Cusp Resonance Research Articles

Damping of Slow Surface Kink Modes in Solar Photospheric Waveguides Modeled by One-dimensional Inhomogeneities

Abstract Given the recent interest in magnetohydrodynamic (MHD) waves in pores and sunspot umbrae, we examine the damping of slow surface kink modes (SSKMs) by modeling solar photospheric waveguides with a cylindrical inhomogeneity comprising a uniform interior, a uniform exterior, and a continuous transition layer (TL) in between. Performing an eigenmode analysis in linear, resistive, gravity-free MHD, our approach is idealized in that, among other things, our equilibrium is structured only in the radial direction. We can nonetheless address two damping mechanisms simultaneously, one being the ohmic resistivity and the other being the resonant absorption of SSKMs in the cusp and Alfvén continua. We find that the relative importance of the two mechanisms depends sensitively on the magnetic Reynolds number (R m). Resonant absorption is the sole damping mechanism for realistically large values of R m, and the cusp resonance in general dominates the Alfvén one unless the axial wavenumbers are at the lower end of the observationally relevant range. We also find that the thin-boundary approximation holds only when the TL-width-to-radius ratios are much smaller than nominally expected. The ohmic resistivity is far more important for realistically small R m. Even in this case, SSKMs are only marginally damped, with damping-time-to-period ratios reaching ∼10 in the parameter range we examine.

Damping of Slow Surface Sausage Modes in Photospheric Waveguides

Abstract There has been considerable interest in sausage modes in photospheric waveguides such as pores and sunspots, and slow surface sausage modes (SSSMs) have been suggested to damp sufficiently rapidly to account for chromospheric heating. Working in the framework of linear resistive magnetohydrodynamics, we examine how efficient electric resistivity and resonant absorption in the cusp continuum can be for damping SSSMs in a photospheric waveguide with equilibrium parameters compatible with recent measurements of a photospheric pore. For SSSMs with the measured wavelength, we find that the damping rate due to the cusp resonance is substantially less strong than theoretically expected with the thin-boundary approximation. The damping-time-to-period ratio (τ/P) we derive for standing modes, equivalent to the damping-length-to-wavelength ratio for propagating modes given the extremely weak dispersion, can reach only ∼180. However, the accepted values for electric resistivity (η) correspond to a regime where both the cusp resonance and resistivity play a role. The values for τ/P attained at the largest allowed η may reach ∼30. We conclude that electric resistivity can be considerably more efficient than the cusp resonance for damping SSSMs in the pore in question, and needs to be incorporated into future studies on the damping of SSSMs in photospheric waveguides in general.

Analytical Solutions for Cusp Resonance in Dissipative MHD

The present paper considers resonant slow waves in 1D non-uniform magnetic flux tubes in dissipative MHD. Analytical solutions are obtained for the Lagrangian displacement and the Eulerian perturbation of the total pressure for both static and stationary equilibrium states. From these analytical solutions we obtain the fundamental conservation law and the jump conditions for resonant slow waves in dissipative MHD. The validity of the ideal conservation law and jump conditions obtained by Sakurai, Goossens, and Hollweg (1991) for static equilibria and Goossens, Hollweg, and Sakurai (1992) for stationary equilibria is justified in dissipative MHD.

Shear flow instability near the cusp resonance in magnetopause boundary layer

Shear flow instability is studied in the planar magnetopause boundary layer region by treating the plasma as compressible. A necessary criterion for instability near the cusp resonance is obtained analytically. The criterion depends on plasmaβ, Alfven Mach numberMAand the ratio of the scale lengths of the gradients in the flow and Alfven velocities. The instability at the cusp resonance layer can be excited rather easily for the lowβ plasma and for shear flow scale length smaller than the typical scale length over which Alfven velocity varies. The growth rate for instability is obtained for anyβ from a cubic equation. The unstable modes may contribute to the ULF wave activity at the magnetopause.

Compressible normal modes of a resistive nonuniform plasma

The theory of resistive dissipation of linear compressible modes in a nonuniform plasma is presented in a unified way, following a normal mode approach. The familiar modes of a homogeneous ideal plasma are shown to be substantially modified by the presence of the resistivity and nonuniformity. It is then argued that the most suitable criterion for mode identification is based on the asymptotic properties of the solutions. The modes are then classified as ideal or resistive according to the asymptotic absence or presence of the resistivity. After a brief summary of the results from previous articles, made for the sake of completeness, the article concentrates on the new results concerning the compressible ideal solutions, the only ones that exhibit a resonant behavior. The properties of the solutions are examined by treating an explicit example of a nonhomogeneous situation. It is shown that the nature of the modes varies in the direction of nonuniformity, being essentially a fast mode in the asymptotic region, a slow mode in the vicinity of the so-called cusp resonance, and an Alfvénic mode near the Alfvénic resonance point.

The behavior of magneto-acoustic-gravity waves near the cusp resonance in a lossless, compressible, isothermal, stratified, electrically conducting and uniformly magnetized atmosphere. II: The valve effect

Abstract In the present paper the behavior of (internal) magneto-acoustic-gravity waves near the cusp resonance in a lossless, compressible, isothermal, stratified, electrically conducting atmosphere that is permeated by a uniform, nearly horizontal magnetic field is re-addressed (Kamp, 1989). The previously analyzed linear conversion of long acoustic-gravity waves into short magneto-acoustic waves that carry off the energy from the resonance region along the magnetic field, is re-analyzed with boundary layer techniques that are based on the smallness of the vertical component of the magnetic field. More specifically the existence of the so-called valve effect for the generated magneto-acoustic mode near the critical level is explicitly demonstrated and shown to be governed by two rivalling effects.

Doubly differential detachment cross sections for 0.5-MeV H- on He including projectile excitation to H(n=2).

Detailed theoretical results are presented for the electron-detachment cross section, doubly differential in both the electron momentum and angle, for the process 0.5-MeV ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$+He\ensuremath{\rightarrow}H(n=2)+${\mathit{e}}^{\mathrm{\ensuremath{-}}}$+${\mathrm{He}}^{\mathrm{*}}$. As discussed briefly elsewhere [C. R. Liu and A. F. Starace, Phys. Rev. Lett. 62, 407 (1989)], the laboratory-frame doubly differential cross sections (DDCS's) for electron detachment in the forward direction are shown to depend sensitively on the low-energy states of the H(n=2)-${e}^{\mathrm{\ensuremath{-}}}$ three-body system. In particular, the angular dependence of characteristic cusp and shape resonance features is presented. We find that the projectile frame DDCS for detached electron energies in the vicinity of the shape resonance peak is nearly isotropic. This is due in part to the $^{1}\mathrm{P}$ symmetry of the resonance feature, which limits the angular distribution to constant and ${\mathrm{cos}}^{2}$\ensuremath{\theta} terms, and in part to cancellation in the integral over momentum transfer on which the coefficient of the ${\mathrm{cos}}^{2}$\ensuremath{\theta} term depends.We also find that the rapid variation of these cusp and shape resonance features with angle in the laboratory frame requires that experimental angular and energy resolutions be accounted for in order to obtain good agreement with the measured results of M. G. Menendez and M. M. Duncan [Phys. Rev. A 36, 1653 (1987)] on the energy spectrum of detached electrons coincident with the formation of H(2p). The DDCS's for 0.5-MeV ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$+He\ensuremath{\rightarrow}H(1s)+${e}^{\mathrm{\ensuremath{-}}}$+${\mathrm{He}}^{\mathrm{*}}$ are also presented. When these latter results are added to those for producing H(n=2), the sum gives good agreement with the experimental data of M. G. Menendez and M. M. Duncan [Phys. Rev. 20, 2327 (1979)], which include all final states of the H atom, thereby confirming the important contribution the H(n=2) states make to the total-detachment cross section. We present evidence on the sensitivity of these results to the choice of the average excitation energy of the helium target. Finally, we pinpoint the origin of our predicted Gailitis-Damburg oscillations in the DDCS near threshold as stemming from a rapid decrease of an analytically known phase appropriate for long-range dipole fields.

Observable effects of poles and shadow poles in coupled-channel systems.

Using separable potentials in a coupled-channel Lippmann-Schwinger equation, we investigate the motion of poles on the different Riemann sheets and their observable effects as the coupling strength is varied. Only in the weak-coupling limit can one determine in which uncoupled channel the pole originated by the sheet on which it lies. Cusp structure versus resonance peak as one crosses a threshold is found to depend on the distance of the pole from the physical sheet. Resonance poles in each uncoupled channel can produce double-loop Argand plots similar to those seen in \ensuremath{\pi}N ${P}_{11}$ amplitude analysis.

The behavior of magneto-acoustic-gravity waves near the cusp resonance in a lossless, compressible, isothermal, stratified, electrically conducting, and uniformly magnetized atmosphere. I. Mode conversion approach

In the present paper the propagation of (internal) magneto-acoustic-gravity waves is analyzed in a compressible, isothermal, stratified, electrically conducting atmosphere that is permeated by a uniform, nearly horizontal magnetic field. The conversion, near the so-called cusp resonance of a long acoustic-gravity wave into a short slow magneto-acoustic wave, is demonstrated by means of boundary layer theory based on the smallness of the vertical component of the magnetic field. The magneto-acoustic wave subsequently carries the energy off upward, which in the limit of a horizontal magnetic field would be fed into the cusp resonance ad infinitum. This gives rise to singular fields. The scaling of the tendency toward singular behavior of a field quantity with the obliqueness of the magnetic field is discussed, as well as the partial reflection of the long acoustic-gravity wave.

Wave transformation at the cusp resonance in a finite conductivity isothermal atmosphere permeated by a nearly horizontal magnetic field

The paper considers wave transformation in the vicinity of the cusp resonance in an isothermic finite conductivity medium with a nearly horizontal magnetic field. It is shown that absorption of magnetogravity waves takes place when the inclination angle of the magnetic field is smaller than the critical angle. When the inclination angle is large then the critical magnetogravity waves are transformed into slow magnetoacoustic waves.

Resonance absorption of compressible magnetohydrodynamic waves at thin “surfaces”

If an MHD surface wave is supported by a true discontinuity, then the total pressure fluctuation, δPtot, is constant across the discontinuity. If the discontinuity is replaced by a thin transition layer, then δPtot will be approximately constant across the transition layer, with a value approximately the same as the value obtained for the case of a true discontinuity. We use this approximation to study the behavior of the plasma and fields in the transition layer. We regard δPtot as known, and write the relevant equations in forms in which δPtot appears as driving terms. Two resonances appear. The cusp resonance affects the density and pressure fluctuations, and the velocity and magnetic field components along the background magnetic field, B0. The Alfven resonance affects the velocity and magnetic field components normal to B0. We concentrate on the Alfven resonance, and show in a simple way how energy is pumped out of the surface wave into thin layers surrounding the resonant field lines. We consider also the effects of three types of viscosity on the Alfven resonance. Only classical shear viscosity is able to absorb the energy which is pumped into the thin resonant layer. In the steady state, the net viscous heating is independent of the viscosity coefficient, if the heating occurs in a sufficiently thin layer. We suggest that the large velocity shears which occur in the vicinity of the resonant field lines can lead to Kelvin‐Helmholtz instabilities, which can in turn lead to an effective eddy viscosity, which we estimate to be large enough in the solar corona to distribute heat throughout large portions of coronal active region loops. We show also that coronal heating by the Alfven resonance is compatible with a variety of coronal data.