...
Hadley Cells in the Atmosphere
Figure 1.92:
(Please do note that the Transient Heat Flux in Petawatts should not feature a 10^-21 factor; that should be omitted).
Using NCDC's and NCEP's data, the Transient Energy Flux across different levels of the atmosphere in January can be plotted with relative ease. As previously mentioned, the Coriolis parameter of the earth only supports a regime of Hadley cell occurrence between 0 and 30º on either side of the equator. As such, the remainder of the globe is dominated by eddy heat transport, which will be detailed later. However, in plotting the heat flux across different levels of the atmosphere (where positive flux describes heat carried northwards), one may confirm the existence of Hadley cells is this region. This is shown in figure 1.92.
Because January features winter in the northern hemisphere, a stronger temperature gradient leads to a more significant poleward heat transport than in the southern hemisphere. This is evident in our findings, given that the peak at northern latitudes in flux is far greater than in southern latitudes. The Hadley cell theory is also supported by the weak positive flux at the surface (1000 millibars) between 0º and 30º north. This is because there is a weak easterly with a slightly northerly component as air returns radially towards the equator; theoretically, there could be negative flux, but some "mixing" of northerly winds likely aids in some weak northerly flow near the surface. The majority of the heat transport occurs aloft, but the heat flux which measures the amount of heat transport at various levels, should be directly proportional to height (for obvious reasons described previously under the theory of Hadley cells) and pressure (as denser air can carry more heat per unit area). Therefore, the location of the "maximum" should be the "happy medium" between height and pressure, which we would estimate to be located around 700-850 millibars, as revealed by the graph. This is coincident with the jet stream, which is essentially the a narrow band of intense westerlies. In addition, the rising air characteristic of the Intertropical Convergence Zone at the equator is tempered by a weak high pressure dome aloft, which can be seen in the extremely weak flux towards the equator at the uppermost levels of the atmosphere. In addition, it was previously mentioned that friction with the surface would reduce the flux close to the ground, which is supported by the lower level of flux at the 1000 millibar level.
// Magical Theory and Stuff
//Verification of Thermal Wind Equation
Eddy Heat Transport in the Tank
Likewise, four thermistors were placed in the tank when the rotational speed was increased and thus the Coriolis parameter augmented to a point when the regime would shift to one of eddy heat transport. Figure 2.0 illustrates the positions of said thermistors:
Figure 2.0:
It would be anticipated that the greatest temperatures would be found at thermistor 1, and the coolest in turn at thermistor 3 or 4. Figure 2.1 depicts the trace of thermistors' reports during the duration of the experiment:
Figure 2.1:
Though difficult to discern due to the technical limitations of Wiki, it becomes clear that the warmest temperatures were, in fact, recorded at thermistor 1, with the chilliest readings located at thermistor 3. This is exactly what would be expected. Unlike the gentler slope within the laminar fluid, however, the eddy transport nature of this second trial of the experiment lead to an oscillating, varying "stair step" pattern in temperature measurements. In addition, an effort was made to determine the velocity of different points using the particle tracking software; the positions of the points sampled is shown in figure 2.2:
Figure 2.2:
The most dramatic eddy sampled using the particle-tracking software, visible in part on bottom left-hand side of figure 2.2, is quite telling apropos to the "heat flux" within the tank. Despite rotation throughout the eddies towards and away from the center, the thermal gradient within the tank suggests that heat should have a net movement towards the center. A side view of the same graph focused on the most intense eddy illustrates the "winds" embedded within this rotating swirl of fluid; positive velocities are towards the center, with negative velocities away from the center. At first a chaotic scatter of points demarcating velocities at given points (with the x and y axes marking position, with the z axis reserved for velocity), a blue parallelogram has been drawn into Figure 2.3, along with a yellow line illustrating the zero level of velocity from the graph's perspective. One can clearly note that the area of the parallelogram containing data points above the zero line is far greater than below the line, indicating a net movement towards, and thus a heat flux in the direction of, the center. This is exactly what is expected in the atmosphere, and what our group observed in the tank.
Figure 2.3:
...
To get a better picture of what the Hadley cell looks like and is doing, we will do some investigation. We know that there is a net north-south heat flux, so we'll look at meridional i.e. north-south winds.
This data is taken for the month of January and is averaged from the years 1948-2016. The positive winds are northward, and the more purple colors are southward. We see that at high altitudes, there is poleward winds, and at low altitudes there are winds heading to the equator. We see air being pushed to the pole up high and returning to the equator at the surface, so we're missing the part that's pushing the air to the surface at higher latitudes and the convection plume at the equator.
The cooler colors are rising columns, and the warmer colors are sinking columns. While there are adjacent rising and sinking columns spanning the whole surface of the Earth, the strongest columns occur between the equator and 30N. The strongest columns appear in the North Hemisphere because it is wintertime there. When it is winter in the southern hemisphere, the strongest columns will be between the equator and 30S. The equatorial column corresponds to convection, and the 30N column corresponds to the point on the Earth at which the warm air has lost sufficient heat and sinks back to the surface. Putting the two pictures together, we can see the full cycle of the Hadley cell.
Eddy Heat Transport in the Tank
Likewise, four thermistors were placed in the tank when the rotational speed was increased and thus the Coriolis parameter augmented to a point when the regime would shift to one of eddy heat transport. Figure 2.0 illustrates the positions of said thermistors:
Figure 2.0:
It would be anticipated that the greatest temperatures would be found at thermistor 1, and the coolest in turn at thermistor 3 or 4. Figure 2.1 depicts the trace of thermistors' reports during the duration of the experiment:
Figure 2.1:
Though difficult to discern due to the technical limitations of Wiki, it becomes clear that the warmest temperatures were, in fact, recorded at thermistor 1, with the chilliest readings located at thermistor 3. This is exactly what would be expected. Unlike the gentler slope within the laminar fluid, however, the eddy transport nature of this second trial of the experiment lead to an oscillating, varying "stair step" pattern in temperature measurements. In addition, an effort was made to determine the velocity of different points using the particle tracking software; the positions of the points sampled is shown in figure 2.2:
Figure 2.2:
The most dramatic eddy sampled using the particle-tracking software, visible in part on bottom left-hand side of figure 2.2, is quite telling apropos to the "heat flux" within the tank. Despite rotation throughout the eddies towards and away from the center, the thermal gradient within the tank suggests that heat should have a net movement towards the center. A side view of the same graph focused on the most intense eddy illustrates the "winds" embedded within this rotating swirl of fluid; positive velocities are towards the center, with negative velocities away from the center. At first a chaotic scatter of points demarcating velocities at given points (with the x and y axes marking position, with the z axis reserved for velocity), a blue parallelogram has been drawn into Figure 2.3, along with a yellow line illustrating the zero level of velocity from the graph's perspective. One can clearly note that the area of the parallelogram containing data points above the zero line is far greater than below the line, indicating a net movement towards, and thus a heat flux in the direction of, the center. This is exactly what is expected in the atmosphere, and what our group observed in the tank.
Figure 2.3:
In addition, our group was able to note at least half a dozen eddies embedded within our fluid visually using food coloring and dye; the red was placed on the outside of the tank, with blue inside the tank. As mixing occurred, the dyes were dragged along with the rotating fluid, and as such the patterns of warm and cold water were able to be shown. Figure 2.4 depicts these illustrative markings:
Figure 2.4:
Thinking back to Project 2, an investigation into the thermal wind balance and formation of the jet stream, one could in essence convert the motion of particles throughout the fluid into a vector field; in doing so, there would exist one continuous path that completely encircles the central cold-dome within the tank. This would indicate the jet stream, with the serpentine pattern waving towards and away from the center and ensnared within eddies and vortices rotating around in the larger body of fluid. One may note that the blue and red eddies rotate in opposite directions, much as is the case within the atmosphere due to regimes of higher and lower pressure.
One item of note in figure 2.1 is that the temperature at each sensor, though subject to slight oscillations due to eddies, slowly decrease with time somewhat uniformly throughout the fluid. This is to be expected, as the thermal energy contained within the relatively warmer is transferred and over time reduced as it in turn warms the ice to above melting point. Thus, though the temperature of the ice/water solution remains at 32 degrees, the increased thermal energy is instead utilized in the form of latent heat, responsible for the change in phase of the liquid. As such, the amount of energy needed to fully melt the block of ice placed in the center should be equal to product of the ice's mass and the specific heat of fusion. Therefore, considering that 771.4 grams of ice were used in the experiment, one would anticipate that roughly 258,000 Joules of energy would be necessary complete this thermal transaction. Spread over a period of approximately 4,000 seconds, the result is in effect the energy required to power a 64-Watt light bulb, and may be practically thought of as a "negative light bulb" placed in the center of the tank according to Dr. John Marshall. An example of a 65-Watt bulb is depicted in figure 2.5.
Figure 2.5:
Eddy Heat Transport in the Atmosphere
The above graphs shows a vertically averaged plot of transient heat flux on Earth. Yellow/blue corresponds to north/southward flow. Overall, there is a pattern of poleward transient heat blux, and everywhere else, it's zero. This comes from transient heat flux data that was collected from 1948-2016.
If we average over longitude, we can get a purely latitude dependent graph, which shows how the mid-latitudes are the only regions of transient heat flux.
Here is a quantitative graph of transient heat flux at different latitudes. Notice that the zonally averaged heat flux map shows that the maximum transient heat flux is ~6 PW, and this graph only shows 1-1.5 PW. The data set filtered out weather systems that were shorter than two weeks, so that could be it. Interesting!
Figure 1.92:
(Please do note that the Transient Heat Flux in Petawatts should not feature a 10^-21 factor; that should be omitted).
Using NCDC's and NCEP's data, the Transient Energy Flux across different levels of the atmosphere in January can be plotted with relative ease. As previously mentioned, the Coriolis parameter of the earth only supports a regime of Hadley cell occurrence between 0 and 30º on either side of the equator. As such, the remainder of the globe is dominated by eddy heat transport, which will be detailed later. However, in plotting the heat flux across different levels of the atmosphere (where positive flux describes heat carried northwards), one may confirm the existence of Hadley cells is this region. This is shown in figure 1.92.
Because January features winter in the northern hemisphere, a stronger temperature gradient leads to a more significant poleward heat transport than in the southern hemisphere. This is evident in our findings, given that the peak at northern latitudes in flux is far greater than in southern latitudes. The Hadley cell theory is also supported by the weak positive flux at the surface (1000 millibars) between 0º and 30º north. This is because there is a weak easterly with a slightly northerly component as air returns radially towards the equator; theoretically, there could be negative flux, but some "mixing" of northerly winds likely aids in some weak northerly flow near the surface. The majority of the heat transport occurs aloft, but the heat flux which measures the amount of heat transport at various levels, should be directly proportional to height (for obvious reasons described previously under the theory of Hadley cells) and pressure (as denser air can carry more heat per unit area). Therefore, the location of the "maximum" should be the "happy medium" between height and pressure, which we would estimate to be located around 700-850 millibars, as revealed by the graph. This is coincident with the jet stream, which is essentially the a narrow band of intense westerlies. In addition, the rising air characteristic of the Intertropical Convergence Zone at the equator is tempered by a weak high pressure dome aloft, which can be seen in the extremely weak flux towards the equator at the uppermost levels of the atmosphere. In addition, it was previously mentioned that friction with the surface would reduce the flux close to the ground, which is supported by the lower level of flux at the 1000 millibar level.
// Magical Theory and Stuff
//Verification of Thermal Wind Equation
...
Figure 2.4:
Thinking back to Project 2, an investigation into the thermal wind balance and formation of the jet stream, one could in essence convert the motion of particles throughout the fluid into a vector field; in doing so, there would exist one continuous path that completely encircles the central cold-dome within the tank. This would indicate the jet stream, with the serpentine pattern waving towards and away from the center and ensnared within eddies and vortices rotating around in the larger body of fluid. One may note that the blue and red eddies rotate in opposite directions, much as is the case within the atmosphere due to regimes of higher and lower pressure.
One item of note in figure 2.1 is that the temperature at each sensor, though subject to slight oscillations due to eddies, slowly decrease with time somewhat uniformly throughout the fluid. This is to be expected, as the thermal energy contained within the relatively warmer is transferred and over time reduced as it in turn warms the ice to above melting point. Thus, though the temperature of the ice/water solution remains at 32 degrees, the increased thermal energy is instead utilized in the form of latent heat, responsible for the change in phase of the liquid. As such, the amount of energy needed to fully melt the block of ice placed in the center should be equal to product of the ice's mass and the specific heat of fusion. Therefore, considering that 771.4 grams of ice were used in the experiment, one would anticipate that roughly 258,000 Joules of energy would be necessary complete this thermal transaction. Spread over a period of approximately 4,000 seconds, the result is in effect the energy required to power a 64-Watt light bulb, and may be practically thought of as a "negative light bulb" placed in the center of the tank according to Dr. John Marshall. An example of a 65-Watt bulb is depicted in figure 2.5.
Figure 2.5:
Eddy Heat Transport in the Atmosphere
The above graphs shows a vertically averaged plot of transient heat flux on Earth. Yellow/blue corresponds to north/southward flow. Overall, there is a pattern of poleward transient heat blux, and everywhere else, it's zero.
If we average over longitude, we can get a purely latitude dependent graph, which shows how the mid-latitudes are the only regions of transient heat flux.