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Spherical Filters

SphericalFilter1D

Bases: Module

1D spectral filter on the Gauss-Legendre latitude grid.

Filters are applied as multiplicative masks in Legendre coefficient space

c_l_filtered = F(l) * c_l

Attributes:

grid : SphericalGrid1D The 1D Gauss-Legendre grid.

Source code in spectraldiffx/_src/spherical/filters.py 
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class SphericalFilter1D(eqx.Module):
    """
    1D spectral filter on the Gauss-Legendre latitude grid.

    Filters are applied as multiplicative masks in Legendre coefficient space:
        c_l_filtered = F(l) * c_l

    Attributes:
    -----------
    grid : SphericalGrid1D
        The 1D Gauss-Legendre grid.
    """

    grid: SphericalGrid1D

    def exponential_filter(
        self,
        u: Float[Array, "N"],
        alpha: float = 36.0,
        power: int = 16,
        spectral: bool = False,
    ) -> Float[Array, "N"]:
        """
        Apply exponential filter in Legendre coefficient space.

            F(l) = exp(-alpha * (l / l_max)^power)

        Near unity for low degrees, falls sharply near l_max.

        Parameters:
        -----------
        u : Float[Array, "N"]
            Physical field or Legendre coefficients (if spectral=True).
        alpha : float
            Damping coefficient. Default is 36.0.
        power : int
            Filter sharpness (even integer). Default is 16.
        spectral : bool
            If True, treat u as Legendre coefficients.

        Returns:
        --------
        Array [N]
            Filtered field or spectral coefficients.
        """
        c = u if spectral else self.grid.transform(u)
        l = self.grid.l
        l_max = float(self.grid.N - 1)
        l_max_safe = jnp.where(l_max == 0, 1.0, l_max)
        mask = jnp.exp(-alpha * (l / l_max_safe) ** power)
        c_f = c * mask
        return c_f if spectral else self.grid.transform(c_f, inverse=True)

    def hyperviscosity(
        self,
        u: Float[Array, "N"],
        nu_hyper: float,
        dt: float,
        power: int = 4,
        spectral: bool = False,
    ) -> Float[Array, "N"]:
        """
        Apply hyperviscous damping in Legendre coefficient space.

            F(l) = exp(-nu_hyper * [l*(l+1)/R^2]^(power/2) * dt)

        where R = grid.L / pi is the sphere radius.  This matches the physical
        Laplacian eigenvalue so that the damping rate is independent of the
        sphere radius for a given nu_hyper.

        Simulates high-order diffusion: du/dt = (-1)^(p+1) * nu_h * nabla^p u.

        Parameters:
        -----------
        u : Float[Array, "N"]
            Physical field or Legendre coefficients.
        nu_hyper : float
            Hyperviscosity coefficient.
        dt : float
            Time step for the damping.
        power : int
            Order of the Laplacian power (e.g., 4 for biharmonic). Default is 4.
        spectral : bool
            If True, treat u as Legendre coefficients.

        Returns:
        --------
        Array [N]
        """
        c = u if spectral else self.grid.transform(u)
        R = self.grid.L / jnp.pi
        l = self.grid.l
        eigenval = l * (l + 1) / (R**2)
        mask = jnp.exp(-nu_hyper * eigenval ** (power / 2.0) * dt)
        c_f = c * mask
        return c_f if spectral else self.grid.transform(c_f, inverse=True)

Functions

exponential_filter(u, alpha=36.0, power=16, spectral=False)

Apply exponential filter in Legendre coefficient space.

F(l) = exp(-alpha * (l / l_max)^power)

Near unity for low degrees, falls sharply near l_max.

Parameters:

u : Float[Array, "N"] Physical field or Legendre coefficients (if spectral=True). alpha : float Damping coefficient. Default is 36.0. power : int Filter sharpness (even integer). Default is 16. spectral : bool If True, treat u as Legendre coefficients.

Returns:

Array [N] Filtered field or spectral coefficients.

Source code in spectraldiffx/_src/spherical/filters.py 
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def exponential_filter(
    self,
    u: Float[Array, "N"],
    alpha: float = 36.0,
    power: int = 16,
    spectral: bool = False,
) -> Float[Array, "N"]:
    """
    Apply exponential filter in Legendre coefficient space.

        F(l) = exp(-alpha * (l / l_max)^power)

    Near unity for low degrees, falls sharply near l_max.

    Parameters:
    -----------
    u : Float[Array, "N"]
        Physical field or Legendre coefficients (if spectral=True).
    alpha : float
        Damping coefficient. Default is 36.0.
    power : int
        Filter sharpness (even integer). Default is 16.
    spectral : bool
        If True, treat u as Legendre coefficients.

    Returns:
    --------
    Array [N]
        Filtered field or spectral coefficients.
    """
    c = u if spectral else self.grid.transform(u)
    l = self.grid.l
    l_max = float(self.grid.N - 1)
    l_max_safe = jnp.where(l_max == 0, 1.0, l_max)
    mask = jnp.exp(-alpha * (l / l_max_safe) ** power)
    c_f = c * mask
    return c_f if spectral else self.grid.transform(c_f, inverse=True)

hyperviscosity(u, nu_hyper, dt, power=4, spectral=False)

Apply hyperviscous damping in Legendre coefficient space.

F(l) = exp(-nu_hyper * [l*(l+1)/R^2]^(power/2) * dt)

where R = grid.L / pi is the sphere radius. This matches the physical Laplacian eigenvalue so that the damping rate is independent of the sphere radius for a given nu_hyper.

Simulates high-order diffusion: du/dt = (-1)^(p+1) * nu_h * nabla^p u.

Parameters:

u : Float[Array, "N"] Physical field or Legendre coefficients. nu_hyper : float Hyperviscosity coefficient. dt : float Time step for the damping. power : int Order of the Laplacian power (e.g., 4 for biharmonic). Default is 4. spectral : bool If True, treat u as Legendre coefficients.

Returns:

Array [N]

Source code in spectraldiffx/_src/spherical/filters.py 
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def hyperviscosity(
    self,
    u: Float[Array, "N"],
    nu_hyper: float,
    dt: float,
    power: int = 4,
    spectral: bool = False,
) -> Float[Array, "N"]:
    """
    Apply hyperviscous damping in Legendre coefficient space.

        F(l) = exp(-nu_hyper * [l*(l+1)/R^2]^(power/2) * dt)

    where R = grid.L / pi is the sphere radius.  This matches the physical
    Laplacian eigenvalue so that the damping rate is independent of the
    sphere radius for a given nu_hyper.

    Simulates high-order diffusion: du/dt = (-1)^(p+1) * nu_h * nabla^p u.

    Parameters:
    -----------
    u : Float[Array, "N"]
        Physical field or Legendre coefficients.
    nu_hyper : float
        Hyperviscosity coefficient.
    dt : float
        Time step for the damping.
    power : int
        Order of the Laplacian power (e.g., 4 for biharmonic). Default is 4.
    spectral : bool
        If True, treat u as Legendre coefficients.

    Returns:
    --------
    Array [N]
    """
    c = u if spectral else self.grid.transform(u)
    R = self.grid.L / jnp.pi
    l = self.grid.l
    eigenval = l * (l + 1) / (R**2)
    mask = jnp.exp(-nu_hyper * eigenval ** (power / 2.0) * dt)
    c_f = c * mask
    return c_f if spectral else self.grid.transform(c_f, inverse=True)

SphericalFilter2D

Bases: Module

2D spectral filter on the full sphere lat-lon grid.

Filters are applied in SHT coefficient space using the spherical harmonic degree l as the spectral index.

Attributes:

grid : SphericalGrid2D The 2D lat-lon grid.

Source code in spectraldiffx/_src/spherical/filters.py 
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class SphericalFilter2D(eqx.Module):
    """
    2D spectral filter on the full sphere lat-lon grid.

    Filters are applied in SHT coefficient space using the spherical harmonic
    degree l as the spectral index.

    Attributes:
    -----------
    grid : SphericalGrid2D
        The 2D lat-lon grid.
    """

    grid: SphericalGrid2D

    def exponential_filter(
        self,
        u: Float[Array, "Ny Nx"],
        alpha: float = 36.0,
        power: int = 16,
        spectral: bool = False,
    ) -> Float[Array, "Ny Nx"]:
        """
        Apply 2D exponential filter using spherical harmonic degree l.

            F(l, m) = exp(-alpha * (l / l_max)^power)

        Parameters:
        -----------
        u : Float[Array, "Ny Nx"]
            Physical field or SHT coefficients.
        alpha : float
            Damping coefficient. Default is 36.0.
        power : int
            Filter sharpness. Default is 16.
        spectral : bool
            If True, treat u as SHT coefficients.

        Returns:
        --------
        Array [Ny, Nx]
            Filtered field or spectral coefficients.
        """
        u_hat = u if spectral else self.grid.transform(u)
        l = self.grid.l  # (Ny,)
        l_max = float(self.grid.Ny - 1)
        l_max_safe = jnp.where(l_max == 0.0, 1.0, l_max)
        mask = jnp.exp(-alpha * (l / l_max_safe) ** power)  # (Ny,)
        u_hat_f = u_hat * mask[:, None]  # broadcast over m
        return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True)

    def hyperviscosity(
        self,
        u: Float[Array, "Ny Nx"],
        nu_hyper: float,
        dt: float,
        power: int = 4,
        spectral: bool = False,
    ) -> Float[Array, "Ny Nx"]:
        """
        Apply 2D hyperviscous damping using l*(l+1)/R^2 eigenvalues.

            F(l) = exp(-nu_hyper * [l*(l+1)/R^2]^(power/2) * dt)

        where R = grid.Ly / pi is the sphere radius.

        Parameters:
        -----------
        u : Float[Array, "Ny Nx"]
            Physical field or SHT coefficients.
        nu_hyper : float
            Hyperviscosity coefficient.
        dt : float
            Time step.
        power : int
            Laplacian power order. Default is 4.
        spectral : bool
            If True, treat u as SHT coefficients.

        Returns:
        --------
        Array [Ny, Nx]
        """
        u_hat = u if spectral else self.grid.transform(u)
        R = self.grid.Ly / jnp.pi
        l = self.grid.l  # (Ny,)
        eigenval = l * (l + 1) / (R**2)
        mask = jnp.exp(-nu_hyper * eigenval ** (power / 2.0) * dt)  # (Ny,)
        u_hat_f = u_hat * mask[:, None]
        return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True)

Functions

exponential_filter(u, alpha=36.0, power=16, spectral=False)

Apply 2D exponential filter using spherical harmonic degree l.

F(l, m) = exp(-alpha * (l / l_max)^power)
Parameters:

u : Float[Array, "Ny Nx"] Physical field or SHT coefficients. alpha : float Damping coefficient. Default is 36.0. power : int Filter sharpness. Default is 16. spectral : bool If True, treat u as SHT coefficients.

Returns:

Array [Ny, Nx] Filtered field or spectral coefficients.

Source code in spectraldiffx/_src/spherical/filters.py 
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def exponential_filter(
    self,
    u: Float[Array, "Ny Nx"],
    alpha: float = 36.0,
    power: int = 16,
    spectral: bool = False,
) -> Float[Array, "Ny Nx"]:
    """
    Apply 2D exponential filter using spherical harmonic degree l.

        F(l, m) = exp(-alpha * (l / l_max)^power)

    Parameters:
    -----------
    u : Float[Array, "Ny Nx"]
        Physical field or SHT coefficients.
    alpha : float
        Damping coefficient. Default is 36.0.
    power : int
        Filter sharpness. Default is 16.
    spectral : bool
        If True, treat u as SHT coefficients.

    Returns:
    --------
    Array [Ny, Nx]
        Filtered field or spectral coefficients.
    """
    u_hat = u if spectral else self.grid.transform(u)
    l = self.grid.l  # (Ny,)
    l_max = float(self.grid.Ny - 1)
    l_max_safe = jnp.where(l_max == 0.0, 1.0, l_max)
    mask = jnp.exp(-alpha * (l / l_max_safe) ** power)  # (Ny,)
    u_hat_f = u_hat * mask[:, None]  # broadcast over m
    return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True)

hyperviscosity(u, nu_hyper, dt, power=4, spectral=False)

Apply 2D hyperviscous damping using l*(l+1)/R^2 eigenvalues.

F(l) = exp(-nu_hyper * [l*(l+1)/R^2]^(power/2) * dt)

where R = grid.Ly / pi is the sphere radius.

Parameters:

u : Float[Array, "Ny Nx"] Physical field or SHT coefficients. nu_hyper : float Hyperviscosity coefficient. dt : float Time step. power : int Laplacian power order. Default is 4. spectral : bool If True, treat u as SHT coefficients.

Returns:

Array [Ny, Nx]

Source code in spectraldiffx/_src/spherical/filters.py 
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def hyperviscosity(
    self,
    u: Float[Array, "Ny Nx"],
    nu_hyper: float,
    dt: float,
    power: int = 4,
    spectral: bool = False,
) -> Float[Array, "Ny Nx"]:
    """
    Apply 2D hyperviscous damping using l*(l+1)/R^2 eigenvalues.

        F(l) = exp(-nu_hyper * [l*(l+1)/R^2]^(power/2) * dt)

    where R = grid.Ly / pi is the sphere radius.

    Parameters:
    -----------
    u : Float[Array, "Ny Nx"]
        Physical field or SHT coefficients.
    nu_hyper : float
        Hyperviscosity coefficient.
    dt : float
        Time step.
    power : int
        Laplacian power order. Default is 4.
    spectral : bool
        If True, treat u as SHT coefficients.

    Returns:
    --------
    Array [Ny, Nx]
    """
    u_hat = u if spectral else self.grid.transform(u)
    R = self.grid.Ly / jnp.pi
    l = self.grid.l  # (Ny,)
    eigenval = l * (l + 1) / (R**2)
    mask = jnp.exp(-nu_hyper * eigenval ** (power / 2.0) * dt)  # (Ny,)
    u_hat_f = u_hat * mask[:, None]
    return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True)