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A fully implicit model for simulating dynamo action in a Cartesian domain

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Ž .
Physics of the Earth and Planetary Interiors 120 2000 339–349www.elsevier.com
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A fully implicit model for simulating dynamo action in aCartesian domain
Jorg Schmalzl
)
, Ulrich Hansen
¨
Institut fur Geophysik, Uni
Õ
ersitat Munster, Munster, Germany
¨ ¨ ¨
Received 4 December 1998; accepted 4 April 2000
Abstract
We present a fully implicit numerical method to solve the incompressible MHD equations in a strongly rotating Cartesiandomain. The equations are solved in a primitive variable formulation using a finite volume discretization. In order to usemassively parallel computers, we applied a domain decomposition approach in space. The performance of this model iscompared with an earlier model, which treated the convective terms of the equations in an explicit manner. Our resultsindicate that although the fully implicit method needs about three times the memory of the implicit–explicit method, it issuperior in terms of computational efficiency. As an application of this model, we investigated the influence of the Prandtlnumber in the range of 0.01–1000 on the dynamics of the dynamo.
q
2000 Elsevier Science B.V. All rights reserved.
Keywords:
Cartesian domain; Dynamo action; Implicit–explicit method
1. Introduction
The dynamo action operating in the molten outercore of the Earth seems today the only vital mecha-nism able to generate and maintain the Earth’s mag-netic field. The underlying idea that a flow in anelectrically conducting fluid maintains a magneticfield by electromagnetic induction, which would de-cay by Ohmic dissipation, otherwise, may appearrather simple. Due to its nonlinear nature, however,the study of stability properties of the magnetohydro-dynamic problem, and especially the investigation of finite amplitude properties, poses one of the most
)
Corresponding author. Fax:
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49-251-833-6100.
Ž .
E-mail address:
joergs@earth.uni-muenster.de J. Schmalzl .
fascinating and likewise difficult problems in geo-physical fluid dynamics.Today, there is almost a consensus that the geody-namo operates in the so-called ‘strong-field regime’
Ž .
Roberts, 1988 . In the ‘weak-field regime’, the dy-namo action is characterized by a balance betweenpressure and Coriolis forces while buoyancy, Lorentz,inertial and frictional forces are understood as small
Ž .
perturbations to the weak-field flow Busse, 1975 .Characteristically, weak field dynamos do generate amagnetic field but the flow is not influenced by thegenerated field. In strong-field dynamos, the mag-
Ž .
netic energy ME is, typically, at least one order of
Ž .
magnitude higher than the kinetic energy KE andthere exists a tight coupling between the magneticfield and the flow where it stems from.
0031-9201
r
00
r
$ - see front matter
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2000 Elsevier Science B.V. All rights reserved.
Ž .
PII: S0031-9201 00 00157-6
( ) J. Schmalzl, U. Hansen
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Physics of the Earth and Planetary Interiors 120 2000 339–349
340
Applying numerical models to the problem of strong-field dynamos has recently led to considerable
Ž .
progress. Glatzmaier and Roberts 1995 were ableto demonstrate the existence of a self-consistent dy-namo, which does not only generate a magnetic fieldbut also resembles some of the typical features of themagnetic field of the Earth. A different but similarmodel has been employed by Kuang and Bloxham,
Ž .
1997 in order to investigate the role of the viscousboundary layers at inner
r
outer core interface. Chris-
Ž .
tensen et al. 1998 observed some of the characteris-tic features even at lower values of the rotation rateand pointed out that the dipole-dominated field seemsto be a robust feature.The recent progress has impressively demon-strated that numerical modeling momentarily formsthe key instrument for research on dynamo problemsof planetary scale. It is also clear that numericalmodeling of the dynamo process is at the edge of today’s generation of computers. Glatzmaier and
Ž .
Roberts, 1995 report on numerical experiments,which need thousands of CPU-hours on a Cray-vectorcomputer. Even these demanding models,though doubtless very impressive, resemble the Earthcore only roughly. Thus, only the most robust con-clusions from dynamo modeling can be regarded asrelevant for the geodynamo problem. Questions like‘Can inertia be neglected?’, or how to treat theboundary between inner and outer core, remainunanswered up to now.Most of the mentioned dynamo models are based
Ž
on a pseudo-spectral formulation Glatzmaier andRoberts, 1995; Kageyama and Sato, 1997; Kuang
.
and Bloxham, 1997 . While spectral methods lead tovery efficient computer codes on vector machines,they cannot easily be parallelized. This drawback will become more important in the future whencluster arrangements based on commodity hardwarewill be available on an unsurpassed price
r
perfor-
Ž .
mance ratio Warren et al., 1997 .We have, therefore, developed a dynamo modelfor parallel computers, which is based on a finite
Ž .
volume discretization Trompert and Hansen, 1999 .As explained below, the finite volume discretizationis particularly well suited for a parallel approach.At the present stage, the model allows the investi-gation of a Childress–Soward dynamo type in Carte-
Ž .
sian coordinates Childress and Soward, 1972 . Therehave already been several models that are able to
Ž
simulate dynamo action in a Cartesian domain e.g.,Soward, 1974; St. Pierre, 1993; Jones and Roberts,
.
1999 . To our knowledge, all these models employspectral approaches in the horizontal direction. Wehave chosen a Cartesian domain in order to be able
Ž
to reach more realistic parameters especially lower
.
values of the Ekman number than the existingspherical codes. The model philosophy is to delin-eate particular physical phenomena rather than toobtain a dynamo that resembles the geodynamo asclosely as possible.In this paper, we address two main points. Atfirst, we will describe the properties of a fully im-plicit time stepping method. We will concentrate onthe question if the increase in memory requirementsand computational effort is met by the larger timestep size. Secondly, we report on experiments thathave been carried out to investigate the importanceof inertia and viscous forces as compared to rotation.Previous work on thermal convection has indicatedthat the Prandtl number plays an important role withrespect to the temporal and spatial behavior of ther-mal convection. In what follows, we present a studyon the role of the Prandtl number on dynamo proper-ties.
2. Model and methods
2.1. Go
Õ
erning equations
The equations describing incompressible magne-tohydrodynamics in a 3D Cartesian domain, which
Ž .
rotates around the vertical
x
axes are given in
˜
3
their non-dimensional form as:
E
u
Pr
2
q
=
P
uu
y
BB
y
Pr
=
u
ž /
E
t pPr
q
=
P
q
x
P
u
y
PrRaT
x
s
0 1
Ž .
˜ ˜
3 3
E
E
B
1
2
q
=
P
uB
y
Bu
y
=
B
s
0 2
Ž . Ž .
E
t p
E
T
2
q
=
P
u
T
y
=
T
s
0 3
Ž . Ž .
E
t
=
P
u
s
0 4
Ž .
=
P
B
s
0 5
Ž .
( ) J. Schmalzl, U. Hansen
r
Physics of the Earth and Planetary Interiors 120 2000 339–349
341
where
p
is the pressure containing parts due to thecentrifugal and Lorentz force but not the hydrostaticpart.
u
,
B
and
T
are the velocity, the magnetic fieldand the temperature, respectively.
x
is the unity
˜
3
vector in vertical direction.
Pr
is the Prandtl numberdefined as
Pr
s
n
r
k
with
n
being the kinematicviscosity and
k
the thermal diffusivity and
p
, theRoberts number as defined by
p
s
k
r
h
where
h
isthe magnetic diffusivity.For the equations, we used the Boussinesq ap-proximation and scaled the length with the height
d
of the box and the time
t
with the thermaldiffusion-scale
d
2
r
k
y
1
.The ratio of the buoyancy forces over viscousforces is given by the Rayleigh-number
a
g
D
Td
3
Ra
s
6
Ž .
kn
where
a
is the thermal diffusivity,
D
T
is the tem-perature difference over the box and
g
is the gravita-tional acceleration in
y
x
direction.
˜
3
The ratio of the viscous forces over the Coriolisforces is measured by the Ekman number,
E
:
n
E
s
7
Ž .
2
2
V
d
where
V
is the angular velocity. In the horizontaldirection, we use periodic boundary conditions. Thevertical boundaries are conducting perfectly the mag-netic field and free-slip for the velocity. The temper-atures for the upper and lower boundaries are set tothe non-dimensional values 0 and 1, respectively.
2.2. Numerical method
The equations are solved in the primitive variableformulation on a uniform collocated grid. A collo-cated, instead of the commonly used staggered ar-
Ž .
rangement Harlow and Welch, 1965 has been usedsince it offers more flexibility with respect to a laterextension to spherical geometry. Discretization isperformed by employing cell-centered finite volumediscretization with a second order central scheme forthe diffusive fluxes and an upwind biased scheme,which is based on a quadratic interpolation, QUICK
Ž .
Leonardo, 1979 . Values that are needed at cellfaces are interpolated using a second order interpola-tion. A general description of the finite volume
Ž .
method is presented in Patankar 1980 .
( )2.2.1. The implicit–explicit method IMEX
In the implicit–explicit version, the solution hasbeen advanced in time by employing a linear multi-
Ž . Ž .
step method to the Eqs. 1 – 3 . Here, the convectiveterms are treated explicitly while diffusive terms are
Ž
treated implicitly. The IMEX method Frank et al.,
.
1997 we have used is defined as follows.Consider an ordinary differential equation of theform,d
wt
s
F t
,
w t
q
G t
,
w t
. 8
Ž . Ž . Ž . Ž .
Ž . Ž .
d
t
Applying the IMEX method leads to the temporaldiscretization1 1
n
q
1
n n n
y
1
w
s
w
q
2
q
d t
F
y
dt
F
Ž .
2 21 2
q
d
1
d
n
q
1
n
y
1
q
t
G
q
t
G
,
ž / ž /
2 1
q
d
2 1
q
d
9
Ž .
t
y
t
n
q
1
n
d
s
,
t
s
t
y
t
. 10
Ž .
n
q
1
n
t
y
t
n n
y
1
The IMEX scheme effectively decouples the sys-tem of equations. Recently, it has become clear thatthe stability of the individual implicit and explicitmethods does not guarantee the stability of the com-
Ž .
bined method Frank et al., 1997 . It can, however,be shown that stability is obtained for a time step,
t
,defined by:
c
t
s
. 11
Ž .
Pr
max
N
u
Nq N
B
N
i i
(
3
ž /
p
Ý
D
x
i
i
s
1
Ž
u
,
B
and
D
x
are the components of the velocity,
i i i
.
magnetic field and gridspacing, respectively with
c
Ž .
being smaller than 0.4 Trompert and Hansen, 1999 .In order to determine a divergence-free velocityfield,
u
, we use a pressure correction method. Insuch a method, first an intermediate velocity
u
)
iscomputed from known pressure values. Then a Pois-
( ) J. Schmalzl, U. Hansen
r
Physics of the Earth and Planetary Interiors 120 2000 339–349
342
son equation is solved for the corrected pressure,
p
n
q
1
and finally a new, divergence-free velocity iscalculated from the corrected pressure.We obtain the pressure correction by subtractingthe equation for
u
)
from the one for
u
n
q
1
butneglecting only the viscous terms, instead of neglect-ing all terms with
u
)
, as done in standard pressurecorrection techniques. This is done since the ratio of
Pr
r
E
can be very large and, therefore, the Coriolisterms should be included.It is also necessary to correct the magnetic field inorder to guarantee that
=
B
s
0 is fulfilled in every
Ž .
time step. Brackbill and Barnes 1980 have demon-strated that a nonzero
=
B
leads to physical incorrectsolutions. Similarly, for the velocity correction, onefirst has to obtain an intermediate solution
B
)
andthen can derive a Poisson equation for an auxiliaryvariable,
f
. From
D
f
s
=
B
)
, the auxiliary variable
f
can be calculated and for the corrected magneticfield
B
n
q
1
s
B
)
y
=
f
, then the relation
=
B
n
q
1
s
0will hold.
2.2.2. The implicit method
The results presented in this paper have beenobtained by employing a fully implicit procedure tothe system of equations. Time stepping is performed
Ž
by a second order two-step implicit BDF Bank,
.
1985 method with variable step sizes:
U
s
a U
n
q
1
q
a U
n
q
a U
n
y
1
, 12
Ž .
t
0 1 2
with
2
1
q
2
a
1 1
q
a
1
Ž .
a
s
,
a
sy
,
0 1
1
q
a
D
t
1
q
a
D
t
a
2
1
D
t a
s
, and
a
s
2
1
q
a
D
t
D
t
old
In the first time step, we use
a
s
0, which results in1
n
y
1
n
U
s
U
y
U
, 13
Ž . Ž .
t
D
t
Ž .
which is the formula for Backward Euler
a
s
0 .The length of the time step is determined by theconvergence of the solver. If there is no convergenceafter 200 iterations, the time step is reduced and thesolution procedure is restarted. If the number of iteration is smaller than 10, the length of the timestep is increased.For each time step, one has to solve a nonlinearset of equations, as given by:
f
A u
B C D
1
Ž .
u
f
0
E u
0 0
Ž .
2
B
s
T f
0 0
F u
0
Ž .
3
0
0
0
P f G
0 0 0
4
where the first row denotes the momentum equationand the second, the equation for the magnetic field,
B
. The third row stands for the energy equation andthe fourth row for the continuity equation. In order tosolve the nonlinear system, we basically follow a
Ž .
SIMPLE strategy Patankar, 1980 . A block-symmet-ric Gauss–Seidel procedure is employed whereblocking is applied over
u
,
B
, and
T
. As such, forevery gridpoint, first a 7
=
7 system is solved andthe pressure is calculated from a Poisson equation. Itis noteworthy that in the pressure correction equa-tion, only the viscous terms are neglected, not thecoriolis terms. For the solution of the pressure equa-
Ž
tion, we used a BICGSTAB procedure van der
.
Vorst, 1992 . Subsequently, the velocity is correctedand if no convergence has been obtained, the systemis further iterated. In case that no convergence can be
Ž .
reached after a number of iterations usually 20 , theSIMPLE procedure is repeated, employing a smallertime step. After a successful series of SIMPLE itera-tions on auxiliary variable, the correction of themagnetic field is calculated. Similarly, for the pres-sure solution procedure, a Poisson equation for anauxiliary variable can be derived. We employBICGSTAB to solve this Poisson equation. Finally,the magnetic field is corrected, which forms theconclusion of a time step.For parallelization, the 3D domain is decomposedinto subblocks and each processor is assigned asubblock. Each subblock has a buffer zone, which istwo cells wide, due to the stencil of the convectionscheme. This buffer zone is used for exchanging datafrom other processors. A sketch of this decomposi-tion is shown in Fig. 1. Explicit message-passing iscarried out using the MPI standard. All the commu-nication between the processors are carried out in aseparate subroutine since this provides a tight syn-chronization between the processors. The scalabilityof the program is demonstrated in Fig. 2. Over awide range, the speed increase is almost linear.
( ) J. Schmalzl, U. Hansen
r
Physics of the Earth and Planetary Interiors 120 2000 339–349
343Fig. 1. Sketch of the decomposition of the Cartesian domain intoeight subblocks as denoted by the dashed lines. The width of thegrey buffer zones is determined by the stencil of the convectionscheme.
2.2.3. Model
Õ
alidation
The implicit–explicit version of the code has beenextensively checked against various cases ranging
Ž .
from a vortex decay problem Chorin, 1968 to the
Ž
thermal convection problem De Vahl and Jones,
. Ž .
1983 . In Trompert and Hansen 1999 , these testsare described in detail. In order to test the fullyimplicit procedure, we have conducted a comparisonbetween the fully implicit code and the implicit–ex-plicit version for a dynamo case. For a value of
Ra
s
10
9
and an Ekman number of 10
y
4
, Fig. 3shows the evolution of the Nusselt number,
Nu
andthe ME
r
KE ratio. Clearly, both codes produce virtu-ally the same values, which we take as a clearindication that the fully implicit version is properlyworking.
3. Efficiency of the fully implicit method
Our efforts to implement a fully implicit methodstem from the well known phenomenon that thevelocity of the Alfven waves increases linearly with
´
the magnetic field. In strong field dynamos, thismeans a severe restriction on the time step, if explicittime stepping is employed. Fully implicit methodscan potentially mean a significant advantage sincethey allow larger time steps, which are not restrictedby the Levi–Courant criterion. On the other hand,
Fig. 2. Scalability of the program for a grid resolution of 32
3
and 64
3
. The calculations have been carried out on the 256-node T3E of theHLRZ Julich
r
Germany. On the
x
-axes, we show the number of processors used, which is increased as a power of two. The program does
¨
not require the number of processors to be a power of two but one does obtain optimal results by doing so, due to the load balancingbetween the processors. The
y
-axes give the time needed per time step averaged over 10 time steps. We did not obtain a time for the 64
3
forless than four processors because the problem did not fit on the local memory of the processors.

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