Resonant Absorption of Magnetohydrodynamic Body Modes under Photospheric Conditions

Observations reveal that surface and body modes exist in solar pores under photospheric conditions. While the effects of resonant absorption on photospheric surface modes are well established, its effect on body modes is not known yet. In this paper, we investigate resonant absorption for the body modes under photospheric conditions. We numerically solve the dispersion relation induced by thin boundary approximation and obtain the wave dispersion curves and damping rates of three arbitrarily chosen body modes for sausage and kink waves, respectively. The results show that resonant absorption for the body modes is weaker than for the slow surface modes in both cusp and Alfvén continua. The damping behavior of body modes is similar to slow surface modes while the higher body mode has stronger resonant absorption.


Introduction
Solar pores (or sunspot umbras) are considered as waveguides for the magnetohydrodynamic (MHD) waves to propagate from the photospheric atmosphere to the upper layers (Cho et al. 2015;Grant et al. 2015;Jess et al. 2015;Felipe et al. 2020).Those waves can transfer a considerable amount of energy into the chromosphere or corona, contributing as a dissipation to heating plasmas (Stangalini 2011;Grant et al. 2018;Riedl et al. 2021;Srivastava et al. 2021;Yuan et al. 2023).They are also test beds for the wave theories.
Photospheric sausage modes in pores are prevalent in observations.Fujimura & Tsuneta (2009) observed propagating sausage and/or kink waves by analyzing the magnetic, velocity, and intensity fluctuations.Morton et al. (2011) observed sausage wave signals of periods from 30 to 450 s with the empirical mode decomposition (EMD) method, which were interpreted as fast modes by Moreels et al. (2013).Dorotovič et al. (2014) detected standing slow and fast modes with wavelet analysis and the EMD method.Keys et al. (2018) first observed surface and body sausage modes in a photospheric pore and addressed that surface modes are more dominant than body modes.Stangalini et al. (2021) reported a direct detection of antisymmetric torsional Aflvén waves that seem to be excited by kink waves via mode conversion (coupling).Which wave mode is to be excited substantially depends on the magnetic field and plasma configurations (Riedl et al. 2019).
Rapid decrements of the measured wave energy flux of propagating sausage waves with atmospheric height were detected first by Grant et al. (2015; surface mode) and later by Gilchrist-Millar et al. (2021; surface and body modes).Analysis of Gilchrist-Millar et al. (2021) demonstrated that the damping length for the surface mode is slightly shorter than the body mode, which implies that the surface mode carries more energy flux than the body mode (Keys et al. 2018).
Resonant absorption had been analytically studied for the mechanism of the rapid damping of the slow surface sausage mode reported in Grant et al. (2015).It was first revealed to be inadequate for explaining the strong wave damping (Yu et al. 2017a(Yu et al. , 2017b)).Later the presence of background flow shear was shown to result in efficient resonant damping (Sadeghi et al. 2021(Sadeghi et al. , 2023)).Chen et al. (2018) investigated another effect, resistivity, and showed that resistivity is more efficient than resonant absorption.Geeraerts et al. (2020) confirmed this result with an analytical study.However, it is still weak to explain the rapid (strong) damping of the slow surface mode.A similar result is obtained for slow surface kink mode by Chen et al. (2021).In 2D numerical simulations, Riedl et al. (2021) found that the observed strong damping in Gilchrist-Millar et al. (2021) could be realized by wave leakage through the spreading geometry provided a driver localized inside the pore is developed.
On the other hand, an analytical description of the spatial distribution of the slow surface waves undergoing resonant absorption in the cusp continuum was demonstrated in Goossens et al. (2021).
While resonant absorption has been thoroughly explored for the surface modes under photospheric conditions, its effect on body modes is unknown and not established yet.In this paper, we analytically study the effect of resonant absorption on the body modes, investigating the phase speed change and damping rate.We compare the results with the previous ones for surface modes.Our model considers the propagating MHD sausage and kink waves in a straight cylindrical flux tube with a circular cross section and radial inhomogeneity as in Yu et al. (2017b).The theory for resonant absorption of MHD body modes is developed and demonstrated in Section 2. The results for the dispersion curves and damping rates are presented in Section 3. The following Section 4 concludes the paper.

Model
We assume an axial background magnetic field.Linearizing the ideal MHD equations with Fourier analysis by a factor leads to the wave equation for the radial Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.displacement ξ r ( = ∂v r /∂t) (e.g., Edwin & Roberts 1983;Sakurai et al. 1991;Yu et al. 2017b): where the Alfvén speed, the cusp speed, and ( ) g r = v p S the sound speed.Here μ 0 is the magnetic permeability, p the background plasma pressure, γ( = 5/3) the adiabatic index, B the background magnetic field, ρ the background density, k z the longitudinal wavenumber, m the azimuthal wavenumber, and ω the angular frequency of the wave.

Dispersion Relation without Transitional Layer
Considering first an axial flux tube with a circular cross section where the outer (e) and inner (i) regions of the flux tube are homogeneous with no transitional layer between them, the equilibrium satisfies the condition For each homogeneous region, Equation (1) has Bessel functions as solutions.From the relation we know that the perturbed total pressure P also has Bessel functions as solutions.
At the tube boundary (r = R), P and ξ r need to satisfy (e.g., Edwin & Roberts 1983;Sakurai et al. 1991;Yu et al. 2017aYu et al. , 2017b) ) which yields the dispersion relation for the body modes where χ = ρ e /ρ i and The prime denotes the differentiation with respect to the entire argument, and J m (K m ) is the (modified) Bessel function of first  (second) kind.In the arguments, k i and k e are defined by We use the parameter values for a (magnetic) pore (Grant et al. 2015;Yu et al. 2017b) such that v Ai = 12 km s −1 , v Si = 7 km s −1 , v Se = 11.5 km s −1 , v Ae = v Ce = 0 km s −1 , and v Ci ≈ 6.05 km s −1 ( ≈ 0.86v Si ). Figure 1 (a) illustrates the dispersion (phase speed, v) curves of three slow body (sbs, sbk) and slow surface (sss, ssk) eigenmode for each sausage (m = 0) and kink (m = 1) wave where Equation (8) is solved.Here we arbitrarily choose three of the body modes for each azimuthal mode that are spread between v Ci and v Si to investigate how their resonant damping depends on their dispersion curve (phase speed).In the following, we call the mode with higher phase speed the higher mode.As shown in Figure 1 (b) the higher mode has smaller spatial oscillations of the radial displacement, implying lower radial harmonic.

Resonant Absorption in the Presence of the Transitional Layer
The presence of gradual variation of the sound and/or Alfvén speeds near the flux tube boundary gives rise to a resonance phenomenon called resonant absorption where the wave frequency matches the local Alfvén or cusp (slow) frequency.This inhomogeneous region is called the transitional layer.Here, we use the model for the transitional layer in Yu where δ = (r − R i )/(R e − R i ) and R i(e) is the inner (outer) boundary of the flux tube.The width of the transitional layer is l = R e − R i .These parameters are assumed to continuously change from r = R i (δ = 0) to r = R e (δ = 1).Here R is defined with R = (R i + R e )/2.Then the parameters v S , v A , and v C can be written in terms of δ: 2 .Figure 2 plots the profiles of v S , v A , and v C in the transitional layer as a function of δ.Comparing with Figure 1 (a), it illustrates that the slow body sausage (sbs) and kink (sbk) modes have resonant absorption in the slow continuum when v Ci < v sbs,sbk v m ≈ 0.9333v Si , while in the Alfvén resonance, the slow body kink modes have resonant absorption when v Ci < v sbk v Si .For an axial magnetic field, there is no resonant absorption for the sausage modes in the Alfvén continuum (Goossens et al. 1992).The position of v m corresponds to δ( = δ m ) = 0.2636.
In the range v Ci < v < v m resonant absorption for the cusp (slow) resonance occurs at two resonance points δ, which are (Yu et al. 2017b) For the Alfvén resonance (v Ae < v < v Ai ), the resonance point δ is obtained as (Yu et al. 2017b) The resonance such as resonant absorption (or mode conversion) gives rise to a singular behavior in the wave Equation (1).Assuming that the width of the transitional layer l is small compared to the flux tube radius R and its profile is simple, an analytical method was developed three decades ago (Sakurai et al. 1991;Goossens et al. 1992), which, for the cusp resonance, leads to the conversion of the Equation (6) into and for the Alfvén resonance into a , the subscript c(a) denotes the position of the cusp (Alfvén) resonance (r = r c(a) ).The imaginary terms newly appear on the right-hand side of the equations.Equation (6) remains the same regardless of resonant absorption.
Here !c and ! a can be written in terms of δ as below (Yu et al. 2017b): Then the dispersion relation (8), D m = 0, needs to be converted into Ai Ae for the cusp resonance, where Ai Ae for the Alfvén resonance.For resonant absorption in the cusp resonance, we have two imaginary terms since slow resonance occurs at two resonance points, δ c1 and δ c2 .Assuming a small wave damping (ω = ω r + iω i , |ω i | = ω r ), an analytical solution for the damping rate γ m ( = ω i /ω r ) can be obtained from r (Goossens et al. 1992;Bellan 2008).For the slow (cusp) resonance (Yu et al. 2017a(Yu et al. , 2017b) ) and for the Alfvén resonance (Yu et al. 2017b) where

Results
In general, multiple solutions (dispersion curves) exist in the range v Ci < v < v Si when solving the Equations (29) and (31).We choose three dispersion curves for body sausage (m = 0) and body kink (m = 1) modes and explore their behaviors as the width of the transitional layer l changes.In the case l/R = 0 the corresponding dispersion curves for both the body and surface modes are displayed in Figure 1 (a).
In Figure 3 we plot phase speed ω r /ω Si (top panel) and damping rate −γ C0 ( = − ω i /ω r ) (bottom panel) versus k z R for the slow body sausage (sbs) modes undergoing cusp (slow) resonance by solving Equation (29).The left, middle, and right columns represent the sausage modes 1, 2, and 3, respectively.As l/R increases there appears a peak of damping rate, which goes to the left with decrement.As the mode number increases (from sbs 1 to sbs 3), on average, the damping becomes stronger regardless of the value of l/R.While the damping behavior is very similar to the slow surface sausage (sss) mode (Yu et al. 2017b), the effect of resonant absorption on the body wave damping is at least 5 times weaker (γ C0 ∼ 10 −3 ).Another difference with the surface mode is that the dispersion curves of body modes have a very small dependence on l/R.
Next, we show the results for the slow body kink (sbk) modes by solving Equation (29). Figure 4 represents the curves of dispersion (top panels) and damping rate (bottom panels) in the cusp resonance.The behavior of phase speed and damping strength (rate) of the slow kink body modes is very similar to those of the slow body sausage modes (γ C1 ∼ 10 −3 ).It is noteworthy that the dispersion curve for sbk 3 crosses over the range of the cusp continuum as l/R increases (no resonant absorption) when k z R > 9.5.
The resonant absorption behavior of body kink modes in the Alfvén continuum is slightly different from the above results.
Here we solved Equation (31). Figure 5 shows that the phase speed (ω r /ω si ) has almost no change with l/R variation and the damping strength is in proportional to l/R, which is different from the cusp resonance.Also, the curve of the damping rate (strength) is wider than the cusp resonance.The effect of resonant absorption on the body wave damping is about an order of 1 weaker (γ A1 ∼ 10 −4 ) than the one for the slow surface kink mode obtained in Yu et al. (2017b).
It is inferred from the above and previous (Yu et al. 2017b) results that resonant absorption in the Alfvén resonance is weaker than that in the cusp resonance for both surface and body modes when these dispersion curves are in a similar range of phase speed (frequency) under photospheric conditions.It is instructive that among the body modes, the higher mode has the stronger resonant damping.This implies that a high phase speed (or frequency) of the wave is crucial to obtaining strong damping.The damping period can be derived by the relation τ m = 1/(2πγ m ).For the resonant damping of the third higher modes (sbs 3, sbk 3) in the cusp and Alfvén continua, τ 0,1 reaches ∼1.59 × 10 2 and ∼1.59 × 10 3 , respectively.
In Figure 6 we compare the numerical results (Equations ( 29) and (31)) with analytical (approximate) formulas (Equations ( 32) and (33), respectively) for (a) the wave mode sbs 3 in the cusp resonance and (b) the wave mode sbk 3 in the Alfvén resonance.The analytical formula for resonant absorption in the Alfvén resonance has good agreement with the numerical solutions.On the other hand, the corresponding one for the sausage mode undergoing the cusp resonance is only valid for small k z R, similar to the slow surface sausage mode (Yu et al. 2017b).This formula is found invalid for the slow body kink modes in the cusp continuum.The other wave modes (sbs 1-2, sbk 1-2) give similar trends.

Conclusions
We studied resonant absorption of MHD body modes in the cusp and Alfvén continua where sausage and kink modes are considered to propagate in a cylindrical flux tube with an axial magnetic field.We arbitrarily chose three body modes among multiple solutions for sausage and kink modes, respectively.
In the cusp continuum, resonant absorption has similar behaviors for sausage and kink modes.First, it inversely depends on the thickness of the transitional layer l/R.Second, it has a maximum (peak) at a certain axial wavenumber k z R. Third, as the wave mode becomes higher (sbs 1 → sbs 3, sbk 1 → sbk 3), resonant absorption becomes stronger and the peak positions shift to smaller ones.Comparison with the slow surface modes (Yu et al. 2017b) reveals that the above features are very similar to the slow surface modes, while the dispersion curves for body modes have little changes and resonant absorption for body modes is less effective.For the Alfvén resonance, which occurs for only kink modes, resonant absorption becomes stronger as l/R increases, opposite to the cusp resonance.The damping rate has also a peak at a certain k z R and gives the same shift behavior with the wave modes (sbk 1 → sbk 3).Its strength is about an order of 1 weaker than the cusp resonance.As for the cusp resonance, these features are similar to the slow surface kink mode (Yu et al. 2017b), while the strength of resonant absorption is again less effective.We point out that high wave frequency or phase speed is needed to obtain strong or efficient resonant absorption.The dispersion curve should be in a high-frequency range.A comparison of the two resonances reveals that resonant absorption in the cusp continuum is stronger than in the Alfvén continuum.
Our study finds that resonant absorption of body modes is inefficient for plasma heating.However, the presence of background shear flow or magnetic field twist may substantially enhance the effect as for the surface modes (Sadeghi et al. 2021(Sadeghi et al. , 2023)).
Resonant absorption of body modes under photospheric conditions is not useful for seismology since its analytical formulas (Equations ( 32) and (33)) are complicated and its effect is small.Instead, we note that the body modes could be useful in themselves for seismology.The existence of body modes or their coexistence with the surface mode may lead to the estimation of v Ci provided their axial wavelengths are sufficiently long (see Figure 1(a)), which can be applied to testing/validating other seismologically obtained values of parameters.

Figure 1 .
Figure 1.(a)The phase speed v/v si ( = ω r /ω si ) vs. k z R for three slow body sausage modes (sbs), three slow body kink modes (sbk), a slow surface sausage mode (sss), and a slow surface kink mode (ssk) under a photospheric condition.(b) For k z R = 1.2, eigenfunctions for each mode are plotted as a function of r/R.For the mode with higher phase speed, the number of spatial oscillations becomes smaller.The used parameter are v Ae = 0 km/s, v Ai = 12 km s −1 , v Se = 11.5 km s −1 , v Si = 7 km s −1 , v Ce = 0 km s −1 , and v Ci ≈ 6.05 km s −1 ( ≈ 0.86v Si ).

Figure 2 .
Figure 2. Profiles of v A , v S , and v C vs. δ in the transitional layer.The parameters are the same as in Figure 1.

Figure 4 .
Figure 4. Resonant absorption of slow body kink modes in the cusp continuum: sbk 1, sbk 2, and sbk 3. The others are the same as in Figure 2.

Figure 5 .
Figure 5. Resonant absorption of slow body kink modes in the Alfvén continuum.The others are the same as in Figure 3.