On the Stabilization of a Twisted Magnetic Flux Tube

The linear magnetohydrodynamic stability of a shielding magnetic flux rope with a surface current under coronal solar conditions is analyzed in the framework of an energy principle. The equation describing the potential energy change induced by disturbances of the equilibrium was derived. It has been shown that the surface reverse current shielding the azimuthal component of the magnetic field lines outside a flux rope stabilizes the development of kink- and flute-type instabilities in the long-wavelength limit independently of the cross-sectional radial profile of current density. Kink modes are the most unstable ones as their generation requires less energy than other modes. Based on the obtained dispersion relation for kink oscillations, we proposed a new expression for the determination of magnetic field components of the twisted loop.


Introduction
The magnetic flux tube and current sheets are the main "building blocks" of solar magnetic configurations (see, e.g., Parker 1979;Priest 1984;Priest & Forbes 2000;Ryutova 2018). The magnetohydrodynamic (MHD) instabilities of magnetic flux ropes (i.e., twisted flux tubes) are partially responsible for plasma heating, particle acceleration, and wave generation at different spatio-temporal scales. Kink instability is a subject of special interest since there is much evidence that this instability can be responsible for flare energy release and coronal mass ejections (see, e.g., Sakurai 1976; Török & Kliem 2005;Myers et al. 2015;Chen 2017). However, the question about the stability conditions of flux ropes is not fully resolved yet (see, e.g., Priest 1984;Hassanin et al. 2016).
Previously, the kink instability of a plasma cylinder with a longitudinal electric current was investigated theoretically by Kruskal & Schwarzschild (1954) and Leontovich & Shafranov (1961). It has been shown that a flux rope with length L, radius a, and azimuthal and longitudinal magnetic field components B j and B z , respectively, becomes unstable with respect to kink perturbations if the total twist angle of magnetic field lines ( ) ( ) F = j LB a aB a z exceeds some critical value Φ c =2π. This value corresponds only to one full rotation of the magnetic field line. For solar coronal loops, which we can consider as straight cylinders, the aspect ratio a/L=1 (see, e.g., Nakariakov et al. 1999), and therefore the azimuthal component of magnetic field B j , is B j =B z . Although the Kruskal-Schafranov condition has been obtained for a flux rope with an electric current surrounded by vacuum, with an accuracy of the coefficient the critical twist angle (Φ c =2.9π) is still valid for solar coronal loops anchored at footpoints (see, e.g., Hood & Priest 1981;Priest 1984). Meanwhile, Srivastava et al. (2010) have found the signature of a highly twisted coronal loop in the active region by using satellite multiwavelength observations. The observed coronal loop showed a strong right-handed twist in the chromospheric and coronal images with a total twist angle for the whole loop Φ∼12π. In turn, Liu et al. (2019), based on the study of twist angles of magnetic field lines released by 30 off-limb rotational solar coronal jets, has found that the maximum Φ≈9.4π and that the kink instability threshold in the solar atmosphere should not be a constant.
A number of theoretical studies have shown that magnetic flux tubes are responsible for the variety of observable solar phenomena that should be isolated in the photosphere, i.e., they are surrounded by the field-free plasma (see, e.g., Ryutov & Ryutova 1976;Parker 1979;Spruit 1981;Priest 1984;Fan 2009;Weber & Browning 2016;Tsap et al. 2018). Parker (1996) has shown that the total current across a cross section of the photospheric magnetic flux tube is equal to zero due to the presence of the surface current. In particular, this suggests that the presence of nonvanishing vertical currents is an artifact caused by the limited spatial resolution of the observing magnetograph. In contrast, Melrose (1991Melrose ( , 1995Melrose ( , 1996 has found arguments against these representations. Some observational and numerical simulation results show evidence in favor of nonneutralized electric currents in the active regions of the solar photosphere (see, e.g., Georgoulis 2018;Schmieder & Aulanier 2018;Russell et al. 2019). At the same time, we cannot exclude the fact that partially isolated magnetic flux ropes with a shielding longitudinal surface current also exist (see also Wilkinson et al. 1992;Wheatland 2000;Schmieder & Aulanier 2018). Sakurai (1976) has applied the method of normal modes to analyze the kink stability of a filament (an isolated twisted magnetic flux tube). However, the external magnetic field and gas pressure were not considered. Linton et al. (1996) have taken into account the external gas pressure and studied the linear stability of an isolated flux rope below the solar photosphere within the linear approximation. Based on an energy principle the authors concluded that kink instability consists mainly of internal motions while the helical translations of the entire tube are stable. It was suggested that the plasma β?1 which is unusual in the lower solar corona. The partially isolated magnetic flux ropes with oppositely directed volume currents were studied previously by Mikic et al. (1990). By using numerical calculations, it was shown that the magnetic flux rope becomes unstable if the twist angle Φ>4.8π. Recently, Cheremnykh et al. (2017) have studied incompressible perturbations by using the normal modes method. It was found that in the long-wavelength limit the eigenfrequency of the kink mode (m=1) is real and completely unaffected by the choice of internal background magnetic twist. Later, Cheremnykh et al. (2018) showed that the stability criterion of the m=1 mode is independent of the background azimuthal components of the plasma velocity and magnetic field. The results of studies mentioned above show that the reverse current inside a flux rope should stabilize the development of kink instability. However, the proposed approaches had some essential restrictions associated with the mathematical difficulties caused by the complexity of either the magnetic configuration (Mikic et al. 1990;Linton et al. 1996) or calculation methodologies (Cheremnykh et al. 2017).
The main aim of this work is to investigate the stability of the partially isolated magnetic flux rope with a shielding surface current under solar coronal conditions on the basis of an energy principle (Bernstein et al. 1958). This method has been used previously for theoretical investigation of the solar coronal loops (e.g., Priest 1984;Hood 1986;Melville et al. 1986;Linton et al. 1996) but shielding longitudinal currents had not been considered.
The paper is organized as follows. In Section 2, we present the main idea of our approach. The procedure of minimization of the potential energy change and the search of the eigenfunctions of a flux rope are described in Section 3 and Section 4, respectively. Section 5 is devoted to the analysis of the stability of kink modes and application of obtained theoretical results. Discussions and conclusions are provided in Section 6.

An Energy Principle for a Flux Rope with Sharp Boundary
In the cgs system of units the equation of plasma motion, induction equation, continuity, and energy balance in a singlefluid ideal MHD can be represented as where ρ is the plasma mass density,v is the plasma velocity, p is the gas pressure,g is the the acceleration of gravity, γ=5/3 is the ratio of specific heat, and the electric current density . 5 LetS be the displacement vector of the elementary plasma volume from the equilibrium. Then, taking into account the equilibrium equation as well as Equations (2)-(5), the linearized Equation (1) can be written as (see, e.g., Bernstein et al. 1958):ˆ{ Here, the differential operatorK depends on the equilibrium values of the gas pressure p, plasma density ρ, and magnetic fieldB:ˆ{ Bernstein et al. (1958) have shown that the differential operatorK is Hermitian, i.e., the following equality should be satisfiedˆ{ for the arbitrary eigenfunctions h and x. By multiplying Equation (7) in a scalar way by the time derivative of the vector displacement  S and by integrating it over the plasma volume V, together with Equation (9), we arrive at where changes in kinetic T and potential W energies arê The equation of energy conservation (10) shows that any arbitrary perturbation, with a decrease of potential energy (W<0), leads to the increase of kinetic energy T and, therefore, to the development of MHD instability. In contrast, the system is stable if a change of energy W>0. This is the main idea of an energy principle (see also Sakurai 1976). Such an approach allows us to obtain useful information about stability conditions without the consideration of eigenfunc-tionS in detail.
By assuming thatS can be represented as instead of Equation (7) we havê Therefore, the problem of MHD stability focuses on finding the eigenvalues λ. Equation (9) suggests that the eigenvalues λ should be real. The negative eigenvalues of λ correspond to the instability growth rate | | l G = while the positive ones describe the frequency l W = of MHD eigenmodes. Investigation of the MHD stability of magnetic plasma configurations reduces to the finding of eigenvalues λ of the linear differential operatorK.
The vectors can be found analytically for a few limited cases by using the method of normal modes (e.g., Priest 1984; Goedbloed & Poedts 2004). Therefore, an energy principle is more convenient for the analysis of instability conditions. By multiplying Equation (11) bys and integrating over the plasma volume, we can find the eigenvalues as From Equation (12), it follows that smallest positive eigenvalues λ correspond to the most feasible oscillations. These eigenvalues can be identified by applying the Rayleigh-Ritz method (see, e.g., Dungey 2016).
For the sharp plasma-plasma boundary which separates internal (i) and external (e) regions by surface σ, from Equation (12) we have (Tsap et al. 2006(Tsap et al. , 2008 , , n wheren is the normal to the boundary surface σ. The equation of the pressure balance should be satisfied at the boundary surface σ By neglecting gravity, from Equation (15) for K we can conclude that the main source of instability is the electric currentj (Ampere force) since terms with the gas pressure p should be equal to zero for the most critical modes. Indeed, the vectors can be expressed in terms of longitudinals P and perpendiculars ⊥ components in respect to the direction of magnetic fieldB  = +ŝ s s .
As a result, is easy to show (see, e.g., Zagorodny & Cheremnykh 2014) thats P is only present in the term The incompressible perturbations are most unstable when the values of W i and W e in Equation (14) are minimal, i.e., when ∇s=0. It can be achieved by choosings P . By using a cylindrical coordinate system (r, j, z), let us consider a magnetic flux rope with a shielding surface current ( Figure 1). The proposed model suggests that the equilibrium magnetic field is As it follows from expression (17) we consider the cylinder boundary as an MHD discontinuity with a surface current. Note that the proposed model reduces to a partially isolated flux rope because the azimuthal component of the magnetic field B j is equal to zero outside of the tube. In our study we take into account the surface potential energy W σ , which is usually not considered in a laboratory plasma. By neglecting gravity, Equations (5) and (6) give: According to the Equations (15) and (18) The displacements can be represented as  (19) and (20) into Equation (14), we obtain Since W σ >0, the surface longitudinal currents stabilize the development of MHD instabilities. Moreover, stabilization depends on the value of the azimuthal component of the magnetic field B j (r) at the boundary only.

Minimization of W i and W e Energies for a Flux Rope
The condition  = s 0 allows us to minimize the potential energies over the internal V i and external V e volumes in respect to s j and s z . By using the equation for a perturbed magnetic field which follows from the induction Equation (2), and Equation (20), we arrive at Then, using Equations (22) and excluding s j and s z by means of Equation (25), from Equation (15) follows This expression has the minimum value at  (14) and (28), the minimization procedure leads to the following (see also Newcomb 1960) Note that the last equation by means of the equilibrium (i.e., Equation (18)) can be reduced to the expression obtained previously by Shafranov (1970)  The potential energy outside a flux rope is

The Eigenfunctions
In order to find the expressions for eigenfunctions we should find solutions of the Euler-Lagrange equations (see e.g., Shafranov 1970): The solution of differential Equation (36), which describes displacement s r outside of the flux rope, is For the kink mode (m = 1), the solution (37) reduces to (see also Cheremnykh et al. 2017) , 38 r r 2 and will be used in the next section. According to Equations (33) and (34), the equation for s r inside the flux rope in the long-wavelength limit, e.g., ka=1, takes the following form It should be stressed that the points wherekB i =0 are usually called resonant points (Shafranov 1970 Note that in the Equation (30), the term proportional to ds dr r under the integral sign is equal to zero in this case.
To determine the condition of the absence of resonant points let us rewrite Equation (40) in the following form If Φ(r) is a monotonously decreasing function of r and by taking into account that the minimum value of n=1, the resonant point inside the magnetic flux tube is absent at and, therefore, the resonance effects can be ignored (see also Cheremnykh et al. 2017). For the general case, when the resonant point r sn inside the magnetic flux rope is present, from Equation (42) it is easy to show that the expression for W i can be minimized if

The Stability of the Kink Mode
Let us obtain the expression for the total potential energy W at m=1 by assuming the arbitrary distribution of the current j(r) over the cross section of the magnetic flux rope. By substituting solution (38) into Equation (32), the potential energy outside a flux rope can be represented as As it follows from Equations (21) and (44), the values of W σ and W e are positive and, therefore, the external and surface perturbations should stabilize the development of kink instability, which can be associated only with the internal potential energy W i .
Since the term with f i under the sign of integral for W i in Equation (30)  where B zi =B zi (a). Thus, the magnetic flux rope will remain stable in respect to the kink modes for arbitrary values of twist angles Φ in the long-wavelength limit.
In accordance with Equation ( Equation (48) coincides with numerical and analytical results obtained previously by Bennett et al. (1999) and Cheremnykh et al. (2017), respectively. However, we did not impose any restriction on the dependence of the longitudinal B zi component (i.e., on j j current component) on r whereas according to Bennett et al. (1999) and Cheremnykh et al. (2017) it should be constant. Equation (48) also coincides with the dispersion relation for the untwisted thin magnetic flux tube (e.g., Spruit 1982;Roberts et al. 1984;Tsap & Kopylova 2001). This equation was proposed by Nakariakov & Ofman (2001) to determine the absolute value of the magnetic field strength B ut in untwisted coronal loops at low plasma beta where T is the observed period of transverse oscillations. For the typical magnetic loop parameters the estimated magnetic field B ut turned out to be equal to 4-30 G (see, e.g., Nakariakov & Ofman 2001;Nakariakov & Verwichte 2005). Equation (49) provides us the value of the magnetic field in an assumption that no magnetic twist is present.
As it follows from our analysis the dispersion relation (48) can also be used for magnetic field diagnostics of twisted coronal loops. In fact, the equation of the pressure balance at the boundary r=a in case of β=1 in accordance with the Equation (16) can be written as  Therefore, according to Equations (49) and (53), the ratio between B ut and B tw is a function of the total twist angle Φ and can be represented as The dependence of ratios for longitudinal (B B zi zi ut tw ) and absolute (B B ut tw ) magnetic fields as the function of angle Φ calculated by using Equations (52) and (54) for the aspect ratios a/L=0.05, 0.1, and 0.2 are presented in Figure 3. It is clearly seen that these ratios depend on Φ significantly. Following observational results reported by Srivastava et al. (2010), for the estimated twist angle Φ≈12π and the aspect ratio a/L=0.05, from Equations (52)  and B ut /B tw ≈1.28.

Discussion and Conclusions
In the present paper, the kink stability of a shielding magnetic flux rope with a surface current has been studied. This type of magnetic configuration may be generated, for example, by the vortex plasma motions (see, e.g., Kitiashvili et al. 2012Kitiashvili et al. , 2013Giagkiozis et al. 2018;Snow et al. 2018;Murawski et al. 2018) accompanied by the slight penetration of magnetic field lines into the external region due to the ohmic diffusion. At the same time, the questions on physical mechanisms responsible for generation of surface neutralized and nonneutralized electric currents remain open (e.g., Dalmasse et al. 2015;Georgoulis 2018;Schmieder & Aulanier 2018).
We have used an energy principle approach which allowed us to analyze the general properties of kink and other MHD modes in the presence of surface current and obtain the exact expressions for eigenfunctions and eigenvalues which usually can be found by using the normal modes method. Following Parker (1996) we limited our study to the model of a magnetic flux rope with a surface current shielding the azimuthal component of magnetic field outside the tube. In particular, an energy principle permitted us to generalize results obtained by Cheremnykh et al. (2017) and conclude that surface currents may stabilize the kink (m = 1) and fluting (m>1) modes under coronal conditions in the long-wavelength limit. This result is independent of the radial profile of the current density. It supports a number of flare models which include connection of the energy release with nonneutralized electric currents (e.g., Kliem & Török 2006;Stepanov & Zaitsev 2016).
The calculated estimates suggest that kink modes should be dominant under coronal conditions since a minimal amount of energy is needed for their generation. In particular, it explains the ubiquitous character of decayless kink oscillations in the solar corona (see, e.g., Nisticò et al. 2013;Anfinogentov et al. 2015). We have also shown that the dispersion relation (48), which is often used for the magnetic field diagnostics in untwisted coronal loops, remains true for twisted ones.
Our findings also indicate that surface currents may play a significant role in the dynamics of magnetic flux tubes. The detection of these currents is an important practical task as they may completely stabilize the development of kink instability. New high-resolution spatial observations of the dynamic magnetic field by the Daniel K. Inouye Solar Telescope should be very helpful for a better understanding of this process.
Y.T., A.S., and Y.K. were partially supported by the Russian Foundation for Basic    As a result, from Equations (13), (21), (A2), and (A3) we get the square of eigenfrequencies for a flux rope with the uniform electric current density j z in the long-wavelength limit:  Cheremnykh et al. (2017) using the method of normal modes and the dispersion relation (48) at m=1. In addition, from Equation (A4) it follows that the excitation of eigenmode with m=1 will be accompanied by the minimum potential energy change.