Homework SourseIn this project you will be examining rotational systems suc
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: Numerical 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.
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: Numerical 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.
Solution
a)The pressure gradient force acts at right angles to isobars in the direction from high to low pressure. The greater the pressure difference over a given horizontal distance, the greater the force and hence the stronger the wind. The central pressure determines the maximum pressure gradient, which must increase inwards to balance the increasing centrifugal force. b)In atmospheric sciences (meteorology, climatology and related fields), the pressure gradient (typically of air, more generally of any fluid) is a physical quantity that describes in which direction and at what rate the pressure changes the most rapidly around a particular location. The pressure gradient is a dimensional quantity expressed in units of pressure per unit length. The SI unit is pascal per metre (Pa/m). c)Geostrophic winds exist in locations where there are no frictional forces and the isobars are striaght. However, such locations are quite rare. Isobars are almost always curved and are very rarely evenly spaced. This changes the geostrophic winds so that they are no longer geostrophic but are instead in gradient wind balance. They still blow parallel to the isobars, but are no longer balanced by only the pressure gradient and Coriolis forces, and do not have the same velocity as geostrophic winds. In the diagram below at point A, the parcel of air will move straight north. The pressure gradient and Coriolis forces are present, but when the isobars are curved, there is a third force -- the centrifugal force. This apparent force, pushes objects away from the center of a circle. The centrifugal force alters the original two-force balance and creates the non-geostrophic gradient wind. In this case, the centrifugal force acts in the same direction as the Coriolis force. As the parcel moves north, it moves slightly away from the center -- decreases the centrifugal force. The pressure gradient force becomes slightly more dominant and the parcel moves back to the original radius. This allows the gradient wind to blow parallel to the isobars. d) The Buffalo Bills lost to the New England Patriots on December 28, 2008. But both lost to Weather. The score was Weather 75 - New England 13 - Buffalo 0. The 75 was the reported peak wind gust at Buffalo Airport, a scant 9 miles north-northeast of Ralph Wilson Stadium, the Bills home field. While strong winds are not unusual in Buffalo, especially in the winter, wind gusts to 75 miles per hour (mph) are unusual almost everywhere (except perhaps Mount Washington, NH and the eastern slopes of the Colorado Rockies). The Mount Washington Observatory (elevation 6,288 feet) still retains the record for the highest reported and recorded wind gust - 231 mph on April 12, 1934. On December 28, 2008, the culprit was a rapidly intensifying and quickly moving surface low-pressure system (which the day before had contributed to more than 130 thunderstorm-related severe weather reports, mostly high winds) across the middle Mississippi River Valley. The storm system also contained a very strong pressure gradient (or large change in pressure across a relatively small distance). This type of pressure pattern, in which isobars or lines of constant pressure are wrapped tightly around a high- or low-pressure system, often brings strong and gusty winds (see Figure 1 - weather map; Figure 2 - station plotting model code). Another contributing factor was the transport of high velocity air from aloft to the ground. On the early morning of December 28, winds were reported at 75 mph at 10,000 feet above the ground in the Buffalo area. As colder, more dense air moved in aloft, it sank, bringing its wind motion with it. Meteorologists refer to this as \"momentum transfer.\" As a result, the intense pressure gradient on the southwest side of the low-pressure system was enhanced further. And the high-speed winds were able to accelerate the front toward and past Buffalo. Based on reports from the National Weather Service, the Buffalo area and surrounding counties suffered widespread power outages. In addition, several jetways were blown into the terminal building at the Buffalo International Airport. There was even damage to the roof of the Buffalo Bills field house. As the winds drove water to the eastern shore of Lake Erie (known as a seiche), water levels at the Buffalo Coast Guard Station rose slightly above flood stage. This is very similar to a hurricane\'s storm surge, except it occurs on a lake rather than a coastline. But, Buffalo was not alone in receiving strong winds. Detroit and other areas in southeast Michigan and also parts northeast Ohio were affected as well. Trees and tree limbs knocked down power lines. More than 400,000 Michigan residents lost power during the storm event. It took several days to restore power to the region. The storm system left almost as quickly as it came. But a series of \"Alberta Clippers\" (fast-moving low pressure systems that originate in Alberta, British Columbia) were expected to follow. One of these brought 100 mph winds to the Boulder, CO area on December 30th. And in the aftermath of yet another, New Year\'s Eve revelers from Washington, DC northeastward shivered in near zero (or colder) wind chills. So, when windy winter weather is expected in your area, check out the weather map. Or even better, check out the weather map and predict the onset of the high winds (it might impress your friends or spouse). Look for those close together isobars and compare their spacing to the forecast and observed winds. You may just discover that you can anticipate the weather.
