In this project, you will be examining rotational systems such as atmospheric pr

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In this project, you will be examining rotational systems such as atmospheric pressure systems and oceanic gyres. Also, I include a discussion so that you can see how some of the concepts presented in this chapter have very real applications and utilizations. For the purposes of the discussion here and the work you will be doing, we will be considering those systems occurring in the Northern Hemisphere only. It is highly recommended that you read Sections 15.1, 15.3, 15.6 and 15.7 in your text book for further background into the significance of the divergence and curl as they are related to rotating systems. (some of these are mentioned in Section 16.5, please review if need be). For any pressure system, atmospheric flow follows the pressure gradient. For a low pressure system, air pressure is higher further out than at the center. Thus, air flow will be directed towards the center. The reverse is true for a high pressure system. Air flow is directed outwards from the center. In addition, there is the Coriolis effect. This effect is the apparent deflection of motion due to the earth's rotation. In the Northern Hemisphere, this apparent deflection is to the right. So, in aiming a projectile at a target, if one aims right at the target and then launches the projectile, the target will not be hit, but the shell will miss to the right since the earth rotated under it while in flight (unless the projectile's path was following the equator exactly, in which case there will be no deflection, and the target is hit) . It is for this reason why air pressure systems (and oceanic gyres) rotate. In the ocean, the oceans are contained within basins. There are temperature and density differences. These differences create an uneven oceanic surface in terms of height. The differences in height establish a pressure gradient where water flows down the pressure gradient. Once again Coriolis then affects the rotation creating circular/elliptical flowing gyres. (Please see accompanying figure that is linked here). As can be seen from the discussion, atmospheric and oceanic flow follow the gradient and in both cases, the pressure gradient. (Please bear in mind that this is a highly simplified explanation). Part I: Based on the explanation above, create a vector field showing wind direction and strength for the following: ( we are looking down at them from above) 1) A low pressure system with a weak gradient 2) A low pressure system with a strong gradient 3) A high pressure system with a weak gradient 4) A high pressure system with a strong gradient Please note: I want vector fields. Do not create what you see on weather maps and newscasts. Those maps showing pressure systems show lines of equal pressure (isobars). These are scalar fields. Remember, vectors have magnitude and direction. Some points to bear in mind. Remember to account for Coriolis and the direction winds will deflect to. Also, the stronger the gradient, the more pronounced the differences will be in the magnitude. (You can indicate this with longer and/or thicker arrows for winds with higher velocities from winds with lower velocities). Finally, due to a phenomenon known as vorticity, rotational velocity (angular momentum) will be faster when closer to the center. (Think of a figure skater who raises her arms while spinning. What happens? She spins faster due to a smaller radius). Show several layers of vectors at least (radiating out from the center). You should also show some differences in the spacing of the vector layers based on the situation you are creating (weak or strong gradient). What do you notice about your graphs? In the northern hemisphere, what can you say about the general rotational direction for high and low air pressure systems? Are they counter clockwise (cyclonic) or clockwise (anti-cyclonic) or neither? In addition to the vector fields you create, please include a short summary explanation. Be sure to answer the question about they being cyclonic or not. Part II: A similar graph that you worked on for part I can be done with respect to oceanic gyres. In this part, however, you will be asked to work with the Navier-Stokes equation that is used to develop the points of this oceanographic academic article On A Wind Driven, Double-Gyre, Quasi-Geostrophic Ocean Model: Numberical Simulations and Structural Analysis.pdf (right click to save the PDF to your machine). Please see that document for explanations of the terms. Read closely pages 387 to 390 in that document. You will be asked to work with equations 2.1 and 2.2. I am also including another document that provides a more general explanation of the equations of motion that are being discussed here. Please be sure to read through that as you should be able to ascertain suggestions, key substitutions for rewriting the equations in the paper linked here so that you can carry out what is asked of you. (By all means, look through the entire research paper included here and check out the resultant graphs of the gyres they created using the models). Recall from your readings that if curl F = 0, then F is a conservative vector field. Also if curl F = 0, then we say the field (fluid field) is irrotational, ie does not rotate. The fact that gyres rotate, what does this tell you about those vector fields? That is, are they conservative or not? Based on this, what would you expect the curl of the Navier-Stokes equation to be? Determine the curl of the Navier-Stokes equation. (Equations 2.1 and 2.2 are the same - in 2.2 there are simply some substitutions for some of the terms in 2.1). Does this support the fact that oceanic gyres do exist. If Div F = 0, then we say the fluid is incompressible. What this means in essence is that mass is conserved. (There are very involved equations that verify this). You will see for the Navier-Stokes equation the statement that div v = 0. I provide for you how you can substitute for v (based on a vector position function). Using this substitution, verify that div v =0. (Hint: you may find that the chain rule is applicable. Also, review chapter 13). Keep in mind that the Navier-Stokes equation was developed for the upper water layers in the ocean where ocean density is rather uniform or does not demonstrate too much variation in density. Because of this, it can be shown that div = 0. However, as one goes deep in the water column, immense pressure and dramatic changes in density occur with the result that the fluid is compressible and divergence does not equal 0. Depending on the direction an oceanic gyre rotates will determine the direction the water deflects to. In oceanography, there is the concept of constant volume (conservation of volume). In other words, if water enters a region, an equal amount of water must leave. Likewise, if water exits a region, an equal amount must enter to maintain the same volume. Based on what you have learned thus far, if an oceanic gyre rotates clockwise, would you expect water to build up in the center (due to Coriolis effect) or deplete the center of the gyre? What about for a gyre that rotates counter-clockwise? Now, think in terms of 3-D. If water piles up in the middle, how would you expect volume to be maintained? Where would the excess water have to go? What about those situations, where water is depleted from the center? Where would you expect water to come from to replenish and maintain the volume? There are observed phenomena in the ocean referred to as a convergence zone and a divergence zone. Convergence zones occur with downwellings. Divergence zones occur with upwellings. Upwellings bring nutrients from below the water surface where the producers can utilize them while carrying out photosynthesis. It is no accident that all the major oceanic productive regions occur where upwellings are found. The middle of the oceans tend to be biological deserts because those regions tend to have convergence zones. So, in which direction does a gyre have to rotate (northern hemisphere) whereby a convergence zone forms? A divergence zone forms? Look at the accompanying figure. Does the above discussion support your determinations? Is there an downwelling or upwelling in the center of the gyre found in the Gulf of Alaska? Why or why not? Think back again in terms of curl, conservative versus non-conservative vector fields. Gyres rotate. Because they rotate, divergence or convergence zones are created which creates changes in the overall volume of water that is then maintained via addition or removal of water. Hence why there are upwellings and downwellings - these compensate for the changing volumes. This should further reinforce your findings from the earlier consideration of the curl and whether or not they are a conservative vector field. Back to divergence. The Divergence Theorem is a powerful analytical tool that can be used in conjunction with such applications as discussed above. If div F (P) > 0, then the net flow is outward and P is referred to as a source. Water flows out from the center at the surface in a divergence zone, so the center where the upwelling is taking place from is referred to as a source (water flows into the center). If div F (P) < 0, then the net flow is inward and P is referred to as a sink (section 16.9). Water flows into a convergence zone at the surface, so the center where the downwelling is taking place is referred to as a sink (the water sinks in the middle and water leaves). Oceanographers can calculate the rate of divergence (convergence is negative divergence), calculate the volume of water being moved and can then calculate, for example, (based on concentrations) how much of a nutrient is being transported in or out of a region. On another note - one can also apply the concepts from sections 13.3 and 13.4 and calculate the curvature as well as the normal and tangential components of the acceleration function for gyres (velocity and acceleration would be based on angular momentum). Simply take the appropriate derivatives of Navier-Stokes and the position function provided for you and apply the dot or cross product as needed. Please include a summary that includes determining the curl and divergence of the Navier-Stokes equation as well as answering ALL the questions raised in part II. I hope you found this discussion and project interesting.

Explanation / Answer

Solving rf = v for f given v Looking back: Recall how antiderivatives were introduced in one variable calculus: to ?nd F with a given derivative f, you learned: 1. For f given by a simple formula, guess and check led to an antiderivative F. 2. Any other anti derivative G had the property G = F + C. 3. When all else fails, any continuous f has anti-derivative F(x) = R x a f(t) dt for any ?xed a. Earlier in this chapter the corresponding result for a single partial derivative was found to be similar, except that instead of adding constants to get new solutions from old ones, functions independent of the single variable could be added. The gradient vector has partial derivatives in components, so we might expect that this result would allow us to solve the system of equations rG = v. This consists of n simultaneous equations in dimension n. This is partially correct, but a new twist emerges: not every vector ?eld v is a gradient ?eld! Mixed partial derivatives must be equal, so if G solves both equations which is not true! Therefore no G exists! In general, the equality of mixed partial derivatives in 2D means that if rG = (v1; v2) is continuously differentiable, For the general linear homogeneous vector ?eld this becomes the following condition, interpreted as describing the range of the gradient operator on purely quadratic functions: Supposing v1 = Ax+By; v2 = Cx+Dy is symmetric! Sound familiar? We already know for purely quadratic functions the gradient is linked to the Hessian, which is symmetric

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