Neutrino viscosity and drag: impact on the magnetorotational instability in protoneutron stars
aa r X i v : . [ a s t r o - ph . H E ] J a n Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 30 December 2017 (MN L A TEX style file v2.2)
Neutrino viscosity and drag: impact on themagnetorotational instability in protoneutron stars
J´erˆome Guilet , , Ewald M¨uller & Hans-Thomas Janka Max-Planck-Institut f¨ur Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany Max Planck/Princeton Center for Plasma Physics
30 December 2017
ABSTRACT
The magnetorotational instability (MRI) is a promising mechanism to amplify themagnetic field in fast rotating protoneutron stars. The diffusion of neutrinos trappedin the PNS induces a transport of momentum, which can be modelled as a viscosity onlength-scales longer than the neutrino mean free path. This neutrino-viscosity can slowdown the growth of MRI modes to such an extent that a minimum initial magneticfield strength of & G is needed for the MRI to grow on a sufficiently short time-scale to potentially affect the explosion. It is uncertain whether the magnetic fieldof fast rotating progenitor cores is strong enough to yield such an initial magneticfield in PNS. At MRI wavelengths shorter than the neutrino mean free path, on theother hand, neutrino radiation does not act as a viscosity but rather induces a dragon the velocity with a damping rate independent of the wavelength. We perform alinear analysis of the MRI in this regime, and apply our analytical results to thePNS structure from a one-dimensional numerical simulation. We show that in theouter layers of the PNS, the MRI can grow from weak magnetic fields at wavelengthsshorter than the neutrino mean free path, while deeper in the PNS MRI growth takesplace in the viscous regime and requires a minimum magnetic field strength.
Key words:
MHD – stars: neutron – stars: rotation – supernovae: general
The explosion mechanism of core collapse supernovae andin particular the role played by rotation and magnetic fieldsis still uncertain. The neutrino driven mechanism aidedby multidimensional hydrodynamical instabilities may beresponsible for explosions with normal energies of 10 − erg, though robust explosions with sufficient energyhave yet to be demonstrated by three-dimensional numericalsimulations including all relevant physics (e.g. Hanke et al.2013; Mezzacappa et al. 2014). A small fraction of corecollapse supernovae, however, have much larger explosionenergies of ∼ erg (”hypernovae” or type Ic BL, e.g.Drout et al. 2011), which most likely require an additionalenergy reservoir beyond neutrinos. Scenarios relying on acombination of fast rotation and strong magnetic fieldsmay be good candidates to explain such extreme explo-sions. The rotation energy contained in a neutron star ro-tating with a period of one millisecond (near break-up ve-locity) is indeed a sufficient energy reservoir, which couldbe efficiently tapped if strong magnetic fields of the orderof ∼ G are present. Axisymmetric simulations assum-ing both a strong poloidal magnetic field and fast differen-tial rotation have for example demonstrated the possibilityof magnetorotational explosions (LeBlanc & Wilson 1970; Bisnovatyi-Kogan et al. 1976; Mueller & Hillebrandt 1979;Symbalisty 1984; Moiseenko et al. 2006; Shibata et al. 2006;Burrows et al. 2007; Dessart et al. 2008; Takiwaki et al.2009; Takiwaki & Kotake 2011), although it remains to bedemonstrated whether the explosion energy can reach thatof hypernova-like explosions. Such magnetorotational explo-sions are furthermore a potential site for the production of r-process elements (Winteler et al. 2012). Note, however, thatthe 3D dynamics of magnetorotational explosions needs tobe explored further, since M¨osta et al. (2014) showed thatnon-axisymmetric instabilities can disrupt the jet before itcan launch an explosion.Another way by which fast rotation and strong mag-netic fields could impact supernovae explosions is throughthe delayed injection of energy due to the spin down of a fastrotating, highly magnetized neutron star (Kasen & Bildsten2010; Woosley 2010), which has been invoked as an expla-nation of some superluminous supernovae like SN 2008 bi(Dessart et al. 2012; Nicholl et al. 2013; Inserra et al. 2013).The birth of such ”millisecond magnetars” is furthermorea potential central engine for long gamma-ray bursts (e.g.Duncan & Thompson 1992; Metzger et al. 2011).The very fast rotation needed by the above scenarios(magnetorotational explosion, millisecond magnetars) may c (cid:13) Guilet et al. not be present in the core of stars following standard stel-lar evolution, as Heger et al. (2005) showed that magnetictorques can slow down the core rotation efficiently. However,Yoon & Langer (2005) and Woosley & Heger (2006) showedthat the fastest rotating stars could follow a chemically ho-mogeneous evolution, in which the core can retain enoughangular momentum to form a neutron star rotating withmilliseconds period.The second crucial ingredient for powerful magne-torotational energy release is an extremely strong, large-scale poloidal magnetic field of the order of 10 G. Thepresence of such a strong magnetic field in some pro-toneutron stars (PNS) is suggested by the observation ofthe most strongly magnetized neutron stars called mag-netars (Woods & Thompson 2006, and references therein).The origin of this strong magnetic field remains, how-ever, uncertain. One hypothesis is a fossil field origin inwhich the magnetic flux is inherited from the progeni-tor star (Ferrario & Wickramasinghe 2006), but it is notclear whether this can explain the population of magne-tars. Given the uncertainty of having a sufficiently strongmagnetic field in the iron core, intense research has beenundertaken on physical mechanisms that could amplifythe magnetic field during core collapse (in addition tothe compression due to magnetic flux conservation dur-ing the collapse). In non-rotating progenitors, the stand-ing accretion shock instability and convection have been in-voked as a source of turbulence giving rise to a small-scaledynamo (Endeve et al. 2010, 2012; Obergaulinger & Janka2011; Obergaulinger et al. 2014). An Alfv´en surface has alsobeen shown to be a potential site of Alfv´en wave and there-fore magnetic field amplification (Guilet et al. 2011). Butthe most promising mechanisms rely on the fast rotationto drive a dynamo in the convective region of the PNS(Thompson & Duncan 1993), or through the magnetoro-tational instability (hereafter MRI; e.g. Balbus & Hawley1991; Akiyama et al. 2003).Since the suggestion by Akiyama et al. (2003) that theMRI could play an important role in core collapse super-novae, the MRI has been the subject of a number of studiesin the context of supernovae with the use of linear analy-sis (Masada et al. 2006, 2007), local (or ”semi-global”) nu-merical simulations representing a small portion of the PNS(Obergaulinger et al. 2009; Masada et al. 2012), and two-dimensional global numerical simulations (Sawai et al. 2013;Sawai & Yamada 2014). Masada et al. (2007) showed thatneutrino radiation in the diffusive regime has several effectson the MRI. On the one hand, neutrino thermal and leptonnumber diffusion alleviates the stabilizing effect of entropyand lepton number gradients in stably stratified regions ofthe PNS. On the other hand, neutrino viscosity slows downMRI growth if the initial magnetic field is weaker than acritical strength, which Masada et al. (2012) estimated tobe ∼ . × G. As a consequence of neutrino viscos-ity, a minimum magnetic field is needed for the MRI togrow on a sufficiently short time-scale to affect the explosion(Masada et al. 2012). In this paper, we revisit this issue bycomputing the neutrino viscosity for the conditions given bythe output of a one-dimensional numerical simulation. Wefind that the effect of neutrino viscosity is even more pro-nounced: MRI growth is slowed down if the magnetic field isweaker than 10 − G (depending on the rotation rate), and becomes too slow to affect the explosion below a mini-mum strength of ∼ G (see Section 3.2). This may be aproblem for the MRI since the initial magnetic field in thePNS is extremely uncertain and could well be below thisminimum strength.The description of the effect of neutrinos by a viscosityis however valid only at length-scales longer than the neu-trino mean free path. We will show that, in the outer partsof the PNS, the viscous prescription is not self-consistent be-cause the MRI grows at wavelengths shorter than the neu-trino mean free path. We therefore provide the first descrip-tion of the effect of neutrinos on the growth of the MRI thatis valid at wavelengths shorter than the neutrino mean freepath (Section 3.3). This allows us to show that in the outerparts of the PNS, the MRI can grow from initially very weakmagnetic fields.The paper is organized as follows. In Section 2, we de-scribe the numerical model of PNS structure. In Section 3,we analyse the different regimes in which the MRI can grow:analytical predictions are obtained, which are then appliedto the PNS model. In Section 4, we discuss and conclude onthe relevant regime of MRI growth as a function of radiusin the PNS and magnetic field strength.
In order to estimate physical quantities relevant forthe growth of the MRI, we use the result of a one-dimensional numerical simulation as a typical structureof the PNS. The calculations were performed with thecode Prometheus-Vertex, which combines the hydrodynam-ics solver Prometheus (Fryxell et al. 1989) with the neu-trino transport module Vertex (Rampp & Janka 2002). Ver-tex solves the energy-dependent moment equations with theuse of a variable Eddington factor closure, and including anup-to-date set of neutrino interaction rates (e.g. M¨uller et al.2012). General relativistic corrections are taken into accountby means of an effective gravitational potential (Marek et al.2006). The model considered hereafter simulates the evolu-tion of the 11 . M ⊙ progenitor of Woosley et al. (2002), us-ing the high-density equation of state of Lattimer & Swesty(1991) with a nuclear incompressibility of K = 220 MeV.Most of the results presented in this paper were obtainedusing a single time frame at t = 170 ms after bounce. The ra-dial profiles of density (upper panel) and temperature (mid-dle panel) at this (arbitrary) reference time are shown inFig. 1. The structure of the PNS evolves in time due toits contraction and energy and lepton emission. We havetherefore performed the same analysis at other times be-tween t = 50 ms and t = 800 ms after bounce, obtainingvery similar results: the main difference is the radius of thePNS (which decreases from 70 km to about 20 km), as willbe discussed in Section 4.As a simplified representation of the rotation profilesobtained by Ott et al. (2006), we further assume that theequatorial plane of the PNS is rotating with the following c (cid:13) , 000–000 mpact of neutrinos on the MRI cm)10 den s i t y ( g . c m - ) cm)110100 t e m pe r a t u r e ( M e V ) rotation angular frequency cm)10100100010000 Ω ( s - ) fast rotationmoderate rotation Figure 1.
Structure of the PNS at time t = 170 ms after bounce.Upper panel: Radial profile of the density. Middle panel: Radialprofile of the temperature. Lower panel: radial profiles of rotationangular frequency defined in equations (1)-(2), with parametersΩ = 2000 s − (black line) or Ω = 200 s − (red line), q = 1 and r = 10 km. rotation profile Ω = Ω , if r < r (1)Ω = Ω (cid:16) rr (cid:17) − q , if r > r (2)where Ω is the angular frequency of the inner solidly rotat- since we restrict our analysis of MRI growth to the equatorialplane, we do not need to specify the angular dependence of therotation frequency ing core (at r < r ), and q is the power law index of the ro-tation profile assumed outside this inner core. Note that theMRI can grow only in regions where the angular frequencyis decreasing outward, such that we only consider the outerregion ( r > r ) in the following analysis. As typical param-eters, we have chosen q = 1, r = 10 km and two differentvalues of the angular frequency: Ω = 2000 s − (fast rota-tion, black line in the lower panel of Fig. 1) or Ω = 200 s − (moderate rotation, red curve in the lower panel of Fig. 1).Assuming shellular rotation, the total angular momentumcontained in the PNS rotating with these angular frequencyprofiles would be: L = 4 . × g cm s − (fast rotation),and L = 4 . × g cm s − (moderate rotation). After cool-ing and contraction to a neutron star with a radius of 12 kmand a moment of inertia of I = 1 . × g cm , the neu-tron star would be rotating with a period (assuming an-gular momentum conservation): P = 2 ms (fast rotation)and P = 20 ms (moderate rotation). The fast rotation iscomparable to the angular momentum of progenitors follow-ing the chemically homogeneous evolution (Yoon & Langer2005; Woosley & Heger 2006), and may be thought of as rep-resentative of the scenario of millisecond magnetar forma-tion (if magnetic field amplification to magnetar strength isindeed achieved), which is one of the central engines consid-ered for gamma-ray bursts and hypernovae explosions. Themoderately rotating model, on the other hand, is comparableto the progenitors of Heger et al. (2005) and is relevant toless extreme supernovae where rotation and magnetic fieldamplification may still play an important role.Note that the PNS structure is taken from a one-dimensional numerical simulation of a non-rotating pro-genitor. Assuming a rotation profile is therefore not self-consistent, but should give the right order of magnitude aslong as the rotation is not too extreme. This will be dis-cussed in Section 4. In this section, we obtain analytical estimates for the ef-fect of neutrino radiation on the MRI growth in differentregimes, corresponding to optically thick or optically thinneutrino transport at the MRI wavelength. In Section 3.1,we recall classical results on the linear growth of the MRIin ideal MHD (neglecting the effects of neutrino radiation).In Section 3.2, we study the effect of neutrino viscosity onthe growth of the MRI, which applies when the wavelengthof the MRI exceeds the neutrino mean free path. In Sec-tion 3.3, we then consider the growth of the MRI at scalesshorter than the neutrino mean free path.In order to highlight these different regimes in a simpleway, we have chosen to make a number of simplifying as-sumptions. First, the growth rates are obtained from the re-sults of a local WKB analysis, which applies when the wave-length is much shorter than the scale of the gradients (i.e.typically the density scaleheight). Secondly, we assume theinitial magnetic field to be purely vertical, which is the mostfavourable configuration for MRI growth. The fastest grow-ing MRI modes are then axisymmetric, with a purely verti-cal wave vector (e.g. Balbus & Hawley 1991), and we there-fore make this assumption in the linear dispersion relationspresented below. Thirdly, we assume that the gas is incom- c (cid:13) , 000–000 Guilet et al. pressible, i.e. we assume that the sound speed is much largerthan the velocities and Alfv´en speed (which should be rea-sonably well justified at least in the linear phase of the MRI)and neglect buoyancy effects due to the presence of entropyand composition gradients. Finally, we neglect the resistiv-ity in all the scaling relations used to estimate the growthrate and wavelength of the MRI in the conditions prevail-ing inside the PNS. This assumption is justified by the verysmall value of the resistivity compared to the neutrino vis-cosity (by a factor of about 10 , e.g. Thompson & Duncan1993; Masada et al. 2006). For completeness, however, resis-tive effects are retained in the dispersion relations presentedin Sections 3.2 and 3.3. Neglecting all diffusion coefficients as well as the effect ofneutrino radiation, the dispersion relation of the axisym-metric MRI modes in the presence of a vertical magneticfield is (e.g. Balbus & Hawley 1991) (cid:0) σ + k v A (cid:1) + κ (cid:0) σ + k v A (cid:1) − k v A = 0 , (3)where σ is the growth rate, k is the wavenumber, v A ≡ B/ √ πρ is the Alfv´en velocity and κ is the epicyclic fre-quency defined by κ ≡ r d( r Ω )d r . The analytical solution ofthis dispersion relation gives the growth rate and wavenum-ber of the fastest growing mode as σ = q , (4) k = p q (1 − q/
4) Ω v A , (5)where q ≡ − d log Ω / d log r (consistent with the definition ofthe rotation profile in Section 2). The growth rate is inde-pendent of the magnetic field strength, and is extremely fastfor rapid rotation σ = 500 q (cid:18) Ω1000 s − (cid:19) s − . (6)The wavelength on the other hand is proportional to themagnetic field strength, such that weak magnetic fields leadto very short MRI wavelength in the ideal MHD case. As-suming q = 1 leads to the following estimate for the wave-length λ = 6 (cid:18) B G (cid:19) (cid:18) ρ g cm − (cid:19) − / (cid:18) Ω1000 s − (cid:19) − m . (7) Neutrinos present in a nascent PNS can transport energy,lepton number and momentum. At length-scales much largerthan the mean free path of neutrinos, their transport canbe well described by diffusive processes. In this regime, thetransport of momentum by neutrinos gives rise to a viscosity ν (van den Horn & van Weert 1984; Burrows & Lattimer1988; Thompson & Duncan 1993). It can be expressed as ν = 215 e ν h l ν i ρc , (8) cm)10 neu t r i no v i sc o s i t y ( c m s - ) ν e - ν e ν x totalscaling(Keil) Figure 2.
Radial profile of the neutrino viscosity, computed byapplying equation (8) to the outputs of the numerical simulation.The solid black line corresponds to electron neutrinos, the dashedline to electron antineutrinos, the dotted line to heavy leptonneutrinos ν x (one representative species) and the solid red line tothe total neutrino viscosity (i.e. the sum over the six neutrinosspecies). The blue solid line shows the approximate analyticalestimate given by equation (10). where c is the speed of light, e ν is the neutrino energy den-sity, and h l ν i is the neutrino mean free path averaged overenergy following h l ν i ≡ (cid:18)Z d e ν d ǫ l ν d ǫ (cid:19) /e ν , (9)with ǫ the neutrino energy. Note that the numer-ical factor in equation (8) is different to that ofvan den Horn & van Weert (1984) due to our different defi-nition of the neutrino mean free path averaged over energy.We compute the viscosity caused by the neutrinos ofdifferent flavors by applying equation (8) to the output ofthe numerical simulation described in Section 2. The sum ofthe contributions from the 6 neutrino species gives the totalneutrino viscosity, which varies between a few 10 cm s − near the inner boundary of the differentially rotating en-velop and 10 cm s − near the neutrinosphere (Fig. 2).An approximate analytical expression for the neutrino vis-cosity as a function of density and temperature has beenobtained by Keil et al. (1996) by considering six species ofnon-degenerate neutrinos in local thermodynamic equilib-rium and assuming that the opacity comes only from scat-tering on to neutrons and protons in non-degenerate nuclearmatter ν = 1 . × (cid:18) T
10 MeV (cid:19) (cid:18) ρ g cm − (cid:19) − cm s − . (10)This analytical formula is compared with the viscosity com-puted from the output of the numerical simulations in Fig. 2.At radii 13 km < r <
23 km, the analytical estimate repro-duces the slope well and is in agreement with the numericalresult within 30% (the difference is due to different prescrip-tions for opacity, and the neglect of degeneracy for electronneutrinos and antineutrinos). At r <
13 km, high-densityeffects such as fermion blocking and nucleon correlation ef-fects increase the mean free path in the numerical modelbut are neglected in equation (10), which therefore underes-timates the viscosity. At r > c (cid:13) , 000–000 mpact of neutrinos on the MRI increases faster than the value computed from the output ofthe simulation. This can be traced back to the fact that theheavy lepton neutrinos and electron antineutrinos are notperfectly in thermal equilibrium with the gas as assumedfor equation (10).It is interesting to compare this result to the litera-ture. Masada et al. (2007, 2012) estimated a typical valueof the neutrino viscosity ν = 10 cm s − at a density of10 g cm − . This is about 10 times smaller than what wefind at the same density (at a radius of ∼
32 km): ν ∼ − × cm s − . Thompson et al. (2005) have computedthe neutrino viscosity due to different species of neutri-nos (their Figure 5). The contribution from muon neutrinosvaries between roughly a few times 10 and 10 cm s − ,while our results show more variation between ∼ cm s − and a few times ∼ cm s − . This difference might be dueto a different PNS structure (for example due to the differenttime (105 ms in their case) and equation of state).In the next subsection, numerical estimates will use asfiducial values representative of a radius at 20 −
25 km fromthe centre: a viscosity: ν = 2 × cm s − , a density ρ =10 g cm − and a rotation angular frequency Ω = 1000 s − (fast rotation). The dispersion relation of the MRI in the presenceof a viscosity ν and a resistivity η can be writ-ten as (Lesur & Longaretti 2007; Pessah & Chan 2008;Masada & Sano 2008) (cid:0) σ ν σ η + k v A (cid:1) + κ (cid:0) σ η + k v A (cid:1) − k v A = 0 , (11)where σ ν ≡ σ + k ν and σ η ≡ σ + k η . The effect of viscosityon the linear growth of the MRI is controlled by the vis-cous Elsasser number E ν ≡ v A ν Ω (e.g. Pessah & Chan 2008;Longaretti & Lesur 2010). For E ν <
1, viscosity affects sig-nificantly the growth of the MRI: as a result the growth rateis decreased, and the wavelength of the most unstable modebecomes longer. Typical conditions inside the PNS lead tothe following estimate of the viscous Elsasser number forfast rotation at a radius ∼ −
25 km E ν ∼ × − (cid:18) B G (cid:19) (cid:18) ρ g cm − (cid:19) − (cid:18) Ω1000 s − (cid:19) − × (cid:18) ν × cm s − (cid:19) − . (12)Viscosity therefore has a large effect on the linear growth ofthe MRI, unless the magnetic field is initially quite strong.The critical strength of the magnetic field below which vis-cous effects become important (at which E ν = 1) is B visc = p πρν Ω (13)= 5 × (cid:18) ρ g cm − (cid:19) / (cid:18) ν × cm s − (cid:19) / × (cid:18) Ω1000 s − (cid:19) / G . (14)This critical magnetic field strength is shown as a functionof radius in the PNS in Fig. 3 (dashed lines). It decreasesoutward but only weakly because the effect of the decreaseof density and angular frequency is partly compensated by cm)10 m agne t i c f i e l d ( G ) idealviscoustoo slow Figure 3.
Radial profile of the critical magnetic field strengthsthat determine the regime of MRI growth. The dashed lines showthe magnetic field strength B visc (defined in equation (13)) belowwhich viscous effects are important. The solid lines show the min-imum magnetic field strength necessary for the MRI to grow ata growth rate faster than σ min = 10 s − (computed using equa-tion (19)). The vertical dotted lines show the radius above whichthe viscous description breaks down because the mean free pathof heavy lepton neutrinos becomes larger than the wavelength ofthe MRI. The two colours represent the two different normaliza-tions of the rotation profile: fast rotation (black) and moderaterotation (red). the increase in neutrino viscosity. The initial magnetic fieldshould be quite large so that MRI growth is not much af-fected by viscosity : B & × − G for fast rotation,and B & − × G for moderate rotation. Note thatthese values are significantly larger than the one estimatedby Masada et al. (2012) ( ∼ . × G), which is due tothe fact that the viscosity we have computed is significantlylarger than the one they have estimated.In order to describe the MRI growth in the viscousregime, useful analytical formulae can be obtained in theasymptotic limit E ν ≪ σ = (cid:18) qE ν ˜ κ (cid:19) / Ω , (15) k = (cid:16) κν (cid:17) / , (16)where ˜ κ ≡ κ/ Ω = p − q ) is the dimensionless epicyclicfrequency. In contrast to the ideal MHD case, the wavelengthof the fastest growing mode is independent of the magneticfield strength (because it is set by the viscous length-scale),while the growth rate is proportional to the magnetic fieldstrength: weak magnetic fields lead to slower growth. Usingequations (15) and (16), we obtain the following estimatesfor the growth rate and wavelength of the fastest growingMRI mode in the viscous regime σ = 17 (cid:18) B G (cid:19) (cid:18) ρ g cm − (cid:19) − / × (cid:18) ν × cm s − (cid:19) − / (cid:18) Ω1000 s − (cid:19) / s − , (17) c (cid:13) , 000–000 Guilet et al. magnetic field (G)1101001000 g r o w t h r a t e ( s - ) magnetic field (G)10 w a v e l eng t h ( c m ) Figure 4.
Growth rate (left) and wavelength (right) of the fastest growing MRI mode in the viscous regime as a function of the magneticfield strength for the following fiducial parameters: ν = 2 × cm s − , ρ = 10 g cm − and Ω = 1000 s − . The solid black line showsthe numerical solution of the dispersion relation (equation 11). The dashed line shows the asymptotic behaviour in the viscous regime( E ν ≪ E ν ≫ λ = 240 (cid:18) Ω1000 s − (cid:19) − / (cid:18) ν × cm s − (cid:19) / m . (18)Compared to the ideal regime described in Section 3.1, thewavelength is much longer and the growth much slower fora moderate magnetic field of 10 G. The growth rate andwavelength of the most unstable MRI mode are shown inFig. 4 as a function of magnetic field strength (and the fidu-cial parameters for the viscosity, density and angular fre-quency representative of a radius at 20 −
25 km). The nu-merical solution of the dispersion relation (black solid line) iscompared to the analytical solution in the asymptotic lim-its of ideal MRI E ν ≫ E ν ≪ E ν > . B > × G (within 10% for E ν > B > G), andwith the viscous limit for E ν < .
05 or
B < G (within10% for E ν < − or B < × G).Fig. 4 shows that, because of neutrino viscosity, theMRI requires a minimum magnetic field strength in orderto grow fast enough to affect the explosion (as was alreadydiscussed by Masada et al. 2012). The minimum magneticfield necessary for the MRI to grow at a minimum growthrate σ min can be expressed using the viscous limit (equa-tion 15) B min = (cid:18) πρ ˜ κνq Ω (cid:19) / σ min , (19)= 6 × (cid:16) σ min
10 s − (cid:17) (cid:18) ρ g cm − (cid:19) / (cid:18) ν × cm s − (cid:19) / (cid:18) Ω1000 s − (cid:19) − / G . (20)Fig. 3 shows this minimum magnetic field strength as a This growth rate corresponds to an e-folding growth time of σ − = 100 ms and is only a very rough estimate of the mini-mum growth rate needed for a significant magnetic field ampli- function of radius, for σ min = 10 s − . Similarly to the crit-ical field B visc , this minimum magnetic field strength B min depends very weakly on the radial position in the PNS (butthis time slightly increasing outward due to the differentscaling with Ω and the same scaling with density and vis-cosity). B min is not so small ( ∼ × G for fast rotation,and ∼ × G for moderate rotation), and we thereforeconclude that neutrino viscosity sets a strong constraint onthe initial magnetic field for the MRI to be able to grow ona sufficiently short time-scale.This constraint, however, only applies if the MRI wave-length is longer than the mean free path of neutrinos, whichmay not be the case everywhere in the PNS. Fig. 5 showsthe mean free path of the different species of neutrinosas a function of radius (the mean free path is averagedover neutrino energy following equation (9)). It increasesby four to five orders of magnitudes from h l ν i ∼ r = 10 km to h l ν i &
10 km near the PNS surface around40 km. This dependence is approximately reproduced by thescaling l ν ∝ ρ − T − (due to the main opacity contribu-tions being proportional to the density and the square ofthe neutrino energy). The wavelength of the MRI in the vis-cous regime is independent of magnetic field strength (equa-tion 18), and can be compared to the neutrino mean freepath to check the consistency of the description (it is shownwith the red curve in Fig. 5 for the case of fast rotation). Be-cause of the strong variation of the neutrino mean free path,the description of the effect of neutrino radiation as a vis-cosity is well justified deep inside the PNS, but breaks downat larger radii where the mean free path becomes longerthan the MRI wavelength. This happens at radii ∼
27 kmfor fast rotation and ∼
34 km for moderate rotation, whichare marked by the vertical dashed lines in Fig. 3. At larger fication. A significant amplification of the magnetic field by theMRI probably requires many e-folding times, and might actu-ally take several seconds with this minimum growth rate. It istherefore possible that a dynamical effect of the MRI in the firstsecond after shock bounce might require larger magnetic fields bya factor of a few. c (cid:13) , 000–000 mpact of neutrinos on the MRI cm)10 m ean f r ee pa t h ( c m ) ν e - ν e ν x scalingdensityscale height λ visc Figure 5.
Radial profile of the neutrino mean free path. Thesolid black line corresponds to electron neutrinos, the dashedline to electron antineutrino and the dotted line to heavy lep-ton neutrinos ν x . The mean free paths are average over neutrinoenergy following equation (9). The blue line shows the scaling l ν = 10 ( ρ/ g cm − ) − ( T/
10 MeV) − . For comparison, thedensity scaleheight is plotted with a green line, and the wave-length of the MRI in the viscous regime in red. For the latterwe use the fast rotation profile defined in equations (1)-(2) withΩ = 2000 s − , q = 1 and r = 10 km. neutrino damping rate cm)10 Γ ( s - ) ν e - ν e ν x totalscaling Figure 6.
Radial profile of the neutrino drag damping rate com-puted according to equation (21). The solid black line correspondsto electron neutrinos, the dashed black line to electron antineutri-nos, the dotted black line to heavy lepton neutrinos ν x (one rep-resentative species), and the solid red line to the total neutrinodamping rate (i.e. the sum over the six neutrino species). Thedashed blue line shows the scaling Γ = 6 × ( T/
10 MeV) s − .For comparison, the green and orange solid lines show the an-gular frequency Ω for the cases of fast rotation (green) and slowrotation (orange). radii, the effect of neutrinos on the MRI cannot be describedby a viscosity because neutrino transport begins to enter thenon-diffusive regime. At length-scales shorter than the neutrino mean free path,the transport of momentum by neutrinos can no longer bedescribed as a viscous process. This momentum transportdoes nevertheless damp velocity fluctuations. Agol & Krolik(1998) and Jedamzik et al. (1998) showed that, in thisregime, radiation induces a drag on the velocity field whichis independent of the wavenumber of the velocity fluctua-tions. This drag is caused by the Doppler effect due to fluidmotion with respect to the background neutrino radiationfield. This creates a neutrino flux in the rest frame of thefluid, which is responsible for the drag upon absorption orscattering of the neutrinos. The neutrino drag can be rep-resented by an acceleration − Γ δ u , where δ u is the velocityperturbation and the damping rate Γ is given byΓ ∼ e ν h κ ν i ρc , (21)where h κ ν i is the neutrino opacity averaged over energy inthe following way h κ ν i ≡ (cid:18)Z d e ν d ǫ κ ν d ǫ (cid:19) /e ν . (22)Agol & Krolik (1998) obtained this result (with a numeri-cal factor of 4 /
3) for photons subject to Thomson scatter-ing, by performing a linearization of the transport equa-tion around an isotropic and steady radiation field, usinga closure model that keeps terms up to the quadrupoleand neglecting higher order (note that this is a higher or-der approximation than the so called Eddington closure).Jedamzik et al. (1998) found a similar result for neutrinosbut did not derive the relevant numerical factor, which ismost likely different from the one obtained by Agol & Krolik(1998) for photons.Fig. 6 shows the radial profile of the damping ratecaused by the different neutrino species, obtained by evalu-ating equation (21) for the output of the numerical simula-tion. The contribution from electron neutrinos is dominantcompared to that of other neutrino species because of theirshorter mean free path. The damping rate decreases outwardby about six orders of magnitudes from 10 s − at 10 km to10 s − at 40 km. This can be explained by the scaling Γ ∝ T (shown with a blue line in Fig. 6), which results from theassumption of thermal equilibrium e ν ∝ T and κ ν ∝ ρT .Thompson et al. (2005) obtained a neutrino damping rate ofΓ ∼
50 s − at a radius of 50 km (they do not show the valuesat smaller radii), which is of the same order of magnitudeas what we obtain at the surface of the PNS.Note that the damping rate Γ is related to the neutrinoviscosity through Γ ∼ ν h κ ν i / (2 h l ν i ) ∼ ν/ h l ν i , i.e. it is(within a numerical factor) the rate at which fluctuationson the same scale as the neutrino mean free path wouldbe damped by the neutrino viscosity. As a consequence thedamping of fluctuations at scales shorter than the neutrinomean free path is less efficient by a factor ∼ k h l ν i (with k the wavenumber of the fluctuations) than what would be ob-tained by applying the viscous formalism at these scales. Itis therefore conceivable that the MRI could grow on length- c (cid:13) , 000–000 Guilet et al. scales shorter than the neutrino mean free path even if theviscous formalism predicts the growth to be too slow.
In this section we study the growth of the MRI in the pres-ence of neutrino drag, which is relevant to scales shorterthan the neutrino mean free path. The equations of in-compressible MHD in the shearing sheet approximation(Goldreich & Lynden-Bell 1965) are written as ∂ t u + u · ∇ u = − ρ ∇ P + 1 µ ρ ( ∇ × B ) × B (23) − Ω × u + 2Ω Sx e x − Γ δu ,∂ t B = ∇ × ( u × B ) + η ∆ B , (24) ∇ · u = 0 , (25) ∇ · B = 0 , (26)where Ω ≡ Ω e z , S ≡ q Ω is the shear rate, and e x , e y , e z are units vector in the radial, azimuthal and vertical direc-tions (we restrict the analysis to the equatorial plane of thePNS). δu ≡ u − u and δB ≡ B − B are the velocity andmagnetic field perturbations with respect to the stationaryequilibrium solution B = B e z , u = − q Ω x e y . The onlynon-standard term in these equations is the neutrino drag − Γ δu discussed in the preceding section. The velocity andmagnetic field perturbations then follow the following set ofequations σδu x = Bµ ρ ikδB x + 2Ω δu y − Γ δu x , (27) σδu y − Sδu x = Bµ ρ ikδB y − δu x − Γ δu y , (28) δu z = 0 , (29) σδB x = Bikδu x − k ηδB x , (30) σδB y = Bikδu y − SδB x − k ηδB y , (31) δB z = 0 , (32)where the perturbations are assumed to have the followingtime and space dependence δA ∝ e σt + ikz , with σ being thegrowth rate of the modes, and k their vertical wave vector.Note that these equations are actually valid for any ampli-tude of the perturbations, as no linearization had to be doneto obtain them (just like the channel modes of the classi-cal MRI are non-linear solutions in the incompressible limit(Goodman & Xu 1994)). These equations can be combinedto obtain the dispersion relation of the MRI in this regime (cid:0) σ v σ η + k v A (cid:1) + κ (cid:0) σ η + k v A (cid:1) − k v A = 0 , (33)where we have defined σ v ≡ σ + Γ . (34)Note that the form of this equation is very similar to thedispersion relation in the viscous-resistive regime given byequation (11), the only difference being that σ ν = σ + k ν has been replaced by σ v = σ + Γ. As we will show, thefact that the neutrino damping rate is independent of thewavenumber makes a big difference for the wavelength andgrowth rate of the fastest growing mode.Fig. 7 shows the growth rate and wavelength of thefastest growing MRI mode (the numerical solution of the dis-persion relation for η = 0 is shown with the solid black line) as a function of the dimensionless parameter Γ / Ω charac-terizing the effect of neutrino drag on the MRI. The growthof MRI channel modes is not much affected by the neu-trino drag as long as Γ < Ω. Interestingly, the damping ratethreshold Γ ∼ Ω is equivalent within a numerical factor tothe limit of applicability of the viscous formalism (at whichthe viscous wavelength equals the neutrino mean free path).This comes from equation (16) giving the viscous wavelength λ ∼ ν/κ and the facts that Γ ∼ ν/ h l ν i (as noted in theprevious subsection) and κ ∼ Ω. As a consequence, in theouter layer of the PNS, where the viscous formalism doesnot apply, the neutrino drag has little impact on the MRIgrowth.When the damping rate Γ is increased further, thegrowth rate of the MRI is reduced significantly whilethe wavelength of the fastest growing mode changes onlyslightly . A useful analytical solution can be obtained in theasymptotic limit Γ ≫ Ω (and therefore Γ ≫ σ since σ < Ω)and η = 0. In this limit, the dispersion relation reduces to (cid:0) Γ σ + k v A (cid:1) = k v A (cid:0) − κ (cid:1) . (35)The growth rate of the MRI as a function of verticalwavenumber is therefore σ = kv A Γ (cid:16)p q Ω − kv A (cid:17) , (36)and the growth rate and wavenumber of the fastest growingmode are σ = q Γ , (37)and k = p q/ v A , (38)respectively. The asymptotic limits Γ ≫ Ω (drag regime)and Γ ≪ Ω (i.e. the ideal regime described in Section 3.1) arecompared to the full numerical solution in Fig. 7. An agree-ment within 10% is obtained for Γ / Ω < . / Ω > / Ω < . / Ω > / Ω. Importantly, and contrary to the viscous regime,the growth rate remains independent of the magnetic fieldstrength. This comes from the fact that the neutrino dragis independent of the wavenumber, and as a consequenceaffects the MRI growing on a weak magnetic field (with It might seem surprising at first sight that the wavelength ofthe most unstable mode varies (although only slightly), while thedamping rate is independent of wavelength. For a given magneticfield strength, the modes at different wavelengths have differentstructures, in particular different ratio of velocity to magneticfield perturbations. Since the damping acts only on the velocityfield (but not on the magnetic field) they are affected in a differ-ent way by the neutrino drag, even if the damping rate itself doesnot depend on the wavelength. The situation is different, how-ever, if we change simultaneously the wavelength and the mag-netic field strength. Indeed, modes with the same dimensionlesswavenumber kv A / Ω have the same (dimensionless) structure andare therefore affected by neutrino drag in exactly the same way.As a consequence, the maximum growth rate is independent ofthe magnetic field strength. c (cid:13) , 000–000 mpact of neutrinos on the MRI growth rate Γ/Ω0.0010.0100.1001.000 σ / Ω wavelength Γ/Ω0.00.51.01.5 Ω / kv A Figure 7.
Growth rate (left) and wavelength (right) of the fastest growing MRI mode at scales shorter than the neutrino mean freepath as a function of the damping rate Γ (assuming zero resistivity). The numerical solution of the dispersion relation (equation (33))is shown with the solid black lines, the strongly damped limit Γ ≫ Ω (equations (37)-(38)) with dashed lines, and the ideal limit Γ = 0(Section 3.1) with dotted lines. All quantities are shown in a non-dimensional way: Γ and σ are normalized by the angular frequency Ω,while Ω / ( kv A ) is a non-dimensional measure of the wavelength. cm)1101001000 g r o w t h r a t e ( s - ) fast rotationmoderate rotation σ min Figure 8.
Radial profile of the maximum growth rate of the MRIat scales shorter than the neutrino mean free path. The growthrate (solid lines) is computed by applying equation (37) whereΓ > Ω, and equation (4) where Γ < Ω. The dotted lines showthe growth rate in the ideal MHD limit (equation (4)). The twocolours represent the two different normalizations of the rotationprofile: fast rotation (black) and moderate rotation (red). Finally,the dashed line shows the minimum growth rate σ min = 10 s − . short wavelength) in the same way as if it were growingon a stronger magnetic field (with longer wavelength). As aconsequence, if the initial magnetic field is weak the MRIis more likely to grow in this regime than in the viscousregime (where the growth rate is proportional to magneticfield strength). The condition for the MRI to grow at a min-imum growth rate σ min is independent of the magnetic fieldstrength, and can be cast as an upper limit on the neutrinodamping rateΓ < q Ω / (2 σ min ) . (39)Applying the analytical results of this section to the nu-merical model of the PNS allows one to compute the growthrate of the MRI as a function of radius in the PNS. Theresult is shown in Fig. 8 for the two rotation profiles consid-ered. Due to the large variation of the neutrino damping rate inside the PNS, different MRI growth regimes are encoun-tered depending on the radius. Deep inside the PNS, thevery large neutrino drag suppresses the growth of the MRI: this is due to the high temperature. Near the PNS surfaceon the contrary, the neutrino drag does not have much effecton the growth of the MRI (because Γ < Ω), which thereforeoccurs in the ideal regime described in Section 3.1. Finally,at intermediate radii, the neutrino drag has a significant im-pact on the MRI growth rate but still allows a sufficientlyfast growth (we call this the drag regime). The extent of thethree different regimes depends on the rotation frequency :the MRI can grow in a larger portion of the PNS for fastrotation than for moderate rotation. Indeed if Ω is largerthe MRI can grow in the ideal regime for higher values of Γ,and therefore smaller radii. The extent of the drag regimealso depends sensitively on Ω: equation (37) shows that inthis regime the MRI growth rate has a steeper dependenceon the rotation rate ( σ max ∝ Ω ) than in the ideal MHDcase ( σ max ∝ Ω). This explains why the region where theMRI grows in the drag regime is less extended in the caseof moderate rotation.Finally, we should determine the condition for the for-malism developed in this section to be self-consistent, i.e.that the wavelength of fastest MRI growth be shorter thanthe neutrino mean free path. As noted above, the wave-length of the fastest growing mode is of the same order ofmagnitude as in the ideal MHD case, and is therefore pro-portional to the magnetic field strength. This is again incontrast to the viscous case, where the wavelength of thefastest growing mode is set by the viscosity and angular fre-quency (independently of the magnetic field strength). Theresults of this section are self-consistent if the wavelength ofthe fastest growing MRI mode is shorter than the neutrinomean free path. Using equation (38), this condition can beexpressed as an upper limit on the magnetic field strength,
B < p ρq/ π Ω / h κ ν i , (40)because the relevant neutrino mean free path is the inverseof the opacity averaged over neutrino energy as defined byequation (22). Fig. 9 shows this maximum magnetic field c (cid:13) , 000–000 Guilet et al. cm)10 m agne t i c f i e l d ( G ) fast rotationmoderate rotation Figure 9.
Magnetic field strength at which the wavelength of thefastest growing MRI mode equals the mean free path of electronneutrinos 1 / h κ ν i (as defined by equation (22)): below this criti-cal strength the formalism used in Section 3.3 is self consistent.The two colours represent the two different normalizations of therotation profile: fast rotation (black) and moderate rotation (red). strength for consistency of MRI growth in the drag regime.It is quite weak (10 − G) deep inside the PNS (wherethe MRI anyway does not grow efficiently due to the strongdrag), and increases by two orders of magnitude towards thesurface of the PNS reaching values of 2 × G for moderaterotation and 2 × G for fast rotation. This shows thatthe growth of the MRI at length-scales shorter than theneutrino mean free path is relevant for weak to moderateinitial magnetic fields in the outer parts of the PNS.
In this paper we have studied the impact of neutrino ra-diation on the growth of the MRI. We have shown that,depending on the physical conditions, the MRI growth canoccur in three different regimes: • Ideal regime (orange colour in Fig. 10) : this is theclassical MRI regime which applies when neutrino viscos-ity or drag are unimportant, i.e. if E ν > / Ω < • Viscous regime (dark blue colour in Fig. 10) : onlength-scales longer than the neutrino mean free path, neu-trino viscosity significantly affects the growth of the MRI if E ν <
1. The growth of the MRI is then slower and takesplace at longer wavelength compared to the ideal regime.In the viscous regime, the wavelength of the most unstablemode is independent of magnetic field strength, while thegrowth rate is proportional to the magnetic field strength.As a result, a minimum magnetic field strength of ∼ Gis required for the MRI to grow on sufficiently short time-scales. • Drag regime (light blue colour in Fig. 10) : on length-scales shorter than the neutrino mean free path, neutrinoradiation exerts a drag on moving fluid elements. This draghas a significant impact on the MRI if the damping rate is larger than the rotation angular frequency (Γ > Ω). In thisregime, the growth rate of the most unstable mode is inde-pendent of the magnetic field strength, but is reduced by afactor Γ / Ω compared to the ideal regime. The wavelength ofthe most unstable mode is not much affected by the neutrinodrag.Fig. 10 shows where in the parameter space these threeregimes apply, as a function of radius and magnetic fieldstrength for the two rotation profiles considered in this pa-per: fast rotation (left-hand panel) and moderate rotation(right-hand panel). Three regions in the PNS can be distin-guished: • Deep inside the PNS, the neutrino mean free path ismuch shorter than the wavelength of the viscous MRI, andΓ ≫ Ω. In this case, the growth of the MRI at scales shorterthan the mean free path is strongly suppressed, and therelevant MRI regime is the viscous MRI described in Sec-tion 3.2. The MRI can grow on sufficiently short time-scalesif the initial magnetic field is above a critical strength givenby equation (19). Viscous effects become unimportant forstrong magnetic fields above B visc given by equation (13). • At intermediate radii, the mean free path of neutrinosis still shorter than the wavelength of the viscous MRI, butΓ is not too large such that the MRI can also grow in thedrag regime (i.e. equation (39) is verified). This is there-fore an intermediate case where MRI growth can take placeboth in the viscous regime at wavelengths longer than theneutrino mean free path, and in the drag regime at length-scales shorter than the mean free path. Since the growth ratein the viscous regime is proportional to the magnetic fieldstrength, the growth is faster in the viscous regime above acritical magnetic field strength, which can be obtained bycombining equations (15) and (37) B visc − drag = √ qπρνκ ΩΓ . (41)Below B visc − drag , the growth is predicted to be faster in thedrag regime. However, this regime is self-consistent only ifthe magnetic field strength is weaker than that given byequation (40) (the MRI wavelength is then shorter than themean free path of electron neutrinos). In between these twocritical strengths, the MRI growth should actually occur ina mixed regime (shown in green in Fig. 10) where electronneutrinos are diffusing and thus induce a viscosity, while theother species are free streaming and exert a drag. • Near the PNS surface, the viscous regime is irrelevantbecause the neutrino mean free path is longer than the wave-length of the MRI. Furthermore, in this region the neutrinodrag does not affect much the growth of the MRI because thedamping rate is smaller than the angular frequency Γ < Ω.As a consequence the MRI growth takes place in the idealregime without much impact of neutrino radiation.For the two rotation rates considered in this paper, theregimes of MRI growth have somewhat different locationsin the parameter space. The region where the MRI can-not grow is more extended for moderate rotation than forfast rotation. The viscous and drag regimes are also less ex-tended. For the ideal regime it is a bit more complicated:it extends to lower magnetic fields deep inside for slowerrotation (because the wavelength is longer and the MRI istherefore less affected by viscosity), but weak field growth c (cid:13) , 000–000 mpact of neutrinos on the MRI fast rotation cm)10 B f i e l d s t r eng t h ( G ) idealviscoustoo slow dragmixO09 moderate rotation cm)10 B f i e l d s t r eng t h ( G ) idealviscoustoo slow dragmixM12 Figure 10.
Different regimes of MRI growth as a function of radius and magnetic field strength in the case of fast rotation (left-hand panel) and moderate rotation (right-hand panel), for a PNS model at t = 170 ms post-bounce. See the text for a description ofthe different regimes. The parameter range used in the simulations by Obergaulinger et al. (2009) is shown in red on the left panel,and the parameters assumed by Masada et al. (2012) are shown with a red cross on the right panel. The black lines separating thedifferent regimes are defined as follows. The vertical line separating the ”too slow” and drag MRI regimes corresponds to equation (39).The vertical line between the drag and ideal MRI regimes corresponds to Γ = Ω (which also represents within a numerical factor thecondition that the viscous MRI wavelength is longer than the neutrino mean free path). The line separating the ”too slow” and viscousregimes corresponds to equation (19). The almost horizontal line separating the viscous and ideal MRI regimes is given by equation (13).The line separating the drag and mixed regimes shows where the MRI wavelength in the drag regime equals the electron neutrino meanfree path (equation (40)). Finally, the line separating the mixed and viscous regimes is defined by the equality of the growth rates in theviscous and the drag regimes (equation (41)). The last criterion is only approximate, based on the assumption that the growth rate inthe mixed regime lies in between those predicted by the drag and viscous formalisms. fast rotation, t=50 ms cm)10 B f i e l d s t r eng t h ( G ) idealviscoustooslow dragmix fast rotation, t=800 ms cm)10 B f i e l d s t r eng t h ( G ) idealviscoustoo slow dragmix Figure 11.
Same as the left panel of Fig. 10 but at two other times : 50 ms after bounce (left-hand panel), and 800 ms after bounce(right-hand panel). Note that the horizontal scales are different because they have been rescaled to the size of the PNS. in the ideal regime can take place deeper inside the PNS forfast rotation (as explained in Section 3.3).The results presented so far correspond to a single timeframe at 170 ms after bounce. In order to study how thedifferent MRI regimes are affected by the PNS contraction,we have performed the same analysis at two other times(shown in Fig. 11 for the case of fast rotation only): 50 msafter bounce (at which time the PNS radius is ∼
70 km),and 800 ms after bounce (the PNS radius has then decreasedto ∼
22 km). Although the size and structure of the PNS isquite different at these different times, the results are strik- The PNS radius is defined here as the radius at which the den-sity equals 10 g cm − . ingly similar: the location of the different MRI regimes isalmost identical once rescaled to the size of the PNS, inparticular with very similar values of B min and B visc delim-itating the viscous regime.Note that the physical conditions inside the PNS (neu-trino viscosity, neutrino damping rate etc.) have been esti-mated using a one dimensional numerical simulation of corecollapse, which considered a non-rotating progenitor. Themoderate rotation is expected to change the PNS structurein a negligible way because the ratio of centrifugal to gravi-tational forces is less than 10 − everywhere in the PNS. Fastrotation, on the other hand, should have a significant influ-ence on the structure of the PNS, as the centrifugal forceamounts to 4 −
8% of the gravitational force in the radiusrange considered. Rotational support is expected to lead to c (cid:13) , 000–000 Guilet et al. a more extended PNS and lower temperatures along theequatorial direction (Kotake et al. 2004; Ott et al. 2006).This would change quantitatively the results presented inthis paper: for example the lower temperature would leadto smaller values of the neutrino damping rate, such thatthe MRI would be less affected by the neutrino drag in theouter envelope of the PNS. The centrifugal force also leadsto an oblate PNS, and the MRI growth regime will there-fore depend on the angular direction in addition to the radialdependence studied in this paper.Numerical simulations of the MRI in core collapse su-pernovae have so far neglected the effects of neutrino radi-ation on the growth of the MRI (Obergaulinger et al. 2009;Masada et al. 2012; Sawai et al. 2013; Sawai & Yamada2014). Our study shows that this assumption is reasonablefor the exponential growth of the MRI only in a limited re-gion of the parameter space, namely in the outer region ofthe PNS or deeper in the PNS but for quite strong initialmagnetic fields ( B & × − G depending on the ro-tation rate ). If the initial magnetic field is not very strong,the growth of the MRI deep inside the PNS is strongly af-fected by neutrino viscosity. The effect of neutrinos shouldtherefore be taken into account in numerical simulations ei-ther by adding a viscous or a drag term (depending on theregime of MRI growth) or by directly computing the neu-trino transport and back reaction on the velocity field. In or-der to properly describe the effects of neutrinos on the MRI(viscosity and drag), a neutrino transport scheme should bemultidimensional (i.e. not ray by ray) and should includevelocity dependent terms.Masada et al. (2012) performed local simulations of theMRI assuming a density ρ = 10 g cm − (this correspondsto a radius of ∼
32 km in our PNS model), an initial mag-netic field strength B = 2 . × G and an angular fre-quency Ω = 100 s − . This lies just at the limit betweenthe viscous, mixed, and no-MRI regimes in the moderaterotation profile (their rotation is only slightly faster). Wetherefore conclude that neutrinos should be taken into ac-count under these conditions, though the prescription to beapplied (viscosity or drag) is not clear. Obergaulinger et al.(2009) considered a box localized at 15 . ρ = 2 . × g cm − , Ω = 1900 s − (i.e. close to the fastrotation profile we considered), and different values of themagnetic field strength varying between B = 4 × and8 × G. Under these conditions, the MRI is actually in theviscous regime and the neutrino viscosity should thereforebe taken into account in the numerical simulation.The large value of the neutrino viscosity has an-other important consequence: since the resistivity is quitesmall in comparison, this leads to a huge value of themagnetic Prandtl number (the ratio of viscosity to resis-tivity): P m ≡ ν/η ∼ (Thompson & Duncan 1993;Masada et al. 2007). Studies in the context of accretion discshave shown that the level of MRI turbulence is very sensi-tive to the magnetic Prandtl number, and that it generallyincreases with this number (e.g. Lesur & Longaretti 2007; Note that this criterion only ensures that the linear growth rateis not much reduced by the viscosity, but the non-linear saturationof the MRI may still be affected by viscosity.
Fromang et al. 2007; Longaretti & Lesur 2010). If viscosityis not explicitly taken into account in numerical simulations,numerical dissipation will give rise to a numerical magneticPrandtl number which may depend on the numerical schemebut which should be of order unity (Fromang & Papaloizou2007; Fromang et al. 2007). This is very far from the trueregime, and could lead to underestimate strongly the finalmagnetic energy and stress. Numerical simulations takingexplicitly into account neutrino viscosity will therefore benecessary to assess its influence on MRI saturation.Let us now discuss the implications of our findings forthe explosion mechanism of core collapse supernovae. Wehave shown that neutrino viscosity and drag have importantconsequences for the growth time-scale of the MRI, whichneed to be taken into account. If the magnetic field is ini-tially weak, the MRI growth can be suppressed by neutrinoviscosity (if B . G in the PNS) or significantly sloweddown (if B . − G). The most important finding ofthis paper is probably that, even if the growth of the MRIfrom very weak magnetic fields is suppressed in the viscousregime deep inside the PNS, the MRI can grow on wave-lengths shorter than the mean free path of neutrinos in theouter parts of the PNS. To have an impact on the explosion,it is probably more important that magnetic field amplifica-tion takes place in the outer parts of the PNS rather thanin its inner parts. In this respect, our findings confirm thatthe growth of the MRI can be fast enough to play a role inthe explosions of fast rotating progenitors. Which impact ithas on the explosion will ultimately depend on the non-linearevolution and saturation of the MRI, which set the efficiencyof magnetic field amplification. How this non-linear evolu-tion is affected by the neutrino drag is currently unknownand should be the subject of future numerical studies.In order to highlight the different regimes of MRIgrowth in a simple way, we have made a number of sim-plifying assumptions, which are discussed below. First, wehave assumed the magnetic field to be purely poloidal.If the azimuthal magnetic field were much stronger thanthe poloidal one as obtained by Heger et al. (2005), thefastest growing perturbations would be non-axisymmetric(Masada et al. 2006). In a local analysis (like this article orMasada et al. (2006)), these shearing waves are only tran-siently growing typically during a few shear time-scales, dueto the fact that their radial wave vector increases linearlywith time (and proportionally to the azimuthal wave vec-tor). If their growth is slowed down by the effect of viscos-ity, it may be difficult for these transiently growing wavesto achieve a significant amplification. A WKB analysis asperformed by Masada et al. (2007) is not valid in that case(these authors assumed the azimuthal magnetic field to bestrong enough such that viscous effects do not reduce thegrowth significantly), but the transient amplification mayinstead be meaningfully studied using a non-modal approach(Squire & Bhattacharjee 2014b,a). It would also be interest-ing to study the non-axisymmetric growth in the presenceof neutrino drag.Secondly, we have neglected buoyancy effects, which canarise due to the presence of entropy and composition gradi-ents. Buoyancy can be either stabilizing or destabilizing de-pending on the location in the PNS. A convective region (i.e.destabilizing buoyancy) is thought to be present in the innerpart of the PNS, at radii between 10 −
15 km and 20 −
30 km c (cid:13) , 000–000 mpact of neutrinos on the MRI (Keil et al. 1996; Dessart et al. 2006; Buras et al. 2006, andreferences therein). The convective motions could play animportant role in amplifying the magnetic field, in particu-lar in the region of parameter space where the MRI growthis too slow. Outside of this convective region, buoyancy hasa stabilizing effect and can potentially have a strong impacton the MRI growth (Balbus & Hawley 1994; Menou et al.2004; Masada et al. 2006, 2007; Obergaulinger et al. 2009).In the neutrino diffusive regime, thermal and lepton num-ber diffusion can, however, alleviate the stabilizing effect ofbuoyancy on the MRI (Acheson 1978; Menou et al. 2004;Masada et al. 2007). This was demonstrated in the contextof a PNS by Masada et al. (2007), who considered the caseof an azimuthal magnetic field that is much stronger thanthe vertical component, and also strong enough so that the(non-axisymmetric) MRI is not much affected by the neu-trino viscosity (i.e. B > B visc in our notations). Under theseconditions, they showed that the MRI can grow at a rateclose to its maximum growth rate despite the stabilizingeffect of buoyancy, if the thermal and lepton number diffu-sivities are much larger than the viscosity (more preciselyif χ & νN/ Ω, where χ represents either thermal or leptonnumber diffusivity, and N is the Brunt-V¨ais¨al¨a frequency).However, they did not much explore the regime of less strongmagnetic fields, where the MRI is strongly affected by neu-trino viscosity. It would be very instructive to investigatethe effect of buoyancy in this viscous regime by applying theformalism of Masada et al. (2007) to the PNS model consid-ered here, but this is beyond the scope of the present workand is therefore left for a future study. Finally, we note thatbuoyancy effects on the MRI growth at wavelengths shorterthan the neutrino mean free path are so far unknown. Theyshould be studied in the future, since they are probably rel-evant in the outer parts of the PNS. ACKNOWLEDGEMENTS
We thank Florian Hanke for providing us with his simulationdata. We thank the anonymous referee for his/her careful re-port, which helped us improve the presentation of the paper.JG acknowledges support from the Max-Planck–PrincetonCenter for Plasma Physics.
REFERENCES
Acheson D. J., 1978, Royal Society of London PhilosophicalTransactions Series A, 289, 459Agol E., Krolik J., 1998, ApJ, 507, 304Akiyama S., Wheeler J. C., Meier D. L., Lichtenstadt I.,2003, ApJ, 584, 954Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214Balbus S. A., Hawley J. F., 1994, MNRAS, 266, 769Bisnovatyi-Kogan G. S., Popov I. P., Samokhin A. A., 1976,Ap&SS, 41, 287Buras R., Janka H.-T., Rampp M., Kifonidis K., 2006,A&A, 457, 281Burrows A., Dessart L., Livne E., Ott C. D., Murphy J.,2007, ApJ, 664, 416Burrows A., Lattimer J. M., 1988, Physics Reports, 163,51 Dessart L., Burrows A., Livne E., Ott C. D., 2006, ApJ,645, 534Dessart L., Burrows A., Livne E., Ott C. D., 2008, ApJL,673, L43Dessart L., Hillier D. J., Waldman R., Livne E., BlondinS., 2012, MNRAS, 426, L76Drout M. R., Soderberg A. M., Gal-Yam A., Cenko S. B.,Fox D. B., Leonard D. C., Sand D. J., Moon D.-S., ArcaviI., Green Y., 2011, ApJ, 741, 97Duncan R. C., Thompson C., 1992, ApJL, 392, L9Endeve E., Cardall C. Y., Budiardja R. D., Beck S. W.,Bejnood A., Toedte R. J., Mezzacappa A., Blondin J. M.,2012, ApJ, 751, 26Endeve E., Cardall C. Y., Budiardja R. D., MezzacappaA., 2010, ApJ, 713, 1219Ferrario L., Wickramasinghe D., 2006, MNRAS, 367, 1323Fromang S., Papaloizou J., 2007, A&A, 476, 1113Fromang S., Papaloizou J., Lesur G., Heinemann T., 2007,A&A, 476, 1123Fryxell B. A., M¨uller E., Arnett, D. 1989, Max-Planck-Institut f¨ur Astrophysik, Preprint, 449Goldreich P., Lynden-Bell D., 1965, MNRAS, 130, 125Goodman J., Xu G., 1994, ApJ, 432, 213Guilet J., Foglizzo T., Fromang S., 2011, ApJ, 729, 71Hanke F., M¨uller B., Wongwathanarat A., Marek A., JankaH.-T., 2013, ApJ, 770, 66Heger A., Woosley S. E., Spruit H. C., 2005, ApJ, 626, 350Inserra C., Smartt S. J., Jerkstrand A., et al. 2013, ApJ,770, 128Jedamzik K., Katalini´c V., Olinto A. V., 1998, Phys. Rev.D, 57, 3264Kasen D., Bildsten L., 2010, ApJ, 717, 245Keil W., Janka H., M¨uller E., 1996, ApJL, 473, L111+Kotake K., Sawai H., Yamada S., Sato K., 2004, ApJ, 608,391Lattimer J. M., Swesty F. D., 1991, Nuclear Physics A,535, 331LeBlanc J. M., Wilson J. R., 1970, ApJ, 161, 541Lesur G., Longaretti P.-Y., 2007, MNRAS, 378, 1471Longaretti P.-Y., Lesur G., 2010, A&A, 516, A51Marek A., Dimmelmeier H., Janka H., M¨uller E., Buras R.,2006, A&A, 445, 273Masada Y., Sano T., 2008, ApJ, 689, 1234Masada Y., Sano T., Shibata K., 2007, ApJ, 655, 447Masada Y., Sano T., Takabe H., 2006, ApJ, 641, 447Masada Y., Takiwaki T., Kotake K., Sano T., 2012, ApJ,759, 110Menou K., Balbus S. A., Spruit H. C., 2004, ApJ, 607, 564Metzger B. D., Giannios D., Thompson T. A., BucciantiniN., Quataert E., 2011, MNRAS, 413, 2031Mezzacappa A., Bruenn S. W., Lentz E. J., Hix W. R.,Messer O. E. B., Harris J. A., Lingerfelt E. J., EndeveE., Yakunin K. N., Blondin J. M., Marronetti P., 2014, inPogorelov N. V., Audit E., Zank G. P., eds, 8th Interna-tional Conference of Numerical Modeling of Space PlasmaFlows (ASTRONUM 2013) Vol. 488 of Astronomical So-ciety of the Pacific Conference Series, Two- and Three-Dimensional Multi-Physics Simulations of Core CollapseSupernovae: A Brief Status Report and Summary of Re-sults from the “Oak Ridge” Group. p. 102Moiseenko S. G., Bisnovatyi-Kogan G. S., Ardeljan N. V.,2006, MNRAS, 370, 501 c (cid:13) , 000–000 Guilet et al.
M¨osta P., Richers S., Ott C. D., Haas R., Piro A. L., Boyd-stun K., Abdikamalov E., Reisswig C., Schnetter E., 2014,ApJL, 785, L29Mueller E., Hillebrandt W., 1979, A&A, 80, 147M¨uller B., Janka H.-T., Marek A., 2012, ApJ, 756, 84Nicholl M., Smartt S. J., Jerkstrand A., et al. 2013, Nature,502, 346Obergaulinger M., Cerd´a-Dur´an P., M¨uller E., Aloy M. A.,2009, A&A, 498, 241Obergaulinger M., Janka H.-T., 2011, preprint(arXiv:1101.1198)Obergaulinger M., Janka H.-T., Aloy M. A., 2014, MNRAS,445, 3169Ott C. D., Burrows A., Thompson T. A., Livne E., WalderR., 2006, ApJS, 164, 130Pessah M. E., Chan C.-k., 2008, ApJ, 684, 498Rampp M., Janka H.-T., 2002, A&A, 396, 361Sawai H., Yamada S., 2014, ApJL, 784, L10Sawai H., Yamada S., Suzuki H., 2013, ApJL, 770, L19Shibata M., Liu Y. T., Shapiro S. L., Stephens B. C., 2006,Phys. Rev. D, 74, 104026Squire J., Bhattacharjee A., 2014a, preprint(arXiv:1407.4742)Squire J., Bhattacharjee A., 2014b, Physical Review Let-ters, 113, 025006Symbalisty E. M. D., 1984, ApJ, 285, 729Takiwaki T., Kotake K., 2011, ApJ, 743, 30Takiwaki T., Kotake K., Sato K., 2009, ApJ, 691, 1360Thompson C., Duncan R. C., 1993, ApJ, 408, 194Thompson T. A., Quataert E., Burrows A., 2005, ApJ, 620,861van den Horn L. J., van Weert C. G., 1984, A&A, 136, 74Winteler C., K¨appeli R., Perego A., Arcones A., Vasset N.,Nishimura N., Liebend¨orfer M., Thielemann F.-K., 2012,ApJL, 750, L22Woods P. M., Thompson C., 2006, Soft gamma repeatersand anomalous X-ray pulsars: magnetar candidates. pp547–586Woosley S. E., 2010, ApJL, 719, L204Woosley S. E., Heger A., 2006, ApJ, 637, 914Woosley S. E., Heger A., Weaver T. A., 2002, Reviews ofModern Physics, 74, 1015Yoon S.-C., Langer N., 2005, A&A, 443, 643This paper has been typeset from a TEX/ L A TEX file preparedby the author. c (cid:13)000