> diff = Difference2D(grid=grid)
>>> psi = jnp.ones((grid.Ny, grid.Nx))
>>> u, v = diff.grad_perp(psi)
Source code in finitevolx/_src/operators/difference.py 
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def grad_perp(
    self,
    psi: Float[Array, "Ny Nx"],
    mask_u: Float[Array, "Ny Nx"] | None = None,
    mask_v: Float[Array, "Ny Nx"] | None = None,
) -> tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]:
    """Perpendicular gradient: T-point streamfunction to geostrophic velocity.

    Maps a streamfunction ψ at T-points to face-centred geostrophic
    velocities (u, v) = (-∂ψ/∂y, ∂ψ/∂x) on an Arakawa C-grid.

    Expanding the T→X bilinear interpolation into the backward X→U / X→V
    differences gives a compact stencil that reads the T-point ghost cells
    of ψ directly:

    u[j, i+1/2] = -(ψ[j+1,i] + ψ[j+1,i+1] - ψ[j-1,i] - ψ[j-1,i+1]) / (4·dy)
    v[j+1/2, i] =  (ψ[j,i+1] + ψ[j+1,i+1] - ψ[j,i-1] - ψ[j+1,i-1]) / (4·dx)

    The resulting (unmasked) velocity field is discretely non-divergent:
    div(u, v) = 0 at T-points. When masks are applied, the divergence-free
    property no longer holds in general.

    Parameters
    ----------
    psi : Float[Array, "Ny Nx"]
        Streamfunction at T-points.
    mask_u : Float[Array, "Ny Nx"] | None, optional
        Binary mask at U-points. If provided, u is zeroed where mask is 0.
    mask_v : Float[Array, "Ny Nx"] | None, optional
        Binary mask at V-points. If provided, v is zeroed where mask is 0.

    Returns
    -------
    tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]
        (u, v) — zonal velocity at U-points and meridional velocity at
        V-points.

    References
    ----------
    .. [1] louity/MQGeometry ``fd.py`` — ``grad_perp`` function.
       https://github.com/louity/MQGeometry/blob/main/fd.py
    .. [2] louity/qgsw-pytorch ``finite_diff.py`` — ``grad_perp`` function.
       https://github.com/louity/qgsw-pytorch/blob/main/src/finite_diff.py

    Examples
    --------
    >>> from finitevolx import ArakawaCGrid2D, Difference2D
    >>> grid = ArakawaCGrid2D.from_interior(8, 8, 1e3, 1e3)
    >>> diff = Difference2D(grid=grid)
    >>> psi = jnp.ones((grid.Ny, grid.Nx))
    >>> u, v = diff.grad_perp(psi)
    """
    # u[j, i+1/2] = -(ψ[j+1,i] + ψ[j+1,i+1] - ψ[j-1,i] - ψ[j-1,i+1]) / (4·dy)
    u = interior(
        -(psi[2:, 1:-1] + psi[2:, 2:] - psi[:-2, 1:-1] - psi[:-2, 2:])
        / (4.0 * self.grid.dy),
        psi,
    )

    # v[j+1/2, i] = (ψ[j,i+1] + ψ[j+1,i+1] - ψ[j,i-1] - ψ[j+1,i-1]) / (4·dx)
    v = interior(
        (psi[1:-1, 2:] + psi[2:, 2:] - psi[1:-1, :-2] - psi[2:, :-2])
        / (4.0 * self.grid.dx),
        psi,
    )

    if mask_u is not None:
        u = u * mask_u
    if mask_v is not None:
        v = v * mask_v

    return u, v

laplacian(h)

Laplacian at T-points.

nabla2_h[j, i] = (h[j, i+1] - 2h[j, i] + h[j, i-1]) / dx^2 + (h[j+1, i] - 2h[j, i] + h[j-1, i]) / dy^2

Parameters:

Name Type Description Default
h Float[Array, 'Ny Nx']

Scalar field at T-points.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Laplacian at T-points.
Source code in finitevolx/_src/operators/difference.py 
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def laplacian(self, h: Float[Array, "Ny Nx"]) -> Float[Array, "Ny Nx"]:
    """Laplacian at T-points.

    nabla2_h[j, i] = (h[j, i+1] - 2*h[j, i] + h[j, i-1]) / dx^2
                   + (h[j+1, i] - 2*h[j, i] + h[j-1, i]) / dy^2

    Parameters
    ----------
    h : Float[Array, "Ny Nx"]
        Scalar field at T-points.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Laplacian at T-points.
    """
    # nabla2_h[j, i] = d^2h/dx^2 + d^2h/dy^2
    d2x = (diff_x_fwd(h) - diff_x_bwd(h)) / self.grid.dx**2
    d2y = (diff_y_fwd(h) - diff_y_bwd(h)) / self.grid.dy**2
    out = interior(d2x + d2y, h)
    return out

finitevolx.Difference3D

Bases: Module

Finite-difference operators on a 3-D Arakawa C-grid.

Operates on the horizontal (y, x) plane for each z-level. The 3-D array shape is [Nz, Ny, Nx].

Parameters:

Name Type Description Default
grid ArakawaCGrid3D

The underlying 3-D grid.

 required
Source code in finitevolx/_src/operators/difference.py 
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class Difference3D(eqx.Module):
    """Finite-difference operators on a 3-D Arakawa C-grid.

    Operates on the horizontal (y, x) plane for each z-level.
    The 3-D array shape is [Nz, Ny, Nx].

    Parameters
    ----------
    grid : ArakawaCGrid3D
        The underlying 3-D grid.
    """

    grid: ArakawaCGrid3D

    def diff_x_T_to_U(self, h: Float[Array, "Nz Ny Nx"]) -> Float[Array, "Nz Ny Nx"]:
        """Forward x-difference over all z-levels: T -> U.

        dh_dx[k, j, i+1/2] = (h[k, j, i+1] - h[k, j, i]) / dx

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Scalar field at T-points.

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            Forward x-difference at U-points.
        """
        out = interior(diff_x_fwd_3d(h) / self.grid.dx, h)
        return out

    def diff_y_T_to_V(self, h: Float[Array, "Nz Ny Nx"]) -> Float[Array, "Nz Ny Nx"]:
        """Forward y-difference over all z-levels: T -> V.

        dh_dy[k, j+1/2, i] = (h[k, j+1, i] - h[k, j, i]) / dy

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Scalar field at T-points.

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            Forward y-difference at V-points.
        """
        out = interior(diff_y_fwd_3d(h) / self.grid.dy, h)
        return out

    def diff_x_U_to_T(self, u: Float[Array, "Nz Ny Nx"]) -> Float[Array, "Nz Ny Nx"]:
        """Backward x-difference over all z-levels: U -> T.

        du_dx[k, j, i] = (u[k, j, i+1/2] - u[k, j, i-1/2]) / dx

        Parameters
        ----------
        u : Float[Array, "Nz Ny Nx"]
            x-velocity at U-points.

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            Backward x-difference at T-points.
        """
        out = interior(diff_x_bwd_3d(u) / self.grid.dx, u)
        return out

    def diff_y_V_to_T(self, v: Float[Array, "Nz Ny Nx"]) -> Float[Array, "Nz Ny Nx"]:
        """Backward y-difference over all z-levels: V -> T.

        dv_dy[k, j, i] = (v[k, j+1/2, i] - v[k, j-1/2, i]) / dy

        Parameters
        ----------
        v : Float[Array, "Nz Ny Nx"]
            y-velocity at V-points.

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            Backward y-difference at T-points.
        """
        out = interior(diff_y_bwd_3d(v) / self.grid.dy, v)
        return out

    def divergence(
        self,
        u: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """Horizontal divergence at T-points over all z-levels.

        delta[k, j, i] = du_dx[k, j, i] + dv_dy[k, j, i]

        Parameters
        ----------
        u : Float[Array, "Nz Ny Nx"]
            x-velocity at U-points.
        v : Float[Array, "Nz Ny Nx"]
            y-velocity at V-points.

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            Divergence at T-points.
        """
        return self.diff_x_U_to_T(u) + self.diff_y_V_to_T(v)

    def laplacian(self, h: Float[Array, "Nz Ny Nx"]) -> Float[Array, "Nz Ny Nx"]:
        """Horizontal Laplacian at T-points over all z-levels.

        nabla2_h[k, j, i] = (h[k, j, i+1] - 2*h[k, j, i] + h[k, j, i-1]) / dx^2
                           + (h[k, j+1, i] - 2*h[k, j, i] + h[k, j-1, i]) / dy^2

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Scalar field at T-points.

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            Laplacian at T-points.
        """
        d2x = (diff_x_fwd_3d(h) - diff_x_bwd_3d(h)) / self.grid.dx**2
        d2y = (diff_y_fwd_3d(h) - diff_y_bwd_3d(h)) / self.grid.dy**2
        out = interior(d2x + d2y, h)
        return out

diff_x_T_to_U(h)

Forward x-difference over all z-levels: T -> U.

dh_dx[k, j, i+1/2] = (h[k, j, i+1] - h[k, j, i]) / dx

Parameters:

Name Type Description Default
h Float[Array, 'Nz Ny Nx']

Scalar field at T-points.

 required

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

Forward x-difference at U-points.
Source code in finitevolx/_src/operators/difference.py 
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def diff_x_T_to_U(self, h: Float[Array, "Nz Ny Nx"]) -> Float[Array, "Nz Ny Nx"]:
    """Forward x-difference over all z-levels: T -> U.

    dh_dx[k, j, i+1/2] = (h[k, j, i+1] - h[k, j, i]) / dx

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Scalar field at T-points.

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        Forward x-difference at U-points.
    """
    out = interior(diff_x_fwd_3d(h) / self.grid.dx, h)
    return out

diff_x_U_to_T(u)

Backward x-difference over all z-levels: U -> T.

du_dx[k, j, i] = (u[k, j, i+1/2] - u[k, j, i-1/2]) / dx

Parameters:

Name Type Description Default
u Float[Array, 'Nz Ny Nx']

x-velocity at U-points.

 required

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

Backward x-difference at T-points.
Source code in finitevolx/_src/operators/difference.py 
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def diff_x_U_to_T(self, u: Float[Array, "Nz Ny Nx"]) -> Float[Array, "Nz Ny Nx"]:
    """Backward x-difference over all z-levels: U -> T.

    du_dx[k, j, i] = (u[k, j, i+1/2] - u[k, j, i-1/2]) / dx

    Parameters
    ----------
    u : Float[Array, "Nz Ny Nx"]
        x-velocity at U-points.

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        Backward x-difference at T-points.
    """
    out = interior(diff_x_bwd_3d(u) / self.grid.dx, u)
    return out

diff_y_T_to_V(h)

Forward y-difference over all z-levels: T -> V.

dh_dy[k, j+1/2, i] = (h[k, j+1, i] - h[k, j, i]) / dy

Parameters:

Name Type Description Default
h Float[Array, 'Nz Ny Nx']

Scalar field at T-points.

 required

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

Forward y-difference at V-points.
Source code in finitevolx/_src/operators/difference.py 
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def diff_y_T_to_V(self, h: Float[Array, "Nz Ny Nx"]) -> Float[Array, "Nz Ny Nx"]:
    """Forward y-difference over all z-levels: T -> V.

    dh_dy[k, j+1/2, i] = (h[k, j+1, i] - h[k, j, i]) / dy

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Scalar field at T-points.

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        Forward y-difference at V-points.
    """
    out = interior(diff_y_fwd_3d(h) / self.grid.dy, h)
    return out

diff_y_V_to_T(v)

Backward y-difference over all z-levels: V -> T.

dv_dy[k, j, i] = (v[k, j+1/2, i] - v[k, j-1/2, i]) / dy

Parameters:

Name Type Description Default
v Float[Array, 'Nz Ny Nx']

y-velocity at V-points.

 required

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

Backward y-difference at T-points.
Source code in finitevolx/_src/operators/difference.py 
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def diff_y_V_to_T(self, v: Float[Array, "Nz Ny Nx"]) -> Float[Array, "Nz Ny Nx"]:
    """Backward y-difference over all z-levels: V -> T.

    dv_dy[k, j, i] = (v[k, j+1/2, i] - v[k, j-1/2, i]) / dy

    Parameters
    ----------
    v : Float[Array, "Nz Ny Nx"]
        y-velocity at V-points.

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        Backward y-difference at T-points.
    """
    out = interior(diff_y_bwd_3d(v) / self.grid.dy, v)
    return out

divergence(u, v)

Horizontal divergence at T-points over all z-levels.

delta[k, j, i] = du_dx[k, j, i] + dv_dy[k, j, i]

Parameters:

Name Type Description Default
u Float[Array, 'Nz Ny Nx']

x-velocity at U-points.

 required
v Float[Array, 'Nz Ny Nx']

y-velocity at V-points.

 required

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

Divergence at T-points.
Source code in finitevolx/_src/operators/difference.py 
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def divergence(
    self,
    u: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """Horizontal divergence at T-points over all z-levels.

    delta[k, j, i] = du_dx[k, j, i] + dv_dy[k, j, i]

    Parameters
    ----------
    u : Float[Array, "Nz Ny Nx"]
        x-velocity at U-points.
    v : Float[Array, "Nz Ny Nx"]
        y-velocity at V-points.

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        Divergence at T-points.
    """
    return self.diff_x_U_to_T(u) + self.diff_y_V_to_T(v)

laplacian(h)

Horizontal Laplacian at T-points over all z-levels.

nabla2_h[k, j, i] = (h[k, j, i+1] - 2h[k, j, i] + h[k, j, i-1]) / dx^2 + (h[k, j+1, i] - 2h[k, j, i] + h[k, j-1, i]) / dy^2

Parameters:

Name Type Description Default
h Float[Array, 'Nz Ny Nx']

Scalar field at T-points.

 required

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

Laplacian at T-points.
Source code in finitevolx/_src/operators/difference.py 
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def laplacian(self, h: Float[Array, "Nz Ny Nx"]) -> Float[Array, "Nz Ny Nx"]:
    """Horizontal Laplacian at T-points over all z-levels.

    nabla2_h[k, j, i] = (h[k, j, i+1] - 2*h[k, j, i] + h[k, j, i-1]) / dx^2
                       + (h[k, j+1, i] - 2*h[k, j, i] + h[k, j-1, i]) / dy^2

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Scalar field at T-points.

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        Laplacian at T-points.
    """
    d2x = (diff_x_fwd_3d(h) - diff_x_bwd_3d(h)) / self.grid.dx**2
    d2y = (diff_y_fwd_3d(h) - diff_y_bwd_3d(h)) / self.grid.dy**2
    out = interior(d2x + d2y, h)
    return out