Investigating the damping rate of phase-mixed Alfvén waves

Abstract

Context. This paper investigates the effectiveness of phase mixing as a coronal heating mechanism. A key quantity is the wave damping rate, γ, defined as the ratio of the heating rate to the wave energy. Aims. We investigate whether or not laminar phase-mixed Alfvén waves can have a large enough value of γ to heat the corona. We also investigate the degree to which the γ of standing Alfvén waves which have reached steady-state can be approximated with a relatively simple equation. Further foci of this study are the cause of the reduction of γ in response to leakage of waves out of a loop, the quantity of this reduction, and how increasing the number of excited harmonics affects γ. Methods. We calculated an upper bound for γ and compared this with the γ required to heat the corona. Analytic results were verified numerically.Results. We find that at observed frequencies γ is too small to heat the corona by approximately three orders of magnitude. Therefore, we believe that laminar phase mixing is not a viable stand-alone heating mechanism for coronal loops. To arrive at this conclusion, several assumptions were made. The assumptions are discussed in Section 2.1. A key assumption is that we model the waves as strictly laminar. We show that γ is largest at resonance. Equation (3.5) provides a good estimate for the damping rate (within approximately 10% accuracy) for resonant field lines. However, away from resonance, the equation provides a poor estimate, predicting γ to be orders of magnitude too large. We find that leakage acts to reduce γ but plays a negligible role if γ is of the order required to heat the corona. If the wave energy follows a power spectrum with slope -5/3 then γ grows logarithmically with the number of excited harmonics. If the number of excited harmonics is increased by much more than 100, then the heating is mainly caused by gradients that are parallel to the field rather than perpendicular to it. Therefore, in this case, the system is not heated mainly by phase mixing.

The effects of driving time scales on heating in a coronal arcade

“…However, even in these two cases, the injected energy could heat the Quiet Sun if the dissipation rate was sufficiently large. This is in contrast to results investigating sinusoidal wave drivers (such as Prokopyszyn & Hood 2019;Howson et al 2020) which show that only high amplitude wave drivers will inject enough energy in low-dissipation regimes. The key difference in these simulations is that the driver will introduce complex tangling into the field.…”

How Transverse Waves Drive Turbulence in the Solar Corona

“…The energy cascade to small scales that develops as a result of these processes is an essential component of many wave heating models. In closed coronal loops, unless waves are driven resonantly, or strongly dissipated, simple sinusoidal drivers do not inject sufficient Poynting flux to balance the expected energy losses (e.g., [71]). As resonant driving quickly excites large-amplitude flows, even in this case, significant damping (and ultimately dissipation) must occur, as very large-amplitude waves have not been observed.…”

“…The solid line shows the results from a numerical simulation and the dashed lines were obtained using Equations ( 6) and (9). More thorough analyses of similar equations relating to Alfvén waves in a variety of media were presented in (e.g., [68][69][70][71][72]).…”

Chromospheric Heating by Magnetohydrodynamic Waves and Instabilities

“…The incompressibility of Alfvén waves is the result of magnetic tension providing the only restoring force when driven by linear perturbations. Hence, to achieve dissipation of the energy embodied in Alfvén waves in order to provide thermal energy to the outer solar atmosphere, specific mechanisms must be invoked, including phase mixing (Heyvaerts and Priest 1983;Ofman and Aschwanden 2002;Ebadi et al 2012;Prokopyszyn and Hood 2019;Van Damme et al 2020), resonant absorption (Ionson 1978;Davila 1987;Poedts et al 1989Poedts et al , 1990Ofman et al 1994Ofman et al ,1995Goossens et al 2006Goossens et al , 2011Giagkiozis et al 2016;Howson et al 2019) 2012) compared the velocity fluctuations corresponding to small-scale magnetic elements in the lower solar atmosphere and found significant Fourier power associated with high-frequency (∼ 20 mHz) horizontal motions, hence providing indirect evidence for the creation of a turbulent environment that can efficiently provide Alfvén wave dissipation. Furthermore, it is also possible for linear Alfvén waves to dissipate their initial energy through the process of parametric decay, which combines a weakly turbulent environment with the coupling of Alfvén waves to other compressible magnetoacoustic modes (Malara and Velli 1996).…”

“…The incompressibility of Alfvén waves is the result of magnetic tension providing the only restoring force when driven by linear perturbations. Hence, to achieve dissipation of the energy embodied in Alfvén waves in order to provide thermal energy to the outer solar atmosphere, specific mechanisms must be invoked, including phase mixing (Ebadi et al, 2012;Heyvaerts & Priest, 1983;Ofman & Aschwanden, 2002;Prokopyszyn & Hood, 2019;Van Damme et al, 2020), resonant absorption (Davila, 1987;Giagkiozis et al, 2016;Goossens et al, 2006Goossens et al, , 2011Howson et al, 2019;Ionson, 1978;Ofman et al, 1994Ofman et al, , 1995Poedts et al, 1989Poedts et al, , 1990, mode conversion (Cally & Khomenko, 2015;Crouch & Cally, 2005;Pagano & De Moortel, 2017;Suzuki & Inutsuka, 2005) Oran et al, 2017;Dinesh Singh & Singh Jatav, 2019). Indeed, Chitta et al (2012) compared the velocity fluctuations corresponding to small-scale magnetic elements in the lower solar atmosphere and found significant Fourier power associated with high-frequency (∼20 mHz) horizontal motions, hence providing indirect evidence for the creation of a turbulent environment that can efficiently provide Alfvén wave dissipation.…”