Linear MHD Sausage Waves in Compressible Magnetically Twisted Flux TubesAbstractIntroductionBasic Equations and AssumptionsGeneral Governing EquationApproximationsUntwisted Incompressible PlasmaTwisted Incompressible PlasmaUntwisted Compressible PlasmaThe Dispersion EquationNumerical SolutionsHigh-Plasma-beta ConditionsPhotospheric ConditionsLow-Plasma-beta EnvironmentConclusionsAcknowledgementsAppendixReferencesSolar Phys (2007) 246: 101–118DOI 10.1007/s11207-007-9022-6Linear MHD Sausage Waves in CompressibleMagnetically Twisted Flux TubesRobert Erdélyi ·Viktor FedunReceived: 22 March 2007 / Accepted: 3 August 2007 / Published online: 12 October 2007© Springer Science+Business Media B.V. 2007Abstract Oscillations of magnetic flux tubes are of great importance as they contain infor-mation about the geometry and fine structure of the flux tubes. Here we derive and analyt-ically solve in terms of Kummer’s functions the linear governing equations of wave propa-gation for sausage surface and body modes (m =0) of a magnetically twisted compressibleflux tube embedded in a compressible uniformly magnetized plasma environment in cylin-drical geometry. A general dispersion relation is obtained for such flux tubes. Numericalsolutions for the phase velocity are obtained for a wide range of wavenumbers and for vary-ing magnetic twist. The effect of magnetic twist on the period of oscillations of sausagesurface modes for different values of the wavenumber and vertical magnetic field strength iscalculated for representative photospheric and coronal conditions. These results generalizeand extend previous studies of MHD waves obtained for incompressible or for compressiblebut nontwisted flux tubes. It is found that magnetic twist may change the period of sausagesurface waves of the order of a few percent when compared to counterparts in straight non-twisted flux tubes. This information will be most relevant when high-resolution observationsare used for diagnostic exploration of MHD wave guides in analogy to solar-interior studiesby means of global eigenoscillations in helioseismology.Keywords Twisted magnetic flux tube · MHD waves · Solar atmosphere ·Magneto-seismology1. IntroductionIn recent years, high-resolution observations (e.g., EIT or SUMER onboard SOHO andTRACE) and modern data-analysis techniques opened the way to a number of novel the-oretical ideas about the physical processes involving MHD waves in the solar atmosphereR. Erdélyi · V. Fedun ()Solar Physics and Space Plasma Research Centre (SP²RC), Department of Applied Mathematics,University of Sheffield, Hounsfield Road, Hicks Building, Sheffield, S3 7RH, UKe-mail:[email protected]. Erdélyie-mail:[email protected] R. Erdélyi, V. Fedun(see Aschwanden, 2004). Observational investigations show that the propagation of MHDwaves in the solar magnetized atmosphere has many peculiarities (e.g., their generation anddissipation) and offer great potential for the diagnostic exploration of the magnetized solaratmosphere. Extensive work concerning wave propagation in magnetic flux tubes embeddedin a magnetically structured atmosphere has been published (see, e.g., the latest reviews byBanerjee et al., 2007 on observations. Various effects on MHD flux tube oscillations wereinvestigated, including longitudinal stratification, radial stratification, curvature, wave leak-age, multifibril structure, and magnetic twist. All of these effects may contribute to changesin periods and may be important, however, with different magnitude. Here we focus on mag-netic twist and its relevance to flux tube oscillations. Previously, linear wave propagation inuniformly twisted magnetic flux tubes embedded in a magnetic-free plasma environmentwas studied in the incompressible limit (e.g., Goedbloed, 1971; Uberoi and Somasundaram,1980; Goossens, 1991; Bennett, Roberts, and Narian, 1999) only. Incompressibility, amongother effects, removes and/or degenerates the fast/slow MHD wave component, giving lim-ited applicability to solar and space plasmas. Here we have attempted to derive the generaldispersion relation that describes the propagation of sausage waves in a uniformly twistedcompressible magnetic flux tube embedded in a compressible, magnetized plasma environ-ment. General solutions to the dispersion equation are obtained numerically for various typ-ical solar plasma parameters, representing conditions of the photosphere or the magnetizedcorona.2. Basic Equations and AssumptionsThe single-fluid, linearized, ideal-MHD equations for a compressible magnetized plasmaread as follows (see, e.g., Kadomtsev, 1966):ρ0∂2ξ∂t2+p +1μ0b ×(×B0) +B0×(×b) = 0, (1)p +ξ ·p0+γp0ξ = 0, (2)b +×(B0×ξ) = 0, (3)where, in a cylindrical coordinate system with the equilibrium state inhomogeneous in theradial direction only, ξ = (ξr,ξϕ,ξz) is the Lagrangian displacement vector denoting per-turbations from the equilibrium position, ρ0(r) and p0(r) are the unperturbed pressure anddensity, B0= (0,B0ϕ(r), B0z(r)) is the equilibrium magnetic field, γ is the adiabatic gasindex, μ0is the magnetic permeability, p is the perturbation of plasma pressure, and b is theperturbation of the magnetic field.Let us introduce the Eulerian perturbation of total pressure: pT=p +B0·b/μ0.Forthenormal-mode analysis in cylindrical geometry the perturbation of all quantities are Fourierdecomposed, for example,ξ,pT∼ expi(kz +mϕ −ωt). (4)Here ω is the mode frequency, m is the azimuthal order of the mode, and k is the longitudinal(axial) wavenumber. Then, Equations (1)–(3) imply that, for example, ξrand pTsatisfyLinear MHD Sausage Waves 103a system of two first-order ordinary differential equations (see, e.g., Appert, Gruber, andVaclavik, 1974):Dddr(rξr) = C1rξr−rC2pT, (5)DddrpT=1rC3rξr−C1pT. (6)The coefficient functions for the general magnetically twisted flux tubes are given in Saku-rai, Goossens, and Hollweg (1991) or Erdélyi and Fedun (2006). The components of thedisplacement vector in the magnetic surfaces perpendicular (ξ⊥) and parallel (ξ) to themagnetic-field lines are related to ξrand pTbyρω2−ω2A ξ⊥=iBgBpT−2fBB0ϕB0zξrμr,ρω2−ω2C ξ=ifBB0C2SC2S+V2ApT−2B20ϕξrμr,V2A=B20μρ0,B20=B20z+B20ϕ,C2S=γp0ρ0,gB=mB0zr−kzB0ϕ,fB=kzB0z+mB0ϕr.Here VAis the Alfvén speed and CSis the sound speed. These equations show that fastwaves (characterized by ξrand pT)