The Gradient Operator, $\nabla$

$\nabla p$ is called the `gradient'or `grad'or `del' of $p$ and is a simple generalisation of the one dimensional gradient, $dp/dx$. It behaves in a similar manner but also obeys both vector laws (adding, subtracting and multiplying vectors) and operator laws (such as differentiating a product, quotient and so on). The basic properties of the gradient operator are listed below.

Properties

1. 
   
   $$\nabla p = {\partial p \over \partial x}{\bf i} + {\partial... ...r \partial y}{\bf j} + {\partial p \over \partial z}{\bf k}$$
   
   

2. 
   
   $$\vert\nabla p\vert = \left [ \left ({\partial p \over \part... ...left ({\partial p \over \partial z}\right )^{2}\right ]^{1/2}$$
   
   

3. 
   
   $${\bf i}\cdot \nabla p = {\partial p \over \partial x}$$
   
   
   gives the rate of change of $p$ in the direction ${\bf i}$.

4. 
   

   $${\bf a} \cdot \nabla p = \vert{\bf a}\vert\vert{\bf\nabla p}\vert \cos \theta$$ (1.14)

   
   
   is the rate of change of $p$ in the direction of ${\bf a}$, i.e. $\partial p/\partial a$. It is also called the directional derivative.

From the above properties, we see that $\nabla p$ is a vector whose magnitude and direction gives the maximum rate of change of $p$ or, equivalently, the direction of the maximum gradient.

For the pressure, $p$, if we plot contours of constant pressure ($p$ = constant), these contours are called ISOBARS and, as shown in Figure 1.7, $\nabla p$ crosses the isobars at right angles.

Figure 1.7: A sketch of the isobars and the associated direction of $\nabla p$. Note that the force, however, acts from high pressure to low pressure whereas $\nabla p$ points from low to high.

The pressure force is $-\nabla p$ and so in Figure 1.7 the air would flow from high pressure to low pressure. However, the Earth's rotation makes the flow spiral as seen on weather maps.

Two final properties of $\nabla p$ are

1. $\nabla p$ points towards larger values of $p$.

2. $\nabla$ is invariant. This means that it is independent of the location and orientation of the coordinate axes. Thus, expressing $\nabla p$ in both cartesian and cylindrical coordinates produces the same vector. Obviously, the components in each coordinate system will be different.

Two examples illustrate the force generated by a non-uniform pressure.

Example 1. .4Consider the pressure $p = x^{2}$. The isobars are shown in Figure 1.8 The force is ${\bf F} = - \nabla p = -dp/dx {\bf i} = -2x{\bf i}$.

Figure 1.8: The isobars and associated direction of the pressure force.

Example 1. .5Consider the pressure $p(x,y) = y - x^{2}$. If $p$ is constant, then the isobars are given by $y = p + x^{2}$, as shown in Figure 1.9.

Figure 1.9: The isobars and associated direction of the pressure force.

The force is given by

$${\bf F} = -\nabla p = \left (-{\partial p \over \partial x}, -{\partial p \over \partial y}, 0\right ) = (2x, -1, 0).$$