Robustness of oscillatory $\alpha^2$ dynamos in spherical wedges

1.2 MB·

by  E. Cole

Astronomy&Astrophysicsmanuscriptno.paper (cid:13)c ESO2016
January21,2016
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Robustness of oscillatory α dynamos in spherical wedges
E.Cole1,2,A.Brandenburg2,3,4,5,P.J.Ka¨pyla¨6,1,2 andM.J.Ka¨pyla¨6
1 DepartmentofPhysics,GustafHa¨llstro¨minkatu2a,POBox64,FI-00014UniversityofHelsinki,Finland
2 Nordita,KTHRoyalInstituteofTechnologyandStockholmUniversity,Roslagstullsbacken23,10691Stockholm,Sweden
3 DepartmentofAstronomy,AlbaNovaUniversityCenter,StockholmUniversity,10691Stockholm,Sweden
4 JILAandDepartmentofAstrophysicalandPlanetarySciences,Box440,UniversityofColorado,Boulder,CO80303,USA
5 LaboratoryforAtmosphericandSpacePhysics,3665DiscoveryDrive,Boulder,CO80303,USA
6 ReSoLVECentreofExcellence,DepartmentofComputerScience,AaltoUniversity,POBox15400,FI-00076Aalto,Finland
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1 January21,2016, Revision:1.106
0
2 ABSTRACT

n
Context. Large-scaledynamo simulationsaresometimesconfinedtosphericalwedgegeometriesbyimposingartificialboundary
a
conditionsathighlatitudes.Thismayleadtospatio-temporalbehavioursthatarenotrepresentativeofthoseinfullsphericalshells.
J
Aims.Westudytheconnectionbetweensphericalwedgeandfullsphericalshellgeometriesusingsimplemean-fieldα2dynamos.
0
Methods.Wesolvetheequationsforaone-dimensionaltime-dependentmean-fielddynamotoexaminetheeffectsofvaryingthepo-
2
larangleθ0betweenthelatitudinalboundariesandthepolesinsphericalcoordinates.Weinvestigatetheeffectsofturbulentmagnetic

diffusivityandαeffectprofilesaswellasdifferentlatitudinalboundaryconditionstoisolateparameterregimeswhereoscillatoryso-
]
lutionsarefound.Finally,weaddshearalongwithadampingtermmimickingradialgradientstostudytheresultingdynamoregimes.
R
Results. Wefindthatthecommonlyusedperfect conductor boundary condition leadstooscillatoryα2 dynamo solutionsonlyif
S
thewedgeboundaryisatleastonedegreeawayfromthepoles.Otherboundaryconditionsalwaysproducestationarysolutions.By
h. varyingtheprofileoftheturbulentmagneticdiffusivityalone,oscillatorysolutionsareachievedwithmodelsextendingtothepoles,
p butthemagneticfieldisstronglyconcentratednearthepolesandtheoscillationperiodisverylong.Bychangingboththeturbulent
- magneticdiffusivityandαprofilessothatbotheffectsaremoreconcentratedtowardtheequator,weseeoscillatorydynamoswith
o equatorward drift,shorter cycles, andmagnetic fieldsdistributedover awider range of latitude.Byintroducing radial shear anda
r dampingtermmimickingradialgradients,weagainseeoscillatorydynamos,andthedirectionofdriftfollowstheParker–Yoshimura
st rule.Oscillatorysolutionsintheweakshearregimearefoundonlyinthewedgecasewithθ0 =1◦andperfectconductorboundaries.
a Conclusions.Areducedαeffectnearthepoleswithaturbulentdiffusivityconcentratedtowardtheequatoryieldsoscillatorydy-
[ namoswithequatorwardmigrationandreproducesbestthesolutionsinsphericalwedges.Forweakshear,oscillatorysolutionsare

obtained only for normal field conditions and negative shear. Oscillatory solutions become preferred at sufficiently strong shear.
1
Recentthree-dimensionaldynamosimulationsproducingsolar-likemagneticactivityareexpectedtolieinthisrange.
v
6 Keywords.turbulence–magnetohydrodynamics(MHD)–hydrodynamics
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2
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1. Introduction arelativevalueof20–30%inlatitude(e.g.Mieschetal.,2000;
0
Ka¨pyla¨etal., 2011).However,whetherornotthisis enoughto
.
1 The Sun’s magnetic field is generally believed to be the result driveanαΩdynamoasopposedtoanα2 dynamo,inwhichthe
0 of a turbulent αΩ dynamo in which differential rotation plays Ωeffectwouldbesubdominant,canonlybedecidedonthebasis
6
animportantrole.Thisis referredtoas theΩ effect,andit has ofquantitativecalculations.
1
long been identified as a robust mechanism for amplifying the
: In the absence of a conclusive answer, one tends to re-
v azimuthalmagneticfieldoftheSunbywindingupthepoloidal
sort to qualitative arguments. One is related to the clear east–
i field (Babcock, 1961; Ulrich&Boyden, 2005; Brownetal.,
X west orientation of bipolar regions in the Sun, which sug-
2010). The production of poloidal field, on the other hand, is
gests that the azimuthal field must be much stronger than the
r more complicated and harder to verify in computer simula-
a poloidal field. Another argument is that αΩ dynamos are usu-
tions, but it is thought to be associated with helical motions
ally cyclic and can display equatorward migration of mag-
intherotating,densitystratifiedconvectionzone(Parker,1955;
netic field either through suitable radial differential rotation
Steenbecketal.,1966).Thisprocessiscommonlyparametrised
(Parker, 1955; Steenbeck&Krause, 1969a) or through suffi-
by an α effect. Although there remain substantial uncertain-
cientlystrongmeridionalcirculationinthepresenceofanαef-
ties regarding the α effect as an important ingredient at large
fect that operates only in the surface layers (Choudhurietal.,
magneticReynoldsnumbers(Cattaneo&Hughes,2006),simu-
1995). However, both arguments are problematic. Although it
lationsofturbulenceandrotatingconvectionhavesubsequently
is probably true that the azimuthal field is stronger than the
confirmed that conventional estimates of α and turbulent dif-
poloidal,theirratiomaynotbelargeenoughtojustifythedom-
fusivity ηt are reasonably accurate up to moderate values of inance of the Ω effect. Furthermore, α2 dynamos may well
the magnetic Reynolds number (Suretal., 2008; Ka¨pyla¨etal.,
beoscillatory(e.g.Ka¨pyla¨etal.,2013a;Masada&Sano,2014)
2009).
andcandisplayequatorwardmigrationundersuitableconditions
Simulations also demonstrate the generation of differential (Mitraetal.,2010).Acompletelydifferentargumentthatmoti-
rotationfromanisotropicrotatingconvection,whichamountsto vatesthestudyofoscillatoryα2dynamosarerecentsimulations
1
E.Coleetal.:Robustnessofoscillatoryα2dynamosinsphericalwedges
of convectivedynamosin spherical wedgesand full shells that 2. Model
also show equatorward migration (Ka¨pyla¨etal., 2012, 2013b;
Weconsiderthemean-fielddynamoequationforthemeanmag-
Warneckeetal., 2013; Augustsonetal., 2015). It is now be-
netic field B with a given mean electromotive force E in the
lievedthattheequatorwardmigrationinthesimulationsisfacil-
form
itatedbyaregionofnegativeshearandpositive(negative)αef-
∂B
fectinthenorthern(southern)hemisphere–inaccordancewith =∇×(cid:0)U ×B+E −ηµ0J(cid:1), (1)
theParker–Yoshimurarule(Warneckeetal.,2014).Recentlyan ∂t
alternative scenario was reported by Duarteetal. (2015), who whereU = φˆ̟Ω isthemeanflowfromangularvelocitywith
foundthatthesignoftheαeffectcanbeinvertedincertainpa-
̟ = rsinθ beingthe distancefromthe axis, Ω(r,θ) isthe in-
rameterrangesallowingequatorwardmigrationalso with posi-
ternalangularvelocity,φˆ istheunitvectorintheazimuthaldi-
tiveradialshear.Althoughitisuncleartowhatextentthosesim-
ulationsrepresentstellar magneticfields, it mightbe helpfulto rection,J = ∇×B/µ0 isthemeancurrentdensity,µ0 isthe
vacuumpermeability,andηisthenon-turbulentmagneticdiffu-
understand first the mechanism operating in those simulations
sion coefficient. In the absence of a memory effect, and under
beforetryingtounderstandrealstars.
theassumptionofisotropicαeffectandturbulentmagneticdif-
Whiletheideaofexplainingequatorwardmigrationthrough fusivityηt,themeanelectromotiveforceisgivenby
α2 dynamo action might work in sphericalwedge simulations,
there is the problem that such solutions have never been seen E =αB−ηtµ0J. (2)
in full shell simulations that extend not just to high latitudes,
but go all the way to the poles. Indeed, α2 dynamos in full We solve Eqs. (1) and (2) numericallyusing sixth-order fi-
nitedifferencesinspaceandathird-orderaccuratetime-stepping
spherical shells are known to be steady (Steenbeck&Krause,
scheme.WeemploythePENCIL CODE1,whichsolvesthegov-
1969b). Exceptions are dynamos with an anisotropic α ten-
erningequationsintermsofthemeanmagneticvectorpotential
sor (Ru¨digeretal., 2003) and the non-axisymmetricoscillatory
A, such that B = ∇×A. It is convenientto use the advec-
solutions found by Jiang&Wang (2006), but for an isotropic
α effect, oscillatory axisymmetric α2 dynamos seem to be an tive gauge (Brandenburgetal., 1995; Candelaresietal., 2011),
inwhichtheelectrostaticpotentialhasacontributionU A ,so
artefact of having imposed a boundary condition at high lati- φ φ
that
tudes.Onecouldchooseanotherboundarycondition;anormal-
∂A
field(pseudo-vacuum)boundaryconditionmightbeanobvious =−̟Aφ∇Ω+E −ηµ0J. (3)
choice,butfromcorrespondingCartesiansimulationsweknow ∂t
thatthiswouldagainleadtooscillatorysolutions,butwithpole- To allow for the use of a one-dimensional model with B =
wardmigration(Brandenburgetal.,2009). B(θ,t),werestrictourselvestoanangularvelocityprofilethat
varieslinearlyinr,i.e.,Ω(r,θ)=rS(θ),sotheangularvelocity
Although the mean-field description of oscillatory α2 dy- gradientbecomes∇Ω = (S,∂ S,0).Themeancurrentdensity
namosseemstofaceaninternalinconsistencyregardingthelimit θ
isthen
to fullsphericalshells, thereremainsthe questionwhethercer-
tainchangesinthesetupofthefullsphericalshellmodelcould J =µ−01R−2(cid:0)DθAθ−Dθ∂θAr, ∂θAr, −∂θDθAφ(cid:1), (4)
lead to oscillatory solutions that are internally consistent and
otherwisesimilarto thesolutionsinsphericalwedges.Thereis whereD =cotθ+∂ isamodifiedθderivative.Toaccountfor
θ θ
apriorinophysicalmotivationforthis,butfromamathematical theneglectofrderivatives,weaddinEq.(3)adampingtermof
pointofview,thisis a naturalchoicewhentryingtoreproduce theform−µ2A,i.e.,wehave
theconditionsencounteredpreviouslywith a perfectconductor
bporoufinldearwyitchonadliatirogne.rOconnedpuocstisvibitiylit(ywiesakaesrumitaabglneetliactitduidffiunsailoηnt) ∂∂At =−̟Aφ∇Ω+E −ηµ0J −µ2A (with∂r =0); (5)
athighlatitudestosimulate thebehaviourofperfectconductor
seeMossetal.(2004)forasurveyofsolutionsfordifferentval-
boundaryconditionsusedinsphericalwedges.
uesofµ.Forαandηt weuselatitudinalprofilefunctionsofthe
In each of those cases, it is important to assess how much form
shear would be needed to change the dynamo mode into an
αΩ type mode. To keep things simple, we employ a one- α=α0cosθ(cid:0)a0+a2sin2θ+...+ansinnθ(cid:1), (6)
dimensional model with only latitudinal extent. However, in
its standard formulation, with radial derivatives simply being ηt =ηt0(cid:0)e0+e2sin2θ+...+ensinnθ(cid:1), (7)
dropped,the first excited mode of such an αΩ dynamois non- where a and e are coefficients denoted by the vectors a =
i i
oscillatory(Jenningsetal.,1990).Thisisanartefactthatiseas- (a0,a2,a4,...,an) and e = (e0,e2,e4,...,en), respectively.
ilyremovedbysubstitutingradialderivativesbyadampingterm However, we often refer to only the three first components as
(Kuzanyan&Sokoloff,1995;Mossetal.,2004),insteadofset- a = (a0,a2,a4) and e = (e0,e2,e4). These expansions can
tingthemtozero. alsobeexpressedintermsofLegendrepolynomials,whichare
orthonormalfunctionsthatobeyregularityatthepoles.Theoc-
Webeginbydescribingourmodelindetail,nextfocusonthe
currence of higher order terms in α has been associated with
analysisofsphericalwedgesofdifferentextentandturnthento higherorderstermsing·Ω,whicharenormallyomittedinthe-
fullsphericalshellswithvariablelatitudinalηt profiles.Inview oreticalcalculations(Ru¨diger&Brandenburg,1995).
oftheaforementionedcomplicationsregardingthepossibilityof
Asusual,theproblemisgovernedbytwodynamonumbers,
oscillatorybehaviourinthecorrespondingαΩdynamos,wealso
discuss the sensitivity of our solutionswith respectto an addi- Cα =α0R/ηt0, CΩ =S0R2/ηt0, (8)
tionaldampingtermthatmimicstheotherwiseneglectedradial
derivativeterms. 1 http://pencil-code.github.com/
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E.Coleetal.:Robustnessofoscillatoryα2dynamosinsphericalwedges
whereS(θ) = S0 isnowaconstant.Weconsiderthefollowing
setsofboundaryconditions:
∂ A =A =A =0 (SAA;regularityonθ =0), (9)
θ r θ φ
Ar =∂θAθ =Aφ =0 (ASA;perf.cond.onθ =θ0),
(10)
∂θAr =Aθ =∂θAφ =0 (SAS;normalfieldonθ =θ0),
(11)
wherethesequenceoflettersSandAreferrespectivelytosym-
metric (∂ = 0) and antisymmetric (vanishing function value)
θ
of A , A , and A across the boundary. The same conditions
r θ φ
arealsoappliedonthecorrespondingboundaryinthesouthern
hemispherewhereπ−θ = θ0.Inthiswork,nosymmetrycon-
dition on the equator is applied,so the parity of the solution is
notconstrained.
Asinitialconditions,weassumea seedmagneticfieldcon-
sisting of low-amplitude Gaussian noise. Such a field is suf-
ficiently complex so that the fastest growing eigenmode of
either parity tends to emerge after a short time. Note that
mixedparitysolutionsareonlypossibleinthenonlinearregime Fig.1. DependenceofCα⋆ onθ0 forthethreeboundarycondi-
tionsASA,SAA,andSAS.
(Brandenburgetal.,1989),butthiswillnotbeconsideredhere.
3. Results
We consider separately the cases where the dynamo is driven
either solely by the α effect(α2 dynamos)or by the combined
actionofαeffectandlarge-scaleshear(α2Ωdynamos).
3.1. α2 dynamos
3.1.1. Varyingθ0
Webeginbyconsideringthesimplestcasewitha=(1,0,0)and
e = (1,0,0)resulting a spatially constantturbulentdiffusivity
and a cosθ profile for α We have calculated the critical value Fig.2. AzimuthalmagneticfieldB foranoscillatorydynamo
of Cα, hereafter Cα⋆, for an oscillatory α2 dynamo, i.e., where withθ0 =5◦ andtheboundarycondφitionASA.
CΩ = 0. We used the boundaryconditionsSAA (Eq. 9), ASA
(Eq.10),andSAS(Eq.11)forselectedvaluesofθ0.
It turns out that C⋆ decreases as we approach the pole
α
(θ0 → 0); see Fig. 1. The SAA and SAS boundaryconditions 3.1.2. Varyinglatitudinalηt profile
resultinverysimilarnon-oscillatorysolutionswithaC⋆ ofonly
approximately40%of thatC⋆ obtainedfortheASA bαoundary Given thatwe have foundthe limit θ0 → 0◦ in the case of the
condition.Oscillatorysolutionαsshowtravellingwavesthatprop- perfect conductor boundary condition not to be an approxima-
tiontoafullsphericalshellmodel,wenowinvestigatewhether
agateequatorward;seeFig.2.Theboundaryconditionwiththe
WgreeaatlessotvfianrdiatthioanttohfeCmα⋆oswteitahsiθl0yiesxtchiteedpedryfencatmcoonmdoudcetocrh,aAnSgAes. pwhityhsicthaellySmAoAtivbaoteudndaaltreyractioonndsitoiofnthecofuulldlspphroedriuccaelsohseclilllmatoodreyl,
from stationary to oscillatory as θ0 increases from zero to one equatorwardsolutionssimilar to those found for θ0 6= 0◦ with
degree in that case. For the case where θ0 = 1◦ we find both the ASA boundarycondition.An obviouspossibility is the use
stationaryandoscillatorysolutionsdependingontheinitialcon- of an ηt profile that corresponds to high conductivity near the
pole. Such a profile could correspond to the possible effect of
ditions. The critical dynamo number is slightly higher for the
oscillatorymodethanthecorrespondingvalueofthestationary rotationonthemagneticdiffusivity(Kitchatinovetal.,1994)at
solution. variouslatitudes.
Theseresultssuggestthatwecannotregardthelimitθ0 →0 One possible alteration to the diffusivity profile is to use
with isotropic α effectand turbulentdiffusionand perfectcon- higher order terms for ηt. In particular, we examine solutions
ductor boundariesas an approximationto a full spherical shell wheretheordersi = 2,4,6,and8areusedforei;seeEq.(7).
model when searching for oscillatory solutions. Extending the Solutionsareexaminedfore0 =η/ηt0 =0.01and0.05.Anon-
model to the poles with the ASA boundary condition changes zero uniformvalue of η is neededto ensurethe stability of the
the resulting dynamo from oscillatory to stationary. The SAA solutionsinthecaseswheretheturbulentmagneticdiffusivityis
andSASboundaryconditionsgivestationarysolutionswithrel- zeroatthepolesduetotheprofilesbeingproportionaltopowers
atively similar valuesfor C⋆, but the ASA boundarycondition ofsinθ,whichvanishesatthepoles.
α
near the poles gives both oscillatory and stationary solutions, Neithervalueofη usedhereleadstospuriousgrowthinthe
depending on the initial conditions of the seed magnetic field. absenceofanα-effect.Furthermore,wecalculatetheoscillation
Whilenostationarysolutionswerefoundforθ0 > 1◦,theirex- frequencyasω =2π/T whereT istheperiodofoscillationfor
istenceisnotruledoutbyourmodels. thelarge-scalemagneticfield.
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E.Coleetal.:Robustnessofoscillatoryα2dynamosinsphericalwedges
Table1. C⋆ forpureα2dynamoswithvariedmagneticdiffusiv-
α
ityandαprofilesandthecorrespondingoscillationsfrequencies
inunitsofηt0/R2.
a
(1,0,0) (0,1,0) (0,0,1)
ei e0 Cα⋆ |ω Cα⋆ |ω Cα⋆ |ω
e2 0.01 0.236|—- 4.063|0.405 9.532|0.562
e2 0.05 0.558|—- 5.308|0.288 11.39|0.654
e4 0.01 0.096|0.008 1.045|0.207 4.144|0.298
e4 0.05 0.326|—- 2.587|0.332 7.039|0.548
e6 0.01 0.070|0.005 0.541|0.184 2.175|0.215
e6 0.05 0.265|—- 1.733|0.258 4.857|0.419
e8 0.01 0.059|0.003 0.403|0.131 1.463|0.165
e8 0.05 0.238|—- 1.384|0.199 3.727|0.364
Values for C⋆ are indicated in Table 1 for cases where the
α
turbulent diffusivity and α effect profiles are expanded up to
orders e8 and a4, respectively. We find that for a = (1,0,0),
the e0 = 0.05 case produces only stationary solutions, but
at e0 = 0.01, only solutions for n = 2 are stationary and
all higher orders oscillate; see Table 1. Some solutions ini-
tiallyshowrapidlyoscillatingbehaviour,exhibitingantisymme-
trywithrespecttotheequator,butthesedisappearlaterandonly
aslower,persistentoscillatorymoderemains:seethetoppanel
of Fig. 3. These low-frequency oscillations have neither equa-
torward nor poleward migration, and are symmetric about the
equator.Cα⋆ increaseswithe0,anddecreasesasn increasesfor
e ,inaccordancewiththetotaldiffusionincreasinganddecreas-
n
ing,respectively.Thefrequencyoftheoscillatorymodesfound
for e0 = 0.01 decreases as n increases. This is also consis-
tent with mean-field theory where the oscillation frequency is
proportionaltothemagneticdiffusioncoefficient.Themagnetic
field is antisymmetric with respect to the equator in all cases,
except for a = (1,0,0) and e0 = 0.01; see the top panel of Fig.3. Azimuthal magnetic field for e4, e0 = 0.01 in Table 1
Fig.3 withθ0 =0◦ andtheSAAcondition.
Theazimuthalmagneticfieldisstronglyconcentratedtoward
the poleswhenthe α effecthasonly the cosθ variationin lati-
tude;seethetoppanelsofFigs.3and4.Inviewoftheequatorial e0 =0.05,theequatorwarddriftbecomesmorepronouncedand
magneticfieldconcentrationintheSunandinthree-dimensional extendstolowerlatitudes(middleandbottompanelsofFig.4).
solar dynamo simulations, where the kinetic helicity is known In summary, extending the model all the way to the poles
to be strongly concentrated toward the equator (Ka¨pyla¨etal., andincludingan ηt profileconcentratedtowardthe equatorre-
2012), it is of interest to consider models with a = (0,1,0) sults in oscillatory behaviour with long cycles but no equator-
anda = (0,0,1),so that the α effectis moreconcentratedto- wardmigration.Includinganα-effectalsoconcentratedatlower
ward lower latitudes. Indications for α being stronger at lower latitudes produces equatorward cycles with shorter cycle peri-
latitudeshavebeenobserved,forexample,inmodelsofrapidly odswith strongestmagneticfieldsappearingatlowerlatitudes.
rotatingconvection(Ka¨pyla¨etal.,2006).ThevaluesforCα⋆ are Theseresultsareinqualitativeagreementwithdirectandlarge-
given in Table 1, columns for a = (0,1,0) and a = (0,0,1). eddy simulations (Ka¨pyla¨etal., 2012, 2013b; Augustsonetal.,
A similar trend as for the case where a = (1,0,0) is seen, 2015;Duarteetal.,2015).
where higher orders of e result in lower values for C⋆, in ac-
i α
cordance with lower total diffusion. Changes in the α profile
have a larger effect on C⋆ than changes in the diffusivity pro- 3.2. α2Ωdynamos
α
file. However,this is simply because, owingto the presence of
3.2.1. Overallbehaviourofdynamosolutions
the cosθ factorin the α profile,itsmaximumvalue diminishes
as higher powers of sinθ are used, while the maximum value We nowaddlarge-scaleradialshearanda dampingtermgiven
ofηt isalwaysunity,irrespectiveoftheprofile.Theoscillation by µR2/ηt and use µ˜ to denote µR2/ηt0. We first explore the
frequenciesofthesolutionsfora = (0,1,0)anda = (0,0,1) dynamo regimes and the dependencyon µ˜ by setting θ0 = 1◦
aretwoordersofmagnitudehigherthanthelow-frequencymode andonceagainusea = (1,0,0)ande = (1,0,0).Thecritical
seenfora=(1,0,0).Itturnsoutthatthemagneticfieldisthen valueC⋆ nowdependsonthevalueofµ˜;seeFig.5,blacksym-
α
moreuniformlydistributedoveralllatitudes;see Figs.3and4. bols. We now concentrate on studying the dynamo modes that
For e0 = 0.01, this distribution is largely uniform with very are excited in the system for values of µ˜ between 0 and 4 and
slightequatorwarddrift(Fig.3, middleandbottom),and when variousvaluesofCΩ.
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E.Coleetal.:Robustnessofoscillatoryα2dynamosinsphericalwedges
Fig.6. Angular frequencyω as a function of CΩ for θ0 = 1◦
(black)andθ0 =0◦ (red)forµ˜ =1.
Table 2. Cα⋆ forrunswith θ0 = 0◦, 1◦, 5◦, and15◦ with a =
(1,0,0)ande=(1,0,0)andµ˜=1.
θ0 Boundary CΩ Cα⋆
Condition
0 SAA 60 5.17
1 ASA 60 4.71
5 ASA 60 4.18
15 ASA 60 4.22
Whenµ˜ = 0,allresultingdynamosarestationary,withthe
exceptionofthecasewhereCΩ = 0whereoscillationsdepend
oninitialconditions,andCα⋆ decreasesasCΩ increases.Forso-
lutions pertaining to µ˜ = 1, two solutions exist in the regime
CΩ >∼ 33.5witheitheroscillatoryorstationarymagneticfields.
Fig.4. Azimuthal magnetic field for e4, e0 = 0.05 in Table 1 When CΩ is less than this value, we find only stationary solu-
withθ0 =0◦andtheSAAcondition. tions. Near this limit, the frequency of oscillations is sensitive
tobothCα⋆ andCΩ andevensmallchangescandoublethefre-
quency.TheC⋆ forstationarydynamosissignificantlylessthan
α
for oscillating solutions. It is possible that for µ˜ > 1 a sim-
ilar bifurcation also exists, as there always appears a jump in
C⋆ asthedynamomodechangesfromstationarytooscillatory.
α
However,atleastinthecasewithµ˜ =2,thestationarysolutions
werefoundtodisappear.Forcaseswhereµ˜ > 2,C⋆ decreases
α
withCΩ,andoscillationsonlyoccurabovecertaincriticalvalues
forCΩ.Intheregimeofnegativeshear(CΩ < 0),allsolutions
foundwereoscillatory.
We calculate the frequency ω of oscillatory solutions as in
the previous section and show the results in Fig. 6. It can be
seen that for positive shear, ω approaches 0 as CΩ → 33.55.
There also exists a jump in frequency around CΩ ∼ 70, cor-
respondingto a changein the symmetryofthe azimuthalfield.
This is demonstratedin Fig. 7 where time-latitude diagramsof
theazimuthalmagneticfieldsareshownforarepresentativese-
lection of CΩ values for models with θ0 = 1◦. The symmetry
change corresponding to the frequency jump in Fig. 6 can be
seeninthechangefromantisymmetricabouttheequator(third
Fanigd.s5t.atVioanluaerysf(ocrroCsαs⋆eass)asofluunticotinosnfoofrCθ0Ω=for1o◦s(cbillalactko)ryan(pdluθ0se=s) CpaΩne=lo8f0F)i.gT.h7e,CmΩag=ne4ti0c)fitoeldsyimsmalesotriscy(mfomuertthricpainnetlhoefoFscigil.la7-,
0◦(red). torysolutionfoundforCΩ =0.
All oscillatory solutions with positive (negative) shear
showpoleward(equatorward)migrationinaccordancewiththe
Parker–Yoshimurarule(Parker,1955;Yoshimura,1975),seethe
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E.Coleetal.:Robustnessofoscillatoryα2dynamosinsphericalwedges
Fig.7. Azimuthal magnetic field for θ0 = 1◦ with the ASA Fig.8. Azimuthal magnetic field for θ0 = 0◦ with the SAA
boundarycondition. boundarycondition.
6
E.Coleetal.:Robustnessofoscillatoryα2dynamosinsphericalwedges
Fig.9. Azimuthal magnetic field for θ0 = 0◦ with the SAA
boundarycondition.
third and fifth panels of Fig. 7, respectively, for representative
results.Thefrequencyoftheoscillationsincreaseswithgreater
CΩ in accordance with linear theory of αΩ dynamos, except
thatthere|ω|∝C1/2(e.g.Brandenburg&Subramanian,2005).
Ω
Mostofthemagneticfieldisconcentratedathighlatitudesabove
|90◦−θ| > 60◦ for cases whereCΩ is positive,Fig. 7(b)–(d).
When CΩ ≤ 0, the field is even more concentrated close to
boundaries,Fig.7(e). Fig.10. Azimuthalmagneticfieldforθ0 = 0◦ withe0 = 0.05,
µ˜ = 1,anda = e = (0,0,1)forCΩ = 40(upperpanel),and
3.2.2. Comparisonbetweenθ0 =0andθ0 =1◦ cases CΩ =−40(lowerpanel).
Themodelisnowextendedtothepolestostudythedifferences
between wedges and full spheres. The boundary condition on frequencyof oscillationsis less by abouta factorof two in the
θ0 = 0◦ is changed to comply with the regularityrequirement formercase,seeFig.6.
(SAA).We focusonthe case whereµ˜ = 1.We considera few Finally,we examinethe effectthat θ0 has on the results by
modelswithµ˜ =0and2toprobewhetherthebehaviourissimi- holdingCΩ constantanddeterminingCα⋆.Theresultsaregiven
lar,asintheθ0 =1◦case.WefindthatthevaluesofCα⋆arefairly in Table 2. We find that there is a dependency on θ0, but the
close to those obtainedfor the correspondingθ0 = 1◦ models; behaviourisconsistentifonegoestothepolesandchangesthe
seetheredsymbolsinFig.5.Similarlyasintheθ0 =1◦case,a boundarycondition;seeTable2wherethechangebetweenθ0 =
bifurcationintostationaryandoscillatorysolutionsexistsinthe 5◦and1◦iscomparabletothedifferencebetween1◦and0◦.All
positiveCΩ regimewithacut-offpointatCΩ ≈ 33.2,whichis solutionsareoscillatorywithpolewardmigration.
slightlylowerthanintheθ0 = 1◦ case.Fornegativeshear,un- Ourresultssuggestthat,atleastinthecaseswhereCΩ >0,
likeforθ0 = 1◦ whereallvaluesproduceoscillatorydynamos, asetupwithθ0 = 1◦ andperfectconductorboundarycondition
theregimeforoscillationsisfoundonlyforCΩ <∼−21.Theos- (ASA)givessimilarresultsasfullspheremodelswithθ0 = 0◦
cillatorymodegraduallydisappearsandonlyastationarymode and the regularity (SAA) condition. Furthermore,solutions for
persists,whichisshowninFig.9. 33.4 < CΩ < 75arealso fairlysimilar. Thisindicatesthatthe
Theoscillationfrequencies(Fig.6,redsymbols)aresimilar wedgesareafairapproximationoffullspheresinthisparameter
ctoasteh,oasejuimnpthiencfarseequoefnpcoysiistivoebssehrevaerd. Swimheinlatrhlyetaozitmheutθh0al=fie1ld◦ rthegeirmeseu.lItfstbheetwsheeeanrθi0s n=eg1a◦tivaen,dthθe0re=is0◦a cqausaelsit.aIttivaepcpheaanrsgethiant
changes symmetry with respect to the equator, as shown in oscillatory solutionsare obtainedonly for the ASA boundaries
Fig. 8(c) and (d) for antisymmetric(CΩ = 40) and symmetric forweaknegativeshear.
(CΩ =80)fieldconfigurations,respectively.Intheantisymmet-
ricregime,theazimuthalfieldisconcentratedatapproximately
the same latitudes as for the case θ0 = 1◦. In the symmetric 3.2.3. Varyingtheαandηt profiles
regime, i.e. for CΩ & 70, the azimuthal field extends to lower Finally, we consider changes to the turbulent magnetic diffu-
latitudes, |90◦ −θ| > 30◦; see Fig. 8(d), The main difference sivity profile. We do not perform a thorough parameter study
occurs at the boundaryitself such that for θ0 = 1◦ (ASA) the but consider a pair of cases corresponding to CΩ = ±40,
magneticfieldpeaksattheboundarywhereasitvanishesatthe a = e = (0,0,1),e0 = 0.05,µ˜ = 1,andθ0 = 0withregular-
pole for θ0 = 0 (SAA). When shear is negative, the field in- ityconditionsforthemagneticfield.Weshowthetime–latitude
steadbecomesconcentratedandsymmetricaroundtheequator, diagramsoftheazimuthalfieldfromthesemodelsinFig.10.
andinaccordancewiththeParker–Yoshimurarule,thedynamo In the case of positive shear, the combination of shear, α
hasanequatorwarddrift.Thecaseofnegativeshearresultsina andηt profiles,createsasteadymigrationpolewardatlatitudes
dramaticallydifferentconcentrationoftheazimuthalfieldwhen above ±45◦. Comparing this to an α2 dynamo with the same
comparedwith the θ0 = 1◦ counterpart;see the bottompanels profilesofαandηt (bottompanelofFig.4),andtoanα2Ωrun
ofFigs.7and8forrunswithCΩ =−40forthetwocases.Even with nosin2nθ contributionsin theprofilesbutthe same value
though the values for Cα⋆ are similar for θ0 = 0◦ and 1◦, the ofCΩ (thirdpanelofFig.8),showsthatthemigrationdirection
7
E.Coleetal.:Robustnessofoscillatoryα2dynamosinsphericalwedges
is reversed in comparison to the α2 run and that the poleward ary conditionto SAA producedqualitativelyandquantitatively
driftismorecoherentthanintheα2Ωmodel.Theseresultsindi- similarresultswhentheshearwaspositive.Resultsarelesssim-
catethatthesheardeterminesthedirectionofthedynamowave ilarifnegativeshearisintroduced.Whenθ0 = 1◦,allsolutions
inthisparameterregime.Theazimuthalfieldinbothofthecom- found where CΩ < 0 were found to oscillate. However, when
parisoncasesisantisymmetric,andthisresultalsocarriesoverto θ0 =0◦,shearhadtoexceedacriticalvalue,|CΩ|=21,forso-
thecasewhenshearisincludedwiththesameαandηt profiles. lutionsto oscillate. Furthermore,the structure of the azimuthal
Thefrequencyoftheoscillationsisω =5.54,andthecriticaldy- field over time was significantly different, showing symmetry
namoparameterisC⋆ =5.62.Thesevaluesaresomewhatclose aboutthe equator and concentrationat the equator.In all cases
α
tothevalues(ω = 3.90andC⋆ = 4.94)obtainedinSect.3.2.1 with shear,the Parker–Yoshimurarule was foundto be obeyed
α
inthecasewithmoreuniformprofilesoftheturbulenttransport where oscillatory solutions with negative shear migrated equa-
coefficients. torwardandpositiveshear,poleward.
We foundearlier that, in the case of negativeshear, the az- When combiningthe ηt profile with shear, the direction of
imuthal field was symmetric aboutthe equator;see the bottom migrationwasdeterminedbythesignofCΩ.Thefrequencyin-
panel of Fig. 8. With more equatorially concentrated turbulent creased in the case of negative shear when using an ηt and α
diffusivity and α profiles we also find solutions with equato- profilewithhigherorderterms.
rial symmetry, see the bottom panel of Fig. 10. Furthermore, There are other possibilities for refining the model and for
the magnetic field now has a minimum aroundlatitudes ±25◦. obtainingoscillatorysolutionstotheα2 dynamo.Onepossibil-
The Parker–Yoshimura rule still holds true, and the migration ityistostudytheeffectofdecreasingthe(microphysical)mag-
is equatorward.However,C⋆ has almostdoubledfrom 5.62 to neticdiffusivityevenfurther.Anotherpossibilityistostudythe
α
10.75, and the frequency of oscillations is much larger, ω = memoryeffect,whichhasrecentlybeenidentifiedasameansto
14.56 in comparison to 5.54. The main effect from the more facilitateoscillatorybehaviour,althoughsofaronlydecayingso-
concentratedprofilesforα and ηt in thecase ofα2Ω dynamos lutionshavebeenfoundtobemodifiedinthatway(Devlenetal.,
isseeninthelatitudinalprofileoftheresultingmagneticfields, 2013). However, under suitable conditions such solutions can
butthequalitativecharacterofthesolutionsremainsunchanged indeed become oscillatory (Rheinhardtetal., 2014) and may
incomparisontomodelswithsimplerlatitudedependenceofthe present a possible solution to the problem where equatorward
turbulenttransportcoefficients. motionobtainedviavaryingtheαprofile(i.e.,a = (0,0,1))is
limitedtocertainlatitudes.
4. Conclusions
Acknowledgements. TheauthorsthankNorditaforhospitalityduringtheirvis-
its.FinancialsupportfromtheVilho,Yrjo¨ andKalleVa¨isa¨la¨Foundation(EC),
Motivatedby earlierresultsofglobalsimulationsin wedgege-
theAcademy ofFinland grants No.136189, 140970 (PJK)andthe Academy
ometry,we have studied the robustness of oscillatory solutions ofFinlandCentreofExcellenceReSoLVE(272157;MJKandPJK),aswellas
in α2 dynamos in simple one-dimensional mean-field dynamo theSwedishResearch Council grants 621-2011-5076 and2012-5797, andthe
models.We foundthattheboundaryconditionson thelatitudi- EuropeanResearchCouncil undertheAstroDynResearchProject 227952are
nalboundariesplayamajorroleintherealisedsolutionsforα2 acknowledged. Weacknowledge CSC – IT Center for Science Ltd., who are
administeredbytheFinnishMinistryofEducation,fortheallocationofcompu-
dynamoswithasimplecosθprofileforαandconstantturbulent
tationalresources.
diffusivity.Imposing the perfectconductorboundarycondition
creates oscillating solutions only for solutions where θ0 >∼ 1◦.
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