Dynamo Transition in Low-dimensional Models

Dynamo T ransition in Lo w-dimensional Mo dels Mahendra K. V erma, 1 Thomas Lessinnes, 2 Daniele Carati, 2 Ioannis Sarris, 3 Krishna Kumar, 4 and Meenakshi Singh 5 1 Dep artment of Physics, IIT Kanpur, India 2 Physique Statistique et Plasmas, Universit´ e Libr e de Bruxel les, B-1050 Bruxel les, Belgium 3 Dep artment of Me chanic al and Industrial Engine ering, University of Thessaly, V olos, Gre e c e 4 Dep artment of Physics, IIT Kharagpur, India 5 Dep artment of Physics, Penn-state University, University Park, USA. Tw o low-dimensional magnetoh ydrodynamic mo dels con taining three velocity and three magnetic mo des are describ ed. One of them (nonhelical model) has zero kinetic and curren t helicity , while the other mo del (helical) has nonzero kinetic and current helicit y . The velocity mo des are forced in b oth these mo dels. These low-dimensional mo dels exhibit a dynamo transition at a critical forcing amplitude that dep ends on the Prandtl num ber. In the nonhelical mo del, dynamo exists only for magnetic Prandtl num b er b ey ond 1, while the helical model exhibits dynamo for all magnetic Prandtl num ber. Although the mo del is far from reproducing all the p ossible features of dynamo mec hanisms, its simplicity allo ws a v ery detailed study and the observ ed dynamo transition is sho wn to b ear similarities with recent n umerical and exp erimen tal results. P ACS n um b ers: 91.25.Cw, 47.65.Md, 05.45.Ac I. INTR ODUCTION The understanding of magnetic field generation, usu- ally referred to as the dynamo effect, in planets, stars, galaxies, and other astrophysical ob jects remains one of the ma jor challenges in turbulence research. There are man y observ ational results from the studies of the Sun, the Earth, and the galaxies [1, 2, 3]. Dynamo has also b een observed recen tly in lab oratory exp erimen ts [4, 5] that hav e made the whole field very exciting. Numerical sim ulations [6, 7, 8, 9, 10] also give access to many useful insigh ts into the physics of dynamo. Ho wev er, the com- plete understanding of the dynamo mec hanisms has not y et emerged. The tw o most imp ortan t nondimensional parameters for the dynamo studies are the Reynolds n um b er Re = U L/ν and the magnetic Prandtl num b er P m = ν /η , where U and L are the large-scale v elo cit y and the large length-scale of the system resp ectiv ely , and ν and η are the kinematic viscosity and the magnetic diffusivity of the fluid. Another nondimensional parameter used in this field is the magnetic Reynolds num ber Re m , defined as U L/η . Clearly Re m = Re P m , hence only tw o among the ab ov e three parameters are indep enden t. Note that galaxies, clusters, and the interstellar medium hav e large P m , while stars, planets, and liquid so dium and mercury (fluids used in laboratory exp eriments) hav e small P m [1, 2]. In a t ypical simulation, the magnetofluid is forced and the dynamo transition is considered to b e observed when a nonzero magnetic field is sustained in the steady-state laminar solution or in the statistically stationary tur- bulen t solution, dep ending on the regime. T ypically , dynamos o ccur for forcing amplitudes b ey ond a critical v alue whic h also defines the critical Reynolds num ber Re c and the critical magnetic Reynolds num b er Re c m . One of the ob jectiv es of b oth the n umerical simulations and the exp erimen ts [4, 5] is the determination of this critical magnetic Reynolds num b er R e c m . It has b een found that Re c m dep ends on both the t ype of forcing and the Prandtl n umber (or Reynolds num b er), yet the range of R e c m ob- serv ed in the numerical simulations is from 10 to 500 for a wide range of P m (from 5 × 10 − 3 to 2500). In recent magnetohydrodynamics (MHD) sim ulations, Sc hekochihin et al. [6, 11] applied nonhelical forcings and observed that the dynamo is active for a magnetic Prandtl num b er larger than a critical Prandtl num b er P c m that is around 1. F or fluids with small Prandtl num- b er P m ( P m < 1), numerical sim ulations [7, 8, 9] indicate that the dynamo can b e pro duced using forcings having lo cal helicit y (the net helicity of the force could still b e zero). The range of the critical magnetic Reynolds num- b er in most of the simulations [7, 8, 9] is 10 to 500. Note that in the V on-Karman-So dium (VKS) exp erimen t, the critical magnetic Reynolds num b er is around 30. There are many attempts to understand the ab ov e ob- serv ations. F or large Prandtl n um b er, the resistiv e length scale is smaller than the viscous scale. F or this regime, Sc hekochihin et al. [6, 11] suggested that the growth rate of the m agnetic field is higher in the small scales b ecause stretc hing is faster at these scales. This kind of magnetic field excitation is referred to as smal l-sc ale turbulent dy- namo . F or low P m Stepano v and Plunian [12] argue for similar gro wth mec hanism. Their results are based on shell model calculations. The n umerical results of Isk ako v et al. [7] how ev er are not conclusiv e in this regard. The main arguments supporting these explanations are based on the inertial range (small-scales) prop erties of turbu- lence [6, 11]. In this pap er, we presen t an alternate view- p oin t. W e show that a low-dimensional dynamical sys- tem containing only large scales prop erties of the fields, three velocity and three magnetic F ourier mo des, is able to repro duce some of the ab ov e numerical results. F or certain types of forcing of velocity , a dynamo transition is observ ed for Re m > R e c m . These observ ations indi- cate that the large-scale eddies ma y also be resp onsible 2 for the dynamo excitation, and sev eral important prop- erties can b e derived from the dynamics of large-scale mo des. These observ ations are consisten t with earlier re- sults of MHD turbulence indicating that the large-scale v elo cit y field provides a significant fraction of the energy (around 40%) con tained in the large-scale magnetic field [13, 14, 15, 16, 17, 18]. I I. DERIV A TION OF LO W DIMENSIONAL MODELS The incompressible MHD equations are: ∂ t u = − n ( u , u ) + n ( b , b ) + ν ∇ 2 u + f − ∇ p tot , (1) ∂ t b = − n ( u , b ) + n ( b , u ) + κ ∇ 2 b , (2) ∇ · u = ∇ · b = 0 , (3) where p tot is the sum of the h ydro dynamic and magnetic pressures divided b y the density . The bilinear operator is defined as n ( a 1 , a 2 ) = a 1 · ∇ a 2 . Only p erio dic solution in a cubic b ox with linear dimension ` will b e consid- ered. The smallest non-zero w av e vector is thus given by k 0 = 2 π /` . The mo del is deriv ed by pro jecting the MHD equations on the subspace S < spanned by a small num- b er of basis vectors that are compatible with the p erio dic b oundary conditions and can be regarded as a subset of a complete basis. Only three vectors will b e considered in this study , so that the pro jection of b oth the equations for u and b on these v ectors leads to a six-dimensional system of equations. The three vectors are defined as follo ws: e 1 = 2 √ 2 + h 2   − sin k 0 x cos k 0 z h sin k 0 x sin k 0 z cos k 0 x sin k 0 z   , (4) e 2 = 2 √ 2 + h 2   h sin k 0 y sin k 0 z − sin k 0 y cos k 0 z cos k 0 y sin k 0 z   , (5) e 3 = 4 √ 6 + 10 h 2     − sin k 0 x cos k 0 y cos 2 k 0 z − cos k 0 x sin k 0 y cos 2 k 0 z cos k 0 x cos k 0 y sin 2 k 0 z   + h   0 − 2 sin k 0 x cos k 0 y sin 2 k 0 z sin k 0 x sin k 0 y cos 2 k 0 z     (6) They dep end on a free parameter h that determines the amoun t of helicity that can b e carried on by the basis v ectors. F or h = 0, each mo de is nonhelical. The three v ectors are also div ergence free ∇ · e α = 0 and are or- thonormal h e α · e β i = δ αβ . The inner pro duct of tw o arbitrary vectors v 1 and v 2 is defined by: h v 1 · v 2 i = 1 ` 3 Z ` 0 dx Z ` 0 dy Z ` 0 dz v 1 ( x, y , z ) · v 2 ( x, y , z ) . (7) F or h = 0, the basis functions e 1 and e 2 ha ve field con- figurations in xz and y z respectively . The pro jection op erator is noted P and its application on a vector v is defined by: P [ v ] ≡ v < = v 1 e 1 + v 2 e 2 + v 3 e 3 (8) where v α = h v · e α i . The pro jected part of v in S < is noted v < and the difference with the original vector is noted v > = v − v < . The pro jection of the velocity , magnetic, and force fields are expressed as follows. P [ u ] = u < = ( u 1 e 1 + u 2 e 2 + u 3 e 3 ) u ? , (9) P [ b ] = b < = ( b 1 e 1 + b 2 e 2 + b 3 e 3 ) u ? , (10) P [ f ] = f < = ( f 1 e 1 + f 2 e 2 + f 3 e 3 ) ( u ? ) 2 k 0 . (11) where the u α ’s, b α ’s and f α ’s are dimensionless real num- b ers and u ? = ν k 0 . The e 3 comp onen t of the forcing is supp osed to b e zero. The system of dynamical equations for these v ariables can b e deriv ed from Eqs. (1-3). F or instance the pro jection of the equation for the velocity on the vector e 1 , h ∂ t u · e 1 i , yields the equation for ˙ u 1 . The resulting system of equations is: ˙ u 1 = r ( h ) (1 − h 2 ) ( u 2 u 3 − b 2 b 3 ) − 2 u 1 + f 1 (12) ˙ u 2 = r ( h ) (1 + 3 h 2 ) ( u 1 u 3 − b 1 b 3 ) − 2 u 2 + f 2 (13) ˙ u 3 = − 2 r ( h ) (1 + h 2 ) ( u 1 u 2 − b 1 b 2 ) − 6 u 3 + f 3 (14) ˙ b 1 = r ( h ) (1 + h 2 ) ( u 2 b 3 − b 2 u 3 ) − 2 P − 1 m b 1 (15) ˙ b 2 = − r ( h ) (1 + h 2 ) ( u 3 b 1 − b 3 u 1 ) − 2 P − 1 m b 2 (16) ˙ b 3 = − 2 r ( h ) h 2 ( u 1 b 2 − b 1 u 2 ) − 6 P − 1 m b 3 . (17) where r ( h ) = 2 /  (2 + h 2 ) √ 6 + 10 h 2  . The structure of this model is, of course, reminiscent of the original MHD equations. In particular, it is interest- ing to study how the conserv ation of the ideal quadratic in v arian ts of the Na vier-Stok es and MHD equations can b e expressed in terms of the v ariables { u α , b α } . The con- serv ation of the kinetic energy E k b y the Na vier-Stok es equations is a direct consequence of h n ( u , u ) · u i = 0 (18) Because this equality holds for an y v elocity , it is also true for u < and, in the absence of magnetic field, the nonlin- ear terms in the system of equations (12-17) conserv e the kinetic energy E < k = ( u 2 1 + u 2 2 + u 2 3 ) / 2 asso ciated to u < . The cross helicity H < c = ( u 1 b 1 + u 2 b 2 + u 3 b 3 ) and the total energy , the sum of the kinetic energy E < k and the magnetic energy E < m = ( b 2 1 + b 2 2 + b 2 3 ) / 2, are conserved by the nonlinear terms for the same reason. Ho wev er, the conserv ation of the kinetic helicity H k = h u · ω i by the Na vier-Stokes equations and the conserv ation the mag- netic helicity H m = h b · a i b y the MHD equations, where ω = ∇ × u is the vorticit y and a is the v ector p otential ( b = −∇ × a ) ha ve no equiv alen t in the system (12-17). Indeed, the conserv ation of the kinetic helicity is a con- sequence of h n ( u , u ) · ω i = 0 (19) 3 Ho wev er, both the nonlinear term and the vorticit y are not fully captured in S < , even when computed from u < : P [ n ( u < , u < )] 6 = n ( u < , u < ) , (20) P [ ∇ × u < ] 6 = ∇ × u < , (21) As a consequence, the kinetic helicity carried on b y u < is not conserv ed by the nonlinear term in the lo w- dimensional mo del and, for the same reason, the mag- netic helicity is not conserved either. F or instance, the non-conserv ation of the kinetic helicity b y the lo w- dimensional mo del can b e expressed as, h n ( u < , u < ) < · ( ∇ × u < ) < i = −h n ( u < , u < ) > · ( ∇ × u < ) > i , (22) and the right hand side in this relation in not represented in the equations (12-17). In the following section we will solv e the truncated MHD equations (12-17) under sp e- cial cases. I II. ANAL YSIS OF THE MODEL Despite its simplicity , the complete analysis of the sys- tem of equations (12-17) is quite difficult. Indeed, the simple search for the fixed points leads to v ery intricate algebraic equations. W e hav e th us fo cused our analysis on tw o limit cases corresp onding to strictly nonhelical equations ( h = 0) and the case h = 1, b oth of which can b e treated analytically . A. Nonhelical mo del The system of equations (12-17) is th us first considered for h = 0. In that case the kinetic helicity captured in subspace S < v anishes ( h u < · ( ∇ × u < ) i = 0). In this first mo del, the forcing is chosen to b e f = f √ 2 ( e 1 + e 2 ) (23) where f = p h f · f i . Hence, f 1 = f 2 = f / √ 2 and f 3 = 0. The fixed p oints can then b e computed analytically . Three fixed p oints corresp ond to a v anishing magnetic ( b 1 = b 2 = b 3 = 0) field and will be referred to as fluid solutions: Fluid A ±                  u 1 = √ 2 8  f ± p f 2 − 1152  u 2 = √ 2 8  f ∓ p f 2 − 1152  u 3 = − 2 √ 6 (24) Fluid B          u 1 = u 2 = ( s 2 − 12) s u 3 = − ( s 2 − 12) 2 √ 54 s 2 (25) where s =  9 f / √ 2 + 3 p 192 + 9 f 2 / 2  1 / 3 . There are t wo additional fixed p oin ts with nonzero magnetic fields: MHD ±                              u 1 = u 2 = √ 2 f P m 4( P m + 1) u 3 = − 2 √ 6 P m b 1 = b 2 = ± P m 4( P m + 1) q f 2 − f 2 c 2 b 3 = 0 (26) 0 1 2 3 4 5 0 5 10 15 20 25 30 35 40 45 50 Pm f Fluid A Fluid B MHD FIG. 1: Plot of critical force f c 2 as a function of P m for the nonhelical mo del. Dynamo is excited for f > f c 2 , only for P m > 1. . The stability of the ab o v e fixed p oin ts can b e estab- lished b y computing the eigenv alues of the stability ma- trix. After some tedious algebra, it can b e shown that the ab o v e fixed points are stable in the three regions shown in Fig. 1(a) and defined b y the follo wing simple equations 4 in the plane ( P m , f ): P m = P c m = 1 (27) f = f c 1 = 24 √ 2 (28) f = f c 2 = 12 √ 2 P m + 1 P 3 / 2 m (29) Since the v elo cit y and magnetic field amplitudes u α and b α m ust b e real num b ers, the solutions Fluid A ± only exist for f > f c 1 . They are stable for P m < 1. The solution Fluid B is stable for P m < 1 and f < f c 1 , and for P m > 1 and f < f c 2 . It is unstable elsewhere. The solution MHD ± is stable for P m > 1 and f > f c 2 and is unstable elsewhere. In summary , we hav e only fluid solutions for P m < 1, while an MHD solution is p ossi- ble only for P m > 1 and f > f c 2 . The ab ov e solutions co ver the entire ( f , P m ) parameter space. The system con verges to one of these states dep ending on the pa- rameter v alues irrespective of its initial conditions, and the system is neither oscillatory nor chaotic. Clearly , the amplitude of b 1 and b 2 close to the dynamo threshold increases as p f − f c 2 . The dynamo transition in this lo w-dimensional model is th us a pitchfork bifurcation. The ab o v e results compare remark ably well with the n umerical findings of Sc hekochihin et al. [6, 11] where a dynamo transition is also found for P m ≥ P c m with P c m near 1. Considering the simplicity of the ab ov e mo del, suc h a quantitativ e ag reemen t may very well b e fortu- itous. Nev ertheless, it is also in teresting to notice that, in our mo del, f c 2 decreases for increasing Prandtl n um- b ers. Hence, it is easier to excite nonhelical dynamo for larger P m or more conductiv e magnetofluid. This feature of our mo del is also in agreement with recent numerical sim ulations [6, 7, 8, 9, 10] where an increase of the critical magnetic Reynolds n um b er Re c m is reported for smaller v alues of P m . It is also interesting to express the ab o ve results in terms of the Reynolds num b er instead of the forcing am- plitude. A Reynolds n um b er can be build b y defining the large scale velocity as U L = p u 2 1 + u 2 2 u ? . Based on this v elo cit y scale, the kinetic Reynolds num ber is given by: Re = U L ν k 0 =                      √ 2 p f 2 − 576 4 Fluid A √ 2 s 2 − 12 s Fluid B f P m 2(1 + P m ) MHD (30) F or large f , the amplitude of the magnetic field is pro- p ortional to the kinetic Reynolds num ber: | b 1 | ≈ | u 1 | ≈ Re/ √ 2 . (31) This result is consistent with the result obtained by Mon- c haux et al. [4, 5, 19]. The critical Reynolds n umber for the dynamo transition b et ween the Fluid B and MHD solutions is easily computed: Re c = 6 r 2 P m (32) and the critical magnetic Reynolds num b er Re c m : Re c m = P m Re = 6 p 2 P m . (33) Since P m > 1, Re c m > 6 √ 2. Note that Re c m Re c = 72. Hence, the Re c − Re c m curv e is a h yp erb ola for Re c ≤ 6 √ 2 since dynamo exists for P m > 1. T o gain further insights into the dynamo mec hanism w e hav e inv estigated the energy exchanges among the mo des of the mo del. The fluid mo des u 1 and u 2 gain en- ergy from the forcing f and give energy to the mo de u 3 . The magnetic mo des b 1 and b 2 gain energy from the mo de u 3 . The net energy transfer to the mo de b 3 through non- linear interaction v anishes. Consequently the mo de b 3 go es to zero due to dissipation. The b 1 and b 2 mo des also lose energy due to dissipation. The energy balances for these mo des reveal in teresting feature of dynamo transi- tion. At the steady-state, the energy input in the mo de b 1 due to nonlinearity ( − b 1 b 2 u 3 / √ 6 = − b 2 1 u 3 / √ 6) matches the Joule dissipation (2 b 2 1 /P m ) [Eq. (15)]. F or f < f c 2 , the v alue of u 3 is less than − 2 √ 6 /P m , hence b 1 → 0 asymptotically , thus sh utting do wn the dynamo. F or f > f c 2 , u 3 = − 2 √ 6 /P m , thus the energy input to the mo de b 1 exactly matc hes with the Joule dissipation. This is the reason wh y b 1 and b 2 are constants asymptotically in the MHD regime. W e also studied a v arian t of the ab o ve mo del in which f 1 = 0 and f 2 = f . F or this mo del, the only solution is u 2 = f / 2 with all the other v ariables b eing zero. Hence this mo del do es not exhibit dynamo. In the next subsec- tion we will discuss another lo w-dimensional mo del that con tains helicit y . B. Helical mo del In this second version of the mo del, the v alue h = 1 has been c hosen. The helicit y captured in the subspace S < is then given by: h u < · ( ∇ × u < ) i =  4 3 ( u 2 1 − u 2 2 ) − 3 2 u 2 3  ( u ? ) 2 k 0 . (34) If w e restrict again the forcing to the large-scale mo des ( f 3 = 0), the amoun t of helicit y carried on by the velocity field is exp ected to increase, at least in absolute v alue, if the forcing act differently on the mo des e 1 and e 2 . W e ha ve c hosen the extreme case for which the forcing acts only on u 2 ( f 1 = f 3 = 0 and f 2 = f ): f = f e 2 (35) 5 where again f = p h f · f i . F or these parameters, a unique fluid stationary solution and tw o stationary MHD solu- tions are found: Fluid      u 1 = u 3 = 0 u 2 = f 2 (36) MHD ±                            u 1 = u 3 = 0 u 2 = 6 √ 3 P m b 1 = √ 3 b 3 = ± √ 3 2  − 72 P m + 2 √ 3 f  1 / 2 b 2 = 0 (37) All these solution ob viously carry a non-zero resolv ed he- licit y as defined by (34). The fluid solution is stable for f < f c 3 ≡ 12 √ 3 /P m , while the MHD solutions are sta- ble for f > f c 3 . The plot of the critical force f c 3 vs. P m for helical mo del is sho wn in Fig. 2. Note that the helical mo del exhibits dynamo for all P m as long as f > f c 3 . Also, the critical forcing for the helical mo del is low er than the corresp onding v alue for the nonhelical mo del. Hence it is easier to excite helical dynamo compared to the nonhelical dynamo consistent with recen t numerical sim ulations [7, 8, 9, 10]. 0 1 2 3 4 5 0 5 10 15 20 25 30 35 40 45 50 Pm f Fluid MHD FIG. 2: Plot of critical force amplitude f c 3 as a function of P m for the helical mo del. The dashed lines repro duce the stabilit y regions of the non-helical mo del. Here again, the Reynolds num ber based on the large scale v elo cit y U L = u ∗ p u 2 1 + u 2 2 can b e defined and yields: Re = U L ν k 0 =        f / 2 Fluid 6 √ 3 P m MHD (38) and the magnetic Reynolds num b er for MHD case is Re m = 6 √ 3, a constant. Using Eq. (34) we can compute the resolved kinetic helicit y H K and current helicity H J defined as H K = h u < · ( ∇ × u < ) i = 4 3 u 2 2 ( u ? ) 2 k 0 = − 144 P 2 m ( u ? ) 2 k 0 (39) H J = h b < · ( ∇ × b < ) i = 5 2 b 2 3 ( u ? ) 2 k 0 = − 5 2 √ 3 f 2 − 18 P m ! ( u ? ) 2 k 0 (40) P ouquet et al. [20], and Chou [21] conjectured the alpha parameter of dynamo to b e of the form α ≈ α u + α b = 1 3 τ ( − u · ∇ × u + b · ∇ × b ) , (41) where τ is the velocity de-correlation time. Clearly α is optimal if H K < 0 and H J > 0. Similar features are observed in flux calculation of V erma [22]. These conditions are satisfied in the helical model indicating a certain in ternal consistency with the results of Pouquet et al. [20], and Chou [21]. The energy exc hange calculation of the helical mo del rev eals that the magnetic mo des b 1 and b 3 receiv e energy from the velocity mo de u 2 ; the rate of energy transfer is proportional to b 1 b 3 u 2 that matc hes exactly with the Joule dissipation rate. Since u 2 = 6 √ 3 /P m for all P m , the rate of energy transfer b 1 b 3 u 2 matc hes with Joule dis- sipation rate ( ∝ b 2 1 /P m ), and the dynamo is p ossible for all P m in case of helical model. In contrast, in nonheli- cal mo del the corresponding velocity mode u 3 v aries as 1 /P m for P m > 1, but saturates at 2 √ 3 for P m < 1, hence the energy transfer cannot matc h the Joule dissi- pation rate for P m < 1 for nonhelical mo del th us shutting off the dynamo for P m < 1. This is one of the main dif- ference in helical and nonhelical models. Also note that as discussed at the end of the previous subsection, the nonhelical mo del with the forcing f 1 = 0 , f 2 = f do es not exhibit dynamo for any parameter. IV. DISCUSSION In this pap er, a class of low-dimensional mo dels that exhibit dynamo transition is deriv ed. These models de- p end on several parameters such as the Prandtl num ber, the forcing amplitude and a parameter, h , that charac- terizes the abilit y of the velocity mo des to carry kinetic helicit y . The fixed p oin ts of tw o simple mo dels corre- sp onding to h = 0 and h = 1 are studied in details. The first model is nonhelical (zero kinetic and curren t helic- it y) and is compatible with a stationary nonzero m ag- netic solution only for P m > 1. The second mo del has nonzero kinetic and current helicities, and it has nonzero 6 stationary magnetic solution for all P m . These findings confirm the idea that b oth kinetic and current helicities ma y play an imp ortan t role in the dynamo transition, esp ecially in helical mo del for P m < 1. These tw o v al- ues of h correspond to the only v alues for whic h one of the nonlinear coupling in the lo w dimensional mo del v an- ishes. With the sp ecific choice of the forcing prop osed in the previous section, these mo dels ha ve then exact and tractable solutions. Arbitrary v alues of h w ould lead to m uch more complex systems. Ob viously , the mo dels presented here are only a small subset of many p ossible MHD mo dels that could exhibit dynamo. F or instance, Rikitake [23] constructed a dy- namo mo del consisting of t w o-coupled disks that sho ws field reversal. Nozi` eres [24] prop osed a mo del inv olv- ing one velocity and tw o magnetic field v ariables. These t wo models are phenomenological. Recently , Donner et al. [25] prop osed a truncation of the MHD equations along the same lines as our mo del, but derived a muc h more complex system of equations for 152 modes. Don- ner et al. [25] analyzed the dynamical evolution of these mo des for P m = 1 and observed steady-state and c haos in their system. The main adv antage of our model is th at it allows a complete analytical treatment. The six v ariables of our mo del are only represen tative of the large-scale mo des of the system, while a realis- tic description of a turbulent system exhibiting dynamo should hav e a large num b er of mo des. Although the small scale v ariables are often quite imp ortant in tur- bulen t flo w, in some cases the large scale v ariables ma y determine some of the flow properties. The surprising success of the very simple mo dels prop osed here in re- pro ducing sev eral feature of the dynamo transition could suggest that we are in suc h a situation. Schek o c hihin et al. [6, 11], Stepanov and Plunian [12], and Isk ak o v et al. [7] ha v e highligh ted the role play ed by inertial range eddies that are absent in our low-dimensional mo dels. Y et in the absence of a definite theory of dynamo, it is in teresting to show that low-dimensional mo dels that fo- cus on the dynamics of the large scale flows may also b e successful. Ac knowledgemen ts This work has b een supp orted in part by the Com- m unaut´ e F ran¸ caise de Belgique (ARC 02/07-283) and by the contract of asso ciation EURA TOM - Belgian state. The conten t of the publication is the sole resp onsibil- it y of the authors and it do es not necessarily represent the views of the Commission or its services. D.C. and T.L. are supp orted by the F onds de la Recherc he Scien- tifique (Belgium). MKV thanks the Ph ysique Statistique et Plasmas group at the Universit y Libre du Brussels for the kind hospitalit y and financial supp ort during his long leav e when this work w as undertak en. 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