Precessional states in a laboratory model of the Earth s core

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2011jb009014, 2012 Precessional states in a laboratory model of the Earth s core S. A. Triana, 1 D. S. Zimmerman, 1 and D. P. Lathrop 1,2 Received 9 November 2011; revised 14 February 2012; accepted 15 February 2012; published 5 April [1] A water-filled three-meter diameter spherical shell, geometrically similar to the Earth s core, shows precessionally forced flows. The precessional torque is supplied by the daily rotation of the laboratory by the Earth. We identify the precessionally forced flow to be primarily the spin-over inertial mode, i.e., a uniform vorticity flow whose rotation axis is not aligned with the sphere s rotation axis. A systematic study of the spin-over mode is carried out, showing that the amplitude depends on the ratio of precession to rotation rates (the Poincaré number), in marginal qualitative agreement with Busse s (1968) laminar theory. We find its phase differs significantly though, likely due to topographic effects. At high rotation rates, free shear layers are observed. Comparison with previous computational studies and implications for the Earth s core are discussed. Citation: Triana, S. A., D. S. Zimmerman, and D. P. Lathrop (2012), Precessional states in a laboratory model of the Earth s core, J. Geophys. Res., 117,, doi: /2011jb Introduction [2] Precession s role in the generation of the geomagnetic field is discussed in several studies dating back to Bullard [1949], and including relatively recent numerical studies [e.g., Tilgner, 2005; Wu and Roberts, 2009]. An interesting link between the Earth s orbital parameters (the Milankovich orbital frequencies) and the geomagnetic reversals is suggested by Consolini and De Michelis [2003]. Even without hydromagnetic effects, the study of precessionally forced flows is essential if we wish to understand the basic flows in planetary cores. [3] The flow of a fluid in a rotating and precessing spheroidal cavity has been considered since the end of the 19th century. Motivated by the question of whether or not the Earth s interior was liquid, Hough [1895] and Poincaré [1910] determined that the flow of an inviscid fluid in a precessing spheroid is a uniform vorticity flow driven by pressure. Bondi and Lyttleton [1953] studied the problem of a fluid in a spherical shell in connection with the Earth s precession and its effect on the liquid outer core, and found singularities in the boundary layers located at the so-called critical latitudes. Viscous effects were then included in the studies by Stewartson and Roberts [1963] although their analysis neglected non-linear effects. Busse [1968] was able to include non-linear advective terms in the boundary layer equations and noted the existence of an axisymmetric cylindrical shear layer connecting the critical latitudes in the flow. Qualitative observations by Malkus [1968] and Vanyo 1 Department of Physics, University of Maryland, College Park, Maryland, USA. 2 Department of Geology, University of Maryland, College Park, Maryland, USA. Copyright 2012 by the American Geophysical Union /12/2011JB et al. [1995] confirmed the existence of such a cylindrical shear layer, although they do not show evidence of conical internal shear layers (launched as well from the critical latitudes and propagating on the surface of characteristics) as suggested by Kerswell [1995] and by the computational work of Hollerbach and Kerswell [1995]. The experiments by Noir et al. [2003] with a precessing spheroid were able to resolve both conical and cylindrical shear layers, and obtained good agreement between the measured fluid rotation axis and that predicted by Busse s theory. Numerical studies of a precessing spheroid by Noir et al. [2003] also largely agree with Busse s theory. [4] In a geophysical context, it is relevant to consider the effect of an inner core. A number of numerical studies have been performed for a spherical shell [Hollerbach and Kerswell, 1995; Tilgner, 1999; Rieutord and Valdettaro, 1997; Buffett, 2010] but experimental data is lacking. Vanyo et al. [1995] performed a few experiments incorporating an inner sphere, but no other systematic experimental investigation, apart from the present study, has been carried out yet. Figure 1 shows the location of different studies, both numerical and experimental, in the (E, W) parameter space (for a definition of E and W see sections 2.1 and 2.2). [5] Regarding studies with geometries different from spherical, one of the earliest experiments using liquid metal in a precessing cylinder was performed by Gans [1970]. More recently, numerical computations geared toward experimental studies of hydromagnetism in precessing flows are exemplified by the work of Krauze [2010] and Nore et al. [2011]. [6] The three-meter experiment at the University of Maryland has been designed as a liquid sodium model of the Earth s core. The initial phase of the experimental campaign used water as the working fluid instead. Part of the experimental results obtained during that phase is presented in this report. Spherical-Couette shear flows have 1of11

2 effects are likely small, so we consider r u =0.Ifwe consider a nearly inviscid fluid and only small flow deviations from uniform rotation, we temporarily set Ro = 0 and E = 0. Then, after applying rron both sides of (1) and cross-differentiating we obtain 2 t 2 r2 u þ 4 2 z 2 u ¼ 0; ð4þ which admits transverse helical plane wave solutions u e i[k (r/l) (w/w o)t], provided that w W o ¼2^k ^z: This peculiar dispersion relation indicates the frequency does not depend on the wavelength but only on the direction of the wave vector. The oscillation frequency is limited to w 2W o. The velocity of the phase fronts c p is ð5þ c p ¼2 ^k ^z ^k; k ð6þ and the group velocity c g = r k w(k) is given by Figure 1. Parameter range (W, E) of various precession experiments (Noir, Vanyo, Malkus, Gans, our 3 m), simulations (Noir, Tilgner) and the Earth s core. All these experiments and simulations possess Earth-like geometries except that of Gans [1970], which consists of a precessing cylinder. While there is uncertainty in the value of viscosity in the Earth s core, its approximate Ekman number is also shown [Tilgner, 2007]. also been characterized by Zimmerman et al. [2011] in this device. 2. Theoretical Background 2.1. Inertial Waves in a Rotating Fluid [7] Consider an incompressible fluid with viscosity n in a container whose outer boundary has size L rotating with constant angular velocity W o = W o^z. Using 1/W o as the unit of time and L as the unit of length, the dimensionless Navier- Stokes equation for the flow velocity field u, as measured in the frame rotating with the boundary is u t þ Roðu rþu þ 2^z u ¼ rpþer2 u; ð1þ where p is the dimensionless reduced pressure defined as p ¼ 1 P W o UL r 1 ð 2 W o rþ 2 : ð2þ E and Ro are the Ekman and Rossby numbers defined respectively as E ¼ n W o L 2 ; Ro ¼ U W o L : Above, U is the typical velocity scale of the flow (which is independent of L or 1/W o ). For our experiment, compressible ð3þ c g ¼ 2 k ^k ^z ^k : ð7þ Therefore c p and c g are perpendicular to each other. These inertial waves are in general dispersive, although it is possible to set up a wave packet composed of plane waves with a single common frequency but different wavelengths, resulting in a wave whose envelope (in the direction perpendicular to the energy propagation) stays unchanged in time [Tilgner, 2000]. [8] Inertial waves incident on a solid surface conserve their oscillation frequency, therefore the angle between the wave vector k and the rotation axis ^z remains constant. This is in contrast to the familiar Snell s law of reflection, where the angle of incidence and reflection are the same with respect to the normal of the surface. Given a particular container shape, the waves can thus be focused after multiple reflections to attractors on closed orbits [Rieutord et al., 2001; Maas, 2001] A Precessing Spheroid [9] Let us consider now a spheroidal boundary with polar radius b polar and equatorial radius b (i.e. ellipticity h =(b b polar )/b), rotating as in section 2.1 with angular velocity W o = W o^z, but this time also precessing with angular velocity W p. In the precessing frame of reference both the rotation axis of the container and the precession axis remain constant in space. If we choose U = W o L as the scale for the fluid velocity (resulting in Ro = 1) the dimensionless Navier- Stokes equation in this frame is u t þ ðu rþu þ 2W u ¼ rpþer2 u; ð8þ where W = W p /W o. The dimensionless quantity W = W p /W o is known in the literature as the Poincaré number. According to Busse [1968] (for an alternate derivation see Noir et al. [2003]) the steady state solution is approximated as a uniform vorticity flow in the fluid interior and a thin boundary 2of11

3 layer that matches the no-slip boundary conditions. The angular velocity vector of the fluid interior w in the precessing frame is determined from the implicit relation w=w o A^z ¼ ^z þ ð W ^z ÞþB ð ^z W Þ 2 ðw=w o Þ A 2 þ B 2 ; ð9þ where A and B are defined as sffiffiffiffiffiffiffiffiffiffiffi E p A ¼ 0:259 þ hw=w ð o Þ 2 þ W ^z; B ¼ 2:62 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ew=W o : w=w o ð10þ pffiffiffi The p ffiffiffi expression above is valid for a shell if E is replaced by E (1 g)/(1 g 5 ), where g = a/b, the radius ratio of the inner to the outer sphere. Expression (9) is expected to be accurate as long as W min p ffiffiffi sina; W 1: ð11þ E h In steady state, the axial torque (i.e. along the fluid rotation axis) acting on the fluid is zero. This ensures that no further spin-up of the fluid will occur. This implies that the component of the angular velocity of the shell along the fluid spin axis is equal to the angular speed of the fluid, as shown by Busse [1968]. Indeed, from equation (9) it can be verified that (w/w o ) ^z =(w/w o ) 2. Note also that there is a steady azimuthal drift as seen in the rotating mantle frame (shell frame), since (w/w o ) ^z 1. [10] In the mantle frame of reference, the uniform vorticity flow described by equation (9) is seen as a uniform rotation around a horizontal axis (hence with an azimuthal wave number m = 1) that precesses around the z axis, plus a small uniform azimuthal flow. Although the path followed by a fluid particle is not simple [Pais and Le Mouël, 2001], it can easily be shown that the velocity profile along a vertical line is a uniform oscillation at the rotation frequency [Triana, 2011]. The flow field u in the rotating (mantle) frame is related to the fluid rotation vector w/w o in the precessing (lab) frame through u =[(w/w o ) ^z ] r. In cylindrical coordinates the z component of u at a point (s 0, z 0, f 0 )is u z ðs 0 ; z 0 ; f 0 w 2 W sin a Þ ¼ s 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinðt f þ f A 2 þ B 2 0 Þ; ð12þ where a is the angle between W o and W p, which is independent of z 0. The dimensional velocity is obtained by multiplying the quantity above by W o L, the velocity scale. The angle q between the fluid rotation vector w and the z axis satisfies tan q ¼ p jw ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin aj : ð13þ A 2 þ B 2 This angle can be determined directly from the amplitude of u z if we set w/w o = 1 in (12), which we are allowed to do, given the range of validity of (9). [11] The thin Ekman boundary layer allows the no-slip boundary condition to be met. It turns out that this boundary layer is not thin everywhere on the boundary; it has a divergence in the inviscid limit at the critical latitudes, which are located near 30 from the equator for small Poincaré numbers. For finite viscosity this manifests itself in the form of oscillating, conical internal shear layers, that are actually inertial waves penetrating into the fluid, spawned by the thickening of the Ekman layer at the critical latitudes. These shear layers are organized around cones aligned with the fluid rotation axis, not the rotation axis of the ellipsoidal container. Internal shear layers of this kind are common in enclosed rotating flows. They can be forced directly by a moving object in the flow [Messio et al., 2008], or by the thickening of the Ekman layer at the critical latitudes [Rieutord and Valdettaro, 1997]. They are not restricted to the spin-over mode excited by precession, but are a general feature for inertial modes in complex geometries. [12] The inclusion of an inner core would have a negligible effect on the solution described above if the core has the same ellipticity as the spheroidal container. Assuming spherical boundaries, the frictional torque of an inner core with diameter 0.35 times the diameter of the core-mantle boundary (or CMB, which corresponds to the outer sphere boundary in our experiment) can be estimated to be only 2% of the total frictional torque [Tilgner, 2007]. 3. The Three-Meter Experiment 3.1. Mechanical Description [13] The experimental device consists of a stainless-steel spherical outer boundary with a 2.54 cm thick wall and 2b = 2.92 meters inner diameter. Centered inside is a 2a = 1.01 meter diameter stainless steel sphere, mounted on a 16.8 cm diameter vertical shaft that extends out of the outer vessel, and is driven directly by a 260 kw electric motor. The outer sphere is belt-driven by an identical motor via a toothed pulley on the top lid. The space between the inner and outer spheres is filled with water at room temperature (viscosity n 10 6 m 2 /s). See the schematic drawing in Figure 2. The outer spherical vessel is designed as a liquid sodium holding vessel for hydromagnetic experiments. Those will be the next phase following the experiments reported here. [14] There is no provision for precessional motion of the vessel other than that provided by the Earth s rotation. This constrains the angle between the precession axis and the experiment rotation axis to be the local geographical colatitude, i.e. a =51, at all times. The fixed precession rate is the rotation rate of the earth, i.e. W p = 1/86400 Hz (see Figure 3). Thus the Ekman and Poincaré numbers could not be varied independently of each other. In the precession experiments reported here, the inner sphere motor was not used and the inner sphere shaft was simply locked to the outer sphere as a rigid body. The outer sphere can be made to rotate around its vertical axis in either direction with a speed ranging from 0.03 Hz up to 4 Hz, although the maximum speed achieved for the present study was 1.5 Hz. The corresponding Ekman and Poincare numbers range from E = n/w o b = , W = W p /W o = (at 0.03 Hz) to E = , W = (at 1.5 Hz). For comparison, the Earth s outer core is expected to have E 10 15, and the year precession of its rotation axis yields W = [Tilgner, 2007]. [15] The outer spherical vessel is not an oblate spheroid, but does have imperfections in its shape. According to our design specifications, the inner surface of the vessel should 3of11

4 Figure 4. Meridional cross sections of the vessel as calculated from the spherical harmonics expansion (deviations in mm from mean radius are exaggerated 20). Each color corresponds to the azimuthal angle between North and the corresponding meridional plane. Figure 2. Schematic diagram of the three-meter experiment. The inner sphere is 1.01 m in diameter while the outer sphere diameter is 2.92 m. Both spheres can rotate independently, powered by two 250 kw electric motors. For the experiments described in this paper they rotated together as a solid body. lie between two concentric spherical surfaces whose diameters differ by at most 1 part in 300. We performed measurements of the vessel s cavity using a laser rangefinder and found that the specification was met (see Figure 4). Since the actual shape involves several components in a spherical harmonic expansion (see Table 1), an ellipticity h is not a useful description to describe the outer vessel s shape. At any rate, the outer vessel distorts when spinning; its equatorial diameter is expected to increase slightly, thus increasing its oblateness, similarly to a planet Instruments [16] There are four stainless steel ports, equally spaced on the top lid fitted with sensing transducers connected to two rotating, on-board computers; these send data wirelessly to a laboratory frame (i.e. precessing frame) computer. The ports are labeled clockwise A through D. We use a Cartesian coordinate system whose z axis coincides with the vessel rotation axis. The x axis is contained in the plane defined by both the rotation and precession axes; it points north when the vessel rotation is prograde and south when rotation is retrograde. See the diagram in Figure 5. [17] Ultrasound transducers on port D and B, at 59 cm of cylindrical radius, are used to measure fluid velocity components along lines parallel to the z axis (labeled as chord D and B respectively). The ultrasound technique relies on Doppler-shifted back-scattered ultrasound pulses from Table 1. Outer Vessel Shape and Its Spherical Harmonic Coefficients l m A lm (mm) Figure 3. The experiment precesses as the Earth rotates. Since we can only vary W o experimentally, our Ekman number E = n/w o L 2 and Poincaré number W = W p /W o are not independent i i i i 4of11

5 sphere. In addition, two rotary encoders mounted on the motor shafts provide rotational speed information. 4. Experimental Results [19] In each experimental run the sphere is set to a predetermined constant angular speed, while wall-shear stress, pressure, temperature and fluid velocity are recorded. Typically, 2 to 3 min after the start of the run the sphere reaches its target speed, while the fluid takes considerably longer to reach steady state. The wall-shear stress probe shows a large signal when the outer sphere first reaches full speed (see plot in Figure 6); as the fluid spins up the signal s magnitude progressively diminishes. After many spin-up p times ffiffiffi elapse (the characteristic timescale of spin-up is (W o E ) 1 ), the fluid still does not rotate with the container as a solid body, as evidenced by the residual wall-shear stress level and fluctuations. Rather, it shows oscillations at the same frequency as the sphere rotation. The ultrasound probe on port D also shows oscillations as can be seen in Figure 7. As the fluid spins up to match the sphere s rotation, the sphere s rotation axis continually changes its orientation in inertial space, due to the rotation of the Earth, and the fluid rotation vector lags behind the sphere s axis. The flow generated this way is a realization of the spin-over mode excited by the precessing sphere. [20] Figure 8 shows the standard deviation of the velocity as measured with the ultrasound Doppler transducer on port D as a function of z for a range of retrograde rotation rates. Note the feature details around z/b = 0.77, common to almost all rotation rate profiles. The oscillations as measured along chord B (not shown), which is azimuthally opposite to chord D, have a relative phase difference of p indicating an Figure 5. The Cartesian coordinate system fixed in the precessing (or laboratory) frame. The z axis is chosen along the rotation axis W o. W p is the precession vector and w is the fluid-spin axis. neutrally buoyant particles in the water. For this purpose, a remotely actuated injector loaded with polyethylene particles ( g/cm 3 density, mm diameter) on port D is used. There are also three dynamic pressure probes mounted flush and directly in contact with the water in ports A, B and C. A static pressure sensor is located very near the shaft at the top of the outer sphere. A wall-shear stress probe is mounted in port C; a thermocouple in contact with the fluid is in port D. [18] In order to obtain phase information, an infrared LED and sensor are used in such a way that an electrical pulse is generated in the sensor every time port D crosses the xz plane. This crossing defines the f = 0 angular position of the Figure 6. Signal t w (uncalibrated) from a wall-shear stress probe on the outer boundary. At t = 0 the sphere spins up from rest to a speed of 0.4 Hz. After the initial transient motions dissipate (about 4000 s in this example), small fluid oscillations persist. These oscillations have the same frequency as the rotating sphere. The small higher harmonic components of the signal are due mainly to the non-linear response of the transducer. 5of11

6 Figure 7. Velocity of the steady state flow in the vertical z direction as measured with an ultrasound doppler transducer on port D. The rotation rate is 0.4 Hz. The ultrasound probe is at z/b = (top of vertical axis) and pointing down. At around z/b = 0.71 there are not enough particle scatterers to get an acceptable ultrasound echo. For z/b < 0.23 the signal starts to degrade as well due to weak echoes from particles and unwanted echoes from solid surfaces. m = 1 azimuthal pattern, consistent with the spin-over mode. The velocity amplitude is generally more uniform at larger Ekman numbers (lower rotation rates), and shows more variation at lower Ekman numbers (higher rotation rates). In particular, amplitude and phase variations sharpen at smaller Ekman numbers. [21] For small rotation rates we were able to obtain fulldepth velocity profiles, while at high rotation rates there were often large gaps where particles were absent, due to the particles slight buoyancy and centrifugal separation. Reliable ultrasound echoes common to all rotation rates were registered only up to 600 mm away from the transducer (0.9 > z/b > 0.5). [22] The mean value of each profile in Figure 8 is a single quantity roughly proportional to the spin-over amplitude at each rotation rate. To compare this quantity with the theory discussed in section 2.2, we need the ellipticity that best represents the outer sphere. As we have seen in section 3.1, the outer sphere shape is not a simple ellipsoid, and also has a shaft supporting the inner sphere. In view of this complication, we compare the measurements with the theory using a range of ellipticities close to the outer sphere s RMS deviation from a true sphere. This translates into an ellipticity h 1/280. A further complication comes from the fact that the outer sphere deforms slightly under rotation. Using linear elasticity theory we can estimate the fractional change in the outer sphere radius to be Db b ¼ s ɛ ; ð14þ where s is the hoop stress and ɛ the modulus of elasticity for type 304L stainless steel. Due to this effect one needs to slightly adjust the ellipticity appearing in equation (9) as the rotation rate is increased. The plot shown in Figure 9 summarizes all velocimetry data for all prograde and retrograde Figure 8. Standard deviation s vz of flow velocity along chord D for different retrograde rotation rates. The velocity is measured using an ultrasound Doppler transducer located at z = pointing in the negative z direction. Departures from the flat profiles observed at low rotation rates are caused by internal shear layers that strengthen at lower Ekman numbers. 6of11

7 Figure 9. Standard deviation s vz (depth averaged) of the z component of the fluid velocity as measured from port D. The error bars give a measure of the variability of the velocity amplitudes along z. Colored curves represent the theoretical prediction computed from equation (9) for the standard deviation of v z given different ellipticities h (at rest). Top and bottom horizontal axes show the corresponding Ekman and Poincaré numbers, respectively. rotation rates, compared against theory (equations (9) and (12)) using a range of initial ellipticities (i.e. the ellipticity of the sphere at rest) between 1/500 and 1/200. The agreement is qualitatively good for the intermediate range of E explored. For E >10 6 the observed spin-over amplitudes are smaller than expected from theory, which is a little unsurprising since equation (9) loses accuracy at this E, W values given the validity range shown in equation (11). More importantly, the theoretical curves do not exhibit a trend downward at the smaller E, W range, in contrast with the measurements, which we hypothesize is due to additional amounts of energy dissipation in the bulk of the fluid caused by internal shear layers. [23] The oscillation amplitude reflects the non-zero angle q between the vessel rotation vector W o and the fluid rotation vector w, as modeled in equation (13). To fully determine the orientation of w, it is necessary to measure the oscillation phase with respect to the lab frame. The measured phase provides the azimuthal coordinate f of w directly; see Figure 5 and equation (12). The measured phase profile for different depths is not constant, especially at high rotation rates, as evidenced in Figures 10 and 11. The phase around z/b = 0.77 changes rapidly with depth and eventually becomes larger than 2p for Ekman numbers smaller than (rotation rates higher than 1.0 Hz). Therefore a single quantity representing the overall phase for a given rotation rate profile is not well defined, given the large variability in the profile at high rotation rates. For low rotation rates an average of the phase is nevertheless useful to estimate of the overall phase. The average of the phases is calculated as a vector sum and not arithmetically (since the average between, e.g. p and p is not 0). Figure 12 shows a comparison of this average and the expected phase as calculated from equation (9). There is qualitative agreement between theory and experiment only for the higher Ekman or Poincaré numbers (slower rotation rates). [24] Let us examine in greater detail the feature around z/b = The space-time velocity diagram in Figure 13 shows a steep gradient (a shear zone) in the z component of the fluid velocity around z/b = Other measurements 7of11

8 Figure 10. The phase of the fluid velocity oscillations f D as measured from chord D at different retrograde rotation rates. Note the feature near z/b = Phase is measured with respect a synchronization signal in the precessing (lab) frame. Figure 11. The phase of the fluid velocity oscillations f B for different prograde rotation rates, measured from port B. Note the rapid change near z/b = 0.77 as the Ekman number decreases. The feature near z/b = 0.57 is an artifact of echoes, reflecting multiple times from solid boundaries, dominating the echoes from tracer particles. 8of11

9 Figure 12. Average phase f of fluid velocity oscillations as a function of Poincaré number for all prograde and all retrograde rotation rates. Colored lines are the predicted phases for different ellipticities. with the ultrasound velocimetry beam originating from different azimuthal positions (using other the other instrumentation ports, which are spaced p/2 radians apart) show the same shear feature at near the same location. Therefore, we believe the shear zone exists at all azimuthal angles, which is consistent with the presence of a conical shear layer as observed in computations and experiment [Hollerbach and Kerswell, 1995; Noir et al., 2003]. The ultrasound velocity measurements intersect these cones at specific heights. There may be more shear layers at larger depths, but this is very difficult to observe due to limitations in the ultrasound velocimetry technique. Back to Figure 13, in the range < z/b < the shear can be described as a phase front propagating down along the profile. This feature was visible only for E < A rough estimate of the feature s wavelength is 58 mm (rotation rate is 1.3 Hz). Assuming this is a shear layer inclined 30 from the vertical (as is expected from equation (5) since the dimensionless oscillation frequency at any point in the profile is 1), then the actual wavelength is l 29 mm. The phase speed c p corresponds to the smallest inclination of the phase fronts in Figure 13, which is 36.5 mm/s as measured from the figure itself. This is in reasonable agreement with equation (6) as the magnitude of the phase velocity is c p = lw o /2p = 37.7 mm/s. 5. Discussion [25] For a cavity with a given shape, the amplitude of the spin-over mode excited by precession depends on the Ekman and Poincaré numbers only. The fact that the measured order of magnitude of the spin-over amplitude is consistent with Busse s theory suggests that the effect of the shaft, or small scale topographical details of the outer boundary, are relatively unimportant when it comes to the energy balance. On the other hand, the measurements indicate that this picture might not hold for E <10 8 if the flow is dominated by internal shear layers spawned from the boundaries. [26] The lower the Ekman number is, the more the flow seems to deviate from a uniform vorticity flow. The flow is still oscillating coherently in space and no turbulent breakdown seems to occur. We hypothesize that the internal shear layers become more and more dominant, significantly altering the original uniform vorticity (spin-over) flow. The phase f (or azimuthal orientation) of the fluid spin axis deviates significantly from the prediction for E < of11

10 Figure 13. An internal shear layer, seen here between z/b = and z/b = 0.775, for E = These layers become more evident at lower Ekman numbers. Note the phase is nearly the same above and below the shear (e.g. along the dashed vertical line). The phase discrepancy might be related to a different torque balance from that of a precessing spheroid as the theory assumes, given that the pressure torque depends primarily on the actual shape of the outer boundary. [27] The kinetic energy is less evenly distributed in space as the Ekman number is reduced, a fact already familiar noted in numerical studies [Rieutord et al., 2001]. Our basic observation is that both the amplitude profile and the phase profile along the z direction develop sharper features as the Ekman number decreases. The observed spin-over amplitude tends to get smaller than theoretically expected, a fact that might be attributed to additional amounts of energy dissipation caused by the increasingly dominant internal shear layers. [28] At small Ekman numbers, topographical features on the boundaries can shed inertial waves, propagating into the interior forming additional free shear layers, just as the singularity in the Ekman layer does at the critical latitudes. The feature observed near z/b = 0.77 could be caused by a topographical feature, in which case its size would be set by the size of the feature, and would therefore be independent of the Ekman number. But Figure 8 suggests that its size decreases as E decreases, just as many numerical simulations and analysis show (e.g. that of Kerswell [1995]). In fact, the width of the internal shear layer, according to those studies, is expected to be of the order of E 1/3, amounting to 11 mm. This compares relatively well to our measured width of 29 mm if the somewhat arbitrary definition of the shear layer s spatial extent is taken into consideration. [29] Figure 14 shows the raypaths emanating from the critical latitudes of both the inner and the outer boundary. The ultrasound beam first crosses one of the rays at around z/b = 0.74, but we can see that a small change in the boundary shape will change the precise location of the crossing. In fact, deviations from a sphere can dramatically change the ray pattern and thus the amplitude and phase profile along any chord. [30] The velocity amplitude of the internal shear layer emanating from the inner sphere, according to Kerswell [1995], should scale as E 1/6 (where is the basic spinover amplitude as seen in the mantle frame) which amounts to 1.2 mm/s for E = This is about an order of magnitude smaller than our measurements suggest. From Figure 8 we estimate a velocity amplitude of 20 mm/s for the flow near the phase discontinuity. Unfortunately, in our measurements the shear layer was detected only for the lowest Ekman numbers, preventing a systematic determination of Ekman number dependence. The discrepancy suggests that the numerical scalings established in previous studies may be inaccurate when considering Ekman numbers smaller than We hope to encourage more research on this issue, since our results leave the extrapolation of velocity amplitudes in the outer core of the Earth on uncertain grounds. 10 of 11

11 Figure 14. Raypaths of inertial waves launched from the critical latitudes up to seven reflections. Green paths emanate from the critical latitudes at the outer boundary and blue paths originate at the critical latitudes on the inner boundary. The two dotted vertical lines show the location of the ultrasound beams used for velocimetry. The first crossing of the ultrasound beams with one of the rays occurs near z/b=0.74. Compare this to the experimentally measured location of the shear layer at z/b=0.77. [31] The Earth s free core nutation is a manifestation of the spin-over mode in the liquid outer core. Dissipation occurs at the CMB as well as at the inner core boundary, but the precise amount is still subject of some debate [Le Mouël et al., 2006]. Our observations suggest that an additional amount of dissipation takes place in the bulk of the fluid, in the form of conical shear layers. In fact, this is consistent with work by Buffett [2010] showing that this might account for the discrepancy between very long baseline interferometry (VLBI) measurements of the Earth s nutation [Herring et al., 2002] and analytical models such as the one described by Mathews et al. [2002]. [32] Acknowledgments. We wish to thank Don Martin for his invaluable technical support over many years of experiment construction. We also gratefully acknowledge financial support from the National Science Foundation under grants EAR , EAR , and EAR References Bondi, H., and R. A. Lyttleton (1953), On the dynamical theory of the rotation of the Earth: The effect of precession on the motion of the liquid core, Proc. Cambridge Philos. Soc., 49(3), Buffett, B. (2010), Tidal dissipation and the strength of the Earth s internal magnetic field, Nature, 468, Bullard, E. C. (1949), The magnetic field within the Earth, Proc. R. Soc. London A, 197(1051), Busse, F. H. (1968), Steady fluid flow in a precessing spheroidal shell, J. Fluid Mech., 33, Consolini, G., and P. De Michelis (2003), Stochastic resonance in geomagnetic polarity reversals, Phys. Rev. Lett., 90(5), , doi: / PhysRevLett Gans, R. F. (1970), On hydromagnetic precession in a cylinder, J. Fluid Mech., 45, Herring, T. A., P. M. Mathews, and B. Buffett (2002), Modeling of nutation-precession: Very long baseline interferometry results, J. Geophys. Res., 107(B4), 2069, doi: /2001jb Hollerbach, R., and R. R. Kerswell (1995), Oscillatory internal shear layers in rotating and precessing flows, J. Fluid Mech., 298, Hough, S. S. (1895), The oscillations of a rotating ellipsoidal shell containing fluid, Philos. Trans. R. Soc. London A, 186, Kerswell, R. R. (1995), On the internal shear layers spawned by the critical regions in oscillatory Ekman boundary-layers, J. Fluid Mech., 298, Krauze, A. (2010), Numerical modeling of the magnetic field generation in a precessing cube with a conducting melt, Magnetohydrodynamics, 46, Le Mouël, J.-L., C. Narteau, M. Greff-Lefftz, and M. Holschneider (2006), Dissipation at the core-mantle boundary on a small-scale topography, J. Geophys. Res., 111, B04413, doi: /2005jb Maas, L. R. M. (2001), Wave focusing and ensuing mean flow due to symmetry breaking in rotating fluids, J. Fluid Mech., 437, Malkus, W. V. R. (1968), Precession of Earth as cause of geomagnetism, Science, 160, Mathews, P. M., T. A. Herring, and B. A. Buffett (2002), Modeling of nutation and precession: New nutation series for nonrigid Earth and insights into the Earth s interior, J. Geophys. Res., 107(B4), 2068, doi: / 2001JB Messio, L., C. Morize, M. Rabaud, and F. Moisy (2008), Experimental observation using particle image velocimetry of inertial waves in a rotating fluid, Exp. Fluids, 44, Noir, J., P. Cardin, D. Jault, and J. P. Masson (2003), Experimental evidence of non-linear resonance effects between retrograde precession and the tilt-over mode within a spheroid, Geophys. J. Int., 154(2), Nore, C., J. Léorat, J.-L. Guermond, and F. Luddens (2011), Nonlinear dynamo action in a precessing cylindrical container, Phys. Rev. E, 84, , doi: /physreve Pais, M. A., and J.-L. Le Mouël (2001), Precession-induced flows in liquid-filled containers and in the Earth s core, Geophys. J. Int., 144(3), Poincaré, H. (1910), Sur la précession des corps déformables, Bull. Astron., 27, Rieutord, M., and L. Valdettaro (1997), Inertial waves in a rotating spherical shell, J. Fluid Mech., 341, Rieutord, M., B. Georgeot, and L. Valdettaro (2001), Inertial waves in a rotating spherical shell: Attractors and asymptotic spectrum, J. Fluid Mech., 435, Stewartson, K., and P. H. Roberts (1963), On the motion of a liquid in a spheroidal cavity of a precessing rigid body, J. Fluid Mech., 17(1), Tilgner, A. (1999), Driven inertial oscillations in spherical shells, Phys. Rev. E, 59(2), Tilgner, A. (2000), Oscillatory shear layers in source driven flows in an unbounded rotating fluid, Phys. Fluids, 12(5), Tilgner, A. (2005), Precession driven dynamos, Phys. Fluids, 17(3), , doi.org/ / Tilgner, A. (2007), Rotational dynamics of the core, in Treatise on Geophysics, vol. 8, edited by G. Schubert, pp , Elsevier, New York. Triana, S. A. (2011), Inertial waves in a laboratory model of the Earth s core, PhD thesis, Univ. Md., College Park. Vanyo, J., P. Wilde, P. Cardin, and P. Olson (1995), Experiments on precessing flows in the Earth s liquid-core, Geophys. J. Int., 121(1), Wu, C. C., and P. H. Roberts (2009), On a dynamo driven by topographic precession. Geophys. Astrophys. Fluid Dyn., 103(6), Zimmerman, D. S., S. A. Triana, and D. P. Lathrop (2011), Bi-stability in turbulent, rotating spherical Couette flow, Phys. Fluids, 23, , doi.org/ / D. P. Lathrop, S. A. Triana, and D. S. Zimmerman, Department of Physics, University of Maryland, College Park, MD 20742, USA. (lathrop@umd.edu; triana@umd.edu; dsz@umd.edu) 11 of 11