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Vector calculus follow-ups from #1663: eager divergence and a user guide fix #1732

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609f80e
Keep divergence lazy on dask-backed input
rajeeja Sep 3, 2026
248bb8c
Teach the spherical operators in the vector calculus user guide
rajeeja Sep 3, 2026
4a8be4d
Merge branch 'main' into rajeeja/vector-calculus-followup
rajeeja Sep 8, 2026
3b3399e
Fix solid-body vorticity sign, define curl notation, reconcile residu…
rajeeja Sep 11, 2026
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151 changes: 63 additions & 88 deletions docs/user-guide/vector_calculus.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -126,7 +126,7 @@
"id": "66caacb5",
"metadata": {},
"source": [
"> **Units.** By default, `gradient()` and `curl()` divide by `uxgrid.sphere_radius` so derivatives carry physical units (e.g. `[data units]/m`, `1/s` for velocity curl). Pass `scale_by_radius=False` to keep results on the unit sphere (per radian). If the grid has no `sphere_radius` attribute, the call falls back to unit-sphere output and emits a `UserWarning`."
"> **Units.** By default, `gradient()`, `curl()`, and `divergence()` divide by `uxgrid.sphere_radius` so derivatives carry physical units (e.g. `[data units]/m`, `1/s` for velocity curl). Pass `scale_by_radius=False` to keep results on the unit sphere (per radian). If the grid has no `sphere_radius` attribute, the call falls back to unit-sphere output and emits a `UserWarning`."
]
},
{
Expand Down Expand Up @@ -228,14 +228,19 @@
"\n",
"### Background\n",
"\n",
"The curl of a vector field **F** = (u, v) measures the local rotation or circulation. In 2D, curl produces a scalar field representing the magnitude of rotation:\n",
"The curl of a vector field **F** = (u, v) measures local rotation. Here $u$, $v$ are the zonal and meridional components, $\\lambda$ is longitude, $\\varphi$ is latitude, $a$ is the sphere radius (`uxgrid.sphere_radius`), and $dx = a\\cos\\varphi\\,d\\lambda$, $dy = a\\,d\\varphi$ are local distances. On a sphere,\n",
"\n",
"$$\n",
"\\text{curl}(\\mathbf{F}) = \\nabla \\times \\mathbf{F} = \\frac{\\partial v}{\\partial x} - \\frac{\\partial u}{\\partial y}\n",
"\\text{curl}(\\mathbf{F}) = \\frac{1}{a\\cos\\varphi}\\left[\\frac{\\partial v}{\\partial \\lambda} - \\frac{\\partial (u\\cos\\varphi)}{\\partial \\varphi}\\right]\n",
"= \\frac{\\partial v}{\\partial x} - \\frac{\\partial u}{\\partial y} + \\frac{u \\tan\\varphi}{a}\n",
"$$\n",
"\n",
"The product rule on $\\partial(u\\cos\\varphi)/\\partial\\varphi$ produces the last term. It is a metric term: longitude lines converge toward the poles, so a zonal flow rotates even when $\\partial u/\\partial y = 0$. It is zero on the equator. In Example 3 it is as large as the derivative term. See Holton and Hakim, *An Introduction to Dynamic Meteorology*, Chapter 4.\n",
"\n",
"`u.curl(v)` evaluates the full expression; `v.gradient()[\"zonal_gradient\"] - u.gradient()[\"meridional_gradient\"]` gives only the first two terms.\n",
"\n",
"- **Positive curl**: Counter-clockwise rotation\n",
"- **Negative curl**: Clockwise rotation \n",
"- **Negative curl**: Clockwise rotation\n",
"- **Zero curl**: No local rotation (irrotational flow)\n",
"\n",
"### Usage\n",
Expand All @@ -244,7 +249,8 @@
"\n",
"| **Input** | **Usage** | **Output** |\n",
"| ---------------------------- | :---------------------: | -------------------- |\n",
"| Vector field (u, v) | `u.curl(v)` | Scalar curl field |"
"| Vector field (u, v) | `u.curl(v)` | Scalar curl field |\n",
""
]
},
{
Expand All @@ -254,7 +260,10 @@
"source": [
"### Constant Fields (Mathematical Validation)\n",
"\n",
"The curl of a constant vector field should be zero everywhere (within numerical precision)."
"On a plane the curl of a constant field is zero. On a sphere the derivative terms vanish but the metric term does not, so `u.curl(v)` $= u\\tan\\varphi/a$ exactly.\n",
"\n",
"This subset spans about ±2° of latitude, so $\\tan\\varphi/a < 5.5\\times10^{-9}\\ \\mathrm{m^{-1}}$ and the result looks like round-off. It is a real signal; the cell below checks it against $u\\tan\\varphi/a$ from the face latitudes.\n",
""
]
},
{
Expand All @@ -268,12 +277,7 @@
"u_constant = uxds[\"face_lat\"] * 0 + 1.0\n",
"v_constant = uxds[\"face_lat\"] * 0 + 2.0\n",
"\n",
"# Compute partials via gradient\n",
"grad_u = u_constant.gradient()\n",
"grad_v = v_constant.gradient()\n",
"du_dy = grad_u[\"meridional_gradient\"]\n",
"dv_dx = grad_v[\"zonal_gradient\"]\n",
"curl_constant = dv_dx - du_dy\n",
"curl_constant = u_constant.curl(v_constant)\n",
"\n",
"finite = np.isfinite(curl_constant.values)\n",
"vals = curl_constant.values[finite]\n",
Expand All @@ -284,7 +288,12 @@
" print(f\"Finite curl range: [{vals.min():.2e}, {vals.max():.2e}]\")\n",
" print(\n",
" f\"Max |curl|: {np.abs(vals).max():.2e}, Mean |curl|: {np.abs(vals).mean():.2e}\"\n",
" )"
" )\n",
"\n",
"# The closed form for a constant u on the sphere is u*tan(lat)/a\n",
"lat_rad = np.deg2rad(uxds.uxgrid.face_lat.values)\n",
"curl_exact = 1.0 * np.tan(lat_rad) / uxds.uxgrid.sphere_radius\n",
"print(f\"Max deviation from u*tan(lat)/a: {np.abs(vals - curl_exact[finite]).max():.2e}\")"
]
},
{
Expand All @@ -306,21 +315,16 @@
"u_gauss = uxds[\"gaussian\"]\n",
"v_gauss = uxds[\"inverse_gaussian\"]\n",
"\n",
"# Compute partials via gradient\n",
"grad_u = u_gauss.gradient()\n",
"grad_v = v_gauss.gradient()\n",
"du_dy = grad_u[\"meridional_gradient\"]\n",
"dv_dx = grad_v[\"zonal_gradient\"]\n",
"curl_gauss = dv_dx - du_dy\n",
"curl_gauss = u_gauss.curl(v_gauss)\n",
"\n",
"finite = np.isfinite(curl_gauss.values)\n",
"vals = curl_gauss.values[finite]\n",
"print(\n",
" f\"Total faces: {curl_gauss.size}, interior: {vals.size}, boundary NaNs: {np.isnan(curl_gauss.values).sum()}\"\n",
")\n",
"if vals.size:\n",
" print(f\"Finite curl range: [{vals.min():.6f}, {vals.max():.6f}]\")\n",
" print(f\"Mean curl (finite): {vals.mean():.6f}\")"
" print(f\"Finite curl range: [{vals.min():.6e}, {vals.max():.6e}]\")\n",
" print(f\"Mean curl (finite): {vals.mean():.6e}\")"
]
},
{
Expand All @@ -332,7 +336,8 @@
"\n",
"In the continuous setting, curl(∇φ) = 0 exactly for any scalar field φ. On an unstructured mesh, however, gradient and curl are independent finite-volume stencils that do not form a discrete de Rham complex, so the identity holds only approximately. The residual is a **discretization error** — not a bug — and shrinks with grid refinement.\n",
"\n",
"The *magnitude* of the residual depends on the units. By default `gradient()`/`curl()` scale by `uxgrid.sphere_radius` (Earth ≈ 6.37×10⁶ m), so each derivative picks up a factor of 1/radius. `curl(∇φ)` applies the gradient stencil twice and therefore carries a factor of 1/radius² (≈ 4×10⁻¹⁴). For the Gaussian field here (φ ~ O(1)) the scaled residual is ~O(10⁻¹³); on the unit sphere (`scale_by_radius=False`) the same residual is ~O(1–10). Either way it shrinks with refinement.\n"
"The *magnitude* of the residual depends on the units. By default `gradient()`/`curl()` scale by `uxgrid.sphere_radius` (Earth ≈ 6.37×10⁶ m), so each derivative picks up a factor of 1/radius. `curl(∇φ)` applies the gradient stencil twice and therefore carries a factor of 1/radius² (≈ 2.5×10⁻¹⁴). For the Gaussian field here (φ ~ O(1)) the scaled residual is a few ×10⁻¹⁵; on the unit sphere (`scale_by_radius=False`) the same residual is ~O(0.1). Either way it shrinks with refinement.\n",
""
]
},
{
Expand All @@ -342,25 +347,18 @@
"metadata": {},
"outputs": [],
"source": [
"# Extract gradient components\n",
"# Extract gradient components and treat them as a vector field\n",
"u_component = grad_gauss.zonal_gradient\n",
"v_component = grad_gauss.meridional_gradient\n",
"\n",
"# Compute partial derivatives via gradient()\n",
"grad_u = u_component.gradient()\n",
"grad_v = v_component.gradient()\n",
"du_dy = grad_u[\"meridional_gradient\"]\n",
"dv_dx = grad_v[\"zonal_gradient\"]\n",
"\n",
"# Curl = ∂v/∂x - ∂u/∂y\n",
"curl_of_gradient = dv_dx - du_dy\n",
"curl_of_gradient = u_component.curl(v_component)\n",
"\n",
"print(\n",
" f\"Curl of gradient range: [{curl_of_gradient.min().values:.2e}, {curl_of_gradient.max().values:.2e}]\"\n",
")\n",
"print(f\"Mean absolute curl: {abs(curl_of_gradient).mean().values:.2e}\")\n",
"\n",
"# Note: values are ~O(1e-13) (per meter^2) with the default radius scaling,\n",
"# Note: values are ~O(1e-15) (per meter^2) with the default radius scaling,\n",
"# so we let the color limits autoscale rather than hardcoding them.\n",
"curl_plot = curl_of_gradient.plot(cmap=\"RdBu_r\", aspect=1).opts(\n",
" title=\"Curl of Gradient Field (Should ≈ 0)\", colorbar=True\n",
Expand Down Expand Up @@ -408,12 +406,8 @@
" v_vortex_data, dims=[\"n_face\"], uxgrid=uxds.uxgrid, name=\"v_vortex\"\n",
")\n",
"\n",
"# Compute curl via gradients (default: scaled by sphere_radius -> per meter)\n",
"grad_u = u_vortex.gradient()\n",
"grad_v = v_vortex.gradient()\n",
"du_dy = grad_u[\"meridional_gradient\"]\n",
"dv_dx = grad_v[\"zonal_gradient\"]\n",
"curl_vortex = dv_dx - du_dy\n",
"# Compute curl (default: scaled by sphere_radius -> per meter)\n",
"curl_vortex = u_vortex.curl(v_vortex)\n",
"\n",
"print(\n",
" f\"Vortex curl range: [{curl_vortex.min().values:.2e}, {curl_vortex.max().values:.2e}]\"\n",
Expand Down Expand Up @@ -462,21 +456,19 @@
"source": [
"### Example 3: Relative Vorticity from Solid-Body Rotation\n",
"\n",
"The 2D curl on the sphere is **relative vorticity** $\\zeta = \\partial v/\\partial x - \\partial u/\\partial y$, a quantity meteorologists and oceanographers care about every day. With the default `scale_by_radius=True`, `u.curl(v)` returns $\\zeta$ in physical units of $s^{-1}$.\n",
"The curl on the sphere is **relative vorticity** $\\zeta$. With the default `scale_by_radius=True`, `u.curl(v)` returns $\\zeta$ in $s^{-1}$.\n",
"\n",
"A clean analytical check is **solid-body rotation about the polar axis** with angular speed $\\Omega$:\n",
"\n",
"$$\n",
"u(\\varphi) = \\Omega R \\cos\\varphi, \\qquad v = 0\n",
"$$\n",
"\n",
"For this flow the relative vorticity on a sphere of radius $R$ is\n",
"\n",
"$$\n",
"\\zeta = -\\frac{1}{R\\cos\\varphi}\\frac{\\partial(u\\cos\\varphi)}{\\partial \\varphi} = -2\\Omega \\sin\\varphi.\n",
"\\zeta = -\\frac{1}{R\\cos\\varphi}\\frac{\\partial(\\Omega R\\cos^2\\varphi)}{\\partial \\varphi} = 2\\Omega \\sin\\varphi\n",
"$$\n",
"\n",
"We construct the field on the existing MPAS subset and compare `u.curl(v)` against this closed form."
"This is the planetary vorticity $f$. The derivative term and the metric term each contribute $\\Omega\\sin\\varphi$, so the planar formula would return half the answer. We compare `u.curl(v)` against $2\\Omega\\sin\\varphi$ on the MPAS subset."
]
},
{
Expand Down Expand Up @@ -509,7 +501,7 @@
")\n",
"\n",
"zeta = u_sbr.curl(v_sbr)\n",
"zeta_analytic = -2.0 * OMEGA * np.sin(lat_rad)\n",
"zeta_analytic = 2.0 * OMEGA * np.sin(lat_rad)\n",
"\n",
"finite = np.isfinite(zeta.values)\n",
"err = np.abs(zeta.values[finite] - zeta_analytic[finite])\n",
Expand All @@ -522,7 +514,9 @@
"print(\n",
" f\"analytic zeta range: [{zeta_analytic.min():.3e}, {zeta_analytic.max():.3e}] 1/s\"\n",
")\n",
"print(f\"max |error|: {err.max():.3e} 1/s (≈ {err.max() / (2 * OMEGA):.1%} of 2Omega)\")"
"print(\n",
" f\"max |error|: {err.max():.3e} 1/s (≈ {err.max() / np.abs(zeta_analytic).max():.2%} of max |zeta|)\"\n",
")"
]
},
{
Expand All @@ -546,10 +540,10 @@
"\n",
"### Background\n",
"\n",
"The divergence of a vector field **F** = (u, v) measures the local expansion or contraction of the field:\n",
"The divergence of a vector field **F** = (u, v) measures the local expansion or contraction of the field. It carries the companion of the metric term in `curl`, with the opposite sign and the meridional component in place of the zonal one:\n",
"\n",
"$$\n",
"\\text{div}(\\mathbf{F}) = \\nabla \\cdot \\mathbf{F} = \\frac{\\partial u}{\\partial x} + \\frac{\\partial v}{\\partial y}\n",
"\\text{div}(\\mathbf{F}) = \\frac{\\partial u}{\\partial x} + \\frac{\\partial v}{\\partial y} - \\frac{v \\tan\\varphi}{a}\n",
"$$\n",
"\n",
"- **Positive divergence**: Expansion (source)\n",
Expand All @@ -562,7 +556,7 @@
"\n",
"| **Input** | **Usage** | **Output** |\n",
"| ---------------------------- | :---------------------: | ----------------------- |\n",
"| Vector field (u, v) | `u.divergence(v)` | Scalar divergence field |"
"| Vector field (u, v) | `u.divergence(v)` | Scalar divergence field |\n"
]
},
{
Expand All @@ -572,7 +566,7 @@
"source": [
"### Constant Fields (Mathematical Validation)\n",
"\n",
"The divergence of a constant vector field should be zero everywhere (within numerical precision)."
"As with curl, a constant field is divergence-free on a plane but not on a sphere. The derivative terms vanish and the metric term leaves $-v\\tan\\varphi/a$, which is twice the curl residual above and of the opposite sign, since $v = 2u$ here.\n"
]
},
{
Expand All @@ -586,12 +580,7 @@
"u_constant = uxds[\"face_lat\"] * 0 + 1.0\n",
"v_constant = uxds[\"face_lat\"] * 0 + 2.0\n",
"\n",
"# Compute partials via gradient\n",
"grad_u = u_constant.gradient()\n",
"grad_v = v_constant.gradient()\n",
"du_dx = grad_u[\"zonal_gradient\"]\n",
"dv_dy = grad_v[\"meridional_gradient\"]\n",
"div_constant = du_dx + dv_dy\n",
"div_constant = u_constant.divergence(v_constant)\n",
"\n",
"finite = np.isfinite(div_constant.values)\n",
"vals = div_constant.values[finite]\n",
Expand All @@ -600,7 +589,11 @@
")\n",
"if vals.size:\n",
" print(f\"Finite divergence range: [{vals.min():.2e}, {vals.max():.2e}]\")\n",
" print(f\"Max |div|: {np.abs(vals).max():.2e}, Mean |div|: {np.abs(vals).mean():.2e}\")"
" print(f\"Max |div|: {np.abs(vals).max():.2e}, Mean |div|: {np.abs(vals).mean():.2e}\")\n",
"\n",
"# The closed form for a constant v on the sphere is -v*tan(lat)/a\n",
"div_exact = -2.0 * np.tan(lat_rad) / uxds.uxgrid.sphere_radius\n",
"print(f\"Max deviation from -v*tan(lat)/a: {np.abs(vals - div_exact[finite]).max():.2e}\")"
]
},
{
Expand All @@ -622,21 +615,16 @@
"u_gauss = uxds[\"gaussian\"]\n",
"v_gauss = uxds[\"inverse_gaussian\"]\n",
"\n",
"# Compute partials via gradient\n",
"grad_u = u_gauss.gradient()\n",
"grad_v = v_gauss.gradient()\n",
"du_dx = grad_u[\"zonal_gradient\"]\n",
"dv_dy = grad_v[\"meridional_gradient\"]\n",
"div_gauss = du_dx + dv_dy\n",
"div_gauss = u_gauss.divergence(v_gauss)\n",
"\n",
"finite = np.isfinite(div_gauss.values)\n",
"vals = div_gauss.values[finite]\n",
"print(\n",
" f\"Total faces: {div_gauss.size}, interior: {vals.size}, boundary NaNs: {np.isnan(div_gauss.values).sum()}\"\n",
")\n",
"if vals.size:\n",
" print(f\"Finite divergence range: [{vals.min():.6f}, {vals.max():.6f}]\")\n",
" print(f\"Mean divergence (finite): {vals.mean():.6f}\")"
" print(f\"Finite divergence range: [{vals.min():.6e}, {vals.max():.6e}]\")\n",
" print(f\"Mean divergence (finite): {vals.mean():.6e}\")"
]
},
{
Expand All @@ -654,15 +642,12 @@
"metadata": {},
"outputs": [],
"source": [
"# Compute divergence of the gradient (Laplacian) via partials\n",
"# Compute divergence of the gradient (Laplacian)\n",
"# ∇²φ = ∂(∇φ_x)/∂x + ∂(∇φ_y)/∂y\n",
"grad_gauss_u = grad_gauss[\"zonal_gradient\"]\n",
"grad_gauss_v = grad_gauss[\"meridional_gradient\"]\n",
"\n",
"gxu = grad_gauss_u.gradient()[\"zonal_gradient\"] # ∂u/∂x\n",
"gyv = grad_gauss_v.gradient()[\"meridional_gradient\"] # ∂v/∂y\n",
"\n",
"div_of_gradient = gxu + gyv\n",
"div_of_gradient = grad_gauss_u.divergence(grad_gauss_v)\n",
"\n",
"print(\n",
" f\"Divergence of gradient range: [{div_of_gradient.min().values:.2e}, {div_of_gradient.max().values:.2e}]\"\n",
Expand All @@ -682,7 +667,8 @@
"source": [
"### Example 2: Divergence of Vortex Field\n",
"\n",
"Pure rotation should have zero divergence (incompressible):"
"Pure rotation is incompressible, so the divergence should be ≈ 0. The metric term $-v\\tan\\varphi/a$ is not identically zero here, so the residual (~4×10⁻⁹) is an order of magnitude above the derivative terms alone (~3×10⁻¹⁰) and four orders below the ~10⁻⁵ signal in the radial case below.\n",
""
]
},
{
Expand All @@ -692,20 +678,15 @@
"metadata": {},
"outputs": [],
"source": [
"# Compute divergence of the vortex via gradients: div = ∂u/∂x + ∂v/∂y\n",
"grad_u = u_vortex.gradient()\n",
"grad_v = v_vortex.gradient()\n",
"du_dx = grad_u[\"zonal_gradient\"]\n",
"dv_dy = grad_v[\"meridional_gradient\"]\n",
"div_vortex = du_dx + dv_dy\n",
"div_vortex = u_vortex.divergence(v_vortex)\n",
"\n",
"print(\n",
" f\"Vortex divergence range: [{div_vortex.min().values:.2e}, {div_vortex.max().values:.2e}]\"\n",
")\n",
"print(f\"Mean absolute divergence: {abs(div_vortex).mean().values:.2e}\")\n",
"print(\"Pure rotation should have zero divergence\")\n",
"\n",
"# Residual is ~O(1e-13) with default radius scaling; let color limits autoscale.\n",
"# Residual is ~O(1e-9) with default radius scaling; let color limits autoscale.\n",
"div_vortex_plot = div_vortex.plot(cmap=\"RdBu_r\", aspect=1).opts(\n",
" title=\"Divergence of Vortex (Should ≈ 0)\", colorbar=True\n",
")\n",
Expand Down Expand Up @@ -740,16 +721,9 @@
" v_radial_data, dims=[\"n_face\"], uxgrid=uxds.uxgrid, name=\"v_radial\"\n",
")\n",
"\n",
"# Compute curl and divergence via gradients (default: scaled by sphere_radius)\n",
"grad_u = u_radial.gradient()\n",
"grad_v = v_radial.gradient()\n",
"du_dy = grad_u[\"meridional_gradient\"]\n",
"dv_dx = grad_v[\"zonal_gradient\"]\n",
"du_dx = grad_u[\"zonal_gradient\"]\n",
"dv_dy = grad_v[\"meridional_gradient\"]\n",
"\n",
"curl_radial = dv_dx - du_dy\n",
"div_radial = du_dx + dv_dy\n",
"# Compute curl and divergence (default: scaled by sphere_radius)\n",
"curl_radial = u_radial.curl(v_radial)\n",
"div_radial = u_radial.divergence(v_radial)\n",
"\n",
"print(\n",
" f\"Radial field curl range: [{curl_radial.min().values:.2e}, {curl_radial.max().values:.2e}]\"\n",
Expand Down Expand Up @@ -863,7 +837,8 @@
"Let's verify some fundamental vector calculus identities using our computed fields:\n",
"\n",
"### Identity 1: Curl of Gradient (Discretization Residual)\n",
"In the continuous setting: ∇ × (∇φ) = 0. UXarray's finite-volume operators are not mimetic, so this holds only approximately. The residual below is discretization error, not a numerical bug. With the default radius scaling it is ~O(10⁻¹³) for this φ ~ O(1) field (the curl stencil applies a 1/radius² factor); on the unit sphere it is ~O(1–10). It shrinks with grid refinement.\n"
"In the continuous setting: ∇ × (∇φ) = 0. UXarray's finite-volume operators are not mimetic, so this holds only approximately. The residual below is discretization error, not a numerical bug. With the default radius scaling it is a few ×10⁻¹⁵ for this φ ~ O(1) field (the curl stencil applies a 1/radius² factor); on the unit sphere it is ~O(0.1). It shrinks with grid refinement.\n",
""
]
},
{
Expand Down
2 changes: 1 addition & 1 deletion uxarray/core/dataarray.py
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -1796,7 +1796,7 @@ def divergence(
# Spherical metric term, the companion of the one in curl(). Omitting
# it is only valid on a plane.
tan_lat = np.tan(np.deg2rad(self.uxgrid.face_lat.values))
metric = other.values * tan_lat
metric = other.data * tan_lat
if scale_by_radius and "sphere_radius" in self.uxgrid._ds.attrs:
metric = metric / self.uxgrid._ds.attrs["sphere_radius"]
divergence = divergence - metric
Expand Down
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