Solar jets are ubiquitous in the solar atmosphere and recent observations have revealed that they are related to small scale filament eruptions. They are continuously observed by the Extreme-ultraviolet Imaging Spectrometer (EIS) (Culhane et al., 2007) on board Hinode (Kosugi et al., 2007) satellite, Atmospheric Imaging Assembly (AIA) (Lemen et al., 2012), on board the Solar Dynamics Observatory (SDO) (Pesnell et al., 2012), as well as from the Interface Region Imaging Spectrograph (IRIS) (De Pontieu et al., 2014) alongside the Earth-based solar telescopes. The physical parameters of various kinds of solar jets have been reported in a series of articles (see for instance, Schmieder et al., 2013; Sterling et al., 2015; Panesar et al., 2016a; Chandra et al., 2017; Joshi et al., 2017, and references cited in). It was established that more of the solar jets possess rotational motion. Such tornado-like jets, termed macrospicules, were firstly detected in the transition region by Pike and Mason (1998) using observations by the Solar and Heliospheric Observatory (SOHO) (Domingo et al., 1995). Rotational motion in macrospicules was also explored by Kamio et al. (2010), Curdt and Tian (2011), Bennett and Erdélyi (2015), Kiss et al. (2017, 2018). Type II spicules, according to De Pontieu et al. (2012) and Martínez-Sykora et al. (2013), along with the coronal hole EUV jets (Liu et al., 2009; Nisticò et al., 2009, 2010; Shen et al., 2011; Chen et al., 2012; Hong et al., 2013; Young and Muglach, 2014a,b; Moore et al., 2015), and X-ray jets (Moore et al., 2013), can rotate, too. Rotating EUV jet emerging from a swirling flare (Zhang and Ji, 2014) or formed during a confined filament eruption (Filippov et al., 2015) confirm once again the circumstance that the rotational motion is a common property of many kinds of jets in the solar atmosphere.

The first scenario for the numerical modeling of hot X-ray jets was reported by Heyvaerts et al. (1997) and the basic idea was that a bipolar magnetic structure emerges into a unipolar pre-existing magnetic field and reconnects to form hot and fast jets that are emitted from the interface between the fields into contact. Later on, by examining many X-ray jets in Hinode/X-Ray Telescope coronal X-ray movies of the polar coronal holes, Moore et al. (2010) found that there is a dichotomy of polar X-ray jets, namely “standard” and “blowout” jets exist. Fang et al. (2014) studied the formation of rotating coronal jets through numerical simulation of the emergence of a twisted magnetic flux rope into a pre-existing open magnetic field. Another scenario for the nature of solar jets was suggested by Sterling et al. (2015), according to which the X-ray jets are due to flux cancellation and/or “mini-eruptions” rather than emergence. An alternative model for solar polar jets due to an explosive release of energy via reconnection was reported by Pariat et al. (2009). Using three-dimensional MHD simulations, the authors demonstrated that this mechanism does produce massive, high-speed jets. In subsequent two articles (Pariat et al., 2015, 2016), Pariat and co-authors, presented several parametric studies of a three-dimensional numerical MHD model for straight and helical solar jets. On the other side, Panesar et al. (2016b) have shown that the magnetic flux cancellation can trigger the solar quiet-region coronal jets and they claim that the coronal jets are driven by the eruption of a small-scale filament, called a “minifilament.” The small-scale chromospheric jets, like microspicules, were first numerically modeled by Murawski et al. (2011). Using the FLASH code, they solved the two-dimensional ideal MHD equations to model a macrospicule, whose physical parameters match those of a solar spicule observed. Another mechanism for the origin of macrospicules was proposed by Kayshap et al. (2013), who numerically modeled the triggering of a macrospicule and a jet.

It is natural to expect, that solar jets, being magnetically structured entities, should support the propagation of different type of MHD waves: fast and slow magnetoacoustic waves and torsional Alfvén waves. All these waves are usually considered as normal MHD modes traveling along the jet. Owing to the presence of a velocity shear near the jet–surrounding plasma interface, every jet can become unstable and the most universal instability which emerges is the Kelvin–Helmholtz (KH) one. The simplest configuration at which one can observe the KHI is the two semi-infinite incompressible magnetized plasmas flowing with different velocities provided that the thin velocity shear at the interface exceeds some critical value (Chandrasekhar, 1961). Recently, Cheremnykh et al. (2018a) theoretically established that shear plasma flows at the boundary of plasma media can generate eight MHD modes, of which only one can be unstable due to the development of the KHI. Ismayilli et al. (2018) investigated a shear instability of the KH type in a plasma with temperature anisotropy under the MHD approximation. The KHI of the magnetoacousic waves propagating in a steady asymmetric slab, and more specifically the effect of varying density ratios was explored by Barbulescu and Erdélyi (2018). A very good review on the KHI in the solar atmosphere, solar wind, and geomagnetosphere in the framework of ideal MHD the reader can find in Mishin and Tomozov, 2016.

In cylindrical geometry, being typical for the solar jets, the KHI exhibits itself as a vortex sheet running on the jet–environment boundary, which like in the flat geometry, is growing in time if the axial velocity of the jet in a frame of reference attached to the surrounding plasma exceeds a threshold value (Ryu et al., 2000). In its non-linear stage, the KHI trigger the wave turbulence which is considered as one of the main heating mechanisms of the solar corona (Cranmer et al, 2015). The development of the KHI in various cylindrical jet–environment configurations has been studied in photospheric jets (Zhelyazkov and Zaqarashvili, 2012), in solar spicules (Zhelyazkov, 2012; Ajabshirizadeh et al., 2015; Ebadi, 2016), in high-temperature and cool solar surges (Zhelyazkov et al., 2015a,b), in magnetic tubes of partially ionized compressible plasma (Soler et al., 2015), in EUV chromospheric jets (Zhelyazkov et al., 2016; Bogdanova et al., 2018), in soft X-ray jets (Vasheghani Farahani et al., 2009; Zhelyazkov et al., 2017), and in the twisted solar wind flows (Zaqarashvili et al., 2014). A review on KHI in the solar atmosphere, including some earlier studies, the reader can find in Zhelyazkov (2015).

The first modeling of the KHI in a rotating cylindrical magnetized plasma jet was done by Bondenson et al. (1987). Later on, Bodo et al. (1989, 1996) carried out a study of the stability of flowing cylindrical jet immersed in constant magnetic field B₀. The authors used the standard procedure for exploring the MHD wave propagation in cylindrical flows considering that all the perturbations of the plasma pressure p, fluid velocity v, and magnetic field B, are ∝exp[i(−ωt + kz + mθ)]. Here, ω is the angular wave frequency, k the propagating wavenumber, and m the azimuthal mode number. Using the basic equations of ideal magnetohydrodynamics, Bodo et al. (1989, 1996) derived a Bessel equation for the pressure perturbation and an expression for the radial component of the fluid velocity perturbation. The found solutions in both media (the jet and its environment) are merged at the perturbed tube boundary through the conditions for continuity of the total (thermal plus magnetic) pressure and the Lagrangian displacement. The latter is defined as the ratio of radial velocity perturbation component and the angular frequency in the corresponding medium. The obtained dispersion relation is used for examining the stability conditions of both axisymmetric, m = 0 (Bodo et al., 1989), and non-axisymmetric, |m| ⩾ 1 modes (Bodo et al., 1996). In a recent article, Bodo et al. (2016) performed a linear stability analysis of magnetized rotating cylindrical jet flows in the approximation of zero thermal pressure. They focused their analysis on the effect of rotation on the current driven mode and on the unstable modes introduced by rotation. In particular, they found that rotation has a stabilizing effect on the current driven mode only for rotation velocities of the order on the Alfvén speed. The more general case, when both the magnetic field and jet flow velocity are twisted, was studied by Zaqarashvili et al. (2015) and Cheremnykh et al. (2018b), whose dispersion equations for modes with m ⩾ 2, represented in different ways, yield practically identical results.

The main goal of this review article is to suggest a way of using the wave dispersion relation derived in Zaqarashvili et al. (2015) to study the possibility for the rising and development of KHI in rotating twisted solar jets. Among the enormous large number of observational studies of rotating jets with different origin or nature, we chose those which provide the magnitudes of axial and rotational speeds, jet width and height alongside the typical plasma parameters like electron number densities and electron temperature of the spinning structure and its environment. Thus, the targets of our exploration are: (i) the spinning coronal hole jet of 2010 August 21 (Chen et al., 2012); (ii) the rotating coronal hole jet of 2011 February 8 (Young and Muglach, 2014a), (iii) the twisted rotating jet emerging from a filament eruption on 2013 April 10–11 (Filippov et al., 2015), and (iv) the rotating macrospicule observed by Pike and Mason (1998) on 1997 March 8.

The paper is organized as follows: in the next section, we discuss the geometry of the problem, equilibrium magnetic field configuration and basic physical parameters of the explored jets. Section 3 is devoted to a short, concise, derivation of the wave dispersion relation. Section 4 deals with numerical results for each of the four jets and contain the available observational data. In the last section 5, we summarize the main findings in our research and outlook the further improvement of the used modeling.

We model whichever jet as an axisymmetric cylindrical magnetic flux tube with radius a and electron number density n_i (or equivalently, homogeneous plasma density ρ_i) moving with velocity U. We consider that the jet environment is a rest plasma with homogeneous density ρ_e immersed in a homogeneous background magnetic field B_e. This field, in cylindrical coordinates (r, ϕ, z), possesses only an axial component, i.e., B_e = (0, 0, B_e) (Note that the label “i” is abbreviation for interior, and the label “e” denotes exterior). The magnetic field inside the tube, B_i, and the jet velocity, U, we assume, are uniformly twisted and are given by the vectors

respectively. We note, that B_iz and U_z, are constant. Concerning the azimuthal magnetic and flow velocity components, we suppose that they are linear functions of the radial position r and evaluated at r = a they correspondingly are equal to B_iϕ(a) ≡ B_ϕ = Aa and U_ϕ = Ωa, where A and Ω are constants. Here, Ω is the jet angular speed, deduced from the observations. Hence, in equilibrium, the rigidly rotating plasma column, that models the jet, must satisfy the following force-balance equation (see, e.g., Chandrasekhar, 1961; Goossens et al., 1992)

where μ is the plasma permeability and pt=pi+Bi2/2μ with Bi2=Biϕ2(r)+Biz2 is the total (thermal plus magnetic) pressure. According to Equation (2), the radial gradient of the total pressure should balance the centrifugal force and the force owing to the magnetic tension. After integrating Equation (2) from 0 to a, taking into account the linear dependence of U_ϕ and B_iϕ on r, we obtain that

where pt(0)=p1(0)+Biz2/2μ. Integrating Equation (2) from 0 to any r one can find the radial profile of p_t inside the tube. Such an expression of p_t(r), obtained, however, from an integration of the momentum equation for the equilibrium variables, have been obtained in Zhelyazkov et al. (2018)—see Equation (2) there. It is clear from a physical point of view that the internal total pressure (evaluated at r = a) must be balanced by the total pressure of the surrounding plasma which implies that

This equation can be presented in the form

where p₁(0) is the thermal pressure at the magnetic tube axis, and p_e denotes the thermal pressure in the environment. In the pressure balance Equation (3), the number ε₁ ≡ B_ϕ/B_iz = Aa/B_iz represents the magnetic field twist parameter. Similarly, we define ε₂ ≡ U_ϕ/U_z as a characteristics of the jet velocity twist. We would like to underline that the choice of plasma and environment parameters must be such that the total pressure balance Equation (3) is satisfied. In our case, the value of ε₂ is fixed by observationally measured rotational and axial velocities while the magnetic field twist, ε₁, has to be specified when using Equation (3). We have to note that Equation (3) is a corrected version of the pressure balance equation used in Zhelyazkov et al. (2018) and Zhelyazkov and Chandra (2018).

From measurements of n and T for similar coronal hole EUV jets (Nisticò et al., 2009, 2010), we take n inside the jet to be ni=1.0×109 cm⁻³, and assume that the electron temperature is T_i = 1.6 MK. The same quantities in the environment are, respectively, ne=0.9×109 cm⁻³ and T_e = 1.0 MK. Note that the electron number density of the blowout jet observed by Young and Muglach (2014a) is in one order lower. The same applies for its environment. We consider that the background magnetic field for both hole coronal jets is B_e = 3 G. The values of n and T of the rotating jet emerging from a filament eruption, observed by Filippov et al. (2015), were evaluated by us and they are ni=4.65×109 cm⁻³ and T_i = 2.0 MK, respectively. From the same data set, we have obtained ne=4.02×109 cm⁻³ and T_e = 2.14 MK. The background magnetic field, B_e, with which the pressure balance Equation (3) is satisfied, is equal to 6 G. For the rotating macrospicule we assume that ni=1.0×1010 cm⁻³ and ne=1.0×109 cm⁻³ to have at least one order denser jet with respect to the surrounding plasma. Our choice for macrospicule temperature is Ti=5.0×105 K, while that of its environment is supposed to be Te=1.0×106 K. The external magnetic field, B_e, was taken as 5 G. All aforementioned physical parameters of the jets are summarized in Table 1. The plasma beta was calculated using (6/5)cs2/vA2, where cs=(γkBT/mion)1/2 is the sound speed (in which γ = 5/3, k_B is the Boltzmann's constant, T the electron temperature, and m_ion the ion or proton mass), and vA=B/(μnionmion)1/2 is the Alfvén speed, in which expression B is the full magnetic field =(Bϕ2+Bz2)1/2, and n_ion is the ion or proton number density.

A dispersion relation for the propagation of high-mode (m ⩾ 2) MHD waves in a magnetized axially moving and rotating twisted jet was derived by Zaqarashvili et al. (2015) and Cheremnykh et al. (2018b). That equation was obtained, however, under the assumption that both media (the jet and its environment) are incompressible plasmas. As seen from the last column in Table 1, plasma beta is greater than 1 in the first, third, and fourth jets which implies that the plasma of each of the aforementioned jets can be considered as a nearly incompressible fluid (Zank and Matthaeus, 1993). It is seen from the same table that the plasma beta of the second jet is less than one as is in each of the jet environments and that is why it is reasonable to treat them as cool media. Thus, the wave dispersion relation, derived, for instance, in Zaqarashvili et al. (2015), has to be modified. In fact, we need two modified versions: one for the incompressible jet–cool environment configuration, and other for the cool jet–cool environment configuration. We are not going to present in details the derivation of the modified dispersion equations on the basis of the governing MHD equations, but will only sketch the essential steps in that procedure. The main philosophy in deriving the wave dispersion equation is to find solutions for the total pressure perturbation, p_tot, and for the radial component, ξ_r of the Lagrangian displacement, ξ, and merge them at the tube perturbed boundary through the boundary conditions for their (p_tot and ξ_r) continuity (Chandrasekhar, 1961). In the case of the first configuration, we start with the linearized ideal MHD equations, governing the incompressible dynamics of the perturbations in the spinning jet

where v = (v_r, v_ϕ, v_z) and B = (b_r, b_ϕ, b_z) are the perturbations of fluid velocity and magnetic field, respectively, and p_tot is the perturbation of the total pressure, pt=pi+Bi2/2μ. The Lagrangian displacement, ξ, can be found from the fluid velocity perturbation, v, using the relation (Chandrasekhar, 1961)

Further on, assuming that all perturbations are ∝exp[i(−ωt + mϕ + k_zz)] and considering that the rotation and the magnetic field twists in the jet are uniform, that is,

where Ω and A are constants, from the above set of Equations (4–8), we obtain the following dispersion equation of the MHD wave with mode number m (for details see Zaqarashvili et al., 2015):

where

In above expressions, the prime means differentiation of the Bessel functions with respect to their arguments,

are the squared wave amplitude attenuation coefficients in the jet and its environment, in which

are the local Alfvén frequencies in both media, and

is the Doppler-shifted angular wave frequency in the jet. We note that in the case of incompressible coronal plasma (Zaqarashvili et al., 2015), κ_e = k_z, because at an incompressible environment the argument of the modified Bessel function of second kind, K_m, and its derivative, Km′, is k_za.

The basic MHD equations for an ideal cool plasma are, generally, the same as the set of Equations (4–8) with Equation (6) replaced by the continuity equation

Recall that for cold plasmas the total pressure reduces to the magnetic pressure only, that is pt=B12/2μ, the z component of the velocity perturbation is zero, i.e., v₁ = (v_1r, v_1ϕ, 0), while B₁ = (B_1r, B_1ϕ, B_1z). The above equation, which defines the density perturbation, is not used in the derivation of the wave dispersion relation because we are studying the propagation and stability of Alfvén-wave-like perturbations of the fluid velocity and magnetic field. Following the standard scenario for deriving the MHD wave dispersion relation (Zhelyazkov and Chandra, 2018), we finally arrive at:

where

Here, the wave attenuation coefficient in the internal medium has the form

while that in the environment, with Ω = 0 and A = 0, is given by

Note that (i) both dispersion relations, (10) and (11), have similar forms—the difference is in the expressions for the wave attenuation coefficient inside the jet, namely κic=κi[1-σ2/ωAi2]1/2; and (ii) the wave attenuation coefficients in the environments are not surprisingly the same, that is, κec=κe≡[1-ω2/ωAe2]1/2.

In studying at which conditions the high (m ⩾ 2) MHD modes in a jet–coronal plasma system become unstable, that is, all the perturbations to grow exponentially in time, we have to consider the wave angular frequency, ω, as a complex quantity: ω ≡ Re(ω) + iIm(ω) in contrast to the wave mode number, m, and propagating wavenumber, k_z, which are real quantities. The Re(ω) is responsible for the wave dispersion while the Im(ω) yields the wave growth rate. In the numerical task for finding the complex solutions to the wave dispersion relation (10) or (11), it is convenient to normalize all velocities with respect to the Alfvén speed inside the jet, defined as vAi=Biz/μρi, and the lengths with respect to a. Thus, we have to search the real and imaginary parts of the non-dimensional wave phase velocity, v_ph = ω/k_z, that is, Re(v_ph/v_Ai) and Im(v_ph/v_Ai) as functions of the normalized wavenumber k_za. The normalization of the other quantities like the local Alfvén and Doppler-shifted frequencies alongside the Alfvén speed in the environment, vAe=Be/μρe, requires the usage of both twist parameters, ε₁ and ε₂, and also of the magnetic fields ratio, b = B_e/B_iz. The non-dimensional form of the jet axial velocity, U_z, is given by the Alfvén Mach number M_A = U_z/v_Ai. Another important non-dimensional parameter is the density contrast between the jet and its surrounding medium, η = ρ_e/ρ_i. Hence, the input parameters in the numerical task of finding the solutions to the transcendental Equation (10) or (11) (in complex variables) are: m, η, ε₁, ε₂, b, and M_A. Zaqarashvili et al. (2015) have established that KHI in an untwisted (A = 0) rotating flux tube with negligible longitudinal velocity can occur if

This inequality says that each MHD wave with mode number m ⩾ 2, propagating in a rotating jet can become unstable. This instability condition can be used also in the cases of slightly twisted spinning jets, provided that the magnetic field twist parameter, ε₁, is a number lying in the range of 0.001–0.005, simply because the numerical solutions, for example, to Equation (10) show that practically there is no difference between the instability ranges at ε₁ = 0, and at 0.001 or 0.005. An important step in our study is the supposition that the deduced from observations jet axial velocity, U_z, is the threshold speed for the KHI occurrence. Then, for fixed values of m, η, U_ϕ = Ωa, v_Ai, and b, the inequality (12) can be rearranged to define the upper limit of the instability range on the k_za-axis

According to the above inequality, the KHI can occur for nondimensional wavenumbers k_za less than (k_za)_rhs. On the other hand, one can talk for instability if the unstable wavelength, λ_KH = 2π/k_z, is shorter than the height of the jet, H, which means that the lower limit of the instability region is given by:

where Δℓ is the jet width. Hence, the instability range in the k_za-space is (k_za)_lhs < k_za < (k_za)_rhs. Note that the lower limit, (k_za)_lhs, is fixed by the width and height of the jet, while the upper limit, (k_za)_rhs, depends on several jet–environment parameters. At fixed U_ϕ, v_Ai, η, and b, the (k_za)_rhs is determined by the MHD wave mode number, |m|. As seen from inequality (13), with increasing the m, that limit shifts to the right, that is, the instability range becomes wider. The numerical solutions to the wave dispersion relation (10) confirm this and for given m one can obtain a series of unstable wavelengths, λ_KH = πΔℓ/k_za, as the shortest one takes place at k_za ≈ (k_za)_rhs. For relatively small mode numbers, when even the shortest unstable wavelengths turn out to be a few tens megameters, that could hardly be associated with the observed KH ones. As observations show, the KHI vortex-like structures running at the boundary of the jet, have the size of the width or radius of the flux tube (see, for instance, Figure 1 in Zhelyazkov et al., 2018). Therefore, we have to look for such an m, whose instability range would accommodate the expected unstable wavelength presented by its nondimensional wavenumber, k_za = πΔℓ/λ_KH. An estimation of the required mode number for an ε₁ = 0.005-rotating flux tube can be obtained by presenting the instability criterion (12) in the form

We will use this inequality for obtaining the optimal m for each of the studied jets by specifying the value of that k_za (along with the other aforementioned input parameters) which corresponds to the expected unstable wavelength λ_KH.

In this article, we have studied the emerging of KHI in four different spinning solar jets (standard and blowout coronal hole jets, jet emerging from a filament eruption, and rotating macrospicule) due to the excitation of high-mode (m ⩾ 2) MHD waves traveling along the jets. First and foremost, we model each jet as a vertically moving with velocity U cylindrical twisted magnetic flux tube with radius a. There are four basic steps in the modeling as follows:

Topology of jet–environment magnetic and velocity fields For simplicity, we assume that the plasma densities of the jet and its environment, ρ_i and ρ_e, respectively, are homogeneous. Generally they are different and the density contrast is characterized by the ratio ρ_e/ρ_i = η. The twisted internal magnetic and velocity fields are supposed to be uniform, that is, represented in cylindrical coordinates, (r, ϕ, z), by the vectors B_i = (0, B_iϕ(r), B_iz) and U = (0, U_ϕ(r), U_z), where their azimuthal components are considered to be linear functions of the radial position r, viz. B_iϕ(r) = Ar and U_ϕ(r) = Ωr, where A and Ω (the azimuthal jet velocity) are constants. We note that B_iz and U_z are also constants. It is convenient the twists of the magnetic field and the flow velocity of the jet to be characterized by the two numbers ε₁ = B_iϕ(a)/B_iz ≡ Aa/B_iz and ε₂ = U_ϕ(a)/U_z ≡ Ωa/U_z, respectively. Note that Ωa is the jet rotational speed U_z. The surrounding coronal or chromospheric plasma is assumed to be immobile and embedded in a constant magnetic field B_e = (0, 0, B_e). In our study, the density contrast, η, varies from 0.1 to 0.9, the magnetic field twist, ε₁, can have a wide range of magnitudes from 0.005 to 0.95 (it has to be less than 1 in order to avoid the rising of the kink instability), while the velocity twist parameter, ε₂, is fixed by the observationally measured rotational and axial speeds.

In this article, we also corrected the total pressure balance equation used in Zhelyazkov et al. (2018) and Zhelyazkov and Chandra (2018), which turns out to be erroneous. The true total balance equation is given by Equation (3). In fact, the corrected pressure balance equation, used here, changes the mode numbers at which the KHI occurs, namely from m = 12 to m = 11 for the coronal hole jet and from m = 18 to m = 10 for the rotating jet emerging from a filament eruption. The computed KHI developing or growth times in the aforementioned articles, nonetheless, are not changed noticeably—they are of the same order with these computed in this paper. In addition, there is another improvement issue, scilicet when studying how the increasing ε₁ shortens the instability region, one has to apply Equation (3) for obtaining the appropriate values of b = B_e/B_iz and vAi=Biz/μρi, and subsequently M_A = U_z/v_Ai. Thus, one can claim that Equation (3) plays an important role in the numerical studies of the KHI in various solar jets.

Our approach in investigating the KHI in rotating twisted solar jets can be improved in the following directions: (i) to assume some radial profile of the plasma density of the jet, which immediately will require additional study on the occurrence of continuous spectra and resonant wave absorption (Goedbloed and Poedts, 2004), alongside to see to what extent these phenomena will influence the instability growth times; (ii) to investigate the impact of the non-linear azimuthal magnetic and velocity fields radial profiles on the emergence of KHI; and (iii) to derive a MHD wave dispersion relation without any simplifications like considering the jet and its environment as incompressible or cool plasmas—this will show how the compressibility will change the picture. We should also not forget that the non-linearity, as Miura and Pritchett (1982) and Miura (1984) claim, can lead to the saturation of the KHI growth, and to formation of non-linear waves. Nevertheless, even in its relatively simple form, our way of investigating the conditions under which the KHI develops is flexible enough to explore that event in any rotating solar jet in case that the basic physical and geometry parameters of the jet are provided by observations.

IZh wrote the substantial parts of the manuscript. RJ wrote the Introduction section and prepared three figures associated with the observations. RC contributed by writing the parts devoted to observations, as well as in the careful proofreading of the text.