FIGURE 6.15. Theoretical trajectory of the puck during one complete rotation period of the table, 2n/Q, in GFD Lab V in the inertial frame (straight line) and in the rotating frame (circle). We launch the puck from the origin of our coordinate system x(0) = 0; y(0) = 0 (chosen to be the center of the rotating parabola) with speed u(0) = 0; v(0) = vo. The horizontal axes are in units of vo/2Q. Observed trajectories are shown in Fig. 6.14.

uin (t) = 0; vin (t) = vo cos Qt Xin (t) = 0; yin (t) = Q sin Qt.

The trajectory in the inertial frame is a straight line shown in Fig. 6.15. Comparing Eqs. 6-35 and 6-38, we see that the length of the line marked out in the inertial frame is twice the diameter of the inertial circle in the rotating frame and the frequency of the oscillation is one half that observed in the rotating frame, just as observed in Fig. 6.14.

The above solutions go a long way to explaining what is observed in GFD Lab V and expose many of the curiosities of rotating versus nonrotating frames of reference. The deflection ''to the right'' by the Coriolis force is indeed a consequence of the rotation

"Note that if there are no frictional forces between the puck and the parabolic surface, then the rotation of the surface is of no consequence to the trajectory of the puck. The puck just oscillates back and forth according to:

where we have used the result that h, the shape of the parabolic surface, is given by Eq. 6-33. This is another (perhaps more physical) way of arriving at Eq. 6-37.

Mid Periphery Retina
FIGURE 6.16. Inertial circles observed by a current meter in the main thermocline of the Atlantic Ocean at a depth of 500 m; 28°N, 54°W. Five inertial periods are shown. The inertial period at this latitude is 25.6 h and 5 inertial periods are shown. Courtesy of Carl Wunsch, MIT.

of the frame of reference: the trajectory in the inertial frame is a straight line!

Before going on we note in passing that the theory of inertial circles discussed here is the same as that of the Foucault pendulum, named after the French experimentalist who in 1851 demonstrated the rotation of the Earth by observing the deflection of a giant pendulum swinging inside the Pantheon in Paris.

Observations of inertial circles Inertial circles are not just a quirk of this idealized laboratory experiment. They are a common feature of oceanic flows. For example, Fig. 6.16 shows inertial motions observed by a current meter deployed in the main thermocline of the ocean at a depth of 500 m. The period of the oscillations is:

Inertial Period =

where p is latitude; the sin p factor (not present in the theory developed here) is a geometrical effect due to the sphericity of the Earth, as we now go on to discuss. At the latitude of the mooring, 28° N, the period of the inertial circles is 25.6 hr.

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