Water resulting from rainfall and snowfall that is not returning to the atmosphere by
means of evapotranspiration, travels from mountains and elevated areas
to the sea or ocean.
During this trip, rivers are formed, enabling the transport of water and sediment
to the sea.
So a river system is a complex network composed of streams that drain earth’s land surface.
The question we are addressing in this submodule on rivers is:
Why does the river look as it does?
The topography defines the limits of the basin or catchment of a river.
Now we will focus on the basin of the Rhine river.
This river system enables transport of water and sediment from the Alps to the North Sea
The water that travels from the mountains to the sea through the river as its highway
also carries sediment.
This sediment is usually eroded over the river’s upper course,
transported through its middle course, and deposited near its mouth.
A river flows over the sediment that it transports at the same time.
It is an alive entity, continuously changing its shape in all directions,
eroding and accreting its banks, changing its course,
abandoning channels, creating new ones,
and aggrading and degrading its bed.
If we analyze the longitudinal profile of the bed elevation of a river system,
we observe that the steepness of the bed (called slope) gradually decreases in downstream direction
leading to an upward concave profile of the river.
This decrease in slope in downstream direction is related to the size of the sediment that
is transported by the river.
At the upstream part, in the mountains,
the sediment is coarse.
In other words the grain size is relatively large.
While travelling, the particle size decreases due to abrasion.
When particles are transported they hit each other and so loose material or turn into small pieces.
Tributaries joining the main river lead to an increase of the water discharge in streamwise direction.
Moreover, the discharge is not constant, varying over time,
for instance, due to local precipitation.
The more downstream the smaller the variability of water discharge.
This is due to the relatively smaller influence of locally high precipitation rates.
Let’s have a look at this river system at different locations.
Here in the Alps, where the water has only traveled a few kilometers
the river is still very steep, has a narrow channel and a limited water discharge.
This discharge varies heavily throughout the year.
During larger flowrates,
the river transports larger particles which will be abraded further downstream.
Downstream from the mountain streams, we usually see braided reaches governed by
high sediment load coming from upstream and resulting from bank erosion.
This means that the river here consists of several channels that bifurcate and join without
a clear main channel.
This part is highly dynamic leading to frequent changes.
Once the river flows through a milder slope area with less energy,
the river usually shows a highly curved single-thread channel.
This pattern is called a meandering planform.
In this part, the river has one main channel and is more
stable.
It shows a sinuous shape.
Although it is less dynamic than the upstream braided reach,
this does not mean it is fixed.
Outer banks erode and inner banks accrete, which leads to bend migration and an increase
in curvature until, during a flood,
a river creates a shortcut and then the process restarts.
The river continuously changes its planform and bed elevation over its alluvial plain.
Let’s have a closer look at an individual bend.
Water is turning in a bend, changing the direction of the flow velocity.
If an object turns, Newton’s second law tells us that there
must be a force making it turn, changing the direction of the velocity.
This force, called the centripetal force, is directed to the center of curvature.
Also in a bend there is a centripetal force, changing the direction of the flow velocity.
In the carousel, the centripetal force is exerted by the chain.
In the case of curved flow in a bend, the centripetal force necessary for changing
the flow direction, originates from a water level gradient in
lateral direction.
This gradient in the water surface ultimately leads to a secondary circulation.
In the upper part flow is directed to the outer bend and in the deeper part flow is
directed towards the inner bend.
This secondary circulation causes degradation in the outer bend and deposition in the inner
bend, resulting in a cross section with a deeper
outer bend and a shallower inner bend.
Let’s go back to the entire river system.
Now, moving downstream, close to the river mouth,
the slope is mildest.
These more quiet conditions have benefited civilizations which initiated many (now large) cities.
Let’s see what the river looks like in this area.
Like this one, Nijmegen, that since the Roman era has been situated along the southern bank of the Rhine.
You can see how, after a 900 kilometer trip, the river slope is much milder and the discharge has increased.
Moreover, this discharge varies much less significantly than in the upper course in the Alps.
And the large sediment size that you could observe in the Alps has now been decreased to sand,
and the river is much wider, which allows for shipping and transportation of goods.
Notice how the ships here are taking the outer part of the bend to stay in the deepest part of the channel.
and remember that this cross-sectional profile is related to the secondary circulation that was discussed before in the lectures.
At the river mouth the flow velocity decreases, enhancing deposition.
The deposited sediment creates a delta.
Here, the conditions of the sea play a decisive role in the dynamics of the river.
Tides and waves govern the shape of the delta.
Now we are going to focus on what happens at a specific cross section of the river.
As we have said, the discharge varies in time.
A cross section is usually characterized by a wide floodplain and a narrower main channel.
During a flood event, the entire cross section is filled with water.
Note how the width of the cross-section is much larger than its depth.
In the main channel, water flows faster than over the floodplains.
During mean flow conditions, the flow is normally constrained to the main
channel with a more limited width.
We are now going to use a schematic cross section in order to define some important
variables in a river.
The geometry can be characterized by a width and a flow depth.
Another important variable is the mean grain size of the sediment that is being transported
by the river at this specific location.
The flow is characterized by the water discharge, which is the volume of water per unit time
that passes through this specific location, and the velocity of the flow.
The amount of sediment that passes through this specific location per unit time is called
the sediment discharge.
The bed and vegetation creates a resistance force to the movement of water.
This is expressed by the friction coefficient that depends on the mean grain size,
the presence of bedforms, vegetation and so on.
Let’s focus on a one meter wide part of the cross section.
Note that the depth, mean grain size,
friction coefficient and velocity are considered the same as in the entire section.
We now define the discharge of water and sediment over this one meter wide.
The water discharge over one meter width is equal to the product of flow velocity and
flow depth.
The slope is a very important parameter because it is the slope that indicates the effectiveness
of gravity in moving the water.
The question we want to answer in the next section is:
what are the flow depth and slope for a given and steady water and sediment discharge?
If we let a river evolve when water and sediment discharge are steady,
it will arrive at a situation in which the gravity force balances with the friction force.
In this situation, the water surface slope is equal to the bed slope.
The question is: How can we predict the normal flow depth that
corresponds to this steady state?
Well, let’s see.
If we analyze a volume of water in steady state,
gravity and friction balance each other.
We know that the force of gravity is the product of mass and the gravitational acceleration.
The component of the gravitational force that is balanced by friction is the one parallel
to the bed surface.
Considering again the one meter wide and now the one meter long section,
the mass is equal to the flow depth times the water density.
The unit of the gravitational force is Newton per square meter.
On the other hand, the friction force is represented by the bed
shear stress, which is defined by the product of water density,
the friction coefficient and the square of flow velocity.
Setting these two forces equal to each other and using the equation of the discharge in
a one meter wide section shown before, we arrive at an expression for the flow depth
depending on the friction coefficient, water discharge, gravity, and slope.
However, the bed is rarely flat.
Depending on the flow conditions, ripples, dunes, antidunes, or bars may appear.
The river bed is not fixed, it is not concrete,
it is evolving, and it is continuously adjusting its shape
in order to transport the sediment coming from upstream.
The amount of sediment that can be transported mainly depends on the flow velocity (the faster
the flow, the more sediment is transported),
and the grain size of the particles (the larger the sediment,
the more difficult it is to transport it).
The river will adjust in order to transport the amount of sediment coming from upstream.
Combining these two equations, we arrive at an expression of the equilibrium
or normal slope.
This is the slope which enables the river to transport the sediment coming from upstream.
If, for example, the amount of sediment delivered to the river through landslides,
bank erosion, bed degradation and so on, increases, the bed slope will increase to enable the
river to transport the larger amount of sediment downstream.
At the same time, if the amount of sediment delivered to the
river and so the slope increases, the flow depth will decrease.
The study of such changes in bed elevation is called river morphodynamics.
As we previously said, the water discharge in the river varies with
time.
After a period of heavy rainfall, the river has to transport more water downstream.
Let’s start in a situation where we are in equilibrium after a long time with a constant
and limited water discharge.
In this situation we can expect the water surface nearly parallel to the bed.
The heavy rainfall initiates a flood wave.
The water surface elevation increases locally due to the flood wave.
At the locations where the water surface elevation is larger,
the river may flood a city.
This flood wave travels downstream over time, potentially increasing the number of towns
and cities affected.
While travelling, the peak is smoothed, decreasing the maximum water surface elevation
but increasing the time a city experiences the flood.