Tidal Bores

Figure 4.7: A tidal bore.

A tidal bore is a sort of hydraulic jump that propagates up (i.e., upstream) a river estuary. The upper part of Figure 4.7 shows such a bore in the local rest frame of the Earth. The bore is propagating at the velocity $V$ up a river of uniform width, and depth $h_1$ , that is flowing downstream at the velocity $u_1$ . The flow behind the bore is of depth $h_2$ , and is flowing upstream at the velocity $u_2$ . The lower part of the figure shows the same phenomenon in the rest frame of the bore. In this frame, we observe a stationary hydraulic jump with an upstream depth and flow velocity $h_1$ and $v_1=V+u_1$ , respectively, and a downstream depth and flow velocity $h_2$ and $v_2=V-u_2$ , respectively. Making use of Equations (4.44) and (4.45), we obtain

$$\left(\frac{h_2}{h_1}\right)^2+\frac{h_2}{h_1}-\frac{2\,(u_1+V)^2}{g\,h_1}=0,$$ (4.62)

which can be rearranged to give

$$V = -u_1 +\left[\frac{g\,h_2\,(h_1+h_2)}{2\,h_1}\right]^{1/2}.$$ (4.63)

Thus, we deduce that the speed of the bore, relative to the unperturbed river, is a simple function of the upstream and downstream depths. Note that, in the limit $h_1\simeq h_2\simeq h$ , the previous equation reduces to $V\simeq -u_1+\sqrt{g\,h}$ . In other words, a weak bore degenerates into an ordinary shallow water gravity wave propagating at the characteristic velocity $\sqrt{g\,h}$ relative to the stream.

Tidal bores are found in river estuaries where a funneling effect causes the speed of the incoming tide to increase to such a point that the flow becomes super-critical. For example, bores can be observed daily on the River Severn in England.