9.1: Introduction

Our third example of the application of the Navier-Stokes equations to natural flows is hydrostatic flows over topography. These occur in many natural settings such as downslope windstorms (Figure \(\PageIndex{1}\)b), tidal ocean currents and flow over dams. They are perhaps most recognizable in the flow of a small stream over a rocky bottom (Figure \(\PageIndex{1}\)a). For this discussion, we abandon the assumption that the amplitude of the disturbance is small. Instead, we assume that the flow is in hydrostatic balance, similar to the long-wave limit discussed above.

Assumptions:

1. The fluid is inviscid: \(ν\) = 0.
2. The fluid is homogeneous: \(\rho = \rho_0\).
3. The flow is confined to the \(x\)−\(z\) plane, so there is no dependence on \(y\) and no motion in the \(y\) direction.
4. The channel width \(W\) is uniform and the walls are vertical.
5. The flow has the character of gravity waves in the long-wave limit:
   1. The horizontal (streamwise) velocity is independent of depth: \(\vec{u}=u(x,t)\hat{e}^{(x)}\).
   2. The pressure is hydrostatic: \(p = \rho_0g(\eta −z)\).

Some of these assumptions will be relaxed as we go along.

Figure \(\PageIndex{1}\): (a) River flow over an obstruction. Photo by Jonathan Ball. (b) “The Bishop Wave”, a storm on the downwind slope of the Sierra Nevada mountain range. Wind is from right to left; clouds and dust at left indicate rapidly rising, turbulent air. Note the similarity to (a). (Photo by Robert Symons, 5 March 1950.) Further details are on page iii.