...
The instability due to high rotation rates is known as "baroclinic instability", and gives rise to this eddy pattern observed. To balance solar radiation, this eddy heat transport, which dominates, must be on the order of 5x1015W, which we will attempt to verify with atmospheric data. The equations governing this calculate eddy heat transport as the difference between the mean and the actual heat transport - the remainder. This is defined in Equation 2 as,
| LaTeX Math Block |
|---|
|
$\overline{v'T'} = \overline{vT\; } - \overline{v}\overline{T\;}$
|
where the overbar indicate a monthly time average,
is the total northward heat flux, and | Mathinline |
|---|
| body | $\overline{v}\overline{T\;}$ |
|---|
|
is the monthly mean heat transport. Using these we may define the zonal average transient heat flux (zonal heat flux due to eddies) to be| LaTeX Math Block |
|---|
|
$[\overline{v'T'] = \frac{1}{x_2 - x_1} \int_{x_1}^{x_2} v'T' dx$
%%should it be v't'? |
where
| Mathinline |
|---|
| body | $x_2 - x_1 = 2\pi a cos(\phi)$ |
|---|
|
, where is latitude and is radius. Therefore heat flux as a function of latitude may be defined as a pressure integral between two levels of the zonal average, as shown in Equation 4,| LaTeX Math Block |
|---|
|
$\mathcal{H} = 2\pi a cos(\phi) \frac(c_p)(g) \int_{0}^{p_s} [\overline{v'T'\;] dp |
The equation in the actual yields the curve as a function of latitude displayed in Figure 2.3, compared to the actual atmospheric and oceanic components.
Image Added
Figure 2.3: Heat flux as a function of latitude, broken up into atmospheric and oceanic components.
As shown in project 2, we expect at the frontal boundary between mid latitude warm air transport and cold polar air, an increase in zonal wind speed with height known as the 'jet stream', that serves both to place a northern bound on efficient eddy transport and maintain the frontal separation. As such we may show in cross section the main features of the global circulation, in Figure 2.4
Image Added
Figure 2.4: A cross-sectional view of the major components of the Global circulation pattern. Note the net radiative influx at the equator and outflux at the poles, balanced by the heat transport in between.
We will use a rotating tank as an an analog to this setup, with low rotation rate mimicking the processes at play in the tropics with low coriolis force, and high rotation rate mimicking the high latitudes. Ice at the center provides the heating imbalance between 'higher' latitudes and 'lower' latitudes. In theory, at high rotation rates, flow will break down and become turbulent. The setup is shown below in Figure 2.5.
Image Added
Figure 2.5: The tank setup, showing the tank and ice in cross section across the diameter, with rotation rate which is varied.
In theory, the flow vs height relation for an incompressible fluid with coefficient of thermal expansion
should be,
| LaTeX Math Block |
|---|
|
$2\Omega \frac{du}{dz} = \alpha g \frac{\partial T}{\partial r}$, |
where
is the radius, and can be approximated by | Mathinline |
|---|
| body | $\frac{\Delta u}{h} = \frac{u_{top} - u_{bottom}}{h}$ |
|---|
|
, where is the height in the tank. The heat flux, for melting ice, is defined by,| LaTeX Math Block |
|---|
|
$L\frac{dm}{dt} \approx L \frac{\Delta m}{\Delta t} = \rho c_p \int \oint \overline{v'T'\;} dz$, |
where
is the mass and is the latent heat of fusion for water. Thus the closed loop integral is analogous to a zonal average in latitude, across pressure levels, dz. The ice heat sink must be balanced by inward radial heat flow, a task accomplished by laminar flow in the low rotation scenario, and eddies in the high rotation case. For the eddies, the Rossby Number may be estimated as a ratio of timescales, or velocity scales,
| LaTeX Math Block |
|---|
|
$R_o = \frac{u}{2\pi L}$,
|
where
is the length scale of eddy rotation, in meters. Thus we in fact may draw analogs between the tank and the atmosphere, verify thermal wind according to the relation in Equations 1 and 5, and obtain a more complete understanding of wind patterns in the atmosphere.
Laminar Flow in Tank: Analysis and Results
...