Some consequences of shear on galactic dynamos with helicity fluxes
MMon. Not. R. Astron. Soc. , 1– ?? () Printed 13 November 2018 (MN L A TEX style file v2.2)
Some consequences of shear on galactic dynamos withhelicity fluxes
Hongzhe Zhou (cid:63) , Eric G. Blackman , , † ] Department of Physics and Astronomy, University of Rochester, Rochester NY, 14627, USA Laboratory for Laser Energetics, University of Rochester, Rochester NY, 14623, USA Kavli Institute for Theoretical Physics, UC Santa Barbara, Santa Barbara, CA, 93106, USA
ABSTRACT
Galactic dynamo models sustained by supernova (SN) driven turbulence and differ-ential rotation have revealed that the sustenance of large scale fields requires a fluxof small scale magnetic helicity to be viable. Here we generalize a minimalist analyticversion of such galactic dynamos to explore some heretofore unincluded contributionsfrom shear on the total turbulent energy and turbulent correlation time, with the he-licity fluxes maintained by either winds, diffusion, or magnetic buoyancy. We constructan analytic framework for modeling the turbulent energy and correlation time as func-tion of SN rate and shear. We compare our prescription with previous approaches thatonly include rotation. The solutions depend separately on the rotation period and theeddy turnover time and not just on their ratio (the Rossby number). We considermodels in which these two time scales are allowed to be independent and also a casein which they are mutually dependent on radius when a radial dependent SN ratemodel is invoked. For the case of a fixed rotation period (or fixed radius) we show thatthe influence of shear is dramatic for low Rossby numbers, reducing the correlationtime of the turbulence, which in turn, strongly reduces the saturation value of thedynamo compared to the case when the shear is ignored. We also show that even inthe absence of winds or diffusive fluxes, magnetic buoyancy may be able to sustainsufficient helicity fluxes to avoid quenching.
Key words: galaxies: magnetic fields; dynamo; turbulence; MHD; galaxies: ISM
In situ galactic dynamo theory has long been a leadingparadigm to explain the ordered large scale magnetic fieldsof galaxies (Ruzmaikin et al. 1988). In this paradigm, a weakseed field, perhaps supplied primordially, is amplified via theaction of turbulence and differential rotation in the galacticinterstellar medium. How such dynamos work in detail, hasbeen a longstanding research enterprise (Ruzmaikin et al.1988; Brandenburg & Subramanian 2005; Shukurov et al.2006; Hanasz et al. 2009; Chamandy et al. 2014; Blackman2015; Kulsrud 2015).Standard (20th century) mean field α − Ω galactic dy-namos typically have at least three key ingredients (1) su-pernova driven turbulence, which in the presence of galacticrotation and stratification produces a kinetic helicity driven (cid:63)
E-mail: [email protected] † E-mail: [email protected] ” α ” effect that converts toroidal to poloidal field and (2)differential rotation that shears the poloidal field into thetoroidal direction and (3) some kind of turbulent diffusionor loss term of the mean field in a thin disk that ensures thethat the net toroidal flux in the disk reflects the observedfield geometry (e.g. quadrupole).A challenge for 20th century galactic dynamo theory hasbeen the absence of a physical understanding of how the dy-namo saturates. That basic theory is kinematic, consideringonly the growth of the large scale field without includingthe dynamics of the field on the driving flow. Intertwinedwith this deficiency has been the realization that standardmean field textbook α − Ω dynamos also do not conservemagnetic helicity (Blackman & Field 2000; Vishniac & Cho2001). (For reviews see Brandenburg & Subramanian (2005)and Blackman (2015)).Principles of dynamically including magnetic helicityconservation in MHD turbulence from Pouquet et al. (1976)and modified lessons from steady-state mean field consid-erations of Gruzinov & Diamond (1994) and Bhattacharjee& Yuan (1995) were synthesized into time-dependent mean c (cid:13) RAS a r X i v : . [ a s t r o - ph . GA ] A p r Zhou & Blackman field dynamical toy models (Blackman & Field 2002) usinga simple closure (now referred to as ”minimal τ ”). In thesemodels, the growth of a helical component of the large scalefield is accompanied by growth of the oppositely signed smallscale helical field which in turn, represents a backreactionthat saturates the dynamo. For dynamos without shear thisleads to a steady state, but for dynamos with shear, this canlead to catastrophic quenching alleviated only when helicityfluxes carry away the excess small scale field. Ultimately thisrequires a dynamo sustained by helicity fluxes (Blackman &Field (2000)). Depending on which terms in the electromo-tive force actually dominate, a complementary perspectiveis that the large scale dynamo is sustained directly via he-licity fluxes even in the absence of any kinetic helicity (e.g.Vishniac & Cho (2001); Vishniac & Shapovalov (2014)). He-licity flux driven dynamos are conceptually related to thesustenance of large scale fields in the different context oflaboratory magnetically dominated plasmas (Strauss 1985;Bhattacharjee & Hameiri 1986).Incorporation of some these principles has led the nu-merical demonstration of the helpful role of magnetic he-licity in numerical simulations of dynamos in stellar con-texts (Brandenburg & Sandin 2004; Chatterjee et al. 2011)as well as practical galactic dynamo models with helicityfluxes (Shukurov et al. 2006; Sur et al. 2007; Chamandy2016).A second challenge of galactic and mean field dynamotheory is to incorporate the influences of rotation and shearon the turbulence, dynamo coefficients, and EMF. One ap-proach is to expand the turbulent quantities into a basestate that is independent of shear and rotation plus correc-tions that depend on them. The resultant mean turbulentEMF (whose curl enters the growth if the mean magneticfield) can then be expanded into a sum of all possible termsthat are linear in the mean magnetic field and linear in themean rotation or shear (Brandenburg & Subramanian 2005;R¨adler & Stepanov 2006). The relevance and interpretationof each of these terms must be assessed independently fora given circumstance. However, this approach does not cap-ture all of the effects of rotation and shear to all orders.Doing so formally is impractical, but physical approachescan provide insight and shortcuts. The influence of rotation can be partly gauged by the ratioof the nonlinear term in the Navier Stokes equation to theCoriolis term in the rotating frame. This dimensionless ratio,the Rossby number, is given by Ro ( τ ed ) = ˜ Ro (Ω) = 1Ω τ ed , (1)where Ω is the rotation speed and τ ed is the eddy turnovertime, presently defined in terms of the turbulence suppliedspecifically by supernovae. The latter is important to keepin mind as we will also utilize a separate correlation time τ cor not necessarily equal to τ ed . The above equation intro-duces our convention of writing Ro for the Rossby numberfor fixed Ω allowing τ ed to vary and ˜ Ro for the Rossby num-ber at fixed τ ed allowing Ω to vary. How dynamos dependseparately on τ ed , Ω and on differential rotation is not com-pletely understood. Even for the basic α − Ω type dynamo, the question of how the kinetic component of the helicitycoefficient contribution α depends on rotation and shearwarrants revisiting for strong shear.There are a few precursors in this context. Ruediger(1978) calculated an effect of rotational quenching on α .Ruzmaikin et al. (1988) considered the effect of the Coriolisforce without shear and their prescription for the effect ofrotation on α can be recast by replacing the correlationtime of the turbulence τ c = τ ed Ro − / when Ro ≥
1, and τ c = τ ed otherwise. In Chamandy et al. (2016), the sameresulting piecewise-defined α was used. Blackman & Thomas(2015) and Blackman & Owen (2016) included an effect ofshear on the correlation time by arguing that τ cor equals τ ed times a factor that depends on Ro and shear.In the present paper we explore and generalize a phys-ical model for the influence of shear and rotation on both τ cor and the turbulent energy density for galactic dynamos.We will see that when Ro, ˜ Ro >>
1, the supernova turbu-lence dominates both the turbulent energy density and itscorrelation time. In the regime
Ro, ˜ Ro << τ ed , and then fix-ing τ ed and changing Ω. Then we consider a model in whichthey are mutually dependent on radius, via their connectionto the star formation rate. The need for this arises becausethe dynamo depends separately on those two parameters notjust in their dimensionless combination of the Rossby num-ber. We explicitly derive τ ed in terms of the SN rate andshow how τ cor changes as a function of rotation and shear.Both the effect of shear on the turbulent correlation timeand as a supplemental source of turbulent energy have notbeen included in galactic dynamo models, although in theabsence of SN, shear is expected to be a source of galacticturbulence (Sellwood & Balbus 1999).We also incorporate a magnetic buoyancy (MB) term(Parker 1966) in the helicity flux term of the dynamo equa-tions, generalizing the corresponding therm of Sur et al.(2007) which included only an advective wind flux term. Thebuoyant speed itself depends on the magnetic field, whichincreases the nonlinearity of the dynamo equations.The paper is arranged as follows. In Sec. 2 we relatethe turbulent velocity and correlation time to the Rossbynumber in both fixed- τ ed and fixed-Ω cases, developing ex-pressions for both the turbulent energy density and corre-lation time as a function of shear, rotation, and SN rate.For a given shear profile we consider three cases: (i) fixedΩ, varying τ ed ; (ii) fixed τ ed , varying Ω; and (iii) mutuallydependent variation of Ω and τ ed . We apply these relationsto the dynamo equations in Sec. 3. The solutions are foundnumerically in Sec. 4, where we show both steady state solu-tions and time evolution of the magnetic fields. We identifywhere the results from our calculations that include the newingredients differ from previous approaches. We also discussthe influence of magnetic buoyancy and the consequencesof our calculations for observed pitch angle. We conclude inSec. 5. c (cid:13) RAS, MNRAS , 1– ?? ome consequences of shear on dynamos The Rossby number is function of two variables ( τ ed , Ω).The value of τ ed can vary for different supernova rates andΩ depends on the details of galaxy formation and the masstherein. In practice, these two quantities could be correlatedbecause a fixed initial mass function for stars, and a baryonmass correlated with total mass would increase both the rateof SN and the rotation rate at a fixed radius. Below we con-sider separately cases where we allow these two quantities tobe independent and then consider a case where they are mu-tually dependent. When they are independent, the dynamothen depends on these two variables independently, not justtheir ratio.We first construct a physical model for the influence ofshear on the turbulent energy and correlation time by fixingΩ and allowing τ ed to change. We then construct the ana-logue where we keep τ ed and allowing Ω to change. We showin the appendix that these two approaches can be unified.In the last subsection of this section we consider the casewhere the two quantites are mutually dependent.In what follows, quantities with a subscript 0 (e.g., τ ed , v , l ed and so on) are evaluated at their fiducial values suchthat Ro = ˜ Ro = ( τ ed Ω ) − = 1. Ω , fixed shear, but different SN rates We distinguish between the turbulent correlation time τ cor and the naked eddy turnover time τ ed determined by SN inthe absence of shear, and τ ed as the fiducial value of thelatter. We define the ratio of the former to latter as y ( Ro ) ≡ τ cor /τ ed , (2)where Ro = Ro ( τ ed ) for fixed Ω in this section. The quantity τ cor must satisfy the physically expected behaviors in thelow and high Ro limits, namely τ cor → τ ed as Ro → ∞ and τ cor → τ s as Ro →
0, where τ s is defined as τ s = ∆ rr ∆Ω = ∆ rr ∆ r∂ r Ω = 1 q Ω = τ ed q (3)along with the rotation profile Ω ∝ r − q . The physical mean-ing of τ s is evident if we consider radially separated pointson two concentric rings orbiting in the galaxy with radii r − ∆ r/ r + ∆ r/ r ∆Ω = r ∆ r∂ r Ω, and τ s characterizes the time scalefor these points to further separate to by ∆ r in the azimuthaldirection. In terms of y , the aforementioned asymptotic lim-its imply y = (cid:40) /q Ro → /Ro Ro → ∞ (cid:41) . (4)Deriving y from first principles is a challenging endeavorbut we can make good progress with a physically motivatedapproach. We posit that quadratic time correlations of tur-bulent quantities decay exponentially in time over a correla-tion time that has separate independent exponential factorsfrom shear and SN turbulence. Then τ − cor = τ − ed + τ − s , (5) or equivalently, y = 1 Ro + q . (6)Eq. (6) satisfies the constraint (4). In the fast rotation limit Ro →
0, we have y = 1 /q so that Eq. (28) predicts a cor-relation time that asymptotically approaches a constant for q >
0, as we will see below.
Next, we consider the effect of the shear on the turbulentenergy. Technically the turbulent energy consists of both en-ergy from supernovae and differential rotation since rotatingMHD shear flows with q > ρv τ cor = Eτ ed l ed + ετ s (7)where ρ is the average density of ISM, v is the mean squareroot velocity of the turbulence, E is the energy input to theISM per supernova, l ed = vτ ed is the eddy scale, and ε is theenergy density input by shear and is taken to be a fractionof the fiducial shear energy density ρv s = ρ ( l ed /τ s ) . Morespecifically, we then have ε = ξρ ( l ed /τ s ) = ξρl ed q τ − ed = ξq ρv (8)where we take ξ = 0 .
1. To provide physical meaning forthe second term in Eq. (7), we note that the energy densitysupplied by SN per unit time can be expressed as E Γ V − where V is the volume of the galaxy, and Γ the rate atwhich SNe are produced in V . Crudely assuming SN occurisotropically, we have Γ τ − ed (cid:39) Vl ed (9)where l ed indicates the turbulent correlation scale from SN.Therefore E Γ V = Eτ ed l ed . (10)We further assume that E is a constant, and that the vari-ation of ρ / with Ro is small compared to v , so that ρ canbe taken as approximately constant as well. For the fiducialpoint values, the ratio between the second term on the RHSto the LHS of Eq. (7) is0 . ρv /τ s ρv /τ cor = 0 . ρv / ( τ ed /q ) ρv / ( τ ed y ) = ξq y = 120 (11)in using (2) and (6). For a flat rotation profile (as in typicalspiral galaxies), q (cid:39) E/ρ (cid:39) v τ ed (12)which can then be used to simplify Eq. (7) to f y = Ro f + q , (13)where f ( Ro ) ≡ v/v . (14) c (cid:13) RAS, MNRAS , 1–, 1–
1. To provide physical meaning forthe second term in Eq. (7), we note that the energy densitysupplied by SN per unit time can be expressed as E Γ V − where V is the volume of the galaxy, and Γ the rate atwhich SNe are produced in V . Crudely assuming SN occurisotropically, we have Γ τ − ed (cid:39) Vl ed (9)where l ed indicates the turbulent correlation scale from SN.Therefore E Γ V = Eτ ed l ed . (10)We further assume that E is a constant, and that the vari-ation of ρ / with Ro is small compared to v , so that ρ canbe taken as approximately constant as well. For the fiducialpoint values, the ratio between the second term on the RHSto the LHS of Eq. (7) is0 . ρv /τ s ρv /τ cor = 0 . ρv / ( τ ed /q ) ρv / ( τ ed y ) = ξq y = 120 (11)in using (2) and (6). For a flat rotation profile (as in typicalspiral galaxies), q (cid:39) E/ρ (cid:39) v τ ed (12)which can then be used to simplify Eq. (7) to f y = Ro f + q , (13)where f ( Ro ) ≡ v/v . (14) c (cid:13) RAS, MNRAS , 1–, 1– ?? Zhou & Blackman
Eq. (13) determines the nonlinear relation between theturbulent speed v and Ro . However its solution has simpleasymptotic behaviors. In the large Ro (cid:29) τ ed → f = ( Ro y ) / . In the Ro (cid:28) f = ( q y/ / . The two terms contributeequally at Ro = 0 .
22 so we approximate f as f = (cid:40) ( Ro y ) / Ro ≥ . q y/ / Ro ≤ . (cid:41) . (15)These relations capture the fact that as SN become scarce,the average turbulent speed of the ISM would decrease, butsince shear provides a fixed baseline of turbulent energy, v approaches to a constant.Given Eq. (15) we are poised to check one more plau-sibility condition for y , namely that the magnitude of α ∼ τ cor (cid:104) (cid:126)v · ∇ × (cid:126)v (cid:105) cannot be larger than v , since the helicalfraction cannot exceed unity. For quasi-isotropic turbulence(Durney & Robinson 1982; Ruzmaikin et al. 1988) α ∼ ξ α τ cor v Ω /h (16)where ξ α is a factor smaller than one and h ∝ v − / ishalf of the scale height of the galaxy in an isothermal self-gravitating slab model (Spitzer 1981). The required inequal-ity is then τ cor v Ω /h ≤ v , or equivalently, y f / ≤ h v τ ed = 5 (17)upon using fiducial values h = 0 . τ ed = 10 s, and v = 10 cm/s, the validity of which can checked using Eqs.(6) and (15).Note also that before turbulent energy is taken over byshear as Ro decreasing, l cor = vτ cor will never exceed thescale height 2 h , since2 hl cor = 2 h f − / l ed fy = 10 f − / y − (18)and it can be verified the quantity above is always greaterthan unity using (15). Complementing the previous subsection, here we instead fixthe SN rate (and thus τ ed ) but allow for different rotationrates. By direct analogy to (6), we define ˜ y by˜ y = τ cor /τ ed . (19)Note that we have τ ed = τ ed here. The asymptotic limitsare now ˜ y = (cid:40) ˜ Ro/q ˜ Ro →
01 ˜ Ro → ∞ (cid:41) . (20)By analogy to Eq. (4) we take˜ y = ˜ Ro ˜ Ro + q . (21)For the turbulent energy, we now generalize the energyinput from shear to allow varying angular velocity, assuming a fixed fraction is available. Thus Eq. (8) is replaced by (with ξ = 0 . ε = ξq ρv ˜ Ro − (22)which gives the correct value of ε at the fiducial point where˜ Ro = Ro = 1. Now ˜ f = v/v is given by˜ f ˜ y = 1˜ f + q
10 ˜ Ro (23)of which the solution is approximately˜ f = (cid:40) (˜ y ) / ˜ Ro ≥ . q ˜ y/
10 ˜ Ro ) / ˜ Ro ≤ . (cid:41) (24)The plausibility analogue to Eq. (17) becomes˜ y ˜ f / / ˜ Ro ≤ , (25)which is also satisfied if we use Eqs. (21) and (24), andsame fiducial values h = 0 . τ ed = 10 s, and v = 10 cm/s as in the last subsection. For rigid rotation, q →
0, and y = Ro − yields τ cor = τ ed as expected in the absence of shear. The effect of rigid rota-tion without shear on α has been previously considered dueto the Coriolis acceleration (see p.163 in Ruzmaikin et al.(1988)). We can interpret this effect as a change to the cor-relation time as follows: Over a time ∆ t , the displacementfrom the Coriolis force can be estimated to be d ∼ Ω v (∆ t ) and then we can set ∆ t = T c as the time inteval for theCoriolis force to rotate an eddy of radius l ed / π/
2, orcause a displacement d = πl ed /
4. This T c is the time scalefor two adjacent eddies to mutually shred from only this in-teraction and if this is the shortest of the eddy destructionmechanisms it would determine the correlation time. Usingthe above expressions for d and ∆ t , we obtain T c = (cid:18) πl ed / v (cid:19) / (cid:39) τ ed Ro / (26)Combining this with case of Sec. 2.2 (fixed τ ed ), we can thenwrite τ cor τ ed = min { ˜ Ro / , ˜ y ( Ro ) } . (27)For the case of Sec. 2.1 (fixed Ω), but with q = 0, we canwrite T c = τ ed Ro − / and then τ cor τ ed = min { Ro − / , y ( Ro ) } . (28)Note that the Eqs. (27) and (28) incorporate the sep-arate influences on the correlation time from pure rotationand shear. Fig. 1 shows the correlation time in our approach.We may express ˜ Ro as a function of the radial coordi-nate r given the rotation profile, i.e.,˜ Ro = 1 τ ed Ω = 1 τ ed Ω ( r/r ) − q = (cid:18) rr (cid:19) q (29)where we have used τ ed Ω = 1. Replacing ˜ Ro by r usingthe relation above and assuming all other variables are in-dependent of r provides us with one of the simplest way towrite down a r -dependent model. c (cid:13) RAS, MNRAS , 1– ?? ome consequences of shear on dynamos Ω both dependon r In the cases considered above, we have assumed that theeddy time and the rotation periods are independent but inpractice, models of star formation rates (SFR) in galaxiesboth depend on radius. We now suppose that τ ed varieswith the radial coordinate r according the the prescrip-tion adopted by Prasad & Mangalam (2016). Specifically,we adopt the relation τ ed = rr τ ed ∝ r (30)where r = 8 kpc and it is determined from the follow-ing argument: if τ − ed is proportional to the SN rate andthe SN rate is proportional to the surface density of theSFR Σ SFR (Shukurov 2004; Rodrigues et al. 2015), we have τ − ed ∝ Σ SFR . Further, we assume a Schmidt-Kennicutt-likepower-law relation Σ
SFR ∝ Σ ξ g g (Schmidt 1959; Kennicutt1989; Heiderman et al. 2010), where Σ g is the gas surfacedensity and typically 1 ≤ ξ g ≤ .
4. For simplicity, we take ξ g = 1 here. The mean galactic gas surface density Σ g ∝ /r if the gas surface density hovers around a fixed fraction oforder unity near the critical Toomre density for gravitationalstability (Toomre 1964; Cowie 1981). Then combining theseabove relations we arrive at Eq. (30) above. If the helicityflux is driven by a galactic fountain, which in turn is drivenby SN (Tenorio-Tagle & Bodenheimer 1988; Shapiro & Field1976; Shukurov 2004; Rodrigues et al. 2015) , we might con-sider that the outflow speed also satisfies U ∝ /τ ed ∝ /r, or , U = r r U . (31)In addition, for a flat rotation curve, Ω ∝ /r .Now since both τ ed and Ω vary with r , we need theunified relations derived in Appx. A, which results in y ( r ) = r/ F ( r ) = max { (1 / r ) / , (1 / r ) / } . (33) The induction equation for the mean field is given by(for reviews Brandenburg & Subramanian (2005); Blackman(2015)) ∂ t B = ∇ × ( U × B + E − β J ) (34)where B and U are the (ensemble or spatial averaged) meanmagnetic field and velocity field, respectively; J = ∇ × B /µ is the mean current (taking µ = 1); β is the Ohmic resis-tive diffusion coefficient; E = α B − β t J is the mean turbulentelectromotive force where β t is the turbulent magnetic dif-fusivity, α ≡ α + α m is the pseudoscalar helicity coefficientseparated into kinetic and magnetic contributions, α and α m , respectively.We adopt cylindrical coordinates and apply the ’no-z’approximation (Subramanian & Mestel 1993; Moss 1995; Phillips 2001; Sur et al. 2007) to reduce the PDEs to a sim-pler set of ODEs; the reduced dynamo equations read ∂ t B r = − π R α (1 + α m ) B φ − (cid:18) R U + π (cid:19) B r (35) ∂ t B φ = R ω B r − (cid:18) R U + π (cid:19) B φ (36) ∂ t α m = − R U α m − β d β t π α m − C (cid:2) (1 + α m )( B r + B φ )+ 38 (cid:114) − π (1 + α m ) R ω R α B r B φ + α m R m (cid:35) + λ V R ω R α ( B r − B φ )(37)where B r and B φ are respectively the radial and azimuthalcomponents of the total magnetic field. The z component isassumed to be much less than these two and is neglected.The second term on the RHS in (37) governs the effectof diffusive fluxes β d ∇ α m (Brandenburg et al. 2009a; Mi-tra et al. 2010; Hubbard & Brandenburg 2010) where β d isthe diffusion coefficient. For most of the discussion of thesolutions in Sec. 4, we take β d = 0 (the case of Sur et al.(2007)) except for Sec. 4.5 where we adopt β d /β t = 1 in amodel using the radial coordinate r as a free parameter andfind that this diffusive helicity flux term raises the magneticthe saturated magnetic energy as it exceeds the wind fluxterm R U for the fiducial parameters chosen, over much ofthe disk. The last term in Eqn. (37) is the Vishniac-Choflux (Vishniac & Cho 2001) with dimensionless coefficient λ V . We find that that, in accordance with Sur et al. (2007)that this flux has an influence only after the field alreadygrows substantially, and has its strongest influence at lowRossby numbers. Even then, the buoyancy flux tempers theinfluence of the Vishniac-Cho flux. In the solutions presentedin the sections below, we focus primarily the case of λ V = 0.The magnetic fields are normalized by the equipartitionfield strength B eq = √ πρv , so that B r = v A,r /v and so onwith v A the Alfv´en speed. Note that B eq is a function of v and thus varies with both the eddy turnover time and thegalactic rotation speed. We normalize the time by the dif-fusion time scale h /β t which again depends on the Rossbynumber. The dimensionless parameters in the above dynamoequations are R α = α hβ t , R U = Uhβ t , R ω = h q Ω β t , C = 2 (cid:18) hl (cid:19) , (38)where U is the buoyancy speed in z direction containing botha convective flow part U and a magnetic buoyancy part U B ; α is normalized by α ; and l = vτ cor is the correlation lengthscale of the turbulence.For the fixed-Ω case of Sec. 2.1, substituting (2) and(14) into those dimensionless parameters gives R α = yR α , R U = y − f − / R U + R U B ,R ω = y − f − qR ω , C = y − f − C (39) Here we are working in the α Ω dynamo approximation. For themore general α Ω dynamo, an extra term − R α (1 + α m ) B r /π would appear on the right hand side in (36). This term is negligi-ble compared to the term R ω B r , since | R α /R ω | ∼ y Ro γ / (cid:28) | R α /R ω | ∼ ˜ y / (cid:28) Ro .c (cid:13) RAS, MNRAS , 1–, 1–
4. For simplicity, we take ξ g = 1 here. The mean galactic gas surface density Σ g ∝ /r if the gas surface density hovers around a fixed fraction oforder unity near the critical Toomre density for gravitationalstability (Toomre 1964; Cowie 1981). Then combining theseabove relations we arrive at Eq. (30) above. If the helicityflux is driven by a galactic fountain, which in turn is drivenby SN (Tenorio-Tagle & Bodenheimer 1988; Shapiro & Field1976; Shukurov 2004; Rodrigues et al. 2015) , we might con-sider that the outflow speed also satisfies U ∝ /τ ed ∝ /r, or , U = r r U . (31)In addition, for a flat rotation curve, Ω ∝ /r .Now since both τ ed and Ω vary with r , we need theunified relations derived in Appx. A, which results in y ( r ) = r/ F ( r ) = max { (1 / r ) / , (1 / r ) / } . (33) The induction equation for the mean field is given by(for reviews Brandenburg & Subramanian (2005); Blackman(2015)) ∂ t B = ∇ × ( U × B + E − β J ) (34)where B and U are the (ensemble or spatial averaged) meanmagnetic field and velocity field, respectively; J = ∇ × B /µ is the mean current (taking µ = 1); β is the Ohmic resis-tive diffusion coefficient; E = α B − β t J is the mean turbulentelectromotive force where β t is the turbulent magnetic dif-fusivity, α ≡ α + α m is the pseudoscalar helicity coefficientseparated into kinetic and magnetic contributions, α and α m , respectively.We adopt cylindrical coordinates and apply the ’no-z’approximation (Subramanian & Mestel 1993; Moss 1995; Phillips 2001; Sur et al. 2007) to reduce the PDEs to a sim-pler set of ODEs; the reduced dynamo equations read ∂ t B r = − π R α (1 + α m ) B φ − (cid:18) R U + π (cid:19) B r (35) ∂ t B φ = R ω B r − (cid:18) R U + π (cid:19) B φ (36) ∂ t α m = − R U α m − β d β t π α m − C (cid:2) (1 + α m )( B r + B φ )+ 38 (cid:114) − π (1 + α m ) R ω R α B r B φ + α m R m (cid:35) + λ V R ω R α ( B r − B φ )(37)where B r and B φ are respectively the radial and azimuthalcomponents of the total magnetic field. The z component isassumed to be much less than these two and is neglected.The second term on the RHS in (37) governs the effectof diffusive fluxes β d ∇ α m (Brandenburg et al. 2009a; Mi-tra et al. 2010; Hubbard & Brandenburg 2010) where β d isthe diffusion coefficient. For most of the discussion of thesolutions in Sec. 4, we take β d = 0 (the case of Sur et al.(2007)) except for Sec. 4.5 where we adopt β d /β t = 1 in amodel using the radial coordinate r as a free parameter andfind that this diffusive helicity flux term raises the magneticthe saturated magnetic energy as it exceeds the wind fluxterm R U for the fiducial parameters chosen, over much ofthe disk. The last term in Eqn. (37) is the Vishniac-Choflux (Vishniac & Cho 2001) with dimensionless coefficient λ V . We find that that, in accordance with Sur et al. (2007)that this flux has an influence only after the field alreadygrows substantially, and has its strongest influence at lowRossby numbers. Even then, the buoyancy flux tempers theinfluence of the Vishniac-Cho flux. In the solutions presentedin the sections below, we focus primarily the case of λ V = 0.The magnetic fields are normalized by the equipartitionfield strength B eq = √ πρv , so that B r = v A,r /v and so onwith v A the Alfv´en speed. Note that B eq is a function of v and thus varies with both the eddy turnover time and thegalactic rotation speed. We normalize the time by the dif-fusion time scale h /β t which again depends on the Rossbynumber. The dimensionless parameters in the above dynamoequations are R α = α hβ t , R U = Uhβ t , R ω = h q Ω β t , C = 2 (cid:18) hl (cid:19) , (38)where U is the buoyancy speed in z direction containing botha convective flow part U and a magnetic buoyancy part U B ; α is normalized by α ; and l = vτ cor is the correlation lengthscale of the turbulence.For the fixed-Ω case of Sec. 2.1, substituting (2) and(14) into those dimensionless parameters gives R α = yR α , R U = y − f − / R U + R U B ,R ω = y − f − qR ω , C = y − f − C (39) Here we are working in the α Ω dynamo approximation. For themore general α Ω dynamo, an extra term − R α (1 + α m ) B r /π would appear on the right hand side in (36). This term is negligi-ble compared to the term R ω B r , since | R α /R ω | ∼ y Ro γ / (cid:28) | R α /R ω | ∼ ˜ y / (cid:28) Ro .c (cid:13) RAS, MNRAS , 1–, 1– ?? Zhou & Blackman where R U B is the magnetic buoyancy term which will beclarified later.For the fixed- τ ed case of Sec. 2.2, we use (19) and ˜ f = v/v to obtain R α = ˜ y ˜ Ro − R α , R U = ˜ y − ˜ f − / R U + R U B ,R ω = ˜ y − ˜ f − ˜ Ro − qR ω , C = ˜ y − ˜ f − C . (40)For the r -dependent model in Sec. 2.4 we use (A7) alongwith U ∝ /r to get R α = R α / , R U = 2 R U / ˜ r F / ,R ω = 2 R ω / ˜ r F , C = 4 C / ˜ r F , (41)where ˜ r = r/r with r = 8 kpc, and we use the followingtypical data for our Galaxy to calculate the fiducial values(same as in Sur et al. (2007), for the comparison later): τ ed = 10 s , v = 10 km/s , r Ω = 200 km/s ,l = 0 . , h = 0 . , U = 1 km/s , which gives (with q = 1) R α ≈ , R U ≈ . , R ω ≈ − , C ≈ , R m ≈ , and the corresponding fiducial Rossby number Ro ≈ γ g and the decay rateterms γ d : D ins ≡ γ g γ d , (42)with γ g = 2 π (1 + α m ) R α | R ω | (43)and γ d = (cid:18) R U + π (cid:19) (44)being respectively, the product of growth and decay termsin (35) and (36). We can define the dynamo growth time,divided by the diffusion time τ diff = h /β t , as τ dyn τ diff = 1 γ g − γ d . (45)The bottom panel of Fig. 2 shows τ dyn (thick blue line)in comparison with the age of the universe τ u ≈ τ ed , τ ed and τ s for our fixed-Ω case, while the dashed purple line(˜ τ dyn ) indicates the dynamo growth time for our fixed- τ ed case. All times in the plot are normalized by τ diff . The ver-tical dot-dashed lines at Ro = 0 .
22 and Ro = 0 .
355 respec-tively, correspond to the transition values of Eqs. (15) and(24) respectively, and marking for each of these cases, thetransition from shear dominated to supernova dominatedturbulent velocities as Ro increases. The top panel of Fig.2 shows the 3-D space that unifies the the cases of Sec. 2.1and 2.2 via Eq. A5.Several interesting features are evident in the bottompanel of Fig. 2. First, in both two case, for either the fixedΩ or fixed τ ed , τ − dyn → Ro approaches ∼ .
2. As aconsequence, for Ro (cid:38) .
2, the initial growth of the magneticfield will be too slow to produce a significant large scale field.Second, τ dyn becomes independent of τ ed when Ro ≤ . τ ed (dashed purple line), the growth time blowsup for Ro (cid:46) .
02 highlighting that field growth becomes in-significant at these values in this case. The top panel of Fig.2 shows how these two different cases are mutually compati-ble in 3-D. The solutions further demonstrating these pointswill be discussed in the next section.For the dynamo to have a significant influence on thelarge scale field, its growth time must be less than the ageof the universe τ u . The associated condition τ dyn ≤ τ u leadsto an upper bound on Ro above which the dynamo solutioncannot produce significant observable large scale fields . Inaddition, we impose a lower bound on Ro for fixed τ ed bythe condition Ω max r = c/
10 with c the speed of light, simplyso that we focus on the the cases where the rotation speedis non-relativistic. Combining these two constraints. we canexpress the physically meaningful range as Ro ≤ .
171 (log Ro ≤ . . ≤ ˜ Ro ≤ .
161 ( − . ≤ log ˜ Ro ≤ . τ ed . For the first 3 subsections below, we focus on the fixed-Ωcase, before addressing a few important features of the fixed- τ ed case in the penultimate subsection. In the last subsectionwe consider solutions for the case when Ω and τ ed mutuallydepend on r . We first consider the case without magnetic buoyancy. Bysolving Eqs. (35)-(37) for a steady state ( ∂ t = 0) we obtainthe darkest blue dotted line in Fig. 3. The y axis, represent-ing the magnetic field strength, is scaled with the equipar-tition field strength B eq = 4 πρv which depends on Ro . Tothe left of the the vertical dot-dashed line at Ro = 0 .
22 theturbulence is mostly driven by shear and right to by SNe.The cusp irregularity at Ro = 0 .
22 occurs because of ourpiecewise-defined (15), which in principle can be removed byrigourously solving f , but not essential for the level of detailexplored here. We used (15), which is sufficient to capturethe asymptotic behavior for large and small Ro . The darkestblue dotted-line solution includes the influence from differ-ential rotation of both τ cor and v and can be compared withthe top dotted line, obtained by taking for the full range of Ro f = ( Ro y ) / ; y = min { Ro − , Ro − / } , (48)which is the expression for f that neglects the effect of shearin the turbulence and correlation time (though shear is stillmaintained for the Ω effect in the B φ equation).In Fig. 4 we show how different components of the mag-netic fields depends on the Rossby number. We define thepitch angle by p ≡ arctan B r B φ = arctan R ω R U + π / c (cid:13) RAS, MNRAS , 1– ?? ome consequences of shear on dynamos where we have used (36) for the last equality. The magni-tude of p decreases with decreasing Ro when the turbulentenergy is mostly provided by SN (region to the right to thevertical line), in agreement with the numerical solution inChamandy et al. (2016) (see their Fig. 2, where they usedthe Coriolis number Co = 1 /Ro ). As expected, the pitchangle goes to a constant as Ro →
0, since without SN, theturbulent energy and the correlation time depend only onthe rotation profile. Then Ro drops out of the equations andthe dynamo saturates to a state purely driven by shear atfixed q . The smallness of the pitch angle is consistent withthe basic observation that galactic magnetic fields are pre-dominantly azimuthal Beck & Wielebinski (2013). We now investigate the inclusion of magnetic buoyancy. Al-though Foglizzo & Tagger (1994) suggest that differentialrotation will stabilize the Parker mode, we neglect this ef-fect in our rough calculations here. We use the buoyancyspeed as calculated in Parker (1979). For a weak magneticfield of sub-equipartiation (with the turbulence) strength, U B ≈ v A /v = v ( B r + B φ ). For a magnetic field compara-ble to equipartition strength, U B ≈ v A = v (cid:113) B r + B φ . Thefield-related buoyancy coefficient (assuming | B φ | (cid:29) | B r | ) isthen R U B = U B hv τ cor = (cid:18) C (cid:19) / y − f − / min { B φ , B φ } (50)for fixed Ω. (For the case of fixed τ ed we would just replace y by ˜ y and f by ˜ f in the above expression.)MB extracts small scale magnetic helicity but also largescale fields. As a consequence, there is a competition betweenthe loss of large scale field and benefit to amplification fromsmall scale magnetic helicity removal. The bottom dottedline of Fig. 3 shows the solution for the fixed Ω case. Here thepresence of MB lowers the overall field strength comparedto the case when R U B = 0. For fixed Ω, we also note thepossibility of dynamo purely supported by only magneticbuoyancy, where U = R U = 0. This solution is representedby the lightest blue curve in Fig. 3.The curves represented by green diamonds and red tri-angles of Fig. 4 show the different behaviors of the toroidaland poloidal magnetic fields. The growth of toroidal field( B φ , blue circles and green diamonds) is suppressed by MB,whereas the poloidal field ( B r , yellow squares and red trian-gles) is amplified by MB. This is understandable by notingthe competing roles of MB mentioned above, and the factthat in Eqs. (35) and (36), MB is more significant for thetoroidal field loss because | B φ | (cid:29) | B r | .The importance of the diffusive helicity flux (secondterm on the right of Eq. (37)) can be assessed by its separateratios to the wind term (first term on the right of Eq. (37)and the MB (third term on the right of Eq. (37)). For β t = β d these are respectively diffwind = π R U yf / (51) and diffMB = (cid:18) π C (cid:19) / yf / min { B φ , B φ } (52)in the case of fixed Ω. Since both ratios are smaller than1 when Ro <
1, keeping or neglecting the diffusive helicityflux term will not change the results significantly.The pitch angle profile under the influence of MB isshown in Fig. 4. this curve explicitly reveals that MB morestrongly suppresses azimuthal fields.For this model, we can predict the tangent of the pitchangle as a function of Ro . The result is shown in Fig. 5where we compare our numerical prediction with that ofChamandy et al. (2016) (who found tan p ∼ τ c ( v/h ) / ( q Ω)).The red part shows a power law ln( − tan p ) = 1 .
13 ln( Ro ) + constant . The limited data in Van Eck et al. (2015) fromtheir Fig. 8 suggests a slope of 0.4-0.5, if we assume that thesurface SFR density ∝ surface SNR density ∝ /τ ed ∝ Ro .This is closer to the predicted value of Chamandy (2016)than ours, but more data and work are ultimately neededto pin down the tightness of these trends and predictions. We now compare the time evolution of magnetic fields fromthe dynamo solutions for different values of Ro in Fig. 6.The time is normalized by the constant τ r = 2 π/ Ω, andthe magnetic fields are normalized by the ( Ro -dependent)equipartition field strength.The two lower curves show the transition from decay-ing solutions to those with an asymptotic sustenance of asteady-state as Ro is dialed below ∼ .
2. As Ro is de-ceased downward from 1.25, the dynamo growth time de-ceases. The growth time reaches a minimum (the dottedcurve, Ro = 0 .
6) and then increases, finally saturating (thesolid curve), in agreement with Fig. 2. The dashed blackcurve indicates the fiducial point Ro = 1. τ ed and changing ΩFig. 7 shows the dynamo solutions using the relation (21)and the corresponding non-dimensional parameters in thedynamo equations for the case of fixed τ ed and varying Ω.The vertical dot-dashed line marks the transition value ˜ Ro =0 .
355 between shear-dominated and SN dominated turbu-lence. The maximum steady-state field strength ∼ . B eq occurs at intermediate ˜ Ro ∼ .
2, and decays with ˜ Ro forboth lower and higher ˜ Ro . This contrasts the saturatedsteady states of Figs. 3 and 4 for small Ro where we fixed Ωand allowed τ ed to vary. ΩFig. 8 shows the result in using the model discussed in Sec.2.4 and (41). The horizontal axis is normalized by r =8 kpc. Here we define e turb = ρv = ρ v f /r and e B = B as the turbulent energy density and magnetic energy density,respectively, and show them in blue curves. The ISM massdensity is assumed to have the same dependence on r as thegalactic surface density, i.e., ρ ∝ /r . Red curves represents c (cid:13) RAS, MNRAS , 1–, 1–
2, and decays with ˜ Ro forboth lower and higher ˜ Ro . This contrasts the saturatedsteady states of Figs. 3 and 4 for small Ro where we fixed Ωand allowed τ ed to vary. ΩFig. 8 shows the result in using the model discussed in Sec.2.4 and (41). The horizontal axis is normalized by r =8 kpc. Here we define e turb = ρv = ρ v f /r and e B = B as the turbulent energy density and magnetic energy density,respectively, and show them in blue curves. The ISM massdensity is assumed to have the same dependence on r as thegalactic surface density, i.e., ρ ∝ /r . Red curves represents c (cid:13) RAS, MNRAS , 1–, 1– ?? Zhou & Blackman the model with (48) being used, i.e., it neglects the effect ofshear on both correlation time and turbulent energy density.Beyond the galactic central region r/r < . e turb (see Eq. (7)).The black curve of Fig. 8 represents the magnetic en-ergy density if we take the diffusive helicity flux term intoconsideration. Here the diffusion coefficient β d is assumedto be equal to the turbulent diffusivity β t , which may be anoverestimate because usually the ratio β d /β t is taken to be (cid:28)
1, e.g. in Brandenburg et al. (2009b) it is 0.05, and inMitra et al. (2010) a value of ∼ . R M compared to what would be appopriate for galax-ies). Using (32), (33) and (51), we find that the contributionfrom the wind term (characterized by R U ) is comparable tothat from the diffusive term (characterized by π/
2) when r < .
3, and π/ (cid:29) R U when r > .
3, showing a domi-nance of this diffusive helicity flux in almost the whole disk.The inclusion of β d increases the saturated value of magneticenergy by nearly an order of magnitude given our fiducialparameter choices. We have generalized a 2-D “no-z” galactic dynamo modelwith helicity fluxes to include two effects of differential ro-tation beyond its role in the Ω-effect which have not beenpreviously combined in galactic dynamo models. First, dif-ferential rotation provides an additional energy source forISM turbulence, beyond that of SN. Second, differential ro-tation can shred turbulent eddies, reducing the correlationtime of the turbulence (Blackman & Thomas 2015).We have incorporated these effects and relaxed the com-monly assumed equality between the correlation time andthe SN driven eddy turnover time. We show that the effectof shear on the correlation time can be important even whenshear does not dominate the turbulent energy. For low SNrates and strong shear, both effects are important. We sep-arately studied the influence of differential rotation on themean field dynamo solutions as a function of the SN inputrate and the rotation period when these quantities are takento be independent and also when they are proportional toeach other. The latter would be expected from correlationsof the SN rate with the star formation rate and in turn, thegalactic surface density and rotation rate (Prasad & Man-galam 2016).Our solutions show that the observable steady-statemean field dynamo field strengths at low Rossby numbersare significantly lower than those when the correlation timeis independent of shear. The model also predicts the pitchangle of the mean field, a measure of radial to toroidal fieldmagnitude, and a clean quantity to compare with obser- vations (Chamandy & Taylor 2015). Unlike previous work,we have also included magnetic buoyancy as a contributorto the helicity fluxes which becomes most important when
Ro, ˜ Ro <
1. We find that dynamos for which the helicityfluxes are entirely determined by buoyancy are possible evenin the absence of advective or diffusive fluxes.We also considered a model (Sec. 2.4) where both τ ed and Ω are functions of r . All dimensionless parameters werethen reinterpreted as functions of r only, as in (41). Whenour model is used in this way to explore radial dependenceof quantities within a galaxy, we derived that the magneticenergy density profile is relatively flat in radius, consistentwith observations (Beck & Wielebinski 2013) and it is aresult that serves as a test/consistency check for the model.The shape of the curve is sensitive to the Schnmdt-Kennicutindex of Sec. 2.4. If we switch it from 1 to 1.4, the radiusat which the steady-state magnetic energy density drops tozero will move from r ∼ . r ∼ . Ro < α (Ruzmaikin et al. 1988), and without explicitlyincluding the role of shear as a source of energy for the tur-bulence. We showed herein that shear causes a further re-duction in the correlation time not captured by the previoustreatments and an α ∝ Ω − / for fast rotation in the fixed τ ed case (Sec 2.2). This is a weaker reduction for fast rota-tors than rotational quenching in the absence of the sheareffects, which predicts α ∝ Ω − Ruediger (1978).Our calculations herein focused only on two specific in-fluences of the role of shear and we do not purport to havecaptured all of the effects of shear on the turbulence and wehave not included all terms in the EMF that depend on rota-tion. There are also other approaches to helicity flux drivenmean field dynamos that bypass the α coefficient altogether.Our point in this paper however to focus on specific effectson shear that have been understudied. Future work shouldincorporate and assess the relevance of lessons learned herein the derivation of other dynamo coefficients not presentlyconsidered. ACKNOWLEDGMENTS
We thank L. Chamandy and E. Vishniac for related stim-ulating discussions. We acknowledge support from grantsHST-AR-13916.002 and nsf-ast1515648. EB also acknowl-edges the Kavli Institute for Theoretical Physics (KITP)USCB and associated support from grant NSF PHY-1125915.
APPENDIX A: UNIFIED TREATMENT OFSEC. 2.1 AND 2.2
We can formally combine the two cases Sec. 2.1 and 2.2 ina single formalism by defining˜ τ ed = τ ed τ ed , ˜ τ r = τ r τ r = Ω Ω , (A1)so that Ro = ˜ τ r / ˜ τ ed . Then we can write y = τ cor /τ ed = 1˜ τ − ed + q ˜ τ − r . (A2) c (cid:13) RAS, MNRAS , 1– ?? ome consequences of shear on dynamos y y f / τ c o r τ ed0 Figure 1.
Plot of τ cor /τ ed versus Ro for different models, and averification of the inequality (17). The blue curve is for τ cor = τ ed ,whereas the combination composed of the red curve below Ro = 1and the blue curve above Ro = 1 gives the overall curve thatincludes only rotational quenching, τ cor = min { Ro − / , Ro − } ,without shear. Our model that includes shear defined by (6) isgiven by the black curve. While it approaches Ro − asymptoti-cally as Ro → ∞ , there is a notable difference when Ro is small,due to the effect of shear which prevents τ cor from becoming ar-bitrarily large without an upper bound. The green line illustratesthat Eq. (17) is satisfied. The the energy rate balance equation is ρv τ cor = Eτ ed l ed + 0 . q ρv (Ω / Ω ) τ s (A3)which, using F = v/v , can then be expressed as F y = 1˜ τ ed F + q τ r . (A4)The solution is approximately F (˜ τ ed , ˜ τ r ) (cid:39) max { ( y/ ˜ τ ed ) / , ( q y/ τ r ) / } , (A5)and it is related to f (Eq. (15)) and ˜ f (Eq. (24))in Sec. 2through f = F (˜ τ ed , , ˜ f = F (1 , ˜ τ r ) . (A6)Non-dimensional parameters (defined in Sec. (3) then ex-hibit the following scalings: R α ∝ y/ ˜ τ r , R U ∝ /F / y, R ω ∝ /F y ˜ τ r , C ∝ /F y . (A7)Fix-Ω and fix- τ ed cases correspond to, respectively, taking˜ τ r = 1 , ˜ τ ed = Ro − and taking ˜ τ ed = 1 , ˜ τ r = Ro in the aboverelations. REFERENCES
Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214Beck R., Wielebinski R., 2013, Magnetic Fields in Galaxies.p. 641Bhattacharjee A., Hameiri E., 1986, Physical Review Let-ters, 57, 206Bhattacharjee A., Yuan Y., 1995, ApJ, 449, 739Blackman E. G., 2015, Sp Sci. Rev., 188, 59Blackman E. G., Field G. B., 2000, MNRAS, 318, 724Blackman E. G., Field G. B., 2002, Physical Review Let-ters, 89, 265007 τ dyn τ s τ ed τ cor τ u τ dyn τ dyn τ u τ ed τ s τ cor - Ro τ τ ed0 ~ Figure 2.
A Comparison between various time scales. The upperplot is obtained by using (A5) which unifies the results of Sec. 2.1and Sec. 2.2 into a single 3-D plot. The dynamo growth time τ dyn (curved blue surface) is compared with the age of the universe τ u (green horizontal flat plane), the eddy turnover time τ ed (redinclined flat plane), the shear time τ s (black inclined flat plane)and the correlation time τ cor (the lowest gray curved surface).The lines in the lower 2-D graph shows the quantities at the slice τ r = τ r corresponding to the case of Sec. 2.1 where we vary τ ed but keep Ω fixed. The the x -axis in the lower plot is givenas the Rossby number. The purple dashed line shows the growthtime, but at a different slice τ ed = τ ed , which corresponds to thesecond case of Sec. 2.2, where τ ed is fixed and Ω varies. The leftand right vertical dot-dashed lines correspond to the transitionvalues of Ro above which supernovae dominate shear turbulencefor the fixed τ r and fixed τ ed respectively. Blackman E. G., Owen J. E., 2016, MNRAS, 458, 1548Blackman E. G., Thomas J. H., 2015, MNRAS, 446, L51Brandenburg A., Candelaresi S., Chatterjee P., 2009a, MN-RAS, 398, 1414Brandenburg A., Candelaresi S., Chatterjee P., 2009b, MN-RAS, 398, 1414Brandenburg A., Sandin C., 2004, A&A, 427, 13Brandenburg A., Subramanian K., 2005, Phys. Rep., 417,1Chamandy L., 2016, MNRAS, 462, 4402Chamandy L., Shukurov A., Subramanian K., Stoker K.,2014, MNRAS, 443, 1867Chamandy L., Shukurov A., Taylor A. R., 2016, arXivpreprint arXiv:1609.05688Chamandy L., Taylor A. R., 2015, ApJ, 808, 28Chatterjee P., Guerrero G., Brandenburg A., 2011, A&A,525, A5Cowie L. L., 1981, ApJ, 245, 66Durney B. R., Robinson R. D., 1982, ApJ, 253, 290Foglizzo T., Tagger M., 1994, A&A, 287, 297Gruzinov A. V., Diamond P. H., 1994, Physical Review c (cid:13) RAS, MNRAS , 1–, 1–
A Comparison between various time scales. The upperplot is obtained by using (A5) which unifies the results of Sec. 2.1and Sec. 2.2 into a single 3-D plot. The dynamo growth time τ dyn (curved blue surface) is compared with the age of the universe τ u (green horizontal flat plane), the eddy turnover time τ ed (redinclined flat plane), the shear time τ s (black inclined flat plane)and the correlation time τ cor (the lowest gray curved surface).The lines in the lower 2-D graph shows the quantities at the slice τ r = τ r corresponding to the case of Sec. 2.1 where we vary τ ed but keep Ω fixed. The the x -axis in the lower plot is givenas the Rossby number. The purple dashed line shows the growthtime, but at a different slice τ ed = τ ed , which corresponds to thesecond case of Sec. 2.2, where τ ed is fixed and Ω varies. The leftand right vertical dot-dashed lines correspond to the transitionvalues of Ro above which supernovae dominate shear turbulencefor the fixed τ r and fixed τ ed respectively. Blackman E. G., Owen J. E., 2016, MNRAS, 458, 1548Blackman E. G., Thomas J. H., 2015, MNRAS, 446, L51Brandenburg A., Candelaresi S., Chatterjee P., 2009a, MN-RAS, 398, 1414Brandenburg A., Candelaresi S., Chatterjee P., 2009b, MN-RAS, 398, 1414Brandenburg A., Sandin C., 2004, A&A, 427, 13Brandenburg A., Subramanian K., 2005, Phys. Rep., 417,1Chamandy L., 2016, MNRAS, 462, 4402Chamandy L., Shukurov A., Subramanian K., Stoker K.,2014, MNRAS, 443, 1867Chamandy L., Shukurov A., Taylor A. R., 2016, arXivpreprint arXiv:1609.05688Chamandy L., Taylor A. R., 2015, ApJ, 808, 28Chatterjee P., Guerrero G., Brandenburg A., 2011, A&A,525, A5Cowie L. L., 1981, ApJ, 245, 66Durney B. R., Robinson R. D., 1982, ApJ, 253, 290Foglizzo T., Tagger M., 1994, A&A, 287, 297Gruzinov A. V., Diamond P. H., 1994, Physical Review c (cid:13) RAS, MNRAS , 1–, 1– ?? Zhou & Blackman w / o MBpure SN w / o MB B w / o MBw / MBpure SN w / o MB pure MB Figure 3.
Magnetic energy B = B φ + B r as a solution of Eqs.(35)-(37). (a) Full 2-D solution surface showing the dependenceon both τ r and τ ed without magnetic buoyancy (blue surface)compared to the 2-D solution surface in the absence of the influ-ence of shear on the turbulence. The solid (dashed) black curvesare obtained by taking a slice at the fiducial value τ r /τ r = 1( τ ed /τ ed = 1), corresponding to the case with fixed Ω = Ω ( τ ed = τ ed ) and varying τ ed (Ω). (b) The 1-D solution for fixed Ωbut varying τ ed , corresponding to the solid red curves in (a). Thetwo darkest dotted curves represent the results with and with-out magnetic buoyancy, respectively. The competition betweenenhanced large scale growth from ejection of buoyant ejectio ofsmall scale magnetic helicity with the buoyant loss of large scalefields can be assessed by comparing the two curves. The top dot-ted curve is obtained by neglecting the contribution from shearin both the turbulent energy and the correlation time. Dynamossupported purely by shear and magnetic buoyancy become pos-sible for small Ro , indicated by the lowest dotted curve. Letters, 72, 1651Hanasz M., Otmianowska-Mazur K., Kowal G., Lesch H.,2009, Ast.. and Astrophys., 498, 335Heiderman A., Evans II N. J., Allen L. E., Huard T., HeyerM., 2010, ApJ, 723, 1019Hubbard A., Brandenburg A., 2010, Geophysical and As-trophysical Fluid Dynamics, 104, 577Kennicutt Jr. R. C., 1989, ApJ, 344, 685Kulsrud R. M., 2015, ArXiv e-prints arxiv:1502.03360Mitra D., Candelaresi S., Chatterjee P., Tavakol R., Bran-denburg A., 2010, Astronomische Nachrichten, 331, 130Moss D., 1995, MNRAS, 275, 191Parker E. N., 1966, ApJ, 145, 811Parker E. N., 1979, Cosmical magnetic fields: Their originand their activityPhillips A., 2001, Geophysical and Astrophysical Fluid Dy-namics, 94, 135Pouquet A., Frisch U., L´eorat J., 1976, Journal of Fluid ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ - - B ○ B ϕ w / o MB □ B r w / o MB ◇ B ϕ w / MB △ B r w / MB ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇ - - - - - -
20 Ro p / ° ○ w / o MB □ w / MB ◇ pure SN w / o MB Figure 4.
Upper: The saturated values of toroidal ( B φ ) andpoloidal ( B r ) components of the field are shown, for cases withand without MB. The toroidal fields are mostly suppressed, whilethe poloidal fields are amplified when MB is included. Lower: Thepitch angle p = arctan (cid:0) B r /B φ (cid:1) is presented, showing the relativestrength of the two components of the magnetic field. The in-crease in | p | when MB is included results from a greater loss inthe toroidal field. ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ - t an ( p ) Figure 5.
The tangent of the pitch angle p as a function of Ro in a fixed-Ω model. In the right half region a power law − tan p ∼ Ro . is found using the red data point. Mechanics, 77, 321Prasad A., Mangalam A., 2016, ApJ, 817, 12R¨adler K.-H., Stepanov R., 2006, Phys. Rev. E, 73, 056311Rodrigues L. F. S., Shukurov A., Fletcher A., Baugh C. M.,2015, MNRAS, 450, 3472Ruediger G., 1978, Astronomische Nachrichten, 299, 217Ruzmaikin A. A., Sokolov D. D., Shukurov A. M., eds,1988, Magnetic fields of galaxies Astrophysics and SpaceScience Library Vol. 133 c (cid:13) RAS, MNRAS , 1– ?? ome consequences of shear on dynamos - - - t B Ro = = = = = Figure 6.
Time evolution of the total fields for different valuesof Ro for the fixed Ω case. Time is normalized by the rotationperiod τ r , and B is normalized by an Ro -dependent equipartitionstrength in which v is determined by (14) and (15). Differentcurves correspond to different values of Ro . A non-trivial steadystate does not exist for large Ro . This plot can be viewed as ageneralization of Fig. 2 of Sur et al. (2007) to include the effectsof shear described in the text. ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ B ○ w / o MB □ w / MB ~ Figure 7.
The solution of the total field for the fixed τ ed case butallowing ˜ Ro to vary by varying Ω. The plot uses the relation (21).We show solutions for cases with and without magnetic buoyancy.This plot shows that at both low and high ˜ Ro (fast and slowrotation) limits, no steady-state solution exists. Schmidt M., 1959, ApJ, 129, 243Sellwood J. A., Balbus S. A., 1999, ApJ, 511, 660Shapiro P. R., Field G. B., 1976, ApJ, 205, 762Shukurov A., 2004, ArXiv Astrophysics e-printsarxiv:astro-ph/0411739Shukurov A., Sokoloff D., Subramanian K., BrandenburgA., 2006, A&A, 448, L33Spitzer Jr. L., 1981, Physical processes in the interstellarmedium.Strauss H. R., 1985, Physics of Fluids, 28, 2786Subramanian K., Mestel L., 1993, MNRAS, 265, 649Sur S., Shukurov A., Subramanian K., 2007, MNRAS, 377,874Tenorio-Tagle G., Bodenheimer P., 1988, ARAA, 26, 145Toomre A., 1964, ApJ, 139, 1217Van Eck C. L., Brown J. C., Shukurov A., Fletcher A.,2015, ApJ, 799, 35Velikhov E. P., 1959, JETP, 36, 995Vishniac E. T., Cho J., 2001, ApJ, 550, 752Vishniac E. T., Shapovalov D., 2014, ApJ, 780, 144 (cid:45) rr e Ρ v e turb, w (cid:144) o DF e B, w (cid:144) o DF e turb,pure SN, w (cid:144) o DF e B,pure
SN, w (cid:144) o DF e B,w (cid:144) DF Figure 8.
Turbulent energy density e turb and magnetic energydensity e B as functions of r in a model where both τ ed and Ωdepend on r only. The vertical dashed line is at r/r = 0 . (cid:13) RAS, MNRAS , 1–, 1–