Cyclostrophic Balance in Surface Gravity Waves: Essay on Coriolis Effects

Transcription

1 Jounal of Oceanogaphy, Vol. 53, pp. 311 to Shot Contibution Cyclostophic Balance in Suface Gavity Waves: Essay on Coiolis Effects KERN E. KENYON 4632 Noth Lane Del Ma, CA , U.S.A. (Received 11 July 1996; in evised fom 12 Decembe 1996; accepted 18 Decembe 1996) When gavity waves of small amplitude pogess ove the suface of deep wate, the paticle obits ae obseved to be closed cicles; theefoe the waves possess obital angula momentum. The paticles expeience a balance of two equal and opposite foces, called the cyclostophic balance: the outwad centifugal foce and an inwad pessue foce. On the Eath s suface the Coiolis foce causes a mino disuption of the nomal cyclostophic balance. Fo plane waves it is possible that the Coiolis foce can be balanced at all times, and the cyclostophic balance can be maintained as well, by a slight change in the oientation and the shape of the paticle obits. In ode to balance all thee foces simultaneously in eithe hemisphee, the shape of the obits should be oval (pehaps elliptical) in geneal, whee the shote axis of the oval is paallel to the mean suface, and the plane of the obits should be tilted to the left of the diection of wave popagation. In the nothen hemisphee the obital planes should also be tilted to the left of the vetical and in the southen hemisphee to the ight of the vetical, facing in the diection of wave popagation. Fo ocean swell the ode of magnitude of the sine of both tilt angles, as well as the eccenticity of the obits, is compaable to the atio of the Coiolis paamete to wave fequency, o one in ten thousand, which is pobably too small to be obseved. If in some paticula cicumstance (pehaps tansient conditions) the Coiolis foce is not balanced, then a Coiolis toque exists that will ty to change the diection, but not the magnitude, of the obital angula momentum of the waves. The geneal fom of the Coiolis toque is woked out in Appendix. Keywods: Suface waves, cyclostophic balance, Coiolis effects, angula momentum. 1. Intoduction When gavity waves of small amplitude pogess ove the suface of deep wate, each fluid paticle, within the depth of wave influence, is obseved to move in a cicula obit about a fixed mean position. In othe wods the waves have obital angula momentum. No matte which way suface gavity waves tavel in the ocean, the fluid paticles have this cicula obital velocity, and the Coiolis foce opeates on any velocity elative to the Eath s suface. What effect does the Coiolis foce have on the obital velocity of the paticle o on the angula momentum of the waves? An answe to this question will be given in the following bief epot. The discussion of the influence of the Coiolis foce on suface gavity waves has had a modeately long histoy (about 50 yeas) but the list of efeences on the topic is quite shot. Usell (1950) aised a puzzle, o a paadox, that may not have been solved in an entiely satisfactoy way even to this day. At finite amplitude obsevations show that the paticle obits ae not quite closed cicles: at the end of each wave peiod the paticles ae displaced slightly in the diection of wave popagation, the net esult being a slow (Stokes) dift velocity o a linea momentum of the paticles. On this small steady dift velocity the Coiolis foce must be acting, fo wave popagation ove the Eath s suface, and thee is appaently no conceivable foce to balance it. So what happens? Can the Stokes dift exist in the ocean o not? (To my knowledge measuements have not yet unambiguously evealed the pesence of the Stokes dift in the ocean.) Hasselmann (1970) used the oppotunity to come up with a theoy fo the geneation of inetial motion by suface gavity waves. In an application to a completely diffeent (much lage) space/time scale Backus (1962) made a calculation of the influence of the Eath s otation on suface wave popagation ove global distances. The computed effects he found wee too small to be detected by the available measuements. Copyight The Oceanogaphic Society of Japan. 311

2 But I am not awae of any pevious studies that deal paticulaly with the influence of the Coiolis foce on the paticle velocity of suface gavity waves duing the complete obital motion. The woking of the Coiolis foce on the obiting fluid paticles has eceived scant attention in the past. In fact, in the vast majoity of theoetical woks on suface gavity waves the Coiolis foce is neglected entiely, and the assumptions behind the omission ae usually not mentioned. A taditional appoximation used in oceanogaphy fo lage-scale motions is to neglect the component of the Coiolis foce that acts on the vetical velocity, o moe geneally to neglect the component of the Eath s otation that is locally paallel to the mean suface of the ocean (Eckat, 1960, p. 36). Howeve this appoximation cannot be made fo suface gavity waves in most cases because the magnitude of the vetical velocity equals that of the hoizontal velocity fo the ciculaly obiting fluid paticles, and the hoizontal and vetical components of the Eath s otation ae compaable at most latitudes (except on o nea the equato). Theefoe both components of the Eath s angula velocity will be taken into consideation hee. One could just as easily extend Usell s puzzle to the whole est of paticle s obit (o to the whole obit in linea waves) and not estict it to the tiny piece of the obit that does not close in finite amplitude waves. And in fact since the paticle velocity is much lage than the Stokes dift velocity, the Coiolis foce is much lage fo the paticle velocity than it is fo the Stokes velocity. Is it possible that the Coiolis foce on a fluid paticle could be balanced at all times duing the obital motion, but we need to undestand how such a balance can be achieved. This consideation is taken up in the main pat of the text. If fo some eason, such as might happen unde tansient conditions, the Coiolis foce cannot be balanced, then thee would exist a Coiolis toque that would cause a change in the obital angula momentum of the waves. Theefoe we need to undestand this pocess as well, and the details of it ae woked out in Appendix. 2. Cyclostophic Balance Take the paticle obits to be cicula and fo the moment neglect the Coiolis foce. At all times as a paticle moves aound in its obit thee is a balance of two equal but opposite foces, and this balance is called the cyclostophic balance (Kenyon, 1991). The outwad diected foce is the centifugal foce and the inwad foce is a pessue foce that is elated to the vaiable height of the wave suface and to the vetical acceleation of the paticles. The Coiolis foce will disupt this balance of foces, but the disuption will be a mino one as will be seen shotly. A balance among the thee foces can be obtained by a slight adjustment in the oientation of the obital plane and in the shape of the obit. Fist, to compae the magnitudes of the centifugal and Coiolis foces on the obital velocity fo typical ocean wave conditions. Consideing fo the moment only the vetical component of the Eath s otation, the atio of the centifugal to the Coiolis foce is centifugal Coiolis = ω f 10 4 () 1 whee f is the Coiolis paamete (twice the angula velocity of the Eath times the sine of the latitude) and ω is the wave fequency. Fo swell with peiods of ode 10 sec the atio of the two foces at mid-latitudes has the ode of magnitude of 10,000, because f itself is ode 10 4 sec 1. The highe the fequency the lage the atio (1) is. If the hoizontal component of the Eath s otation is also taken into account, the atio of foces will have the same ode of magnitude as just stated. Fom this fact alone a logical guess can be made that it would only take a vey small eaangement of the obital paametes of the fluid paticle in ode to keep the main balance of foces (the cyclostophic one) in tact and to balance the Coiolis foce at the same time. 3. Coiolis Effects Now the Coiolis foce will be bought diectly into the pictue and the assumption will be etained tempoaily that the only component of the Eath s otation that is opeating is the one locally pependicula to the mean suface (o as an equivalent idealization one could think of waves popagating at o nea the noth pole). The vetical component of otation poduces a Coiolis foce that acts exclusively on the hoizontal component of the paticle velocity. Let the wave tavel in the positive y-diection, which can have any oientation in the hoizontal plane tangent to the Eath s suface. The z-axis points up anti-paallel to gavity. Figue 1 gives the obit of a suface paticle in the vetical (xz-)plane, edge on so to speak. The view in the yzplane (not shown) would potay a cicle of adius equal to the wave amplitude, but the Coiolis effect could not be seen in such a view. Without the Coiolis foce Fig. 1 would show a vetical line of length equal to twice the wave amplitude, which is the pojection of the cicula obit onto this plane. In ode that the Coiolis foce can be balanced thee must be a slight tilt of the vetical line to the left in the nothen hemisphee, whee the Coiolis foce acts to the ight of the velocity. The amount of tilt is geatly exaggeated in Fig. 1, fo ease of illustation, as is the size of the Coiolis foce in elation to the centifugal foce. How the balance of foces can occu is indicated in Fig. 1. The thee foces ae shown fo a paticle at the suface of a cest. At the top of the tilted line segment the paticle velocity is hoizontal and theefoe diected into the pape. When the obit is tilted, the outwad centifugal foce no longe acts vetically but thee is a slight hoizontal com- 312 K. E. Kenyon

3 Fig. 1. Vetical section (xz-plane) nomal to the diection of wave popagation (positive y-diection) showing the pojection of the cicula obit of a suface paticle and the balance of thee foces at the wave cest, whee the paticle velocity is hoizontal and diected into the pape. The pojected obit is a staight line segment of length equal to twice the wave amplitude (2a) that is tilted to the left of the vetical by the angle α. The centifugal foce F ce points up paallel to the tilted line, and its component F ce sinα is equal and opposite to the hoizontal Coiolis foce F co, which is due to the vetical component of the Eath s otation and acts to the ight of the hoizontal paticle velocity in the nothen hemisphee. The vetical component of the centifugal foce F ce cosα balances the downwad pessue foce F p, and this is the cyclostophic balance. Fig. 2. Simila to Fig. 1 but a hoizontal plan view (xy-plane) showing the pojection of the cicula obit of a suface paticle and the balance of thee foces at the back of the wave cest, whee the paticle velocity is vetically downwad and diected into the pape. The pojected obit is a staight line segment, of length equal to twice the wave amplitude (2a), tilted to the left of the wave popagation diection (positive y-diection, which is assumed to be due noth) by the angle β. The centifugal foce F ce points up paallel to the tilted line, and its component F ce sinβ is equal and opposite to the Coiolis foce F co, which is due to the hoizontal component of the Eath s otation acting to the ight of the vetical paticle velocity. The othe component of the centifugal foce F ce cosβ is equal and opposite to the hoizontal pessue foce F p, which points antipaallel to the wave popagation diection. ponent to it that can be made to balance the Coiolis foce by adjusting the amount of tilt (angle α). Also the vetical component of the centifugal foce can equal in magnitude the downwad pessue foce, which emains stictly vetical, so that the cyclostophic balance is maintained. As the second example conside only the hoizontal component of the Eath s otation, o think of waves taveling at the equato. The Coiolis foce associated with the hoizontal component of the Eath s otation only acts on the vetical component of the paticle velocity. Figue 2 shows the balance of foces at the tailing edge of the cest whee the paticle velocity is downwad. Fo this pupose the hoizontal plan view is selected (the xy-plane), and the wave tavels in the positive y-diection, which to stat with is assumed to be due noth. At the top end of the tilted line segment the paticle velocity is down and theefoe into the pape. The slight tilt (angle β) of the obital plane is to the left of the diection of wave popagation, and the amount of tilt is adjusted so that the Coiolis foce is balanced. The majo balance of foces in the wave diection, the cyclostophic balance, is also still in effect. Assume now that the wave tavels due east (a wave taveling due noth will not expeience the following effect). At the tailing edge of the cest whee the paticle velocity is vetically downwad the Coiolis foce, due to the hoizontal component of the Eath s otation, points in the diection of wave popagation, which is in the diection of the centifugal foce and in the opposite diection to the pessue foce. Now due to the colineaity of the thee foces thee is no possibility of balancing the Coiolis foce by a change in the oientation of the obital plane. Howeve, by a slight change in the shape of the obit, the Coiolis foce could be balanced. What is needed is a little smalle centifugal foce, because the given pessue foce has to balance Cyclostophic Balance in Suface Gavity Waves: Essay on Coiolis Effects 313

4 both the Coiolis and centifugal foces. This could be accomplished by an oval o elliptical obit, whee the shote axis of the ellipse is paallel to the mean suface and the longe axis emains equal twice the wave amplitude, fo the suface paticles. Then the adius of cuvatue would be lage, at the ends of the shot axis, than fo a cicle, which would give a smalle centifugal foce. The eccenticity of such an ellipse would be so close to one, due to the small atio of Coiolis to centifugal foces, that it pobably would not be measuable. (In this bief epot an attempt is not made to pove mathematically whethe o not the obits ae indeed ellipses.) If fo some eason the Coiolis foce is not balanced (e.g. the obital planes ae not tilted), then each fluid paticle will ty to move (acceleate) in the diection of the Coiolis foce. In the fist example fo waves taveling in any diection nea the noth pole, the paticles at the cest would ty to move to the ight of the diection of wave popagation and to the left at the tough. The net esult at both cest and tough would be a toque pointing in the diection of wave tavel (see Appendix). Such a toque would attempt to change the diection of the angula momentum without alteing its magnitude. In the last example of the wave taveling east at the equato, if the paticle obits emain cicula so that the Coiolis foce is not balanced, then a paticle will ty to move in the diection of wave popagation on the back face of the wave and in the opposite diection on the font face. Again the esult will be a toque that ties to change the diection of the angula momentum vecto. By the natue of the way a toque opeates on the angula momentum, if the Coiolis foce is not initially balanced by anothe foce, then the toque caused by the Coiolis foce will not tend to bing it into balance because the toque will only ty to change the diection of the angula momentum. A change in the diection of the angula momentum will leave the Coiolis foce still unbalanced. (One could think of the classical poblem of a spinning bicycle wheel suppoted by a point on its axis that pecesses in the hoizontal plane due to the toque caused by the vetical foce of gavity.) 4. Discussion Fo wave popagation at an abitay latitude in the nothen hemisphee, and assuming all the foces ae balanced, both types of tilts of the obital plane shown in Figs. 1 and 2 will occu simultaneously (it may be difficult to visualize the two effects in a single diagam, howeve). Of couse the tilt angles ae vey small and pobably not measuable: fo example sinα α f/ω 10 4, and the angle β will have a simila ode of magnitude. In addition when the waves tavel in an abitay diection (except due noth) ove the ocean suface, and all the foces ae balanced, then one expects the shape of the paticle obits to be slightly elliptical in eithe hemisphee. The type of ellipticity caused by the Coiolis foce is qualitatively diffeent fom that obseved when waves shoal, whee the long (not the shot) axes of the ellipses ae paallel to the mean fee suface. In the southen hemisphee the tilt angle α will have the opposite sign to that in the nothen hemisphee, because the sign of the hoizontal component of the Coiolis foce changes when the latitude changes sign. The sign of the tilt angle β, howeve, is independent of hemisphee. Since it has been found that the Coiolis foce on the paticle velocity can easily be balanced at all times in most cases, by only a mino adjustment to the obital paametes, when the waves have infinitesimal amplitude, then we conclude that in all pobability this is what will take place, with the possible exception of tansient o apidly vaying conditions. Theefoe the geneal expectation is that thee will be no toque caused by the Coiolis foce that will change the wave angula momentum. At finite amplitude the paadox of the Stokes dift emains to be discussed, i.e. how can the Coiolis foce on the Stokes dift be balanced? Now possibly the Stokes dift itself could be the esult of a slight imbalance in the cyclostophic pai of foces. One way to think about it is to conside the gavity toque on long cested waves of finite amplitude (Kenyon, 1996). Even if the font and back faces of the wave emain symmetic at finite amplitude, as in the classical Stokes wave, the gavity toque will cause the hoizontal pessue foce to ovebalance the centifugal foce on the font face of the wave and the centifugal foce to ovebalance the pessue foce on the back face. The net esult is an acceleation of the paticles in the diection of wave popagation, which would poduce a net displacement of the paticles in the diection of wave popagation afte each wave peiod. This would then qualitatively and physically explain the existence of the Stokes dift, which was explained mathematically so many yeas ago by Stokes (1847). The Coiolis foce on the paticle velocity could still be balanced at finite amplitude and at all times by the appopiate tilting of the obital planes and change in shape of the obit. Then the Coiolis foce on the Stokes dift will automatically be taken cae of in the pocess, that is it will be balanced by a small component of the centifugal foce. In othe wods balancing the Coiolis foce on the Stokes dift is not a sepaate poblem; it is all pat of balancing the thee foces (pessue, centifugal, Coiolis) thoughout a wave peiod as the paticles pefom thei complete obital motion, whethe the amplitude is infinitesimal o finite. Also when the Coiolis foce is in balance thoughout the wave cycle, then it will not be possible fo the waves to geneate inetial motion. Acknowledgments The anonymous eviewes and the edito ae thanked fo thei specific comments on the text. 314 K. E. Kenyon

5 Appendix: Coiolis Toque The geneal expession fo the Coiolis toque can be woked out most easily using vecto notation. Othewise it is difficult to keep tack of all the special cases that can occu. Conside a pogessive suface gavity wave at latitude θ popagating in the diection α, measued counteclockwise fom the east. The Coiolis toque at a fixed position is τ C = F ( A1) whee is the adius vecto of an obiting paticle and F is the Coiolis foce on that paticle. The Coiolis foce is F = 2Ω u ( A2) whee Ω is the angula velocity of the Eath and u is the obital velocity of a paticle. Now the angula velocity of the Eath can be epesented as Ω=Ω( ẑ sin θ + ŷ cosθ) ( A3) whee Ω is the angula speed of the Eath s otation and ẑ and ŷ ae unit vectos in the vetical and noth diections, espectively. Also the paticle velocity can be expessed fo cicula obital motion as u = ω ˆl A4 whee ω is the wave fequency and ˆl is a unit vecto paallel to the angula momentum of the obiting paticle. Finally the adius vecto needs to be stated in component fom = a( ẑ cosωt + ˆk sin ωt) ( A5) whee a is the adius of a paticle at the suface (wave amplitude), and ˆk is a unit vecto in the diection of wave popagation (i.e. in the wave numbe diection and pependicula to ˆl ). With the help of equations (A2) (A5) equation (A1) can be woked out in detail. Fo this pupose it is convenient to pesent the time mean and fluctuating pats of the Coiolis toque sepaately. The time mean pat is τ C =Ωωa 2 [ ˆk ( sin θ ) ẑ ( cosθ sin α ) ] A6 and the fluctuating pat is τ C =Ωωa 2 [ ˆk cosθ sin α sin 2ωt + sin θ cos 2ωt +ẑ( cosθ sin α cos 2ωt sin θ sin 2ωt)] ( A7) and τ C = τ C + τ C. It can be seen fom (A6) and (A7) that both the mean and fluctuating pats of the Coiolis toque have two components, a hoizontal component diected paallel to the wave numbe, and a vetical component. Also (A7) shows that the fequency of the fluctuating Coiolis toque is double the wave fequency. The tendency of the hoizontal mean toque in (A6) is to otate the angula momentum vectos clockwise in the nothen hemisphee and counteclockwise in the southen hemisphee (the toque changes sign between hemisphees due to the latitude dependence). The vetical mean toque changes sign if the waves tavel noth o south but the sign is independent of hemisphee. Refeences Backus, G. E. (1962): The effect of the Eath s otation on the popagation of ocean waves ove long distances. Deep-Sea Res., 9, Eckat, C. (1960): Hydodynamics of Oceans and Atmosphees. Pesimmon Pess, New Yok, 290 pp. Kenyon, K. E. (1991): The cyclostophic balance in suface gavity waves. J. Oceanog. Soc. Japan, 47, Kenyon, K. E. (1996): Gavity toques fo suface waves. J. Oceanog., 53, Hasselmann, K. (1970): On wave diven inetial oscillations. Geophys. Fluid Dyn., 1, Stokes, G. G. (1847): On the theoy of oscillatoy waves. Tans. Camb. Phil. Soc., 8, Usell, F. (1950): On the theoetical fom of ocean swell on a otating Eath. Mon. Not. R. Ast. Soc., Geophys. Suppl., 6, 1 8. Cyclostophic Balance in Suface Gavity Waves: Essay on Coiolis Effects 315