Skip to content

Vorticity Operators

Relative vorticity and Jacobian operators on Arakawa C-grids.

finitevolx.Vorticity2D

Bases: Module

2-D vorticity and PV-flux operators.

Parameters:

Name Type Description Default
grid ArakawaCGrid2D
required
Source code in finitevolx/_src/operators/vorticity.py 
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 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
130
131
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
class Vorticity2D(eqx.Module):
    """2-D vorticity and PV-flux operators.

    Parameters
    ----------
    grid : ArakawaCGrid2D
    """

    grid: ArakawaCGrid2D
    diff: Difference2D
    interp: Interpolation2D

    def __init__(self, grid: ArakawaCGrid2D) -> None:
        self.grid = grid
        self.diff = Difference2D(grid=grid)
        self.interp = Interpolation2D(grid=grid)

    def relative_vorticity(
        self,
        u: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """Relative vorticity at X-points (corners).

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

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

        Returns
        -------
        Float[Array, "Ny Nx"]
            Relative vorticity at X-points.
        """
        return self.diff.curl(u, v)

    def potential_vorticity(
        self,
        u: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
        h: Float[Array, "Ny Nx"],
        f: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """Potential vorticity at X-points (corners).

        q[j+1/2, i+1/2] = (zeta[j+1/2, i+1/2] + f_on_q[j+1/2, i+1/2])
                         / h_on_q[j+1/2, i+1/2]

        Parameters
        ----------
        u : Float[Array, "Ny Nx"]
            x-velocity at U-points.
        v : Float[Array, "Ny Nx"]
            y-velocity at V-points.
        h : Float[Array, "Ny Nx"]
            Layer thickness at T-points.
        f : Float[Array, "Ny Nx"]
            Coriolis parameter at T-points.

        Returns
        -------
        Float[Array, "Ny Nx"]
            Potential vorticity at X-points.
        """
        zeta = self.relative_vorticity(u, v)  # zeta at X-points
        f_on_q = self.interp.T_to_X(f)  # f interpolated to X-points
        h_on_q = self.interp.T_to_X(h)  # h interpolated to X-points
        # q[j+1/2, i+1/2] = (zeta + f) / h  at X-points
        num = zeta[1:-1, 1:-1] + f_on_q[1:-1, 1:-1]
        den = h_on_q[1:-1, 1:-1]
        out = interior(jnp.where(den == 0, jnp.nan, num / den), h)
        return out

    def pv_flux_energy_conserving(
        self,
        q: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> tuple:
        """Energy-conserving PV flux.

        Interpolate q and velocity independently to faces, then multiply.

        qu[j, i+1/2] = q_on_u[j, i+1/2] * u[j, i+1/2]
        qv[j+1/2, i] = q_on_v[j+1/2, i] * v[j+1/2, i]

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

        Returns
        -------
        tuple
            (qu at U-points, qv at V-points)
        """
        q_on_u = self.interp.X_to_U(q)  # q_on_u[j, i+1/2] = avg in y
        q_on_v = self.interp.X_to_V(q)  # q_on_v[j+1/2, i] = avg in x
        # qu[j, i+1/2] = q_on_u[j, i+1/2] * u[j, i+1/2]
        qu = interior(q_on_u[1:-1, 1:-1] * u[1:-1, 1:-1], u)
        # qv[j+1/2, i] = q_on_v[j+1/2, i] * v[j+1/2, i]
        qv = interior(q_on_v[1:-1, 1:-1] * v[1:-1, 1:-1], v)
        return qu, qv

    def pv_flux_enstrophy_conserving(
        self,
        q: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> tuple:
        """Enstrophy-conserving PV flux.

        Multiply q*u at corners/faces, then interpolate to faces.

        qu[j, i+1/2] = X_to_U(q * U_to_X(u))
        qv[j+1/2, i] = X_to_V(q * V_to_X(v))

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

        Returns
        -------
        tuple
            (qu at U-points, qv at V-points)
        """
        u_on_q = self.interp.U_to_X(u)  # u_on_q[j+1/2, i+1/2]
        v_on_q = self.interp.V_to_X(v)  # v_on_q[j+1/2, i+1/2]
        # Multiply at corners
        # qu_at_q[j+1/2, i+1/2] = q[j+1/2, i+1/2] * u_on_q[j+1/2, i+1/2]
        qu_at_q = interior(q[1:-1, 1:-1] * u_on_q[1:-1, 1:-1], q)
        # qv_at_q[j+1/2, i+1/2] = q[j+1/2, i+1/2] * v_on_q[j+1/2, i+1/2]
        qv_at_q = interior(q[1:-1, 1:-1] * v_on_q[1:-1, 1:-1], q)
        # Interpolate back to faces
        qu = self.interp.X_to_U(qu_at_q)  # qu[j, i+1/2]
        qv = self.interp.X_to_V(qv_at_q)  # qv[j+1/2, i]
        return qu, qv

    def pv_flux_arakawa_lamb(
        self,
        q: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
        alpha: float = 1.0 / 3.0,
    ) -> tuple:
        """Arakawa-Lamb PV flux: weighted blend of energy and enstrophy.

        flux = alpha * energy_conserving + (1 - alpha) * enstrophy_conserving

        Parameters
        ----------
        q : Float[Array, "Ny Nx"]
            Potential vorticity at X-points.
        u : Float[Array, "Ny Nx"]
            x-velocity at U-points.
        v : Float[Array, "Ny Nx"]
            y-velocity at V-points.
        alpha : float
            Blending weight.  Default 1/3 gives Arakawa-Lamb scheme.

        Returns
        -------
        tuple
            (qu at U-points, qv at V-points)
        """
        qu_e, qv_e = self.pv_flux_energy_conserving(q, u, v)
        qu_s, qv_s = self.pv_flux_enstrophy_conserving(q, u, v)
        # Weighted blend
        qu = alpha * qu_e + (1.0 - alpha) * qu_s
        qv = alpha * qv_e + (1.0 - alpha) * qv_s
        return qu, qv

potential_vorticity(u, v, h, f)

Potential vorticity at X-points (corners).

q[j+1/2, i+1/2] = (zeta[j+1/2, i+1/2] + f_on_q[j+1/2, i+1/2]) / h_on_q[j+1/2, i+1/2]

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
h Float[Array, 'Ny Nx']

Layer thickness at T-points.

 required
f Float[Array, 'Ny Nx']

Coriolis parameter at T-points.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Potential vorticity at X-points.
Source code in finitevolx/_src/operators/vorticity.py 
59
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
def potential_vorticity(
    self,
    u: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
    h: Float[Array, "Ny Nx"],
    f: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """Potential vorticity at X-points (corners).

    q[j+1/2, i+1/2] = (zeta[j+1/2, i+1/2] + f_on_q[j+1/2, i+1/2])
                     / h_on_q[j+1/2, i+1/2]

    Parameters
    ----------
    u : Float[Array, "Ny Nx"]
        x-velocity at U-points.
    v : Float[Array, "Ny Nx"]
        y-velocity at V-points.
    h : Float[Array, "Ny Nx"]
        Layer thickness at T-points.
    f : Float[Array, "Ny Nx"]
        Coriolis parameter at T-points.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Potential vorticity at X-points.
    """
    zeta = self.relative_vorticity(u, v)  # zeta at X-points
    f_on_q = self.interp.T_to_X(f)  # f interpolated to X-points
    h_on_q = self.interp.T_to_X(h)  # h interpolated to X-points
    # q[j+1/2, i+1/2] = (zeta + f) / h  at X-points
    num = zeta[1:-1, 1:-1] + f_on_q[1:-1, 1:-1]
    den = h_on_q[1:-1, 1:-1]
    out = interior(jnp.where(den == 0, jnp.nan, num / den), h)
    return out

pv_flux_arakawa_lamb(q, u, v, alpha=1.0 / 3.0)

Arakawa-Lamb PV flux: weighted blend of energy and enstrophy.

flux = alpha * energy_conserving + (1 - alpha) * enstrophy_conserving

Parameters:

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

Potential vorticity at X-points.

 required
u Float[Array, 'Ny Nx']

x-velocity at U-points.

 required
v Float[Array, 'Ny Nx']

y-velocity at V-points.

 required
alpha float

Blending weight. Default 1/3 gives Arakawa-Lamb scheme.

 1.0 / 3.0

Returns:

Type Description
tuple

(qu at U-points, qv at V-points)
Source code in finitevolx/_src/operators/vorticity.py 
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
def pv_flux_arakawa_lamb(
    self,
    q: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
    alpha: float = 1.0 / 3.0,
) -> tuple:
    """Arakawa-Lamb PV flux: weighted blend of energy and enstrophy.

    flux = alpha * energy_conserving + (1 - alpha) * enstrophy_conserving

    Parameters
    ----------
    q : Float[Array, "Ny Nx"]
        Potential vorticity at X-points.
    u : Float[Array, "Ny Nx"]
        x-velocity at U-points.
    v : Float[Array, "Ny Nx"]
        y-velocity at V-points.
    alpha : float
        Blending weight.  Default 1/3 gives Arakawa-Lamb scheme.

    Returns
    -------
    tuple
        (qu at U-points, qv at V-points)
    """
    qu_e, qv_e = self.pv_flux_energy_conserving(q, u, v)
    qu_s, qv_s = self.pv_flux_enstrophy_conserving(q, u, v)
    # Weighted blend
    qu = alpha * qu_e + (1.0 - alpha) * qu_s
    qv = alpha * qv_e + (1.0 - alpha) * qv_s
    return qu, qv

pv_flux_energy_conserving(q, u, v)

Energy-conserving PV flux.

Interpolate q and velocity independently to faces, then multiply.

qu[j, i+1/2] = q_on_u[j, i+1/2] * u[j, i+1/2] qv[j+1/2, i] = q_on_v[j+1/2, i] * v[j+1/2, i]

Parameters:

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

Potential vorticity at X-points.

 required
u Float[Array, 'Ny Nx']

x-velocity at U-points.

 required
v Float[Array, 'Ny Nx']

y-velocity at V-points.

 required

Returns:

Type Description
tuple

(qu at U-points, qv at V-points)
Source code in finitevolx/_src/operators/vorticity.py 
 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 pv_flux_energy_conserving(
    self,
    q: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> tuple:
    """Energy-conserving PV flux.

    Interpolate q and velocity independently to faces, then multiply.

    qu[j, i+1/2] = q_on_u[j, i+1/2] * u[j, i+1/2]
    qv[j+1/2, i] = q_on_v[j+1/2, i] * v[j+1/2, i]

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

    Returns
    -------
    tuple
        (qu at U-points, qv at V-points)
    """
    q_on_u = self.interp.X_to_U(q)  # q_on_u[j, i+1/2] = avg in y
    q_on_v = self.interp.X_to_V(q)  # q_on_v[j+1/2, i] = avg in x
    # qu[j, i+1/2] = q_on_u[j, i+1/2] * u[j, i+1/2]
    qu = interior(q_on_u[1:-1, 1:-1] * u[1:-1, 1:-1], u)
    # qv[j+1/2, i] = q_on_v[j+1/2, i] * v[j+1/2, i]
    qv = interior(q_on_v[1:-1, 1:-1] * v[1:-1, 1:-1], v)
    return qu, qv

pv_flux_enstrophy_conserving(q, u, v)

Enstrophy-conserving PV flux.

Multiply q*u at corners/faces, then interpolate to faces.

qu[j, i+1/2] = X_to_U(q * U_to_X(u)) qv[j+1/2, i] = X_to_V(q * V_to_X(v))

Parameters:

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

Potential vorticity at X-points.

 required
u Float[Array, 'Ny Nx']

x-velocity at U-points.

 required
v Float[Array, 'Ny Nx']

y-velocity at V-points.

 required

Returns:

Type Description
tuple

(qu at U-points, qv at V-points)
Source code in finitevolx/_src/operators/vorticity.py 
131
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
def pv_flux_enstrophy_conserving(
    self,
    q: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> tuple:
    """Enstrophy-conserving PV flux.

    Multiply q*u at corners/faces, then interpolate to faces.

    qu[j, i+1/2] = X_to_U(q * U_to_X(u))
    qv[j+1/2, i] = X_to_V(q * V_to_X(v))

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

    Returns
    -------
    tuple
        (qu at U-points, qv at V-points)
    """
    u_on_q = self.interp.U_to_X(u)  # u_on_q[j+1/2, i+1/2]
    v_on_q = self.interp.V_to_X(v)  # v_on_q[j+1/2, i+1/2]
    # Multiply at corners
    # qu_at_q[j+1/2, i+1/2] = q[j+1/2, i+1/2] * u_on_q[j+1/2, i+1/2]
    qu_at_q = interior(q[1:-1, 1:-1] * u_on_q[1:-1, 1:-1], q)
    # qv_at_q[j+1/2, i+1/2] = q[j+1/2, i+1/2] * v_on_q[j+1/2, i+1/2]
    qv_at_q = interior(q[1:-1, 1:-1] * v_on_q[1:-1, 1:-1], q)
    # Interpolate back to faces
    qu = self.interp.X_to_U(qu_at_q)  # qu[j, i+1/2]
    qv = self.interp.X_to_V(qv_at_q)  # qv[j+1/2, i]
    return qu, qv

relative_vorticity(u, v)

Relative vorticity at X-points (corners).

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

Parameters:

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

x-velocity at U-points.

 required
v Float[Array, 'Ny Nx']

y-velocity at V-points.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

Relative vorticity at X-points.
Source code in finitevolx/_src/operators/vorticity.py 
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
def relative_vorticity(
    self,
    u: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """Relative vorticity at X-points (corners).

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

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

    Returns
    -------
    Float[Array, "Ny Nx"]
        Relative vorticity at X-points.
    """
    return self.diff.curl(u, v)

finitevolx.Vorticity3D

Bases: Module

3-D vorticity operators (horizontal plane per z-level).

Parameters:

Name Type Description Default
grid ArakawaCGrid3D
required
Source code in finitevolx/_src/operators/vorticity.py 
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
class Vorticity3D(eqx.Module):
    """3-D vorticity operators (horizontal plane per z-level).

    Parameters
    ----------
    grid : ArakawaCGrid3D
    """

    grid: ArakawaCGrid3D

    def relative_vorticity(
        self,
        u: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """Relative vorticity at X-points over all z-levels.

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

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

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            Relative vorticity at X-points.
        """
        out = eqx.filter_vmap(
            lambda u_k, v_k: _curl_2d(u_k, v_k, self.grid.dx, self.grid.dy)
        )(u, v)
        # Zero z-ghost slices to match 3D ghost-ring convention.
        return zero_z_ghosts(out)

relative_vorticity(u, v)

Relative vorticity at X-points over all z-levels.

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

Parameters:

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

x-velocity at U-points.

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

y-velocity at V-points.

 required

Returns:

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

Relative vorticity at X-points.
Source code in finitevolx/_src/operators/vorticity.py 
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
def relative_vorticity(
    self,
    u: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """Relative vorticity at X-points over all z-levels.

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

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

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        Relative vorticity at X-points.
    """
    out = eqx.filter_vmap(
        lambda u_k, v_k: _curl_2d(u_k, v_k, self.grid.dx, self.grid.dy)
    )(u, v)
    # Zero z-ghost slices to match 3D ghost-ring convention.
    return zero_z_ghosts(out)

Jacobian

finitevolx.arakawa_jacobian(f, g, dx, dy)

Arakawa (1966) discretization of J(f, g).

Computes the Jacobian J(f, g) = ∂f/∂x·∂g/∂y − ∂f/∂y·∂g/∂x using the energy- and enstrophy-conserving three-term Arakawa scheme on a collocated grid. The inputs must include a one-point boundary halo on each side so that the returned interior array has shape (..., Ny-2, Nx-2) (i.e. Ny_i = Ny - 2, Nx_i = Nx - 2).

Parameters:

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

First scalar field (including one halo cell on each side).

 required
g Float[Array, '... Ny Nx']

Second scalar field (same shape as f).

 required
dx float

Grid spacing in the x-direction (last array axis).

 required
dy float

Grid spacing in the y-direction (second-to-last array axis).

 required

Returns:

Type Description
Float[Array, '... Ny_i Nx_i']

Jacobian evaluated on the interior grid points, where Ny_i = Ny - 2 and Nx_i = Nx - 2. Boundary points are consumed by the stencil and are not included in the output.
Notes

The Arakawa scheme averages three discrete forms:

  • J⁺⁺ (standard centred form):

.. code-block:: text

 Jpp[j,i] = ( (f[j,i+1] - f[j,i-1]) * (g[j+1,i] - g[j-1,i])
            - (f[j+1,i] - f[j-1,i]) * (g[j,i+1] - g[j,i-1]) ) / (4 dx dy)
  • J⁺× (advective form):

.. code-block:: text

 Jpx[j,i] = ( f[j,i+1] * (g[j+1,i+1] - g[j-1,i+1])
            - f[j,i-1] * (g[j+1,i-1] - g[j-1,i-1])
            - f[j+1,i] * (g[j+1,i+1] - g[j+1,i-1])
            + f[j-1,i] * (g[j-1,i+1] - g[j-1,i-1]) ) / (4 dx dy)
  • J×⁺ (divergence form):

.. code-block:: text

 Jxp[j,i] = ( g[j+1,i] * (f[j+1,i+1] - f[j+1,i-1])
            - g[j-1,i] * (f[j-1,i+1] - f[j-1,i-1])
            - g[j,i+1] * (f[j+1,i+1] - f[j-1,i+1])
            + g[j,i-1] * (f[j+1,i-1] - f[j-1,i-1]) ) / (4 dx dy)

Together: J = (Jpp + Jpx + Jxp) / 3

This triple average conserves energy (∫∫ f·J dA = 0), enstrophy (∫∫ g·J dA = 0), satisfies J(f, f) = 0, and ∫∫ J(f, g) dA = 0 at the discrete level.

The function is JAX-compatible and jit-able. Batch dimensions (...) are supported via standard broadcasting.

Examples:

>>> import jax.numpy as jnp
>>> from finitevolx import arakawa_jacobian
>>> Ny, Nx = 12, 10
>>> x = jnp.linspace(0, 1, Nx)
>>> y = jnp.linspace(0, 1, Ny)
>>> dx, dy = x[1] - x[0], y[1] - y[0]
>>> X, Y = jnp.meshgrid(x, y)
>>> J = arakawa_jacobian(X, Y, float(dx), float(dy))
>>> J.shape
(10, 8)
Source code in finitevolx/_src/operators/jacobian.py 
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 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
130
131
def arakawa_jacobian(
    f: Float[Array, "... Ny Nx"],
    g: Float[Array, "... Ny Nx"],
    dx: float,
    dy: float,
) -> Float[Array, "... Ny_i Nx_i"]:
    """Arakawa (1966) discretization of J(f, g).

    Computes the Jacobian J(f, g) = ∂f/∂x·∂g/∂y − ∂f/∂y·∂g/∂x using the
    energy- and enstrophy-conserving three-term Arakawa scheme on a collocated
    grid.  The inputs must include a one-point boundary halo on each side so
    that the returned interior array has shape ``(..., Ny-2, Nx-2)``
    (i.e. ``Ny_i = Ny - 2``, ``Nx_i = Nx - 2``).

    Parameters
    ----------
    f : Float[Array, "... Ny Nx"]
        First scalar field (including one halo cell on each side).
    g : Float[Array, "... Ny Nx"]
        Second scalar field (same shape as *f*).
    dx : float
        Grid spacing in the x-direction (last array axis).
    dy : float
        Grid spacing in the y-direction (second-to-last array axis).

    Returns
    -------
    Float[Array, "... Ny_i Nx_i"]
        Jacobian evaluated on the interior grid points, where
        ``Ny_i = Ny - 2`` and ``Nx_i = Nx - 2``.
        Boundary points are consumed by the stencil and are not included
        in the output.

    Notes
    -----
    The Arakawa scheme averages three discrete forms:

    * J⁺⁺ (standard centred form):

      .. code-block:: text

         Jpp[j,i] = ( (f[j,i+1] - f[j,i-1]) * (g[j+1,i] - g[j-1,i])
                    - (f[j+1,i] - f[j-1,i]) * (g[j,i+1] - g[j,i-1]) ) / (4 dx dy)

    * J⁺× (advective form):

      .. code-block:: text

         Jpx[j,i] = ( f[j,i+1] * (g[j+1,i+1] - g[j-1,i+1])
                    - f[j,i-1] * (g[j+1,i-1] - g[j-1,i-1])
                    - f[j+1,i] * (g[j+1,i+1] - g[j+1,i-1])
                    + f[j-1,i] * (g[j-1,i+1] - g[j-1,i-1]) ) / (4 dx dy)

    * J×⁺ (divergence form):

      .. code-block:: text

         Jxp[j,i] = ( g[j+1,i] * (f[j+1,i+1] - f[j+1,i-1])
                    - g[j-1,i] * (f[j-1,i+1] - f[j-1,i-1])
                    - g[j,i+1] * (f[j+1,i+1] - f[j-1,i+1])
                    + g[j,i-1] * (f[j+1,i-1] - f[j-1,i-1]) ) / (4 dx dy)

    Together: ``J = (Jpp + Jpx + Jxp) / 3``

    This triple average conserves energy (∫∫ f·J dA = 0), enstrophy
    (∫∫ g·J dA = 0), satisfies J(f, f) = 0, and ∫∫ J(f, g) dA = 0 at the
    discrete level.

    The function is JAX-compatible and ``jit``-able.  Batch dimensions
    (``...``) are supported via standard broadcasting.

    Examples
    --------
    >>> import jax.numpy as jnp
    >>> from finitevolx import arakawa_jacobian
    >>> Ny, Nx = 12, 10
    >>> x = jnp.linspace(0, 1, Nx)
    >>> y = jnp.linspace(0, 1, Ny)
    >>> dx, dy = x[1] - x[0], y[1] - y[0]
    >>> X, Y = jnp.meshgrid(x, y)
    >>> J = arakawa_jacobian(X, Y, float(dx), float(dy))
    >>> J.shape
    (10, 8)
    """
    # J++ (standard centred form)
    # Jpp[j,i] = (df/dx * dg/dy - df/dy * dg/dx)
    # where df/dx ~ (f[j, i+1] - f[j, i-1]) / (2dx), etc.
    Jpp = (f[..., 1:-1, 2:] - f[..., 1:-1, :-2]) * (
        g[..., 2:, 1:-1] - g[..., :-2, 1:-1]
    ) - (f[..., 2:, 1:-1] - f[..., :-2, 1:-1]) * (g[..., 1:-1, 2:] - g[..., 1:-1, :-2])

    # J+x (advective form)
    # f evaluated at off-centre x-neighbours, g differenced in y at those neighbours
    Jpx = (
        f[..., 1:-1, 2:] * (g[..., 2:, 2:] - g[..., :-2, 2:])
        - f[..., 1:-1, :-2] * (g[..., 2:, :-2] - g[..., :-2, :-2])
        - f[..., 2:, 1:-1] * (g[..., 2:, 2:] - g[..., 2:, :-2])
        + f[..., :-2, 1:-1] * (g[..., :-2, 2:] - g[..., :-2, :-2])
    )

    # Jx+ (divergence form)
    # g evaluated at off-centre y-neighbours, f differenced in x at those neighbours
    Jxp = (
        g[..., 2:, 1:-1] * (f[..., 2:, 2:] - f[..., 2:, :-2])
        - g[..., :-2, 1:-1] * (f[..., :-2, 2:] - f[..., :-2, :-2])
        - g[..., 1:-1, 2:] * (f[..., 2:, 2:] - f[..., :-2, 2:])
        + g[..., 1:-1, :-2] * (f[..., 2:, :-2] - f[..., :-2, :-2])
    )

    return (Jpp + Jpx + Jxp) / (12.0 * dx * dy)