A past lunar dynamo thermally driven by the precession of its inner core

Abstract

The Cassini state equilibrium associated with the precession of the Moon predicts that the mantle, fluid core and solid inner core precess at different angles. We present estimates of the dissipation from viscous friction associated with the differential precession at the core-mantle boundary (CMB), $Q_{cmb}$, and at the inner core boundary (ICB), $Q_{icb}$, as a function of the evolving lunar orbit. We focus on the latter and show that, provided the inner core was larger than 100 km, $Q_{icb}$ may have been as high as $10^{10}-10^{11}$ W for most of the lunar history for a broad range of core density models. This is larger than the power required to maintain the fluid core in an adiabatic state, therefore the heat released by the differential precession at the ICB can drive a past lunar dynamo by thermal convection. This dynamo can outlive the dynamo from precession at the CMB and may have shutoff only relatively recently. Estimates of the magnetic field strength at the lunar surface are of the order of a few $\mu$T, compatible with the lunar paleomagnetic intensities recorded after 3 Ga. We further show that it is possible that a transition of the Cassini state associated with the inner core may have occurred as a result of the evolution of the lunar orbit. The heat flux associated with $Q_{icb}$ can be of the order of a few mW m$^{-2}$, which should slow down inner core growth and be included in thermal evolution models of the lunar core.

Full text

Confidential manuscript submitted to JGR-Planets
A past lunar dynamo thermally driven by the
precession of its inner core
Christopher Stys1and Mathieu Dumberry1
1Department of Physics, University of Alberta, Edmonton, Alberta, Canada.
Key Points:
•Viscous heating from the differential precession between the Moon’s fluid and solid
cores was high enough in the past to power a dynamo.
•The surface magnetic field that it produces is of the order of a few microTeslas,
compatible with the lunar field recorded after 3 Ga.
•The associated heat flux at the inner core boundary is an important contribution
to the thermal evolution of the lunar core.
Corresponding author: Mathieu Dumberry, dumberry@ualberta.ca
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arXiv:2201.00795v1 [astro-ph.EP] 3 Jan 2022

Confidential manuscript submitted to JGR-Planets
Abstract
The Cassini state equilibrium associated with the precession of the Moon predicts that
the mantle, fluid core and solid inner core precess at different angles. We present esti-
mates of the dissipation from viscous friction associated with the differential precession
at the core-mantle boundary (CMB), Qcmb, and at the inner core boundary (ICB), Qicb,
as a function of the evolving lunar orbit. We focus on the latter and show that, provided
the inner core was larger than 100 km, Qicb may have been as high as 1010 −1011 W
for most of the lunar history for a broad range of core density models. This is larger than
the power required to maintain the fluid core in an adiabatic state, therefore the heat
released by the differential precession at the ICB can drive a past lunar dynamo by ther-
mal convection. This dynamo can outlive the dynamo from precession at the CMB and
may have shutoff only relatively recently. Estimates of the magnetic field strength at the
lunar surface are of the order of a few µT, compatible with the lunar paleomagnetic in-
tensities recorded after 3 Ga. We further show that it is possible that a transition of the
Cassini state associated with the inner core may have occurred as a result of the evo-
lution of the lunar orbit. The heat flux associated with Qicb can be of the order of a few
mW m−2, which should slow down inner core growth and be included in thermal evo-
lution models of the lunar core.
Plain language summary: While the Moon today no longer has a large scale
magnetic field generated by dynamo action in its small iron core, magnetic studies on
lunar rocks collected during the Apollo missions suggest that it did in the past. How-
ever, the mechanism responsible for this dynamo is still debated. In this study, we in-
vestigate whether the precession motion of the Moon may have been capable to sustain
a past lunar dynamo. The mantle, fluid core, and solid inner core of the Moon precess
at different angles today. These precession angles were larger in the past when the Moon
was orbiting closer to Earth. We calculate the dissipation generated by the viscous fric-
tion from the differential precession at the boundaries between the fluid core and the man-
tle (CMB) and between the fluid and solid cores (ICB). We focus of the latter and show
that dissipation at the ICB in the past was high enough to drive thermal convection and
sustain a dynamo. A lunar dynamo driven by this mechanism is long-lived and may have
shutoff as late as 1 billion years ago and perhaps even more recently.
1 Introduction
The Moon does not currently possess a global magnetic field generated by dynamo
action. However, remanent magnetization measured in the crust by satellites [e.g. Mitchell
et al., 2008; Purucker and Nicholas, 2010] and on lunar rock samples collected during
the Apollo missions [e.g. Weiss and Tikoo, 2014] both suggest that a dynamo was op-
erating in the past. Paleomagnetic analyses on Apollo samples indicate that a dynamo
characterized by high surface intensities of several tens of µT to perhaps as high as 120
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µT operated early in the lunar history between about 4.25 and 3.56 Ga [Garrick-Bethell
et al., 2009, 2017; Courn`ede et al., 2012; Shea et al., 2012; Tikoo et al., 2012; Suavet et al.,
2013], although the accuracy of the very high paleointensity values have been called into
question [e.g. Lepaulard et al., 2019]. This high-field epoch is followed by weaker pale-
ointensities below 4 µT from 3.2 Ga onwards [Tikoo et al., 2014, 2017]. The lunar pa-
leomagnetic record is more spotty in this weak-field epoch and we do not know with high
accuracy when the lunar dynamo turned off, though there is good evidence that it may
have persisted until as recently as 1 Ga [Tikoo et al., 2017; Mighani et al., 2020].
No single dynamo mechanism has so far been shown to be capable of explaining
both the large paleointensities of the high-field epoch and the longevity of the weak-field
epoch. Explaining the large surface field recorded during the high-field epoch is partic-
ularly challenging. A long lasting dynamo driven by thermo-chemical convection in the
liquid core may explain the paleomagnetic record of the low-field epoch [Laneuville et al.,
2014; Scheinberg et al., 2015], but not the large intensities of the high-field epoch [Evans
et al., 2018]. Other possibilities that have been suggested to explain parts of the pale-
omagnetic record include a short-lived early core dynamo following a mantle overturn
event [Stegman et al., 2003], a dynamo generated in a magma ocean at the base of the
mantle [Scheinberg et al., 2018] and a mechanically forced dynamo induced either by im-
pacts [Le Bars et al., 2011] or mantle precession [Williams et al., 2001; Dwyer et al., 2011;
C´ebron et al., 2019; ´
Cuk et al., 2019].
A lunar dynamo powered by precession motion is the scenario that we further ex-
plore in our study. The basis for this idea stems from the rotational dynamics of the Moon
which is characterized by a Cassini state, and in which the orbit normal and spin-symmetry
axis remain coplanar with, and are precessing about, the normal to the ecliptic [Colombo,
1966; Peale, 1969]. The precession is retrograde, with a present-day period of 18.6 yr.
Lunar Laser Ranging (LLR) observations [e.g. Dickey et al., 1994; Williams et al., 2001]
indicate a present-day tilt of the spin-symmetry axis of 1.543◦with respect to the eclip-
tic, though this only applies for the solid outer shell of the Moon comprised of its man-
tle and crust. The spin axis of the fluid core should also lie in the plane that defines the
Cassini state, however we do not expect it to be aligned with the symmetry axis of the
mantle. This is because the amplitude of the pressure torque exerted by the pole-to-equator
core-mantle boundary (CMB) flattening on a misaligned fluid core is too small for the
fluid core to be locked into synchronous precession with the mantle [Poincar´e, 1910; Gol-
dreich, 1967]. An equivalent and complementary way to express this is to consider the
free precession period of a misaligned rotation vector of the fluid core with the symme-
try axis of the CMB sustained by this pressure torque. This rotation mode is referred
to as the free core nutation (FCN) and its period, though not directly observed, is ex-
pected to be a few hundred years [e.g. Viswanathan et al., 2019], much longer than the
18.6 yr period of forced mantle precession. This implies that the fluid core does not have
time to adjust to the precession motion of the mantle and is not efficiently entrained with
it. The rotation vector of the fluid core should therefore remain in close alignment with
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the normal to the ecliptic, though its precise angle is not observed directly and is un-
known.
The flow motion of a fluid core precessing at a different angle than the mantle can-
not be represented by a simple rigid body rotation [e.g. Tilgner , 2015]. This is, first, be-
cause a secondary flow must exist to satisfy the no-penetration condition at the ellip-
tically shaped CMB, and second, because finite viscosity requires the additional pres-
ence of a boundary layer flow underneath the CMB. Based on the amplitude of the dif-
ferential velocity across this boundary layer at present-day, this boundary layer flow is
expected to be turbulent [Yoder, 1981; Williams et al., 2001]. An estimate of the vis-
cous dissipation associated with this turbulent flow is inferred by LLR and, although the
mechanical stirring is not sufficiently large to power a dynamo today, it may have been
in the past, when the Moon was closer to Earth [Williams et al., 2001]. Indeed, the past
Cassini state of the Moon featured a larger mantle tilt angle [Ward, 1975] and a fluid
core spin axis remaining closely aligned with the ecliptic normal [Meyer and Wisdom,
2011]. Estimates of the viscous dissipation associated with this past, larger differential
velocity at the CMB suggest that it may have been sufficiently large to power a dynamo
by mechanical forcing [Dwyer et al., 2011; C´ebron et al., 2019; ´
Cuk et al., 2019].
It is uncertain whether an inner core is present at the centre of the Moon today.
Its presence has been suggested by seismic observations [Weber et al., 2011], but this is
not universally accepted [e.g. Garcia et al., 2011]. If an inner core is present, like the man-
tle and fluid core, it is also forced to precess at a period of 18.6 yr and its spin-symmetry
axis should also lie in the plane that defines the Cassini state [Williams , 2007]. Its an-
gle of tilt is unknown, but it is determined by the period of the free inner core nutation
(FICN), a free mode of rotation similar to the FCN but associated with the inner core
[Dumberry and Wieczorek, 2016; Stys and Dumberry, 2018, henceforth referred to as DW16
and SD18, respectively]. In analogy with the FCN, the FICN period depends of the am-
plitude of the pressure torque applied on the inner core when its elliptical surface is mis-
aligned with the spin axis of the fluid core. But in addition, the FICN period also de-
pends of the gravitational torque between the misaligned figures of the inner core and
mantle, and for the Moon, it is the latter that dominates [DW16, SD18].
Just as the tilt angle of the spin axis of the fluid core depends on the FCN period
relative to the forcing period of 18.6 yr, the tilt angle of the spin-symmetry of the in-
ner core is set by how the period of the FICN compares with this forcing period [DW16,
SD18]. The FICN period is not known, but for reasonable models of the interior den-
sity structure of the Moon it is expected to be in the range of 10 to 40 yr. Because the
FICN period is close to the 18.6 yr forcing period, a large tilt of the inner core with re-
spect to the mantle can result by resonant amplification [DW16, SD18]. Since the spin
axis of the fluid core is expected to remain closely aligned with the ecliptic normal, this
implies that there could be a large misalignment between the rotation vectors of the fluid
and solid cores, and a differential velocity at the inner core boundary (ICB). If an in-
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ner core has been present for a good portion of the lunar past, then mechanical stirring
in the fluid core caused by differential velocity at the ICB may have been capable of gen-
erating a dynamo. Furthermore, the heat generated by viscous friction at the ICB is avail-
able to drive a dynamo by thermal convection.
The main objective of our study is to explore the latter scenario. The possibility
of dynamos driven by precession is an active area of research [e.g. Tilgner, 2005; Lin et al.,
2016; C´ebron et al., 2019]. The morphology of the resulting magnetic field depends on
whether the turbulent flows are confined to a boundary layer or whether they trigger large
scale instabilities destabilizing the whole of the fluid core. Here, we do not present a dy-
namical model of a dynamo sustained by core flows entrained by the precession of an in-
ner core. Instead, we focus on whether a thermally driven, convective dynamo may be
powered by the heat released at the ICB from the viscous friction associated with a dif-
ferentially precessing inner core. To do so, we approach the problem from an energy bal-
ance perspective [e.g. Nimmo, 2015]. We seek to determine whether viscous dissipation
at the ICB can overcome ohmic dissipation in the core and hence be sufficiently high to
sustain a dynamo.
This was the strategy followed by Dwyer et al. [2011]. They estimated the dissi-
pation produced by viscous friction from the differential precession at the CMB as a func-
tion of the evolving lunar orbit. They showed that earlier in the lunar history, this dis-
sipation exceeded by a large amount the power required to maintain the fluid core in an
adiabatic state, hence that the remaining power was available to drive a dynamo. The
heat dissipated at the CMB is not available to drive thermal convection, as it flows up-
wards into the mantle or pools at the top of the core. Hence, a lunar dynamo driven by
precession at the CMB invariably depends on whether the vigour and geometry of the
core flows forced by the mantle precession can generate and sustain a magnetic field.
In this work, we extend the idea of a dynamo generated by differential precession
to the ICB. We investigate whether the viscous friction at the ICB from the differential
precession between the inner core and fluid core may have dissipated enough heat in the
past to sustain a dynamo. One key difference is that, in contrast to the heat released at
the CMB, that released at the ICB is available to power a thermally driven convective
dynamo in the fluid core. Hence, a dissipation at the ICB higher than the power required
to maintain the fluid core in an adiabatic state provides a more robust condition for the
presence of a dynamo than the equivalent statement at the CMB. Core flows forced by
a precessing elliptical inner core may further help (or oppose) the generation of a mag-
netic field, but we do not consider their dynamo capability in our study, nor their influ-
ence on a thermally driven dynamo.
We build predictions of the differential rotation at both the ICB and CMB using
the Cassini state model presented in SD18. This model allows one to calculate the mis-
alignment between the rotation vectors of the mantle, inner core and fluid core for a given
interior model of the Moon and a set of orbital parameters. The calculation of the dis-
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sipation at the CMB presented in Dwyer et al. [2011] assumed that the spin vector of
the fluid core remained aligned with the ecliptic normal. But the rotational model of SD18
allows us to calculate more precisely the orientation of the spin vector of the fluid core.
A secondary objective is thus to recalculate the power dissipated at the CMB on the ba-
sis of this more complete model.
2 Theory
2.1 Interior models of the Moon
We follow SD18 and assume a simple model of the lunar interior comprised of four
layers made up of a solid inner core, a fluid outer core, a mantle, and a thin crust. The
outer radii of each of these layers, in the same sequence, are denoted by rs,rf,rm, and
R, and their densities, assumed uniform, by ρs,ρf,ρm, and ρc. Each layer is triaxial in
shape, specified by its polar and equatorial flattenings. For all interior models in the present
study, we use a fixed crustal thickness of h=R−rm= 38.5 km with density ρc=
2550 kg m−3[e.g. Wieczorek et al., 2013] and an inner core density fixed at ρs= 7700
kg m−3[Matsuyama et al., 2016]. To build our interior models, we follow the strategy
detailed in SD18: for a given set of rsand rf, the density of the fluid core and mantle
are set by matching the lunar mass M= (4π/3)¯ρR3, where ¯ρ= 3345.56 kg m−3is
the mean density and R= 1737.151 km is the mean radius, and the moment of iner-
tia of the solid Moon Ism = 0.393112 ·M R2[Williams et al., 2014], comprised here of
the mantle and crust. The polar and equatorial flattenings at each boundary are con-
strained by matching the degree 2 gravitational potential coefficients J2and C22 as well
as the observed surface polar and equatorial flattenings. We further assume that the ICB
and CMB are both at hydrostatic equilibrium with the imposed gravitational potential
from the mantle and crust.
2.2 Extrapolating the Cassini state and differential rotation in the past
From the perspective of the rotational dynamics, the mantle and crust are welded
together and rotate as a single body, a body which we refer to as the “mantle” in the
context of the lunar rotation. Hence, the Moon has three independently rotating regions,
this “mantle”, the fluid core and the inner core. Assuming a Cassini state equilibrium,
the rotation and figure axes of the mantle and inner core, and the rotation axis of the
fluid core can be misaligned from one another, but should all lie in a common plane (the
Cassini plane) which also includes the ecliptic and orbit normals. The Cassini state model
developed in SD18 allows one to calculate the mutual orientations of each of these axes.
More specifically, it gives the tilt angles of: the mantle symmetry axis with respect to
the ecliptic normal (θp); the rotation vector of the mantle (θm) and the symmetry axis
of the inner core (θn), both with respect to the symmetry axis of the mantle; and the
rotation vectors of the fluid core (θf) and solid inner core (θs), both with respect to the
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mantle rotation vector. (See Figure 2 of SD18 for a visual representation of each of these
angles.)
For a given interior model, the solution depends on a set of orbital parameters which
include the inclination I, the eccentricity eL, the precession frequency Ωpof the lunar
orbit and the sidereal frequency of lunar rotation Ωo. The ratio of the latter two form
the Poincar´e number, δω = Ωp/Ωo. Present-day values for these quantities are I= 5.145◦,
eL= 0.0549, Ωp= 2π/18.6 yr−1, Ωo= 2π/27.322 day−1, and δω = 4.022 ·10−3.
Once a solution is obtained, the differential rotation at the CMB and ICB can be
deduced. Viewed by an observer in the mantle frame, the misaligned rotation axis of the
fluid core is precessing in a retrograde direction at a frequency of ω= Ωo+Ωp, so the
amplitude and orientation of the differential velocity at the CMB varies with location
and time. Likewise, for the differential velocity at the ICB. A useful measure of the dif-
ferential motion is given by the maximum amplitude of the differential angular veloc-
ity in the equatorial direction at each of the CMB and ICB. We denote these as ∆ωcmb
and ∆ωicb, respectively, and they are related to θfand θsby
∆ωcmb = Ωo
sin θf
,(1a)
∆ωicb = Ωo
sin(θf−θs)
.(1b)
The dissipation at the CMB and ICB can be cast as a function of ∆ωcmb and ∆ωicb, re-
spectively, as we show in the next subsection.
An order of magnitude for ∆ωcmb at present day is readily obtained by assuming
a fluid core rotation vector perfectly aligned with the ecliptic normal, and so θf=−θp=
−1.543◦, giving ∆ωcmb = 7.17×10−8s−1. Taking a CMB radius of rf= 400 km gives
a differential velocity at the CMB of the order of 3 cm s−1and an associated Reynolds
number Re =r2
f∆ωcmb/ν of the order of 1011 for a kinematic viscosity of ν= 10−6
m2s−1. Such a large Reynolds number indicates that the viscous friction between the
fluid core and mantle should induce turbulent flows. It is based on this argument that
viscous coupling at the CMB of the Moon is assumed to be in a turbulent regime [Yo-
der, 1981; Williams et al., 2001]. Although the radius of the ICB is smaller, ∆ωicb is typ-
ically larger than ∆ωcmb because |θs|>|θf|, so the Reynolds number associated with
differential precession at the ICB is of similar order and viscous coupling at the ICB is
also expected to be in a turbulent regime. With Ωoincreasing going back in time, and
likewise for the magnitudes of θsand θfas we show in our results, it is safe to assume
that viscous coupling at both the CMB and ICB has remained in a turbulent regime for
the whole of the lunar history.
Each of the orbital parameters I,eL, Ωpand Ωohad different values in the past
when the Moon was closer to Earth. To build a history of the differential rotation at the
CMB and ICB, we must first determine how these orbital parameters have evolved through
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time, or as we do here, as a function of the semi-major axis of the lunar orbit aL. For
simplicity, we will often refer to aLas the lunar orbit radius.
Assuming the Moon to be tidally locked into a 1:1 spin orbit resonance, using Ke-
pler’s 3rd law, the rotational frequency of the Moon in the past (Ωo(aL)) as a function
of aLis given by
Ωo(aL) = ao
aL3/2
Ωo(ao),(2)
where aois the present-day semi-major axis equal to 60.3RE, where REis the Earth’s
mean spherical radius. A numerical integration of the tidal evolution of the Earth-Moon
system must be carried out in order to determine how I,eLand Ωphave varied as a func-
tion of aL. Examples of such computations can be found in Touma and Wisdom [1994]
and more recently in ´
Cuk et al. [2016].
We restrict our investigation to lunar orbital radii greater than 34RE, hence after
the Cassini state transition that occurred at approximately 29RE[´
Cuk et al., 2016]. We
take the variation of Ωpas a function of aLpresented in Figure 19 of Touma and Wis-
dom [1994]. It is often assumed that changes in Ihave not been significant after aL>
34REbased on the results of Touma and Wisdom [1994] (e.g. their Figure 16), However,
´
Cuk et al. [2016] have shown that tidal dissipation have lead to substantial changes in
I, from approximately 18◦at aL= 34REto its present-day value of 5.145◦(see their
Figure 4a). We use the following model for the evolution of I,
I=c1+c2ao
aL6
,(3)
with coefficients c1= 4.71976◦and c2= 0.425237◦; this gives a good approximation
to the evolution of Ipresented in ´
Cuk et al. [2016]. The eccentricity of the orbit has also
varied with aLalthough, for simplicity, we assume a fixed eccentricity equal to today’s
value of eL= 0.0549.
An important caveat of our model is that, while we make predictions of the dis-
sipation at the ICB and CMB based on how Ωo, Ωpand Ihave changed as a function
of aL, we do not take into account the feedback that this internal dissipation may have
on the evolution of the lunar orbit. Instead, we follow a simplified approach whereby we
evaluate a posteriori whether these predictions are consistent with the lunar orbit his-
tory model that we have used.
2.3 Viscous coupling from turbulent flow at the CMB
Assuming a turbulent boundary layer, the shear stress acting on the solid bound-
ary can be written as
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τ=fρf|u|u,(4)
where ρfis the density of the fluid core, uis the flow velocity outside the boundary layer,
and fis a dimensionless coefficient of friction which depends, among other things, on
surface roughness. Integrated over the CMB, the amplitude of the torque at the CMB
resulting from this turbulent viscous shear stress can be written in the form
Γcmb =fcmb ¯
Cf
∆ωcmb

2,(5)
where ∆ωcmb is given by Equation (1a) and where ¯
Cf= (8π/15)ρfr5
fis the mean mo-
ment of inertia of an entirely fluid core. fcmb is a coefficient of friction, though differ-
ent in numerical value from fas it takes into account the integration of the stress over
the whole surface of the CMB. Constraints on the amplitude of the viscous friction at
the CMB of the Moon at present-day can be derived from LLR observations [Williams
et al., 2001, 2014; Williams and Boggs, 2015], and provide an estimate of fcmb. The ro-
tational model of the Moon used to fit LLR data consists of a rigid mantle and a fluid
core (it does not include an inner core). Viscous dissipation is incorporated into the model
by prescribing a viscous torque on the mantle in the form
Γcmb =K · ∆ωcmb ,(6)
where Kis a coupling coefficient. A recent estimate of Kis [e.g. Williams and Boggs,
2015]
K
¯
C= (1.41 ±0.34) ×10−8days−1= (1.63 ±0.39) ×10−13 s−1,(7)
where ¯
C= (8π/15) ¯ρ R5is the mean moment of inertia of the whole Moon. Equating
Equations (5) and (6), an estimate of fcmb at present-day is then given by
fcmb =K
¯
C ¯
C
¯
Cf1

∆ωcmb
today
,(8)
where the subscript today emphasizes that it is based on the present-day value of the dif-
ferential rotation at the CMB. For a given interior density model of the Moon, we can
calculate the ratio ¯
Cf/¯
Cand determine 
∆ωcmb
today using the Cassini state model of
SD18. Hence, we can readily calculate fcmb. To present an estimate, using θf=−1.6◦
[e.g. SD18], ¯
Cf/¯
C= 7 ·10−4[e.g. Williams et al., 2014] and K/¯
Cfrom Equation (7)
gives fcmb = 0.0314.
Viscous dissipation at the CMB (Qcmb) can be calculated as the product of the torque
Γcmb and the angular velocity difference ∆ωcmb,
Qcmb = Γcmb ·∆ωcmb =fcmb ¯
Cf
∆ωcmb

3.(9)
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Based on the estimate of fcmb at present-day from Equation (8), and assuming that fcmb
has remained constant, the dissipation in the past, when Ωoand θfwere both different,
can be obtained from
Qcmb =¯
CK
¯
C
Ωosin θf

3
past

Ωosin θf
today
.(10)
A more proper evaluation of Qcmb should take into account the fact that fcmb de-
pends on Ωo[e.g. C´ebron et al., 2019]. However, the dependence is weak, and since our
primary objective is to derive an order of magnitude estimate of how Qcmb has evolved,
we neglect this effect here.
It is important to emphasize that the estimate of Kobtained from LLR observa-
tions is based on a rotational model of the Moon that does not include an inner core.
If an inner core is present, the coefficient Kin this rotational model captures the com-
bined effect of friction at both the CMB and ICB [Williams and Boggs, 2009]. Moreover,
the manner in which the delayed tidal response of the Moon varies with different forc-
ing frequencies – ultimately the method by which an estimate of Kseparate from tidal
deformation is obtained [e.g. Williams et al., 2001; Williams and Boggs, 2015] – can also
be different if an inner core is present. Additionally, viscous deformation within the in-
ner core may also contribute to a part of the observed dissipation in lunar rotational en-
ergy. These caveats are mentioned to keep the reader alert to the fact that the estimate
of the present-day viscous dissipation at the CMB (and at the ICB) remain not well con-
strained by observations. Since our results are ultimately tied to value of Kgiven by Equa-
tion 7, they must be interpreted as order of magnitude estimates.
2.4 Viscous coupling from turbulent flow at the ICB
By analogy with the torque at the CMB, the amplitude of the torque at the ICB
resulting from turbulent viscous shear stress is given by
Γicb =ficb ¯
Csρf
ρs
∆ωicb

2,(11)
where ¯
Cs= (8π/15)ρsr5
sis the mean moment of inertia of the inner core, the ratio ρf/ρs
accounts for the fact that it is the density of the fluid core which is involved in the shear
stress, and ficb is a friction coefficient for the ICB.
It is not possible to get an independent estimate of ficb based on LLR observations.
To move forward and build an estimate of the dissipation at the ICB, we must make an
assumption on ficb and we simply assume that it is equal to fcmb. There is no reason
a priori why this should be the case, but this is the simplest assumption one can make.
All our results depend on this assumption so they must be viewed with this caveat in
mind.
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The viscous dissipation at the ICB is
Qicb = Γicb ·∆ωicb =ficb ¯
Csρf
ρs
∆ωicb

3.(12)
Setting ficb =fcmb and using fcmb as prescribed by Equation (8), the dissipation at
the ICB in the past is given by
Qicb =¯
Crs
rf5K
¯
C
Ωosin(θf−θs)

3
past

Ωosin θf
today
.(13)
2.5 Magnetic field strength from dissipation at the CMB and ICB
To convert dissipation at the CMB into magnetic field intensity at the lunar sur-
face, Dwyer et al. [2011] used a scaling law derived in Christensen et al. [2009], based
on numerical dynamo models powered by convection. This scaling may not be entirely
suitable for a dynamo generated by mechanical stirring at the CMB, but no equivalent
scaling law is available yet for precessional dynamos. We thus proceed similarly here. We
use the notation B(cmb)to denote the amplitude of the magnetic field at the lunar sur-
face resulting from mechanical forcing due to precession at the CMB. The relationship
between B(cmb)(in units of µT) and the dissipation available to power the dynamo (Qdyn)
used in Dwyer et al. [2011] is
B(cmb)≈6drf
rfo 3Qdyn
cmb
¯
Q1/3
,(14)
where ¯
Q= 3×1011 W and dis the ratio of the dipolar magnetic field to the total field
at the CMB. For simplicity, we set dequal to 1, which represents an upper bound for
B(cmb). The factor (rf/rfo) takes into account a different choice of core radius than the
reference rfo = 350 km used by Dwyer et al. [2011]. Qdyn
cmb is the dissipation readily avail-
able to power the dynamo and in Dwyer et al. [2011] it was taken as
Qdyn
cmb =Qcmb −Qth ,(15)
where Qcmb is given by Equation (10) and Qth is a threshold value below which no dy-
namo can exists. Qth was taken in Dwyer et al. [2011] as the adiabatic heat flow at the
CMB, estimated at 4.7×109W, which represents the minimum heat flow out of the core
required in order to sustain a dynamo driven by thermal convection alone [e.g. Nimmo,
2015]. While this may be a valid threshold for convective dynamos, it is less clear that
it applies for a mechanical dynamo driven by precession. Even if no heat flow escapes
the core, and the latter is thermally stratified, tidal instabilities can still be generated
[e.g. C´ebron et al., 2010; Vidal et al., 2018, 2019] and the additional mechanical forcing
at the boundary from the precessing mantle may be capable to drive a dynamo. Nev-
ertheless, there must be a threshold value otherwise a precession dynamo would still be
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Confidential manuscript submitted to JGR-Planets
operating in the Moon today. For simplicity, and in the absence of a different estimate,
we also use Qth = 4.7×109W.
We use the same scaling law given by Equation (14) to determine the amplitude
of the magnetic field at the lunar surface resulting from a precession driven dynamo at
the ICB, which we denote by B(icb), and is given by, in units of µT,
B(icb)≈6drf
rfo 3Qdyn
icb
¯
Q1/3
,(16)
where again we set dequal to 1, and with
Qdyn
icb =Qicb −Qth ,(17)
where Qicb is given by Equation (13). The adiabatic heat flow at the ICB is different than
at the CMB, but the criteria to maintain a dynamo driven by thermal convection in the
fluid core remains tied to the total heat flow escaping the core, so we take again Qth =
4.7×109W. It should be noted that, while heat dissipated at the CMB is not available
to drive convection, as it flows upwards into the mantle or pools at the top of the core,
heat generated by friction at the ICB is available to power a thermally driven convec-
tive dynamo. Hence, the scaling law of Equation (16) may be more justified for a dy-
namo thermally driven by precession at the ICB. We note that it is likely a lower bound
since, in addition to the thermal energy available to drive convective flows, the preces-
sion motion of an elliptical inner core also generates flow by mechanical stirring. If such
flows lead to global instabilities and large scale eddies [e.g. Lin et al., 2016], they may
further contribute to (though they might also suppress) dynamo action. In addition to
the uncertainty in estimating Qicb, as detailed in the previous section, this further adds
to the uncertainty of estimating B(icb). Hence, our calculation of B(icb)from Equation
(16) must be viewed as an order of magnitude estimate.
3 Results
3.1 Evolution of the Cassini state
We first show an example of how the Cassini state equilibrium of the different in-
terior regions of the Moon changes as a function of orbital radius, and thus how the dif-
ferential velocity at both the CMB and ICB may evolve. Figure 1 shows the evolution
of θp,θfand θsas a function of the lunar orbit radius for a particular Moon model with
a fluid outer core radius of rf= 350 km and a solid inner core radius of rs= 250 km.
When the Moon was at 34RE,θpwas close to 40◦, consistent with the results shown in
Ward [1975]. The magnitude of the variation of θfis slightly larger and tracks the changes
in θp, though with the reverse sign. Recall that θfis measured with respect to the spin
vector of the mantle θm, and the latter always remains closely aligned with the symme-
try axis of the mantle. Hence, Figure 1 shows that, for aL>34RE, the rotation axis
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Confidential manuscript submitted to JGR-Planets
−70
−60
−50
−40
−30
−20
−10
0
10
20
30
40
Tilt angle (degrees)
35 40 45 50 55 60 65
Semimajor axis (RE)
state B
state A
state C
θ
f
θ
s
θ
p
θ
s
Figure 1. Evolution of the Cassini state of a specific Moon model (rf= 350 km, rs= 250
km) as a function of lunar orbital radius in units of Earth radii (RE). Shown are the tilt angles
of: the mantle symmetry axis with respect to the ecliptic normal (θp, red); the rotation vectors
of the fluid core (θf, green) and solid inner core (θs, blue), both with respect to the mantle ro-
tation vector. States A, B and C refer to the different Cassini states of the inner core. Today
corresponds to 60.3RE.
of the fluid core always remain closer to an alignment with the ecliptic normal than to
an alignment with the mantle. However, it is important to point out that the spin axis
of the fluid core is always misaligned with the ecliptic normal, in the opposite direction
than the mantle tilt, and that the offset gets larger the smaller aLis. This is consistent
with the recent results of ´
Cuk et al. [2019]. For aL= 34RE, the offset is as large as 10◦.
The reason why the rotation vector of the fluid core never lines up closely with the man-
tle symmetry axis is because, for all values of aLin Figure 1, the FCN frequency is al-
ways much smaller than the precession frequency Ωp[e.g. Meyer and Wisdom, 2011].
For all values of aLon Figure 1, the spin axis of the inner core θs(which remains
closely aligned with the symmetry axis of the inner core θn) is significantly offset from
the mantle symmetry axis. This is because, for all aL, the FICN frequency remains suf-
ficiently close to the forcing (precession) frequency (see Figure 2) and a large tilt of the
inner core results from resonant amplification. Moreover, Figure 1 shows that the solid
inner core can occupy different Cassini states. The states labelled A, B and C follow the
convention introduced in SD18. For this specific Moon model, only state B was possi-
ble for aL<51RE, but all three states are possible solutions for aL>51RE. The ori-
entation of the spin vector of the fluid core is slightly different in each of these three Cassini
states (see SD18) and for a large inner core, as is the case here, the shift in θfis suffi-
ciently large than it can be seen in Figure 1.
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Confidential manuscript submitted to JGR-Planets
Assuming that the lowest energy state is favoured (i.e. the state with the small-
est inner core tilt), Figure 1 shows that, as the Moon moved away from the Earth, it is
possible that a transition in the Cassini state of the inner core from state B to state A
may have occurred. Whether such a transition did take place depends on the core den-
sity model. As shown in Figure 4 of SD18, the Cassini state that the inner core occu-
pies today is determined by how the FICN frequency compares with the precession fre-
quency Ωp. The FICN frequency is retrograde (as is Ωp) and we use here the notation
Ωficn to denote its amplitude, as it is seen by an observer in a space-fixed frame. The
transition between states B and A does not occur exactly at Ωf icn = Ωp, but instead
at Ωficn = Ωp+δΩ, where δΩ is a correction that involves the tilt angles θnand θp.
Denoting this “transition” frequency by Ωt= Ωp+δΩ, if Ωficn <Ωt, the inner core
occupies state B; if Ωf icn >Ωt, it occupies state A. For the present-day Moon, Ωt=
2π/16.4 yr−1(SD18).
This rule applies at any moment in the lunar history, so a Cassini state transition
from states B to A implies an intersection between Ωficn and Ωtas they both evolve.
Figure 2 shows how Ωficn changes as a function of aLfor a Moon model with rs= 250
km and different choices of fluid outer core radii, as well as how the transition frequency
Ωtvaries as a function of aL. The lunar model with rf= 350 km whose tilt angles’ evo-
lution are shown in Figure 1 corresponds to the light blue curve in Figure 2: for this model,
Ωficn intersects Ωtat aL= 51RE, which marks the point at which the Cassini state
transition from state B to state A occurs.
To understand how Ωficn changes with aL, to a good approximation it is given by
Equation 24 of SD18 multiplied by Ωo(and with a reversed sign, since we define here
a retrograde frequency as being positive),
Ωficn =−Ωoesα1+ Ωoesαg1−α1+3
2Ωoβs1−α1cos2I−sin2I,(18)
where we have assumed eL= 0 to simplify and where α1=ρf/ρs,es= (Cs−¯
As)/¯
As,
βs= (Cs−As)/¯
As, with Cs,Asand ¯
Asbeing respectively the polar, minimum and
mean equatorial moments of inertia of the inner core. The parameter αgis given by Equa-
tion (18) of SD18 and represents the ratio of the gravitational to the centrifugal (or in-
ertial) pressure torque exerted on the inner core. For a non-evolving lunar density struc-
ture, αgwas smaller in the past because the rotation rate Ωowas higher and thus the
centrifugal torque was relatively more important. In fact, αgis proportional to Ω−2
o. The
second term on the right-hand side of Equation (18), which features αg, dominates the
two other terms. Hence, Ωf icn is proportional to Ωoαg, and thus is inversely proportional
to Ωo. Since Ωodecreases with aL, Ωficn increases with aLas seen in Figure 2. αgde-
pends on the density contrast at the ICB, so the Ωficn curves for different rfshown in
Figure 2 would be displaced with a different choice of inner core radius. However, note
that the transition frequency is to first order independent of the density structure in the
lunar core; for any choice of core density model, the Cassini state transition is determined
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Confidential manuscript submitted to JGR-Planets
by when the Ωficn curve for this particular model intersects the Ωtcurve that is shown
in Figure 2.
0.1
0.2
0.3
0.4
0.5
0.6
FICN Frequency (yr −1)
35 40 45 50 55 60 65
Semimajor axis (RE)
CMB radius
340 km
380 km
370 km
360 km
350 km
Ωp
Ωt
state B
state A
Figure 2. Evolution of the FICN frequency Ωficn as a function of lunar orbital radius in
units of Earth radii (RE), for different choices of outer core radii, and for a solid inner core radius
of 250 km. The precession frequency Ωpis indicated by the dashed line. The Cassini state transi-
tion frequency Ωtof the inner core is indicated by the black solid curve. The inner core occupies
state B when Ωficn <Ωt, and state A when Ωficn >Ωt. Today corresponds to 60.3RE.
A Cassini state transition has important implications for a dynamo driven by in-
ner core precession. First, the change from states B to A is not instantaneous, and the
large change in inner core tilt that it involves (from approximately −42◦to +21◦on Fig-
ure 1) should generate instabilities and flows in the fluid core. For a relatively short pe-
riod of time, these flows may be capable of generating a dynamo. After the transition,
and once instabilities in the fluid core have attenuated, the differential velocity at the
ICB associated with state A is much smaller, implying a sudden drop in the power dis-
sipated at the ICB and the potential for dynamo action.
3.2 Power dissipation at the CMB and ICB versus lunar orbit radius
The dissipation at the CMB and ICB can be expressed as a function of the lunar
orbit by Equations (10) and (13), respectively. Figure 3 shows how the viscous dissipa-
tion at both the CMB and ICB vary as a function of aLfor the same lunar model with
rf= 350 km and rs= 250 km, whose tilt angles’ evolution is shown in Figure 1. For
comparison, we also show on Figure 3 the dissipation at the CMB estimated by Dwyer
et al. [2011], using the same core radius of rf= 350 km, and computed from
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Confidential manuscript submitted to JGR-Planets
107
108
109
1010
1011
1012
1013
1014
Viscous dissipation (Watts)
35 40 45 50 55 60 65
Semimajor axis (RE)
Qicb
Qcmb
Qcmb
Dwyer
Figure 3. Power available to drive a dynamo estimated from viscous dissipation at the CMB
(Qcmb, red) and at the ICB (Qicb , blue) as a function of lunar orbit radius in units of Earth radii
(RE). The dissipation at the CMB from the model presented in Dwyer et al. [2011] is shown in
orange (QDwyer
cmb ). The evolution of the differential velocities at the CMB and ICB correspond
to those shown on Figure 1. The power threshold to sustain a dynamo (Qth = 4.7×109W) is
indicated by the grey horizontal dashed line. Today corresponds to 60.3RE.
QDwyer
cmb ≈3×1020W×sin 3θp
(aL(t)/RE)(9/2) .(19)
Our estimate of Qcmb differs from that of Dwyer et al. [2011] for two reasons. First, Equa-
tion (19) is based on a dissipation at present day of Qcmb ≈(5.8±1.3)×107W which
is itself derived from a viscous coupling coefficient of K/¯
C≈1.122±0.257×10−8day−1
estimated in Williams et al. [2001]. Using instead the updated value of K/¯
Cgiven in Equa-
tion (7) gives a larger present-day dissipation of 7.3±1.8×107W and our higher esti-
mates of Qcmb are in part due to this. Second, the reconstruction of Qcmb in Equation
(19) makes the implicit assumption that the spin axis of the fluid core has remained aligned
with the ecliptic normal. As we have shown above (and in DW16 and SD18), this is in-
correct: the spin axis of the fluid core is offset from the ecliptic normal, in the reverse
direction than the mantle offset, resulting in a larger angle of offset between the rota-
tion vectors of the fluid core and mantle. This difference is larger the further we go back
in time and this also contributes to make our estimates of Qcmb larger. Note that as a
result of the Cassini state transition associated with the inner core at aL= 51RE, there
is a drop in the differential velocity at the CMB, and thus a drop in Qcmb. The dashed
grey line on Figure 3 corresponds to the power required to maintain an adiabat in the
fluid core. Dwyer et al. [2011] used this value as the threshold power for dynamo action,
Qth. We have already pointed out that this may not be an appropriate lower bound for
a mechanically forced dynamo, but if we adopt this specific choice, the intersection be-
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Confidential manuscript submitted to JGR-Planets
tween Qcmb and Qth occurs at aL≈50.4RE, slightly before the inner core Cassini state
transition, at which point the dynamo from Qcmb ceases.
We also show on Figure 3 the power dissipated at the ICB, Qicb. Not only is Qicb
higher than Qcmb for aL>42 RE, it remains above the dynamo threshold for a much
longer period of time, therefore allowing for a dynamo that may have persisted to more
recent epochs. The sudden drop in Qicb at aL≈51 REis due to the transition of the
Cassini state of the inner core, from state B to state A, the latter featuring a smaller dif-
ferential rotation at the ICB. Note that a large scale flow reorganization in the core may
accompany this Cassini transition, which would lead to a spike in Qicb (and also Qcmb),
before settling to the lower energy state. However, we cannot model this with our ide-
alized Cassini state equilibrium model.
Different combinations of rsand rf, and thus, in general, different interior density
models of the Moon, lead to changes in the predicted variations of θfand θswith aL.
Changes in θfremain modest for different interior models, with differences not exceed-
ing 1 degree from the evolution scenario shown in Figure 1. These lead to modifications
in the predicted amplitude of Qcmb versus aLdepicted in Figure 3, but not by more than
approximately 10%.
In contrast, the way in which θs(and thus Qicb) vary with aLis highly sensitive
to the choice of lunar interior density model. To illustrate this, Figure 4 shows how Qicb
vary with aLfor five different choices of outer radii (340, 350, 360, 370 and 380 km) and
two different choices of inner core radii (250 km for panel a; and 200 km for panel b).
Since Qicb is proportional to r5
s(see Equation 13), the size of the inner core is of cru-
cial importance for the power available to drive a dynamo, and by reducing rsfrom 250
to 200 km, the maximum Qicb has dropped by approximately a factor of 4. In addition,
different interior density models lead to different histories of Qicb. This is because the
different combinations of rsand rfimply a different fluid core density in each of these
interior models in order to match the lunar mass. In turn, the different density struc-
ture affects the FICN frequency Ωficn of the lunar model. Since the tilt angle of the in-
ner core is determined by the relative difference between Ωf icn and the forcing frequency
Ωp, the differential velocity at the ICB, and thus Qicb, shows very different histories for
different lunar models. Moreover, the point in time at which Ωficn intersects the tran-
sition frequency Ωtcan be substantially changed, and hence so does the timing of a Cassini
transition. For a given inner core size, the smaller the fluid outer core radius, the later
in lunar history the transition occurs (see Figure 2). For the model with rs= 250 km
and rf= 340 km (Figure 4a), and for the models with rs= 200 km and rf= 340
and 350 km (Figure 4b), the transition has not yet occurred.
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Confidential manuscript submitted to JGR-Planets
a) ICB radius = 250 km b)
ICB radius = 200 km
107
108
109
1010
1011
1012
1013
Viscous dissipation (Watts)
35 40 45 50 55 60
65
Semimajor axis (RE)
CMB radius
340 km
380 km
370 km
360 km
350 km
107
108
109
1010
1011
1012
1013
Viscous dissipation (Watts)
35 40 45 50 55 60
65
Semimajor axis (RE)
CMB radius
340 km
380 km
370 km
360 km
350 km
Figure 4. Power available to drive a dynamo estimated from viscous dissipation at the ICB
(Qicb) as a function of lunar orbit radius in units of Earth radii (RE), for different choices of
outer core radii, and for a solid inner core radius of (a) 250 km and (b) 200 km. The power
threshold to sustain a dynamo (Qth = 4.7×109W) is indicated by the grey horizontal dashed
line. Today corresponds to 60.3RE.
0
5
10
15
20
25
30
Magnetic field intensity (
μ
T)
35 40 45 50 55 60
65
Semimajor axis (RE)
0
5
10
15
20
25
30
Magnetic field intensity (
μ
T)
35 40 45 50 55 60
65
Semimajor axis (RE)
a) ICB radius = 250 km b)
ICB radius = 200 km
CMB radius
340 km
380 km
370 km
360 km
350 km
CMB radius
340 km
380 km
370 km
360 km
350 km
Figure 5. Paleomagnetic field intensities at the Moon’s surface from viscous dissipation at
the CMB (B(cmb)) and ICB (B(icb)) as a function of lunar orbit radius in units of Earth radii
(RE), for different choices of outer core radii, and for a solid inner core radius of (a) 250 km and
(b) 200 km. Today corresponds to 60.3RE.
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Confidential manuscript submitted to JGR-Planets
3.3 Paleomagnetic intensity from power dissipation
For the same set of models as in Figure 4, Figure 5 shows the magnetic field strengths
B(cmb)and B(icb)at the lunar surface predicted from dissipation at the CMB and at the
ICB, respectively, and as a function of aL. Dissipation at the CMB leads to a B(cmb)field
in the range of 20-30 µT for aL= 34RE, although its amplitude drops rapidly with in-
creasing aL. Note that the change in B(cmb)between the different models is dominantly
caused by the factor of rfin Equation (14). For all cases shown in Figure 5, Qcmb drops
below Qth between aL= 49REand 50.4RE, marking the point at which the dynamo
from dissipation at the CMB ceases. Note also that when a transition in the Cassini state
associated with the inner core occurs, there is a drop in Qcmb , leading to a smaller B(cmb)
and an earlier dynamo shutoff; this is most evidently seen in Figure 5a for the largest
inner core.
Dissipation at the ICB leads to a surface field amplitude B(icb)which can be as high
as 10 µT. We recall that our estimates of the magnetic field strength are based on d=
1 in Equation (16), in other words that all the magnetic energy is assumed to be in the
dipole part. Since B(icb)scales linearly with d, the predictions on Figure 5 would decrease
in proportion with a smaller choice for d. However, we also recall that the scaling law
that we use for B(icb)is based on convective dynamos, and the flows induced by the pre-
cession of an elliptically shaped inner core may further contribute to dynamo action and
thus lead to an increased B(icb). For all cases shown in Figure 5, the dynamo from Qicb
shuts off after that from Qcmb. While not as strong earlier in lunar history, the dynamo
from precession at the ICB may have persisted for a much longer period.
The dissipation and magnetic field strength at the ICB shown in Figures 4 and 5
are for relatively large inner core radii of 250 and 200 km. Since Qicb scales with r5
s(see
Equation 13) and thus B(icb)scales with r5/3
s(from Equation 16), it is clear that smaller
rswould yield smaller Qicb and B(icb). There is a critical inner core size below which Qicb
is smaller than the dynamo threshold Qth even at small aL. This is explored in the next
subsection.
Figures 4 and 5 illustrate how the histories of Qicb and B(icb)are sensitive to the
density structure of the lunar core. The evolution scenarios presented in these figures
only take into account the changes in the lunar orbit through its effect on Ωo, Ωpand
I, and assume a non-changing core density structure. However, the presence of an in-
ner core is due to growth from crystallization, and hence the radius of the inner core should
increase as a function of time. As the inner core grows, assuming it is composed primar-
ily of Fe, the proportion of lighter elements in the fluid core increases. The changing den-
sity contrast at the ICB implies a change in FICN frequency which can affect the evo-
lution scenarios presented above.
The precise history of the inner core growth depends among other things on the
initial composition of the fluid core and on the evolution of the heat flux at the CMB
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Confidential manuscript submitted to JGR-Planets
[e.g. Laneuville et al., 2014] which are not well known. To add to this difficulty, attach-
ing a precise time-history to the lunar orbit is challenging. Instead of presenting a pos-
sible evolution scenario that takes into account inner core growth, here we simply men-
tion how our above results would need to be adapted. Since Qicb is highly sensitive to
rs, taking inner core growth into account would make Qicb in Figure 4 weaker at the ear-
liest aLthat we have considered. Not only B(icb)would be reduced, since Qicb may fall
below Qth for smaller aL, a dynamo powered by differential precession at the ICB may
only have started later in lunar history.
3.4 Power dissipation at the ICB and magnetic field strength for a suite
of interior Moon models
Figure 6 further illustrates how the interior density structure of the Moon affects
Qicb and B(icb). It shows how Qicb and B(icb)vary as a function inner core radius rsand
fluid core radius rf, at three specific choices of lunar orbit radius, or equivalently three
specific epochs. These are snapshots in time so are independent of the history of inner
core growth. The three epochs that are chosen are: aL= 42RE(panels a-b), approx-
imately where Qicb becomes larger than Qcmb in Figure 4, and coinciding also to when
Qicb reaches its largest value; aL= 50RE(panels c-d), coinciding approximately with
the shutoff of the dynamo from differential precession at the CMB; and aL= 60.3RE
(panels e-f), corresponding to today.
As expected, Qicb generally increases with inner core size. However, the complete
picture is more intricate, as Qicb also depends on the Cassini state occupied by the in-
ner core and how close the FICN frequency Ωficn is to Cassini transition frequency Ωt.
The discontinuity in the Qicb contours (identified by a white dashed line) marks the lo-
cation in the rs−rfspace where Ωficn is equal to Ωt(Ωtis equal to 2π/19.6 yr−1, 2π/19.3
yr−1and 2π/16.4 yr−1for aL= 42RE, 50REand 60.3RE, respectively). This discon-
tinuity marks the boundary between models for which the inner core is in Cassini state
A (rfvalues above the discontinuity) versus those in state B (rfvalues below the dis-
continuity). For a given combination of rsand rf, the closer Ωficn is to Ωt, the larger
Qicb is. The largest absolute inner core tilt angles, and thus the largest Qicb, are achieved
in state B. Qicb amplitudes are highest at aL= 42REand decrease with increasing aL,
consistent with the behaviour shown in Figure 4.
The red contour line on the Qicb panels corresponds to Qth = 4.7×109W, our
chosen threshold for dynamo action. For Qicb > Qth, a magnetic field is generated, and
its strength at the lunar surface, B(icb), is shown in panels b-d-f of Figure 6. The largest
magnetic field strengths coincide with the largest values of Qicb. Being mindful of the
caveats on our estimates of the magnetic field strength, the maximum B(icb)at aL=
42RE, 50REand 60.3REare, respectively, 11.4µT, 6.4µT and 3.4µT.
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320
340
360
380
400
420
outer core radius (km)
50 100 150 200 250
inner core radius (km)
0
1
2
3
4
5
6
7
8
9
10
11
12
320
340
360
380
400
420
outer core radius (km)
50 100 150 200 250
inner core radius (km)
0
1
2
3
4
5
6
7
8
9
10
11
12
320
340
360
380
400
420
outer core radius (km)
50 100 150 200 250
inner core radius (km)
0
1
2
3
4
5
6
7
8
9
10
11
12
320
340
360
380
400
420
outer core radius (km)
50 100 150 200 250
inner core radius (km)
1012
1011
1010
109
108
107
106
105
104
103
102
101
100
Q
icb
(W)
320
340
360
380
400
420
outer core radius (km)
50 100 150 200 250
inner core radius (km)
1012
1011
1010
109
108
107
106
105
104
103
102
101
100
Q
icb
(W)
320
340
360
380
400
420
outer core radius (km)
50 100 150 200 250
inner core radius (km)
1012
1011
1010
109
108
107
106
105
104
103
102
101
100
Q
icb
(W)
a) aL = 42 RE aL = 42 RE
aL = 50 RE aL = 50 RE
aL = 60.3 RE aL = 60.3 RE
b)
d) c)
e) f)
state A
state A
state B
state B
state B
state A
B(icb) (µT)
B(icb) (µT)
B(icb) (µT)
Figure 6. Dissipation at the ICB (Qicb, in Watts, left column) and magnetic field strength at
the lunar surface (B(icb), in µT, right column) as a function of rsand rf, at three different lunar
orbit radii: aL= 42RE(top row), aL= 50RE(middle row) and aL= 60.3RE(bottom row).
Qth = 4.7×109W is shown by the red line on the Qicb plots. The white dashed lines show where
the FICN frequency is equal to the Cassini transition frequency.
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Confidential manuscript submitted to JGR-Planets
One conclusion that emerges from Figure 6 is that the inner core of the Moon must
be sufficiently large to sustain a dynamo by differential precession at the ICB. At aL=
42RE, the minimum inner core size is approximately 100 km, and the requirement on
inner core size increases with aL. A second conclusion from Figure 6 is that at a given
epoch Qicb is above Qth only for a specific range of core density models, and this range
gets narrower the further away the Moon is from Earth.
Interestingly, Figure 6f suggests that there is still a range of rsand rfthat allow
for dynamo action today, more specifically models for which the inner core is larger than
approximately 150 km, is in Cassini state B and with a FICN frequency close to the Cassini
transition frequency. Obviously, this depends directly on our assumption of a power thresh-
old of Qth = 4.7×109W. A higher threshold would further restrict the range of mod-
els for which a dynamo is possible; a weaker threshold would extend it. But if the thresh-
old that we have used is approximately correct, the fact that the Moon does not have
an on-going dynamo today implies then that: 1) the inner core radius is smaller than
approximately 150 km; or 2) that the inner core currently occupies state A; or 3) the in-
ner core is in state B, but the FICN frequency Ωficn is not close to the present-day tran-
sition frequency of Ωt= 2π/16.4 yr−1(Ωf icn >2π/19 yr−1for rs= 200 km; Ωficn >
2π/20 yr−1for rs= 250 km). If the present-day Moon has an inner core radius larger
than 150 km, and if its FICN frequency is close but outside the interval 2π/16.4−2π/20
yr−1, a lunar dynamo powered by precession at the ICB may have shut down only very
recently.
The viscous dissipation at the ICB shown in the Qicb panels on Figure 6 consti-
tutes a source of heat at the bottom of the fluid core. It is this heat that can drive ther-
mal convection and generate dynamo action. Figure 7 shows the associated heat flux at
the ICB, qicb, as a function of rsand rf, calculated from
qicb =Qicb
4πr2
s
.(20)
At aL= 42REand 50RE, a large portion of the rs−rfmodel space features a heat
flux larger than 1 mW/m2. The heat flux is considerably weaker today (aL= 60.3RE)
although there is still a large portion of the rs−rfspace where it is above 0.1 mW/m2.
For comparison, typical adiabatic heat flux values for the lunar core are of the order of
3-10 mW/m2[Dwyer et al., 2011; Laneuville et al., 2014] and the heat flux associated
with the latent heat release from inner core crystallization is of the order of 2 mW/m2
[e.g. Laneuville et al., 2014]. Hence, in addition to considerations of the past dynamo of
the Moon, the amplitude of the viscous dissipation at the ICB from the differential pre-
cession between the fluid and solid cores provides an important source of heat which can-
not be neglected in thermal evolution models.
Lastly, the Qicb,B(icb)and qicb contour maps of Figures 6 and 7 are tied to the choices
we have made for the density (ρc) and thickness (h) of the crust and for the density of
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Confidential manuscript submitted to JGR-Planets
320
340
360
380
400
420
outer core radius (km)
50 100 150 200 250
inner core radius (km)
100
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
320
340
360
380
400
420
outer core radius (km)
50 100 150 200 250
inner core radius (km)
100
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
320
340
360
380
400
420
outer core radius (km)
50 100 150 200 250
inner core radius (km)
100
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
aL = 50 RE
aL = 60.3 RE
qicb (W m-2)
qicb (W m-2)
qicb (W m-2)
a)
b)
c)
aL = 42 RE
Figure 7. Heat flux from viscous dissipation at the ICB from (qicb, in W m−2) as a function
of rsand rffor a) aL= 42RE, b) aL= 50RE, and c) aL= 60.3RE. Black lines identify contours
for which qicb = 1 mW m−2.
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Confidential manuscript submitted to JGR-Planets
the inner core (ρs). These choices influence the densities of the mantle and fluid core that
can match Ism and ¯ρ, and in turn, this affects the frequency of the FICN for a given com-
bination of rsand rf. With different assumptions on h,ρcand ρs, the contours of Qicb,
B(icb)and qicb would be shifted in rs−rfspace. Our general conclusions remain un-
altered, but one should be careful in extracting specific values of Qicb,B(icb)and qicb as
a function of rsand rffrom Figures 6 and 7.
4 Discussion and Conclusions
In agreement with Dwyer et al. [2011], we find that the dissipation at the CMB of
the Moon, Qcmb, from viscous friction generated by the differential precession between
the mantle and fluid core was large in the past, reaching values above 1013 W when the
Moon was orbiting closer to Earth. Our estimates of Qcmb differ slightly from those pre-
sented in Dwyer et al. [2011] because we have used an updated measure of the present-
day turbulent dissipation at the CMB, and also because our rotational model takes into
account the misalignment of the rotation vector of the fluid core with respect to the eclip-
tic normal. Using the power required to keep the fluid core in an adiabatic state as a guide-
line for the threshold of dynamo action, Qcmb was sufficient to power a dynamo when
the Moon was closer to Earth than approximately 50RE. Given the lack of scaling laws
associated with precession driven dynamos, obtaining estimates of the magnetic field strength
from such a dynamo is difficult. Based on a scaling law derived from convective dynamos,
the strength of the lunar surface magnetic field may have been as high as 20 −30µT,
although this is likely an overestimate as we explain further ahead. The amplitude of
Qcmb, the magnetic field strength associated with it, and the precise timing of the dy-
namo shutoff are only weakly dependent on the density structure of the lunar core and
the presence of an inner core.
The novel contribution from our study is the computation of estimates of the dis-
sipation from turbulent friction at the ICB of the Moon, Qicb, caused by the differen-
tial precession between the inner core and fluid core. We find that Qicb can be as high
as approximately 1010 W in the present-day Moon for an inner core radius of approx-
imately 200 km and for a tight window of fluid core density models. Qicb may have reached
amplitudes in the range of 1010−1011 W in the past for a broad range of core density
models. The largest source of error in these calculations is the unknown numerical con-
stant involved in our viscous coupling model, and our estimates of Qicb may be errant
by an order of magnitude.
The amplitude of Qicb and its evolution through time are highly sensitive to the
interior density structure of the core, notably the size of the inner core, but also the den-
sity contrast at the ICB. Both are influencing the FICN frequency, and it is how the lat-
ter compares with the precession frequency that determines the tilt angle of the inner
core in its Cassini state (SD18). For sensible density models of the core, the FICN fre-
quency has remained within the resonant band of the precession frequency for the whole
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Confidential manuscript submitted to JGR-Planets
range of orbital radius that we have covered in our study (>34RE) and large inner core
tilt angles with respect to the mantle result by resonant excitation. The closer the FICN
frequency is to the transition frequency between Cassini states A and B of the inner core,
the larger the inner core tilt angle is. Inner core tilt angles, and consequently Qicb, are
typically larger when the inner core occupies state B.
The heat generated by the differential precession between the solid and fluid core
is released at the ICB. This heat may drive a dynamo by thermal convection in the fluid
core provided Qicb is larger than the adiabatic heat flow out of the core. This power thresh-
old is likely a lower bound because flows mechanically forced by the precession of the in-
ner core can potentially further assist dynamo action. On the basis of this threshold, we
have shown that Qicb may have been sufficiently large in the past to power a lunar dy-
namo. A key requirement is that the inner core radius must be larger than approximately
100 km. Interestingly, Qicb can remain above the dynamo threshold for a lunar orbit ra-
dius larger than 50RE, and thus can outlive a dynamo generated by differential preces-
sion at the CMB. In fact, a range of core density models are compatible with a dynamo
persisting until very recently. The magnetic field amplitude at the lunar surface that we
predict from such a dynamo is of the order of a few µT. Although this should be viewed
as an order of magnitude estimate at best, it is nevertheless compatible with the lunar
paleointensities recorded after 3 Ga [e.g. Weiss and Tikoo, 2014].
We have presented our results in terms of lunar orbit radius aLand have not at-
tempted to connect aLto a specific time-history. A series of models relating aLto time
before present [Ooe et al., 1990; Walker et al., 1983; Webb, 1982] are summarized in Dwyer
et al. [2011]. Differences between them are important, but we can use the mean of these
models [presented in Figure S2 of Dwyer et al., 2011] to obtain approximate yardsticks.
The smaller aLthat we considered, 34RE, corresponds to approximately 4.2 Ga; aL=
42RE, the point beyond which Qicb can exceed Qcmb, corresponds to approximately 3.5
Ga; and aL= 50RE, the shutoff point of the dynamo from precession at the CMB, cor-
responds to approximately 2.2 Ga. Using this rough mapping, the large (>20µT) field
strengths produced by precession between the mantle and the fluid core are consistent
(although weaker) with the paleointensities of the high-field epoch between 4.2-3.5 Ga
[e.g. Weiss and Tikoo, 2014]. This mantle driven precession dynamo would operate un-
til 2.2 Ga, though with weaker field intensities, consistent with the paleointensities of
the weak-field epoch. This was the main conclusion of Dwyer et al. [2011].
Complementing this picture, our results show that a dynamo driven by the pre-
cession of the inner core can outlive the dynamo driven by mantle precession. Hence, not
only this inner core driven precession dynamo may further explain a part of the lunar
magnetic field recorded in the weak-field epoch, it can also explain a lunar dynamo per-
sisting until as recently as 1 Ga [e.g. Mighani et al., 2020], long after the mantle driven
precession dynamo would have shutoff. Hence, a combination of precession driven dy-
namos, by the mantle and the inner core, may be consistent with at least a part of the
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Confidential manuscript submitted to JGR-Planets
lunar paleomagnetic record. Obviously, this does not preclude that a dynamo driven by
thermo-chemical convection in the core [e.g. Laneuville et al., 2014] or the lower man-
tle [e.g. Scheinberg et al., 2018] may have coexisted with these precessionally driven dy-
namos.
The onset time of the dynamo driven by inner core precession depends on the his-
tory of inner core growth, so it may have been delayed by a fraction to a couple of Gyr
after the Moon formed. This dynamo ceased when Qicb eventually dropped below the
power threshold to maintain the core adiabat. This may have occurred smoothly as the
lunar orbit evolved, or it may be connected to a transition from state B to state A of
the Cassini state occupied by the inner core. The tilt angle of the inner core is smaller
in state A, and hence Qicb would have dropped significantly after such a transition, though
large scale flows and a possibly more energetic temporary dynamo may have resulted in
the process of the transition. Perhaps offering support for such a scenario, the most re-
cent estimate of the CMB radius from LLR is 381±12 km [Viswanathan et al., 2019]:
this places the inner core in Cassini state A at present-day (see Figure 6ef) but relatively
close to state B and consistent with a recent transition.
There are important feedback effects that we have not taken into account in our
rotational model which may significantly alter our results. First, once a magnetic field
is present in the lunar core, electromagnetic (EM) coupling at the ICB acts to reduce
the differential rotation between the fluid and solid cores (see for instance DW16). On
the one hand, the reduced differential rotation at the ICB implies a weaker viscous dis-
sipation. On the other hand, the shearing of the radial magnetic field at the ICB would
introduce EM dissipation. Ultimately, the source of energy remains the amplitude of the
differential precession at the ICB in the absence of a dynamo. Hence, the total of the
viscous and EM dissipation may not be wholly different from the Qicb values we have
estimated, just separated in different pools. Without an actual theoretical or numeri-
cal model it is difficult to predict precisely how taking into account EM coupling would
alter the estimates of Qicb that we have presented.
A second feedback that we have neglected is how viscous friction at the CMB and
ICB may alter the mutual orientations of the rotation vectors of the mantle, fluid core
and inner core. The Cassini state model of SD18 that we have used assumes no dissi-
pation. However, the large viscous friction at the CMB and ICB, especially at earlier times
in the lunar history, may limit the misalignment between the different rotation vectors.
A third feedback effect that is missing is how Qicb and Qcmb may alter the evolu-
tion of Iand Ωpas a function of orbit radius. Viscous friction at the CMB and ICB of
the Moon act to dissipate the rotational and orbital energy of the Moon. As an exam-
ple, a rate of energy dissipation of Qwithin the Moon, regardless of its nature, leads to
a reduction of the inclination Iof the lunar orbit according to [e.g. Equation (17) of Chen
and Nimmo, 2016]
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Confidential manuscript submitted to JGR-Planets
dI
dt =−1
tan I
aLQ
GMEM,(21)
where Mand MEare the masses of the Moon and Earth and Gis the gravitational con-
stant. For small I, a typical attenuation timescale τIof Iis then
τI≈I2GMEM
aLQ.(22)
To give an estimate of τI, let us use as a guide an inclination of I≈10◦at aL= 40RE.
For Q= 1012 W, this gives τI≈100 Myr. Energy dissipation of this magnitude in-
side the Moon should have rapidly driven the inclination close to zero and this is incon-
sistent with a present-day residual value of I= 5.145◦. This simple order of magnitude
estimate neglects the rotation of the Earth in the energy and angular momentum bud-
gets and also neglects the possibility of inclination re-excitation events [e.g Pahlevan and
Morbidelli , 2015]. Nevertheless, it illustrates that some of the high dissipation values that
we predict are likely not compatible with the time-history of Ithat we have used. In or-
der to lengthen the attenuation timescale τIto a more realistic estimate of 1 Gyr, vis-
cous dissipation in the core of the Moon should be limited to approximately 1011 W.
This problem was pointed by Dwyer et al. [2011] as they realized that the very large
Qcmb exceeding 3×1011 W (when aL<43REin our Figure 2) would lead to widespread
mantle melting, suggesting that such large dissipation never occurred. If so, then the large
magnetic field amplitudes at the surface in excess of 10 µT at earlier times are likely also
overestimated. This makes it more difficult to explain the lunar paleointensities of the
order of 100 µT in the high-field epoch by a precession dynamo. Likewise, viscous dis-
sipation at the ICB in excess of 1011 W that we have calculated in our study are unlikely
realistic. As illustrated by Figure 6, predictions of Qicb larger than 1011 W are associ-
ated with an inner core radius larger than 200 km. Hence, this limits the validity of our
results to inner core radii smaller than 200 km. Our general conclusions are not altered,
except that the largest surface magnetic field amplitudes that we predict are limited to
approximately 5 µT.
Ideally, viscous dissipation at the CMB and ICB should be included in calculations
of the orbital evolution of the Moon. The power available to drive a dynamo, either me-
chanically of thermally, could thus be estimated in a self-consistent manner. Recent ef-
forts have been made in this direction [e.g. ´
Cuk et al., 2019] although in a limited way
and in the absence of an inner core, as challenges remain important. If viscous dissipa-
tion at the ICB throughout the Moon’s history must be limited so that it is is not in con-
flict with the present-day lunar orbit inclination, this could place constraints on the max-
imum inner core size. We hope that our study may serve as an additional motivation to
include the presence of an inner core and viscous friction at the fluid core boundaries in
lunar orbital evolution models.
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Confidential manuscript submitted to JGR-Planets
Our results indicate that the heat flux qicb associated with Qicb can be of the or-
der of a few mW m−2. This heat flux may drive convective flows and power a dynamo.
But the mantle would still act as a bottleneck for this extra heat. A higher core tem-
perature decreases the radius at which the adiabatic temperature intersects the melt-
ing temperature of the iron alloy, hence the onset and rate of inner core growth is ulti-
mately controlled by how much heat can escape the core. Thermal evolution models for
the core of the Earth and planets typically do not include a contribution drawn from ro-
tational or orbital energy [e.g. Nimmo, 2015]. Tidal heating has been considered in the
thermal budget for many moons of the solar system [e.g Breuer and Moore, 2015; Nimmo
and Pappalardo, 2016], including the Moon [e.g. Peale and Cassen, 1978; Meyer et al.,
2010], but viscous heating associated with precession is usually ignored. Our results sug-
gest that the latter may play a first order role in the thermal evolution of the lunar core.
To put our results into perspective, the latent heat released from inner core crystalliza-
tion – the largest contribution to the heat budget in the absence of precession – is of the
order of 2 mW m−2[e.g. Laneuville et al., 2014]. Once the inner core radius reaches about
50 km, the heat flow from viscous friction connected to the precession of the inner core
is of the same order and cannot be neglected. The rate of inner core growth may then
be significantly slowed down by the additional heat at the ICB induced by inner core pre-
cession. A slower inner core growth would further reduce the latent heat released at the
ICB and thus further decrease its importance compared to qicb.
Lastly, if the heat released at the ICB is subcritical for a thermally driven dynamo,
core flows mechanically forced by the precession of an elliptically shaped inner core may
be capable of generating dynamo action by themselves. Whether this is possible is un-
known at present. Ultimately, the conditions for the onset of such a dynamo, and the
form and strength that its associated magnetic field may take inside the core and at the
lunar surface, can only be answered by an actual dynamical model. These questions are
particularly relevant given the recent paleomagnetic intensities weaker than 0.1µT recorded
in two Apollo samples and dated at 0.44±0.01 and 0.91±0.11 Ga [Mighani et al., 2020].
Since such magnetic field intensities are much smaller than those typically expected from
convective dynamos, Mighani et al. [2020] conclude that the lunar dynamo must have
likely ceased before ∼0.8 Ga. However, a magnetic field generated by mechanically forced
flows from precession at the ICB could have a r.m.s. strength of the order of 10−100µT
inside the core but with its energy dominantly in small length scales features. Because
of the sharp spatial attenuation of this field outside the core, the larger length scale part
would dominate the field recorded at the lunar surface but it may only amount to a frac-
tion of a µT. Indeed, numerical models of dynamos generated by precession at the CMB
suggest this is the case [C´ebron et al., 2019]. Hence, a lunar dynamo from precession at
the ICB generating surface field strengths of a fraction of a µT may be compatible with
the weak paleointensities recorded after 1 Ga, which would push the end of the lunar dy-
namo to more recently than 0.44 Ga. We hope that our results may serve as a motiva-
tion to modellers to attempt to address these questions.
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Confidential manuscript submitted to JGR-Planets
Acknowledgments
We thank Francis Nimmo and an anonymous reviewer for their constructive comments
which helped to improve this paper. The model used in this research is presented in de-
tail in Stys and Dumberry [2018]. Input parameters that are different than those used
in Table 1 of Stys and Dumberry [2018] are described in the text. Figures were created
using the GMT software [Wessel et al., 2013]. The source codes, GMT scripts and data
files to reproduce all figures are freely accessible at https://doi.org/10.7939/DVN/T1DTCM.
This work was supported by an NSERC/CRSNG Discovery Grant.
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