Introduction
In all thermal systems, inequities imbalances in radiative heating lead to convective or conductive processes in fluid or solid bodies respectively in an effort to evenly distribute heat energy throughout the object. The earth is no exception, and the impetus of myriads of meteorological and oceanological phenomena that exhibit turbulence in this system can often be traced back to attempts of nature to evenly distribute the energy obtained through incident solar radiation.
Conceptual Visualization: The Earth
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Figure 1.3: At its simplest, a thematic of a Hadley cell. http://www.elgheko.us/meteo.htm
Theory: Laminar or Turbulent Fluid, and Resultant Atmospheric Effects
Given that air should rise near the equator and sink in the vicinity of the polar regions, in a non-rotating system one would expect the motion of fluid to be purely meridional, towards the poles in the upper levels, and away from the poles and directed equatorially at the surface. However, the earth is a rotating system, with the atmosphere roughly in solid-body rotation. Therefore, we must take this into consideration, and do so using the Coriolis parameter. Because the Coriolis deflection is weakest at the equator and considering that fluid at the equator orbits the globe where its radius is the greatest, its angular momentum is greater than at any other point on earth since the earth is a solid body. Therefore, by the conservation of angular momentum, we would expect that, to conserve momentum, a parcel of air from the equator attempting to move towards the pole would be deflected eastward in direction since its zonal velocity must increase. Therefore, at the upper levels, a westerly jet should form, strongest as one heads away from the equator. For the sinking air later in its poleward journey, the opposite should be true; as it returns towards the equator, its weakening horizontal velocity due to the increase in radius should lead to surface easterly forming, strongest at the equator. This gives rise to the Intertropical Convergence Zone, where both Hadley cells converge. The rising motion here, coupled with the easterly winds (and the subtropical easterly jet, which, due to surface friction, is considerably weaker than other jets) leads to the development of tropical waves, and is an integral portion of cyclone forecasting.
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//This is where somebody better than me does their equation magic...
Analogues in the Tank...
Several key differences exist between the earth and our tank experiment; after all, one is a three dimensional rotating sphere comprised of solid, liquid, and gas, complete with pressure gradients amidst the fluid as well as several means of thermal heat fluxes. Meanwhile, the tank is a two-dimensional representation, and the incompressible nature of the fluid leads to further behavioral differences.
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One distinction between the tank experiment and the earth is that while the Coriolis parameter changes at each location of the earth (due to a constantly changing angle phi, or the angle equivalent to the earth's latitude), whereas it remains constant in the tank because the Coriolis parameter remains unchanged. Therefore, in order to simulate the change in regime from a Hadley cell-dominated means of heat transport to one tyrannized by rotating eddies, the rotation rate of the tank must be manually changed. At some threshold of rotation rate, the fluid will transition from being laminar in solid-body rotation to turbulent, resulting in changes in the means of heat transport.
Theory
We can understand the general circulation of the atmosphere by formulating a general theory of the Hadley Cell and its development and progression. Due to the uneven heating of the earth, the equator receives significantly more solar radiation and thermal energy than do higher latitudes. As a result, rising motion and convection occur in this the region known as the intertropical convergence zone (ITCZ). As a result of this rising motion, at the surface air moves equator-ward to replace the air displaced by convection, and air moves poleward aloft before sinking again at about 30o latitude. Thus the formation of the atmospheric "Hadley Cell". Conservation of angular momentum with this northward and southward motion dictates the formation of westerly and easterly winds, and above 30o latitude, a different regime sets in, in which northward flow combined with the coriolis effect and the thermal gradient with the cold polar regions leads to a more inconsistent regime in which eddies play a major role. The main features of the Global Circulation can be seen below in Figure 2.1.
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where
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Laminar Flow in Tank: Analysis and Results
To begin, we first executed the experiment at a low rotation rate to instigate laminar, Hadley cell heat transport. Prior to initializing tank rotation, five thermistors were installed within the tank, as is visible in Figure 1.4:
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Using the above plots, we can confirm the thermal wind relationship using the Margulis equation; though solving as displayed in figure 1.91 yields a somewhat unbalanced equation, in the absence of thermistor errors, etc., the fact that the results share the same order of magnitude is enough to lend confidence that the thermal wind balance applies.
Figure 1.91:
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).
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//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:
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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:
