A Mechanism for the Triple-ridge Emission Structure of AGN Jets

We assume that the force-free condition is valid in the funnel region, where particle inertia and pressure are not important for dynamics, as shown in GRMHD simulations and implied from the current observational data of the M87 jet (Kino et al. 2014, 2015). We then activate the steady axisymmetric condition, and then the poloidal magnetic field and electric field measured in the lab frame are described as

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{{\rm{p}}}=\displaystyle \frac{1}{R}{\rm{\nabla }}{\rm{\Psi }}\times \hat{\phi },\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\boldsymbol{E}}=-\displaystyle \frac{1}{c}{{\rm{\Omega }}}_{{\rm{F}}}{\rm{\nabla }}{\rm{\Psi }}=-\displaystyle \frac{R{{\rm{\Omega }}}_{{\rm{F}}}}{c}\hat{\phi }\times {\boldsymbol{B}}.\end{eqnarray} \tag{ 2 }$$

(R, ϕ, z) are the standard cylindrical coordinates, with the z axis set to be the jet axis. $\hat{\phi }$ is the azimuthal unit vector. Ψ is the magnetic flux function and Ω_F corresponds to the angular velocity. We use a parametrically controlled form of Ψ,

$$\begin{eqnarray}&&{\rm{\Psi }}={{Ar}}^{\nu }(1\mp \cos \theta ),\end{eqnarray} \tag{ 3 }$$

where (r, θ, ϕ) are the standard spherical coordinates. A is a constant that normalizes the magnetic field strength, and ν controls the jet shape. The plus and minus signs stand for z < 0 and z > 0, respectively. Note that we focus on the emission structure far from the central BH, where we can neglect the GR effects. For this form of Ψ, the toroidal magnetic field is approximately given by (Tchekhovskoy et al. 2008)

$$\begin{eqnarray}&&{B}_{\phi }=\mp \displaystyle \frac{2{{\rm{\Omega }}}_{{\rm{F}}}{\rm{\Psi }}}{{Rc}}.\end{eqnarray} \tag{ 4 }$$

Equations (3) and (4) are not exact solutions of the Maxwell equations under steady axisymmetric condition (i.e., Grad-Shafranov equation), but they fit well to GRMHD simulation results (Tchekhovskoy et al. 2010; Nakamura et al. 2018). Ψ and Ω_F are quantities conserved along each field line and the electromagnetic field is described by their distribution. The cases of ν = 0 and 1 correspond to the radial and parabolic shapes of ${{\boldsymbol{B}}}_{{\rm{p}}}$, respectively. For ν = 2, the cylindrical structure is obtained in the far zone. As explained in Section 1, only the model with field lines penetrating the rapidly rotating BH can produce the nearly symmetric limb-brightened images as observed (T18). We focus on this case, in which Ω_F(Ψ) is given as Ω_F(Ψ) ≈ 0.5 Ω_H, where Ω_H = ac/2r₊, a is a spin parameter of the BH, and r₊ is the horizon radius of the rotating BH. We set a = 0.998.

We may not define the fluid velocity in principle in the force-free model, but practically we can use the drift velocity (Tchekhovskoy et al. 2008; Beskin 2010),

$$\begin{eqnarray}&&{\boldsymbol{v}}=c\displaystyle \frac{{\boldsymbol{E}}\times {\boldsymbol{B}}}{{B}^{2}}=R{{\rm{\Omega }}}_{{\rm{F}}}\hat{\phi }-R{{\rm{\Omega }}}_{{\rm{F}}}\displaystyle \frac{{B}_{\phi }}{{B}^{2}}{\boldsymbol{B}}.\end{eqnarray} \tag{ 5 }$$

This is consistent with the velocity in the Poynting-dominated cold ideal MHD formalism at the zone where RΩ_F ≫ c (see Equations (45) and (46) of T18). For RΩ_F ≪ c, the drift velocity has ${v}_{\phi }\sim R{{\rm{\Omega }}}_{{\rm{F}}}\gg {v}_{{\rm{p}}}$ (Equations (42) and (43) of T18) for the electromagnetic field that we consider (as discussed later in Section 3.1), which is also consistent with the cold ideal MHD velocity. Cold ideal GRMHD calculations also show v_ϕ ≫ v_p within the outer light cylinder (Pu et al. 2015), which supports our treatment of the fluid velocity. We do not treat the case in which the jet particles are injected along the magnetic field lines with a large Lorentz factor at the outflow base. We briefly discuss this case in Section 4.3.

We give the density distribution from the sourceless equation of continuity,

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot (n{\boldsymbol{v}})=0,\end{eqnarray} \tag{ 6 }$$

where n is the particle number density measured in the lab frame. This equation, combined with Equation (5), leads to another conserved quantity,

$$\begin{eqnarray}&&\displaystyle \frac{n}{{B}^{2}}=\mathrm{const}.\ \mathrm{along}\ \,\ \mathrm{each}\ \,\ \mathrm{field}\ \,\ \mathrm{line}.\end{eqnarray} \tag{ 7 }$$

When we give the density value at a certain point on each field line, we can obtain the density distribution everywhere. We note that Equation (7) is consistent with nv_p/B_p = const., that is, the mass flux per unit magnetic flux conserved in the cold ideal MHD formalism. Indeed, one can obtain Equation (7) by substituting Equation (5) to nv_p/B_p = const.

We set the fluid density at z = z₁ just for simplicity with

$$\begin{eqnarray}&&n(R,{z}_{1})={n}_{0}\exp \left(-\displaystyle \frac{{\left(R-{R}_{{\rm{p}}}\right)}^{2}}{2{{\rm{\Delta }}}^{2}}\right),\end{eqnarray} \tag{ 8 }$$

where z₁, R_p, Δ are free parameters. We assume that a constant fraction of the electrons are non-thermal⁵ and their energy distribution is described as

$$\begin{eqnarray}f(\gamma ^{\prime} )\propto \left\{\begin{array}{ll}\gamma {{\prime} }^{-p} & \mathrm{for}\ \gamma ^{\prime} \gt {\gamma }_{m}^{{\prime} }\\ 0 & \mathrm{for}\ \gamma ^{\prime} \lt {\gamma }_{m}^{{\prime} },\end{array}\right.\end{eqnarray} \tag{ 9 }$$

where γ' is the Lorentz factor of electrons measured in the fluid frame, ${\gamma }_{m}^{{\prime} }$ is the minimum value of γ', and p is the power-law index. p and ${\gamma }_{m}^{{\prime} }$ are set to 1.1 and 100, respectively, throughout this paper. We set the non-thermal electron density as zero along the field lines that do not penetrate the horizon, i.e., for Ψ > Ψ(r = r₊, θ = π/2).

The synchrotron emissivity in the fluid frame j_ω'^' is written by (Rybicki & Lightman 1986)

$$\begin{eqnarray}\begin{array}{rcl}{j}_{\omega ^{\prime} }^{{\prime} }(\hat{{\boldsymbol{n}}}^{\prime} ) & = & \displaystyle \frac{\sqrt{3}(p-1){e}^{3}n^{\prime} B^{\prime} \sin \alpha ^{\prime} }{8{\pi }^{2}{{\gamma }_{m}^{{\prime} }}^{1-p}{m}_{e}{c}^{2}(p+1)}{\left(\displaystyle \frac{{m}_{e}c\omega ^{\prime} }{3{eB}^{\prime} \sin \alpha ^{\prime} }\right)}^{-\tfrac{p-1}{2}}\\ & & \times \bar{{\rm{\Gamma }}}\left(\displaystyle \frac{p}{4}+\displaystyle \frac{19}{12}\right)\bar{{\rm{\Gamma }}}\left(\displaystyle \frac{p}{4}-\displaystyle \frac{1}{12}\right),\end{array}\end{eqnarray} \tag{ 10 }$$

where e, m_e, and $\bar{{\rm{\Gamma }}}(x)$ are the elementary charge, the mass of the electron, and the gamma function, respectively. $B^{\prime} \equiv | {\boldsymbol{B}}^{\prime} |$, and α' is the pitch angle of the electrons defined by $\cos \alpha ^{\prime} =(\hat{{\boldsymbol{n}}}^{\prime} \cdot {\boldsymbol{B}}^{\prime} )/| {\boldsymbol{B}}^{\prime} |$, where $\hat{{\boldsymbol{n}}}^{\prime}$ is the unit vector directing toward the observer in the fluid frame. Then, the emissivity at each point in the jet that is received by the observer is given by (Rybicki & Lightman 1986; Shibata et al. 2003)

$$\begin{eqnarray}&&{j}_{\omega }(\hat{{\boldsymbol{n}}})=\displaystyle \frac{1}{{{\rm{\Gamma }}}^{2}{\left(1-\beta \mu \right)}^{3}}{j}_{\omega ^{\prime} }^{{\prime} }(\hat{{\boldsymbol{n}}}^{\prime} ),\end{eqnarray} \tag{ 11 }$$

where β is the bulk speed normalized by the speed of light, and Γ is its corresponding Lorentz factor. μ is the cosine of the angle between the direction of the bulk flow and the direction of the line of sight in the lab frame.

The jet is optically thin for synchrotron radiation in the region where the triple-ridge structure is observed (∼10–30 mas from the radio core) and at the wavebands of those observations (Asada et al. 2016; Hada 2017). The observed intensity is then calculated by

$$\begin{eqnarray}&&{I}_{\omega }(X,Y)=\int {j}_{\omega }(\hat{{\boldsymbol{n}}},X,Y,Z){dZ},\end{eqnarray} \tag{ 12 }$$

where dZ is the line element along the line of sight. (X, Y) are the coordinates of the sky with the Y axis being the projected jet axis. The viewing angle Θ is defined as the angle between the line of sight (Z axis) and the jet axis (z axis). We first obtain the intensity map on the X–Y plane by the computation of Equation (12) with the spatial resolution of 3 R_S, and obtain the simulated VLBI image after the convolution with a Gaussian beam kernel.

Our model parameters are summarized in Table 1. a, ${\gamma }_{m}^{{\prime} }$, and p are set to their fiducial values for which T18 obtained limb-brightened images. While T18 used the same fiducial values for ν, Θ, and M_BH as in Broderick & Loeb (2009) to compare their calculation results, we replace them with the values roughly consistent with recent observations of the M87 jet. Nakamura et al. (2018) show that the width of the observed jet of M87 can be fitted by z ∝ R^1.6 (see also Asada & Nakamura 2012; Hada et al. 2013; Nakamura & Asada 2013). Since Equation (3) indicates that the shape of each field line obeys z ∝ R^2/(2−ν) asymptotically, we adopt ν = 0.75, assuming that the observed jet width reflects the magnetic field structure. The BH mass and the viewing angle are set by M_BH = 6.2 × 10⁹ M_⊙ (Gebhardt et al. 2011, rescaled for D = 16.7 Mpc) and Θ = 15° (Wang & Zhou 2009), respectively. Then, 1 mas corresponds to ≈136 R_S for D = 16.7 Mpc. We use the beam size of 1.14 mas × 0.55 mas, for which a triple-ridge image of the M87 jet was obtained (Hada 2017). Note that the beam size is larger than that in T18.

Table 1.  Model Parameters

Quality	Symbol	Value
Mass of the BH	MBH	6.2 × 109M⊙
Dimensionless Kerr parameter of the BH	a	0.998
Rotational frequency of the magnetic field	ΩF	0.5ΩH
Viewing angle	Θ	15°
Jet Shape (Equation (3))	ν	0.75
Radius where n peaks at z = ± z1	Rp	0
Height of the plane where n is given by Equation (8)	z1	5 RS
Width of n distribution (Equation (8))	Δ	2 RS
Energy spectral index of the non-thermal electrons	p	1.1
Minimal Lorentz factor of the non-thermal electrons	${\gamma }_{m}^{{\prime} }$	100
Luminosity distance to the jet	D	16.7 Mpc
Inclination between the jet axis and the major axis of the beam kernel		16°
Beam size		1.14 mas × 0.55 mas

Download table as:  ASCIITypeset image

For R_p, Δ, and z₁, T18 showed the dependence of the image on R_p with fixed values of Δ and z₁. In the case of R_p = 0, the jet image has one component around the axis, while in the case of R_p > 0 (e.g., ${R}_{{\rm{p}}}=40\,{R}_{{\rm{S}}}$, which is the fiducial value in their paper), limb-brightened images are obtained. They focused on the region ∼1–4 mas from the radio core, which is nearer than the ∼10–30 mas where the triple-ridge structure of the M87 jet is observed. Even though we compute images with the same parameters as T18 for the far region, the triple-ridge structure cannot be obtained, as shown in the Appendix. However, we found that the intensity map before the convolution for R_p = 0 has a distinct bright component along the jet axis that may correspond to the observed inner ridge (see Figure 7). Then we fixed R_p = 0, while setting large Δ to have a brighter jet edge, and found that the triple-ridge image can be obtained with Δ = 2 R_S and z₁ = 5 R_S. We show the calculation results with these parameter values in the next section.

Note that A and n₀ should be specified to obtain absolute intensity. T18 showed that reasonable values for them can lead to the observed level of intensity. However, the absolute intensity also depends on other uncertain parameters (p and ${\gamma }_{m}^{{\prime} }$), as well as the physical processes that we do not take into account in our current model, such as the radiative cooling and reacceleration of electrons. In this paper, we treat A and n₀ arbitrarily and focus on the relative intensity along the transverse direction of jets in order to simply illustrate our novel idea on the mechanism for the triple-ridge images.

Figure 1 shows the resultant image with the computational resolution (left panel) and the one convolved with the Gaussian beam (right panel). The Gaussian beam size is represented as the gray circle in the bottom left corner of the right panel. For both of the images, the intensity is normalized by each maximum value and the contours represent the intensity at 2^−k (k = 1, 2, 3, ...). In addition to the nearly symmetric two outer ridges, the inner ridge emerges in both images. In the left panel we find that the inner ridge extends along the jet axis and has an asymmetric shape. The right panel shows that the triple-ridge structure is highly smoothed by the convolution but still remains in Y > 15 mas. The counter jet appears in Y < 0 in the left panel and is overwhelmed by the bright core in the right panel.

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The physical reason for this triple-ridge image can be explained by focusing on the dependence of j_ω on n', B' and the relativistic boosting factor 1/Γ²(1 − βμ)³ (see Equations (10) and (11)):

$$\begin{eqnarray}&&{j}_{\omega }\propto \displaystyle \frac{n^{\prime} B{{\prime} }^{(p+1)/2}}{{{\rm{\Gamma }}}^{2}{\left(1-\beta \mu \right)}^{3}}.\end{eqnarray} \tag{ 13 }$$

We show the distribution of the three factors, as well as v_p/v_ϕ, in Figure 2. Note that the distribution of $\sin \alpha ^{\prime}$ is not relevant to the image structure in this problem (but see Shibata et al. 2003, for the problem of images of pulsar wind nebulae). We confirmed that the shapes of computed images when artificially setting $\sin \alpha ^{\prime} =1$ do not change from those shown in Figure 1.

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The upper left panel of Figure 2 shows the profile of B', which is roughly understood by writing down three magnetic field components (for z > 0) from Equations (1) and (4):

$$\begin{eqnarray}\begin{array}{rcl}{B}_{r} & = & {{Ar}}^{\nu -2}=\displaystyle \frac{{\rm{\Psi }}}{{R}^{2}}(1+\cos \theta ),\\ {B}_{\theta } & = & -\nu {{Ar}}^{\nu -2}\displaystyle \frac{1-\cos \theta }{\sin \theta }=-\nu \displaystyle \frac{{\rm{\Psi }}}{{R}^{2}}\sin \theta ,\\ {B}_{\phi } & = & -\displaystyle \frac{2{\rm{\Psi }}}{{R}^{2}}\displaystyle \frac{R{{\rm{\Omega }}}_{{\rm{F}}}}{{c}^{2}}.\end{array}\end{eqnarray} \tag{ 14 }$$

For a given R, each component is roughly proportional to Ψ, so it decreases for larger z. The Lorentz transformation from the lab frame to the fluid frame does not significantly change the profile. The upper right panel of Figure 2 shows the density distribution. At high z, the density around the axis becomes much smaller than the jet edge even though we set the density distribution concentrated around the axis at z = z₁. This is because the distribution along the field line is derived from Equation (7) and the magnetic field is stronger around the jet edge. Therefore, one can see that the bright outer ridges are produced by the strong magnetic field and high density around the jet edge.

The magnetic field is weaker and the electron density is lower around the axis, but the strong relativistic beaming makes the bright inner ridge. As seen in Equation (14) the magnetic field is poloidal-dominant at R ≪ c/Ω_F and toroidal-dominant at R ≫ c/Ω_F ($c/{{\rm{\Omega }}}_{{\rm{F}}}\simeq 2.13\,{R}_{{\rm{S}}}$). Correspondingly, the fluid velocity is toroidal-dominant at R ≪ c/Ω_F and poloidal-dominant at R ≫ c/Ω_F (see the lower left panel of Figure 2). In between, there is a region where the fluid velocity is almost parallel to the line of sight, i.e., μ ≈ 1. This enhances the factor $1/{{\rm{\Gamma }}}^{2}{(1-\beta \mu )}^{3}$, as shown in the lower right panel of Figure 2. Although the magnetic field, electron density, and Lorentz factor are small near the jet axis, the beaming effect produces the bright sharp inner ridge.

The transverse width of the relativistically beamed area, ∼10 R_S ∼ 0.1 mas, shown in the bottom right panel of Figure 2, corresponds to the width of the inner ridge of the left panel of Figure 1. We expect that observations with better resolution can illustrate the narrower inner ridge. Besides, the inner ridge appears only on one side of the jet axis. Which side the inner ridge appears on depends on the sign of Ω_F and tells us the direction of the BH rotation.

We calculate the images with different Θ, ν, Δ, and z₁ to examine the parameter dependence on the image structure. For illustration, we take Y = 25 mas, for which we plot the resultant transverse intensity profiles in Figure 3. Note that we normalize the intensity by the value at X = 0 of each line in Figure 3. The upper left panel represents the intensity profiles in the case of Θ = 7°, 10°, 15°, 20°, and 75°, with the other parameters unchanged. For smaller Θ, the relation $z=Y/\sin {\rm{\Theta }}$ for a fixed value of Y means that the line of sight passes points farther from the BH, and the jet width appears to be larger. For larger Θ, the outer ridges are debeamed because the line of sight and the beaming cone of the outer ridges are misaligned, while the inner ridge remains.

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The upper right panel of Figure 3 shows the intensity profiles in the case of ν = 0.7, 0.75, and 0.8 with the other parameters unchanged. When ν increases, the field lines and density concentrate on the axis. This leads to an image shaped like a candle flame. On the other hand, when ν decreases, the field lines and density distribution expand. This results in the bright outer ridges, which dominate the inner-ridge component.

As for the dependence on the parameters related to the density distribution, we examine only the dependence on Δ and z₁ because the dependence on R_p was discussed in T18. As Δ increases or z₁ decreases, the density in the jet edge increases and the outer ridges become brighter, as shown in the lower panels in Figure 3. The positions X of the outer ridge peaks do not change between the cases of Δ = 2 R_S and Δ = 3 R_S, and between the cases of z₁ = 2.5 R_S and z₁ = 5 R_S because the jet width is limited by the outermost magnetic field line threading the BH. Although our setup for the electron density distribution is a toy model, this analysis indicates that the observed image structure can strongly constrain the spatial electron density distribution when Θ and ν are estimated by other observational information such as the blob pattern speed, the brightness ratio between the approaching and counter jets, and the width profile of the jet.

As seen in Figures 1 and 3, the inner ridge and outer ridges produced by our model are not so clearly separate as reported in Hada (2017). This is because the jet edges have a sheath-like three-dimensional structure and the emission from the sheath enhances the brightness of the valleys between the ridges.

For more details, we show j_ω distributions along lines of sight passing (X, Y) = (±0.5 mas, 25 mas), (±1 mas, 25 mas), and (±2 mas, 25 mas) as functions of z in Figure 4 to examine which parts of the jet contribute to the transverse intensity profile ${I}_{\omega }(X,Y=25\,\mathrm{mas})$ (the black lines in Figure 3). The jet emission is composed of the sheath-like jet-edge component and the beamed inner-ridge component. For the line of sight of (X, Y) = (−0.5 mas, 25 mas), the inner-ridge component appears in addition to the jet-edge component, while for the line of sight of (X, Y) = (0.5 mas, 25 mas), the emission around the axis is debeamed. This leads to the asymmetry of the inner ridge in our computed image, which we have pointed out for the left panel of Figure 1 in Section 3.

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The lines of sight passing (X, Y) = (±1 mas, 25 mas) and (±2 mas, 25 mas), for which the valleys and outer ridges are seen, respectively (see Figures 1 and 3), penetrate only the jet edge. Interestingly, j_ω at the rear part of jet edge z ≲ 1.25 × 10⁴ R_S is comparable to that at the front part of jet edge z ≳ 1.4 × 10⁴ R_S for each of these sight-lines. That is because the fluid motion at the rear part directs away from the line of sight and then the emission is debeamed, but n' and B' are larger there than those at the front part with higher z. Figure 4 implies that the intensities ${I}_{\omega }=\int {j}_{\omega }{dZ}$ for (X, Y) = (±1 mas, 25 mas) and (±2 mas, 25 mas) should be comparable, which means that the valleys are not deep.

The observed deep valleys may indicate more complex structure of the jet. For example, if the non-thermal electrons distributed separately at the spine and a thin layer of the jet edge, then the brightness for (X, Y) = (±1 mas, 25 mas) would become lower. A thin layer with dense non-thermal electrons would produce bright outer ridges with deep valleys.⁶ We also consider that a non-axisymmetric jet may lead to deep valleys. For example, if only the rear part of the jet edge at $-1\,\mathrm{mas}\lesssim X\lesssim 1\,\mathrm{mas}$ is intrinsically dim, the intensities at the valleys halve when keeping the outer ridges bright.

We assume that the force-free approximation is valid even at the far zone from the BH. Then the bulk Lorentz factor Γ keeps increasing. Figure 5 represents the Lorentz factor profiles along the magnetic field lines that satisfy Ψ = Ψ (r₊, π/2), Ψ = Ψ (r₊, π/2)/2, and Ψ = Ψ (r₊, π/2)/10 in the model of Figure 1. These obey the analytical relation $1/{{\rm{\Gamma }}}^{2}=1/{{\rm{\Gamma }}}_{1}^{2}+1/{{\rm{\Gamma }}}_{2}^{2}$, where ${{\rm{\Gamma }}}_{1}^{2}={B}^{2}/{B}_{{\rm{p}}}^{2}\propto {R}^{2}$ and ${{\rm{\Gamma }}}_{2}^{2}={B}^{2}/({B}_{\phi }^{2}-{E}^{2})\propto ({R}_{{\rm{c}}}/R)$ at the far region $R{{\rm{\Omega }}}_{F}\gg c$, where R_c is the curvature radius of the poloidal field line. This asymptotic relation is shown in Tchekhovskoy et al. (2008) for the force-free case⁷ and in Komissarov et al. (2009) and Lyubarsky (2009) for the Poynting-dominated cold ideal MHD case.

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The Lorentz factor increases up to ∼80 in the computed area along each field line shown in Figure 5. This is much higher than the Lorentz factors deduced from blob motions observed in the VLBI observations (Mertens et al. 2016; Hada 2017, and references there in), Γ ≲ 10, although it is possible that faster blobs are not identified due to the apparent decrease of the number of fast blobs (Komissarov & Falle 1997) and the low time resolution of the current monitoring (Nakamura et al. 2018). If the Lorentz factors of currently identified blob motions are similar to those of the steady flows in jets, we require MHD models with Poynting to kinetic energy flux ratio (i.e., σ parameter) at the jet base as low as ∼10, which gives lower-saturation Lorentz factors.

The inner ridge of our computed image arises because there is a region where the direction of the velocity is parallel to the line of sight between the toroidal-dominant region (${v}_{\phi }\gg {v}_{{\rm{p}}}$, RΩ_F/c ≪ 1) and poloidal-dominant region (v_p ≫ v_ϕ, RΩ_F/c ≫ 1), as explained in Section 3.1. We have demonstrated this property by setting the fluid velocity to be the drift velocity (Equation (5)). The cold ideal MHD velocity also has the same property as the drift velocity, i.e., it is toroidal-dominant at RΩ_F/c ≪ 1, while it is poloidal-dominant at RΩ_F/c ≫ 1 (Beskin 2010; Toma & Takahara 2013; Pu et al. 2015, T18).

However, if the jet particles are injected along the magnetic field lines with the large Lorentz factor, Γ_in, at the jet base, the velocity structure will change, i.e., the velocity at the region R Ω_F/c ≪ 1 where B_p ≫ B_ϕ will become dominated by the poloidal component (Beskin 1997; Beskin & Malyshkin 2000). We plot the Lorentz factor close to the axis for our fiducial model in Figure 6, which shows that 5 ≲ Γ ≲ 9 around the region that produces the inner ridge (c/Ω_F ≲ 10R_S ≲ R ≲ 20R_S; see the bottom right panel in Figure 2). If Γ_in is smaller than 5, the velocity structure at R ≳ 10R_S will not be changed, so that the triple-ridge structure will be still observed. For a smaller viewing angle, the inner-ridge radius is larger (as seen in the upper left panel of Figure 3) and the Lorentz factor at that radius is larger, so that the triple-ridge structure will still appear in cases of even larger Γ_in. We note that the large Γ_in will also change the field configuration because of the particle inertia. These effects have not been taken into account either in GRMHD simulations. We need a more sophisticated model to consider them in detail.

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The lower right panel of Figure 2 shows that the inner-ridge radius does not depend on z. In contrast, Asada et al. (2016) indicates that the inner-ridge width of the M87 jet varies as a function of the distance from the BH. They showed that farther from the BH, the inner ridge becomes wider at 5 GHz, while it becomes narrower at 1.6 GHz. This might be caused by the synchrotron cooling, as mentioned in Asada et al. (2016), or the time variation of the jet, neither of which are included in our model.