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Problem 1 When does a spring tide occur? When does a neap tide occur? A really heavy cargo ship is coming in to port with significant tidal activity

Physics

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Problem 1

1. When does a spring tide occur? When does a neap tide occur?
2. A really heavy cargo ship is coming in to port with significant tidal activity. The captain is concerned about getting grounded. Approximately how many days after a new moon would the risk be the greatest? (i.e. when will the under-keel clearance be the least?)
3. What is the general direction of the tide producing force at point X in Fig. 2.2 of your notes?
4. What is the direction and magnitude of the tide producing force at the earth’s center?
5. If a high tide occurs in Eastport, Maine, at 11 am, roughly when will the next two high tides occur? What is the basis for your answer?
6. The harmonic constituents for Eastport, Maine, are given as follows: https://tidesandcurrents.noaa.gov/harcon.html?id=8410140

 

 

 

 

Disregard the constituents after the first three; they are of much smaller amplitude than the first three. Use your notes/textbooks/ to find the periods of the first three constituents. The amplitudes (IN FEET) and phases (IN DEGREES) are given. The mean water level is 11 feet above MLLW.

 

Write a formula to estimate the water level at any time in the future (for example, at T= 4 hours).  You may use Excel or a computer program to produce plots of the water levels for 48 hours. Best to write everything in terms of hours! 

 

 

 

 

Problem 2

 

A breakwater design was to be designed using the Hudson equation for a K_d value of 4 (fixed). The limit state equation (Hudson’s equation turned around) can be written in terms of resistance and loading, as:

 

A D D_n (K_d Cota)^1/3 - H = 0

 

where D = ((r_s/r) - 1)). This is the same formula you used earlier in the course, except an uncertainty parameter A (nominally close to 1) has been included. D_n is the stone diameter, Cota is the slope of breakwater face, and H is the design wave height. The various parameters (except K_d) have been obtained from multiple sources and assumed to be random, with mean and standard deviations as follows:

 

 

Estimate the probability of failure if these parameters are used, using the Mean Value Approach.

 

 

 

 

Problem 3

 

1. Use your diffusion code, for the 1 d channel of width 0.25 m and depth 1 m. Use dx =0.125 m for the 3 m domain with zero derivative boundary conditions (as before).

Use the same source of 1 gm/s, but apply it at x = 1.5 m (middle of the channel for the sake of symmetry). Turn off advection and decay. Apply the source for 1 second and then turn it off.  (This is what you did for comparing with the analytical solution). Run the model for 10 s.

 

Now use the solutions for all grid points for t = 7 s and 10 s, use these numbers to confirm the part of the analytical solution (check your notes if needed) which says

 

M = ? c(x, t) dx

 

(obviously, you’ll have to do the integral numerically).

 

What does this prove?

 

(b) In the diffusion equation, why is the decay term represented by – aC? What does this type of decay mean? What does a mean? If a = 0.1 per sec, what does that mean?

 

 

Problem 4

 

Consider the 400 km by 800 km basin for which you estimated water levels for a constant wind stress. Use the same parameters as you did before.

 

Instead of the constant wind, now consider a storm. We will describe the storm with the Rankine Vortex model as follows:

 

 

 

where Vm is the maximum windspeed = 40 m/s (about 80 knots), and take Rm = radius to maximum winds to be ~45km. B is a shape factor which may nominally be taken as ~0.5.

 

 

1. Your first task is to develop windfields and wind stresses (denoted by Fs and Gs in your notes and previous HW code) for this windfield.  Assume the storm is centered in the middle of your 400km by 800 km grid at (xc, yc). Further assume it just sits there, i.e. there is no time-related variation of the winds. (In reality it will move and you’ll need new windfields every time step).

 

Take a total of 18 x 10 grid points. For all the grid points in your domain, calculate the windspeed based on the above formula, with the storm center at the middle of this grid. To do this, write code to find the x and y coordinates of each point, determine “R” (distance of each point from the storm center, using dx = x –xc, dy = y-yc, etc). Use the formula above and estimate the wind speed V at each point.  Then use the formula in van Dorn’s formula in Sorenson’s book (for example, as in Assignment 8), and estimate the wind stress t at each grid point.

 

2. Now we need directions for the wind speed/stress. Say the RV model gives you an ANTI CLOCKWISE vortex (see https://demonstrations.wolfram.com/RankineVortexASimpleHurricaneModel/).

For each grid point you’ve found the radius vector (R was estimated), Pretend you drew a circle for this R. Then the wind would be tangential. So draw a line at right angles to the R vector and the anticlockwise sense will be the direction of the stress. Now find its x and y components. So Fs = t*cos or sine (of some angle), and the cosine and sine will be related to dx/R or dy/R.  Similarly for Gs. Not hard to find.

 

3. Prepare a table of the 18x10 resulting windspeeds, and also the Fs and Gs. Check the signs/directions. They should correspond to the anticlockwise wind motion!

 

4. When you are satisfied the wind stresses Fs( i,j)and Gs (i,j) look right, use them with your circulation model code. Don’t worry too much about getting the exact match between the location of the u point on the grid and and the wind stress (I,j). For the u equation just use Fs(I,j) as computed above. And the same for v(I,j), as long as your origin etc were the same.

 

(There are two ways to do this. You can import the new code into your old code in the appropriate manner to calculate the winds and then the flow. Or you can modify your old code to simply “read” the ouput containing Fs and Gs info).

 

Run the flow model for 300 time steps and prepare a table of the resulting zeta values, u values, and v values. Examine the velocity plots. What type of circulation is being created?