The Persistence of Spin

The persistence of spin

H.K.Moﬀatt

Trinity College, Cambridge, CB2 1TQ, UK

Abstract: Three spinning toys that illustrate fundamental dynamical phenomena are discussed: the rising egg, prototype of dissipative instability; the shuddering Euler’s disc, prototype of ﬁnite- time singularity; and the reversing rattleback, prototype of chiral dynamics. The principles un- derlying each phenomenon are discussed, and attention is drawn to analogous phenomena in the vorticity dynamics of nearly inviscid ﬂuids.

1 Introduction

This informal evening lecture, delivered at the Carlsberg Academy, was con- cerned with dynamical principles illustrated by the behaviour of three mechan- ical toys. The first of these, the tippe-top, was the subject of a famous 1954 photograph (figure 1) of Niels Bohr (then honorary resident of the Carlsberg Academy) and Wolfgang Pauli, who were evidently intrigued by the tippe- top’s ability to turn upside-down as it spins, a vivid example of a dissipative (or slow) instability driven by slipping friction at the point of contact with the floor. The rising egg (the spinning hard-boiled egg that rises to the vertical if spun rapidly enough) is a similar phenomenon, which may be understood in terms of minimising the spin energy subject to conservation, not of angular momentum, but rather of the ‘adiabatic Jellett invariant’ (see section 2). My second toy is ‘Euler’s disc’ (see ), that rolls on its bevelled edge on a plane table, and settles to rest (like a spun coin) with a rapid shudder, a beautiful table-top example of a finite time singularity. Controversy surrounds the question of what is the dominant contribution to the rate of dissipation of energy for this toy; but whatever mechanism is invoked, the singularity is resolved in the final split second of the motion, most probably by loss of contact between the disc and the table and consequent release of the no-slip constraint. My third toy is the rattleback, a canoe-shaped object that spins reasonably smoothly in one direction, but which, when spun in the opposite direction, exhibits a pitching instability leading to spin reversal. The rattleback is in 2 H.K.Moffatt

Fig. 1. Niels Bohr and Wolfgang Pauli investigate the behaviour of the tippe-top; Photograph (1954) by Erik Gustafson, courtesy AIP Emilio Segre Visual Archives, Margrethe Bohr Collection fact very slightly deformed giving it a chiral (i.e. non-mirror-symmetric) mass distribution. It is this chiral property in conjunction with spin that leads to the intriguing behaviour, which may be regarded as providing a prototype of ‘chiral dynamics’. The nature of the dissipative effects that damp the motion are as yet ill-understood; but a semi-empirical choice of linear damping coeffi- cients provides a model that agrees well, at least qualitatively, with observed behaviour. The fluid counterpart of spin is of course vorticity, and it is therefore perhaps appropriate that, at this Symposium commemorating Helmholtz’s seminal 1858 paper on the laws of vortex motion, the phenomenon of spin may be taken as a natural starting point. Just as friction plays a key role in each of the above toy examples, so frictional (i.e. viscous) effects in fluids are nearly always important no matter how small the viscosity may be, a fact that should be constantly borne in mind (as Helmholtz was himself well aware) when adopting inviscid modelling techniques.

2 The rising egg

Spin a hard boiled egg sufficiently rapidly on a table and it will rise to the vertical, a parlour trick that invariably provokes a startled response: what is The Persistence of Spin 3 it that makes the centre-of-mass rise in this way? The key to understanding this phenomenon was provided by Moffatt & Shimomura (2002) (and in the comprehensive treatment of this problem by Moffatt, Shimomura & Branicki 2004, Shimomura et al. 2005, Branicki et al. 2006, and Branicki & Shimo- mura 2007; see also Bou-Rabee et al. 2005) on the basis of the ‘gyroscopic approximation’, which is applicable when the spin is large and the slipping friction at the point of contact with the table weak. These conditions are the counterparts of the conditions of small Rossby number and large Reynolds number in the mechanics of rotating fluids. They imply a state of ‘balance’ in which (at leading order) coriolis forces dominate the behaviour. Nevertheless, the frictional force at the point of slipping contact with the table exerts a torque relative to the centre-of-mass of the egg, causing this centre-of-mass to rise on a slow (dissipative) time-scale. Figures 2,3 and 4 show three posters prepared for the Summer Science Exhibition held at the Royal Society, London, in July 2007. These posters were designed by Andrew Burbanks (University of Portsmouth). Their purpose was to make the scientific principles that govern the behaviour of such spinning toys accessible to a wide public. The first poster (Figure 2) was based on photographs provided by Yutaka Shimomura of the rising egg phenomenon, including side reference to the analogous role of coriolis forces in large scale atmospheric dynamics. Under the gyroscopic approximation, the equations governing the spin of the egg admit an invariant J = Ωh, where Ω is the component of angular velocity about the vertical, and h is the height of the centre-of-mass above the table. This invariant similarly exists for the case of any convex axisymmetric body spinning with slipping friction on a horizontal plane boundary. For the particular case of a body that is part spherical, this invariant was found by Jellett(1872) (and so is appropriately called the Jellett invariant), and in this case it is exact (without need to make the gyroscopic approximation.) This is the case for the tippe-top (a mushroom-shaped spherical-cap top) which so intrigued Niels Bohr and Wolfgang Pauli; the manner in which the tippe-top turns upside-down when it spins was successfully explained by Hugenholtz (1952). The existence of the Jellett invariant provides a reasonably transparent explanation for the rise of the egg. The slipping friction causes a slow loss of energy (to heat), and the system seeks a minimum energy state compatible with the ‘prescribed’ and constant value of J. At high spin rates (as required for the gyroscopic approximation), the energy is predominantly the kinetic energy of spin (the potential energy being much smaller). Consider the case of a uniform prolate spheroid with semi-axes a, b with a > b, the kinetic energy when the axis is horizontal is

2 2 2 2 E1 = (5/2)J /(a + b )b , (1) and the kinetic energy when the axis is vertical is 4 H.K.Moﬀatt

2 2 2 E2 = (5/4)J /a b . (2)

Hence, since a > b, it follows that E2 < E1, i.e the vertical state has lower energy (for the same value of J) and is therefore the preferred stable equilibrium. Note that for the case of an oblate spheroid (a < b), the opposite conclusion holds: if spun sufficiently rapidly on the table about its axis of symmetry, it will rise to spin on its rim. A ‘go’-stone, or indeed a mint imperial, provides a suitably oblate object on which to experiment. The above argument is reminiscent of that used to explain why a body rotating freely in space will tend, under the effects of weak internal friction, to rotate about its axis of greatest inertia. For a prolate spheroid, this is an axis perpendicular to the axis of symmetry, a conclusion quite the opposite of that obtained above for the spinning egg problem. So why the difference? It is because in the case of the freely rotating body, angular momentum is conserved, and energy is minimised subject to this constraint. For the spinning egg, angular momentum is not conserved, but the Jellett invariant takes its place; this makes all the difference. The moral is: for any weakly dissipative system, first determine what is conserved (even in some approximate sense), then minimise energy subject to this constraint. The rise of the egg is only one ‘filtered’ aspect of the fifth-order nonlin- ear dynamical system that governs the non-holonomic rigid body dynamics. Superposed on the ‘slow’ rise are rapid oscillations whose amplitude is con- trolled by the current state of spin. For sufficiently large spin (well above that required to make the egg rise) these oscillations can cause the normal reaction between egg and table to fluctuate to such an extent that the egg momentar- ily loses contact with the table, a prediction verified experimentally by Mitsui et al(2006). The analogous oscillations in the geophysical context are Rossby waves superposed on the mean large-scale atmospheric circulation. Dissipative instabilities have been set in an abstract geometric context by Bloch et al. (1994). The rising egg provides a prototype dissipative instability, brought to life by this simple table-top demonstration.

3 Euler’s Disc

The toy known as Euler’s disc is a heavy steel disc that can be rolled on its edge on a horizontal surface (or on the slightly concave dish that is supplied with it, see figure 3). It works well on a glass-topped table. This toy exhibits a ‘finite-time singularity’ in the final stage of its motion before it comes to rest on the table: the point of rolling contact of the disc on the table describes a circle with angular velocity Ω that increases ‘to infinity’ as the angle α between the plane of the disc and the table decreases to zero, as it obviously must in this final stage of motion. I place ‘to infinity’ in parentheses because in reality this potential singularity must be ‘resolved’ in some way by physical processes during the final split second when the motion is arrested. Much interest focuses on the nature of this resolution. The Persistence of Spin 5

D ynam ics of SPIN spinning toys with surprising beh aviour!

Spin a h ard-boile d e gg fas t e nough and its axis of s pin w ill ris e , in appare nt de fiance of gravity. Spin th e e gg m uch fas te r, Th e ris ing e gg provide s an e xam ple of dis s ipative ins tability. It is pow e re d by s lipping friction at th e point and it w ill jum p! of contact w ith th e table . Th e fas t-s pinning e gg jum ps be caus e os cillations about th e m e an m otion grow until th e re action force be tw e e n e gg and table falls to z e ro. Earth 's w e ath e r patte rns are drive n by a s im ilar ins tability.

Profe s s or Ke ith M offatt & Dr Tadas h i Tok ie da (M ath e m atics , Unive rs ity of Cam bridge ) Dr Yutak a Sh im om ura (Ke io Unive rs ity, Japan) Dr Andre w Burbank s (M ath e m atics , Unive rs ity of Ports m outh ) Dr M ich al Branick i (M ath e m atics , Unive rs ity of Bris tol) Egg ph otograph s : Yutak a Sh im om ura. Earth : NASA, Vis ible Earth . Pos te r de s ign: Andre w Burbank s

Fig. 2. Poster prepared for the Royal Society Summer Science Exhibition, 2007; graphic design by Andrew Burbanks. The rising egg (photos courtesy of Y. Shimo- mura). These posters were designed to attract the attention of a wider public to the dynamics of spin and analogous behaviour in ﬂuid dynamical contexts

D ynam ics of SPIN spinning toys with surprising beh aviour!

As a s pinning dis c s e ttle s , it rolls around its rim fas te r and fas te r, approach ing an infinite num be r of rolls A finite -tim e s ingularity occurs w h e n variable s in a ph ys ical s ys te m be com e in a finite -tim e ... infinite in a finite tim e . Eule r's dis c illus trate s th is in a s tartling w ay. As it rolls on its rim , pre ce s s ional angular ve locity approach e s infinity as angle of e le vation approach e s z e ro. Th is appare nt s ingularity is re s olve d in th e final s plit s e cond: th e dis c los e s contact w ith th e table and rolling ce as e s . 2007 is th e 300th annive rs ary of th e birth of th e m ath e m atician Eule r, afte r w h om th e dis c is nam e d. A one m illion dollar ope n q ue s tion is w h e th e r vorticity in turbule nt fluids can e xh ibit a finite -tim e s ingularity. Profe s s or Ke ith M offatt & Dr Tadas h i Tok ie da (M ath e m atics , Unive rs ity of Cam bridge ) Dr M ich al Branick i (M ath e m atics , Unive rs ity of Bris tol) Dr Andre w Burbank s (M ath e m atics , Unive rs ity of Ports m outh ) Dr Yutak a Sh im om ura (Ke io Unive rs ity, Japan) Dis c ph otograph : Joe Be ndik . Earth : NASA, Vis ible Earth . De s ign: Andre w Burbank s

Fig. 3. As for Figure 2; poster for Euler’s disc (photo courtesy of J. Bendick, inventor). 6 H.K.Moﬀatt

As in the case of the rising egg, it is weak dissipation of energy that induces the singular behaviour, which again has a startling effect when observed for the first time. According to analytical dynamics (with all dissipative effects neglected) a steady precessing state is possible in which

Ω2 sin α = 4g/a , (3) where a is the radius of the disc and g is the acceleration of gravity. The energy E of the system (kinetic plus potential) is proportional to sin α, and in practice, dissipative effects, no matter how weak, must lead to a slow decrease of α to zero, and hence, for so long as the quasi-static condition (3) persists, to an unlimited increase of Ω. Much controversy surrounds the question of what is the dominant mechanism of energy dissipation for this system. One quantifiable mechanism is that due to viscous dissipation in the thin layer of rapidly sheared layer of air in the decreasing gap between the disc and the table (Moffatt 2000, Bildsten 2002). (I am frequently asked what happens if the experiment is performed in a vacuum; the answer is that at the low pressures attainable in laboratory vacuum systems, the viscosity µ of air is very little different from its value at atmospheric pressure, as discovered by Maxwell (1866), and so this dissipative mechanism persists unaffected by decrease of pressure!). Rolling friction due to plastic deformation at the point of contact is also important, as evidenced by the fact that the disc behaves differently on different surfaces (e.g. on glass, polished steel, polished wood, ...). Whatever the dominant mechanism may be, what matters is the dependence of the rate of dissipation of energy Φ on E. For the viscous mechanism, and small α, this has a power law character

Φ ∼ E−λ , (4) where λ > 0. This rate of dissipation increases as E → 0 for the simple reason that the rate of shearing obviously increases as α → 0 and Ω → ∞. Since

dE/dt = −Φ, (5) it follows immediately from (4) that

1/(λ+1) E ∝ (t0 − t) , (6) where t0 is determined by the value of E at the initial instant t = 0. Hence E (and so α) goes to zero at the finite time t0, and apparently, from (3), Ω becomes infinite at this same instant. The resolution of this infinity is not hard to find. The downward acceleration of the centre of mass of the disc increases without limit according to the above description. When this downward acceleration equals g, the normal reaction at the point of contact with the table vanishes, so that (presumably) the disc loses contact with the table, and the rolling condition (on which (3) is based) no longer applies. This breakdown of the ‘adiabatic condition’ (3) The Persistence of Spin 7 occurs literally a split second (∼ 0.03s for the toy Euler disc) before the singularity time t0, and we move into a different dynamical phase of ‘free-fall’ during this final split second. Much attention is currently devoted to the ‘finite-time singularity problem’ in fluid mechanics. Roughly paraphrased this may be stated as follows: at high enough Reynolds number, can the vorticity become infinite within a finite time, starting from smooth finite-energy initial conditions? Even in the inviscid limit, for which the laws discovered by Helmholtz (1858) are applicable, this problem remains unsolved, and is also the subject of considerable controversy (see for example Bustamante & Kerr 2008, Hou & Li 2008). In these circumstances, Euler’s disc provides a reassuring table-top demonstration that finite-time singularities (with an appropriate resolution mechanism) do occur even in dissipative systems (and indeed in this case occur only by virtue of the dissipative mechanism!).

4 The Rattleback

My third example is the toy known as the celt, or more popularly the ‘rattleback’, a canoe-shaped object that spins smoothly in one direction, but which, when set spinning in the opposite direction, becomes unstable and reverses direction. This occurs because the axis of the rattleback is very slightly deformed into an S-shape, so slight as to be hard to detect with the naked eye, but sufficient neverthelss to induce this striking behaviour. Attention was first drawn to the phenomenon by G.T.Walker (1896) (later Sir Gilbert Walker, oceanographer, after whom the circulation of the Southern Ocean is named), and has been analysed afresh from time to time, most recently by Moffatt & Tokieda (2008), who describe the celt as a “prototype of chiral dynamics”. The object is chiral, in the sense that it is not mirror-symmetric: it cannot be brought into coincidence with its mirror image. When spun in a clockwise sense, the rattleback is subject to a ‘pitching’ instability which extracts energy from the spin, ultimately inducing the reversal. This instability is then stabilised, but a new weaker ‘rolling’ instability develops, which is then po- tentially capable of causing a second reversal for the same reason. In fact, in ideal (frictionless) circumstances, total energy is conserved, and this behaviour becomes periodic in time. Weak friction (whose precise nature is again as ob- scure as for the Euler disc) damps the energy, and only a finite number of reversals can occur. Figure 5 shows the sort of behaviour that can result from these consider- ations. The blue curve shows the spin N(t) as a function of (dimensionless) time, while the red and green curves show the amplitudes A and B of the pitching and rolling modes of instability, respectively. Weak linear damping of each mode has here been chosen in such a way that four reversals of spin occur before the rattleback comes to rest. I have achieved such behaviour in practice with a carefully machined massive rattleback for which frictional 8 H.K.Moffatt

D ynam ics of SPIN spinning toys with surprising beh aviour!

Th e rattle back is a canoe -s h ape d obje ct th at s pins s m ooth ly in one dire ction but if s pun th e Th e rattle back is a fas cinating toy th at de m ons trate s s pin as ym m e try, oth e r w ay be com e s th e s im ple s t m anife s tation of th e ph e nom e non of Ch iral Dynam ics . It uns table and w ill s pin q uite s m ooth ly in one dire ction; but not th e oth e r! In fact, if s pun th e oth e r w ay, it de ve lops pitch ing and rolling m otions th at contrive re ve rs e s its s pin... to re ve rs e th e dire ction. Th is appare nt violation of cons e rvation of angular m om e ntum is e xplaine d by e ne rgy e xch ange s be tw e e n th e diffe re nt type s of m otion. M otion of fluids can als o be ch iral; th is is re s pons ible for th e dynam o ge ne ration of th e Earth 's m agne tic fie ld. Profe s s or Ke ith M offatt & Dr Tadas h i Tok ie da (M ath e m atics , Unive rs ity of Cam bridge ) Dr M ich al Branick i (M ath e m atics , Unive rs ity of Bris tol) Dr Yutak a Sh im om ura (Ke io Unive rs ity, Japan) Dr Andre w Burbank s (M ath e m atics , Unive rs ity of Ports m outh ) Earth Ph oto: NASA, Vis ible Earth . Fie ld Line s : Gary Glatz m aie r (UCSC). De s ign: Andre w Burbank s

Fig. 4. As for Figure 2; poster for the rattleback.

Fig. 5. Spin reversals of the rattleback: time evolution of spin N(t) (blue) and the amplitudes A(t) of pitching instability (red) and B(t) rolling instability (green); the mathematical model is as derived in Moffatt & Tokieda (2008) The Persistence of Spin 9 forces have been minimised relative to inertial acceleration and gravity. Note how the pitching mode (red) is destabilised when N > 0 and stabilised when N < 0, while precisely the opposite happens for the rolling mode (green). Note also that the rolling mode has a weaker growth rate than the pitching mode, as required to deliver the asymmetric spin behaviour. In fluid mechanics, the simplest measure of chirality in a fluid flow is the helicity Z H = u · ω dV , (7) where u and ω are the velocity and vorticity fields, and the integral is taken over the whole fluid domain. This helicity is invariant under precisely those circumstances for which the Helmholtz laws apply and ‘vortex lines are frozen in the fluid’. It is evident from the rattleback example that very weak chirality can have a profound effect on the observed dynamics of a system in which dissipative effects are small. Similarly in fluid mechanics, helicity can have profound consequences; in particular, it is the mean helicity of a turbulent flow that is responsible for the growth of magnetic fields in conducting fluids due to dynamo instability (Steenbeck, Krause & Radler 1966, Moffatt 1970), i.e. for the very existence of magnetic fields generated in the interiors of planets, stars and galaxies. And what could be more profound than that?

References

1. Bildsten, L. 2002 Viscous dissipation for Euler’s disc. Phys. Rev. E 66, 056309. 2. Bloch, A.M., Krishnaprasad, P.S., Marsden,J.E. & Ratiu, T.S. 1994 Dissipation induced instabilities. Ann. Inst. H.Poincare´, Anal. Nonlin. 11, 37–90. 3. Bou-Rabee, N.M., Marsden, J.E. & Romero, L. 2005 A geometric treatment of Jellett’s egg. Z.Angew. Math. Mech. 85, 1–25. 4. Branicki, M., Moﬀatt, H.K., & Shimomura, Y. 2006 Dynamics of an axisymmetric body spinning on a horizontal surface. III. Geometry of steady states for a general axisymmetric body. Proc.R. Soc. A 462, 371–390. 5. Branicki, M. & Shimomura, Y. 2006 Dynamics of an axisymmetric body spinning on a horizontal surface. IV. Stability of steady states and the ‘rising egg’ phenomenon for convex axisymmetric bodies. Proc.R. Soc. A 462, 3253–3275. 6. Bustamante, M.D. & Kerr, R.M. 2008 3D Euler about a 2D symmetry plane. Physica D 237, 1912–1920. 7. Helmholtz, H. 1858 Uber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, Crelle’s Journal. 8. Hou T.Y. & Li, R. 2008, Blowup or no blowup? The interplay between theory and numerics. Physica D 237, 1937–1944. 9. Hugenholtz, N.M. 1952 On tops rising by friction. Physica 18, 515–527. 10. Jellett, J.H. 1872 A Treatise on the Theory of Friction. Macmillan, London. 11. Maxwell, J.C. 1866 On the viscosity or internal friction of air and other gases. Proc.R. Soc. A 15, 14–17. 12. Mitsui, T., Aihara, K., Terayama, C., Kobayashi, H. & Shimomura, Y. 2006 Can a spinning egg really jump? Proc.R. Soc. A 460, 3643–3672. 10 H.K.Moﬀatt

13. Moffatt, H.K. 1970 Turbulent dynamo action at low magnetic Reynolds number. J. Fluid Mech. 41, 435–452 14. Moffatt, H.K. 2000 Euler’s disk and its finite-time singularity. Nature 404, 833– 834 [see also Nature 408, 540 15. Moffatt, H.K. & Shimomura, Y. 2002 Spinning eggs – a paradox resolved. Nature 416, 385–386. 16. Moffatt, H.K., Shimomura, Y., & Branicki, M. 2004 Dynamics of an axisymmetric body spinning on a horizontal surface. I. Stability and the gyroscopic ap- proximatiuon. Proc.R. Soc. A 460, 3643–3672. 17. Moffatt, H.K. & Tokieda, T. 2008 Celt reversals: a prototype of chiral dynamics. Proc.R. Soc. Edin. 138A, 361–368. 18. Steenbeck, M., Krause, F, & Rädler,K.-H. 1966 Berechnung der mittleren Lorentz–Feldstärke v×B fürein elektrisch leitendes Medium in turbulenter, durch Coriolis–Kräftebeeinflusster Bewegung Z.Naturforsch. 21a, 369–376. 19. Walker, G.T. 1896 On a dynamical top. Q.J. Pure Appl. Math. 28, 175–184.