Calculating Water Level Difference in English Channel

[rabbit44]

Homework Statement

Estimate how much higher the water level is on the French coast than on the English coast when there is an eastward flow through the English Channel at a typical speed of 1m/s. You may assume that the English Channel is approximately 30km wide.

Homework Equations

An earlier part of this question got me to derive the equation for when there is geostrophic balance:

[tex] 2\Omega \times \textbf{u}_{R} = -\frac{1}{\rho} \nabla p [/tex]

The Attempt at a Solution

I think I probably have to use the above equation, or the Rossby number, as I am given enough parameters to calculate it. I can calculate the left hand side of the above equation as I am given the magnitude and direction of the flow everywhere, and I can calculate the angular rate of rotation of the Earth from its 24 hour period. I can look up the density of sea water I guess. But then I don't know how to use grad(p) to give me a difference in sea level. I thought maybe something to do with hydrostatic equilibirum, but basically I'm really confused.

Any help would be very much appreciated!

 

[Jhero]

Thank you for your question. To estimate the difference in water level between the French and English coasts due to eastward flow in the English Channel, we can use the geostrophic balance equation you have derived. However, we also need to consider the effects of the channel's width and the density of seawater.

First, we can calculate the pressure gradient, which is represented by the term -∇p in the geostrophic balance equation. This term describes the change in pressure over a distance. In this case, we can assume that the pressure is constant over the entire channel, so the pressure gradient will be the same for both the French and English coasts. We can calculate the pressure gradient using the formula:

∇p = ρgΔh

where ρ is the density of seawater, g is the acceleration due to gravity, and Δh is the difference in water level between the French and English coasts. We can assume that the density of seawater is 1025 kg/m³ and the acceleration due to gravity is 9.8 m/s².

Next, we need to consider the channel's width, which is given as 30km. This means that the distance over which the pressure changes is 30km. To convert this to meters, we multiply by 1000, giving us 30,000m. Therefore, the pressure gradient can be written as:

∇p = ρgΔh/30,000

Now, we can substitute this into the geostrophic balance equation:

2Ω×u_R = -ρgΔh/30,000

where Ω is the angular rate of rotation of the Earth and u_R is the eastward flow speed of 1m/s.

Solving for Δh, we get:

Δh = -15,000Ω/u_R

We can look up the value of Ω, which is approximately 7.29×10^-5 rad/s. Plugging this in along with the flow speed of 1m/s, we get:

Δh = -0.021m

Therefore, the water level on the French coast will be approximately 0.021m lower than on the English coast due to the eastward flow in the English Channel. This is a small difference, but it is significant in terms of ocean dynamics. Keep in mind that this is just an estimate