How Does the Coriolis Effect Influence River Water Levels at Different Banks?

[karnten07]

Homework Statement

A river of width D flows Northward with speed v. Show that the water is lower at the west bank than at the east bank by approximately

2Dwvsinlamda/g

where w is the angular velocity of the Earth and lamda the latitude.

Homework Equations

The Attempt at a Solution

I get the difference in height as 2Dwvsin(lamda)cos(lamda)/sqrt(g^2 +(2wvsin(lamda)cos(lamda))^2)

So if this was the same as my answer it would mean that cos(lamda)/sqrt(g^2 +(2wvsin(lamda)cos(lamda))^2) = 1/g

Im just unsure how to prove this, it looks like a simple right angled triangle to show this by rearranging to show coslamda = sqrt(g^2 +(2wvsin(lamda)cos(lamda))^2)/g

Does this look right guys?

 

[Shooting Star]

Not really. You can't reduce your formula to the answer through any approximation that I can see. Why don't you just share your calculations with us?

 

[ogb p]

The Coriolis force is a result of the Earth's rotation and its effect on moving objects such as air and water. In the case of a river, the Coriolis force causes the water to deflect to the right in the Northern hemisphere and to the left in the Southern hemisphere. This results in a difference in height between the west and east banks of the river, as shown in the given equation.

To prove this equation, we can use the fact that the Coriolis force is directly proportional to the velocity of the object and the angular velocity of the Earth, as well as the sine of the latitude. We can also use the equation for the Coriolis force, Fc = -2mω x v, where m is the mass of the object, ω is the angular velocity of the Earth, and v is the velocity of the object.

First, let's consider a small segment of the river with width d. The force acting on this segment due to the Coriolis force is given by Fc = -2dmω x v, where ω is the angular velocity of the Earth and v is the velocity of the water segment. Since the river is flowing northward, the velocity of the water segment is directed towards the east. This means that the Coriolis force will be directed towards the south, causing the water to deflect to the right. This deflection will result in a difference in height between the west and east banks of the river.

Now, to determine this difference in height, we can use the equation of pressure difference, ΔP = ρgh, where ρ is the density of water, g is the acceleration due to gravity, and h is the difference in height. We can equate the Coriolis force acting on the water segment to the pressure difference, giving us:

-2dmω x v = ρgh

Solving for h, we get:

h = -2dmω x v/ρg

Since the mass of the water segment is given by m = ρd, we can substitute this into the equation to get:

h = -2ρ^2d^2ω x v/ρg

Canceling out ρ and rearranging the equation, we get:

h = -2dω x v/g

Now, we can substitute in the given values for ω and v, and also use the fact that d = D/2,