Long-term evolution of large-scale magnetic fields in rotating stratified convection

S2-1 Publ. Astron. Soc. Japan (2014) 66 (SP1), S2 (1–7) doi: 10.1093/pasj/psu081 Advance Access Publication Date: 2014 October 28 Long-term evolution of large-scale magnetic fields in rotating stratified convection Youhei MASADA1,∗ and Takayoshi SANO2,∗ 1 Department of Computational Science, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe, Hyogo 857-8501, Japan 2 Institute of Laser Engineering, Osaka University, 2-6 Yamadaoka, Suita, Osaka 565-0871, Japan *E-mail: , Received 2014 February 17; Accepted 2014 March 25 Abstract Convective dynamo simulations are performed in local Cartesian geometry. We report the first successful simulation of a large-scale oscillatory dynamo in rigidly rotating convection without stably stratified layers. A key requirement for exciting the large-scale dynamo is a sufficiently long integration time comparable to the ohmic diffusion time. By comparing two models with and without stably stratified layers, their effect on the largescale dynamo is also studied. The spatiotemporal evolution of the large-scale magnetic field is similar in both models. However, it is intriguing that the magnetic cycle is much shorter in the model without the stable layer than with the stable layer. This suggests that the stable layer impedes the cyclic variations of the large-scale magnetic field. Key words: convection — stars: magnetic fields — Sun: magnetic fields — turbulence 1 Introduction A grand challenge in astrophysics is to understand a self-organizing property of magnetic fields in highly turbulent flows. Solar magnetism is the front line of this area. The solar magnetic field shows a remarkable spatiotemporal coherence even though it is generated by turbulent convection operating within its interior. Our understanding of solar magnetism has been accelerated over the past decade in response to the broadening, deepening, and refining of numerical dynamo models (Charbonneau 2010; Brandenburg et al. 2012; Miesch 2012). However, it is still unclear what dynamo mode is excited in the solar interior and how it regulates the magnetic cycle. Various geometries have been applied to the numerical dynamo modeling: global spherical shell geometry (e.g., Gilman & Miller 1981; Brun et al. 2004; Ghizaru et al. 2010; Masada et al. 2013), spherical-wedge geometry (e.g., Brandenburg et al. 2007; Käpylä et al. 2010), and local Cartesian geometry (e.g., Cattaneo & Vainshtein 1991; Brandenburg et al. 1996). Among them, local Cartesian geometry is the most simplified and is often used for distilling the physical essence of the convective dynamo process by more accurately resolving convective eddies. A long-standing goal in numerical dynamo modeling in local Cartesian geometry is to realize the successful simulation of self-organized and self-sustained large-scale magnetic fields, so-called large-scale dynamos, by rotating convection alone without mean shear flow. The mean-field dynamo theory predicts that rigidly rotating convection can generate net helicity and then excite the large-scale  C The Author 2014. Published by Oxford University Press on behalf of the Astronomical Society of Japan. All rights reserved. For Permissions, please email: S2-2 Publications of the Astronomical Society of Japan, (2014), Vol. 66, No. SP1 dynamo even without the mean shear effect via a stochastic process, which is known as the α-effect (Moffatt 1978; Krause & Raedler 1980). However, no evidence of the large-scale dynamo was found in earlier studies of rigidly rotating convection (e.g., Cattaneo & Hughes 2006; Tobias et al. 2008). Käpylä, Korpi, and Brandenburg (2009) provided a breakthrough in dynamo modeling in the local system. They were the first to demonstrate that rigidly rotating convection can excite the large-scale dynamo in the local system that consists of the convection layer and the stably stratified layers. Subsequently, the oscillatory behavior of the largescale magnetic field was reported in Käpylä, Mantere, and Brandenburg (2013). Since the large-scale dynamo can be excited only when the Coriolis number is large, they concluded that the absence of the large-scale dynamo in earlier studies is caused by slow rotation speed. However, even in a sufficiently rapidly rotating convection, Favier and Bushby (2013) could not find evidence for the large-scale dynamo in the local system with the convection zone alone. They suggested that the essential part for the large-scale dynamo might be the stably stratified layer assumed in the model of Käpylä, Korpi, and Brandenburg (2009) rather than the rapid rotation. Therefore, at present, the key requirement for the largescale dynamo is still controversial. The purpose of this work is to find evidence of the large-scale magnetic field in the system only with the convection zone in order to demonstrate that rigidly rotating convection is a sufficient condition for the large-scale dynamo. In addition, by comparing two convective dynamo models with and without stably stratified layers, we will discuss their effect on the large-scale dynamo. 2 Model setup We numerically solve two convective dynamo systems in the local Cartesian domain: a one-layer system (model A) only with convection zone of thickness d (z1 ≤ z < z2 ), and a three-layer system (model B) consisting of an upper isothermal cooling layer of depth 0.15 d (z0 ≤ z ≤ z1 ), a middle convection layer of depth d (z1 ≤ z < z2 ), and a lower stably stratified layer of depth 0.85 d (z2 ≤ z < z3 ), where the x- and y-axes are taken to be horizontal and the z-axis is pointing downward. The aspect ratio between the thickness of the convection layer and the box width (W) is set to be W/d = 4 for both models. The setups in models A and B are similar to those used in Favier and Bushby (2013) and Käpylä, Korpi, and Brandenburg (2009), respectively. The basic equations are the compressible MHD equations in a frame of reference rotating with a constant angular velocity:  = −0 ez ∂ρ = −∇ · (ρu), ∂t (1) ∇P J ×B ∇ · Du =− + − 2 × u + + g, Dt ρ ρ ρ (2) P∇ · u  − 0 D =− + Qheat − , Dt ρ τ (z) (3) ∂B = ∇ × (u × B − η0 J ), ∂t (4) where J = ∇ × B/ μ0 is the current density, g = g0 ez is the gravity, and  is the specific internal energy. The viscosity, magnetic diffusivity, and thermal conductivity are represented by ν 0 , η0 , and κ 0 , respectively. The last term in equation (3) works only in model B and describes a cooling at the top of the domain with cooling time τ (z) which has a smooth profile connecting to the convection layer, where τ (z1 ) = ∞. The viscous stress  is written by  = 2ρν0 S with the strain rate tensor Si j = 1 2  ∂u j ∂ui 2 ∂ui + − δi j ∂ xj ∂ xi 3 ∂ xi  . (5) The heating term Qheat consists of thermal conduction, viscous heating, and Joule heating: Qheat = ∇ · (κ0 ∇) μ0 η0 J 2 + 2ν0 S2 + . ρ ρ (6) We assume a perfect gas law P = (γ − 1)ρ with γ = 5/3. The initial hydrostatic balance is described by a piecewise polytropic (...truncated)