Skip to content

Spherical Harmonic Solvers

SphericalPoissonSolver

Bases: Module

Spectral Poisson solver on the sphere: ∇²φ = f.

In SHT-coefficient space the mode-by-mode inversion is

φ̂(l, m) = −f̂(l, m) · [l(l+1)/R²]⁻¹    (l ≥ 1)

The l=0 mode is set to zero when zero_mean=True (default) since ∇² annihilates constants on the sphere.

Attributes

grid : SphericalGrid1D or SphericalGrid2D Underlying spherical grid.

Examples

import jax.numpy as jnp grid = SphericalGrid2D.from_N_L(Nx=32, Ny=16) solver = SphericalPoissonSolver(grid=grid) PHI, THETA = grid.X

Laplacian of cos(θ) is −2 cos(θ)/R², so Poisson RHS is that:

R = grid.Ly / jnp.pi f = -2.0 * jnp.cos(THETA) / R**2 phi = solver.solve(f) # ≈ cos(θ) up to an additive constant

Source code in spectraldiffx/_src/spherical/solvers.py 
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
class SphericalPoissonSolver(eqx.Module):
    """Spectral Poisson solver on the sphere:  ∇²φ = f.

    In SHT-coefficient space the mode-by-mode inversion is

        φ̂(l, m) = −f̂(l, m) · [l(l+1)/R²]⁻¹    (l ≥ 1)

    The l=0 mode is set to zero when ``zero_mean=True`` (default) since
    ∇² annihilates constants on the sphere.

    Attributes
    ----------
    grid : SphericalGrid1D or SphericalGrid2D
        Underlying spherical grid.

    Examples
    --------
    >>> import jax.numpy as jnp
    >>> grid = SphericalGrid2D.from_N_L(Nx=32, Ny=16)
    >>> solver = SphericalPoissonSolver(grid=grid)
    >>> PHI, THETA = grid.X
    >>> # Laplacian of cos(θ) is −2 cos(θ)/R², so Poisson RHS is that:
    >>> R = grid.Ly / jnp.pi
    >>> f = -2.0 * jnp.cos(THETA) / R**2
    >>> phi = solver.solve(f)  # ≈ cos(θ) up to an additive constant
    """

    grid: SphericalGrid1D | SphericalGrid2D

    def solve(
        self,
        f: Num[Array, ...],
        zero_mean: bool = True,
        spectral: bool = False,
    ) -> Float[Array, ...]:
        """Solve ∇²φ = f on the sphere.

        Parameters
        ----------
        f : Num[Array, ...]
            Source field.  Shape ``(N,)`` for 1D or ``(Nlat, Nlon)`` for 2D.
        zero_mean : bool
            If ``True``, pin the l=0 mode of φ to zero (gauge fix).  Required
            for well-posedness of Poisson on the sphere.
        spectral : bool
            If ``True``, ``f`` is already a DLT/SHT coefficient array.

        Returns
        -------
        Float[Array, ...]
            Solution in physical space (same shape as ``f``).
        """
        R = _sphere_radius(self.grid)
        f_hat = f if spectral else self.grid.transform(f)
        l = self.grid.l
        eigenval = l * (l + 1) / (R**2)

        if isinstance(self.grid, SphericalGrid1D):
            # Guard l=0 division; the mode is overwritten immediately below.
            denom = jnp.where(eigenval == 0.0, 1.0, eigenval)
            phi_hat = -f_hat / denom
            if zero_mean:
                phi_hat = jnp.where(l == 0.0, 0.0, phi_hat)
        else:
            denom = jnp.where(eigenval[:, None] == 0.0, 1.0, eigenval[:, None])
            phi_hat = -f_hat / denom
            if zero_mean:
                phi_hat = jnp.where(l[:, None] == 0.0, 0.0, phi_hat)

        return self.grid.transform(phi_hat, inverse=True)

Functions

solve(f, zero_mean=True, spectral=False)

Solve ∇²φ = f on the sphere.

Parameters

f : Num[Array, ...] Source field. Shape (N,) for 1D or (Nlat, Nlon) for 2D. zero_mean : bool If True, pin the l=0 mode of φ to zero (gauge fix). Required for well-posedness of Poisson on the sphere. spectral : bool If True, f is already a DLT/SHT coefficient array.

Returns

Float[Array, ...] Solution in physical space (same shape as f).

Source code in spectraldiffx/_src/spherical/solvers.py 
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
def solve(
    self,
    f: Num[Array, ...],
    zero_mean: bool = True,
    spectral: bool = False,
) -> Float[Array, ...]:
    """Solve ∇²φ = f on the sphere.

    Parameters
    ----------
    f : Num[Array, ...]
        Source field.  Shape ``(N,)`` for 1D or ``(Nlat, Nlon)`` for 2D.
    zero_mean : bool
        If ``True``, pin the l=0 mode of φ to zero (gauge fix).  Required
        for well-posedness of Poisson on the sphere.
    spectral : bool
        If ``True``, ``f`` is already a DLT/SHT coefficient array.

    Returns
    -------
    Float[Array, ...]
        Solution in physical space (same shape as ``f``).
    """
    R = _sphere_radius(self.grid)
    f_hat = f if spectral else self.grid.transform(f)
    l = self.grid.l
    eigenval = l * (l + 1) / (R**2)

    if isinstance(self.grid, SphericalGrid1D):
        # Guard l=0 division; the mode is overwritten immediately below.
        denom = jnp.where(eigenval == 0.0, 1.0, eigenval)
        phi_hat = -f_hat / denom
        if zero_mean:
            phi_hat = jnp.where(l == 0.0, 0.0, phi_hat)
    else:
        denom = jnp.where(eigenval[:, None] == 0.0, 1.0, eigenval[:, None])
        phi_hat = -f_hat / denom
        if zero_mean:
            phi_hat = jnp.where(l[:, None] == 0.0, 0.0, phi_hat)

    return self.grid.transform(phi_hat, inverse=True)

SphericalHelmholtzSolver

Bases: Module

Spectral Helmholtz solver on the sphere: (∇² − α) φ = f.

In SHT-coefficient space:

φ̂(l, m) = −f̂(l, m) / [l(l+1)/R² + α]

Non-singular for α > 0; for α = 0 this reduces to Poisson and the l=0 gauge (zero-mean) is enforced by default.

Attributes

grid : SphericalGrid1D or SphericalGrid2D Underlying spherical grid.

Examples

import jax.numpy as jnp grid = SphericalGrid2D.from_N_L(Nx=32, Ny=16) solver = SphericalHelmholtzSolver(grid=grid) PHI, THETA = grid.X R = grid.Ly / jnp.pi alpha = 4.0

For φ = cos θ: (∇² − α) φ = (−2/R² − α) cos θ

f = (-2.0 / R**2 - alpha) * jnp.cos(THETA) phi = solver.solve(f, alpha=alpha, zero_mean=False) # ≈ cos(θ)

Source code in spectraldiffx/_src/spherical/solvers.py 
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
class SphericalHelmholtzSolver(eqx.Module):
    """Spectral Helmholtz solver on the sphere:  (∇² − α) φ = f.

    In SHT-coefficient space:

        φ̂(l, m) = −f̂(l, m) / [l(l+1)/R² + α]

    Non-singular for α > 0; for α = 0 this reduces to Poisson and the
    l=0 gauge (zero-mean) is enforced by default.

    Attributes
    ----------
    grid : SphericalGrid1D or SphericalGrid2D
        Underlying spherical grid.

    Examples
    --------
    >>> import jax.numpy as jnp
    >>> grid = SphericalGrid2D.from_N_L(Nx=32, Ny=16)
    >>> solver = SphericalHelmholtzSolver(grid=grid)
    >>> PHI, THETA = grid.X
    >>> R = grid.Ly / jnp.pi
    >>> alpha = 4.0
    >>> # For φ = cos θ: (∇² − α) φ = (−2/R² − α) cos θ
    >>> f = (-2.0 / R**2 - alpha) * jnp.cos(THETA)
    >>> phi = solver.solve(f, alpha=alpha, zero_mean=False)  # ≈ cos(θ)
    """

    grid: SphericalGrid1D | SphericalGrid2D

    def solve(
        self,
        f: Num[Array, ...],
        alpha: float = 0.0,
        zero_mean: bool = True,
        spectral: bool = False,
    ) -> Float[Array, ...]:
        """Solve (∇² − α) φ = f on the sphere.

        Parameters
        ----------
        f : Num[Array, ...]
            Source field (1D ``(N,)`` or 2D ``(Nlat, Nlon)``).
        alpha : float
            Helmholtz parameter (≥ 0).  α=0 falls back to Poisson.
        zero_mean : bool
            If ``True`` and α = 0, enforce the l=0 gauge by zeroing the
            mean of φ.  Ignored effectively when α > 0 (non-singular).
        spectral : bool
            If ``True``, ``f`` is already a DLT/SHT coefficient array.

        Returns
        -------
        Float[Array, ...]
            Solution in physical space.
        """
        if alpha < 0:
            raise ValueError(f"alpha must be >= 0, got {alpha}")
        R = _sphere_radius(self.grid)
        f_hat = f if spectral else self.grid.transform(f)
        l = self.grid.l
        eigenval = l * (l + 1) / (R**2)

        if isinstance(self.grid, SphericalGrid1D):
            denom = eigenval + alpha
            denom_safe = jnp.where(denom == 0.0, 1.0, denom)
            phi_hat = -f_hat / denom_safe
            if zero_mean:
                phi_hat = jnp.where(l == 0.0, 0.0, phi_hat)
        else:
            denom = eigenval[:, None] + alpha
            denom_safe = jnp.where(denom == 0.0, 1.0, denom)
            phi_hat = -f_hat / denom_safe
            if zero_mean:
                phi_hat = jnp.where(l[:, None] == 0.0, 0.0, phi_hat)

        return self.grid.transform(phi_hat, inverse=True)

Functions

solve(f, alpha=0.0, zero_mean=True, spectral=False)

Solve (∇² − α) φ = f on the sphere.

Parameters

f : Num[Array, ...] Source field (1D (N,) or 2D (Nlat, Nlon)). alpha : float Helmholtz parameter (≥ 0). α=0 falls back to Poisson. zero_mean : bool If True and α = 0, enforce the l=0 gauge by zeroing the mean of φ. Ignored effectively when α > 0 (non-singular). spectral : bool If True, f is already a DLT/SHT coefficient array.

Returns

Float[Array, ...] Solution in physical space.

Source code in spectraldiffx/_src/spherical/solvers.py 
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
def solve(
    self,
    f: Num[Array, ...],
    alpha: float = 0.0,
    zero_mean: bool = True,
    spectral: bool = False,
) -> Float[Array, ...]:
    """Solve (∇² − α) φ = f on the sphere.

    Parameters
    ----------
    f : Num[Array, ...]
        Source field (1D ``(N,)`` or 2D ``(Nlat, Nlon)``).
    alpha : float
        Helmholtz parameter (≥ 0).  α=0 falls back to Poisson.
    zero_mean : bool
        If ``True`` and α = 0, enforce the l=0 gauge by zeroing the
        mean of φ.  Ignored effectively when α > 0 (non-singular).
    spectral : bool
        If ``True``, ``f`` is already a DLT/SHT coefficient array.

    Returns
    -------
    Float[Array, ...]
        Solution in physical space.
    """
    if alpha < 0:
        raise ValueError(f"alpha must be >= 0, got {alpha}")
    R = _sphere_radius(self.grid)
    f_hat = f if spectral else self.grid.transform(f)
    l = self.grid.l
    eigenval = l * (l + 1) / (R**2)

    if isinstance(self.grid, SphericalGrid1D):
        denom = eigenval + alpha
        denom_safe = jnp.where(denom == 0.0, 1.0, denom)
        phi_hat = -f_hat / denom_safe
        if zero_mean:
            phi_hat = jnp.where(l == 0.0, 0.0, phi_hat)
    else:
        denom = eigenval[:, None] + alpha
        denom_safe = jnp.where(denom == 0.0, 1.0, denom)
        phi_hat = -f_hat / denom_safe
        if zero_mean:
            phi_hat = jnp.where(l[:, None] == 0.0, 0.0, phi_hat)

    return self.grid.transform(phi_hat, inverse=True)