**4. Dispersion diagrams of MHD surface waves in spicules**

Before starting solving the wave dispersion relation (34), we have to specify some input parameters, characterizing both media (the jet and its surrounding). Bearing in mind, as we have already mention in the beginning of Sec. 2, the mass density of the environment is much less (50–100 times); thus we take the density contrast – the ratio of equilibrium plasma density outside to that inside of spicule – to be *η* = 0.02. Our choice of the sound and Alfvén speeds in the jet is *c*si = 10 km s−<sup>1</sup> and *v*Ai = 80 km s−1, respectively, while those speeds in the environment are correspondingly *c*se ∼= 488 km s−<sup>1</sup> and *v*Ae = 200 km s−1. All these values are in agreement with the condition for the balance of total pressures at the flux tube interface – that condition can be expressed in the form

$$p\_{\mathbf{i}} + \frac{B\_{\mathbf{i}}^2}{2\mu} = p\_{\mathbf{e}} + \frac{B\_{\mathbf{e}}^2}{2\mu}.$$

which yields (Edwin & Roberts (1983)

12 Will-be-set-by-IN-TECH

After applying the boundary conditions (we recall that for the ambient medium *U* = 0) finally we arrive at the required dispersion relation of the normal MHD modes propagating along the

(*ω* − **k** · **U**)

For the azimuthal mode number *m* = 0 the above equation describes the propagation of so called *sausage* waves, while with *m* = 1 it governs the propagation of the *kink* waves (Edwin & Roberts, 1983). As we have already seen, the wave frequency, *ω*, is Doppler-shifted inside the jet. The two quantities *κ*<sup>i</sup> and *κ*e, whose squared magnitudes are given by Eq. (31) are termed *wave attenuation coefficients*. They characterize how quickly the wave amplitude having its maximal value at the interface, *r* = *a*, decreases as we go away in both directions. Depending on the specific sound and Alfvén speeds in a given medium, as well as on the density contrast, *η* = *ρ*e/*ρ*e, and the ratio of the embedded magnetic fields, *b* = *B*e/*B*e, the attenuation coefficients can be real or imaginary quantities. In the case when both *κ*<sup>i</sup> and *κ*<sup>e</sup> are real, we have a *pure surface wave*. The case *κ*<sup>i</sup> imaginary and *κ*<sup>e</sup> real corresponds to *pseudosurface waves* (or *body waves* according to Edwin & Roberts terminology (Edwin & Roberts, 1983)). In that case the modified Bessel function inside the jet, *I*0, becomes the spatially periodic Bessel function *J*0. In the opposite situation the wave energy is carried away from the flux tube – then the wave is called *leaky wave* (Cally, 1986). The waves, which propagate in spicules and X-ray jets, are generally pseudosurface waves, that can however, at some flow speeds become

For the kink waves one defines the *kink speed* (Edwin & Roberts, 1983)

Ai <sup>+</sup> *<sup>ρ</sup>*e*v*<sup>2</sup>

*ρ*<sup>i</sup> + *ρ*<sup>e</sup>

Ae

theory. Our linear approach can determine just the instability threshold only.

1/2

= *v*2

which is independent of sound speeds and characterizes the propagation of transverse

Our study of the dispersion characteristics of kink and sausage waves, as well as their stability status will be performed in two steps. First, at given sound and Alfvén speeds inside the jet and its environment and a fixed flow speed *U*, we solve the transcendental dispersion Eq. (34) assuming that the wave angular frequency, *ω*, and the wave number, *kz*, are real quantities. In the next step, when studying their stability/instability status, we assume that the wave frequency and correspondingly the wave phase velocity, *v*ph = *ω*/*kz*, become complex. Then, as the imaginary part of the complex frequency/phase velocity at a given wave number, *kz*, and a critical jet speed, *U*crt, has some non-zero positive value, one says that the wave becomes unstable – its amplitude begins to grow with time. In this case, the linear theory is no longer applicable and one ought to investigate the further wave propagation by means of a nonlinear

In the next two section we numerically derive the dispersion curves of kink and sausage waves

Ai + (*ρ*e/*ρ*i)*v*<sup>2</sup>

1 + *ρ*e/*ρ*<sup>i</sup>

Ae

1/2

, (35)

*<sup>ω</sup>* <sup>−</sup> **<sup>k</sup>** · **<sup>U</sup>**, has also to be continuous (Chandrasekhar, 1961).

<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> *zv*2 Ai *κ*e *K*� *<sup>m</sup>*(*κ*e*a*)

*Km*(*κ*e*a*) <sup>=</sup> 0. (34)

• *δp*tot has to be continuous across the interface,

jet (Nakariakov, 2007; Terra-Homen et al., 2003)

• the perturbed interface, *<sup>δ</sup>vr*

*ρ*e *ρ*i *<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> *zv*2 Ae *κ*i *I*� *<sup>m</sup>*(*κ*i*a*) *Im*(*κ*i*a*) <sup>−</sup>

pure surface modes.

perturbations.

*c*<sup>k</sup> =

 *ρ*i*v*<sup>2</sup>

running along spicules and X-ray jets, respectively.

$$\frac{\rho\_{\rm e}}{\rho\_{\rm i}} = \frac{c\_{\rm si}^2 + \frac{\gamma}{2} v\_{\rm Ai}^2}{c\_{\rm se}^2 + \frac{\gamma}{2} v\_{\rm Ae}^2}. \tag{36}$$

The two tube speeds (look at Eq. (32)) are *c*Ti = 9.9 km s−<sup>1</sup> and *c*Te = 185 km s−1. The kink speed, associated with the kink waves, in our case (see Eq. (35)) is 84 km s−1.

It is obvious that dispersion Eq. (34) of either mode can be solved only numerically. Before starting that job, we normalize all velocities to the Alfvén speed *v*Ai inside the jet thus defining the dimensionless phase velocity *V*ph = *v*ph/*v*Ai and the *Alfvén–Mach number M*<sup>A</sup> = *U*/*v*Ai. The wavelength is normalized to the tube radius *a*, which means that the dimensionless wave number is *K* = *kz a*. The calculation of wave attenuation coefficients requires the introduction of three numbers, notably the two ratios *β*¯ = *c*<sup>2</sup> s/*v*<sup>2</sup> <sup>A</sup> correspondingly in the jet and its environment, and the ratio of the background magnetic field outside to that inside the flow, *b* = *B*e/*B*i, in addition to the density contrast, *η*. We recall that the two *β*¯s are 1.2 times smaller than the corresponding plasma betas in both media – the latter are given by the expressions

$$
\beta\_{\mathbf{i},\mathbf{e}} = 2\vec{\beta}\_{\mathbf{i},\mathbf{e}}/\gamma.
$$

Thus, the input parameters in the numerical procedure are

$$
\eta = 0.02, \quad \vec{\beta}\_{\rm i} \cong 0.016, \quad \vec{\beta}\_{\rm e} \cong 5.96, \quad b \cong 0.35, \quad \text{and} \quad M\_{\rm A}.
$$

The value of the Alfvén–Mach number, *M*A, naturally depends on the value of the streaming velocity, *U*. Our choice of this value is 100 km s−<sup>1</sup> that yields *M*<sup>A</sup> = 1.25. With these input values, we calculate the dispersion curves of first kink waves and then sausage ones.

#### **4.1 Kink waves in spicules**

We start by calculating the dispersion curves of kink waves assuming that the angular wave frequency, *ω*, is real. As a reference, we first assume that the plasma in the flux tube is static, i.e., *M*<sup>A</sup> = 0. The dispersion curves, which present the dependence of the normalized wave phase velocity on the normalized wave number, are in this case shown in Fig. 5. One can recognize three types of waves: a sub-Alfvénic slow magnetoacoustic wave (in

a forward propagating wave that has, however, a lower normalized phase velocity than that

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 149

The most interesting waves especially for the Type II spicules seems to be the waves labelled ck. It would be interesting to see whether these modes can become unstable at some, say critical, value of the Alfvén–Mach number, *M*A. To study this, we have to assume that the wave frequency is complex, i.e., *ω* → *ω* + i*γ*¯, where *γ*¯ is the expected instability growth rate. Thus, the dispersion equation becomes complex (complex wave phase velocity and real wave number) and the solving a transcendental complex equation is generally a difficult task

Before starting to derive a numerical solution to the complex version of Eq. (34), we can simplify that equation. Bearing in mind that the plasma beta inside the jet is very small (*β*<sup>i</sup> ∼= 0.02) and that of the surrounding medium quite high (of order 7), we can treat the jet as a cool plasma and the environment as a hot incompressible fluid. We point out that according to the numerical simulation of spicules by Matsumoto & Shibata (Matsumoto & Shibata, 2010) the plasma beta at heights greater than 2 Mm is of that order (0.03–0.04) – look at Fig. 4 in their paper. For cool plasma, *c*<sup>s</sup> → 0; hence the normalized wave attenuation

and the corresponding attenuation coefficient is simply equal to *kz*, i.e., *κ*e*a* = *K*. Under these

<sup>1</sup>(*κ*i*a*) *<sup>I</sup>*1(*κ*i*a*) <sup>−</sup>

1/2

*V*ph, is a complex number. We note that this simplified version of the dispersion equation of

To investigate the stability/instability status of kink waves we numerically solve Eq. (37) using the Müller method (Muller, 1956) for finding the complex roots at fixed input parameters *η* = 0.02 and *b* = 0.35 and varying the Alfvén–Mach number, *M*A, from zero to some reasonable numbers. Before starting any numerical procedure for solving the aforementioned dispersion equation, we note that for each input value of *M*<sup>A</sup> one can get two *c*k-dispersion curves one of which (for relatively small magnitudes of *M*A) has normalized phase velocity roughly equal to *M*<sup>A</sup> − 1 and a second dispersion curve associated with dimensionless phase velocity equal

The results of the numerical solving Eq. (37) are shown in Fig. 7. For *M*<sup>A</sup> = 0, except for the dispersion curve with normalized phase velocity approximately equal to 1, one can find a dispersion curve with normalized phase velocity close to −1 – that curve is not plotted in Fig. 7. Similarly, for *M*<sup>A</sup> = 2 one obtains a curve at *V*ph = 1 and another at *V*ph = 3, and so on. With increasing the magnitude of the Alfvén–Mach number kink waves change their structure – for small numbers being pseudosurface (body) waves and for *M*<sup>A</sup> 4 becoming pure surface modes. Another effect associated with the increase in *M*A, is that, for instance at *M*<sup>A</sup> 6, the shapes of pairs of dispersion curves begin to visibly change as can be seen in Figs. 7 and 8. The most interesting observation is that for *M*<sup>A</sup> 8 both curves begin to

the kink waves closely reproduces the dispersion curves labelled c<sup>k</sup> in Figs. 5 and 6.

*V*ph − *M*<sup>A</sup>

similarly be considered as a mirror image of the high-harmonic super-Alfvénic waves.

<sup>k</sup>-labelled dispersion curve. Moreover, there appears to be a family of generally

<sup>k</sup>-labelled curve) plotted in blue colour that can

*K*, while for the incompressible environment *c*<sup>s</sup> → ∞

2 − 1 *K*� <sup>1</sup>(*K*)

*K*, and the normalized wave phase velocity,

*<sup>K</sup>*1(*K*) <sup>=</sup> 0, (37)

<sup>k</sup> and <sup>c</sup><sup>h</sup>

<sup>k</sup> in Fig. 6.

of its sister c<sup>h</sup>

(Acton, 1990).

coefficient *κ*i*a* =

 *V*2 ph*<sup>η</sup>* <sup>−</sup> *<sup>b</sup>*<sup>2</sup>  1 − 

where, we recall, that *κ*i*a* =

<sup>1</sup> <sup>−</sup> (*V*ph <sup>−</sup> *<sup>M</sup>*A)<sup>2</sup>

*V*ph − *M*<sup>A</sup>

1/2

2 1/2 *I*�

<sup>1</sup> <sup>−</sup> (*V*ph <sup>−</sup> *<sup>M</sup>*A)<sup>2</sup>

to *M*<sup>A</sup> + 1. These curves are similar to the dispersion curves labelled c<sup>l</sup>

circumstances the simplified dispersion equation of kink waves takes the form

backward propagating waves (below the c<sup>l</sup>

Fig. 5. Dispersion curves of kink waves propagating along the flux tube at *M*<sup>A</sup> = 0.

magenta colour) labelled cTi (which is actually the normalized value of *c*Ti to *v*Ai), an almost Alfvén wave labelled c<sup>k</sup> (the green curve), and a family of super-Alfvénic waves (the red dispersion curves). We note that one can get by numerically solving Eq. (34) the mirror images (with respect to the zeroth line) of the ck-labelled dispersion curve, as well as of the fast super-Alfvénic waves – both being backward propagating modes that are not plotted in Fig. 5. The next Fig. 6 shows how all these dispersion curves change when the plasma inside the

Fig. 6. Dispersion curves of kink waves propagating along the flux tube at *M*<sup>A</sup> = 1.25.

tube flows. One sees that the flow first shifts upwards the almost Alfvén wave now labelled ch <sup>k</sup>, as well as high-harmonic super-Alfvénic waves. Second, the slow magnetoacoustic wave (cTi in Fig. 5) is replaced by two, now, super-Alfvénic waves, whose dispersion curves (in orange and cyan colours) are collectively labelled cTi. These two waves have practically constant normalized phase velocities equal to 1.126 and 1.374, respectively, which are the (*M*<sup>A</sup> <sup>∓</sup> *<sup>c</sup>*<sup>0</sup> Ti)-values, where *<sup>c</sup>*<sup>0</sup> Ti is the normalized magnitude of the slow magnetoacoustic wave at *M*<sup>A</sup> = 0. Unsurprisingly, one gets a c<sup>l</sup> <sup>k</sup>-labelled curve, which is the mirror image of the ch <sup>k</sup>-labelled curve. That is why this curve is plotted in green and, as can be seen, it is now 14 Will-be-set-by-IN-TECH

Fig. 5. Dispersion curves of kink waves propagating along the flux tube at *M*<sup>A</sup> = 0.

Fig. 6. Dispersion curves of kink waves propagating along the flux tube at *M*<sup>A</sup> = 1.25.

ch

ch

(*M*<sup>A</sup> <sup>∓</sup> *<sup>c</sup>*<sup>0</sup>

Ti)-values, where *<sup>c</sup>*<sup>0</sup>

at *M*<sup>A</sup> = 0. Unsurprisingly, one gets a c<sup>l</sup>

tube flows. One sees that the flow first shifts upwards the almost Alfvén wave now labelled

<sup>k</sup>, as well as high-harmonic super-Alfvénic waves. Second, the slow magnetoacoustic wave (cTi in Fig. 5) is replaced by two, now, super-Alfvénic waves, whose dispersion curves (in orange and cyan colours) are collectively labelled cTi. These two waves have practically constant normalized phase velocities equal to 1.126 and 1.374, respectively, which are the

<sup>k</sup>-labelled curve. That is why this curve is plotted in green and, as can be seen, it is now

Ti is the normalized magnitude of the slow magnetoacoustic wave

<sup>k</sup>-labelled curve, which is the mirror image of the

magenta colour) labelled cTi (which is actually the normalized value of *c*Ti to *v*Ai), an almost Alfvén wave labelled c<sup>k</sup> (the green curve), and a family of super-Alfvénic waves (the red dispersion curves). We note that one can get by numerically solving Eq. (34) the mirror images (with respect to the zeroth line) of the ck-labelled dispersion curve, as well as of the fast super-Alfvénic waves – both being backward propagating modes that are not plotted in Fig. 5. The next Fig. 6 shows how all these dispersion curves change when the plasma inside the a forward propagating wave that has, however, a lower normalized phase velocity than that of its sister c<sup>h</sup> <sup>k</sup>-labelled dispersion curve. Moreover, there appears to be a family of generally backward propagating waves (below the c<sup>l</sup> <sup>k</sup>-labelled curve) plotted in blue colour that can similarly be considered as a mirror image of the high-harmonic super-Alfvénic waves.

The most interesting waves especially for the Type II spicules seems to be the waves labelled ck. It would be interesting to see whether these modes can become unstable at some, say critical, value of the Alfvén–Mach number, *M*A. To study this, we have to assume that the wave frequency is complex, i.e., *ω* → *ω* + i*γ*¯, where *γ*¯ is the expected instability growth rate. Thus, the dispersion equation becomes complex (complex wave phase velocity and real wave number) and the solving a transcendental complex equation is generally a difficult task (Acton, 1990).

Before starting to derive a numerical solution to the complex version of Eq. (34), we can simplify that equation. Bearing in mind that the plasma beta inside the jet is very small (*β*<sup>i</sup> ∼= 0.02) and that of the surrounding medium quite high (of order 7), we can treat the jet as a cool plasma and the environment as a hot incompressible fluid. We point out that according to the numerical simulation of spicules by Matsumoto & Shibata (Matsumoto & Shibata, 2010) the plasma beta at heights greater than 2 Mm is of that order (0.03–0.04) – look at Fig. 4 in their paper. For cool plasma, *c*<sup>s</sup> → 0; hence the normalized wave attenuation coefficient *κ*i*a* = <sup>1</sup> <sup>−</sup> (*V*ph <sup>−</sup> *<sup>M</sup>*A)<sup>2</sup> 1/2 *K*, while for the incompressible environment *c*<sup>s</sup> → ∞ and the corresponding attenuation coefficient is simply equal to *kz*, i.e., *κ*e*a* = *K*. Under these circumstances the simplified dispersion equation of kink waves takes the form

$$\frac{1}{2}\left(V\_{\text{ph}}^2 \eta - b^2\right) \left[1 - \left(V\_{\text{ph}} - M\_{\text{A}}\right)^2\right]^{1/2} \frac{I\_1'(\kappa\_i a)}{I\_1(\kappa\_i a)} - \left[\left(V\_{\text{ph}} - M\_{\text{A}}\right)^2 - 1\right] \frac{K\_1'(\text{K})}{K\_1(\text{K})} = 0,\tag{37}$$

where, we recall, that *κ*i*a* = <sup>1</sup> <sup>−</sup> (*V*ph <sup>−</sup> *<sup>M</sup>*A)<sup>2</sup> 1/2 *K*, and the normalized wave phase velocity, *V*ph, is a complex number. We note that this simplified version of the dispersion equation of the kink waves closely reproduces the dispersion curves labelled c<sup>k</sup> in Figs. 5 and 6.

To investigate the stability/instability status of kink waves we numerically solve Eq. (37) using the Müller method (Muller, 1956) for finding the complex roots at fixed input parameters *η* = 0.02 and *b* = 0.35 and varying the Alfvén–Mach number, *M*A, from zero to some reasonable numbers. Before starting any numerical procedure for solving the aforementioned dispersion equation, we note that for each input value of *M*<sup>A</sup> one can get two *c*k-dispersion curves one of which (for relatively small magnitudes of *M*A) has normalized phase velocity roughly equal to *M*<sup>A</sup> − 1 and a second dispersion curve associated with dimensionless phase velocity equal to *M*<sup>A</sup> + 1. These curves are similar to the dispersion curves labelled c<sup>l</sup> <sup>k</sup> and <sup>c</sup><sup>h</sup> <sup>k</sup> in Fig. 6. The results of the numerical solving Eq. (37) are shown in Fig. 7. For *M*<sup>A</sup> = 0, except for the dispersion curve with normalized phase velocity approximately equal to 1, one can find a dispersion curve with normalized phase velocity close to −1 – that curve is not plotted in Fig. 7. Similarly, for *M*<sup>A</sup> = 2 one obtains a curve at *V*ph = 1 and another at *V*ph = 3, and so on. With increasing the magnitude of the Alfvén–Mach number kink waves change their structure – for small numbers being pseudosurface (body) waves and for *M*<sup>A</sup> 4 becoming pure surface modes. Another effect associated with the increase in *M*A, is that, for instance at *M*<sup>A</sup> 6, the shapes of pairs of dispersion curves begin to visibly change as can be seen in Figs. 7 and 8. The most interesting observation is that for *M*<sup>A</sup> 8 both curves begin to

Fig. 9. Growth rates of unstable kink waves propagating along the flux tube at values of *M*<sup>A</sup>

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 151

velocity, *U*, plays the role of the necessary velocity difference across the interface between the

The big question that immediately springs to mind is whether one can really observe such an instability in spicules. The answer to that question is obviously negative – to register the onset of a Kelvin–Helmholtz instability of kink waves travelling on a Type II spicule one would need to observe jet velocities of the order of or higher than 712 km s−1! If we assume that the density contrast, *η*, possesses the greater value of 0.01 (which means that the jet mass density is 100 times larger than that of the ambient medium) and the ratio of the background magnetic fields, *b*, is equal to 0.36 (which may be obtained from a slightly different set of characteristic sound and Alfvén speeds in both media), the critical Alfvén–Mach number at which the instability starts is even much higher (equal to 12.6) – in that case the corresponding jet speed is *U*crt = 882 km s−<sup>1</sup> – too high to be registered in a spicule. The value of 882 was

We note that very similar dispersion curves and growth rates of unstable kink waves like those shown in Figs. 8 and 9 were obtained for cylindrical jets when both media were treated as incompressible fluids. In that case, dispersion Eq. (37) becomes a quadratic equation

that provides solutions for the real and imaginary parts of the normalized wave phase velocity

*<sup>V</sup>*ph <sup>=</sup> <sup>−</sup>*M*A*<sup>B</sup>* <sup>±</sup> <sup>√</sup>*<sup>D</sup>*

*<sup>m</sup>*(*K*)/*Im*(*K*), *B* = *K*�

*V*ph − *M*<sup>A</sup>

*<sup>η</sup><sup>A</sup>* <sup>−</sup> *<sup>B</sup>* ,

2 − 1 *K*� *<sup>m</sup>*(*K*)

*<sup>m</sup>*(*K*)/*Km*(*K*),

*Km*(*K*) <sup>=</sup> 0, (38)

computed under the assumption that the Alfvén speed inside the jet is 70 km s−1.

 *I*� *<sup>m</sup>*(*K*) *Im*(*K*) <sup>−</sup>

equal to 8.9, 8.95, 9, and 9.05, respectively.

 *V*2 ph*<sup>η</sup>* <sup>−</sup> *<sup>b</sup>*<sup>2</sup>

where

in closed forms, notably (Zhelyazkov, 2010; 2011)

*A* = *I* �

spicule and its environment.

Fig. 7. Dispersion curves of kink waves propagating along the flux tube at various values of *M*A.

Fig. 8. Dispersion curves of kink waves propagating along the flux tube for relatively large values of *M*A.

merge and at *M*<sup>A</sup> = 8.5 they form a closed dispersion curve. The ever increasing of *M*<sup>A</sup> yields yet smaller closed dispersion curves – the two non-labelled ones depicted in Fig. 8 correspond to *M*<sup>A</sup> = 8.8 and 8.85, respectively. All these dispersion curves present stable propagation of the kink waves. However, for *M*<sup>A</sup> 8.9 we obtain a new family of wave dispersion curves that correspond to an unstable wave propagation. We plot in Fig. 8 four curves of that kind that have been calculated for *M*<sup>A</sup> = 8.9, 8.95, 9, and 9.05, respectively. The growth rates of the unstable waves are shown in Fig. 9. The instability that arises is of the Kelvin–Helmholtz type. We recall that the Kelvin–Helmholtz instability, which is named after Lord Kelvin and Hermann von Helmholtz, can occur when velocity shear is present within a continuous fluid, or when there is a sufficient velocity difference across the interface between two fluids (Chandrasekhar, 1961). In our case, we have the second option and the relative jet

Fig. 9. Growth rates of unstable kink waves propagating along the flux tube at values of *M*<sup>A</sup> equal to 8.9, 8.95, 9, and 9.05, respectively.

velocity, *U*, plays the role of the necessary velocity difference across the interface between the spicule and its environment.

The big question that immediately springs to mind is whether one can really observe such an instability in spicules. The answer to that question is obviously negative – to register the onset of a Kelvin–Helmholtz instability of kink waves travelling on a Type II spicule one would need to observe jet velocities of the order of or higher than 712 km s−1! If we assume that the density contrast, *η*, possesses the greater value of 0.01 (which means that the jet mass density is 100 times larger than that of the ambient medium) and the ratio of the background magnetic fields, *b*, is equal to 0.36 (which may be obtained from a slightly different set of characteristic sound and Alfvén speeds in both media), the critical Alfvén–Mach number at which the instability starts is even much higher (equal to 12.6) – in that case the corresponding jet speed is *U*crt = 882 km s−<sup>1</sup> – too high to be registered in a spicule. The value of 882 was computed under the assumption that the Alfvén speed inside the jet is 70 km s−1.

We note that very similar dispersion curves and growth rates of unstable kink waves like those shown in Figs. 8 and 9 were obtained for cylindrical jets when both media were treated as incompressible fluids. In that case, dispersion Eq. (37) becomes a quadratic equation

$$\left(V\_{\rm ph}^{2}\eta - b^{2}\right)\frac{I\_{\rm m}^{\prime}(K)}{I\_{\rm m}(K)} - \left[\left(V\_{\rm ph} - M\_{\rm A}\right)^{2} - 1\right]\frac{K\_{\rm m}^{\prime}(K)}{K\_{\rm m}(K)} = 0,\tag{38}$$

that provides solutions for the real and imaginary parts of the normalized wave phase velocity in closed forms, notably (Zhelyazkov, 2010; 2011)

$$V\_{\rm ph} = \frac{-M\_{\rm A}B \pm \sqrt{D}}{\eta A - B} \prime$$

where

16 Will-be-set-by-IN-TECH

Fig. 7. Dispersion curves of kink waves propagating along the flux tube at various values of

Fig. 8. Dispersion curves of kink waves propagating along the flux tube for relatively large

merge and at *M*<sup>A</sup> = 8.5 they form a closed dispersion curve. The ever increasing of *M*<sup>A</sup> yields yet smaller closed dispersion curves – the two non-labelled ones depicted in Fig. 8 correspond to *M*<sup>A</sup> = 8.8 and 8.85, respectively. All these dispersion curves present stable propagation of the kink waves. However, for *M*<sup>A</sup> 8.9 we obtain a new family of wave dispersion curves that correspond to an unstable wave propagation. We plot in Fig. 8 four curves of that kind that have been calculated for *M*<sup>A</sup> = 8.9, 8.95, 9, and 9.05, respectively. The growth rates of the unstable waves are shown in Fig. 9. The instability that arises is of the Kelvin–Helmholtz type. We recall that the Kelvin–Helmholtz instability, which is named after Lord Kelvin and Hermann von Helmholtz, can occur when velocity shear is present within a continuous fluid, or when there is a sufficient velocity difference across the interface between two fluids (Chandrasekhar, 1961). In our case, we have the second option and the relative jet

*M*A.

values of *M*A.

$$A = I\_m'(K) / I\_m(K), \qquad B = K\_m'(K) / K\_m(K),$$

Fig. 11. Growth rates of unstable kink waves calculated from Eq. (38) at values of *M*<sup>A</sup> equal

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 153

The dispersion curves of sausage waves both in a static and in a flowing plasma shown in Figs. 12 and 13 are very similar to those of kink waves (compare with Figs. 5 and 6). The latter curves were calculated from dispersion Eq. (34) with azimuthal mode number *m* = 0 for the

Fig. 12. Dispersion curves of sausage waves propagating along the flux tube at *M*<sup>A</sup> = 0.

same input parameters as in the case of kink waves. The main difference is that the ck-labelled green dispersion curve is replaced by a curve corresponding to the Alfvén wave inside the jet. We note that the dispersion curve in Fig. 13 corresponding to a normalized phase velocity 0.25

that can be derived from the dispersion equation. As in the case of kink waves, the dispersion

dispersionless curves collectively labelled cTi (in the same colours, orange and cyan, as in

curve corresponding to the higher speed has the label v<sup>h</sup>

Fig. 6) with normalized wave phase velocities equal to 1.126 and 1.374.

Ai because it can be considered as the one dispersion curve of the (1.25 ∓ 1)-curves

Ai. Here we also get the two almost

to 8.87, 8.9, 8.95, and 9, respectively.

**4.2 Sausage waves in spicules**

is labelled v<sup>l</sup>

.

and the discriminant *D* is

$$D = M\_\text{A}^2 B^2 - (\eta A - B) \left[ \left( 1 - M\_\text{A}^2 \right) B - A b^2 \right]$$

Obviously, if *D* 0, then

$$\operatorname{Re}(V\_{\rm ph}) = \frac{-M\_{\rm A}B \pm \sqrt{D}}{\eta A - B}, \qquad \operatorname{Im}(V\_{\rm ph}) = 0,$$

else

$$\operatorname{Re}(V\_{\rm ph}) = -\frac{M\_{\rm A}B}{\eta A - B}, \qquad \operatorname{Im}(V\_{\rm ph}) = \frac{\sqrt{D}}{\eta A - B}.$$

We note that our choice of the sign of <sup>√</sup>*<sup>D</sup>* in the expression of Im(*V*ph) is plus although, in principal, it might also be minus – in that case, due to the arising instability, the wave's energy is transferred to the jet.

It is interesting to note that for our jet with *b* = 0.35 and *η* = 0.02 the quadratic dispersion Eq. (38) yields a critical Alfvén–Mach number for the onset of a Kelvin–Helmholtz instability equal to 8.87, which is lower than its magnitude obtained from Eq. (37). With this new critical Alfvén–Mach number, the required jet speed for the instability onset is ∼=710 km s−1. The most astonishing result, however, is the observation that the dispersion curves and the corresponding growth rates, when kink waves become unstable, – look at Figs. 10 and 11 – are very similar to those shown in Figs. 8 and 9. It is worth mentioning that for the

Fig. 10. Dispersion curves of kink waves derived from Eq. (38) for relatively large values of *M*A.

same *η* = 0.02, but for *b* = 1 (equal background magnetic fields), the quadratic equation yields a much higher critical Alfvén–Mach number (=11.09), which means that the critical jet speed grows up to 887 km s−1. This consideration shows that both the density contrast, *η*, and the ratio of the constant magnetic fields, *b*, are equally important in determining the critical Alfvén–Mach number. Moreover, since Eq. (37) and its simplified form as quadratic Eq. (38) yield almost similar results (both for dispersion curves and growth rates when kink waves become unstable) firmly corroborates the correctness of the numerical solutions to the complex dispersion Eq. (37).

Fig. 11. Growth rates of unstable kink waves calculated from Eq. (38) at values of *M*<sup>A</sup> equal to 8.87, 8.9, 8.95, and 9, respectively.

#### **4.2 Sausage waves in spicules**

18 Will-be-set-by-IN-TECH

*<sup>η</sup><sup>A</sup>* <sup>−</sup> *<sup>B</sup>*, Im(*V*ph) =

We note that our choice of the sign of <sup>√</sup>*<sup>D</sup>* in the expression of Im(*V*ph) is plus although, in principal, it might also be minus – in that case, due to the arising instability, the wave's energy

It is interesting to note that for our jet with *b* = 0.35 and *η* = 0.02 the quadratic dispersion Eq. (38) yields a critical Alfvén–Mach number for the onset of a Kelvin–Helmholtz instability equal to 8.87, which is lower than its magnitude obtained from Eq. (37). With this new critical Alfvén–Mach number, the required jet speed for the instability onset is ∼=710 km s−1. The most astonishing result, however, is the observation that the dispersion curves and the corresponding growth rates, when kink waves become unstable, – look at Figs. 10 and 11 – are very similar to those shown in Figs. 8 and 9. It is worth mentioning that for the

Fig. 10. Dispersion curves of kink waves derived from Eq. (38) for relatively large values of

same *η* = 0.02, but for *b* = 1 (equal background magnetic fields), the quadratic equation yields a much higher critical Alfvén–Mach number (=11.09), which means that the critical jet speed grows up to 887 km s−1. This consideration shows that both the density contrast, *η*, and the ratio of the constant magnetic fields, *b*, are equally important in determining the critical Alfvén–Mach number. Moreover, since Eq. (37) and its simplified form as quadratic Eq. (38) yield almost similar results (both for dispersion curves and growth rates when kink waves become unstable) firmly corroborates the correctness of the numerical solutions to the

<sup>1</sup> <sup>−</sup> *<sup>M</sup>*<sup>2</sup> A 

*<sup>η</sup><sup>A</sup>* <sup>−</sup> *<sup>B</sup>* , Im(*V*ph) = 0,

*<sup>B</sup>* <sup>−</sup> *Ab*<sup>2</sup>

<sup>√</sup>*<sup>D</sup> <sup>η</sup><sup>A</sup>* <sup>−</sup> *<sup>B</sup>*.

 .

<sup>A</sup>*B*<sup>2</sup> <sup>−</sup> (*η<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*)

Re(*V*ph) = <sup>−</sup>*M*A*<sup>B</sup>* <sup>±</sup> <sup>√</sup>*<sup>D</sup>*

Re(*V*ph) = <sup>−</sup> *<sup>M</sup>*A*<sup>B</sup>*

and the discriminant *D* is

Obviously, if *D* 0, then

is transferred to the jet.

else

*M*A.

complex dispersion Eq. (37).

*D* = *M*<sup>2</sup>

The dispersion curves of sausage waves both in a static and in a flowing plasma shown in Figs. 12 and 13 are very similar to those of kink waves (compare with Figs. 5 and 6). The latter curves were calculated from dispersion Eq. (34) with azimuthal mode number *m* = 0 for the

Fig. 12. Dispersion curves of sausage waves propagating along the flux tube at *M*<sup>A</sup> = 0.

same input parameters as in the case of kink waves. The main difference is that the ck-labelled green dispersion curve is replaced by a curve corresponding to the Alfvén wave inside the jet. We note that the dispersion curve in Fig. 13 corresponding to a normalized phase velocity 0.25 is labelled v<sup>l</sup> Ai because it can be considered as the one dispersion curve of the (1.25 ∓ 1)-curves that can be derived from the dispersion equation. As in the case of kink waves, the dispersion curve corresponding to the higher speed has the label v<sup>h</sup> Ai. Here we also get the two almost dispersionless curves collectively labelled cTi (in the same colours, orange and cyan, as in Fig. 6) with normalized wave phase velocities equal to 1.126 and 1.374.

the (*M*<sup>A</sup> + 1)-value) for *M*<sup>A</sup> = 0 coincides with the lower-speed dispersion curve (i.e., that associated with the (*M*<sup>A</sup> − 1)-value) for *M*<sup>A</sup> = 2. In contrast to the kink waves, which for *M*<sup>A</sup> 4 are pure surface modes, the sausage waves can be both pseudosurface and pure surface modes, or one of the pair can be a surface mode while the other is a pseudosurface one. For example, all dispersion curves for *M*<sup>A</sup> = 0 and 8 correspond to the pseudosurface waves while the curves' pair associated with *M*<sup>A</sup> = 4 describes the dispersion properties of pure surface waves. For the other Alfvén–Mach numbers, one of the wave is a pseudosurface and the other is a pure surface. However, there is a 'rule': if, for instance, the higher-speed wave with *M*<sup>A</sup> = 10 is a pseudosurface mode, the lower-speed wave for *M*<sup>A</sup> = 12 is a pure surface wave. We finish the discussion of sausage waves with the following conclusion: with increasing the Alfvén–Mach number *M*<sup>A</sup> the initially independent high-harmonic waves and their mirroring counterparts begin to merge – this is clearly seen in Fig. 14 for *M*<sup>A</sup> = 12 – the resulting dispersion curve is in red colour. A similar dispersion curve can be obtained, for example, for *M*<sup>A</sup> = 10; the merging point of the corresponding two high-harmonic dispersion curves moves, however, to the right – it lies at *kz a* = 1.943. It is also evident that in the long wavelength limit the bottom part of the red-coloured dispersion curve describes a backward propagating sausage pseudosurface wave. Another peculiarity of the same dispersion curve is the circumstance that for the range of dimensionless wave numbers between 0.7 and 1.23, one can have two different wave phase velocities. Which one is detected, the theory cannot

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 155

**5. Dispersion diagrams of MHD surface waves in soft X-ray jets**

The geometry model of solar X-ray jets is the same as for the spicules – straight cylinder with radius *a*. Before starting the numerical calculations, we have to specify, as before, the input parameters. The sound and Alfvén speed that are typical for X-ray jets and their environment are correspondingly *c*si = 200 km s−1, *v*Ai = 800 km s−1, *c*se = 120 km s−1, and *v*Ae = 2300 km s−1. With these speeds the density contrast is *η* = 0.13. The same *η* (calculated from a slightly different set of sound and Alfvén speeds) Vasheghani Farahani et al. (Vasheghani Farahani et al., 2009) used in studying the propagation of transfer waves in soft X-ray coronal jets. Their analysis, however, is restricted to the long-wavelength limit, |*k*|*a* � 1 in their notation, while our approach considers the solving the exact dispersion relation without any limitations for the wavelength – such a treating is necessary bearing in mind that the wavelengths of the propagating along the jets fast magnetoacoustic waves might be of the order of X-ray jets radii. We remember that the soft X-ray coronal jets are much ticker than the

With our choice of sound and Alfvén speeds, the tube velocities in both media (look at Eq. (32)), respectively, are *c*Ti ∼= 194 km s−<sup>1</sup> and *c*Te = 119.8 km s−1. The kink speed (see Eq. (35)) turns out to be rather high, namely ∼=1078 km s−1. To compare our result of the critical jet speed for triggering the Kelvin–Helmholtz instability with that found by Vasheghani Farahani et al. (Vasheghani Farahani et al., 2009), we take the same jet speed as theirs, notably *U* = 580 km s−1, which yields Alfvén–Mach number equal to 0.725. (For simplicity we assume that the ambient medium is static, i.e., *U*<sup>e</sup> = 0.) Thus, our input parameters for the numerical

<sup>i</sup> ∼= 0.06, *β*¯e ∼= 0.003, *b* = 1.035, and *M*<sup>A</sup> = 0.725.

predict.

Type II spicules.

computations are

*η* = 0.13, *β*¯

Fig. 13. Dispersion curves of sausage waves propagating along the flux tube at *M*<sup>A</sup> = 1.25.

When examining the stability properties of sausage waves as a function of the Alfvén–Mach number, *M*A, we use the same Eq. (37) while changing the order of the modified Bessel functions from 1 to 0. As in the case of kink waves, we are interested primarily in the behaviour of the waves whose phase velocities are multiples of the Alfvén speed. The results of numerical calculations of the complex dispersion equation are shown in Fig. 14. It turns out that for all reasonable Alfvénic Mach numbers the waves are stable. This is unsurprising

Fig. 14. Dispersion curves of sausage waves propagating along the flux tube at various values of *M*A.

because the same conclusion was drawn by solving precisely the complex dispersion equation governing the propagation of sausage waves in incompressible flowing cylindrical plasmas (Zhelyazkov, 2010; 2011). In Fig. 14 almost all dispersion curves have two labels: one for the (M<sup>A</sup> − 1)-labelled curve at given *M*<sup>A</sup> (the label is below the curve), and second for the (M� <sup>A</sup> + 1)-labelled curve associated with the corresponding (*M*� <sup>A</sup> = *M*<sup>A</sup> − 2)-value (the label is above the curve). This labelling is quite complex because for all *M*<sup>A</sup> we find dispersion curves that overlap: for instance, the higher-speed dispersion curve (i.e., that associated with 20 Will-be-set-by-IN-TECH

Fig. 13. Dispersion curves of sausage waves propagating along the flux tube at *M*<sup>A</sup> = 1.25.

Fig. 14. Dispersion curves of sausage waves propagating along the flux tube at various

<sup>A</sup> + 1)-labelled curve associated with the corresponding (*M*�

because the same conclusion was drawn by solving precisely the complex dispersion equation governing the propagation of sausage waves in incompressible flowing cylindrical plasmas (Zhelyazkov, 2010; 2011). In Fig. 14 almost all dispersion curves have two labels: one for the (M<sup>A</sup> − 1)-labelled curve at given *M*<sup>A</sup> (the label is below the curve), and second for the

is above the curve). This labelling is quite complex because for all *M*<sup>A</sup> we find dispersion curves that overlap: for instance, the higher-speed dispersion curve (i.e., that associated with

<sup>A</sup> = *M*<sup>A</sup> − 2)-value (the label

values of *M*A.

(M�

When examining the stability properties of sausage waves as a function of the Alfvén–Mach number, *M*A, we use the same Eq. (37) while changing the order of the modified Bessel functions from 1 to 0. As in the case of kink waves, we are interested primarily in the behaviour of the waves whose phase velocities are multiples of the Alfvén speed. The results of numerical calculations of the complex dispersion equation are shown in Fig. 14. It turns out that for all reasonable Alfvénic Mach numbers the waves are stable. This is unsurprising the (*M*<sup>A</sup> + 1)-value) for *M*<sup>A</sup> = 0 coincides with the lower-speed dispersion curve (i.e., that associated with the (*M*<sup>A</sup> − 1)-value) for *M*<sup>A</sup> = 2. In contrast to the kink waves, which for *M*<sup>A</sup> 4 are pure surface modes, the sausage waves can be both pseudosurface and pure surface modes, or one of the pair can be a surface mode while the other is a pseudosurface one. For example, all dispersion curves for *M*<sup>A</sup> = 0 and 8 correspond to the pseudosurface waves while the curves' pair associated with *M*<sup>A</sup> = 4 describes the dispersion properties of pure surface waves. For the other Alfvén–Mach numbers, one of the wave is a pseudosurface and the other is a pure surface. However, there is a 'rule': if, for instance, the higher-speed wave with *M*<sup>A</sup> = 10 is a pseudosurface mode, the lower-speed wave for *M*<sup>A</sup> = 12 is a pure surface wave. We finish the discussion of sausage waves with the following conclusion: with increasing the Alfvén–Mach number *M*<sup>A</sup> the initially independent high-harmonic waves and their mirroring counterparts begin to merge – this is clearly seen in Fig. 14 for *M*<sup>A</sup> = 12 – the resulting dispersion curve is in red colour. A similar dispersion curve can be obtained, for example, for *M*<sup>A</sup> = 10; the merging point of the corresponding two high-harmonic dispersion curves moves, however, to the right – it lies at *kz a* = 1.943. It is also evident that in the long wavelength limit the bottom part of the red-coloured dispersion curve describes a backward propagating sausage pseudosurface wave. Another peculiarity of the same dispersion curve is the circumstance that for the range of dimensionless wave numbers between 0.7 and 1.23, one can have two different wave phase velocities. Which one is detected, the theory cannot predict.
