A Simulation Study of Low-power Relativistic Jets: Flow Dynamics and Radio Morphology of FR-I Jets

The RHD equations describing the jet development with mass loading can be written as follows (e.g., L. D. Landau & E. M. Lifshitz 1959):

$$\begin{eqnarray}&&\displaystyle \frac{\partial D}{\partial t}+{\boldsymbol{\nabla }}\cdot (D{\boldsymbol{v}})={S}_{D},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\frac{\partial {\boldsymbol{M}}}{\partial t}+{\boldsymbol{\nabla }}\cdot ({\boldsymbol{M}}{\boldsymbol{v}}+p)={{\boldsymbol{S}}}_{M},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial E}{\partial t}+{\boldsymbol{\nabla }}\cdot [(E+p){\boldsymbol{v}}]={S}_{E},\end{eqnarray} \tag{ 3 }$$

where D = Γρ, M = Γ² ρ(h/c²) v , and E = Γ² ρ h − p are the mass, momentum, and total energy densities in the computational frame, respectively. Here, c is the speed of light, ${\rm{\Gamma }}=1/\sqrt{1-{\left(v/c\right)}^{2}}$ is the Lorentz factor, v is the flow speed, and h = (e + p)/ρ is the specific enthalpy, with e = + ρ c² being the sum of the internal energy density, , and the rest-mass energy density. Since we consider a single-species baryonic fluid, we assume that both the jet inflow and the ambient gas consist of electrons and protons.

The source terms on the right-hand side incorporate mass loading from stellar winds, as well as the effects of background stratification. Assuming that the loaded mass possesses negligible momentum and internal energy, as compared to the jet, it contributes only to the mass and rest-mass energy terms. The external gravity, which is required to balance the pressure gradient in the stratified background medium, contributes to the momentum and energy source terms. Combining these, the source terms are written as

$$\begin{eqnarray}&&{S}_{D}={q}_{D},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{{\boldsymbol{S}}}_{M}=\rho {\boldsymbol{g}},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{S}_{E}={q}_{D}{c}^{2}+\rho {\boldsymbol{v}}\cdot {\boldsymbol{g}},\end{eqnarray} \tag{ 6 }$$

where q_D is the mass-loading rate per unit volume, and the external gravity, g = (∇ p_b )/ρ_b , is set up with the initial density and pressure profiles of the background medium (see Section 2.2). These source terms are applied to the entire computational box.

As in Paper I, simulations are performed using the HOW-RHD code based on the WENO and SSPRK schemes. The Ryu–Chattopadhyay version of the EOS is employed to accurately follow the thermodynamics of relativistic fluids (D. Ryu et al. 2006), along with several treatments to improve the accuracy and stability of the simulations.

As the fiducial model for stratified background media, we adopt the King profile of the density,

$$\begin{eqnarray}&&{\rho }_{b}(r)={\rho }_{c}{\left[1+{\left(\displaystyle \frac{r}{{r}_{c}}\right)}^{2}\right]}^{-3{\beta }_{K}/2},\end{eqnarray} \tag{ 7 }$$

where r is the radial distance from the center of the host galaxy. We choose the setup of M. Perucho et al. (2014), which is designed to model the FR-I radio galaxy 3C 31 in the host galaxy, NGC 383 (M. J. Hardcastle et al. 2002): β_K = 0.73, r_c = 1.2 kpc, and ρ_c = 3.0 × 10⁻²⁵ g cm⁻³. The background medium is assumed to be isothermal with T_b = 4.9 × 10⁶ K. Assuming the background gas of fully ionized hydrogen, the pressure is then given as p_b (r) = 2ρ_b (r)k_B T_b /m_p , where k_B is the Boltzmann constant, and m_p is the proton mass: p_c = 2ρ_c k_B T_b /m_p = 2.4 × 10⁻¹⁰ dyn cm⁻².

It is worth noting that in uniform background media, jets form scale-free systems, which can be scaled up and down to address jets at different length scales. However, in stratified media, jets are not scale-free, and simulations depend on the radius of the jet at injection, r_j , and the injection position, r_inj, from which the jet is launched. We consider two cases: r_j = 10 pc and r_inj = 80 pc as the fiducial case (M. Perucho et al. 2014), and r_j = 100 pc and r_inj = 250 pc for larger scale FR-I jets, which are tracked beyond the galactic core. In the former, ρ_b and p_b at r_inj are 99.3% of ρ_c and p_c , while in the latter, they are 95.5% of ρ_c and p_c .

We also consider the background profile of a simple power-law distribution, specifically the one designed to model Centaurus A in the elliptical galaxy NGC 5128 (S. Wykes et al. 2019): ρ_b (r), p_b (r) ∝ r^−3/2 with the density and pressure at r_inj matching those of the fiducial model, that is, 99.3% of ρ_c and p_c .

For the source term, q_D (r), representing mass loading from stellar winds (denoted by the subscript "sw"), we adopt the prescription of M. Perucho et al. (2014):

$$\begin{eqnarray}&&{q}_{D}(r)={q}_{\mathrm{sw}}{\left[1+{\left(\displaystyle \frac{r}{{r}_{\mathrm{sw}}}\right)}^{2}\right]}^{-{\beta }_{\mathrm{sw}}},\end{eqnarray} \tag{ 8 }$$

where β_sw = 0.23, r_sw = 265 pc, and q_sw = 9.5 ×10²³ g pc⁻³yr⁻¹ = 3.2 × 10⁻²⁶ g cm⁻³ Gyr⁻¹. This particular profile was derived from the surface brightness of an elliptical galaxy (T. R. Lauer et al. 2007). We choose a larger q_sw, compared to those used in M. Perucho et al. (2014), in order to consider the case where mass loading may lead to significant dynamical effects.

As in Paper I, the computational domain is a 3D rectangular box for z ≥ 0 that contains only the upward-moving jet; no counterjet is explicitly included. Jet material is injected along the z-axis through a circular nozzle located at the center of the bottom surface (see Figures 1 and 2 below).

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The nozzle radius, r_j , determines the cross-sectional area of the jet inflow and sets the overall scale of the jet-induced structures. As mentioned above, in Table 1, two sets of models are considered: the "r10" models, with r_j = 10 pc, present the early development of the jets on scales of ∼200−500 pc, while the "r100" models, with r_j = 100 pc, depict the later stage of the jets that extends up to ∼10 kpc.

Table 1. Parameters of Jet Models ^a

Model Name	Qj	rj	η	$\dot{{M}_{j}}$	$\tfrac{{v}_{j}}{c}$	Γj	[〈Γ〉spine] b	ηr	$\tfrac{{v}_{\mathrm{head}}* }{c}$	[$\tfrac{{v}_{\mathrm{head}}}{c}$] c	tcross	$\tfrac{{t}_{\mathrm{end}}}{{t}_{\mathrm{cross}}}$
	(erg s−1)	(pc)		(dyne)							(yr)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)
Q42-η5-r10	2.20E+42	10	1.E-05	9.34E+31	0.95	3.2	2.07	1.27E-04	0.0106	0.005	3.07E+3	50
Q43-η5-r10	1.33E+43	10	1.E-05	4.94E+32	0.99	7.1	4.42	6.28E-04	0.0242	0.014	1.35E+3	50
Q44-η5-r10	1.45E+44	10	1.E-05	5.00E+33	0.999	22.4	8.68	6.25E-03	0.0732	0.052	4.46E+2	50
Q42-η5-r10-P	2.20E+42	10	1.E-05	9.34E+31	0.95	3.2	2.08	1.27E-04	0.0106	0.016	3.07E+3	28
Q42-η5-r10-M	2.20E+42	10	1.E-05	9.34E+31	0.95	3.2	1.68	1.27E-04	0.0106	0.005	3.07E+3	50
Q44-η5-r100	3.50E+44	100	1.E-05	1.35E+34	0.9664	3.9	2.77	1.89E-04	0.0131	0.024	2.49E+4	66
Q45-η5-r100	3.50E+45	100	1.E-05	1.18E+35	0.996	11.2	5.65	1.55E-03	0.0379	0.089	8.61E+3	38
Q45-η3-r100	3.52E+45	100	1.E-03	2.01E+35	0.85	1.9	1.82	3.61E-03	0.0482	0.146	6.77E+3	41

Notes.

^a Here, η ≡ ρ_j /ρ_c , ${{\rm{\Gamma }}}_{j}\equiv {\left[1-{\left({v}_{j}/c\right)}^{2}\right]}^{-1/2}$, ${\eta }_{r}\equiv ({\rho }_{j}{h}_{j}/{\rho }_{c}{h}_{c}){{\rm{\Gamma }}}_{j}^{2}$, and ${v}_{\mathrm{head}}* \approx {v}_{j}{\eta }_{r}^{1/2}$. For all models, ζ ≡ p_j /p_c = 1. ^b The mean Lorentz factor estimated by averaging Γ along the jet axis in the region of z ≤ 2/3 l_jet, where the jet propagation length, l_jet, is defined by the location of the contact discontinuity along the z-axis. ^c The jet-head advance speed estimated using the simulation data at t_end.

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The jets are specified by the velocity, v_j , radius, r_j , density, ρ_j , and pressure, p_j , of the injected jet inflow. This set of the jet parameters are often translated into the jet power Q_j , the jet-to-background density contrast, η ≡ ρ_j /ρ_c , and pressure contrast, ζ ≡ p_j /p_c , for given background conditions (ρ_c and p_c ) (e.g., P. Rossi et al. 2020, Paper I). Here, the jet power is defined as

$$\begin{eqnarray}&&{Q}_{j}=\pi {r}_{j}^{2}{v}_{j}\left({{\rm{\Gamma }}}_{j}^{2}{\rho }_{j}{h}_{j}-{{\rm{\Gamma }}}_{j}{\rho }_{j}{c}^{2}\right),\end{eqnarray} \tag{ 9 }$$

where ${{\rm{\Gamma }}}_{j}=1/\sqrt{1-{\left({v}_{j}/c\right)}^{2}}$ and h_j are the initial bulk Lorentz factor and the specific enthalpy of the jet inflow, respectively. The jet parameters can also be combined to express the momentum injection rate or the jet thrust as (e.g., M. Perucho et al. 2014; M. J. Hardcastle & J. H. Croston 2020, Paper I)

$$\begin{eqnarray}&&{\dot{M}}_{j}=\pi {r}_{j}^{2}\left({{\rm{\Gamma }}}_{j}^{2}{\rho }_{j}\displaystyle \frac{{h}_{j}}{{c}^{2}}{v}_{j}^{2}+{p}_{j}\right).\end{eqnarray} \tag{ 10 }$$

For kinetically dominated, light, relativistic jets with v_j ≈ c, Equations (9) and (10) can be approximated as follows:

$$\begin{eqnarray}&&\displaystyle \frac{{Q}_{j}}{\pi {r}_{j}^{2}}\sim {{\rm{\Gamma }}}_{j}^{2}{\rho }_{j}{h}_{j}{v}_{j}\sim \eta {{\rm{\Gamma }}}_{j}^{2}{h}_{j}{\rho }_{c}c\sim {\eta }_{r}{\rho }_{c}{h}_{c}c,\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\dot{M}}_{j}}{\pi {r}_{j}^{2}}\sim {{\rm{\Gamma }}}_{j}^{2}{\rho }_{j}{h}_{j}{\left(\displaystyle \frac{{v}_{j}}{c}\right)}^{2}\sim \eta {{\rm{\Gamma }}}_{j}^{2}{h}_{j}{\rho }_{c}\sim {\eta }_{r}{\rho }_{c}{h}_{c}.\end{eqnarray} \tag{ 12 }$$

Here, the relativistic density contrast is given as

$$\begin{eqnarray}&&{\eta }_{r}\equiv \displaystyle \frac{{\rho }_{j}{h}_{j}}{{\rho }_{c}{h}_{c}}{{\rm{\Gamma }}}_{j}^{2}=\eta {{\rm{\Gamma }}}_{j}^{2}\left(\displaystyle \frac{{h}_{j}}{{h}_{c}}\right)\propto \eta {{\rm{\Gamma }}}_{j}^{2}.\end{eqnarray} \tag{ 13 }$$

In this study, we assume p_j = p_c (i.e., ζ = 1); hence, h_j = c² + (_j + p_j )/ρ_j ∼ c². Then, for identical background conditions, ${{\rm{\Gamma }}}_{j}^{2}\propto ({Q}_{j}/\eta \pi {r}_{j}^{2})\propto ({\dot{M}}_{j}/\eta \pi {r}_{j}^{2})\propto {\eta }_{r}/\eta$. So, Γ_j is larger for greater ${Q}_{j}/\pi {r}_{j}^{2}$ and for lighter jets with smaller η.

Initially, the jet propagation into the surrounding medium can be approximated by the advance speed of the jet head, ${v}_{\mathrm{head}}* \approx {v}_{j}\cdot \sqrt{{\eta }_{r}}/(\sqrt{{\eta }_{r}}+1)$ (J. M. Martí et al. 1997). This expression is derived from the balance between the jet's ram pressure and the background pressure in one-dimensional (1D) planar geometry. With the jet injected along the z-axis, this would provide a reasonable approximation for the initial velocity of the jet head. In our model setup with η_r ≪ 1, the initial estimate for the jet advance speed can be approximated as

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{\mathrm{head}}* }{c}\approx \left(\displaystyle \frac{{v}_{j}}{c}\right){\eta }_{r}^{1/2}\approx {\eta }^{1/2}{{\rm{\Gamma }}}_{j}\propto {\left(\displaystyle \frac{{Q}_{j}}{\pi {r}_{j}^{2}}\right)}^{1/2}.\end{eqnarray} \tag{ 14 }$$

Again, for identical background conditions, the jet-head advance speed is faster for greater ${\left({Q}_{j}/\pi {r}_{j}^{2}\right)}^{1/2}$.

Thus, the initial flow dynamics of our simulated jets are primarily determined by Γ_j and v_head*, which depend on the energy injection flux, ${Q}_{j}/\pi {r}_{j}^{2}$, and the density contrast, η, in our simulation setup with ζ = 1. In general, however, the characteristics of simulated jets are governed by four key parameters: r_j , defining the length scale; Q_j , specifying the energy input rate; η, specifying the density contrast; and ${\dot{M}}_{j}$, defining the momentum thrust of the jet inflow.

Table 1 shows the ranges of these model parameters. Column (1) lists the model name; following the nomenclature of Paper I, it includes three elements, the exponents of Q_j and η, and r_j in units of parsec. We assign η ≡ ρ_j /ρ_c = 10⁻⁵ for the default setting. The three r10 models in the first group are the fiducial models, all sharing the same background profile described by Equation (7). In the second group, the Q42 model labeled "P" features a background profile following a −3/2 power law, while the Q42 model labeled "M" incorporates mass loading from stellar winds. The r100 models in the third group are designed to explore the jet morphology on larger scales.

At the beginning, Γ_j and v_head*, which are listed in columns (7) and (10) of Table 1, control the flow dynamics. However, as the jet flow expands and decelerates, these two quantities gradually decrease with time. For comparison, we present the mean Lorentz factor of the jet-spine flow, 〈Γ〉_spine, and the actual jet-head advance speed, v_head, in columns (8) and (11), respectively. They are estimated using the simulation data at the end time of simulations, t_end. Additionally, in order to quantify the degree of the jet-flow deceleration, we define the "deceleration factor" as ${{ \mathcal R }}_{\mathrm{dec}}\equiv {v}_{\mathrm{head}}/{v}_{j}$, which is listed in column (2) of Table 2. It is worth noting that a smaller ${{ \mathcal R }}_{\mathrm{dec}}$ indicates a stronger deceleration.

Table 2. Morphological Parameters for Simulated Jets

Model Name	[${{ \mathcal R }}_{\mathrm{dec}}$] a	[θj (z2)] b	[vz (R = 0, z2)/c] c	[$\mathrm{log}{ \mathcal F }{| }_{{\theta }_{\mathrm{obs}}={90}^{^\circ }}$] d	[$\tfrac{{d}_{b}}{{d}_{f}}{| }_{{\theta }_{\mathrm{obs}}={90}^{^\circ }}$] e	$\tfrac{{d}_{b}}{{d}_{f}}{| }_{{\theta }_{\mathrm{obs}}={60}^{^\circ }}$	$\tfrac{{d}_{b}}{{d}_{f}}{| }_{{\theta }_{\mathrm{obs}}={30}^{^\circ }}$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
Q42-η5-r10	5.4E-3	19.7	0.75	0.0	0.94	0.81(0.88)	0.24(0.79)
Q43-η5-r10	1.4E-2	14.2	0.89	1.7	0.98	0.95	0.93
Q44-η5-r10	5.2E-2	11.8	0.96	4.1	∼1	∼1	0.93
Q42-η5-r10-P	1.7E-2	22.7	0.46	−0.4	0.47	0.45(0.92)	0.07(0.62)
Q42-η5-r10-M	5.5E-3	26.4	0.35	−0.6	0.25	0.09	0.07
Q44-η5-r100	2.5E-2	15.0	0.40	−0.1	0.58	0.61(0.91)	0.34(0.77)
Q45-η5-r100	9.0E-2	13.5	0.98	2.9	0.98	0.96	0.89
Q45-η3-r100	1.7E-1	11.4	0.76	2.4	0.96	0.98	0.96

Notes.

^a The deceleration factor is defined as ${{ \mathcal R }}_{\mathrm{dec}}\equiv {v}_{\mathrm{head}}/{v}_{j};$ a smaller ${{ \mathcal R }}_{\mathrm{dec}}$ indicates a stronger deceleration. ^b The jet-spine spreading angle, ${\theta }_{j}({z}_{2})\equiv 2\arctan [{\rm{\Delta }}R({z}_{2})/{z}_{2}]$, at z = z₂ ≡ 2l_jet/3. Here, ΔR(z₂) is the transverse width of the jet-spine region with the positive vertical velocity, v_z > 0, at z = z₂. ^c v_z (R = 0, z₂)/c is the vertical velocity along the jet axis at z = z₂. ^d ${ \mathcal F }{| }_{{\theta }_{\mathrm{obs}}={90}^{^\circ }}={I}_{\nu }^{\max }/{I}_{\nu ,N}^{{\rm{Q}}42-\eta 5-{\rm{r}}10}$, where ${I}_{\nu }^{\max }$ is the peak value in the face-on map (θ_obs = 90^°) for each model and ${I}_{\nu ,N}^{{\rm{Q}}42-\eta 5-{\rm{r}}10}$ is the peak value in the face-on map of the Q42-η5-r10 model. ^e The ratio of the projected distances to the brightest spot, d_b , and the faintest edge, d_f , obtained from the surface brightness maps of Figures 8–9. The values for the counterjets are also given in the parenthesis for selected models.

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The jet crossing timescale is defined as t_cross ≡ r_j /v_head*, and listed in column (12) of Table 1. The end time of simulations given in units of t_cross is listed in column (13). The spatial resolution is controlled by the number of grid zones across the initial jet radius, N_j = r_j /Δx. For all models, N_j = 8, except for the Q44-η5-r100 model with N_j = 5. Hence, for instance, ${\left(400\right)}^{3}$ grid zones employed for the r10 models make up a box of volume ${\left(500\ \mathrm{pc}\right)}^{3}$.

The boundary condition of the jet nozzle plane (z = 0) could affect the properties of simulated jets (e.g., M. Perucho et al. 2014; J. Donohoe & M. D. Smith 2016). Here, we impose the continuous outflow condition to all six faces of the simulation domain, including the z = 0 plane, except at the jet nozzle. The jet flow is injected mostly along the z-axis (i.e., v_z ≈ v_j ); yet, we introduce a slow, small-angle precession with period τ_prec = 3 t_cross, and angle θ_prec = 1°, to break the rotational symmetry (see Paper I for details).

To generate synthetic maps of simulated jets, we estimate the synchrotron emission by adopting simple modelings: we utilize physically motivated prescriptions for the magnetic field strength, $B^{\prime}$, and the cosmic-ray (CR) electron population, ${ \mathcal N }{{\prime} }_{e}({\gamma }_{e}^{\prime} )$, which are not explicitly modeled within our RHD simulations. Here, ${\gamma }_{e}^{\prime}$ is the Lorentz factor of CR electrons. Throughout the paper, the primed variables, such as $B^{\prime}$ and ${\gamma }_{e}^{\prime}$, represent the quantities defined in the local fluid frame, while unprimed variables are used for the quantities defined in the computational or observer frame.

2.4.1. Magnetic Field Models

2.4.2. CR Electron Population

The jet ejected from the central engine is thought to be composed of ionized plasma, which includes relativistic electrons, protons, and possibly positrons (e.g., M. C. Begelman et al. 1984). Electrons are expected to undergo further acceleration to CRs, due to shocks and MHD turbulence within the jet-induced flows (e.g., A. R. Bell 1978; L. O. Drury 1983; G. Brunetti & A. Lazarian 2007). While a comprehensive theoretical modeling of all the relevant processes would be quite complex, we adopt a simplified phenomenological approach available in the literature, for instance, J. L. Gomez et al. (1995) to model the CR electron population. The number and energy densities of CR electrons in the fluid frame are assumed to be proportional to the rest-mass density ρ and the internal energy density of the fluid, respectively. In addition, the energy spectrum of CR electrons is represented by a power-law form, as follows:

$$\begin{eqnarray}&&{ \mathcal N }{{\prime} }_{e}({\gamma }_{e}^{\prime} )={{ \mathcal N }}_{0}^{\prime} (x,y,z)\cdot {\gamma }_{e}^{{\prime} -\sigma },\end{eqnarray} \tag{ 15 }$$

where the energy of CR electrons is given by the Lorentz factor, ${\gamma }_{e}^{\prime}$. We employ a single power-law slope of σ = 2.2, which corresponds to the representative slope for the energy spectrum of CR electrons accelerated at relativistic shocks. Electron cooling due to synchrotron and inverse-Compton losses is not considered.

The normalization factor, ${ \mathcal N }{{\prime} }_{0}$, depends on ρ and of the local fluid as follows:

$$\begin{eqnarray}&&{{ \mathcal N }}_{0}^{\prime} \propto {f}_{\mathrm{jet}}\cdot {\left(\displaystyle \frac{\epsilon (\sigma -2)}{1-{C}_{E}^{2-\sigma }}\right)}^{\sigma -1}{\left(\displaystyle \frac{1-{C}_{E}^{1-\sigma }}{\rho (\sigma -1)}\right)}^{\sigma -2},\end{eqnarray} \tag{ 16 }$$

where ${C}_{E}={\gamma }_{\max }/{\gamma }_{\min }$ is the ratio of the maximum and minimum energies of CR electrons (J. L. Gomez et al. 1995). This parameter is kept constant at C_E = 10³ (e.g., ${\gamma }_{\min }\approx {10}^{3}$ and ${\gamma }_{\max }\approx {10}^{6}$) for all models. An ad hoc weight function, ${f}_{\mathrm{jet}}=9{\left(5/3-\gamma \right)}^{2}$, is introduced in Equation (16) to assign the CR electron population preferentially to the relativistic plasma contained in the jet flow: for relativistic fluid with γ = 4/3, f_jet = 1, while for thermal fluid with γ = 5/3, f_jet = 0. We point out that inside the cocoon, the relativistic electrons of the jet material are mixed with the nonrelativistic thermal electrons of the background medium, causing the adiabatic index of the fluid to range between 1.47 ≲ γ ≲ 5/3 in our jet models (see Figure 3). In our model calculations, the normalization factor scales as ${N}_{0}^{\prime} \propto {f}_{\mathrm{jet}}\cdot \epsilon {\left(\epsilon /\rho \right)}^{\sigma -2}$, while its absolute amplitude is arbitrary.

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2.4.3. Synchrotron Emission

The synchrotron power emitted by the CR electrons in Equation (15) with an isotropic pitch angle distribution can be estimated as follows (G. B. Rybicki & A. P. Lightman 1979):

$$\begin{eqnarray}&&P{{\prime} }_{\nu ^{\prime} }=\displaystyle \frac{2\pi \sqrt{3}{q}_{e}^{2}{\nu }_{L}}{c}{\int }_{{\gamma }_{\min }}^{{\gamma }_{\max }}d{\gamma }_{e}^{\prime} {{ \mathcal N }}_{0}^{\prime} {\gamma }_{e}^{{\prime} -\sigma }\cdot \xi {\int }_{\xi }^{\infty }{K}_{5/3}(\chi )d\chi ,\end{eqnarray} \tag{ 17 }$$

where ${\nu }_{L}={q}_{e}B^{\prime} /2\pi {m}_{e}c$ is the Larmor frequency, ${\nu }_{c}=(3/2){\gamma }_{e}^{{\prime} 2}{\nu }_{L}$ is the critical frequency, q_e is the electron charge, m_e is the electron mass, $\xi \equiv \nu ^{\prime} /{\nu }_{c}$, and K_5/3(ξ) is a modified Bessel function. For ${\gamma }_{\max }\gg {\gamma }_{\min }\gg 1$, the synchrotron volume emissivity can be expressed as

$$\begin{eqnarray}&&j{{\prime} }_{\nu ^{\prime} }=\displaystyle \frac{P{{\prime} }_{\nu ^{\prime} }}{4\pi }\propto {f}_{\mathrm{jet}}\cdot \epsilon {\left(\displaystyle \frac{\epsilon }{\rho }\right)}^{\sigma -2}\cdot {B}^{{\prime} (\sigma +1)/2}{\nu }^{{\prime} -(\sigma -1)/2}.\end{eqnarray} \tag{ 18 }$$

Then, the observed intensity, or surface brightness, can be calculated by integrating along the line of sight (LOS). For example, when the y-axis represents the LOS, the intensity is given by I_ν (x, z) = ∫j_ν (x, y, z)dy.

In the computational frame, the observed frequency shifts to $\nu ={ \mathcal D }{\nu }^{{\prime} }$, where ${ \mathcal D }=1/({\rm{\Gamma }}[1-(v/c)\cos \theta ])$ is the Doppler factor, and θ is the angle between the fluid velocity vector, ${\boldsymbol{v}}$, and the LOS. With relativistic beaming, the volume emissivity is given as ${j}_{\nu }={{ \mathcal D }}^{2}{j}_{{\nu }^{{\prime} }}^{{\prime} }(\nu /{ \mathcal D })$ in the computational frame. For "observed" images, the inclination angle, θ_obs, with respect to the jet axis, if nonzero, could have important consequences. It controls not only the relativistic Doppler factor ${ \mathcal D }$, but also the number of bright points with high ${j}_{{\nu }^{{\prime} }}^{{\prime} }$ values along the path of integration. To incorporate such effects, we tilt the simulation box by the angle of π/2 − θ_obs around the x-axis, and then calculate the integration along the LOS.

We ignore the synchrotron self-absorption because we expect it to be insignificant in the diffuse plasmas in radio jets.

Figures 1 and 2 display 2D slice images of the density, $\mathrm{log}(\rho /{\rho }_{c})$, and the vertical components of the flow velocity, v_z /c, along with 1D profiles of the vertical velocity averaged over the azimuthal angle ϕ, 〈v_z (R, z)/c〉_ϕ , across the transverse $R={\left({x}^{2}+{y}^{2}\right)}^{1/2}$ direction for the five r10 models and three r100 models, respectively (see Table 1). In addition, the vertical velocity along the jet axis, v_z (R = 0, z)/c, is displayed across the z-distance in the small insets.

Consistent with earlier numerical studies of FR-I jets mentioned in the introduction, the jets in Figures 1 and 2 have the following parts in common: the bow shock, the shocked background medium, the backflow, and the jet-spine flow. The backflow is identifiable as blue regions in the 2D distribution of v_z (x, z)/c, while the jet-spine flow appears red. Surrounded by the bow shock, the shocked background medium would appear as the surface of the X-ray cavity in observation. Inside, the plume-like cocoon, filled with turbulence, develops. The shape of the cocoon changes from laterally expanded to longitudinally elongated, as the jet power increases. Significant "mixing" is observed between the jet-spine flow and the backflow, as well as between the cocoon material and the shocked background medium. Instabilities, such as Kelvin–Helmholtz instability at the shear interfaces, contribute to the mixing. In the lower-power models, the mixing at the boundary between the jet-spine flow and the backflow is more extended, inducing wider cocoons filled with more substantial turbulence. We note that the evolution timescale, t_cross, is shorter for higher ${v}_{\mathrm{head}}^{* }$, as listed in Table 1.

The jet flow is kinematically relativistic at injection, but its specific enthalpy is only mildly relativistic with h_j /c² ≈ 1.3, and its adiabatic index is γ_j ≈ 1.6, in our jet models. However, the kinetic energy is converted to the internal energy in the post-shock regions of relativistic shocks, including recollimation shocks. As a result, the fluid's adiabatic index decreases, as shown in Figure 3. The formation and growth of turbulent shear layers mix the shocked jet material (dark blue) with the ambient gas that has nonrelativistic enthalpy (dark red). We see the mixing to be more efficient beyond recollimation shocks at 200 pc for the Q42-η5-r10-P case and beyond 6 kpc for the Q44-η5-r100 case.

The r10 models produce the jet-induced flows confined within core regions (r < r_c = 1.2 kpc) of typical elliptical galaxies, possibly reflecting the flow structures of typical radio jets in their early phase (M. Perucho & J. M. Martí 2007; M. Perucho et al. 2014). In Figures 1(a)–(c) and (f)–(h), the three fiducial models, Q42-η5-r10, Q43-η5-r10, and Q44-η5-r10, demonstrate how the jet structures evolve depending on the jet power; higher Q_j (or higher ${v}_{\mathrm{head}}^{* }$) leads to faster penetration of the jet head into the background medium, resulting in a more elongated cocoon. These findings align closely with previous studies (e.g., J. M. Martí et al. 1997; M. Perucho & J. M. Martí 2007; P. Rossi et al. 2008; Y. Li et al. 2018).

The red regions with v_z > 0 in Figures 1(f)–(j) show the degree of the jet-head spreading, which is governed by the relative flow speeds in the longitudinal and transverse directions. We define the jet-head spreading angle as ${\theta }_{j}(z)\equiv 2\arctan [{\rm{\Delta }}R(z)/z]$, where ΔR(z) is the transverse width of the jet-spine region with v_z > 0 and z is the distance from the injection point. Column (2) of Table 2 lists the value of θ_j (z₂) at z₂ = 2l_jet/3, where the length of the jet, l_jet, is the distance between the injection point and the contact discontinuity along the z-axis in the simulated jet models at t_end. Among the five models shown in Figure 1, the jet-head spreading is larger in models with lower Q_j because the longitudinal deceleration is greater for lower Γ_j .

Figure 4(c) shows the overall relation between θ_j (z₂) and v_head*. It is the largest in Q42-η5-r10-M (magenta triangle), in which the mean Lorentz factor of the jet-spine flow, 〈Γ〉_spine, is the smallest due to the mass loading. On the other hand, θ_j (z₂) is slightly larger in Q42-η5-r10-P (green-filled square) than that in Q42-η5-r10, due to the larger transverse spreading of the jet in the power-law gradient of the external density/pressure. Section 3.2 discusses the last two cases in detail.

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In Figures 1(k)–(m), the profile of 〈v_z (R, z)/c〉_ϕ across the transverse R direction displays a stronger deceleration in the models with lower Q_j , whereas deeper dips with negative values indicate faster downward-moving velocity of the backflow in the models with higher Q_j . The velocity curves at different z-locations also show spreading of upward-moving velocity with increasing z, implying stronger decollimation in the models with lower Q_j . Column (4) of Table 2 also confirms that the vertical velocity at z₂ = 2l_jet/3 in the z-axis, v_z (R = 0, z₂)/c, increases with increasing Q_j . Such trends of deceleration and decollimation agree with the visual impression of the 2D distribution of v_z (x, z) displayed in Figures 1(f)–(h).

In the insets of Figures 1(k)–(m), the vertical velocity along the jet axis, v_z (R = 0)/c, exhibits a substantial deceleration in the models with low Q_j . This deceleration results from mixing at the jet-backflow interface and the entrainment of the background medium. Among the fiducial r10 models, the deceleration factor is the smallest, at ${{ \mathcal R }}_{\mathrm{dec}}=5.3\times {10}^{-3}$ in the Q42 case, indicating the most significant deceleration. In contrast, ${{ \mathcal R }}_{\mathrm{dec}}$ is largest at 5.0 × 10⁻² in the Q44 case. Figures 4(a)–(b) illustrate that, overall, ${{ \mathcal R }}_{\mathrm{dec}}$ is smaller (i.e., stronger deceleration) for smaller Γ_j or smaller 〈Γ〉_spine (see also Table 1).

As a result, in the low-power Q42 models, the vertical flow diffuses relatively smoothly toward the jet head (see Figure 1(f)), while in the high-power Q44 model, it comes to an abrupt stop at the jet head (see Figure 1(h)). The mean Lorentz factor of the jet-spine flow is estimated as 〈Γ〉_spine ≈ 2.1, 4.4, and 8.7 for the Q42, Q43, and Q45 models, respectively (see Table 1). Overall, the vertical velocity of the backflow is faster for higher Q_j : ∣v_z ∣ ∼ 0.1c for the Q42 models and ∣v_z ∣ ∼ 0.5c for the Q44 model.

Along the jet-spine flow, multiple recollimation shocks appear. In each model, the first recollimation shock is located at the initial dip in the v_z (R = 0)/c profile, which is deepest in the Q42-η5-r10-M model and shallowest in the Q44-η5-r10 model. This reflects that the induced shocks have higher Mach numbers (or higher speeds in the shock rest frame) in higher-power jets. A few additional recollimation shocks follow along the jet-spine flow. These shocks are also manifested as discontinuous jumps in the 2D distribution of ρ and v_z . Apart from the oscillations arising from instabilities and jet precession, the recollimation shocks are almost stationary in the computational frame for a steady jet inflow.

3.2.1. Q42-r10 Models in Different Environments

3.2.2. Q44 Models on Different Scales

3.2.3. Q45-r100 Models with Different Density Contrasts

3.2.4. Evolution of Lobe Shape

Figure 5 illustrates the time evolution of the actual jet-head speed, v_head, the axial ratio ${ \mathcal A }={l}_{\mathrm{jet}}/{ \mathcal W }$, and the lobe's lateral width ${ \mathcal W }$ in our jet models. In the fiducial r10 models, v_head is smaller than v_head*, with v_head/v_head* ∼ 0.5−0.7 at t_end (see Table 1). In contrast, in models where the jet propagates into backgrounds of decreasing density and pressure, such as Q42-η5-r10-P and the three r100 models, v_head/v_head* ∼ 1.5−3.0 at t_end. In the r100 models, v_head(t) increases after the jet head crosses the core, as shown with the dark green, light blue, and dark blue lines in Figure 5(a). M. Perucho et al. (2019) demonstrated in their 3D simulations that the propagation of jet heads can accelerate in backgrounds with decreasing pressure and density, and also can be influenced by helical perturbations at the jet injection. In our jet models, we add a small amplitude precession to the jet inflow in all cases. However, the r10 models do not show jet-head acceleration, suggesting that the impact of precession in the models is likely marginal.

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In Figure 5, we observe the following points: (1) Among the fiducial r10 models, v_head is larger with higher ${\eta }_{r}\propto \eta {{\rm{\Gamma }}}_{j}^{2}$. (2) Among the r10 models, the axial ratio ${ \mathcal A }$ is larger with higher Q_j . (3) The differences in the Q42-η5-r10 and Q42-η5-r10-P models with the same η_r indicate that the deceleration of the jet flow is less pronounced in a background with faster-decreasing density/pressure. (4) The differences in the Q45-η5-r100 and Q45-η3-r100 models demonstrate that a heavier jet with a larger η_r propagates faster and generates a more elongated cocoon.

In Paper I, we investigated the properties of nonlinear flow structures, such as shocks, velocity shear, and turbulence, generated in the simulated FR-II jets. The primary objective of this analysis was to understand various particle acceleration processes operating in radio jets. Although the particle acceleration is not the main topic of this paper, we still present the following quantities utilizing the simulation data: shocks with their speed β_s = u_s /c and Mach number M_s , velocity shear Ω_shear ≡ ∣∂v_z /∂r∣, relativistic shear coefficient ${{ \mathcal S }}_{r}\,=({{\rm{\Gamma }}}_{z}^{4}/15)\cdot {{\rm{\Omega }}}_{\mathrm{shear}}^{2}$ where ${{\rm{\Gamma }}}_{z}={\left[1-{\left({v}_{z}/c\right)}^{2}\right]}^{-1/2}$, vorticity Ω_t = ∣∇× v ∣, and vorticity excluding the shear Ω₋ = ∣Ω_t $+(\partial {v}_{z}/\partial r)\hat{{\boldsymbol{\theta }}}$∣. For a comprehensive description of these variables and the numerical procedures employed, readers are directed to the details provided in Paper I.

In Figures 6(a)–(f), we show the 2D distributions of shock zones, velocity shear parameters, and vorticity factors in the x–z plane with y = 0, for the Q42-η5-r10 and Q42-η5-r10-M models. Overall, the bow shock surface and recollimation shocks exhibit higher Mach numbers compared to the shocks formed in the backflow and the shocked background medium. In the models considered here, typically, the volume-averaged shock speeds are relativistic with 〈β_s 〉_JS ≈ 0.1−0.3 in the jet spine, subrelativistic with 〈β_s 〉_BF ≈ 0.04−0.07 in the backflow, and nonrelativistic with 〈β_s 〉_BS ≈ (1−7) × 10⁻³ in the bow shock surface.

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The velocity shear appears in the jet spine and backflow. Especially, the relativistic shear coefficient is strongly concentrated along the interface between the jet-spine flow and the backflow owing to the ${{\rm{\Gamma }}}_{z}^{2}$ factor: $\langle {{ \mathcal S }}_{r}/{\left(c/{r}_{j}\right)}^{2}{\rangle }_{\mathrm{JS}}\approx 0.01-0.06$ in the jet-spine flow, whereas $\langle {{ \mathcal S }}_{r}/{\left(c/{r}_{j}\right)}^{2}{\rangle }_{\mathrm{BF}}\approx (3-5)\times {10}^{-3}$ in the backflow. The vorticity factors Ω_t and Ω₋ spread out inside the cocoon. The volume-averaged vorticity excluding the shear, which is a more direct measure of turbulence, ranges as 〈Ω₋/(c/r_j )〉_JS ≈ 0.5−1.1 in the jet-spine flow, and 〈Ω₋/(c/r_j )〉_BF ≈ 0.04−0.17 in the backflow. The flow dynamics in Figures 6(a)–(f) are mostly consistent with those in Figures 8, 11, and 13 of Paper I, which depicted high-power FR-II jets.

The comparison of Q42-η5-r10 and Q42-η5-r10-M, in Figures 6(a)–(c) and (d)–(f), respectively, shows relatively minor differences, except that the jet head advances more slowly in the model with mass loading. This indicates that the amount of mass loading, q_D , adopted here does not substantially affect the flow dynamics in this specific example.

Furthermore, the probability distribution functions (PDFs) of M_s and β_s are presented in Figure 6(g)–(i) and (j)–(l), respectively, for six selected jet models. The bow shock, recollimation shocks, and turbulent shocks are generally stronger, with higher M_s and larger β_s , when induced by a jet with a larger initial Lorentz factor, Γ_j . Panels (g) and (j) confirm such trend: the volume-averaged values are 〈β_s 〉_BS ≈ 10⁻³, 〈M_s 〉_BS ≈ 2.7, 〈β_s 〉_JS ≈ 0.17, and 〈M_s 〉_JS ≈ 1.1 in Q42-η5-r10, while 〈β_s 〉_BS ≈ 2 × 10⁻³, 〈M_s 〉_BS ≈ 4.6, 〈β_s 〉_JS ≈ 0.27, and 〈M_s 〉_JS ≈ 1.3 in the higher-power Q43-η5-r10 model. Panels (h) and (k) display the effects of mass loading and declining density/pressure background, respectively. On average, the shock speeds and Mach numbers are slightly smaller, with 〈β_s 〉_BS ≈ 10⁻³, 〈M_s 〉_BS ≈ 2.6, 〈β_s 〉_JS ≈ 0.12, and 〈M_s 〉_JS ≈ 1.1 in the Q42-η5-r10-M model. On the other hand, they are substantially higher with 〈β_s 〉_BS ≈ 3 × 10⁻³, 〈M_s 〉_BS ≈ 6.7, 〈β_s 〉_JS ≈ 0.15, and 〈M_s 〉_JS ≈ 1.1 in the Q42-η5-r10-P model.

The PDFs of M_s and β_s for the two r100 models on larger scales of ∼10 kpc can be seen in Figures 6(i) and (l). As expected, the higher-power jet, Q45-η5-r100, induces shocks with higher M_s and larger β_s on average, compared to the lower-power jet, Q44-η5-r100, as expected. In these two r100 models, the PDFs of β_s and M_s exhibit broader ranges than those in the fiducial r10 models shown in panels (g) and (j). These broader distributions arise because the jets in the r100 models escape the dense core and propagate through the low-density regions of the stratified background, leading to shock formation in less dense environments. This results in PDFs that extend to higher M_s and larger β_s .

The synchrotron surface brightness, or intensity, depends on factors such as magnetic fields, CR electron distributions, relativistic beaming, and relativistic Doppler shift. It determines the morphological characteristics of optically thin, diffuse sources like FR-type radio galaxies. We employ models for magnetic fields and CR electrons that use hydrodynamic variables from simulations, and estimate the synchrotron volume emissivity, j_ν (x, y, z), from which the synthetic intensity map, I_ν (x, z), is generated, as described in Section 2.4. We point out that our synthetic maps may not fully capture realistic radio morphology due to limitations in our modeling, such as the omission of MHD effects and the use of simplified methods for estimating magnetic field strength and CR electron populations. Yet, as discussed in Paper I, our RHD simulations show that even high-power FR-II jets with Q_j ≈ 3 × 10⁴⁷ erg s⁻¹ form chaotic, turbulent structures near the jet head, rather than a well-defined termination shock (hot spot), acting as a stable working surface (see Figure 2 in that paper). None of the synthetic maps of the low-power models considered here exhibit distinct hot spots. Despite these limitations, Figures 7–9 illustrate the general dependence of radio morphology on various model parameters, as described in detail below.

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Figures 7(a) and (b) display the 2D distributions of magnetic field strength in two different frames: $B^{\prime} (x,z)$ in the local fluid frame and B(x, z) in the computational (or observer) frame. The results at t_end are shown for the Q42-η5-r10 model. With our magnetic field recipes, both ${B}^{{\prime} }$ and B are a few × 10 μG in the subrelativistic backflow, while $B^{\prime} \sim \,10\sim 100\,\mu {\rm{G}}$ and B ≳ 100 μG in the relativistic jet-spine flow. These are in good agreement with the magnetic field strength inferred from X-ray and radio observations of radio galaxies (see, e.g., J. Kataoka & ŁStawarz 2005; S. Ito et al. 2021), as well as the values extrapolated from the studies of objects on more compact scales (M. Zamaninasab et al. 2014).

Figures 7(c)–(d) depict the 2D slice views of the synchrotron volume emissivity, $j{{\prime} }_{\nu ^{\prime} }(x,z)$ in the fluid frame and j_ν (x, z) in the observer frame, respectively. The regions near the first and second recollimation shocks, along with those around the expanding boundary/shear layer, appear brighter, whereas the jet spine appears to be hollow. When viewed from the computational frame (j_ν ), faster-moving regions on the jet spine become fainter due to the Doppler dimming effect; hence, only the shear layer and the region near the jet head dominate the image. These features fit well with the expanding "spine/shear-layer model" proposed by R. A. Laing & A. H. Bridle (2002a). In this model, Doppler dimming, shock-mediated brightening, and strong emission from the shear/turbulent region can be seen.

The surface brightness map is a superposition of such slices, integrated along the LOS. As an illustration, a radio map for Q42-η5-r10-P is displayed in Figure 7(f). Here, the simulation box is tilted by the angle of π/2 − θ_obs around the x-axis with the inclination angle θ_obs = 60°, as demonstrated in Figure 7(e). This example illustrates the pair of the boosted approaching jet and the deboosted receding jet.

Figure 8 shows how the morphology of radio jets at 150 MHz varies with the inclination angle, θ_obs, and the power-law index, σ, of the CR electron spectrum in the Q42-η5-r10, Q42-η5-r10-M, Q42-η5-r10-P, Q44-η5-r10, and Q44-η5-r100 models. We note that with our modeling of synchrotron emission, only the relative intensity is meaningful in the surface brightness maps; thus, the intensity is normalized by its peak value, ${I}_{\nu }^{\max }$, to highlight the characteristic features in each panel. The relative intensity ratio, ${ \mathcal F }={I}_{\nu }^{\max }/{I}_{\nu ,N}^{{\rm{Q}}42{\rm{r}}10}$, is provided in each panel, where ${I}_{\nu ,N}^{{\rm{Q}}42{\rm{r}}10}$ represents the peak value of the face-on map (θ_obs = 90°) for the Q42-η5-r10 model with σ = 2.2. For instance, ${ \mathcal F }$ is higher for greater Q_j and for smaller θ_obs. In addition, the values of ${ \mathcal F }{| }_{{\theta }_{\mathrm{obs}}=90^\circ }$ are listed in column (5) of Table 2.

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Figure 8(a) exhibits an "edge-brightened" morphology with a bright but diffuse jet head in the face-on case (θ_obs = 90°). Additionally, mixing layers around the jet spine are clearly visible. In contrast, Figure 8(c) shows a "center-brightened" morphology with a fainter diffuse lobe for θ_obs = 30° due to strong relativistic beaming effects. Note that ${ \mathcal F }=10$ in panel (c), so the region of the recollimation shocks is much brighter than that in panel (a). As expected, the morphology at θ_obs = 60° is intermediate, with both the center and edge displaying comparable brightness.

Following the original criterion established by B. L. Fanaroff & J. M. Riley (1974), we estimate the FR ratio, d_b /d_f , by measuring the projected distances from the injection point to the brightest spot, d_b , and to the faintest edge, d_f , as defined in Figure 7(e). We point out that the first recollimation shock is excluded in the estimation of d_b . The comparison of Figures 8 (a)–(c) demonstrates the transition from edge-brightened to center-brightened as θ_obs decreases from 90^◦ to 30^◦, resulting in a decrease in the d_b /d_f ratio from 0.94 to 0.24. The FR ratios for inclination angles, θ_obs = 90°, 60°, and 30°, are given in columns (6–8) of Table 2, respectively. For Q44-η5-r10, which displays a hot spot at the jet head, d_b ≈ d_f , so we set d_b /d_f ∼ 1 (see Figure 8(i)–(j)).

On the other hand, the comparison of Figures 8(a), (d), and (e) shows that, as the power-law slope σ increases, the intensity map becomes progressively dominated by the jet material, characterized by higher f_jet, greater , and lower ρ. This is consistent with the scaling of the normalization factor in our model, ${{ \mathcal N }}_{0}^{{\prime} }\propto {f}_{\mathrm{jet}}\cdot \epsilon {\left(\epsilon /\rho \right)}^{\sigma -2}$. We note that for radio maps with different σ's, the normalization of the synchrotron emissivity depends somewhat arbitrarily on how we model the CR electron population, $N{{\prime} }_{e}({\gamma }_{e}^{\prime} )$. Therefore, the relative intensity ratio ${ \mathcal F }$ is not explicitly provided in Figures 8(d)–(e) for σ = 3 and 4.

The comparison between Figures 8(c) and (f) reveals the effects of mass loading in Q42-η5-r10-M. In panel (f), the absence of a visible arch at the jet head makes the morphology akin to a tailed FR-I source (e.g., B. Mingo et al. 2019, Figure 2). In Figures 8(g)–(h), the morphology of the Q42-η5-r10-P jet with θ_obs = 60° and 30^◦ is seen to be dominated by the loss of pressure confinement, resulting in a bulge caused by what seems to be like a flaring activity.

To analyze the asymmetry in the radio morphology between the approaching jet and the receding counterjet, we generate a counterjet in the region z < 0 by mirroring the simulated jet in z > 0, that is, by reversing the vertical velocity component, v_z → −v_z . Both the jet and the counterjet are tilted at different inclination angles, and surface brightness maps are generated, as illustrated in Figure 7(e). In this configuration, the approaching jet brightens while the receding counterjet dims, due to the relativistic bulk motion of the jet-spine flow.

In Figure 9, we present the surface brightness maps of jet-counterjet pairs at two different inclination angles for the Q42-η5-r10, Q42-η5-r10-P, and Q44-η5-r100 models. The maps clearly show an asymmetry between the jet-counterjet pairs, characterized by the boosted approaching jet and the deboosted receding jet. This asymmetry becomes more pronounced as θ_obs decreases, leading to stronger center brightening in the approaching jet and more noticeable edge brightening in the receding counterjet. These results align well with the established correlation between core dominance and inclination angle in 3CRR sources (e.g., F. Marin & R. Antonucci 2016).

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As shown in Figures 9(b), (d), and (f), at θ_obs = 30° the receding counterjets may look like edge-brightened jets, whereas the approaching jets behave as BL Lac-like FR-I jets. For instance, Figure 9(f) resembles the radio image of the low-luminosity radio galaxy 3C 31 (R. A. Laing & A. H. Bridle 2002a). Moreover, such pairs are similar to those seen in HYbrid MOrphology Radio Sources (HyMoRS) reported by W. P. J. Gopal-Krishna (2000). Recent observations by J. J. Harwood et al. (2020) also found such boosted FR-I/deboosted FR-II type hybrid sources, as we see in our synthetic maps. It is thought that the formation of HyMoRS is controlled by complicated processes, possibly involving the central engine and the properties of the host galaxy (W. P. J. Gopal-Krishna 2000), apart from the inclination angle. However, our results indicate that the jet inclination angle may play a significant role in the FR-I/II hybrid sources.