**3. Wave dispersion relations**

It is well-known that in infinite magnetized plasmas there exist three types of MHD waves (Chen, 1995), namely the Alfvén wave and the fast and slow magnetoacoustic waves. Alfvén wave (Alfvén, 1942; Gekelman et al., 2011), is a transverse wave propagating at speed *v*<sup>A</sup> = *B*0/(*μ*0*ρ*0)1/2, where *B*<sup>0</sup> and *ρ*<sup>0</sup> are the equilibrium (not perturbed) magnetic field and mass density, respectively. The propagation characteristics of magnetoacoustic waves depend upon their plasma beta environment. In particular, in high-beta plasmas (*β* � 1) the fast magnetoacoustic wave behaves like a sound wave travelling at sound speed *c*<sup>s</sup> = (*γp*0/*ρ*0)1/2, while in low-beta plasmas (*<sup>β</sup>* � 1) it propagates roughly isotropically and across

Assuming that each perturbation is presented as a plain wave *g*(*r*) exp [i(−*ωt* + *mϕ* + *kzz*)] with its amplitude *g*(*r*) being just a function of *r*, and that in cylindrical coordinates the nabla

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 143

*∂ ∂z z*ˆ,

− i(*ω* − **k** · **U**)*δp* + *γp*0∇ · *δ***v** = 0. (20)

(*ω* − **k** · **U**)*δBr* − *kzB*0*δvr* = 0, (21) (*ω* − **k** · **U**)*δB<sup>ϕ</sup>* − *kzB*0*δv<sup>ϕ</sup>* = 0, (22)

> *ω* − **k** · **U** *kz*

> > *m r*

(*<sup>ω</sup>* <sup>−</sup> **<sup>k</sup>** · **<sup>U</sup>**)<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup>

*kzc*<sup>2</sup> s *δv<sup>ϕ</sup>* + i*kzδvz*

*z c*2 s  .

*δB<sup>ϕ</sup>* + i*kzδBz* = 0. (24)

− i(*ω* − **k** · **U**)*δBz* − i*kzB*0*δvz* + *B*0∇ · *δ***v** = 0. (23)

*B*0*δBr* = 0, (17)

*B*0*δB<sup>ϕ</sup>* = 0, (18)

*B*0*δBz* = 0. (19)

*δvz*, (25)

*δvz* (26)

*δBr* = 0. (27)

∇ ≡ *<sup>∂</sup> ∂r r*ˆ + 1 *r ∂ ∂ϕ <sup>ϕ</sup>*<sup>ˆ</sup> <sup>+</sup>

> d d*r δp* + 1 *μ*0 *B*0*δBz* − i*kz* 1 *μ*0

*m r δp* + 1 *μ*0 *B*0*δBz* − *kz* 1 *μ*0

 *δp* + 1 *μ*0 *B*0*δBz* − *kz* 1 *μ*0

> *m r*

*δp* or *δp* = *ρ*<sup>0</sup>

*<sup>ω</sup>* <sup>−</sup> **<sup>k</sup>** · **<sup>U</sup>** *<sup>γ</sup>p*0∇ · *<sup>δ</sup>***v**,

− i*ρ*0(*ω* − **k** · **U**)*δvr* +

− *ρ*0(*ω* − **k** · **U**)*δv<sup>ϕ</sup>* +

− *ρ*0(*ω* − **k** · **U**)*δvz* + *kz*

d d*r δBr* + 1 *r δBr* + i

*<sup>δ</sup>vz* <sup>=</sup> *kz*

*<sup>δ</sup>vz* <sup>=</sup> <sup>−</sup><sup>i</sup> *kz*

d d*r δvr* + 1 *r δvr* + i

Let us now differentiate Eq. (17) with respect to *r*:

<sup>−</sup> <sup>i</sup>*ρ*0(*<sup>ω</sup>* <sup>−</sup> **<sup>k</sup>** · **<sup>U</sup>**) <sup>d</sup>

*ω* − **k** · **U**

(*<sup>ω</sup>* − **<sup>k</sup>** · **<sup>U</sup>**)<sup>2</sup>

d*r δvr* +

After some rearranging this expression can be rewritten in the form

1 *ρ*0

*<sup>δ</sup><sup>p</sup>* <sup>=</sup> <sup>−</sup><sup>i</sup> <sup>1</sup>

*γp*<sup>0</sup> *ρ*0

> *m r*

d2 d*r*<sup>2</sup>  *δp* + 1 *μ*0 *B*0*δBz* − 1 *μ*0 *B*0i*kz* d d*r*

 d d*r δvr* + 1 *r δvr* + i

*δv<sup>ϕ</sup>* = i

operator has the form

Accordingly Eq. (13) yields

Induction Eq. (12) gives

Finally Eq. (14) yields

From Eq. (19) we obtain

while Eq. (20) gives

which means that

Eq. (11) reads

the magnetic field lines at Alfvén speed, *v*A. The slow magnetoacoustic wave in high-beta plasmas is guided along the magnetic field **B**<sup>0</sup> at Alfvén speed, *v*<sup>A</sup> – in the opposite case of low-beta plasmas it is a longitudinally propagating along **B**<sup>0</sup> wave at sound speed, *c*s. A question that immediately raises is how these waves will change when the magnetized plasma is spatially bounded (or magnetically structured) as in our case of spicules or X-ray jets. The answer to that question is not trivial – we actually have to derive the *normal modes* supported by the flux tube, which models the jets.

As we will study a linear wave propagation, the basic MHD variables can be presented in the form

$$\rho = \rho\_0 + \delta \rho, \quad p = p\_0 + \delta p, \quad \mathbf{v} = \mathbf{U} + \delta \mathbf{v}, \quad \text{and} \quad \mathbf{B} = \mathbf{B}\_0 + \delta \mathbf{B}\_1$$

where *ρ*0, *p*0, and **B**<sup>0</sup> are the equilibrium values in either medium, **U**<sup>i</sup> and **U**<sup>e</sup> are the flow velocities inside and outside the flux tube, *δρ*, *δp*, *δ***v**, and *δ***B** being the small perturbations of the basic MHD variables. For convenience, we chose the frame of reference to be attached to the ambient medium. In that case

$$\mathbf{U}^{\text{rel}} = \mathbf{U}\_{\text{i}} - \mathbf{U}\_{\text{e}}$$

is the *relative* flow velocity whose magnitude is a non-zero number inside the jet, and zero in the surrounding medium. For spicules, *U*<sup>e</sup> ≈ 0; which is why the relative flow velocity is indeed the jet velocity, which we later denote as simply **U**.

With the above assumptions, the basic set of MHD equations for the perturbations of the mass density, pressure, fluid velocity, and magnetic field become

$$\frac{\partial}{\partial t}\delta\rho + (\mathbf{U}\cdot\nabla)\delta\rho + \rho\_0 \nabla\delta\mathbf{v} = 0,\tag{10}$$

$$
\rho\_0 \frac{\partial}{\partial t} \delta \mathbf{v} + \rho\_0 \left( \mathbf{U} \cdot \nabla \right) \delta \mathbf{v} + \nabla \left( \delta p + \frac{1}{\mu\_0} \mathbf{B}\_0 \cdot \delta \mathbf{B} \right) - \frac{1}{\mu\_0} (\mathbf{B}\_0 \cdot \nabla) \delta \mathbf{B} = 0,\tag{11}
$$

$$\frac{\partial}{\partial t}\delta \mathbf{B} + (\mathbf{U} \cdot \nabla)\delta \mathbf{B} - (\mathbf{B}\_0 \cdot \nabla)\delta \mathbf{v} + \mathbf{B}\_0 \nabla \cdot \delta \mathbf{v} = 0,\tag{12}$$

$$\frac{\partial}{\partial t}\delta p + (\mathbf{U}\cdot\nabla)\delta p + \gamma p\_0 \nabla \cdot \delta \mathbf{v} = 0,\tag{13}$$

$$
\nabla \cdot \delta \mathbf{B} = 0.\tag{14}
$$

We note that the gravity force term in momentum Eq. (11) has been omitted because one assumes that the mass density of the jet does not change appreciably in the limits of the spicule's length of order 10–11 Mm.

From Eq. (10) we obtain that

$$\nabla \cdot \delta \mathbf{v} = -\frac{1}{\rho\_0} \left[ \frac{\partial}{\partial t} \delta \rho + (\mathbf{U} \cdot \nabla) \delta \rho \right]. \tag{15}$$

Inserting this expression into Eq. (13) we get

$$
\left[\frac{\partial}{\partial t} + (\mathbf{U} \cdot \nabla)\right] \delta p - c\_s^2 \left[\frac{\partial}{\partial t} + (\mathbf{U} \cdot \nabla)\right] \delta \rho = 0,
$$

which means that the pressure's and density's perturbations are related via the expression

$$
\delta p = c\_s^2 \delta \rho\_\prime \qquad \text{where} \qquad c\_s = \left(\gamma p\_0 / \rho\_0\right)^{1/2} . \tag{16}
$$

Assuming that each perturbation is presented as a plain wave *g*(*r*) exp [i(−*ωt* + *mϕ* + *kzz*)] with its amplitude *g*(*r*) being just a function of *r*, and that in cylindrical coordinates the nabla operator has the form

$$\nabla \equiv \frac{\partial}{\partial r}\mathfrak{H} + \frac{1}{r}\frac{\partial}{\partial \varphi}\mathfrak{H} + \frac{\partial}{\partial z}\mathfrak{H}$$

Eq. (11) reads

8 Will-be-set-by-IN-TECH

the magnetic field lines at Alfvén speed, *v*A. The slow magnetoacoustic wave in high-beta plasmas is guided along the magnetic field **B**<sup>0</sup> at Alfvén speed, *v*<sup>A</sup> – in the opposite case of low-beta plasmas it is a longitudinally propagating along **B**<sup>0</sup> wave at sound speed, *c*s. A question that immediately raises is how these waves will change when the magnetized plasma is spatially bounded (or magnetically structured) as in our case of spicules or X-ray jets. The answer to that question is not trivial – we actually have to derive the *normal modes* supported

As we will study a linear wave propagation, the basic MHD variables can be presented in the

*ρ* = *ρ*<sup>0</sup> + *δρ*, *p* = *p*<sup>0</sup> + *δp*, **v** = **U** + *δ***v**, and **B** = **B**<sup>0</sup> + *δ***B**, where *ρ*0, *p*0, and **B**<sup>0</sup> are the equilibrium values in either medium, **U**<sup>i</sup> and **U**<sup>e</sup> are the flow velocities inside and outside the flux tube, *δρ*, *δp*, *δ***v**, and *δ***B** being the small perturbations of the basic MHD variables. For convenience, we chose the frame of reference to be attached to

**<sup>U</sup>**rel <sup>=</sup> **<sup>U</sup>**<sup>i</sup> <sup>−</sup> **<sup>U</sup>**<sup>e</sup> is the *relative* flow velocity whose magnitude is a non-zero number inside the jet, and zero in the surrounding medium. For spicules, *U*<sup>e</sup> ≈ 0; which is why the relative flow velocity is

With the above assumptions, the basic set of MHD equations for the perturbations of the mass

We note that the gravity force term in momentum Eq. (11) has been omitted because one assumes that the mass density of the jet does not change appreciably in the limits of the

**B**<sup>0</sup> · *δ***B**

*δρ* + (**U** · ∇)*δρ*

+ (**U** · ∇)

 − 1 *μ*0

*δ***B** + (**U** · ∇)*δ***B** − (**B**<sup>0</sup> · ∇)*δ***v** + **B**0∇ · *δ***v** = 0, (12)

*δp* + (**U** · ∇)*δp* + *γp*0∇ · *δ***v** = 0, (13)

*δρ* = 0,

∇ · *δ***B** = 0. (14)

 *δp* + 1 *μ*0

*δρ* + (**U** · ∇)*δρ* + *ρ*0∇*δ***v** = 0, (10)

(**B**<sup>0</sup> · ∇)*δ***B** = 0, (11)

. (15)

1/2 . (16)

by the flux tube, which models the jets.

the ambient medium. In that case

*ρ*0 *∂ ∂t*

spicule's length of order 10–11 Mm.

Inserting this expression into Eq. (13) we get

 *∂ ∂t*

*δp* = *c*<sup>2</sup>

From Eq. (10) we obtain that

indeed the jet velocity, which we later denote as simply **U**.

density, pressure, fluid velocity, and magnetic field become

*δ***v** + *ρ*<sup>0</sup> (**U** · ∇) *δ***v** + ∇

*∂ ∂t*

∇ · *<sup>δ</sup>***<sup>v</sup>** <sup>=</sup> <sup>−</sup> <sup>1</sup>

+ (**U** · ∇)

*ρ*0

*<sup>δ</sup><sup>p</sup>* <sup>−</sup> *<sup>c</sup>*<sup>2</sup> s *∂ ∂t*

which means that the pressure's and density's perturbations are related via the expression

<sup>s</sup> *δρ*, where *c*<sup>s</sup> = (*γp*0/*ρ*0)

 *∂ ∂t*

*∂ ∂t*

*∂ ∂t*

form

$$-\mathbf{i}\rho\_0(\omega-\mathbf{k}\cdot\mathbf{U})\delta v\_r + \frac{\mathbf{d}}{\mathbf{d}r}\left(\delta p + \frac{1}{\mu\_0}B\_0\delta B\_z\right) - \mathbf{i}k\_z\frac{1}{\mu\_0}B\_0\delta B\_r = 0,\tag{17}$$

$$-\rho\_0(\omega - \mathbf{k} \cdot \mathbf{U})\delta v\_\varphi + \frac{m}{r}\left(\delta p + \frac{1}{\mu\_0}B\_0\delta B\_z\right) - k\_z \frac{1}{\mu\_0}B\_0\delta B\_\varphi = 0,\tag{18}$$

$$-\rho\_0(\omega - \mathbf{k} \cdot \mathbf{U})\delta v\_z + k\_z \left(\delta p + \frac{1}{\mu\_0} B\_0 \delta B\_z\right) - k\_z \frac{1}{\mu\_0} B\_0 \delta B\_z = 0. \tag{19}$$

Accordingly Eq. (13) yields

$$-\mathbf{i}(\omega - \mathbf{k} \cdot \mathbf{U})\delta p + \gamma p\_0 \nabla \cdot \delta \mathbf{v} = 0. \tag{20}$$

Induction Eq. (12) gives

$$(\omega - \mathbf{k} \cdot \mathbf{U})\delta B\_r - k\_z B\_0 \delta v\_r = 0,\tag{21}$$

$$(\omega - \mathbf{k} \cdot \mathbf{U})\delta B\_{\varphi} - k\_z B\_0 \delta v\_{\varphi} = 0,\tag{22}$$

$$-\mathbf{i}(\omega - \mathbf{k} \cdot \mathbf{U})\delta B\_z - \mathbf{i}k\_z B\_0 \delta v\_z + B\_0 \nabla \cdot \delta \mathbf{v} = 0. \tag{23}$$

Finally Eq. (14) yields

$$\frac{\mathbf{d}}{\mathbf{d}r}\delta B\_{\mathbf{r}} + \frac{1}{r}\delta B\_{\mathbf{r}} + \mathbf{i}\frac{m}{r}\delta B\_{\mathbf{q}} + \mathbf{i}k\_{z}\delta B\_{z} = 0.\tag{24}$$

From Eq. (19) we obtain

$$
\delta v\_z = \frac{k\_z}{\omega - \mathbf{k} \cdot \mathbf{U}} \frac{1}{\rho\_0} \delta p \quad \text{or} \quad \delta p = \rho\_0 \frac{\omega - \mathbf{k} \cdot \mathbf{U}}{k\_z} \delta v\_z. \tag{25}
$$

while Eq. (20) gives

$$\delta p = -\mathbf{i}\frac{1}{\omega - \mathbf{k}\cdot\mathbf{U}}\gamma p\_0 \nabla \cdot \delta \mathbf{v}\_{\prime\prime}$$

which means that

$$\delta v\_z = -\mathbf{i} \frac{k\_z}{(\omega - \mathbf{k} \cdot \mathbf{U})^2} \frac{\gamma p\_0}{\rho\_0} \left( \frac{\mathbf{d}}{\mathbf{d}r} \delta v\_r + \frac{1}{r} \delta v\_r + \mathbf{i} \frac{m}{r} \delta v\_\varphi + \mathbf{i} k\_z \delta v\_z \right).$$

After some rearranging this expression can be rewritten in the form

$$\frac{\mathbf{d}}{r}\delta v\_r + \frac{1}{r}\delta v\_r + \mathbf{i}\frac{m}{r}\delta v\_\phi = \mathbf{i}\frac{(\omega - \mathbf{k}\cdot\mathbf{U})^2 - k\_z^2 c\_s^2}{k\_z c\_s^2}\delta v\_z\tag{26}$$

Let us now differentiate Eq. (17) with respect to *r*:

$$-\mathbf{i}\rho\_0(\omega-\mathbf{k}\cdot\mathbf{U})\frac{\mathbf{d}}{\mathbf{d}r}\delta v\_r + \frac{\mathbf{d}^2}{\mathbf{d}r^2}\left(\delta p + \frac{1}{\mu\_0}B\_0\delta B\_z\right) - \frac{1}{\mu\_0}B\_0\mathbf{i}k\_z\frac{\mathbf{d}}{\mathbf{d}r}\delta B\_r = 0.\tag{27}$$

where, we remember, *v*<sup>A</sup> = *B*0/(*μ*0*ρ*0)1/2 is the Alfvén speed. After inserting in the above equation *δp* expressed in terms of *δvz* – see Eq. (25) – and performing some straightforward

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 145

*k*2 *z c*2 s

<sup>A</sup> <sup>−</sup> (*<sup>ω</sup>* <sup>−</sup> **<sup>k</sup>** · **<sup>U</sup>**)<sup>2</sup>

(*<sup>ω</sup>* <sup>−</sup> **<sup>k</sup>** · **<sup>U</sup>**)<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup>

*c*2 <sup>s</sup> + *v*<sup>2</sup> A *<sup>δ</sup>p*tot.

(*<sup>ω</sup>* <sup>−</sup> **<sup>k</sup>** · **<sup>U</sup>**)<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup>

*z c*2 T

*<sup>δ</sup>p*tot <sup>=</sup> 0. (30)

1/2 (32)

, (31)

*δp*tot. (33)

*zv*2 A  *<sup>z</sup> δp*-term with

algebra we obtain that

where

*r*) of *δp*tot

and

boundary conditions are:

surrounding medium.

*<sup>δ</sup>vz* <sup>=</sup> <sup>−</sup> <sup>1</sup>

*<sup>κ</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>

Here, *κ*<sup>2</sup> is given by the expression

*ρ*0

 d<sup>2</sup> <sup>d</sup>*r*<sup>2</sup> <sup>+</sup>

*ω* − **k** · **U** *kz*

*k*2 *z c*2 s *v*2

Next step is to insert above expression of *<sup>δ</sup>vz* into Eq. (29) and combine the <sup>−</sup>*k*<sup>2</sup>

 *κ*<sup>2</sup> + *m*2 *r*2

> *z c*2 s

*<sup>c</sup>*<sup>T</sup> <sup>=</sup> *<sup>c</sup>*s*v*<sup>A</sup> *c*2 <sup>s</sup> + *v*<sup>2</sup> A

is the so-called *tube velocity* (Edwin & Roberts, 1983). It is important to notice that both *κ*<sup>2</sup> (respectively *κ*) and the tube velocity, *c*T, *have different values inside and outside the jet* due to the different sound and Alfvén speeds, which characterize correspondingly the jet and its

As can be seen, Eq. (30) is the equation for the modified Bessel functions *Im* and *Km* and,

*<sup>δ</sup>p*tot(*r*) = *<sup>A</sup>*i*Im*(*κ*i*r*) for *<sup>r</sup> <sup>a</sup>*,

From Eq. (17) one can obtain an expression of *δvr* and inserting it in the expression of *δBr* deduced from Eq. (21) one gets a formula relating *δvr* with the first derivative (with respect to

> *ω* − **k** · **U** (*<sup>ω</sup>* <sup>−</sup> **<sup>k</sup>** · **<sup>U</sup>**)<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup>

> > *ω* − **k** · **U**

*ω <sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> *zv*2 Ae

respectively. Now it is time to apply some boundary conditions, which link the solutions of total pressure and fluid velocity perturbations at the interface *r* = *a*. The appropriate

<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> *zv*2 Ai

*κ*e*A*e*K*�

It is clear that we have two different expressions of *δvr*, which, bearing in mind the solutions

(*ω* − **k** · **U**)

*A*e*Km*(*κ*e*r*) for *r a*.

*zv*2 A

d d*r*

> *κ*i*A*i*I* � *<sup>m</sup>*(*κ*i*r*)

*<sup>m</sup>*(*κ*e*r*),

the last member in the same equation to get a new form of Eq. (29), notably

(*<sup>ω</sup>* <sup>−</sup> **<sup>k</sup>** · **<sup>U</sup>**)<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup>

 *c*2 <sup>s</sup> + *v*<sup>2</sup> A

accordingly, its solutions in both media (the jet and its environment) are:

*<sup>δ</sup>vr* <sup>=</sup> <sup>−</sup> <sup>i</sup>

to the ordinary second order differential Eq. (30), read

*<sup>δ</sup>vr*(*<sup>r</sup> <sup>a</sup>*) = <sup>−</sup> <sup>i</sup>

*<sup>δ</sup>vr*(*<sup>r</sup> <sup>a</sup>*) = <sup>−</sup> <sup>i</sup>

*ρ*0

*ρ*i

*ρ*e

1 *r* d d*r* −

But according to Eqs. (26) and (24)

$$\frac{\mathbf{d}}{\mathbf{d}r}\delta v\_r = -\frac{1}{r}\delta v\_r - \mathbf{i}\frac{m}{r}\delta v\_\phi - \mathbf{i}k\_z\left[1 - \frac{(\omega - \mathbf{k}\cdot\mathbf{U})^2}{k\_z^2c\_s^2}\right]\delta v\_z\rho$$

$$\frac{\mathbf{d}}{\mathbf{d}r}\delta B\_r = -\frac{1}{r}\delta B\_r - \mathbf{i}\frac{m}{r}\delta B\_\phi - \mathbf{i}k\_z\delta B\_z.$$

Then Eq. (27) becomes

$$\mathrm{i}\rho\_{0}(\omega-\mathbf{k}\cdot\mathbf{U})\frac{1}{r}\delta v\_{r}-\rho\_{0}(\omega-\mathbf{k}\cdot\mathbf{U})\frac{m}{r}\delta v\_{\varphi}-\rho\_{0}(\omega-\mathbf{k}\cdot\mathbf{U})\left[1-\frac{(\omega-\mathbf{k}\cdot\mathbf{U})^{2}}{k\_{z}^{2}c\_{s}^{2}}\right]k\_{z}\delta v\_{z}$$

$$+\frac{\mathbf{d}^{2}}{\mathrm{d}r^{2}}\left(\delta p+\frac{1}{\mu\_{0}}B\_{0}\delta B\_{z}\right)+\frac{1}{\mu\_{0}}B\_{0}\mathrm{i}k\_{z}\frac{1}{r}\delta B\_{r}-\frac{1}{\mu\_{0}}B\_{0}k\_{z}\frac{m}{r}\delta B\_{\varphi}-\frac{1}{\mu\_{0}}B\_{0}k\_{z}^{2}\delta B\_{z}=0\tag{28}$$

In order to simplify notation we introduce a new variable, namely the perturbation of the total pressure, *δp*tot = *δp* + <sup>1</sup> *<sup>μ</sup>*<sup>0</sup> *B*0*δBz*. From Eqs. (17) to (19) one can get that

$$\frac{1}{\mu\_0} B\_0 \mathbf{i} k\_z \frac{1}{r} \delta B\_r = -\mathbf{i} \rho\_0 (\omega - \mathbf{k} \cdot \mathbf{U}) \frac{1}{r} \delta v\_r + \frac{1}{r} \frac{\mathbf{d}}{\mathbf{d} r} \delta p\_{\text{tot}}.$$

$$-\frac{1}{\mu\_0} B\_0 \mathbf{i} k\_z \frac{m}{r} \delta B\_\vartheta = \rho\_0 (\omega - \mathbf{k} \cdot \mathbf{U}) \frac{m}{r} \delta v\_\vartheta - \frac{m^2}{r^2} \delta p\_{\text{tot}}.$$

$$-\frac{1}{\mu\_0} B\_0 \mathbf{i} k\_z^2 \delta B\_z = \rho\_0 (\omega - \mathbf{k} \cdot \mathbf{U}) k\_z \delta v\_z - k\_z^2 \delta p\_{\text{tot}}.$$

Inserting these expressions into Eq. (28), we obtain

$$
\left[\frac{\mathbf{d}^2}{\mathbf{d}r^2} + \frac{1}{r}\frac{\mathbf{d}}{\mathbf{d}r} - \left(k\_z^2 + \frac{m^2}{r^2}\right)\right]\delta p\_{\rm tot} + \rho\_0 \frac{(\boldsymbol{\omega} - \mathbf{k} \cdot \mathbf{U})^3}{k\_z c\_s^2} \delta v\_z = 0. \tag{29}
$$

Bearing in mind that according to Eq. (15)

$$\nabla \cdot \delta \mathbf{v} = \mathbf{i} (\omega - \mathbf{k} \cdot \mathbf{U}) \frac{\delta \rho}{\rho\_0} \prime$$

from Eq. (23) we get

$$(\boldsymbol{\omega} - \mathbf{k} \cdot \mathbf{U}) \delta B\_z - k\_z B\_0 \delta v\_z + B\_0 (\boldsymbol{\omega} - \mathbf{k} \cdot \mathbf{U}) \frac{\delta \rho}{\rho\_0} = 0.1$$

On using Eq. (16) we express *δρ* in the above equation as *δp*/*c*<sup>2</sup> <sup>s</sup> , multiply it by

$$-\frac{1}{\mu\_0}B\_0 \frac{1}{\omega - \mathbf{k} \cdot \mathbf{U}}$$

to get after some algebra that

$$
\delta p + \frac{1}{\mu\_0} B\_0 \delta B\_z = -\frac{k\_z \rho\_0}{\omega - \mathbf{k} \cdot \mathbf{U}} \frac{B\_0^2}{\mu\_0 \rho\_0} \delta v\_z + \delta p \left( 1 + \frac{v\_\mathbf{A}^2}{c\_\mathbf{s}^2} \right),
$$

where, we remember, *v*<sup>A</sup> = *B*0/(*μ*0*ρ*0)1/2 is the Alfvén speed. After inserting in the above equation *δp* expressed in terms of *δvz* – see Eq. (25) – and performing some straightforward algebra we obtain that

$$\delta v\_z = -\frac{1}{\rho\_0} \frac{\omega - \mathbf{k} \cdot \mathbf{U}}{k\_z} \frac{k\_z^2 c\_s^2}{k\_z^2 c\_s^2 v\_\mathcal{A}^2 - (\omega - \mathbf{k} \cdot \mathbf{U})^2 \left(c\_s^2 + v\_\mathcal{A}^2\right)} \delta p\_{\text{tot}}.$$

Next step is to insert above expression of *<sup>δ</sup>vz* into Eq. (29) and combine the <sup>−</sup>*k*<sup>2</sup> *<sup>z</sup> δp*-term with the last member in the same equation to get a new form of Eq. (29), notably

$$
\left[\frac{\mathbf{d}^2}{\mathbf{d}r^2} + \frac{1}{r}\frac{\mathbf{d}}{\mathbf{d}r} - \left(\kappa^2 + \frac{m^2}{r^2}\right)\right]\delta p\_{\rm tot} = 0.\tag{30}
$$

Here, *κ*<sup>2</sup> is given by the expression

$$\chi^2 = -\frac{\left[ (\omega - \mathbf{k} \cdot \mathbf{U})^2 - k\_z^2 c\_s^2 \right] \left[ (\omega - \mathbf{k} \cdot \mathbf{U})^2 - k\_z^2 v\_A^2 \right]}{\left( c\_s^2 + v\_A^2 \right) \left[ (\omega - \mathbf{k} \cdot \mathbf{U})^2 - k\_z^2 c\_\Gamma^2 \right]} \,, \tag{31}$$

where

10 Will-be-set-by-IN-TECH

*δv<sup>ϕ</sup>* − i*kz*

In order to simplify notation we introduce a new variable, namely the perturbation of the total

*<sup>μ</sup>*<sup>0</sup> *B*0*δBz*. From Eqs. (17) to (19) one can get that

*δBr* = −i*ρ*0(*ω* − **k** · **U**)

*δB<sup>ϕ</sup>* = *ρ*0(*ω* − **k** · **U**)

∇ · *δ***v** = i(*ω* − **k** · **U**)

(*ω* − **k** · **U**)*δBz* − *kzB*0*δvz* + *B*0(*ω* − **k** · **U**)

*ω* − **k** · **U**

− 1 *μ*0 *B*0

*<sup>B</sup>*0*δBz* <sup>=</sup> <sup>−</sup> *kzρ*<sup>0</sup>

*<sup>z</sup>δBz* <sup>=</sup> *<sup>ρ</sup>*0(*<sup>ω</sup>* <sup>−</sup> **<sup>k</sup>** · **<sup>U</sup>**)*kzδvz* <sup>−</sup> *<sup>k</sup>*<sup>2</sup>

*δp*tot + *ρ*<sup>0</sup>

1 *ω* − **k** · **U**

> *B*2 0 *μ*0*ρ*0

*δvz* + *δp*

 1 + *v*2 A *c*2 s

*m r*

*δv<sup>ϕ</sup>* − *ρ*0(*ω* − **k** · **U**)

*<sup>δ</sup>Br* <sup>−</sup> <sup>1</sup> *μ*0 *B*0*kz m r*

> 1 *r δvr* + 1 *r* d d*r δp*tot,

*m r*

*<sup>δ</sup>v<sup>ϕ</sup>* <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

(*<sup>ω</sup>* <sup>−</sup> **<sup>k</sup>** · **<sup>U</sup>**)<sup>3</sup> *kzc*<sup>2</sup> s

*δρ ρ*0 , *<sup>r</sup>*<sup>2</sup> *<sup>δ</sup>p*tot,

*<sup>z</sup>δp*tot.

*δρ ρ*0 = 0.

<sup>s</sup> , multiply it by

 ,

<sup>1</sup> <sup>−</sup> (*<sup>ω</sup>* <sup>−</sup> **<sup>k</sup>** · **<sup>U</sup>**)<sup>2</sup> *k*2 *z c*2 s

*δB<sup>ϕ</sup>* − i*kzδBz*.

 *δvz*,

<sup>1</sup> <sup>−</sup> (*<sup>ω</sup>* <sup>−</sup> **<sup>k</sup>** · **<sup>U</sup>**)<sup>2</sup> *k*2 *z c*2 s

> *<sup>δ</sup>B<sup>ϕ</sup>* <sup>−</sup> <sup>1</sup> *μ*0 *B*0*k*<sup>2</sup>

 *kzδvz*

*δvz* = 0. (29)

*<sup>z</sup>δBz* = 0 (28)

*m r*

> *m r*

*<sup>δ</sup>Br* <sup>=</sup> <sup>−</sup><sup>1</sup> *r δBr* − i

But according to Eqs. (26) and (24)

Then Eq. (27) becomes

i*ρ*0(*ω* − **k** · **U**)

+ d2 d*r*<sup>2</sup>

pressure, *δp*tot = *δp* + <sup>1</sup>

from Eq. (23) we get

to get after some algebra that

*δp* + 1 *μ*0

d d*r*

1 *r*

> *δp* + 1 *μ*0 *B*0*δBz* + 1 *μ*0 *B*0i*kz* 1 *r*

1 *μ*0 *B*0i*kz* 1 *r*

− 1 *μ*0 *B*0i*kz m r*

> − 1 *μ*0 *B*0i*k*<sup>2</sup>

Inserting these expressions into Eq. (28), we obtain

1 *r* d d*r* −  *k*2 *<sup>z</sup>* + *m*2 *r*2

On using Eq. (16) we express *δρ* in the above equation as *δp*/*c*<sup>2</sup>

 d<sup>2</sup> <sup>d</sup>*r*<sup>2</sup> <sup>+</sup>

Bearing in mind that according to Eq. (15)

*<sup>δ</sup>vr* <sup>=</sup> <sup>−</sup><sup>1</sup> *r δvr* − i

> d d*r*

*δvr* − *ρ*0(*ω* − **k** · **U**)

$$\mathcal{L}\_{\rm T} = \frac{c\_{\rm s} v\_{\rm A}}{\left(c\_{\rm s}^2 + v\_{\rm A}^2\right)^{1/2}}\tag{32}$$

is the so-called *tube velocity* (Edwin & Roberts, 1983). It is important to notice that both *κ*<sup>2</sup> (respectively *κ*) and the tube velocity, *c*T, *have different values inside and outside the jet* due to the different sound and Alfvén speeds, which characterize correspondingly the jet and its surrounding medium.

As can be seen, Eq. (30) is the equation for the modified Bessel functions *Im* and *Km* and, accordingly, its solutions in both media (the jet and its environment) are:

$$\delta p\_{\rm tot}(r) = \begin{cases} A\_{\rm i} I\_{\rm m}(\kappa\_{\rm i} r) & \text{for } r \leqslant a\_{\star} \\ A\_{\rm e} K\_{\rm m}(\kappa\_{\rm e} r) & \text{for } r \geqslant a\_{\star} \end{cases}$$

From Eq. (17) one can obtain an expression of *δvr* and inserting it in the expression of *δBr* deduced from Eq. (21) one gets a formula relating *δvr* with the first derivative (with respect to *r*) of *δp*tot

$$\delta v\_{\mathbf{r}} = -\frac{\mathbf{i}}{\rho\_0} \frac{\omega - \mathbf{k} \cdot \mathbf{U}}{(\omega - \mathbf{k} \cdot \mathbf{U})^2 - k\_z^2 v\_{\mathbf{A}}^2} \frac{\mathbf{d}}{\mathbf{d}r} \delta p\_{\text{tot}}.\tag{33}$$

It is clear that we have two different expressions of *δvr*, which, bearing in mind the solutions to the ordinary second order differential Eq. (30), read

$$\delta v\_r(r \lessapprox a) = -\frac{\rm i}{\rho\_{\rm i}} \frac{\omega - \mathbf{k} \cdot \mathbf{U}}{\left(\omega - \mathbf{k} \cdot \mathbf{U}\right)^2 - k\_z^2 v\_{\rm Ai}^2} \kappa\_{\rm i} A\_{\rm i} l\_m'(\kappa\_{\rm i} r)$$

and

$$\delta v\_r(r \gg a) = -\frac{\mathrm{i}}{\rho\_{\mathrm{e}}} \frac{\omega}{\omega^2 - k\_z^2 v\_{\mathrm{Ae}}^2} \kappa\_{\mathrm{e}} A\_{\mathrm{e}} K\_m'(\kappa\_{\mathrm{e}} r)\_{\mathrm{e}}$$

respectively. Now it is time to apply some boundary conditions, which link the solutions of total pressure and fluid velocity perturbations at the interface *r* = *a*. The appropriate boundary conditions are:

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

*p*<sup>i</sup> + *B*2 i <sup>2</sup>*<sup>μ</sup>* <sup>=</sup> *<sup>p</sup>*<sup>e</sup> <sup>+</sup>

*ρ*e *ρ*i

speed, associated with the kink waves, in our case (see Eq. (35)) is 84 km s−1.

introduction of three numbers, notably the two ratios *β*¯ = *c*<sup>2</sup>

Thus, the input parameters in the numerical procedure are

*η* = 0.02, *β*¯

**4.1 Kink waves in spicules**

<sup>=</sup> *<sup>c</sup>*<sup>2</sup> si <sup>+</sup> *<sup>γ</sup>* 2 *v*2 Ai

*c*2 se <sup>+</sup> *<sup>γ</sup>* <sup>2</sup> *<sup>v</sup>*<sup>2</sup> Ae

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

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

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

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

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

values, we calculate the dispersion curves of first kink waves and then sausage ones.

i,e/*γ*.

<sup>i</sup> ∼= 0.016, *β*¯e ∼= 5.96, *b* ∼= 0.35, and *M*A.

*β*i,e = 2*β*¯

– that condition can be expressed in the form

which yields (Edwin & Roberts (1983)

expressions

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

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 147

*B*2 e 2*μ* ,

. (36)

<sup>A</sup> correspondingly in the jet

s/*v*<sup>2</sup>


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

$$\frac{\rho\_{\rm e}}{\rho\_{\rm i}} \left(\omega^2 - k\_z^2 v\_{\rm Ae}^2\right) \kappa\_{\rm i} \frac{I\_m'(\kappa\_{\rm i} a)}{I\_m(\kappa\_{\rm i} a)} - \left[\left(\omega - \mathbf{k} \cdot \mathbf{U}\right)^2 - k\_z^2 v\_{\rm Ai}^2\right] \kappa\_{\rm e} \frac{K\_m'(\kappa\_{\rm e} a)}{K\_{\rm m}(\kappa\_{\rm e} a)} = 0. \tag{34}$$

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 pure surface modes.

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

$$\mathbf{c\_{k}} = \left(\frac{\rho\_{\rm i}v\_{\rm Ai}^{2} + \rho\_{\rm e}v\_{\rm Ae}^{2}}{\rho\_{\rm i} + \rho\_{\rm e}}\right)^{1/2} = \left(\frac{v\_{\rm Ai}^{2} + (\rho\_{\rm e}/\rho\_{\rm i})v\_{\rm Ae}^{2}}{1 + \rho\_{\rm e}/\rho\_{\rm i}}\right)^{1/2},\tag{35}$$

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

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

In the next two section we numerically derive the dispersion curves of kink and sausage waves running along spicules and X-ray jets, respectively.
