### Grad, Div and Curl on Weather Maps: a Gateway to Vector Analysis

Vector analysis can be daunting for students. The theory can appear abstract, and operators like Grad, Div and Curl seem to be introduced without any obvious motivation. Concrete examples can make things easier to understand. Weather maps, easily obtained on the web, provide real-life applications of vector operators.

Fig. 1. An idealized scalar field representing the mean sea-level atmospheric pressure over the North Atlantic area.

Weather charts provide great examples of scalar and vector fields, and they are ideal for illustrating the vector operators called the gradient, divergence and curl. We will look at some weather maps and describe how these operators arise and what they mean in a practical context.

The Surface Pressure

Let us consider a map of the sea-level pressure over the North Atlantic region. We have two independent variables, latitude and longitude, but it is simpler to use cartesian coordinates. We can do this by projecting a region on the spherical globe onto a flat plane, using a stereographic projection, and define ${x}$ and ${y}$ coordinates on a rectangular area. Then the pressure is a function of ${x}$ and ${y}$: ${p = p (x, y)}$. Of course, pressure also varies with time, but we will focus on a particular moment.

Pressure is a scalar field: it takes a single value for each point ${(x, y)}$ in the region. There are many other examples of scalar fields: temperature, density, relative humidity and so on. The atmosphere moves in response to forces acting upon it. Pressure is one of the most important factors, which explains the popularity of isobaric maps, maps showing lines of constant pressure.

If the pressure is constant, taking the same value everywhere, it does not give rise to any force. But when pressure varies from place to place, the air is pushed from regions of high pressure to regions where the pressure is low. Fig. 1 shows an idealized scalar field representing the sea-level atmospheric pressure over the North Atlantic area. There is a deep depression in the east of the region, and the `Azores anticyclone’ to the south-west. This pressure field is defined by a simple analytical function, making it easy to compute spatial derivatives.

It is the gradient of pressure, the spatial variation, that is dynamically important. Thus, we are led to consider spatial derivatives of pressure in different directions. For the cartesian coordinates, the fundamental derivatives are the changes in the ${x}$ and ${y}$ directions, namely ${\partial p / \partial x}$ and ${\partial p / \partial y}$. We combine these into a vector called the gradient of pressure:

$\displaystyle \boldsymbol{\nabla} p = \frac{\partial p}{\partial x}\mathbf{i} + \frac{\partial p}{\partial x}\mathbf{j} \,.$

Here ${\mathbf{i}}$ and ${\mathbf{j}}$ are unit vectors in the ${x}$ and ${y}$ directions. Actually, we have defined the horizontal component of the gradient. That is because the atmosphere is normally close to hydrostatic balance, and the vertical component of pressure is in balance with the force of gravity (${\partial p/\partial z = - g\rho}$, where ${\rho}$ is the density).

Fig. 2. Left: Gradient of the atmospheric pressure near the depression (this is a vector field). Right: Absolute value of the gradient (a scalar field).

We plot the vector field ${\boldsymbol{\nabla} p}$ in Fig. 2 (left panel), for a region around the depression. We see how the gradient points outwards towards higher pressure. Plotted in the right panel is the magnitude of this vector field, ${|\boldsymbol{\nabla} p|}$, a scalar field. The largest magnitude of the gradient is to the west of the low pressure centre, where the isobars are packed most tightly together.

We note that the wind does not blow down-gradient; the pressure force is balanced by the Coriolis force arising from the Earth’s rotation, so that the wind blows more-or-less along the isobars, perpendicular to the pressure gradient. This is called the geostrophic wind.

Divergence and Vorticity

The divergence and vorticity measure how the winds change from place to place. They involve partial derivatives of the wind components. We denote the 3D wind as ${\boldsymbol{U}}$ and the horizontal component as ${\boldsymbol{V} = u \boldsymbol{i} + v \boldsymbol{j}}$ with ${u}$ and ${v}$ the eastward and northward components of wind.

We are most interested in horizontal variations of the wind. Thus, we define two scalar quantities,

$\displaystyle \delta = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \qquad\qquad \zeta = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \,.$

These are the horizontal divergence ${\delta}$ and the vertical component of vorticity ${\zeta}$.

A Closer Look at Divergence

The divergence ${\boldsymbol{\nabla\cdot U}}$ is a measure of the expansion of a parcel of air. Its negative is convergence, a measure of contraction. The divergence can be evaluated as the limit of an integral of the outgoing component of wind through the boundary ${S}$ of a volume ${V}$, as the volume goes to zero:

$\displaystyle \delta = \boldsymbol{\nabla\cdot V} = \lim_{V\rightarrow 0} \frac{1}{V} \int\!\!\!\!\int_\mathbf{S} \boldsymbol{V\cdot n}\,\mathrm{d}\sigma \,,$

(this follows from Gauss’s Theorem). The divergence has units of inverse time, ${s^{-1}}$.

Winds Near the Surface

Due to surface friction, the low-level winds tend to blow across the isobars towards the low pressure. There is a three-way balance between the pressure gradient, Coriolis force and friction force. The result is divergence in anticyclones and convergence in depressions. Using the pressure field shown above, we have defined a vector field to simulate the surface winds corresponding to it (Fig. 3).

Fig. 3. A modelled surface wind with ${20^\circ}$ cross-isobar flow.

The strongest winds are close to the main depression, blowing in an anticlockwise (cyclonic) direction but crossing the isobars at an angle of ${20^\circ}$. The corresponding divergence field is shown in Fig. 4, and we see that the largest (absolute) values are in the depression.

Fig. 4. The horizontal divergence field.

If the eastward component of wind increases towards the east and the northward component towards the north, both terms of ${\delta}$ are positive so the divergence is positive. The effect is to stretch or expand a parcel of air horizontally. Frequently, the two terms are of opposite sign, so the evaluation of ${\delta}$ involves a delicate balance.

Another factor is the cancellation between horizontal divergence and the vertical element ${\partial w/\partial z}$. This results in the overall size of divergence being smaller than the horizontal component alone, and is known as Dines compensation. Low-level convergence, found in depressions, is coupled with rising motion and precipitation.

A Closer Look at Vorticity

Vorticity is a vector field that measures the instantaneous rotation of a fluid parcel. It is analogous to angular velocity in solid body dynamics. The vorticity vector is the curl of the velocity vector ${\boldsymbol{U}}$. Thus, ${\boldsymbol{\zeta} = \boldsymbol{\nabla\times U}}$.

In meteorology, the vertical component of vorticity is especially interesting:

$\displaystyle \zeta = \boldsymbol{k\cdot\nabla\times V} = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \,.$

Like divergence, vorticity has units of inverse time, ${s^{-1}}$. Vorticity is related to the fluid circulation, a measure of rotation for a finite (horizontal) area.

$\displaystyle \zeta = \lim_{A\rightarrow 0} \frac{1}{A} \oint_\mathbf{C} \boldsymbol{V\cdot }\,\mathrm{d}\boldsymbol{s} \,,$

where ${\mathrm{d}\boldsymbol{s}}$ is tangent to ${C}$. This is a consequence of Stokes’s Theorem.

The vorticity ${\zeta}$ due to fluid motion is called the relative vorticity. The atmosphere also has vorticity due to the Earth’s rotation; this is called planetary vorticity. The vector planetary vorticity is parallel to the rotation axis of Earth. Its vertical component is ${f = 2\Omega\sin\phi}$ with ${\Omega}$ the Earth’s angular velocity and ${\phi}$ the latitude. The absolute vorticity is the sum of the relative vorticity and planetary vorticity: ${\eta = \zeta + f}$.

Fig. 5. The vertical component of (relative) vorticity, ${\zeta}$.

Using the vector wind field again, we compute the vorticity ${\zeta}$ and plot it in Fig. 5. The pattern is quite similar to the divergence, but this is a result of the simple way the pressure field and corresponding wind field were constructed. Note that the largest positive values of ${\zeta}$ — in the depression — correspond to the largest negative values of ${\delta}$.

More Realistic Weather Charts from www1.wetter3.de

To give a more realistic impression of divergence and vorticity fields, two charts were downloaded from the site http://www1.wetter3.de/. They show the fields of ${\delta}$ and ${\zeta}$ for the 300 hPa surface (high in the troposphere). Fig. 6 shows a low pressure trough over Ireland (no surprise!). Pronounced divergence east of the trough is consistent with its eastward movement. There is a strong maximum of vorticity over Ireland (Fig. 7) associated with strong NW winds to the west and strong south-westerlies to the east.

Using charts like these, meteorologists gain valuable information about the atmospheric flow. They can also detect warning signals of severe or extreme weather.

Fig. 6. 300hPa Height (black), Divergence (white) and Windspeed (colours).

Fig. 7. 300hPa Wind Speed and Direction (arrows) and Relative Vorticity (colours). [Image from \url{http://www1.wetter3.de/}].