Cross-correlation Aided Transport in Stochastically Driven Accretion Flows
CCross-correlation Aided Transport in Stochastically DrivenAccretion Flows
Sujit Kumar Nath
Department of Physics, Indian Institute of Science, Bangalore 560 012, India ∗ Amit K Chattopadhyay
Aston University, Nonlinearity and Complexity Research Group,Engineering and Applied Science, Birmingham B4 7ET, UK † a r X i v : . [ a s t r o - ph . H E ] D ec bstract Origin of linear instability resulting in rotating sheared accretion flows has remained a controver-sial subject for long. While some explanations of such non-normal transient growth of disturbancesin the Rayleigh stable limit were available for magnetized accretion flows, similar instabilities in ab-sence of magnetic perturbations remained unexplained. This dichotomy was resolved in two recentpublications by Chattopadhyay, et al where it was shown that such instabilities, especially for non-magnetized accretion flows, were introduced through interaction of the inherent stochastic noise inthe system (even a “cold” accretion flow at 3000K is too “hot” in the statistical parlance and iscapable of inducing strong thermal modes) with the underlying Taylor-Couette flow profiles. Bothstudies, however, excluded the additional energy influx (or efflux) that could result from nonzerocross-correlation of a noise perturbing the velocity flow, say, with the noise that is driving thevorticity flow (or equivalently the magnetic field and magnetic vorticity flow dynamics). Throughthe introduction of such a time symmetry violating effect, in this article we show that nonzeronoise cross-correlations essentially renormalize the strength of temporal correlations. Apart froman overall boost in the energy rate (both for spatial and temporal correlations, and hence in theensemble averaged energy spectra), this results in mutual competition in growth rates of affectedvariables often resulting in suppression of oscillating Alfven waves at small times while leading tofaster saturations at relatively longer time scales. The effects are seen to be more pronounced withmagnetic field fluxes where the noise cross-correlation magnifies the strength of the field concerned.Another remarkable feature noted specifically for the autocorrelation functions is the removal ofenergy degeneracy in the temporal profiles of fast growing non-normal modes leading to fastersaturation with minimum oscillations. These results, including those presented in the previoustwo publications, now convincingly explain subcritical transition to turbulence in the linear limitfor all possible situations that could now serve as the benchmark for nonlinear stability studies inKeplerian accretion disks.
PACS numbers: 47.35.Tv,95.30.Qd,05.20.Jj,98.62.Mw ∗ [email protected] † [email protected] . INTRODUCTION Transition to turbulence notwithstanding the linear stability of laminar flow profiles hasbeen the subject of increasing attention in the realm of sheared turbulence [3–6]. While be-ing unconventional in connection with turbulence that is generally associated with growingnonlinear instabilities, such a phenomenology is not altogether unknown, since equivalentpredictions in the context of Taylor-Couette twist instability vortices were already made wayback in the early nineties [7]. The new surge in interest though is in keeping with experimen-tal projections in relation to the physics of hot accretion flows [8–14, 17–19] where rotatingshear flows in the Rayleigh stable regime have shown unmistakable signatures of transitionto turbulence with a combination of angular momentum increase inspite of decreasing an-gular speed profiles. Often referred to as “subcritical turbulence”, especially in the contextof Keplerian accretion disks, the subject remained largely controversial since the emergenceof such strong power-law driven instabilities leading to huge energy bursts (optimal tran-sient energy ∼ Re / [3, 19]) was difficult to explain on the basis of purely hydrodynamicmechanisms. While certain remarkable theoretical headways were made in understandingrotating accretion flow dynamics perturbed by magnetic fluctuations [15–18], the interpre-tation proved grossly insufficient in systems without the presence of such magnetic fluxes.These theories were, however, accurate only within a “shearing sheet” approximation, anew age acronym for the “small gap” approximation previously employed [7], as correctlypointed out [20, 21]. Such theories also failed to establish any particular critical threshold asa function of relevant system parameters (Reynolds’ number, magnetic strength, etc) beyondwhich such magnetism induced transition to turbulence would take place. Complementarytheoretical strategies in absence of magnetic perturbations [19] provided some vital clues asto the nature of the dynamical scaling of the energy growth rate. Essentially, it was shownthat the presence of non-normal transient eigenmodes were the absolute pre-requisites forexplaining such axisymmetric energy bursts in Taylor-Couette flows in presence of a Coriolisforce. But once again the system approaches remained reclusive to a simultaneous presenceof shear, rotation and magnetic flux.This theoretical impasse was broken only recently, as shown in two recent publicationsby Chattopadhyay et al [1, 2]. In these works, the authors introduced noise as a stochasti-cally perturbing (field theoretically) relevant variable alongside the magnetically perturbed3eplerian flows (other types of flow, like constant angular momentum, flat rotation curveand solid body rotation were considered too). Chattopadhyay et al showed that noise drivenhydrodynamic instabilities (vital for cold accretion disks at temperatures of 3000K andabove) led to dynamical cross-overs from laminar to turbulent profiles, both in the pres-ence and absence of magnetic perturbations. Close to the linearly stable limit, this led toa new (dynamic) universality class with interesting statistical properties as were detailed inthese publications [1, 2]. The mathematical foundation also accounted for the large scale(stochastically driven) instabilities in autocorrelation functions in three dimensions, leadingto identical instabilities in the energy growth spectrum, revealing large energy growth oreven instability in the system as were previously predicted in slightly different contexts [22].Starting with the exploration of the effects of a Gaussian distributed white stochastic noiseon Rayleigh stable flows (Taylor-Couette flow in presence of Coriolis effects), focusing on theregime governed by transition to turbulence [1], we employed this theoretical architectureto investigate the amplification of linear magnetohydrodynamic perturbations in Rayleighstable rotating, hot, shear flows in the presence of stochastic noise in three dimensions [2].While [1] analyzed a noise driven Rayleigh flow close to the turbulent regime, [2] focused onthe effects of noise in the simultaneous presence of a Coriolis perturbed Taylor-Couette flowacted on by a magnetic field. Notably, a case of a linear non-rotating, non-magnetized shearflow was previously studied [23] that highlighted a small subset of the more detailed resultsderived in these works. Both of these works[1, 2], including the earlier truncated version[23], though relied on a drastic symmetry assumption related to the nature of the noisedistribution. All modes of noise correlations were assumed to be strictly correlated onlywithin their respective configuration hyper spaces, thereby neglecting the effects of cross-correlations that could arise out of sheared cross-sections. The problem with such a majorapproximation was already shown to be most vital in the understanding of boundary layersheared flow profiles in the context of regular (non-accretion type) Taylor-Couette flows [24].This is not too difficult to envisage either. An approximation of the type essentially impliesthat the noisy part of (for example) the velocity profile does not, for example, perturbthe vorticity or the magnetic flow profiles at all. This might as well be true for a specificparametric regime but there is no ad hoc proof of such a limitation being tenable across theentire parametric space, more importantly in the manifold close to the laminar-turbulenttransition regime. The present work extends the scope of the previously laid mathematical4oundation even further, now by introducing noise cross-correlations and by analyzing suchsymmetry violating effects on large (linear) instabilities.In what follows, section II will outline the basic model where the flow equations wouldremain identical to [2], with the only changes being in the noise cross-correlation profiles.Section III would focus on the effects of non-zero noise cross-correlation on temporal (PartA) and spatial (Part B) autocorrelation functions. Section IV too would be subdivided into two parts: Part A would analyze temporal cross-correlation functions of the variables(velocity u , vorticity ζ , magnetic field B and magnetic vorticity ζ B ) while Part B wouldanalyze the spatial cross-correlation functions of the same variables. This would be followedup by a summary and discussion in section V. In order to avoid multiplicity of reference, wewould allude to the previous published references [1, 2] as far as is practicable while tryingto avoid repetitions. In a slight departure from the previous two works in the series, thepresent article would not focus on the effects of colored noise, this being already known notto show any qualitative difference in outcomes. II. FLOW EQUATIONS: PERTURBED MAGNETIZED ROTATING SHEAR FLOWSIN PRESENCE OF CROSS-CORRELATED NOISE
As mentioned already, we would adopt a model identical to [2] and use the same notationsin order to retain continuity in discussions. All quantities would be expressed in dimension-less units where length has been normalized with respect to the system size L , time to bemeasured in units of the inverse of background angular flow velocity Ω along z , velocity tobe measured in units of q Ω L (1 ≤ q < q = 3 / q = 2 represents a disk with aconstant angular momentum while q = 1 represents a disk with a flat rotation curve. Insuch notations and within the ambits of the “small gap approximation” − L/ ≤ x ≤ L/ , − x, , B , B being a constant and the Coriolis angular velocity profile would as pre-viously be defined as Ω ∝ r − q , where r is the radial distance measured. Along with theincompressibility constraint ( ∇ · u = 0) and zero magnetic charge ( ∂ B ∂t = 0) imposed ona magnetized Orr-Sommerfeld and Squire flow system that is acted on by a Coriolis force5ogether with a Gaussian distributed white stochastic noise, we get [2] (cid:18) ∂∂t − x ∂∂y (cid:19) ∇ u + 2 q ∂ζ∂z − π (cid:18) B ∂∂y + ∂∂z (cid:19) ∇ B x = 1 R e ∇ u + η ( x, t ) , (1a) (cid:18) ∂∂t − x ∂∂y (cid:19) ζ + ∂u∂z − q ∂u∂z − π (cid:18) B ∂∂y + ∂∂z (cid:19) ζ B = 1 R e ∇ ζ + η ( x, t ) , (1b) (cid:18) ∂∂t − x ∂∂y (cid:19) B x − B ∂u∂y − ∂u∂z = 1 R m ∇ B x + η ( x, t ) , (1c) (cid:18) ∂∂t − x ∂∂y (cid:19) ζ B − ∂ζ∂z − B ∂ζ∂y − ∂B x ∂z = 1 R m ∇ ζ B + η ( x, t ) , (1d)where the velocity and magnetic field perturbations are given by ( u, v, w ) and ( B x , B y , B z )respectively, with R e and R m being the respective hydrodynamic and magnetic Reynoldsnumbers. p tot is the total pressure perturbation (including that due to the magnetic field)which has been eleminated from the equations [2] and the most general form of the (colored)noise components η , , , could be given by < η i ( x , t ) η j ( x , t ) > = D ij ( x ) δ ( x − x ) δ ( t − t ) . (2)The asymptotic large distance, long time behavior of the noise statistics have been encap-sulated in a pioneering work by Forster, Nelson & Stephen [25]. It has also been previouslyproved that the effects of finite sized (power law) spatiotemporal correlations would only bevital for nonlinear (stochastically driven) sheared flows [24] or otherwise for flows affectedby multiplicative noise [26]. Given that our focal point is a linearly stable Rayleigh flowprofile driven by an additive noise, such colored noise statistics would be irrelevant for theuniversality class under consideration and hence we would restrict ourselves to a white noiseitself, that without any loss of generality could be assumed to have the same noise strengthfor all correlations ( D ij = D ).As discussed in details in [1, 2], we would restrict ourselves to the “small gap” limit [7],in which, the Fourier expanded flows could be resolved as follows6 , ζ , B x , ζ B and η i as u ( x , t ) = Z ˜ u ( k , ω ) e i ( k . x − ωt ) d k dω,ζ ( x , t ) = Z ˜ ζ ( k , ω ) e i ( k . x − ωt ) d k dω,B x ( x , t ) = Z ˜ B x ( k , ω ) e i ( k . x − ωt ) d k dω,ζ B ( x , t ) = Z ˜ ζ B ( k , ω ) e i ( k . x − ωt ) d k dω,η i ( x , t ) = Z ˜ η i ( k , ω ) e i ( k . x − ωt ) d k dω, (3)and substituting them into equations (1a), (1b), (1c) and (1d) we obtain ˜ u ( k , ω )˜ ζ ( k , ω )˜ B ( k , ω )˜ ζ B ( k , ω ) = M − ˜ η ( k , ω )˜ η ( k , ω )˜ η ( k , ω )˜ η ( k , ω ) , (4)where M = M M M M M M M M M M M M M M M M , (5) M = ik ω + ilk k y − k R e , M = 2 ik z q , M = ik π ( B k y + k z ) , M = 0 , M = ik z (cid:18) − q (cid:19) , M = − iω − ilk y + k R e , M = 0 , M = − i π ( B k y + k z ) , M = ( − iB k y − ik z ) , M = 0 , M = (cid:18) − iω − ilk y + k R m (cid:19) , M = 0 , M = 0 , M = ( − iB k y − ik z ) , M = − ik z , M = (cid:18) − iω − ilk y + k R m (cid:19) , (6)where ˜ η i ( k , ω ) ( i = 1 , , ,
4) are the components of noise in k − ω space ( k = p k x + k y + k z )with noise correlations given as follows < η i ( k , ω ) η j ( k , ω ) > = 2 D δ ( k + k ) δ ( ω + ω ) . (7)7 II. AUTOCORRELATION FUNCTIONS IN PRESENCE OF NON-ZERO NOISECROSS-CORRELATION
In this section, we would analyze the spatiotemporal autocorrelations of the perturba-tion flow fields u , ζ , B x and ζ B for very large R e and R m in presence of non-zero noisecross-correlation. As mentioned earlier, this would imply the consideration of cross-variableaffectation of flows; in other words, how symmetry structure could change due to non-trivialmagnetohydrodynamical fluctuations. The choice of individually large R e and R m but afinite R e R m is quite meaningful for accretion flows in that this implies the presence of a finitePrandtl number . A. Temporal autocorrelations
The quantities of interest here are the following < u ( x , t ) u ( x , t + τ ) > = C uu ( τ ) = Z d k dω e − iωτ < ˜ u ( k , ω ) ˜ u ( − k , − ω ) >, (8a) < ζ ( x , t ) ζ ( x , t + τ ) > = C ζζ ( τ ) = Z d k dω e − iωτ < ˜ ζ ( k , ω ) ˜ ζ ( − k , − ω ) >, (8b) < B x ( x , t ) B x ( x , t + τ ) > = C BB ( τ ) = Z d k dω e − iωτ < ˜ B x ( k , ω ) ˜ B x ( − k , − ω ) >, (8c) < ζ B ( x , t ) ζ B ( x , t + τ ) > = C ζ B ζ B ( τ ) = Z d k dω e − iωτ < ˜ ζ B ( k , ω ) ˜ ζ B ( − k , − ω ) >, (8d)calculated over the projected hyper-surface for which k x = k y = k z = | k |√ , as in [2]. Thiscorresponds to a special choice of initial perturbation. As our principal interest is in probingthe scaling regime at which non-zero noise cross-correlations contribute to create (or other-wise destroy at times) sudden surges in energy activity, this restriction would only alter themagnitude of the probability density function (PDF) at worst while retaining the qualitativestructure, including scaling exponents, if any, unchanged. This also adds up to the noise in-compressibility constraint while still retaining the non-trivial nature of the cross-correlationsunaffected.In line with [2], we now perform the ω -integration of the integrands in equation (8d) bycomputing the four second order poles of the kernel which are functions of k . The form of8ll the integrands in equation (8d) is given by f ( k, ω ) = p ( k, ω )[ ω − ω ( k )] [ ω − ω ( k )] [ ω − ω ( k )] [ ω − ω ( k )] , which clearly reveals second order imaginary poles at ω , ω , ω and ω . We choose therange of k in such a way that the poles lie in the upper-half of the complex plane. Theresidue theorem is then used to calculate the frequency contributions at the poles in thewave vector range k to k m , where k = πL max , k m = πL min with L = L max − L min being the sizeof the chosen small section of the flow (“small gap approximation”) in the radial direction(arbitrarily chosen to be 2 units throughout for our evaluations), we obtain C uu ( τ ), C ζζ ( τ ), C BB ( τ ) and C ζ B ζ B ( τ ).The two-point temporal autocorrelation functions that we get individually for velocity-velocity (Fig 1), vorticity-vorticity (Fig 2), magnetic field-magnetic field (Fig 3) and mag-netic vorticity-magnetic vorticity (Fig 4) are most suggestive, especially when compared tosimilar results but in absence of any noise cross-correlation (detailed in [2]). τ C uu FIG. 1: (Color online) Temporal autocorrelations of velocity ( C uu ) upto τ = 10. The solidline represents q ≈
2, the dashed line is for q = 1 .
7, the dotted line represents q = 1 . q = 1. While oscillation amplitudes are lesser compared tothe case without cross-correlations in noise [2], each q-valued correlation spectrum comesup with a separate plot at varying magnitudes. This is remarkably different compared tothe zero cross-correlation in noise situation where velocity-autocorrelation functions for allq-valued data collapsed over each other (Fig 3 in [2]).9s discussed in details in [2], all values of q that are sufficiently close to the q → q = 1 . q = 1 . q = 1 . q closer to the limit q = 2. In order to refrain fromrepetition, we avoid τ × × × × × × C ζζ FIG. 2: (Color online) Temporal autocorrelations of vorticity ( C ζζ ) upto τ = 10. Plot stylesand symbol conventions are same as in Fig 1. Once again the effects of noise cross-correlationcan be clearly seen in that the plots for the different q-values do not collapse over each other.The other remarkable difference lies in the increasing profiles of vorticity correlations for thelarger q-values ( q ≈ q = 1 .
7) while q = 1 qualitatively matches with the profile with zerocross-correlation in noise. This indicates that neglecting noise cross-correlations implies adomination of the flat rotation profile over other available modes like Keplerian disc andconstant angular momentum.A remarkable impact of non-zero noise cross-correlation in the spatiotemporal dynamics isin the development of low frequency oscillatory waves, called Alfven waves [27, 28], dueto interaction of the magnetic flux with the extra inertia generated by the noise coupling,essentially leading to (oscillatory) instability due to additional energy influx. The resultantrenormalized phase velocity pumps even more energy in to the system that adds up tothe quantitative influx. This is the reason why even though no new phenomenological10hanges are observed for the case of the spatial autocorrelation functions (next section),unlike that for temporal dynamics, the energy fluxes are hugely revitalized resulting inorders of magnitude difference (spatial change in the energy flux is proportional to the thespatial two-point autocorrelation function, as was previously shown in [1]) in energy flows.In Fig 1, apart from the flat rotation curve ( q = 1), all higher q-valued correlation profilesshow increasingly smaller amplitudes and larger wave lengths for the resultant Alfven wavesat low k-values. Apart from an overall difference of up to two orders of magnitude for higherq-valued correlation spectra (higher with non-zero noise correlation), the small k-profiles( C uu ( k → q = 1 where due to velocity conservation (instead of angular momentum conservationor Keplerian rotation), the qualitative profile remains relatively unchanged with the onlychange showing up in the number of oscillating cycles (that is number of Alfven waves) andtheir wave lengths which are now stretched due to non-zero noise cross-correlation.For all values of q = 1, the vorticity-vorticity autocorrelation spectra (Fig 2) assume remark-ably different qualitative forms compared to the case without any noise cross-correlation.What the noise here does is to initially aggravate the Coriolis flux resulting in increasinggradients for each of the correlations functions ( C ζζ ) for q = 1 that then saturate at slightlyhigher magnitudes, at which point the zero noise-correlation dynamics takes over. Onceagain, q = 1 turns out to be a special case in that it practically replicates the zero noisecorrelation structure. This is not difficult to perceive and could be attributed to an effectivelack of momentum conservation whose structure changes with increasing angular momentumflux due to increased Coriolis forces. 11 τ × × × C BB FIG. 3: (Color online) Temporal autocorrelations of the magnetic field ( C BB ) upto τ = 10.Plot styles and symbol conventions are same as in Fig 1. As opposed to the two otherautocorrelation plots in presence of non-zero cross-correlation in noise, as shown above, theorder of magnitude remains the same although higher q-values ( q ≈ q = 1 .
7) decidedlyshow faster and stronger time oscillating profiles (Alfven waves) unlike in the previous (zeronoise-correlation) case. 12 τ C ζ B FIG. 4: (Color online) Temporal autocorrelations of magnetic vorticity ( C ζ B ) upto τ = 10.Plot styles and symbol conventions are same as in Fig 1. The oscillation profiles clearlyindicate similarity with the velocity autocorrelation function shown in Fig 1 above. Whileoscillations are much lesser compared to the case without cross-correlations in noise, each q-value comes up with a separate plot with periodic oscillations appearing for the flat rotationcurve (q=1). This is remarkably different compared to the zero cross-correlation in noisesituation where velocity-autocorrelation functions for all q-valued data collapsed over eachother.The magnetic field and magnetic vorticity autocorrelation functions too show distinctlydifferent qualitative and quantitative characteristics (compared to the zero noise cross-correlation case as in [2]). Evidently more energy coming from noise cross-correlations inducestronger Alfven waves as are evident from comparatively fast oscillating profiles in the mag-netic field correlation function (Fig 3). The magnetic vorticity autocorrelation though doesnot show much qualitative change compared to the zero noise cross-correlation case sincehere the fixation of the Prandtl number contrives to balance this extra rotational energybeing pitched in to the dynamics (Fig 4). 13 . Spatial autocorrelations The quantities of interest here are the following < u ( x , t ) u ( x + r , t ) > = S uu ( r ) = Z d k dω e i k . r < ˜ u ( k , ω ) ˜ u ( − k , − ω ) >, (9a) < ζ ( x , t ) ζ ( x + r , t ) > = S ζζ ( r ) = Z d k dω e i k . r < ˜ ζ ( k , ω ) ˜ ζ ( − k , − ω ) >, (9b) < B x ( x , t ) B x ( x + r , t ) > = S BB ( r ) = Z d k dω e i k . r < ˜ B x ( k , ω ) ˜ B x ( − k , − ω ) >, (9c) < ζ B ( x , t ) ζ B ( x + r , t ) > = S ζ B ζ B ( r ) = Z d k dω e i k . r < ˜ ζ B ( k , ω ) ˜ ζ B ( − k , − ω ) > . (9d)The three dimensional spatial integration may be reduced to the radial component only.Following is an example: S uu ( r ) = 2 π Z k m k dk k Z π dθ e ikr cos θ Z dω < ˜ u ( k , ω ) ˜ u ( − k , − ω ) >, (10)The other autocorrelations would follow suit. - r S uu FIG. 5: (Color online) Spatial autocorrelations of velocity ( S uu ) upto r = 1 .
0. Plot stylesand symbol conventions are same as in Fig 1.Apart from noting the obvious similarity with the zero noise cross-correlation case, introduc-tion of noise cross-correlation does not seem to change much in the dynamics qualitatively.14he qualitative similarity is not so very difficult to predict given the fact that a whiteGaussian noise is not expected to renormalize the spatial autocorrelation spectrum. - r S ζζ FIG. 6: (Color online) Spatial autocorrelations of vorticity ( S ζζ ) upto r = 1 .
0. Plot stylesand symbol conventions are same as in Fig 1.The quantitative comparison is more interesting though. While the magnitude of energyspurt (10 to 10 ) remains roughly unchanged, what the noise cross-correlation now doesis to define the spatial universality class while also clearly indicating that the transition fromsmall-r to large-r spectrum (resulting from radial movement across the sheared surface) leadsonly to a “crossover” and is not a phase transition, a question that was not entirely settled inabsence of noise cross-correlations [2]. Two additional spatial autocorrelation functions fornon-trivial cross-correlated noise have been added in the supplemental material (Figures 1and 2) [29]. The first of these figures show the spatial autocorrelation for magnetic vorticitywhile the second shows the same for magnetic field, each for a set of q-values as detailed inthe legend to these figures.The other similarity to be noted in between the two cases (noise cross-correlated versuszero cross-correlation in noise) is the fact that the autocorrelation functions in the log-logscale show almost parallel lines indicating identical gradients on both sides of the crossoverin both cases. 15 V. CROSS-CORRELATION FUNCTIONS IN PRESENCE OF NON-ZERO NOISECROSS-CORRELATION
In this section, we would analyze the spatiotemporal cross-correlations of the perturbationflow fields u , ζ , B x and ζ B for very large R e and R m in presence of non-zero noise cross-correlation. The essential implication of such an estimation is to ascertain how much of afluctuation in any one variable affects a different variable. Contrary to the drastic assumptionof dropping all noise cross-correlations as in [2], now we would consider contributions fromsuch symmetry violating terms as well. As shown in the context of a sheared boundary layerflow [24], such quantities may dramatically alter both qualitative and quantitative outcomesin variable cross-correlations. Using [2] as our benchmark (with zero noise cross-correlation),the results analyzed in this section would be contrasted against the report in [2]. A. Temporal cross-correlations
The quantities of interest here are the following < u ( x , t ) B x ( x , t + τ ) > = C uB ( τ ) = Z d k dω e − iωτ < ˜ u ( k , ω ) ˜ B x ( − k , − ω ) > (11a) < u ( x , t ) ζ ( x , t + τ ) > = C uζ ( τ ) = Z d k dω e − iωτ < ˜ u ( k , ω ) ˜ ζ ( − k , − ω ) > (11b) < u ( x , t ) ζ B ( x , t + τ ) > = C uζ B ( τ ) = Z d k dω e − iωτ < ˜ u ( k , ω ) ˜ ζ B ( − k , − ω ) > (11c) < ζ ( x , t ) ζ B ( x , t + τ ) > = C ζζ B ( τ ) = Z d k dω e − iωτ < ˜ ζ ( k , ω ) ˜ ζ B ( − k , − ω ) > (11d) < B x ( x , t ) ζ B ( x , t + τ ) > = C Bζ B ( τ ) = Z d k dω e − iωτ < ˜ B x ( k , ω ) ˜ ζ B ( − k , − ω ) > (11e) < ζ ( x , t ) B x ( x , t + τ ) > = C Bζ ( τ ) = Z d k dω e − iωτ < ˜ ζ ( k , ω ) ˜ B x ( − k , − ω ) > . (11f)The magnitude of the two-point temporal cross-correlation function is higher than beforeand oscillation is less compared to the zero noise cross-correlation case. For cross-correlatedvariables, the extra energy generated due to noise cross-correlation often leads to effectiveboost in energy in one of the variables while effectively suppressing the dynamics of theother through slower growth, resulting in a dynamical “push-pull” mechanism.16 τ C u ζ FIG. 7: (Color online) Temporal cross-correlation of velocity and vorticity ( C uζ ). Plot stylesand symbol conventions are same as in Fig 1.In the case of the velocity-vorticity cross-correlation, it is obvious that the comparative boostin vorticity energy is higher than that for velocity. As a result of this “competition”, after aninitially time growing profile, the (velocity-vorticity) cross-correlation function shows a clearsaturation unlike the case without any noise cross-correlation. An indirect confirmation ofthis lies in the magnitude of the saturation regime that shows an order of magnitude lowervalue compared to the case without any noise cross-correlation. For the same reason, theAlfven oscillations for large time scales are suppressed here. For small enough time scales,up to a specific cut-off, the rate of energy dissipation for the velocity part outperformsthe energy growth rate due to vorticity. This is the reason for the initial dip in the cross-correlation profiles (for all values of q ). Beyond a critical time scale though, the vorticityfactor starts dominating the energy picture which then accounts for the follow-up growth.17 τ C uB FIG. 8: (Color online) Temporal cross-correlation of velocity and magnetic field ( C uB ).Magnitude is higher than the zero noise cross-correlation counterpart and (Alfven [28])oscillation is also there. Extra cross-correlation in noise is equivalent to addition of extraenergy in the system in presence of velocity-vorticity cross-correlation. This extra energydownplays the previous energy by destroying the eddies previously generated. This resultsin lower oscillations. Plot styles and symbol conventions are same as in Fig 1.While being significantly different to their equivalent zero noise cross-correlation cases,the rest of the cross-correlation functions mostly demonstrate traditional profiles for tempo-ral correlation functions with a sharp increase in value for small times followed by respectivesaturation at larger time scales. In line with the explanation provided earlier, the differencein the comparative relaxation time scales of the cross-correlating variables lead to compe-titions in their mutual rates of growth, thereby ensuring lower saturation regimes for eachnoise cross-correlated case compared to their noise-cross-correlation-free equivalents. In allthese plots the q = 1 structure provides the strongest signature of oscillating Alfven waves,that once again agrees with the extra energy input due to nonzero noise cross-correlationhypothesis propounded earlier. 18 τ C u ζ B FIG. 9: (Color online) Temporal cross-correlation of velocity and magnetic vorticity ( C uζ B ).Magnitude is almost at the same range as in the zero noise cross-correlation counterpart butalmost without any oscillation. Plot styles and symbol conventions are same as in Fig 1. τ C ζζ B FIG. 10: (Color online) Temporal cross-correlation of vorticity and magnetic vorticity ( C ζζ B ).Magnitude is higher than zero noise cross-correlation counterpart and oscillation is there asprevious. Plot styles and symbol conventions are same as in Fig 1.19 τ × × × × × × C B ζ B FIG. 11: (Color online) Temporal cross-correlation of magnetic field and magnetic vorticity( C Bζ B ). Magnitude is almost same (may be little higher but not significant) and (Alfven[28]) oscillation is also not that much different than zero noise cross-correlation counterpart.Plot styles and symbol conventions are same as in Fig 1. τ × × × C ζ B FIG. 12: (Color online) Temporal cross-correlation of vorticity and magnetic field ( C ζB ).Plot styles and symbol conventions are same as in Fig 1.In order to address the issue of increasing proximity to the q → q : q = 1 . q = 1 . = 1 . q = 1 . q = 1 . q = 1 .
0. This figure convincingly clarifies that apartfrom linear shifts in the values of the cross-correlation function, the q → B. Spatial cross-correlations
The quantities of interest here are the following < u ( x , t ) B x ( x + r , t ) > = S uB ( r ) = Z d k dω e i k . r < ˜ u ( k , ω ) ˜ B x ( − k , − ω ) > (12a) < u ( x , t ) ζ ( x + r , t ) > = S uζ ( r ) = Z d k dω e iωτ < ˜ u ( k , ω ) ˜ ζ ( − k , − ω ) > (12b) < u ( x , t ) ζ B ( x + r , t ) > = S uζ B ( r ) = Z d k dω e iωτ < ˜ u ( k , ω ) ˜ ζ B ( − k , − ω ) > (12c) < ζ ( x , t ) ζ B ( x + r , t ) > = S ζζ B ( r ) = Z d k dω e iωτ < ˜ ζ ( k , ω ) ˜ ζ B ( − k , − ω ) > (12d) < B x ( x , t ) ζ B ( x + r , t ) > = S Bζ B ( r ) = Z d k dω e iωτ < ˜ B x ( k , ω ) ˜ ζ B ( − k , − ω ) > (12e) < ζ ( x , t ) B x ( x + r , t ) > = S Bζ ( r ) = Z d k dω e iωτ < ˜ ζ ( k , ω ) ˜ B x ( − k , − ω ) > . (12f)As before, the three dimensional spatial integration may be reduced to the radial componentonly. Following is an example (one of the six possible): S uB ( r ) = 2 π Z k m k dk k Z π dθ e ikr cos θ Z dω < ˜ u ( k , ω ) ˜ B x ( − k , − ω ) >, (13)As like the case with the spatial autocorrelation functions, the spatial cross-correlations toodo not show any significant qualitative difference compared to their zero noise-correlatedcounterparts. While in a way this makes the spatial profiles “uninteresting”, in the senseof being less exciting, they immensely help in retaining the sanctity of the physical logicpresented earlier. 21 - r S u ζ FIG. 13: (Color online)Spatial cross-correlation of velocity and vorticity ( S uζ ) upto τ = 10.Huge difference in magnitude with zero cross-correlated noise counterpart [2]. At least 9orders of magnitude difference. Plot styles and symbol conventions are same as in Fig 1.Only one spatial correlation plot is presented in the main text to indicate the quantitativevariation brought about by the noise cross correlation in spatial dynamics. Other spatialplots while not being qualitatively any different than previous work [2], they still do indicatevarying levels of quantitative difference of previous work [2] as functions of system parameters( R e , R m , q , etc.). Five additional spatial cross-correlation functions for non-trivial cross-correlated noise have been added in the supplemental material (Figures 3 to 7) [29]. Onceagain, they do not indicate any qualitative or behavioral change, rather all changes arelimited to quantitative variations.A non-zero value for noise cross-correlation implies a violation of time translation symme-try making it imperative that only temporal correlation functions, both autocorrelation andcross-correlations, should be the primarily affected ones. This is what we have seen already.In order to enact similar qualitative changes in spatial correlation profiles, we needed tohave a spatially correlated colored noise that would have ensured a space reflection symme-try violation. This is outside of our present remit as well as interest too.22 . SUMMARY AND CONCLUSIONS In this paper, our primary focus has been the extension of the groundwork laid down inour previous publications [1, 2] where we established an alternative theoretical hypothesis(compared to [15, 16, 19]) that is capable of explaining the origin of instability in the Rayleighstable limit often leading to turbulence both in magnetized, rotating and non-rotating shearflows supplemented by Coriolis force), now also incorporating the effects of nonzero noisecross-correlation across all four variables (velocity, vorticity, magnetic field and magneticvorticity). The target application is in the transport properties of accretion flows, both coldand hot.The importance of thermal noise as a (field theoretically) “relevant” order parameter wasalready established in these previous works (as also in [30–32], the latter focusing on generalthermal perturbations and is not expressly stochastic though) but what was not addressed ineither publication is the role of time symmetry violation (both translational and reflectional)in the energy growth dynamics, that could potentially obviate or otherwise nullify the impactof the noise induced dynamics through (constructive or destructive, as the case might be)interference of additional energy flux introduced into the system by time symmetry violatingnonzero noise cross-correlations. While we have no claim towards a unique mechanism ofstudying subcritical transition to turbulence in time symmetry violating Keplerian accretiondisks [3, 5], the mechanism introduced here is consistent with our approach established earlier[1, 2] in effecting such symmetry violations through stochastic noise distributions.By introducing noise cross-correlation as a time symmetry violating measure, what wehave essentially done is to create additional influx or efflux of energy into the magnetizedrotating accretion flows such that at any transient time scale, there is either a drop or increasein the total energy of the system on top of sheared dissipation. While such a cross-correlationin noise boosts the effective Coriolis force expressing itself as an added energy flux, there isan effective deconstruction with respect to the linear velocity flow component. The combinedeffect of these two factors can be seen in the velocity-vorticity temporal correlation whereafter an initial dip, the correlation function shows a saturation at larger time scales (for all q-values other than q = 1). At q = 1, while the dip is even more pronounced, the profile showslarge wavelength Alfven waves [1] with a saturating envelope for the correlation function.Similar trends could be seen in connection with all temporal correlation functions, and hence23or the ensemble averaged energy growth too. The fact that there is an additional energyinput associated with a nonzero noise cross-correlation comes out across all these correlationfunctions in that the relative magnitudes are always one to two orders of magnitude higherthan their equivalent zero cross-correlated noise cases [1, 2]. The qualitative summary forthe temporal autocorrelation functions of the different variables (Figs 1-4) too portray verysimilar patterns. Apart from an overall increase in transient energy, the same saturationtendency (all q-values) with large wavelength oscillations for q = 1 are omnipresent in allof velocity, vorticity and magnetic vorticity related autocorrelations. The magnetic fieldautocorrelation is a special and most reassuring case in that profiles for all q-values nowshow oscillating Alfven waves that could be easily attributed to an effective increase in themagnetic field energy due to this time symmetry violating effect. A remarkable differencewith the nonzero cross-correlation could also be seen from the fact that the profiles forthe different q-values are now distinct. This can be visualized as a classical analogue ofthe Zeeman effect where energy degeneracy is removed through additional magnetic energyinflux in the system.Compared to the temporal correlations (both autocorrelation and cross-correlations), thecases for the spatial correlations are rather “less interesting” in that the results are obviousand analogous to their zero-noise-cross-correlated counterparts. Apart from quantitative dif-ferences in energy magnitudes, the reason for which has already been explained, the decayingprofiles reassert the initial hypothesis of nonzero noise cross-correlation as being predomi-nantly a time symmetry effect. To summarize, a nonzero noise cross-correlation (and hencetime symmetry violation too) boosts the energy levels in the system while simultaneouslyneutralizing the oscillating effects generally attributed to Alfven waves whose major effectscould be seen in all forms of temporal statistics.Many issues in relation to such subcritical (linear) turbulence in rotating accretion flowsstill remain unclear. Although it is now rather well acknowledged that transient linearturbulence is essentially caused by non-normal disturbances [3, 19] (indirectly shown in [1]too), the implication of the resultant power law (energy growth rate ∼ Re / ) with referenceto stochasticity is a question that is pretty much unresolved as yet. Also, how all of thesepatterns evolve and modify the dynamic bifurcation structures in relevant phase diagrams[19], especially in the non-equilibrium nonlinear regime, are exciting questions that needto be delved with. Through this paper what we have now shown is that a nonzero noise24ross-correlation is effectively another independent (field theoretically) “relevant” variablewhose effects are expected to affect all such situations through time symmetry violationsthat needs to be appropriately dealt with in all accretion flow related situations. ACKNOWLEDGMENTS
Both authors acknowledge discussions with Banibrata Mukhopadhyay. AKC also thanksthe Royal Society, U.K., research grant number RG110622, for partial support. [1] Mukhopadhyay, B., & Chattopadhyay, A. K., J. Phys. A , 035501 (2013).[2] Nath, S. K., Mukhopadhyay, B. and Chattopadhyay, A. K., Phys. Rev. E , 013010 (2013).[3] Maretzke, S., Bj¨orn, H. & Avila, M., J. Fluid Mech. , 254 (2014).[4] Avila, M., Phys. Rev. Lett. , 124501 (2012)[5] Rincon, F., Ogilvie, G. I., & Cossu, C., Astron. Astrophys. , 817 (2007).[6] Grassberger, P. & Procaccia, I., Phys. Rev. Lett. , 346 (1983).[7] Weisshaar, E., Busse, F. H. & Nagata, M., J. Fluid Mech. , 549 (1991).[8] Gu, P.-G., Vishniac, E. T., & Cannizzo, J. K., ApJ , 380 (2000).[9] Kim, W.-T., & Ostriker, E. C., ApJ , 372 (2000).[10] Mahajan, S. M., & Krishan, V., ApJ , 602 (2008).[11] Rudiger, G., & Zhang, Y., A&A , 302 (2001).[12] Dauchot, O., & Daviaud, F., Phys. Fluids , 335 (1995).[13] Richard, D., & Zahn, J.-P., A&A , 734 (1999).[14] Klahr, H. H., & Bodenheimer, P., ApJ , 869 (2003).[15] Balbus, S. A., Hawley, J. F., & Stone, J. M., ApJ , 76 (1996).[16] Hawley, J. F., Balbus, S. A., & Winters, W. F., ApJ , 394 (1999).[17] Dubrulle, B., Dauchot, O., Daviaud, F., & Longaretti, P.-Y., Richard, D., & Zahn, J.-P.,Phys. Fluids , 095103 (2005).[18] Dubrulle, B., Mari, L., Normand, C., Hersant, F., Richard, D., & Zahn, J.-P. A&A , 1(2005).[19] Mukhopadhyay, B., Afshordi, N., & Narayan, R., ApJ , 383 (2005).
20] Pumir, A., Phys. Fluids , 3112 (1996).[21] Fromang, S., & Papaloizou, J., A&A , 1113 (2007).[22] Mukhopadhyay, B, Mathew, R., & Raha, S., New J. Phys. , 023029 (2011).[23] Eckhardt, B., & Pandit, R., Eur. Phys. J. B , 373 (2003).[24] Chattopadhyay, A. K., & Bhattacharjee, J. K., Phys. Rev. E , 016306 (2000).[25] Forster, D., Nelson, D. R., & Stephen, M. J., Phys. Rev. A , 732 (1977).[26] Chattopadhyay, A. K., Basu, A. and Bhattacharjee, J. K., Phys. Rev. E , 2086 (2000).[27] Lesur, G., & Longaretti, P.-Y., A&A , 25 (2005).[28] Alfv´en, H., Nature (3805): 405 (1942).[29] See Supplemental Material at [URL will be inserted by publisher] for additional auto andcross-correlation plots for a range of values of q . The first 2 plots are for spatial autocorrela-tion functions, followed by 6 spatial cross-correlation plots and a temporal cross-correlationfunction plot focusing on the q → , A64 (2012).[31] Matsakos, T., Chi` e ze, Stehl´ e , C., J.-P., Gonz´ a lez, M., Ibgui, L., de S´ a , L., Lanz, T., Orlando,S., Bonito, R., Argiroffi, C., Reale, F. & Peres, G., A&A , A69 (2013).[32] Proga, D., Astrophys. J.,
693 (2007). upplementary information for the manuscript: “Cross-correlation Aided Transportin Stochastically Driven Accretion Flows” Sujit Kumar Nath
Department of Physics, Indian Institute of Science, Bangalore 560 012, India ∗ Amit K Chattopadhyay
Aston University, Nonlinearity and Complexity Research Group,Engineering and Applied Science, Birmingham B4 7ET, UK † I. SUPPLEMENTARY SECTION
A non-zero noise cross-correlation implies a violation of time translation symmetry making it imperative that onlytemporal correlation functions, both for autocorrelation and cross-correlations, should be the primarily affected ones.This is what we find too as have been explained in details in the main text (article). This noise cross-correlationbrings about some perceptible quantitative changes in the spatial correlation functions too. Although there is notmuch change in the spatial physics part, in view of the fact that quantitative changes still occur, here we present allother spatial correlation plots which are not presented in the main text for the convenience of the more interestedreaders. The implication of the q-values remains the same as in the main text: q = 3 / q = 2 represents a disk with a constant angular momentum while q = 1 represents a disk with a flat rotation curve. - r S ζ B FIG. 1: (Color online) Spatial autocorrelations of magnetic vorticity ( S ζ B ) upto r = 1 .
0. The solid line represents q ≈
2, the dashed line is for q = 1 .
7, the dotted line represents q = 1 . q = 1. ∗ [email protected] † [email protected] a r X i v : . [ a s t r o - ph . H E ] D ec - r S BB FIG. 2: (Color online) Spatial autocorrelations of magnetic field ( S BB ) upto r = 1 .
0. Plot styles are same as in Fig 1. - r S uB FIG. 3: (Color online) Spatial correlations of velocity and magnetic field ( S uB ) upto r = 1 .
0. Plot styles are same asin Fig 1. - r S u ζ B FIG. 4: (Color online) Spatial correlations of velocity and magnetic vorticity ( S uζ B ) upto r = 1 .
0. Almost no differencein magnitude compared to the zero noise cross-correlation case. Plot styles are same as in Fig 1. - r S ζ B FIG. 5: (Color online) Spatial correlations of vorticity and magnetic field ( S ζB ) upto r = 1 .
0. Similar in magnitudecompared to the zero noise cross-correlation case. Plot styles are same as in Fig 1. - r S ζζ B FIG. 6: (Color online) Spatial correlations of vorticity and magnetic vorticity ( S ζζ B ) upto r = 1 .
0. 8 orders ofmagnitude higher than the zero noise cross-correlation case. Plot styles are same as in Fig 1. - r S B ζ B FIG. 7: (Color online) Spatial correlations of magnetic field and magnetic vorticity ( S Bζ B ) upto r = 1 .
0. 8 order ofmagnitude higher than the zero noise cross-correlation case. Plot styles are same as in Fig 1. τ C uB FIG. 8: (Color online) Temporal cross-correlation of the velocity with the magnetic field ( C uB ) plotted against timeseparation τ . This plot shows the effect of the limit q → q becomes increasingly closer to 2. The thick line showsthe result for q = 1 . q = 1 . q = 1 . q = 1 .
7, the dotted line represents q = 1 . q = 1. Clearly, a progressiveincrease in the q-value approaching 2 only causes a quantitative shift in the correlation function while keeping thequalitative result (Alfven oscillation) unchanged. In the plot shown, the q = 1 . q = 1 ..