The flow of water in a river obeys three basic principles of mechanics: conservation of mass, conservation of momentum, and conservation of energy.
We can use these principles, plus a bit of fluid dynamics, to learn a lot
about how rivers behave. Here, we’ll start with mass conservation.
11.1 Mass conservation in a river channel
Engineers and geoscientists have invested a great deal of e↵ort in understanding how, when, and why rivers flood. In principle, flooding is not hard to understand: it occurs when the channel takes on more water than it can handle, and as a result the water rises above the level of the banks. But this observation just raises another question: how much water can a river carry when the water level is just up to its banks?
The water flow in a river, in volume of water per unit time, is its discharge,
often expressed symbolically with the letter Q. The discharge is
equal to the product of the average velocity, U, and the cross-sectional area
of the channel, A. If we want to calculate the discharge that a river is able
to carry, we need to find out the cross-sectional area of its channel (which
we can measure). We also need to know the velocity of the water, averaged
over the cross-section. We can sometimes measure that too, but often it is
useful to be able to calculate what the velocity would be given a particular
combination of factors, such as depth, slope, and channel roughness.
A first step is to establish a relationship between the depth of water and
the discharge. A good place to start is by applying the principle of mass

FLOW IN RIVERS AND STREAMS

EXERCISE:

Consider a cross-sectional “slice” across a river channel.
The width is W (constant), the depth of water is H, and the length of the
slice in the direction of flow is x.
Let Q(x) be the discharge, which is a
function of downstream distance, x. Assume there is no lateral inflow, and
no precipitation on the channel. Use the principle of mass conservation to
derive an equation for the time rate of change of water depth.
11.2 Rivers as engines
In the most general case, the flow of water in a river channel is described
by the Navier-Stokes equations (which we met in a previous chapter), with
the added constraint of boundary conditions related to the shape, size, and
roughness of the channel boundary. However, the Navier-Stokes equations
are very difficult to deal with in their full form. Therefore, fluvial geomorphologists and hydrologists typically rely on simpler equations that still capture the essence of flow dynamics in channels.
One simplification is to consider the case of a flow that is relatively
steady: not changing very rapidly through time. Flows in many natural
channels meet this criterion. It turns out that for quasi-steady flow, we can
describe the flow of a river in terms of its energy budget. Using the same
approach that we applied when we derived the Bernoulli equation, we can
express the energy (per unit weight) at a particular channel cross-section
as the sum of kinetic, gravitational potential, and pressure potential:
E = U2.