Winds and Disk Turbulence Exert Equal Torques on Thick Magnetically Arrested Disks

We use the H-AMR code (Liska et al. 2022b), which evolves the equations of ideal GRMHD on a spherical polar grid in modified Kerr-Schild coordinates (Gammie et al. 2003) using the piecewise parabolic method for spatial reconstruction (Colella & Woodward 1984) and second-order time-stepping. H-AMR uses various computational speed-ups, including both static and adaptive mesh refinement (SMR and AMR, respectively), local adaptive time-stepping, and acceleration by graphical processing units (GPUs). We adopt dimensionless units G = M = c = 1, where M is the mass of the BH, such that the gravitational radius r_g = GM/c² = 1.

In this paper, we study a long-duration (out to 120, 000r_g /c) simulation that was originally presented in Liska et al. (2020). We consider a rapidly rotating BH of dimensionless spin a = 0.9 at the center of an equilibrium Chakrabarti (1985) torus with a sub-Keplerian angular momentum profile, l ∝ λ^1/4 (where λ is the radius of the relativistic von Zeipel cylinder, which is asymptotically the cylindrical radius), with an inner edge located at r_in = 6r_g and a pressure maximum at ${r}_{\max }\,=13.792{r}_{g}$. The outer edge of the torus, which is determined by our angular momentum distribution, lies at r_out ≈ 4 × 10⁴ r_g . We use a polytropic equation of state with an adiabatic index γ = 5/3, which results in a torus scale height that ranges from H/r ∼ 0.2 at ${r}_{\max }$ to H/r ∼ 0.5 at r_out.

The outer edge of the grid is located at r = 10⁵ r_g , and the inner edge is sufficiently far below the event horizon such that our inner boundary is causally disconnected from the BH exterior. The grid is uniformly spaced in $\mathrm{log}r$, θ, and φ for the ranges $0.1127\,\lt \mathrm{log}r/{r}_{{\rm{g}}}\lt 5$, 0 < θ < π, and 0 < φ < 2π. We use transmissive boundary conditions (BCs) at the poles, periodic BCs in the φ-direction, and absorbing BCs at the inner and outer radial boundaries. We select a base resolution of N_r × N_θ × N_φ = 1872 × 624 × 128 with four levels of static mesh refinement in φ. This means that the φ resolution reaches 1024 cells at the midplane and decreases at higher latitudes until it reaches 128 cells near the pole. This results in a total of over one billion cells and 70–90 cells per disk scale height (taking H/r ≈ 0.4).

Within the torus, we initialize a toroidal magnetic field with a (nearly) uniform plasma β ≡ p_gas/p_mag = p_gas/b² = 5, where p and b² are the fluid-frame gas and magnetic pressure, respectively. Via dynamo action, the initially toroidal magnetic field evolves into a large-scale poloidal magnetic field at ∼10,000r_g /c and soon thereafter pushes the disk into the MAD regime (see Liska et al. 2020, and references therein for a full discussion).

One of the main goals of this work is to take inventory of the angular momentum transport mechanisms in MADs. To measure the angular momentum flux, we use the mixed stress-energy-momentum tensor,

$$\begin{eqnarray}&&{T}_{\nu }^{\mu }=\left(\rho +u+p+{b}^{2}\right){u}^{\mu }{u}_{\nu }+\left(p+\displaystyle \frac{1}{2}{b}^{2}\right){\delta }_{\nu }^{\mu }-{b}^{\mu }{b}_{\nu },\end{eqnarray} \tag{ 1 }$$

where ρ is the gas density, u is the internal energy, and p is the gas pressure, each measured in the fluid-frame. b^μ is the contravariant magnetic field four-vector (such that b² = b^μ b_μ ), u^μ is the contravariant four-velocity, and ${\delta }_{\nu }^{\mu }$ is the Kronecker delta. We also use the relativistic enthalpy, w ≡ ρ + u + p + b². In our decomposition, we separate the first term in Equation (1), which represents hydrodynamic stresses (except for a small contribution from the magnetic pressure),

$$\begin{eqnarray}&&{\left({T}_{\nu }^{\mu }\right)}_{u}={{wu}}^{\mu }{u}_{\nu },\end{eqnarray} \tag{ 2 }$$

and the third term, which represents magnetic stresses,

$$\begin{eqnarray}&&{\left({T}_{\nu }^{\mu }\right)}_{{EM}}=-{b}^{\mu }{b}_{\nu }.\end{eqnarray} \tag{ 3 }$$

Since we only are concerned with off-diagonal terms (μ ≠ ν) of ${T}_{\nu }^{\mu }$ in this work, the second term in Equation (1) is always zero, and we do not consider it further. We average our stresses in space and time using the following procedure. First, we perform Reynolds decompositions for the various terms. If a stress equals the product XY, then we define the Reynolds decomposition as

$$\begin{eqnarray*}&&\left\langle {XY}\right\rangle \,\equiv \left\langle X\right\rangle \left\langle Y\right\rangle +\left\langle \delta X\delta Y\right\rangle ,\end{eqnarray*}$$

where we have defined our averages over space and time as

$$\begin{eqnarray}&&\left\langle X\right\rangle \,\equiv \iint \sqrt{-g}X\,{dt}\,d\varphi \,/\iint \sqrt{-g}\,{dt}\,d\varphi .\end{eqnarray} \tag{ 4 }$$

Here, quantities of the form 〈X〉〈Y〉 represent the laminar large-scale behavior of the flow, while quantities of the form 〈δ X δ Y〉 represent the turbulent small-scale behavior of the flow. Our time averages are taken over the period 80,000–120,000r_g /c. When calculating the turbulent δ X term, we first calculate 〈X〉 on the entire data set, then iterate through each time step to find the difference between the instantaneous data and the averaged data (i.e., δ X = X − 〈X〉).

We perform our Reynold's decomposition on Equations (2) and (3) to derive the laminar and turbulent components of the hydrodynamic and magnetic stresses, respectively. The decomposition of Equation (2) is

$$\begin{eqnarray}&&{\left({T}_{\nu }^{\mu }\right)}_{\delta u}=\left\langle \left({{wu}}^{\mu }-\langle {{wu}}^{\mu }\rangle \right)\left({u}_{\nu }-\langle {u}_{\nu }\rangle \right)\right\rangle =\left\langle \delta ({{wu}}^{\mu })\delta {u}_{\nu }\right\rangle ,\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\left({T}_{\nu }^{\mu }\right)}_{\langle u\rangle }=\langle {{wu}}^{\mu }\rangle \langle {u}_{\nu }\rangle .\end{eqnarray} \tag{ 6 }$$

Moving forward, we refer to Equation (5) as the turbulent Reynolds stress and to Equation (6) as the advective flow of angular momentum. We decompose Equation (3) in a similar fashion,

$$\begin{eqnarray}&&{\left({T}_{\nu }^{\mu }\right)}_{\delta b}=-\left\langle \left({b}^{\mu }-\langle {b}^{\mu }\rangle \right)\left({b}_{\nu }-\langle {b}_{\nu }\rangle \right)\right\rangle =-\left\langle \delta {b}^{\mu }\delta {b}_{\nu }\right\rangle ,\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\left({T}_{\nu }^{\mu }\right)}_{\langle b\rangle }=-\langle {b}^{\mu }\rangle \langle {b}_{\nu }\rangle .\end{eqnarray} \tag{ 8 }$$

We refer to Equation (7) as the turbulent Maxwell stress and to Equation (8) as the large-scale Maxwell stress. We return to these calculations when we present them in Section 3.3.

Time- and φ-averages are necessary to understand the long-term behavior of MADs. This, however, smooths over many interesting transient structures in the accretion flow that we would like to highlight first.

Figure 1 depicts the turbulent nature of the accretion disk by plotting the plasma β of the fluid at t = 120,000 r_g /c. In Figure 1(a), we show a slice of plasma β in the x–z plane. Within the equatorial accretion disk, we see regions that are dominated by gas pressure (blue) and by magnetic pressure (red). These regions are turbulent and irregular, except for one large-scale vertically extended structure at x ∼ −50 r_g /c. This is a magnetic flux eruption event, which is a standard feature of MADs (Tchekhovskoy et al. 2011). Magnetic flux eruptions travel radially outward through the accretion disk and can be identified as large-scale magnetic flux bundles characterized by relatively lower values of β. Chatterjee & Narayan (2022) suggest that these flux eruption events provide an important form of angular momentum transport in MADs.

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In Figure 1(b), we show an equatorial slice through the system. We now observe both small-scale turbulence and highly magnetized spiral modes, which is a standard feature of MADs and arises from the magnetic interchange instability (Kaisig et al. 1992; Lubow & Spruit 1995; Spruit et al. 1995; Marshall et al. 2018; Mishra et al. 2019). Indeed, we see the x–y projection of the magnetic flux eruption event in Figure 1(a) at x ∼ − 50r_g , y ∼ −10r_g .

In this section, we study the angular momentum flow in three regions of our system: in the accretion disk, in the disk winds, and in jets. We define each region using our time-averaged axisymmetrized data. Then, we plot these definitions in Figure 2, which shows x-z slices of the flow that we φ- and time-averaged from 80,000 r_g/c to 120,000 r_g/c, where the accretion flow has reached a quasi-steady state.

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Figures 2(a) and (b) show the gas density ρ, and panels (c) and (d) show the plasma β. At the origin of each panel [x = 0, z = 0], we have plotted the BH event horizon. The left column (Figures 2(a), (c)) zooms in on the right column (Figures 2(b), (d)) from 100r_g to 25r_g . We plot streamlines over the density and plasma β contours to show the flow of the angular momentum flux, as defined by the four-vector ${T}_{\varphi }^{\mu }$ projected onto the Cartesian basis vectors $\hat{x}$ and $\hat{z}$. We use these streamlines to define the three regions of our system: jets, winds, and the accretion disk.

The jet regions exist at the poles of the BH and form an hourglass shape (at x = y = 0 and extending vertically in z). Here, the magnetic pressure increases to over 100 times the gas pressure (e.g., ${\mathrm{log}}_{10}\beta \lesssim -2$) ¹⁰ as the jet is magnetically launched and angular momentum is extracted from the BH through the Blandford–Znajek (BZ; Blandford & Znajek 1977) mechanism. For each of the two jets, we define their boundaries as the last radially outward angular momentum streamline (counting from the pole) to originate from the BH event horizon. Figure 2 shows the jet boundaries with dashed black and white lines. We adopt this robust definition as it conserves the radial time-averaged angular momentum flux within the jet. We also note that there are angular momentum streamlines anchored to the accretion disk within the BH ergosphere. However, the ergosphere lies within the innermost stable circular orbit (ISCO) for our choice of BH spin (a = 0.9) (Bardeen et al. 1972; Carroll 2019). Therefore, the disk within the ergosphere has a negligible specific angular momentum compared to the magnitude of the angular momentum extracted within the jet and from the disk outside of the ergosphere. Additionally, any angular momentum transport measured would be measured as a Blandford–Payne (i.e., wind) outflow, just as intended.

We define all other radially outward angular momentum flux streamlines (${T}_{\varphi }^{r}$), which are anchored to the disk, as part of the winds. We define the disk-wind boundary by the set of turn-around points of the angular momentum streamlines, e.g., where the radial angular momentum flux vanishes (${T}_{\varphi }^{r}$). We show this boundary in Figure 2 with solid green lines. This is a somewhat novel way of describing the vertical extent of the accretion disk. In the innermost regions of the disk, r ≲ (10–20)r_g, the wind is squeezed between the jet and the disk and subtends a smaller solid angle as compared to at larger radii (e.g., r ≥ 50 r_g). We also note that the surfaces ${T}_{\varphi }^{r}=0$ and ρ v^r = 0 (turning point of the accretion flow) are not identical. At r < 20r_g , the difference is negligible. However, at 20 r_g < r < 100r_g , the vertical height of the surfaces as measured from the x–y plane differs by up to ∼4r_g , which is a difference of about a 5%. Using mass flux surfaces would likely produce the same results.

We define the accretion disk as the region where angular momentum streamlines have a negative radial component (${T}_{\varphi }^{r}\lt 0$). The disk resides roughly within z ≲ (0.4–0.5)r, where the gas density peaks at ρ ∼ 10², but is not significantly denser than the wind region.

The typical definition of the disk thickness is set by the midplane temperature of the accretion disk,

$$\begin{eqnarray}&&\displaystyle \frac{H}{r}\sim \displaystyle \frac{{c}_{s}}{{v}_{\mathrm{rot}}},\end{eqnarray} \tag{ 9 }$$

where c_s is the sound speed, and v_rot/r = u^φ /u^t , and u^μ is the four-velocity of the fluid. We plot this definition in Figure 2 with dashed green lines. We can see that it underpredicts the disk thickness for r ≲ 20r_g and overpredicts it for r ≳ 20r_g compared to our streamline-based definition. Furthermore, this definition assumes vertical hydrostatic equilbrium, which is not satisfied near the BH as the disk is squeezed vertically between the two jets. Thus, we opt for our definition instead as it is specifically constructed to separate regions of angular momentum inflow and outflow.

Figures 2(b) and (d) show that the angular momentum flow becomes asymmetric across the midplane for large radii (r > 50 r_g ). While this may seem surprising, this asymmetry is not unprecedented, as in totally energy-conserving MADs the turbulent eddies have length scales ${ \mathcal O }(H)$ that are ${ \mathcal O }(r)$ since H/r ≲ 0.4–0.5. These eddies can disrupt the current sheet at the midplane of the disk and cause asymmetries over long timescales (e.g., Tchekhovskoy 2015; Chatterjee & Narayan 2022).

In this section, we discuss the vertical profile of the angular momentum transport in our simulation. Here, we use the Reynolds decomposition of the stress-energy tensor (given by Equations (5)–(8)). Figure 3 shows the terms contributing to the radial angular momentum transport as a function of θ at r = 30r_g . Positive values indicate radially outward flow. We have normalized the y-axis to the average angular momentum flux onto a sphere at 2 r_g,

$$\begin{eqnarray}&&{({G}_{\varphi }^{r})}_{2\,{r}_{{\rm{g}}}}=2\pi {\left[\int \langle {T}_{\varphi }^{r}\rangle \sqrt{-g}\,d\theta \right]}_{r=2{r}_{g}}.\end{eqnarray} \tag{ 10 }$$

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Here, the factor of 2π results from a trivial φ integral over the axisymmetrized fluxes. The streamlines in Figure 2 show the total angular momentum flux, which is also given by the solid gray line in Figure 3. It is directed outward in the wind regions and inward in the accretion disk. The advective transport (〈wu^r 〉〈u_φ 〉; Equation (6)) is radially outward in the winds and inward in the accretion disk. The Reynolds stresses (〈δ wu^r δ u_φ 〉; Equation (5)) are negligible in comparison to the other modes of transport. We break down the radial angular momentum transport driven by Maxwell stresses (Equations (7) and (8)) into turbulent, −〈δ b^r δ b_φ 〉, and laminar, −〈b^r 〉〈b_φ 〉, components. Within the disk, the turbulent Maxwell stresses dominate the outward angular momentum transport. As we cross into the wind regions, the large-scale magnetically driven angular momentum transport begins to dominate (e.g., Blandford & Payne 1982). This is consistent with other simulations of highly magnetized disks (Jacquemin-Ide et al. 2021; Scepi et al. 2024).

In Figure 4(a), we present the radially outward transport of angular momentum within the three regions—the jets, the winds, and the disk—as a function of radius. We decompose the radial angular momentum fluxes in the wind and disk regions into their turbulent and laminar Maxwell and Reynolds stress components (see Equation (11)). Then, we integrate over the radially oriented surfaces of the disk and the wind regions,

$$\begin{eqnarray}&&{({G}_{\varphi }^{r})}_{\mathrm{region}}=2\pi {\int }_{{\theta }_{\min }(r)}^{{\theta }_{\max }(r)}\left\langle {T}_{\varphi }^{r}\right\rangle \,\sqrt{-g}\,d\theta ,\end{eqnarray} \tag{ 11 }$$

where ${\theta }_{\min }$ and ${\theta }_{\max }$ are the minimum and maximum θ boundaries of the disk or wind regions, respectively. We ignore the turbulent Reynolds stress as it is insignificant compared to the other stresses (see Figure 3). On the x-axis, we plot the radius out to 100r_g , and on the y-axis, we plot the normalized angular momentum flux. We normalize the flux to the angular momentum flux through the sphere at 2r_g (Equation (10)).

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The horizontal gray line shows that the total angular momentum flux within the jet is constant $({\left({G}_{\varphi }^{r}\right)}_{\mathrm{jet}}=\mathrm{const}.)$, which is by construction. The solid blue and red lines describe the total angular momentum fluxes in the wind and disk regions, respectively. The transport of angular momentum in the wind and by the disk stresses are in equipartition at all radii. Furthermore, we see that the total angular momenta transported by the disk and wind regions both follow the same power law in radius, ∝r^0.9. It is unclear which physical processes result in this power-law dependence. We link this power law to the mass ejection rate in Section 3.4.

We plot the various mechanisms of angular momentum transport within each region with the dashed (turbulent Maxwell stress; −〈δ b^r δ b_φ 〉), dotted (large-scale Maxwell stress; −〈b^r 〉〈b_φ 〉), and dash–dotted (advective flow; −〈u^r 〉〈u_φ 〉) lines. We break down the radially outward angular momentum flux within the disk into the turbulent Maxwell and large-scale Maxwell stresses. We do not include the inward radial advective term within the disk fluxes as it describes the net radially inward accretion due to outward vertical or radial angular momentum transport processes. We see that the turbulent Maxwell stress (〈δ b^r δ b_φ 〉) closely follows the extracted total angular momentum and is the dominant mode of angular momentum transport within the disk. The large-scale Maxwell stress (〈b^r 〉〈b_φ 〉) is highly subdominant within the disk, which is as expected because the disk is turbulent. We also note that since the plasma β in the accretion disk is roughly constant out to ∼80 r_g , we expect the relative strength of the magnetic and velocity stress-energy terms to remain constant. In disks that are only dominated by magnetic pressure in the inner regions (i.e., the plasma β increases with radius), we do not expect our radial $\dot{L}$ profiles to hold. A disk like that might be better described by a broken power law where the inner magnetically saturated regions are described by our $\dot{L}\propto {r}^{0.9}$ profile, and the outer regions are not.

We break down the angular momentum transport within the winds into the turbulent Maxwell, large-scale Maxwell, and advective terms. For r ≲ 20 r_g , the large-scale Maxwell stresses are the dominant mechanism for radially outward angular momentum transport. For 20r_g ≲ r ≲ 70r_g , the advective term dominates. The transition between these transport mechanisms could be due to the acceleration of the wind by the magnetic forces, effectively redistributing the angular momentum transport from magnetic to advective transport. The advective component could also be driven by different forces, such as an enthalpy gradient, which would be naturally present in our hot total energy-conserving MAD. Below, we try to disentangle these two possibilities by looking at the angular momentum fluxes throughout the disk surface.

At distances r > 70r_g , the turbulent Maxwell angular momentum flux exceeds the large-scale Maxwell angular momentum flux. However, at these radii and beyond, we only sample ≲10 orbits in our averages, and the disk might not have achieved quasi-steady state at these large radii. This may not be enough time to average over all variability timescales, and our averages at these radii must therefore be taken with a grain of salt.

In Figure 4(b), we show the cumulative integral in r of the angular momentum flux through the disk-wind boundary and its decomposition, analogous to panel (a). Namely, here we decompose the angular momentum flux in the θ direction, ${T}_{\varphi }^{\theta }$, rather than ${T}_{\varphi }^{r}$. Our specific steps are as follows: we perform our decomposition and averaging procedure outlined in Section 2.2, we find the values of our decomposed $\langle {T}_{\varphi }^{\theta }\rangle$ at the north and south disk-wind boundaries, and we cumulatively integrate them in radius. We then sum the result from the top and bottom surfaces of the disk,

$$\begin{eqnarray}\begin{array}{rcl}{G}_{\varphi }^{\theta } & = & 2\pi {\int }_{{r}_{h}}^{r}\sqrt{-g}\,\langle {T}_{\varphi }^{\theta }\rangle (r^{\prime} ,\theta ={\theta }_{\mathrm{bottom}})\,{dr}^{\prime} \\ & & -2\pi {\int }_{{r}_{h}}^{r}\sqrt{-g}\,\langle {T}_{\varphi }^{\theta }\rangle (r^{\prime} ,\theta ={\theta }_{\mathrm{top}})\,{dr}^{\prime} ,\end{array}\end{eqnarray} \tag{ 12 }$$

where ${G}_{\varphi }^{\theta }$ is the cumulative angular momentum flux through the surface of the accretion disk, θ_bottom and θ_top are the bottom and top disk-wind boundaries (solid green lines in Figure 2), and r_h is the event horizon of the BH. As in Equations (10) and (11), the factor of 2π arises from a trivial φ integral over the axisymmetrized fluxes. The flux through the top of the accretion disk is in the negative θ direction, and therefore, we place a negative sign in front of the θ_top term to find the net flow of angular momentum out of the accretion disk.

By definition, no angular momentum crosses the wind-jet boundary on average, and the divergence theorem tells us that in steady state, this integral, Equation (12), evaluated at radius r, equals the surface integral of ${T}_{\varphi }^{r}$ over the wind region at the same radius. Indeed, Figure 4(b) shows that the total disk surface flux (solid green line) is almost equal to the wind flux (solid light blue line) and follows the same ∝r^0.9 radial scaling. The slight discrepancy between the two is likely due to the wind-jet boundary close to the BH. Close to the BH, the wind and disk are squeezed by the strong magnetic field attached to the BH. In the inner regions, the winds become a thin vertical sheath that could easily be contaminated by the angular momentum being extracted from the BH. This discrepancy may be further exacerbated by the density floors applied to the low-density regions at the base of the jet, close to the BH. This artificially adds mass, and therefore, momentum, to the inner jet and wind regions. We see this uptick in the total mass accretion rate of the system, which we depict with a thick solid black line in Figure 5. Nonetheless, we stress that the agreement between the radial wind flux and the vertical disk surface flux is remarkable.

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We decompose the angular momentum flux into its components at the disk surface (as detailed in Figure 4(b)) and show that the large-scale Maxwell torques, 〈b^θ 〉〈b_φ 〉, dominate the angular momentum transport for r < 20 r_g , beyond which the advective term, 〈wu^θ 〉〈u_φ 〉, takes over. ¹¹ To make these comparisons between flux components, we compare the gradient of each component because this describes the instantaneous strength of each flux at a radius (see Figure 6, Appendix). Note that the large-scale Maxwell torques remain dominant up to a similar radius when compared to panel (a).

In panel (b), we show that the advective torque, 〈wu^θ 〉〈u_φ 〉, becomes dynamically important in the outer regions of the accretion disk, r > 20r_g . The advective torque is related to the vertical gradient of angular momentum and should be negligible for negligible vertical gradients of 〈u_φ 〉 (see the fourth term of Equation (17) of Zhu & Stone (2018)). However, radiatively inefficient accretion disks such as the one modeled here can have sharp vertical angular momentum gradients related to their high thermal pressure. These sharp gradients can then lead to an efficient advective torque, 〈wu^θ 〉〈u_φ 〉, because gas that is displaced in the $+\hat{z}$ direction will carry its angular momentum into a region with lower angular momentum. This is in contrast with Scepi et al. (2024), who find that this term is always negligible even for their thicker disks. We emphasize that Scepi et al. (2024) employed explicit cooling to establish a thick disk, whereas our approach involves evolving the disk without any cooling. Despite yielding a similar geometric thickness, these methods may not produce identical dynamics, potentially influencing the wind launching (Casse & Ferreira 2000).

It is clear that the advective (〈wu^θ 〉〈u_φ 〉) and laminar magnetic (〈b^θ 〉〈b_φ 〉) torques both contribute to the total surface angular momentum transport. Hence, the winds in geometrically thick MADs should be understood as magneto-thermal (Casse & Ferreira 2000) and not purely driven by the Blandford & Payne (1982) mechanism. This is not surprising as the thermal energy of a geometrically thick MAD is considerable.

Furthermore, Scepi et al. (2024) find that the large-scale vertical torque, 〈b^θ 〉〈b_φ 〉, dominates the angular momentum transport in the inner regions of the accretion disk, r < 20r_g , which we confirm with our findings in this study. However, unlike Scepi et al. (2024), our analysis extends to 20 r_g < r ≲ 70r_g , where we find that the turbulent radial Maxwell torque within the accretion disk, 〈δ b^r δ b_φ 〉 (Figure 4(a)), is in equipartition with the total vertical torque at the disk surface. Last, we note that the vertical turbulent Maxwell term, 〈δ b^θ δ b_φ 〉, is subdominant throughout.

In panel (b) of Figure 4, we plot the cumulative integral of the surface fluxes along the disk-wind boundaries. This was done to properly conserve the angular momentum fluxes through our Gaussian surfaces. However, this can make it difficult to assess by eye the radial dependence of the various terms contributing to ${G}_{\varphi }^{\theta }$. To remedy this, we have included Figure 6 in the Appendix, where we depict $\tfrac{d}{{dr}}{G}_{\varphi }^{\theta }$ as a function of radius. This shows that the advective surface flux (〈wu^θ 〉〈u_φ 〉) locally dominates the vertical angular momentum transport for 20r_g < r ≲ 70r_g , even though they only become globally dominant for r > 70r_g (see Figure 4(b)). We refer to the Appendix for further discussion.

In this section, we assess how efficiently our system can drive mass inflow. To do this, we compare the measured mass accretion rate in the winds (${\dot{M}}_{\mathrm{wind}}$) and the disk (${\dot{M}}_{\mathrm{disk}}$), and in the system as a whole ($\dot{M}$) to what classical α viscosity theory predicts (Shakura & Sunyaev 1973; Pringle 1981). The α prescription predicts the following mass accretion rate:

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{SS}}=\alpha \displaystyle \frac{4\pi {{HP}}_{\mathrm{gas}}}{{\rm{\Omega }}}.\end{eqnarray} \tag{ 13 }$$

Here, H(r) is the vertical scale height of the accretion disk, and Ω(r) is the orbital frequency of the gas, which we have measured in the frame of the the zero angular momentum observer (ZAMO; see, e.g., definitions in Takahashi 2007). This frame is chosen to eliminate rotation due to frame-dragging by the central BH. P(r) is the gas pressure in the disk. We evaluate the quantities Ω(r) and P(r) using density-weighted averages of the solid angle, i.e., for instance, $P(r)=\int \langle p\rangle \langle \rho \rangle \sqrt{-g}\,{{dA}}_{\theta }/\int \langle \rho \rangle \sqrt{-g}\,{{dA}}_{\theta }$.

We can also define a mass ejection index (Ferreira & Pelletier 1995) to quantify the radial dependence of the mass accretion rate,

$$\begin{eqnarray}&&\xi =d\mathrm{ln}{\dot{M}}_{\mathrm{disk}}/d\mathrm{ln}r,\end{eqnarray} \tag{ 14 }$$

where ${\dot{M}}_{\mathrm{disk}}$ is the mass accretion rate within the disk, and r is the radial coordinate. We then use Equation (13) to compute ${\dot{M}}_{\mathrm{SS}}$ for a range of values, α = {0.01, 0.1, 1}, and we show the resulting radial profiles of ${\dot{M}}_{\mathrm{SS}}$ in Figure 5 with the solid gray lines. This allows us to compare our simulated mass accretion rate, shown with the thick black line, with the prediction of the α disk. First, we see that the total time-averaged mass accretion rate (thick solid black line) is roughly constant out to ∼80r_g , with a deviation of ∼10%. This is despite the asymmetries in the accretion flow in Figure 2, which suggests that the system has reached inflow equilibrium out to that radius. We also note that the uptick in $\dot{M}$ at r ≲ 5r_g is due to the density floors, which add mass in the jets. We then separate the mass flux into the wind and disk components using the definitions of Section 3.2.

The mass accretion rates in the winds (dash–dotted blue line) and disk (dashed red line), however, are not constant because the disk is losing gas to the wind. This results in a roughly ∝r^0.4 radial dependence of the accretion rate in the disk. Due to mass conservation, the wind should exhibit the same power law. This power law is altered at small radii, however, due to the boundary condition on the accretion rate in each region: The disk accretion rate must approach the BH accretion rate, while the wind accretion rate must approach zero. In other words, in the inner regions (r ≳ r_isco), the wind region carries almost no mass (see Figure 5). In the outer regions (r ≳ 25r_g ), the wind ejection rate recovers roughly the same power law as the accretion rate (see Figure 4).

If the system is in statistical steady state, the radial mass ejection power law should be related to the angular momentum power law we measured in Section 3.2,

$$\begin{eqnarray}&&{\dot{L}}_{\mathrm{tot}}\sim {\dot{M}}_{\mathrm{disk}}\langle {u}_{\varphi }\rangle ,\end{eqnarray} \tag{ 15 }$$

where ${\dot{L}}_{\mathrm{tot}}$ is the angular momentum accretion rate, and 〈u_φ 〉 is the averaged φ-velocity. If we assume Keplerian orbits in the accretion disk, we know that 〈u_φ 〉 ∝ r^0.5. Therefore, we can use our radial dependence in Figure 4 to recover the radial dependence of the mass accretion rate we measure in Figure 5. Indeed, we see that $\dot{L}\propto {r}^{0.9}$ in Figure 4 and confirm that ${\dot{M}}_{\mathrm{disk}}\propto {r}^{0.4}$ in Figure 5, as expected. The coherence of the angular momentum and mass flux power laws is correctly predicted by self-similar theory (Ferreira & Pelletier 1995; Ferreira 1997; Jacquemin-Ide et al. 2019). The measured power law for the accretion rate, ξ = 0.4, is consistent with a heavily mass-loaded outflow and is in the upper bound of what can be achieved with self-similar models (Jacquemin-Ide et al. 2019). The value we measure is in contrast with Scepi et al. (2024), who measure ξ = 1.04, but it is consistent with McKinney et al. (2012), who measure ξ = 0.4.

We can see from Figure 5 that the effective value of α is ≳1 at r ≲ 10r_g , but it decreases toward unity with radius. The overall magnitude of α ∼ 1 is high, but not surprising. Wind transport, which cannot be captured by the α formalism, is significant and in equipartition with turbulent transport, suggesting that our effective α could be limited to ≲1–2. Additionally, shearing-box MRI (Salvesen et al. 2016) and global MAD simulations (Scepi et al. 2024) both suggest that the turbulent α is ∝β⁻¹. Throughout our disk, we measure β ∼ 1 (see Figure 1), which is consistent with an order-of-unity effective α (McKinney et al. 2012).