High-frequency Spicule Oscillations Generated via Mode Conversion

The basic equations are as follows:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}(\rho A)+\displaystyle \frac{\partial }{\partial z}(\rho {v}_{z}A)=0,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}(\rho {v}_{z}A)+\displaystyle \frac{\partial }{\partial z}\left[\left(\rho {{v}_{z}}^{2}+p+\displaystyle \frac{{{{\boldsymbol{B}}}_{\perp }}^{2}}{8\pi }\right)A\right]\\ &&\quad =\,\left(p+\displaystyle \frac{\rho {{{\boldsymbol{v}}}_{\perp }}^{2}}{2}\right)\displaystyle \frac{{dA}}{{dz}}-\rho {gA},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}(\rho {{\boldsymbol{v}}}_{\perp }{A}^{3/2})+\displaystyle \frac{\partial }{\partial z}\left[\left(\rho {v}_{z}{{\boldsymbol{v}}}_{\perp }-\displaystyle \frac{{B}_{z}{{\boldsymbol{B}}}_{\perp }}{4\pi }\right){A}^{3/2}\right]=0,\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}(\sqrt{A}{{\boldsymbol{B}}}_{\perp })+\displaystyle \frac{\partial }{\partial z}[({{\boldsymbol{B}}}_{\perp }{v}_{z}-{B}_{z}{{\boldsymbol{v}}}_{\perp })\sqrt{A}]=0,\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left[\left(e+\displaystyle \frac{1}{2}\rho {{\boldsymbol{v}}}^{2}+\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{8\pi }\right)A\right]\\ &&\quad +\,\displaystyle \frac{\partial }{\partial z}\left[\left(e+p+\displaystyle \frac{1}{2}\rho {{\boldsymbol{v}}}^{2}+\displaystyle \frac{{{{\boldsymbol{B}}}_{\perp }}^{2}}{4\pi }\right){v}_{z}A\right.\\ &&\qquad \left.-\,{B}_{z}\displaystyle \frac{{{\boldsymbol{B}}}_{\perp }\cdot {{\boldsymbol{v}}}_{\perp }}{4\pi }A\right]\\ &&=\,-{L}_{\mathrm{rad}}A-\displaystyle \frac{\partial }{\partial z}({q}_{\mathrm{cond}}A)-\rho {{gv}}_{z}A,\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&e=\displaystyle \frac{1}{\gamma -1}p\end{eqnarray} \tag{ 6 }$$

A generalized form of a spherical coordinate system is used such that the super radial expansion of a flux tube is considered (Hollweg et al. 1982; Suzuki & Inutsuka 2005) (see the Appendix for derivation). The xy plane is defined as perpendicular to the flux tube, while the z axis is curved along the flux tube. Specifically, A denotes the cross section of the flux tube that satisfies the divergence-free condition of a magnetic field as follows:

$$\begin{eqnarray}&&{B}_{z}A=\mathrm{const}.\end{eqnarray} \tag{ 7 }$$

$g=2.74\times {10}^{4}\,\mathrm{cm}\ {{\rm{s}}}^{-2}$ is the gravitational acceleration, $\gamma =5/3$ corresponds to the specific heat ratio of the adiabatic gas, ${L}_{\mathrm{rad}}$ is the radiative cooling, and ${q}_{\mathrm{cond}}$ denotes the thermal conductive flux.

Following Kopp & Holzer (1976) and Suzuki & Inutsuka (2005), the chromospheric flux tube expansion is modeled as

$$\begin{eqnarray}&&A(z)=\displaystyle \frac{{A}_{\max }\exp \left(\tfrac{z-{z}_{1}}{{\sigma }_{1}}\right)+{A}_{1}}{\exp \left(\tfrac{z-{z}_{1}}{{\sigma }_{1}}\right)+1},\end{eqnarray} \tag{ 8 }$$

where

$$\begin{eqnarray}&&{A}_{1}=1-({A}_{\max }-1)\exp \left(-\displaystyle \frac{{z}_{1}}{{\sigma }_{1}}\right).\end{eqnarray} \tag{ 9 }$$

In this study, we apply ${A}_{\max }=20$ and ${z}_{1}={\sigma }_{1}=1\,\mathrm{Mm}$. As for thermal conduction, Spitzer–Härm-type flux is employed (Spitzer & Härm 1953).

$$\begin{eqnarray}&&{q}_{\mathrm{cond}}=-{\kappa }_{0}{T}^{5/2}\displaystyle \frac{\partial T}{\partial z},\end{eqnarray} \tag{ 10 }$$

where ${\kappa }_{0}={10}^{-6}$ in the CGS–Gaussian unit. Approximated cooling functions are included for both optically thick and thin radiations as follows:

$$\begin{eqnarray}&&{L}_{\mathrm{rad}}=(1-\xi ){L}_{\mathrm{thick}}+\xi {L}_{\mathrm{thin}},\end{eqnarray} \tag{ 11 }$$

where ξ denotes the coefficient that controls the contribution of each cooling function. The chromosphere is dominated by optically thick cooling, while the corona is dominated by optically thin cooling, and thus, ξ is set as follows:

$$\begin{eqnarray}&&\xi =\exp \left[-\displaystyle \frac{\rho }{{\rho }_{\mathrm{tr}}}\right],\end{eqnarray} \tag{ 12 }$$

where ${\rho }_{\mathrm{tr}}={10}^{-14}\,{\rm{g}}\ {\mathrm{cm}}^{-3}$ denotes mass density near the transition region. Following Gudiksen & Nordlund (2005), as opposed to directly solving the radiative transfer (Carlsson & Stein 1997; Guerreiro et al. 2013), we approximate the optically thick cooling by Newtonian cooling. This is formulated as

$$\begin{eqnarray}&&{L}_{\mathrm{thick}}=\displaystyle \frac{1}{{\tau }_{\mathrm{thick}}}(e-{e}_{0}),\end{eqnarray} \tag{ 13 }$$

where ${\tau }_{\mathrm{thick}}$ is a timescale of the cooling, and e₀ denotes an internal energy distribution with a reference temperature model. ${\tau }_{\mathrm{thick}}$ denotes a function of density that is given as

$$\begin{eqnarray}&&{\tau }_{\mathrm{thick}}=1.0\times {\left(\displaystyle \frac{\rho }{{\rho }_{\odot }}\right)}^{-0.4}\ {\rm{s}},\end{eqnarray} \tag{ 14 }$$

where ${\rho }_{\odot }$ is the mass density at the photosphere. With respect to ${L}_{\mathrm{thin}}$, the following expression is used:

$$\begin{eqnarray}&&{L}_{\mathrm{thin}}={n}_{i}{n}_{e}{\rm{\Lambda }}(T),\end{eqnarray} \tag{ 15 }$$

where ${\rm{\Lambda }}(T)$ is an approximated radiative loss function (Sutherland & Dopita 1993; Matsumoto & Suzuki 2014). Additionally, n_i and n_e are calculated by assuming a certain ionization degree as a function of temperature.

Photospheric flux tube intensity ${B}_{z,\odot }$ is fixed to $200\,{\rm{G}}$. Radial velocity ${v}_{z,\odot }$ is given as a monochromatic function in time, while mass density ${\rho }_{\odot }$ is determined such that upward waves are excited on the photosphere as follows:

$$\begin{eqnarray}&&{v}_{z,\odot }=\sqrt{2}\delta v\sin (2\pi \,{f}_{0}t),\ {\rho }_{\odot }={10}^{-7}\left(1+\displaystyle \frac{{v}_{z,\odot }}{{c}_{\odot }}\right)\,{\rm{g}}\ {\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 16 }$$

where $\delta v$ denotes the photospheric root-mean-square velocity of v_z, f₀ denotes the input frequency of longitudinal waves, and ${c}_{\odot }$ denotes the sound speed at the photosphere. We set $\delta v$ and f₀ to $0.4\,\mathrm{km}\ {{\rm{s}}}^{-1}$ and $5\,\mathrm{mHz}$, respectively. Furthermore, f₀ is set close to the most dominant frequency of the chromospheric longitudinal waves (Tian et al. 2014; Kanoh et al. 2016).

The transverse velocity fluctuations ${{\boldsymbol{v}}}_{\perp ,\odot }$ are assumed to pose a broadband spectrum, while the transverse magnetic field ${{\boldsymbol{B}}}_{\perp ,\odot }$ is given such that the downward Elsässer variables (Elsässer 1950) disappear at the photosphere. The expression is as follows:

$$\begin{eqnarray}{{\boldsymbol{v}}}_{\perp ,\odot } & \propto & {\displaystyle \int }_{{f}_{\min }}^{{f}_{\max }}{f}^{-1}{e}^{2\pi {if}}{df},\\ {{\boldsymbol{B}}}_{\perp ,\odot } & = & -\sqrt{4\pi {\rho }_{\odot }}{{\boldsymbol{v}}}_{\perp ,\odot },\end{eqnarray} \tag{ 17 }$$

where ${f}_{\min }=1\,\mathrm{mHz}$ and ${f}_{\max }=10\,\mathrm{mHz}$, and the root-mean-square velocity of each component is $0.4\,\mathrm{km}\ {{\rm{s}}}^{-1}$. The vanishing downward Elsässer variables result from the brevity of the numerical calculation. Several previous studies show the standing waves on the photosphere (Fujimura & Tsuneta 2009; Kanoh et al. 2016) while Morton et al. (2011) indicates that the upward propagating mode potentially explains the observation. The numerical result should be independent of the boundary condition as long as sufficient energy is injected into the atmosphere.

Basic Equations (1)–(6) are solved from the photosphere ($z=0\,\mathrm{Mm}$) to the corona ($z=12\,\mathrm{Mm}$). We use 2400 uniform grid points to resolve the computational domain. The outgoing (transmitting) boundary condition (Del Zanna et al. 2001; Suzuki & Inutsuka 2006) is applied for the top boundary such that unphysical wave reflection is excluded. Furthermore, an additional heating is imposed near the top boundary to maintain the coronal temperature (Iijima & Yokoyama 2015). The HLLD approximated Riemann solver (Miyoshi & Kusano 2005) is used to solve nonlinear wave propagation. The fifth-order accurate WENOZ scheme (Borges et al. 2008) is used to reduce the numerical dissipation, while the third-order SSP Runge–Kutta method (Shu & Osher 1988) is used for time integration. The super-time-stepping method (Meyer et al. 2012, 2014) is used to solve the thermal conduction, and this significantly reduces the numerical costs.

In Figure 1, we use a probability distribution function (pdf) to show the structure of the chromosphere and the corona in a quasi-steady state. The mass density, temperature, and transverse velocity are plotted as functions of height. Dark colors denote high probability with a logarithmic color table. The obtained density, temperature, and wave amplitude are sufficiently realistic to discuss the chromospheric wave dynamics.

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Wave amplitudes should be normalized with respect to the wave action to discuss the energy flux variation. With respect to the WKB approximation, wave action conservation is expressed as (Bretherton & Garrett 1968)

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}\left(\displaystyle \frac{E}{{\omega }^{{\prime} }}\right)+({\rm{\nabla }}\cdot {\boldsymbol{c}})\left(\displaystyle \frac{E}{{\omega }^{{\prime} }}\right)=0,\end{eqnarray} \tag{ 18 }$$

where E denotes the wave energy, ${\boldsymbol{c}}$ denotes the group velocity, and $d/{dt}=\partial /\partial t+{\boldsymbol{c}}\cdot {\rm{\nabla }}$ denotes the material time derivative. ${\omega }^{{\prime} }$ is the intrinsic frequency that is defined as

$$\begin{eqnarray}&&{\omega }^{{\prime} }=\omega -{\boldsymbol{k}}\cdot {\boldsymbol{U}},\end{eqnarray} \tag{ 19 }$$

where ω and ${\boldsymbol{k}}$ denote the wave frequency and wave number, and ${\boldsymbol{U}}$ denotes the mean flow. In our study, mean flow is negligible $({\boldsymbol{U}}\approx 0)$, and therefore, ${\omega }^{{\prime} }=\omega$ is a constant. The action conservation then yields

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}E+{\rm{\nabla }}\cdot ({{\boldsymbol{F}}}_{\mathrm{wave}})=0,\,\ {{\boldsymbol{F}}}_{\mathrm{wave}}=E{\boldsymbol{c}},\end{eqnarray} \tag{ 20 }$$

where ${{\boldsymbol{F}}}_{\mathrm{wave}}$ denotes the wave energy flux. Specifically, in a quasi-steady state, $\partial E/\partial t\approx 0$ and

$$\begin{eqnarray}&&{F}_{\mathrm{wave}}A=\mathrm{const}.\end{eqnarray} \tag{ 21 }$$

The energy flux of upward transverse and longitudinal waves are as follows:

$$\begin{eqnarray}&&{F}_{\mathrm{tran}}=\displaystyle \frac{1}{4}\rho {{{\boldsymbol{z}}}_{\perp }^{+}}^{2}a,\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{F}_{\mathrm{long}}=\displaystyle \frac{1}{2}\rho {v}_{z}^{2}c,\end{eqnarray} \tag{ 23 }$$

where ${{\boldsymbol{z}}}_{\perp }^{\pm }={{\boldsymbol{v}}}_{\perp }\mp {{\boldsymbol{B}}}_{\perp }/\sqrt{4\pi \rho }$ denote Elsässer variables and $a={B}_{z}/\sqrt{4\pi \rho }$ and $c=\sqrt{\gamma p/\rho }$ denote the Alfvén and sound speed, respectively. It should be noted that the Elsässer variables are characteristic variables of Alfvén waves in incompressible plasma (Elsässer 1950). They are not exact characteristic variables in compressible plasma. However, they are expected to yield a good approximation of transverse wave amplitude (Marsch & Mangeney 1987; Stone et al. 2008). Equations (21)–(23) yield

$$\begin{eqnarray}&&\displaystyle \frac{1}{4}\rho {{{\boldsymbol{z}}}_{\perp }^{+}}^{2}\ {aA}={\rm{const.}},\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}\rho {{v}_{z}}^{2}{cA}=\mathrm{const}.\end{eqnarray} \tag{ 25 }$$

The conservation of normalized Elsässer variable ${\zeta }^{+}$ is derived from Equation (24) as follows:

$$\begin{eqnarray}{{\zeta }^{+}}^{2} & = & \mathrm{const}.\ \ \ \ \mathrm{where}\\ {\zeta }^{+} & = & {\left(\displaystyle \frac{\rho }{{\rho }_{\odot }}\right)}^{1/4}\sqrt{{{{\boldsymbol{z}}}_{\perp }^{\pm }}^{2}},\end{eqnarray} \tag{ 26 }$$

while the normalized longitudinal velocity ν is derived from Equation (25) as follows:

$$\begin{eqnarray}{\nu }^{2} & = & \mathrm{const}.\ \ \ \mathrm{where}\\ \nu & = & {\left(\displaystyle \frac{\rho }{{\rho }_{\odot }}\right)}^{1/4}{\left(\displaystyle \frac{p}{{p}_{\odot }}\right)}^{1/4}{\left(\displaystyle \frac{{B}_{z}}{{B}_{z,\odot }}\right)}^{-1/2}{v}_{z}.\end{eqnarray} \tag{ 27 }$$

Dissipation and reflection that are neglected in the WKB approximation decrease ${\zeta }^{+}$ and ν with respect to the height. ${\zeta }^{+}$ and ν increase only when wave energy supply exists.

Figure 2 shows the spacetime plot of ${\zeta }^{+}$ and ν. The solid white line represents the equipartition layer with adiabatic sound speed (${c}_{\mathrm{adi}}=\sqrt{\gamma p/\rho }$), while the dotted white denotes the equipartition layer with isothermal sound speed (${c}_{\mathrm{iso}}=\sqrt{p/\rho }$). The sound speed ranges between ${c}_{\mathrm{adi}}$ and ${c}_{\mathrm{iso}}$ based on the timescale of the Newtonian cooling and wave period, and thus the equipartition layer lies between the solid and dotted lines. ${\zeta }^{+}$ is clearly amplified near the equipartition layer, and this is especially evident when high ν exists near the white line. This directly implies that longitudinal-to-transverse mode conversion occurs near the equipartition layer, and this is consistent with the results of previous studies (Bogdan et al. 2003; Schunker & Cally 2006; Khomenko & Cally 2012). The most important parameter for mode conversion, the angle between wave vector and magnetic field line (the attacking angle α), is zero in the absence of transverse waves because waves are assumed to propagate along the background field line in our system. The mode conversion observed in the simulation is triggered by wave–wave interaction, and thus the efficiency of each mode conversion event is never predictable because α changes relative to time based on the amplitude and phase of the transverse wave. Hence, the pdf of the transverse velocity (Figure 1(c)) is vertically broadened when compared with the density and temperature. This differs from the results of the previous studies that consider the interaction between waves and background magnetic fields.

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In order to perform a detailed analysis, we applied frequency decomposition into each normalized variable. With respect to arbitrary variable $\eta (z,t)$, the decomposition is conducted in the following procedure. First, the Fourier transformation is applied for each z as

$$\begin{eqnarray}&&\tilde{\eta }(z,\omega )=\displaystyle \frac{1}{2\pi T}{\int }_{0}^{T}\eta (z,t)\exp (-i\omega t){dt},\end{eqnarray} \tag{ 28 }$$

where T = 480 minutes, corresponding to the total simulation time. From $\tilde{\eta }(z,\omega )$, the decomposed values are calculated as follows:

$$\begin{eqnarray*}{\eta }_{{LL}}(z) & = & \sqrt{{\displaystyle \int }_{| \omega | \lt 2\pi {f}_{1}}| \tilde{\eta }(z,\omega ){| }^{2}\ d\omega },\\ {\eta }_{L}(z) & = & \sqrt{{\displaystyle \int }_{2\pi {f}_{1}\lt | \omega | \lt 2\pi {f}_{2}}| \tilde{\eta }(z,\omega ){| }^{2}\ d\omega },\end{eqnarray*}$$

$$\begin{eqnarray}{\eta }_{H}(z) & = & \sqrt{{\displaystyle \int }_{2\pi {f}_{2}\lt | \omega | \lt 2\pi {f}_{3}}| \tilde{\eta }(z,\omega ){| }^{2}\ d\omega },\\ {\eta }_{{HH}}(z) & = & \sqrt{{\displaystyle \int }_{2\pi {f}_{3}\lt | \omega | }| \tilde{\eta }(z,\omega ){| }^{2}\ d\omega },\end{eqnarray} \tag{ 29 }$$

where ${f}_{1}=2.5\,\mathrm{mHz}$, ${f}_{2}=5\,\mathrm{mHz}$, and ${f}_{3}=10\,\mathrm{mHz}$. We refer to ${\eta }_{{HH}}$, ${\eta }_{H}$, ${\eta }_{L}$, and ${\eta }_{{LL}}$ as the very-high-frequency component, high-frequency component, low-frequency component, and very-low-frequency component, respectively. The normalized Elsässer variable ${\zeta }^{+}$ and longitudinal velocity ν are decomposed in this manner.

Figure 3 shows the decomposed ${\zeta }^{+}$ (solid line) and $4\times \nu$ (dashed line) as functions of height. It should be noted that we focus on $4\times \nu$ as opposed to ν to provide a better description. Red, orange, green, and blue lines correspond to the very-high-frequency component, high-frequency component, low-frequency component, and very-low-frequency component, respectively. In order to demonstrate the role of mode conversion, the results with (upper panel) and without (lower panel) longitudinal wave inputs are simultaneously shown. The lower panel shows a natural character of Alfvén wave propagation in which the low-frequency mode experiences reflection (Velli 1993; Cranmer & van Ballegooijen 2005; Verdini & Velli 2007), while the high-frequency mode conserves its energy flux (Heinemann & Olbert 1980). The profiles of the low-frequency components (red and orange lines) in upper and lower panels are similar to each other, and this suggests that they are not influenced by the longitudinal waves while the high-frequency waves are amplified in the upper panel. This behavior is consistent with the mode conversion scenario because a higher mode conversion rate is observed (transmission rate is smaller) for high-frequency waves (Schunker & Cally 2006; Cally & Goossens 2008). In fact, ${\zeta }_{{HH}}^{+}$ increases when ${\nu }_{{HH}}$ decreases in $z=1.2\mbox{--}1.8\,\mathrm{Mm}$, and this demonstrates the energy transport from ν to ${\zeta }^{+}$. The selective amplification of high-frequency transverse waves occurs in this manner.

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In the previous subsection, we have shown that, via mode conversion, high-frequency longitudinal waves (${\nu }_{{HH}}$) are converted to high-frequency transverse waves (${\zeta }_{{HH}}^{+}$). Therefore, the origin of ${\nu }_{{HH}}$ is the key in our process. To clarify it, using Fourier transformation, we decompose ν into low-frequency and high-frequency parts with cut-off frequency of $10\,\mathrm{mHz}$. In Figure 4, we show the spacetime plots of the low-frequency part (a) and high-frequency part (b). Panel (a) in Figure 4 is similar with panel (b) in Figure 2 because the input frequency ${f}_{0}=5\,\mathrm{mHz}$ is lower than the cut-off frequency. Panel (b) indicates that there are two origins of high-frequency longitudinal waves. The first one lies in $z=0.5\mbox{--}1.0\,\mathrm{Mm}$ and this corresponds to the upward-wave origin, while the second one in $z=1.5\mbox{--}2.0\,\mathrm{Mm}$ where downward waves are generated. Considering the wave momentum conservation, only upward longitudinal waves are converted to upward transverse waves. Figure 4 indicates that high-frequency upward longitudinal waves are generated in $z=0.5\mbox{--}1.0\,\mathrm{Mm}$, which is below the conversion region ($z=1.0\mbox{--}1.5\,\mathrm{Mm}$). This is consistent with Figure 3. The possible physical mechanism of this high-frequency wave generation is wave steepening, because high-frequency components are always accompanied by low-frequency components. However, we cannot rule out the possibility that the reflected waves play a role, because high-frequency upward waves are amplified after they collide with reflected high-frequency waves.

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The physical processes in our simulation is summarized in Figure 5. First, longitudinal waves from the photosphere steepen and high-frequency longitudinal waves are generated. Such high-frequency longitudinal waves efficiently convert their mode to transverse waves by mode conversion, because the transmission rate is smaller for higher-frequency waves (Schunker & Cally 2006; Cally & Goossens 2008). By the collision between longitudinal waves and transition region, (type-I) spicule is generated (Hollweg 1982; Iijima & Yokoyama 2015), while high-frequency transverse waves probably appear as high-frequency spicule oscillations (He et al. 2009; Okamoto & De Pontieu 2011).

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Transverse velocities at $z=2\,\mathrm{Mm}$ are shown in Figure 6. The middle and right panels depict the low-frequency and high-frequency components, respectively, as decomposed by Fourier analysis with a cut-off period corresponding to $150\,{\rm{s}}$, which approximately equals the upper limit of the lifetime of spicules (Pereira et al. 2012) observed by Hinode. It should be noted that recent observations by IRIS (De Pontieu et al. 2014) indicate a longer lifetime of spicules (Skogsrud et al. 2015). Nevertheless, they are beyond the scope of the present study because the aim of this study includes a comparison with the Hinode observation. The rms velocities of the high-frequency components correspond to $5.29\,\mathrm{km}\ {{\rm{s}}}^{-1}({v}_{x})$ and $4.97\,\mathrm{km}\ {{\rm{s}}}^{-1}({v}_{y})$, respectively. In terms of amplitude, they correspond to $7\mbox{--}8\,\mathrm{km}\ {{\rm{s}}}^{-1}$, and this is consistent with the observed value (Okamoto & De Pontieu 2011). Additionally, as shown in the lower left panel of Figure 6, the typical period (duration time) of the high-frequency transverse waves is to $40\,{\rm{s}}$. It should be noted that the appearance of pulse-like fluctuation has a frequency of f₀. The typical period of the pulse is also in accordance with Okamoto & De Pontieu (2011). A natural interpretation is that this period represents the duration time of the mode conversion. Specifically, the sound-crossing time of the equipartition layer ${\tau }_{\mathrm{MC}}$ is given as follows:

$$\begin{eqnarray}&&{\tau }_{\mathrm{MC}}={\left.\displaystyle \frac{h}{c}\right|}_{a=c}={\left.\displaystyle \frac{1}{c}{\left[\displaystyle \frac{d}{{dz}}\left(\displaystyle \frac{{a}^{2}}{{c}^{2}}\right)\right]}^{-1}\right|}_{a=c}\sim 40\,{\rm{s}}.\end{eqnarray} \tag{ 30 }$$

This supports the interpretation.

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We require a careful interpretation when we compare our results with observation. Transverse waves in our simulation are a mixture of fast and Alfvén waves. Alfvén waves are restricted such that they propagate along the field line, while fast waves are refracted due to the high Alfvén speed gap between the chromosphere and the corona (Rosenthal et al. 2002; Bogdan et al. 2003). The refraction is not considered in our simulation, and therefore our calculation potentially overestimates the amplitude of the transverse velocity.

Okamoto & De Pontieu (2011) argue that the energy flux estimated from observation is slightly lower than the amount required for coronal heating when the filling factor is considered. The observed feature is consistent with high-frequency components of the simulation, and thus the fore-mentioned study could potentially omit the low-frequency wave contribution that is not observed by spicule oscillation. In fact, in our calculation, the energy flux of the high-frequency components are a few times lower than the total energy flux. Thus, the observed flux is sufficiently high when both the low-frequency waves (De Pontieu et al. 2007a; McIntosh et al. 2011; Thurgood et al. 2014) and high-frequency waves are considered.