Universal scaling behavior of resonant absorption

TL;DR

This paper demonstrates that the universal scaling behavior of resonant absorption, previously observed in planar geometries, also applies to cylindrical geometries across various density profiles, suggesting a fundamental property of wave absorption.

Contribution

The study generalizes previous findings by showing that the universal scaling behavior of resonant absorption applies to multiple density profiles in cylindrical geometry, extending its applicability.

Findings

Scaling behavior exists for multiple density profiles in cylindrical geometry.

Universal scaling behavior is likely independent of mode conversion type.

Results support the hypothesis of a universal scaling law in wave absorption.

Abstract

Mode conversion and resonant absorption are crucial mechanisms for wave transport and absorption. Scaling behavior of mode conversion or resonant absorption is well-known for electromagnetic and MHD waves in planar geometry. Our recent study showed that such a scaling behavior of resonant absorption could also exist for coronal loop oscillations with cylindrical geometry, but it was only tested for one density profile. Here we generalise our previous study on the scaling behavior of resonant absorption by considering multiple density profiles. Applying an invariant imbedding method to the ideal MHD wave equations, we show that the scaling behavior also exists for these density models. We thus generalise our earlier results and show that such a universal scaling exists in cylindrical geometry, too. Given these results and the earlier results in planar geometry, we formulate a hypothesis that a universal scaling behavior exists regardless of the type of mode conversion or resonant absorption.

Equations

$$\displaystyle\frac{d^{2}P}{dr^{2}}+\bigg{[}\frac{1}{r}-$$

$$\displaystyle\rho_{0}(r)=\left\{\begin{array}[]{ll}\rho_{i}&\mbox{ if $r\leq R_{0}-l/2$ }\\ \rho_{t}(r)&\mbox{ if $R_{0}-l/2<r<R_{0}+l/2$ }\\ \rho_{e}&\mbox{ if $r\geq R_{0}+l/2$}\end{array}\right.,$$

$${\rho }_{li}(r)$$

$${\rho }_p(r)$$

$${\rho }_s(r)$$

$$P(r,\varphi )$$

$$\frac{d{r}_m(R)}{dR}$$

$$A=-\sum _{m=\pm 1}\text{Re}(r_m+|r_m{|}^2),$$

$$\frac{d^{2}P}{d{\bar{r}}^2}$$

Full text

Mode conversion and resonant absorption can often occur in inhomogeneous plasmas such that two different waves are coupled with an exchange of energy. This mechanism has been considered as crucial for plasma heating and transport (Chen and Hasegawa, 1974; Ionson, 1978; Kivelson and Southwood, 1986; Davila, 1987; Poedts, Goossens, and Kerner, 1990; Steinolfson and Davila, 1993; Ofman, Davila, and Steinolfson, 1995; Nakariakov et al., 1999; Johnson, Cheng, and Song, 2001; Kim, Cairns, and Robinson, 2007, 2008; Okamoto et al., 2015). The previous literature is mainly concerned with mode conversion between electromagnetic waves and electrostatic waves or between magnetoacoustic waves and Alfvén waves (e.g., Kivelson and Southwood, 1986; Yu and Van Doorsselaere, 2016). From theoretical and numerical studies, it is well established that a scaling behavior exists for the mode conversion coefficient in a planar geometry  (Forslund et al., 1975; Kivelson and Southwood, 1986; Hinkel-Lipsker, Fried, and Morales, 1989; Mjølhus, 1990; Hinkel-Lipsker, Fried, and Morales, 1991, 1992, 1993; Willes and Cairns, 2001; Kim and Lee, 2005, 2006; Kim, Cairns, and Robinson, 2008; Yu, Kim, and Lee, 2013; Schleyer, Cairns, and Kim, 2014). The scaling parameter has in general a certain relation with the density gradient, wave vector, and wave frequency (see, e.g., Willes and Cairns, 2001; Kim and Lee, 2005, 2006; Kim, Cairns, and Robinson, 2008; Schleyer, Cairns, and Kim, 2014). The mode conversion coefficient, which denotes the ratio of energy transformation from one mode to another mode, has a maximum value of about 0.5 for the linear or parabolic density profile. When the density profile is more complex, the scaling behavior is broken and the conversion coefficient can vary greatly (Willes and Cairns, 2003; Yu, Kim, and Lee, 2010; Yu and Kim, 2013, 2016). While this scaling behavior is well-known in planar geometry, its behavior in cylindrical geometry was unknown until recently. In our recent study, we considered kink modes in coronal loops, where the coronal loop was assumed as a infinitely long, straight, and axisymmetric cylinder with a radial inhomogeneity. We found that resonant absorption of the kink modes in the Alfvén resonance also show a certain scaling behavior. We called this behavior sub-universal therein. However, we only considered a sinusoidal density profile in the transitional layer (Yu and Van Doorsselaere, 2016). Given that it was studied only for a particular density profile, it was unclear whether this behavior would be generally valid, as it is for planar geometry. This motivates the question if the results from our previous paper can be generalised to other density profiles. If they can be generalised, it would imply that the scaling behavior is universal in inhomogeneous plasma, regardless of the geometry, density profile, or mode conversion.

In this work, to answer this question if the scaling law also occurs for other density profiles, we extend our previous study on resonant absorption by comparing linear, sinusoidal, and parabolic density profiles. Adding to our previous results on the sinusoidal profile, we show that this kind of scaling behavior exists also for linear and parabolic density profiles. Thus, we show that the scaling behavior also exists in cylindrical geometry, for monotonic density profiles. Based on these results, we speculate that a scaling feature of mode conversion (or resonant absorption) can generally occur regardless of the geometry, by synthesizing the newly-found scaling behaviors in cylindrical geometry with the previous ones in planar geometry.

The governing equation for the wave amplitude that describes the mode conversion usually contains a singularity in it, which makes it difficult to get exact solutions. The invariant imbedding method (IIM) transforms the concerned differential equations with boundary conditions into differential equations with initial conditions. It is an efficient and powerful tool for this kind of problems when one considers a one-dimensional inhomogeneity in the wave equations (Babkin and Klyatskin, 1980; Doucot and Rammal, 1987; Rammal and Doucot, 1987; Klyatskin, 1994; Kim, 1998; Kim, Lim, and Lee, 2001; Kim and Lee, 2005; Klyatskin, 2005). We apply the recently used invariant imbedding method (see Yu and Van Doorsselaere, 2016, and references therein) to resonant absorption of externally-driven standing kink modes in the Alfvén continuum.

From the ideal MHD equations, the wave equation for the perturbed total pressure $P$ under coronal conditions is written as, by assuming a dependence $\exp[i(m\phi+k_{z}z-\omega t)]$ in cylindrical coordinates, (e.g., Soler et al., 2013; Yu and Van Doorsselaere, 2016)

[TABLE]

where $P\approx\mathbf{B}\cdot\mathbf{b}/\mu_{0}=B_{0}b_{z}/\mu_{0}$, $b_{z}$ is the perturbed magnetic field, the background magnetic field is $\mathbf{B}=(0,0,B_{0})$, $\omega_{A}^{2}(r)=k_{z}^{2}v_{A}^{2}=k_{z}^{2}B_{0}^{2}/\rho_{0}(r)\mu_{0}$, $k_{z}$ is the longitudinal wave number, $m$ is the azimuthal wavenumber, and $\mu_{0}$ is the magnetic permeability.

We assume that the density varies from $\rho_{i}$ inside the loop to $\rho_{e}$ outside the loop through the transitional layer of thickness $l$ ($\rho_{t}$):

[TABLE]

where $R_{0}$ represents the radius of the loop. For the density $\rho_{t}(r)$ of the transitional layer we consider linear $(li)$, parabolic $(p)$, and sinusoidal $(s)$ profiles (Soler et al., 2013):

[TABLE]

which are shown in Fig. 1.

For mode conversion and the related resonant absorption to occur, the wave frequency should reside in the Aflvén continuum ($\omega_{Ae}<\omega<\omega_{Ai}$) of the loop. We consider that an external wave is incident on a coronal loop with the frequency of the fundamental standing kink mode, $\omega_{k}=k_{z}B_{0}\sqrt{2/\mu_{0}(\rho_{i}+\rho_{e})}$ with $k_{z}=\pi/L_{0}$, where $L_{0}$ is the loop length. Thus mode conversion occurs where $\omega_{A}=\omega_{k}$ or, in other words, $\rho(r)=(\rho_{e}+\rho_{i})/2$. We consider the frequency dependence of mode conversion afterward. The wave outside the loop can be described with an incident wave and a scattered wave, which is

[TABLE]

where $J_{m}(H_{m}^{(1)})$ is the Bessel (Hankel) function of the first kind, $r_{m}$ the scattering coefficient, $c$ a constant equivalent to $kR$, and $k(=\sqrt{(\omega^{2}/v_{A}^{2})-k_{z}^{2}})$ the radial wave number for $r>R$. A constant $c$ is used both to apply the invariant imbedding method and for Eq. (9) to be eigenfunctions for $r>R$ (Yu and Van Doorsselaere, 2016). The value $a_{m}$ depends on the form of the incident wave. For instance, when a plane wave is incident to the $x$ direction, $a_{m}=i^{m}$ (Stratton, 2007). Applying IIM to Eq. (LABEL:eq:1) with Eq. (9) and using the thickness $R$ as a new variable (imbedding parameter), we obtain for $r_{m}$ (Yu and Van Doorsselaere, 2016)

[TABLE]

where $\mu(r)=\rho_{0}(r)(\omega^{2}-\omega_{A}^{2}(r))$, $\epsilon(r)=\mu_{0}/k_{0}^{2}B_{0}^{2}$, $k_{0}=\omega/c_{0}$, and $c_{0}$ is the speed of the electromagnetic wave in vacuum, and $\mu_{1}$ and $\epsilon_{1}$ are the values of $\mu$, $\epsilon$ for $r>R$, prime denotes $df(x)/dx$ for $f(x)$, $J_{m}=J_{m}(c)$, and $H_{m}^{(1)}=H_{m}^{(1)}(c)$. Eq. (Universal scaling behavior of resonant absorption) is a first order ordinary differential equation with $R$ as an integration variable. We integrate Eq. (Universal scaling behavior of resonant absorption) from [math] to $R$ using initial conditions $r_{m}(0)=0$ to obtain $r_{m}(r=R)$ (Yu and Van Doorsselaere, 2016), where the $1/kR$ term gives rise to a singularity at $R=0$ which we avoid by letting the values of parameters for $R<\delta$ equal those for $R=\delta\ll 1$. This singularity is due to the cylindrical geometry we consider here. The value of $\delta$ needs to be sufficiently small to not affect the results. We also set $\omega=\omega_{k}+i\gamma$ to avoid the singularity at the resonance position ($\mu=0$) where $\gamma\ll 1$, not affecting the results. We define the mode conversion (absorption) coefficient $A$ as (Hanson and Cally, 2014; Yu and Van Doorsselaere, 2016)

[TABLE]

where two kink modes ($m=\pm 1$) are considered.

In Fig. 2, we show the mode conversion coefficient $A$ versus $l/R_{0}$ for each density profile when $\Delta\rho(=\rho_{e}/\rho_{i})=$ 2 and 10, and $L_{0}/R_{0}=$ 50, 100, and 150, while other fixed parameters are $\omega=\omega_{k}$, $k_{z}=\pi/L_{0}$, $R=3R_{0}$, $\rho_{i}=1.67353\times 10^{-12}kg/m^{3}$, $B_{0}=10^{-3}T$, $R_{0}=2\times 10^{6}m$, $\gamma=10^{-8}s^{-1}$, and $\delta=10^{-6}$. We find that the mode conversion is strong when $l/R_{0}$ is small, depending on the loop length and density contrast. The peak positions shift towards a lower value of $l/R_{0}$ as $\Delta\rho$ decreases and $L_{0}/R_{0}$ increases. The figure shows that mode conversion for the parabolic profile is less effective than the other two density profiles, which is due to the difference of the resonance point.

In Fig. 3 (a)-(c), we plot the mode conversion coefficient $A$ versus $L_{0}/R_{0}$ for each density profile when $l/R_{0}=$ 0.002, 0.2 and 2.0, and $\Delta\rho=$ 2, 5, and 20. For each $l/R_{0}$, the peak value of $A$ for each density profile has the same value regardless of $\Delta\rho$, which draws us to a finding below. The mode conversion coefficient can be plotted as a single curve when expressed with a scaling factor for some simple density (or Alfvén speed) profiles (Forslund et al., 1975; Kivelson and Southwood, 1986; Hinkel-Lipsker, Fried, and Morales, 1992, 1993; Kim and Lee, 2005). Approximating the density profile near the resonance (mode conversion) position $r=R_{*}$ as a linear function $\rho_{t}(r)=a(r-R_{*})+b$ ($a$ and $b$ are constants) and including $\omega=\omega_{k}+i\gamma$, the wave equation reduces to (Yu and Van Doorsselaere, 2016)

[TABLE]

where $\bar{r}=\kappa r$, $q=\kappa R_{*}$, $\bar{\gamma}=2(a/b)^{-2/3}{k_{z}}^{2/3}(\eta/\omega_{k})$ and $\kappa=(a/b)^{1/3}(k_{z})^{2/3}$. In the limit $\bar{\gamma}~{}(\gamma)\to 0$, this equation only depends on the value of $q$ for a given $m$ (Kim and Lee, 2005; Kim, Lee, and Lim, 2008; Yu and Van Doorsselaere, 2016). Thus one can expect a single curve of the mode conversion coefficient in terms of $q$ for simple density profiles. For the linear, sinusoidal, and parabolic density profiles, we obtain that $q$ equals $[(\pi R_{0}/l)(\Delta\rho-1)/(\Delta\rho+1)]^{1/3}(k_{z}R_{0})^{2/3}$, $[(2R_{0}/l)(\Delta\rho-1)/(\Delta\rho+1)]^{1/3}(k_{z}R_{0})^{2/3}$, and $[(2\sqrt{2}R_{0}/l)(\Delta\rho-1)/(\Delta\rho+1)]^{1/3}(k_{z}R_{0})^{2/3}$, respectively (cf., Yu and Van Doorsselaere, 2016). With those scaling factors, the mode conversion coefficient $A$ in Fig. 3 (a)-(c) reduces to a single curve in Fig. 3 (d)-(f) for each density profile. A difference from the results in planar geometry is that $A$ has some dependence on $l/R_{0}$ despite $q$ is a function of $l/R_{0}$. However, Fig. 3 points out that the scaling behavior is a general property if the density profile can approximately be linearized at the resonance point, reinforcing and extending the results from planar geometry. It is straightforward to apply these results to the mode conversion of electromagnetic waves into electrostatic modes by using the mathematical analogy of the two governing wave equations (Kim and Lee, 2005; Kivelson and Southwood, 1986; Yu and Van Doorsselaere, 2016). Comparing the scaling behavior in planar and cylindrical geometries, we propose that a scaling behavior can exist regardless of the geometry for density configurations that are approximately linear around the resonant point.

In contrast with previous results, we here consider the frequency dependence of the mode conversion. Fig. 4 presents the dependence of the mode conversion on the wave frequency. We draw ${A}$ versus $\omega/\omega_{k}$ for $\Delta\rho$=5, $l/R_{0}$=0.002, 0.2 and 2.0, and $L_{0}/R_{0}$=20 and 100. Mode conversion is strongest when $\omega\approx\omega_{k}$ and $l/R_{0}$ is small, but the peak position shifts downwards as $l/R_{0}$ increases. Comparing the top and bottom rows, the shape of the mode conversion coefficient is almost independent of $L_{0}/R_{0}$.

In this paper, we have generalised our earlier results on universal scaling in resonant absorption to systems with different density profiles. We have done this using the invariant embedding approach. We found, irrespective of the particular density profile, that mode conversion (resonant absorption) is efficient when the thickness of the transitional layers ($l/R_{0}$) is small and the loop length ($L_{0}/R_{0}$) is relatively large. From this dependence irrespective of the density profile (as long as it is approximately linear near the resonant point), we thus infer that a universal scaling behavior of mode conversion is found for coronal loops exhibiting kink oscillations. In this particular case, the scaling formulas is useful, because it could be possible to infer the wave energy absorption due to resonant absorption from the observed wave parameters (Arregui, 2018). We expect that a similar scaling behavior is expected for resonant absorption of usual coronal loop oscillations for which the density contrast $\Delta\rho<1$.

The results of our study also have implications beyond the field of solar physics. Our results show that the universal scaling behaviour for mode conversion in resonant absorption extends to cylindrical geometry, adding to the previous knowledge on planar geometry. The universal scaling is thus truely universal, and happens irrespective of geometry. It is generally applicable to other forms of mode conversion as well. In particular, we can infer that the universal scaling also applies to mode conversion of electromagnetic waves into electrostatics waves in unmagnetized plasmas or in metamaterials (Ding, Chan, and Wang, 2013; Kim, Lee, and Lim, 2008; Sun et al., 2015) in cylindrical (or spherical) geometry, because that is also described with the same mathematical analogy of the wave equations (Yu and Van Doorsselaere, 2016) as here. Furthermore, going beyond mode conversion of electromagnetic or magnetohydrodynamic waves, other kind of mode conversion like e.g. between gravitational waves and electromagnetic waves are also possible (Brodin and Marklund, 1999; Marklund, Brodin, and Dunsby, 2000). Our results allow us to postulate that, even for these exotic cases, a scaling behavior also exists. We can thus conclude the paper by formulating the hypothesis: “a scaling behavior of the mode conversion coefficient is possible for monotonic density profiles regardless of the geometry and the type of mode conversion.”