Radiative zone solar magnetic fields and g modes

T. I. Rashba 1 2 V. B. Semikoz 0 1 J. W. F. Valle 0 0 AHEP Group, Instituto de Fsica Corpuscular - C.S.I.C./Universitat de Vale`ncia, Edificio Institutos de Paterna , Apt 22085 E-46071, Vale`ncia, Spain 1 Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation of the Russian Academy of Sciences , IZMIRAN, Troitsk, Moscow Region 142190, Russia 2 Max-Planck-Institut fur Physik (Werner-Heisenberg-Institut) , Fohringer Ring 6, D-80805 Munchen, Germany A B S T R A C T We consider a generalized model of seismic-wave propagation that takes into account the effect of a central magnetic field in the Sun. We determine the g-mode spectrum in the perturbative magnetic field limit using a 1D magnetohydrodynamics picture. We show that central magnetic fields of about 600-800 kG can displace the pure g-mode frequencies by about 1 per cent, as hinted by the helioseismic interpretation of GOLF (Global Oscillations at Low Frequencies) observations. 1 I N T R O D U C T I O N Currently, there is very little direct information about the structure and strength of magnetic fields in the radiative zone (RZ) of the Sun, for a short review see Introduction of the paper (Burgess et al. 2004a). Some authors argue that for the young Sun ( 330 Myr) relatively small fields, 10 kG (Moss 2003) and 1 G (Kitchatinov, Jardine & Collier-Cameron 2001) could survive, being relic fields captured from the primordial ones in the protostar plasma. For the Sun at the present epoch there is an upper bound of 23 MG near the tachocline obtained from the magnetic splitting of acoustic oscillations (Ruzmaikin & Lindsey 2002). However, some authors have considered very strong magnetic fields in the RZ, up to 30 MG (Couvidat, Turck-Chie`ze & Kosovichev 2003). Here, we suggest a new way to estimate the magnetic field strengths in the RZ of the Sun by relating them to the frequency shifts of g-mode candidates suggested by the first observations made with the GOLF (Global Oscillations at Low Frequencies) experiment (Turck-Chie`ze et al. 2004). We discuss some effects of RZ magnetic fields which could explain the displacement of g-mode frequencies with respect to the theoretical frequencies calculated in the absence of magnetic field. Indeed, the existence of such shifts are hinted in GOLFs data. If eventually confirmed by further data, the idea that RZ magnetic fields cause such frequency shifts would provide us with a useful tool to estimate their magnitude. In order to find spectra of seismic waves accounting for the magnetic field in the RZ a number of assumptions is required. For example: (i) We consider ideal magnetohydrodynamics (MHD) neglecting both the heat conductivity and viscosity contributions to energy losses, as well as the ohmic dissipation. E-mail: (TIR); (VBS); (JWFV) (ii) We linearize the MHD equations about a static background configuration, i.e. a background configuration which is time independent and for which the background fluid velocity vanishes, v0 = 0. (iii) We assume the fluctuations to be adiabatic, with the contributions of fluctuations to the heat source vanishing: Q = 0. (iv) Moreover, we consider a fully ionized ideal gas, so that the thermodynamic quantity, first adiabatic exponent = cp/cV, is time independent and uniform. For numerical estimates we will take = 5/3 for hydrogen plasma. (v) We adopt the Cowling approximation, which amounts to the neglect of perturbations of the gravitational potential (i.e. = 0). (vi) We assume a rectangular geometry with Cartesian coordinates: x, y and z, where z corresponds to the solar radial direction. The background quantities vary along the z direction only (which implies the local gravitational acceleration, g, is directed along the z axis, but in opposite direction). We also take a constant, uniform background magnetic field, B0, pointing along the x axis. (vii) The background massdensity profile is assumed to be exponential, 0 = c exp[z/H ], for constant c and H. The conditions of hydrostatic equilibrium for the background then determine the profiles of thermodynamic quantities, and in particular imply is a constant. We assume that the BruntVaisala frequency is zero in the convective zone (CZ) and non-zero, but constant in the RZ. In what follows, we shall again specify the assumptions used, as they are needed, in order to keep clear which results rely on which assumptions. Note that, deep within the RZ, the last approximation above holds to very good approximation for real massdensity profiles obtained by standard solar models, provided we identify the z direction with the radial direction. The constancy of in this region is also expected since the highly ionized plasma satisfies an ideal gas equation-of-state to good approximation. The rectangular geometry provides a reasonable approximation so long as we do not examine too close to the solar centre. What is important about our choice for B0 is that it is slowly varying in the region of interest, and it is perpendicular to both g and all background gradients, 0, p0, etc. As suggested in (Burgess et al. 2004a) such 1D picture can be fully described in analytical terms in contrast to the 3D case. There are two parameters which describe the spectra of magnetogravity waves (Burgess et al. 2004a): (i) strength of the background magnetic field B0 and (ii) the dimensionless transversal wavenumber K = kxH. Here H is the density scaleheight and kx is the projection of the wavevector on to the x-axis. Let us estimate the value of the transversal wavenumber that could be relevant for the g-mode candidates observed on the photosphere. Since g modes decay in the CZ as eKz/H , only modes with low transversal wavenumber K 14 (long wavelengths) could be seen at the photosphere. This follows from the simple estimate for the longitudinal fluid velocity v z (z) which is directed along the Sun Earth line and causes the Doppler shifts of optic lines registered by the GOLF experiment: vz (z = R ) = 2 mm s1. This formula comes from equation (14) (equivalent to our equations 10) and (30) of (Burgess et al. 2004a) for the decaying solution B(z1) (z) = bz eKz/H , where B(z1) is the z-component of the magnetic field perturbation. Here, in the right-hand side we substituted the sensitivity of the GOLF instrument to the minimum fluid speed, v z = 2 mm s1, while in the left-hand side we substituted the frequency estimate N and the wavelength through the CZ: R z RZ = 3H = 0.3 R . For instance, substituting for the magnetic field perturbation, bz /B 0 = 0.01, N = 2.8 103 rad s1 for the BruntVaisala frequency in the RZ, H = 0.1 R = 7 109 cm (Bahcall 1988) for the density scaleheight, one obtains e3K/K 106, from which the estimate K = K max 4 comes. We organize our presentation as follows. In Section 2 we formulate the MHD model for an ideal plasma. In Section 3, we linearize the full set of MHD equations and then derive a single master equation for the z-component (...truncated)