**2.1 Basic equations of ideal magnetohydrodynamics**

4 Will-be-set-by-IN-TECH

and more specifically their stability/instability status. If while propagating along the jets MHD waves become unstable and the expected instability is of the Kelvin–Helmholtz type, that instability can trigger the onset of wave turbulence leading to an effective plasma jet heating and the acceleration of the charged particles. We note that the Alfvénic turbulence is considered to be the most promising source of heating in the chromosphere and extended corona (van Ballegooijen et al., 2011). In this study, we investigate these travelling wave properties for a realistic, cylindrical geometry of the spicules and X-ray jets considering appropriate values for the basic plasma jet parameters (mass density, magnetic fields, sound, Alfvén, and jet speeds), as well as those of the surrounding medium. For detailed reviews of the oscillations and waves in magnetically structured solar spicules we refer the reader to (Zaqarashvili & Erdélyi, 2009) and (Zaqarashvili, 2011). Our research concerns the dispersion curves of kink and sausage modes for the MHD waves travelling primarily along the Type II spicules and X-ray jets for various values of the jet speed. In studying wave propagation characteristics, we assume that the axial wave number *kz* (*z*ˆ is the direction of the embedded constant magnetic fields in the two media) is real, while the angular wave frequency, *ω*, is complex. The imaginary part of that complex frequency is the wave growth rate when a given mode becomes unstable. All of our analysis is based on a linearized set of equations for the adopted form of magnetohydrodynamics. We show that the stability/instability status of the travelling waves depends entirely on the magnitudes of the flow velocities and the values of two important control parameters, namely the so-called density contrast (the ratio of the mass density inside to that outside the flux tube) and the ratio of the background magnetic field of

The simplest model of spicules is a straight vertical cylinder (see Fig. 4) with radius *a*

Fig. 4. Geometry of a spicule flux tube containing flowing plasma with velocity **U**.

the environment to that of the spicules and X-ray jets.

**2. Geometry and basic magnetohydrodynamic equations**

Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas and liquid metals. The field of MHD was initiated in 1942 by the Swedish physicist Hannes Alfvén (1908–1995), who received the Nobel Prize in Physics (1970) for "fundamental work and discoveries in magnetohydrodynamics with fruitful applications in different parts of plasma physics." The fundamental concept behind MHD is that magnetic fields can induce currents in a moving conductive fluid, which in turn creates forces on the fluid and also changes the magnetic field itself. The set of equations, which describe MHD are a combination of the equations of motion of fluid dynamics (Navier–Stokes equations) and Maxwell's equations of electromagnetism. These partial differential equations have to be solved simultaneously, either analytically or numerically.

Magnetohydrodynamics is a macroscopic theory. Its equations can in principle be derived from the kinetic Boltzmann's equation assuming space and time scales to be larger than all inherent scale-lengths such as the Debye length or the gyro-radii of the charged particles (Chen, 1995). It is, however, more convenient to obtain the MHD equations in a phenomenological way as an electromagnetic extension of the hydrodynamic equations of ordinary fluids, where the main approximation is to neglect the displacement current ∝*∂***E**/*∂t* in Ampère's law.

Finally, the equation of the thermal energy is given by

usually is written as an equation for the pressure, *p*,

arises because ions and electrons contribute equally.

MHD fixes the topology of the magnetic field in the fluid.

**3. Wave dispersion relations**

fluid.

*∂p*

d d*t p <sup>ρ</sup><sup>γ</sup>* <sup>=</sup> 0,

where *γ* = 5/3 is the ratio of specific heats for an adiabatic equation of state. This equation

Review of the Magnetohydrodynamic Waves and Their Stability in Solar Spicules and X-Ray Jets 141

Equation (9) implies that the equation of state of the ideal fully ionized gas has the form

*p* = 2(*ρ*/*m*i)*k*B*T*,

where *T* is the temperature, *m*<sup>i</sup> the ion mass, *k*<sup>B</sup> is the Boltzmann constant, and the factor 2

In total the ideal MHD equations thus consist of two vector equations, (7) and (8), and two scalar equations, (1) and (9), respectively. Occasionally, when studying wave propagation in magnetized plasmas, one might also be necessary to use Eq. (6). We note that the basic variables of the ideal MHD are the mass density, *ρ*, the fluid bulk velocity, **v**, the pressure, *p*,

In MHD there is a few dimensionless numbers, which are widely used in studying various phenomena in magnetized plasmas. Such an important dimensionless number in MHD theory is the plasma beta, *β*, defined as the ratio of gas pressure, *p*, to the magnetic pressure,

> *<sup>β</sup>* <sup>=</sup> *<sup>p</sup> B*2/2*μ*<sup>0</sup> .

When the magnetic field dominates in the fluid, *β* � 1, the fluid is forced to move along with the field. In the opposite case, when the field is weak, *β* � 1, the field is swirled along by the

We finish our short introduction to MHD recalling that in ideal MHD Lenz's law dictates that the fluid is in a sense tied to the magnetic field lines, or, equivalently, magnetic filed lines are *frozen into the fluid*. To explain, in ideal MHD a small rope-like volume of fluid surrounding a field line will continue to lie along a magnetic field line, even as it is twisted and distorted by fluid flows in the system. The connection between magnetic field lines and fluid in ideal

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

and the magnetic induction, **B**; the electric field, **E**, has been excluded via Ohm's law.

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> **<sup>v</sup>** · ∇*<sup>p</sup>* <sup>+</sup> *<sup>γ</sup>p*∇ · **<sup>v</sup>** <sup>=</sup> 0. (9)

In the standard nonrelativistic form the MHD equations consist of the basic conservation laws of mass, momentum, and energy together with the induction equation for the magnetic field. Thus, the MHD equations of our magnetized quasineutral plasma with singly charged ions (and electrons) are

$$\frac{\partial \rho}{\partial t} + \nabla \cdot \rho \mathbf{v} = 0,\tag{1}$$

where *ρ* is the mass density and **v** is the bulk fluid velocity. Equation (1) is the so called *continuity equation* in our basis set of equations.

The momentum equation is

$$\frac{\partial(\rho \mathbf{v})}{\partial t} + \rho(\mathbf{v} \cdot \nabla)\mathbf{v} = \mathbf{j} \times \mathbf{B} - \nabla p + \rho \mathbf{g}\_{\prime} \tag{2}$$

where **j** × **B** (with **j** being the current density and **B** magnetic field induction) is the *Lorentz force* term, −∇*p* is the pressure-gradient term, and *ρ***g** is the gravity force.

Faraday's law reads

$$\frac{\partial \mathbf{B}}{\partial t} = -\nabla \times \mathbf{E} \,\tag{3}$$

where **E** is the electric field. The ideal Ohm's law for a plasma, which yields a useful relation between electric and magnetic fields, is

$$
\mathbf{E} + \mathbf{v} \times \mathbf{B} = 0.\tag{4}
$$

The low-frequency Ampère's law, which neglects the displacement current, is given by

$$
\mu\_0 \mathbf{j} = \nabla \times \mathbf{B} \,\tag{5}
$$

where *μ*<sup>0</sup> is the permeability of free space.

The magnetic divergency constraint is

$$
\nabla \cdot \mathbf{B} = 0.\tag{6}
$$

By determining the current density **j** from Ampère's Eq. (5), the expression of the Lorentz force can be presented in the form

$$\mathbf{j} \times \mathbf{B} = \frac{1}{\mu\_0} (\mathbf{B} \cdot \nabla) \mathbf{B} - \nabla \left( \frac{B^2}{2\mu\_0} \right) \mathbf{j}$$

where the first term on the right hand side is the magnetic tension force and the second term is the magnetic pressure force. Thus, momentum Eq. (2) can be rewritten in a more convenient form, notably

$$\frac{\partial(\rho \mathbf{v})}{\partial t} + \rho(\mathbf{v} \cdot \nabla)\mathbf{v} = -\nabla\left(p + \frac{B^2}{2\mu\_0}\right) + \frac{1}{\mu\_0}(\mathbf{B} \cdot \nabla)\mathbf{B} + \rho \mathbf{g}.\tag{7}$$

On the other hand, on using Ohm's law (4) the Faraday's law (or induction equation) takes the form

$$\frac{\partial \mathbf{B}}{\partial t} = -\nabla \times (\mathbf{v} \times \mathbf{B}).\tag{8}$$

Finally, the equation of the thermal energy is given by

6 Will-be-set-by-IN-TECH

In the standard nonrelativistic form the MHD equations consist of the basic conservation laws of mass, momentum, and energy together with the induction equation for the magnetic field. Thus, the MHD equations of our magnetized quasineutral plasma with singly charged ions

where *ρ* is the mass density and **v** is the bulk fluid velocity. Equation (1) is the so called

where **j** × **B** (with **j** being the current density and **B** magnetic field induction) is the *Lorentz*

where **E** is the electric field. The ideal Ohm's law for a plasma, which yields a useful relation

By determining the current density **j** from Ampère's Eq. (5), the expression of the Lorentz

(**B** · ∇)**B** − ∇

where the first term on the right hand side is the magnetic tension force and the second term is the magnetic pressure force. Thus, momentum Eq. (2) can be rewritten in a more convenient

On the other hand, on using Ohm's law (4) the Faraday's law (or induction equation) takes

 *p* + *B*2 2*μ*<sup>0</sup>  *B*<sup>2</sup> 2*μ*<sup>0</sup>

 + 1 *μ*0  ,

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> −∇ × (**<sup>v</sup>** <sup>×</sup> **<sup>B</sup>**). (8)

**<sup>j</sup>** <sup>×</sup> **<sup>B</sup>** <sup>=</sup> <sup>1</sup>

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>ρ</sup>*(**<sup>v</sup>** · ∇)**<sup>v</sup>** <sup>=</sup> −∇

*μ*0

*∂***B**

The low-frequency Ampère's law, which neglects the displacement current, is given by

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> ∇ · *<sup>ρ</sup>***<sup>v</sup>** <sup>=</sup> 0, (1)

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> −∇ × **<sup>E</sup>**, (3)

**E** + **v** × **B** = 0. (4)

*μ*0**j** = ∇ × **B**, (5)

∇ · **B** = 0. (6)

(**B** · ∇)**B** + *ρ***g**. (7)

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>ρ</sup>*(**<sup>v</sup>** · ∇)**<sup>v</sup>** <sup>=</sup> **<sup>j</sup>** <sup>×</sup> **<sup>B</sup>** − ∇*<sup>p</sup>* <sup>+</sup> *<sup>ρ</sup>***g**, (2)

*∂ρ*

(and electrons) are

The momentum equation is

Faraday's law reads

*continuity equation* in our basis set of equations.

between electric and magnetic fields, is

where *μ*<sup>0</sup> is the permeability of free space. The magnetic divergency constraint is

force can be presented in the form

*∂*(*ρ***v**)

form, notably

the form

*∂*(*ρ***v**)

*force* term, −∇*p* is the pressure-gradient term, and *ρ***g** is the gravity force.

*∂***B**

$$\frac{\mathbf{d}}{\mathbf{d}t} \frac{p}{\rho^{\gamma}} = 0,$$

where *γ* = 5/3 is the ratio of specific heats for an adiabatic equation of state. This equation usually is written as an equation for the pressure, *p*,

$$
\frac{
\partial p
}{
\partial t
} + \mathbf{v} \cdot \nabla p + \gamma p \nabla \cdot \mathbf{v} = 0. \tag{9}
$$

Equation (9) implies that the equation of state of the ideal fully ionized gas has the form

$$p = \mathcal{Z}(\rho/m\_{\mathrm{i}})k\_{\mathrm{B}}T\_{\prime}$$

where *T* is the temperature, *m*<sup>i</sup> the ion mass, *k*<sup>B</sup> is the Boltzmann constant, and the factor 2 arises because ions and electrons contribute equally.

In total the ideal MHD equations thus consist of two vector equations, (7) and (8), and two scalar equations, (1) and (9), respectively. Occasionally, when studying wave propagation in magnetized plasmas, one might also be necessary to use Eq. (6). We note that the basic variables of the ideal MHD are the mass density, *ρ*, the fluid bulk velocity, **v**, the pressure, *p*, and the magnetic induction, **B**; the electric field, **E**, has been excluded via Ohm's law.

In MHD there is a few dimensionless numbers, which are widely used in studying various phenomena in magnetized plasmas. Such an important dimensionless number in MHD theory is the plasma beta, *β*, defined as the ratio of gas pressure, *p*, to the magnetic pressure,

$$\beta = \frac{p}{B^2/2\mu\_0}.$$

When the magnetic field dominates in the fluid, *β* � 1, the fluid is forced to move along with the field. In the opposite case, when the field is weak, *β* � 1, the field is swirled along by the fluid.

We finish our short introduction to MHD recalling that in ideal MHD Lenz's law dictates that the fluid is in a sense tied to the magnetic field lines, or, equivalently, magnetic filed lines are *frozen into the fluid*. To explain, in ideal MHD a small rope-like volume of fluid surrounding a field line will continue to lie along a magnetic field line, even as it is twisted and distorted by fluid flows in the system. The connection between magnetic field lines and fluid in ideal MHD fixes the topology of the magnetic field in the fluid.
