Geosci. Model Dev., 5, 277–287, 2012www.geosci-model-dev.net/5/277/2012/doi:10.5194/gmd-5-277-2012© Author(s) 2012. CC Attribution 3.0 License.

GeoscientificModel Development

Verification of SpacePy’s radial diffusion radiation belt model

D. T. Welling1,*, J. Koller1, and E. Camporeale1

1Los Alamos National Laboratory, P.O. Box 1663, Los Alamos, NM 87545, USA* now at: University of Michigan Department of Atmospheric, Oceanic, and Space Sciences, 2455 Hayward St., Ann Arbor,MI 48109-2143, USA

Correspondence to:D. T. Welling (dwelling@umich.edu)

Received: 17 August 2011 – Published in Geosci. Model Dev. Discuss.: 8 September 2011Revised: 30 January 2012 – Accepted: 8 February 2012 – Published: 6 March 2012

Abstract. Model verification, or the process of ensuring thatthe prescribed equations are properly solved, is a necessarystep in code development. Careful, quantitative verificationguides users when selecting grid resolution and time stepand gives confidence to code developers that existing code isproperly instituted. This work introduces the RadBelt radia-tion belt model, a new, open-source version of the DynamicRadiation Environment Assimilation Model (DREAM) anduses the Method of Manufactured Solutions (MMS) to quan-titatively verify it. Order of convergence is investigated for aplethora of code configurations and source terms. The abilityto apply many different diffusion coefficients, including timeconstant and time varying, is thoroughly investigated. Themodel passes all of the tests, demonstrating correct imple-mentation of the numerical solver. The importance ofDLL

and source term dynamics on the selection of time step andgrid size is also explored. Finally, an alternative method toapply the source term is examined to illustrate additionalconsiderations required when non-linear source terms areused.

1 Introduction

The terrestrial radiation belts are regions of near-Earth outerspace where relativistic electrons and ions are electromag-netically orbiting the planet. The belts naturally organizeinto two tori: a stable inner belt lying within≈2RE and afar more dynamic outer belt located outside of≈3RE (vanAllen and Frank, 1959). Since their discovery, the belts havebeen the focus of intense research due to the innumerableunknowns concerning their behavior and their damaging ef-fects on spacecraft, both transient (Baker, 2000; Feynmanand Gabriel, 2000; Koons et al., 1999; Pirjola et al., 2005,etc.) and accumulated over a satellite’s lifetime (Gubby andEvans, 2002; Welling, 2010).

During slowly changing conditions, radiation belt particlesundergo three types of periodic motion, each with its owncorresponding adiabatic invariant: gyration about field lines,bounce along field lines, and drift about the Earth. Duringactive times, when conditions change on time scales shorterthan the periods of motion, adiabaticity can be broken andparticle motion can no longer be described by a simple sumof these three. Casting the particle evolution in terms of thethree invariants allows non-adiabatic motion to be modeleddiffusively via the Fokker-Planck equation, yielding a pow-erful description of belt dynamics (Schulz and Lanzerotti,1974).

The third invariant represents the total magnetic flux en-closed within a full particle orbit. It is common to use a nor-malized form of this,L∗ (referred to as simplyL herein),described byRoederer(1970). It is analogous to radial dis-tance from the center of the Earth (in Earth radii) to the equa-torial crossing point of the bouncing particle, exactly so if theterrestrial magnetic field is adiabatically relaxed to a simpledipole geometry. Diffusion inL alone (the other invariantsshall be considered conserved) accounts for the capture andinward radial transport of radiation belt particles (Roederer,1970; Falthammar, 1965). The long period of a particle or-bit makes this invariant the most readily broken. For thesereasons, modeling of phase space density inL space is bothfruitful (a great deal of dynamics are captured) and easilyachieved (the long particle orbit time allows for larger timesteps).

The evolution of the phase space density,f , of a radiationbelt population of constantµ andK (values correspondingto the first and second adiabatic invariants, respectively) for agiven diffusion rate,DLL, is expressed as (Lyons and Schulz,1989),

∂f

∂t= L2 ∂

∂L

(DLL

L2

∂f

∂L

)+Q (1)

Published by Copernicus Publications on behalf of the European Geosciences Union.

278 D. T. Welling et al.: RadBelt verification

whereQ represents combined sources and losses. Solvingthis equation forf has been the focal point of many impor-tant radiation belt studies.Brautigam and Albert(2000) andMiyoshi et al.(2003), using different boundary conditions athighL and different loss processes, demonstrated the impor-tance of radial diffusion in radiation belt dynamics.Shpritset al.(2006) showed that radial diffusion and magnetopauseshadowing (particle loss as the magnetopause intersects driftpaths) can account for observations of rapid flux dropoutsat the onset of particular storms.Lam et al.(2007) used aradial diffusion model to explore the importance of plasmas-pheric hiss driven losses. The results compared favorably tomeasurements of the radiation belts aroundL ≈ 4. These arebut a few examples; a more complete review can be foundin Shprits et al.(2008). Clearly, radial diffusion representa-tion of the belts is a powerful tool for investigation of theirdynamics.

Many models and studies have moved beyond simple one-dimensional diffusion. Three-dimensional diffusion modelshave arose, which allow for not only radial but also pitch an-gle and energy space diffusion. The Salammbo model (Beu-tier and Boscher, 1995) is an early example of such. More re-cent 3-D diffusion models now include cross-diffusion termsand have shown that such terms can be important to the evo-lution of the radiation belts (Xiao et al., 2010; Subbotin et al.,2010). The Dynamic Radiation Environment AssimilationModel (DREAM, Reeves et al., 2005) incorporates data as-similation to drive radial diffusion results towards more re-alistic values (Koller et al., 2007). Other models, such asthe Radiation Belt Environment (RBE) model (Fok et al.,2008), use bounce-averaged kinetic representations of thebelts along with time-accurate magnetic and electric fieldspecifications instead of simpler diffusion equations. Despitethese important modeling advances, radial diffusion still re-mains a core part of radiation belt investigations.

This work introduces a new, open source version of theDREAM radial diffusion radiation belt code and, as part ofthe code’s development, verifies the model. Code verifica-tion is the process of testing for proper implementation ofa code’s numerics and other processes.Verificationasks, “Isthe model solving the prescribed equations correctly?” and isseparate from modelvalidation, which asks, “Are the equa-tions solved representative of the real world?”; examples ofquantitative model validation can be found inWang et al.(2008) andWelling and Ridley(2010). Verification studiesare key steps in code development and guide user decisionsin terms of code configuration. Proper verification also ex-pedites future development; as additional features and ex-panded physics are added to a model, it is critical to ensurethat the existing features are instituted properly. BecauseRadBelt is currently being expanded to include diffusion inpitch angle and energy space, verification at this early stageis crucial.

This study leverages the Method of Manufactured Solu-tions, described below, to quantitatively assess the code’s

performance and ensure that the results are accurate to typ-ical machine limitations. Code exercises are repeated for aplethora of configurations to demonstrate proper implemen-tation and convergence. Alternative source functions and nu-merical source applications are applied to test all facets ofthe model.

2 Methodology

2.1 Model description

The code being investigated here is the RadBelt module ofthe SpacePy software library (Morley et al., 2011). Thismodel solves Eq. (1) for a singleµ andK. The bulk of thecode is written in Python, making the model unique amongothers in terms of flexibility and capability; the model isinitialized, configured, executed, and visualized all througha Python interface. Code domain (typicallyL = [1 : 10]),sources and losses,DLL, time step and grid size are all spec-ified in an object-oriented manner, e.g., setting object at-tributes. Required input data, such as Kp indices, are ob-tained automatically by SpacePy from theQin et al.(2007)input set housed at the Virtual Radiation Belt Observatory(ViRBO). Results can be quickly visualized through built-inobject methods. The RadBelt module is currently being ex-panded to include full three-dimensional diffusion and dataassimilation capabilities.

Several built-in empirical relationships act as defaultsources and losses in RadBelt. The default electron sourceis an empirical acceleration term given by,

S = γ (Kp)2e

−(L−Lcenter)2

2L2width (2)

where γ is the magnitude of the source (typically0.1 days−1), Lcenter is the center of the source curve (typi-cally L = 5.6), andLwidth is the width of the source curve(typically 0.3). This function represents electrons being ac-celerated into the currentµ−K slice as a function of mag-netospheric activity. This simple source model was institutedto provide users with a quick, built-in source of phase spacedensity; more accurate, physics-based source models will bedeveloped in the future. Two loss mechanisms are currentlyemployed: magnetospheric shadowing losses (loss due toparticles diffusing into open drift shells, strongly controlledby the magnetopause location) and loss due to plasmaspherichiss. The former is applied by enforcing rapid decay of phasespace density in regions outside of the last closed drift shell,Lmax. In RadBelt, the following empirical relationship yieldsthe defaultLmax values:

Lmax= 6.07×10−5Dst2+0.0436Dst+9.37 (3)

This relationship, introduced byKoller and Morley(2010),is the result of fitting a second order polynomial toLmaxversus Dst values from July 2002 to December 2002 at a

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D. T. Welling et al.: RadBelt verification 279

300 310 320 3301

2

3

4

5

6

7

8

9

10

L*

Summary Plot

1001011021031041051061071081091010

Phase

Space

Densi

ty

300 310 320 330DOY in 2003

500

400

300

200

100

0

100

Dst

0

1

2

3

4

5

6

7

8

9

Kp

Fig. 1. Example output from a RadBelt simulation of the wellknown “Halloween Storms”. The top frame shows phase space den-sity with Lmax location over plotted in white. The bottom frameshows the geomagnetic indices used to driveDLL and the empiri-cal source and loss functions.

five minute time resolution.Lmax was calculated using theTsyganenko 2001 storm-time empirical magnetic field model(Tsyganenko, 2002; Tsyganenko et al., 2003). This magneticfield model was selected to balance accuracy with speed ofexecution, making the six-month calculation time feasiblewhile returning reasonable results. The root-mean-squarederror of this relationship is 0.64RE. Loss due to plasmas-pheric hiss (Millan and Thorne, 2007), important for devel-oping the slot region, is included by using a loss lifetime of10 days. The radial extent of the plasmapause is obtained viathe Kp-dependent relationship defined inCarpenter and An-derson(1992). Though these relationships are the defaults,RadBelt is flexible and any arbitrary functions can be used.

An example of RadBelt output is shown in Fig.1, whichshows the results from a simulation of the well known “Hal-loween Storms” occurring from 20 October 2003 to 5 De-cember 2003. The simulation was run using theBrautigamand Albert(2000) specification ofDLL, the default sourceand losses described above, no density at the boundaries, andinitially empty radiation belts. The top frame shows elec-tron phase space density forµ = 2083 MeV G−1 and K =

0.03√

G RE over theL domain. These values will be usedthroughout this study. The over-plotted white line shows thelocation of the last closed drift shell for this particularµ andK combination. The bottom frame shows the Dst and Kp in-dices during the simulated period. These values drive boththeDLL and the empirical functions described above.

Throughout the simulated period, the last closed drift shellconfines the outer radiation belt as phase space density out-side of Lmax decays quickly (Fig.1, top frame). Duringquieter periods,Lmax grows and the belts diffuse outwardto fill the newly closed drift shells. The limitations of thesimple source model are evident, as there is little variability

1 2 3 4 5 6 7 8 9 10Roederer L ∗

10-10

10-8

10-6

10-4

10-2

100

102

104

106

DLL (

1Days

)

B&A 2000 (KP=0)

U 2008

B&A 2000 (KP=5)

F&C 2006

B&A 2000 (KP=9)

S 1997

Fig. 2. Comparison of the four formulations ofDLL used in thisstudy. Because the formulation ofBrautigam and Albert(2000) isKp dependent, it is shown for several different Kp values.

Table 1. A comparison of the differentDLL formulations used inthis study in terms ofα andβ.

Formulation α β

S1997 1.9×10−10 11.7BA2000 100.506Kp−9.325 10.0FC2006 1.5×10−6 8.5U2008 7.7×10−6 6.0

within the belts and phase space density values remain ele-vated throughout the simulation.

The driving physics of any radial diffusion model is con-tained in the diffusion coefficient,DLL. Recognizing this,RadBelt presently includes several differentDLL formula-tions. Adding a new formulation is as simple as defin-ing a new function and assigning it as a RadBelt object at-tribute. The currently includedDLL models are taken fromSelesnick et al.(1997), Brautigam and Albert(2000), Feiet al. (2006), and Ukhorskiy and Sitnov(2008), denotedas S1997, BA2000, FC2006, and U2008 herein for conve-nience. Each of these has the generic form,

DLL = αLβ (4)

The values ofα andβ are summarized in Table1; each iscompared in Fig.2 over the typical spatial domain of Rad-Belt.

RadBelt uses a modified Crank-Nicolson implicit finitedifference solver which is second-order accurate in spaceand time (Crank and Nicolson, 1947). Equation (1) has sev-eral characteristics that make the standard Crank-Nicolsonapproach inappropriate, namely a space (and potentiallytime) dependent diffusion coefficient as well as a right handside that contains factors ofL2 in both the numerator and

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280 D. T. Welling et al.: RadBelt verification

Fig. 3. Behavior of the manufactured solution (top frame),QMMSwhen S1997DLL is employed (center frame), andQMMS whenBA2000DLL(Kp = 9) is used (bottom frame). Following Eq. (8),for a smallerα factor, the source closely resembles the analytic so-lution, but scaled and phase-shifted (center frame). Asα becomesstronger, the source becomes more complicated to counteract thestrong diffusion at highL (bottom frame).

denominator. The traditional solver is modified by combin-ing the factor ofL−2 with DLL and discretizing the righthand side via a second order difference scheme that takesinto account the spatial dependence ofDLL, outlined inPresset al.(2007). The possibility of a time-dependentDLL is ac-counted for by computing it at the midpoint betweent andt +1t . This approach maintains unconditional stability ofthe scheme as demonstrated byTadjeran(2007). The result-ing discretized form of Eq. (1) is,

f n+1j −f n

j

1t=Q

n+12

j (5)

+L2

j

2

Dn+

12

j+12

(f n

j+1−f nj

)−D

n+12

j−12

(f n

j −f nj−1

)1L2

+

Dn+

12

j+12

(f n+1

j+1 −f n+1j

)−D

n+12

j−12

(f n+1

j −f n+1j−1

)1L2

where indexesn andj represent discretization in time andspace, respectively. The derivation then follows the usualpath of separatingf n+1 terms fromf n terms, resulting inthe familiar arrangement,

Af n+1= Bf n

+1tQn+12 (6)

whereA andB are tridiagonal matrices of coefficients thatinclude factors ofL, 1t , 1L, andDLL. In RadBelt, routinesto invert A via standard LU tridiagonal matrix decomposi-tion (Press et al., 2007) to advance the solution forward intime are written in the C programming language to obtain thefastest possible execution speed. The portions of SpacePywritten in C are compiled to a shared object library whichPython interfaces using its built-in C-Types module.

2.2 Verification technique

The prototypical code verification approach begins by ob-taining an analytical solution to the equation or system ofequations. The known solution is compared to the numericsolution in a systematic way covering a range of grid spac-ings and time steps, demonstrating the code’s ability to prop-erly converge to the correct solution as expected given theimplemented solver. For complicated systems, however, ob-taining an analytical solution becomes either prohibitivelydifficult or outright impossible.

One way to overcome this difficulty is to employ theMethod of Manufactured Solutions (MMS) (Roache, 2002).This method begins by selecting a solution independent ofthe governing equations. The manufactured solution is sub-stituted into the governing equation to produce a source termthat, when added to the original governing equation, yields anew equation for which the manufactured solution is an exactsolution. The manufactured solution can now be applied toconvergence studies of the code, which, if passes, can be con-sidered verified – it correctly solves the governing equationsand properly applies the given numerical scheme (Roache,1998)

This process is illustrated clearly for the situation at hand.The arbitrary solution is selected with no regard for Eq. (1)outside of the variables we wish to exercise (t andL):

f = sin(aL−a)sin(bt) (7)

wherea andb will be set to satisfy prescribed boundary con-ditions. This “solution” is>2 times differentiable in bothtandL, which is required once substituted into Eq. (1). Sub-stituting Eqs. (7) and (4) into Eq. (1) yields the source term,

QMMS = bcos(bt)sin(aL−a) (8)

−αa(β −2)Lβ−1cos(aL−a)sin(bt)

+αa2Lβ sin(aL−a)sin(bt)

For this exercise, Dirichlet boundary conditions will be usedin L (set to zero for convenience using a typical domain ofL = [1 : 10]) while the initial conditions will be no phasespace density. This restricts the choice ofa to 2π

9 . Theconstantb is then chosen to ensure a source that has a short

Geosci. Model Dev., 5, 277–287, 2012 www.geosci-model-dev.net/5/277/2012/

D. T. Welling et al.: RadBelt verification 281

Fig. 4. A demonstration of the code converging towards the analytical solution as the time step is refined. The left column shows thenumerical solution while the right column shows the corresponding error. Because the maximum error, listed in the title of each plot, reducesas the square of the time step, this figure hints at the expected convergence rate given the numerical scheme used.

period so that the verification simulations can be performedquickly. Given a simulation run tot = 600 s,b is set to 2π

300such that two full periods are completed in one simulation.

The behavior of both the manufactured solution and theassociated source term for the chosen values ofa and b

are shown in Fig.3. As desired, the solution shows plentyof variation in just a short time period (Fig.3, top frame).For a moderateDLL (S1997, whose behavior is shown as ablack dashed-dotted line in Fig.2), the first term of Eq. (8)dominates and the artificial source is similar to the manufac-tured solution, but phase-shifted and scaled (Fig.3, centerframe). However, for a strongDLL, the pattern in the sourceterm changes drastically (bottom frame). This occurs for theBA2000DLL when Kp is high. Asα increases by several or-ders of magnitude, theL-dependent second and third termsof QMMS (Eq.8) become dominant. The strong source termvalues at highL overcomes the strong,L-dependent stormtime diffusion of electrons and maintains the shape of themanufactured solution and enforces the prescribed boundaryconditions.

With this manufactured solution and source term, the Rad-Belt model was run thousands of times with different com-binations of1t and1L. Each run covered ten minutes ofsimulation time, or two full periods of the manufactured so-lution, over the typical radial domain ofL = [1 : 10]. Thiswas repeated for eachDLL model currently included in thecode.

3 Results

3.1 Time constantDLL

Figure4 illustrates the model converging towards the analyt-ical solution as1t is refined for a constant1L = 0.1 withS1997DLL. The left column displays the solution for fivedifferent values of1t , decreasing from top to bottom, whilethe right column shows the corresponding error (defined asthe magnitude of the difference between the analytical so-lution and the numerical solution) throughout the domain.The maximum error for each is listed at the top of each error

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282 D. T. Welling et al.: RadBelt verification

10-2 10-1 100 101 102

Timestep (seconds)

10-5

10-4

10-3

10-2

10-1

100

101

||Yanalytic−Ynumeric||

Asym

ptotic

Regim

e

Slope=2.000194

Time Convergence (∆L=0.1)

10-2 10-1 100 101

Grid Size (∆L)

10-6

10-5

10-4

10-3

||Yanalytic−Ynumeric||

Asym

ptot

ic R

egim

e

Slop

e=2.

0058

58

Grid Convergence (∆t=0.1)

Fig. 5. Top frame (bottom frame): convergence as1t (1L) is re-fined. Each star represents the results from an individual simulation.The approximate location of the asymptotic regime is delimited byvertical red dashed lines. The slope of the line indicates the orderof convergence; for each frame the correct, expected behavior isobserved.

plot. A quick comparison between the time steps used andthe corresponding maximum error suggests that the code isconverging at the expected order (second) given the selectionof solver: when the time step is decreased by a factor ofx,the error decreases by a factor ofx2.

Figure 5 illustrates this clearly and quantitatively. Timesteps and associated error values are plotted on a log-logscale for 100 separate simulations (top frame). These resultsare naturally sorted into three distinct regimes. On the farright is the regime where the time step is too large to prop-erly capture the dynamics. This is illustrated in the top row ofFig. 4, where1t = 300 s misses most of the dynamics of theanalytical solution and the error is greater than the amplitudeof the forcing source term. Because the domain is resolvedso poorly, the error varies wildly with spurious peaks andvalleys developing. Such oscillations are caused by a varietyof factors, such as the time step or grid resolution harmoniz-ing with the spatial and temporal frequencies ofQMMS, butare ultimately the result of very poor resolution and shouldbe disregarded. On the far left is the regime where the er-ror is dominated by floating point error. Though the timestep is refined throughout this regime, the maximum error isnot reduced as it is either of machine origin or requires fur-ther refinement in1L. This is observed in the bottom two

10-2 10-1 100

∆L

10-1

100

101

102

∆T

1.0E-101.0E-091.0E-081.0E-07

1.0E-07

1.0E-06

1.0E-06

1.0E-05

1.0E-05

1.0E-04

1.0E-04

1.0E-03

1.0E-03

1.0E-02

1.0E-02

1.0E-01

1.0E-01

Results for DLL=S1997

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Max E

rror

Fig. 6. Error map for 8000 simulations as a function of time step(vertical axis) and grid size (horizontal axis). White lines denotecontours of constant error at each order-of-magnitude. These resultswere obtained using the S1997DLL model. A cut of this map alonga line of constant1L (1t) will yield a curve similar to what is seenin the top (bottom) frame of Fig.5 with the slope indicating theorder of convergence. The proper behavior is observed here over alarge range of time steps and grid sizes.

rows in Fig.4 where the error reduction is less than expectedgiven the second order solver. The center region of Fig.5,delimited by the vertical dashed red lines, is known as theasymptotic regime. Here, the time step and grid size are bothappropriate for the conditions being simulated and the errordrops off as expected given the selected numerical scheme.The slope of this region is nearly equal to the order of thesolver, indicating that the code is correctly solving the sys-tem as intended. The lower frame of Fig.5 shows that thesame is true when the grid resolution is refined.

Figure6 takes these results one step further. The numberof simulations is now increased to 8000, and the error foreach1t-1L combination is sorted into a two-dimensionalcolor map. The white contour lines show order-of-magnitudeboundaries clearly. Figure6 not only confirms the resultsfrom above, but also reveals more about the code’s behaviorfor a particularDLL andQMMS. For 1t > 1 s, refining thegrid has almost no effect on the overall error. In this region,the contours are parallel to the1L axis. This is because fora moderate value ofDLL and a quickly varying source func-tion, it is much more important to have a properly refinedtime grid as opposed to a fineL grid. Only when the peri-odic source term is properly resolved does reducing the gridsize improve results. The curve that passes through the in-tersection of the vertical and horizontal contours (not shown)traces the most efficient combinations of1t and1L. Thisinformation is important to code users who want to maximizecode performance in terms of controlling error and maintain-ing fast run speeds; understanding the balance between therate of variation in time and space is key.

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D. T. Welling et al.: RadBelt verification 283

10-1

100

101

102

∆T

1.0E-101.0E-091.0E-081.0E-071.0E-06

1.0E-06

1.0E-05

1.0E-05

1.0E-04

1.0E-04

1.0E-03

1.0E-03

1.0E-02

1.0E-02

1.0E-01

1.0E-01

Results for DLL=FC20061.0E-101.0E-091.0E-081.0E-071.0E-06

1.0E-06

1.0E-05

1.0E-05

1.0E-04

1.0E-04

1.0E-03

1.0E-03

1.0E-02

1.0E-02

1.0E-01

1.0E-01

Results for DLL=U2008

10-1

100

101

102

∆T

1.0E-101.0E-091.0E-081.0E-071.0E-06

1.0E-06

1.0E-05

1.0E-05

1.0E-04

1.0E-04

1.0E-03

1.0E-03

1.0E-02

1.0E-02

1.0E-01

1.0E-01

Results for DLL=BA2000, Kp=1.01.0E-101.0E-091.0E-081.0E-071.0E-061.0E-05

1.0E-05

1.0E-04

1.0E-04

1.0E-03

1.0E-03

1.0E-02

1.0E-02

1.0E-01

1.0E-01

Results for DLL=BA2000, Kp=3.0

10-2 10-1 100

∆L

10-1

100

101

102

∆T

1.0E-101.0E-091.0E-081.0E-071.0E-061.0E-051.0E-04

1.0E-04

1.0E-03

1.0E-03

1.0E-02

1.0

E-0

2

1.0E-01

1.0E-01

Results for DLL=BA2000, Kp=6.0

10-2 10-1 100

∆L

1.0E-101.0E-091.0E-081.0E-071.0E-061.0E-051.0E-04

1.0

E-0

4

1.0E-03

1.0E-03

1.0E-021

.0E-0

21.0E-01

1.0

E-0

1

Results for DLL=BA2000, Kp=9.0

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102

Max Error

Fig. 7. Similar to Fig.6, but for otherDLL models.

This process was repeated for all otherDLL models. ForBA2000, the simulations were performed for Kp= [0 : 9] fora total of fourteen separate verification sets. The results aresummarized in Fig.7. Returning to Fig.2, it would be ex-pected that the error maps for FC2006, U2008, and BA2000for low Kp should appear similar to the error map for S1997(Fig. 6) as theDLL models are similar in shape and haveweak to moderate values. As seen in the top four frames ofFig. 7, this is indeed the case. As Kp increases, however,BA2000DLL becomes far stronger than any other values in-vestigated to this point. This is evident in the bottom twoframes of Fig.7, as the region where1L refinement is mostimportant (contour lines are dominantly parallel to the1t

axis) grows as Kp increases from 6 (lower left frame) to 9(lower right frame). Again, it is important for code users tounderstand the balance between time dynamics and diffusiveL dynamics when selecting time step and grid resolution.

3.2 Time dependentDLL

Real world applications of RadBelt will involve a time-varying Kp. When using the BA2000 model, which is Kpdependent,DLL will vary throughout the simulation time do-main. Verification of RadBelt’s ability to properly capturethis new complexity must be performed.

0 100 200 300 400 500 600Time (seconds)

123456789

KP

KP & DLL

0

200

400

600

800

1000

1200

DLL (Days−

1)

10-2 10-1 100 101 102 103

Timestep (seconds)

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

||Yanalytic−Ynumeric||

Asymptotic

Regime

Slope=2.004162

Time Convergence - Time Varying DLL

Fig. 8. Results for time verification when using a time-varyingDLL. 1L = 0.01 for this set. The top frame shows the values of Kp(green line) and BA2000DLL (blue line). The bottom frame showsthe expected convergence of the model in the same manner as thetop frame of Fig.5.

The new test case is summarized in the top frame of Fig.8.Kp is allowed to change from 9 to 1 in a smooth, linear fash-ion over the simulated temporal domain (shown in green withthe scale on the left). This results in an exponentially de-creasingDLL (shown in blue with the scale to the right). RealKp values vary as a step function with a three-hour window;it is often linearly interpolated to create a smoothly varyingfunction when used as a model input. As such, the situa-tion selected in the top frame of Fig.8 represents conditionsfound in science applications.

The bottom frame of Fig.8 illustrates that the code indeedconverges properly as1t decreases (constant1L = 0.01)when a time-varyingDLL is used. Because the maximumerror is bounded by what can accumulate whenDLL is at itsmaximum, the curve closely matches what is observed for aconstant BA2000DLL(KP = 9) (Fig. 7, lower right frame,values along1L = 0.01). Similar tests for an impulsive Kpthat jumps from 1 to 9 att = 300 s were performed with sim-ilar, positive results (not shown). These results demonstratethat RadBelt properly handles a time-varyingDLL model.

3.3 Alternative source function

Though unlikely (Roache, 1998), false positives (results thatindicate the code converges properly even though it is not)may be possible when using the MMS method. To test theveracity of the results to this point, portions of the above ex-ercises are repeated using a new manufactured solution. This

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284 D. T. Welling et al.: RadBelt verification

Fig. 9. Similar to Fig.4, but showing results when the second, non-periodic manufactured solution is used (Eq.9).

is the last step in ensuring proper verification of the RadBeltmodel.

The second solution selected is,

f = sin(aL−a)(ebt−1) (9)

wherea andb are again constants set to agree with bound-ary conditions. Unlike the original solution (Eq.7), there isno longer periodicity in time. The constanta is again cho-sen to accommodate Dirichlet boundary conditions whileb

is set to bind the maximum value of the solution to a reason-able value within the time domain of the simulations. Theserequirements yield,

a =2π

9(10)

b =ln(2)

tfinal(11)

which binds the absolute maxima of the solution to±1 whentfinal is the total amount of time simulated. When substitutedinto Eq. (1), the new source term is,

Q2ndMMS = bsin(aL−a)ebt (12)

−aα(β −2)Lβ−1(ebt−1)cos(aL−a)

+a2αLβ(ebt−1)sin(aL−a)

Figures9 and10 summarize the results when using the sec-ond manufactured solution and S1997DLL. Figure9, similarto Fig.4, demonstrates that the code does indeed converge onthe analytic solution as1t is progressively refined. The er-ror values are much lower than before, likely a function ofQ2nd

MMS (Eq.12) varying far more slowly in time than the ini-tial QMMS (Eq.8). With the overall error drastically reduced,residual error at highL (lower right frame) becomes evident.This occurs in the region whereDLL is strongest. Becausethe new source term varies more slowly in time compared tothe old, theseL-dependent features can be seen at a higher1t than before. In other words, the results for this newsource term are more sensitive to1L refinement than1t .This feature is illustrated clearly in Fig.10, where the regionof contours that follow constant1T values is reduced com-pared to what was seen in Fig.6. The results demonstrateproper convergence, but emphasize that the source term must

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D. T. Welling et al.: RadBelt verification 285

10-2 10-1 100

∆L

10-1

100

101

102

∆T

1.0

E-0

8

1.0

E-0

7

1.0

E-0

6

1.0E-05

1.0E-04

1.0E-03

Results for DLL=S1997 (2nd Source)

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Max E

rror

Fig. 10. Similar to Fig.6 but showing results when the second,non-periodic manufactured solution is used (Eq.9).

be taken into consideration when selecting time and spacediscretizations.

3.4 Code performance

Figure11 benchmarks the code performance using a singlecore on an Intel Xeon 2.6 GHz CPU. The color map showsthe time to complete each of the 8000 simulations for theBA2000DLL(Kp = 9) set. The black contour lines show thesame results in terms of simulation time (always ten minutes)divided by total CPU time. RadBelt runs very quickly on amodern machine, with most simulations finishing at>100-times real time. These results demonstrate that it is possibleto combine the benefits of a modern interpreted programminglanguage with the light weight, fast number-crunching abili-ties of a compiled language.

3.5 Source term application considerations

The source term used thus far has been simple enough toinclude in the derivation of the Crank-Nicolson solver. Thisleaves nothing to the imagination when the source term is ap-plied. In real-world, non-verification situations, more com-plex source terms may arise that require a more sophisticatedapplication, for example, a non-linear source term that de-pends on the phase space density. A popular approach is tosplit out the source term as one would when applying Strangsplitting (Strang, 1968) to a multi-dimensional problem. Thisapproach splits the differential equation into two parts: a par-tial differential equation and an ordinary differential equa-tion. The two subproblems, the first representing the diffu-sion off and the second representing the source off , can besolved in an alternating fashion that is mathematically equiv-alent to solving the whole system (Toro, 1999). A secondorder accurate method for this approach can be written as,

(sec

onds

)

(sec

onds

)

Fig. 11. Run speed benchmark for all 8000 simulations performedusing BA2000DLL(Kp = 9). The color map shows raw simulationcompletion time (seconds) while black contour lines show the ratioof simulation time to CPU time. For example, for a total simulationtime of 600 s and a CPU run time of 6 s (near-white on the colorscale), the code simulates the radiation belts 100 times faster thanreal time and has a simulation-to-CPU time ratio (black contourlines) of 100X.

f t+1t= S

1t2 C1tS

1t2 f t (13)

wheref is the phase space density,S1t2 is the operator that

advances the source subproblem forward in time by one-half time step, andC1t is the Crank-Nicolson operator thatadvances the diffusion subproblem forward in time a fulltime step. Such splitting yields a powerful, generalized al-gorithm for tackling complicated source terms but requiresadditional considerations not previously taken into account,namely careful selection of the source term solver.

Figure12 illustrates the impact of the selection of sourcesolver if splitting were applied to the artificial source termsused in this work. Source splitting is not necessary forQMMSas it is only a function oft andL, but is applied to illustratethe complications that arise when it is used. The blue line la-beled “Case 1” shows that employing a simple, second orderaccurate trapezoidal integration to solve the source subprob-lem yields the desired results: a model that converges cleanlywith the prescribed accuracy. But what if a higher accuracyquadrature method is used? Case 2 uses the fourth-order ac-curate Simpson’s Rule to integrate the source term; the re-sults are displayed as the green line in Fig.12. The error isreduced from Case 1, however the code still converges in asecond order manner. This is because less error is introducedfrom the integration of the source term but the total error isstill dominated by the Crank-Nicholson solver and converges

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286 D. T. Welling et al.: RadBelt verification

101 102

Timestep (seconds)

10-7

10-6

10-5

10-4

10-3

10-2

10-1

||Yanalytic−Ynumeric||

Comparison of Source Applications

Case 1Case 2Case 3

Fig. 12.Convergence curves for three separate simulations that splitout the source term from the Crank-Nicolson solver. Case 1 uses asimple second-order accurate source solver, while Cases 2 and 3use a fourth order source solver. The variety in the curves demon-strate the importance of properly selecting a solver along with anappropriate grid size and time step.

along that lower-order rate. Finally, Case 3 (red line) is iden-tical to Case 2 but the magnitude ofDLL is reduced by anorder of magnitude. Because the rate of diffusion is smaller,a larger time step is better suited to the problem and the over-all error should drop. This proves true for1t < 30.0 s inFig. 12, but for larger time steps, the error quickly rises tomatch Case 2. What is observed is the two solution operatorstrading the bulk of the error. When the source term solver ac-counts for the greater portion of the error, the code convergeswith fourth order accuracy. Because this drops below the er-ror inherent in the diffusion operator quickly, the code thenbegins to converge more slowly, and all three lines becomeparallel. All three of these curves result from slight variationsof a Crank-Nicolson, split-source term implementation.

This example illustrates the additional complexities thatmust be taken under consideration when more complicatedsource and loss terms are included. The selected source in-tegration scheme has dramatic implications on the overallerror. Changing diffusion rates, which happens commonlywhen the BA2000 model is used, can push the model intodifferent error regimes and change the behavior of the model.These effects must be kept in mind as the model is developedin the future.

4 Conclusions

This work introduced and verified the RadBelt radial diffu-sion model that is part of the SpacePy software library. Themethod of manufactured solutions was employed to providean analytical solution where one could not be trivially ob-tained. The model’s flexibility, especially in terms of choiceof DLL and sources, was put to many rigorous tests. It was

found that the solver and many code features have been im-plemented correctly and robustly.

The results here are of special interest to code users. Re-peatedly, it was demonstrated that code convergence is be-holden to the choice ofDLL and the rate of change of thesource function. While selecting1t and1L, it is easy to findoneself in a regime where refining one has little impact onthe final error. How the source and diffusion coefficient af-fect these semi-stagnant convergence regimes is vital knowl-edge for code users who want to maximize code performancewhile minimizing error.

Finally, it must be emphasized that performance in termsof convergence regimes and overall error is situation specific.The time and grid discretizations used in this study are farsmaller than the typical scientific user will require. It is sug-gested that, when employing new source, loss or diffusioncoefficient models, the user perform a quick study to ensureproper refinement for the most extreme situation they chooseto encounter.

Acknowledgements.The authors would like to thank Burton Wen-droff and Charles Kiyanda for their helpful discussions concerningnumerical methods. Work at Los Alamos was conducted underthe auspices of the US Department of Energy. This researchwas conducted as part of the Dynamic Radiation EnvironmentAssimilation Model (DREAM) project at Los Alamos NationalLaboratory. We are grateful to the sponsors of DREAM forfinancial and technical support.

Edited by: J. Annan

References

Baker, D.: The Occurrence of Operational Anomalies in Spacecraftand Thei Relationship to Space Weather, IEEE Transactions onPlasma Science, 28, 2007–2016, 2000.

Beutier, T. and Boscher, D.: A three-dimensional analysis of theelectron radiation belt by the Salammbo code, J. Geophys. Res.,100, 14853–14862,doi:10.1029/94JA03066, 1995.

Brautigam, D. H. and Albert, J. M.: Radial diffusion anal-ysis of outer radiation belt electrons during the October9, 1990, magnetic storm, J. Geophys. Res., 105, 291–310,doi:10.1029/1999JA900344, 2000.

Carpenter, D. L. and Anderson, R. R.: An ISEE/Whistler modelof equatorial electron density in the magnetosphere, J. Geophys.Res., 97, 1097–1108,doi:10.1029/91JA01548, 1992.

Crank, J. and Nicolson, P.: A practical method for numerical eval-uation of solutions of partial differential equations of the heat-conduction type, Proc. Cambridge Philos. Soc., pp. 50–67, 1947.

Falthammar, C.-G.: Effects of Time-Dependent Electric Fieldson Geomagnetically Trapped Radiation, J. Geophys. Res., 70,2503–2516,doi:10.1029/JZ070i011p02503, 1965.

Fei, Y., Chan, A. A., Elkington, S. R., and Wiltberger, M. J.:Radial diffusion and MHD particle simulations of relativis-tic electron transport by ULF waves in the September 1998storm, J. Geophys. Res. (Space Physics), 111, A12209,doi:10.1029/2005JA011211, 2006.

Geosci. Model Dev., 5, 277–287, 2012 www.geosci-model-dev.net/5/277/2012/

D. T. Welling et al.: RadBelt verification 287

Feynman, J. and Gabriel, S. B.: On space weather conse-quences and predictions, J. Geophys. Res., 105, 10543–10564,doi:10.1029/1999JA000141, 2000.

Fok, M.-C., Horne, R. B., Meredith, N. P., and Glauert, S. A.: Ra-diation Belt Environment model: Application to space weathernowcasting, J. Geophys. Res. (Space Physics), 113, A03S08,doi:10.1029/2007JA012558, 2008.

Gubby, R. and Evans, J.: Space environment effects and satellitedesign, J. Atmos. Solar-Terr. Phys., 64, 1723–1733, 2002.

Koller, J. and Morley, S.: Magnetopause shadowing effects for ra-diation belt models during high-speed solar wind streams, AGUFall Meeting Abstracts, pp. A1787+, 2010.

Koller, J., Chen, Y., Reeves, G. D., Friedel, R. H. W., Cayton,T. E., and Vrugt, J. A.: Identifying the radiation belt source re-gion by data assimilation, J. Geophys. Res. (Space Physics), 112,A06244,doi:10.1029/2006JA012196, 2007.

Koons, H. C., Mazur, J. E., Selesnick, R. S., Blake, J. B., and Fen-nell, J. F.: The Impact of the Space Environment on Space Sys-tems, NASA STI/Recon Technical Report N, pp. 69036–69041,1999.

Lam, M. M., Horne, R. B., Meredith, N. P., and Glauert, S. A.:Modeling the effects of radial diffusion and plasmaspheric hisson outer radiation belt electrons, Geophys. Res. Lett., 34,L20112,doi:10.1029/2007GL031598, 2007.

Lyons, L. R. and Schulz, M.: Access of energetic particles tostorm time ring current through enhanced radial ‘diffusion’, J.Geophys. Res., 94, 5491–5496,doi:10.1029/JA094iA05p05491,1989.

Millan, R. M. and Thorne, R. M.: Review of radiation belt rela-tivistic electron losses, J. Atmos. Solar-Terr. Phys., 69, 362–377,doi:10.1016/j.jastp.2006.06.019, 2007.

Miyoshi, Y., Morioka, A., Misawa, H., Obara, T., Nagai, T., andKasahara, Y.: Rebuilding process of the outer radiation belt dur-ing the 3 November 1993 magnetic storm: NOAA and Exos-D observations, J. Geophys. Res. (Space Physics), 108, 1004,doi:10.1029/2001JA007542, 2003.

Morley, S. K., Koller, J., Welling, D. T., and Larsen, B. A.: Spacepy– A Python-based library of tools for the space sciences, in: Pro-ceedings of the 9th Python in science conference (SciPy 2010),in press, Austin, TX, 2011.

Pirjola, R., Kauristie, K., Lappalainen, H., and Viljanen,A.: Space weather risk, Space Weather, 3, S02A02,doi:10.1029/2004SW000112, 2005.

Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P.:Numerical Recipes, Cambridge University Press, New York, NY,3rd edn., 2007.

Qin, Z., Denton, R. E., Tsyganenko, N. A., and Wolf, S.: Solar windparameters for magnetospheric magnetic field modeling, SpaceWeather, 5, S11003,doi:10.1029/2006SW000296, 2007.

Reeves, G. D., Friedel, R. H. W., Bourdarie, S., Thomsen, M. F., Za-haria, S., Henderson, M. G., Chen, Y., Jordanova, V. K., Albright,B. J., and Winske, D.: Toward Understanding Radiation Belt Dy-namics, Nuclear Explosion-Produced Artificial Belts, and ActiveRadiation Belt Remediation: Producing a Radiation Belt DataAssimilation Model, Washington D.C. American GeophysicalUnion Geophysical Monograph Series, 159, 221–235, 2005.

Roache, P. J.: Verification and Validation in Computational Scienceand Engineering, Hermosa, Albuquerque, NM, 1998.

Roache, P. J.: Code Verification by the Method of Manufactured So-

lutions, J. Fluids Eng., 124, 4–10,doi:10.1115/1.1436090, 2002.Roederer, J. G.: Dynamics of geomagnetically trapped radiation,

Springer, Berlin, Germany, 1970.Schulz, M. and Lanzerotti, L.: Particle Diffusion in the Radiation

Belts, Spring-Verlag, New York, 1974.Selesnick, R. S., Blake, J. B., Kolanski, W. A., and Fritz, T. A.: A

quiescent state of 3 to 8 MeV radiation belt electrons, Geophys.Res. Lett., 24, 1343,doi:10.1029/97GL51407, 1997.

Shprits, Y. Y., Thorne, R. M., Friedel, R., Reeves, G. D., Fen-nell, J., Baker, D. N., and Kanekal, S. G.: Outward radial diffu-sion driven by losses at magnetopause, J. Geophys. Res. (SpacePhysics), 111, A11214,doi:10.1029/2006JA011657, 2006.

Shprits, Y. Y., Elkington, S. R., Meredith, N. P., and Sub-botin, D. A.: Review of modeling of losses and sourcesof relativistic electrons in the outer radiation belt I: Ra-dial transport, J. Atmos. Solar-Terr. Phys., 70, 1679–1693,doi:10.1016/j.jastp.2008.06.008, 2008.

Strang, G.: On the Construction and Comparison of DifferenceSchemes, SIAM J. Numer. Anal., 5, 506–517, 1968.

Subbotin, D., Shprits, Y., and Ni, B.: Three-dimensional VERBradiation belt simulations including mixed diffusion, J. Geophys.Res. (Space Physics), 115, A03205,doi:10.1029/2009JA015070,2010.

Tadjeran, C.: Stability analysis of the Crank-Nicholson methodfor variable coefficient diffusion equation, Communica-tions in Numerical Methods in Engineering, 23, 29–34,doi:10.1002/cnm.879, 2007.

Toro, E. F.: Riemann solvers and numerical models for fluid dy-namics, Springer-Verlag, New York, 1999.

Tsyganenko, N. A.: A model of the near magnetosphere with adawn-dusk asymmetry - 1. Mathematical Structure, J. Geophys.Res., 107, 1179–1195,doi:10.1029/2001JA000219, 2002.

Tsyganenko, N. A., Singer, H. J., and Kasper, J. C.: Storm-time distortion of the inner magnetosphere: How severecan it get?, J. Geophys. Res. (Space Physics), 108, 1209,doi:10.1029/2002JA009808, 2003.

Ukhorskiy, A. Y. and Sitnov, M. I.: Radial transport inthe outer radiation belt due to global magnetospheric com-pressions, J. Atmos. Solar-Terr. Phys., 70, 1714–1726,doi:10.1016/j.jastp.2008.07.018, 2008.

van Allen, J. A. and Frank, L. A.: Radiation Around the Earthto a Radial Distance of 107 400 km, Nature, 183, 430–434,doi:10.1038/183430a0, 1959.

Wang, H., Ridley, A. J., and Luhr, H.: Validation ofthe Space Weather Modeling Framework using observa-tions from CHAMP and DMSP, Space Weather, 6, S03002,doi:10.1029/2007SW000355, 2008.

Welling, D. T.: The long-term effects of space weather on satelliteoperations, Ann. Geophys., 28, 1361–1367,doi:10.5194/angeo-28-1361-2010, 2010.

Welling, D. T. and Ridley, A. J.: Validation of SWMFmagnetic field and plasma, Space Weather, 8, S03001,doi:10.1029/2009JA014596, 2010.

Xiao, F., Su, Z., Zheng, H., and Wang, S.: Three-dimensional simu-lations of outer radiation belt electron dynamics including cross-diffusion terms, J. Geophys. Res. (Space Physics), 115, A05216,doi:10.1029/2009JA014541, 2010.

www.geosci-model-dev.net/5/277/2012/ Geosci. Model Dev., 5, 277–287, 2012