zotero-db/storage/KJX6LNF3/.zotero-ft-cache

CONCURRENCY AND COMPUTATION: PRACTICE AND EXPERIENCE Concurrency Computat.: Pract. Exper. 2011; 23:38–56 Published online 27 September 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cpe.1593
Scalability of pseudospectral methods for geodynamo simulations
Christopher J. Davies1,∗,†, David Gubbins1 and Peter K. Jimack2
1School of Earth and Environment, University of Leeds, Leeds LS2 9JT, U.K. 2School of Computing, University of Leeds, Leeds LS2 9JT, U.K.
SUMMARY
The problem of understanding how Earth’s magnetic ﬁeld is generated is one of the foremost challenges in modern science. It is believed to be generated by a dynamo process, where the complex motions of an electrically conducting ﬂuid provide the inductive action to sustain the ﬁeld against the effects of dissipation. Current dynamo simulations, based on the numerical approximation to the governing equations of magnetohydrodynamics, cannot reach the very rapid rotation rates and low viscosities (i.e. low Ekman number) of Earth due to limitations in available computing power. Using a pseudospectral method, the most widely used method for simulating the geodynamo, computational requirements needed to run simulations in an ‘Earth-like’ parameter regime are explored theoretically by approximating operation counts, memory requirements and communication costs in the asymptotic limit of large problem size. Theoretical scalings are tested using numerical calculations. For asymptotically large problems the spherical transform is shown to be the limiting step within the pseudospectral method; memory requirements and communication costs are asymptotically negligible. Another limitation comes from the parallel implementation, however, this is unlikely to be threatened soon and we conclude that the pseudospectral method will remain competitive for the next decade. Extrapolating numerical results based upon the code analysis shows that simulating a problem characterizing the Earth with Ekman number E = 10−9 would require at least 13 000 days per magnetic diffusion time with 54 000 available processors, a formidable computational challenge. At E = 10−8 an allocation of around 350 million CPU hours would compute a single diffusion time, many more CPU hours than are available in current supercomputing allocations but potentially reachable in the next decade. Exploration of the 10−6 ≤ E ≤ 10−7 regime could be performed at the present time using a substantial share of national supercomputing facilities or a dedicated cluster. Copyright ᭧ 2010 John Wiley & Sons, Ltd.
Received 19 August 2009; Revised 16 March 2010; Accepted 17 March 2010
KEY WORDS: geodynamo; pseudospectral method; scalability
1. INTRODUCTION
Geodynamo theory asserts that the Earth’s magnetic ﬁeld is continually generated and destroyed by motions of a vigorously convecting, electrically conducting ﬂuid (liquid iron plus a small percentage of lighter elements) conﬁned to the outer core, a 2260 km thick spherical shell some 2800 km below Earth’s surface. In this dynamo process, the movement of ﬂuid across magnetic ﬁeld lines induces electric currents that sustain the magnetic ﬁeld against the effects of electrical resistance. Not all ﬂows act as dynamos and the highly nonlinear magnetohydrodynamic equations that govern the problem make understanding the physics of dynamo action a formidable task.
∗Correspondence to: Christopher J. Davies, Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, La Jolla, CA 92093-0225, U.S.A.
†E-mail: cjdavies@ucsd.edu
Contract/grant sponsor: Leverhulme; contract/grant number: F/00 122/AD
Copyright ᭧ 2010 John Wiley & Sons, Ltd.

GEODYNAMO SCALABILITY

39

The equations must be solved numerically, which has led to many successful dynamo solutions (e.g. [1–9]). These solutions reproduce certain features of the observed ﬁeld: dipole-dominance, westward drift of magnetic features and complete polarity reversals. However, simulations operate in parameter regimes far removed from that believed to be appropriate for the Earth, which remains inaccessible using the current computing power.
The mathematical problem consists of solving three advection–diffusion partial differential equations in a spherical shell representing the Earth’s liquid core. The dependent variables are the ﬂuid ﬂow, magnetic ﬁeld and density. Two of these equations describe the conservation of momentum and heat; the third, a combination of Maxwell’s laws of electromagnetism and Ohm’s law, governs the evolution of the magnetic ﬁeld.
For any choice of parameter values, boundary conditions and basic state, the computational task involves representing dependent variables with sufﬁcient spatial resolution to obtain converged solutions (solutions that do not change upon increasing resolution) while using sufﬁcient temporal resolution to resolve the intrinsic timescales of the system. Great disparity of scales causes the main numerical difﬁculties: intrinsic timescales range from the rotation period, one day, to the reversal timescale, a few hundred thousand years. The numerical timestep must, therefore, be small enough to resolve the shortest timescales, but the equations must be integrated for at least one magnetic diffusion time (≈ 25000 years, the time taken for the dipole ﬁeld to decay in the absence of a generation mechanism) to demonstrate a successful dynamo. Polarity reversals [10] require an even longer integration time. Turbulence in the core will likely lead to a broad spectrum of lengthscales; numerical simulations will never be able to resolve the smallest lengthscales without some simplifying assumptions about the turbulence. The spatial resolution for any particular model must be sufﬁcient for that model to have converged, which puts ‘Earth-like’ parameters out of reach at present. As simulations move towards more ‘Earth-like’ parameter regimes, spatial resolution must increase whereas the timestep must decrease.
This paper assesses the scalability of a parallel, distributed memory pseudospectral code, by far the most common numerical implementation for solving the dynamo equations [5]. In particular, the following questions will be addressed:
1. How does the method scale with increasing problem size? 2. What are the limits on the size of problem the method can address using current available
hardware? 3. How big a computer is needed to perform an ‘ideal’ calculation that runs in an ‘Earth-like’
parameter regime?
These objectives are achieved by ﬁrst specifying the simplest physical problem that might be representative of the Earth. This problem will be called the ideal problem as it represents a goal not achieved by any geodynamo simulation to date. Despite its relative simplicity this problem remains well beyond the capacity of current computers. Next, estimates of the spatial and temporal resolutions required to undertake the ideal problem are determined. Naïve operation counts, memory requirements and communication costs for asymptotically large problems are then obtained for the pseudospectral method. Numerical calculations are then presented for comparison with theoretical scalings. Finally, theoretical scalings are used to extrapolate numerical calculations to Earth-like values, obtaining computational requirements for simulating the ideal problem.
Numerical calculations are conducted on a parallel computer cluster located at the University of Leeds. The cluster comprises 128 dual-core nodes connected by a Myrinet 2000 network. Each core has a clock speed of 2.4 Ghz and each node has 2 GB of memory. As we show later, the computational costs outweigh memory and communication requirements and hence this computer cluster is satisfactory for the calculations pursued here.
The numerical solutions reproduce Case 1 of the dynamo benchmark [5]. This solution is chosen because it incorporates all the essential physics and reaches a ﬁnal state that is precisely deﬁned in term of energies, drift rate and local properties of the dependent variables, making it accurately reproducible by any dynamo code. Moreover, Case 1 of the dynamo benchmark converges at modest

Copyright ᭧ 2010 John Wiley & Sons, Ltd.

Concurrency Computat.: Pract. Exper. 2011; 23:38–56 DOI: 10.1002/cpe

40

C. J. DAVIES, D. GUBBINS AND P. K. JIMACK

resolutions, allowing a wide range of resolutions to be explored with the computing resources available. The parameter values deﬁning Case 1 are many orders of magnitude removed from those of the Earth and therefore from the ideal problem discussed in this paper, but this is true of any parameter set that is tractable on current computers.
Scalability of the pseudospectral method will be assessed numerically in two ways. First, for a ﬁxed problem size the number of processors, N p, will be incrementally increased from a serial calculation with N p = 1. This type of scalability will be called strong scalability, assessed by measuring the speedup, S = T1/Tp, where T1 is the time in seconds to complete a timestep on one processor and Tp is the time in seconds to complete a timestep on p processors. Ideal strong scalability occurs when S doubles as N p doubles. Second, the problem size and the number of processors will be proportionally increased from a serial calculation in such a way so as to keep the work per processor constant. This type of scalability will be called weak scalability, assessed using asymptotic scaling laws (derived in Section 3 below) for the total work, i.e. number of computations for a simulation, and the work per timestep. Weak scalability is the more relevant to dynamo modellers because larger simulations generally run on larger clusters; the goal is primarily to increase the problem size, a necessity as parameter values are pushed to more realistic values, rather than solve existing problems faster.
The physical and mathematical problem is described in Section 2, where the required spatial and temporal resolutions for simulating the ideal problem are determined. Approximate operation counts, memory requirements and communication costs for asymptotically large problems are obtained in Section 3. Section 4 shows the results from numerical simulations. Section 5 uses the results of numerical simulations and the operation counts to extrapolate to large clusters and more ‘realistic’ parameter values. Summary and discussion are presented in Section 6.

2. PROBLEM FORMULATION

2.1. The ideal problem

We seek to deﬁne what might be deemed an ‘ideal’ geodynamo simulation: one that includes the

most important effects, i.e. those that are physically essential and/or may lead to predictions of

existing observations. We consider an electrically conducting ﬂuid conﬁned to a rapidly rotating

spherical shell of radial extent d =ro −ri and aspect ratio ri/ro = 0.35. Here, ri corresponds to the

inner boundary and ro to the outer boundary. The ﬂuid rotates about the vertical z-axis with angular

velocity . Gravity, denoted g, is assumed to vary linearly with radius, a close approximation

to the Earth [11]. The ﬂuid is assumed to be incompressible and Boussinesq (e.g. [12]): density

variations are neglected other than when they modify gravity.

The standard model for driving convective ﬂuid motions in the geodynamo is via a mix of

thermal and compositional buoyancy sources. The Earth’s solid inner core grows slowly over time

as the heavy component of the outer core ﬂuid alloy freezes onto it, providing a source of thermal

buoyancy [13–15]. Thermal sources may also be internal, perhaps due to the presence of radioactive

elements in the core (e.g. [16–18]). Inner core freezing can also release light elements, which rise

and provide a source of compositional buoyancy to drive the convection [19]. We assume that a

single energy source drives the convection as this is unlikely to produce a drastic increase in the

computational task.

We follow common practice by assuming constant thermal diffusivity, , constant coefﬁcient of

thermal expansion, , constant viscosity, , and constant electrical conductivity, . The magnetic

diffusivity, deﬁned as 1/( 0 ), where 0 is the permeability of free space, is also assumed constant. Temperature is ﬁxed at To + T and To on the inner and outer boundaries, respectively. With the

aforementioned assumptions we scale the magnetohydrodynamic equations by the shell depth, d,

the magnetic diffusion time, d2/ , T as the unit of temperature and B = (2

1
0 ) 2 as a measure

of the magnetic ﬁeld strength to obtain the following equations for modelling the dynamo process

Copyright ᭧ 2010 John Wiley & Sons, Ltd.

Concurrency Computat.: Pract. Exper. 2011; 23:38–56 DOI: 10.1002/cpe

GEODYNAMO SCALABILITY

41

in spherical polar coordinates, (r, , 
):

E *u −u×(∇ ×u) +z×u = −∇ P +q RaT r+(∇ ×B)×B+ E∇2u,

(1)

q Pr *t

*T +(u·∇)T = q∇2T,

(2)

*t

*B = ∇ ×(u×B)+∇2B,

(3)

*t

∇ ·u = 0,

(4)

∇ ·B = 0,

(5)

where u is the ﬂuid velocity, B is the magnetic ﬁeld, T is the temperature deviation from the basic state temperature, P is the modiﬁed pressure, and is the density (see [12] for details). Equation (1) is an expression of Newton’s second law of motion. The term (*/*t −u×∇×)u is the motional derivative (the time derivative of momentum following the ﬂuid) and the term z×u is the Coriolis force. The right-hand side is the sum of the applied forces that cause changes in momentum; from left to right these are the pressure gradient, buoyancy, Lorentz and viscous forces. Equation (2) is an energy conservation equation for the thermal energy source that drives the convection. Equation (3) governs the magnetic ﬁeld evolution and is nonlinearly coupled to Equation (1) through the induction term ∇ ×(u×B). The ﬁnal two solenoidal equations describe respectively conservation of mass for an incompressible ﬂuid and the fact that no magnetic monopoles have been observed.
The inner and outer boundaries are assumed to co-rotate in our simulations. We assume that the velocity vanishes at both boundaries (the no-slip condition), which are both electrical insulators. The latter condition is essentially non-local and requires the solution of Laplace’s equation outside the spherical shell. The pseudospectral method involves an expansion in spherical harmonics which are themselves solutions of Laplace’s equation, and so the non-local boundary condition is converted to a local condition on each radial function [20]. The inner boundary is assumed to be held at a ﬁxed temperature. The thermal boundary condition on ro is likely to be inhomogeneous (e.g. [21]) but we neglect this effect, as it is unlikely to increase the computational task, and prescribe a ﬁxed temperature on ro.
In Equations (1)–(3) the Ekman number, E, measuring the rotation rate, the Prandtl number, Pr, the ratio of viscous and thermal diffusivities, the Rayleigh number, Ra, measuring the strength of the applied temperature difference across the shell, and Roberts number, q, the ratio of thermal and magnetic diffusivities, are given respectively by

E= 2

d2 ,

Pr = ,

Ra = g T d3 , q = .

(6)

2

We complete our speciﬁcation of the ideal problem by assigning geophysical values to these parameters.
Recent theoretical and experimental work on iron at high temperatures and pressures (see [22] and references therein) has yielded an understanding of the molecular, thermal and chemical properties of the core ﬂuid, although the electrical and thermal conductivities are still uncertain by a factor of up to 3 [23]. The Prandtl number, Pr , is O(1), but the Roberts number and the magnetic Prandtl number, Pm = q Pr , are 10−6 or less; such extreme values will be impossible to incorporate into numerical simulations for the indeﬁnite future. This problem is commonly addressed [24] by assuming a simple turbulence model that brings the thermal and viscous diffusivities up to the value of the magnetic diffusivity (so-called ‘turbulent’ diffusivity values), implying Prandtl numbers of order unity. We believe that the problem of core turbulence is better studied outside

Copyright ᭧ 2010 John Wiley & Sons, Ltd.

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C. J. DAVIES, D. GUBBINS AND P. K. JIMACK

the context of the full geodynamo problem and hence adopt Prandtl numbers equal to unity for the ideal problem.
The Ekman number is the major computational challenge. Large-scale motions are controlled by the spherical geometry [24]; however, the smallest scales, where energy is dissipated, are determined by the low viscosity [25, 26]. Hence the Ekman number controls the lengthscale of the ﬂow and, in turn, the spatial resolution that is required in numerical simulations. The lowest value of E achieved in any simulation is ∼ 5×10−7 [8]; very little work has been done in this regime and the calculations are computationally intensive. This is still signiﬁcantly higher than the ‘turbulent’ Ekman number for the core, E = 10−9 (e.g. [27]), obtained when turbulent diffusivities are invoked. Using molecular values for the diffusivities gives E = 10−16, an enormous challenge. To combat the problem some authors have used ‘hyperdiffusivity’ (e.g. [1, 2, 28]) a form of scale-dependent viscosity that artiﬁcially damps high wavenumbers. This allows a low ‘headline’ Ekman number, applicable to the large scales, without the need to resolve very small scales [29]. Hyperdiffusivity has been shown to prevent formation of small scale structures that may play a crucial role in the dynamics [30]. Here we restrict ourselves to constant turbulent viscosity.
Thermal buoyancy forces in the core are determined by the Rayleigh number. The thermal Rayleigh number (Ra in Equation (1)) may not be too large to simulate [31, 32], perhaps less than 1000 times the critical value for onset of non-magnetic convection. Current dynamo simulations have used Ra up to O(100) times supercritical [7–9]; the highly supercritical regime is very poorly understood in both physical and numerical respects. A low Ra will therefore be relevant. For the ideal problem we therefore assume that Ra is not too far above the critical value for the onset of non-magnetic convection.
From the foregoing discussion, we suggest an ‘ideal’ geodynamo simulation having E = 10−9, Prandtl numbers equal to unity, and a Rayleigh number not too far above critical. A run length of one magnetic diffusion time is required to demonstrate dynamo action. Achieving such a calculation would represent a huge advance. Many other complications could be explored with relatively little additional computational expense.

2.2. Numerical representation

The pseudospectral method used in this work is described in [33]; it is similar to other pseudospectral methods [1, 28]. Vector dependent variables u and B are expanded in toroidal and poloidal scalars, T and P,

{u, B} = ∇ ×(Tr)+∇ ×∇ ×(Pr),

(7)

(e.g. [20]), thus satisfying the solenoidal conditions (4), (5) exactly. Scalars are then expanded as

Nl

{T(r, , 
, t), P(r, , 
, t)} =

Slm(r, t)Ylm( , 
),

(8)

l=1 m=0

where N is a suitably chosen to be determined and the fully

truncation point for the inﬁnite series, normalized spherical harmonics, Ylm(

the Slm(r, t) are coefﬁcients , 
), are given by

Ylm ( , 
) = Plm (cos ) expim
,

(9)

where Plm(cos ) are radial dependence of

associated Legendre functions of degree l the Slm(r, t) is represented on a grid of Nr

and order m [34]. In our code the points with variable spacing. Grid

points are clustered near the boundaries, allowing ﬁner resolution of the small-scale structures

that appear in the boundary layers. Angular derivatives of dependent variables are found using

the corresponding derivatives of the spherical harmonics; radial derivatives are computed using

high-order ﬁnite differences.

The toroidal and poloidal parts of the governing equations are timestepped in the spectral domain

(i.e. by operating on spherical harmonic coefﬁcients); the right-hand sides of Equations (1)–(3)

must therefore be evaluated in spectral space. Our code uses a semi-implicit predictor-corrector

method: diffusion terms are treated implicitly whereas all nonlinear terms and the Coriolis term

Copyright ᭧ 2010 John Wiley & Sons, Ltd.

Concurrency Computat.: Pract. Exper. 2011; 23:38–56 DOI: 10.1002/cpe

GEODYNAMO SCALABILITY

43

are treated explicitly to reduce the size of the resulting matrices. The spherical harmonics satisfy the differential equation

∇2 Slm(r, t)Ylm ( , 
) = Dl Slm(r, t)Ylm ( , 
),

(10)

(e.g. [12]), where the Dl operator is given by

Dl

=

1 r2

d dr

r2 d dr

−

l(l + r2

1)

,

(11)

and hence the Laplacian does not couple spherical harmonic modes. Hence these terms in Equations (1)–(3) can be treated separately for each harmonic and the timestepping equations take the form

*Slm (r, *t

t

)

+

Dl

Slm

(r,

t

)

=

Nlm

(r,

t

),

(12)

where Nlm represents the result of evaluating the nonlinear terms in any of Equations (1)–(3). Curls and gradients are evaluated in spectral space (i.e. by operating on spherical harmonic coefﬁcients). As with the Laplacians, these operations do not couple harmonic modes. Computing nonlinear terms in spectral space is very slow and hence the spherical transform method is used [35] in which the nonlinear terms are formed by multiplication in physical space (i.e. evaluated on gridpoints in a spherical coordinate system).
In this paper we will focus on the parallel decomposition used in our code; discussion of alternative decompositions is deferred to Section 5.1. Our code uses two parallelization strategies depending on the operations being performed. For linear operations, harmonic coefﬁcients are split across processors with each processor having access to the whole radial grid. Before undertaking the spherical transform the logical grid is reorganized so that all harmonics for a given radial grid point are on the same processor, and hence radial shells are split across processors. The vector transpose that accomplishes this change in data distribution is the major communication step in our code. Because the spherical transforms are not split across processors the number of processors that our code can use is limited to be no greater than the number of radial points. This matter is revisited in Section 5.

2.3. Anticipated resolution
Fluid motions at the onset of non-magnetic convection in a rapidly rotating spherical shell take the form of columns with an O(E1/3) azimuthal lengthscale [36, 37], much smaller than their radial and axial lengthscales e.g. [38]. The required spatial resolution will therefore scale as x = E1/3.
Magnetic forces and low Prandtl numbers are known to increase the azimuthal lengthscale near
onset (e.g. [16, 39]); these effects are therefore unlikely to increase the required resolution. Higher
Rayleigh numbers increase the turbulence, changing the lengthscale in all directions, however, the smallest scales are ultimately dictated by the viscosity, i.e. E (Equation (6)). Thin (O(E1/2)) struc-
tures may appear in boundary layers, however, these can be accommodated in the pseudospectral
code by using a non-uniform grid (e.g. [33]). Detached shear layers, such as Stewartson layers [40] may form, but these are likely thicker than E1/3 [41].
We, therefore, propose that E1/3 is the smallest lengthscale to be resolved. In practice, a suite
of calculations at progressively lower E is required. If any important structures with lengthscales smaller than E1/3 exist they should become evident before any calculations are performed with insufﬁcient resolution. With this assumption, spatial resolution requires N = k Nr = Ks(E−1/3) points, or spherical harmonics, for each dimension, where k is a number that allows N to differ from Nr and Ks = O(1) cannot be determined more accurately without undertaking simulations. Asymptotically, Ks is assumed not to be of primary signiﬁcance.
Temporal resolution is determined from the Courant–Friedrichs–Lewy (CFL) conditions [42] arising from the use of explicit timestepping to advance the nonlinear terms (u·∇)T , u×(∇ ×u), B×(∇ ×B) and ∇ ×(u×B) in Equations (1)–(3). The time restriction for all terms except the

Copyright ᭧ 2010 John Wiley & Sons, Ltd.

Concurrency Computat.: Pract. Exper. 2011; 23:38–56 DOI: 10.1002/cpe

44

C. J. DAVIES, D. GUBBINS AND P. K. JIMACK

Lorentz force is t< x/|u|, where |u| is the local amplitude of ﬂuid motions. The Lorentz force

is quadratic in B and errors can grow due to the coupling between Equations (1) and (3). These

equations travel at a

support many different speed V . A reasonable

types of estimate

mfoargVnewtoihllydbreotdhyenAamlfivcénwavveelosc[i2ty5]V, Ath=eBfa/s√test0of,

which where

B is a characteristic amplitude of the magnetic ﬁeld (e.g. [43]). The timestep is assumed here to

be determined by the smaller of x/|u| or x/|VA|. The important point is that an increase in

spatial resolution results in a proportional decrease in the maximum stable timestep.

The ﬂuid velocity must be great enough to regenerate magnetic ﬁeld, which is usually measured

by the magnetic Reynolds number, Rm = |u|d/ , which must exceed a critical value for dynamo

action. Rm is an output parameter from a numerical simulation because it depends on the solution.

Formal bounds on the minimum Rm for dynamo action are O(10) [44, 45], but these are only lower bounds. Taking |u| = 10−4 ms−1 (e.g. [46]), d = 2260 km and = 1.6 m2 s−1 for the Earth [22] gives

Rm = O(100). Using the non-dimensionalization of Section 2.1, Rm is a dimensionless measure of the ﬂuid velocity and hence |u| is likely to be O(100).

Estimates of VA depend on the ﬁeld strength, B, which itself depends on many factors, most

notably the Rayleigh number. Hence B cannot be estimated a priori. In dimensionless units the ratio VA/|u| = d 2 / ≈ 8×104, so that the timestep restriction will be determined by the magnetic

ﬁeld rather than by the ﬂuid velocity. A dimensional argument based on Earth-like values gives

a similar result: core ﬂuid velocities inferred from time variations in the geomagnetic ﬁeld are O(10−4) m s−1 whereas Alfvén speeds for magnetic ﬁeld strengths of 1 mT are O(10−2) m s−1.

Faster magnetohydrodynamic waves than the Alfvén waves may exist in the system but this does

not alter the conclusion that the timestep restriction comes from the magnetic ﬁeld.

Another timestep restriction can come from the Coriolis force. This restriction is not signiﬁcant

at the values of E currently employed in geodynamo simulations, but will become overwhelming when E is reduced to geophysical values, say 10−9. The term is linear in u and can be treated

implicitly, as already undertaken by some authors (e.g. [47, 48]), which removes the timestep

restriction at the expense of solving larger matrix systems [29]. We do not, therefore, consider the

Coriolis term’s impact on the choice of t in this paper. Because t< x/ max [|u|, |VA|] and x = E1/3 has been assumed, the number of timesteps,
Nt , for a given run must scale as Nt = Kt (E−1/3), where Kt depends on |u| and VA and cannot be

determined more accurately without undertaking simulations. We assume that Kt is not of primary

signiﬁcance for asymptotically small E, i.e. asymptotically large problem size. In practice, our

code uses a semi-implicit timestepping algorithm that ensures stability of the timestep. By changing

the spatial resolution it is therefore possible to measure the scaling of t in the computationally

accessible parameter regime, which can be compared to the theoretical scaling.

In the following sections scalings will be derived per timestep, with the extra cost associated

with the decrease in t with x accommodated empirically when extrapolations are undertaken.

3. APPROXIMATE SCALINGS FOR THE PSEUDOSPECTRAL METHOD

3.1. Operation count
The evaluation of nonlinear terms by the spherical transform is the rate-determining part of the pseudospectral method; to obtain an approximate operation count this section therefore focuses solely on calculation of the nonlinear terms.
Given spherical harmonic coefﬁcients, flm and glm, of two scalar functions, f ( , 
) and g( , 
), the task is to ﬁnd the spherical harmonic coefﬁcients of the product f g. The procedure is to individually forward transform the functions from the spectral domain to the physical space domain, carry out the multiplications, and inverse transform the result back. In the physical domain the scalar functions are represented on a lateral grid ( i , 
 j ) at each radial grid point. The -points are chosen to lie at the Gauss points, i.e. the zeros of a high-order polynomial that cluster near the poles [49]. The 
-points are equally spaced. Using Gaussian quadrature (e.g. [49]) for the integration gives an exact discrete transform between physical and spectral domains with the

Copyright ᭧ 2010 John Wiley & Sons, Ltd.

Concurrency Computat.: Pract. Exper. 2011; 23:38–56 DOI: 10.1002/cpe

GEODYNAMO SCALABILITY

45

minimum number of -points for a given truncation in spectral space [35, 50]. The transform pair takes the form

N
f ( i,
j)=

l

N
flm Ylm ( i , 
 j ) =

l

flm Plm ( i ) expim
 j ,

(13)

l=1 m=0

l=1 m=0

3N /2 N
flm =

3N /2 N
f ( i , 
 j )wi Ylm∗( i , 
 j ) =

f ( i , 
 j )wi Plm ( i ) exp−im
 j ,

(14)

i=1 j=1

i=1 j=1

where wi are Gaussian weights and ∗ denotes complex conjugation. Because physical quantities are real, fl−m is the complex conjugate of flm, a fact that is used to reduce storage and computational requirements for the complex coefﬁcients.
The m- and j-sums in the transform pair constitute discrete Fourier transforms and hence the
FFT algorithm can be used. Separating the sums as

l
f ( i,
j)=

N flm Plm ( i ) expim
 j ,

(15)

m=0 l=1

3N /2

flm =

wi Plm ( i )

N f ( i , 
 j ) exp−im
 j ,

(16)

i =1

j =1

shows that Equation (15) is a Fourier transform of the l-sum whereas Equation (16) is an i-sum of the Fourier transform of the f ( i , 
 j ) over j. The l-sum in (15) requires O(N 2) operations for each i and all m, whereas the i-sum in (16) requires O(N 2) operations for each l and all
m. Here an operation is taken to be a multiplication followed by an addition. The subsequent
and embedded Fourier transforms take only O(N log2 N ) operations [49], which is negligible for asymptotically large N . Each sum must be computed for all -points and all radial points, Nr , giving an asymptotic operation count of O(N 3) per radial point or, assuming that N = k Nr , O(N 4) per scalar transform. Note that there may be scope for improving this asymptotic estimate very
slightly (e.g. using algorithms such as that of Strassen [51], which may become competitive when
N is sufﬁciently large), however, we do not consider such improvements in this work.
The total number of scalar transforms required at each timestep is obtained by considering Equations (1)–(3). For the forward transform it is necessary to transform u and B to calculate the term ∇ ×(u×B), but it is also necessary to transform ∇ ×u and ∇ ×B so that the nonlinear terms u×(∇ ×u) and B×(∇ ×B) can be computed. T must be transformed and the ﬁnal nonlinear term, (u·∇)T , requires that ∇T is also transformed. For the inverse transform the nonlinear terms u×(∇ ×u) and B×(∇ ×B) in the momentum equation can be added together in physical space and hence only the result is transformed back. The terms ∇ ×(u×B) and (u·∇)T must also be transformed back to spectral space. Hence, the forward transform requires 4 vector transforms
and 2 scalar transforms, whereas the inverse transform requires 2 vector transforms and 1 scalar
transform.
The number of scalar transforms comprising a vector transform can be identiﬁed from the
components of the toroidal and poloidal scalars. When expanded in spherical harmonics, these
components are

Ar (r,

,


)

=

l ,m

l (l

+ 1) r

Plm

(r

)

Plm

(

) expim
,

A (r, , 
) =
l ,m

im sin

Tlm

(r

)

+

1 r

d

Plm d

(

)

d dr

[r

Plm

(r

)]

expim
,

A
(r, , 
) =
l ,m

−Tlm

(r

)

d

Plm d

(

) +

im

r sin

d dr

[r

Plm

(r

)]

Plm

(

)

expim
,

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46

C. J. DAVIES, D. GUBBINS AND P. K. JIMACK

where A ∈ {u, B}. Ar requires a single scalar transform, A a further two, and A
 only a further one because the last term in the sum involves the same l-sum as that computed for Ar ; the additional radial differentiations and multiplications do not add to the leading term in the asymptotic operation count. Hence each vector transform requires 4 scalar transforms. Thus the forward transform requires 18 scalar transforms whereas the inverse transform requires 9 scalar transforms, giving a total of 27 scalar transforms per timestep. Transforms must be computed at each radial point yielding an approximate operation count, O, of

O = 27C1 N 4,

(17)

for a ﬁxed time unit, where C1 is a constant reﬂecting the leading asymptotic term for a single scalar transform. As previously discussed, the timestep is assumed to scale with N and hence the total operation count is O(N 5).
An estimate of the operation count per processor involves replacing a factor of N in (17) by N /N p, thus accounting for parallel decomposition in the radial direction only. This gives an operation count of

O = 27C1 N 4

(18)

Np

per processor per timestep.

3.2. Memory requirements
Our code uses dynamic array allocation so that memory is only allocated when needed. Hence the amount of memory required at any stage during a given timestep may vary. In the current implementation of the code the point of maximum memory usage occurs when integrating the solution forward in time as this requires storage of all matrices and vectors for the inversions. Releasing memory once an inversion has been calculated may enable a reduction in memory requirements but this has not been attempted.
The timestepping equations take the form of a set of linear algebraic equations

Ax = N,

(19)

where the vector N represents the spherical harmonic expansion of the nonlinear terms, which have been evaluated at each radial point by the spherical transform method. In addition, x is an unknown vector of spherical harmonic coefﬁcients at the new timestep, and A is a matrix of known coefﬁcients. Equation (19) can be considered separately for each harmonic (see Section 2.2). When timestepping in spectral space, this also allows harmonics to be split across processors in the parallel implementation, with all radial points for a given harmonic located on a single processor.
Radial derivatives are calculated using fourth order ﬁnite differences with a stencil width of 10. For Nr radial points Equation (19) is a system of Nr equations with the banded matrix A having dimensions (10× Nr ). Equation (19) is solved for each of the Nh = O(N 2) harmonic coefﬁcients, which are split across N p processors. All numbers are of double precision; each vector or matrix element occupies 8 bytes in memory. The number of bytes of memory, M, needed for one inversion is then approximated by

M

=

8(2

Nr

+

10

Nr

)

Nh Np

.

(20)

Here the factor 2Nr is the storage required for the two vectors, N and x, whereas the factor 10Nr is the storage required for the matrix A.

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GEODYNAMO SCALABILITY

47

Equations of the form (19) are inverted at each timestep for each of the scalar variables: uT , u P , T, BT , BP , where subscripts T and P represent, respectively, the toroidal and poloidal scalars for the velocity u and magnetic ﬁeld B. The equation for poloidal velocity is solved using a Green’s function method [33, 50], which adds another matrix inversion. Furthermore, because a complex representation of the spherical harmonics is used, each scalar requires two inversions; one for the real part and one for the imaginary part. Hence there are 12 matrix inversions per timestep. The ﬁnal memory estimate is therefore

M

=

1152 Nr

×

Nh Np

.

(21)

Assuming once more that N = k Nr , this gives M = O(N 3).

3.3. Communication costs
A simple model for the time, T , taken to send a single message of length L across a network can be written as

T =F+ L,

(22)

where F is proportional to the network latency and is inversely proportional to the bandwidth. Note that such a model assumes that the switching capability of the network scales sufﬁciently well that packet collisions do not dominate as the number and length of messages grow. Furthermore, F and are properties of the network that can be estimated on a speciﬁc parallel computer. A theoretical estimate of communication costs, therefore, requires estimating the number of messages sent per timestep and the length of each message.
Spherical harmonic coefﬁcients are split across processors in spectral space. Before undertaking the spherical transform the logical grid is reorganized so that all harmonic coefﬁcients corresponding to a given radial point are located on the same processor; parallelization is achieved in radius with each processor holding all harmonics for a subset of the radial grid. The change between these two parallel decompositions is accomplished by a transpose, which is the main communication step. For the forward transpose each processor initially holds a subset of the total number of harmonics on all radial levels. After the forward transpose each processor holds all harmonics on a subset of the radial grid. Hence each processor must send all its harmonics, Nh/N p, for a given number of radial points, Nr /N p. The reverse is true of the inverse transpose. The length of a single message for forward or inverse transposes in bytes is

L = 16

Nr × Nh

N

2 p

,

(23)

where the prefactor contains a factor 2 because the coefﬁcients are complex and a factor 8 to give the expression in bytes.
The total communication time depends on how many messages of length L are sent. Source and destination processors must be chosen with care because the destination processor only requires harmonics that correspond to radial points within that processor’s radial subdomain. Hence the communication is undertaken in N p −1 steps. At each step, each processor sends and receives a message of length L.
The number of variables that need to be transposed is the same as the number that must be transformed using the spherical transform method. Hence there are 18 forward transposes and

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48

C. J. DAVIES, D. GUBBINS AND P. K. JIMACK

9 inverse transposes per timestep. Each of these transposes is accomplished in N p −1 steps. The ﬁnal estimated communication time is therefore

T = 27(N p −1) F +16

Nr Nh

N

2 p

.

(24)

Assuming N = k Nr therefore gives T = O(N 3).
For asymptotically large problems the limiting factor is, therefore, the computational work of the spherical transform, which scales as O(N 4) per timestep, whereas communication scales only as O(N 3). Memory requirements also scale as O(N 3).

4. NUMERICAL TESTS

The non-dimensional parameter values for the dynamo benchmark Case 1 are: Ra=32.5, E=5×10−4, Pr = 1 and q = 5. Numerical integration from the initial condition prescribed in [5] using 1 processor and N = Nr = 48 shows that kinetic and magnetic energy reach stationary values in under 1 magnetic diffusion time (Figure 1). The solution in this ﬁnal state has dimensionless kinetic energy (KE) 30.77 and dimensionless magnetic energy (ME) 626.3, which are deﬁned as

KE

=

1 2V q2

u2dV ,
V

(25)

ME

=

1 2V q2

B2dV ,
V

(26)

Kinetic energy Magnetic energy

45

1300

40

1200

35 1100
30

25

1000

20

900

15 800
10

5

700

0

600

0

1

2

3

4

5

0

1

2

3

4

5

Time (magnetic diffusion times)

Time (magnetic diffusion times)

Figure 1. Solution for the dynamo benchmark Case 1. Top left: Kinetic energy time series; top right magnetic energy time series; bottom: snapshot of the radial component of magnetic ﬁeld, Br, in Mollweide projection at the outer boundary at t = 5. The solution displays a fourfold azimuthal symmetry and magnetic
features drift with a constant frequency in the westward direction.

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GEODYNAMO SCALABILITY

49

Table I. Global properties of the solution for the dynamo benchmark Case 1 using different numerical resolutions.

N

t

KE

ME

40

4.08 × 10−5

48

3.94 × 10−5

72

2.91 × 10−5

96

2.39 × 10−5

120

2.09 × 10−5

144

1.92 × 10−5

32.892 30.779 30.774 30.781 30.781 30.781

730.26 626.34 626.15 626.37 626.34 626.29

The timestep increment, t, is determined by an adaptive timestepping algorithm.

where V is the volume of the shell. Table I lists the global properties of the benchmark solution
at different numerical resolutions, showing that the solution has converged. A snapshot of Br at the outer boundary is also displayed in Figure 1. The pattern has a fourfold symmetry in longitude
and drifts westward at a constant rate.
Figure 2 shows the strong scalability curves for Case 1 at different numerical resolutions. The
top curve shows speedup as a function of the number of processors. The time taken to complete one timestep in serial mode is 3.9 s for N = 48, 48.2 s for N = 96 and 213.4 s for N = 144. All curves depart from the ideal linear scaling as N p is increased because of communication overhead; for a ﬁxed problem size the ratio of computational work to communication overhead decreases
as N p increases. Larger problems (higher N ) follow the linear trend to higher N p because the computational work is greater for a serial calculation‡.
Figure 2 also shows the scaling of memory requirements with N p, which approaches a constant value for each resolution. This is because, for N p large and a ﬁxed problem size, signiﬁcant memory requirements come from information held locally by each processor. For ﬁxed problem
size the arrays that are split across processors contribute a smaller amount to the total memory
requirement as N p increases. The difference between observed and theoretical memory requirements [Equation(21)] is also shown in Figure 2. The differences also asymptote to constant values
as N p increases. This is again due to the static parts of the code that are not split across processors. Arrays that are not split across processors grow in size as the problem size is increased and hence
each curve asymptotes to a different value; larger problem sizes asymptote at higher memory.
Weak scalability can be investigated in two ways, both of which require the use of an approximate asymptotic scaling. The ﬁrst is to assume an operation count scaling as O(N 5), meaning that
the total work per processor over the course of a simulation is constant as the problem size is
increased. This is probably the most useful for practical applications. Doubling the resolution in each direction would then require 25 = 32 times more processors, which is impractical given the number of processors available. The second option is to assume an N 4 scaling; work per processor
per timestep is constant, however the run will require more timesteps to reach the same ﬁnal state. An N 4 scaling for weak scalability is used here, but can be easily extended to an N 5 scaling as
shown below. Figure 3 shows weak scalability for Case 1 using an N 4 operation count, i.e. asymptotically ﬁxed
work per processor per timestep. The time to compute a single timestep is relatively unaffected as N 4 and N p are proportionally increased and hence communication costs are seen to be unimportant so long as the problem size is large enough. Furthermore, this demonstrates that the pseudospectral method does indeed follow the O(N 4) scaling predicted by Equation (17). Figure 3 also shows that the memory requirement per processor increases as N 4 and N p are proportionally increased.

‡The rather poor strong scaling performance shown in Figure 2 (top) is a consequence of the size of the ﬁxed problem not being sufﬁciently large for perfect strong scalability to occur. The current implementation of the code may be improved by passing fewer, but longer, messages, however, we use this original version here primarily because it illustrates that good weak scalability can be achieved even when the work per processor is not sufﬁcient to allow such good strong scalability to be observed.

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50

C. J. DAVIES, D. GUBBINS AND P. K. JIMACK

Figure 2. Strong scalability curves for the dynamo benchmark Case 1. Top: Speedup, S, plotted
against number of processors. The dot-dashed line shows the ideal linear scaling. Middle: Observed
memory requirement per core, Mobs, as a function of the number of processors. Bottom: Difference between Mobs and the predicted memory requirements, Mpred, as a function of the number of
processors. The theoretical prediction uses Equation (21).

This observation disagrees with our O(N 3) asymptotic memory scaling and will be discussed in Section 5.1.
Allowing N p to grow in proportion to N 4 does not account for changes in the time increment, t, as the spatial resolution changes. Figure 4 shows the estimated number of timesteps, Nt , as a function of N . The observed number of timesteps required to simulate one magnetic diffusion time has a shallower gradient than the prediction as resolution is increased; the predicted scaling represents an overestimate. It is not entirely clear why this should be, however we note that the variable timestep selection algorithm is based upon a simple local truncation error estimate and

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GEODYNAMO SCALABILITY

51

(a)
(b)
Figure 3. Weak scalability for the dynamo benchmark Case 1 where N p is chosen to be proportional to N 4. Top: Time taken to complete one timestep as a function of the problem size (bottom x-axis) and number of processors (top x-axis). Bottom: Memory requirement per processor as a function of the
problem size (bottom x-axis) and number of processors (top x-axis).

Figure 4. Number of timesteps required to simulate one magnetic diffusion time, Nt , plotted against problem size, N , on a log–log scale. The two lines represent the observed scaling using Case 1 of the dynamo benchmark and the predicted scaling that the timestep decreases with increasing resolution
proportional to N (Section 2.3).

hence for large x (small N ) it is very possible that the timestep is less then the maximum stable step size. Whatever the explanation, assuming an O(N 5) scaling for the total operation count
certainly represents a conservative estimate.

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52

C. J. DAVIES, D. GUBBINS AND P. K. JIMACK

5. EXTRAPOLATION OF NUMERICAL RESULTS
5.1. Alternative domain decompositions
Our weak scalability analysis applies to a one-dimensional parallel decomposition of the pseudospectral method that does not split spherical transforms across processors. This is the strategy implemented in our parallel solver and it enables us to (1) derive simple expressions for operation counts, memory requirements and communication costs that do not depend critically on the underlying network architecture and (2) determine the impact of communication costs by testing our asymptotic scalings using a weak scalability analysis. Moreover, simple choices of the truncation points in each dimension ensured a balanced load per timestep.
Alternative parallel decompositions are possible and would allow the number of usable processors to be increased by splitting the Fourier and/or Legendre transforms across processors as well as across the radial domain. In principle it is possible to decompose all three physical dimensions across processors. Communication is required between the individual processors that compute Fourier transforms for a given latitude and then between processors that compute Legendre transforms for a given azimuthal wavenumber. The number of usable processors is then potentially O(N 3) although this is likely to be unachievable in practice because of the amount of interprocessor communication required. Other possibilities involve reorganizing the physical or spectral grid so that the Fourier and Legendre transforms are each performed on a single processor. Communication is then required to reorganize the grid via a vector transpose, but not when computing the transform. For example, each processor performs the Fourier transform for a given latitude and then performs the Legendre transform for a given azimuthal wavenumber, as done in the dynamo context by [52]. In this conﬁguration the number of usable processors is potentially O(N 2).
Many communication strategies based on these approaches are available for the spectral transform (see [53] for a discussion of some possibilities). It is, therefore, possible that an alternative parallel decomposition to the one that we have implemented and studied could demonstrate weak scalability while improving the overall balance between computation and communication. It is of course theoretically possible to derive asymptotic scaling laws for such decomposition strategies; however, without an efﬁcient implementation against which to benchmark, there is no way of either verifying their correctness or of quantifying the multiplicative constants that are implicit in such scalings. This last point is especially important when bandwidth limitations and communication latencies become important due to the increased volume of communications. The resulting communication costs then depend critically on the network architecture and the logical layout of processors, making asymptotic formulae much less useful; empirical scalings based on particular network conﬁgurations must be used to gain meaningful predictions of communication overheads [53]. Furthermore, not all domain decompositions will display the ideal weak scalability that we demonstrate for the one-dimensional decomposition (i.e. time scales in proportion to the work per processor). Such weak scalability is essential to our goal of using scaling laws to extrapolate to high resolution. For these reasons the remainder of this section is restricted to further consideration of the one-dimensional parallel decomposition employed in our code as described above.

5.2. Extrapolation for 1D parallelization
The results from weak scalability analysis suggest that the asymptotic scaling for operation counts [Equation (17)] is adequately followed by numerical calculations for the number of processors currently available. Conversely, the theoretical memory scaling [Equation (21)] is not borne out in the numerical calculations. This occurs because certain working arrays, e.g. those required to hold the Legendre polynomials and their derivatives, must be replicated on each processor. This is only problematic if extrapolation of the true results to larger resolutions leads to memory requirements that are beyond the capacity of modern computers.
To extrapolate to higher resolutions we assume N = Ks E−1/3 (Section 2.3). The estimate of Ks depends on many factors and is highly uncertain, but we note from Table 1 that the dynamo benchmark solution has converged with N ≈ 40, giving Ks = 3. A dynamo model with similar parameters

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Table II. Computing time required for runs with the different of E and Np using Ks = 3.

E
5 × 10−7 10−7 10−8 10−9

Np
6803 11 634 25 064 54 000

Time (days)
3.3 29 620 13 392

Time (CPU hours)
540 000 8 × 106 3.8 × 108 1.7 × 1010

to the benchmark but using E = 10−5 gives a converged solution when Ks=3.1. Throughout this section we therefore take Ks = 3 and note that an improved estimate of Ks can only be obtained by

undertaking many simulations over a wide range of parameters and resolutions. The ideal problem, where E = 10−9, then requires N = 3000. Extrapolating to N = 3000 using Figure 3, which appears to allow linear extrapolation beyond N = 172, reveals that for E = 10−9 each core would require

≈ 2.5 GB of RAM: many current supercomputing facilities can provide this. It is, therefore, unlikely

that memory issues will be important at lower E.

Before proceeding we note a parallel limitation of the current version of our pseudospectral

code: the maximum number of processors cannot exceed the number of radial points. This is an

unnecessary restriction because at each timestep there are 18 independent forward scalar transforms

and 9 independent inverse scalar transforms that could also be split across processors without

signiﬁcant complication or overhead. Consequently, for the purposes of this extrapolation we

assume the maximum number of usable processors to be 18Nr ; 27Nr processors is not possible

because forward and inverse transforms do not occur concurrently.

To ﬁnd the time T required to complete a simulation of one diffusion time undertaken at

resolution N using N p processors, we extrapolate from a solution undertaken at resolution NB

using

N

B p

processors

that

required

a

time

TB

to

compute

one

diffusion

time

using

the

formula

N5 T = TB × N B

Np

N

B p

.

(27)

The

solution

with

NB

= 96

and

N

B p

= 16

required

TB

= 36.4

hours

to

simulate

one

diffusion

time.

To simulate E = 10−9 requires N = 3000 points or spherical harmonics with a maximum number of

18Nr = 54 000 available processors, which, upon substituting into Equation (27) gives T ≈ 13 000 days of computer time, or 1.7×1010 CPU hours, a formidable computational task.
Computational requirements for simulating various Ekman numbers based on our N 5 scaling are summarized in Table II. Simulating E = 10−8, where N = 3×(10−8)−1/3 ≈ 1390, gives a maximum

usable number of processors 18N ≈ 25 064. Using 25 064 processors requires, using (27), over 350

million CPU hours; such an allocation may be available sometime in the next ten years. Simulating E = 10−7 is predicted to require 8 million CPU hours; such a simulation is possible at present although it may be out of reach for current allocations of computing time. Simulating E = 5×10−7

requires 540 000 hours, hence a few runs could be accomplished within a single allocation of

computing time. We therefore conclude that meaningful parameter studies in the Ekman number range 10−6 ≤ E ≤ 10−7 are possible at the present time.

6. SUMMARY AND DISCUSSION

This paper has analysed the scalability of the pseudospectral method for geodynamo simulations. We have derived scaling laws for operation counts, memory requirements, and communication costs in the asymptotic limit of large problem size and ﬁnd that the method follows an N 5 scaling. The scalings represent best estimates; if our scalings are optimistic or pessimistic it will become apparent as the simulations proceed to lower values of E.

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54

C. J. DAVIES, D. GUBBINS AND P. K. JIMACK

The excellent efﬁciency of the pseudospectral method shows that it will compete with any other method that performs comparably in serial mode: only small studies are therefore needed to evaluate relative efﬁciencies with large clusters. Finite element methods are unlikely to be faster in serial because they require a large number of matrix inversions. Finite difference methods could outperform the pseudospectral method but may require more points for the same resolution everywhere on the sphere. The question of relative accuracy of the various representations is a complex issue that cannot be addressed without a full comparison of the different methods across a wide range of parameters.
For the number of processors presently available, the limiting factor in the pseudospectral method is the spherical transform; memory requirements and communication costs are asymptotically negligible relative to the operation count. Currently memory resources and network interconnects are large/fast enough to demonstrate this on a modest parallel computer. Another issue comes from the parallel implementation: it is currently only possible to use a maximum of 18Nr processors for a given resolution because only the radial domain is split across processors. This limit is unlikely to be threatened in the near future and can be extended by discretizing in the direction [52], but this would need to be achieved without compromising the weak scalability.
We have also outlined an ideal geodynamo problem that operates in a parameter regime relevant to the Earth. Using our theoretical and empirical scaling relationships it has been argued that to simulate this problem using the pseudospectral method would require approximately four years of computing time using 54 000 processors. This challenge is so formidable that it is unlikely to be accomplished in the next ten years. Indeed, more aggressive parallelization strategies are likely to be essential in order to permit an increase in the number of processors so as to reduce the elapsed time. Simulations with 10−6 ≤ E ≤ 10−7 could be performed at the present time, whereas simulations in the next few years appear capable of moving towards the E = 10−8 regime. This would represent a signiﬁcant advance.

ACKNOWLEDGEMENTS
C. Davies is supported by an NERC E-Science research studentship. D Gubbins is partially supported by Leverhulme grant F/00 122/AD.

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