Physical Oceanography Problem Set

  Paraphrase below to be more understandable or rewrite it in your own words. If you feel like any answer isn't correct, please don't use it. In this problem set, we explore the relationships between winds, shear stress, Ekman transport, and Ekman pumping. We will first generate some hypothetical maps of the wind shear stress (a vector field) and then calculate the map pattern of Ekman transport, and finally, the Ekman pumping. Note that some of the questions ask for plots and copies of your scripts that you create — just copy and paste them into this document. Part A: Bi-directional shear stresses Our map will be a square 3000 km on a side, with the x-axis going from -1500 to 1500 and the same on the y-axis. The first thing to do then is to create a map grid as follows: L=1500*1e3; % dimensions of the grid in m step=100*1e3; x=-L:step:L; y=-L:step:L; [X,Y]=meshgrid(x,y); Note that the units here are in meters — this makes the units work out right later. The meshgrid command creates a grid whose axes are defined by x and y. The result of this is actually two matrices, X and Y, each of which is a 31 x 31 matrix; together, they give the x and y values of every point on the grid. The next step is to create a wind shear stress vector for each point on the grid; each vector will have an x and a y component, and all together these make up a vector field — think of a field with arrows scattered all over it. In this first case, we will create a very simple pattern of shear stress that is constant in the x (E-W) direction and varies as a sinusoidal function in the y direction (N-S) such that the shear stress in the upper part of our map will go to the east and in the lower part of the map, it will go to the west. To make things simple, we’ll say that the y-component of the shear stress is 0. Rather than entering the values for the x and y components of shear stress one by one, we’ll use a function to define the x-component as follows: τ_x=1.0*sin(Y*π/2L) To implement this in Matlab, you have to remember that Y is actually a matrix, and so the multiplier in front of the sin has to have a period before it, like this: taux=1.*sin(Y*(pi/(2*L))); This will create a new matrix called taux that will have the same dimensions (rows and columns) as Y. We also need a matrix for the y-component of the shear stress; here is one way of doing that: tauy=zeros(size(Y)); Now that we have our shear stress components, it is a good idea to plot them to see if they are what we expect. Matlab has a nice plot command to do this. Make a figure (type figure(1);) and then type in: quiver(X,Y,taux,tauy); Add a title, label your axes, and you should see a bunch of vectors pointing to the right (east) at the top and then shrinking in size toward the mid-point, and then reversing direction (pointing west) and growing larger in the bottom part. 1. [10] Paste your script and plot here. L=1500*1e3; % dimensions of the grid in m step=100*1e3; x=-L:step:L; y=-L:step:L; [X,Y]=meshgrid(x,y); taux=1.*sin(Y*(pi/(2*L))); tauy=zeros(size(Y)); figure(1); quiver(X,Y,taux,tauy); xlabel('x-axis'); ylabel('y-axis'); title('shear stresses map'); The next step is to calculate the components of the Ekman transport resulting from this shear stress field, following the equations: U_ek=τ_y/ρf V_ek=-τ_x/ρf U-momentum equation -ρfv = (∂τ^x)/∂z → -ρf∫_(-∞)^0▒〖vdz=〗 ∫_(-∞)^0▒ (∂τ^x)/∂z dz= τ^x → -ρfVE = τ^x → VE = τ^x/ρf in m2/s V-momentum equation ρfu = (∂τ^y)/∂z → UE = τ^y/ρf in m2/s Note that since the shear stress components above are actually matrices, you have to stick a period in front of the divide sign, like this: tauy./(rho*f). All we need then is the density (1030 kg/m3) and f, the Coriolis parameter: f=2Ωsinλ  is just the rotation rate (use 7.3e-5 s-1), and  is the latitude. To get the components of the Ekman flow, we first need to define a matrix for the Coriolis parameter and to do that, we have to convert our Y grid points from km into degrees latitude. We’ll take the center of the grid to be at 30°N, and you should recall that there are 111km/° latitude. You could create a matrix of latitudes like this: lat=30+(Y/111000); f=2*omega.*sind(lat); (having first defined omega) You should then be able to forge ahead and calculate the Ekman transport components (label them Ue and Ve), and then plot them using the quiver command again (quiver(X,Y,Ue,Ve);), which will give you a map of the Ekman transport. Be sure to add axis labels and a title. 2. [10] Paste a copy of your figure below. omega=7.3*1e-5; lat=30+(Y/111000); f=2*omega.*sind(lat); rho=1030; Ue=tauy./(rho*f); Ve=-taux./(rho*f); figure (2); quiver(X,Y,Ue,Ve); xlabel('x-axis'); ylabel('y-axis'); title('Map of the Ekman transport'); 3. [5] If you’ve done this first part correctly, you should see that the Ekman transport is not symmetrical about the y=0 line, even though the shear stress did have symmetry about the y=0 line. Explain why this is the case. The Ekman transport is not symmetrical as a result of the concurrent breeze power inside the Ekman stream and wind shear pressure. To elaborate more, it is not symmetrical because of the convergent wind forces within the Ekman flow and wind shear stress. This causes the wind flow and the Ekman transport to move perpendicularly away from each other. The perpendicular motion relative to the wind flow makes the Ekman transport not symmetrical which is prolonged out of equilibrium due to the rotation of the planet. Furthermore, the shear stress has symmetry because they migrate together around the circumference of the planet following its relative motion which further induces the Ekman flow. 4. [10] You should be able to study the Ekman transport vectors and make some guesses about where you will get upwelling and downwelling. Explain what you expect to see in terms of upwelling and downwelling in the top part of your map, in the middle of the map, and in the lower part of the map, including why you have these expectations. The Ekman transport stream combined with the breeze stream would bring about anticyclonic breezes that would thus cause surface converging and a downwelling. Moreover, at the lower and upper piece of the guide; cyclonic breezes would be normal, which would initiate surface dissimilarity and upwelling sea momentum designs The Ekman transport is not symmetrical due to the convergence of the water within the layer of the Ekman while the shear stress has symmetry because it has the divergence in the upwelling into the Ekman layer. Convergence results in the downwelling of water out of eth Ekman layer. Next, we move on to the Ekman pumping, which is defined as: W_ek=1/ρf ((∂τ_y)/∂x-(∂τ_x)/∂y) Negative values mean downwelling, and positive values mean upwelling, and the units are in m2/s if you use the units I suggest for the other parameters. The term in the parentheses is called the curl of the shear stress vector field, and since we only have two dimensions here (x and y), the curl is a vector in the z dimension. We can calculate this using the curl command of Matlab, as follows: [curlz,cav]=curl(X,Y,taux,tauy); Wek=curlz./(rho*f); Then plot Wek as a colorized map along with the Ekman transport vectors like this: figure(3); pcolor(x,y,Wek); shading interp; colorbar; hold on; quiver(X,Y,Ue,Ve); hold off; Add labels to the axes and a title here. 5. [10] Paste your script to this point, and this last figure, and then briefly summarize the pattern of upwelling and downwelling resulting from this pattern of shear stress. [curlz,cav]=curl(X,Y,taux,tauy); Wek=curlz./(rho*f); figure(3); pcolor(x,y,Wek); shading interp; colorbar; hold on; quiver(X,Y,Ue,Ve,'r'); hold off; xlabel('x-axis'); ylabel('y-axis'); title('the Ekman pumping and the Ekman transport') The tinge of the figure delineates a z-axis in which the downwelling and upwelling can be watched. The lower esteems at the best and base of the guide bring about downwelling and mid in positive Next, let’s try this in the southern hemisphere. Start with the same shear stress field, but make the latitude at the center of the map be -30° (and make the latitude grow less negative toward the top of your map (the +y half) and more negative toward the bottom. L=1500*1e3; % dimensions of the grid in m step=100*1e3; step2=50*1e3; x=-L:step:L; y=[-L:step:0,step2:step2:L]; [X,Y]=meshgrid(x,y); taux=1.*sin(Y*(pi/(2*L))); tauy=zeros(size(Y)); figure(1); quiver(X,Y,taux,tauy,'r'); xlabel('x-axis'); ylabel('y-axis'); title('shear stresses map'); omega=7.3*1e-5; lat=-30+(Y/111000); f=2*omega.*sind(lat); rho=1030; Ue=tauy./(rho*f); Ve=-taux./(rho*f); figure (2); quiver(X,Y,Ue,Ve,'r'); xlabel('x-axis'); ylabel('y-axis'); title('Map of the Ekman transport'); [curlz,cav]=curl(X,Y,taux,tauy); Wek=curlz./(rho*f); figure(3); pcolor(x,y,Wek); shading interp; colorbar; hold on; quiver(X,Y,Ue,Ve,'r'); hold off; xlabel('x-axis'); ylabel('y-axis'); title('the Ekman pumping and the Ekman transport'); 6. How does the pattern of Ekman transport in this case compare with the one in the northern hemisphere? The pattern of the Ekman transport seems to be similar to the northern hemisphere. 7. How does the pattern of Ekman pumping in this case compare with the one in the northern hemisphere? The pattern of the Ekman pumping in the southern hemisphere seems to follow the pattern of the northern hemisphere. Part B. A more complex shear stress pattern Now we move on to a more complex case, where the shear stress varies in the both directions. Use the same map dimensions and northern hemisphere location as at the beginning, but now make the y-component of the shear stress vary as a function of x as follows: tauy=1*sin(X*(pi/(2*L))); 8. [10] What is the pattern of Ekman transport generated by this pattern of shear stress? Paste a copy of the figure here. In the Ekman transport develops a quadrupole pattern which strictly allow to the perpendicularly to the shear stress. The pattern leads in the Ekman transport dipole of the vertical velocity across a vortex with the reduced magnitude. Pattern in the Ekman transport should give the changes in the magnitude of the balanced flow which along the meandering current. 9. [10] What is the pattern of Ekman pumping generated by this pattern of shear stress? Paste a copy of the figure here. The Ekman pumping generated the differeiental seasonal patterns of the shear stress , where as the summer pattern is charcteristized by the Ekman transport. The Ekman pumping will be senstive for the particular velocity of the vortex structure where the contrary solution is depend on the value of the zeta. Ekman pumping have the horzontialy velocity with the pattern of the wind stress. Thus, the pattern is aligned with the shear stress.