Icarus 172 (2004) 305–315
www.elsevier.com/locate/icarus
Planetary dynamos: effects of electrically conducting flows overlying
turbulent regions of magnetic field generation
Gerald Schubert a,∗ , Kit H. Chan b , Xinhao Liao c , Keke Zhang d
a Department of Earth and Space Sciences, Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90095-1567, USA
b Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China
c Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China
d School of Mathematical Sciences, University of Exeter, EX4 4QE, United Kingdom
Received 26 September 2003; revised 25 March 2004
Available online 7 August 2004
Abstract
A fully three-dimensional, nonlinear, time-dependent, multi-layered spherical kinematic dynamo model is used to study the effect on the
observable external magnetic field of flow in an electrically conducting layer above a spherical turbulent dynamo region in which the α effect
generates the magnetic field. It is shown that the amplitude and structure of an observable planetary magnetic field are largely determined by
the magnitude and structure of the flow in the overlying layer. It is also shown that a strong-field planetary dynamo can be readily produced
by the effect of an electrically conducting flow layer at the top of a convective core. The overlying layer and the underlying convective
region constitute a magnetically strongly coupled system. Such overlying layers might exist at the top of the Earth’s core due to chemical or
thermal causes, in the cores of other terrestrial planets for similar reasons, and in Saturn due to the differentiation of helium from hydrogen.
An electrically conducting and differentially rotating layer could exist above the metallic hydrogen region in Jupiter and affect the jovian
magnetic field similar to the overlying layers in other planets. Lateral temperature gradients resulting in thermal winds drive the flow in the
overlying layers. All planetary magnetic fields could be maintained by similar turbulent convective dynamos in the field-generation regions
of planets with the differences among observable magnetic fields due to different circulations in the overlying electrically conducting layers.
 2004 Elsevier Inc. All rights reserved.
Keywords: Geophysics; Magnetic fields
1. Introduction
It is generally accepted that planetary magnetic fields are
generated through magnetohydrodynamic processes in their
electrically conducting fluid cores by thermal or compositional convection (Moffatt, 1978; Fearn, 1998). An essential
condition for a planet to possess an intrinsic magnetic field
is that it has a sufficiently large magnetic Reynolds number
Rm
Rm =
VD
O(10),
λ
* Corresponding author.
E-mail address: [email protected] (G. Schubert).
0019-1035/$ – see front matter  2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.icarus.2004.06.007
where V is the typical amplitude of convective flows within
the core, D is the thickness (radius) of the core and λ is
the magnetic diffusivity (Moffatt, 1978). For Earth, we estimate that V ≈ 10−4 m s−1 , λ ≈ 1 m2 s−1 and D = 3 × 106 m
(Gubbins and Roberts, 1987) and Rm is about 300. For
Jupiter, if we take V = 3 × 10−2 m s−1 based on mixing
length theory, D ≈ 3 × 107 m and the magnetic diffusivity λ ≈ 30 m2 s−1 , Rm is about 3 × 104 (Stevenson, 2003).
A fundamental parameter that determines the properties of
a convective flow, whether it is laminar or turbulent, is the
Reynolds number Re
VD
ν
(or the Rayleigh number Ra ), in the convective region of a
planet, where ν is the kinematic viscosity. For the Earth’s
Re =
306
G. Schubert et al. / Icarus 172 (2004) 305–315
core, ν ≈ 10−6 m2 s−1 (Gubbins and Roberts, 1987) and
Re = O(108); for Jupiter, if we take a similar value for
ν, Re is about O(109). A large Re (Ra ) usually suggests
a strongly turbulent convective region (Sumita and Olson,
2000), though the effect of planetary rotation and magnetic
fields would modify the features of convection. Though the
precise values of these parameters are uncertain, they clearly
suggest that there exist turbulent flows in Earth and Jupiter
that can readily drive and sustain dynamos.
For a rapidly rotating planet with sufficiently large values of Rm and Re (or Ra ), it is reasonable to assume that
the core turbulence is associated with an α effect and that
the form of α has a symmetry reflecting the rapid rotation.
In this case, we would expect to observe a planetary magnetic field dominated by a dipole positioned at the planet’s
center and aligned with the axis of rotation (Roberts, 1972).
Nevertheless, the observed magnetic fields of the planets are
significantly different (Connerney, 1993). We explain this by
proposing that the observed structure of a planetary magnetic
field is largely determined by the amplitude and pattern of
flow in an electrically conducting flow layer that lies above
the turbulent convective magnetic field generation region.
The existence of a stably stratified electrically conducting
layer at the top of the Earth’s outer core has been suggested
by a number of authors (e.g., Braginsky, 1984, 1998, 1999,
2000; Whaler, 1980; Lister and Buffett, 1998). During the
general cooling of the Earth, the inner core grows and the
light constituents excluded from the inner core rise upward
through the outer core, perhaps as blobs. Some fraction of
the light constituents might survive remixing to create and
sustain a stable layer at the top of the core (Braginsky, 1984;
Moffatt and Loper, 1994). Because of the small density difference between the stable layer and the outer core, the existence of the layer would be difficult to detect seismically.
As a result, this stable layer has been referred to as “the
hidden ocean” (Braginsky, 1998, 1999, 2000; Shearer and
Roberts, 1997). An alternative argument for the existence of
a stably stratified conducting shell was put forward by Lister
and Buffett (1998). They suggested that a negative buoyancy
flux across the core–mantle boundary (the heat flux imposed
by the slowly convecting mantle is less than the value that
could be conducted up the core adiabat) can form a stable
layer of sufficiently strong stratification to prevent substantial penetration of rising blobs from below (see also Labrosse
et al., 1997). However, there is no conclusive evidence to
support the existence of a stably stratified conducting layer
at the top of the Earth’s outer core (Lay and Young, 1990).
The important influence of the inhomogeneous mantle on
core convection and the geodynamo has been recognized for
a long time and there is a large literature on possible observational effects and on the possible existence of toroidal
flows in the stable layer that could arise from lateral heterogeneity on the core–mantle boundary (e.g., Gubbins, 1987;
Gubbins and Bloxham, 1987; Gubbins and Richards, 1986;
Bloxham and Gubbins, 1985; Bloxham, 2000; Olson and
Christensen, 2002; Christensen and Olson, 2003). The the-
oretical problem of how an inhomogeneous mantle drives
a dominantly toroidal flow by the thermal-wind mechanism
near the core–mantle boundary was studied by Zhang and
Gubbins (1992). Experiments on convection driven by an externally imposed inhomogeneous heat flux were carried out
by Sumita and Olson (2000).
For Saturn, Stevenson (1980, 1982) suggested that there
exists a stable conducting fluid shell above the interior dynamo generation region as a result of the ongoing gravitational differentiation of helium from hydrogen (see also
Fortney and Hubbard, 2003). To explain the axisymmetry of
Saturn’s magnetic field, Stevenson (1980, 1982) considered
a multilayered Saturn model with a stably stratified conducting fluid shell between the dynamo generation interior and
the external region (see also Love, 2000). In Stevenson’s
(1982) model, a toroidal flow is driven in the shell by an
equator-to-pole temperature difference arising from the temperature condition of Saturn’s atmosphere. He argued that
the conducting layer is stably stratified and the axisymmetric toroidal flow in the layer tends to make the external
field of Saturn axisymmetric. For Jupiter, shock wave experiments (Nellis et al., 1996) indicate that magnetic diffusivity in the jovian interior increases gradually from the
field-generation metallic core to the outer semi-conducting
flow region. The electrically conducting and differentially
rotating layer above the metallic hydrogen region in Jupiter
would affect the jovian magnetic field similar to the overlying layers in other planets.
The primary aim of this paper is to understand the general effects of an electrically conducting flow layer on the
amplitude and structure of a planetary magnetic field generated by dynamo action in an underlying turbulent convecting
core. It is not our purpose to model the magnetic field of
a particular planet. We investigate the following scenario
through numerical simulations. An electrically conducting,
thin flow layer at the top of a core or regions of magnetic
field generation by turbulent convection is thermally coupled with the lateral thermal heterogeneity of the overlying
region (the lower mantle in the case of the Earth). This thermal coupling drives thermal winds in the thin conducting
layer. The conducting layer is, in turn, magnetically coupled
to the turbulent convective core through the magnetic interface conditions at the base of the layer, which influences the
whole process of magnetic field generation. This scenario is
fundamentally different from that discussed by Zhang and
Gubbins (1992) and Olson and Christensen (2002) for the
Earth. These authors considered how the convective flow
and the generated magnetic field in the outer core are directly affected by the nonuniform temperature or heat flux
boundary conditions in the absence of an overlying layer.
The influence of this direct thermal coupling between the
lower mantle and outer core is quite weak when convection
is fully turbulent and has small scales incompatible with the
large lateral scale of the core–mantle boundary.
We consider a fully three-dimensional, nonlinear, timedependent, spherical kinematic dynamo consisting of four
Planetary dynamos with overlying layers
different regions: an electrically conducting solid inner core,
a turbulent convective outer core that generates a planetary magnetic field by convection, an electrically conducting
stable layer overlying the dynamo region that changes the
observable magnetic field, and a nearly insulating exterior.
The four spherical regions are coupled magnetically through
matching conditions at the interfaces. Since the key issue
we address in this paper is not how a planetary magnetic
field is generated in the convective core, but how an overlying electrically conducting flow layer with zonal thermal
winds alters the planetary magnetic field, we simply assume
a parameterized, nonlinear α 2 kinematic dynamo in the convective outer core. It should be emphasized that the choice
of this simple α formulation is motivated mainly by mathematical convenience rather than by its physical importance.
We demonstrate that the structure and amplitude of the magnetic field that would be generated by the convection-driven
dynamo in the deep core, acting by itself, is not observable. Instead, the amplitude and structure of the observed
planetary magnetic field is determined by the magnitude and
planform of the zonal thermal winds in the overlying layer.
The remainder of the paper is organized as follows. After discussing our planetary model and mathematical formulation in Section 2, we discuss the corresponding finiteelement model of the dynamo in Section 3. We present the
results of our numerical simulations in Section 4 and close
the paper with a summary and some remarks in Section 5.
2. Model and mathematical formulation
Our dynamo model is fully three-dimensional and consists of four different spherical regions; the geometry of the
model is shown in Fig. 1. The inner sphere in 0 < r < ri with
constant magnetic diffusivity λi is solid and is governed by
the equations
∂Bi
+ λi ∇ × ∇ × Bi = 0,
∂t
∇ · Bi = 0.
(1)
(2)
The magnetic field Bi cannot be generated in the inner core.
However, an electrically conducting core may be important
for the generation mechanism (Hollerbach and Jones, 1993).
We assume fully turbulent convection in the core or region of dynamo action ri < r < ro . In this region with constant magnetic diffusivity λo , a magnetic field Bo is generated by an α-effect in connection with turbulent convection.
The nonlinear α 2 dynamo in the convection zone is described by
∂Bo
= αo ∇ × α r, θ, |Bo /Beq |2 Bo + λo ∇ 2 Bo ,
∂t
ri < r ro ,
(3)
∇ · Bo = 0,
(4)
ri < r ro ,
307
Fig. 1. Geometry of the three-dimensional, four-zone, interface dynamo
model: 0 < r ri , the uniformly rotating, electrically conducting solid inner core with magnetic diffusivity λi ; ri r r0 , the uniformly rotating
convection zone with magnetic diffusivity λo ; r0 r rt , the overlying
electrically conducting flow layer with magnetic diffusivity λt ; and r > r0 ,
the exterior with large magnetic diffusivity λe .
where Beq is the typical magnetic field used for the scaling,
αo is a positive parameter and the α distribution is confined
in ri < r ro and will be discussed later.
Overlying the region of dynamo generation is an electrically conducting fluid shell, a region of strong, mainly
toroidal flow in the form of thermal winds ut , driven either by the thermally nonuniform lower mantle in the case
of an Earth-like planet (e.g., Gubbins, 1987; Lloyd and Gubbins, 1990) or driven by the atmosphere boundary condition
in the case of a Saturn-like planet (Stevenson, 1982). The
structure of the toroidal flow in the overlying layer is externally imposed and is independent of the nature of thermal
convection in the turbulent core. Of course, there may be
some flows from the turbulent region that overshoot into the
overlying layer that will be neglected here. In the overlying
electrically conducting shell ro < r < rt , the toroidal flow
shears the poloidal magnetic field that is generated in the
turbulent convective region and penetrates into the overlying
layer. This important process is described by the equations
∂Bt
= ∇ × (ut × Bt ) − λt ∇ × ∇ × Bt , ro < r rt ,
∂t
∇ · Bt = 0, ro < r rt ,
1
2Ω × ut = − ∇P + α̂grT , ro < r rt ,
ρ
∇ · ut = 0, ro < r rt ,
(5)
(6)
(7)
(8)
where ut is a flow driven by lateral thermal heterogeneity
in the region above the conducting shell (the lower mantle in
the case of the Earth), α̂ is the thermal expansion coefficient,
g is the acceleration of gravity, T represents the deviation
of the temperature from its static distribution, and λt is the
magnetic diffusivity of the overlying layer. In this kinematic
dynamo study, Eq. (7), the thermal wind equation, is diagnostic and we do not solve Eq. (7). The purpose of Eq. (7)
308
G. Schubert et al. / Icarus 172 (2004) 305–315
here is to show that (i) the flow can be generated by the lateral variation of temperature in a rotating system and (ii) the
effect of the Lorentz force on the flow ut is neglected in our
model. The toroidal flow in the overlying layer alone cannot
generate a magnetic field. It is the magnetic coupling at the
base of the overlying electrically conducting layer that plays
a key role in controlling the whole process of a planetary
dynamo.
We nondimensionalize length by the thickness of the dynamo region (ro − ri ), magnetic field by Beq , and time by the
magnetic diffusion time (ro − ri )2 /λo of the convective dynamo region. There are four-nondimensional quantities that
characterize the dynamo problem: the magnetic diffusivity
ratios βi , βt , and the magnetic Reynolds numbers Rα and
Rm , defined by
λi
λt
,
βt = ,
λo
λo
(ro − ri )α0
U (ro − ri )
Rα =
(9)
,
Rm =
,
λo
λo
where U is a typical amplitude of the flow in the overlying layer. The governing equations are solved subject to a
number of crucial matching and boundary conditions at the
interfaces. In particular, at the base of the overlying layer, all
components of the magnetic field and the tangential component of the electrical field are continuous at the interface r = ro
βi =
(Bt − Bo ) = 0 at r = ro ,
r × Rm (ut × Bt ) − βt ∇ × Bt − Rα αBo + ∇ × Bo = 0
at r = ro ,
To overcome this difficulty, we construct a mathematically equivalent planetary dynamo problem that is not overdetermined. This can be achieved by introducing a Lagrange
multiplier into the dynamo equation (for example, Assous
et al., 1993). The finite element formulation of our dynamo
model is then given by the following four sets of modified dimensionless equations. In the solid inner core, the equations
are
∂Bi
= −βi ∇ × ∇ × Bi + ∇pi , 0 < r < ri ,
(11)
∂t
∇ · Bi = 0, 0 < r < ri ,
(12)
where the Lagrange multiplier pi is the fourth variable
needed for closure of the problem. In the convective outer
core, the dimensionless governing equations are
∂Bo
= Rα ∇ × α r, θ, φ, |Bo |2 Bo − ∇ × ∇ × Bo
∂t
+ ∇po , ri < r ro ,
(13)
∇ · Bo = 0,
ri < r < ro ,
where po is the Lagrange multiplier in the outer core. For a
given flow in the overlying layer, the governing dimensionless equations are
∂Bt
= Rm ∇ × (ut × Bt ) − βt ∇ × ∇ × Bt + ∇pt ,
∂t
ro < r < rt ,
(15)
∇ · Bt = 0,
ro < r < rt ,
pi = pt
at r = ri ,
where r is the position vector. At the top of the overlying
layer, the magnetic field matches the nearly potential external field which is of secondary importance for this particular
problem. Of primary importance are the magnetic field conditions at the interface between the overlying layer and the
turbulent convective region that pass the structural signature
of the overlying layer onto the whole deep dynamo process.
In our model, we choose ri /ro = 0.4; the thickness of
the overlying flow layer is (rt − ro )/rt = 0.1. The dynamo
model is fully three-dimensional, kinematic, nonlinear and
time-dependent; there is no imposition of any spatial or temporal symmetries.
pt = po
at r = rt ,
po = pe
at r = ro ,
We use a finite element method based on the threedimensional tetrahedralization of the whole spherical system. The primitive variables for the magnetic field are
adopted by solving the dynamo equation together with the
solenoidal condition. The method is particularly efficient
when the dynamo system involves discontinuous layers like
the present problem. However, with this approach, the dynamo problem is over-determined since there are only three
unknowns (Br , Bθ , Bφ ) but four equations in each region.
(16)
and pt is the Lagrange multiplier in the overlying layer. The
Lagrange multipliers must be continuous at the interfaces
(10)
3. Finite element formulation
(14)
(17)
where pe is the Lagrange multiplier in the exterior region.
On the outer spherical surface r = rm of the dynamo solution
domain, we impose
pe = 0 at r = rm .
(18)
Further details about the treatment of the magnetic field
boundary conditions can be found in Chan et al. (2001).
Since the Lagrange multipliers satisfy Laplace’s equation
and condition (18), they are analytically exactly zero or numerically very small everywhere. The Lagrange multipliers
used in the dynamo equations have no physical significance.
The dynamo problem defined by Eqs. (1)–(8) is mathematically equivalent to that defined by Eqs. (11)–(16).
The mathematical formulation using a finite element approximation is given by the following sets of equations. In
the solid interior region
∂
Wi · Bi dV
∂t
Vi
=−
βi ∇ × Bi · ∇ × Wi + pi ∇ · Wi dV
Vi
Planetary dynamos with overlying layers
−
βi r̂ × ∇ × B − pi r̂ · Wi dS,
(19)
∂Vi
qi ∇ · Bi dV = 0,
(20)
where r̂ is a unit vector in the radial direction, Wi and qi are
the weighting functions, Vi denotes the domain 0 r ri
and ∂Vi denotes the interface between the inner core and
the convective outer core at r = ri . In the convective outer
region, we have
∂
Wo · Bo dV
∂t
Vo
=
Rα α |Bo |2 Bo − ∇ × Bo · ∇ × Wo
Vo
+ po ∇ · Wo dV
Rα α |Bo |2 Bo − r̂ × ∇ × Bo + po r̂ · Wo dS
+
∂Vt
−
written as
B = [Bi , Bt , Bo , Be ]
j
j
j
Bi (t)Nj (r),
Bt (t)Nj (r),
Bo (t)Nj (r),
=
j
j
Vi
Rα α |Bo |2 Bo − r̂ × ∇ × Bo + po r̂ · Wo dS,
∂Vo
(21)
qo ∇ · Bo dV = 0,
309
j
Be (t)Nj (r) ,
j
(25)
j
p = [pi , pt , po , pe ]
j
j
j
=
pi (t)Mj (r),
pt (t)Mj (r),
po (t)Mj (r),
j
j
j
pe (t)Mj (r) ,
j
(26)
j
where Mj (r) and Nj (r) are shape functions defined in each
j
j
tetrahedral element, Bi and pi , for example, denote the
value of Bi and pi at the j th node. The shape functions
Mj (r) in a tetrahedron are defined by
M1 = L1 ,
M2 = L2 ,
M3 = L3 ,
M4 = L4 ,
(27)
where (Lj , j = 1, 2, 3, 4) are the volume coordinates for a
tetrahedron, and satisfy
Mj (ri ) = δij ,
(22)
4
Mj (r) = 1.
(28)
j =1
Vo
The shape functions Nj (r) in a tetrahedron are defined by
where Wo and qo are the weighting functions, Vo denotes the
domain ri r ro and ∂Vo denotes the interface between
the convection zone and the overlying layer at r = ro . In the
overlying conducting flow shell, we have
∂
Wt · Bt dV
∂t
Vt
=
(Rm ut × Bt − βt ∇ × Bt ) · ∇ × Wt + pt ∇ · Wt dV
N1 = L1 (2L1 − 1),
N2 = L2 (2L2 − 1),
N3 = L3 (2L3 − 1),
N4 = L4 (2L4 − 1),
Vt
+
−
Rm r̂ × (u × Bt ) + βt r̂ × ∇ × Bt + pt r̂ · Wt dS
Rm r̂ × (u × Bt ) + βt r̂ × ∇ × Bt + pt r̂ · Wt dS,
∂Vi
qt ∇ · Bt dV = 0,
N6 = 4L1 L3 ,
N7 = 4L1 L4 ,
N8 = 4L2 L3 ,
N9 = 4L2 L4 ,
N10 = 4L3 L4 ,
(23)
(24)
Vt
where Wt and qt are the weighting functions, Vt denotes the
domain ro r rt and ∂Vt denotes the interface between
the overlying layer and the exterior at r = rt (Fig. 1).
After the three-dimensional tetrahedralization of the
whole spherical system, the finite element approximation
for the magnetic field and the Lagrange multiplier can be
(29)
and satisfy
Nj (ri ) = δij ,
∂Vt
N5 = 4L1 L2 ,
10
Nj (r) = 1.
(30)
j =1
In other words, there are four nodes for the Lagrange multiplier in each tetrahedron (the four corners), and ten nodes for
the magnetic field (the four vertices and six middle points of
the six edges of each tetrahedron). More details can be found
in Chan et al. (2001). The weight functions are selected to
be the same as the expansions used in Eqs. (25)–(26). After carrying out the standard procedure for the finite-element
analysis, we obtain a system of nonlinear ordinary differential equations, representing a typical saddle-point, initial
value problem. A Crank–Nicholson scheme is then used for
time integration wherein the nonlinear term is treated explicitly by a second-order extrapolation while all the linear terms
are treated implicitly. The resulting linear system is solved
by an iterative method.
310
G. Schubert et al. / Icarus 172 (2004) 305–315
4. The effect of an overlying layer on planetary
dynamos
4.1. A nonlinear dynamo without flow in the overlying layer
It is well known that the interaction of the small-scale velocity and magnetic field in the fluid shell produces a largescale magnetic field B through an α-effect (Moffatt, 1978).
Since the issue we address in this paper is how the overlying layer alters the planetary magnetic field generated in the
convective outer core, we simply choose a parameterized alpha effect
r − ri
cos θ
,
Bo sin π
α=
(31)
ro − ri
(1 + |Bo |2 )
where (r, θ, φ) are spherical polar coordinates with θ = 0 at
the axis of rotation, and Bo is the generated magnetic field
in the outer convective region. While antisymmetry (cos θ )
about the equatorial plane reflects the influence of rotation,
the factor (1+|B|2) represents the nonlinear process of alpha
quenching which saturates the growing magnetic field (e.g.,
Hagee and Olson, 1991; Roberts and Soward, 1992).
For providing the reference dynamo solution, we first
switch off the effect of the shear flow in the overlying layer
by setting Rm ≡ 0 so that the layer becomes an electrically
conducting layer (Bullard and Gubbins, 1977). The resulting dynamo corresponds to a well-understood conventional
α 2 dynamo (e.g., Roberts, 1972; Hagee and Olson, 1991).
Our calculations show that any initial magnetic field decays with increasing time if Rα is less than 20 and grows
if Rα > 20 (with βi = βt = 1). Furthermore, the properties
of the nonlinear dynamo are not dependent on initial conditions. The onset of dynamo action takes place at about
Rα = 20. The nonlinear dynamo is always stationary, axisymmetric and equatorially dipolar even though numerical
simulations are always fully three-dimensional and time dependent. The planform of the generated magnetic field at
Rα = 40 is shown in Fig. 2 for a typical dynamo with
βi = βt = 1. The magnetic energy Em of the generated magnetic field
Em = |B|2 dV ,
V
as a function of time, where V denotes the fluid region, is
shown in Fig. 3. The maximum level of contours for Bφ (top
panel in Fig. 2) is 1.6 while the maximum level of contours
for Br (the middle panel) is 2.5. The strength of the toroidal
magnetic field is comparable with (slightly weaker than) that
of the poloidal field. There is also a weak toroidal magnetic
field in the motionless overlying layer because βt = 1.
4.2. A nonlinear dynamo with axisymmetric flow in the
overlying layer
We now introduce an axisymmetric toroidal flow in
the overlying layer while keeping all other parameters unchanged. The thermal wind equation is diagnostic: there is a
Fig. 2. Contours of the azimuthal field Bφ in a meridional plane (top panel)
and the radial field at the middle of the convection shell (middle panel)
and at the top of the overlying layer (lower panel) viewed from the north
pole. The parameters are Rα = 40 with βi = βt = 1. Solid contours indicate
azimuthal field lines with Bφ < 0 (Br < 0) and dashed contours correspond
to fields with Bφ > 0 (Br > 0). The contours in the middle and lower panels
are projections of the hemispherical surfaces on the equatorial plane. The
sizes of these projected surfaces are drawn with equal radii for convenience
of presentation.
thermal-wind flow ut for an imposed temperature T . Since
our purpose is to examine how the flow in the overlying layer
affects the action of a planetary dynamo, we first choose a
simple axisymmetric shear flow in the form
ut (r, θ, φ) = (uθ , ur , uφ )
r − ro
= 0, 0, sin θ sin π
rt − ro
(32)
in the layer ro < r rt , where (uθ , ur , uφ ) denote the three
components of the velocity in spherical polar coordinates.
This shear flow is confined in the overlying fluid shell that
is hydrodynamically detached from the turbulent convection
Planetary dynamos with overlying layers
311
Fig. 3. Magnetic energy Em as a function of time for four nonlinear dynamo
solutions with βi = 1 and Rα = 40. The onset of dynamo action occurs at
about Rα ≈ 20 for βi = βt = 1.
region, but magnetically coupled with it. Clearly, the flow
satisfies condition (8).
There are two main parameters that characterize the overlying flow layer: the magnetic Reynolds number Rm and the
diffusivity ratio βt . If the overlying layer is stably stratified,
the fluid layer may have a smaller value of magnetic diffusivity compared with the turbulent (eddy) magnetic diffusivity
in the convective region, i.e., βt 1. The magnetic energies of two typical axisymmetric dynamos are also shown
in Fig. 3 for Rm = 100 with βt = 0.1 and βt = 1. The generated magnetic field at Rm = 100 is shown in Fig. 4 for
βi = βt = 1. The maximum level of contours for Bφ (top
panel in Fig. 4) is 4.2 while the maximum level of contours for Br (middle panel in Fig. 4) is 3.1. The nonlinear
dynamos with axisymmetric differential rotation in the overlying layer remain stationary, axisymmetric and equatorially
dipolar. The shear flow in the overlying layer greatly amplifies the azimuthal magnetic field in the layer and strengthens
the toroidal magnetic field compared with the poloidal field.
The strongest toroidal magnetic field occurs in the overlying
layer as a result of the shear even though βt = 1. The total
magnetic energy of the dynamo increases from Em = 8 to
15, indicating that the amplitude of the generated magnetic
field is significantly increased. At βt = 0.1, the structure
of the generated magnetic field remains similar, but the total magnetic energy of the dynamo increases to Em = 28
with |Bφ | |Br |. It is remarkable that a planet can have a
strong field dynamo just because of an overlying electrically
conducting flow layer at the top of its turbulent convective
dynamo region.
4.3. A nonlinear dynamo with nonaxisymmetric flow in the
overlying layer
The most interesting case is related to nonaxisymmetric flow in the layer driven by large-scale nonaxisymmetric
Fig. 4. Same as Fig. 1 except Rm = 100 and βt = 1 with the presence of an
axisymmetric flow in the overlying layer.
thermal conditions above (in the lower mantle, for example, in the case of the Earth). We seek to determine how
such nonaxisymmetric flow affects the generated magnetic
field because of the magnetic coupling of the overlying layer
with the outer core. It is to be emphasized that, in the absence of the overlying layer, the turbulent flow in the outer
core is associated with an α effect given by Eq. (31) that is
axisymmetric and has the symmetry reflecting the rapid rotation and that generates an axisymmetric dipole magnetic
field positioned at the planet’s center and aligned with the
axis of rotation.
For simplicity, we choose a simple nonaxisymmetric
toroidal flow in the form
ut (r, θ, φ) = (uφ , ur , uθ )
M
sin 2θ cos Mφ, 0, sin θ 3 sin2 θ − 2
=
2
312
G. Schubert et al. / Icarus 172 (2004) 305–315
Fig. 5. Contours of the azimuthal field Bφ for a nonlinear dynamo in the
presence of nonaxisymmetric flow in the overlying layer plotted at two
meridional planes, φ = 0 (left panel) and φ = π/2 (on the right panel) for
Rα = 40, Rm = 100, M = 2 with βi = βt = 1. Solid contours indicate azimuthal field lines with Bφ > 0 and dashed contours correspond to fields
with Bφ < 0.
r − ro
× sin Mφ sin π
(33)
rt − ro
in ro < r rt , where M is a parameter (we choose M = 2 in
our numerical simulations). The flow in the overlying layer
satisfies condition (8). The structure of a typical nonlinear
dynamo solution for Rm = 100 and Rα = 40 (βi = β1 = 1)
is shown Figs. 5 and 6. The magnetic energy Em of the generated magnetic field as a function of time is given in Fig. 7.
Insight into how the dynamo is influenced by the overlying flow layer can be obtained by comparing the dynamo solution shown in Fig. 5 and 6 to the dynamo solution displayed in Fig. 2. Both dynamos have exactly the
same parameters for the convective outer region. As clearly
shown in Fig. 5, the azimuthal magnetic field is fully threedimensional and nonaxisymmetric in the whole domain (the
convective zone and the overlying layer) due to the nonaxisymmetric flow in the overlying layer. Figure 6 shows
contours of Br at three different spherical surfaces, from
the middle of the convective core to the top of the overlying
shell. The simple axisymmetric dipolar field that would exist
without the overlying layer is replaced by two concentrated
regions of poloidal magnetic field at diametrically opposite
longitudes where the nonaxisymmetric layer flow is imposed
on the coupled system. The nonaxisymmetric magnetic field
in the convective core is characterized by the azimuthal wave
number m = 2, reflecting the pattern of the toroidal flow in
the overlying layer. Evidently, the purely toroidal flow in the
overlying layer cannot generate a magnetic field by itself.
However, it affects not only the observable external field, but
also the structure of the generated magnetic field in the convective outer region below. Comparison of Fig. 1 to Figs. 5
Fig. 6. Contours of the radial magnetic field for the solution of Fig. 5 plotted
at three spherical surfaces, the middle of the convection zone (top panel),
the interface between the convection zone and the overlying layer (middle
panel) and the interface between the stable layer and the exterior (lower
panel). The contours in the middle and lower panels are projections of the
hemispherical surfaces on the equatorial plane. The sizes of these projected
surfaces are drawn with equal radii for convenience of presentation.
and 6 suggests that the overlying layer, through the magnetic
matching condition at the interface between the overlying
layer and the convective region, plays a controlling role on
the main features of a planetary dynamo. The influence of
the layer is substantial; it completely alters the generated
nonlinear dynamo.
4.4. A nonaxisymmetric convective dynamo with
axisymmeric flow in the overlying layer
Finally, we consider the opposite case in which the convective dynamo is nonaxisymmetric while the flow in the
overlying layer is axisymmetric. This is the case studied by
Planetary dynamos with overlying layers
Fig. 7. Magnetic energy Em as a function of time for two nonlinear dynamo
solutions with Rα = 40 and Rm = 100 (βi = βt = 1). Em indicates an
axisymmetric α in the convection zone.
313
which generates a nonaxisymmetric dynamo when m > 1.
In our simulations, we take m = 2 in Eq. (34) while the
flow in the overlying layer is axisymmetric and given by
Eq. (32). The result is a weakly nonaxisymmetric magnetic
field, which is shown in Fig. 8 for Rα = 40, Rm = 100 with
βi = βt = 1. At the middle surface of the outer core, the
magnetic field is nonaxisymmetric but it becomes nearly
axisymmetric at the top of the overlying layer. This is consistent with the previous investigation of a similar problem
by Stevenson (1980, 1982) and Love (2000).
The nonlinear nonaxisymmetric dynamos are also stationary. We do not fully understand why we have always
produced time-stationary magnetic fields. It is well known
that a time-independent α 2 dynamo would produce a stationary magnetic field while some α − ω dynamos generate
oscillatory solutions. Moreover, the inclusion of the dynamical feedback of the generated magnetic field usually leads
to time-dependent magnetic fields. The time-stationarity of
our dynamo solutions could be associated with the particular
nature of our kinematic dynamo model such as the strongly
nonlinear quenching that may suppress the time dependence
of a nonlinear α − ω dynamo.
5. Concluding remarks
Fig. 8. Magnetic field generated by the nonaxisymmetric alpha in the convective zone with an axisymmetric flow in the overlying layer. The top left
gives contours of the azimuthal field Bφ in a meridional plane; the top
right (lower left and lower right) shows the radial field at the middle of
the outer core (and the bottom and top of the stable layer). The parameters
are Rα = 40 and Rm = 100 with βi = βt = 1. The sizes of these projected
surfaces are drawn with equal radii for convenience of presentation.
Stevenson (1980, 1982). However, he did not consider the
magnetic interaction between the core and the stable layer.
We simply choose a nonaxisymmetric alpha effect in the
form
cos θ (1 + cos mφ)
r − ri
α=
(34)
sin
π
,
B
o
ro − ri
(1 + |Bo |2 )
In an effort to understand the effect of an overlying electrically conducting flow layer at the top of the
core on planetary magnetic fields, we have investigated a
fully three-dimensional, nonlinear, time-dependent, multilayered, spherical kinematic dynamo. It is shown that the
effect of the overlying layer can change a planetary dynamo
in a fundamental way. It completely alters the magnetic field
that would be generated by the turbulent convective region
in the absence of the layer. When the thin stable shell and
the underlying convection are not coupled dynamically, we
show that the magnetic field observed outside the planet is
controlled mainly by the flow pattern in the overlying layer.
A major difficulty in any theory of planetary dynamos
is to explain why the observed magnetic fields of planets are different. For example, a general planetary dynamo
theory or model should be capable of explaining why the
Earth’s magnetic field is different from Saturn’s field or
why Saturn’s field is different from Jupiter’s. In fact, all
well-understood dynamo models such as turbulent dynamo
models (Roberts, 1972) or Braginsky’s nearly axisymmetric
dynamo model (Braginsky, 1975) reveal nearly axisymmetric magnetic fields as a result of rotational symmetry. The
axisymmetry of a planetary dynamo should be regarded as
the natural state of a planetary magnetic field if the planet
is rotating rapidly and has a turbulent dynamo region. The
difficulty is to explain nonaxisymmetric planetary magnetic
fields.
Stably stratified layers plausibly exist in the Earth’s outer
core (Braginsky, 1984, 1999, 2000; Lister and Buffett, 1998)
and in Saturn (Stevenson, 1980, 1982). Such layers might
314
G. Schubert et al. / Icarus 172 (2004) 305–315
exist in the outer cores of other terrestrial planets for similar
reasons to the Earth. Differentially rotating layers could exist above the regions of magnetic field generation in Jupiter,
Neptune, and Uranus if near surface zonal flows extend to
sufficient depth. Our dynamo calculations suggest that the
observable magnetic field of a planet is sensitively influenced by the effect of the flow in an overlying conducting
flow layer. In the cases of terrestrial planets the overlying
layer flow reflects the lateral thermal heterogeneity of the
lower mantle. In the cases of the outer planets the large flow
possibly reflects thermal conditions at the planet’s surface.
Accordingly, the strong departure from axisymmetry of a
planetary magnetic field could be caused by the presence of
nonaxisymmetric flow in the layer. All the well-established
dynamo theories can be applied within the convection-driven
dynamo region without the need of modification. It is the influence of the stable layer at the top of the Earth’s fluid core
that largely determines the structure of the observable geomagnetic field as supported by observations (Gubbins, 1987;
Bloxham, 2000). The axisymmetry of Saturn’s magnetic
field indicates either the presence of an axisymmetric flow
in the stable shell (Stevenson, 1980, 1982) or the absence of
such a stable layer in Saturn. Meanwhile the larger tilt of the
dipole in Jupiter’s magnetic field may be indicative of the
presence of the strong flow overlying the metallic dynamo
region. The magnetic field of Mercury may be significantly
nonlinear and nonaxisymmetric because of stable layer flow
in the outer core imposed by small-scale lateral thermal
heterogeneity in Mercury’s mantle. The highly nondipolar
magnetic fields of Uranus and Neptune may also be related
to flow patterns above their regions of dynamo field generation. In this case, nonaxisymmetric flows in stably-stratified
shells below the strongly axisymmetric zonal flows observed
at the planets’ surfaces may be a result of shear flow instabilities of the axisymmetric flows.
It is significant to note that our dynamo models are purely
kinematic and that the large-scale circulation in the stable
layer is prescribed without considering the influence of the
underlying convection. In other words, there is no dynamical compatibility between the large-scale circulation in the
stable shell and the underlying convection. When the shell
is strongly stably stratified and the underlying convection is
highly turbulent, the pattern of the large-scale circulation in
the stable shell is likely determined by the nonuniform overlying layer. However, the amplitude of the large-scale circulation, i.e., the size of the magnetic Reynolds number Rm
for the stable shell, can be strongly influenced by the breaking effect of the Lorentz force which is coupled dynamically
with the underlying convection. Because this effect is neglected in our kinematic dynamo model, the influence of the
large-scale circulation may be overestimated.
Acknowledgments
G.S. is supported by grants from the NASA Planetary Geology and Geophysics and Planetary Atmospheres programs
and from NSF under NSF EAR 00105945. X.L. is supported
by China NSFC and MOST grants and K.Z. is supported by
UK NERC and PPARC grants.
References
Assous, F., Degond, P., Heintze, E., Raviart, P.A., Segre, J., 1993. On a
finite-element method for solving the three-dimensional Maxwell equations. J. Comput. Phys. 109, 222–237.
Bloxham, J., 2000. Sensitivity of the geomagnetic axial dipole to thermal
core–mantle interactions. Nature 405, 63–65.
Bloxham, J., Gubbins, D., 1985. The secular variation of the Earth’s magnetic field. Nature 317, 777–781.
Braginsky, S.I., 1975. An almost axially symmetric model of the hydrodynamic dynamo of the Earth. Geomagn. Aeronomy 15, 149–156.
Braginsky, S.I., 1984. Short-period geomagnetic secular variation. Geophys. Astrophys. Fluid Dynam. 30, 1–78.
Braginsky, S.I., 1998. Magnetic Rossby waves in the stratified ocean at the
top of the core, and topographic core–mantle coupling. Earth Planets
Space 50, 641–649.
Braginsky, S.I., 1999. Dynamics of the stably stratified ocean at the top of
the core. Phys. Earth Planet. Inter. 111, 21–34.
Braginsky, S.I., 2000. Effect of the stratified ocean of the core upon the
Chandler wobble. Phys. Earth Planet. Inter. 118, 195–203.
Bullard, E.C., Gubbins, D., 1977. Generation of magnetic fields by fluid
motions of global scale. Geophys. Astrophys. Fluid Dynam. 8, 43–56.
Chan, K.H., Zhang, K., Zou, J., Schubert, G., 2001. A nonlinear, 3-d spherical α 2 dynamo using a finite element method. Phys. Earth Planet.
Inter. 128, 35–50.
Christensen, U.R., Olson, P., 2003. Secular variation in numerical geodynamo models with lateral variations of boundary heat flow. Phys. Earth
Planet. Inter. 138, 39–54.
Connerney, J.E.P., 1993. Magnetic fields of the outer planets. J. Geophys.
Res. 98, 18659–18679.
Fearn, D.R., 1998. Hydromagnetic flow in planetary cores. Rep. Prog.
Phys. 61, 175–235.
Fortney, J.J., Hubbard, W.B., 2003. Phase separation in giant planets: inhomogeneous evolution of Saturn. Icarus 164, 228–243.
Gubbins, D., 1987. Mechanism for geomagnetic polarity reversals. Nature 326, 167–169.
Gubbins, D., Bloxham, J., 1987. Morphology of the geomagnetic field and
implications for the geodynamo. Nature 325, 509–511.
Gubbins, D., Richards, M., 1986. Coupling of the core dynamo and mantle:
thermal or topographic? Geophys. Res. Lett. 13, 1521–1524.
Gubbins, D., Roberts, P.H., 1987. Magnetohydrodynamics of the Earth’s
core. In: Geomagnetism, vol. 2. Jacobs, J.A. (Ed.). Academic Press,
London, pp. 1–183.
Hagee, V.L., Olson, P., 1991. Dynamo models with permanent dipole fields
and secular variation. J. Geophys. Res. 96, 11673–11687.
Hollerbach, R., Jones, C.A., 1993. Influence of the Earth’s inner core on
geomagnetic fluctuations and reversals. Nature 365, 541–543.
Labrosse, S., Poirier, J.P., Le Mouel, J., 1997. On cooling of the Earth’s
core. Phys. Earth Planet. Inter. 99, 1–17.
Lay, T., Young, C.J., 1990. The stable stratified outermost core revisited.
Geophys. Res. Lett. 17, 2001–2004.
Lister, J.R., Buffett, B.A., 1998. Stratification of the outer core at the core–
mantle boundary. Phys. Earth Planet. Inter. 105, 5–19.
Lloyd, D., Gubbins, D., 1990. Toroidal fluid motion at the top of Earth’s
core. Geophys. J. Inter. 100, 455–467.
Love, J.J., 2000. Dynamo action and the nearly axisymmetric magnetic field
of Saturn. Geophys. Res. Lett. 27, 2889–2892.
Moffatt, H.K., 1978. Magnetic Field Generation in Electrically Conducting
Fluids. Cambridge Univ. Press, Cambridge.
Moffatt, H.K., Loper, D.E., 1994. Hydromagnetics of the Earth’s core. I.
The rise of a buoyant blob. Geophys. J. Inter. 117, 394–402.
Planetary dynamos with overlying layers
Nellis, W.J., Weir, S.T., Mitchell, A.C., 1996. Metallization and electrical
conductivity of hydrogen in Jupiter. Science 273, 936–938.
Olson, P., Christensen, U.R., 2002. The time-averaged magnetic field in
numerical dynamos with nonuniform boundary heat flow. Geophys. J.
Inter. 151, 809–823.
Roberts, P.H., 1972. Kinematic dynamo model. Philos. Trans. R. Soc. London Ser. A 272, 663–698.
Roberts, P., Soward, A.M., 1992. Dynamo theory. Annu. Rev. Fluid
Mech. 24, 459–512.
Shearer, M.J., Roberts, P.H., 1997. The hidden ocean at the top of Earth’s
core. Dynam. Atmos. Oceans 27, 631–647.
Stevenson, D.J., 1980. Saturn’s luminosity and magnetism. Science 208,
746–748.
315
Stevenson, D.J., 1982. Reducing the nonaxisymmetry of a planetary dynamo and an application to Saturn. Geophys. Astrophys. Fluid Dynam. 21, 112–127.
Stevenson, D.J., 2003. Planetary magnetic fields. Earth Planet. Sci.
Lett. 208, 1–11.
Sumita, I., Olson, P., 2000. Laboratory experiments on high Rayleigh number thermal convection in a rapidly rotating hemispherical shell. Phys.
Earth Planet. Inter. 117, 153–170.
Whaler, K., 1980. Does the whole of the Earth’s core convect? Nature 287,
528–529.
Zhang, K., Gubbins, D., 1992. On convection in the Earth’s core forced
by lateral temperature variations in the lower mantle. Geophys. J. Inter. 108, 247–255.