**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 Type II spicules.

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 computations are

$$
\eta = 0.13, \quad \vec{\rho}\_{\text{i}} \cong 0.06, \quad \vec{\rho}\_{\text{e}} \cong 0.003, \quad b = 1.035, \quad \text{and} \quad M\_{\text{A}} = 0.725.
$$

Fig. 16. Dispersion curves of kink waves propagating along a flux tube modelling X-ray jet at

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 157

this case, the simplified dispersion equation of kink waves (in complex variables!) takes the

*V*ph − *M*<sup>A</sup>

<sup>1</sup> <sup>−</sup> *<sup>V</sup>*<sup>2</sup> ph*η K*� <sup>1</sup>(*κ*e*a*)

We numerically solve this equation by varying the magnitude of the Alfvén–Mach number, *M*A, using as before the Müller method and the dispersion curves of both stable and unstable kink waves are shown in Fig. 17. In this figure, we display only the most interesting, upper, part of the dispersion diagram, where one can observe the changes in the shape of the dispersion curves related to the corresponding *c*k-speeds. First and foremost, the shape of the merging dispersion curves (labelled 4, 4.1, 4.2, and 4.23 in Fig. 17) is distinctly different from that of the similar curves in Fig. 8. Here, the curves, which are close to the dispersion curves corresponding to an unstable wave propagation (the first one is with label 4.25) are *semi-closed* in contrast to the closed curves in Fig. 8. The wave growth rates corresponding to Alfvén–Mach numbers 4.25, 4.3, 4.35, and 4.4 are shown in Fig. 18. As can be seen, the shape of those curves is completely different to that of the wave growth rates shown in Figs. 9 and 11. We note that all dispersion curves for *M*<sup>A</sup> 4 correspond to pure surface kink waves. It is clear from Figs. 17 and 18 that the critical Alfvén–Mach number, which determines the onset of a Kelvin–Helmholtz instability of the kink waves, is equal to 4.25 – the corresponding flow speed is 3400 km s−1, that is much higher than the value we have used for calculating the dispersion curves in Fig. 16. The critical Alfvén–Mach number evaluated by Vasheghani Farahani et al. (Vasheghani Farahani et al., 2009), is 4.47, that means the corresponding flow speed must be at least 3576 km s−1. If we use our Eq. (39) with the same *ρ*e/*ρ*<sup>i</sup> = 0.13, but

*K* and *κ*e*a* =

2 1/2 *I*�

<sup>1</sup>(*κ*i*a*) *I*1(*κ*i*a*)

 <sup>1</sup> <sup>−</sup> *<sup>V</sup>*<sup>2</sup> ph*η* 1/2 *K*.

*<sup>K</sup>*1(*κ*e*a*) <sup>=</sup> 0, (39)

 1 − 

2 − 1 

2 1/2

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

*V*ph − *M*<sup>A</sup>

*V*ph − *M*<sup>A</sup>

− 

*κ*i*a* = 1 − 

*M*<sup>A</sup> = 0.725.

form

where

We note that *b* = 1.035 means that the equilibrium magnetic fields inside and outside the X-ray coronal jet are almost identical. Moreover, due to the relatively small plasma betas, *β*<sup>e</sup> = 0.0033 and *β*<sup>i</sup> = 0.075, respectively, the magnetic pressure dominates the gas one in both media and the propagating waves along X-ray jets should accordingly be predominantly transverse.

#### **5.1 Kink waves in soft X-ray coronal jets**

The dispersion diagrams of kink waves propagating along a static-plasma (*U* = 0) flux tube are shown in Fig. 15. They, the dispersion curves, have been obtained by numerically finding

Fig. 15. Dispersion curves of kink waves propagating along a flux tube modelling X-ray jet at *M*<sup>A</sup> = 0.

the solutions to dispersion Eq. (34) with mode number *m* = 1 and input data listed in the introductory part of this section with *M*<sup>A</sup> = 0. The dispersion curves are very similar to those for spicules (look at Fig. 5). Here, there is, however, one distinctive difference: the cTe-labelled dispersion curve (blue color) lies below the curve corresponding to the tube velocity inside the jet (magenta coloured line labelled cTi). The dispersion curves of the high-harmonic super-Alfvénic waves (red colour) lye, as usual, above the green curve associated with the kink speed. What actually does the flow change when is taken into account? The answer to this question is given in Fig. 16. As in the case with spicules, the flow duplicates the cTi-labelled dispersion curve in Fig. 15. The two, again collectively labelled cTi dispersion curves, are sub-Alfvénic waves having normalized phase velocities equal to 0.482 and 0.968 in correspondence to the (*M*<sup>A</sup> <sup>∓</sup> *<sup>c</sup>*<sup>0</sup> Ti)-rule. All the rest curves have the same behaviour and notation as in Fig. 6. The only difference here is the circumstance that the lower-speed *c*k-curve lies below the zero line, i.e., it describes a *backward* propagating kink pseudosurface wave. This is because the Alfvén–Mach number now is less than one. We note also that the cTe-labelled dispersion curve is unaffected by the presence of flow.

The most intriguing question is whether the ck-labelled wave can become unstable at any reasonable flow velocity. Before answering that question, we have, as before, to simplify dispersion Eq. (34). Since the two plasma betas, as we have already mentioned, are much less that one, we can treat both media (the X-ray jet and its environment) as cool plasmas. In

Fig. 16. Dispersion curves of kink waves propagating along a flux tube modelling X-ray jet at *M*<sup>A</sup> = 0.725.

this case, the simplified dispersion equation of kink waves (in complex variables!) takes the form

$$
\left(V\_{\rm ph}^2 \eta - b^2\right) \left[1 - \left(V\_{\rm ph} - M\_{\rm A}\right)^2\right]^{1/2} \frac{I\_1'(\kappa\_{\rm i} a)}{I\_1(\kappa\_{\rm i} a)}
$$

$$
$$

where

22 Will-be-set-by-IN-TECH

We note that *b* = 1.035 means that the equilibrium magnetic fields inside and outside the X-ray coronal jet are almost identical. Moreover, due to the relatively small plasma betas, *β*<sup>e</sup> = 0.0033 and *β*<sup>i</sup> = 0.075, respectively, the magnetic pressure dominates the gas one in both media and the propagating waves along X-ray jets should accordingly be predominantly

The dispersion diagrams of kink waves propagating along a static-plasma (*U* = 0) flux tube are shown in Fig. 15. They, the dispersion curves, have been obtained by numerically finding

Fig. 15. Dispersion curves of kink waves propagating along a flux tube modelling X-ray jet at

the solutions to dispersion Eq. (34) with mode number *m* = 1 and input data listed in the introductory part of this section with *M*<sup>A</sup> = 0. The dispersion curves are very similar to those for spicules (look at Fig. 5). Here, there is, however, one distinctive difference: the cTe-labelled dispersion curve (blue color) lies below the curve corresponding to the tube velocity inside the jet (magenta coloured line labelled cTi). The dispersion curves of the high-harmonic super-Alfvénic waves (red colour) lye, as usual, above the green curve associated with the kink speed. What actually does the flow change when is taken into account? The answer to this question is given in Fig. 16. As in the case with spicules, the flow duplicates the cTi-labelled dispersion curve in Fig. 15. The two, again collectively labelled cTi dispersion curves, are sub-Alfvénic waves having normalized phase velocities equal to 0.482 and 0.968

notation as in Fig. 6. The only difference here is the circumstance that the lower-speed *c*k-curve lies below the zero line, i.e., it describes a *backward* propagating kink pseudosurface wave. This is because the Alfvén–Mach number now is less than one. We note also that the

The most intriguing question is whether the ck-labelled wave can become unstable at any reasonable flow velocity. Before answering that question, we have, as before, to simplify dispersion Eq. (34). Since the two plasma betas, as we have already mentioned, are much less that one, we can treat both media (the X-ray jet and its environment) as cool plasmas. In

cTe-labelled dispersion curve is unaffected by the presence of flow.

Ti)-rule. All the rest curves have the same behaviour and

transverse.

*M*<sup>A</sup> = 0.

in correspondence to the (*M*<sup>A</sup> <sup>∓</sup> *<sup>c</sup>*<sup>0</sup>

**5.1 Kink waves in soft X-ray coronal jets**

$$\kappa\_{\rm i}a = \left[1 - \left(V\_{\rm ph} - M\_{\rm A}\right)^2\right]^{1/2} \,\mathrm{K} \qquad \text{and} \qquad \kappa\_{\rm e}a = \left(1 - V\_{\rm ph}^2 \eta\right)^{1/2} \,\mathrm{K}.$$

We numerically solve this equation by varying the magnitude of the Alfvén–Mach number, *M*A, using as before the Müller method and the dispersion curves of both stable and unstable kink waves are shown in Fig. 17. In this figure, we display only the most interesting, upper, part of the dispersion diagram, where one can observe the changes in the shape of the dispersion curves related to the corresponding *c*k-speeds. First and foremost, the shape of the merging dispersion curves (labelled 4, 4.1, 4.2, and 4.23 in Fig. 17) is distinctly different from that of the similar curves in Fig. 8. Here, the curves, which are close to the dispersion curves corresponding to an unstable wave propagation (the first one is with label 4.25) are *semi-closed* in contrast to the closed curves in Fig. 8. The wave growth rates corresponding to Alfvén–Mach numbers 4.25, 4.3, 4.35, and 4.4 are shown in Fig. 18. As can be seen, the shape of those curves is completely different to that of the wave growth rates shown in Figs. 9 and 11. We note that all dispersion curves for *M*<sup>A</sup> 4 correspond to pure surface kink waves.

It is clear from Figs. 17 and 18 that the critical Alfvén–Mach number, which determines the onset of a Kelvin–Helmholtz instability of the kink waves, is equal to 4.25 – the corresponding flow speed is 3400 km s−1, that is much higher than the value we have used for calculating the dispersion curves in Fig. 16. The critical Alfvén–Mach number evaluated by Vasheghani Farahani et al. (Vasheghani Farahani et al., 2009), is 4.47, that means the corresponding flow speed must be at least 3576 km s−1. If we use our Eq. (39) with the same *ρ*e/*ρ*<sup>i</sup> = 0.13, but

trigger the onset of a Kelvin–Helmholtz instability of the kink surface waves running along

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 159

The dispersion diagram of sausage waves in a static-plasma flux tube should be more or less similar to that of the kink waves under the same circumstances. Here, however, the green curve in Fig. 15, associated with the kink speed *c*k, is now replaced by a dispersionless line

Fig. 19. Dispersion curves of sausage waves propagating along a flux tube modelling X-ray

Fig. 20. Dispersion curves of sausage waves propagating along a flux tube modelling X-ray

related to the Alfvén speed – see the green curve in Fig. 19. Another difference is the number of the red-coloured high-harmonic super-Alfvénic waves – here it is 3 against 2 in Fig. 15. The dispersion diagram of the same mode in a flow with *M*<sup>A</sup> = 0.725 (*U* = 580 km s−1) is also predictable – the presence of the flow is the reason for splitting the green vAi-labelled curve

the jets.

jet at *M*<sup>A</sup> = 0.

jet at *M*<sup>A</sup> = 0.725.

**5.2 Sausage waves in soft X-ray coronal jets**

Fig. 17. Dispersion curves of kink waves propagating along a flux tube modelling X-ray jets for relatively large values of *M*A.

Fig. 18. Growth rates of unstable kink waves propagating along a flux tube modelling X-ray jets at values of *M*<sup>A</sup> equal to 4.25, 4.3, 4.35, and 4.4, respectively.

with a little bit higher *B*e/*B*<sup>i</sup> = 1.1132, we get that the critical jet speed for triggering the Kelvin–Helmholtz instability in a soft X-ray coronal get would be 4.41*v*Ai = 3528 km s−1. It is necessary, however, to point out that the correct density contrast that can be calculated from Eq. (36) with *c*si = 360 km s−1, *v*Ai = 800 km s−1, *c*se = 120 km s−1, and *v*Ae = 2400 km s−<sup>1</sup> (the basic speeds in Vasheghani Farahani et al. paper) is *ρ*e/*ρ*<sup>i</sup> = 0.137698, which is closer to 0.14 rather than to 0.13. The solving Eq. (39) with the exact value of the density contrast (=0.1377) and the same *B*e/*B*<sup>i</sup> as before (=1.1132) yields a critical flow speed equal to 4.31*v*Ai = 3448 km s−1. All these calculations show that even small variations in the two ratios *ρ*e/*ρ*<sup>i</sup> and *B*e/*B*<sup>i</sup> lead to visibly different critical Alfvén–Mach numbers – our choice of the sound and Alfvén speeds gives the smallest value of the critical *M*A. According to the more recent observations (Madjarska, 2011; Shimojo & Shibata, 2000), the soft X-ray coronal jets can have velocities above 10<sup>3</sup> km s−<sup>1</sup> and it remains to be seen whether a speed of 3400 km s−<sup>1</sup> can trigger the onset of a Kelvin–Helmholtz instability of the kink surface waves running along the jets.

#### **5.2 Sausage waves in soft X-ray coronal jets**

24 Will-be-set-by-IN-TECH

Fig. 17. Dispersion curves of kink waves propagating along a flux tube modelling X-ray jets

Fig. 18. Growth rates of unstable kink waves propagating along a flux tube modelling X-ray

with a little bit higher *B*e/*B*<sup>i</sup> = 1.1132, we get that the critical jet speed for triggering the Kelvin–Helmholtz instability in a soft X-ray coronal get would be 4.41*v*Ai = 3528 km s−1. It is necessary, however, to point out that the correct density contrast that can be calculated from Eq. (36) with *c*si = 360 km s−1, *v*Ai = 800 km s−1, *c*se = 120 km s−1, and *v*Ae = 2400 km s−<sup>1</sup> (the basic speeds in Vasheghani Farahani et al. paper) is *ρ*e/*ρ*<sup>i</sup> = 0.137698, which is closer to 0.14 rather than to 0.13. The solving Eq. (39) with the exact value of the density contrast (=0.1377) and the same *B*e/*B*<sup>i</sup> as before (=1.1132) yields a critical flow speed equal to 4.31*v*Ai = 3448 km s−1. All these calculations show that even small variations in the two ratios *ρ*e/*ρ*<sup>i</sup> and *B*e/*B*<sup>i</sup> lead to visibly different critical Alfvén–Mach numbers – our choice of the sound and Alfvén speeds gives the smallest value of the critical *M*A. According to the more recent observations (Madjarska, 2011; Shimojo & Shibata, 2000), the soft X-ray coronal jets can have velocities above 10<sup>3</sup> km s−<sup>1</sup> and it remains to be seen whether a speed of 3400 km s−<sup>1</sup> can

jets at values of *M*<sup>A</sup> equal to 4.25, 4.3, 4.35, and 4.4, respectively.

for relatively large values of *M*A.

The dispersion diagram of sausage waves in a static-plasma flux tube should be more or less similar to that of the kink waves under the same circumstances. Here, however, the green curve in Fig. 15, associated with the kink speed *c*k, is now replaced by a dispersionless line

Fig. 19. Dispersion curves of sausage waves propagating along a flux tube modelling X-ray jet at *M*<sup>A</sup> = 0.

Fig. 20. Dispersion curves of sausage waves propagating along a flux tube modelling X-ray jet at *M*<sup>A</sup> = 0.725.

related to the Alfvén speed – see the green curve in Fig. 19. Another difference is the number of the red-coloured high-harmonic super-Alfvénic waves – here it is 3 against 2 in Fig. 15. The dispersion diagram of the same mode in a flow with *M*<sup>A</sup> = 0.725 (*U* = 580 km s−1) is also predictable – the presence of the flow is the reason for splitting the green vAi-labelled curve

Fig. 22. Zoomed part of the dispersion diagram in Fig. 21 where two dispersion curves of

their shapes and one may occur to observe the merging of, for instance, the first curves of each family as this has been shown in Fig. 14 (see the red curve there). Here, however, the situation is über-complicated – instead of merging we encounter a new phenomenon, notably the *touching* of two dispersion curves – see the green and red curves in Fig. 21 (calculated for *M*<sup>A</sup> = 5), and with more details in Fig. 22. The tip of the horizontal spike lies at *kz a* = 1.395. Another peculiarity of this complex curve is the inverted-s shape of the red curve between the dimensionless wave numbers 1.28 and 1.29. Across that range, at a fixed *kz a*, one can 'detect' four different normalized wave phase velocities. Similar sophisticated dispersion curves might also be obtained for *M*<sup>A</sup> = 4 or *M*<sup>A</sup> = 6. Maybe nowadays the sausage mode is not too interesting for the spacecrafts' observers but, who knows, it can sometime become

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 161

We now summarize the main findings of our chapter. We have studied the dispersion properties and the stability of the MHD normal modes running along the length of Type II spicules and soft X-ray coronal jets. Both have been modelled as straight cylindrical jets of ideal cool plasma surrounded by a warm/hot fully ionized medium (for spicules) or as flux tubes of almost cool plasma surrounded by a cool medium (for the X-ray jets). The wave propagation has been investigated in the context of standard magnetohydrodynamics by using linearized equations for the perturbations of the basic quantities: mass density, pressure, fluid velocity, and wave magnetic field. The derived dispersion equations describe the well-known kink and sausage mode influenced by the presence of spicules' or X-ray jets' moving plasma. The streaming plasma is characterized by its velocity **U**, which is directed along the background magnetic fields **B**<sup>i</sup> and **B**<sup>e</sup> inside the jet and in its environment. An alternative and more convenient way of specifying the jet is by defining the Alfvén–Mach number: the ratio of jet speed to the Alfvén speed inside the jet, *M*<sup>A</sup> = *U*/*v*Ai. The key parameters controlling the dispersion properties of the waves are the so-called density contrast, *η* = *ρ*e/*ρ*i, the ratio of the two background magnetic fields, *b* = *B*e/*B*i, and the two

se/*v*<sup>2</sup>

Ae and *<sup>β</sup>*¯

<sup>i</sup> = *c*<sup>2</sup> si/*v*<sup>2</sup>

Ai. How does the

super-Alfvénic sausage waves (at *M*<sup>A</sup> = 5) are touching each other.

important in interpretation observational data.

ratios of the squared sound and Alfvén speeds, *β*¯e = *c*<sup>2</sup>

**6. Conclusion**

in Fig. 19 into two sister curves labelled, respectively, v<sup>l</sup> Ai and <sup>v</sup><sup>h</sup> Ai – look at Fig. 20. Observe that the normalized speeds of those two waves are, as expected, equal to *M*<sup>A</sup> ∓ 1 – in our case the lower-speed Alfvén wave is a backward propagating one. The two sub-Alfvénic waves, whose dispersion curves are in orange and cyan colours and collectively labelled cTi have practically the same normalized phase velocities as the corresponding curves in Fig. 16. We note that one of the aforementioned curve is slightly decreasing (the orange curve) whilst the other, cyan-coloured, curve is slightly increasing when the normalized wave number *kz a* becomes larger – the same holds for the analogous waves in spicules. One can see in Fig. 20 a symmetry between the upper and bottom parts of the dispersion diagram – the 'mirror line' lies somewhere between the orange and cyan dispersion curves.

The 'evolution' of the green vAi-labelled curves in Fig. 20 with the increase in the Alfvén–Mach number is illustrated in Fig. 21. It is unsurprising that the sausage surface waves in soft X-ray

Fig. 21. Dispersion curves of sausage waves propagating along a flux tube modelling X-ray jets at various values of *M*A.

coronal jets (like in spicules) are unaffected by the Kelvin–Helmholtz instability. Similarly as in Fig. 14, we have an overlapping of the dispersion curves associated with different Alfvén–Mach numbers. The labelling of dispersion curves in Fig. 21 is according to the previously discussed in Sec. 4.2 rule, namely each horizontal dispersion curve possesses two labels: one for the (M<sup>A</sup> − 1)-curve at given *M*<sup>A</sup> (the label is below the curve), and second for the (M� <sup>A</sup> + 1)-curve associated with the corresponding (*M*� <sup>A</sup> = *M*<sup>A</sup> − 2)-value (the label is above the curve). Interestingly, even for the relatively low *M*<sup>A</sup> = 1 both the lower- and the high-speed curves describe pure surface sausage waves. The same is also valid for the dispersion lines corresponding to Alfvén–Mach numbers equal to 4 and 5. The lower-speed Alfvénic curve at *M*<sup>A</sup> = 6 is related to a pseudosurface sausage wave while the higher-speed one (with normalized phase velocity equal to 7) corresponds to a pure surface mode. (With *M*<sup>A</sup> = 2 we have just the opposite situation.) At *M*<sup>A</sup> 7 all waves are pseudosurface ones. Each choice of the Alfvén–Mach number indeed requires separate studying of the wave's proper mode. Apart from Alfvénic waves and the pair of sub-Alfvénic modes (orange and cyan curves in Fig. 20), there appear to be families of high-speed harmonic waves (with red and blue colours of their dispersion curves), which also change with the increase of *M*A. Initially being independent, with the growing of the Alfvén–Mach number, they change

Fig. 22. Zoomed part of the dispersion diagram in Fig. 21 where two dispersion curves of super-Alfvénic sausage waves (at *M*<sup>A</sup> = 5) are touching each other.

their shapes and one may occur to observe the merging of, for instance, the first curves of each family as this has been shown in Fig. 14 (see the red curve there). Here, however, the situation is über-complicated – instead of merging we encounter a new phenomenon, notably the *touching* of two dispersion curves – see the green and red curves in Fig. 21 (calculated for *M*<sup>A</sup> = 5), and with more details in Fig. 22. The tip of the horizontal spike lies at *kz a* = 1.395. Another peculiarity of this complex curve is the inverted-s shape of the red curve between the dimensionless wave numbers 1.28 and 1.29. Across that range, at a fixed *kz a*, one can 'detect' four different normalized wave phase velocities. Similar sophisticated dispersion curves might also be obtained for *M*<sup>A</sup> = 4 or *M*<sup>A</sup> = 6. Maybe nowadays the sausage mode is not too interesting for the spacecrafts' observers but, who knows, it can sometime become important in interpretation observational data.

#### **6. Conclusion**

26 Will-be-set-by-IN-TECH

that the normalized speeds of those two waves are, as expected, equal to *M*<sup>A</sup> ∓ 1 – in our case the lower-speed Alfvén wave is a backward propagating one. The two sub-Alfvénic waves, whose dispersion curves are in orange and cyan colours and collectively labelled cTi have practically the same normalized phase velocities as the corresponding curves in Fig. 16. We note that one of the aforementioned curve is slightly decreasing (the orange curve) whilst the other, cyan-coloured, curve is slightly increasing when the normalized wave number *kz a* becomes larger – the same holds for the analogous waves in spicules. One can see in Fig. 20 a symmetry between the upper and bottom parts of the dispersion diagram – the 'mirror line'

The 'evolution' of the green vAi-labelled curves in Fig. 20 with the increase in the Alfvén–Mach number is illustrated in Fig. 21. It is unsurprising that the sausage surface waves in soft X-ray

Fig. 21. Dispersion curves of sausage waves propagating along a flux tube modelling X-ray

coronal jets (like in spicules) are unaffected by the Kelvin–Helmholtz instability. Similarly as in Fig. 14, we have an overlapping of the dispersion curves associated with different Alfvén–Mach numbers. The labelling of dispersion curves in Fig. 21 is according to the previously discussed in Sec. 4.2 rule, namely each horizontal dispersion curve possesses two labels: one for the (M<sup>A</sup> − 1)-curve at given *M*<sup>A</sup> (the label is below the curve), and second

is above the curve). Interestingly, even for the relatively low *M*<sup>A</sup> = 1 both the lower- and the high-speed curves describe pure surface sausage waves. The same is also valid for the dispersion lines corresponding to Alfvén–Mach numbers equal to 4 and 5. The lower-speed Alfvénic curve at *M*<sup>A</sup> = 6 is related to a pseudosurface sausage wave while the higher-speed one (with normalized phase velocity equal to 7) corresponds to a pure surface mode. (With *M*<sup>A</sup> = 2 we have just the opposite situation.) At *M*<sup>A</sup> 7 all waves are pseudosurface ones. Each choice of the Alfvén–Mach number indeed requires separate studying of the wave's proper mode. Apart from Alfvénic waves and the pair of sub-Alfvénic modes (orange and cyan curves in Fig. 20), there appear to be families of high-speed harmonic waves (with red and blue colours of their dispersion curves), which also change with the increase of *M*A. Initially being independent, with the growing of the Alfvén–Mach number, they change

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

Ai and <sup>v</sup><sup>h</sup>

Ai – look at Fig. 20. Observe

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

in Fig. 19 into two sister curves labelled, respectively, v<sup>l</sup>

lies somewhere between the orange and cyan dispersion curves.

jets at various values of *M*A.

for the (M�

We now summarize the main findings of our chapter. We have studied the dispersion properties and the stability of the MHD normal modes running along the length of Type II spicules and soft X-ray coronal jets. Both have been modelled as straight cylindrical jets of ideal cool plasma surrounded by a warm/hot fully ionized medium (for spicules) or as flux tubes of almost cool plasma surrounded by a cool medium (for the X-ray jets). The wave propagation has been investigated in the context of standard magnetohydrodynamics by using linearized equations for the perturbations of the basic quantities: mass density, pressure, fluid velocity, and wave magnetic field. The derived dispersion equations describe the well-known kink and sausage mode influenced by the presence of spicules' or X-ray jets' moving plasma. The streaming plasma is characterized by its velocity **U**, which is directed along the background magnetic fields **B**<sup>i</sup> and **B**<sup>e</sup> inside the jet and in its environment. An alternative and more convenient way of specifying the jet is by defining the Alfvén–Mach number: the ratio of jet speed to the Alfvén speed inside the jet, *M*<sup>A</sup> = *U*/*v*Ai. The key parameters controlling the dispersion properties of the waves are the so-called density contrast, *η* = *ρ*e/*ρ*i, the ratio of the two background magnetic fields, *b* = *B*e/*B*i, and the two ratios of the squared sound and Alfvén speeds, *β*¯e = *c*<sup>2</sup> se/*v*<sup>2</sup> Ae and *<sup>β</sup>*¯ <sup>i</sup> = *c*<sup>2</sup> si/*v*<sup>2</sup> Ai. How does the

For the X-ray jets, the dispersion curves' reorganization, because the environment has been considered as a cool plasma, is different – now, at high enough flow speeds, the merging lower- and higher-speed *c*k-dispersion curves take the form of semi-closed loops (see Fig. 17). As we increase the flow speed (or equivalently the Alfvén–Mach number), the semi-closed loops shrink and at some critical flow speed the kink wave becomes unstable and the instability is of the Kelvin–Helmholtz type. We note that the shapes of the waves' growth rates of kink waves in spicules and soft X-ray coronal jets are distinctly different –

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 163

• We have found that the sausage waves are unaffected by the Kelvin–Helmholtz instability. This conclusion was also previously drawn for the sausage modes in flowing solar-wind

As we have seen, very high jet speeds are required to ensure that the Kelvin–Helmholtz instability occurs for kink waves propagating in Type II spicules associated with a subsequent triggering of Alfvén-wave turbulence, hence the possibility that this mechanism is responsible for chromospheric/coronal heating has to be excluded. However, a twist in the magnetic field of the flux tube or its environment may have the effect of lowering the instability threshold (Bennett et al., 1999; Zaqarashvili et al., 2010) and eventually lead to the triggering of the Kelvin–Helmholtz instability. According to Antolin & Shibata (Antolin & Shibata, 2010), a promising way to ensure spicules'/coronal heating is by means of the mode conversion and parametric decay of Alfvén waves generated by magnetic reconnection or driven by the magneto-convection at the photosphere. However, spicules can be considered as Alfvén wave resonant cavities (Holweg, 1981; Leroy, 1981) and as Matsumoto & Shibata (Matsumoto & Shibata, 2010) claim, the waves of the period around 100–500 s can transport a large amount of wave energy to the corona. Zahariev & Mishonov (Zahariev & Mishonov, 2011) state that the corona may be heated through a self-induced opacity of high-frequency Alfvén waves propagating in the transition region between the chromosphere and the corona owing to a considerable spectral density of the Alfvén waves in the photosphere. Another trend in explaining the mechanism of coronal heating is the dissipation of Alfvén waves' energy by strong wave damping due to the collisions between ions and neutrals (Song & Vasyliunas, 2011; Tsap et al., 2011). In particular, Song & Vasyli ¯ unas, by analytically solving a ¯ self-consistent one-dimensional model of the plasma–neutral–electromagnetic system, show that the damping is extremely strong for weaker magnetic field and less strong for strong field. Under either condition, the high-frequency portion of the source power spectrum is strongly damped at the lower altitudes, depositing heat there, whereas the lower-frequency perturbations are nearly undamped and can be observed in the corona and above when the

The idea that Alfvén waves propagating in the transition region can contribute to the coronal heating was firmly supported by the observational data recorded on April 25, 2010 by NASA's *Solar Dynamics Observatory* (see Fig. 2). As McIntosh et al. (McIntosh et al., 2011) claim, "*SDO* has amazing resolution, so you can actually see individual waves. Previous observations of Alfvénic waves in the corona revealed amplitudes far too small (0.5 km s−1) to supply the energy flux (100–200 W m−2) required to drive the fast solar wind or balance the radiative losses of the quiet corona. Here we report observations of the transition region (between the chromosphere and the corona) and of the corona that reveal how Alfvénic motions permeate the dynamic and finely structured outer solar atmosphere. The ubiquitous outward-propagating Alfvénic motions observed have amplitudes of the order of 20 km s−<sup>1</sup>

compare Figs. 9 and 18.

field is strong.

plasma (Zhelyazkov, 2010; 2011).

jet change the dispersion curves of both modes (kink and sausage waves) in a static-plasma flux tube? The answers to that question are as follows:


A rough criterion for the appearance of the Kelvin–Helmholtz instability of kink waves is the satisfaction of an inequality suggested by Andries & Goossens (Andies & Goossens, 2001), which in our notation reads

$$M\_{\mathbf{A}} > 1 + b/\sqrt{\eta}.$$

This criterion provides more reliable predictions for the critical *M*<sup>A</sup> when *b* ≈ 1 (Zhelyazkov, 2010). In particular, for a X-ray jet with *η* = 0.13 and *b* = 1.035 the above criterion yields *M*<sup>A</sup> > 3.87, which is lower than the numerically found value of 4.25.

• The onset of the Kelvin–Helmholtz instability for kink surface waves running along a cylindrical jet, modelling a Type II spicule, is preceded by a substantial reorganization of wave dispersion curves. As we increase the Alfvén–Mach number, the pairs of highand low-speed curves (look at Fig. 8) begin to merge transforming into closed dispersion curves. After a further increase in *M*A, these closed dispersion curves become smaller – this is an indication that we have reached the critical *M*<sup>A</sup> at which the kink waves are subjected to the Kelvin–Helmholtz instability – the unstable waves propagate across the entire *kz a*-range having growth rates depending upon the value of the current *M*A. We note that this behaviour has been observed for kink waves travelling on flowing solar-wind plasma (Zhelyazkov, 2010; 2011).

28 Will-be-set-by-IN-TECH

jet change the dispersion curves of both modes (kink and sausage waves) in a static-plasma

• The flow shifts upwards the specific dispersion curves, the kink-speed curve for kink waves and Alfvén-speed curve for sausage waves, as well as the high-harmonic fast waves of both modes. The sub-Alfvénic tube speed inside the jet, *c*Ti, belongs to two waves with normalized phase velocities equal to *M*<sup>A</sup> ∓ *c*Ti/*v*Ai. One also observes such a duplication of the *c*k- or *v*Ai-speed curve of kink or sausage waves. Below the lower-speed *c*k- or *v*Ai-curve there appears to be a set of dispersion curves, which are a mirror image of the high-harmonic fast waves. We note that the flow does not affect the *c*Te-speed dispersion

• For a typical set of characteristic sound and Alfvén speeds in both media (the jet and its environment) at relatively small Alfvén–Mach numbers both modes are pseudosurface waves. With increasing *M*A, some of them become pure surface waves. For kink waves,

• The kink waves running along the jet can become unstable when the Alfvén–Mach number, *M*A, exceeds some critical value. That critical value depends upon the two input parameters, *η* and *b*; the increase in the density contrast, *ρ*e/*ρ*i, decreases the magnitude of the critical Alfvén–Mach number, whilst the increase in the background magnetic fields ratio, *B*e/*B*i, leads to an increase in the critical *M*A. For our choice of parameters for Type II spicules (*η* = 0.02 and *b* = 0.35) the value of the critical *M*<sup>A</sup> is 8.9. This means that the speed of the jet must be at least 712 km s−<sup>1</sup> for the onset of the Kelvin–Helmholtz instability of the propagating kink waves. Such high speeds of Type II spicules have not yet been detected. For the soft X-ray coronal jets, due to the greater density contrast (*η* = 0.13) and almost equal background magnetic fields (*b* = 1.035), the critical Alfvén–Mach number is approximately twice smaller (=4.25), but since the jet Alfvén speed is 10 times larger than that of spicules, the critical flow speed, *U*crt, is much higher, namely 3400 km s−1. Such

A rough criterion for the appearance of the Kelvin–Helmholtz instability of kink waves is the satisfaction of an inequality suggested by Andries & Goossens (Andies & Goossens,

*M*<sup>A</sup> > 1 + *b*/

This criterion provides more reliable predictions for the critical *M*<sup>A</sup> when *b* ≈ 1 (Zhelyazkov, 2010). In particular, for a X-ray jet with *η* = 0.13 and *b* = 1.035 the above criterion yields *M*<sup>A</sup> > 3.87, which is lower than the numerically found value of 4.25. • The onset of the Kelvin–Helmholtz instability for kink surface waves running along a cylindrical jet, modelling a Type II spicule, is preceded by a substantial reorganization of wave dispersion curves. As we increase the Alfvén–Mach number, the pairs of highand low-speed curves (look at Fig. 8) begin to merge transforming into closed dispersion curves. After a further increase in *M*A, these closed dispersion curves become smaller – this is an indication that we have reached the critical *M*<sup>A</sup> at which the kink waves are subjected to the Kelvin–Helmholtz instability – the unstable waves propagate across the entire *kz a*-range having growth rates depending upon the value of the current *M*A. We note that this behaviour has been observed for kink waves travelling on flowing solar-wind

√*η*.

high jet speeds can be in principal registered in soft X-ray coronal jets.

flux tube? The answers to that question are as follows:

this finding is valid for *M*<sup>A</sup> 4.

2001), which in our notation reads

plasma (Zhelyazkov, 2010; 2011).

curve associated with the tube velocity in the environment.

For the X-ray jets, the dispersion curves' reorganization, because the environment has been considered as a cool plasma, is different – now, at high enough flow speeds, the merging lower- and higher-speed *c*k-dispersion curves take the form of semi-closed loops (see Fig. 17). As we increase the flow speed (or equivalently the Alfvén–Mach number), the semi-closed loops shrink and at some critical flow speed the kink wave becomes unstable and the instability is of the Kelvin–Helmholtz type. We note that the shapes of the waves' growth rates of kink waves in spicules and soft X-ray coronal jets are distinctly different – compare Figs. 9 and 18.

• We have found that the sausage waves are unaffected by the Kelvin–Helmholtz instability. This conclusion was also previously drawn for the sausage modes in flowing solar-wind plasma (Zhelyazkov, 2010; 2011).

As we have seen, very high jet speeds are required to ensure that the Kelvin–Helmholtz instability occurs for kink waves propagating in Type II spicules associated with a subsequent triggering of Alfvén-wave turbulence, hence the possibility that this mechanism is responsible for chromospheric/coronal heating has to be excluded. However, a twist in the magnetic field of the flux tube or its environment may have the effect of lowering the instability threshold (Bennett et al., 1999; Zaqarashvili et al., 2010) and eventually lead to the triggering of the Kelvin–Helmholtz instability. According to Antolin & Shibata (Antolin & Shibata, 2010), a promising way to ensure spicules'/coronal heating is by means of the mode conversion and parametric decay of Alfvén waves generated by magnetic reconnection or driven by the magneto-convection at the photosphere. However, spicules can be considered as Alfvén wave resonant cavities (Holweg, 1981; Leroy, 1981) and as Matsumoto & Shibata (Matsumoto & Shibata, 2010) claim, the waves of the period around 100–500 s can transport a large amount of wave energy to the corona. Zahariev & Mishonov (Zahariev & Mishonov, 2011) state that the corona may be heated through a self-induced opacity of high-frequency Alfvén waves propagating in the transition region between the chromosphere and the corona owing to a considerable spectral density of the Alfvén waves in the photosphere. Another trend in explaining the mechanism of coronal heating is the dissipation of Alfvén waves' energy by strong wave damping due to the collisions between ions and neutrals (Song & Vasyliunas, 2011; Tsap et al., 2011). In particular, Song & Vasyli ¯ unas, by analytically solving a ¯ self-consistent one-dimensional model of the plasma–neutral–electromagnetic system, show that the damping is extremely strong for weaker magnetic field and less strong for strong field. Under either condition, the high-frequency portion of the source power spectrum is strongly damped at the lower altitudes, depositing heat there, whereas the lower-frequency perturbations are nearly undamped and can be observed in the corona and above when the field is strong.

The idea that Alfvén waves propagating in the transition region can contribute to the coronal heating was firmly supported by the observational data recorded on April 25, 2010 by NASA's *Solar Dynamics Observatory* (see Fig. 2). As McIntosh et al. (McIntosh et al., 2011) claim, "*SDO* has amazing resolution, so you can actually see individual waves. Previous observations of Alfvénic waves in the corona revealed amplitudes far too small (0.5 km s−1) to supply the energy flux (100–200 W m−2) required to drive the fast solar wind or balance the radiative losses of the quiet corona. Here we report observations of the transition region (between the chromosphere and the corona) and of the corona that reveal how Alfvénic motions permeate the dynamic and finely structured outer solar atmosphere. The ubiquitous outward-propagating Alfvénic motions observed have amplitudes of the order of 20 km s−<sup>1</sup>

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Notwithstanding, as we have already mentioned in the end of Sec. 5.1, the possibility for the onset of a Kelvin–Helmholtz instability of kink waves running along soft X-ray coronal jets should not be excluded – at high enough flow speeds, which in principal are reachable, one can expect a dramatic change in the waves' behaviour associated with an emerging instability, and subsequently, with an Alfvén-wave-turbulence heating.

In all cases, the question of whether large coronal spicules can reach coronal temperatures remains open – for a discussion from an observational point of view we refer to the paper by Madjarska et al. (Madjarska et al., 2011).
