Direct numerical simulations of helical dynamo action: MHD and beyond

Magnetohydrodynamic dynamo action is often invoked to explain the existence of magnetic fields in several astronomical objects. In this work, we present direct numerical simulations of MHD helical dynamos, to study the exponential growth and saturation of magnetic fields. Simulations are made within the framework of incompressible flows and using periodic boundary conditions. The statistical properties of the flow are studied, and it is found that its helicity displays strong spatial fluctuations. Regions with large kinetic helicity are also strongly concentrated in space, forming elongated structures. In dynamo simulations using these flows, we found that the growth rate and the saturation level of magnetic energy and magnetic helicity reach an asymptotic value as the Reynolds number is increased. Finally, extensions of the MHD theory to include kinetic effects relevant in astrophysical environments are discussed. 1 Introduction In magnetohydrodynamic (MHD) dynamos, an initially small magnetic field is amplified and sustained by currents induced solely by the motion of a conducting fluid (Moffatt, 1978). Dynamo action is often invoked to explain the existence of magnetic fields in several astronomical objects, including the Sun (Larmor, 1919; Parker, 1955; Leighton, 1969; Dikpati and Charbonneau, 1999), late-type stars (Brandenburg et al., 1998; Tobias, 1998), accretion disks (Blandford and Payne, 1982; Matsumoto et al., 1996; Casse and Keppens, 2004), and the Earth (Bullard, 1949; Braginsky, 1964; Glatzmaier et al., 1999; Buffett, 2000). Together with rotation, turbulence is ubiquitous in all these objects, and the generation of a magnetic field by a turbulent flow has become a crucial aspect of dynamo theory. Evidence of the Correspondence to: D. O. Gómez ([email protected]) existence of turbulent flows in astrophysics can be found in a large number of objects, such as the interstellar medium (Armstrong et al., 1995; Minter and Spangler, 1996) or the solar convective region (Espagnet et al., 1993; Krishan et al., 2002). In the last decades dynamo theory made significant advances, pushed forward by the positive feedback from direct numerical simulations (DNS), and the growing power of computers. The linear (or kinematic) regime of dynamo action was first studied theoretically within the framework of mean field magnetohydrodynamics (Steenbeck et al., 1966; Krause and Rädler, 1980). Later, its nonlinear regime was studied using MHD closures, such as the EDQNM (Pouquet et al., 1976). These studies led to the conclusion that a helical flow can under general conditions generate a large scale magnetic field through the α-effect or the inverse cascade of magnetic helicity. Since then, direct simulations of MHD turbulence have given an important insight into the problem. The first numerical simulation of dynamo action was made by Meneguzzi et al. (1981). Although computer resources were insufficient at the time to span magnetic diffusion timescales, the results made clear that a helical flow can amplify an initially small magnetic field exponentially fast, as well as significantly increase its spatial correlation. Later, several turbulent simulations were carried using mechanic helical forces (e.g. ABC type force), most of them with idealized conditions (Galanti et al., 1991; Brandenburg, 2001; Archontis et al., 2003; Mininni et al., 2003). These simulations are often made in the incompressible MHD limit (see however Meneguzzi and Pouquet (1989); Cattaneo (1999); Balsara and Pouquet (1999) for studies of the effect of compressibility or stratification), and using periodic boundary conditions. Extensions of these models to include idealized boundaries can be found e.g. in Brandenburg and Dobler (2001). These conditions are far from astrophysical or geophysical applications, but allow drastic 620 D. O. Gómez and P. D. Mininni: Simulations of helical dynamo action simplifications in the equations and numerical methods. In turn, these simplifications allow to resolve a larger separation of scales, which is crucial to study turbulent dynamo action. Actually, the strongest limitation for DNS is related with the extreme computational resources needed to ensure this scale separation. A systematic study of different flows, long simulations, or simulations in the range of parameters found in astrophysics are impossible in the foreseeable future. As a result, it seems evident that theory and simulations must evolve together to reach a deeper understanding of the problem. On the other hand, in DNS the conditions that the flow must satisfy to amplify and sustain a magnetic field can be studied directly. In this work we follow this path and review recent results in the study of helical dynamo action. In Sect. 2 we introduce the MHD equations, and in Sect. 3 we present results for the time evolution of the MHD helical dynamo. In the last years, new effects relevant in several astrophysical scenarios have been introduced into the theory and simulations. Section 4 reviews a few of these extensions, and discuss recent results found within the framework of HallMHD. Finally, in Sect. 5 we present the conclusions of this work. 2 The equations Incompressible MHD is described by the induction and the Navier-Stokes equations, ∂B ∂t = ∇× (U×B)+ η∇2B (1) ∂U ∂t = − (U · ∇)U + (B · ∇)B − − ∇ ( P + B2 2 ) + F + ν∇2U , (2) with the additional constraints ∇·U=∇·B=0. The external force F is assumed to be solenoidal, and the velocity U and magnetic field B are expressed in units of a characteristic speed U0. The pressure P was divided by the constant fluid density. Here, η and ν are respectively the magnetic diffusivity and the kinematic viscosity (assumed constant and uniform). The MHD system has three ideal quadratic invariants (i.e. for η=ν=0) E = 1 2 V ∫ V (U2 + B2) dV = 1 2 < U2 + B2 > , (3) Hm = 1 2 V ∫ V A · B dV = 1 2 < A · B > , (4) Hc = 1 2 V ∫ V B · U dV = 1 2 < B · U > . (5) Here E is the mean energy density, Hm is the mean magnetic helicity, and Hc is the mean cross helicity. It is standard practice in numerical simulations to compute the volume average of the ideal invariants over the total volume V (indicated by < . . .>), rather than their total values. The vector potential A is defined by B=∇×A. The simulations discussed in the following sections were made using a parallel pseudospectral code (Mininni et al., 2003, 2004; Gómez et al., 2004). Equations (1) and (2) are integrated in a cubic box with periodic boundary conditions. The code implements Runge-Kutta of several orders to evolve the equations in time, but most of the simulations were performed using Runge-Kutta of order two. The total pressure PT=P+B/2 is computed solving a Laplace equation in Fourier space at each time step, to ensure the incompressibility condition ∇·U=0 (Canuto et al., 1988). To satisfy the divergence-free condition for the magnetic field, the induction equation (1) is replaced by an equation for the vector potential ∂A ∂t = U×B −∇φ + η∇2A , (6) where φ (the self-consistent electrostatic potential) is computed at each time step to satisfy the gauge ∇·A=0, by solving another Laplace equation with the same method used to obtain the pressure. The code uses the 2/3-rule to control aliasing error (Canuto et al., 1988), and therefore the largest wavenumber kmax that can be resolved in a spatial grid of N3 points is given by kmax=N/3. In this paper, we show simulations up to resolutions of 2563 grid points. 3 MHD helical dynamos It is clear from the equations presented in the previous section, that effects such as stratification, magnetic buoyancy, or rotation are neglected. As a result, no natural mechanism is present to break the mirror symmetry in the flow. Therefore, a net kinetic helicity must be injected into the fluid by the body force F . Usually, eigenfunctions of the curl operator are used to force the fluid, ensuring both injection of energy and kinetic helicity at large scales. To help scale separation, the energy injection band is often restricted to a few wave numbers around the wavenumber kf orce. In our simulations we use an ABC force F with A=B=C (Childress, 1970), acting at kf orce (see Mininni et al. (2003) for details). The amount of helicity in the flow is measured by the kinetic helicity (Moffatt, 1978) Hk = 1 2 V ∫ V U · ω dV = 1 2 < U · ω > , (7) where ω=∇×U is the vorticity. It is also useful to normalize this quantity between 1 and −1 introducing the relative helicity 2Hk/( 〈 U2 〉 〈 ω2 〉 )1/2. The main question is whether a turbulent flow can amplify and sustain a magnetic field. As a result, no source of magnetic field is used except for a small amplitude random seed at scales smaller than the energy injection band. 3.1 Helical flows Before introducing the magnetic seed, a fully developed turbulent flow is needed. To this end, a forced hydrodynamic D. O. Gómez and P. D. Mininni: Simulations of helical dynamo action 621 D. O. Gómez and P. D. Mininni: Simulations of helical dynamo action 3 Fig. 1. (a) Kinetic energy spectrum at different Reynolds numbers. (b) Spectrum of kinetic energy, and kinetic helicity divided by }V~€ik‚„ƒ . The Kolmogorov’s law is showed as a reference. simulation is carried out until it reaches a statistically steady state. The resulting flow has, say, positive helicity (in all the simulations discussed here this will be the case). The r.m.s value of the velocity 243 …w