Skip to content

Advection Operators

Flux-form advection operators for scalar transport on staggered C-grids.

finitevolx.Advection1D

Bases: Module

1-D advection operator.

Parameters:

Name Type Description Default
grid ArakawaCGrid1D
required
Source code in finitevolx/_src/advection/advection.py 
 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
class Advection1D(eqx.Module):
    """1-D advection operator.

    Parameters
    ----------
    grid : ArakawaCGrid1D
    """

    grid: ArakawaCGrid1D
    recon: Reconstruction1D

    def __init__(self, grid: ArakawaCGrid1D) -> None:
        self.grid = grid
        self.recon = Reconstruction1D(grid=grid)

    def __call__(
        self,
        h: Float[Array, "Nx"],
        u: Float[Array, "Nx"],
        method: str = "upwind1",
    ) -> Float[Array, "Nx"]:
        """Advective tendency -d(h*u)/dx at T-points.

        dh[i] = -(fe[i+1/2] - fe[i-1/2]) / dx

        Parameters
        ----------
        h : Float[Array, "Nx"]
            Scalar at T-points.
        u : Float[Array, "Nx"]
            Velocity at U-points.
        method : str
            Reconstruction method: ``'naive'``, ``'upwind1'``, ``'upwind2'``,
            ``'upwind3'``, ``'weno3'``, ``'weno5'``, ``'weno7'``, ``'weno9'``,
            or a flux-limiter TVD scheme: ``'minmod'``, ``'van_leer'``,
            ``'superbee'``, ``'mc'``.

        Returns
        -------
        Float[Array, "Nx"]
            Advective tendency at T-points.
        """
        if method == "naive":
            fe = self.recon.naive_x(h, u)
        elif method == "upwind1":
            fe = self.recon.upwind1_x(h, u)
        elif method == "upwind2":
            fe = self.recon.upwind2_x(h, u)
        elif method == "upwind3":
            fe = self.recon.upwind3_x(h, u)
        elif method == "weno3":
            fe = self.recon.weno3_x(h, u)
        elif method == "weno5":
            fe = self.recon.weno5_x(h, u)
        elif method == "weno7":
            fe = self.recon.weno7_x(h, u)
        elif method == "weno9":
            fe = self.recon.weno9_x(h, u)
        elif method in _TVD_LIMITERS:
            fe = self.recon.tvd_x(h, u, limiter=method)
        else:
            raise ValueError(f"Unknown method: {method!r}")

        out = jnp.zeros_like(h)
        # dh[i] = -(fe[i+1/2] - fe[i-1/2]) / dx
        # fe[i] represents the flux at the east face of cell i (at i+1/2)
        # For cell i, we need fe[i] (east) and fe[i-1] (west)
        # Only use face fluxes that are defined by the reconstruction scheme,
        # avoiding the ghost-ring entries fe[0] and fe[-1].
        out = out.at[2:-2].set(-(fe[2:-2] - fe[1:-3]) / self.grid.dx)
        return out

__call__(h, u, method='upwind1')

Advective tendency -d(h*u)/dx at T-points.

dh[i] = -(fe[i+1/2] - fe[i-1/2]) / dx

Parameters:

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

Scalar at T-points.

 required
u Float[Array, 'Nx']

Velocity at U-points.

 required
method str

Reconstruction method: 'naive', 'upwind1', 'upwind2', 'upwind3', 'weno3', 'weno5', 'weno7', 'weno9', or a flux-limiter TVD scheme: 'minmod', 'van_leer', 'superbee', 'mc'.

 'upwind1'

Returns:

Type Description
Float[Array, 'Nx']

Advective tendency at T-points.
Source code in finitevolx/_src/advection/advection.py 
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
def __call__(
    self,
    h: Float[Array, "Nx"],
    u: Float[Array, "Nx"],
    method: str = "upwind1",
) -> Float[Array, "Nx"]:
    """Advective tendency -d(h*u)/dx at T-points.

    dh[i] = -(fe[i+1/2] - fe[i-1/2]) / dx

    Parameters
    ----------
    h : Float[Array, "Nx"]
        Scalar at T-points.
    u : Float[Array, "Nx"]
        Velocity at U-points.
    method : str
        Reconstruction method: ``'naive'``, ``'upwind1'``, ``'upwind2'``,
        ``'upwind3'``, ``'weno3'``, ``'weno5'``, ``'weno7'``, ``'weno9'``,
        or a flux-limiter TVD scheme: ``'minmod'``, ``'van_leer'``,
        ``'superbee'``, ``'mc'``.

    Returns
    -------
    Float[Array, "Nx"]
        Advective tendency at T-points.
    """
    if method == "naive":
        fe = self.recon.naive_x(h, u)
    elif method == "upwind1":
        fe = self.recon.upwind1_x(h, u)
    elif method == "upwind2":
        fe = self.recon.upwind2_x(h, u)
    elif method == "upwind3":
        fe = self.recon.upwind3_x(h, u)
    elif method == "weno3":
        fe = self.recon.weno3_x(h, u)
    elif method == "weno5":
        fe = self.recon.weno5_x(h, u)
    elif method == "weno7":
        fe = self.recon.weno7_x(h, u)
    elif method == "weno9":
        fe = self.recon.weno9_x(h, u)
    elif method in _TVD_LIMITERS:
        fe = self.recon.tvd_x(h, u, limiter=method)
    else:
        raise ValueError(f"Unknown method: {method!r}")

    out = jnp.zeros_like(h)
    # dh[i] = -(fe[i+1/2] - fe[i-1/2]) / dx
    # fe[i] represents the flux at the east face of cell i (at i+1/2)
    # For cell i, we need fe[i] (east) and fe[i-1] (west)
    # Only use face fluxes that are defined by the reconstruction scheme,
    # avoiding the ghost-ring entries fe[0] and fe[-1].
    out = out.at[2:-2].set(-(fe[2:-2] - fe[1:-3]) / self.grid.dx)
    return out

finitevolx.Advection2D

Bases: Module

2-D advection operator.

Parameters:

Name Type Description Default
grid ArakawaCGrid2D
required
Source code in finitevolx/_src/advection/advection.py 
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
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
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
class Advection2D(eqx.Module):
    """2-D advection operator.

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

    grid: ArakawaCGrid2D
    recon: Reconstruction2D

    def __init__(self, grid: ArakawaCGrid2D) -> None:
        self.grid = grid
        self.recon = Reconstruction2D(grid=grid)

    def __call__(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
        method: str = "upwind1",
        mask: ArakawaCGridMask | None = None,
    ) -> Float[Array, "Ny Nx"]:
        """Advective tendency -div(h * u_vec) at T-points.

        dh[j, i] = -( (fe[j, i+1/2] - fe[j, i-1/2]) / dx
                    + (fn[j+1/2, i] - fn[j-1/2, i]) / dy )

        Parameters
        ----------
        h : Float[Array, "Ny Nx"]
            Scalar at T-points.
        u : Float[Array, "Ny Nx"]
            x-velocity at U-points.
        v : Float[Array, "Ny Nx"]
            y-velocity at V-points.
        method : str
            Reconstruction method: ``'naive'``, ``'upwind1'``, ``'upwind2'``,
            ``'upwind3'``, ``'weno3'``, ``'weno5'``, ``'weno7'``, ``'weno9'``,
            ``'wenoz5'``, or a flux-limiter TVD scheme: ``'minmod'``,
            ``'van_leer'``, ``'superbee'``, ``'mc'``.
        mask : ArakawaCGridMask | None
            When provided and *method* supports mask dispatch (``'weno3'``,
            ``'weno5'``, ``'wenoz5'``, or any TVD limiter), stencil-width
            fallback is applied via :func:`upwind_flux`.  ``None`` (default)
            uses the standard unmasked path.

        Returns
        -------
        Float[Array, "Ny Nx"]
            Advective tendency at T-points.
        """
        # ── masked path: route through upwind_flux ──────────────────────
        if mask is not None and method in _MASK_DISPATCHABLE_2D:
            rfx, rfy, sizes = _rec_funcs_for_method_2d(self.recon, method)
            mask_x = mask.get_adaptive_masks(direction="x", stencil_sizes=sizes)
            mask_y = mask.get_adaptive_masks(direction="y", stencil_sizes=sizes)
            fe = upwind_flux(h, u, dim=1, rec_funcs=rfx, mask_hierarchy=mask_x)
            fn = upwind_flux(h, v, dim=0, rec_funcs=rfy, mask_hierarchy=mask_y)
        # ── unmasked path: existing dispatch ────────────────────────────
        elif method == "naive":
            fe = self.recon.naive_x(h, u)
            fn = self.recon.naive_y(h, v)
        elif method == "upwind1":
            fe = self.recon.upwind1_x(h, u)
            fn = self.recon.upwind1_y(h, v)
        elif method == "upwind2":
            fe = self.recon.upwind2_x(h, u)
            fn = self.recon.upwind2_y(h, v)
        elif method == "upwind3":
            fe = self.recon.upwind3_x(h, u)
            fn = self.recon.upwind3_y(h, v)
        elif method == "weno3":
            fe = self.recon.weno3_x(h, u)
            fn = self.recon.weno3_y(h, v)
        elif method == "weno5":
            fe = self.recon.weno5_x(h, u)
            fn = self.recon.weno5_y(h, v)
        elif method == "wenoz5":
            fe = self.recon.wenoz5_x(h, u)
            fn = self.recon.wenoz5_y(h, v)
        elif method == "weno7":
            fe = self.recon.weno7_x(h, u)
            fn = self.recon.weno7_y(h, v)
        elif method == "weno9":
            fe = self.recon.weno9_x(h, u)
            fn = self.recon.weno9_y(h, v)
        elif method in _TVD_LIMITERS:
            fe = self.recon.tvd_x(h, u, limiter=method)
            fn = self.recon.tvd_y(h, v, limiter=method)
        else:
            raise ValueError(f"Unknown method: {method!r}")

        # dh[j, i] = -( (fe[j, i+1/2] - fe[j, i-1/2])/dx
        #             + (fn[j+1/2, i] - fn[j-1/2, i])/dy )
        # fe[j,i] is flux at east face of cell [j,i], fn[j,i] is flux at north face
        # For cell [j,i], we need fe[j,i] (east) and fe[j,i-1] (west),
        #                      and fn[j,i] (north) and fn[j-1,i] (south)
        # Only use face fluxes that are defined by the reconstruction scheme,
        # avoiding ghost-ring flux entries.
        out = interior(
            -(
                (fe[2:-2, 2:-2] - fe[2:-2, 1:-3]) / self.grid.dx
                + (fn[2:-2, 2:-2] - fn[1:-3, 2:-2]) / self.grid.dy
            ),
            h,
            ghost=2,
        )
        return out

__call__(h, u, v, method='upwind1', mask=None)

Advective tendency -div(h * u_vec) at T-points.

dh[j, i] = -( (fe[j, i+1/2] - fe[j, i-1/2]) / dx + (fn[j+1/2, i] - fn[j-1/2, i]) / dy )

Parameters:

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

Scalar at T-points.

 required
u Float[Array, 'Ny Nx']

x-velocity at U-points.

 required
v Float[Array, 'Ny Nx']

y-velocity at V-points.

 required
method str

Reconstruction method: 'naive', 'upwind1', 'upwind2', 'upwind3', 'weno3', 'weno5', 'weno7', 'weno9', 'wenoz5', or a flux-limiter TVD scheme: 'minmod', 'van_leer', 'superbee', 'mc'.

 'upwind1'
mask ArakawaCGridMask | None

When provided and method supports mask dispatch ('weno3', 'weno5', 'wenoz5', or any TVD limiter), stencil-width fallback is applied via :func:upwind_flux. None (default) uses the standard unmasked path.

 None

Returns:

Type Description
Float[Array, 'Ny Nx']

Advective tendency at T-points.
Source code in finitevolx/_src/advection/advection.py 
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
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
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
def __call__(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
    method: str = "upwind1",
    mask: ArakawaCGridMask | None = None,
) -> Float[Array, "Ny Nx"]:
    """Advective tendency -div(h * u_vec) at T-points.

    dh[j, i] = -( (fe[j, i+1/2] - fe[j, i-1/2]) / dx
                + (fn[j+1/2, i] - fn[j-1/2, i]) / dy )

    Parameters
    ----------
    h : Float[Array, "Ny Nx"]
        Scalar at T-points.
    u : Float[Array, "Ny Nx"]
        x-velocity at U-points.
    v : Float[Array, "Ny Nx"]
        y-velocity at V-points.
    method : str
        Reconstruction method: ``'naive'``, ``'upwind1'``, ``'upwind2'``,
        ``'upwind3'``, ``'weno3'``, ``'weno5'``, ``'weno7'``, ``'weno9'``,
        ``'wenoz5'``, or a flux-limiter TVD scheme: ``'minmod'``,
        ``'van_leer'``, ``'superbee'``, ``'mc'``.
    mask : ArakawaCGridMask | None
        When provided and *method* supports mask dispatch (``'weno3'``,
        ``'weno5'``, ``'wenoz5'``, or any TVD limiter), stencil-width
        fallback is applied via :func:`upwind_flux`.  ``None`` (default)
        uses the standard unmasked path.

    Returns
    -------
    Float[Array, "Ny Nx"]
        Advective tendency at T-points.
    """
    # ── masked path: route through upwind_flux ──────────────────────
    if mask is not None and method in _MASK_DISPATCHABLE_2D:
        rfx, rfy, sizes = _rec_funcs_for_method_2d(self.recon, method)
        mask_x = mask.get_adaptive_masks(direction="x", stencil_sizes=sizes)
        mask_y = mask.get_adaptive_masks(direction="y", stencil_sizes=sizes)
        fe = upwind_flux(h, u, dim=1, rec_funcs=rfx, mask_hierarchy=mask_x)
        fn = upwind_flux(h, v, dim=0, rec_funcs=rfy, mask_hierarchy=mask_y)
    # ── unmasked path: existing dispatch ────────────────────────────
    elif method == "naive":
        fe = self.recon.naive_x(h, u)
        fn = self.recon.naive_y(h, v)
    elif method == "upwind1":
        fe = self.recon.upwind1_x(h, u)
        fn = self.recon.upwind1_y(h, v)
    elif method == "upwind2":
        fe = self.recon.upwind2_x(h, u)
        fn = self.recon.upwind2_y(h, v)
    elif method == "upwind3":
        fe = self.recon.upwind3_x(h, u)
        fn = self.recon.upwind3_y(h, v)
    elif method == "weno3":
        fe = self.recon.weno3_x(h, u)
        fn = self.recon.weno3_y(h, v)
    elif method == "weno5":
        fe = self.recon.weno5_x(h, u)
        fn = self.recon.weno5_y(h, v)
    elif method == "wenoz5":
        fe = self.recon.wenoz5_x(h, u)
        fn = self.recon.wenoz5_y(h, v)
    elif method == "weno7":
        fe = self.recon.weno7_x(h, u)
        fn = self.recon.weno7_y(h, v)
    elif method == "weno9":
        fe = self.recon.weno9_x(h, u)
        fn = self.recon.weno9_y(h, v)
    elif method in _TVD_LIMITERS:
        fe = self.recon.tvd_x(h, u, limiter=method)
        fn = self.recon.tvd_y(h, v, limiter=method)
    else:
        raise ValueError(f"Unknown method: {method!r}")

    # dh[j, i] = -( (fe[j, i+1/2] - fe[j, i-1/2])/dx
    #             + (fn[j+1/2, i] - fn[j-1/2, i])/dy )
    # fe[j,i] is flux at east face of cell [j,i], fn[j,i] is flux at north face
    # For cell [j,i], we need fe[j,i] (east) and fe[j,i-1] (west),
    #                      and fn[j,i] (north) and fn[j-1,i] (south)
    # Only use face fluxes that are defined by the reconstruction scheme,
    # avoiding ghost-ring flux entries.
    out = interior(
        -(
            (fe[2:-2, 2:-2] - fe[2:-2, 1:-3]) / self.grid.dx
            + (fn[2:-2, 2:-2] - fn[1:-3, 2:-2]) / self.grid.dy
        ),
        h,
        ghost=2,
    )
    return out

finitevolx.Advection3D

Bases: Module

3-D advection operator (horizontal plane per z-level).

Parameters:

Name Type Description Default
grid ArakawaCGrid3D
required
Source code in finitevolx/_src/advection/advection.py 
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
class Advection3D(eqx.Module):
    """3-D advection operator (horizontal plane per z-level).

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

    grid: ArakawaCGrid3D
    recon: Reconstruction3D

    def __init__(self, grid: ArakawaCGrid3D) -> None:
        self.grid = grid
        self.recon = Reconstruction3D(grid=grid)

    def __call__(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
        method: str = "upwind1",
        mask: ArakawaCGridMask | None = None,
    ) -> Float[Array, "Nz Ny Nx"]:
        """Advective tendency -div(h * u_vec) at T-points over all z-levels.

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Scalar at T-points.
        u : Float[Array, "Nz Ny Nx"]
            x-velocity at U-points.
        v : Float[Array, "Nz Ny Nx"]
            y-velocity at V-points.
        method : str
            Reconstruction method: ``'naive'``, ``'upwind1'``, ``'weno3'``,
            ``'weno5'``, ``'weno7'``, ``'weno9'``, or a flux-limiter TVD
            scheme: ``'minmod'``, ``'van_leer'``, ``'superbee'``, ``'mc'``.
        mask : ArakawaCGridMask | None
            When provided and *method* supports mask dispatch (``'weno3'``,
            ``'weno5'``, or any TVD limiter), the masked reconstruction
            variants are used.  The 2-D mask is broadcast over the
            z-dimension.  ``None`` (default) uses the standard unmasked path.

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            Advective tendency at T-points.
        """
        # ── masked path: route to Reconstruction3D.*_masked methods ─────
        if mask is not None and method in _MASK_DISPATCHABLE_3D:
            if method == "weno3":
                fe = self.recon.weno3_x_masked(h, u, mask)
                fn = self.recon.weno3_y_masked(h, v, mask)
            elif method == "weno5":
                fe = self.recon.weno5_x_masked(h, u, mask)
                fn = self.recon.weno5_y_masked(h, v, mask)
            elif method in _TVD_LIMITERS:
                fe = self.recon.tvd_x_masked(h, u, mask, limiter=method)
                fn = self.recon.tvd_y_masked(h, v, mask, limiter=method)
            else:
                raise ValueError(f"Unknown masked method: {method!r}")
        # ── unmasked path: existing dispatch ────────────────────────────
        elif method == "naive":
            fe = self.recon.naive_x(h, u)
            fn = self.recon.naive_y(h, v)
        elif method == "upwind1":
            fe = self.recon.upwind1_x(h, u)
            fn = self.recon.upwind1_y(h, v)
        elif method == "weno3":
            fe = self.recon.weno3_x(h, u)
            fn = self.recon.weno3_y(h, v)
        elif method == "weno5":
            fe = self.recon.weno5_x(h, u)
            fn = self.recon.weno5_y(h, v)
        elif method == "weno7":
            fe = self.recon.weno7_x(h, u)
            fn = self.recon.weno7_y(h, v)
        elif method == "weno9":
            fe = self.recon.weno9_x(h, u)
            fn = self.recon.weno9_y(h, v)
        elif method in _TVD_LIMITERS:
            fe = self.recon.tvd_x(h, u, limiter=method)
            fn = self.recon.tvd_y(h, v, limiter=method)
        else:
            raise ValueError(f"Unknown method: {method!r}")

        out = jnp.zeros_like(h)
        # dh[k, j, i] = -( (fe[k,j,i+1/2] - fe[k,j,i-1/2])/dx
        #                 + (fn[k,j+1/2,i] - fn[k,j-1/2,i])/dy )
        # Reconstruction writes to [1:-1, 1:-1, 1:-1]; the west flux at i=0
        # and south flux at j=0 are ghost cells (value 0, not filled).
        # Consistent with 1D/2D operators, skip the ghost-adjacent interior
        # ring in the horizontal plane so we never read ghost flux cells.
        # All z-levels are independent, so z uses the full interior [1:-1].
        out = out.at[1:-1, 2:-2, 2:-2].set(
            -(
                (fe[1:-1, 2:-2, 2:-2] - fe[1:-1, 2:-2, 1:-3]) / self.grid.dx
                + (fn[1:-1, 2:-2, 2:-2] - fn[1:-1, 1:-3, 2:-2]) / self.grid.dy
            )
        )
        return out

__call__(h, u, v, method='upwind1', mask=None)

Advective tendency -div(h * u_vec) at T-points over all z-levels.

Parameters:

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

Scalar at T-points.

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

x-velocity at U-points.

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

y-velocity at V-points.

 required
method str

Reconstruction method: 'naive', 'upwind1', 'weno3', 'weno5', 'weno7', 'weno9', or a flux-limiter TVD scheme: 'minmod', 'van_leer', 'superbee', 'mc'.

 'upwind1'
mask ArakawaCGridMask | None

When provided and method supports mask dispatch ('weno3', 'weno5', or any TVD limiter), the masked reconstruction variants are used. The 2-D mask is broadcast over the z-dimension. None (default) uses the standard unmasked path.

 None

Returns:

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

Advective tendency at T-points.
Source code in finitevolx/_src/advection/advection.py 
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
def __call__(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
    method: str = "upwind1",
    mask: ArakawaCGridMask | None = None,
) -> Float[Array, "Nz Ny Nx"]:
    """Advective tendency -div(h * u_vec) at T-points over all z-levels.

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Scalar at T-points.
    u : Float[Array, "Nz Ny Nx"]
        x-velocity at U-points.
    v : Float[Array, "Nz Ny Nx"]
        y-velocity at V-points.
    method : str
        Reconstruction method: ``'naive'``, ``'upwind1'``, ``'weno3'``,
        ``'weno5'``, ``'weno7'``, ``'weno9'``, or a flux-limiter TVD
        scheme: ``'minmod'``, ``'van_leer'``, ``'superbee'``, ``'mc'``.
    mask : ArakawaCGridMask | None
        When provided and *method* supports mask dispatch (``'weno3'``,
        ``'weno5'``, or any TVD limiter), the masked reconstruction
        variants are used.  The 2-D mask is broadcast over the
        z-dimension.  ``None`` (default) uses the standard unmasked path.

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        Advective tendency at T-points.
    """
    # ── masked path: route to Reconstruction3D.*_masked methods ─────
    if mask is not None and method in _MASK_DISPATCHABLE_3D:
        if method == "weno3":
            fe = self.recon.weno3_x_masked(h, u, mask)
            fn = self.recon.weno3_y_masked(h, v, mask)
        elif method == "weno5":
            fe = self.recon.weno5_x_masked(h, u, mask)
            fn = self.recon.weno5_y_masked(h, v, mask)
        elif method in _TVD_LIMITERS:
            fe = self.recon.tvd_x_masked(h, u, mask, limiter=method)
            fn = self.recon.tvd_y_masked(h, v, mask, limiter=method)
        else:
            raise ValueError(f"Unknown masked method: {method!r}")
    # ── unmasked path: existing dispatch ────────────────────────────
    elif method == "naive":
        fe = self.recon.naive_x(h, u)
        fn = self.recon.naive_y(h, v)
    elif method == "upwind1":
        fe = self.recon.upwind1_x(h, u)
        fn = self.recon.upwind1_y(h, v)
    elif method == "weno3":
        fe = self.recon.weno3_x(h, u)
        fn = self.recon.weno3_y(h, v)
    elif method == "weno5":
        fe = self.recon.weno5_x(h, u)
        fn = self.recon.weno5_y(h, v)
    elif method == "weno7":
        fe = self.recon.weno7_x(h, u)
        fn = self.recon.weno7_y(h, v)
    elif method == "weno9":
        fe = self.recon.weno9_x(h, u)
        fn = self.recon.weno9_y(h, v)
    elif method in _TVD_LIMITERS:
        fe = self.recon.tvd_x(h, u, limiter=method)
        fn = self.recon.tvd_y(h, v, limiter=method)
    else:
        raise ValueError(f"Unknown method: {method!r}")

    out = jnp.zeros_like(h)
    # dh[k, j, i] = -( (fe[k,j,i+1/2] - fe[k,j,i-1/2])/dx
    #                 + (fn[k,j+1/2,i] - fn[k,j-1/2,i])/dy )
    # Reconstruction writes to [1:-1, 1:-1, 1:-1]; the west flux at i=0
    # and south flux at j=0 are ghost cells (value 0, not filled).
    # Consistent with 1D/2D operators, skip the ghost-adjacent interior
    # ring in the horizontal plane so we never read ghost flux cells.
    # All z-levels are independent, so z uses the full interior [1:-1].
    out = out.at[1:-1, 2:-2, 2:-2].set(
        -(
            (fe[1:-1, 2:-2, 2:-2] - fe[1:-1, 2:-2, 1:-3]) / self.grid.dx
            + (fn[1:-1, 2:-2, 2:-2] - fn[1:-1, 1:-3, 2:-2]) / self.grid.dy
        )
    )
    return out