Resonant behaviour of MHD waves on magnetic flux tubes

Resonant absorption of MHD waves on a nonuniform flux tube is investigated as a driven problem for a 1D cylindrical equilibrium. The variation of the fractional absorption is studied as a function of the frequency and its relation to the eigenvalue problem of the MHD radiating eigenmodes of the nonuniform flux tube is established. The optimal frequencies producing maximal fractional absorption are determined and the condition for total absorption is obtained. This condition defines an impedance matching and is fulfilled for an equilibrium that is fine tuned with respect to the incoming wave. The variation of the spatial wave solutions with respect to the frequency is explained as due to the variation of the real and imaginary parts of the dispersion relation of the MHD radiating eigenmodes with respect to the real driving frequency.     

Methylmercury production in sediments of Chesapeake Bay and the mid-Atlantic continental margin

Methylmercury (MeHg) concentration and production rates were studied in bottom sediments along the mainstem of Chesapeake Bay and on the adjoining continental shelf and slope. Our objectives were to 1) observe spatial and temporal changes in total mercury (HgT) and MeHg concentrations in the mid-Atlantic coastal region, 2) investigate biogeochemical factors that affect MeHg production, and 3) examine the potential of these sediments as sources of MeHg to coastal and open waters. Estuarine, shelf and slope sediments contained on average 0.5 to 1.5% Hg as MeHg (% MeHg), which increased significantly with salinity across our study site, with weak seasonal trends. Methylation rate constants (k_meth), estimated using enriched stable mercury isotope spikes to intact cores, showed a similar, but weaker, salinity trend, but strong seasonality, and was highly correlated with % MeHg. Together, these patterns suggest that some fraction of MeHg is preserved thru seasons, as found by others [Orihel, D.M., Paterson, M.J., Blanchfield, P.J., Bodaly, R.A., Gilmour, C.C., Hintelmann, H., 2008. Temporal changes in the distribution, methylation, and bioaccumulation of newly deposited mercury in an aquatic ecosystem. Environmental Pollution 154, 77] Similar to other ecosystems, methylation was most favored in sediment depth horizons where sulfate was available, but sulfide concentrations were low (between 0.1 and 10 μM). MeHg production was maximal at the sediment surface in the organic sediments of the upper and mid Bay where oxygen penetration was small, but was found at increasingly deeper depths, and across a wider vertical range, as salinity increased, where oxygen penetration was deeper. Vertical trends in MeHg production mirrored the deeper, vertically expanded redox boundary layers in these offshore sediments. The organic content of the sediments had a strong impact on the sediment:water partitioning of Hg, and therefore, on methylation rates. However, the HgT distribution coefficient (K_D) normalized to organic matter varied by more than an order of magnitude across the study area, suggesting an important role of organic matter quality in Hg sequestration. We hypothesize that the lower sulfur content organic matter of shelf and slope sediments has a lower binding capacity for Hg resulting in higher MeHg production, relative to sediments in the estuary. Substantially higher MeHg concentrations in pore water relative to the water column indicate all sites are sources of MeHg to the water column throughout the seasons studied. Calculated diffusional fluxes for MeHg averaged  1 pmol m^− 2 day^− 1. It is likely that the total MeHg flux in sediments of the lower Bay and continental margin are significantly higher than their estimated diffusive fluxes due to enhanced MeHg mobilization by biological and/or physical processes. Our flux estimates across the full salinity gradient of Chesapeake Bay and its adjacent slope and shelf strongly suggest that the flux from coastal sediments is of the same order as other sources and contributes substantially to the coastal MeHg budget.     

Alfvén waves in the solar atmosphere

We examine the propagation of Alfvén waves in the solar atmosphere. The principal theoretical virtues of this work are: (i) The full wave equation is solved without recourse to the small-wavelength eikonal approximation (ii) The background solar atmosphere is realistic, consisting of an HSRA/VAL representation of the photosphere and chromosphere, a 200 km thick transition region, a model for the upper transition region below a coronal hole (provided by R. Munro), and the Munro-Jackson model of a polar coronal hole. The principal results are:

1. If the wave source is taken to be near the top of the convection zone, where n _H = 5.2 × 10¹⁶ cm^?3, and if B _⊙ = 10.5 G, then the wave Poynting flux exhibits a series of strong resonant peaks at periods downwards from 1.6 hr. The resonant frequencies are in the ratios of the zeroes of J ₀, but depend on B _⊙, and on the density and scale height at the wave source. The longest period peaks may be the most important, because they are nearest to the supergranular periods and to the observed periods near 1 AU, and because they are the broadest in frequency.
2. The Poynting flux in the resonant peaks can be large enough, i.e. P _⊙ ≈ 10⁴–10⁵ erg cm^?2s^?1, to strongly affect the solar wind.
3. ¦δv¦ and ¦δB¦ also display resonant peaks.
4. In the chromosphere and low corona, ¦δv ≈ 7–25 kms^?1 and ¦δB¦ ≈0.3–1.0 G if P _⊙≈10⁴-10⁵ erg cm^?2s^?1.
5. The dependences of ¦δv¦ and ¦δB¦ on height are reduced by finite wavelength effects, except near the wave source where they are enhanced.
6. Near the base, ¦δB¦ ≈ 350–1200 G if P _⊙ ～- 10⁴–10⁵. This means that nonlinear effects may be important, and that some density and vertical velocity fluctuations may be associated with the Alfvén waves.
7. Below the low corona most wave energy is kinetic, except near the base where it becomes mostly magnetic at the resonances.
8. ?₀ < δv ² > v _A or < δB ² > v _A/4π are not good estimators of the energy flux.
9. The Alfvén wave pressure tensor will be important in the transition region only if the magnetic field diverges rapidly. But the Alfvén wave pressure can be important in the coronal hole.

     

A Kinetic Model of Coronal Heating and Acceleration by Ion-Cyclotron Waves: Preliminary Results

We present a kinetic model of the heating and acceleration of coronal protons by outward-propagating ion-cyclotron waves on open, radial magnetic flux tubes. In contrast to fluid models which typically insist on bi-Maxwellian distributions and which spread the wave energy and momentum over the entire proton population, this model follows the kinetic evolution of the collisionless proton distribution function in response to the combination of the resonant wave-particle interaction and external forces. The approximation is made that pitch-angle scattering by the waves is faster than all other processes, resulting in proton distributions which are uniform over the resonant surfaces in velocity space. We further assume, in this preliminary version, that the waves are dispersionless so these resonant surfaces are portions of spheres centered on the radial sum of the Alfvén speed and the proton bulk speed. We incorporate the fact that only those protons with radial speeds less than the bulk speed will be resonant with outward-propagating waves, so this rapid interaction acts only on the sunward half of the distribution. Despite this limitation, we find that the strong perpendicular heating of the resonant particles, coupled with the mirror force, results in substantial outward acceleration of the entire distribution. The proton distribution evolves towards an incomplete shell in velocity space, and appears vastly different from the distributions assumed in fluid models. Evidence of these distinctive distributions should be observable by instruments on Solar Probe.     

The solar wind: Our current understanding and how we got here

In the original theory for the solar wind, the electron pressure gradient was the principal accelerating force. This was soon recognized to be insufficient to drive the high-speed streams. Subsequently, the discovery of Alfvén waves in the solar wind led to a long series of models in which wave pressure provided additional acceleration, but these wavedriven models ultimately failed to explain the rapid acceleration of the fast wind close to the Sun. An alternate view was that the pressure of hot protons close to the Sun could explain the rapid acceleration, with the proton heating coming from the cyclotron resonance. SOHO has provided remarkable data which have verified some of the predictions of this view, and given impetus to ongoing studies of the ion-cyclotron resonance in the fast wind. After a historical review, we discuss the basic ideas behind current research, emphasizing the importance of particle kinetics. We conclude with some guesses as to how work might proceed in the future.     

Alfvén waves in sunspots

The propagation of Alfvén waves in a simple model of a sunspot is considered. The vertical structure near the center of the umbra is modelled realistically, but the horizontal structure is not considered. The full wave equation is solved, without recourse to the WKB approximation. Only wave propagation in the vicinity of the central field line in an axially symmetric spot is examined, and it is assumed that this field line is open. By taking wave reflections into account, we find that the observations of non-thermal motions near the temperature minimum (Beckers, 1976) and in the corona (Beckers and Schneeberger, 1977) are both consistent with an upward-propagating Alfvénic energy flux density of a few times 10⁷ erg cm^–2 s^–1. This flux density is too small to cool the sunspot, but it is large enough to supply the energy requirements of the transition region and corona above a sunspot. This conclusion depends on the assumptions that the observed motions are indeed Alfvénic with periods near 180 s.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Dissipative MHD solutions for resonant AlfvÉn waves in 1-dimensional magnetic flux tubes

The present paper extends the analysis by Sakurai, Goossens, and Hollweg (1991) on resonant Alfvén waves in nonuniform magnetic flux tubes. It proves that the fundamental conservation law for resonant Alfvén waves found in ideal MHD by Sakurai, Goossens, and Hollweg remains valid in dissipative MHD. This guarantees that the jump conditions of Sakurai, Goossens, and Hollweg, that connect the ideal MHD solutions for _r , andP across the dissipative layer, are correct. In addition, the present paper replaces the complicated dissipative MHD solutions obtained by Sakurai, Goossens, and Hollweg for _r , andP in terms of double integrals of Hankel functions of complex argument of order with compact analytical solutions that allow a straightforward mathematical and physical interpretation. Finally, it presents an analytical dissipative MHD solution for the component of the Lagrangian displacement in the magnetic surfaces perpendicular to the magnetic field lines which enables us to determine the dominant dynamics of resonant Alfvén waves in dissipative MHD.     

Nonlinear development of phase-mixed alfvén waves

Abstract

We derive an equation governing the nonlinear propagation of a linearly polarized Alfvén wave in a two-dimensional, anisotropic, slightly compressible, highly magnetized, viscous plasma, where nonlinearities arise from the interaction of the Alfvén wave with fast and slow magnetoacoustic waves. The phase mixing of such a wave has been suggested as a mechanism for heating the outer solar atmosphere (Heyvaerts and Priest, 1983).

We find that cubic wave damping dominates shear linear dissipation whenever the Alfvén wave velocity amplitude δv_y exceeds a few times ten metres per second. In the nonlinear regime, phase-mixed waves are marginally stable, while non-phase-mixed waves of wavenumber k_a are damped over a timescale k_uRe ₀|δ v_y/v_A |^?2, Re ₀ being the Reynolds number corresponding to the Braginskij viscosity coefficient η₀ and v_A the Alfvén speed. Dissipation is most effective where β = (v_s /v_A) ² ≈ 1, v_s being the speed of sound.     

Resonant behaviour of MHD waves on magnetic flux tubes

The resonances that appear in the linear compressible MHD formulation of waves are studied for equilibrium states with flow. The conservation laws and the jump conditions across the resonance point are determined for 1D cylindrical plasmas. For equilibrium states with straight magnetic field lines and flow along the field lines the conserved quantity is the Eulerian perturbation of total pressure. Curvature of the magnetic field lines and/or velocity field lines leads to more complicated conservation laws. Rewritten in terms of the displacement components in the magnetic surfaces parallel and perpendicular to the magnetic field lines, the conservation laws simply state that the waves are dominated by the parallel motions for the modified slow resonance and by the perpendicular motions for the modified Alfvén resonance.The conservation laws and the jump conditions are then used for studying surface waves in cylindrical plasmas. These waves are characterized by resonances and have complex eigenfrequencies when the classic true discontinuity is replaced by a nonuniform layer. A thin non-uniform layer is considered here in an attempt to obtain analytical results. An important result related to earlier work by Hollweg et al. (1990) for incompressible planar plasmas is found for equilibrium states with straight magnetic field lines and straight velocity field lines. For these equilibrium states the incompressible and compressible surface waves have the same frequencies at least in the long wavelength limit and there is an exact correspondence with the planar case. As a consequence, the conclusions formulated by Hollweg et al. still hold for the straight cylindrical case. The effects of curvature are subsequently considered.     

A new resonance in the solar atmosphere

We consider a horizontally stratified isothermal model of the solar atmosphere, with vertical and uniform B ₀, and v _A ² v _s ² . The equations of motion are linearized about a background which is in hydrostatic equilibrium. A homogeneous wave equation results for the motions perpendicular to B ₀; this wave equation is similar to the equation for the MHD fast mode. On the other hand, the equation for the parallel motions is inhomogeneous, containing driving terms which arise from the presence of the fast mode; the homogeneous form of this equation is identical to the equation describing vertically-propagating gravity-modified acoustic waves. We demonstrate that a resonance can exist between the (driving) fast wave and the (driven) gravity-modified acoustic wave, in such a way that very large parallel velocities can be driven by small perpendicular velocities. Applications of this resonance to solar spicules, jets, and other phenomena are discussed.The National Center for Atmospheric Research is sponsored by the National Science Foundation.