Vertical motions in an intense magnetic flux tube

Abstract

The recent discovery of localised intense magnetic fields in the solar photosphere is one of the major surprises of the past few years. Here we consider the theoretical nature of small amplitude motions in such an intense magnetic flux tube, within which the field strength may reach 2 kG. We give a systematic derivation of the governing 'expansion' equations for a vertical, slender tube, taking into account the dependence upon height of the buoyancy, compressibility and magnetic forces. Several special cases (e.g., the isothermal atmosphere) are considered as well as a more realistic, non-isothermal, solar atmosphere. The expansion procedure is shown to give good results in the special case of a uniform basic-state (in which gravity is negligible) and for which a more exact treatment is possible.The form of both pressure and velocity perturbations within the tube is discussed. The nature of pressure perturbations depends upon a critical 'transition' frequency, top, which in turn is dependent upon depth, field strength, pressure and density in the basic (unperturbed) state of the tube. At a given depth in the tube pressure oscillations are possible only for frequencies greater than tot,; for frequencies below top exponentially decaying (evanescent) pressure modes occur. In a similar fashion the nature of motions within the flux tube depends upon a 'transition' frequency, to,. At a giver~ depth within the tube vertically propagating waves are possible only for frequencies greater than too ; for frequencies below too exponentially decaying (evanscent) motions occur. The dependence of both tot, and to~ on depth is determined for each of the special cases, and for a realistic solar atmosphere. It is found that the use of an isothermal atmosphere, instead of a more realistic temperature profile, may well give misleading results.For the solar atmosphere it is found that toy is zero at about 12 km above optical depth ~'5o0o = 1, thereafter rising to a maximum of 0.04 s -1 at some 600 km above 75o0o = 1. Below 750o0 = 1, in the convection zone, toy has a maximum of 0.013 s -I. The transition frequency, top, for the pressure perturbations, is peaked at 0.1 s -1 just below zsooo = 1, falling to a minimum of 0.02 s -1 at about one scale-height deeper in the tube Solar Physics 56 (1978) 5-35. All Rights Reserved Copyright @~ 1978 by D. Reidel Publishing Company, Dordrecht, Holland * Note that the z-scale refers to the surrounding photosphere, not to the tube's interior. VERTICAL MOTIONS IN AN INTENSE MAGNETIC FLUX TUBE

Expansion of magnetic flux concentrations: a comparison of Hinode SOT data and models

“…The thin flux tube model is based on the assumption that the flux tube is much thinner than the pressure and density scale heights. The plasma and magnetic properties inside the tube can then be expanded in terms of a low order polynomial (Defouw 1976;Roberts & Webb 1978). Here we use the lowest order polynomial possible: 0th order in radius, r, for the vertical component of the magnetic field, B z , and 1st order in the radial component of the field, B r .…”

Comparison of the thin flux tube approximation with 3D MHD simulations

“…For an axisymmetric vertical flux tube, we adopt cylindrical coordinates (r, θ, z), with the z-axis pointing in the vertical direction. Physical quantities are regular at the axis (r = 0), so that they can be described in terms of a Taylor expansion in the radial coordinate (Roberts & Webb 1978;Spruit 1981;Pneuman et al 1986; The properties of the axisymmetric MHD equations (FerrizMas & Schüssler 1989) imply that only even orders are nonzero in the above-mensioned expansions for scalar quantities (such as temperature or density) and for z-components of vectors, whereas for the radial and θ-components of vectors only the odd orders remain.…”

Moving magnetic tubes: fragmentation, vortex streets and the limit of the approximation of thin flux tubes

“…In the following, we consider how a rising magnetic flux tube behaves in the context of this approximation (Roberts & Webb 1978;Spruit 1981).…”

“…This assumption corresponds to retaining only the zeroth-order term in the axis-centered Taylor-expansion of the quantities in the tube. Higher-order treatments can also be derived (Roberts & Webb 1978;Ferriz-Mas et al 1989). In the following, we develop a model based on the zeroth-order approximation, which is already sufficient for modelling how the physical properties near the tube centre evolve.…”

“…To model the rise of magnetic tubes through the whole convection zone, typically with a view to understanding the origin of active regions and some of their global features, one has to resort to drastic simplifications. The approach commonly applied makes use of the slender (or thin) flux tube approximation (Roberts & Webb 1978;Spruit 1981), which models a flux tube as a quasi-1D string of mass elements under the assumption that its diameter is small compared to any other relevant length scale of the problem. The comparatively simple 1D equations derived through this approximation allow to study the evolution of a flux tube in a spherical shell with a stratification taken from the mixinglength models of the solar convection zone.…”