Overview
In order to understand the different regimes in the atmosphere, we performed two experiments on rotating tanks of water. Both experiments used the same sized tank, with a metal bucket filled with ice at the center. This setup acted as an analog to the atmosphere on our rotating earth, with the cold polar region reproduced by the ice bucket. Varying the rotation rate of the tank was akin to altering the latitudinal location on the earth. In both experiments, temperature sensors were placed radially along the bottom the the tank to track the movement of heat over the course of the experiment, as well as two 10 cm above - one taped to the ice bucket and one to the edge of the larger tank. Particles were also dropped onto the water to track surface motion, and drops of dye used to visualize flows.
Hadley
hadley hadley
Tank Experiment
The first experiment was performed with a very slowly rotating tank - moving at just 1 revolution per minute in order to replicate the low latitudes (30ºN and 30ºS).
The positions of the particles at the surface could be monitored using a particle tracking software. Data showed that they followed circular paths around the center of the tank. Azimuthal velocities of the particles were calculated to be around 2 cm/s.
The temperature data from the thermometers revealed that an overall trend of decreasing temperature in the tank, as a result of the melting of the ice. The sensors at the bottom of the tank maintained a near-constant radial temperature over time, of about 4ºC. The temperatures higher up in the tank evolved differently however. The sensor on the edge of the tank measured a high temperature of about 21.2 ºC throughout the experiment, and the one on the ice bucket, gave a reading higher than all but the furthest sensor from the ice bucket at the bottom. These high temperatures are a result of the overturning circulation seen in the non-rotating case as well. Cold water near the ice bucket sinks and spreads along the bottom, leaving the surface water much warmer than that below.
We recall the thermal wind equation, which describes how a horizontal temperature gradient affects vertical wind shear:
Discretizing the equation and solving for u gives
where the coriolis parameter, f, has been replaced by 2Ω, L is the radial distance between the ice bucket and the edge of the tank, and H is the height of the water.
We can use the temperature data to calculate the right hand side of the equation, and compare it to the observed surface velocities of the paper dots.
The result is close to the particle velocities found of about 2 cm/s, showing that the theory explains the observed phenomena well within margin of measurement error.
Atmospheric Data
To find T’, we first fit the temperature data with a loglog fit. This was chosen instead of a simple linear fit because when the hadley temperature data is plotted in loglog coordinates the line appears linear (see below), and the eddy data seems to follow a similar downward curve.
The eddy temperature data is plotted in loglog coordinates with the corresponding fit below (left). This fit was used as the mean temperature value, T. We then subtracted this value from the raw data, which gave the figure on the right. The loglog fit seems to be a good one, because once the mean downward trend is subtracted out, the residuals remain near zero and do not exhibit further upwards or downwards trends.
The root mean square was then taken, such that the full calculation process can be described by:
This gave a result of ~1ºC.
Eddy
While the Hadley cell is critical for heat transport at low latitudes, the atmospheric mean data reveals that meridional overturning decreases significantly at higher latitudes. However, data show that poleward energy flux peaks beyond the edge of the Hadley cell, near 40ºN, and remains high even further towards the poles (see figure below).
(Marshall & Plumb, 2008)
The mean, zonally symmetric solution is therefore insufficient to explain the continued heat transport to high latitudes. At mid-high latitudes, potential energy that is stored in the horizontal temperature gradient can be converted to kinetic energy through the poleward transport of momentum; an increased Coriolis force results in stronger winds. Baroclinic instabilities in the flow create longitudinally asymmetric motions, which are commonly referred to as eddies.
(Marshall & Plumb, 2008)
Tank Experiment
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Atmospheric Data
While atmospheric mean data could be used to study the Hadley circulation, eddies are shorter timescale phenomena and are calculated through deviations from the mean.
We were given data of v’t’ - the heat flux from the mean deviations for all latitudes, longitudes, and heights up to 100 mbar in the atmosphere. Looking at the vertical average of the deviations in January, we see particularly strong northward flux in regions 40ºN, over the Pacific and Atlantic Oceans. As the mean data revealed, the Hadley cell descends near 30º, with little There are also similar, though not nearly as strong, negative deviations around 40ºS as well. The values here are negative because they imply transport to the South Pole from the equator, and are not quite as large as those in the Northern Hemisphere due at least in part to the fact that the temperature gradient is larger in the northern hemisphere January, when it is winter there.
We can also plot the zonal average with height. The maximums at around 40ºN and S are evident here as well. An interesting feature is that for both latitudes, there exists a maximum near the surface, and one higher up in the atmosphere. The lower maximum, centered around 800 mbar, is a result of the large temperature gradient between the poles and equator near the surface. At higher altitudes, the temperature gradient lessens, but the meridional winds increase. The heat flux maximums near 200 mbars are a result of these stronger winds.
Finally, averaging both vertically and zonally gives the total northward flux at each latitude. Again, peaks occur just past 40 N and S. The peak in the Southern Hemisphere is very regular, while that in the northern hemisphere drops off less quickly at high latitudes. This lack of symmetry is likely a result of the fact that the majority of the continental land mass is located in the northern hemisphere, and the uneven surface leads to further atmospheric instabilities and eddies.
It is also useful to change the units of heat flux to Watts in order to compare our findings to the total energy flux of the earth. To do so, we vertically integrate our v’T’ zonal average and insert it into the following equation for energy flux:
where a is the radius of the earth, ɸ is the latitude, Cp is the specific heat of air, g is gravitational acceleration. For a latitude of 40ºN, the equation gives a maximum energy flux of 1.1 PW (10^15 Watts).
Comparing this figure to the graph of total energy flux in the atmosphere (the first figure in this section), we can see that it is on the same order of magnitude as the total transport of 5.5 PW at 40ºN; from our numbers, eddy transport would constitute 20% of the total at this latitude. This number seems slightly low however, considering that eddies are the predominant transport mechanism in mid-high latitudes. Our data has been filtered somewhat, to only account for the eddies that exist on timescales on the order of a week, but some eddies last only a couple days. There are also stationary eddies that remain relatively constant over several months. These are especially prominent in the Northern Hemisphere during the wintertime.
The figure below shows the total northward flux of sensible heat (not including latent heat), with breakdowns for transient eddies, stationary eddies, and mean meridional circulation. The units here are in K m/s the original units of our calculations, and winter (thin), summer (dashed) and yearly averages (thick) are plotted for each. The review here found a maximum transport for transient eddies of about 8 K m/s, twice as much as our calculations. Stationary eddies account for another 12 K m/s in the winter in the northern hemisphere, but disappear in the summertime. The mean meridional circulation, or Hadley circulation, is also plotted, and provides a maximum of over 30 K m/s near 10ºN and S. This circulation has a much smaller effect at high latitudes however, as our mean data also showed. One other significant source of energy transport is through latent heat, a topic not discussed in this project, but important to mention. Water that evaporates near the equator and is lifted and transported poleward condenses, releasing latent heat and further warming the higher latitudes.
(Peixoto & Oort, 1992)
Bibliography:
Marshall, J., & Plumb, R. A. (2008). Atmosphere, Ocean and Climate Dynamics: An Introductory Text. Burlington, MA: Elsevier Academic Press.
Peixoto, J.P., and A.H. Oort. (1992). Physics of Climate. United States: New York, NY (United States); American Institute of Physics.