Modelling p-Mode Interaction with a Spreading Sunspot Field

Sunspots absorb and scatter incident p modes. The dominant mechanism is still uncertain. One possibility, mode conversion to slow magneto-acoustic waves, has been shown to yield results in agreement with observations for the f mode only. Absorption of p modes in simple vertical magnetic field models is too weak by an order of magnitude or more. Here we report on numerical calculations of p modes encountering a simple sunspot model with field which spreads with height. It is found that p-mode absorption is greatly enhanced by field spread, to a level consistent with observations, and it appears that it occurs preferentially in the outer regions of the spot, in line with recent results from acoustic holography.     

Absorption of acoustic waves by sunspots

Observations of the scattering of acoustic waves by sunspots show a substantial deficit in scattered power relative to incident power. A number of calculations have attempted to model this process in terms of absorption at the magnetohydrodynamic Alfvén resonance. The results presented here extend these calculations to the case of a highly structured axisymmetric translationally invariant flux-tube embedded in a uniform atmosphere. The fractional energy absorbed is calculated for models corresponding to flux-tubes of varying radius, mean flux-density and location below the photosphere. The effects of twist are also included.It is found that absorption can be very efficient even in models with low mean magnetic flux density, provided the flux is concentrated into intense slender annuli. Twist is found to increase the range of wave numbers over which absorption is efficient, but it does not remove the low absorption at low azimuthal orders which is a feature of resonance absorption calculations in axisymmetric geometry, and which is in conflict with observation.These results suggest that resonance absorption could be an efficient mechanism in plage fields and fibril sunspots as well as in monolithic sunspots. At present it is too early to make any definite deductions about sunspot structure from the observations, but the possible future use of sunspot seismology to resolve open questions in the theory of sunspots is briefly discussed.     

Resonant Absorption of Alfvén Waves in Steady Coronal Loops

The effect of equilibrium flow on linear Alfvén resonances in coronal loops is studied in the compressible viscous MHD model. By means of a finite element code, the full set of linearised driven MHD equations are solved for a one-dimensional equilibrium model in which the equilibrium quantities depend only on the radial coordinate. Computations of resonant absorption of Alfvén waves for two classes of coronal loop models show that the efficiency of the process of resonant absorption strongly depends on both the equilibrium parameters and the characteristics of the resonant wave. We find that a steady equilibrium shear flow can also significantly influence the resonant absorption of Alfvén waves in coronal magnetic flux tubes. The presence of an equilibrium flow may therefore be important for resonant Alfvén waves and coronal heating. A parametric analysis also shows that the resonant absorption can be strongly enhanced by the equilibrium flow, even up to total dissipation of the incoming wave.     

Nonlinearity effects on resonant absorption of surface Alfvén waves in incompressible plasmas

The resonant absorption of small amplitude surface Alfvén waves is studied in nonlinear incompressible MHD for a viscous and resistive plasma. The reductive perturbation method is used to obtain the equation that governs the spatial and temporal behaviour of small amplitude nonlinear surface Alfvén waves. Numerical solutions to this equation are obtained under the initial condition that att = 0 the spatial variation is purely sinusoidal. The numerical results show that nonlinearity accelerates the wave damping due to resonant absorption. Resonant absorption is a more efficient wave damping mechanism than can be anticipated on the basis of linear theory.     

Total resonant absorption of acoustic oscillations in sunspots

The question of total resonant absorption of acoustic oscillations in sunspots is studied for cylindrical 1-D flux tubes that are stratified only in the radial direction and surrounded by a uniform, non-magnetic plasma. The numerical investigation of Goossens and Poedts (1992) in linear resistive MHD is taken further by increasing the strength of the azimuthal magnetic field in the equilibrium flux tubes. For relatively strong azimuthal magnetic fields, total absorption is found over a relatively wide range of spot radii.     

Numerical simulation of coronal heating by resonant absorption of Alfvén waves

The heating of coronal loops by resonant absorption of Alfvén waves is studied in compressible, resistive magnetohydrodynamics. The loops are approximated by straight cylindrical, axisymmetric plasma columns and the incident waves which excite the coronal loops are modelled by a periodic external driver. The stationary state of this system is determined with a numerical code based on the finite element method. Since the power spectrum of the incident waves is not well known, the intrinsic dissipation is computed. The intrinsic dissipation spectrum is independent of the external driver and reflects the intrinsic ability of the coronal loops to extract energy from incident waves by the mechanism of resonant absorption.The numerical results show that resonant absorption is very efficient for typical parameter values occurring in the loops of the solar corona. A considerable part of the energy supplied by the external driver, is actually dissipated Ohmically and converted into heat. The heating of the plasma is localized in a narrow resonant layer with a width proportional to ^1/3. The energy dissipation rate is almost independent of the resistivity for the relevant values of this parameter. The efficiency of the heating mechanism and the localization of the heating strongly depend on the frequency of the external driver. Resonant absorption is extremely efficient when the plasma is excited with a frequency near the frequency of a so-called collective mode.     

Resonant Absorption of Nonlinear Slow MHD Waves in Isotropic Steady Plasmas – II. Application: Resonant Acoustic Waves

Nonlinear theory of driven magnetohydrodynamic (MHD) waves in the slow dissipative layer in isotropic steady plasmas developed by Ballai and Erdélyi (Solar Phys. 180 (1998)) is used to study the nonlinear interaction of sound waves with one-dimensional isotropic steady plasmas. An inhomogeneous magnetic slab with field-aligned plasma flow is sandwiched by a homogeneous static magnetic-free plasma and by a homogeneous steady magnetic plasma. Sound waves launched from the magnetic-free plasma propagate into the inhomogeneous region interacting with the localised slow dissipative layer and are partially reflected, dissipated or transmitted by this region. The nonlinearity parameter, introduced by Ballai and Erdélyi, is assumed to be small and a regular perturbation method is used to obtain analytical wave solutions. Analytical studies of resonant absorption of sound waves show that the efficiency of the process of resonant absorption strongly depends on both the equilibrium parameters and the characteristics of the resonant wave. We also find that a steady equilibrium shear flow can significantly influence the nonlinear resonant absorption in the limits of thin inhomogeneous layer and weak nonlinearity. The presence of an equilibrium flow may therefore be important for the nonlinear resonant MHD wave phenomena. A parametric analysis also shows that the nonlinear part of resonant absorption can be strongly enhanced by the equilibrium flow.     

Mode Conversion of Solar <Emphasis Type="Italic">p</Emphasis>-Modes in Non-Vertical Magnetic Fields

Sunspots absorb and scatter incident f- and p-modes. Until recently, the responsible absorption mechanism was uncertain. The most promising explanation appears to be mode conversion to slow magnetoacoustic-gravity waves, which carry energy down the magnetic field lines into the interior. In vertical magnetic field, mode conversion can adequately explain the observed f-mode absorption, but is too inefficient to account for the absorption of p-modes. In the first paper of the present series we calculated the efficiency of fast-to-slow magnetoacoustic-gravity wave conversion in uniform non-vertical magnetic fields. We assumed two-dimensional propagation, where the Alfvén waves decouple. In comparison to vertical field, it was found that mode conversion is significantly enhanced in moderately inclined fields, especially at higher frequencies. Using those results, Cally, Crouch, and Braun showed that the resultant p-mode absorption produced by simple sunspot models with non-vertical magnetic fields is ample to explain the observations. In this paper, we further examine mode conversion in non-vertical magnetic fields. In particular, we consider three-dimensional propagation, where the fast and slow magnetoacoustic-gravity waves and the Alfvén waves are coupled. Broadly speaking, the p-mode damping rates are not substantially different to the two-dimensional case. However, we do find that the Alfvén waves can remove similar quantities of energy to the slow MAG waves.     

磁场中p模/s模的转换及黑子对声波的吸收

在柱坐标下将黑子周围的环形区域（黑子除外）内的振荡分解为朝向黑子传播的（入射的）波和离开黑子传播的（出射的）波。对无黑子的环形区域内的振荡也进行了同样的分解。将黑子周围的入射波看成是被黑子磁流管磁化了的介质（介质内的磁场基本是水平的）中的波。而无黑子区的入射波看成是非磁化介质中的波。比较这两种波在固定波数下功率随频率的分布发现，在磁化介质中不同径向除ｎ的声波（ｐ模）频率系统降低，同时功率也降低，降低的功率最高达非磁化介质中波的功率的３０％。而比较在固定频率下功率随波数的分布发现，磁场中ｆ模及ｎ＝１，２，３的ｐ模的脊向高波数方向位移，功率的降低受频率调制，即声波在某些有限的频带中被吸收。这些观测表明，在磁场中ｐ模与磁声重力波（ＭＡＧ）产生了模式混合或耦合。模式混合的存在支持了模式转换作为ｐ模式被黑子吸收的机制的解释。此外，本文还分析了转换的ＭＡＧ波进入黑子磁流管（其中的磁场基本上是垂直的）后进一步被吸收，吸收的功率最高达ＭＡＧ波的２０％。在磁流管内没有进一步观测到模式的转换     

On the linear transformation and resonant absorption of Alfvén p-modes in sunspots

We analyse the linear transformation of Alfvén p-modes into quasisurface waves and the resonant absorption of Alfvén p-waves in a slowly varying medium with a density gradient, parallel to the ambient homogeneous magnetic field. It is pointed out, that the energy transfer from Alfvén p-waves to quasisurface waves appear to take place in sunspots. The results obtained also suggest that Alfvén p-waves generated by overstability in sunspots may be absorbed in deep layers under sunspots. Moreover, it is very likely that part of the downward wave flux is reflected and hence it is possibble that essentially the energy from sunspots is transported outward by magnetogravitational waves, to which Alfvén p-waves are transformed.     

Probing sunspot magnetic fields with p-mode absorption and phase shift data

Long-standing observations of incoming and outgoing f- and p-modes in annuli around sunspots reveal that the spots partially absorb and substantially shift the phase of waves incident upon them. The commonly favoured absorption mechanism is partial conversion to slow magneto-acoustic waves that disappear into the solar interior channelled by the magnetic field of the sunspot. However, up until now, only f-mode absorption could be accounted for quantitatively by this means. Based on vertical magnetic field models, the absorption of p-modes was insufficient. In this paper, we use the new calculations of Crouch & Cally for inclined fields, and a simplified model of the interaction between spot interior and exterior. We find excellent agreement with phase shift data assuming field angles from the vertical in excess of 30° and Alfvén/acoustic equipartition depths of around 600–800 km. The absorption of f-modes produced by such models is considerably larger than is observed, but consistent with numerical simulations. On the other hand, p-mode absorption is generally consistent with observed values, up to some moderate frequency dependent on radial order. Thereafter, it is too large, assuming absorbing regions comparable in size to the inferred phase-shifting region. The excess absorption produced by the models is in stark contrast with previous calculations based on a vertical magnetic field, and is probably due to finite mode lifetimes and excess emission in acoustic glories. The excellent agreement of phase shift predictions with observational data allows some degree of probing of subsurface field strengths, and opens up the possibility of more accurate inversions using improved models. Most importantly, though, we have confirmed that slow mode conversion is a viable, and indeed the likely, cause of the observed absorption and phase shifts.     

Effect of Flow on Resonant Absorption of Slow MHD Waves in Coronal Arcades

Resonant absorption of slow MHD waves is studied numerically by using the SGH method and is applied to a model of a coronal arcade in the presence of equilibrium plasma flows. The arcade is approximated by a 1D horizontal magnetic slab that is non-uniform along the vertical direction and which is surrounded by two homogeneous media. While propagating from the photosphere upwards into the corona, the magneto-acoustic waves can be resonantly absorbed in the inhomogeneous region of the arcade. Computational results show that the resonant absorption of the impinging waves strongly depends on the equilibrium model and on the characteristics of the driving wave. The results also indicate that the presence of an equilibrium plasma flow along the magnetic field of the arcade reduces the resonant absorption for the flow speed parameters considered.     

On the propagation of magneto-acoustic-gravity waves

An analysis of magneto-acoustic-gravity waves in the case of an isothermal atmosphere permeated by a uniform magnetic field is presented. The general solution is expressed in terms of generalized hypergeometric functions. It can be used in numerical simulation of oscillations in a magnetic atmosphere.

It is shown that the elliptically polarized magneto-acoustic-gravity waves consist of a pair of surface waves and a pair of body waves above the cut-off frequency. The body waves along the magnetic field are similar to acoustic waves in an atmosphere and their cut-off frequency is unaffected by magnetic field. The transverse oscillation decreases with height. For the usual boundary condition, the longitudinal oscillation decreases with height; however, in some cases, it may contain terms that increase with height. The solution is singular on a family of ellipses in the frequency - horizontal wave number plane. Near these ellipses, the wave components grow indefinitely.     

Ion acoustic solitary waves in a weakly relativistic plasma with <Emphasis Type="Italic">q</Emphasis>-nonextensive electrons and thermal positrons

Nonlinear ion acoustic solitary waves (IASWs) are addressed in a weakly relativistic plasma consisting of cold ion fluid, q-nonextensive electron velocity distribution and Boltzmann distributed positron. The Korteweg-de Vries- (KdV) equation is derived by reductive perturbation method. We investigate the effect of nonextensive electrons on solitary waves in this medium. It is found that only compressive solitons can be appeared in the existence of nonextensive electrons. It is shown that the structure of soliton depend sensitively on the q-nonextensive parameter.     

On Absorption and Scattering of P Modes by Small-Scale Magnetic Elements

The solar surface is characterised everywhere by the presence of small-scale magnetic structures. Their collective behaviour in the form of active regions is known to have strong influence on p-mode power. For example, sunspots and plages are strong absorbers of acoustic waves. This paper studies the effects of individual small-scale magnetic elements to understand the details of absorption of p-mode power. For this, we consider a thin magnetic flux tube and calculate the phase shifts and the absorption coefficients by numerically solving the linearised MHD equations. The phase shifts calculated from the Born Approximation are then compared for the same range of degrees. The results are discussed with a view to understanding the physical mechanism.     

Arbitrary Amplitude Ion Acoustic Solitary Waves in An Unmagnetized Two Electron Population Ultra-Relativistic Dense Plasmas

The nonlinear wave structure of arbitrary amplitude ion acoustic solitary waves (IASWs) are studied in the Sagdeev’s pseudopotential framework for an ultra-relativistic degenerate dense plasma comprising cold and hot electrons and inertial ultra-cold ions. By employing standard normal-mode analysis the dispersion relation for linear waves is studied. The numerical results are presented to understand the features of ion acoustic solitary wave structures. It is shown that the present plasma model supports IASWs having positive potential well. Also, it is found that the small amplitude rarefactive double layer solution can exist in such a plasma system in some parametric region. It is shown that solitary structures and double layers are affected by relevant plasma parameters.     

Heating of Jupiter’s thermosphere by the dissipation of upward propagating acoustic waves

Thunderstorms in Jupiter’s atmosphere are likely to be prodigious generators of acoustic waves, as are thunderstorms in Earth’s atmosphere. Accordingly, we have used a numerical model to study the dissipation in Jupiter’s thermosphere of upward propagating acoustic waves. Model simulations are performed for a range of wave periods and horizontal wavelengths believed to characterize these acoustic waves. The possibility that the thermospheric waves observed by the Galileo Probe might be acoustic waves is also investigated. Whereas dissipating gravity waves can cool the upper thermosphere through the effects of sensible heat flux divergence, it is found that acoustic waves mainly heat the Jovian thermosphere through effects of molecular dissipation, sensible heat flux divergence, and Eulerian drift work. Only wave-induced pressure gradient work cools the atmosphere, an effect that operates at all altitudes. The sum of all effects is acoustic wave heating at all heights. Acoustic waves and gravity waves heat and cool the atmosphere in fundamentally different ways. Though the amplitudes and mechanical energy fluxes of acoustic waves are poorly constrained in Jupiter’s atmosphere, the calculations suggest that dissipating acoustic waves can locally heat the thermosphere at a significant rate, tens to a hundred Kelvins per day, and thereby account for the high temperatures of Jupiter’s upper atmosphere. It is unlikely that the waves detected by the Galileo Probe were acoustic waves; if they were, they would have heated Jupiter’s thermosphere at enormous rates.     

Viscous computations of resonant absorption of MHD waves in flux tubes by fem

A numerical code is presented for computing the stationary state of resonant absorption of MHD waves in cylindrical flux tubes in linear, compressible, and viscous MHD. The full viscosity stress tensor is included in the code with the five viscosity coefficients as given by Braginskii (1965). Also non-zero plasma pressure effects are taken into account, and the finite elements discretization with the Galerkin method has been used. The implementation of the stress tensor and the numerical accuracy of the tensorial viscous MHD code are scrutinized in test case. The test case involves the absorption of waves in cylindrical flux tubes considered by Lou (1990) and Goossens and Poedts (1992) in the context of absorption of acoustic oscillations. The results for the absorption rates obtained with the tensorial viscous code agree completely with the results obtained by Lou in a scalar viscous MHD and by Goossens and Poedts in resistive MHD. This verifies not only the complicated tensor viscous code but again proves that the absorption rate is independent of the actual dissipation mechanism.     

关于磁声重力波传播性质的探讨

本文对充满垂直均匀磁场的等温大气内的磁声重力波做了严格的解析分析,并将其通解表述成广义超几何函数的形式。该解可用于对磁大气内振荡现象的进一步数值模拟研究。对解的分析澄清了若干磁声重力波的传播性质。