Dissipative Instability of Magnetohydrodynamic Sausage Waves in a Compressional Cylindrical Plasma: Effect of Flow Shear and Viscosity Shear

When there are steady background flow ( U ₀ = (0, 0, U )) and magnetic field ( B ₀ = (0, 0, B _i )) in the axial direction, we have two coupled differential equations for the perturbed total pressure P and the radial Lagrangian displacement ξ_r (e.g., Goossens et al. 1992) as

$$\begin{eqnarray}&&{D}_{1}\displaystyle \frac{d(r{\xi }_{\mathrm{ri}})}{{dr}}=-{{rC}}_{1}{P}_{i},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{D}_{1}\displaystyle \frac{{{dP}}_{i}}{{dr}}={C}_{2}{\xi }_{\mathrm{ri}},\end{eqnarray} \tag{ 8 }$$

where

$$\begin{eqnarray}&&{D}_{1}={\rho }_{i}({{\boldsymbol{v}}}_{\mathrm{si}}^{2}+{{\boldsymbol{v}}}_{\mathrm{Ai}}^{2})({{\rm{\Omega }}}^{2}-{\omega }_{\mathrm{Ai}}^{2})({{\rm{\Omega }}}^{2}-{\omega }_{\mathrm{ci}}^{2}),\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{C}_{1}={{\rm{\Omega }}}^{4}-{k}_{z}^{2}({{\boldsymbol{v}}}_{\mathrm{si}}^{2}+{{\boldsymbol{v}}}_{\mathrm{Ai}}^{2})({{\rm{\Omega }}}^{2}-{\omega }_{\mathrm{ci}}^{2}),\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{C}_{2}={\rho }_{i}^{2}{({{\boldsymbol{v}}}_{\mathrm{si}}^{2}+{{\boldsymbol{v}}}_{\mathrm{Ai}}^{2})({{\rm{\Omega }}}^{2}-{\omega }_{\mathrm{Ai}}^{2})}^{2}({{\rm{\Omega }}}^{2}-{\omega }_{\mathrm{ci}}^{2}),\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\rm{\Omega }}=\omega -{k}_{z}{\boldsymbol{U}},\end{eqnarray} \tag{ 12 }$$

and ω_Ai = v _Ai k_z , ω_ci = v _ci k_z , ${{\boldsymbol{v}}}_{\mathrm{Ai}}={{\boldsymbol{B}}}_{i}/\sqrt{{\rho }_{i}{\mu }_{0}}$ is the Alfvén speed, ${{\boldsymbol{v}}}_{\mathrm{si}}=\sqrt{{\gamma }_{0}{p}_{i}/{\rho }_{i}}$ is the sound speed, and ${{\boldsymbol{v}}}_{\mathrm{ci}}\,={{\boldsymbol{v}}}_{\mathrm{si}}{{\boldsymbol{v}}}_{\mathrm{Ai}}/\sqrt{{{\boldsymbol{v}}}_{\mathrm{si}}^{2}+{{\boldsymbol{v}}}_{\mathrm{Ai}}^{2}}$. The radial displacement ξ_ri has the relation with the perturbed radial velocity v _ri (Goossens et al. 1992):

$$\begin{eqnarray}&&{\xi }_{\mathrm{ri}}=\displaystyle \frac{i}{{\rm{\Omega }}}{{\boldsymbol{v}}}_{\mathrm{ri}}.\end{eqnarray} \tag{ 13 }$$

Equations (7) and (8) have Bessel functions as solutions.

For the outside of the flux tube we assume viscosity terms and no background flow where B ₀ = (0, 0, B _e ). Instead of deriving wave equations for the displacement ξ and the perturbed total pressure P, we derive a governing wave equation for the divergence of the perturbed velocity $\bigtriangleup \equiv {\rm{\nabla }}\cdot {{\boldsymbol{v}}}_{1}=\psi (r)\exp [-i(\omega t-{k}_{z}z)]$ (see Appendix A):

$$\begin{eqnarray}\begin{array}{rcl}{\omega }^{4}{\bigtriangleup }_{e} & + & \left[({\bar{{\boldsymbol{v}}}}_{\mathrm{se}}^{2}+{{\boldsymbol{v}}}_{\mathrm{Ae}}^{2}){\omega }^{2}-\left({{\boldsymbol{v}}}_{\mathrm{Ae}}^{2}{\bar{{\boldsymbol{v}}}}_{\mathrm{se}}^{2}{k}_{z}^{2}+\displaystyle \frac{i\mu }{{\rho }_{e}}{\omega }^{3}\right)\right]{{\rm{\nabla }}}^{2}{\bigtriangleup }_{e}\\ & - & \displaystyle \frac{i\mu \omega }{{\rho }_{e}}({\bar{{\boldsymbol{v}}}}_{\mathrm{se}}^{2}+{{\boldsymbol{v}}}_{\mathrm{Ae}}^{2}){{\rm{\nabla }}}^{4}{\bigtriangleup }_{e}=0,\end{array}\end{eqnarray} \tag{ 14 }$$

where ∇² is Laplacian and ${\bar{{\boldsymbol{v}}}}_{\mathrm{se}}^{2}={{\boldsymbol{v}}}_{\mathrm{se}}^{2}-i\omega \tfrac{\mu +\varsigma }{{\rho }_{e}}$. This equation can be further reduced to

$$\begin{eqnarray}&&\begin{array}{l}A\left[\displaystyle \frac{{d}^{4}\psi (r)}{{{dr}}^{4}}+\displaystyle \frac{2}{r}\displaystyle \frac{{d}^{3}\psi (r)}{{{dr}}^{3}}\right]-\left[A\left(2{k}_{z}^{2}+\displaystyle \frac{1}{{r}^{2}}\right)-1\right]\displaystyle \frac{{d}^{2}\psi (r)}{{{dr}}^{2}}\\ \,\,-\left[A\left(2{k}_{z}^{2}-\displaystyle \frac{1}{{r}^{2}}\right)-1\right]\displaystyle \frac{1}{r}\displaystyle \frac{d\psi (r)}{{dr}}-{n}^{2}\psi (r)=0,\end{array}\end{eqnarray} \tag{ 15 }$$

where

$$\begin{eqnarray}&&A=\displaystyle \frac{i\mu \omega }{{\rho }_{e}({\bar{\omega }}_{\mathrm{ce}}^{2}-{\omega }^{2})},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{n}^{2}=\displaystyle \frac{{\omega }^{4}-{k}_{z}^{2}({\bar{{\boldsymbol{v}}}}_{\mathrm{se}}^{2}+{{\boldsymbol{v}}}_{\mathrm{Ae}}^{2})\left({\omega }^{2}-{\bar{\omega }}_{\mathrm{ce}}^{2}+\tfrac{i\mu \omega }{{\rho }_{e}}{k}_{z}^{2}\right)}{({\bar{{\boldsymbol{v}}}}_{\mathrm{se}}^{2}+{{\boldsymbol{v}}}_{\mathrm{Ae}}^{2})({\bar{\omega }}_{\mathrm{ce}}^{2}-{\omega }^{2})},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{\bar{\omega }}_{\mathrm{ce}}^{2}=\displaystyle \frac{\left({\bar{{\boldsymbol{v}}}}_{\mathrm{se}}^{2}{{\boldsymbol{v}}}_{\mathrm{Ae}}^{2}{k}_{z}^{2}+\tfrac{i\mu }{{\rho }_{e}}{\omega }^{3}\right)}{({\bar{{\boldsymbol{v}}}}_{\mathrm{se}}^{2}+{{\boldsymbol{v}}}_{\mathrm{Ae}}^{2})}.\end{eqnarray} \tag{ 18 }$$

Equation (15) has Bessel functions of order zero as analytical solutions (e.g., Geeraerts et al. 2020). In the present paper we use for the solution the modified Bessel function of the second kind, K₀(kr), where

$$\begin{eqnarray}&&k={k}_{\pm }=\sqrt{\displaystyle \frac{-1+2{{Ak}}_{z}^{2}\pm \sqrt{{(1-2{{Ak}}_{z}^{2})}^{2}+4{{An}}^{2}}}{2A}}.\end{eqnarray} \tag{ 19 }$$

When μ(ς) = 0, Equation (15) reduces to the well-known Bessel's differential equation. As k_± includes imaginary terms, the solution itself is a complex function. It was illustrated by Geeraerts et al. (2020) that the solution function behaves like body-like or surface-like mode depending on the values of the complex eigenfrequency obtained from the dispersion relation.

We first consider the situation with no viscosity. We assume the background flow is inside the flux tube as in Section 2.1 and no flow for the outside region. Using the matching conditions at r = R (e.g., Goossens et al. 1992; Yu et al. 2017; Sadeghi et al. 2021),

$$\begin{eqnarray}&&{\xi }_{ri}={\xi }_{re},\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{P}_{i}={P}_{e},\end{eqnarray} \tag{ 21 }$$

we derive the dispersion relation as

$$\begin{eqnarray}\begin{array}{rcl}{\rho }_{i}({{\rm{\Omega }}}^{2}-{\omega }_{\mathrm{Ai}}^{2})-{\rho }_{e}({\omega }^{2}-{\omega }_{\mathrm{Ae}}^{2})\displaystyle \frac{{k}_{i}}{{k}_{e}}{Q}_{0} & = & 0,\end{array}\end{eqnarray} \tag{ 22 }$$

where

$$\begin{eqnarray}&&{k}_{i}^{2}=f\displaystyle \frac{({{\rm{\Omega }}}^{2}-{\omega }_{\mathrm{si}}^{2})({{\rm{\Omega }}}^{2}-{\omega }_{\mathrm{Ai}}^{2})}{({{\boldsymbol{v}}}_{\mathrm{si}}^{2}+{{\boldsymbol{v}}}_{\mathrm{Ai}}^{2})({{\rm{\Omega }}}^{2}-{\omega }_{\mathrm{ci}}^{2})},\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{k}_{e}^{2}=-\displaystyle \frac{({\omega }^{2}-{\omega }_{\mathrm{se}}^{2})({\omega }^{2}-{\omega }_{\mathrm{Ae}}^{2})}{({{\boldsymbol{v}}}_{\mathrm{se}}^{2}+{{\boldsymbol{v}}}_{\mathrm{Ae}}^{2})({\omega }^{2}-{\omega }_{\mathrm{ce}}^{2})},\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{Q}_{0}={F}_{0}({k}_{i}R)\displaystyle \frac{{K}_{0}({k}_{e}R)}{{K}_{0}^{\prime} ({k}_{e}R)},\end{eqnarray} \tag{ 25 }$$

and

$$\begin{eqnarray}f=\left\{\begin{array}{ll}-1 & {\rm{for}}\,{\rm{surface}}\,{\rm{modes}}\\ 1 & {\rm{for}}\,{\rm{body}}\,{\rm{modes}}\end{array},\,\,\,\,\,\right.\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}{F}_{0}({k}_{i}R)=\left\{\begin{array}{ll}\displaystyle \frac{{I}_{0}^{\prime} ({k}_{i}R)}{{I}_{0}({k}_{i}R)} & {\rm{for}}\,{\rm{surface}}\,{\rm{modes}}\\ \displaystyle \frac{{J}_{0}^{\prime} ({k}_{i}R)}{{J}_{0}({k}_{i}R)} & {\rm{for}}\,{\rm{body}}\,{\rm{modes}}\end{array}\right..\,\,\,\,\,\end{eqnarray} \tag{ 27 }$$

Here J₀(I₀) is the (modified) Bessel function of the first kind, and the prime denotes the derivative with respect to the entire argument.

The threshold of flow shear for NEW is called the critical speed. When the flow shear is higher than the critical speed, the wave energy of the backward wave becomes negative. The criterion for the NEW is given as (Cairns 1979; Joarder et al. 1997; Marcu 2007; Yu & Nakariakov 2020)

$$\begin{eqnarray}&&{ \mathcal C }=\omega \displaystyle \frac{\partial { \mathcal D }}{\partial \omega }\lt 0.\end{eqnarray} \tag{ 28 }$$

The dispersion function ${ \mathcal D }$ has to be suitably defined for the characteristic function ${ \mathcal C }$ to be positive when U = 0. We may define ${ \mathcal D }$ as

$$\begin{eqnarray}&&{ \mathcal D }=\chi ({\tilde{{\rm{\Omega }}}}^{2}-{\tilde{\omega }}_{\mathrm{Ai}}^{2})-({\tilde{\omega }}^{2}-{\tilde{\omega }}_{\mathrm{Ae}}^{2})\displaystyle \frac{{k}_{i}}{{k}_{e}}{Q}_{0},\end{eqnarray} \tag{ 29 }$$

where χ = ρ_i /ρ_e and tilde means the parameters are normalized by ω_si. Similarly, the characteristic function ${ \mathcal C }$ can be defined as ${ \mathcal C }={{\boldsymbol{v}}}_{p}\tfrac{\partial { \mathcal D }}{\partial {{\boldsymbol{v}}}_{p}}$ where v_p = ω/k_z . We note that an analytical expression of ${ \mathcal C }$ is possible when following the procedure to obtain $\tfrac{\partial { \mathcal D }}{\partial \omega }$ in Yu et al. (2017) or in Sadeghi et al. (2021).

The phase speed of the backward wave changes its sign when the flow shear crosses the critical speed. Therefore, one may obtain the critical speed by setting ω = 0 in the dispersion relation (Equation (22); e.g., Yu & Nakariakov 2020). See Figure 5.

Now we consider the situation that there exist flow inside and viscosity outside the flux tube. When the viscosity terms are included, two wave solutions can exist outside the flux tube, so the wave solution ψ(r) has in general the following form

$$\begin{eqnarray}\psi (r)=\left\{\begin{array}{ll}{A}_{1}{{ \mathcal F }}_{0}({k}_{i}r) & {\rm{for}}\,r\lt R\\ {A}_{2}{K}_{0}({k}_{+}r)+{A}_{3}{K}_{0}({k}_{-}r) & {\rm{for}}\,r\gt R\end{array},\right.\end{eqnarray} \tag{ 30 }$$

where A₁, A₂, and A₃ are arbitrary complex constants, and

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal F }}_{0}({k}_{i}r) & = & \left\{\begin{array}{ll}{I}_{0}({k}_{i}r) & {\rm{for}}\,{\rm{surface}}\,{\rm{modes}}\\ {J}_{0}({k}_{i}r) & {\rm{for}}\,{\rm{body}}\,{\rm{modes}}\end{array}\right..\end{array}\end{eqnarray} \tag{ 31 }$$

In the presence of viscosity outside the flux tube, the boundary condition in Equation (21) changes to

$$\begin{eqnarray}&&{P}_{i}={P}_{e}-2\mu \displaystyle \frac{\partial {{\boldsymbol{v}}}_{{re}}}{\partial r}-\left(\zeta -\displaystyle \frac{2\mu }{3}\right){\bigtriangleup }_{e}.\end{eqnarray} \tag{ 32 }$$

Since we have three unknown variables, A₁, A₂, and A₃, we need another boundary condition for the perturbed magnetic field in z direction, b_z , (e.g., Geeraerts et al. 2022):

$$\begin{eqnarray}&&{b}_{\mathrm{zi}}={b}_{\mathrm{ze}}.\end{eqnarray} \tag{ 33 }$$

Using Equations (20), (32), and (33), we derive a matrix equation

$$\begin{eqnarray}\left(\begin{array}{ccc}{X}_{1} & {X}_{2} & {X}_{3}\\ {X}_{4} & {X}_{5} & {X}_{6}\\ {X}_{7} & {X}_{8} & {X}_{9}\end{array}\right)\left(\begin{array}{c}{A}_{1}\\ {A}_{2}\\ {A}_{3}\end{array}\right)={\boldsymbol{X}}{\boldsymbol{A}}=0,\end{eqnarray} \tag{ 34 }$$

where X₁, X₂, ..., X₉ are shown in Appendix B.

To have a nontrivial solution for Equation (34). It should be $\det ({\boldsymbol{X}})=0$, which yields the dispersion relation

$$\begin{eqnarray}&&\begin{array}{l}{X}_{1}({X}_{5}{X}_{9}-{X}_{6}{X}_{8})-{X}_{2}({X}_{4}{X}_{9}-{X}_{6}{X}_{7})\\ +\,{X}_{3}({X}_{4}{X}_{8}-{X}_{5}{X}_{7})=0.\end{array}\end{eqnarray} \tag{ 35 }$$

Similarly to the solution for κ₊ in Geeraerts et al. (2020), the solution for k₋ represents the electromagnetic boundary layer (Hartmann layer) at r = R. In the limit μ → 0 and ζ → 0, ∣k₋∣ → ∞ . The solution for k₊ corresponds to the wave part. For the present problem, the numerical results including the solution for k₋ are found to be physically unacceptable. Thus, by ignoring the term for k₋ (A₃ = 0), we derive the following dispersion relation, using Equations (20) and (32),

$$\begin{eqnarray}&&{X}_{1}{X}_{5}-{X}_{2}{X}_{4}=0.\end{eqnarray} \tag{ 36 }$$

The use of Equation (33) for the boundary condition also gives wrong results. Equation (36) reduces to Equation (22) when μ = ζ = 0.

We first consider the behavior of the dispersion curves for the sausage modes depending on the flow speed U in the absence of the viscosity. The dispersion relation (Equation (22)) allows two phase speeds v _p = v _±. The phase speed v ₊( v ₋) corresponds to propagation in the positive (negative) z direction with the relation v ₋ = − v ₊ in the absence of the steady flow. In Figure 1 we plot the phase speed of the forward sausage modes, v ₊/ v _si, versus k_z R under a photospheric condition in the presence of no flow shear, where v _se = 1.5 v _si, v _Ai = 2.0 v _si, v _Ae = 0.5 v _si, v _ci ≈ 0.8944 v _si, and v _ce ≈ 0.4743 v _si (Edwin & Roberts 1983). We use these parameter values throughout the paper. There appear three kinds of eigenmodes: fast surface (fss), slow surface (sss), and slow body (sbs) modes. Three body modes are plotted among multiple solutions.

Zoom In Zoom Out Reset image size

Since the NEWI or DI occurs for the backward wave ( v ₋), we focus on the backward wave modes. Figure 2(a) represents the dependence of the dispersion curve of the backward slow surface sausage (sss) mode, v ₋, on the flow speed U . The phase speed v ₋ tends to shift upward as U increases, while a gap appears when 0.3 < U / v _si < 1.5. The Doppler-shifted phase speed in Figure 2(b) shows that in this range Ω < − ω_ci and the slow surface mode is not allowed. When Ω > ω_ci, the slow surface mode also does not exist.

Zoom In Zoom Out Reset image size

In Figure 3 we plot U dependence of (a) the phase speed v ₋ and (b) Doppler-shifted phase speed Ω₋ for the backward body mode 1 (sbs 1). The phase speed v ₋ goes upward as U increases without gap appearance. The Doppler-shifted phase speed Ω₋ shows its change of direction at U / v _si ≈ 0.9.

Zoom In Zoom Out Reset image size

In Figure 4 we present the characteristic function ${ \mathcal C }$ versus k_z R for the slow surface sausage mode, using Equations (28) and (29). This mode becomes NEW when the flow shear is above the gap regime. On the other hand, it is not possible to obtain the critical speed due to the gap appearance.

Zoom In Zoom Out Reset image size

For the body modes, the situation is opposite. The characteristic function ${ \mathcal C }$ using the dispersion function in Equation (29) is incorrect, and a suitable form was not found due to its singular behavior in the concerned range of v _p . Instead, we obtain the critical speed U _c from the dispersion relation in Equation (22) with the condition ω = 0, as explained in Section 2.4. Figure 5 gives U _c and v ₋ versus k_z R for the slow body mode 1 (sbs 1). It reveals that for the DI (NEWI) to occur, the flow shear has to be at least higher than the phase speed. If U _c is fixed, the corresponding k_z is also fixed. By denoting this value k_c , there will be a sign change of the phase speed v ₋ in both real and imaginary parts when k_z goes through k_c . When U / v _si = 0.9, k_c R ≈ 0.6.

Zoom In Zoom Out Reset image size

In this subsection, we solve the dispersion relation in Equation (36), considering both flow and viscosity. We first consider the slow surface sausage mode. In Figure 6, we plot (a) Doppler-shifted real part and (b) imaginary part of the phase speed v ₋, and (c) their ratio, γ (=Im( v ₋)/Re( v ₋)), as a function of k_z R with varying U / v _si, where $\tilde{\mu }(=\mu /({\rho }_{e}{{\boldsymbol{v}}}_{\mathrm{se}}R))\,=\tilde{\zeta }(=\zeta /({\rho }_{e}{{\boldsymbol{v}}}_{\mathrm{se}}R))={10}^{-4}$. Figures 6(a) and (b) show that both the real and imaginary parts of phase speed v ₋ shift upward as U increases. Figure 6(c) indicates that γ tends to increase with the increment of U and k_z in the NEW regime (${ \mathcal C }\lt 0$). The same trend was previously shown for the incompressible plasma (Yu & Nakariakov 2020). The order of magnitude is ∼ − 6 ($\gamma /\tilde{\mu }\sim {10}^{-2}$).

Zoom In Zoom Out Reset image size

It may be more useful to use $\gamma /\tilde{\mu }$ for estimating the strength of the instability. Yu & Nakariakov (2020) showed that γ of the surface mode in an incompressible plasma is a linear function of $\tilde{\mu }$ (${\tilde{\nu }}_{e}$ therein). Its order of magnitude is ∼1 for underdense and ∼ − 1 for overdense plasmas. This implies that the strength of DI in the compressible plasma is weaker than in the incompressible plasma.

We examine whether the linear relation between γ and $\tilde{\mu }$ obtained in an incompressible plasma (Yu & Nakariakov 2020) is valid or not for the compressional plasma. In Figure 7 we compare two situations, U / v _si = 1.5 (left column) and U/ v _si = 1.6 (right column), with variation of $\tilde{\mu }(=\tilde{\zeta })$. In the first row (panels (a) and (b)), the effect of $\tilde{\mu }$ on Doppler-shifted phase speed is shown to be weak for small $\tilde{\mu }$, while the middle row (panels (c) and (d)) reveals that γ is in proportion to $\tilde{\mu }$ and U . From the bottom row (panels (e) and (f)), we know that the linear relation is valid for small $\tilde{\mu }$ and k_z R and for low U . When U is sufficiently high and $\tilde{\mu }$ is relatively large, one may anticipate that γ deviates from the linear trend as k_z R increases.

Zoom In Zoom Out Reset image size

A similar behavior is obtained for the slow body mode (sbs) 1. In Figure 8, we plot (a) real and (b) imaginary parts of the phase speed v ₋ and (c) their ratio γ (=Im( v ₋)/Re( v ₋)) versus k_z R for slow body mode 1, with varying U / v _si where $\tilde{\mu }(=\tilde{\zeta })={10}^{-4}$. Both parts of the phase speed shift upward as U increases and becomes positive when crossing the critical speed. The value of γ is always positive except at k_z = k_c , and its order of magnitude is about −6. Figure 8(c) presents that $\gamma /\tilde{\mu }\approx {10}^{-2}$, similar to the slow surface sausage mode. It has a peak at k_z R ≈ 1.4, contrary to the slow surface mode. The axial wavelength at the peak is comparable to the diameter of the flux tube.

Zoom In Zoom Out Reset image size

In Figure 9 we display two flow cases for the slow body mode 1, U / v _si = 1.0 (left column) and U / v _si = 1.2 (right column), by varying $\tilde{\mu }(=\tilde{\zeta })$. The top row, (a) and (b), shows the variation of Re( v ₋) versus k_z R. Its dependence on $\tilde{\mu }$ is tiny but tends to increases as U increases. The middle row, (c) and (d), describes the growth rate γ versus k_z R while the bottom row, (e) and (f), represents $\gamma /\tilde{\mu }$ versus k_z R. In two cases, the behavior of γ looks similar, but its normalized value $\gamma /\tilde{\mu }$ is different. The linear trend is valid only for small $\tilde{\mu }$. After $\tilde{\mu }$ is over a certain point, it turns into a nonlinear regime, and the peak at k_z R ≈ 1.4 shifts to the right. The threshold of $\tilde{\mu }$ for the deviation to the nonlinear trend becomes smaller as U increases.

Zoom In Zoom Out Reset image size

In Figure 10 we show the effect of $\tilde{\zeta }$ on γ for $\tilde{\mu }={10}^{-6}$ ((a) and (b)) and $\tilde{\mu }={10}^{-4}$ ((c) and (d)). The left (right) column denotes the results for the slow surface mode (slow body mode 1) when U / v _si = 1.6 (1.2). The $\tilde{\zeta }$-dependent behavior of the $\gamma /\tilde{\mu }$ looks similar for both modes. When the value of $\tilde{\zeta }$ is less than that of $\tilde{\mu }$, its effect on γ is negligible. When its order of magnitude is greater than that of $\tilde{\mu }$, $\gamma /\tilde{\mu }$ gives a noticeable increment. With a further increment of $\tilde{\zeta }$, $\gamma /\tilde{\mu }$ reaches a maximum in the plot range of k_z R, which depends on the value of $\tilde{\mu }$. It is found that when the value of $\tilde{\zeta }$ is sufficiently large, a new peak appears for the surface mode. Its position shifts to the left (to smaller k_z R) as $\tilde{\zeta }$ increases. On the other hand, for the body mode, the peak position shifts, at first, to the right and then to the left as $\tilde{\zeta }$ increases (panel (b)). When $\tilde{\mu }$ is further increases, the motion of the peak is more complicated. Panel (d) shows that the peak has a zigzag motion. Comparing the top and bottom rows, one may notice that the effect of $\tilde{\zeta }$ on the increment of $\gamma /\tilde{\mu }$ becomes weaker as $\tilde{\mu }$ increases. When $\tilde{\mu }={10}^{-2}$, the variation range of $\gamma /\tilde{\mu }$ is tiny, and $\gamma /\tilde{\mu }\sim {10}^{-2}$ (see Figures 7(f) and 9(f)). The maximum growth rate is obtained when $\tilde{\zeta }/\tilde{\mu }\sim {10}^{3}-{10}^{6}$, depending on the wave mode and $\tilde{\mu }$. We point out that $\tilde{\zeta }$ rather may give a negative effect on γ as shown in panel (c) (the curves for $\tilde{\zeta }\geqslant {10}^{0}$).

Zoom In Zoom Out Reset image size

Cowley's theoretical treatment (1990) showed that μ ∼ 4.1 × 10⁻⁴ g cm^-1s⁻¹, and the order of magnitude of ζ/μ can reach 7 in the upper photosphere. Adopting ρ_e ≈ 4.9 × 10⁻⁹ g cm⁻³ and v _se ∼ 10 km s⁻¹, we obtain $\tilde{\mu }\sim 8.4\times \,{10}^{-4}{\rm{m}}/{\rm{R}}$. Thus, the smaller the R, the stronger the viscosity. Since $\tilde{\mu }$ has inverse relation with the radius R, the waves in the small scale flux tube can be more unstable and more easily excited due to the DI in photospheric regions (see, e.g., Section 3.4 in Jess et al. 2023), contributing to the enhancement of the wave propagations into the upper atmosphere.