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Diagnostic Quantities

Pointwise and domain-integrated diagnostic quantities on Arakawa C-grids.

Kinetic Energy

finitevolx.kinetic_energy(u, v, mask=None)

Kinetic energy at T-points (cell centers) on an Arakawa C-grid.

Eq: ke[j, i] = 0.5 * (u²_on_T[j, i] + v²_on_T[j, i])

where u² and v² are averaged from face-points to T-points: u²_on_T[j, i] = 0.5 * (u[j, i+1/2]² + u[j, i-1/2]²) = 0.5 * (u[j, i]² + u[j, i-1]²) v²_on_T[j, i] = 0.5 * (v[j+1/2, i]² + v[j-1/2, i]²) = 0.5 * (v[j, i]² + v[j-1, i]²)

Args: u (Array): x-velocity at U-points (east faces), shape [Ny, Nx]. v (Array): y-velocity at V-points (north faces), shape [Ny, Nx]. mask (Array | None): optional binary mask at T-points. If provided, KE is zeroed where mask is 0.

Returns: ke (Array): kinetic energy at T-points, shape [Ny, Nx]. Ghost ring is zero; interior is [1:-1, 1:-1].

Source code in finitevolx/_src/operators/diagnostics.py 
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def kinetic_energy(
    u: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
    mask: Float[Array, "Ny Nx"] | None = None,
) -> Float[Array, "Ny Nx"]:
    """Kinetic energy at T-points (cell centers) on an Arakawa C-grid.

    Eq:
        ke[j, i] = 0.5 * (u²_on_T[j, i] + v²_on_T[j, i])

    where u² and v² are averaged from face-points to T-points:
        u²_on_T[j, i] = 0.5 * (u[j, i+1/2]² + u[j, i-1/2]²)
                       = 0.5 * (u[j, i]² + u[j, i-1]²)
        v²_on_T[j, i] = 0.5 * (v[j+1/2, i]² + v[j-1/2, i]²)
                       = 0.5 * (v[j, i]² + v[j-1, i]²)

    Args:
        u (Array): x-velocity at U-points (east faces), shape [Ny, Nx].
        v (Array): y-velocity at V-points (north faces), shape [Ny, Nx].
        mask (Array | None): optional binary mask at T-points.  If provided,
            KE is zeroed where mask is 0.

    Returns:
        ke (Array): kinetic energy at T-points, shape [Ny, Nx].
            Ghost ring is zero; interior is [1:-1, 1:-1].
    """
    dtype = jnp.result_type(u, v, 0.0)
    u_float = jnp.asarray(u, dtype=dtype)
    v_float = jnp.asarray(v, dtype=dtype)
    u2 = u_float**2
    v2 = v_float**2
    # u²_on_T[j, i] = 0.5 * (u²[j, i] + u²[j, i-1])  (east + west U-faces)
    # v²_on_T[j, i] = 0.5 * (v²[j, i] + v²[j-1, i])  (north + south V-faces)
    u2_on_T = avg_x_bwd(u2)
    v2_on_T = avg_y_bwd(v2)
    ke_int = 0.5 * (u2_on_T + v2_on_T)
    if mask is not None:
        ke_int = ke_int * mask[1:-1, 1:-1]
    out = interior(ke_int, u_float)
    return out

Bernoulli Potential

finitevolx.bernoulli_potential(h, u, v, gravity=GRAVITY)

Bernoulli potential at T-points on an Arakawa C-grid.

Eq: p[j, i] = ke[j, i] + g * h[j, i]

where ke is the kinetic energy at T-points.

Args: h (Array): layer thickness at T-points, shape [Ny, Nx]. u (Array): x-velocity at U-points (east faces), shape [Ny, Nx]. v (Array): y-velocity at V-points (north faces), shape [Ny, Nx]. gravity (float): gravitational acceleration. Default = 9.81.

Returns: p (Array): Bernoulli potential at T-points, shape [Ny, Nx]. Ghost ring is zero; interior is [1:-1, 1:-1].

Example: >>> u, v, h = ... >>> p = bernoulli_potential(h=h, u=u, v=v)

Source code in finitevolx/_src/operators/diagnostics.py 
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def bernoulli_potential(
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
    gravity: float = GRAVITY,
) -> Float[Array, "Ny Nx"]:
    """Bernoulli potential at T-points on an Arakawa C-grid.

    Eq:
        p[j, i] = ke[j, i] + g * h[j, i]

    where ke is the kinetic energy at T-points.

    Args:
        h (Array): layer thickness at T-points, shape [Ny, Nx].
        u (Array): x-velocity at U-points (east faces), shape [Ny, Nx].
        v (Array): y-velocity at V-points (north faces), shape [Ny, Nx].
        gravity (float): gravitational acceleration. Default = 9.81.

    Returns:
        p (Array): Bernoulli potential at T-points, shape [Ny, Nx].
            Ghost ring is zero; interior is [1:-1, 1:-1].

    Example:
        >>> u, v, h = ...
        >>> p = bernoulli_potential(h=h, u=u, v=v)
    """
    dtype = jnp.result_type(h, u, v, 0.0)
    h_float = jnp.asarray(h, dtype=dtype)
    ke = kinetic_energy(u=u, v=v)
    out = interior(ke[1:-1, 1:-1] + gravity * h_float[1:-1, 1:-1], h_float)
    return out

Relative Vorticity

finitevolx.relative_vorticity_cgrid(u, v, dx, dy)

Relative vorticity at X-points (corners) on an Arakawa C-grid.

zeta[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

This is a standalone functional form of the class-based :meth:Vorticity2D.relative_vorticity. Both share the same underlying implementation (:func:finitevolx._src.operators.difference._curl_2d).

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
dx float

Grid spacing in x.

 required
dy float

Grid spacing in y.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Relative vorticity at X-points. Ghost ring is zero.
Source code in finitevolx/_src/operators/diagnostics.py 
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def relative_vorticity_cgrid(
    u: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
    dx: float,
    dy: float,
) -> Float[Array, "Ny Nx"]:
    """Relative vorticity at X-points (corners) on an Arakawa C-grid.

    zeta[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

    This is a standalone functional form of the class-based
    :meth:`Vorticity2D.relative_vorticity`.  Both share the same
    underlying implementation
    (:func:`finitevolx._src.operators.difference._curl_2d`).

    Parameters
    ----------
    u : Float[Array, "Ny Nx"]
        x-velocity at U-points.
    v : Float[Array, "Ny Nx"]
        y-velocity at V-points.
    dx : float
        Grid spacing in x.
    dy : float
        Grid spacing in y.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Relative vorticity at X-points.  Ghost ring is zero.
    """
    return _curl_2d(u, v, dx, dy)

Potential Vorticity

finitevolx.potential_vorticity(omega, f, h)

Potential vorticity (pointwise).

q = (omega + f) / h

All inputs must live on the same grid points (typically X-points after interpolating f and h from T-points).

Parameters:

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

Relative vorticity.

 required
f Float[Array, 'Ny Nx']

Coriolis parameter.

 required
h Float[Array, 'Ny Nx']

Layer thickness (or depth).

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Potential vorticity. Where h == 0 the result is NaN.
Source code in finitevolx/_src/operators/diagnostics.py 
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def potential_vorticity(
    omega: Float[Array, "Ny Nx"],
    f: Float[Array, "Ny Nx"],
    h: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """Potential vorticity (pointwise).

    q = (omega + f) / h

    All inputs must live on the same grid points (typically X-points
    after interpolating ``f`` and ``h`` from T-points).

    Parameters
    ----------
    omega : Float[Array, "Ny Nx"]
        Relative vorticity.
    f : Float[Array, "Ny Nx"]
        Coriolis parameter.
    h : Float[Array, "Ny Nx"]
        Layer thickness (or depth).

    Returns
    -------
    Float[Array, "Ny Nx"]
        Potential vorticity.  Where ``h == 0`` the result is NaN.
    """
    return jnp.where(h == 0, jnp.nan, (omega + f) / h)

QG Potential Vorticity

finitevolx.qg_potential_vorticity(psi, f0, beta, dx, dy, y, y0)

QG potential vorticity at T-points (single layer).

q = nabla^2(psi) / f0 + beta * (y - y0) / f0

This computes the per-layer (barotropic) QG PV for a single [Ny, Nx] streamfunction. For multi-layer QG, lift this function with :func:~finitevolx.multilayer and subtract the cross-layer stretching term separately::

from finitevolx import multilayer, qg_potential_vorticity

# Per-layer PV (vmapped over the layer axis)
q = multilayer(lambda p: qg_potential_vorticity(p, f0, beta, dx, dy, y, y0))(
    psi
)  # psi: [nl, Ny, Nx] -> q: [nl, Ny, Nx]

# Cross-layer stretching (A couples layers, cannot be vmapped)
q = q - stretching_term(A, psi)

Parameters:

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

Streamfunction at T-points.

 required
f0 float

Reference Coriolis parameter.

 required
beta float

Meridional gradient of the Coriolis parameter.

 required
dx float

Grid spacing in x.

 required
dy float

Grid spacing in y.

 required
y Float[Array, 'Ny Nx']

Meridional coordinate at T-points.

 required
y0 float

Reference latitude.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

QG potential vorticity at T-points. Ghost ring is zero; interior is [1:-1, 1:-1].
Source code in finitevolx/_src/operators/diagnostics.py 
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def qg_potential_vorticity(
    psi: Float[Array, "Ny Nx"],
    f0: float,
    beta: float,
    dx: float,
    dy: float,
    y: Float[Array, "Ny Nx"],
    y0: float,
) -> Float[Array, "Ny Nx"]:
    """QG potential vorticity at T-points (single layer).

    q = nabla^2(psi) / f0 + beta * (y - y0) / f0

    This computes the per-layer (barotropic) QG PV for a single ``[Ny, Nx]``
    streamfunction.  For multi-layer QG, lift this function with
    :func:`~finitevolx.multilayer` and subtract the cross-layer stretching
    term separately::

        from finitevolx import multilayer, qg_potential_vorticity

        # Per-layer PV (vmapped over the layer axis)
        q = multilayer(lambda p: qg_potential_vorticity(p, f0, beta, dx, dy, y, y0))(
            psi
        )  # psi: [nl, Ny, Nx] -> q: [nl, Ny, Nx]

        # Cross-layer stretching (A couples layers, cannot be vmapped)
        q = q - stretching_term(A, psi)

    Parameters
    ----------
    psi : Float[Array, "Ny Nx"]
        Streamfunction at T-points.
    f0 : float
        Reference Coriolis parameter.
    beta : float
        Meridional gradient of the Coriolis parameter.
    dx : float
        Grid spacing in x.
    dy : float
        Grid spacing in y.
    y : Float[Array, "Ny Nx"]
        Meridional coordinate at T-points.
    y0 : float
        Reference latitude.

    Returns
    -------
    Float[Array, "Ny Nx"]
        QG potential vorticity at T-points.
        Ghost ring is zero; interior is ``[1:-1, 1:-1]``.
    """
    # Laplacian at T-points via centred second differences:
    d2x = (diff_x_fwd(psi) - diff_x_bwd(psi)) / dx**2
    d2y = (diff_y_fwd(psi) - diff_y_bwd(psi)) / dy**2
    # q[j,i] = (d2psi/dx2 + d2psi/dy2) / f0 + beta * (y - y0) / f0
    q_int = (d2x + d2y) / f0 + beta * (y[1:-1, 1:-1] - y0) / f0
    out = interior(q_int, psi)
    return out

Stretching Term (Multi-Layer QG)

finitevolx.stretching_term(A, psi)

Cross-layer stretching term for multi-layer QG models.

Computes A . psi — the vertical coupling that mixes information across layers. This is a cross-layer operation and cannot be vmapped; it should be used alongside :func:qg_potential_vorticity (which handles the per-layer Laplacian + beta terms via :func:~finitevolx.multilayer).

Parameters:

Name Type Description Default
A Float[Array, 'nl nl']

Coupling (stretching) matrix. Shape (nl, nl) where nl is the number of layers.

 required
psi Float[Array, 'nl Ny Nx']

Streamfunction at T-points for all layers.

 required

Returns:

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

Stretching contribution, same shape as psi. Ghost ring is zero; interior is [:, 1:-1, 1:-1].
Source code in finitevolx/_src/operators/diagnostics.py 
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def stretching_term(
    A: Float[Array, "nl nl"],
    psi: Float[Array, "nl Ny Nx"],
) -> Float[Array, "nl Ny Nx"]:
    """Cross-layer stretching term for multi-layer QG models.

    Computes ``A . psi`` — the vertical coupling that mixes information
    across layers.  This is a cross-layer operation and cannot be vmapped;
    it should be used alongside :func:`qg_potential_vorticity` (which
    handles the per-layer Laplacian + beta terms via
    :func:`~finitevolx.multilayer`).

    Parameters
    ----------
    A : Float[Array, "nl nl"]
        Coupling (stretching) matrix.  Shape ``(nl, nl)`` where ``nl``
        is the number of layers.
    psi : Float[Array, "nl Ny Nx"]
        Streamfunction at T-points for all layers.

    Returns
    -------
    Float[Array, "nl Ny Nx"]
        Stretching contribution, same shape as ``psi``.
        Ghost ring is zero; interior is ``[:, 1:-1, 1:-1]``.
    """
    # (A . psi)_k = sum_j A[k,j] * psi_j  for each spatial point
    # Flatten spatial dims, matmul along layer axis, reshape back.
    nl = A.shape[0]
    Ny, Nx = psi.shape[-2], psi.shape[-1]
    psi_flat = psi.reshape(nl, -1)  # [nl, Ny*Nx]
    Ap_flat = A @ psi_flat  # [nl, Ny*Nx]
    Ap = Ap_flat.reshape(nl, Ny, Nx)  # [nl, Ny, Nx]
    out = jnp.zeros_like(psi)
    out = out.at[:, 1:-1, 1:-1].set(Ap[:, 1:-1, 1:-1])
    return out

Strain

finitevolx.shear_strain(u, v, dx, dy)

Shear strain at X-points (corners) on an Arakawa C-grid.

Ss[j+1/2, i+1/2] = dv/dx + du/dy = (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
dx float

Grid spacing in x.

 required
dy float

Grid spacing in y.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Shear strain at X-points. Ghost ring is zero.
Source code in finitevolx/_src/operators/diagnostics.py 
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def shear_strain(
    u: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
    dx: float,
    dy: float,
) -> Float[Array, "Ny Nx"]:
    """Shear strain at X-points (corners) on an Arakawa C-grid.

    Ss[j+1/2, i+1/2] = dv/dx + du/dy
        = (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.
    dx : float
        Grid spacing in x.
    dy : float
        Grid spacing in y.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Shear strain at X-points.  Ghost ring is zero.
    """
    out = interior(diff_x_fwd(v) / dx + diff_y_fwd(u) / dy, u)
    return out

finitevolx.tensor_strain(u, v, dx, dy)

Tensor (normal) strain at T-points on an Arakawa C-grid.

Sn[j, i] = du/dx - dv/dy = (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
dx float

Grid spacing in x.

 required
dy float

Grid spacing in y.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Tensor strain at T-points. Ghost ring is zero.
Source code in finitevolx/_src/operators/diagnostics.py 
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def tensor_strain(
    u: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
    dx: float,
    dy: float,
) -> Float[Array, "Ny Nx"]:
    """Tensor (normal) strain at T-points on an Arakawa C-grid.

    Sn[j, i] = du/dx - dv/dy
        = (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.
    dx : float
        Grid spacing in x.
    dy : float
        Grid spacing in y.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Tensor strain at T-points.  Ghost ring is zero.
    """
    out = interior(diff_x_bwd(u) / dx - diff_y_bwd(v) / dy, u)
    return out

finitevolx.strain_magnitude_squared(sn, ss)

Squared strain magnitude (pointwise).

sigma^2 = Sn^2 + Ss^2

Inputs must live on the same grid points. If sn is at T-points and ss at X-points, interpolate one to the other before calling.

Parameters:

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

Tensor (normal) strain.

 required
ss Float[Array, 'Ny Nx']

Shear strain.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Squared strain magnitude.
Source code in finitevolx/_src/operators/diagnostics.py 
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def strain_magnitude_squared(
    sn: Float[Array, "Ny Nx"],
    ss: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """Squared strain magnitude (pointwise).

    sigma^2 = Sn^2 + Ss^2

    Inputs must live on the same grid points.  If ``sn`` is at T-points
    and ``ss`` at X-points, interpolate one to the other before calling.

    Parameters
    ----------
    sn : Float[Array, "Ny Nx"]
        Tensor (normal) strain.
    ss : Float[Array, "Ny Nx"]
        Shear strain.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Squared strain magnitude.
    """
    return sn**2 + ss**2

Okubo-Weiss Parameter

finitevolx.okubo_weiss(sn, ss, omega)

Okubo-Weiss parameter (pointwise).

OW = Sn^2 + Ss^2 - omega^2

All inputs must live on the same grid points.

Parameters:

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

Tensor (normal) strain.

 required
ss Float[Array, 'Ny Nx']

Shear strain.

 required
omega Float[Array, 'Ny Nx']

Relative vorticity.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Okubo-Weiss parameter. Positive = strain-dominated, negative = vorticity-dominated.
Source code in finitevolx/_src/operators/diagnostics.py 
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def okubo_weiss(
    sn: Float[Array, "Ny Nx"],
    ss: Float[Array, "Ny Nx"],
    omega: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """Okubo-Weiss parameter (pointwise).

    OW = Sn^2 + Ss^2 - omega^2

    All inputs must live on the same grid points.

    Parameters
    ----------
    sn : Float[Array, "Ny Nx"]
        Tensor (normal) strain.
    ss : Float[Array, "Ny Nx"]
        Shear strain.
    omega : Float[Array, "Ny Nx"]
        Relative vorticity.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Okubo-Weiss parameter.  Positive = strain-dominated,
        negative = vorticity-dominated.
    """
    return sn**2 + ss**2 - omega**2

Enstrophy

finitevolx.enstrophy(omega, mask=None)

Enstrophy (pointwise).

Z = 0.5 * omega^2

Parameters:

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

Relative vorticity (typically at X-points).

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

Binary mask. If provided, result is zeroed where mask is 0.

 None

Returns:

Type Description
Float[Array, 'Ny Nx']

Enstrophy at the same grid points as omega.
Source code in finitevolx/_src/operators/diagnostics.py 
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def enstrophy(
    omega: Float[Array, "Ny Nx"],
    mask: Float[Array, "Ny Nx"] | None = None,
) -> Float[Array, "Ny Nx"]:
    """Enstrophy (pointwise).

    Z = 0.5 * omega^2

    Parameters
    ----------
    omega : Float[Array, "Ny Nx"]
        Relative vorticity (typically at X-points).
    mask : Float[Array, "Ny Nx"] | None, optional
        Binary mask.  If provided, result is zeroed where mask is 0.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Enstrophy at the same grid points as *omega*.
    """
    out = 0.5 * omega**2
    if mask is not None:
        out = out * mask
    return out

finitevolx.potential_enstrophy(q, h, mask=None)

Potential enstrophy (pointwise).

PE = 0.5 * q^2 * h

Inputs must live on the same grid points.

Parameters:

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

Potential vorticity.

 required
h Float[Array, 'Ny Nx']

Layer thickness (or depth).

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

Binary mask. If provided, result is zeroed where mask is 0.

 None

Returns:

Type Description
Float[Array, 'Ny Nx']

Potential enstrophy.
Source code in finitevolx/_src/operators/diagnostics.py 
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def potential_enstrophy(
    q: Float[Array, "Ny Nx"],
    h: Float[Array, "Ny Nx"],
    mask: Float[Array, "Ny Nx"] | None = None,
) -> Float[Array, "Ny Nx"]:
    """Potential enstrophy (pointwise).

    PE = 0.5 * q^2 * h

    Inputs must live on the same grid points.

    Parameters
    ----------
    q : Float[Array, "Ny Nx"]
        Potential vorticity.
    h : Float[Array, "Ny Nx"]
        Layer thickness (or depth).
    mask : Float[Array, "Ny Nx"] | None, optional
        Binary mask.  If provided, result is zeroed where mask is 0.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Potential enstrophy.
    """
    out = 0.5 * q**2 * h
    if mask is not None:
        out = out * mask
    return out

Available Potential Energy

finitevolx.available_potential_energy(h, H, g_prime)

Available potential energy at T-points.

APE = 0.5 * g' * (h - H)^2

Parameters:

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

Layer thickness at T-points.

 required
H Float[Array, 'Ny Nx']

Mean (reference) layer thickness at T-points.

 required
g_prime float

Reduced gravity.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Available potential energy at T-points.
Source code in finitevolx/_src/operators/diagnostics.py 
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def available_potential_energy(
    h: Float[Array, "Ny Nx"],
    H: Float[Array, "Ny Nx"],
    g_prime: float,
) -> Float[Array, "Ny Nx"]:
    """Available potential energy at T-points.

    APE = 0.5 * g' * (h - H)^2

    Parameters
    ----------
    h : Float[Array, "Ny Nx"]
        Layer thickness at T-points.
    H : Float[Array, "Ny Nx"]
        Mean (reference) layer thickness at T-points.
    g_prime : float
        Reduced gravity.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Available potential energy at T-points.
    """
    eta = h - H
    return 0.5 * g_prime * eta**2

Domain-Integrated Quantities

finitevolx.total_energy(ke, ape, dx, dy)

Domain-integrated total energy.

E = sum_{interior}(KE + APE) * dx * dy

Parameters:

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

Kinetic energy at T-points.

 required
ape Float[Array, 'Ny Nx']

Available potential energy at T-points.

 required
dx float

Grid spacing in x.

 required
dy float

Grid spacing in y.

 required

Returns:

Type Description
Float[Array, '']

Scalar total energy.
Source code in finitevolx/_src/operators/diagnostics.py 
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def total_energy(
    ke: Float[Array, "Ny Nx"],
    ape: Float[Array, "Ny Nx"],
    dx: float,
    dy: float,
) -> Float[Array, ""]:
    """Domain-integrated total energy.

    E = sum_{interior}(KE + APE) * dx * dy

    Parameters
    ----------
    ke : Float[Array, "Ny Nx"]
        Kinetic energy at T-points.
    ape : Float[Array, "Ny Nx"]
        Available potential energy at T-points.
    dx : float
        Grid spacing in x.
    dy : float
        Grid spacing in y.

    Returns
    -------
    Float[Array, ""]
        Scalar total energy.
    """
    return jnp.sum(ke[1:-1, 1:-1] + ape[1:-1, 1:-1]) * dx * dy

finitevolx.total_enstrophy(ens, dx, dy)

Domain-integrated enstrophy.

Z = sum_{interior}(0.5 * omega^2) * dx * dy

Parameters:

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

Enstrophy field (e.g. from :func:enstrophy).

 required
dx float

Grid spacing in x.

 required
dy float

Grid spacing in y.

 required

Returns:

Type Description
Float[Array, '']

Scalar total enstrophy.
Source code in finitevolx/_src/operators/diagnostics.py 
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def total_enstrophy(
    ens: Float[Array, "Ny Nx"],
    dx: float,
    dy: float,
) -> Float[Array, ""]:
    """Domain-integrated enstrophy.

    Z = sum_{interior}(0.5 * omega^2) * dx * dy

    Parameters
    ----------
    ens : Float[Array, "Ny Nx"]
        Enstrophy field (e.g. from :func:`enstrophy`).
    dx : float
        Grid spacing in x.
    dy : float
        Grid spacing in y.

    Returns
    -------
    Float[Array, ""]
        Scalar total enstrophy.
    """
    return jnp.sum(ens[1:-1, 1:-1]) * dx * dy

Vertical Velocity

finitevolx.vertical_velocity(u, v, dx, dy, dz, mask=None)

Vertical velocity from the continuity equation.

Integrates the horizontal divergence from bottom (w=0) to top:

w[k+1/2] = w[k-1/2] - (du/dx + dv/dy)[k] * dz

where the horizontal divergence is computed at T-points for each z-level.

Parameters:

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

x-velocity at U-points, with ghost cells in all three dimensions.

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

y-velocity at V-points, with ghost cells in all three dimensions.

 required
dx float

Grid spacing in x.

 required
dy float

Grid spacing in y.

 required
dz float

Grid spacing in z.

 required
mask Float[Array, 'Nz Ny Nx'] | None

Binary mask at T-points. If provided, the horizontal divergence is zeroed where mask is 0 before vertical integration.

 None

Returns:

Type Description
Float[Array, 'Nzp1 Ny Nx']

Vertical velocity at W-points (cell interfaces in z). Shape is (Nz + 1, Ny, Nx). w[0] is the bottom boundary (zero) and w[Nz] is the top. Only the horizontal interior [1:-1, 1:-1] contains meaningful values.
Source code in finitevolx/_src/operators/diagnostics.py 
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def vertical_velocity(
    u: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
    dx: float,
    dy: float,
    dz: float,
    mask: Float[Array, "Nz Ny Nx"] | None = None,
) -> Float[Array, "Nzp1 Ny Nx"]:
    """Vertical velocity from the continuity equation.

    Integrates the horizontal divergence from bottom (w=0) to top:

        w[k+1/2] = w[k-1/2] - (du/dx + dv/dy)[k] * dz

    where the horizontal divergence is computed at T-points for each
    z-level.

    Parameters
    ----------
    u : Float[Array, "Nz Ny Nx"]
        x-velocity at U-points, with ghost cells in all three dimensions.
    v : Float[Array, "Nz Ny Nx"]
        y-velocity at V-points, with ghost cells in all three dimensions.
    dx : float
        Grid spacing in x.
    dy : float
        Grid spacing in y.
    dz : float
        Grid spacing in z.
    mask : Float[Array, "Nz Ny Nx"] | None, optional
        Binary mask at T-points.  If provided, the horizontal divergence
        is zeroed where mask is 0 before vertical integration.

    Returns
    -------
    Float[Array, "Nzp1 Ny Nx"]
        Vertical velocity at W-points (cell interfaces in z).
        Shape is ``(Nz + 1, Ny, Nx)``.  ``w[0]`` is the bottom boundary
        (zero) and ``w[Nz]`` is the top.  Only the horizontal interior
        ``[1:-1, 1:-1]`` contains meaningful values.
    """
    Nz, Ny, Nx = u.shape
    # Horizontal divergence at T-points for each z-level
    # du/dx[k, j, i] = (u[k, j, i] - u[k, j, i-1]) / dx
    # dv/dy[k, j, i] = (v[k, j, i] - v[k, j-1, i]) / dy
    du_dx = jnp.zeros_like(u)
    du_dx = du_dx.at[:, 1:-1, 1:-1].set((u[:, 1:-1, 1:-1] - u[:, 1:-1, :-2]) / dx)
    dv_dy = jnp.zeros_like(v)
    dv_dy = dv_dy.at[:, 1:-1, 1:-1].set((v[:, 1:-1, 1:-1] - v[:, :-2, 1:-1]) / dy)
    div_h = du_dx + dv_dy  # [Nz, Ny, Nx]

    if mask is not None:
        div_h = div_h * mask

    # Integrate bottom-to-top: w[0] = 0 (bottom)
    # w[k+1] = w[k] - div_h[k] * dz  for k = 0, ..., Nz-1
    w = jnp.zeros((Nz + 1, Ny, Nx), dtype=u.dtype)
    cumsum = jnp.cumsum(div_h, axis=0)  # [Nz, Ny, Nx]
    w = w.at[1:, :, :].set(-cumsum * dz)
    return w