On Viscosity Estimates for the Earth's Fluid Outer Core and Core-Mantle Coupling. L. Ian LUMS and Keith D. ALDRIDGE

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1 J. Geomag. Geoelectr., 43, ,1991 On Viscosity Estimates for the Earth's Fluid Outer Core and Core-Mantle Coupling L. Ian LUMS and Keith D. ALDRIDGE Department of Earth and Atmospheric Science, York University, North York, Ontario M3J 1P3, Canada (Received December 11,1989; Revised August 14, 1990) The kinematic viscosity of the Earth's fluid core is uncertain because estimates inferred from studies of liquid metals and from real-earth measurements differ significantly. The role of dynamic effects is invoked in an attempt to reconcile this conflict. The implications of these additional effects for the coupling between a fluid and its container are discussed with respect to the nearly-diurnal free wobble and the free core nutation, which depend on the properties of the fluid core near the core-mantle boundary. 1. Overview The viscosity of a fluid is a fundamental property whose approximate knowledge is critical for understanding dynamical processes within that fluid, as well as interactions between the fluid and its container. Reconciliation of viscosity estimates for the Earth's fluid outer core, with values ranging over some twelve orders of magnitude (e.g. POIRIER, 1988), forms the primary motivation for the current work. In Section 2, the discussion begins by reviewing estimates of the core's viscosity, and an attempt is made towards understanding these widely-varying estimates by considering the importance of dynamical effects. Part of the resolution rests on the use of either an effective viscosity or an eddy viscosity, the latter of which is used commonly in atmospheric dynamics (e.g. STULL, 1988). In Section 3, the strength and usefulness of this interpretation is considered with the use of an illustrative example from laboratory inertial wave studies. In Section 4, an application of this work concerns the coupling of the outer core and mantle in the Earth's free core nutation (FCN) or corresponding nearly-diurnal free wobble (NDFW). Since the dynamical response of the core at nearly-diurnal periods is expected to be dominated by nonmagnetic effects (e.g. CROSSLEY, 1984), it is reasonable to apply the present findings augmented by knowledge from laboratory studies of inertial waves, to suggest reasons for the 26-day discrepancy between the observed and theoretical values for this rotational mode. In Section 5, the paper concludes by stressing the importance of using an effective or eddy viscosity in dynamical considerations of internal core modes and core-mantle coupling. 2. Viscosity Estimates Estimates for the viscosity of the fluid core have been derived by a wide variety of unrelated methods, and a compilation of these values is provided in Table 1. It is beyond the scope of this paper to review the manner in which each of these determinations was 93

2 94 L. I. LUMB and K. D. ALDRIDGE Table 1. Viscosity estimates for the Earth's fluid outer core. Sources: AL ALDRIDGE and LUMB (1987), Ba BACKUS (1968), BK BUKOWINSKI and KNOPOFF (1975), Bu BULLARD (1949), G GANS (1972), GHS GWINN et al. (1986), H HIDE (1971), J JEFFREYS (1976), Mi MIKI (1952), Mo MOLODENSKIY (1981) SE SATO and ESPINOSA (1967), NHZ NEUBERG et al. (1990), OFFICER (1986),P POIRIER (1988), R RIEUTORD (1990),, SJ SCHLOESSIN and JACOBS (1980), SS SUZUKI and SATO (1970), SAAB SVENDSEN.et al. (1989), T66 TOOMRE (1966), T74 TOOMRE (1974), WK WON and KUo (1973), and Ys YATSIV and SASAO (1975). made, and the interested reader is referred to the original articles for the details. Listed in the first column of Table 1 is a brief description of the method used to obtain the fluid's viscosity. The superscripted letters define the source of the estimate, and correspond to the references given at the bottom of the table. Shown in the second column of the table is the inferred kinematic viscosity, v, in units of cm2/s, and in order of increasing value. Using the viscosity it is possible to calculate the corresponding Ekman number for the liquid core, which is given in the third column of Table 1. Used in the context of a contained rotating fluid (e.g. GREENSPAN, 1968), the Ekman number, E, is a gross measure of the importance of the viscous force relative to the Coriolis force, (1) where is the rotation rate of the fluid and a is the radius of the container. There are a number of comments and observations to which Table 1 lends itself.

3 On Viscosity Estimates for the Earth's Fluid Outer Core and Core-Mantle Coupling 95 The diverse methods can be broadly categorized as studies involving: laboratory analogs of the core fluid; the theory of the molecular viscosity of liquid metals; the rotation of the Earth; the passage of seismic waves through the Earth; observations from long-period gravimetry (LPG) and very-long-baseline interferometry (VLBI); correlations between the Earth's gravitational and magnetic fields. As a distressing consequence of the number and distinctiveness of the measurements, viscosity values for the fluid core range over some twelve orders of magnitude, as was noted by POIRIER (1988). Traditionally, the higher estimates have been dismissed as upper bounds, while the lower estimates have been taken as reliable. Thus a prejudice towards a kinematic viscosity comparable to that of water, i.e. 0.01cm2/s, has led to the widespread use of the CANS (1972) value. Geodynamicists have faithfully used this estimate in the modeling of core-mantle coupling, core hydrodynamics, geodynamo theory, and secular core flows. One exception was the effort of OFFICER (1986) to justify the use of a seismicallydetermined core viscosity in his model of the geodynamo. There is a natural division of the estimates presented in Table 1 into two groups. With the exception of the sixth, the first ten values are based on theoretical and laboratory studies of liquid metals (LM), some of which have been extrapolated to sensible core temperatures and pressures. The remainder of the values in the table were inferred from measurements of the real Earth (RE). While the LM estimates show a fair degree of consistency amongst themselves, the range apparent in the RE estimates is presumably a result of differences in both the method and time scale of the sampling process. This simple division has been noted elsewhere in the literature, and the high estimates have been explained in various ways. GUO (1989) has noted the LM/ RE distinction, but reverts to the traditional approach of treating the LM and RE estimates, respectively, as lower and upper bounds. Although VOORHIES (1988) has also noted the LM/ RE distinction, he comments on the explainable large, seismically-determined viscosities; but does not discuss the results from the remainder of the RE measurements. Although shear viscosity is important in the attenuation of longitudinal waves in liquid metals, bulk viscosity will be of similar size in liquid metals at low pressure and relevant to compressional waves (e.g. ANDERSON, 1989, p. 295). This suggests that a deeper understanding of the seismically-derived values may be gathered by decomposing the shear-stress tensor into its isotropic and nonisotropic parts (e.g. BATCHELOR,1970, p. 142). If we further assume a simple linear relationship between each of these parts, and their associated rate-of-strain tensors, then constant coefficients define the ratio of stress to strain. Thus the isotropic part defines a bulk viscosity, while the nonisotropic part defines a shear viscosity (e.g. BATCHELOR, 1970, pp. 144 and 154), each of which may take on significantly different values for the same fluid. Indeed, ANDERSON (1980) has noted such a case where he appeals to structural or compositional fluctuations as being responsible for the excessive bulk-viscosity absorption of radial modes. Also STEVENSON (1983) has suggested that the source of these "anomalous bulk viscosities" may be due to the presence of two compositional phases in the Earth's core. These arguments may justify the large bulk viscosity estimates from P-wave studies (e.g. JEFFREYS, 1976), through their effect on the rate-of-expansion tensor. S wave, shear viscosity estimates (e.g. SATO and ESPINOSA, 1967; SUZUKI and SATO, 1970; OFFICER, 1986) may be similarly explainable, since they depend on a difference between the rate-of-strain and rate-of-expansion tensors. In any event,

4 96 L. I. LUMB and K. D. ALDRIDGE unmodeled physical processes appear to be responsible for the disparity between these seismic estimates for core viscosity and those found from LM studies. The collection of estimates presented in Table 1 clearly indicates the poor grasp of a reasonable value for this very important physical property of the liquid core. Therefore, an obvious priority involves the reconciliation of these estimates in some geophysicallymeaningful way. The following simple explanation is proposed: Since the LM group of results are based on theoretical and laboratory studies of the properties of liquid metals, these results are representative of the molecular viscosity of a fluid. In contrast, the RE group of results are inferred from the real-earth measurements where there are various unmodeled dynamical processes involved. Therefore, the RE values should be considered as effective or eddy viscosities, which are properties of some hypothetical flow. An effective or eddy viscosity simply recognizes that the unmodeled dynamics of the flow are so complicated as to seemingly modify the molecular viscosity in some fashion. However, some justification needs to be given for the context in which these measures for the viscosity of a flow are intended for the case of the fluid core. Laboratory inertial wave experiments on a rotating fluid in a spheroidal shell (e.g. LUMB and ALDRIDGE, 1988), which have shown the importance of viscous and nonlinear effects, have been applied to the dynamics of the Earth's fluid core (e.g. ALDRIDGE et al., 1989). Turbulent flow, which could result from shearing of the fluid, boundary roughness (e.g. STULL,1988) or large-amplitude forcing (e.g. LUMB and ALDRIDGE,1991), has also been suggested for the Earth's fluid core (e.g. TOOMRE, 1966; ROCHESTER, 1970; RIEUTORD, 1990). In these examples, nonlinear wave interaction and turbulent flow, respectively, are both described by a modified viscosity, i.e. an effective or eddy viscosity. The following illustrative example explores this connection in greater detail. 3. Inertial-Wave Experiments The above interpretation of the viscosity estimates of Table 1, as a measure of fluid or flow property, may be better understood through the following example. An analogy may be drawn from the viscosity estimates derived from laboratory studies of inertial waves. Since the details of these experiments have been given elsewhere (e.g. STERGIOPOULOS and ALDRIDGE, 1984; LUMB and ALDRIDGE, 1988), only a brief summary is presented here. A spheroidal inner boundary perturbs at angular rate co a fluid-filled spherical cavity which rotates on a turntable at angular rate Q. When the semi-major axis of the inner spheroid is inclined from the equatorial plane of the cavity, inertial waves of azimuthal number unity are excited through normal pressure forces. The tilt of the inner spheroid scales as the Rossby number for the flow, where this dimensionless number is a gross measure of the convective acceleration relative to the Coriolis acceleration (GREENSPAN,.1968). Inertial waves in a contained, rotating fluid exist as a result of the near-balance between pressure-gradient and Coriolis forces. Any member of the infinite suite of modes can be excited by forcing the system near a resonant point. One experimental method is to force this system near the location of a resonance, switch off the perturbation, and observe the excited mode as it decays freely. The results of such a free-decay experiment are shown in Fig. 1, which is a time series of the dimensionless disturbance pressure. The expectation that a linear system would produce a simple envelope of exponential decay clearly is not satisfied here. In contrast, the observed decay is seen to be much more

5 On Viscosity Estimates for the Earth's Fluid Outer Core and Core-Mantle Coupling 97 Free decay near ƒö/ƒ =1.000 ƒã =0.14 rad, E=8.8e-5, T=13.61 sec. Fig. 1. Disturbance pressure time series of freely decaying inertial waves for forcing at ƒér=1.000, with ƒã =0.14 rad, and E=8.8 ~10-5. complex. A significant nonlinear interaction among modes during free decay, which gives rise to the finding of a suite of modes after a least-squares analysis, is apparently the cause of this nonmonotonic decay (e.g. LUMB and ALDRIDGE, 1988). The nature of this nonlinear interaction can be better understood if we follow SMITH (1977) and plot the recovered eigenfrequencies as a function of the degree of the Legendre polynomial in the relevant eigenfrequency equation (see Eq from GREENSPAN, 1968). In Fig.2, it is clear that the free decay of the "spin-over" mode leads to the excitation of its higher-order harmonics. These modes are higher-order relative to the "spin-over" mode because of increased spatial complexity, an observation that can readily be made by comparison with the eigenfunctions presented in ALDRIDGE et al. (1988, 1989). Based on this evidence, it appears that the primary nonlinear effect is manifested through wave-wave interactions throughout the interior of the fluid; the source of these interactions is the advective term in the momentum equations (see e.g. Eq. (1) of GREENSPAN, 1969), whose amplitude is scaled by the Rossby number. Although no velocity measurements are available in the above experiments, it is expected that the interaction of inertial waves

6 98 L.I. LUMS and K. D. ALDRIDGE Splitting of retrograde arid prograde k=1 inertial waves (sphere) Fig. 2. "Splitting" of azimuthal-number-unity (k=1) inertial waves. The star marks the excited mode, while the diamonds show the modes recovered from the free-decay experiment illustrated in Fig. 1. The plus signs indicate the remaining k=1 inertial modes which were not excited during this experiment. would give rise to a time-varying zonal flow (e.g. GREENSPAN, 1969; ALDRIDGE et al., 1989), which in turn will interact with the inertial oscillations. Shown in the first and second columns of Table 2 are the measured values of the amplitudes and eigenfrequencies (plotted in Fig. 2 as retrieved by the least-squares analysis together with the formal error involved in such an analysis). In the third column of the table are values of these eigenfrequencies as derived from a theoretical expression for a spherical cavity (see again Eq from GREENSPAN, 1968), while the fourth column details the effect of the first-order ((-E1/2) viscous correction using values from Table 2.1 in Greenspan and Eq. (6). Presented in the last column of the table are the modal numbers which correspond to a particular eigenfrequency. The modal numbering scheme adopted here is that of ALDRIDGE and LUMB (1987) in which k is the azimuthal wavenumber, n is the degree of the associated Legendre polynomial of the frequency expression, and m is a measure of the radial complexity of a particular mode. The modes are tentatively identified by matching the observed and predicted frequencies.

7 On Viscosity Estimates for the Earth's Fluid Outer Core and Core-Mantle Coupling 99 Since the amplitude of forcing in these experiments is fairly large, the excitation of other modes during free decay is an expected nonlinear effect. By modeling the time series as a sum of decaying sinusoids, the eigenfrequencies of these modes have been retrieved by a linearized least-squares method which is described in detail elsewhere (e.g. ALDRIDGE and STERGIOPOULOS, 1989). Even though the effect of the inner boundary and viscosity have been neglected in GREENSPAN's (1968) linear theory, it is clear that the agreement between measured and predicted frequencies is quite good. Although the inclusion of the viscous correction (see column four of Table 2) marginally improves the agreement between measured and calculated eigenfrequencies, it is clear that there still exist some small but significant discrepancies. When the same linear theory is used to interpret the decay rates of these modes, the comparison appears as in Table 3. In the first column, the experimentally-derived decay rates, are presented, while the second column presents theoretical values from GREENSPAN (Table 2.1, 1968). The modal identifications based on frequency matching are presented in the last column. Comparison of columns one and two indicates that the linear theory is insufficient to explain the observed decay rates. Even if the large uncertainties in the measured values are considered, the interaction among the decaying waves leads to estimates of decay rates for individual modes which are greatly distorted. Such anomalous apparent decay has already been noted by STERGIOPOULOS and Table 2. Dimensionless amplitudes and eigenfrequencies for forcing at,ċ=0.14, E=8.8 ~10-5 (after LUMB and ALDRIDGE, 1988). Table 3. Dimensionless decay rates for forcing at, with e=0.14 rad, E=8.8 ~10-5 (after LUMB and ALDRIDGE, 1988), and effective viscosity as implied by decaying inertial waves.

8 100 L. I. LUMB and K. D. ALDRIDGE ALDRIDGE (1984). Suppose, however, that our a priori knowledge of the complexities of this system is ignored. Then, using the decay rates given in Table 3, estimates of the viscosity of the fluid would be inferred from (2) Such estimates, which have been tabulated in the third column of Table 3, exhibit a variation over two orders of magnitude. These results can be contrasted with the viscosity of cm2/sec, which was measured by a viscometer. Hence this latter measure of the viscosity, and the viscosities implied by the freely-decaying inertial waves cannot be sensibly compared. The comparison fails since the model used does not fully account for the relevant dynamics. The results obtained from the decaying inertial waves are directly the result of an incorrect application of the linear theory for a flow where aevection is probably important, whereas those measured by the viscometer represent a true molecular viscosity. The analogy between this laboratory inertial wave situation and the interpretation provided for Table 1 should now be self-evident. A similar illustrative example is available from the work of CROsSLEY and SMYLIE (1975) who considered the laminar, viscous damping of spheroidal free oscillations by a compressible liquid core. Using the CANS (1972) estimate they performed a forward calculation for the Q value of viscous shear dissipation for several modes. Their results (presented in their Table 4) vary over some four orders of magnitude in Q, which in turn implies an effective viscosity which varies over a similar range. The use of a linear theory to understand these more complex situations leads to meaningless results. 4. The Nearly-Diurnal Free Wobble An important application, which will serve to illustrate the differences between the molecular and effective viscosities of a fluid, concerns the coupling of a fluid with its container. This type of coupling, such as between the outer core and mantle, is considered in the geophysical context of the NDFW. This retrograde motion of the instantaneous rotation axis relative to the figure axis, as seen from Earth, has a period that is close to 1 sidereal day (e.g. NEUBERG et al., 1987). The associated motion, as seen from space, is known as the FCN. Although the FCN is expected to have an amplitude some 460 times larger than that of the NDFW (To0MRE,1974), the detection of the FCN by VLBI (e.g. GwINN et al., 1986; HERRING et al., 1988) has been questioned (e.g. ALDRIDGE et al., 1989). Since the luni-solar gravitational attraction results in deformation of the Earth's mantle, there exists the possibility of measuring the frequency of the smaller NDFW indirectly. Ongoing theoretical studies (e.g. HINDERER et al., 1987; HINDERER and LEGROS, 1989) have determined that this elastic deformation is resonant at nearlydiurnal periods, due to the proximity in period of these modes and the NDFW. Thus, the proximity to resonance leads to an amplification of the tidally-induced nutations at nearly-diurnal periods, which can be detected by measuring its effect on the tidal

9 On Viscosity Estimates for the Earth's Fluid Outer Core and Core-Mantle Coupling 101 gravimetric factor. Such a test of this theory was made by NEUBERG et al. (1987) who stacked records from several LPG, to retrieve the complex eigenfrequency of the NDFW, by measuring its effect on several tidally-induced nutations with periods close to 1 sidereal day. They obtained an estimate of (1+1/434.2 }7) for the eigenfrequency of the NDFW, and a rate of fractional dissipation, Q, of 2800 }500. Comparable results have been reported by RICHTER and ZURN (1988) who obtained a period of (1+1/431.2 }3.2) and a Q of 3120 }323. The values cited above from LPG agree very well with the day estimate for the FCN from VLBI (GwINN et al., 1986). Both these observations are in contrast with the predicted 460 day value (e.g. ROCHESTER et al., 1974) by some 26 days. Since the eigenfrequency of the NDFW, or equivalently the eigenperiod of the FCN, depends on a number of real-earth parameters, much effort has been devoted towards trying to explain this 26-day discrepancy. More specifically, the complex eigenfrequency of the NDFW, QNDFW, can be expressed (e.g. NEUBERG et al., 1987) as (3) where Q is the rotation rate of the Earth, A and Am are the moments of inertia of the Earth and mantle, respectively, plus ƒ c is the dynamic ellipticity of the core-mantle boundary (CMB). is the ratio of the centrifugal to gravitational forces at the Earth's surface for a core of length scale a, while hc refers to the elastic behavior of the CMB. The viscomagnetic coupling constants, K and K' due to LoPER (1975) and ROCHESTER (1976), have viscous contributions (4) and where E is the Ekman number as before. The form of these coupling parameters has been derived from earlier work (e.g. STEWARTSON and ROBERTS, 1963; BUSSE, 1968) which was concerned with fluid flow in a precessing spheroidal shell. Since the excitation of the NDFW involves an exchange of angular momentum between the mantle and outer core, an expected dynamical consequence of the instantaneous misalignment of the angular momentum vectors of the mantle and outer core is the excitation of internal modes of oscillation in the contained fluid (e.g. STERGIOPOULOS and ALDRIDGE, 1982). If the outer core is perturbed in a retrograde sense (as seen by an observer in the rotating frame) at precisely the frequency of rotation of the mantle, this misalignment would lead to the excitation of the real-earth counterpart of the fundamental nonaxisymmetric inertial mode. This (2,1,1) or "spinover" mode is easily amenable to laboratory study, and has been excited by a boundary deformation with a unit azimuthal wavenumber perturbation (e.g. LUMB and ALDRIDGE, 1988), which is analogous to the misalignment perturbation mentioned above. Thus there exists an intimate relationship between this inertial mode and the NDFW which has been (5)

10 102 L. I. LUMB and K. D. ALDRIDGE alluded to in the literature (ALDRIDGE et al., 1989). To explore this connection in greater detail, consider the following viscous corrections, 2211,1, to the complex eigenfrequency of the (2,1,1) mode, 2211,0, from the rotating fluids literature (e.g, after GREENSPAN, Eq , 1968), i.e. (6) where all terms have been made dimensionless by scaling with the rotation rate of the container, Q. Viscous corrections, which are applied to the inviscid eigenfrequency are given in Table 2.1 from GREENSPAN (1968) as 2.62 and (cf. Eqs. (4) and (5)), respectively, for 21,1 and The results tabulated by GREENSPAN (1968) were obtained previously by BUSSE (1968) as well as STEWARTSON and ROBERTS (1963) all of whom were motivated by the role of precession as energy source for the geodynamo. Thus the result from rotating fluids, when scaled by the ratio of the moments of inertia A/Am, yields the viscous corrections for the real Earth as seen in Eq. (3). Not only does this demonstrate the connection between the rotating fluids literature and the study of Earth's deep interior, it also serves to illustrate the relevance and importance of our own inertial-wave experiments to global geodynamics. NEUBERG et al. (1990) considered the problem of evaluating the effects of various geophysical contributions to andfw in an attempt to explain the 26-day discrepancy. To this end they defined a frequency shift, ƒ QNDFW, between the observations and theory, to represent 100% of the effects unaccounted for. Their procedure was to see how much of this discrepancy they could account for by any one of various geophysical effects, and a summary of their findings is presented in Table 4. Although anelasticity was expected to account for the 26-day difference, recent calculations (e.g. WAHR and BERGEN, 1986; DEHANT,1990) suggest that this factor tends to increase the difference by 20%. Reasonable variations in mantle rigidity lead to a }9% compensation for the 26-day difference. The lack of participation of the inner core in the NDFW, which leads to a change in the inertia of the outer core, can account for only 2% of the discrepancy and the change is in the wrong direction (NEUBERG et al., 1990). Coupling between the inner core, outer core and mantle also yields an additional mode, the "free inner core nutation" (FICN) (DE VRIES and WAHR,1988, 1989), for which some observational evidence from VLBI exists (e.g. MATHEWS et al., 1988). For a hydrostatic Earth, the FICN is a prograde, rigid-rotational motion of the inner core relative to the outer core about an equatorial axis with a period of 471 days (DE VRIES and WAHR,1990). The presence of this mode at Table 4. Frequency of the NDFW: Theory-Observation Discrepancy (after NEUBERG et al.,1990).

11 On Viscosity Estimates for the Earth's Fluid Outer Core and Core-Mantle Coupling 103 nearly-diurnal periods, has been assessed to have little effect on the FCN theoryobservation discrepancy (DE VRIES and WAHR, 1990). Viscous coupling constants defined in Eqs. (4) and (5), with similar expressions for the magnetic couple, together comprise the viscomagnetic couple. The magnetic couple depends in part on the poorly-constrained lower mantle conductivity (e.g. LAMBECK, 1988, p. 571). NEUBERG et al. (1990) used a lower mantle conductivity of order 102 (ƒ m)-1 and explained less than a fraction of a percent of the noted discrepancy. While the GANs (1972) value of v can account for only a fraction of a percent of ƒ ƒðndfw, NEUBERG et al. (1990) used their own estimate (see Table 1) based on the Q for the NDFW (NEUBERG et al., 1987)-which could account for 11% of the discrepancy. Contributions similar to those suggested by NEUBERG et al. (1990), would be inferred from the data from RICHTER and ZURN (1988). NEUBERG et al. (1990) used spherical-harmonic coefficients from MORELLI and DZIEWONSKI's (1987) expansions for long-wavelength CMB topography to estimate the effect of the irregularity of this interface on the 26-day discrepancy. The net effect of the calculations is to modify the dynamical ellipticity of the CMB. In so doing, up to 78% of the theory-observation discrepancy can be accounted for, although NEUBERG et al. (1990) question the accuracy of the results that they obtained themselves. The commonly-preferred explanation for the above 26-day difference appeals to a departure of the value of CMB ellipticity from that of its hydrostatic value. According to NEUBERG et al. (1990), a departure of m can account for 100% of the 26-day difference between theory and observation. This estimate is not inconsistent with the earlier estimate by GWINN et al. (1986) who required a 450 m deviation to account for this difference. While the nonhydrostatic flattening of the whole Earth marginally exceeds that of its hydrostatic reference figure, observations from both VLBI and LPG have been interpreted in terms of a more significant departure of this type. The comparison is given explicitly in Table 5 for each of the lithosphere-ocean (LO) and CM boundaries (see the first column). In the second and third columns of this table, the reciprocal values for the hydrostatic and nonhydrostatic flattenings are given respectively. The last column details the nonhydrostatic relative to hydrostatic radial difference, which has both a polar and equatorial component (e.g. SOURIAU and SOURIAU, 1989). Although proper adjustment of the nonhydrostatic flattening can be used to explain the 26-day shift, neither GWINN et al. (1986) nor NEUBERG et al. (1990) supply reasonable justification of the geodynamical basis for such a departure from hydrostatic equilibrium. Furthermore, it is in principle unacceptable to make an arbitrary adjustment to the value of the flattening, given that the governing equations assume a hydrostatically-pre-stressed state in the first place (DE VRIES and WAHR, 1990). Thus it seems difficult to justify such a large difference between Table 5. Inverse flattening of the mantle's boundaries. (Sources are as for Table 1 with the addition of L LAMBECK, 1988.)

12 104 L. I. LUMB and K. D. ALDRIDGE the LPG and VLBI "observed" and hydrostatic flattenings for the CMB, when compared with the smaller difference obtained for the whole Earth, and we therefore suggest that other dynamical explanations for the 26-day shift are possible. Given the wide range in estimates (see Table 1) of the kinematic viscosity of the liquid core, it is appropriate to ask which value of kinematic viscosity would account for 100% of the noted discrepancy. This particular viscosity can be calculated from (7) (after NEUBERG et al.,1990), where AC DFW is the real part of the complex-eigenfrequency shift of 1.3x 10.4 cycles/ sidereal day (cpsd). Typical values for A and Am are ~1037 and x 1037 kg Em2, respectively (e.g. LAMBECK, 1988). Using standard values for a and Q, yields a formal kinematic viscosity of approximately 2 ~106 cm2/s. This is consistent with those in Table 1, and may indicate the importance of effective-viscous corrections to the eigenfrequencies of modes which arise due the coupling between the core and mantle. Furthermore, this value of effective viscosity compares very closely with that obtained by ALDRIDGE and LUMB (1987) who determined the decay rate for the inertial waves that they claimed to have identified in the LPG data of MELCHIOR and DUCARME (1986). In an independent study using a spherical-harmonics method, RIEUTORD (1990) has obtained a similar effective or eddy viscosity by matching the width of the spectral peaks from his theoretical calculations with those observed by MELCHIOR and DUCARME (1986). The near balance between pressure-gradient and Coriolis forces in a contained, rotating fluid, gives rise to an infinite suite of internal fluid oscillations (GREENSPAN, 1968). That any or several such internal modes of the Earth's core can exchange angular momentum with the mantle through normal-pressure forces or viscous stresses, to produce a change in the rotational spectrum of the Earth, has been recently suggested (e.g. ALDRIDGE et a!.,1989). In particular Table 6 illustrates that there are other inertial modes, in addition to the (2,1,1) mode, with eigenperiods close to that of the NDFW whose theoretical, TNDFW, and "observed", TNDFW, values are given. In the first column of this table the common name for each of the tesseral Earth tides is tabulated, while the inertial waves are indicated by their modal numbers as (n, m, k) with the details of this calculation given elsewhere (ALDRIDGE and LUMB, 1987; ALDRIDGE et al., 1989). The second column lists the corresponding periods, where a hydrostatically-flattened CMB has been assumed for both the inertial waves and the NDFW. Based on the GANs (1972) estimate for core viscosity, viscous corrections to the inertial-wave and NDFW periods can be ignored since they are of order. `(10.8). However in the case of coupling between the core and mantle, a much larger effective viscosity like ƒò100%, shows that the effectiveviscous corrections to these eigenfrequencies are significant enough to account for 100% of the 26 day shift. Accordingly the existence of inertial modes with periods close to those of tidal modes, and that of the NDFW, should change the response of the Earth at nearly-diurnal periods. If the results of our own laboratory inertial-wave experiments (e.g. LUMB and ALDRIDGE, 1988) do "track over" to the real Earth, then viscous and nonlinear effects may significantly alter the dynamics of the coupling process between the core and mantle, and thus play a role in the shift observed for the NDFW/FCN.

13 On Viscosity Estimates for the Earth's Fluid Outer Core and Core-Mantle Coupling 105 Table 6. The tesseral Earth tides and k=1 inertial modes with nearly-diurnal periods (after MELCHIOR,1978 LAMBECK, 1988). Note that the superscripted daggers indicate modes used in the study by NEVBERG et al (1987). 5. Discussion In contrast to previous efforts aimed at justifying large viscosity values for the Earth's outer core, the present work does not consider RE estimates to be simply dismissible as extreme upper bounds of the molecular viscosity. Rather identification of estimated core viscosities from LM and RE, as molecular and effective viscosities respectively, provides a reasonable framework for reconciling the disparate values given in Table 1. Clearly viscosities inferred without allowance for the appropriate dynamical context can be meaningless, and a cursory examination of Table 1 lends more than sufficient support to this conclusion. The RE values obtained, which derive from a wide variety of ways of sampling a wide variety of fluid properties, are affected by the dynamics relevant to the particular sampling method. For these real-earth determinations, the information furnished suggests that the physics involved is more complex than surmised, but the details of the physics have not yet been deduced from the data. The plausibility of this interpretation has been illustrated with an example from a laboratory study of the inertial waves in a rotating fluid. In this example, it was shown

14 106 L. I. LUMB and K. D. ALDRIDGE that had linear theory been improperly used to measure decay rates of excited inertial modes, no apparent relation to the actual molecular viscosity as measured by a viscometer would have been possible. Indeed in our experiments the source of the nonlinearity was intentionally introduced via an inclined, spheroidal inner boundary, and this exaggerated oblateness and tilt is not meant in any way to mimic directly the motions of the inner core. Our approach has been to study and quantify the nonlinear phenomena relevant to a rotating fluid, as revealed by our experiments on real fluids. As such, nonlinear effects in the Earth's core may be mechanically generated through boundary irregularities or radial shear, as well as through large-amplitude forcing as in the case of the tidally-excited core resonance (LUMB and ALDRIDGE, 1991). It is not intended that these findings be applied directly to the Earth's core, rather it is hoped that the idea presented will serve to caution anyone attempting to interpret the real-earth measurements naively in terms of molecular viscosity. The invocation of a nonhydrostatic flattening for the CMB has been noted as an arbitrary means for explaining the 26-day discrepancy in the period of the FCN. And in this context, Table 7 suggests that the closest agreement between the "observed" period of the NDFW/ FCN and that of the counterpart of the (2, 1, 1) inertial mode in the fluid core, will be achieved for a hydrostatic CMB flattening. Furthermore, this illustrates the primary importance of rotation and boundary geometry, facts well established by laboratory experiments on inertial waves (e.g. ALDRIDGE et al., 1989), for the dynamics of contained, rotating fluids like the Earth's core. The remaining disparity between the periods of the (2, l,1) inertial mode and that of the NDFW will be resolved by more realistic treatments of core-mantle coupling at nearly-diurnal periods. Thus an alternative explanation for the theory-observation discrepancy is required, and it is on these grounds that we have demonstrated the use of an effective viscosity, which could account for the 26-day discrepancy through a viscous shift in the eigenfrequency of the NDFW. Molecular-viscous corrections to the eigenfrequencies of contained, rotating fluids can generally be ignored on the basis of magnitude, and this has been verified experimentally (e.g. ALDRIDGE and T00MRE, 1969; LUMB and ALDRIDGE, 1988). However, it has been shown that this is not the case for the coupling between a fluid and its container, where an effective viscosity is needed to describe the unmodeled dynamics in the viscous boundary layer. Furthermore, the existence and close proximity in frequency of several k=1 inertial modes at nearly-diurnal periods, suggests that the concept of a "core resonance" needs to be generalized to allow for the full range of internal motions possible in the fluid core (LUMB and ALDRIDGE,1991). Through this analysis of coupling, a connection between the fundamental (2, 1, l) inertial mode of a Table 7. Inverse flattening of the CMB and the corresponding period of the (2,1,1) inertial mode.

15 On Viscosity Estimates for the Earth's Fluid Outer Core and Core-Mantle Coupling 107 spheroidal shell of rotating fluid and the NDFW has been established. Since the (2,1,1) mode is readily accessible in the laboratory, much may be gained from inertial-wave experiments and carried over to the real Earth. Other sources, which include oceanic damping (e.g. GOODKIND, 1983) and pressure variations at the CMB (e.g. DEHANT, 1990), are also being considered. Translational motion of the inner core (SLIGHTER, 1961) is another possibility whose contribution needs to be determined, although the effect of these motions at nearly-diurnal periods appears uncertain (e.g. MELCHIOR et al., 1988). DE VRIES and WAHR (1990) have recently shown that complex nonhydrostatic structure for the CMB has no appreciable effect on the period of the FCN, so that simple nonhydrostatic-flattening departures seem to be acceptable. However, the effects of an irregular CM interface will have a profound influence on the internal motions in the fluid core, and studies directed at eventually quantifying these effects have been considered (ALDRIDGE et al., 1988). It is worth noting that effective or eddy viscosities of size similar to that of ƒò100% were required to explain the 26-day shift reported by ALDRIDGE and LUMB (1987) in their analysis of the decay of the inertial waves that they identified in the LPG data of MELCHIOR and DUCARME (1986). In a related effort, calculations by RIEUTORD (1990) are also supportive of an effective or eddy viscosity of roughly this magnitude. Although the identification remains a matter of debate (e.g. ZURN et al., 1987; MELCHIOR et al., 1988; ALDRIDGE et al., 1989; CUMMINS et al., 1989; MANSINHA et al., 1990), the comparable effective viscosities (or more appropriately Q's) suggest that the observations of MELCHIOR and DUCARME (1986) and those obtained by other LPG (e.g. NEUBERG et al., 1987; RICHTER and ZURN, 1988) may be connected to the same dynamical process. Corroborative results from the analysis of VLBI data for short period polar motions (e.g. ALDRIDGE, 1989, 1990; ALDRIDGE and CANNON, 1991) are also suggestive that motions in the Earth's core are being observed indirectly through variations in the Earth's rotation. The importance of the concept and use of an effective or eddy viscosity has been demonstrated for the case of the Earth. From this one can conclude that the widespread use of the GANS (1972) viscosity estimate is reasonable for describing internal oscillations of the fluid core, which take place in the essentially inviscid interior, but insufficient to account for exchanges of angular momentum between the fluid core and mantle. Although core-mantle coupling has received considerable attention for time scales on the order of the secular variation (e.g. LE MOUEL,1991), it is clear that a re-examination for nearly-diurnal periods is warranted. Of course the use of an effective or eddy viscosity is usually cautioned, since it represents only a first-order closure to a more-complex problem (e.g. STULL, 1988). Nonetheless, at the present level of understanding in core dynamics, the untried eddy viscosity approach is appealing and accessible. However, it is wise to test the validity of this approach using similarity theory, higher-order closure, or large-eddy simulation methods, all of which are commonly used in the more developed atmospheric applications (e.g. STULL, 1988). The authors would like to thank: G. A. Henderson, J. Hinderer, G. T. Jarvis, G. P. Klaassen, J. Neuberg, D. E. Smylie and P. A. Taylor for various discussions; H. H. Schloessin who supplied additional references for the viscosities of liquid metals; V. Dehant, D. de Vries, J. Neuberg, and M. Rieutord for preprints of their papers; and an anonymous referee for several useful suggestions.

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