Total and Linearly Polarized Synchrotron Emission from Overpressured Magnetized Relativistic Jets

Magnetohydrodynamical models have been computed following the approach developed by Komissarov et al. (2015) that allows to study the structure of steady, axisymmetric relativistic (magnetized) flows using 1D time-dependent simulations. This approach is based on the fact that for narrow jets (quasi-1D approach) with axial velocities close to the light speed the steady-state equations of RMHD can be accurately approximated by the 1D time-dependent equations with the axial coordinate acting as the temporal coordinate. Hence, our models are time-independent, 2D (radial-, axial-dependent) models, but they are computed as time-dependent, 1D (radial-dependent) models. Once the model has been computed, the axial dependence is recovered from the time dependence, taking into account that for highly relativistic jets, z ≈ ct, where t and z are the time and the axial coordinate, and c is the speed of light. Appendix A summarizes the main characteristics of Komissarov et al. (2015) approach.

Despite its approximate nature, using the quasi-1D approach to generate the axisymmetric steady jet models has many advantages for the present study. The most obvious is that because the models are computed at the cost of 1D calculations, the space of parameters can be swept densely. Moreover, the quasi-1D approach circumvents the inherent difficulties found by 2D time-dependent codes to reach steady-state solutions. In the case of the results presented by Martí et al. (2016) (MPG16, from now on) these difficulties led to (i) limit the length of the steady models (to, e.g., 40 jet radii in low Mach number models), and (ii) introduce a wide shear layer to damp the growth of instabilities in the transient phase in both magnetically dominated and kinetically dominated jet models. Using the quasi-1D approach allowed us to compute long enough jet models (specifically, 100 jet radii long) and avoid the use of wide shear layers. On the other hand, among the main drawbacks of using the quasi-1D approximation is that it cannot properly describe the flow dynamics at the jet/ambient medium interface as this interface is handled artificially to fix the boundary conditions every integration step.

The stationary models have been generated according to the procedure described in MPG16 (see also Martí 2015b). Axially symmetric, non-rotating, steady jet models are characterized by five functions, namely the jet density and pressure (ρ(r), and p(r), respectively); the jet axial velocity, v(r); and the toroidal and axial components of the jet magnetic field (B^ϕ(r), and B^z(r), respectively), whereas the static unmagnetized ambient medium is characterized by a constant pressure, p_a, and a constant density, ρ_a. As discussed in the previous references, the equation of transversal equilibrium allows one to find the equilibrium profile for any of the functions, in particular the jet pressure, in terms of the others. As in these references, we have chosen top-hat profiles for the density, axial flow velocity and axial magnetic field and considered a particular profile for the toroidal component of the magnetic field. Taking all of this into account, once fixed the equation of state (which we assume to be the one corresponding to a perfect gas with constant adiabatic index 4/3), the jet models are characterized by six parameters, namely the constant values of the jet density (ρ_j), and axial flow velocity (v_j), and the averaged values of the jet overpressure factor, K, the internal (relativistic) magnetosonic Mach number, ${{ \mathcal M }}_{\mathrm{ms},j}$, the jet magnetization, β_j, and the magnetic pitch angle, ϕ_j. The selection of the parameters defining the jet models is justified by their role in the characterization of the jet internal structure. Appendix B summarizes the procedure to build the jet models from the chosen set of parameters. Parameters ρ_j, v_j, K, and ϕ_j have the same fixed values as in MPG16 (0.005ρ_a, 0.95c, 2, and 45° respectively). Table 1 lists the values of the remaining parameters defining the models which are also displayed on the magnetosonic Mach number-specific internal energy diagram shown in Figure 1. The magnetosonic Mach number covers the same range of variation (2.0–10.0) as in MPG16 whereas the interval of the jet magnetization has been expanded to cover models with passive magnetic fields (β_j = 0.01), as well as models with the maximum allowed magnetizations (compatible with a positive gas pressure at the jet surface; β_j ≈ 17.5 for the current choice of jet parameters). Hot jets, magnetically dominated jets, and kinetically dominated jets occupy different regions in the Mach number-specific internal energy diagram (see Figure 1 caption). According to this, models M2B1 and M3B1 are hot; M1B1, M1B2 and M1B3 are magnetically dominated; and M4B2, M4B3, M4B4, M5B1, M5B2, M5B3, and M5B4 are kinetically dominated. The remaining models are hybrid: M2B2, between hot and magnetically dominated jets; M2B3 and M2B4, between magnetically dominated and kinetically dominated jets; and M4B1, between hot and kinetically dominated jets. As in MPG16, the transition between the jet and the ambient medium is smoothed by means of a shear layer of width Δr_sl by convolving the sharp jumps at the jet surface with the function sech(r^m) for some integer m. However, unlike that paper, where uncomfortably wide shear layers had to be enforced to stabilize the jets against pinch instabilities, a thin shear layer (m = 16, Δr_sl ≈ 0.12) has been imposed in all the present models.

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Table 1.  Parameters Defining the Overpressured Jet Models

Model	${{ \mathcal M }}_{\mathrm{ms},j}$	βj	${\varepsilon }_{j}\,[{c}^{2}]$	${\beta }_{j}{\varepsilon }_{j}\,[{c}^{2}]$	K1	${p}_{a}\,[{\rho }_{a}{c}^{2}]$
M1B1	2.0	2.77	10.0	27.7	1.87	3.31 × 10−2
M1B2	2.0	5.0	1.34	6.7	1.85	6.72 × 10−3
M1B3	2.0	17.5	0.230	4.03	1.83	3.55 × 10−3
M2B1	3.5	0.45	10.0	4.5	1.94	1.21 × 10−2
M2B2	3.5	1.0	1.72	1.72	1.91	2.87 × 10−3
M2B3	3.5	5.0	0.243	1.22	1.85	1.22 × 10−3
M2B4	3.5	17.5	0.0661	1.16	1.83	1.02 × 10−3
M3B1	4.505	0.01	10.0	0.1	2.00	8.42 × 10−3
M4B1	6.0	0.01	1.16	0.0116	2.00	9.72 × 10−4
M4B2	6.0	1.0	0.291	0.291	1.91	4.86 × 10−4
M4B3	6.0	5.0	0.0724	0.362	1.85	3.62 × 10−4
M4B4	6.0	17.5	0.0216	0.378	1.83	3.33 × 10−4
M5B1	10.0	0.01	0.251	0.00251	2.00	2.11 × 10−4
M5B2	10.0	1.0	0.0900	0.0900	1.91	1.50 × 10−4
M5B3	10.0	5.0	0.0250	0.125	1.85	1.25 × 10−4
M5B4	10.0	17.5	0.00770	0.135	1.83	1.19 × 10−4

Note. Tabulated data denote jet model, (relativistic) magnetosonic Mach number, magnetization, specific internal energy, specific magnetic energy, overpressure factor at the jet surface, and ambient medium pressure, in that order.

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It is important to note that as all of the models have identical jet rest-mass density and axial flow velocity, all have the same kinetic energy flux in the limit of zero internal energy (cold jets) and zero magnetization. This means that the jets' total energy flux is different from model to model, and increases for those with increasing internal and magnetic energy densities (in practice, increasing ε_j and increasing ε_j β_j). The ambient pressure follows the same trend. Because it sets the condition for the jet transversal equilibrium (for fixed jet average overpressure factor), the ambient pressure increases for increasing jet total pressure (or, again, for increasing ε_j and increasing ε_j β_j).

Finally, two out of the 16 models analyzed in the present paper coincide with those in MPG16, namely M1B1 (PH02) and M2B1 (HP03). The comparison of the original 2D models with the corresponding quasi-1D models helps us to gauge the quality of the approximation considered in the present paper (see Appendix A).

In all of the models, the equilibrium of the jet against the underpressured ambient medium is established by a series of standing oblique shocks (recollimation shocks) and gentle expansions and compressions of the jet flow. On the other hand, a jet propagating through a pressure-decreasing atmosphere with a steep enough gradient would lose the internal shock structure after few periods, due to the sideways jet expansion and the corresponding decrease in energy flux per unit area (e.g., Gómez et al. 1995). However, the fact that in our models the ambient medium is homogeneous helps to keep this internal structure periodic. The expansions and compressions produce a net toroidal component of the Lorentz force, which causes the growth of nonzero toroidal flow speeds (of the order of a few percent³ ). Superimposed to these periodical structures, as a result of both the magnetic pinch exerted by the toroidal magnetic field and the gradient of the magnetic pressure, models with large magnetizations tend to concentrate most of their internal energy in a thin, hot spine around the axis.

For fixed overpressure factor, the properties of the recollimation shocks (i.e., strength, obliquity) and those of the radial oscillations (amplitude, wavelength) are governed by the magnetosonic Mach number, which controls the angle at which waves penetrate into the jet (Mach angle) whose steepening forms the recollimation shocks, and the specific internal energy, which establishes the amount of energy that can be exchanged into kinetic energy at shocks/radial oscillations. Figures 2–4 correspond to models M1B3, M3B1, and M5B2, which have been chosen as representative of magnetically dominated, hot, and kinetically dominated models, respectively. The figures include panels of the rest-mass density and pressure (in logarithmic scale); azimuthal flow velocity and Lorentz factor; and toroidal and axial components of the magnetic field. Both recollimation shocks and radial oscillations are clearly seen in all the panels.⁴

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The following paragraphs describe in a more quantitative way the properties of recollimation shocks and radial oscillations, and the jet transversal structure, as functions of three scalar quantities defining the jet models; namely, the magnetosonic (relativistic) Mach number, the specific internal energy in units of the specific rest-mass energy, and the magnetization.

In contrast to the results shown in MPG16, the use of a thin shear layer in the present calculations allows the formation of recollimation shocks in all the models. Crossing a shock is an irreversible process. The irreversibility manifests in the increase in specific entropy of the fluid parcels going through the shock. Because the specific entropy of the fluid parcels can never decrease along the evolution, the pre-shock flow conditions cannot be recovered downstream and the sequence of recollimation shocks in the overpressured jet models cannot be exactly periodic. However, the fact that shocks appear so alike is evidence that the change in entropy across them is small and that the shocks are weak. Besides this (small) difference between shocks due to the net increase of specific entropy along the streamlines, the imposed boundary conditions at the jet's inlet sets an additional difference between the first shock and the rest (see Figures 2–7).

The energy involved in the shocks can be estimated through the averages of gas pressure and magnetic pressure⁵ across the shock, respectively, ${\bar{p}}_{s}=({p}_{1}+{p}_{2})/2$, ${\bar{p}}_{m,s}=({p}_{m,1}+{p}_{m,2})/2$. In these expressions, subindex 1 (2) refers to pre-(post-)shock quantities. A criterion to determine the shock strength is the magnitude of the jumps of gas pressure, Δp_s = p₂/p₁, and magnetic pressure, Δp_m,s = p_m,2/p_m,1, equal, respectively, to the jumps of internal and magnetic energy densities. Finally, connected to the jump in gas pressure is the jump in the flow Lorentz factor. Quantities ΔΓ = Γ₁/Γ₂ (note the change in the definition with respect to Δp_s and Δp_m,s) and ${\bar{{\rm{\Gamma }}}}_{s}=({{\rm{\Gamma }}}_{1}+{{\rm{\Gamma }}}_{2})/2$ define the jump in Lorentz factor and the average between the pre- and post-shock values, respectively.

Table 2 collects the values of all these quantities calculated at some particular radius close to the axis for the shocks of the models in Table 1.⁶ As seen from the table and the top panel of Figure 8, there is a correlation between both the average gas pressure and the average magnetic pressure involved at the shocks, and the ambient pressure. In the case of the average gas pressure, it is always an almost fixed fraction (50%–60% for high-magnetization models; close to 80% for low-magnetization ones) of the ambient pressure of the corresponding model. In the case of the average magnetic pressure, it is almost zero in the lowest-magnetized models and increases up to 75% of the ambient pressure in the models with the largest magnetizations.⁷

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Table 2.  Properties of the Recollimation Shocks

Model	${\bar{p}}_{s}\,[{\rho }_{a}{c}^{2}]$	Δps	${\bar{p}}_{m,s}\,[{\rho }_{a}{c}^{2}]$	Δpm,s	${\bar{{\rm{\Gamma }}}}_{s}$	ΔΓs	${\phi }_{s}\,(^\circ )$
M1B1	1.9 × 10−2	21 ± 11	9.5 × 10−3	20 ± 9	3.6	2.2 ± 0.4	13.0
M1B2	4.3 × 10−3	18 ± 10	3.1 × 10−3	20 ± 10	3.4	1.9 ± 0.2	12.0
M1B3	2.3 × 10−3	10 ± 5	2.2 × 10−3	18 ± 11	3.2	1.5 ± 0.1	12.5
M2B1	6.7 × 10−3	23 ± 11	1.2 × 10−3	22 ± 10	3.9	2.1 ± 0.3	7.0
M2B2	1.8 × 10−3	17 ± 7	6.5 × 10−4	23 ± 10	3.7	1.6 ± 0.1	8.5
M2B3	6.9 × 10−4	11 ± 3	6.5 × 10−4	18 ± 6	3.4	1.3 ± 0.1	8.5
M2B4	5.9 × 10−4	9 ± 2	7.0 × 10−4	17 ± 5	3.3	1.3 ± 0.1	9.5
M3B1	5.6 × 10−3	28 ± 15	2.8 × 10−4	31 ± 15	4.3	2.2 ± 0.3	5.5
M4B1	7.7 × 10−4	22 ± 11	3.5 × 10−5	40 ± 20	3.6	1.5 ± 0.1	4.0
M4B2	2.4 × 10−4	12 ± 4	1.3 × 10−4	26 ± 11	3.3	1.2 ± 0.1	5.0
M4B3	2.0 × 10−4	7 ± 2	2.1 × 10−4	15 ± 5	3.2	1.1 ± 0.1	5.5
M4B4	1.8 × 10−4	7 ± 2	2.5 × 10−4	13 ± 4	3.2	1.1 ± 0.1	5.5
M5B1	1.6 × 10−4	21 ± 10	7.5 × 10−7	60 ± 40	3.2	1.2 ± 0.1	3.0
M5B2	9.1 × 10−5	10 ± 3	4.4 × 10−5	30 ± 11	3.2	1.1 ± 0.1	3.0
M5B3	6.7 × 10−5	6 ± 2	1.5 × 10−4	11 ± 5	3.2	1.1 ± 0.1	3.5
M5B4	6.2 × 10−5	5 ± 2	1.8 × 10−4	10 ± 4	3.2	1.1 ± 0.1	3.5

Note. Tabulated data denote jet model; averages ($\bar{x}$) and jumps (Δx) of gas pressure; magnetic pressure and Lorentz factor across shocks (see the text for definitions); and shock angle.

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The jumps presented in Table 2 are calculated at a particular radius close to the jet axis where the shocks can be more planar and hence stronger than at larger radii. The large values of the gas pressure jumps (in the range ≈5–30) and the magnetic pressure jumps (in the range ≈10–60) indicate that the shocks are strong close to the axis; however, as said in a previous paragraph, the small deviation of the shock sequence from periodicity is an indication of the shocks' overall weakness. From the data shown in the table and also in the bottom panel of Figure 8, it can be concluded that the jump in gas pressure decreases for decreasing ε_j (i.e., colder models). This tendency holds for fixed Mach number (and increasing magnetization), as well as for increasing Mach number and constant magnetization. This means that the strength of the shocks in terms of the internal energy density jumps is smaller for colder jets or, alternatively, for kinetically dominated/magnetically dominated jets. The jump in the flow Lorentz factor follows the same tendency. Larger jumps are found in low magnetization, low Mach number jets (i.e., hot jets). Also deduced from the bottom panel of Figure 8 is the trend of the gas pressure jump to increase for fixed internal energy and increasing Mach number (decreasing magnetization). The magnitude of the jumps of magnetic pressure (or, equivalently, magnetic energy density), Δp_m,s, for kinetically dominated jets (models M4Bx and M5Bx) show a remarkable tendency to decrease with increasing magnetization for constant Mach number. However this trend is less solid in the case of hot/magnetically dominated jets (models M1Bx and M2Bx). For increasing Mach number and constant magnetization, Δp_m,s increases for low-magnetization models and decreases for highly magnetized ones. As in the case of the gas pressure, the jumps of magnetic pressure tend to increase for fixed internal energy and increasing Mach number (decreasing magnetization). All of these trends are consistent with the the fact that the strength of the shocks depends on the internal energy of the jet, which establishes the amount of energy that can be exchanged into kinetic energy at shocks (as advanced earlier in this section), and is in general reduced for increasing jet magnetizations (probably as a consequence of the magnetic tension). The shock obliquity (as determined by ϕ_s, the angle between the shock and the jet axis; see last column in Table 2) and the shock separation follows a remarkable correlation with the relativistic (magnetosonic) Mach number.

Together with the sequence of recollimation shocks, the jets exhibit a series of radial oscillations with the same periodicity. Table 3 lists the relative average variation of several relevant jet quantities along the jet as a result of the radial expansions and compressions. The same quantities are shown in Figure 9 as a function of the magnetosonic Mach number of the jet and for models with similar internal energy. The jet radii change by ≈35%–50%, with a slight tendency of the oscillation amplitude to increase with the Mach number for constant internal energy (decreasing magnetization), and to decrease with increasing magnetization (decreasing internal energy) for constant Mach number. The pressure follows the changes dictated by the adiabatic expansions/compressions of the flow (as a consequence of the variations in jet radius) plus the jumps at internal shocks and display variations of a few (≈3) units. A similar behavior is found in the magnetic pressure, dominated by the axial component of the magnetic field, which also experiences changes by a factor of 3–4. Since both magnetic and gas pressures behave similarly at compressions/expansions and shocks, the changes in magnetization are more limited (below ≈30%). Associated with the changes in the axial and toroidal magnetic field components is also a change in the magnetic pitch angle, which changes as little as a ≈37% (i.e., between 45° and 60°).⁸ The changes in the Lorentz factor reflects the potential of the jet flow to exchange internal and kinetic energies and is larger in the hotter models, and becomes negligible in cold, kinetically dominated jets in spite of the large radial oscillations. It is interesting to highlight the trend of the variations of the jet radius, gas pressure, magnetic pressure, and magnetic pitch angle to decrease with increasing magnetization and constant internal energy, which is a consequence of the increasing magnetic pinch of the jet for higher magnetization models.

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The last column in Table 3 records the axial wavelength of the oscillations in the different models. It shows the expected correlation with the Mach number. For a fixed Mach number, only hot (low Mach number) jets display a slight variation of the oscillation wavelength with the internal energy, whereas it is almost constant for kinetically dominated jets despite the broad spread of magnetization.

As a consequence of the profile of the magnetic pressure across the jet and the pinch exerted by the toroidal component of the magnetic field, the thermal pressure is not constant across the jet. For a given Mach number, models with increasing magnetization tend to concentrate most of their internal energy in a thin hot spine around the axis. (Compare the thermal pressure panels of Figure 2 corresponding to model M1B3, a highly magnetized jet, with those of Figure 3, a purely hydrodynamic jet with a passive magnetic field, and Figure 4, a jet in equipartition.)

To calculate the synchrotron emission from the previous RMHD jet models, we need to establish some assumptions. While the radio continuum emission we are interested in is being produced by a population of non-thermal electrons (and maybe positrons), the RMHD simulations discussed previously account only for the evolution of the thermal electrons present in the jet. Establishing a relationship between the thermal and non-thermal populations requires a detailed prescription for the particle acceleration processes that connect both populations, presumably taking place in strong shocks or in magnetic reconnection events (see e.g., Sironi et al. 2015). A proper treatment of particle acceleration/injection in shocks (e.g., Kirk et al. 2000) or magnetic reconnection (e.g., Lyubarsky 2005) requires a microscopic description of the fluid, such as in particle-in-cell (PIC) simulations (e.g., Nishikawa et al. 2016), and its implementation in macroscopic RMHD models, such as the one used here, still falls outside current computing capabilities, given the vastly different scales involved. Nevertheless, as a first-order approximation, we consider that the internal energy of the non-thermal population is a constant fraction of the thermal electrons considered in the RMHD simulations (e.g., Gómez et al. 1995, 1997; Komissarov & Falle 1997; Broderick & McKinney 2010; Porth et al. 2011). Alternatively, the non-thermal population can also be considered to be proportional to the magnetic energy density (e.g., Porth et al. 2011), which determines the particle acceleration efficiency in shocks and magnetic reconnection events. No significant differences are found in our emission calculations when considering the latter approach for particle acceleration, given the similarities between the gas pressure and magnetic energy density distributions in our RMHD simulations, except for the particular case of jet spine brightening discussed in more detail in Section 3.3. On the other hand, we note that particle acceleration at shock fronts is probably the most important ingredient for computing the expected non-thermal emission from our RMHD simulations. Our results should therefore be considered in these cases as a first-order approximation, which could be used as a base model to test different prescriptions for in situ particle acceleration in future modeling.

We consider the usual power law for distributing the total energy computed by the RMHD simulations among the relativistic non-thermal electrons using N(E)dE = ${N}_{0}{E}^{-\gamma }$dE, where ${E}_{\min }\leqslant E\leqslant {E}_{\max }$. Neglecting radiative losses, the ratio C_E = E_max/E_min will remain constant throughout the jet, and can be treated as a parameter in our model. In this case, assuming that the internal energy of the non-thermal population is a constant fraction of the thermal one, the power law for the electron energy distribution is fully determined by the equations (Gómez et al. 1995)

$$\begin{eqnarray}&&{N}_{0}={\left[\displaystyle \frac{U(\gamma -2)}{1-{C}_{E}^{2-\gamma }}\right]}^{\gamma -1}{\left[\displaystyle \frac{1-{C}_{E}^{1-\gamma }}{N(\gamma -1)}\right]}^{\gamma -2},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{E}_{\min }=\displaystyle \frac{U}{N}\displaystyle \frac{\gamma -2}{\gamma -1}\displaystyle \frac{1-{C}_{E}^{1-\gamma }}{1-{C}_{E}^{2-\gamma }},\end{eqnarray} \tag{ 2 }$$

where U and N are a constant fraction of the internal energy density and rest-mass density calculated by the RMHD code. Note that the fraction between the thermal and non-thermal populations provides a scale factor for the emission in our models (expressed in arbitrary units), but otherwise our simulations are not affected by the particular value chosen.

To compute the emission and absorption coefficients for the synchrotron radiation is convenient to establish two different reference frames, the observer's and emitting fluid frames (see Figure 10). The radiation coefficients are computed in the fluid's frame (1, 2), where the direction of the magnetic field in the plane of the sky defines the axis 2, which together with the axis 1 and the direction toward the observer form a right-handed orthogonal system. For the case of our assumed power-law energy distribution the emission and absorption coefficients, for a given polarization (i), are given by

$$\begin{eqnarray}{\varepsilon }^{(i)} & = & \displaystyle \frac{1}{2}{c}_{5}(\gamma ){N}_{0}{({B}^{{\prime} }\sin {\vartheta }^{{\prime} })}^{(\gamma +1)/2}\\ & & \cdot {\left(\displaystyle \frac{\nu }{2{c}_{1}}\right)}^{(1-\gamma )/2}\left[{(-1)}^{i+1}\displaystyle \frac{\gamma +1}{\gamma +7/3}+1\right],\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}{\kappa }^{(i)} & = & {c}_{6}(\gamma ){N}_{0}{({B}^{{\prime} }\sin {\vartheta }^{{\prime} })}^{(\gamma +2)/2}\\ & & \cdot {\left(\displaystyle \frac{\nu }{2{c}_{1}}\right)}^{-(\gamma +4)/2}\left[{(-1)}^{i+1}\displaystyle \frac{\gamma +2}{\gamma +10/3}+1\right],\end{eqnarray} \tag{ 4 }$$

being i = 1, 2; ${B}^{{\prime} }$ the modulus of the magnetic field calculated by the RMHD code in the lab frame and transformed to the fluid frame; ${\vartheta }^{{\prime} }$ the angle between the magnetic field and the line of sight; ν the observing frequency; ${c}_{1}=(3e)/(4\pi {m}^{3}{c}^{5});$ and c₅, c₆ dimensionless functions of γ which are defined and tabulated by Pacholczyk (1970).

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If the orientation of the magnetic field is not uniform within the jet the fluid frame will change from computational cell to computational cell. Thus, the integration of the of transfer equations is more conveniently formulated in the observer's frame (a, b), with χ_B defining the angle between the axis 2 (for each particular computational cell) and a. The radiation field is characterized by the four Stokes parameters I, Q, U, and V, or alternatively I^(a), I^(b), U, and V, where I = I^(a) + I^(b) and Q = I^(a) − I^(b). Given the small amount of circular polarization observed in blazar jets we consider V  =  0. The transfer equations in the observer's frame, characterizing the radiation passing a volume element ds are given by (e.g., Pacholczyk 1970)

$$\begin{eqnarray}\displaystyle \frac{{{dI}}^{(a)}}{{ds}} & = & {I}^{(a)}\left[-{\kappa }^{(1)}{\sin }^{4}{\chi }_{B}-{\kappa }^{(2)}{\cos }^{4}{\chi }_{B}-\displaystyle \frac{1}{2}\kappa {\sin }^{2}2{\chi }_{B}\right]\\ & & +U\left[\displaystyle \frac{1}{4}({\kappa }^{(1)}-{\kappa }^{(2)})\sin 2{\chi }_{B}+\displaystyle \frac{d{\chi }_{F}}{{ds}}\right]\\ & & +{\varepsilon }^{(1)}{\sin }^{2}{\chi }_{B}+{\varepsilon }^{(2)}{\cos }^{2}{\chi }_{B},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}\displaystyle \frac{{{dI}}^{(b)}}{{ds}} & = & {I}^{(b)}\left[-{\kappa }^{(1)}{\cos }^{4}{\chi }_{B}-{\kappa }^{(2)}{\sin }^{4}{\chi }_{B}-\displaystyle \frac{1}{2}\kappa {\sin }^{2}2{\chi }_{B}\right]\\ & & +U\left[\displaystyle \frac{1}{4}({\kappa }^{(1)}-{\kappa }^{(2)})\sin 2{\chi }_{B}-\displaystyle \frac{d{\chi }_{F}}{{ds}}\right]\\ & & +{\varepsilon }^{(1)}{\cos }^{2}{\chi }_{B}+{\varepsilon }^{(2)}{\sin }^{2}{\chi }_{B},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}\displaystyle \frac{{dU}}{{ds}} & = & {I}^{(a)}\left[\displaystyle \frac{1}{2}({\kappa }^{(1)}-{\kappa }^{(2)})\sin 2{\chi }_{B}-2\displaystyle \frac{d{\chi }_{F}}{{ds}}\right]\\ & & +{I}^{(b)}\left[\displaystyle \frac{1}{2}({\kappa }^{(1)}-{\kappa }^{(2)})\sin 2{\chi }_{B}+2\displaystyle \frac{d{\chi }_{F}}{{ds}}\right]\\ & & -\kappa U-({\varepsilon }^{(1)}-{\varepsilon }^{(2)})\sin 2{\chi }_{B},\end{eqnarray} \tag{ 7 }$$

where κ = (κ⁽¹⁾ + κ⁽²⁾)/2, and dχ_F/ds is the polarization plane variation per unit distance due to Faraday rotation. Multi-frequency VLBI images of blazar jets commonly show regions of enhanced Faraday rotation which can be used to determine the line-of-sight component of the magnetic field, as well as probes of the thermal population of electrons in the jet sheaths (e.g., Gómez et al. 2008, 2011, 2016; Hovatta et al. 2012; Gabuzda et al. 2015, 2017; Lico et al. 2017). We have, however, decided to ignore Faraday rotation effects in our simulations presented here (assuming dχ_F/ds = 0) to study the polarization in our models as a function of the dominant type of energy in the jet, disentangled from any possible Faraday rotation effects (which in turn would depend also on the physical parameters chosen for the jet sheath adding extra free parameters in our simulations). Hence, our models can also be used as a testbed case for future modeling of Faraday rotation effects in AGN jets.

Figures 11–14 show the total intensity, linearly polarized intensity, and degree of polarization plots at viewing angles θ = 2°, 5°, 10°, and 20° computed for models M1B3, M3B1, and M5B2, chosen as representative for each dominant type of energy in the jet. Note that all of the models are computed at a frequency at which the emission is optically thin to discard any opacity effects in our analyses. This is a good approximation for our study of the stationary knots commonly observed at parsec scales in AGN jets (Jorstad et al. 2017), where we expect the emission to be optically thin, but we note that closer to the VLBI core opacity effects are a necessary ingredient to understand the radio emission, specially in polarization (e.g., Gómez et al. 1994a, 1994b; Porth et al. 2011). Figures 11–18 contain the emission plots for the whole set of jet models and viewing angles considered. The most salient feature in the emission plots is the presence of knots associated with the recollimation shocks.

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The threaded helical magnetic field produces a well-known emission asymmetry between the jet top and bottom halves (e.g., Aloy et al. 2000; Lyutikov et al. 2005; Clausen-Brown et al. 2011). This effect is maximized for a magnetic pitch angle of ϕ = 45°, which is the case we consider in all our models. The enhanced emitting half reverts from top to bottom when the viewing angle in the fluid frame reaches ${\theta }_{r}^{{\prime} }=90^\circ$. This can be related to the viewing angle in the observer's frame by using the light aberration equations (e.g., Rybicki & Lightman 1979)

$$\begin{eqnarray}&&\sin {\theta }^{{\prime} }=\displaystyle \frac{\sin \theta }{{\rm{\Gamma }}(1-{v}_{j}\cos \theta )},\quad \cos {\theta }^{{\prime} }=\displaystyle \frac{\cos \theta -{v}_{j}}{(1-{v}_{j}\cos \theta )},\end{eqnarray} \tag{ 8 }$$

from which we obtain that the flip in the dominant section of the jet is obtained when $\cos$ θ_r = v_j. Given that we are considering an axial flow velocity v_j = 0.95c at injection, we expect that the jet cross section asymmetry will reverse at an approximate value of θ_r ≈ 18°. For lower values of the viewing angle (θ < θ_r), the emission in the top half of the jet will dominate over the bottom half. This is clearly visible in the total intensity panels of Figures 11–12 and 15–16, where θ = 2° and 5°, and to a lesser extent in Figures 13 and 17 where θ = 10°. When θ approaches θ_r the emission becomes qualitatively axially symmetric, as can be seen in Figures 14 and 18 where θ = 20°. Although larger viewing angles θ > θ_r are not shown, in these cases the bulk of the emission moves progressively to the bottom half of the jet.

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The jet cross section asymmetry produced by the helical magnetic field is more clearly seen in Figures 19–20, where transverse profiles of the total and polarized intensity integrated along the jet are shown for each jet model at viewing angles of θ = 2° and 20°. For a viewing angle of θ = 2°, not only the bulk of the jet emission is concentrated on the top half of the jet, but also the peak intensity is displaced from the jet axis. This offset is progressively larger for magnetically dominated, kinetically dominated, and hot jets. At a viewing angle of θ = 20° the bottom half of the jet starts to dominate the emission, as expected for our choice of jet flow velocity and magnetic field pitch angle.

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As discussed in Section 2, jet models with large magnetizations concentrate the majority of their internal energy in a hot spine, due to the larger magnetic pressure gradient and magnetic tension. As shown in Figures 11–18 following our prescription for particle acceleration, in which the internal energy of the non-thermal population is a constant fraction of the thermal one, this translates into a spine brightening in both, total and polarized intensity, which is more clearly seen in the magnetically models M1B3 and M2B4, and in the kinetically dominated models M4B4 and M5B4 (with magnetizations β = 17.5). For comparison, Figure 21 shows the emission plots for the M1B3 model computed considering a non-thermal particle injection based on the magnetic energy density, in which no significant spine brightening is seen. The detection of spine brightening in actual observations of AGN jets can therefore be considered as a good indication for originating in a jet that is magnetically dominated, and in which the internal energy of the non-thermal population of emitting particles is proportional to the internal energy of the thermal gas. Alternatively, spine brightening can also arise through differential Doppler boosting in jets with a significant stratification in velocity across the jet width, a situation that has not been considered in the present simulations.

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By looking at the bottom panels of Figures 19–20, with a more symmetric emission structure across the jet width, we can observe that the models tend to cluster for similar magnetizations, which is the dominant factor determining the spine brightening. Model M1B3, as well as models MxB4, presenting the highest magnetizations have their emission more concentrated around the jet axis; on the other hand, pure hydrodynamic models (β = 0) present a more evenly distributed emission across the jet width.

To quantify the spine brightening, we have computed the distance (in jet radius units) from the axis at which the emission adds to the 50% and 70% of the total jet emission. For this, we have selected the models at a viewing angle of 20° (with a more symmetric emission) and also considered the small displacements in the peak emission with respect to the jet axis discussed previously. We also note that for the case of optically thin emission, as considered in these models, the integrated emission along a given integration column is directly proportional to column length; hence for a homogeneous jet model we expect that 50% (70%) of the emission will be concentrated within 0.4R_j (0.59R_j) from the jet axis. The results are presented in Table 4, confirming the higher spine brightening with increasing jet magnetization. We also find that for a given jet magnetization the spine brightening increases with Mach number. Model M1B3 presents the largest spine brightening, reaching 50% (70%) of the integrated emission at 0.16R_j (0.29R_j) from the peak emission. For the pure hydrodynamic model M5B1, the results agree with the expected values in case there is no significant spine brightening.

One of the main characteristics of the radio emission from the RMHD jet models is the presence of a variable number of bright knots both, in total and polarized emission (see Figures 11–18). These are associated with the recollimation shocks, and are a consequence of the increase in density and gas pressure produced by the shocks. These knots can be associated with the stationary features that appear commonly in blazar jets near the VLBI core (e.g., Jorstad et al. 2005, 2017; Gómez et al. 2016).

To characterize these stationary knots as a function of the dominant type of energy in the jet, we have computed their relative strength, measured as the mean value of the ratio (in percentage) between the peak intensity in the knots and the underlying jet emission once the emission is integrated across the jet width into a 1D profile. The results are tabulated in Table 5 and plotted in Figure 22. We have not analyzed the models with θ = 2°, as the emission from multiple recollimation shocks is overlapped in the integration column. The same is also true for models M1B2, M2B1, and M2B2 at a viewing angle of θ = 5°. We find an overall trend of increasing relative knots intensity with increasing viewing angle, which is due to the variable Doppler factor with viewing angle and to an increase in the ratio between unshocked and shocked cells in the integration column with decreasing viewing angle, as discussed below.

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For stationary jet models, as those considered here, the observed emission is enhanced by a factor δ², where $\delta \,={{\rm{\Gamma }}}^{-1}{(1-\beta \cos \theta )}^{-1}$ is the Doppler factor and depends on the flow Lorentz factor Γ and viewing angle θ. These change along the jet as the emitting particles go first through the rarifying, expanding pre-shock state and then through the compressing, recollimating post-shock state forming the recollimation shock structure, leading to a variable δ that will modulate the observed emission. To obtain a better understanding on how the final radiation reaching the observer depends on the jet emissivity and Doppler boosting, we have analyzed in more detail the variability range of θ and Γ, and their relative contribution to δ along the jet for the representative models M1B3, M3B1, and M5B2. These are analyzed for viewing angles 2°, 5°, 10° and 20° (see also Figures 11–14), hereafter referred to as models v02, v05, v10, and v20, respectively.

The results are presented in Figure 23, which shows the change of δ as a function of Γ within the four regions (I–IV) where θ takes values for each viewing angle model vXX. Each color represents also the variability of Γ depending on the model. These values are also detailed in Table 6. By looking at Figure 23, we observe that Γ is the main parameter contributing to δ when θ oscillates around 2° and 5°, particularly in the hot jet model M3B1 and the magnetically dominated model M1B3, and to a less extent in the kinetically dominated model M5B2. As θ increases, δ is progressively more influenced by the local variations in θ, more strongly in the case of M5B2 and to a lesser extent in M1B3.

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Table 6.  Variations of θ and Γ along the Jet Axis

vXX	${\rm{\Delta }}\theta \,{[}^{\circ }]$	Model	$\overline{{\rm{\Gamma }}}$	ΔΓ
v02	[2.6, 2.9]	M1B3	3.8	[2.4, 5.3]
v05	[4.5, 5.9]	M3B1	5.3	[2.6, 8.1]
v10	[8.7, 11.4]	M5B2	3.3	[3.1, 3.4]
v20	[18.0, 22.3]

Note. Tabulated data denote the initial viewing angle and its variability range (for all considered models), jet model, and mean Lorentz factor with its variability range.

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Therefore, for viewing angles smaller than 10°, the contribution of the Doppler boosting to the observed emission is larger in the rarefactions (where the flow accelerates and expands) than in the recollimating post-shock states, leading to a reduction of the relative intensity of the shocks with respect to the underlying jet emission. The opposite is true for larger viewing angles.

As described in Section 2, kinetically dominated jets have weaker shocks; however, Figure 22 shows that for small viewing angles (θ = 5°, 10°) these models present stronger knots in the emission than hot jets. This is due to the relative number of shocks present in the different models, so that for jet models with a lower Mach number (i.e., larger number of shocks) and small viewing angles we barely observe the underlying jet emission, leading to a larger ratio between the unshocked and shocked cells in the integration columns, and therefore smaller relative intensity in shocks. At larger viewing angles (θ = 20°) magnetically dominated and hot jets present a higher ratio in the Doppler factor between the shocks and rarefactions (∼1.6 and ∼1.4, respectively) than for kinetically dominated jets (∼1.1), causing the observed increase in the relative intensity of the knots at this viewing angle (see Figure 22 zoom-in).

Finally, we should also note that the knot intensity in our emission simulations will depend on the expected particle acceleration in the shock fronts of the recollimation shocks, which in turn depends on the magnetic field configuration and magnetization, among other parameters (e.g., Sironi et al. 2015). We expect that particle acceleration should have an overall effect of increasing the relative intensity of the knots with respect to that of the underlying jet. Comparison with future simulations including a parametrized description of particle acceleration could serve to obtain a better understanding of shock acceleration and its implication in the radio knots observed in AGN jets.

The axial symmetry of our models and the helical magnetic field considered lead to a bimodal distribution of the EVPAs (e.g., Lyutikov et al. 2005), being either perpendicular or parallel to the jet. This is also modulated by the viewing angle, and its Lorentz transformation into the fluid frame, determining what is the dominant magnetic field component projected onto the plane of the sky. For our chosen magnetic field pitch angle of ϕ = 45° (in the lab frame) and jet flow bulk Lorentz factor, the poloidal component of the magnetic field B^z dominates over the toroidal component B^ϕ for viewing angles larger than 5°, as observed in Figures 13–14 and 17–18. At smaller viewing angles, these projection effects yield to a bimodal distribution in the EVPAs across the jet width, with a flip in the EVPAs in the bottom half of the jet when the projected toroidal component of the magnetic field dominates (see Figures 11–12 and 15–16).

Polarized intensity images in Figures 11–18 show small variations in the polarization angle of up to ∼26° around stationary components regardless of the viewing angle. For this, it is necessary to break down the symmetry in our models between the back and front sections of the jet along the integration column to generate some Stokes U.

Figure 24 shows, in normalized units, the Stokes U profile along the integration column plotted in red color in the bottom panel, which corresponds to the jet density of the kinetically dominated model M5B2 for a viewing angle of 5°. The chosen integration column, contained between 31R_j and 55R_j from the beginning of the jet, maximizes the variation of the polarization angle for this particular model and viewing angle. Note that the column is constrained to a jet width of [−1, 1]R_j (black dashed lines in the bottom panel), i.e., as if the jet was perfectly cylindrical, to assure both parts of the column have the same number of computational cells. Each color of the Stokes U profiles represents a different configuration of the parameters involved in the calculations of Stokes U.

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If we consider an idealized jet configuration, with a uniform distribution of non-thermal particles, velocity, and magnetic field, i.e., as if there were no internal structure or recollimation shocks present inside the jet, we would obtain for Stokes U an integrated value of zero, as it is shown in blue, yielding a polarization angle of 180° (or 90°, depending on the viewing angle and magnetic pitch angle). When the actual RMHD values of the model are considered the jet symmetry is broken, leading to some generation of Stokes U along the integration column (plotted in red) and a final polarization angle of χ = 154°; that is, a variation of ∼26° with respect to the idealized homogeneous jet model. The underlying process involved in the break of the Stokes U symmetry, and the subsequent change in the polarization angle χ, is the presence of a recollimation shock in the jet that modifies the distribution of the jet flow velocity, magnetic field, and energy density of non-thermal electrons along the integration column, as can be seen in the bottom panel of Figure 24.

In an attempt to determine which of the RMHD parameters is more affected by the recollimation shock, and therefore has a larger contribution to the jet asymmetry and generation of Stokes U, we have considered other models setting ad hoc values in some of these parameters. We find that for the case of a uniform magnetic field, corresponding to the green profile of Figure 24, the variation in the gas pressure leads to some generation of Stokes U, resulting in a final polarization angle of ∼160°. A similar value is obtained when the gas pressure is set to be homogeneous (orange profile), confirming that both, the magnetic field and gas pressure variations in the recollimation shocks contribute similarly to the generation of Stokes U. Finally, we find that the velocity field changes produced by the recollimation shock do not affect significantly the generation of Stokes U.

As discussed previously, in situ particle acceleration in recollimation shocks should increase the relative contribution to the emission of the shocked cells with respect to that of the underlying flow, which in turn may result in larger variations in the polarization angle and degree of polarization in the associated emission knots.

Figures 11–18 also show the distribution of the degree of polarization in the jet for the different models analyzed in this work. Given that in our simulations we are considering fully uniform magnetic fields, the maximum value of the degree of polarization is of the order of 70%, which corresponds to the expected value for optically thin synchrotron emission. We note, however, that polarimetric VLBI observations of AGN jets rarely show linear polarization degrees in excess of few tens of percent (e.g., Jorstad et al. 2005; Hovatta et al. 2012), which suggests the presence of a randomly oriented component of the magnetic field (Burn 1966; Gómez et al. 1994b; Wardle & Homan 2003) in turbulent flows (Marscher 2014). The inclusion of turbulence in RMHD models is, however, particularly difficult, as this requires connecting the scales resolved with the RMHD code with the unresolved turbulent ones by accounting for the kinetic and magnetic energy transfers between them. This connection could be made through the addition of new terms (tensor components) in the dynamical equations whose strength have to be calibrated by direct numerical simulations with varying numerical resolutions (see, e.g., Kessar et al. 2016, on incompressible, non-relativistic, MHD turbulence), or the comparison of direct numerical simulations with PIC simulations. This approach leads to models of turbulence that are dynamically consistent and have a limited number of free parameters but at the cost of a very expensive a priori tuning of the transfer terms which is, at present, beyond the current computational capabilities. Turbulence in AGN jets will not only reduce the degree of polarization with respect to that obtained in our simulations, but also produce a more variable polarization throughout the jet.

By looking at the degree of polarization plots we also observe the top-down asymmetry produced by the helical magnetic field (see also Aloy et al. 2000). The recollimation shocks also leave a clear signature in the degree of polarization, presenting variations between the knots and the underlying jet. It is therefore possible to discern whether the stationary jet features present in VLBI images of blazar jets are produced by bends in the jet orientation—through differential Doppler boosting (e.g., Gómez et al. 1993, 1994a, 1994b)—or by recollimation shocks by looking for these distinctive polarization signatures.

The helical structure of the magnetic field produces also a clear stratification in the degree of polarization across the jet width. By looking at Figures 13–14 and 17–18 we observe a progressive increase in the degree of polarization with distance from the jet axis (more relevant in the underlying jet emission than in the knots) that is more pronounced as the jet Mach number and viewing angle increase, and magnetization decreases. A similar stratification in degree of polarization across the jet width was observed previously in VLBI images of the radio galaxy 3C 120 (Gómez et al. 2008), which on the light of these simulations may be interpreted as produced by a large scale helical magnetic field field in a jet with relatively low magnetization and high Mach number seen at moderate viewing angles, consistent with previous estimations for this source (e.g., Gómez et al. 2000).

Magnetohydrodynamical models have been computed following the approach developed by Komissarov et al. (2015), which allows one to study the structure of steady, axisymmetric relativistic (magnetized) flows using 1D time-dependent simulations. The approach is based on the fact that for narrow jets with axial velocities close to the light speed the steady-state equations of RMHD can be accurately approximated by the 1D time-dependent equations with the axial coordinate acting as the temporal coordinate.

The approximation is valid as long as the radial dimension of the flow is much smaller than the axial one, and simple, quasi-1D flows are considered with the radial and azimuthal components of the flow velocity much smaller than the axial one, which approaches light speed (i.e., ${v}^{r},{v}^{\phi }\ll {v}^{z}$). Consistency with the 1D version of the divergence free condition forces to consider configurations with very small radial components of the magnetic field (${B}^{r}\ll {B}^{\phi },{B}^{z}$). All of these constraints can be verified a posteriori, once the approximate 2D solution has been obtained.

The more delicate point of the approximation consists in the implementation of the boundary conditions at the jet surface, which in the 1D approximation become a kind of time-dependent boundary conditions. Special actions should be undertaken to mimic the effect of the 2D, steady boundary conditions at the jet surface: (i) the jet surface should be tracked along the time and (ii) the ambient gas parameters are reset every computational time step according the prescribed functions of time. Following Komissarov et al. (2015), the jet surface is tracked from the injection at the jet base using a passive scalar which is advected with the continuity equation. Second, to keep the jet surface to behave as a contact, the radial velocity of the ambient gas is reset not to zero but to its value at the last jet cell.

From a numerical point of view, the code used in these simulations is the 1D, radial-cylindrical, time-dependent version of the RMHD code used in Martí (2015a) and Martí et al. (2016). It is a second-order conservative, finite-volume code based on high-resolution shock-capturing techniques. An overview of the specific algorithms used in the code and an analysis of its performance can be found in Appendices A and B, respectively, of Martí (2015a). Also in that paper this 1D version of the code was used to test the code's ability to keep rotating and non-rotating configurations of axially symmetric relativistic magnetized flows in equilibrium (see their Section 5.1).

Figure 25 reproduces Figure 6 of Komissarov et al. (2015) corresponding to the so-called model A (see their Section 4.3). In this test, a moderately magnetized, relativistic jet with a purely azimuthal magnetic field is injected into an atmosphere with a power-law pressure distribution from a nozzle located at some distance of the jet base. As the jet enters the pressure-decreasing atmosphere, it expands rapidly and a rarefaction wave propagates toward the axis. Once the jet overexpands, it starts to recollimate and a reconfinement shock sets in. The shock reaches the axis at z ≈ 450, reflects and then returns to the jet boundary at z ≈ 700, from where the jet re-expands again. Figure 25 shows the same rest-mass density contours as in the original Komissarov et al.'s plot. As it can be seen our simulation captures not only the essential features of their calculation (the fast expansion of the jet reaching a maximum radius of r_max ≈ 12 at z ≈ 300, the recollimation shock reaching the axis at z ≈ 450, and the jet re-expansion beyond z ≈ 700), but also the tiniest details of the contour lines.

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Our simulation was performed with a numerical resolution of 128 cells per initial jet radius (320 cells per initial jet radius in the original simulation of Komissarov and colls.) We used a piecewise-linear reconstruction of the spatial grid with MC limiter, and the HLLC Riemann solver (Mignone & Bodo 2006). The advance in time was done using the third-order TVD-preserving Runge–Kutta of Shu & Osher (1988), Shu (1989) with CFL = 0.3.

Finally, we can compare the stationary 2D solutions found in Martí et al. (2016) with the thinnest shear layers (models PH02, HP03) with the corresponding 1D approximations used in the present paper. Figure 26 displays this comparison for model HP03 with a shear layer corresponding to m = 12. The differences in the extrema of the distributions of the rest-mass density, thermal pressure, axial flow velocity, and magnetic field components within the jet between the two simulations are small (of a few percent in relative terms). The discrepancies in the shock separation are of the same order (≈3.3%). The differences for model PH02 are similar. The rest of simulations of Martí et al. (2016), with wider shear layers, do not admit a fair comparison with their corresponding 1D models, as the shear layers in these cases cannot be treated consistently within the 1D approximation.

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Axially symmetric, non-rotating, steady jet models are characterized by five functions. Using cylindrical coordinates (referred to an orthonormal cylindrical basis {e_r, e_ϕ, e_z}) in which the jets propagate along the z axis, these functions are the jet density and pressure (ρ(r), p(r), respectively), the jet axial velocity, v^z(r), and the toroidal and axial components of the jet magnetic field (B^ϕ(r), B^z(r), respectively), whereas the static unmagnetized ambient medium is characterized by a constant pressure, p_a and a constant density, ρ_a. The equation of transversal equilibrium establishing the radial balance between the total pressure gradient and the magnetic tension, allows to find the equilibrium profile of one of the variables in terms of the others. We shall fix the radial profiles of ρ, v^z, B^ϕ, and B^z, and solve for the profile of the gas pressure, p. We use top-hat profiles for ρ, v^z, and B^z:

$$\begin{eqnarray}\rho (r)=\left\{\begin{array}{ll}{\rho }_{j}, & 0\leqslant r\leqslant {R}_{j}\\ {\rho }_{a}, & r\gt {R}_{j},\end{array}\right.\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}{v}^{z}(r)=\left\{\begin{array}{ll}{v}_{j}, & 0\leqslant r\leqslant {R}_{j}\\ 0, & r\gt {R}_{j},\end{array}\right.\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}{B}^{z}(r)=\left\{\begin{array}{ll}{B}_{j}^{z}, & 0\leqslant r\leqslant {R}_{j}\\ 0, & r\gt {R}_{j},\end{array}\right.\end{eqnarray} \tag{ 11 }$$

(where ρ_j, v_j and ${B}_{j}^{z}$ are constants) and a particular profile for the toroidal component of the magnetic field

$$\begin{eqnarray}{B}^{\phi }(r)=\left\{\begin{array}{ll}\displaystyle \frac{2{B}_{j,m}^{\phi }(r/{R}_{{B}^{\phi },m})}{1+{(r/{R}_{{B}^{\phi },m})}^{2}}, & 0\leqslant r\leqslant {R}_{j}\\ 0, & r\gt {R}_{j},\end{array}\right.\end{eqnarray} \tag{ 12 }$$

with ${R}_{{B}^{\phi },m}$, the radius at which the toroidal magnetic field reaches its maximum, B_j,m, equal to 0.37R_j in all the models.

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In the general case, the equation of transversal equilibrium establishes the radial balance between the total pressure gradient, the centrifugal force and the magnetic tension,

$$\begin{eqnarray}&&\displaystyle \frac{{{dp}}^{* }}{{dr}}=\displaystyle \frac{\rho {h}^{* }{W}^{2}{({v}^{\phi })}^{2}-{({b}^{\phi })}^{2}}{r}.\end{eqnarray} \tag{ 13 }$$

In this equation, ${p}^{* }$ and ${h}^{* }$ stand for the total pressure and the specific enthalpy including the contribution of the magnetic field

$$\begin{eqnarray}&&{p}^{* }=p+\displaystyle \frac{{b}^{2}}{2}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{h}^{* }=1+\varepsilon +p/\rho +{b}^{2}/\rho ,\end{eqnarray} \tag{ 15 }$$

where p is the fluid pressure, ρ its density and ε its specific internal energy. b^μ (μ = t, r, ϕ, z) are the components of the 4-vector representing the magnetic field in the fluid rest frame and b² stands for ${b}^{\mu }{b}_{\mu }$, where summation over repeated indices is assumed. vⁱ (i = r, ϕ, z) are the components of the fluid 3-velocity in the laboratory frame, which are related to the flow Lorentz factor, W, according to

$$\begin{eqnarray}&&W=\displaystyle \frac{1}{\sqrt{1-{v}^{i}{v}_{i}}}.\end{eqnarray} \tag{ 16 }$$

The following relations hold between the components of the magnetic field 4-vector in the comoving frame and the three vector components Bⁱ measured in the laboratory frame:

$$\begin{eqnarray}&&{b}^{0}={{WB}}^{i}{v}_{i},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{b}^{i}=\displaystyle \frac{{B}^{i}}{W}+{b}^{0}{v}^{i}.\end{eqnarray} \tag{ 18 }$$

The square of the modulus of the magnetic field, defining the magnetic energy density, can be written as

$$\begin{eqnarray}&&{b}^{2}=\displaystyle \frac{{B}^{2}}{{W}^{2}}+{({B}^{i}{v}_{i})}^{2}\end{eqnarray} \tag{ 19 }$$

with ${B}^{2}={B}^{i}{B}_{i}$.

For a non-rotating flow with constant axial velocity v_j and axial magnetic field, the equation of transversal equilibrium can be rewritten as

$$\begin{eqnarray}&&\displaystyle \frac{{dp}}{{dr}}=-\displaystyle \frac{{({B}^{\phi })}^{2}}{{{rW}}_{j}^{2}}-\displaystyle \frac{{B}^{\phi }}{{W}_{j}^{2}}\displaystyle \frac{{{dB}}^{\phi }}{{dr}},\end{eqnarray} \tag{ 20 }$$

(where ${W}_{j}={(1-{v}_{j}^{2})}^{-1/2}$) which can be integrated by separation of variables to give

$$\begin{eqnarray}&&p(r)=2{\left(\displaystyle \frac{{B}_{j,m}^{\phi }}{{W}_{j}(1+{(r/{R}_{{B}^{\phi },m})}^{2})}\right)}^{2}+C\quad (0\leqslant r\leqslant {R}_{j}),\end{eqnarray} \tag{ 21 }$$

where C is an integration constant set by the boundary condition at R_j.

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Equations (9)–(12) and (21) define the jet models for given values of ρ_j, v_j, ${B}_{j}^{z}$, ${B}_{j,m}^{\phi }$ and C. However, some of the parameters of this set are not specially useful (in particular ${B}_{j}^{z}$, ${B}_{j,m}^{\phi }$ and C) and we are going to replace them by some others better suited for our purposes: ϕ_j, the average magnetic pitch angle; β_j, the average jet magnetization; and ${{ \mathcal M }}_{\mathrm{ms},j}$, the average magnetosonic Mach number.

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Together with other parameters (significantly the jet overpressure factor, K), the relativistic magnetosonic Mach number

$$\begin{eqnarray}&&{{ \mathcal M }}_{\mathrm{ms}}=\displaystyle \frac{W}{{W}_{\mathrm{ms}}}\displaystyle \frac{v}{{c}_{\mathrm{ms}}},\end{eqnarray} \tag{ 22 }$$

governs the properties of internal conical shocks in overpressured magnetized jets in the same way as the Mach number does in purely hydrodynamic, overpressured jets. In the previous expression, W_ms is the flow Lorentz factor associated to the magnetosonic speed, c_ms,

$$\begin{eqnarray}&&{c}_{\mathrm{ms}}=\sqrt{{c}_{A}^{2}+{c}_{s}^{2}(1-{c}_{A}^{2})}.\end{eqnarray} \tag{ 23 }$$

which, in turn, is defined in terms of the sound speed, c_s, and the Alfvén speed, c_A,

$$\begin{eqnarray}&&{c}_{A}=\sqrt{\displaystyle \frac{{b}^{2}}{\rho h+{b}^{2}}}.\end{eqnarray} \tag{ 24 }$$

Finally, the magnetization, β, is defined as

$$\begin{eqnarray}&&\beta =\displaystyle \frac{{b}^{2}}{2p}.\end{eqnarray} \tag{ 25 }$$

For fixed values of the jet flow velocity, v_j, and the jet rest-mass density, ρ_j, the averaged values of the magnetosonic Mach number and the magnetization allows one to fix the averaged values of the jet gas pressure, p_j, and the magnetic energy density, ${b}_{j}^{2}$. Then, the averaged value of the jet gas pressure determines C, whereas the averaged value of the jet magnetic energy density and the averaged magnetic pitch angle,

$$\begin{eqnarray}&&\phi =\arctan \left(\displaystyle \frac{{B}^{\phi }}{{B}^{z}}\right),\end{eqnarray} \tag{ 26 }$$

allows one to fix the remaining two parameters, ${B}_{j}^{z}$ and ${B}_{j,m}^{\phi }$.

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The set of parameters is completed with K, the averaged jet overpressure factor, which together with p_j and ${b}_{j}^{2}$, fixes the ambient pressure p_a.

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