## Differentiation Following The Motion

When we apply the laws of motion and thermodynamics to a fluid to derive the equations that govern its motion, we must remember that these laws apply to material elements of fluid that are usually mobile. We must learn, therefore, how to express the rate of change of a property of a fluid element, following that element as it moves along, rather than at a fixed point in space. It is useful to consider the following simple example.

Consider again the situation sketched in Fig. 4.13 in which a wind blows over a hill. The hill produces a pattern of waves in its lee. If the air is sufficiently saturated in water vapor, the vapor often condenses out to form a cloud at the ''ridges'' of the waves as described in Section 4.4 and seen in Figs. 4.14 and 4.15.

Let us suppose that a steady state is set up so the pattern of cloud does not change in time. If C = C(x, y, z, t) is the cloud amount, where (x, y) are horizontal coordinates, z is the vertical coordinate, and t is time, then (-) =

\ dt / fixed point in space in which we keep at a fixed point in space, but at which, because the air is moving, there are constantly changing fluid parcels. The derivative ( -ยง ) is called the Eulerian

\ dt / fixed point derivative after Euler.1

But C is not constant following along a particular parcel; as the parcel moves upwards into the ridges of the wave, it cools, water condenses out, a cloud forms, and so C increases (recall GFD Lab 1, Section 1.3.3); as the parcel moves down into the troughs it warms, the water goes back in to the gaseous phase, the cloud disappears and C decreases. Thus fixed particle

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