From: linden@positive.Eng.Sun.COM (Peter van der Linden) Subject: The pure truth on the Coriolis Farce (sic). Date: 7 Dec 1992 20:19:13 GMT Here, specially for that thrusting young Antipodean Wayne, is the straight dope on why the Coriolis force doesn't bend the water flowing out of a tub. --------- reposted from 1 year, 4 months ago --------------- >>>>> On 30 Aug 91 04:34:40 GMT, moroney@ramblr.enet.dec.com said: mike> The story that water in a bathtub spirals in a certain way due mike> to the coriolis force came up again. Can anyone provide me with mike> a formula that, given the diameter of a bucket and the diameter mike> of a small hole in the center of its bottom, and it's filled mike> with water to a given depth, and the latitude of the bucket (on mike> Earth) and any other variables necessary, what the observed mike> rotation rate of the water will be as it drains? Can this rate mike> ever exceed one revolution per day? After all, Earth rotate mike> only once per (sidereal) day. It is possible to make an experimental apparatus that actually measures the influence of the Coriolis force, but it is not easy! The governing equations for a homogeneous, incompressible inviscid fluid are the Euler equations. When you add the vertical component of the Coriolis force, you get: du du du dh -- + u -- + v -- - f*v = -g -- dt dx dy dx dv dv dv dh -- + u -- + v -- + f*u = -g -- dt dx dy dy dh dh dh du dv -- + u -- + v -- + h -- + h -- = 0 dt dx dy dx dy Where u and v are the two horizontal components of the velocity and h is the thickness of the fluid. 'f' is 2*Omega*sin(Phi), with Omega being 2 Pi/(1 day) and Phi being the latitude. When you assume that the velocity scales like U, and the horizontal length scale like L, then the ratio of the nonlinear terms to the Coriolis terms is U ---- f L For a bathtub, we have U=O(0.1 m/s), L=O(0.1 m), and at mid-latitudes we have f=O(.0001/s). So the ratio is O(10,000), meaning that the nonlinear terms are 4 orders of magnitude bigger than the Coriolis terms. So for a quasi-steady swirling flow, the dominant balance is going to be the nonlinear (centrifugal) terms against the pressure gradient. The Coriolis force will be utterly negligible.... An alternate scaling contrasts the size of the Coriolis term with the size of the acceleration term. The ratio of du/dt over fv is (1/(Tf)), where T is the time scale of the flow. In order for the Coriolis terms to be O(1), the time scale would have to be of the order of (1/f) or 10000 seconds (3 hours). Most of us don't put up with bathtub drains that slow! -- John D. McCalpin mccalpin@perelandra.cms.udel.edu Assistant Professor mccalpin@brahms.udel.edu College of Marine Studies, U. Del. J.MCCALPIN/OMNET From exodus!cronkite.Central.Sun.COM!sun-barr!cs.utexas.edu!swrinde!zaphod.mps.ohio-state.edu!cis.ohio-state.edu!rutgers!zodiac!tiscione Sun Sep 1 15:01:27 PDT 1991 In article

, mccalpin@perelandra.cms.udel.edu (John D. McCalpin) writes: > [Impressive quantitative analysis deleted] > For a bathtub, we have U=O(0.1 m/s), L=O(0.1 m), and at mid-latitudes > we have f=O(.0001/s). So the ratio is O(10,000), meaning that the > nonlinear terms are 4 orders of magnitude bigger than the Coriolis > terms. So for a quasi-steady swirling flow, the dominant balance is > going to be the nonlinear (centrifugal) terms against the pressure > gradient. The Coriolis force will be utterly negligible.... This reminds me of a letter to the editor of Science News, when there was a real furor over how the Coriolis force interacts with this "sensitive dependence on initial conditions" water drainage thing. Some guy who was a tourist in Africa wrote that he was at the equator. One of the native folk had a pan full of water, and some leaves floating on the top. He would hold the pan in the air WITH HIS HANDS and let the water drain out the bottom. When he did this twenty feet from the equator, the leaves swirled clockwise, and twenty feet on the other side, it swirled counterclockwise, much to the astonishment of the tourists. When he stood directly on the equator, the leaves and water did not swirl around but just flowed radially into the hole. Someone who responded to this letter made the calculation that unless the guy held it to within one millionth of an arc second to the horizontal, the Coriolis force could not possibly be responsible for the direction of the drainage. He speculated that he just imperceptibly twisted the pan one way or the other, and let the water do the rest. (Anything for the tourists.) -- * THIS SPACE FOR RENT!!! * * Imagine YOUR ad reaching THOUSANDS of USENET readers! * * For details contact Jason Tiscione, tiscione@zodiac.rutgers.edu * From exodus!cronkite.Central.Sun.COM!sun-barr!cs.utexas.edu!uunet!munnari.oz.au!bruce!monu0.cc.monash.edu.au!monu1.cc.monash.edu.au!map Tue Sep 3 12:34:26 PDT 1991 In article <1991Sep2.210401.23439@athena.mit.edu>, aeosawa@athena.mit.edu (Atsushi E Osawa) writes: > The Coriolis effect can be observed in a bath-tub vortex if you wait long > enough for the residual momentum of the filling action to die down. In > an article in Nature (196:1080-1081, Dec. 15, 1962), Ascher Shapiro reports > that a counter-clockwise rotation of 0.25-0.33 sec-1 was observed when > a circular tank of water (filled clockwise) was drained after allowing the > water to sit for about 24 hrs. If the tank was drained after only 1 or 2 hrs, > the vortex went clockwise. His tank was 6 ft in diameter, 6 in high, with > a 3/8 in diameter drain, for the experimentally minded of you out there. He > was at MIT, approximately 42 N latitude. > > Another fun fact from the "goofy articles" file of > > -the Edster I've been waiting to see if anyone actually answers the guy's original question and at last someone comes close. John McCalpin gave us a good outline of scale analysis (see Pedlosky's Geophysical Fluid Dynamics for more mathematical detail) but still no formula. So here goes. If we assume that the fluid is inviscid (dubious on long time time scales, but the order of magnitude should be correct) then it can be shown that the vorticity zeta=du/dy-dv/dx of the flow satisfies D( (zeta+f)/H )/Dt = 0 where f = 2*Omega*sin(theta) in terms of the angular velocity Omega = 2*pi/day of the earth and latitude theta, and where H is the depth of the fluid. For a fluid initially at rest in a bath of depth H0 we have that (zeta+f)/H = f/H0 for every fluid particle in the bath. If we drain a bath of area roughly 1 m^2 through a hole of area (0.05)^2 m^2 = 2.5E-3 m^2 then we are effectively rearranging this volume into a tube of height H0/2.5E-3 = 400*H0. Since (zeta+f)/H is conserved for fluid particles (see D/Dt equation above), we conclude that the vorticity zeta of the fluid in the tube of height 400*H0 is (zeta+f)/(400*H0) = f/H0 => zeta = 399*f (call it 400*f). This corresponds to an angular velocity of 0.5*zeta = 200*f, which has a period of 1/200 of a day when theta = 30 degrees. That's about a 7.5 minute rotation period, which is indistinguishable by most of us. It is smallest near the poles, where it is half that, but gets longer as you approach the equator (infinite as theta -> 0). As has been pointed out earlier, the observed swirl is due to residual rotation from when the bath is filled, which can take a LONG time to completely die out. It's easy to rig the results, particularly in a bath with two separate outlets (for hot and cold). To make the earth's rotation noticeable you would need to make the bath larger by a factor of 500 or so, by extending the linear dimensions by 20 or so AND leaving the whole thing to settle for a long time, as they do for the experiments. Sure, there is no formula above, but it's easy to work it out from the information above (for any sized bath and plug hole). -- +------------------------------------------------------------------------------+ Michael Page, Maths Dept, Monash University, Clayton, Victoria, AUSTRALIA 3168 email: map@monu1.cc.monash.edu.au | phone: +61 3 565 4486 | FAX: +61 3 565 4403 +------------------------------------------------------------------------------+ ----------------- end of included guff ----------- Peter

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