Non-axisymmetric vertical shear and convective instabilities as a mechanism of angular momentum transport
aa r X i v : . [ a s t r o - ph . E P ] J u l Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 9 June 2018 (MN L A TEX style file v2.2)
Non-axisymmetric vertical shear and convectiveinstabilities as a mechanism of angular momentumtransport
Francesco Volponi ⋆ Graduate School of Frontier Science, The University of Tokyo, Chiba 277-8561, Japan
ABSTRACT
Discs with a rotation profile depending on radius and height are subjectto an axisymmetric linear instability, the vertical shear instability. Herewe show that non-axisymmetric perturbations, while eventually stabilized,can sustain huge exponential amplifications with growth rate close to theaxisymmetric one. Transient growths are therefore to all effects genuineinstabilities. The ensuing angular momentum transport is positive. Thesegrowths occur when the product of the radial times the vertical wavenum-bers (both evolving with time) is positive for a positive local vertical shear,or negative for a negative local vertical shear.We studied, as well, the interaction of these vertical shear induced growthswith a convective instability. The asymptotic behaviour depends on therelative strength of the axisymmetric vertical shear ( s vs ) and convective( s c ) growth rates.For s vs > s c we observed the same type of behaviour described above -large growths occur with asymptotic stabilization. When s c > s vs the sys-tem is asymptotically unstable, with a growth rate which can be slightlyenhanced with respect to s c .The most interesting feature is the sign of the angular momentum trans-port. This is always positive in the phase in which the vertical shear driventransients growths occur, even in the case s c > s vs .Thermal diffusion has a stabilizing influence on the convective instability,specially for short wavelengths. Key words: accretion, accretion discs - hydrodynamics - convection -instabilities ⋆ email: [email protected] (cid:13) Francesco Volponi
The evolution of accretion discs is determined by theoutward transport of angular momentum, which in-duces infall of matter toward the orbital center. Itis generally assumed that the accretion rate dependson an effective viscosity (Shakura & Sunyaev 1973).In magnetized discs this mechanism is triggered by alinear axisymmetric destabilization, the magnetorota-tional instability, which occurs when a weak poloidalmagnetic field is coupled to rotation (Balbus & Haw-ley 1991).In hydrodynamic Keplerian discs, instead, the pres-ence of an efficient outward transport of angular mo-mentum is still an open question. In this context thereis no linear axisymmetric mechanism of growth, sinceKepler rotation is stable according to Rayleigh crite-rion.A possibility is given by the bypass scenario (Ioannou& Kakouris 2001, Chagelishvili et al. 2003), where ar-bitrarily large amplifications of perturbations can oc-cur due to the leading-trailing evolution of shearwisewavenumbers. These growths occur independently ofthe presence of a linear instability drive. This linearmechanism and non-linear interactions, providing apositive feedback loop, could trigger a subcritical tran-sition into turbulence. However, Lesur & Longaretti(2005) showed that, while subcritical transition to tur-bulence indeed occurs in non-stratified discs, values ofthe Shakura-Sunyaev α parameter (Shakura & Sun-yaev 1973) are much too low to give rise to an efficientturbulent transport.If radial stratification is taken into account ra-dial entropy gradients can drive non-linear subcriti-cal baroclinic destabilization leading to positive angu-lar momentum transport (Klahr & Bodenheimer 2003,Lesur & Papaloizou 2010) and formation of vortices.However, discs are in general radially and verticallystratified and the angular velocity is a function ofboth radial and vertical coordinates (Kippenhahm &Thomas 1982, Urpin 1984). Vertical velocity sheardrives an axisymmetric instability (Goldreich & Schu-bert 1967; Urpin 2003) which can result in positivetransport of angular momentum in the non-linearregime (Arlt & Urpin 2004). The linear destabilizationmechanism was derived by means of a local dispersionrelation in the full cylindrical geometry. Instability oc-curs when k r ≫ k z .In a recent study Nelson, Gressel & Umurhan (2012)found that in non-axisymmetric non-linear simula-tions outward transport can occur with α ∼ − .Vertical convection is another possible path toturbulence. Ryu & Goodman (1992) showed, by meansof a linear analysis, that an unstable vertical strati-fication is not favourable for accretion; angular mo-mentum transport occurs, but inward. Stone & Bal-bus (1996) extended the analysis to non-linear regimeswith analogous results. Departures from this pictureare however possible as shown by Lesur & Ogilvie(2010), in the strongly non-linear regime of highRayleigh number, and by Volponi (2010), for linear non-axisymmetric Boussinesq perturbations in discswith radial and vertical stratification.In the present study we will concentrate on thevertical shear and vertical convective instabilities. Pre-cisely, first, we will extend the linear analysis of thevertical shear instability to non-axisymmetric pertur-bations. Results will be derived by solving the shear-ing sheet equations in the presence of vertical veloc-ity shear (Knobloch & Spruit 1986) and in the shortwavelength regime. We will show that, while non-axisymmetric perturbations eventually decay, they ex-perience transient exponential growths which are in-distinguishable from a genuine instability.Second, we will study the interaction between the ver-tical shear and convective instabilities. This is the cen-tral part of the present study. We will show that thepresence of vertical shear changes the direction of thevertical convection induced angular momentum trans-port from inward to outward depending on the sign ofthe product of vertical and radial wavenumbers (i.e.positive for A z > A z <
0, where A z is the local vertical shear). Thermal diffusion has astabilizing influence on the convective instability, spe-cially for short wavelengths. In the shearing sheet approximation, the equationsgoverning the dynamics of a 3-dimensional disc are ∂ t D + ∇ · D V = 0 , (1) ∂ t V + V · ∇ V = − ∇ P D − Ω × V + 2 q Ω x ˆ x − Ω z ˆ z , (2) ∂ t (ln S ) + V · ∇ (ln S ) = 0 , (3)where D and P are density and pressure, V is the fluidvelocity, S = P D − γ is a measure of the fluid entropy, γ is the adiabatic index, Ω is the local rotation frequencyand q is the shear parameter ( q = 1 . − Ω × V is the Coriolis term,2 q Ω x ˆ x is the tidal expansion of the effective potentialand − Ω z ˆ z is the vertical gravitational acceleration.The equations are expressed in terms of the pseudo-Cartesian coordinates x = r − r , y = r ( φ − φ ) and z ( r and φ are reference radius and angle).The above equations are more manageable than thefull equations in cylindrical geometry due to the ne-glect of curvature effects. The simplification holds ifthe length scale of radial gradients L ≪ r , where r isthe cylindrical radial coordinate (Knobloch & Spruit1986).We consider fully stratified (i.e. radially and verti-cally) baroclinic discs. P e ( x, z ) and ρ e ( x, z ) are respec-tively the equilibrium pressure and density. Equilib-rium in general requires the angular velocity to be x and z dependent.The equilibrium velocity field is then given by[ − q Ω x + ∂ x P e ( x, ρ e ( x,
0) + a V z ]ˆ y , (4)which is obtained from the expansion of the angularvelocity profile about the midplane as given in Kley c (cid:13) , 000–000 on-axisymmetric vertical shear and convective instabilities & Lin (1992) (see as well Urpin 1984), in the limit L/r ≪
1. The dimensional coefficient a V can be de-rived from equation (A23) of Kley & Lin (1992).The z dependence gives rise to vertical shear at theorigin of the axisymmetric vertical shear instability.Notice that at the midplane the vertical shear goes tozero, so the instability drive is present away from it.This is the most interesting region for the present anal-ysis. Away from the midplane we can approximate thelast z -quadratic term, a V z , in (4) with a linear one,¯ A z z . Therefore in the following our reference equilib-rium flow will be V e ( x, z ) = [ − q Ω x + ∂ x P e ( x, ρ e ( x,
0) + ¯ A z z ]ˆ y , (5)bearing in mind that our conclusions will not hold atthe midplane. The second term on the RHS of theabove equation will be dealt with as in Johnson &Gammie (2005) by considering the background flowas giving an effective shear rate˜ q ( x )Ω = − dV e ( x ) dx , (6)that varies with x .Vertical equilibrium gives ∂ z P e ρ e = − Ω z ≡ − g z . (7)By defining 1 /L P z = ∂ z P e /γP e we obtain from (7) g z = − c s L P z , (8)where c s = γP e /ρ e . We decompose the physical variables in equilibriumand perturbation parts V = V e + v ′ , D = ρ e + ρ ′ , P = P e + P ′ , (9)and consider the linearized equations. Localized on the x and z dependent flow (see Johnson & Gammie 2005)and on the x and z dependent density and pressurebackgrounds, we consider short wavelength Eulerianperturbations of the type δ ′ ( t, x, y, z ) = ˆ δ ′ ( t ) e i R ˜ K x ( t,x ) dx + iK y y + i ˜ K z ( t ) z , (10)where˜ K x ( t, x ) = K x + ˜ q ( x )Ω K y t, ˜ K z ( t ) = K z − ¯ A z K y t. (11)Notice that due to the presence of vertical shear thevertical wavenumber as well evolves with time. The x dependence in ˜ q ( x ) is very weak and in the rest of thepaper we will consider the effective shear parameterconstant and very close to its Keplerian value.As previously stated to be consistent with the ne-glect of the curvature terms we consider a backgroundwith radial and vertical length scales L ≪ r and H ≪ r . The short wavelength perturbations are suchthat ˜ K x L ≫ K z H ≫ ∂ t ˆ ρ ′ ρ e + ˆ v ′ x L ρx + ˆ v ′ z L ρz + i ˜ K x ˆ v ′ x + iK y ˆ v ′ y + i ˜ K z ˆ v ′ z = 0 , (12) ∂ t ˆ v ′ x = 2Ω ˆ v ′ y − i ˜ K x ˆ P ′ ρ e + c s L P x ˆ ρ ′ ρ e , (13) ∂ t ˆ v ′ y = − (2 − ˜ q )Ω ˆ v ′ x − ¯ A z ˆ v ′ z − iK y ˆ P ′ ρ e , (14) ∂ t ˆ v ′ z = − i ˜ K z ˆ P ′ ρ e − ˆ ρ ′ ρ e g z , (15) ∂ t ˆ P ′ ρ e − c s ∂ t ˆ ρ ′ ρ e + c s ˆ v ′ x L Sx + c s ˆ v ′ z L Sz = 0 . (16)The radial and vertical length scales for pressure, den-sity and entropy are defined by1 L P x ≡ ∂ x P e γP e = 1 L ρx + 1 L Sx ≡ ∂ x ρ e ρ e + ∂ x S e γS e , (17)1 L P z ≡ ∂ z P e γP e = 1 L ρz + 1 L Sz ≡ ∂ z ρ e ρ e + ∂ z S e γS e . (18)In the Boussinesq approximation equations (12)and (16) become˜ K x ˆ v ′ x + K y ˆ v ′ y + ˜ K z ˆ v ′ z = 0 , (19) ∂ t ˆ ρ ′ ρ e = ˆ v ′ x L Sx + ˆ v ′ z L Sz , (20)Equations (13)-(15), (19) and (20) are equations (59)-(63) of Knobloch & Spruit (1986) expressed in termsof the short wavelength shearing modes (10). The onlydifference is given by the presence of the radial strat-ification term in (13). We will see in the following,however, that it has essentially no influence on theperturbations evolution.Now by deriving with respect to time the incom-pressibility condition (19) and then, in the equationobtained, expressing ∂ t ˆ v ′ x , ∂ t ˆ v ′ y and ∂ t ˆ v ′ z with equa-tions (13), (14) and (15), we can express ˆ P ′ in termsof ˆ ρ ′ , ˆ v ′ x , ˆ v ′ y and ˆ v ′ z i ˆ P ′ ρ e = 1˜ K [( ˜ K x c s L P x − g z ˜ K z ) ˆ ρ ′ ρ e +2(˜ q − k y ˆ v ′ x + 2Ω ˜ K x ˆ v ′ y − A z K y ˆ v ′ z ] , (21)where ˜ K = ˜ K x + K y + ˜ K z . By means of equation (21)we obtain the system ∂ t ˆ v ′ x = − q − K y ˜ K x ˜ K ˆ v ′ x + 2Ω(1 − ˜ K x ˜ K ) ˆ v ′ y +2 ¯ A z ˜ K x K y ˜ K ˆ v ′ z + c s L P x (1 − ˜ K x ˜ K ) ˆ ρ ′ ρ e + g z ˜ K z ˜ K x ˜ K ˆ ρ ′ ρ e , (22) ∂ t ˆ v ′ y = Ω[˜ q − − q − K y ˜ K ] ˆ v ′ x − K y ˜ K x ˜ K ˆ v ′ y +2 ¯ A z K y ˜ K ˆ v ′ z − ¯ A z ˆ v ′ z − K y ˜ K ( ˜ K x c s L P x − g z ˜ K z ) ˆ ρ ′ ρ e , (23) ∂ t ˆ v ′ z = − q − K y ˜ K z ˜ K ˆ v ′ x −
2Ω ˜ K x ˜ K z ˜ K ˆ v ′ y +2 ¯ A z K y ˜ K z ˜ K ˆ v ′ z − g z (1 − ˜ K z ˜ K ) ˆ ρ ′ ρ e − c s L P x ˜ K z ˜ K x ˜ K ˆ ρ ′ ρ e , (24) c (cid:13) , 000–000 Francesco Volponi ∂ t ˆ ρ ′ ρ e = ˆ v ′ x L Sx + ˆ v ′ z L Sz . (25)Normalizing time with Ω − , velocities with L Sz Ω anddensity with ρ e we obtain for the evolution of the non-dimensional variables v x , v y , v z , ρ the system ∂ t v x = − q − k y ˜ k x ˜ k v x + 2(1 − ˜ k x ˜ k ) v y +2 A z ˜ k x k y ˜ k v z + L Sx L Sz Ri x ( ˜ k x ˜ k − ρ + Ri z ˜ k z ˜ k x ˜ k ρ, (26) ∂ t v y = [˜ q − − q − k y ˜ k ] v x − k y ˜ k x ˜ k v y +2 A z k y ˜ k v z − A z v z + k y ˜ k (˜ k x L Sx L Sz Ri x + Ri z ˜ k z ) ρ, (27) ∂ t v z = − q − k y ˜ k z ˜ k v x − k x ˜ k z ˜ k v y +2 A z k y ˜ k z ˜ k v z + Ri z ( ˜ k z ˜ k − ρ + L Sx L Sz Ri x ˜ k z ˜ k x ˜ k ρ, (28) ∂ t ρ = L Sz L Sx v x + v z , (29)where ( k x , k y , k z ) ≡ L Sz ( K x , K y , K z ), ˜ k x ≡ L Sz ˜ K x ,˜ k z ≡ L Sz ˜ K z and ˜ k ≡ L S z ˜ K . We introduced, as well, A z ≡ ¯ A z Ω and the Richardson numbers Ri x ≡ N x Ω , Ri z ≡ N z Ω . (30) N x and N z are the Brunt-V¨ais¨al¨a frequencies N x ≡ − c s L Sx L P x , N z ≡ g z L Sz = − c s L Sz L P z . (31)We notice that the above equations are scale invariantin the sense that results pertaining to wavenumbers k x , k y and k z hold as well for wavenumbers βk x , βk y and βk z , where β is a real number. This symmetry isbroken if we introduce the effect of thermal diffusivityin the system above. For axisymmetric perturbations (i.e. k y = 0) equa-tions (26)-(29) become ∂ t v x = 2 k z k v y − L Sx L Sz Ri x k z k ρ + Ri z k z k x k ρ, (32) ∂ t v y = (˜ q − v x − A z v z , (33) ∂ t v z = − k x k z k v y − Ri z k x k ρ + L Sx L Sz Ri x k z k x k ρ, (34) ∂ t ρ = L Sz L Sx v x + v z . (35)In the above equations there are no time dependentcoefficients and therefore we can assume an exponen-tial form of the type e st for the velocity and densityperturbation fields. Neglecting radial stratification thedispersion relation reads s + 2(2 − ˜ q ) k z k + Ri z k x k − k x k z A z k = 0 . (36)This is the shearing sheet equivalent of equation (9) inUrpin (2003) in the case of zero diffusivity. For finite Ri z the vertical shear term is dominated either bythe epycyclic frequency term or by the Ri z term. Inthe case of weak vertical stratification (i.e. Ri z ≪ q = 3 / k x k z = A z ± q A z + 42 , (37)which is the same as equation (34) of Urpin (2003)since A z ≫ A z = r∂ z Ω. The growth rate isgiven by s max = vuut √ A z A z + A z q A z + 4 . (38)For A z ≪
1, a condition to be expected in astrophys-ical discs, we have therefore s max ≈ | A z | .Next we consider the opposite limit by neglecting thevertical stratification. In this case the dispersion rela-tion becomes s + k z k [2(2 − ˜ q ) + Ri x ] − k x k z k A z = 0 . (39)Being radial stratification typically weak, stability isnot influenced.When both radial and vertical gradients are presentthe dispersion relation reads s + k z k [2(2 − ˜ q ) + Ri x ] − k x k z k (2 A z + L Sx L Sz Ri x + L Sz L Sx Ri z ) + Ri z k x k = 0 . (40)In this case additional destabilization is possible inprinciple due to the term Σ = L Sx L Sz Ri x + L Sz L Sx Ri z .However its effect will be dominated either by the sta-bilizing effect of the epicyclic frequency (for k z > k x )or by the last term on the LHS of (40) (for k x ≫ k z ).We notice that for a barotropic equilibrium ( A z = 0)we have Σ = ± √ Ri x Ri z and Ri x , Ri z are bound tohave the same sign (Volponi 2010) and s = − k z − ˜ q ) + ( k z √ Ri x ± k x √ Ri z ) k (41)In the presence of diffusion the only modification ofthe evolution equations occurs in (20), which becomes ∂ t ˆ ρ ′ ρ e = ˆ v ′ x L Sx + ˆ v ′ z L Sz − χ ˜ K ˆ ρ ′ ρ e , (42)where χ is the thermal diffusion coefficient.Once normalized the above equation reads ∂ t ρ = L Sz L Sx v x + v z − ˜ k P e ρ, (43)where
P e = L Sz Ω χ is the Peclet number.The axisymmetric dispersion relation then becomes s + k P e − s + s (cid:20) k z k [2(2 − ˜ q ) + Ri x ] − c (cid:13) , 000–000 on-axisymmetric vertical shear and convective instabilities k x k z k (2 A z + L Sx L Sz Ri x + L Sz L Sx Ri z ) + Ri z k x k (cid:21) + P e − [2 k z (2 − ˜ q ) − k x k z A z ] = 0 . (44)The above equation is completely equivalent to thedispersion relation (7) in Urpin (2003) when neglect-ing the effect of viscosity and the same instability con-ditions derived in his paper follow. We briefly sum uphis results. By casting equation (44) in the form s + a s + a s + a = 0 , (45)where a = k P e − ,a = (cid:20) k z k [2(2 − ˜ q ) + Ri x ] − k x k z k (2 A z + L Sx L Sz Ri x + L Sz L Sx Ri z ) + Ri z k x k (cid:21) ,a = P e − [2 k z (2 − ˜ q ) − k x k z A z ] , (46)instability conditions read (see Urpin 2003 and refer-ences therein) a < , a a < a , a < . (47)The first of the inequalities above (equation (17) inUrpin 2003), P e − [2 k z (2 − ˜ q ) − k x k z A z ] < , (48)shows that the presence of thermal diffusion relaxesthe instability condition with respect to the ideal case.The second inequality (equation (14) in Urpin 2003)reads (cid:20) k z k Ri x − k x k z k ( L Sx L Sz Ri x + L Sz L Sx Ri z )+ Ri z k x k (cid:21) < k x /k z = 10,which pertains to the maximal vertical shear growthrate for A z = 0 . Ri z = − . P e = 10 . Wenotice that the growth rate is the convective one( √− Ri z ), but differently from conventional under-standing W xy ≡ ( v x v y ) /v >
0, where v = v x + v y + v z . To ascertain the origin of the positive sign of W xy we set A z = 0. As can be seen in Fig. 2, this resultsin W xy < W xy is determined by verticalshear.Linear inviscid axisymmetric perturbations cannottransport angular momentum in hydrodynamic discs(Ruden, Papaloizou & Lin 1988), however we will seein the next section that for non-axisymmetric pertur-bations, which induce transport, the same type of be-haviour described above holds.We determined as well the effect of thermal diffusion tv x tv y tv z tW xy Figure 1.
Evolution of velocities and normalized xy -Reynolds stress (i.e. W xy ≡ ( v x v y ) /v , where v = v x + v y + v z ) for P e = 10 , Ri z = − . Ri x = 0 . L Sz /L Sx = 0 . A z = 0 . k x = 500, k z = 50, k y = 0. tv x tv y tv z tW xy Figure 2.
Same as previous figure but with A z = 0. on the growth rate s . In Table 1 we report the de-pendence of s on P e . We notice that s is drasticallyreduced down to s ∼ A z . Further decreasing of P e does not have effect on s . Table 1.
Dependence of mixed (convective + verticalshear) axisymmetric growth rate on Peclet number (rel-ative to the case in Fig. 1)
P e ∞ s .
44 0 .
44 0 .
35 0 .
14 0 . . (cid:13) , 000–000 Francesco Volponi tv x tv y tv z tW xy Figure 3.
Evolution of velocities and normalized xy -Reynolds stress in the case of very weak stratification for A z = 0 . k x = 1000, k z = 100, k y = 1. We consider here the evolution of ideal non-axisymmetric perturbations in the presence of a con-vectively stable vertical stratification.With a reference stratification of Ri z = 0 . Ri x =0 .
01 and a realistic A z , perturbations always decay forany value of the wavenumbers. In this case the verti-cal shear destabilizing drive is never able to overcomethe stabilizing influence of stratification and epicyclicfrequency.If we allow for weaker stratification the vertical shearcan drive huge transient growths in the perturbations.Fixing A z to a positive value, the interval of growthis connected to ˜ k x and ˜ k z having the same sign. Wecan see this in Fig. 3 for the case ˜ k x > k z > k x /k z ratio to the value pertain-ing to the maximal axisymmetric growth rate A z . Byincreasing k y clearly the interval of growth is short-ened and therefore the largest amplification reacheddiminishes. We stress that the transients are indistin-guishable from a full-fledged instability. The growthsare in fact exponential with growth rate close to A z .Varying the ratio k x /k z , as well, causes a decrease inthe maximum amplification reached.In the case A z < k x and ˜ k z have opposite signs.We notice as well that during the phase of growth,radial transport is always positive.As stated before eventually all perturbations decay.Heuristically this can be shown neglecting the strat-ification in equations (26)-(29). By using the incom-pressibility condition (19) we obtain for the x and y velocity components the evolution equations ∂ t v x = − q − A z ˜ k x ˜ k z ) k y ˜ k x ˜ k v x +2(1 − ˜ k x ˜ k − A z k y ˜ k x ˜ k ˜ k z ) v y , (50) ∂ t v y = [˜ q − − q − k y ˜ k − A z k y ˜ k x ˜ k ˜ k z + A z ˜ k x ˜ k z ] v x + ( A z k y ˜ k z − k y ˜ k x ˜ k − A z k y ˜ k ˜ k z ) v y , (51)which in the limit t −→ ∞ become ∂ t v x = 2 A z q + A z v y , (52) ∂ t v y = − v x . (53)These can be reduced to the equation ∂ t v x = − A z q + A z v x , (54)which implies stability.For sake of completeness, we considered as wellthe case of stable weak vertical and unstable strongradial stratifications. The vertical shear growth rateis only slightly enhanced. We examine here the evolution of ideal perturbationsin the presence of an unstable vertical stratification.This is the central part of this study, because the inter-action of vertical shear and convective drives resultsin evolutions which are determined by the relativestrength of the vertical shear growth rate ( s vs ∼ | A z | )and the vertical convective growth rate ( s c ∼ √− Ri z ).For s vs > s c we observed the same type of behaviourdescribed in Case 1 - large exponential amplificationswith growth rate s vs occur with asymptotic stabiliza-tion. For s c > s vs the system is asymptotically unsta-ble with growth rate s c . For s vs ∼ s c a variety of mixedevolutions is possible: these include transient growths,exponential instability and algebraic instability.In the following we will concentrate on the case s c >s vs , for which the system is asymptotically unstable.Similarly to what we found in the axisymmetric case,the main difference with a purely convective insta-bility is given by the sign of the angular momentumtransport, which is positive exactly in the same timeintervals in which transient growth occured in Case1. In other words for A z > <
0) positive transportoccurs when ˜ k x ˜ k z > < k x /k z ∼
10 pertaining to maximal axisymmetric vertical sheargrowth rate for A z = 0 . s vs ∼ . Ri z = − .
2. As can be seen, the growth rate isgiven by s c ∼ √− Ri z and transport is positive in theregimes described above. This is important because, inthe absence of vertical shear, convection has the ten-dency to transport angular momentum inward ratherthan outward (Ryu & Goodman 1992). To see this weswitched off the vertical drive (i.e. A z = 0) and foundthat this results in negative transport (Fig. 6).We stress that this mechanism occurs for any value ofthe vertical shear growth rate, however small.We noticed as well that when s c > s vs , but their val-ues are close, the growth rate of the mixed instabil-ity, s mix , is enhanced of a fraction of s c . For example, c (cid:13) , 000–000 on-axisymmetric vertical shear and convective instabilities tv x tv y tv z tW xy Figure 4.
Evolution of velocities and normalized xy -Reynolds stress (up to t = 400) for Ri z = − . Ri x =0 . L Sz /L Sx = 0 . A z = 0 . k x = 500, k z = 50, k y = 1. tv x tv y tv z tW xy Figure 5.
Same as previous picture up to t = 800. in the case k x /k z ∼ A z = 0 . s vs ∼ . Ri z = − . s c ∼ .
45) and k y = 0 . s mix ∼ . Two are the main differences with respect to the idealcase.The first regards the vertical shear instability and con-sists in the fact, predicted by the axisymmetric theory(Urpin 2003), that the condition for instability is re-laxed. Indeed for non-axisymmetric perturbations aswell, we noticed that the presence of a stable verticalstratification does not have influence on growth, whichoccurs for any value of Ri z . In this case also, instabilitytransforms into a transient amplification with growthrate close to the axisymmetric one, as shown in Fig. 7.Again growth is connected to the signs of ˜ k x and ˜ k z exactly in the same way discussed in Case 1. tv x tv y tv z tW xy Figure 6.
Evolution of velocities and normalized xy -Reynolds stress (up to t = 400) for Ri z = − . Ri x =0 . L Sz /L Sx = 0 . A z = 0 and k x = 500, k z = 50, k y = 1. tv x tv y tv z tW xy Figure 7.
Evolution of velocities and normalized xy -Reynolds stress for P e = 10 , Ri z = 0 . Ri x = 0 . L Sz /L Sx = 0 . A z = 0 . k x = 1000, k z = 100, k y = 1. The second difference consists in the fact that, as forthe axisymmetric case, thermal diffusion has a sta-bilizing effect on the convective instability. This canbe seen in Fig. 8, where we considered Ri z = − . A z = 0 . k x = 500, k y = 1, k z = 50, with P e = 10 .Convective growth is suppressed and the growth ratein the amplification phase is the one pertaining to thevertical shear destabilization. We can however increasethe growth rate toward the one pertaining to a convec-tive instability in two ways. The first is by increasingthe Peclet number as done in Fig. 9, where parametershave the same values as in Fig. 8 but with P e = 10 .The second is by increasing the wavelength of the per-turbations as done in Fig. 10 which is the same asFig. 8 except that now k x = 100, k z = 10, k y = 0 .
2. Itcan be seen that the growth rate is strongly enhanced.We caution however that here we are reaching a bor- c (cid:13) , 000–000 Francesco Volponi tv x tv y tv z tW xy Figure 8.
Evolution of velocities and normalized xy -Reynolds stress for P e = 10 , Ri z = − . Ri x = 0 . L Sz /L Sx = 0 . A z = 0 . k x = 500, k z = 50, k y = 1. tv x tv y tv z tW xy Figure 9.
Evolution of velocities and normalized xy -Reynolds stress for P e = 10 , Ri z = − . Ri x = 0 . L Sz /L Sx = 0 . A z = 0 . k x = 500, k z = 50, k y = 1. derline regime for our short wavelength theory.We can sum up the effects of thermal diffusivity as fol-lows. Diffusivity acts as a stabilization mechanism forconvection and not for the vertical shear instability.By decreasing the Peclet number the growth rate de-creases until the vertical shear instability growth rateis reached. There the diffusion induced stabilizationstops. On the other side if we keep P e fixed and de-crease wavenumbers, the growth rate increases towardthe convective instability value ( ∼ √− Ri z ). Thereforewe can conclude that the stabilizing effect of thermaldiffusion affects convection for short wavelengths. tv x tv y tv z tW xy Figure 10.
Evolution of velocity and normalized xy -Reynolds stress for P e = 10 , Ri z = − . Ri x = 0 . L Sz /L Sx = 0 . A z = 0 . k x = 100, k z = 10, k y = 0 . We presented the evolution of non-axisymmetric per-turbations in discs whose rotation profile is x and z dependent. In the shearing sheet limit and moving to areference frame corotating with the background flowthe presence of vertical shear induces the evolutionof the vertical wavenumber ˜ k z . We considered shortwavelength perturbations.Perturbations are always stabilized asymptoti-cally; however, huge exponential growths can oc-cur which are virtually indistinguishable from a full-fledged instability. The condition for the occurrenceof such amplifications is ˜ k x ˜ k z > A z > k x ˜ k z < A z <
0. We summarized this in Table 2.We examine more in detail this condition. Let’s con-sider A z > k x > k z >
0. In this case ˜ k x is always positive, ˜ k z instead evolves from positive tonegative values. Therefore we observe the transient ex-ponential phase of growth when k z − A z k y t > t < k z A z k y ≡ t g ). We stress that this is different fromthe usual transient growth paradigm (Chagelishvili etal. 2003) since here the growth is exponential and thephysical origin is the vertical shear drive. Similar typeof exponential transient growths were described byBalbus & Hawley (1992) in the context of the non-axisymmetric magnetorotational instability, Korycan-sky (1992) in hydrodynamic convection and Volponi,Yoshida & Tatsuno (2000) in the stabilization of plas-mas kink modes. In all these studies, though, the flowwas just radially sheared.For t ∼ t g our short wavelength approximation breaksdown since ˜ k z −→
0; however, huge amplificationvalues are reached at times t well before t g , when˜ K z ≫ H .The growth rate is close to the axisymmetric growthrate. The transport is positive in the phase of growth.As for the axisymmetric theory the most easily desta-bilizable modes occur in the non-adiabatic case of fi- c (cid:13) , 000–000 on-axisymmetric vertical shear and convective instabilities Table 2.
Vertical Shear (Ideal, Ri z ≪
1, or Non ideal) A z > A z < k x ˜ k z > W xy >
0) Decay˜ k x ˜ k z < W xy > Table 3.
Vertical Shear and Convection (Ideal, s c > s vs ) A z > A z < A z = 0˜ k x ˜ k z > W xy >
0) Instability( W xy <
0) Instability( W xy < k x ˜ k z < W xy <
0) Instability( W xy >
0) Instability( W xy < nite thermal diffusivity. In the ideal case the presenceof a stable vertical stratification prevents the occur-rence of growth. Growth is possible only for small val-ues of the vertical Richardson number.This first set of results extends the axisymmet-ric theory to non-axisymmetric perturbations, provid-ing as well a linear local mechanism to explain thenon-axisymmetric non-linear results in Nelson et al.(2012). The linear non-axisymmetric mechanism is es-sentially the axisymmetric one which is still active forshort to intermediate times.We considered as well the concomitant action ofthe vertical shear and vertical convective instabilities.The asymptotic behaviour depends on the relativestrength of s vs and s c .For s vs > s c we observed the same type of behaviourdescribed above - large growths occur with asymptoticstabilization. When s c > s vs the system is asymptoti-cally unstable. For s vs ∼ s c a variety of mixed evolu-tions is possible.The most interesting feature for the present study isconnected with the sign of the angular momentumtransport. In fact we observed that in the phase inwhich the vertical shear driven transients growths oc-cur (i.e. ˜ k x ˜ k z > A z > k x ˜ k z < A z <
0) transport is always positive, even in the case s c > s vs . When the vertical shear drive is switched off(i.e. A z = 0), the convective transport turns negative.Table 3 summarizes these results.We conclude that, for s c > s vs , the interaction of ver-tical shear growths and convective instability resultsin a destabilization whose strength (i.e. growth rate)is governed mainly by convection and whose transportby vertical shear. This mechanism holds for any valueof s vs ( < s c ), however small. Again in the case s c > s vs ,but when their values are close, the growth rate of themixed instability is increased of a fraction of s c withrespect to s c .Thermal diffusion has a stabilizing influence onthe convective instability. This is specially strong forshort wavelengths. Instead it has no influence on the vertical shear growth rate. In the case s c > s vs , bydecreasing the Peclet number, we observed the transi-tion from convection to vertical shear dominated evo-lution. By increasing the Peclet number or decreas-ing the wavenumbers huge growths occur. The regimeof very long wavelengths is outside the scope of thepresent theory.We can say that whereas non-axisymmetricity sta-bilizes the axisymmetric vertical shear instabilityasymptotically for t −→ ∞ (not for intermediate timesthough) but has weak influence on the convective one,non-adiabaticity stabilizes the convective instabilitybut has no influence on the vertical shear growth rate.This second set of results instead points at a newpossible path to explain outward angular momentumtransport in astrophysical discs: interaction of verticalshear and convective instabilities. ACKNOWLEDGEMENTS
The author would like to express his gratitude to Prof.Zensho Yoshida for his suggestions, advice and con-tinuous support and to Prof. Ryoji Matsumoto forthe many discussions and encouragement. He wouldalso like to acknowledge very stimulating conversa-tions with Prof. Alexander Tevzadze.
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