arXiv:/0005028v1 [physics.ed-ph] 10 May 2000 Ω celeration, ne.(.) h em2 term the (2.1), eq. In a ln ietoso xdsas h frame The stars. fixed of directions along center the at origin in h nrilframe inertial The tion. oteErh thsa nua angular an has It . the to ewe h ceeain fapoint a of the between at (Ω Earth to xmlso cai iclto:wn-rvncirculation . -driven two Kelvin presents and circulation: asso- section oceanic are last of The examples strong waves. these Rossby that with Corio- also ciated the show by We stabilized . are lis which Streams, circula- Jets atmospheric the of tion, example second third a The presents hurricanes. section become may ) how not their show (but and describe approximation, Physics we geostrophic basic the the in cyclones: discuss the and we of anticyclones section, presentation second of the the to In devoted equations. is section first at place scale. a taking geophysical phenomena the only consider is must one force dominant, In the . perturbation Foucault examples, the the two and examples being these projectile standard Earth deflected the a rotating to of restricted the often for are paradigm, effects force Coriolis a R e sitouetoframes two introduce us Let The paper. present the by followed line the is This the frames, non-inertial in of study The ∧ R R ′ .Nve-tkseuto narttn frame rotating a in equation Navier-Stokes A. ( ( M ( where , M ⋆ Ω aoaor ePyiu el Mati`ere Condens´ee, la Coll`eg de Physique de Laboratoire in ) = ) ∧ ≃ oilsfrei epyis neeetr nrdcina introduction elementary an : in force Coriolis OM R a v nodrt aeteCroi oc effects force Coriolis the make to order In . ⋆⋆⋆ 7 omdlte navr ipewyt rn u h hsclphe physical the out bring pedagogic to way of simple reason very that a for LPENSL-TH-06/2000 in are them and in model interested with, to then familiar are We are we effects. ena force Coriolis of study the Ω R R eso o epyismyilsrt n hsipoeclass improve thus and illustrate may Geophysics how show We ′ . 29 ′ ′ a aiyb band[1]: obtained be easily may I AI EQUATIONS BASIC II. ´ te -rne NMGM/eif 2aeu .Coriolis, G. M´et´eo-France, avenue 42 CNRM/GMME/Relief, stecnrfglaclrto.I the In . centrifugal the is ) ( ( steaglrvlct frtto fthe of rotation of velocity angular the is M M · .INTRODUCTION I. ⋆⋆ 10 2 + ) stevlct of velocity the is ) aoaor ePyiu,ESLo n NS 6all´ee d’Ita 46 CNRS, and ENS-Lyon Physique, de Laboratoire O − 5 fteErh n hs xsare axes whose and Earth, the of rad Ω Ω R ∧ ∧ · v s stegoeti n,with one, geocentric the is v − R R 1 ′ R .Tefloigrelation following The ). ( ′ M ( .Vandenbrouck F. M and + ) . M steCroi ac- Coriolis the is ) R Ω a ′ R M Ω ∧ nrltv mo- relative in ( R M ( ihrespect with Ω ′ in in ) sfastened is ∧ R OM R Fbur ,2008) 2, (February ′ ⋆ and , .Berthier L. , small and , (2.1) ) . 1 eFac,1 lc .Brhlt 50 ai,France , 75005 Berthelot, M. place 11 France, de e oaigframe rotating con h bv nriltrsadras[2]: reads and terms inertial above the account xenlfre fte exist, they if external force The and t.Tonndmninlnmesmyte ederived be scales. then these may from numbers non-dimensional Two and ity. velocity angular the Ω locity, ert h terms the rewrite that ieetfo h rvttoa ed hc nytakes [1]. only attraction which Earth’s field, the account gravitational into the from different scnevtv.I hsi lotecs for case the also where is this force If centrifugal The conservative. clarity. for is terms the all in removed yitouigtedffrn hrceitcsae fthe done of is scales This characteristic different flow: simplifications. to the terms further necessary introducing different make hence by the to is of It order importance in general. relative in the solve evaluate to difficult it ν osdffso.Frteamshrcflw tde here, vis- studied flows and are: atmospheric advection values the through typical For fluid momen- diffusion. the the cous of in importance transport relative tum the characterizes It p rm,tehdottceulbimeuto is: equation equilibrium 0 hydrostatic the frame, ′ h yaia pressure dynamical The . i h enlsnme sdfie as: defined is number Reynolds The (i) h olnaiyo h airSoe qainmakes equation Navier-Stokes the of nonlinearity The ∂ ∼ = ⋆⋆ v ∂t p n .Gheusi F. and , g R p L 10 h rsuefil.Tedpnec on dependence The field. pressure the ′ nldstecnrfgltr,adi hsslightly thus is and term, centrifugal the includes p topei swl soencphenom- oceanic as well as atmospheric + ′ ( + eoe h yia pta extension, spatial typical the denotes − scalled is ρgz 5 ladpatclitrs.Oramis aim Our interest. practical and al .Ryod n osynumbers Rossby and Reynolds B. f R v clMcaislcue concerning lectures Mechanics ical m 15 OLUECdx France Cedex, TOULOUSE 31057 R e oeata r involved. are that nomena where , nldstegaiainlfreadother and force gravitational the includes 2 ′ = R · · i,607Lo,France Lyon, 69007 lie, ∇

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and = p Ω ρ ρ 1 dexamples nd ′ ∇ νU/L stedniyo h fluid the of density the is ed,wti constant, a within reads, ∧ U ρ Ω · p 2 ( ν s Ω /L + − ∧ h ieai viscos- kinematic the 2 1 ∧ ρ 1 ( , Ω = f nterotating the In . OM L ∧ − UL ∼ ν OM 2 + ) Ω M . 0k and km 10 f ∧ n can one , U ν as ) a been has v ∆ ∇ h ve- the R v ′ p (2.2) (2.3) R ∇ ′ ′ p . = ′ , about 1010. A large value of the Reynolds number is In these equations, f 2Ω sin λ is the Coriolis param- ≡ also obtained for oceanic flows [3]. Hence, the Navier- eter. In equation (3.1c), the term 2ρ(Ω⊥ v) uz is Stokes equation reduces, for geophysical flows, to the Eu- small compared to ρg (ΩU/g 10−5). Equation∧ · (3.1c) ler equation: therefore reduces to the hydrostatic∼ equilibrium equation ∂vR′ 1 1 ∂p/∂z = ρg. + (vR′ ∇)vR′ = ∇p + f If we consider− the case of an in the Northern ∂t · − ρ ρ hemisphere and assume that the velocity field is tangen- 2Ω vR′ Ω (Ω OM). − ∧ − ∧ ∧ tial (u = 0), then, v < 0 (clockwise rotation) implies (2.4) ∂p/∂r < 0. The pressure is higher at the eddy center Moreover, geophysical flows are turbulent (high Reynolds than outside: it is an . A would cor- number) [4]. For the sake of simplicity, we ignore this respond to an anticlockwise rotation. Both situations are complication in what follows. A simple way of taking represented in figure 1. The rotation senses are opposite into account the relevant effects of turbulence will be pre- in the . sented in the last section. (See section V A.) Centrifugal force (ii) The is defined as: Pressure forces Pressure forces (v ∇)v U 2/L U R = · = = . (2.5) o 2Ω v ΩU LΩ

∧ It compares the advection and the rotation effects. The

Coriolis force dominates if Ro 1. A geophysical flow, D characterized by a large spatial≪ extension, may easily be A influenced by the Earth’s rotation, as one typically has −2 Ro 10 1. On the other hand, an emptying bath- ∼ ≪ −1 −1 5 tub with U 1 m s , and L 10 m, has Ro 10 . Such a flow∼ is more· strongly influenced∼ by the advection∼ in the fluid, and thus by the initial conditions, than by Coriolis force the Earth’s rotation. (a) (b) FIG. 1. Anticyclone (a) and cyclone (b) in the . III. ATMOSPHERIC EDDIES We end this section with two concluding remarks about A. Anticyclones and cyclones the presence of the Coriolis force: (i) Without this force, an eddy center is always a pres- We consider first the situation when the Rossby num- sure minimum. However, in the case of the anticyclone, ber is negligible. This is the case for anticyclones the Coriolis force stabilizes the inverse situation, with the and cyclones since one typically has U 10 m s−1, eddy center being a pressure maximum. ∼ · L 1000 km, which yields Ro 0.1. In the Euler (ii) In its vectorial form, the geostrophic equilibrium equation∼ (2.4), we only have to keep∼ the gravity, pres- equation reads: ∇p′ = 2ρΩ v. This implies that sure and Coriolis terms. This hypothesis constitutes the the pressure p′ is constant− along∧ a streamline. When the geostrophic approximation. For each point M of the usual Bernoulli equation is valid, pressure variations are, Earth, we define a vertical axis (Mz), and a cylindri- on the contrary, associated with velocity variations along cal (r,θ,z). The vertical component a streamline. of the velocity field is supposed to be zero, which implies that the movements of the fluid are locally horizontal. u is the radial component of the velocity field and v the B. Hurricanes tangential one. The Earth’s Ω is writ- ten as Ω = Ω + Ω where Ω Ω sin λ uz and Ω k ⊥ k ≡ ⊥ Let us consider an eddy (anticyclone or cyclone) whose is Ω’s projection on the plane (r, θ); λ is the . angular velocity and radius are respectively ω and R. The flow is supposed to be stationary. In this system of The Rossby number characterizing this eddy can be writ- coordinates, the Euler equation can be rewritten, under ten as Ro = U/LΩ = ω/Ω. Therefore, the geostrophic the geostrophic approximation, as: equilibrium corresponds to a small angular velocity of the ∂p eddy, i.e. ω Ω. We shall now consider the case where = ρvf, (3.1a) ≪ ∂r the eddy’s angular velocity is not small compared to the 1 ∂p Earth’s rotation. This means that the centrifugal force = ρuf, (3.1b) due to the eddy rotation has to be taken into account. r ∂θ − ∂p In this case, the Rossby number is of order unity. In the = ρg 2ρ(Ω v) uz. (3.1c) ∂z − − ⊥ ∧ ·

2 frame ′, the fluid particle has a uniform circular mo- does not exist for a cyclone (G> 0). When the angular tion. ForcesR acting on it are the Coriolis force and the velocity grows, the radial follows this radial pressure gradient. The equation of for a evolution and becomes more and more important. This fluid particle, located at the eddy’s periphery reads, in explains why hurricanes are always associated with very ′: low pressure. R We note in conclusion that the balance between the 2 1 dp r0ω = + r0fω, (3.2) centrifugal force and the radial pressure gradient is pos- − −ρ dr sible whatever the sense of rotation. Thus, the existence 2 where r0 is the eddy radius. The term r0ω corresponds of clockwise hurricanes in the northern hemisphere can- to the centrifugal acceleration of the− fluid particle, and not be excluded. However, most of the hurricanes ob- r0fω is the Coriolis term. served in the northern hemisphere are anticlockwise and An anticyclone in the northern hemisphere is shown result from the amplification of earlier tropical cyclones, in figure 1a. For such an equilibrium, the Coriolis force the amplification mechanism being the conversion of the compensates both pressure and centrifugal forces. If the latent heat of evaporating tropical warm into ro- angular velocity of the anticyclone grows, the Coriolis tational kinetic . force is not sufficient to counterbalance these two forces since the centrifugal force grows faster than the Corio- lis force with increasing ω. This is not the case for the IV. JET STREAMS AND ROSSBY WAVES cyclone depicted in the figure 1b. The pressure and cen- trifugal forces may counterbalance each other when the The difference in solar heating between the equatorial rotation of the cyclone becomes faster. This qualitative and polar regions drives a convective cell at the plane- approach shows that there is no limit to the kinetic en- tary length scale, the , which extends in both ergy of rotation for a cyclone. hemispheres from the up to the sub-tropical lat- More quantitatively, equation (3.2) can be solved to itudes. The heated equatorial air rises, moves toward find: the poles where it cools, then sinks and comes back to the equator. When coming back, the air is deflected f G ω± = 1 1+ , (3.3) toward the west by the Coriolis force, generating east- 2 G "− ± r 0 # erlies at the tropical which are known as the . Conversely, the upper-troposphere tra- where G dp/dr and G ρr f 2/4. Figure 2 gives the ≡ 0 ≡ 0 jectories toward the poles are deflected toward the east. evolution of an eddy angular velocity as a function of the Because of the thermal structure of the atmosphere [5], radial pressure gradient. In this figure, the geostrophic these upper-level concentrate in narrow tubes situation can be found around the origin (small pressure of very strong winds up to 80 m s−1, the Jet Streams. gradient and angular velocity). In the northern hemi- The Jet Streams are typically found· at altitudes of about sphere, the sign of the angular velocity is the same as 10 km and at latitudes between 30◦ and 40◦. However, that of the pressure gradient. One can even obtain the their strength and location may depart significantly from angular velocity of an eddy by developing the expres- these mean values, depending on the season, the longi- sion (3.3) around zero: ω fG/4G . ≈ 0 tude, and the day-to-day thermal structure of the atmo- sphere at mid latitudes. It can be noted that B. Piccard 2 and B. Jones took great advantage of the Jet Streams for ω+ ω− their recent travel around the world in a balloon. The Jet 1 Streams are also useful to the planes flying from America Extratropical cyclones Tropical hurricanes to Europe. 0 0) Extratropical In this section, we propose to show how a zonal wind anticyclones f > (i.e. along the parallels) may be stabilized by the Coriolis

( -1 force. A M of air near the Earth’s surface is reduced ω/f

2 -2 to a point G. Its coordinates are the usual spherical ones (R,θ,ϕ), θ being the colatitude and R the radius of the -3 Earth. The velocity of G can then be explicitly written: High pressure centre (G < 0) Low pressure centre (G < 0) vR′ (G) = Rθ˙ uθ + Rϕ˙ sin θ uϕ. The quantity Rϕ˙ sin θ -4 -4 -3 -2 -1 0 1 2 3 4 is the drift velocity u0 of the point G along a parallel. G/G0 We deduce the following expression of the Coriolis force FIG. 2. Normalized angular velocity as a function of the about the centre of the Earth (point O): normalized pressure gradient. 2 MO = 2MR Ωθ˙ uθ +2MRΩu0 cos θ uϕ. (4.1) The condition G> G0, for the existence of the above solutions, gives a limit− to the angular velocity of an anti- The computation of the angular of G about cyclone (G < 0). One finds ωmax = 2Ω sin λ. This limit

3 O, in the frame ′, yields: are concerned, the kinematic viscosity is neglected, since R 5 typically νturb/ν 10 for oceanic flows, and νturb/ν is 2 2 L ′ (O)= (MR )ϕ ˙ sin θ uθ + (MR )θ˙ uϕ. (4.2) 7 ∼ R − about 10 for atmospheric flows. Let us write the Navier-Stokes equation in projection The theorem of for the point G, on (Oxyz), where (Oxy) is the surface of the globe, (Oz) about O and projected on uϕ gives: the ascendant vertical, and (u,v,w) are the velocity com- Ωu ponents: λ¨ = 2 0 sin λ, (4.3) − R du 1 ∂p ∂2u ∂2u ∂2u = + fv + νturb + + , where λ π/2 θ is the latitude. This equation is dt −ρ ∂x ∂x2 ∂y2 ∂z2 ≡ −   linearized for small deviations around a given latitude dv 1 ∂p ∂2v ∂2v ∂2v λ , leading to = fu + νturb + + . 0 dt −ρ ∂y − ∂x2 ∂y2 ∂z2   Ωu (5.1) δλ¨ + 2 0 cos λ δλ =0, (4.4) R 0   For a stationary situation with large Rossby number, the acceleration terms are negligible: the velocity depends where δλ λ λ0. The meridional motion of G re- ≡ − then only on . The horizontal pressure gradient mains bounded, only if u0 > 0, which corresponds to a drift velocity from west to east. This motion is char- terms can also be neglected since the equations have been acterized by small oscillations around the mean latitude linearized and one can consider the real physical situa- tion as the superposition of a (taking λ0 with ω0 = 2Ωu0 cos λ0/R. These oscillations correspond to the stationary case of a Rossby into account the pressure terms) and a wind-driven cur- [6]. More generally, Rossbyp waves in the atmosphere rent, which will now be described. We consider a solution are guided by strong westerlies. depending on space only through the coordinate z. The boundary conditions are the following: the velocity has to be finite both as z and at the free surface, → −∞ V. OCEANIC CIRCULATION the stress is proportional to ∂v/∂z and parallel to the wind flow, assumed to be in the (Oy) direction. One can solve eq.(5.1) and find the velocity field (the solution is Oceanic circulation is, of course, described by the straightforward defining W (z) u(z)+ iv(z)): same equations as . For large ≡ scale oceanic currents, like e.g. the , the π z z u(z)= V cos + exp , geostrophic approximation (see section III A) is relevant: ± 0 4 δ δ the Coriolis force compensates the horizontal pressure π z  z  (5.2) v(z)= V sin + exp , gradient, which is related to the slope of the free surface, 0 4 δ δ which is not necessarily horizontal [7].     We shall be interested here in a slightly different case where δ 2νturb/ f is a distance called the Ekman for which the interaction between the wind and the depth. Typical≡ values| | for δ are δ 10 100m. “+” gives rise to a current. stands forp the northern hemisphere∼ and− “ ” for the southern one. −

A. Wind-driven circulation : z y The wind induces a at the ocean surface, trans- WIND mitted through turbulence to the deeper layers of the O . There is a supplementary difficulty that we can- not ignore here. The flow is not laminar, but essentially turbulent. The fluid viscosity is related to molecular ag- x itation, dissipating the energy of a fluid particle. A dif- fusive momentum transport is associated with this phe- nomenon. In a turbulent flow, agitation dissipates the energy associated with the mean velocity of the current. This analogy allowed Prandtl to introduce the notion of an eddy viscosity [4]. In this approximation, considering vR′ as the mean flow velocity, the Navier-Stokes equation (2.2) remains unchanged, the eddy viscosity νturb being added to the kinematic viscosity ν. It must be remarked that the former is a property of the flow while the latter FIG. 3. . The surface is generated by the is a property of the fluid. As far as geophysical flows velocity field v(0, 0,z).

4 Close to the surface (z = 0), the current deviates 45◦, waves is plotted in figure 4. One can notice that the sur- and the direction of the velocity rotates clockwise (anti- face undulation is trapped in the vicinity of the coast, clockwise) in the northern (southern) hemisphere. The and its spatial extension in the direction of the ocean is amplitude of the velocity decreases exponentially on a typically of order L. length scale δ, which represents the characteristic depth over which the influence of the wind is significant. This velocity field, the so-called Ekman spiral, is plotted in figure 3. The mean effect of the wind, over a depth δ, is z the fluid motion in a direction perpendicular to it: this COAST effect is called the Ekman transport. x

y

B. Kelvin waves

The main difference between atmospheric flows and oceanic flows occurs near in the coastline, limiting the waters motion. This is the origin of Kelvin waves. If one considers the deformation of the free surface of the , one can see that gravity acts as a restoring force, MEAN CURRENT giving rise to a “” [8]. When influenced by the Earth’s rotation, these waves are called “modified waves” [9]. FIG. 4. Surface shape generated by a . Let us consider the following geometry: a south-north current, with a coast on its right (east). The coast is The Kelvin waves are in fact easily observed, since the supposed to be a vertical wall, the height being de- currents generated by are influenced by the Coriolis noted h0 + h(x,y,t). The Coriolis force usually deflects force and give rise to them. As a consequence, the coast is a south-north current toward the east, i.e. toward the always to the right of the flow direction (in the northern coast. Hence water gathers close to the coast, and gives hemisphere). On the scale, mean move- rise to an west-east horizontal pressure gradient counter- ments are in this case an anticlockwise rotation around a balancing the Coriolis force. The equations describing point called . This geometry is found the gravity waves are the linearized Euler and continuity in many places over the globe, the rotation being clock- equations [8]: wise in the southern hemisphere [9]. ∂u ∂h = g + fv, ∂t − ∂x VI. CONCLUSION ∂v ∂h = g fu, (5.3) ∂t − ∂y − Coriolis force effects become important as soon as the ∂h ∂u ∂v spatial extension of the flow is important (Ro 1/L). = h0 + . ∝ ∂t − ∂x ∂y This is the reason why the Earth’s rotation considerably   influences the atmosphere and oceans . We have Taking (Oy) perpendicular to the coast, and considering presented in this paper several simple examples of geo- a solution describing the above situation, i.e. v = 0, physical fluid dynamics. We hope it will be helpful for u = ξ(y)exp i(ωt kx), h = η(y)exp i(ωt kx), one Mechanics teachers to illustrate inertial effects with sim- − − obtains: ple but physically relevant examples. fy u = u0 exp exp i(ωt kx), −√gh0 − ACKNOWLEDGMENTS   (5.4) h0 h = u . We thank P. C. W. Holdsworth for his kind help during s g the preparation of the manuscript. 2 2 The relation is given by ω = gh0k , as for usual gravity waves. The characteristic length L ≡ √gh0/f is the Rossby radius of deformation. At a mid- latitude λ 45◦, one finds L 2200 km for h 5 km, ∼ ∼ 0 ∼ while for a shallow sea, i.e. h0 100 m, one rather has L 300 km. The surface shape∼ generated by the Kelvin ∼

5 [1] H. Goldstein, (Addison-Wesley, 1980), pp. 177-188. [2] H.P. Greenspan, The theory of rotating fluids (Cambridge University Press, 1969), pp. 5-10. 11 −1 −6 2 [3] For oceans, Re ∼ 10 , with U ∼ 1 m · s , ν ∼ 10 m · − s 1: the approximation is thus the same. [4] M. Lesieur, Turbulence in fluids (Kluwer, 1997), pp. 315- 320. [5] J.R. Holton, An introduction to dynamic (Academic Press, 1992), pp. 141-149. [6] C.G. Rossby, Relation between variations in the intensity of the zonal circulation of the atmosphere and the displace- ments of the semi-permanent , J. Mar. Res., 2, (1939). [7] Ocean circulation (The Open University, Pergamon, 1989), pp. 40-58. [8] L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Perga- mon Press, 1987), pp. 31-37. [9] S. Pond, G.L. Pickard, Introductory dynamical Oceanog- raphy (Pergamon Press, 1983), pp. 273-276.

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