Protoneutron star dynamos for generating magnetar-strength magneticfields

Peter B. Rau

Cornell University

(Dated: December 7, 2018)

I. INTRODUCTION

Magnetars are a class of neutron stars with ex-tremely strong magnetic fields, with surface fields reach-ing strengths of order 1014–1015 G, a factor of 100 ormore larger than those of usual radio pulsars. This dif-ference is observable from their electromagnetic emis-sion: while radio pulsars emit mostly radio waves andare powered by their rotation, magnetar emission ishighly variable. Two classes of objects now believedto be magnetars are anomalous X-ray pulsars (AXPs),which have persistent pulsar-like emission of soft X-rays< 10 keV and with long periods1 P = 2–12 s and largespin-down rates Ṗ ∼ 10−12–10−10 s s−1, and soft gammarepeaters (SGRs), which emit bursts of hard X-rays/softgamma-rays. Later identification of persistent, AXP-likeX-ray emission from SGRs and gamma-ray bursts fromAXPs has meant that these sources are now believedto be the same class of objects. Explanations for theiremission require stronger magnetic fields than those ofnormal pulsars, as does their large spin-down rate– forthe naive rotating magnetic dipole model in vacuum,this model implies2 B ∼ 1014–1015 G.

The source of the extreme magnetic fields that wouldexplain AXP and SGR emission is understood to re-quire an explanation beyond that for the already strongradio pulsar fields. The fossil field hypothesis, in whichflux conservation preserves and concentrates the mag-netic field of the neutron star progenitor to a strongerfield in a neutron star after core collapse, is the most sim-ple explanation. However, the strongest observed mainsequence (MS) star magnetic fields3 are of order 104 G,and when accounting for the huge loss of mass of the starduring supernova, would result in surface magnetic fieldstrengths of only 1014 in a most optimistic calculation,just at the threshold required for magnetars. Magnetarsalso appear to be created frequently, and the populationof MS stars with fields near 104 G appears insufficient toexplain the magnetar birth rate1. Thus the fossil fieldhypothesis, while seemingly sufficient to explain radiopulsar field strengths, is unsatisfactory to explain AXPsand SGRs.

The most widely-accepted theory to explain AXPsand SGRs is the magnetar model, first expounded by

Duncan and Thompson4;5. In this model, a dynamoresulting from neutrino-led convection in the protoneu-tron star phase amplifies a magnetic field by convert-ing turbulent kinetic energy into magnetic energy. Themagnetic field is then locked into the surface by sta-ble stratification at the end of convection. Magneticstresses from the strong field create cracks in the neu-tron star crust, releasing outbursts of gamma-rays thatare observed as SGRs, while charged particles acceler-ated along field lines onto the surface lead to localizedsoft X-ray surface thermal emission that is observed asAXPs1. This model has been followed by other modelsfor the dynamo mechanism, including those using themagnetorotational instability (MRI) and Tayler insta-bility. These share the common feature that they occurin the protoneutron star formed immediately after corecollapse and act only ephemerally, since the dynamo pro-cesses are suppressed by stable stratification of the staras it cools and the formation of the crust, which anchorsthe field lines.

In this review, we first discuss basic dynamo theory, inparticular kinematic dynamo theory, a type of mean fieldtheory, and introduce the two basic mean field dynamomechanisms, the α-and ω-effects. The seminal Duncan–Thompson magnetar model, with neutrino-driven con-vection in protoneutron stars responsible for the dy-namo, is then reviewed. Two recent papers on alterna-tive dynamo mechanisms are then discussed. The firstis a numerical result demonstrating a magnetorotationalinstability (MRI)-driven dynamo in a rapidly-rotatingprotoneutron star. We then discuss a proposal that ther-mode instability, in conjunction with the Tayler insta-bility of toroidal magnetic fields, could serve as a pro-toneutron star dynamo.

II. INTRODUCTION TO STELLAR DYNAMO THEORY

A stellar dynamo is a mechanism that converts ki-netic energy from fluid motion in a star into magneticenergy, thus sustaining a magnetic field or increasing itsstrength. The evolution of the magnetic field in a staris controlled by the equations of magnetohydrodynamics(MHD), most importantly the induction equation, which

is derived by replacing the electric field in Faraday’s Lawusing Ohm’s Law. The latter (in cgs units) is

J = σ(E +

v

c×B

)(1)

for electrical conductivity σ and fluid flow velocity v.This gives as the induction equation

∂B

∂t= ∇×

(v ×B− c

σJ). (2)

Usually the displacement current term in Ampère’s Lawcan be ignored, so it can be used to replace J in theinduction equation

∂B

∂t= ηm∇2B +∇× (v ×B) , (3)

where ηm = c2/(4πσµ) is the magnetic diffusivity, and

we have assumed a uniform permeability µ. ηm is ameasure of magnetic field dissipation, so the first termEq. (3) in is an Ohmic dissipative term. The second termis responsible for dynamo action. The relative effects ofthe two terms is characterized by the magnetic Reynoldsnumber

Rem =vL

ηm, (4)

where v and L are a characteristic fluid velocity andlength scale respectively. Rem is usually large in as-trophysical applications, and so the diffusion term ofthe induction equation is often ignored, correspondingto perfect conduction and the ideal MHD limit.

To self-consistently solve the induction equation, weneed to solve the full system of equations of MHD,since the fluid velocity will itself be affected by theLorentz force from the magnetic field that it helps gen-erate. This involves simultaneously solving the continu-ity equation, the Navier-Stokes equation (including theLorentz force), energy conservation equation and Pois-son equation (or Einstein field equations in a generalrelativistic calculation). A numerical simulation of thefull system of equations of general relativistic MHD ina protoneutron star dynamo is discussed in Section V.For now, we examine solutions of the induction equationgiven a particular flow velocity v- this is known as kine-matic dynamo theory. In kinematic dynamo theory, theback-reaction acting on the fluid flow due to the Lorentzforce from the generated magnetic field is ignored, so theresults are unrealistic, but it is still useful to help un-derstand the processes involved in a dynamo.

Kinematic dynamo theory6;7 involves solving themean-field induction equation- that is, the inductionequation where the magnetic field and flow velocity arereplaced by their Reynolds-decomposed versions

v = 〈v〉+ V , B = 〈B〉+ b (5)

where the quantities in angle brackets are the mean(time-averaged) flow and mean magnetic field, and theother quantities are fluctuations with mean zero. Thefluctuations are due to small-scale velocity flows andmagnetic fields associated with turbulence. Taking themean of the induction equation then gives

∂〈B〉∂t

= ∇× (〈v〉 × 〈B〉+ 〈E〉) + ηm∇2〈B〉 (6)

where 〈E〉 = 〈V × b〉 is the mean EMF. To proceed, wesuppose that a closure relation exists linking the meanEMF to the mean magnetic field of the form7

〈E〉 = α · 〈B〉 − η · (∇× 〈B〉) + ... (7)where α and η are tensors. Assuming for simplicity thatboth are isotropic, the mean field induction equationbecomes

∂〈B〉∂t

= ∇× (〈v〉 × 〈B〉+ α〈B〉) + (ηm+ηt)∇2〈B〉 (8)

where we used α = αI, η = ηtI where I is the identitytensor.

The turbulent transport coefficients α and ηt can becalculated from the average properties of the turbulence.A method for doing so is the first-order smoothing ap-proximation (FOSA), in which only terms linear in thefluctuating quantities are retained in the induction equa-tion. Subtracting Eq. (6) from Eq. (2) gives a time-evolution equation for b, and integrating this with re-spect to time after dropping the quadratic terms in band V gives an equation for b which can be insertedinto E = 〈V ·b〉. By comparing this result to Eq. (6), αand ηt are found. This calculation gives

6

α = − 13τcor〈V · ω〉, (9)

ηt =1

3τcor〈V · V〉, (10)

where ω = ∇×V is the vorticity of the turbulence andτcor is the correlation time (roughly the period of motionof the largest turbulent eddies). α depends on the helic-ity h = V · ω of the turbulence, while ηt, the turbulentcontribution to the magnetic diffusivity, depends on itskinetic energy density.

The non-dissipative term in Eq. (8) has two contri-butions, both of which can lead to a dynamo. In theabsence of a mean flow 〈v〉 = 0 and assuming constantα, we have

∂〈B〉∂t

= α∇× 〈B〉+ ηtot∇2〈B〉, (11)

where ηtot = ηm + ηt is the total magnetic diffusivity.We look for solutions of the form6

〈B〉 = 〈B(k)〉eik·x+λt, (12)

2

J

BV

ω 〈h〉 = 〈V · ω〉 > 0⇒ α < 0

FIG. 1: Illustration of the α-effect, showing how thehelicity of a turbulent fluid parcel generates a loop inthe initial magnetic field B and induces a current J in

the opposite direction to the field.

where λ > 0 is required for a growing magnetic fieldand thus a successful dynamo, and k is the wave vectorof the large-scale (mean) magnetic field. Inserting intoEq. (11) gives

λ0 = −ηtotk2, λ± = −ηtotk2 ± |αk|. (13)

λ0 corresponds to a non-solenoidal (k · 〈B〉 6= 0) eigen-function, so we ignore it. λ > 0 thus requires

1 <|α|

ηtot|k|. (14)

The dynamo mechanism in which a magnetic field is sus-tained or amplified by turbulence in this way is calledthe α-effect. FIG. 1 shows an illustration of this mech-anism: a fluid parcel moving upward and rotating suchthat its helicity h > 0 (so α < 0 by Eq. (9)) generates aloop in the initial magnetic field B, corresponding to aninduced current J antiparallel to B.

Now suppose that there is a mean flow given in theform of a linear shear

〈v〉 = ωxŷ, (15)

where we use Cartesian coordinates for simplicity (ŷ be-

ing locally analogous to φ̂ in spherical coordinates). Ina star, the quantity ω (not to be confused with the vec-tor quantity vorticity), representing the strength of theshear, is related to the differential rotation ∂Ω/∂r

ω = r∂Ω

∂r, (16)

where Ω is the rotational frequency of the star. Thelinear shear gives a term of the form

∇× (〈v〉 × 〈B〉) = 〈Bx〉ωŷ, (17)

which shows that a magnetic field in the same direc-tion as the flow will be generated. This mechanism isknown as the ω-effect. FIG. 2 a–d shows how in the ω-effect, differential rotation twists the poloidal magneticfield, like that expected from the simple dipole modelof a pulsar, to form a toroidal field. Note that the ω-effect cannot regenerate a magnetic field, only transfer itfrom one direction to (poloidal) into another (toroidal).The “dynamo loop” thus must be completed by turbu-lence (like in the α-effect), which regenerates and am-plifies the poloidal field (FIG. 2 e–f). This combineddynamo is called an α–ω dynamo. For purely axisym-metric (ky = 0) solutions, the solutions λ to Eq. (8) foran α–ω dynamo are6

λ0 = −ηtotk2, λ± = −ηtotk2 ±√α2k2 − iαkzω. (18)

λ0 again corresponds to a non-solenoidal solution, whileλ± ∈ C, so the condition for a growing magnetic field isRe(λ) > 0: for small k, this is

2 <|αkzω|η2totk

4. (19)

The α-effect itself can also generate a toroidal fieldstarting from a poloidal field, then regenerate thepoloidal field– this is an α2 dynamo. The action ofboth types of dynamo are illustrated schematically inFIG. 3. Often in α–ω dynamos, the α-effect also con-tributes to the generation of the toroidal field, so some-times these dynamos are called α2–ω dynamos. We alsomake clear that we have only discussed how the mag-netic field is transformed or amplified kinematically andhave not discussed how it saturates. The mechanismsdiscussed above are also only mean-field solutions, anddo not necessarily describe all physical stellar dynamos,though they do generally contribute to some extent indynamo action and are often good descriptions for whatwe believe happens– for example, the solar dynamo iswell-described as an α–ω dynamo6.

How the turbulence which powers the α-effect is gen-erated has not yet been discussed. In Sections (IV–VI),we review specific examples of instabilities which gen-erate the turbulence necessary to complete the dynamoprocess.

III. STABILITY OF THE MAGNETIC FIELD

The stability of a magnetic field formed or amplifiedby a dynamo is important- if the fields formed are not ina stable configuration, they will not survive long enoughto be observed through their effect on electromagneticemission from the star. For field strengths under 1013

G, the neutron star crust, formed after ∼ 100 s, can pin

3

FIG. 2: Illustration of the α–ω dynamo: an initiallypoloidal magnetic field (a) is twisted into a toroidalfield by the differential rotation of the star (ω-effect,b–d). Turbulencence then leads to the formation ofmagnetic field loops attached to the toroidal field

which regenerate the poloidal field (α-effect, e–f)8.

Bpol Btor Bpol Btor

ω

α

α

α

FIG. 3: Schematic illustration of an α-ω dynamo (left)and an α2 dynamo (right). Bpol and Btor are the

poloidal and toroidal components of the magnetic field,respectively.

magnetic field lines to the surface3. However, magnetar-strength fields are likely too strong to be anchored to thecrust. Instead, two effects combine to stabilize the mag-netic fields generated by the dynamo in magnetars. Thefirst of these is stable stratification of the star– absentthis, buoyancy will carry the field lines through the sur-face and into the plasma beyond, where they diffuse.

The second effect which stabilizes magnetar-strengthmagnetic fields is conservation of magnetic helicity. Themagnetic helicity H is a property of an entire magneticfield configuration dependent on the field itself and themagnetic vector potential A, and is represented mathe-matically as

H =∫V

B ·AdV, (20)

where V is a volume at whose boundaries B · n̂ = 0 (n̂being an outward-pointing unit vector to V ) or at whichthe fields go to zero. Taking the partial derivative of thiswith respect to time6

∂H∂t

=

∫V

[−E ·B + B · ∇φ−A · ∇ ×E] dV

= − 2∫V

E ·BdV +∮∂V

(A×E + φB) · n̂dS

= − 2σC, (21)

for current helicity C =∫

B · JdV , and where the sur-face integral goes to zero since the fields are assumedto go to zero far from the star. In the perfect conduc-tivity limit, σ → ∞, this says that magnetic helicityis conserved, and realistic finite conductivities are usu-ally high enough (or Rem high enough) to mean thatH changes slowly. This says that a magnetic field con-figuration with nonzero magnetic helicity is stabilizedby the (approximate) conservation of magnetic helic-ity. Such stable configurations usually consist of poloidalmagnetic field lines threaded through a roughly axisym-metric toroidal magnetic– such a field configuration isdepicted in FIG. 4. Other, more simple field configura-tions are ruled out: a purely poloidal field is not stable10,and a purely axisymmetric field that vanishes at infin-ity, like an axisymmetric purely toroidal field, cannot bemaintained by a dynamo due to Cowling’s theorem11.

IV. CONVECTION-DRIVEN DYNAMO: THEDUNCAN–THOMPSON MODEL

The original protoneutron star dynamo model is theDuncan–Thompson model4;5, in which neutrino-drivenconvection in the outer layers of a protoneutron star isresponsible for amplification of a seed magnetic field.This dynamo does not correspond to any of the mean-field dynamos discussed in Section II, but instead is aturbulent dynamo6, in which small-scale, chaotic stretch-ing of magnetic field lines results in the formation oflarge-scale magnetic structures.

Convection occurs in a region of the protoneutron starformed after core collapse if

ds

dr+

(∂s

∂Yl

)p,ρ

dYldr

< 0, (22)

where s is the entropy per baryon, Yl the lepton numberper baryon, p is the pressure and ρ the mass density.In the outer third to half of a protoneutron star whichhas been shock heated after a core bounce, Eq. (22) issatisfied, as the shock establishes a radially decreasingentropy gradient by dissociating heavy nuclei. This is re-inforced by the large neutrino flux (energy ∼ 1053 erg)

4

FIG. 4: a: Illustrations of a stable magnetic fieldconfiguration with finite magnetic helicity, consisting ofpoloidal field lines threaded through an approximately

axisymmetric toroidal field. The toroidal field lines,which do not emerge from the stellar surface, areshown in blue, while the poloidal lines which do

penetrate the surface are in red. The thick grey circlerepresents the axis of the torus. b: A

toroidally-averaged magnetic field diagram9.

within the first ∼ 3 s of the protoneutron star’s life,which moves heat and leptons outward. This convectioncontinues for the first ∼ 10 s of the protoneutron star’slife, when the outer layers of the star become transparentto neutrinos. Vigorous convective cells that are roughlythe pressure scale height lp = p/ρg ∼ 1 km in size andwith convective overturn time τcon = vcon/lp ∼ 0.1–1 msare established. Here g the magnitude of the local grav-itational acceleration and vcon is the convective velocity.

Since most of the heat and momentum is trans-ported by the neutrinos in this convection, the relevantReynolds number for determining whether the convec-tion is laminar or turbulent is

Re[ν] =lpvconν[ν]

≈ 6× 103, (23)

where ν[ν] is the neutrino shear viscosity. The cutoffbetween laminar and turbulent flow is Re ≈ 103, so theconvection here is turbulent. The kinetic energy of theconvective turbulence is transferred from large to smallscales in a turbulent energy cascade which terminates atthe Kolmogorov length scale, where viscosity dominates

and the turbulent kinetic energy is dissipated as heat.For neutrino viscosity damping, this scale is

lvis[ν] =

(ν[ν]3

�̇

)3/4= Re[ν]−3/4lp ≈ 200 cm, (24)

where �̇ is the rate of turbulent kinetic energy dissipationper unit mass. Neutrino viscosity will only effectivelydamp the turbulence if many neutrino collisions can oc-cur within lvis[ν] i.e. if the neutrino mean free pathis much less than lvis. However, typical neutrino meanfree paths in the young protoneutron star are ∼ 200 cm∼ lvis[ν], and so the turbulent cascade will not be ter-minated by neutron viscosity damping but by electronviscosity damping, and so it ends at a length scale

lvis[e] = Re[e]−3/4lp ≈ 8 µm. (25)

How does the turbulence amplify the magnetic field?The turbulent convective cells tend to expel magneticfield lines, concentrating magnetic flux into flux ropes–loops of twisted flux lines– which are stretched by theconvective motion, further concentrating the flux intoa smaller radius rope. This concentration of magneticflux is opposed by magnetic diffusivity, which tends toweaken the field. However, the magnetic Reynolds num-ber for the convective motion is

Rem =lpvconηm

∼ 1017, (26)

and so for efficient Ohmic dissipation of the field, thelast term in Eq. (2) must be non-negligibly small, whichrequires extremely small-scale field gradients for suchsmall ηm. The stretching of the field lines tends tosmooth the field, reducing Ohmic dissipation.

Flux conservation during core collapse provides a large(∼ 1011–1012 G) seed field, and this will suppress small-scale turbulent diffusion of the flux when

vA =B√

4πρµ> v(l), (27)

where vA is the Alfvén velocity, µ the magnetic perme-ability and v(l) a typical turbulent velocity at lengthscale l. As the field is further stretched by the tur-bulent convection, the field in the flux ropes is furtherstrengthened, which suppresses turbulent diffusion at in-creasingly longer length scales. There is thus an inversecascade of energy from small-scale magnetic structuresof size ldiss = lp/

√Rem, the length scale for efficient

Ohmic dissipation, into large- scale, high field flux ropes.The saturation magnetic field strength Bsat inside a

flux rope is determined by the balance of drag forceacting on the rope as it moves through the fluid andthe magnetic tension within it, and is reached when theAlfvén velocity equals the convective velocity5, or

Bsat ≈√

4πρµv2con = 6× 1015vcon,8(ρ14

3

)1/3G, (28)

5

where vcon,8 = vcon/(108 cm s−1) and ρ14 =

ρ/(1014 g cm−3). This says that magnetar strengthfields can be generated if the dynamo is efficient andthe field is amplified to its saturated value or near it.An inefficient dynamo will still maintain or amplify theseed field from flux conservation to values expected innormal radio pulsars.

The flux ropes will rise to the surface through mag-netic buoyancy, and can even do so after the neutrino-driven convection ends and the fluid becomes stablystratified. A magnetized fluid is unstable to convectionif12

0 >N2

v2A+

g

Γc2s

d

drln

(B

ρ

), (29)

where N is the Brunt–Väisälä (buoyant) frequency andΓ = cp/cv for specific heats cp and cv. Even if N ispositive (that is, displaced fluid parcels would oscillatestably in the radial direction), a radially weakening mag-netic field can destabilize the fluid. Eventually stablestratification and the establishment of a magnetic fieldconfiguration with nonzero helicity will result in fluxrope loops which partially penetrate through the sur-face.

While we have spent the most time in this section de-scribing the convective turbulent dynamo, α–ω and α2

dynamo processes may also be active in some protoneu-tron stars. Both α and ω-effects are active on timescalesof the rotation period P , but the fields they generate aredissipated by turbulent diffusion on the convective over-turn time τcon ≈ P/Ro, where Ro is the Rossby numbercharacterizing the relative strength of inertial forces tothe Coriolis force. Ro decreases with decreasing rotationperiod, so protoneutron stars with initial rotation peri-ods P < 3 ms will sustain efficient α–ω dynamos5. Theconvective overturn time also decreases by nearly an or-der of magnitude in the latter stages of neutrino-drivenconvection, so α–ω dynamo action is likely during atleast part of the protoneutron star’s lifetime for P . 30ms. However, there is likely to be more energy availableto be converted into magnetic energy from the convec-tive motion than from differential rotation, especially forstars without very short periods, as the relative size ofthe available energy from the two sources scales as

Ediff. rot.Econv.

∼ Ro−2. (30)

It may also be difficult to obtain the maximum field val-ues of order Bsat because a turbulent dynamo is unlikelyto operate at equipartition between turbulent kineticand magnetic energy, which could reduce the maximumattainable field by an order of magnitude or more1, andstable stratification of the outer layers of the star canweaken convection. This has led to the considering dif-ferent mechanisms for dynamo action in protoneutron

stars since the Duncan–Thompson model was first pro-posed.

V. MAGNETOROTATIONAL INSTABILITY-DRIVENDYNAMO

The magnetorotational instability (MRI)-driven dy-namo is an example of an α–ω -like dynamo in whichdifferential rotation winds a poloidal field into a toroidalfield while unstable MRI modes transfer energy into tur-bulence which regenerates and amplifies the poloidalfield, thus closing the dynamo loop. This mecha-nism was demonstrated to be a viable dynamo processthrough a nonrelativistic numerical simulation13, andlater through a fully general-relativistic MHD simulationfor a protoneutron star14. We review the results of thesecond reference (Mösta et al. 2015, hence MORRSH18)below.

The MORRSH18 simulation takes as its initial condi-tions a simulation of collapse of a rapidly-spinning (ironcore period P0 = 2.25 s) progenitor star with a purelypoloidal large-scale magnetic field. The resulting pro-toneutron star has an initial period P = 1.18 ms imme-diately after core bounce, and a poloidal field with mag-nitude B = 1010 G at this time. As the protoneutronstar contracts and rotates, it strengthens its magneticfield and winds up the poloidal field into a toroidal field.Thus, at time tmap = 20 ms after core bounce, the fieldhas poloidal and toroidal components of roughly equalmagnitude B = 7 × 1014 G. At this time, MORRSH18map the simulation to a higher-resolution grid in orderto resolve the MRI which develops in the shear layersurrounding the core of the protoneutron star. Four dif-ferent grid resolutions were used: dx = 500m, 200 m,100 m and 50 m. This was done in order to ensure thatthe fastest-growing mode (FGM) of the MRI, the mostimportant mode in the turbulence which transfers en-ergy to the magnetic field, was resolved.

The MRI is an instability of a rotating magnetizedfluid to radial perturbations if its rotational frequencyΩ decreases as a function of radius r. In a cylindricalcoordinate system, consider a fluid with background flowv = Ω×r and a magnetic field applied in the z-direction.Assuming axisymmetric perturbations with exponentialspace and time dependence

exp(iσt+ ikzz), (31)

and ignoring curvature terms, the radial displacementfield ξr of the modes is determined by

15

d2ξrdr2

=

(k2z

σ2 − Ω2)(

σ2 − Ω2A − Φ(r)−4Ω2Ω2Aσ2 − Ω2A

)ξr,

(32)

6

where Ω2A = v2Ak

2z and Φ(r) = 4Ω

2 + rdΩ2/dr isthe Rayleigh discriminant. If we further assume thatξr(r) ∝ exp(ikrr), then we obtain a quartic equation forthe mode frequency σ

σ4 − σ2(

2Ω2A +k2rk2

Φ(r)

)+ Ω2A

(Ω2A +

k2rk2rdΩ2

dr

)= 0

(33)where k2 = k2r +k

2z . A quartic equation of this form will

have a negative imaginary route, corresponding to anunstably growing mode, if the σ0 coefficient is negative,or if

−r dΩ2

dr>B2k2

4πρµ

(k2zk2r

). (34)

This says that as long as dΩ2

dr < 0, their will be un-stable modes even for vanishingly small magnetic fields,and that even for strong fields, long wavelength modescan still be unstable. Stable stratification opposes theinstability, and so in the weak-field limit the stabilitycriterion is given by

CMRI =N2

Ω2+

r

Ω2dΩ2

dr, (35)

with N being the Brunt–Väisälä as before. For CMRI <0, the background fluid flow is unstable to the MRI.

FIG. 5 a–b show CMRI throughout the protoneutronstar at tmap = 20 ms after core bounce, indicating thata shear layer around the star is initially unstable to theMRI. The local wavelength λFGM and the growth timeτFGM of the FGM of the MRI for MORSSH18’s sim-ulation are also shown in panels c–d. This indicatesthat in the protoneutron star’s shear layer around itscore, the FGM grows fast enough to reach its satura-tion amplitude and have its energy converted into tur-bulent kinetic energy (and then into magnetic energy)many times within a few ms after core bounce. The MRImodes’ growth times do not depend on the field strength,so it should be able to drive an efficient dynamo evenfor smaller seed magnetic fields.

The Fourier spectra of the turbulent kinetic energyEkin and magnetic energy Emag as a function of dimen-sionless wave number k ≡ L/λ (L = length of simula-tion domain) are shown in FIG. 6. Panel a shows thathigh resolutions are needed to Panel b shows that for10 . k . 100 and dx = 50 m, the magnetic energyapproaches (within a factor of 3) equipartition with theturbulent kinetic energy as the turbulent energy is con-verted into magnetic energy. Panel c shows the growthin the magnetic field energy with time, notably thatthere is an inverse cascade of energy to large-scale/smallwave number magnetic features, with the dimensionlesswave number k = 4 (length scale 5 km) component ofthe magnetic energy Emag growing approximately expo-nentially with growth time τ ≈ 3.5 ms.

The evolution of the poloidal magnetic field as a func-tion of time from tmap is shown in FIG. 7. The toroidalfield evolution, not shown, is very similar to that of thepoloidal component. Notably, the field is amplified onlya small amount or not at all for the lowest two reso-lutions, indicating that these were not small enough toresolve the fastest-growing mode (FGM) of the MRI.The two highest-resolution simulations show initially ex-ponential growth of the magnetic field with growth timeτ ≈ 0.5 ms, with the magnetic field reaching its fully tur-bulent saturation strength after only 3 ms, though thesaturated field differs between resolutions. This growthtime is similar to that expected for the FGM of the MRIin the shear layer, τFGM, from FIG. 5 panel d. The ra-dial component of the magnetic field is itself plotted inFIG. 8 at a time t − tmap = 7.6 ms, showing fluctua-tions at the length scale of the FGM ∼ λFGM. Thesesimulations thus show that magnetar-strength magneticfields of & 1015 G are quite feasibly generated by anMRI-driven dynamo for rapidly-rotating protoneutronstars.

VI. r-MODE INSTABILITY-DRIVEN DYNAMO

The r-mode instability–driven dynamo is another ex-ample of an α–ω dynamo, where the differential rota-tion which twists the poloidal field into a toroidal fieldis generated by unstable r-modes. The dynamo loop isthen closed by the Tayler instability of a toroidal field,which acts like an α-effect to regenerate and amplify thepoloidal field. We first review the r-mode instability andthe Tayler instability involved in the dynamo processbefore discussing the results of a calculation by Chengand Yu 201416 (hence CY14) to determine the viabilityof this as a mechanism to generate magnetar-strengthmagnetic fields in a protoneutron star.

The r-mode instability17 occurs in rotating stars whena r-mode– an oscillation with the Coriolis force as itsrestoring force– is driven unstable by the emission ofgravitational radiation. For a star with rotational fre-quency Ω and r-mode frequencies in the co-rotatingframe ±ωr, the mode frequencies in the inertial frameare

ω′r,± = Ω± ωr. (36)If Ω > ωr, both modes will be prograde in the inertialframe, even as the −ωr mode is retrograde in the co-rotating frame (FIG. 9). The r-mode generates a massquadrupole moment for the star, and so the star emitsgravitational radiation that carries away energy and an-gular momentum. Since the −ωr mode already has neg-ative angular momentum in the co-rotating frame, ifΩ > ωr, the magnitude of the −ωr mode’s angular mo-mentum will thus increase with time, and so its energy

7

FIG. 5: Panels a–b: MRI stability criterion CMRI throughout the protoneutron star at 20 ms after core bounce inthe equatorial plane (z = 0) and y = 0 plane, respectively. The protoneutron star core initially has a radius ≈ 20km. Panels c–d: Wavelength λFGM and growth time τFGM of the fastest-growing mode (FGM) of the MRI in the

equatorial plane of the protoneutron star 20 ms after core bounce14.

grows unstably, and so an r-mode being prograde in theinertial frame while retrograde in the co-rotating frameis the condition for the r-mode instability. From an en-ergy perspective, the energy of the mode in each frameis related by

Er = Ei −Ω · J (37)

where i/r indicates inertial/co-rotating frame and J isthe angular momentum. If Ei and J are both negative,Er can still increase. The instability can be quenchedby various viscosity sources, and in young neutron starsthe formation of the crust could also suppress it16. Thel = m = 2 r-mode is the most important for the r-modeinstability, and is the mode on which CY18 focus.

The Tayler instability18 occurs in toroidal magneticfields once they reach a threshold strength comparedto the poloidal field, after which the instability causesthe toroidal field lines to reorientate. For a toroidalfield created by differential rotation and the winding up

of a poloidal field, this instability thus regenerates thepoloidal field. The two conditions for the instability tooccur are19

∂ lnBtor∂ ln r

>

{1, m = 0m2

2 − 1, m 6= 0(38)

(vAr

)3>ηmN

2

L2> Ω3, (39)

where m is the azimuthal order of the unstable modeand L is a length scale representing the cylindrical ra-dial extent of the field. The first condition specifies therequired rate of change of the toroidal magnetic fieldin the radial direction required for a mode of order mto be unstable, whereas the second specifies the toroidalmagnetic field strength Btor ∝ vA required for the insta-bility, since stable stratification (positive Brunt-Väisäläfrequency N) and magnetic diffusion suppress the insta-bility. The second condition also indicates that insta-bility is aided by the increase of Btor and decrease by

8

FIG. 6: Panel a: Magnetic energy Emag as a functionof the dimensionless wave number k = L/λ for different

resolutions dx at t− tmap = 10 ms. Emag at 50 mresolution at t− tmap = 0 ms is also shown. The

turbulent kinetic Ekin for 50 ms resolution att− tmap = 10 ms is shown for comparison, as is

Ekin(k) ∝ k−5/3, which is expected from Kolmogorovturbulence theory. Panel b: like panel a, but only

showing Emag(k) for 50 m resolution at varying timest− tmap. Panel c: Emag for fixed k as a function of

time t− tmap, showing the inverse cascade of magneticenergy into large-scale fields (small k) through the

growth of the k = 4 component of Emag14.

FIG. 7: Evolution of the poloidal magnetic field Bp asa function of time after the map to the high resolution

domain. Panel a shows the global maximum of thefield, while panel b shows the maximum value within a

15 km layer centered on the equatorial plane. Theresolution of the simulations is labeled, as is an

exponentially-growing fit14.

gravitational radiation of Ω through the action of theunstable r-mode. The growth time of the Tayler insta-bility is of order R/vA for stellar radius R.

CY18 analyzed the viability of the r-mode instabil-ity dynamo mechanism by solving a system of coupleddifferential equations for the r-mode angular momentumJr, the volume-averaged toroidal magnetic field Btor andthe total angular momentum of the star Jtot = Jr + IΩfor moment of inertia I:

dJrdt

= 2Jr

(1

τGW,r− 1τv− 1τtor

), (40)

dBtordt

=3

2π

√5

14Bpol(2δ + 3)α

2Ω, (41)

dJtotdt

= − 3α2I∗Ω

τGW,r− IΩτGW,tor

− IΩτdipole

. (42)

Here α is the dimensionless amplitude of the r-mode, δis a parameter which sets the initial amount of differen-tial rotation in the protoneutron star and the saturationamplitude αsat of the r-mode, Bpol the surface dipole

9

FIG. 8: Radial component of the magnetic field att− tmap = 7.6 ms after core bounce for the four

resolutions used in the simulation (top right of eachpanel)14.. When the FGM of the MRI is resolved, the

radial field oscillates on a scale λFGM ∼ 1 km

FIG. 9: Illustration of condition for r-mode instability:in the co-rotating frame (left), the motion of ther-mode (represented by an l = m = 2 spherical

harmonic) is retrograde, while in the inertial frame(right), the motion of the r-mode is prograde17.

(poloidal) field of the star and I∗ an effective moment ofinertia for the r-mode. The various timescales τ whichdetermine the rate of change of angular momentum arespecified in Table I. To model the Tayler instabilitywhich amplifies the surface dipole field, the field is heldat its initial value until the Tayler instability condition

vA/R > Ω is satisfied. When this is true,

Bpol = γBtor,vAR

> Ω, (43)

where CY18 adopt γ = 0.01, since it is likely that manypoloidal field lines are closed within the protoneutronstar interior, so the surface dipole field is weaker thanthe toroidal field.

Eqs. (40) and (42) can be combined to give separatedifferential equations for the time evolution of the ro-tational frequency Ω and the dimensionless r-mode am-plitude α. These equations were solved for three differ-ent initial rotational frequencies Ω0, and the results areshown in FIG. 10. This calculation used a seed poloidalfield Bpol,i = 10

11 G and an initial dimensionless ampli-tude αi = 10

−10.FIG: 10 indicates that internal toroidal fields of 1016−

1017 G can be generated within 102 − 103 s of the on-set of the r-mode instability for Ω0 & 3ΩK/4, withweaker surface poloidal fields of 1015 G generated onthe same timescales. For Ω0 = ΩK/2, both the toroidalfield growth times and spin-down are slower, delayingthe onset of the Tayler instability and hence the latergrowth of the poloidal field. For the Tayler instabilityto occur at all here requires spin-down via gravitationalradiation due to the magnetic deformation of the starby the toroidal field on timescale τGW,tor. CY18 founda critical initial period Pi,c = 1.7 ms for which a star canspin down fast enough to generate poloidal fields ∼ 1015G through r-mode induced GW alone. They also findthat the r-mode instability is damped by viscosity forPi > 3 ms, so this dynamo mechanism could only gen-erate magnetar-strength surface fields for very rapidlyrotating protoneutron stars.

VII. SUMMARY

We have reviewed dynamo mechanisms which maybe responsible for the generation of magnetar-strengthmagnetic fields in protoneutron stars, focusing onthe convective turbulent dynamo, the MRI-driven dy-namo and an r-mode instability-driven dynamo. Theneutrino-driven convective dynamo is general enough tooccur in most protoneutron stars, while nearly Keplerianinitial periods are required for the r-mode instability-driven dynamo to generate surface poloidal fields of1015 G. The model of the poloidal field used in CY14in which the field is simply set to be proportional tothe toroidal field once the Tayler instability is acti-vated could also be improved, which may yield differ-ent surface poloidal field strengths. The study of theMRI instability-driven dynamo discussed here only useda single, rapidly-rotating protoneutron star (Pi = 1.2ms) in its simulation, so additional simulations could

10

TABLE I: Timescales appearing in system of equations governing the r-mode instability-driven dynamo in CY18.

Timescale Description

τGW,r Timescale of gravitational radiation induced by r-mode

τv Timescale of viscous damping of r-mode

τtor Timescale of formation of toroidal field ∝ Er/Ėtor for r-mode energy Er and Etor ∝ B2tor

τGW,tor Timescale of gravitational radiation induced by quadrupole deformation of star by the toroidal field

τdipole Timescale of magnetic dipole radiation ∝ B−2pol

FIG. 10: Time evolution of magnetic field B (solid lineis internal toroidal field, dashed is surface poloidal

field), rotational frequency ν = Ω/2π and dimensionlessr-mode amplitude α for three different initial

rotational frequencies Ω0. Red, green and blue linesindicate Ω0 = ΩK, 3ΩK/4 and ΩK/2, respectively

16.

help determine if this mechanism alone could gener-ate magnetar-strength fields in slower-rotating stars. A

combination of dynamo mechanisms can also act in con-junction to generate strong fields.

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