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Fluid Dynamics and the Navier-Stokes Equations


High and Low Pressure Cells

When cool air mass roams over a warm surface of the Earth, it sinks downward as shown in Figure 10. There is a temporary build-up of air at the central core before it can flow away. The congestion increases the air density and results in a
High Pressure Ridge Low Pressure Cell relatively high, central-pressure zone. As the air diverges from the central region, it is deflected by the Coriolis force in a clockwise circulation (Figures 10 & 12). Thus, most Highs are generally elliptical in shape following their formation. But as they interact with other air masses and topography, and are distorted by forces of the upper atmosphere, high pressure cells often become long and

Figure 10 High Pressure Ridge

Figure 11 Low Pressure Cell
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narrow in shape, and is referred to as high pressure ridge in the weather map. Since the air at high altitude is dry, the High is usually associated with fair weather.
At the hot spots on the Earth's surface, warm air rises up triggering surface air to rush in toward the core (Figure 11). The Coriolis force now deflects the converging air in a counter clockwise circulation. Thus, a Low will develop when there is not enough infalling air to replace the rising air at the center. The rising air eventually dissipates at high altitude as high level wind or returning to the surface in cyclic motion. When the circular region of low pressure elongates to a long and narrow band, it is referred to as a low pressure trough. Since the warm air contains lot of moisture, Lows are usually associated the stormy weather as the vapor condensed at upper level. Low pressure cells that travel long distances across the Earth are called cyclones. In extreme cases over warm tropical waters, they become hurricanes or typhoons.

Coriolis Force Mathematically, the sinking and rising air can be explained by Archimedes' principle as discussed in the section of Hydrostatics. Cooler air mass will sink as it is denser than the surrounding air, and vice versa for the warmer air mass. The swirling motion of air on the horizontal plane is determined by the Navier-Stokes Equations in Eq.(2). Since the radial velocity ur is usually much smaller than the circular velocity v = r u in the core region, the radial component of the equation (in cylindrical coordinates) can be reduced to:

Figure 12 Coriolis Force
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2 v sin = (1/)p/r ---------- (14a)

where is the angular velocity of the Earth's rotation, and is the angle of the latitude (Figure 12). Eq.(14a) shows that for the low pressure cell, the Coriolis force on the left-hand side is balanced by the pressure gradient force on the right-hand side, and the air circulates in the counter-clock wise direction as depicted in Figure 11. When the radial velocity diverges from the center as for the case of high pressure cell, both forces change sign leaving the equation in exactly the same form with the circular velocity moving in clock wise direction.

If we further assume that the pressure p does not depend on , and v does not depend on z, then the circular component of Eq.(2) (in cylindrical coordinates) can be simplified to:

(v/r) (v/) = - 2 ur sin ---------- (14b)

which shows that another component of the Coriolis force bumps up the circular velocity in a counter-clock wise direction if the air is moving inward, but in a clock wise direction when the radial flow is reversed. Since = 0 at the equator, the effect of the Coriolis force vanishes there as shown in both Eqs.(14a) and (14b).

Red Spot The Great Red Spot (Figure 13) in Jupiter provides a very good example to illustrate the Coriolis force at work. It is a high pressure cell located 22o South of the equator. Thus, the rotational vector is pointing inward to the center (instead of pointing outward as for the case in the Northern hemisphere), and the swirling gas is circulating counter-clock wise. This system of anticyclonic storm has existed for up to 400 years. The long lifetime cannot be attributed entirely to the higher rotational speed (about twice as much as that for the Earth), and hence the stronger Coriolis force. It is suggested that the lack of solid surface to provide friction may play a part contrary to the hurricane on Earth, which always break up shortly after landfall. Note that the oval shape is caused by the constriction from the neighboring cloud bands.

Figure 13 Great Red Spot
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