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Spectral Derivative Operators

SpectralDerivative1D

Bases: Module

1D Spectral derivative operator using the Fast Fourier Transform (FFT).

This class provides methods to differentiate fields periodically using spectral accuracy. It leverages the property that the Fourier Transform (FT) turns differentiation into multiplication by the wavenumber vector.

Mathematical Formulation:

Given a periodic field u(x) on a domain of length L with N points: u(x) = Σ u_hat(k) * exp(i * k * x)

The n-th derivative is: d^n u / dx^n = Σ (i * k)^n * u_hat(k) * exp(i * k * x)

where k are the discrete wavenumbers: k = 2 * pi * n / L.

Dealiasing

Linear operators (__call__, gradient, laplacian, biharmonic, hyperviscosity, inverse_laplacian) act on every resolved mode and never apply the grid's 2/3 mask, so e.g. laplacian(inverse_laplacian(u)) == u. Use :meth:apply_dealias to truncate nonlinear products (gh-91).

Dtypes

Physical-space inputs must be real; complex input raises TypeError (gh-93). With spectral=True the input is complex Fourier coefficients. Results are returned in physical space as real arrays.

Attributes:

Name Type Description
grid FourierGrid1D

The 1D grid object containing wavenumbers k [N].
Source code in spectraldiffx/_src/fourier/operators.py 
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class SpectralDerivative1D(eqx.Module):
    """
    1D Spectral derivative operator using the Fast Fourier Transform (FFT).

    This class provides methods to differentiate fields periodically using
    spectral accuracy. It leverages the property that the Fourier Transform (FT)
    turns differentiation into multiplication by the wavenumber vector.

    Mathematical Formulation:
    -------------------------
    Given a periodic field u(x) on a domain of length L with N points:
        u(x) = Σ u_hat(k) * exp(i * k * x)

    The n-th derivative is:
        d^n u / dx^n = Σ (i * k)^n * u_hat(k) * exp(i * k * x)

    where k are the discrete wavenumbers: k = 2 * pi * n / L.

    Dealiasing
    ----------
    Linear operators (``__call__``, ``gradient``, ``laplacian``,
    ``biharmonic``, ``hyperviscosity``, ``inverse_laplacian``) act on every
    resolved mode and never apply the grid's 2/3 mask, so e.g.
    ``laplacian(inverse_laplacian(u)) == u``.
    Use :meth:`apply_dealias` to truncate nonlinear products (gh-91).

    Dtypes
    ------
    Physical-space inputs must be real; complex input raises ``TypeError``
    (gh-93). With ``spectral=True`` the input is complex Fourier
    coefficients. Results are returned in physical space as real arrays.

    Attributes
    ----------
        grid : FourierGrid1D
            The 1D grid object containing wavenumbers k [N].
    """

    grid: FourierGrid1D

    def __call__(
        self, u: Array, order: int = 1, spectral: bool = False
    ) -> Float[Array, "N"]:
        """
        Compute the n-th derivative of a field.

        Parameters
        ----------
        u : Array [N]
            The input field. If spectral=False, this is physical space.
            If spectral=True, this is complex Fourier coefficients.
        order : int, optional
            The order of the derivative (1=first, 2=second, etc.). Default is 1.
        spectral : bool, optional
            Whether the input 'u' is already in Fourier space. Default is False.

        Returns
        -------
        du_dx : Array [N]
            The n-th derivative in physical space.
        """
        # 1. Obtain spectral coefficients u_hat [N]
        u_hat = u if spectral else self.grid.transform(_real(u))

        # 2. Multiply by (i*k)^order [N]. Linear operators keep every
        # resolved mode; only products are 2/3-truncated (gh-91).
        k = self.grid.k
        du_hat = (1j * k) ** order * u_hat

        # 3. Inverse Transform back to physical space [N]
        return self.grid.transform(du_hat, inverse=True).real

    def gradient(self, u: Array, spectral: bool = False) -> Array:
        """
        Compute the first derivative (gradient) du/dx.

        Parameters
        ----------
        u : Array [N]
            Input field.
        spectral : bool, optional
            If True, treats u as Fourier coefficients.
        """
        return self(u, order=1, spectral=spectral)

    def laplacian(self, u: Array, spectral: bool = False) -> Array:
        """
        Compute the second derivative (Laplacian) d^2u/dx^2.

        Operation in Fourier Space:
            lap_hat = -(k^2) * u_hat
        """
        u_hat = u if spectral else self.grid.transform(_real(u))
        k = self.grid.k
        lap_hat = -(k**2) * u_hat
        return self.grid.transform(lap_hat, inverse=True).real

    def biharmonic(self, u: Array, spectral: bool = False) -> Float[Array, "N"]:
        """Compute the biharmonic operator: d^4u/dx^4.

        In Fourier space: k^4 * u_hat.
        """
        u_hat = u if spectral else self.grid.transform(_real(u))
        k = self.grid.k
        bih_hat = k**4 * u_hat
        return self.grid.transform(bih_hat, inverse=True).real

    def hyperviscosity(
        self, u: Array, nu: float, order: int = 2, spectral: bool = False
    ) -> Float[Array, "N"]:
        """Compute the hyperviscosity term: (-1)^{n+1} * nu * d^{2n}u/dx^{2n}.

        This is always dissipative: the spectral multiplier is
        ``-nu * |k|^{2n}``, regardless of *order*.

        Parameters
        ----------
        u : Array [N]
            Input field.
        nu : float
            Hyperviscosity coefficient.
        order : int
            Order n of the hyperviscosity (n=1 is regular diffusion,
            n=2 is biharmonic diffusion). Default: 2.
        spectral : bool
            Whether u is in Fourier space.

        Raises
        ------
        ValueError
            If ``order < 1`` or ``nu < 0``.
        """
        _validate_hyperviscosity(nu, order)
        u_hat = u if spectral else self.grid.transform(_real(u))
        k = self.grid.k
        hyp_hat = -nu * k ** (2 * order) * u_hat
        return self.grid.transform(hyp_hat, inverse=True).real

    def inverse_laplacian(self, u: Array, spectral: bool = False) -> Float[Array, "N"]:
        """Compute the inverse Laplacian: nabla^{-2} u (1D Poisson solve).

        On a periodic domain, d^2 psi / dx^2 = u has a solution only if u
        has zero spatial mean.  This implementation removes the k=0 component
        of u and returns the unique zero-mean solution psi, i.e. it solves
        d^2 psi / dx^2 = u - mean(u) with psi_hat[k=0] = 0.
        """
        u_hat = u if spectral else self.grid.transform(_real(u))
        k = self.grid.k
        k2 = k**2
        k2_safe = jnp.where(k2 == 0, 1.0, k2)
        psi_hat = -u_hat / k2_safe
        psi_hat = jnp.where(k2 == 0, 0.0, psi_hat)
        return self.grid.transform(psi_hat, inverse=True).real

    def apply_dealias(self, u: Array, spectral: bool = False) -> Array:
        """
        Apply the 2/3 dealiasing rule mask to a field.

        This zeros out the top 1/3 of the spectrum to prevent aliasing
        errors in nonlinear products (e.g., u * du/dx).
        """
        u_hat = u if spectral else self.grid.transform(_real(u))
        mask = self.grid.dealias_filter()
        u_hat_f = u_hat * mask
        return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True).real

Functions

__call__(u, order=1, spectral=False)

Compute the n-th derivative of a field.

Parameters:

Name Type Description Default
u Array[N]

The input field. If spectral=False, this is physical space. If spectral=True, this is complex Fourier coefficients.

 required
order int

The order of the derivative (1=first, 2=second, etc.). Default is 1.

 1
spectral bool

Whether the input 'u' is already in Fourier space. Default is False.

 False

Returns:

Name Type Description
du_dx Array[N]

The n-th derivative in physical space.
Source code in spectraldiffx/_src/fourier/operators.py 
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def __call__(
    self, u: Array, order: int = 1, spectral: bool = False
) -> Float[Array, "N"]:
    """
    Compute the n-th derivative of a field.

    Parameters
    ----------
    u : Array [N]
        The input field. If spectral=False, this is physical space.
        If spectral=True, this is complex Fourier coefficients.
    order : int, optional
        The order of the derivative (1=first, 2=second, etc.). Default is 1.
    spectral : bool, optional
        Whether the input 'u' is already in Fourier space. Default is False.

    Returns
    -------
    du_dx : Array [N]
        The n-th derivative in physical space.
    """
    # 1. Obtain spectral coefficients u_hat [N]
    u_hat = u if spectral else self.grid.transform(_real(u))

    # 2. Multiply by (i*k)^order [N]. Linear operators keep every
    # resolved mode; only products are 2/3-truncated (gh-91).
    k = self.grid.k
    du_hat = (1j * k) ** order * u_hat

    # 3. Inverse Transform back to physical space [N]
    return self.grid.transform(du_hat, inverse=True).real

gradient(u, spectral=False)

Compute the first derivative (gradient) du/dx.

Parameters:

Name Type Description Default
u Array[N]

Input field.

 required
spectral bool

If True, treats u as Fourier coefficients.

 False
Source code in spectraldiffx/_src/fourier/operators.py 
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def gradient(self, u: Array, spectral: bool = False) -> Array:
    """
    Compute the first derivative (gradient) du/dx.

    Parameters
    ----------
    u : Array [N]
        Input field.
    spectral : bool, optional
        If True, treats u as Fourier coefficients.
    """
    return self(u, order=1, spectral=spectral)

laplacian(u, spectral=False)

Compute the second derivative (Laplacian) d^2u/dx^2.

Operation in Fourier Space: lap_hat = -(k^2) * u_hat

Source code in spectraldiffx/_src/fourier/operators.py 
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def laplacian(self, u: Array, spectral: bool = False) -> Array:
    """
    Compute the second derivative (Laplacian) d^2u/dx^2.

    Operation in Fourier Space:
        lap_hat = -(k^2) * u_hat
    """
    u_hat = u if spectral else self.grid.transform(_real(u))
    k = self.grid.k
    lap_hat = -(k**2) * u_hat
    return self.grid.transform(lap_hat, inverse=True).real

biharmonic(u, spectral=False)

Compute the biharmonic operator: d^4u/dx^4.

In Fourier space: k^4 * u_hat.

Source code in spectraldiffx/_src/fourier/operators.py 
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def biharmonic(self, u: Array, spectral: bool = False) -> Float[Array, "N"]:
    """Compute the biharmonic operator: d^4u/dx^4.

    In Fourier space: k^4 * u_hat.
    """
    u_hat = u if spectral else self.grid.transform(_real(u))
    k = self.grid.k
    bih_hat = k**4 * u_hat
    return self.grid.transform(bih_hat, inverse=True).real

hyperviscosity(u, nu, order=2, spectral=False)

Compute the hyperviscosity term: (-1)^{n+1} * nu * d^{2n}u/dx^{2n}.

This is always dissipative: the spectral multiplier is -nu * |k|^{2n}, regardless of order.

Parameters:

Name Type Description Default
u Array[N]

Input field.

 required
nu float

Hyperviscosity coefficient.

 required
order int

Order n of the hyperviscosity (n=1 is regular diffusion, n=2 is biharmonic diffusion). Default: 2.

 2
spectral bool

Whether u is in Fourier space.

 False

Raises:

Type Description
ValueError

If order < 1 or nu < 0.
Source code in spectraldiffx/_src/fourier/operators.py 
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def hyperviscosity(
    self, u: Array, nu: float, order: int = 2, spectral: bool = False
) -> Float[Array, "N"]:
    """Compute the hyperviscosity term: (-1)^{n+1} * nu * d^{2n}u/dx^{2n}.

    This is always dissipative: the spectral multiplier is
    ``-nu * |k|^{2n}``, regardless of *order*.

    Parameters
    ----------
    u : Array [N]
        Input field.
    nu : float
        Hyperviscosity coefficient.
    order : int
        Order n of the hyperviscosity (n=1 is regular diffusion,
        n=2 is biharmonic diffusion). Default: 2.
    spectral : bool
        Whether u is in Fourier space.

    Raises
    ------
    ValueError
        If ``order < 1`` or ``nu < 0``.
    """
    _validate_hyperviscosity(nu, order)
    u_hat = u if spectral else self.grid.transform(_real(u))
    k = self.grid.k
    hyp_hat = -nu * k ** (2 * order) * u_hat
    return self.grid.transform(hyp_hat, inverse=True).real

inverse_laplacian(u, spectral=False)

Compute the inverse Laplacian: nabla^{-2} u (1D Poisson solve).

On a periodic domain, d^2 psi / dx^2 = u has a solution only if u has zero spatial mean. This implementation removes the k=0 component of u and returns the unique zero-mean solution psi, i.e. it solves d^2 psi / dx^2 = u - mean(u) with psi_hat[k=0] = 0.

Source code in spectraldiffx/_src/fourier/operators.py 
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def inverse_laplacian(self, u: Array, spectral: bool = False) -> Float[Array, "N"]:
    """Compute the inverse Laplacian: nabla^{-2} u (1D Poisson solve).

    On a periodic domain, d^2 psi / dx^2 = u has a solution only if u
    has zero spatial mean.  This implementation removes the k=0 component
    of u and returns the unique zero-mean solution psi, i.e. it solves
    d^2 psi / dx^2 = u - mean(u) with psi_hat[k=0] = 0.
    """
    u_hat = u if spectral else self.grid.transform(_real(u))
    k = self.grid.k
    k2 = k**2
    k2_safe = jnp.where(k2 == 0, 1.0, k2)
    psi_hat = -u_hat / k2_safe
    psi_hat = jnp.where(k2 == 0, 0.0, psi_hat)
    return self.grid.transform(psi_hat, inverse=True).real

apply_dealias(u, spectral=False)

Apply the 2/3 dealiasing rule mask to a field.

This zeros out the top 1/3 of the spectrum to prevent aliasing errors in nonlinear products (e.g., u * du/dx).

Source code in spectraldiffx/_src/fourier/operators.py 
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def apply_dealias(self, u: Array, spectral: bool = False) -> Array:
    """
    Apply the 2/3 dealiasing rule mask to a field.

    This zeros out the top 1/3 of the spectrum to prevent aliasing
    errors in nonlinear products (e.g., u * du/dx).
    """
    u_hat = u if spectral else self.grid.transform(_real(u))
    mask = self.grid.dealias_filter()
    u_hat_f = u_hat * mask
    return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True).real

SpectralDerivative2D

Bases: Module

2D Spectral derivative operators for doubly periodic rectangular domains.

Mathematical Formulation:

For a 2D field u(x, y): u(x, y) = ΣΣ u_hat(kx, ky) * exp(i * (kxx + kyy))

Partial derivatives: ∂u/∂x ↔ ikx * u_hat ∂u/∂y ↔ iky * u_hat

Gradient vector: ∇u = (∂u/∂x, ∂u/∂y) Laplacian: ∇^2 u = ∂^2u/∂x^2 + ∂^2u/∂y^2 ↔ -(kx^2 + ky^2) * u_hat

Dealiasing

Linear operators (gradient, divergence, curl, laplacian, biharmonic, hyperviscosity, inverse_laplacian, velocity_from_streamfunction, project_vector) act on every resolved mode and never apply the grid's 2/3 mask, so e.g. laplacian(inverse_laplacian(u)) == u. Only the product operators jacobian and advection_scalar truncate (inputs and result); :meth:apply_dealias applies the mask explicitly (gh-91).

Dtypes

Physical-space inputs must be real; complex input raises TypeError (gh-93). With spectral=True the input is complex Fourier coefficients. Results are returned in physical space as real arrays.

Source code in spectraldiffx/_src/fourier/operators.py 
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class SpectralDerivative2D(eqx.Module):
    """
    2D Spectral derivative operators for doubly periodic rectangular domains.

    Mathematical Formulation:
    -------------------------
    For a 2D field u(x, y):
        u(x, y) = ΣΣ u_hat(kx, ky) * exp(i * (kx*x + ky*y))

    Partial derivatives:
        ∂u/∂x ↔ i*kx * u_hat
        ∂u/∂y ↔ i*ky * u_hat

    Gradient vector: ∇u = (∂u/∂x, ∂u/∂y)
    Laplacian: ∇^2 u = ∂^2u/∂x^2 + ∂^2u/∂y^2 ↔ -(kx^2 + ky^2) * u_hat

    Dealiasing
    ----------
    Linear operators (``gradient``, ``divergence``, ``curl``, ``laplacian``,
    ``biharmonic``, ``hyperviscosity``, ``inverse_laplacian``,
    ``velocity_from_streamfunction``, ``project_vector``) act on every
    resolved mode and never apply the grid's 2/3 mask, so e.g.
    ``laplacian(inverse_laplacian(u)) == u``. Only the product operators
    ``jacobian`` and ``advection_scalar`` truncate (inputs and result);
    :meth:`apply_dealias` applies the mask explicitly (gh-91).

    Dtypes
    ------
    Physical-space inputs must be real; complex input raises ``TypeError``
    (gh-93). With ``spectral=True`` the input is complex Fourier
    coefficients. Results are returned in physical space as real arrays.
    """

    grid: FourierGrid2D

    def gradient(
        self, u: Array, spectral: bool = False
    ) -> tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]:
        """Compute the gradient vector [du/dx, du/dy]."""
        u_hat = u if spectral else self.grid.transform(_real(u))
        KX, KY = self.grid.KX

        du_dx = self.grid.transform(1j * KX * u_hat, inverse=True).real
        du_dy = self.grid.transform(1j * KY * u_hat, inverse=True).real
        return du_dx, du_dy

    def divergence(
        self, vx: Array, vy: Array, spectral: bool = False
    ) -> Float[Array, "Ny Nx"]:
        """
        Compute the divergence of a 2D vector field: div(V) = ∂vx/∂x + ∂vy/∂y.
        """
        vx_hat = vx if spectral else self.grid.transform(_real(vx))
        vy_hat = vy if spectral else self.grid.transform(_real(vy))
        KX, KY = self.grid.KX

        div_hat = 1j * KX * vx_hat + 1j * KY * vy_hat
        return self.grid.transform(div_hat, inverse=True).real

    def curl(
        self, vx: Array, vy: Array, spectral: bool = False
    ) -> Float[Array, "Ny Nx"]:
        """
        Compute the 2D scalar curl (vorticity): ζ = ∂vy/∂x - ∂vx/∂y.
        """
        vx_hat = vx if spectral else self.grid.transform(_real(vx))
        vy_hat = vy if spectral else self.grid.transform(_real(vy))
        KX, KY = self.grid.KX

        curl_hat = 1j * KX * vy_hat - 1j * KY * vx_hat
        return self.grid.transform(curl_hat, inverse=True).real

    def laplacian(self, u: Array, spectral: bool = False) -> Float[Array, "Ny Nx"]:
        """
        Compute the 2D Laplacian: ∇^2 u = ∂^2u/∂x^2 + ∂^2u/∂y^2.
        """
        u_hat = u if spectral else self.grid.transform(_real(u))
        K2 = self.grid.K2
        lap_hat = -K2 * u_hat
        return self.grid.transform(lap_hat, inverse=True).real

    def biharmonic(self, u: Array, spectral: bool = False) -> Float[Array, "Ny Nx"]:
        """Compute the 2D biharmonic: nabla^4 u = (kx^2 + ky^2)^2 * u_hat."""
        u_hat = u if spectral else self.grid.transform(_real(u))
        K2 = self.grid.K2
        bih_hat = K2**2 * u_hat
        return self.grid.transform(bih_hat, inverse=True).real

    def hyperviscosity(
        self, u: Array, nu: float, order: int = 2, spectral: bool = False
    ) -> Float[Array, "Ny Nx"]:
        """Compute the 2D hyperviscosity: (-1)^{n+1} * nu * nabla^{2n} u.

        This is always dissipative: the spectral multiplier is
        ``-nu * |k|^{2n}``, regardless of *order*.

        Parameters
        ----------
        u : Array [Ny, Nx]
            Input field.
        nu : float
            Hyperviscosity coefficient.
        order : int
            Order n (n=1 is diffusion, n=2 is biharmonic). Default: 2.
        spectral : bool
            Whether u is in Fourier space.

        Raises
        ------
        ValueError
            If ``order < 1`` or ``nu < 0``.
        """
        _validate_hyperviscosity(nu, order)
        u_hat = u if spectral else self.grid.transform(_real(u))
        K2 = self.grid.K2
        hyp_hat = -nu * K2**order * u_hat
        return self.grid.transform(hyp_hat, inverse=True).real

    def inverse_laplacian(
        self, u: Array, spectral: bool = False
    ) -> Float[Array, "Ny Nx"]:
        """Compute the 2D inverse Laplacian: nabla^{-2} u (Poisson solve).

        On a periodic domain, nabla^2 psi = u is solvable only if u has zero
        spatial mean.  This implementation removes the (0,0) Fourier mode
        of u and returns the unique zero-mean solution psi.
        """
        u_hat = u if spectral else self.grid.transform(_real(u))
        K2 = self.grid.K2
        K2_safe = jnp.where(K2 == 0, 1.0, K2)
        psi_hat = -u_hat / K2_safe
        psi_hat = jnp.where(K2 == 0, 0.0, psi_hat)
        return self.grid.transform(psi_hat, inverse=True).real

    def velocity_from_streamfunction(
        self, psi: Array, spectral: bool = False
    ) -> tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]:
        """Compute velocity from a streamfunction: u = -dpsi/dy, v = dpsi/dx."""
        psi_hat = psi if spectral else self.grid.transform(_real(psi))
        KX, KY = self.grid.KX
        u = self.grid.transform(-1j * KY * psi_hat, inverse=True).real
        v = self.grid.transform(1j * KX * psi_hat, inverse=True).real
        return u, v

    def jacobian(
        self, f: Array, g: Array, spectral: bool = False
    ) -> Float[Array, "Ny Nx"]:
        """Compute the Jacobian: J(f,g) = df/dx * dg/dy - df/dy * dg/dx.

        Uses the pseudo-spectral method: derivatives in Fourier space,
        products in physical space, with dealiasing.
        """
        f_hat = f if spectral else self.grid.transform(_real(f))
        g_hat = g if spectral else self.grid.transform(_real(g))
        KX, KY = self.grid.KX
        dealias = self.grid.dealias_filter()

        df_dx = self.grid.transform(1j * KX * f_hat * dealias, inverse=True).real
        df_dy = self.grid.transform(1j * KY * f_hat * dealias, inverse=True).real
        dg_dx = self.grid.transform(1j * KX * g_hat * dealias, inverse=True).real
        dg_dy = self.grid.transform(1j * KY * g_hat * dealias, inverse=True).real

        jac = df_dx * dg_dy - df_dy * dg_dx
        return self.apply_dealias(jac)

    def apply_dealias(self, u: Array, spectral: bool = False) -> Array:
        """Apply the 2D spectral dealiasing filter mask [Ny, Nx]."""
        u_hat = u if spectral else self.grid.transform(_real(u))
        mask = self.grid.dealias_filter()
        u_hat_f = u_hat * mask
        return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True).real

    def project_vector(self, vx: Array, vy: Array) -> tuple[Array, Array]:
        """
        Perform Leray projection to extract the divergence-free component.

        Solves the decomposition: V = V_solenoidal + V_irrotational
        where div(V_solenoidal) = 0 and curl(V_irrotational) = 0.

        Physics:
        --------
        In incompressible flows, we project the velocity onto the divergence-free
        manifold by solving for a scalar potential φ:
            div(V - grad(φ)) = 0  =>  ∇^2 φ = div(V)

        Then: V_solenoidal = V - grad(φ)
        """
        # 1. Transform components to Fourier space [Ny, Nx]
        vx_hat = self.grid.transform(_real(vx))
        vy_hat = self.grid.transform(_real(vy))
        KX, KY = self.grid.KX

        # 2. Compute divergence in Fourier space
        div_hat = 1j * (KX * vx_hat + KY * vy_hat)

        # 3. Solve Poisson: -|k|^2 * φ_hat = div_hat
        K2 = self.grid.K2
        K2_safe = jnp.where(K2 == 0, 1.0, K2)
        phi_hat = -div_hat / K2_safe
        phi_hat = jnp.where(K2 == 0, 0.0, phi_hat)  # Force mean to zero

        # 4. Correct velocity: V_sol_hat = V_hat - i*k*φ_hat
        vx_sol_hat = vx_hat - 1j * KX * phi_hat
        vy_sol_hat = vy_hat - 1j * KY * phi_hat

        return (
            self.grid.transform(vx_sol_hat, inverse=True).real,
            self.grid.transform(vy_sol_hat, inverse=True).real,
        )

    def advection_scalar(
        self, vx: Array, vy: Array, q: Array, spectral: bool = False
    ) -> Float[Array, "Ny Nx"]:
        """
        Compute scalar advection (u·∇)q using the pseudo-spectral method.

        Both factors are truncated with the grid's 2/3 mask, the products
        formed in physical space, and the result truncated again, exactly as
        :meth:`jacobian` does. With ``(vx, vy)`` from
        :meth:`velocity_from_streamfunction` the two agree (gh-90).

        Out = dealias(vx * ∂q/∂x + vy * ∂q/∂y)

        Parameters
        ----------
        vx, vy : Array [Ny, Nx]
            Velocity components in physical space.
        q : Array [Ny, Nx]
            Advected scalar, physical (or spectral if ``spectral``).
        spectral : bool
            If True, ``q`` is given as spectral coefficients. The result is
            always returned in physical space.

        Returns
        -------
        Array [Ny, Nx]
            The dealiased advection term in physical space.
        """
        q_hat = q if spectral else self.grid.transform(_real(q))
        KX, KY = self.grid.KX
        dealias = self.grid.dealias_filter()

        dq_dx = self.grid.transform(1j * KX * q_hat * dealias, inverse=True).real
        dq_dy = self.grid.transform(1j * KY * q_hat * dealias, inverse=True).real

        vx, vy = self.apply_dealias(vx), self.apply_dealias(vy)
        return self.apply_dealias(vx * dq_dx + vy * dq_dy)

Functions

gradient(u, spectral=False)

Compute the gradient vector [du/dx, du/dy].

Source code in spectraldiffx/_src/fourier/operators.py 
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def gradient(
    self, u: Array, spectral: bool = False
) -> tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]:
    """Compute the gradient vector [du/dx, du/dy]."""
    u_hat = u if spectral else self.grid.transform(_real(u))
    KX, KY = self.grid.KX

    du_dx = self.grid.transform(1j * KX * u_hat, inverse=True).real
    du_dy = self.grid.transform(1j * KY * u_hat, inverse=True).real
    return du_dx, du_dy

divergence(vx, vy, spectral=False)

Compute the divergence of a 2D vector field: div(V) = ∂vx/∂x + ∂vy/∂y.

Source code in spectraldiffx/_src/fourier/operators.py 
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def divergence(
    self, vx: Array, vy: Array, spectral: bool = False
) -> Float[Array, "Ny Nx"]:
    """
    Compute the divergence of a 2D vector field: div(V) = ∂vx/∂x + ∂vy/∂y.
    """
    vx_hat = vx if spectral else self.grid.transform(_real(vx))
    vy_hat = vy if spectral else self.grid.transform(_real(vy))
    KX, KY = self.grid.KX

    div_hat = 1j * KX * vx_hat + 1j * KY * vy_hat
    return self.grid.transform(div_hat, inverse=True).real

curl(vx, vy, spectral=False)

Compute the 2D scalar curl (vorticity): ζ = ∂vy/∂x - ∂vx/∂y.

Source code in spectraldiffx/_src/fourier/operators.py 
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def curl(
    self, vx: Array, vy: Array, spectral: bool = False
) -> Float[Array, "Ny Nx"]:
    """
    Compute the 2D scalar curl (vorticity): ζ = ∂vy/∂x - ∂vx/∂y.
    """
    vx_hat = vx if spectral else self.grid.transform(_real(vx))
    vy_hat = vy if spectral else self.grid.transform(_real(vy))
    KX, KY = self.grid.KX

    curl_hat = 1j * KX * vy_hat - 1j * KY * vx_hat
    return self.grid.transform(curl_hat, inverse=True).real

laplacian(u, spectral=False)

Compute the 2D Laplacian: ∇^2 u = ∂^2u/∂x^2 + ∂^2u/∂y^2.

Source code in spectraldiffx/_src/fourier/operators.py 
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def laplacian(self, u: Array, spectral: bool = False) -> Float[Array, "Ny Nx"]:
    """
    Compute the 2D Laplacian: ∇^2 u = ∂^2u/∂x^2 + ∂^2u/∂y^2.
    """
    u_hat = u if spectral else self.grid.transform(_real(u))
    K2 = self.grid.K2
    lap_hat = -K2 * u_hat
    return self.grid.transform(lap_hat, inverse=True).real

biharmonic(u, spectral=False)

Compute the 2D biharmonic: nabla^4 u = (kx^2 + ky^2)^2 * u_hat.

Source code in spectraldiffx/_src/fourier/operators.py 
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def biharmonic(self, u: Array, spectral: bool = False) -> Float[Array, "Ny Nx"]:
    """Compute the 2D biharmonic: nabla^4 u = (kx^2 + ky^2)^2 * u_hat."""
    u_hat = u if spectral else self.grid.transform(_real(u))
    K2 = self.grid.K2
    bih_hat = K2**2 * u_hat
    return self.grid.transform(bih_hat, inverse=True).real

hyperviscosity(u, nu, order=2, spectral=False)

Compute the 2D hyperviscosity: (-1)^{n+1} * nu * nabla^{2n} u.

This is always dissipative: the spectral multiplier is -nu * |k|^{2n}, regardless of order.

Parameters:

Name Type Description Default
u Array[Ny, Nx]

Input field.

 required
nu float

Hyperviscosity coefficient.

 required
order int

Order n (n=1 is diffusion, n=2 is biharmonic). Default: 2.

 2
spectral bool

Whether u is in Fourier space.

 False

Raises:

Type Description
ValueError

If order < 1 or nu < 0.
Source code in spectraldiffx/_src/fourier/operators.py 
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def hyperviscosity(
    self, u: Array, nu: float, order: int = 2, spectral: bool = False
) -> Float[Array, "Ny Nx"]:
    """Compute the 2D hyperviscosity: (-1)^{n+1} * nu * nabla^{2n} u.

    This is always dissipative: the spectral multiplier is
    ``-nu * |k|^{2n}``, regardless of *order*.

    Parameters
    ----------
    u : Array [Ny, Nx]
        Input field.
    nu : float
        Hyperviscosity coefficient.
    order : int
        Order n (n=1 is diffusion, n=2 is biharmonic). Default: 2.
    spectral : bool
        Whether u is in Fourier space.

    Raises
    ------
    ValueError
        If ``order < 1`` or ``nu < 0``.
    """
    _validate_hyperviscosity(nu, order)
    u_hat = u if spectral else self.grid.transform(_real(u))
    K2 = self.grid.K2
    hyp_hat = -nu * K2**order * u_hat
    return self.grid.transform(hyp_hat, inverse=True).real

inverse_laplacian(u, spectral=False)

Compute the 2D inverse Laplacian: nabla^{-2} u (Poisson solve).

On a periodic domain, nabla^2 psi = u is solvable only if u has zero spatial mean. This implementation removes the (0,0) Fourier mode of u and returns the unique zero-mean solution psi.

Source code in spectraldiffx/_src/fourier/operators.py 
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def inverse_laplacian(
    self, u: Array, spectral: bool = False
) -> Float[Array, "Ny Nx"]:
    """Compute the 2D inverse Laplacian: nabla^{-2} u (Poisson solve).

    On a periodic domain, nabla^2 psi = u is solvable only if u has zero
    spatial mean.  This implementation removes the (0,0) Fourier mode
    of u and returns the unique zero-mean solution psi.
    """
    u_hat = u if spectral else self.grid.transform(_real(u))
    K2 = self.grid.K2
    K2_safe = jnp.where(K2 == 0, 1.0, K2)
    psi_hat = -u_hat / K2_safe
    psi_hat = jnp.where(K2 == 0, 0.0, psi_hat)
    return self.grid.transform(psi_hat, inverse=True).real

velocity_from_streamfunction(psi, spectral=False)

Compute velocity from a streamfunction: u = -dpsi/dy, v = dpsi/dx.

Source code in spectraldiffx/_src/fourier/operators.py 
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def velocity_from_streamfunction(
    self, psi: Array, spectral: bool = False
) -> tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]:
    """Compute velocity from a streamfunction: u = -dpsi/dy, v = dpsi/dx."""
    psi_hat = psi if spectral else self.grid.transform(_real(psi))
    KX, KY = self.grid.KX
    u = self.grid.transform(-1j * KY * psi_hat, inverse=True).real
    v = self.grid.transform(1j * KX * psi_hat, inverse=True).real
    return u, v

jacobian(f, g, spectral=False)

Compute the Jacobian: J(f,g) = df/dx * dg/dy - df/dy * dg/dx.

Uses the pseudo-spectral method: derivatives in Fourier space, products in physical space, with dealiasing.

Source code in spectraldiffx/_src/fourier/operators.py 
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def jacobian(
    self, f: Array, g: Array, spectral: bool = False
) -> Float[Array, "Ny Nx"]:
    """Compute the Jacobian: J(f,g) = df/dx * dg/dy - df/dy * dg/dx.

    Uses the pseudo-spectral method: derivatives in Fourier space,
    products in physical space, with dealiasing.
    """
    f_hat = f if spectral else self.grid.transform(_real(f))
    g_hat = g if spectral else self.grid.transform(_real(g))
    KX, KY = self.grid.KX
    dealias = self.grid.dealias_filter()

    df_dx = self.grid.transform(1j * KX * f_hat * dealias, inverse=True).real
    df_dy = self.grid.transform(1j * KY * f_hat * dealias, inverse=True).real
    dg_dx = self.grid.transform(1j * KX * g_hat * dealias, inverse=True).real
    dg_dy = self.grid.transform(1j * KY * g_hat * dealias, inverse=True).real

    jac = df_dx * dg_dy - df_dy * dg_dx
    return self.apply_dealias(jac)

apply_dealias(u, spectral=False)

Apply the 2D spectral dealiasing filter mask [Ny, Nx].

Source code in spectraldiffx/_src/fourier/operators.py 
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def apply_dealias(self, u: Array, spectral: bool = False) -> Array:
    """Apply the 2D spectral dealiasing filter mask [Ny, Nx]."""
    u_hat = u if spectral else self.grid.transform(_real(u))
    mask = self.grid.dealias_filter()
    u_hat_f = u_hat * mask
    return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True).real

project_vector(vx, vy)

Perform Leray projection to extract the divergence-free component.

Solves the decomposition: V = V_solenoidal + V_irrotational where div(V_solenoidal) = 0 and curl(V_irrotational) = 0.

Physics:

In incompressible flows, we project the velocity onto the divergence-free manifold by solving for a scalar potential φ: div(V - grad(φ)) = 0 => ∇^2 φ = div(V)

Then: V_solenoidal = V - grad(φ)

Source code in spectraldiffx/_src/fourier/operators.py 
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def project_vector(self, vx: Array, vy: Array) -> tuple[Array, Array]:
    """
    Perform Leray projection to extract the divergence-free component.

    Solves the decomposition: V = V_solenoidal + V_irrotational
    where div(V_solenoidal) = 0 and curl(V_irrotational) = 0.

    Physics:
    --------
    In incompressible flows, we project the velocity onto the divergence-free
    manifold by solving for a scalar potential φ:
        div(V - grad(φ)) = 0  =>  ∇^2 φ = div(V)

    Then: V_solenoidal = V - grad(φ)
    """
    # 1. Transform components to Fourier space [Ny, Nx]
    vx_hat = self.grid.transform(_real(vx))
    vy_hat = self.grid.transform(_real(vy))
    KX, KY = self.grid.KX

    # 2. Compute divergence in Fourier space
    div_hat = 1j * (KX * vx_hat + KY * vy_hat)

    # 3. Solve Poisson: -|k|^2 * φ_hat = div_hat
    K2 = self.grid.K2
    K2_safe = jnp.where(K2 == 0, 1.0, K2)
    phi_hat = -div_hat / K2_safe
    phi_hat = jnp.where(K2 == 0, 0.0, phi_hat)  # Force mean to zero

    # 4. Correct velocity: V_sol_hat = V_hat - i*k*φ_hat
    vx_sol_hat = vx_hat - 1j * KX * phi_hat
    vy_sol_hat = vy_hat - 1j * KY * phi_hat

    return (
        self.grid.transform(vx_sol_hat, inverse=True).real,
        self.grid.transform(vy_sol_hat, inverse=True).real,
    )

advection_scalar(vx, vy, q, spectral=False)

Compute scalar advection (u·∇)q using the pseudo-spectral method.

Both factors are truncated with the grid's 2/3 mask, the products formed in physical space, and the result truncated again, exactly as :meth:jacobian does. With (vx, vy) from :meth:velocity_from_streamfunction the two agree (gh-90).

Out = dealias(vx * ∂q/∂x + vy * ∂q/∂y)

Parameters:

Name Type Description Default
vx Array[Ny, Nx]

Velocity components in physical space.

 required
vy Array[Ny, Nx]

Velocity components in physical space.

 required
q Array[Ny, Nx]

Advected scalar, physical (or spectral if spectral).

 required
spectral bool

If True, q is given as spectral coefficients. The result is always returned in physical space.

 False

Returns:

Type Description
Array[Ny, Nx]

The dealiased advection term in physical space.
Source code in spectraldiffx/_src/fourier/operators.py 
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def advection_scalar(
    self, vx: Array, vy: Array, q: Array, spectral: bool = False
) -> Float[Array, "Ny Nx"]:
    """
    Compute scalar advection (u·∇)q using the pseudo-spectral method.

    Both factors are truncated with the grid's 2/3 mask, the products
    formed in physical space, and the result truncated again, exactly as
    :meth:`jacobian` does. With ``(vx, vy)`` from
    :meth:`velocity_from_streamfunction` the two agree (gh-90).

    Out = dealias(vx * ∂q/∂x + vy * ∂q/∂y)

    Parameters
    ----------
    vx, vy : Array [Ny, Nx]
        Velocity components in physical space.
    q : Array [Ny, Nx]
        Advected scalar, physical (or spectral if ``spectral``).
    spectral : bool
        If True, ``q`` is given as spectral coefficients. The result is
        always returned in physical space.

    Returns
    -------
    Array [Ny, Nx]
        The dealiased advection term in physical space.
    """
    q_hat = q if spectral else self.grid.transform(_real(q))
    KX, KY = self.grid.KX
    dealias = self.grid.dealias_filter()

    dq_dx = self.grid.transform(1j * KX * q_hat * dealias, inverse=True).real
    dq_dy = self.grid.transform(1j * KY * q_hat * dealias, inverse=True).real

    vx, vy = self.apply_dealias(vx), self.apply_dealias(vy)
    return self.apply_dealias(vx * dq_dx + vy * dq_dy)

SpectralDerivative3D

Bases: Module

3D Spectral derivative operators for triply periodic domains.

Mathematical Framework:

Field shapes: (Nz, Ny, Nx) following 'ij' indexing. Wavenumbers: (kz, ky, kx).

Gradient vector: ∇u = (∂u/∂z, ∂u/∂y, ∂u/∂x)

Dealiasing

Linear operators (gradient, divergence, curl, laplacian, biharmonic, hyperviscosity, inverse_laplacian, velocity_from_streamfunction, project_vector) act on every resolved mode and never apply the grid's 2/3 mask, so e.g. laplacian(inverse_laplacian(u)) == u. Only the product operators jacobian and advection_scalar truncate (inputs and result); :meth:apply_dealias applies the mask explicitly (gh-91).

Dtypes

Physical-space inputs must be real; complex input raises TypeError (gh-93). With spectral=True the input is complex Fourier coefficients. Results are returned in physical space as real arrays.

Source code in spectraldiffx/_src/fourier/operators.py 
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class SpectralDerivative3D(eqx.Module):
    """
    3D Spectral derivative operators for triply periodic domains.

    Mathematical Framework:
    -----------------------
    Field shapes: (Nz, Ny, Nx) following 'ij' indexing.
    Wavenumbers: (kz, ky, kx).

    Gradient vector: ∇u = (∂u/∂z, ∂u/∂y, ∂u/∂x)

    Dealiasing
    ----------
    Linear operators (``gradient``, ``divergence``, ``curl``, ``laplacian``,
    ``biharmonic``, ``hyperviscosity``, ``inverse_laplacian``,
    ``velocity_from_streamfunction``, ``project_vector``) act on every
    resolved mode and never apply the grid's 2/3 mask, so e.g.
    ``laplacian(inverse_laplacian(u)) == u``. Only the product operators
    ``jacobian`` and ``advection_scalar`` truncate (inputs and result);
    :meth:`apply_dealias` applies the mask explicitly (gh-91).

    Dtypes
    ------
    Physical-space inputs must be real; complex input raises ``TypeError``
    (gh-93). With ``spectral=True`` the input is complex Fourier
    coefficients. Results are returned in physical space as real arrays.
    """

    grid: FourierGrid3D

    def gradient(self, u: Array, spectral: bool = False) -> tuple[Array, Array, Array]:
        """Compute the 3D gradient vector field."""
        u_hat = u if spectral else self.grid.transform(_real(u))
        KZ, KY, KX = self.grid.KX

        du_dz = self.grid.transform(1j * KZ * u_hat, inverse=True).real
        du_dy = self.grid.transform(1j * KY * u_hat, inverse=True).real
        du_dx = self.grid.transform(1j * KX * u_hat, inverse=True).real
        return du_dz, du_dy, du_dx

    def divergence(
        self, vz: Array, vy: Array, vx: Array, spectral: bool = False
    ) -> Float[Array, "Nz Ny Nx"]:
        """Compute 3D divergence: ∂vz/∂z + ∂vy/∂y + ∂vx/∂x."""
        vz_hat = vz if spectral else self.grid.transform(_real(vz))
        vy_hat = vy if spectral else self.grid.transform(_real(vy))
        vx_hat = vx if spectral else self.grid.transform(_real(vx))
        KZ, KY, KX = self.grid.KX

        div_hat = 1j * KZ * vz_hat + 1j * KY * vy_hat + 1j * KX * vx_hat
        return self.grid.transform(div_hat, inverse=True).real

    def curl(
        self, vz: Array, vy: Array, vx: Array, spectral: bool = False
    ) -> tuple[Array, Array, Array]:
        """
        Compute the 3D curl vector: ω = ∇ x V.

        Components:
            ωz = ∂vy/∂x - ∂vx/∂y
            ωy = ∂vx/∂z - ∂vz/∂x
            ωx = ∂vz/∂y - ∂vy/∂z
        """
        vz_hat = vz if spectral else self.grid.transform(_real(vz))
        vy_hat = vy if spectral else self.grid.transform(_real(vy))
        vx_hat = vx if spectral else self.grid.transform(_real(vx))
        KZ, KY, KX = self.grid.KX

        wz_hat = 1j * KX * vy_hat - 1j * KY * vx_hat
        wy_hat = 1j * KZ * vx_hat - 1j * KX * vz_hat
        wx_hat = 1j * KY * vz_hat - 1j * KZ * vy_hat

        return (
            self.grid.transform(wz_hat, inverse=True).real,
            self.grid.transform(wy_hat, inverse=True).real,
            self.grid.transform(wx_hat, inverse=True).real,
        )

    def laplacian(self, u: Array, spectral: bool = False) -> Float[Array, "Nz Ny Nx"]:
        """Compute 3D Laplacian: ∇^2 u = ∂^2u/∂z^2 + ∂^2u/∂y^2 + ∂^2u/∂x^2."""
        u_hat = u if spectral else self.grid.transform(_real(u))
        K2 = self.grid.K2
        lap_hat = -K2 * u_hat
        return self.grid.transform(lap_hat, inverse=True).real

    def biharmonic(self, u: Array, spectral: bool = False) -> Float[Array, "Nz Ny Nx"]:
        """Compute the 3D biharmonic: nabla^4 u."""
        u_hat = u if spectral else self.grid.transform(_real(u))
        K2 = self.grid.K2
        bih_hat = K2**2 * u_hat
        return self.grid.transform(bih_hat, inverse=True).real

    def hyperviscosity(
        self, u: Array, nu: float, order: int = 2, spectral: bool = False
    ) -> Float[Array, "Nz Ny Nx"]:
        """Compute the 3D hyperviscosity: (-1)^{n+1} * nu * nabla^{2n} u.

        Raises
        ------
        ValueError
            If ``order < 1`` or ``nu < 0``.
        """
        _validate_hyperviscosity(nu, order)
        u_hat = u if spectral else self.grid.transform(_real(u))
        K2 = self.grid.K2
        hyp_hat = -nu * K2**order * u_hat
        return self.grid.transform(hyp_hat, inverse=True).real

    def inverse_laplacian(
        self, u: Array, spectral: bool = False
    ) -> Float[Array, "Nz Ny Nx"]:
        """Compute the 3D inverse Laplacian: nabla^{-2} u (Poisson solve).

        On a periodic domain, nabla^2 psi = u is solvable only if u has zero
        spatial mean.  This implementation removes the (0,0,0) Fourier mode
        of u and returns the unique zero-mean solution psi.
        """
        u_hat = u if spectral else self.grid.transform(_real(u))
        K2 = self.grid.K2
        K2_safe = jnp.where(K2 == 0, 1.0, K2)
        psi_hat = -u_hat / K2_safe
        psi_hat = jnp.where(K2 == 0, 0.0, psi_hat)
        return self.grid.transform(psi_hat, inverse=True).real

    def velocity_from_streamfunction(
        self, psi: Array, spectral: bool = False
    ) -> tuple[Float[Array, "Nz Ny Nx"], Float[Array, "Nz Ny Nx"]]:
        """Compute horizontal velocity from a streamfunction.

        Returns (u, v) where u = -dpsi/dy, v = dpsi/dx (horizontal components).
        """
        psi_hat = psi if spectral else self.grid.transform(_real(psi))
        _KZ, KY, KX = self.grid.KX
        u = self.grid.transform(-1j * KY * psi_hat, inverse=True).real
        v = self.grid.transform(1j * KX * psi_hat, inverse=True).real
        return u, v

    def jacobian(
        self, f: Array, g: Array, spectral: bool = False
    ) -> Float[Array, "Nz Ny Nx"]:
        """Compute the horizontal Jacobian: J(f,g) = df/dx * dg/dy - df/dy * dg/dx.

        Operates on the horizontal (x, y) components only — the z direction
        is treated as a batch dimension.
        """
        f_hat = f if spectral else self.grid.transform(_real(f))
        g_hat = g if spectral else self.grid.transform(_real(g))
        _KZ, KY, KX = self.grid.KX
        dealias = self.grid.dealias_filter()

        df_dx = self.grid.transform(1j * KX * f_hat * dealias, inverse=True).real
        df_dy = self.grid.transform(1j * KY * f_hat * dealias, inverse=True).real
        dg_dx = self.grid.transform(1j * KX * g_hat * dealias, inverse=True).real
        dg_dy = self.grid.transform(1j * KY * g_hat * dealias, inverse=True).real

        jac = df_dx * dg_dy - df_dy * dg_dx
        return self.apply_dealias(jac)

    def apply_dealias(self, u: Array, spectral: bool = False) -> Array:
        """Apply the 3D periodic dealiasing filter mask [Nz, Ny, Nx]."""
        u_hat = u if spectral else self.grid.transform(_real(u))
        mask = self.grid.dealias_filter()
        u_hat_f = u_hat * mask
        return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True).real

    def project_vector(
        self, vz: Array, vy: Array, vx: Array
    ) -> tuple[Array, Array, Array]:
        """Project a 3D vector field onto its solenoidal (divergence-free) component."""
        vz_hat = self.grid.transform(_real(vz))
        vy_hat = self.grid.transform(_real(vy))
        vx_hat = self.grid.transform(_real(vx))
        KZ, KY, KX = self.grid.KX

        div_hat = 1j * (KZ * vz_hat + KY * vy_hat + KX * vx_hat)

        K2 = self.grid.K2
        K2_safe = jnp.where(K2 == 0, 1.0, K2)
        phi_hat = -div_hat / K2_safe
        phi_hat = jnp.where(K2 == 0, 0.0, phi_hat)

        vz_sol_hat = vz_hat - 1j * KZ * phi_hat
        vy_sol_hat = vy_hat - 1j * KY * phi_hat
        vx_sol_hat = vx_hat - 1j * KX * phi_hat

        return (
            self.grid.transform(vz_sol_hat, inverse=True).real,
            self.grid.transform(vy_sol_hat, inverse=True).real,
            self.grid.transform(vx_sol_hat, inverse=True).real,
        )

    def advection_scalar(
        self, vz: Array, vy: Array, vx: Array, q: Array, spectral: bool = False
    ) -> Float[Array, "Nz Ny Nx"]:
        """Compute the dealiased 3D scalar advection term (u·∇)q.

        Both factors are truncated with the grid's 2/3 mask, the products
        formed in physical space, and the result truncated again, as in 2D
        (gh-90).

        Parameters
        ----------
        vz, vy, vx : Array [Nz, Ny, Nx]
            Velocity components in physical space.
        q : Array [Nz, Ny, Nx]
            Advected scalar, physical (or spectral if ``spectral``).
        spectral : bool
            If True, ``q`` is given as spectral coefficients. The result is
            always returned in physical space.

        Returns
        -------
        Array [Nz, Ny, Nx]
            The dealiased advection term in physical space.
        """
        q_hat = q if spectral else self.grid.transform(_real(q))
        KZ, KY, KX = self.grid.KX
        dealias = self.grid.dealias_filter()

        dq_dz = self.grid.transform(1j * KZ * q_hat * dealias, inverse=True).real
        dq_dy = self.grid.transform(1j * KY * q_hat * dealias, inverse=True).real
        dq_dx = self.grid.transform(1j * KX * q_hat * dealias, inverse=True).real

        vz, vy, vx = (self.apply_dealias(v) for v in (vz, vy, vx))
        return self.apply_dealias(vz * dq_dz + vy * dq_dy + vx * dq_dx)

Functions

gradient(u, spectral=False)

Compute the 3D gradient vector field.

Source code in spectraldiffx/_src/fourier/operators.py 
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def gradient(self, u: Array, spectral: bool = False) -> tuple[Array, Array, Array]:
    """Compute the 3D gradient vector field."""
    u_hat = u if spectral else self.grid.transform(_real(u))
    KZ, KY, KX = self.grid.KX

    du_dz = self.grid.transform(1j * KZ * u_hat, inverse=True).real
    du_dy = self.grid.transform(1j * KY * u_hat, inverse=True).real
    du_dx = self.grid.transform(1j * KX * u_hat, inverse=True).real
    return du_dz, du_dy, du_dx

divergence(vz, vy, vx, spectral=False)

Compute 3D divergence: ∂vz/∂z + ∂vy/∂y + ∂vx/∂x.

Source code in spectraldiffx/_src/fourier/operators.py 
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def divergence(
    self, vz: Array, vy: Array, vx: Array, spectral: bool = False
) -> Float[Array, "Nz Ny Nx"]:
    """Compute 3D divergence: ∂vz/∂z + ∂vy/∂y + ∂vx/∂x."""
    vz_hat = vz if spectral else self.grid.transform(_real(vz))
    vy_hat = vy if spectral else self.grid.transform(_real(vy))
    vx_hat = vx if spectral else self.grid.transform(_real(vx))
    KZ, KY, KX = self.grid.KX

    div_hat = 1j * KZ * vz_hat + 1j * KY * vy_hat + 1j * KX * vx_hat
    return self.grid.transform(div_hat, inverse=True).real

curl(vz, vy, vx, spectral=False)

Compute the 3D curl vector: ω = ∇ x V.

Components: ωz = ∂vy/∂x - ∂vx/∂y ωy = ∂vx/∂z - ∂vz/∂x ωx = ∂vz/∂y - ∂vy/∂z

Source code in spectraldiffx/_src/fourier/operators.py 
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def curl(
    self, vz: Array, vy: Array, vx: Array, spectral: bool = False
) -> tuple[Array, Array, Array]:
    """
    Compute the 3D curl vector: ω = ∇ x V.

    Components:
        ωz = ∂vy/∂x - ∂vx/∂y
        ωy = ∂vx/∂z - ∂vz/∂x
        ωx = ∂vz/∂y - ∂vy/∂z
    """
    vz_hat = vz if spectral else self.grid.transform(_real(vz))
    vy_hat = vy if spectral else self.grid.transform(_real(vy))
    vx_hat = vx if spectral else self.grid.transform(_real(vx))
    KZ, KY, KX = self.grid.KX

    wz_hat = 1j * KX * vy_hat - 1j * KY * vx_hat
    wy_hat = 1j * KZ * vx_hat - 1j * KX * vz_hat
    wx_hat = 1j * KY * vz_hat - 1j * KZ * vy_hat

    return (
        self.grid.transform(wz_hat, inverse=True).real,
        self.grid.transform(wy_hat, inverse=True).real,
        self.grid.transform(wx_hat, inverse=True).real,
    )

laplacian(u, spectral=False)

Compute 3D Laplacian: ∇^2 u = ∂^2u/∂z^2 + ∂^2u/∂y^2 + ∂^2u/∂x^2.

Source code in spectraldiffx/_src/fourier/operators.py 
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def laplacian(self, u: Array, spectral: bool = False) -> Float[Array, "Nz Ny Nx"]:
    """Compute 3D Laplacian: ∇^2 u = ∂^2u/∂z^2 + ∂^2u/∂y^2 + ∂^2u/∂x^2."""
    u_hat = u if spectral else self.grid.transform(_real(u))
    K2 = self.grid.K2
    lap_hat = -K2 * u_hat
    return self.grid.transform(lap_hat, inverse=True).real

biharmonic(u, spectral=False)

Compute the 3D biharmonic: nabla^4 u.

Source code in spectraldiffx/_src/fourier/operators.py 
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def biharmonic(self, u: Array, spectral: bool = False) -> Float[Array, "Nz Ny Nx"]:
    """Compute the 3D biharmonic: nabla^4 u."""
    u_hat = u if spectral else self.grid.transform(_real(u))
    K2 = self.grid.K2
    bih_hat = K2**2 * u_hat
    return self.grid.transform(bih_hat, inverse=True).real

hyperviscosity(u, nu, order=2, spectral=False)

Compute the 3D hyperviscosity: (-1)^{n+1} * nu * nabla^{2n} u.

Raises:

Type Description
ValueError

If order < 1 or nu < 0.
Source code in spectraldiffx/_src/fourier/operators.py 
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def hyperviscosity(
    self, u: Array, nu: float, order: int = 2, spectral: bool = False
) -> Float[Array, "Nz Ny Nx"]:
    """Compute the 3D hyperviscosity: (-1)^{n+1} * nu * nabla^{2n} u.

    Raises
    ------
    ValueError
        If ``order < 1`` or ``nu < 0``.
    """
    _validate_hyperviscosity(nu, order)
    u_hat = u if spectral else self.grid.transform(_real(u))
    K2 = self.grid.K2
    hyp_hat = -nu * K2**order * u_hat
    return self.grid.transform(hyp_hat, inverse=True).real

inverse_laplacian(u, spectral=False)

Compute the 3D inverse Laplacian: nabla^{-2} u (Poisson solve).

On a periodic domain, nabla^2 psi = u is solvable only if u has zero spatial mean. This implementation removes the (0,0,0) Fourier mode of u and returns the unique zero-mean solution psi.

Source code in spectraldiffx/_src/fourier/operators.py 
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def inverse_laplacian(
    self, u: Array, spectral: bool = False
) -> Float[Array, "Nz Ny Nx"]:
    """Compute the 3D inverse Laplacian: nabla^{-2} u (Poisson solve).

    On a periodic domain, nabla^2 psi = u is solvable only if u has zero
    spatial mean.  This implementation removes the (0,0,0) Fourier mode
    of u and returns the unique zero-mean solution psi.
    """
    u_hat = u if spectral else self.grid.transform(_real(u))
    K2 = self.grid.K2
    K2_safe = jnp.where(K2 == 0, 1.0, K2)
    psi_hat = -u_hat / K2_safe
    psi_hat = jnp.where(K2 == 0, 0.0, psi_hat)
    return self.grid.transform(psi_hat, inverse=True).real

velocity_from_streamfunction(psi, spectral=False)

Compute horizontal velocity from a streamfunction.

Returns (u, v) where u = -dpsi/dy, v = dpsi/dx (horizontal components).

Source code in spectraldiffx/_src/fourier/operators.py 
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def velocity_from_streamfunction(
    self, psi: Array, spectral: bool = False
) -> tuple[Float[Array, "Nz Ny Nx"], Float[Array, "Nz Ny Nx"]]:
    """Compute horizontal velocity from a streamfunction.

    Returns (u, v) where u = -dpsi/dy, v = dpsi/dx (horizontal components).
    """
    psi_hat = psi if spectral else self.grid.transform(_real(psi))
    _KZ, KY, KX = self.grid.KX
    u = self.grid.transform(-1j * KY * psi_hat, inverse=True).real
    v = self.grid.transform(1j * KX * psi_hat, inverse=True).real
    return u, v

jacobian(f, g, spectral=False)

Compute the horizontal Jacobian: J(f,g) = df/dx * dg/dy - df/dy * dg/dx.

Operates on the horizontal (x, y) components only — the z direction is treated as a batch dimension.

Source code in spectraldiffx/_src/fourier/operators.py 
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def jacobian(
    self, f: Array, g: Array, spectral: bool = False
) -> Float[Array, "Nz Ny Nx"]:
    """Compute the horizontal Jacobian: J(f,g) = df/dx * dg/dy - df/dy * dg/dx.

    Operates on the horizontal (x, y) components only — the z direction
    is treated as a batch dimension.
    """
    f_hat = f if spectral else self.grid.transform(_real(f))
    g_hat = g if spectral else self.grid.transform(_real(g))
    _KZ, KY, KX = self.grid.KX
    dealias = self.grid.dealias_filter()

    df_dx = self.grid.transform(1j * KX * f_hat * dealias, inverse=True).real
    df_dy = self.grid.transform(1j * KY * f_hat * dealias, inverse=True).real
    dg_dx = self.grid.transform(1j * KX * g_hat * dealias, inverse=True).real
    dg_dy = self.grid.transform(1j * KY * g_hat * dealias, inverse=True).real

    jac = df_dx * dg_dy - df_dy * dg_dx
    return self.apply_dealias(jac)

apply_dealias(u, spectral=False)

Apply the 3D periodic dealiasing filter mask [Nz, Ny, Nx].

Source code in spectraldiffx/_src/fourier/operators.py 
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def apply_dealias(self, u: Array, spectral: bool = False) -> Array:
    """Apply the 3D periodic dealiasing filter mask [Nz, Ny, Nx]."""
    u_hat = u if spectral else self.grid.transform(_real(u))
    mask = self.grid.dealias_filter()
    u_hat_f = u_hat * mask
    return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True).real

project_vector(vz, vy, vx)

Project a 3D vector field onto its solenoidal (divergence-free) component.

Source code in spectraldiffx/_src/fourier/operators.py 
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def project_vector(
    self, vz: Array, vy: Array, vx: Array
) -> tuple[Array, Array, Array]:
    """Project a 3D vector field onto its solenoidal (divergence-free) component."""
    vz_hat = self.grid.transform(_real(vz))
    vy_hat = self.grid.transform(_real(vy))
    vx_hat = self.grid.transform(_real(vx))
    KZ, KY, KX = self.grid.KX

    div_hat = 1j * (KZ * vz_hat + KY * vy_hat + KX * vx_hat)

    K2 = self.grid.K2
    K2_safe = jnp.where(K2 == 0, 1.0, K2)
    phi_hat = -div_hat / K2_safe
    phi_hat = jnp.where(K2 == 0, 0.0, phi_hat)

    vz_sol_hat = vz_hat - 1j * KZ * phi_hat
    vy_sol_hat = vy_hat - 1j * KY * phi_hat
    vx_sol_hat = vx_hat - 1j * KX * phi_hat

    return (
        self.grid.transform(vz_sol_hat, inverse=True).real,
        self.grid.transform(vy_sol_hat, inverse=True).real,
        self.grid.transform(vx_sol_hat, inverse=True).real,
    )

advection_scalar(vz, vy, vx, q, spectral=False)

Compute the dealiased 3D scalar advection term (u·∇)q.

Both factors are truncated with the grid's 2/3 mask, the products formed in physical space, and the result truncated again, as in 2D (gh-90).

Parameters:

Name Type Description Default
vz Array[Nz, Ny, Nx]

Velocity components in physical space.

 required
vy Array[Nz, Ny, Nx]

Velocity components in physical space.

 required
vx Array[Nz, Ny, Nx]

Velocity components in physical space.

 required
q Array[Nz, Ny, Nx]

Advected scalar, physical (or spectral if spectral).

 required
spectral bool

If True, q is given as spectral coefficients. The result is always returned in physical space.

 False

Returns:

Type Description
Array[Nz, Ny, Nx]

The dealiased advection term in physical space.
Source code in spectraldiffx/_src/fourier/operators.py 
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def advection_scalar(
    self, vz: Array, vy: Array, vx: Array, q: Array, spectral: bool = False
) -> Float[Array, "Nz Ny Nx"]:
    """Compute the dealiased 3D scalar advection term (u·∇)q.

    Both factors are truncated with the grid's 2/3 mask, the products
    formed in physical space, and the result truncated again, as in 2D
    (gh-90).

    Parameters
    ----------
    vz, vy, vx : Array [Nz, Ny, Nx]
        Velocity components in physical space.
    q : Array [Nz, Ny, Nx]
        Advected scalar, physical (or spectral if ``spectral``).
    spectral : bool
        If True, ``q`` is given as spectral coefficients. The result is
        always returned in physical space.

    Returns
    -------
    Array [Nz, Ny, Nx]
        The dealiased advection term in physical space.
    """
    q_hat = q if spectral else self.grid.transform(_real(q))
    KZ, KY, KX = self.grid.KX
    dealias = self.grid.dealias_filter()

    dq_dz = self.grid.transform(1j * KZ * q_hat * dealias, inverse=True).real
    dq_dy = self.grid.transform(1j * KY * q_hat * dealias, inverse=True).real
    dq_dx = self.grid.transform(1j * KX * q_hat * dealias, inverse=True).real

    vz, vy, vx = (self.apply_dealias(v) for v in (vz, vy, vx))
    return self.apply_dealias(vz * dq_dz + vy * dq_dy + vx * dq_dx)