Convective Quenching of Field Reversals in Accretion Disc Dynamos

4.7 MB·

by  Matthew S. B. Coleman

Mon.Not.R.Astron.Soc.000,1–12(2016) Printed31January2017 (MNLATEXstylefilev2.2)
Convective Quenching of Field Reversals in Accretion Disc
Dynamos
Matthew S. B. Coleman1 , Evan Yerger1,2, Omer Blaes1,3, Greg Salvesen1 , and
∗ †
Shigenobu Hirose4
1 Department of Physics, University of California, Santa Barbara, CA 93106, USA
2 Department of Astrophysical Sciences, Program in Plasma Physics, Princeton, NJ 08540, USA
3 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
4 DepartmentofMathematicalScienceandAdvancedTechnology,JapanAgencyforMarine-EarthScienceandTechnology,Yokohama,
Kanagawa 236-0001, Japan
7
1 Accepted—.Received—;inoriginalform—
0
2
ABSTRACT
n
Vertically stratified shearing box simulations of magnetorotational turbulence com-
a
J monlyexhibitaso-calledbutterflydiagramofquasi-periodicazimuthalfieldreversals.
However, in the presence of hydrodynamic convection, field reversals no longer occur.
7
2 Instead, the azimuthal field strength fluctuates quasi-periodically while maintaining
the same polarity, which can either be symmetric or antisymmetric about the disc

] midplane. Using data from the simulations of Hirose et al. (2014), we demonstrate
E
that the lack of field reversals in the presence of convection is due to hydrodynamic
H mixing of magnetic field from the more strongly magnetized upper layers into the
. midplane,whichthenannihilatefieldreversalsthatarestartingthere.Ourconvective
h
simulations differ in several respects from those reported in previous work by others,
p
in which stronger magnetization likely plays a more important role than convection.
-
o
Key words: accretion, accretion discs, dynamos, convection — MHD — turbulence
r
t — stars: dwarf novae.
s
a
[

1
v 1 INTRODUCTION previously had trouble with matching observations (King
7 etal.2007;Kotko&Lasota2012).Thisisbecausehydrody-
As a cloud of gas contracts under the influence of gravity,
7 namicconvectionoccursinthevicinityofthehydrogenion-
it is likely to reach a point where net rotation dominates
1 ization regime, and enhances the time-averaged Shakura &
8 the dynamics and becomes a bottleneck restricting further Sunyaev (1973) alpha-parameter (Hirose et al. 2014; Hirose
0 collapse. This scenario naturally leads to a disc structure,
2015). Enhancement of MRI turbulent stresses by convec-
. thus explaining why accretion discs are so prevalent in as-
1 tion was also independently claimed by Bodo et al. (2012,
trophysics.Thequestionofhowthesediscstransportangu-
0 2015). Whether this enhancement is enough to reproduce
lar momentum to facilitate accretion still remains. For suf-
7 observed dwarf nova light curves remains an unanswered
1 ficiently electrically conducting discs, there is a reasonable question, largely due to uncertain physics in the quiescent
: consensus that the magnetorotational instability (MRI) is
v state and in the propagation of heating and cooling fronts
thepredominatemeansoftransportingangularmomentum,
i (Coleman et al. 2016).
X at least on local scales (Balbus & Hawley 1998). There has
alsobeensignificantworkonunderstandingnon-localmech- The precise reason as to why convection enhances the
r
a anismsofangularmomentumtransport,suchasspiralwaves turbulentstressesresponsibleforangularmomentumtrans-
(Juetal.2016).Whiletheseglobalstructuresmaybeimpor- port is still not fully understood. Bodo et al. (2012, 2013)
tantindiscswithlowconductivity,localMRIsimulationsof conducted simulations with fixed thermal diffusivity and
fully ionized accretion discs produce values of α consistent impenetrable vertical boundary conditions, and found that
with those inferred from observations for accretion discs in when this diffusivity was low, the time and horizontally-
binarysystems.Thisiseventrueforthecaseofdwarfnovae, averaged vertical density profiles became very flat or even
forwhichMRIsimulationslackingnetverticalmagneticflux slightly inverted, possibly due to hydrodynamic convection
taking place. They suggested that either these flat profiles,
or the overturning convective motions themselves, might
makethemagnetohydrodynamicdynamomoreefficient.Hi-
E-mail:[email protected]
∗ NSFAstronomy&AstrophysicsPostdoctoralFellow. roseetal.(2014)usedradiationMHDsimulationswithout-
†
c 2016RAS
(cid:13)
2 M. S. B. Coleman et al.
flow vertical boundary conditions, and the hydrogen ion- This paper is organized as follows. In Section 2 we dis-
ization opacities and equation of state that are relevant to cussthebutterflydiagramindetailandhowitchangeschar-
dwarfnovae.Theyfoundintermittentepisodesofconvection acterwhenconvectionoccurs.InSection3wedescribemag-
separated by periods of radiative diffusion. The beginning neticbuoyancyandtheroleitplaysinestablishingthebut-
oftheconvectiveepisodeswereassociatedwithanenhance- terflydiagram,andtherelatedthermodynamics.Weexplain
mentofenergyinverticalmagneticfieldrelativetothehor- how convection acts to alter these effects in Section 4. The
izontal magnetic energy, and this was then followed by a implicationsofthisworkarediscussedinSection5,andour
rapid growth of horizontal magnetic energy. These authors results are summarized in Section 6.
therefore suggested that convection seeds the axisymmetric
magnetorotationalinstability,albeitinamediumthatisal-
ready turbulent. In addition, the phase lag between stress 2 THE BUTTERFLY DIAGRAM
build up and heating which causes pressure to build up
To construct the butterfly diagram and explore its physical
also contributes to an enhancement of the alpha parame-
origin,itisusefultodefinethefollowingquantitiesrelatedto
ter.Themerepresenceofverticalhydrodynamicconvection
somefluidvariablef:thehorizontalaverageofthisquantity,
is not sufficient to enhance the alpha parameter, however;
the variation with respect to this horizontal average, and a
the Mach number of the convective motions also has to be
version of the variable that is smoothed in time over one
sufficientlyhigh,andinfactthealphaparameterappearsto
orbit. These are defined respectively by
be better correlated with the Mach number of the convec-
tive motions than with the fraction of heat transport that 1 (cid:90) Lx/2 (cid:90) Ly/2
is carried by convection (Hirose 2015). (cid:104)f(cid:105)(t,z)≡ dx dyf(t,x,y,z) (1a)
L L
Hydrodynamicconvectiondoesnotsimplyenhancethe x y −Lx/2 −Ly/2
δf ≡f −(cid:104)f(cid:105), (1b)
turbulentstresstopressureratio,however.Italsofundamen-
tally alters the character of the MRI dynamo. In the stan- (cid:90) t+1/2 (cid:44)
dardweak-fieldMRI,verticallystratifiedshearingboxsimu- {f}t(t)≡ f(t(cid:48))dt(cid:48) 1 orbit. (1c)
lationsexhibitquasi-periodicfieldreversalsoftheazimuthal t−1/2
magnetic field (B ) with periods of ∼ 10 orbits (Branden- Here L , L , and L are the radial, azimuthal and vertical
y x y z
burg et al. 1995; Davis et al. 2010). These reversals start extentsofthesimulationdomain,respectively(listedinTa-
near the midplane and propagate outward making a pat- ble1).Additionallywedefinethequantityf asameans
conv
tern (see top-left panel of Figure 1 below) which resembles to estimate the fraction of vertical energy transport which
a time inverse of the solar sunspot butterfly diagram. The is done by convection:
means by which these field reversals propagate away from (cid:26)(cid:82) {(cid:104)(e+E)v (cid:105)} sign(z)(cid:104)P (cid:105)dz(cid:27)
themidplaneislikelythebuoyantadvectionofmagneticflux fconv(t)≡ (cid:82) {(cid:104)F (cid:105)}z sitgn(z)(cid:104)P (cid:105)thdz , (2)
tubes(Gressel2010),andmanystudieshavealsosuggested tot,z t th t
thatmagneticbuoyancyisimportantinaccretiondiscs(e.g. where e is the gas internal energy density, E is the radi-
Galeev et al. 1979; Brandenburg & Schmitt 1998; Miller & ation energy density, v is the vertical velocity, P is the
z th
Stone2000;Hiroseetal.2006;Davisetal.2010;Blaesetal. thermalpressure(gasplusradiation),andF isthetotal
tot,z
2011). Magnetic buoyancy is consistent with the Poynting energyfluxintheverticaldirection,includingPoyntingand
fluxwhichtendstobeorientedoutwards(seetop-rightpanel radiationdiffusionflux.Thesequantitieswillassistusinan-
ofFig.1),andwegivefurtherevidencesupportingthisthe- alyzing and discussing the interactions between convection
orybelow.Whilethisexplainshowfieldreversalspropagate and dynamos in accretion discs.
through the disc, it does not explain how these magnetic The butterfly diagram is obtained by plotting (cid:104)B (cid:105) as
y
field reversals occur in the first place, and despite numer- afunctionoftimeanddistancefromthediscmidplane(see
ous dynamo models there is currently no consensus on the leftframesofFig.1).Theradiativesimulationws0429(from
physicalmechanismdrivingthereversals(e.g.Brandenburg Hiroseetal.2014,andlistedinTable1)showsthestandard
et al. 1995; Gressel 2010; Shi et al. 2010, 2016; Squire & pattern of field reversals normally associated with the but-
Bhattacharjee 2015). terfly diagram, which appear to start at the midplane and
However, in the presence of convection, the standard propagateoutwards.Thisoutwardpropagationofmagnetic
pattern of azimuthal field reversals is disrupted. Periods of fieldisconsistentwiththePoyntingflux(alsoshowninFig.
convectionappeartobecharacterizedbylongertermmain- 1),whichgenerallypointsoutwardsawayfromthemidplane.
tenanceofaparticularazimuthalfieldpolarity,andthisper- When simulations with convection are examined (e.g.
sistent polarity can be of even (Bodo et al. 2015) or odd ws0446 listed in Table 1), however, the butterfly diagram
parity with respect to the disk midplane. As we discuss in looks completely different (see bottom left frame of Fig. 1),
this paper, the simulations of Hirose et al. (2014) also ex- as first discussed by Bodo et al. (2015). Similar to the lack
hibitthispatternofpersistentmagneticpolarityduringthe oftheazimuthalmagneticfieldreversalsfoundbytheseau-
intermittentperiodsofconvection,butthefieldreversalsas- thors,wefindthatwhenconvectionispresentintheHirose
sociated with the standard butterfly diagram return during et al. (2014) simulations, there is also a lack of field re-
theepisodesofradiativediffusion(seeFig.1below).Herewe versals. Additionally, we find that the azimuthal magnetic
exploitthisintermittencytotryandunderstandthecauseof field in the high altitude “wings” of the butterfly diagram
the persistent magnetic polarity in the convective episodes. is better characterized by quasi-periodic pulsations, rather
We demonstrate that this is due to hydrodynamic mixing than quasi-periodic field reversals. These pulsations have
of magnetic field from strongly magnetized regions at high roughlythesameperiodasthefieldreversalsfoundinradia-
altitude back toward the midplane. tive epochs. For example, the convective simulation ws0446
c 2016RAS,MNRAS000,1–12
(cid:13)
Convective Quenching of Field Reversals 3
ws0429 103 ws0429 1011
2 ×2.0 2 ×6.0
m 1.5 m 4.5
910c× 1 01..50 910c× 1 13..50 Flux
.Height/141−10 −−−0.0110...505By .Height/141−10 −−−0.0431...505Poynting
2.0 6.0
2 − 2 −
− 0 50 100 150 − 0 50 100 150
Time(Orbits) Time(Orbits)
ws0446 ws0446 1010
m 2 fconv 680000 m 2 fconv ×46..50
810c× 1 240000 810c× 1 13..50 Flux
.Height/951−210 −−−−0864200000000By .Height/951−210 −−−−0.06431....0505Poynting
− 0 20 40 60 80 100 120 140 160 − 0 20 40 60 80 100 120 140 160
Time(Orbits) Time(Orbits)
Figure1.Horizontally-averagedazimuthalmagneticfield By (leftframes),and1orbitsmoothedhorizontally-averagedverticalPoynt-
(cid:8)(cid:10) (cid:11)(cid:9) (cid:104) (cid:105)
ingflux FPoynt,z t (rightframes)asafunctionoftimeandheightforaradiativesimulation(ws0429,topframes)andasimulation
which exhibits convective epochs (ws0446, bottom frames). For the Poynting flux frames, we have purposely allowed some of the data
to lie outside the colorbar range (which then shows up as saturated regions) in order to increase the color contrast in the midplane
regions. The normalizations indicated in the vertical axes are the respective simulation length units. The dashed black lines show the
time-dependent heights of the photospheres in the horizontally averaged structures. For simulation ws0446, which exhibits convection,
theconvectivefractionfconv 2(seeEqn.2)isplottedinmagenta.Notethatfconv 2usesthesameverticalscaleasBy,i.e.whenthe
− −
magenta line is near 1 then fconv 1. Focusing on By, the radiative simulation ws0429 shows the standard pattern of field reversals
− ≈
normallyassociatedwiththebutterflydiagram.Insimulationws0446wherefconv ishigh,thefieldmaintainsitssignandallchangesin
sign/parityareassociatedwithadipinfconv.
Simulation h0 Σ Teff α Nx Ny Nz Lx/h0 Ly/h0 Lz/h0 Lz/hp tth
ws0446(conv) 9.51E+08 127 7490 0.1062 32 64 256 0.500 2.000 4.000 12.0 5.06
ws0429(rad) 1.41E+09 1030 13352 0.0332 32 64 256 0.500 2.000 4.000 8.54 9.49
Table 1. Simulation parameters for the convective simulation ws0446 and radiative simulation ws0429. The units of time averaged
surfacedensities(Σ),effectivetemperatures(Teff),height(h0),andthermaltime(tth)are,respectively,gcm−2,K,cm,andorbits.Lx,
Ly, and Lz are the lengths, and Nx, Ny, and Nz are the numbers of cells, in the x, y, and in z directions, respectively. The pressure
(cid:82)
scaleheightofthesteadystateiscomputedashp [ pthermal ]dz/2max([ pthermal ]).
≡ (cid:104) (cid:105) (cid:104) (cid:105)
showninFig.1hasaradiativeepochwherefieldreversalsoc- toorbits70-100inthebottomleftpanelofFig.1).Thisphe-
cur(centerednear55orbits),andthebehaviorofthisepoch nomenon of the parity of B being held fixed throughout
y
resemblesthatoftheradiativesimulationws0429.However, a convective epoch shall henceforth be referred to as par-
duringconvectiveepochswheref ishigh,thefieldmain- ity locking.Duringevenparityepochs(e.g.orbits10-40and
conv
tains its polarity and pulsates with a period of ∼10 orbits. 120-140ofws0446),therearefieldreversalsinthemidplane,
Infact,theonlytimefieldreversalsoccuriswhenf dips buttheyarequicklyquenched,andwhatfieldconcentrations
conv
to low values1, indicating that radiative diffusion is domi- are generated here do not migrate away from the midplane
nating convection. astheydoduringradiativeepochs.Alsoduringevenparity
This lack of field reversals during convective epochs convective epochs, the Poynting flux tends to be oriented
locks the vertical structure of B into either an even parity inwardsroughlyhalfwaybetweenthephotospheresandthe
y
oroddparitystate,whereB maintainssignacrossthemid- midplane. For odd parity convective epochs the behavior of
y
planeoritchangessign,respectively.(Compareorbits10-40 thePoyntingfluxismorecomplicatedbutislikelylinkedto
the motion of the (cid:104)B (cid:105)=0 surface.
y
In summary, we seek to explain the following ways in
which convection alters the butterfly diagram:
1 AsdiscussedinHiroseetal.(2014)fconv canbeslightlynega-
tive.Thiscanhappenwhenenergyisbeingadvectedinwardsto (i) Magneticfieldreversalsnearthemidplanearequickly
thediscmidplane. quenched during convective epochs.
c 2016RAS,MNRAS000,1–12
(cid:13)
4 M. S. B. Coleman et al.
therefore never present, temperature variations (δT) at a
ws0429 given height are rapidly suppressed by radiative diffusion.
0.6 102 Thiscauseshorizontalvariationsingaspressure(δPgas)and
mass density (δρ) to be highly correlated (see Fig. 2), and
allowsustosimplifyouranalysisbyassumingδT =0.This
0.4 should be contrasted with convective simulations (see Fig.
101
3)whichshowamuchnoisierrelationandshowatendency
0.2 towards adiabatic fluctuations during convective epochs.
asi 100 By computing rough estimates of the thermal time we
Pg can see how isothermal and adiabatic behaviour arise for
/h 0.0 radiativeandconvectiveepochs,respectively.Thetimescale
Pgas 10−1 to smooth out temperature fluctuations over a length scale
δ ∆L is simply the photon diffusion time times the ratio of
0.2
− gas internal energy density e to photon energy density E,
10 2
− 3κ ρ(∆L)2 e
0.4 t (cid:39) R . (3)
− th c E
10 3 Forthemidplaneregionsoftheradiativesimulationws0429
0.6 − at times 75−100 orbits, the density ρ (cid:39) 7×10−7g cm−3,
−
0.6 0.4 0.2 0.0 0.2 0.4 e (cid:39) 2 × 107 erg cm−3, E (cid:39) 9 × 105 erg cm−3, and
− − −
δρ/ ρ the Rosseland mean opacity κR (cid:39) 10 cm2 g−1. Hence,
h i t (cid:39) 30(∆L/H)2 orbits. Radiative diffusion is therefore
th
extremely fast in smoothing out temperature fluctuations
Figure 2. A 2D histogram of horizontal variations (see Eqn. on scales of order several tenths of a scale height, and thus
1b)ofmassdensity(δρ,horizontalaxis)andgaspressure(δPgas, horizontal fluctuations are roughly isothermal.
verticalaxis)forthefullyradiativesimulationws0429.Eachgrid
Isothermality (T = (cid:104)T(cid:105)) in combination with pressure
zone saved to file from this simulation with 90 (cid:54) t (cid:54) 190 and
equilibrium (P =(cid:104)P (cid:105)) leads to the following equation:
z (cid:54) 0.5 simulation units is placed into one of 100 100 bins tot tot
| | × ρ
basedonitslocalproperties.Theresultingnormalizedprobabil- (cid:104)P (cid:105)= k(cid:104)T(cid:105)+P , (4)
itydensityisshownherewiththegivencolorbar.Theprobability tot µmp mag
(cid:82)
density,p,isnormalizedsothat p(x˜,y˜)dx˜dy˜=1wherex˜andy˜
whereradiationpressurehasbeenneglected,asP (cid:28)P .
arethevariablesusedforthehorizontalandverticalaxes,respec- rad gas
Thus, during radiative epochs, it is clear that regions of
tively.ThetightlinearcorrelationbetweenδρandδPgas signifies
highlyconcentratedmagneticfield(e.g.fluxtubes)mustbe
that temperature variations are small at any fixed height, i.e.
under-dense.Figure4confirmsthisfortheradiativesimula-
horizontalfluctuationsareisothermal.
tion ws0429 by depicting a 2D histogram of magnetic pres-
sureanddensityfluctuations.Aclearanticorrelationisseen
(ii) Magnetic field concentrations do not migrate away which extends up to very nonlinear concentrations of mag-
fromthemidplaneduringconvectiveepochsastheydodur- neticfield,allofwhichareunderdense.Thisanticorrelation
ing radiative epochs. was also observed in radiation pressure dominated simula-
(iii) During convective epochs, the magnetic field in the tions appropriate for high luminosity black hole accretion
wings of the butterfly diagram is better characterized by discs in Blaes et al. (2011). This anti-correlation causes the
quasi-periodicpulsations,ratherthanquasi-periodicfieldre- buoyant rise of magnetic field which would explain the out-
versals, with roughly the same period. ward propagation seen in the butterfly diagram and is also
(iv) By isheldfixedineitheranoddorevenparitystate consistentwiththeverticallyoutwardPoyntingflux(seetop
during convective epochs. panels of Figure 1).
Ontheotherhand,forthemidplaneregionsofthecon-
vective simulation ws0446 at the times 80−100 orbits, ρ(cid:39)
2×10−7gcm−3,e(cid:39)3×106ergcm−3,E (cid:39)1×103ergcm−3,
3 THERMODYNAMICS AND MAGNETIC
and κ (cid:39) 7×102 cm2 g−1. Hence, t (cid:39) 4×104(∆L/H)2
BUOYANCY R th
orbits. All fluctuations in the midplane regions that are re-
Much like in Blaes et al. (2011), we find that that during solvable by the simulation are therefore roughly adiabatic.
radiative epochs,nonlinear concentrations ofmagnetic field Perhaps somewhat coincidentally, Γ ≈1.3 in the midplane
1
form in the midplane regions, and these concentrations are regionsoftheconvectivesimulation,sothepressure-density
underdense and therefore buoyant. The resulting upward fluctuations, even though adiabatic, are in any case close
motionofthesefieldconcentrationsisthelikelycauseofthe to an isothermal relationship2. However, the biggest differ-
verticallyoutwardmovingfieldpatternobservedinthestan- ence between the radiative and convective cases is caused
dard butterfly diagram. In our simulations, this magnetic
buoyancyappearstobemoreimportantwhenradiativedif-
2 This reduction in the adiabatic gradient within the hydrogen
fusion, rather than convection, is the predominate energy
ionization transition actually contributes significantly to estab-
transport process. This is due to the different opacities and
lishingaconvectivelyunstablesituationinourdwarfnovasimu-
ratesofradiativediffusionbetweenthesetworegimes,which
lations.Wetypicallyfindthattheadiabatictemperaturegradient
alter the thermodynamic conditions of the plasma. adwithinthehydrogenionizationtransitionissignificantlyless
∇
When the disc is not overly opaque, and convection is thanthevalue0.4foramonatomicgas.Infact,thegaspressure
c 2016RAS,MNRAS000,1–12
(cid:13)
Convective Quenching of Field Reversals 5
ws0446
Radiative Convective
2.0 2.0 101
1.5 1.5
100
i
as 1.0 1.0
g
P
/h 10−1
as 0.5 0.5
g
P
δ
0.0 0.0 10−2
0.5 0.5
− − 10−3
0 1 2 3 4 0 1 2 3 4
δρ/ ρ δρ/ ρ
h i h i
Figure 3. 2D histograms of horizontal variations of mass density (δρ, horizontal axis) and gas pressure (δPgas, vertical axis) in the
convective simulation ws0446. The left panel shows data from a radiative epoch (49 (cid:54) t (cid:54) 64), and the right panel shows data from
a convective epoch (70(cid:54)t(cid:54)100). The two dotted black lines in each figure show the expected relations for isothermal (slope 1) and
monatomicadiabatic(slope5/3)gases.DuringtheradiativeepochthereisasignificantpopulationofcellsthatlieonthelineδPgas=δρ
justasinFig.2.Howeverthereisalsoasignificantamountofdispersionwhichresemblestheprobabilitydistributionoftheconvective
epoch, therefore it is likely that this dispersion is a remnant of the previous convective epoch. During the convective epoch the yellow
core of the probability distribution follows a linear trend with a slope steeper than 1. This is indicative of perturbations which are
adiabaticandthespreadinslopesoverthisyellowregionarelikelyduetothevariationinΓ1 (∂lnPgas/∂lnρ)s fromourequationof
≡
state, which includes in particular the effects of ionizing hydrogen. Values of Γ1 range from about 1.15 to 5/3 in this simulation. The
populationbelowδPgas=δρislikelyaresultofverticalmixing.
by the departure from isothermality in convective epochs, overdense material from the wing down towards the mid-
allowing for the possibility of highly magnetized regions to plane. As is typical of stratified shearing box simulations
beoverdense.Thisleadstomuchlargerscatterintheproba- of MRI turbulence (e.g. Miller & Stone 2000; Krolik et al.
bilitydistributionofthedensityperturbationsinconvective 2007), the horizontally and time-averaged magnetic energy
epochs.Howthisaffectsmagneticbuoyancyinradiativeand densitypeaksawayfromthemidplane,andthesurfacepho-
convectiveepochswillbediscussedindetailinthenextsec- tospheric regions are magnetically dominated. Dense fluid
tion. parcels that sink down toward the midplane are therefore
likelytocarrysignificantmagneticfieldinward.Thesefluid
parcels that originated from high altitude can actually be
identifiedinthesimulationsbecausethehighopacity,which
4 EFFECTS OF CONVECTION
contributes to the onset of convection, prevents cold fluid
In this section we lay out the main mechanisms by which parcels from efficiently thermalizing with their local sur-
convection acts to modify the dynamics of the dynamo and roundings. Hence they retain a lower specific entropy com-
thereby fundamentally alter the large scale magnetic field paredtotheirsurroundingsastheyarebroughttothemid-
structure in the simulations. plane by convective motions. We therefore expect negative
specific entropy fluctuations in the midplane regions to be
correlated with high azimuthal magnetic field strength of
4.1 Mixing from the Wings thesamepolarityasthephotosphericregionsduringacon-
vective epoch. Figures 5 and 6 show that this is indeed the
As a convective cell brings warm underdense plasma from case.
themidplaneoutwardtowardsthephotosphere(i.e.thewing
region of the butterfly diagram), it must also circulate cold Figure 5 shows a 2D histogram of entropy fluctuations
andazimuthalfieldstrengthB inthemidplaneregionsfor
y
the even parity convective epoch 15(cid:54)t(cid:54)40 in simulation
ws0446(seeFig.1).Theyellowverticalpopulationisindica-
weightedaveragevalueof ad canbeaslowas0.18.For 60%
∇ ∼ tive of adiabatic fluctuations (i.e. δs = 0) at every height
oftheconvectivesimulationsofHiroseetal.(2014)andColeman
et al. (2016), the temperature gradient is superadiabatic but which are largely uncorrelated with By. However, the up-
∇
lessthan0.4duringconvectiveepochs. per left quadrant of this figure shows a significant excess of
c 2016RAS,MNRAS000,1–12
(cid:13)
6 M. S. B. Coleman et al.
byconsideringthelineartheoryofisothermal,isobaricfluc-
ws0429 tuations at a particular height. Such fluctuations have per-
101 turbations in entropy given by
15
k B2−(cid:10)B2(cid:11)
δs= . (5)
100 µmp 2(cid:104)Pgas(cid:105)
This is shown as the dotted line in Figure 7, and fits the
magi10 10−1 obserTvheedicnowrraerldatfliounxvoefrmyawgenlle.tic energy from high altitude
P is also energetically large enough to quench field reversals
h
/mag 10−2 itnhethdeivmerigdepnlaceneofretghieonPso.yTnotidnegmflounxstarnadtectohmisp,awreedexitamtointehde
P
δ 5 time derivative of the magnetic pressure. During radiative
10 3 epochswhenthemagneticfieldisgrowingafterafieldrever-
−
sal,typicalvaluesford(cid:104)P (cid:105)/dtinthemidplaneareabout
mag
half of the typical value of −d(cid:104)F (cid:105)/dz near the mid-
Poynt,z
10 4 plane during convective epochs. This shows that the mag-
0 −
netic energy being transported by the Poynting flux during
0.6 0.4 0.2 0.0 0.2 0.4 convectiveepochsisstrongenoughtoquenchthefieldrever-
− − − sals that would otherwise exist. The sign of the divergence
δρ/ ρ
h i ofthePoyntingfluxduringconvectiveepochsisalsoconsis-
tentwithmagneticenergybeingremovedfromhighaltitude
Figure 4. A2Dhistogramofhorizontalvariationsofmassden- (positive) and deposited in the midplane (negative).
sity(δρ,horizontalaxis)andmagneticpressure(δPmag,vertical To conclude, by using specific entropy as a proxy for
axis), using data from the same vertical and temporal ranges as whereafluidparcelwaslastinthermalequilibrium,wehave
Fig.2.Theroughanti-correlationbetweenδρandδPmagsignifies shown that convection advects field inward from high alti-
that highly magnetized regions are likely to be underdense and tude,whichisconsistentwiththeinwardPoyntingfluxseen
thusbuoyant. during even parity convective epochs (see Fig. 1). The lack
of such clear inward Poynting flux during the odd parity
convective epochs is likely related to the movement of the
cells with lower than average entropy for their height and (cid:104)B (cid:105) = 0 surface by convective overshoot across the disc
y
large positive By. It is important to note that this corre- midplane.However,inbothevenandoddconvectiveepochs
sponds to the sign of the azimuthal field at high altitude, d(cid:104)F (cid:105)/dz is typically a few times d(cid:104)P (cid:105)/dt and is
Poynt,z mag
even though near the midplane (cid:104)By(cid:105) is often negative (see consistent with enough magnetic energy being deposited in
bottompanelofFig.1).Thisisstrongevidencethatconvec- themidplanetoquenchfieldreversalsthatwouldotherwise
tion is advecting low entropy magnetized fluid parcels from take place. This further suggests that regardless of parity,
the near-photosphere regions into the midplane. thisconvectivemixingfromhighaltitudetothemidplaneis
Figure 6 shows the same thing for the odd parity con- quenching field reversals in the midplane by mixing in field
vective epoch 80(cid:54)t(cid:54)100. Because the overall sign of the of a consistent polarity.
horizontally-averaged azimuthal magnetic field flips across
the midplane, cells that are near but above the midplane
areshownintheleftpanel,whilecellsthatarenearbutbe-
4.2 Disruption of Magnetic Buoyancy
low the midplane are shown on the right. Like in Figure 5,
there is a significant excess of negative entropy fluctuations In addition to quenching magnetic field reversals, convec-
that are correlated with azimuthal field strength and have tion and the associated high opacities act to disrupt mag-
the same sign as the field at higher altitudes on the same netic buoyancy which transports field away from the mid-
sideofthemidplane.Again,theselowentropyregionsrepre- plane,therebypreventinganyreversalswhichdooccur,from
sent fluid parcels that have advected magnetic field inward propagating vertically outwards. The large opacities which
from higher altitude. The correlation between negative en- contribute to the onset of convection also allow for ther-
tropy fluctuation and azimuthal field strength is somewhat mal fluctuations on a given horizontal slice (δT) to per-
weaker than in the even parity case (Fig. 5), but that is sist for several orbits. This breaks one of the approxima-
almost certainly due to the fact that inward moving fluid tions which lead to the formulation of Eqn. 4, allowing for
elements can overshoot across the midplane. the possibility for large magnetic pressures to be counter-
In contrast, Figure 7 shows a similar histogram of balancedbylowtemperatures,reducingtheanti-correlation
entropy fluctuations and B for the radiative simulation between density and temperature. Additionally, convective
y
ws0429, and it completely lacks this correlation between turbulencealsogeneratesdensityperturbationsthatareun-
high azimuthal field strength and negative entropy fluctu- correlatedwithmagneticfields,andcombinedwiththelack
ation. This is in part due to the fact that fluid parcels are of isothermality can cause fluid parcels to have both a high
no longer adiabatic, but isothermal. But more importantly, magneticpressureandbeoverdense(seeFig.8).Theseover-
it isbecausethere isno mixingfromthe highlymagnetized denseover-magnetizedregionscanbeseenintheright(con-
regions at high altitude down to the midplane. Instead, the vective) frame of Fig. 8 where the probability density at
tight crescent shaped correlation of Figure 7 arises simply δρ ≈ 0.5(cid:104)ρ(cid:105), δP ≈ (cid:104)P (cid:105) is only about one order
mag mag
c 2016RAS,MNRAS000,1–12
(cid:13)
Convective Quenching of Field Reversals 7
we provided any explanation for the quasi-periodicities ob-
103 ws0446 served in both the field reversals of the standard diagram
1.0 ×
10 2 and the pulsations that we observe at high altitude dur-
−
ing convective epochs. However, the outward moving pat-
terns in the standard butterfly wings combined with the
factthatthehorizontally-averagedPoyntingfluxisdirected
0.5 outward strongly suggests that field reversals are driven
10−3 in the midplane first and then propagate out by magnetic
buoyancy. On the other hand, we continue to see the same
quasi-periodicity at high altitude in convective epochs as
y
B we do in the field reversals in the radiative epochs. More-
0.0
over,itisclearfromFigure1thatfieldreversalsoccasionally
10 4
− start to manifest in the midplane regions during convective
epochs, but they simply cannot be sustained because they
areannihilatedbyinwardadvectionofmagneticfieldofsus-
0.5
tained polarity. This suggests perhaps that there are two
−
10 5 spatiallyseparateddynamoswhichareoperating.Thismod-
−
ification of the dynamo by convection presents a challenge
todynamomodels.However,potentiallypromisingdynamo
0.3 0.2 0.1 0.0 0.1 0.2 0.3
− − − mechanisms have recently been discovered in stratified and
δs/ s unstratified shearing boxes.
h i
Recently,Shietal.(2016)foundthatquasi-periodicaz-
imuthal field reversals occur even in unstratified, zero net
Figure 5. A 2D histogram of azimuthal magnetic field (By, fluxshearingboxsimulations,providedtheyaresufficiently
vertical axis) and horizontal variations of specific entropy (δs,
tall (L /L (cid:62)2.5). Furthermore, they found that the mag-
horizontal axis) for an even parity convective epoch. Each cell z x
savedtofilefromws0446with15(cid:54)t(cid:54)40and z (cid:54)0.25simula- neticshear-currenteffect(discussedinSquire&Bhattachar-
tion units is placed into one of 100 100 bins b|a|sed on its local jee 2015) was responsible for this dynamo; however, they
properties.Thelightgreenregionat×lowentropy( 0.2(cid:46)δs<0) were not able to explain why the reversals occurred. The
and high magnetic field (By (cid:38) 500) signifies high−ly magnetized shear-current effect can also apparently be present during
fluid parcels which have been mixed towards the midplane by hydrodynamic convection (Rogachevskii & Kleeorin 2007),
convection. implying that this dynamo mechanism might be capable of
persisting through convective epochs.
Most of the work on the MRI dynamo has been done
of magnitude below its peak value. This should be con-
with vertically stratified shearing box simulations. One of
trasted with the left (radiative) frame of the same figure
the earliest examples of this is Blackman & Tan (2004),
and with Fig. 4 where the probability density at this co-
who found that multiple dynamos can work in conjunction
ordinate is very small or zero respectively. Hence while the
withtheMRIondifferentscales.Thisisconsistentwiththe
overallanti-correlationbetweenmagneticpressureandden-
findingsofGressel(2010),thatan“indirect”largerscaledy-
sity still exists in convective epochs, indicating some mag-
namo should coexist with the MRI, and they propose two
neticbuoyancy,thecorrelationisweakenedbythepresence
candidates: a Parker-type dynamo (i.e. the α-effect; Parker
ofoverdensehighmagneticfieldregions.Magneticbuoyancy
1955; Ruediger & Kichatinov 1993), and a “buoyant” dy-
is therefore weakened compared to radiative epochs.
namo caused by the Lorentz force. Furthermore, Gressel
& Pessah (2015) find that the αΩ dynamo produces cy-
clefrequenciescomparabletothatofthebutterflydiagram,
4.3 Parity Locking
andthatthereisanon-localrelationbetweenelectromotive
The effects described above both prevent magnetic field re- forces and the mean magnetic field which varies vertically
versalsinthemidplaneandreducethetendencyforanyre- throughout the disc. This sort of non-local description may
versals which manage to occur from propagating outwards. benecessarytounderstandhowthemidplaneandhighalti-
Thereforeconvectioncreatesanenvironmentwhichprevents tude regions differ from each other, and we hope to pursue
field reversals, and leads to the parity of the field being such an analysis in future work.
locked in place. Due to the variety of parities seen, it ap-
pears likely that it is simply the initial conditions when a
5.1 Departures from Standard Disc Dynamo:
convective epoch is initiated that set the parity for the du-
Comparison with Other Works
ration of that epoch.
We find that some properties of the dynamo during con-
vective epochs are similar to the convective simulations
of Bodo et al. (2012, 2015), such as prolonged states of
5 DISCUSSION
azimuthal magnetic field polarity and an enhancement of
Itisimportanttonotethatwestilldonotunderstandmany Maxwell stresses compared to purely radiative simulations
aspectsoftheMRIturbulentdynamo.Ouranalysisherehas (see, e.g. Figure 6 of Hirose et al. 2014). However, there
not shed any light on the actual origin of field reversals in are some dynamo characteristics observed by Bodo et al.
the standard (non-convective) butterfly diagram, nor have (2015)thatarenotpresentinoursimulations.Forexample,
c 2016RAS,MNRAS000,1–12
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8 M. S. B. Coleman et al.
ws0446
103 Above Midplane 103 Below Midplane
1.0 × 1.0 ×
10 2
−
0.5 0.5
10 3
−
y
B 0.0 0.0
10 4
−
0.5 0.5
− −
0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.2 0.1 0.0 0.1 0.2
− − − − − −
δs/ s δs/ s
h i h i
Figure 6. 2D histograms of azimuthal magnetic field (By, vertical axis) and horizontal variations of specific entropy (δs, horizontal
axis).Eachcellsavedtofilefromws0446with80(cid:54)t(cid:54)100and0(cid:54)z(cid:54)0.25simulationunits(left)and 0.25(cid:54)z(cid:54)0simulationunits
−
(right) are used to create the respective histograms. It is important to note that for both panels there is a low entropy, high By tail
| |
which corresponds to the sign of the respective wings of the butterfly diagram (see bottom panel of Fig. 1), indicating that convection
ismixingregionsofhighmagnetizationfromthewingsintothemidplane.
theirsimulationstypicallyevolvedtoastronglymagnetized WefirstconsidertheBodoetal.(2015)simulationswith
state, something which we never find. We also find that in net vertical magnetic flux. Figure 10 of Bodo et al. (2015)
our simulations, which exhibit intermittent convection, the shows that for increasing net vertical flux, the strength of
time-averagedMaxwellstressinthemidplaneregionsisap- the azimuthal field increases and field reversals decrease in
proximately the same in both the convective and radiative frequencywithlong-lived(short-lived/transitionary)epochs
epochs3, and is independent of the vertical parity of the ofeven(odd)parity.Nodynamoflipsintheazimuthalfield
azimuthal field. In contrast, Bodo et al. (2015) find sub- were seen for the strongest net flux case. Figure 9 shows
stantiallylessMaxwellstressduringoddparityepochs.Az- thattheisothermalnetverticalfluxsimulationsofSalvesen
imuthal field reversals occasionally occur during their con- et al. (2016b) reproduce all features of the butterfly dia-
vectivesimulations,whereasweneverseesuchreversalsdur- grams in the convective simulations of Bodo et al. (2015),
ingourconvectiveepochs.Finally,theirsimulationsexhibit forthesamerangeofinitialplasma-β.Thisremarkablesim-
a strong preference for epochs of even parity, and a lack of ilarity between these simulations with and without convec-
quasi-periodic pulsations in the wings of the butterfly dia- tiveheattransportsuggeststhatstrongmagnetization(i.e.,
gram. β ∼1 at the disc mid-plane) is responsible for the conflicts
Remarkably, all of these aforementioned properties of listedinthepreviousparagraphwiththeHiroseetal.(2014)
their simulations arise in strongly magnetized shearing box simulations under consideration.
simulations (Bai & Stone 2013; Salvesen et al. 2016b). We We now seek to understand the role of convection on
suggestherethatthesepropertiesofthedynamothatBodo dynamobehaviorinthezeronetverticalmagneticfluxsim-
et al. (2015) attribute to convection are actually a mani- ulationsofBodoetal.(2015).Thesesimulationsalsodevel-
festation of strong magnetization. To demonstrate this, we oped into a strongly magnetized state and exhibited simi-
startbynotingthatthesimulationspresentedinBodoetal. larly dramatic departures from the standard butterfly pat-
(2015) adopted: (1) impenetrable vertical boundary condi- tern as their net flux counterparts. Gressel (2013) demon-
tionsthatpreventedoutflows;thus,trappingmagneticfield strated that zero net vertical flux shearing box simulations
withinthedomain,and(2)initialconfigurationswitheither with constant thermal diffusivity and the same impenetra-
zero or non-zero net vertical magnetic flux. ble vertical boundaries adopted by Bodo et al. (2015) lead
to the following: (1) A butterfly pattern that is irregular,
yet still similar to the standard pattern that is recovered
for outflow boundaries. However, the box size in Gressel
3 Although the Maxwell stress is approximately the same be-
(2013) was comparable to the smallest domain considered
tweenradiativeandconvectiveepochsinagivensimulation,the
in Bodo et al. (2015), which was not a converged solution.
α parameter is enhanced during convective epochs because the
medium is cooler and the time-averaged midplane pressure is (2) Maxwell stresses that are enhanced by a factor of ∼ 2
smaller. compared to the simulation with outflow boundaries. This
c 2016RAS,MNRAS000,1–12
(cid:13)
Convective Quenching of Field Reversals 9
weakly magnetized state with a conventional butterfly pat-
103 ws0429 tern.
×
However, we note that Gressel (2013) found that the
10 1 standard butterfly diagram is recovered when replacing im-
−
2 penetrable boundaries with outflow boundaries. Similarly,
Salvesen et al. (2016a) initialized two zero net vertical flux
simulations with a purely azimuthal magnetic field corre-
10 2
1 − sponding to β =1. In simulation ZNVF-P, which adopted
0
periodic boundary conditions that prevent magnetic field
from buoyantly escaping, the butterfly pattern was not
By 0 10 3 present and the azimuthal field locked into a long-lived,
−
even parity state. However, for simulation ZNVF-O, which
adopted outflow vertical boundaries, the initially strong
1
− magnetic field buoyantly escaped and the disc settled down
10 4
− to a weakly magnetized configuration with the familiar dy-
namoactivity(seeFigure2of Salvesenetal.2016a).There-
2
− fore,verticalboundariesthatconfinemagneticfieldmaydic-
10 5 tate the evolution of shearing box simulations without net
−
poloidal flux.
1 0 1 2 3
− Basedonthediscussionabove,wesuggestthatthedy-
δs/hsi ×10−2 namo behavior in the Bodo et al. (2015) simulations with
net vertical magnetic flux is a consequence of strong mag-
netizationandnotconvection.Forthezeronetverticalflux
Figure 7. A 2D histogram of azimuthal magnetic field (By, simulationswithimpenetrableverticalboundaries,therela-
vertical axis) and horizontal variations of specific entropy (δs,
tiverolesofconvectionvs.strongmagnetizationininfluenc-
horizontal axis) for the radiative simulation ws0429. Note the
ingthedynamoislessclear.Themainresultofthispaper—
symmetryandhowtheprobabilitydistributioncurvestohighen-
tropyathighmagneticfieldstrength.Thiscurvatureisindicative that convection quenches azimuthal field reversals in accre-
of isothermal, isobaric fluctuations and matches well with linear tiondiscdynamos—appliestothecaseofzeronetvertical
theory(blackdottedline)whichiscomputedfromEqn.5assum- magnetic flux and realistic outflow vertical boundaries. Fu-
ing that B2 = By2. This trend should be contrasted with what ture simulations in this regime with larger domain size will
we observe during convective epochs inFigs. 5and 6. Notethat help to determine the robustness of this result.
the entropy fluctuations here are an order of magnitude smaller
comparedtothoseoftheconvectivesimulationws0446.
5.2 Quasi-Periodic Oscillations
In addition to the outburst cycles observed in dwarf novae,
is likely because impenetrable boundaries confine magnetic cataclysmic variables in general exhibit shorter timescale
field, which would otherwise buoyantly escape the domain variabilitysuchasdwarfnovaoscillations(DNOs)on∼10s
(Salvesenetal.2016a).(3)Substantialturbulentconvective time scales and quasi-periodic oscillations (QPOs) on ∼
heat flux, which is significantly reduced when using out- minute to hour time scales (Warner 2004). A plausible ex-
flow boundaries. Therefore, perhaps the enhanced convec- planation for DNOs in CVs involving the disk/white dwarf
tionresultingfromusingimpenetrableboundariesisindeed boundary layer has been proposed (Woudt & Warner 2002;
responsible for the strongly magnetized state and dynamo Warner & Woudt 2002). However, a substantial number of
activity seen in the “Case D” zero net flux simulation of QPOs remain unexplained. It is possible that the quasi-
Bodo et al. (2015). periodic magnetic field reversals seen in the MRI butterfly
Despite starting with identical initial and boundary diagram are responsible for some of these QPOs and other
conditions,thezeronetverticalfluxsimulationM4ofGres- variability.Temporally,onewouldexpectvariationsfromthe
sel (2013) does not evolve to the strongly magnetized state butterflydiagramtooccuronminutestoanhourtimescales:
seen in the Bodo et al. (2015) simulations. The reason (cid:18) M (cid:19)−1/2 τ
for this discrepancy is unclear. In an attempt to repro- τ =223s×r3/2 bf , (6)
duce Case D in Bodo et al. (2015) and following Salvesen bf 9 0.6M(cid:12) 10τorb
et al. (2016a), we ran an isothermal, zero net vertical flux whereτ istheperiodofthebutterflycycle,r istheradial
bf 9
shearing box simulation that had an initial magnetic field, location of variability in units of 109 cm, M is the mass of
B=B sin(2πx/L ), where B corresponded to β =1600. the white dwarf primary, and τ is the orbital period.
0 x 0 0 orb
This simulation, labeled ZNVF-βZ1600, had domain size Indeed, this suggestion has already been made in the
(L ,L ,L ) = (24H ,18H ,6H ) with H being the initial context of black hole X-ray binaries (O’Neill et al. 2011;
x y z 0 0 0 0
scale height due to thermal pressure support alone, reso- Salvesenetal.2016b),however,noplausibleemissionmech-
lution 24 zones/H in all dimensions, and periodic vertical anismtoconvertthesefieldreversalsintoradiationhasbeen
0
boundariesthattrapmagneticfield,whichwebelievetobe identified.However,itisnoteworthythatquasi-periodicaz-
thesalientfeatureoftheimpenetrableboundariesdiscussed imuthal field reversals have also been seen in global accre-
above. Figure 10 shows that the space-time diagram of the tion disk simulations with substantial coherence over broad
horizontally-averagedazimuthalmagneticfieldforthissim- ranges of radii (e.g. O’Neill et al. 2011; Flock et al. 2012;
ulationdoesnotreproduceCaseD,butinsteadevolvestoa Jiang et al. 2014), indicating that this phenomena is not
c 2016RAS,MNRAS000,1–12
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10 M. S. B. Coleman et al.
ws0446
Radiative Convective
12 12
100
10 10
i 8 8 10−1
g
a
m
P
h 6 6
/
mag 10−2
P 4 4
δ
2 2 10−3
0 0
10 4
−
0 1 2 3 4 0 1 2 3 4
δρ/ ρ δρ/ ρ
h i h i
Figure8. 2Dhistogramsofhorizontalvariationsofmassdensity(δρ,horizontalaxis)andmagneticpressure(δPmag,verticalaxis).The
leftpanelshowsdatafromaradiativeepoch(49(cid:54)t(cid:54)64)ofsimulationws0446andtherightpanelshowsdatafromaconvectiveepoch
(70(cid:54)t(cid:54)100)ofthesamesimulation.Whilethereisstillaroughanti-correlationbetweenδρandδPmagasseeninFig.4thiscorrelation
is much weaker here. Also, note that the scale of the x-axis here is an order of magnitude higher than that of Fig. 4. The left frame is
similartothedatafromws0429buthasanextrasourceofdispersionwhichislikelyconnectedtopreviousepisodesofconvection.The
rightframehasasignificantlyhigherpopulationofoverdenseandhighlymagnetizedfluidparcelscomparedtotheradiativeepoch.This
signifiesthatmagneticbuoyancyisweakenedduringconvectiveepochs.
uniquetoshearingboxsimulations.IfQPOsindwarfnovae wings) down into the midplane. This mixing was identified
are in fact associated with azimuthal field reversals, then throughcorrelationsbetweenentropyandB (seeFigs.5,6).
y
our work here further suggests that these QPOs will dif- Duetothehighopacityoftheconvectiveepochs,perturbed
fer between quiescence and outburst due to the fact that fluid parcels maintain their entropy for many dynamical
theconvectionquenchingoffieldreversals(i.e.thebutterfly times. Hence the observed low-entropy highly-magnetized
diagram) only occurs in outburst, and this quenching may fluid parcels found in the midplane must have been mixed
leave an observable mark on the variability of dwarf novae. infromthewings.ThesignofB fortheseparcelsalsocor-
y
respondtothewingwhichisclosestandtendtoopposethe
sign found in the midplane, quenching field reversals there.
The high opacity which allows for the fluid parcels to
6 CONCLUSIONS
preserve their entropy also allows for thermal fluctuations
We analyzed the role of convection in altering the dynamo to be long lived. This combined with the turbulence gen-
in the shearing box simulations of Hirose et al. (2014). erated by convection weakens the anti-correlation between
Throughout this paper we explained how convection acts magneticpressureanddensityfoundinradiativesimulations
to: (contrast Figs. 4 and 8) and creates an environment where
some flux tubes can be overdense. This acts to weaken, but
(i) quickly quench magnetic field reversals near the mid-
notquench,magneticbuoyancytherebypreventingtheweak
plane;
andinfrequentfieldreversalswhichdooccurfrompropagat-
(ii) weaken magnetic buoyancy which transports mag-
ing outwards to the wings.
netic field concentrations away from the midplane;
It is through these mechanisms that convection acts to
(iii) prevent quasi-periodic field reversals, leading to
disrupt the butterfly diagram and prevent field reversals,
quasi-periodic pulsations in the wings of the butterfly di-
eventhoughitisclearthattheMRIturbulentdynamocon-
agram instead; and
tinuestotryanddrivefieldreversalsinthemidplaneregions.
(iv) hold the parity of B fixed in either an odd or even
y
ThisresultsinthesignofB anditsparityacrossthemid-
state. y
plane to be locked in place for the duration of a convective
Allofthesearedramaticdeparturesfromhowthestandard epoch.Thequenchingoffieldreversalsandthemaintenance
quasi-periodicfieldreversalsandresultingbutterflydiagram of a particular parity (odd or even) across the midplane is
work during radiative epochs. a hallmark of convection in our simulations, and we hope
Theprimaryroleofconvectionindisruptingthebutter- that it may shed some light on the behaviour of the MRI
flydiagramistomixmagneticfieldfromhighaltitude(the turbulent dynamo in general.
c 2016RAS,MNRAS000,1–12
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