Physics · Electricity and magnetism · Year 3

Vector fields: div and curl

Where does a paddle wheel in a river turn, and where does cyclone wind gather?

The situation

You are standing by the Chirchiq: in the middle of the river the water flows fast, near the bank slowly, and floating leaves turn slowly. On a weather map the wind circles round a cyclone and converges on its centre; in summer a dust devil in the desert lifts dust into the sky. Whichever way a skier comes down the Chimgan slope, the difference in height is the same — yet the speed at the bottom differs. Where in a field is there a “source”, where a “vortex” — and how can this be measured?

Topic

Vector fields: gradient, divergence, curl. River flow and wind map: sources, vortices, Gauss's and Stokes' theorems

Key ideas

Gradient, divergence, curl
Gauss's and Stokes' theorems
Potential and rotational fields

Formulas

\nabla U = (\frac{\partial U}{\partial x}, \frac{\partial U}{\partial y}, \frac{\partial U}{\partial z}), \vec{F} = - \nabla U

Gradient — a vector pointing in the direction in which a scalar field increases fastest; its length is the rate of increase in that direction. A conservative force is opposite to the gradient of potential energy.

div \vec{v} = \nabla \cdot \vec{v} = \lim_{V \to 0} \frac{1}{V} \oint_{S} \vec{v} \cdot d \vec{S}

Divergence — the ratio of the flux leaving a small volume to that volume (source density).

{(rot \vec{v})}_{n} = \lim_{S \to 0} \frac{1}{S} \oint_{L} \vec{v} \cdot d \vec{l}

Curl — the ratio of the circulation around a small loop to its area. A small paddle wheel rotates with angular velocity $ω = \frac{1}{2} {(curl \vec{v})}_{n}$ .

\oint_{S} \vec{v} \cdot d \vec{S} = \int_{V} \nabla \cdot \vec{v} d V

Ostrogradsky–Gauss theorem: the flux out of a closed surface equals the integral of the divergence over the volume inside the surface (the sum of the sources).

\oint_{L} \vec{v} \cdot d \vec{l} = \int_{S} (\nabla \times \vec{v}) \cdot d \vec{S}

Stokes' theorem. In polar coordinates ${(\nabla \times \vec{v})}_{z} = \frac{1}{r} \partial_{r} (r v_{θ}) - \frac{1}{r} \partial_{θ} v_{r}$ and $\nabla \cdot \vec{v} = \frac{1}{r} \partial_{r} (r v_{r}) + \frac{1}{r} \partial_{θ} v_{θ}$ ; point source: $\nabla \cdot (Q \hat{r} / 2 π r) = Q δ (\vec{r})$ .

Theory

Vector field, gradient and conservative force

If a vector is assigned to every point — a vector field (river velocity, wind). If a number is assigned to every point — a scalar field (height, temperature, pressure). The gradient $\nabla h$ of a scalar field points in the direction of steepest ascent, and its length is the slope; it is perpendicular to the contour lines (lines of equal height). In “Chimgan” mode the violet arrows point uphill, while the red ones — $- m g \nabla h$ (on a gentle slope close to the component of gravity along the slope) — point downhill. Potential energy and force are related by $\vec{F} = - \nabla U$ , so the work of gravity depends only on the initial and final height: $\oint \vec{F} \cdot d \vec{l} = 0$ , $curl \vec{F} = 0$ — this is a conservative force. Friction is non-conservative (dissipative): its work depends on the length of the path and turns mechanical energy into heat. Conservation of energy — in the “Roller coaster” experiment.

Divergence: source and sink

$div \vec{v} = \partial v_{x} / \partial x + \partial v_{y} / \partial y (+ \partial v_{z} / \partial z)$ — the flux leaving a unit volume. The small square in “River” mode measures this directly from the definition: as the square shrinks, $Φ / S \to div \vec{v}$ . For incompressible water (river, whirlpool) $div \vec{v} = 0$ : as much flows out as flows in. For a “spring” $\vec{v} = k (x, y)$ , $div \vec{v} = 2 k$ — water is welling up from the bottom. In the weather, near the ground air converges into a cyclone ( $div < 0$ , the air rises and forms cloud) and spreads out from an anticyclone ( $div > 0$ , the air sinks and the sky is clear).

Curl: vortex and paddle wheel

${(curl \vec{v})}_{z} = \partial v_{y} / \partial x - \partial v_{x} / \partial y$ — the local rotation of the field: a small wheel or leaf rotates at $ω = \frac{1}{2} curl \vec{v}$ . For rotation like a rigid body, $\vec{v} = Ω (- y, x)$ , $curl \vec{v} = 2 Ω$ . Two surprises: (1) in a straight river the speed changes across the stream near the bank — $curl \vec{v} \neq 0$ and the wheel turns; (2) outside the core of a vortex ( $v_{θ} = Γ / 2 π r$ ) the water flows round in circles, yet $curl \vec{v} = 0$ — this is a potential vortex (the Rankine model). The velocity gradient tensor at a point is $J_{i j} = \partial v_{i} / \partial x_{j}$ : its trace is $tr J = div \vec{v}$ , while its antisymmetric part gives the rotation: $J_{21} - J_{12} = curl \vec{v}$ ; its symmetric part is the rate of deformation.

Gauss's and Stokes' theorems

Vector algebra works inside the integrals: flux is the scalar product $\vec{v} \cdot \vec{n}$ , circulation is $\vec{v} \cdot d \vec{l}$ , and curl is written as a “vector product” $\nabla \times \vec{v}$ . Ostrogradsky–Gauss theorem: the flux out of a closed surface equals the integral of the divergence over the volume inside the surface. Check this with the cube in “Whirlwind” mode: for incompressible air ( $div = 0$ ), wherever the cube is, $Φ = 0$ — the air entering through the sides leaves through the top; for expanding air ( $div \vec{v} = 3 k$ ), $Φ = 3 k V$ does not depend on the position of the cube. Stokes' theorem: the circulation around a loop equals the flux of the curl through a surface spanning the loop. In a wind that strengthens with height, $curl \vec{v}$ is horizontal: in a horizontal ring $Γ = 0$ , while in a vertical ring whose normal points along the curl $Γ$ is greatest. In electrodynamics both theorems link the integral and differential forms of Maxwell's equations — in the “Electric field and potential” and “Magnetic field” experiments.

Curvilinear coordinates and the Dirac delta function

Rotational fields are convenient to write in polar (cylindrical) coordinates: the div and curl formulas for $v_{r}$ and $v_{θ}$ are in the “Key formulas” section (under Stokes' theorem). For example, for $v_{θ} = Γ / 2 π r$ the product $r v_{θ}$ is constant, so curl = 0. For a point source $\vec{v} = Q \hat{r} / 2 π r$ , div = 0 at $r \neq 0$ , but the flux through any loop enclosing it equals $Q$ : the divergence is concentrated at a single point, $div \vec{v} = Q δ (x) δ (y)$ . In “Wind” mode, reduce the radius $a$ of the centre: div and curl concentrate at the centre, while the flux and circulation do not change — that is the delta function. Expanding a field in harmonics (a Fourier series if it is periodic, otherwise a Fourier integral) — in the “Fourier: the spectrum of sound” experiment.

What the model simplifies

The velocity profile in the river is parabolic (a laminar idealisation; in a real river the flow is turbulent and the profile is flatter), and only the two-dimensional flow at the surface is considered; the leaves and the wheel do not disturb the flow. The weather is a model of source-vortices with a Gaussian core; the map is not real geography, and Coriolis dynamics are not calculated. The whirlwind is a Burgers vortex (an exact solution of the Navier–Stokes equations); the layer near the ground is ignored. The hill is a Gaussian relief resembling Chimgan (not a real map), the skier is a point mass, $μ$ is constant, and air resistance is $k v^{2}$ ( $k$ = 0.22 kg/m, which can be switched off). The integrals are computed numerically (Gauss–Legendre and midpoint methods); the difference between the two sides is usually below 0.1 %, but may be larger when the loop passes over a thin centre (small $a$ ).

Did you know?

The general proof of the divergence theorem was given in 1826 by M. V. Ostrogradsky while studying the spread of heat; Gauss and Green had proved special cases of it. That is why it is also called the Ostrogradsky–Gauss theorem.
Stokes' theorem first appears in a letter from William Thomson (Lord Kelvin) to Stokes dated 2 July 1850; Stokes set it as a question in the Smith's Prize examination at Cambridge in 1854.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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