ANGULAR MOMENTUM TRANSPORT BY ACOUSTIC MODES GENERATED IN THE BOUNDARY LAYER. II. MAGNETOHYDRODYNAMIC SIMULATIONS

For our simulations, we use Athena (Stone et al 2008) to solve the MHD equations with an isothermal equation of state in cylindrical geometry. These equations are

$$\begin{equation} \frac{\partial \rho }{\partial t} + \boldsymbol{\nabla }\cdot (\rho \boldsymbol{v}) = 0 \end{equation} \tag{ 1 }$$

$$\begin{eqnarray} \frac{\partial (\rho \boldsymbol{v})}{\partial t} + \boldsymbol{\nabla }\cdot (\rho \boldsymbol{v}\boldsymbol{v}) &=& -\boldsymbol{\nabla }\left(P + \frac{B^2}{2\mu } \right) + \frac{1}{\mu } (\boldsymbol{B}\cdot \boldsymbol{\nabla }) \boldsymbol{B}- \rho \boldsymbol{\nabla }\Phi \nonumber\\ \end{eqnarray} \tag{ 2 }$$

$$\begin{equation} \frac{\partial \boldsymbol{B}}{\partial t} = \boldsymbol{\nabla }\times (\boldsymbol{v}\times \boldsymbol{B}) \end{equation} \tag{ 3 }$$

$$\begin{equation} P = \rho s^2. \end{equation} \tag{ 4 }$$

Here, $\boldsymbol{v}$ is the velocity, ρ is the density, P is the pressure, Φ is the gravitational potential, s is the sound speed, and $\boldsymbol{B}$ is the magnetic field. We also use ϖ throughout this work to denote the cylindrical radius.

Adoption of an isothermal equation of state is equivalent to assuming fast cooling to a set temperature. Although this is not a realistic assumption, it allows us to simply attain a quasi-steady state and focus on exploring the effect of a magnetic field on the dynamics of the BL.

We nondimensionalize quantities by setting the radius of the star to ϖ_⋆ = 1, the Keplerian angular velocity in the disk just above the surface of the star to Ω_K(ϖ_⋆) = 1, the density in the disk, initially a constant, to ρ = 1, and the magnetic permeability to μ = 1. Finally, we define a characteristic Mach number for our simulations as M = 1/s. This corresponds to the true Mach number of fluid rotating at the Keplerian velocity at ϖ = ϖ_⋆.

The simulation setup consists of a radially stratified, unrotating star attached to a Keplerian disk via a thin interfacial region. The simulations are performed in cylindrical geometry with an isothermal equation of state and are not vertically stratified. The interfacial region smoothly connects the rotation profile of the star to that of the disk and is chosen to be as thin as possible, while still being resolved with ∼10 cells. The highest density in the star is approximately 4 × 10⁶ the density in the disk. Thus, the inner regions of the star have a high enough inertia to avoid being spun up over the course of a simulation.

We do not incorporate self-gravity into our runs, but instead use a fixed gravitational potential V = −1/ϖ. We initialize each of our simulations to be in hydrostatic equilibrium and add random perturbations to the radial velocity from which instabilities develop. For additional information on aspects of our numerical model that do not pertain to the magnetic field see Belyaev et al. (2013).

We now discuss the initial magnetic field strengths and geometries used in our simulations. For all our simulations, we set $\boldsymbol{B}= 0$ for ϖ < ϖ_⋆ (an unmagnetized star), and initialize one of three possible magnetic field geometries for ϖ ⩾ ϖ_⋆:

$$\begin{equation} \boldsymbol{B}(\varpi \ge \varpi _\star ,z) = \left\lbrace \begin{array}{@{}lc}\dsty\frac{B_0}{\varpi } \hat{\boldsymbol{z}}, & {\rm NVF} \\ B_0 \hat{\boldsymbol{\phi }}, & {\rm NAF}\\ \dsty\frac{B_0}{\varpi } \sin \left[\frac{2 \pi }{\lambda _\varpi } (\varpi - \varpi _\star)\right] \hat{\boldsymbol{z}}, & {\rm ZNF} \end{array} \right.. \end{equation} \tag{ 5 }$$

NVF simulation have a constant vertical magnetic flux through any radius, NAF simulations have a constant azimuthal magnetic flux, and ZNF simulations have on average zero net flux. For the ZNF simulation, we set λ_ϖ = Δz/2, where Δz is the extent of the simulation domain in the vertical direction. Thus, we have a rapid variation of the field in the radial direction, which allows comparison with shearing box ZNF simulations.

In all of our runs, we use an initial value of B₀ = .002 in units for which the magnetic permeability is μ = 1. Defining the plasma parameter β as

$$\begin{equation} \beta \equiv \frac{\rho s^2}{B^2/2\mu } \end{equation} \tag{ 6 }$$

the initial value of β using B₀ for a baseline is β₀ ≈ 6200. Note that in the NVF and ZNF simulations, β varies as a function of position and β₀ specifies the minimum initial value of β in the disk.

The particular value of B₀ = .002 is motivated by two considerations. The first is the need to resolve the wavelength of the fastest growing mode of the MRI in the case of a pure vertical field (NVF simulations), and the second is the need to fit several wavelengths of the fastest growing mode within the simulation domain.

For a global disk simulation in which the domain has constant width and resolution in the z direction, these two criteria are generally at odds with one another. This is because the z-wavelength of the fastest growing axisymmetric mode for constant vertical field in a Keplerian disk is given by Balbus & Hawley (1991)

$$\begin{eqnarray} &&\lambda _{z,{\rm MRI}} = \sqrt{\frac{16}{15}} \frac{2 \pi B}{\Omega \sqrt{\rho }}. \end{eqnarray} \tag{ 7 }$$

For a Keplerian disk with constant B_z field and constant ρ, λ_{z, MRI}∝ϖ^3/2. For our NVF field geometry, the magnetic field strength falls off as 1/ϖ so λ_{z, MRI}∝ϖ^1/2. This is a weaker dependence than for a constant field and is the main reason we use a constant flux criterion in the NVF case, rather than simply using a constant field.

For the ZNF case, we no longer have to worry about fitting the vertical wavelength of the fastest growing mode within the simulation domain, since the field strength varies as a function of radius and the MRI grows on all scales. The same goes for NAF simulations because instability manifests itself differently if the initial field is azimuthal rather than vertical. In the former case, the most unstable modes are non-axisymmetric ones that undergo transient exponential growth in the shearing sheet approximation (Balbus & Hawley 1992; Tagger et al. 1992) or pure exponential growth in a global disk analysis (Matsumoto & Tajima 1995; Coleman et al. 1995; Ogilvie & Pringle 1996; Terquem & Papaloizou 1996). In a shearing sheet analysis, the modes which undergo the most vigorous growth are the ones that have the largest values of k_z for a given value of k_x and k_y (Balbus & Hawley 1992; Hawley et al. 1995). Thus, resolution of these modes in a simulation is limited by the grid resolution, but their vertical wavelength is always able to fit within the vertical extent of the simulation domain.

The only physical parameter varied across simulations in this study is the initial field geometry. In principle, other parameters such as the Mach number, the vertical extent of the simulation domain, the presence of vertical stratification, etc., would also have an important effect on the dynamics of the BL. However, it is computationally expensive to undertake an exhaustive parameter study, so we limit ourselves to varying only the initial field geometry in this exploratory work. This allows us to explore the major effects a magnetic field has on BL dynamics with no added frills. We mention that Belyaev et al. (2012, 2013) have already explored how varying the dimensionality (2D/3D), stratification (stratified/unstratified), and Mach number affects the outcome of hydrodynamic BL simulations.

Some important parameters that are fixed across different runs are the Mach number, set to M = 9; the vertical extent of the simulation domain, set to Δz = s/Ω_K(ϖ_⋆) = 1/9; the ϕ extent of the simulation domain, set to Δϕ = 2π/7; and the ϕ resolution set to N_ϕ = 384 cells. Table 1 lists important numerical parameters that vary from simulation to simulation.

The specific choice of Δϕ = 2π/7, represents a compromise between computational resources and simulating a wedge that spans a wide enough azimuthal angle. Also, setting Δϕ = 2π/7, we resolve all modes having m a multiple of seven. The m = 14 mode happens to be close to a geometric resonance (Equations (44) and (45) of Belyaev et al. 2013) for M = 9, which means its amplitude is enhanced relative to modes with a similar m number. Therefore, the m = 14 mode typically peaks in our simulations, simplifying the analysis, since we can study a single dominant mode rather than a superposition of modes having similar amplitudes. Note that the finite extent of our simulations in the azimuthal direction precludes us from exploring the Tayler–Spruit dynamo (Tayler 1973; Spruit 1999, 2002) in the BL, which would require long azimuthal wavelength to be resolved. However, the very existence in the BL of this dynamo previously shown to operate only in the incompressible limit should be seriously questioned in a highly dynamic, compressible environment such as the BL.

The boundary conditions we use are periodic in the z and ϕ directions, and do-nothing at the inner radial boundary. A do-nothing boundary condition means that the fluid quantities retain their initial values on the boundary for the duration of the simulation. This is preferred over an open (outflow) boundary condition for the inner boundary, since it ensures that the star does not slowly "drift" out of the simulation domain. It is also preferred to a reflecting boundary condition, since it is (partially) transparent to outgoing waves (Belyaev et al. 2013).

At the outer radial boundary, we implement two different types of boundary conditions. The first type, which we call OBC0, is simply an open (outflow) boundary condition. The second type, OBC1, is a do-nothing boundary condition, just like at the inner radial boundary. In simulations implementing an outer do-nothing boundary condition, we initialize a nonmagnetic "buffer zone" between a fiducial radius in the disk ϖ_{B, max} and the outer boundary. The magnetic field is initialized to be zero in the region ϖ_{B, max} ⩽ ϖ ⩽ ϖ_out, where ϖ_out denotes the radius of the outer boundary. In all of our simulations implementing OBC1, ϖ_{B, max} = 4.

To demonstrate the buffer zone more clearly, we plot the magnetic field magnitude from simulation M9d at t = 0 and t = 500 in panels (a) and (b) of Figure 1. In panel (a), the field decays ∝ 1/ϖ between the star at ϖ = 1 and the buffer zone at ϖ = ϖ_{B, max} = 4. In the region ϖ_{B, max} < ϖ < ϖ_out, the field is initially set to zero. In panel (b), magnetic field has penetrated into the buffer zone, as a result of the natural evolution of the MHD equations, but has not yet reached the outer boundary. In general, simulations implementing OBC1 are stopped before the magnetic field can diffuse through the buffer zone and reach the outer boundary. This allows us to run simulations for a long time, without having to worry about the effect of magnetic field at the outer boundary on the rest of the simulation domain.

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The primary diagnostic we use to check that the disk reaches a resolved quasi-steady state of MRI turbulence is the magnetic pitch angle

$$\begin{eqnarray} &&\theta _B \equiv \frac{\sin ^{-1}\left(\alpha _B \beta \right)}{2}. \end{eqnarray} \tag{ 21 }$$

Equation (21) gives an approximation for the angle of the magnetic field with respect to the ϕ direction if B_ϕ ≫ B_ϖ and B_ϖ ≫ B_z (Guan et al. 2009). Sorathia et al. (2012) found that θ_B was a particularly good indicator of convergence, since it varies monotonically with resolution and converges to a value of θ_B ≈ 13° in resolved simulations. A similar indicator of convergence is the relation αβ ≈ 1/2 (Guan et al. 2009; Hawley et al. 2011; Sorathia et al. 2012). However, α includes both hydrodynamical and magnetic stresses, whereas α_B is calculated using only magnetic stresses. We find in our simulations that the hydrodynamical stress due to waves is non-negligible in the disk, but waves do not contribute to the magnetic stress. Thus, a diagnostic based on α_B is strongly preferred to one based on α, when assessing the convergence of MRI turbulence in a simulation that includes a BL.

Figure 2 shows radius time plots of θ_B for simulations M9d, e, and f in the annulus 1 ⩽ ϖ ⩽ 3. Ignoring the region near ϖ  =  ϖ_⋆  =  1 where the rotation profile is strongly non-Keplerian, we see that θ_B ≈ 12°–14° in the disk proper for the NAF and ZNF simulations (panels (b) and (c)) once MRI turbulence is fully developed. In the NVF simulations, θ_B ≈ 12°–14° in the inner disk, but rises with radius. Nevertheless, the values of θ_B observed in our simulations are in good agreement with Sorathia et al. (2012). This gives us confidence that we have properly resolved the MRI in the disk.

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The second diagnostic of convergence we perform is a measurement of the rms volume weighted average of B_ϖ, B_ϕ, and B_z for simulations with the same initial magnetic field geometry, but having a different resolution. Panels (a), (b), and (c) of Figure 3 show rms averages of B_ϖ, B_ϕ, and B_z at t = 400 for NVF simulations (M9a, d), NAF simulations (M9b, e), and ZNF simulations (M9c, f), respectively. We see that the NVF and NAF simulations are converged, although the magnetic field in the outer disk of the NAF simulation is still approaching a steady state. The reason why we did not perform the convergence study at a later time is that it was too expensive to run the high resolution simulations beyond a time of t = 450.

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In the ZNF case, magnetic field components for the low resolution simulation have a higher amplitude than for the high resolution simulation, and we have checked that this is true at all times, not just at t = 400. However, in Section 4.1 we showed that MRI turbulence is resolved in ZNF simulations based on the pitch angle indicator.

Taken together, these two pieces of information suggest that MRI turbulence in our ZNF simulations is resolved, but that for a given simulation, the rms field strength in the saturated state of MRI turbulence is influenced by the effective numerical viscosity and resistivity, which depend on the grid resolution (Fromang & Papaloizou 2007). In order to make the results of ZNF simulations independent of resolution, we could add explicit viscosity and resistivity (Fromang et al. 2007).

However, given the exploratory nature of this paper, the exact value of the viscous and resistive coefficients is of secondary importance. What is more important is that for a given value of these coefficients, MRI turbulence is properly resolved, which is indeed the case according to the pitch angle indicator of Section 4.1.

The most important parameter that characterizes transport by MRI turbulence in the disk is α. However, α in our simulations is not a constant and is a function of both time and radius. One reason for this is that the timescale for instability is longer in the outer parts of the disk than in the inner parts, so the inner parts reach a quasi-steady state faster. However, even in the quasi-steady state, we find α∝ϖ^−3/2, which is not a surprising scaling, and implies a constant value of α_ν (Equation (16)). The reason α and α_ν have a different radial dependence in simulations is that the height of the simulation domain, Δz is fixed, whereas the scale height in a Keplerian disk goes as H = s/Ω∝ϖ^3/2 for constant s. Thus, we define

$$\begin{eqnarray} &&\alpha _{\rm eff} \equiv \alpha \varpi ^{3/2}. \end{eqnarray} \tag{ 22 }$$

Since, s/Ω  =  Δz at ϖ  =  ϖ_⋆  =  1, α_eff can be thought of as the "effective" value of α for a Keplerian disk, based on our simulation results. The assumption here is that the angular momentum transport rate is proportional to the vertical scale of the problem, which is fixed for our simulations but goes as H∝ϖ^3/2 for a Keplerian disk with a constant value of s.

Another complication to measuring the α due to MRI turbulence is the presence of waves emitted from the BL, which contribute to the value of α in the disk. Therefore, rather than showing α_eff directly, we show α_{B, eff}, which is simply the component of α_eff that is due to magnetic stresses (i.e., α_{B, eff} = α_Bϖ^3/2). As mentioned in Section 4.1, α_B is insensitive to waves, which contribute primarily to the hydrodynamical stress. Studies of the MRI typically find that α ≈ 5/4α_B (Sorathia et al. 2012), which also roughly holds for our simulations.

Figure 4 shows radius time plots of α_{B, eff} for simulations M9d, e, and f. Once a quasi-steady state has been reached, the value of α_{B, eff} for the NVF simulation (M9d) is α_{B, eff} ≈ .02 − .03, for the NAF simulation (M9e) is α_{B, eff} ≈ .0075 − .0125, and for the ZNF simulation (M9f) is α_{B, eff} ≈ .004 − .01. These values of α are consistent with those obtained by Steinacker & Papaloizou (2002) in their BL simulations. However, the value of α in ZNF simulations does depend on the resolution, for the same reasons as the rms magnetic field value depends on the resolution (Section 4.2).

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Belyaev et al. (2013) found that for the isothermal BL, there were three branches of acoustic modes, and they found dispersion relations that matched simulation results for each of these three branches. However, their results were for the pure hydro case, and we comment on how they should be modified to account for the presence of a magnetic field in the disk.

It is natural to expect that the major modification to the dispersion relation of the acoustic modes in the limit of β ≫ 1 is the replacement of the sound speed with the magnetosonic speed:

$$\begin{equation} v_{\rm ms} = \sqrt{s^2 + v_A^2} \end{equation} \tag{ 23 }$$

$$\begin{equation} = s \sqrt{1 + \frac{2}{\gamma } \beta ^{-1}}, \end{equation} \tag{ 24 }$$

where v_A is the Alfvén speed, and γ = 1 for an isothermal equation of state.

In reality, the situation is slightly more complicated, and the dispersion relation depends on the jump in magnetic field across the BL. We show this explicitly in the Appendix, where we derive an analytic dispersion relation for the 2D modes of a compressible vortex sheet (Miles 1958; Gerwin 1968; Belyaev & Rafikov 2012) in the presence of a magnetic field that is perpendicular to the plane of the modes. This is not a realistic setup, but it highlights how the hydrodynamical dispersion is modified when a magnetic field is introduced.

The main result of the analysis, in the context of astrophysical BLs, is that in the limit of β ≫ 1, the dispersion relation for the model system is only modified by terms of $\mathcal {O} (\beta ^{-1} )$. This result, which may be expected to hold for more general field configurations, shows that when gas pressure dominates magnetic pressure, acoustic modes should still be present, and their properties should only be slightly modified as compared to the hydro case. In a typical astrophysical context, such as in MRI turbulence, the magnetic field is subthermal, meaning β > 1. As we have shown in Section 5, the same is also true inside the BL, meaning that the modes present in the layer and its vicinity are essentially acoustic modes, weakly modified by a magnetic field.

In addition to magnetosonic waves, an ideal fluid in the presence of a magnetic field also supports Alfvén and slow waves. One may wonder whether there are also modes of the system that correspond to a coupling of these waves in the disk across the BL to either gravity or sound waves in the star. This subject is beyond the scope of the present work. However, we mention that in our simulations, the modes that show up most clearly are the analogs of the acoustic modes discussed by Belyaev et al. (2012, 2013). Thus, it is natural to expect that these magnetosonic modes are the most important ones for transporting angular momentum in the presence of a magnetic field, when β ≫ 1.

We observe excitation of magnetosonic modes in our simulations, which persist, typically, for the duration of the simulation. Figure 8 shows both the simple vertical average and the cross-section through the z = 0 plane of both $\varpi \sqrt{\rho } v_\varpi$ and the magnetic field magnitude for simulation M9f at time t = 1000. The dominant mode in Figure 8 is the m = 14 lower branch mode.¹ It appears prominently in $\varpi \sqrt{\rho } v_\varpi$ (panels (a) and (b)), but only weakly in the magnetic field magnitude (panels (c) and (d)). The measured pattern speed of the m = 14 mode from simulations is Ω_P ≈ .4, which is in good agreement with the theoretically predicted pattern speed for the m = 14 lower branch mode, Ω_P = .415 (Belyaev et al. 2013).

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We also point out that the mode depicted in Figure 8 is effectively two-dimensional (2D), since there is not much difference between the vertically averaged image of $\varpi \sqrt{\rho } v_\varpi$ (panel (a)) and the cross-sectional slice (panel (b)). The magnetic field, on the other hand, exhibits much more vertical structure (panels (c) and (d)), which is indicative of turbulence.

The lower branch has an evanescent region in the disk around the corotation radius and waves incident on the evanescent region are reflected back toward the BL. The edges of this evanescent region are located approximately at the Lindblad radii, which are implicitly defined by the condition Ω(ϖ_LR) = Ω_P  ±  κ(ϖ_LR)/m, where κ is the epicyclic frequency. The dashed lines in Figure 8 show the edges of the evanescent region, as predicted by hydrodynamical theory assuming a Keplerian rotation profile, and the solid line shows the corotation radius of the mode in the disk. Even though we now have MRI turbulence in the disk, shocks still reflect off inner edge of the evanescent region. This fact, together with the agreement between the measured pattern speed and that from hydrodynamical theory, means hydrodynamical theory provides a good description of the properties of magnetosonic modes even in the presence of MRI turbulence in the disk. As argued earlier, this can be attributed to the fact that β⁻¹ ≪ 1 in the simulations, so the magnetic field does not significantly affect the properties of acoustic modes.

Although the hydrodynamical and MHD cases exhibit many similarities, some differences are apparent. The first difference is that MHD simulations are typically "noisier" than hydro runs (Belyaev et al. 2012, 2013) with a superposition of modes present. The second difference is that unlike the hydrodynamical case, there can be significant wave action tunneling through the evanescent region in MHD runs.

The presence of a superposition of acoustic modes in the disk is a generic feature, independent of initial magnetic field geometry. In addition to the lower branch, which is persistent in all of our simulations for t ≳ 60, we also observe the upper branch at the beginning of some simulations, but only in transience. Figure 9 shows the upper branch at t = 40 in simulation M9c. The mode depicted in the figure has m = 14 and a measured pattern speed of Ω_P = .84. This agrees well with the theoretically predicted pattern speed for this mode, which is Ω_P = .87.

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We begin our discussion of wave transport of angular momentum by showing the evolution of the density profile. Figure 10 shows radius time plots of Σ₀(ϖ, t) in simulations M9d, e, and f. In each case, a gap in the density develops in the inner disk in the vicinity of the BL (ϖ ≳ 1). The gap is opened up relatively rapidly, over a time of ≲ 10 orbits, and in panels (a) and (c), gap formation is followed at a later time by gap deepening. Concentrating on panel (c), after the initial gap opening event at t ≈ 200, the density profile of the gap stays approximately constant, until t ≈ 1100 when it undergoes substantial gap deepening. For t > 1100, the gap is gradually filled in by the action of turbulent viscosity in the disk.

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Belyaev et al. (2013) also observed gap formation in their hydrodynamical simulations, which was driven by angular momentum transport by acoustic modes. Consistent with their results, we find that magnetosonic modes are responsible for opening up the gap in our MHD simulations, and we discuss the link between acoustic modes and the gap in the inner disk in more detail in Section 7.2.

Material gets removed from the inner part of the disk between the BL and the wave reflection point at the inner Lindblad resonance by the action of magnetosonic modes. In this region gas orbits the star faster than the pattern of acoustic modes, and every time a fluid element encounters a weak shock (into which modes evolve) it loses angular momentum and moves toward the star, giving rise to continuing accretion. Figure 11 shows plots of the instantaneous mass accretion rate through radius ϖ, $\dot{M}(\varpi)$, for simulations M9d, e, and f (panels (a), (b), and (c)) and for a M = 9 hydro simulation from Belyaev et al. (2013) (panel (d)). The solid curves correspond to periods when magnetosonic/acoustic modes are excited to a high amplitude (high shock accretion rate) and the dashed curves to periods when their amplitude is not as high (low shock accretion rate). Each of the solid curves in panels (a)–(c) corresponds to one of the gap formation or opening events in Figure 10.

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In panels (a)–(c), the dashed and solid curves converge to approximately the same value of $\dot{M}$ in the outer disk in Figure 11, but the solid curves have a significantly higher amplitude than the dashed ones in the inner disk (ϖ ≳ 1). The extra contribution to $\dot{M}$ for the solid curves in the inner disk in panels (a)–(c) is provided by magnetosonic modes, and the base level of $\dot{M}$ in the outer disk is due to MRI turbulence.

Evidence for this assertion is provided by considering panel (d) of Figure 11. Since the simulation corresponding to panel (d) is hydrodynamical, there is no MRI in the disk. This explains why the dashed line is everywhere approximately zero, since in the absence of waves excited to a high amplitude or MRI, there is no efficient mechanism for angular momentum transport in the disk. On the other hand, the shape and amplitude of the solid curve in the inner disk in panel (d) are similar to those of the solid curves in panels (a)–(c). This shows that in the MHD case, the mass accretion rate in the inner disk due to shocks can dominate that due to MRI, during periods when modes are excited to a high amplitude.

Indeed, we can write the following expression for the combined mass accretion rate close to the star (interior to the evanescent region in the disk) driven by both the MRI and shocks:

$$\begin{eqnarray} \dot{M} &=& \frac{\Sigma s^2}{\Omega } \left[4\pi \alpha q\frac{\partial \ln (\alpha q s^2\Sigma \varpi ^2)}{\partial \ln \varpi } \right. \nonumber\\ && \left. +\frac{m}{3(2-q)}\left(1 + \frac{\Delta \Sigma }{\Sigma } \right)^{-2} \left(\frac{\Delta \Sigma }{\Sigma }\right)^3\right], \end{eqnarray} \tag{ 25 }$$

where q(ϖ) ≡ −dln Ω/dln ϖ is the local shear rate (q = 3/2 for a Keplerian disk but is a function of ϖ for a general rotation profile), m is the azimuthal wavenumber of the mode, and ΔΣ/Σ is the density contrast across the shock (mode) fronts. The first term in brackets is the contribution from MRI (α here is solely due to MRI) and can derived using the results of Lynden-Bell & Pringle (1974) and Rafikov (2012). The second term is the contribution from magnetosonic waves following from Belyaev et al. (2012), namely from their Equation (B7). For a characteristic azimuthal wavenumber of m ∼ 15–30 and MRI viscosity α ≲ 10⁻² (see Figure 4) measured in our simulations, one finds that both transport mechanisms provide comparable contribution to $\dot{M}$ if ΔΣ/Σ ∼ 0.2.

However, during periods of strong wave excitation ΔΣ/Σ increases and the wave term strongly dominates $\dot{M}$ because of the steep (cubic) dependence of $\dot{M}$ upon ΔΣ/Σ. For instance, the amplitude of the m = 14 magnetosonic mode changes by a factor of ≈5 in simulation M9f between t = 900 and t = 1100. This results in a predicted mass accretion rate due to shocks in the inner disk that is higher by approximately two orders of magnitude at t = 1100 as compared to t = 900.

This effect of enhanced mass accretion rate by shocks during periods when magnetosonic modes are excited to a high amplitude is demonstrated by the solid curves in Figure 11. Equation (25) also explains the characteristic spatial dependence of $\dot{M}$ on ϖ: approximate conservation of the wave angular momentum flux (neglecting for the moment dissipation at the shock fronts) results in (see Equation (25) of Belyaev et al. 2012)

$$\begin{eqnarray} &&\frac{\Delta \Sigma }{\Sigma }\propto \left[\frac{|\Omega (\varpi)-\Omega _P|}{\Sigma s^3\varpi }\right]^{1/2}. \end{eqnarray} \tag{ 26 }$$

This expression shows that ΔΣ/Σ is highest near the BL, where |Ω(ϖ) − Ω_P| is large and goes to zero in the evanescent region where Ω = Ω_P. This explains why during the high wave amplitude episodes, when mass accretion is clearly dominated by dissipation of acoustic modes, $\dot{M}$ is largest near the BL, but rapidly decays to small value as ϖ approaches the evanescent region in the disk.

Note that inside the BL MRI does not operate, α = 0, and the first term in Equation (25) vanishes, leaving wave dissipation as the only means of mass transport. On the other hand, in the disk outside the corotation radius, waves do not contribute to transport and second term in Equation (25) disappears. Thus, MRI remains the only mass transport mechanism outside the resonant cavity in which the pattern of standing acoustic modes exists, in agreement with the picture outlined in Belyaev et al. (2012) and Belyaev et al. (2013).

In this section, we connect the density gap in the inner disk to magnetosonic modes and discuss angular momentum transport in the star, BL, and inner disk more generally.

Figure 12 shows radius time plots of the angular momentum current, C_S, from simulations M9d, e, and f. In the disk, far away from the BL, angular momentum transport is due primarily to MRI turbulence (Section 4), and C_S is approximately constant at a given radius once the MRI has saturated. We note that the striations appearing in C_S are caused by waves. The striations in the panels of Figure 12 are predominantly downward sloping in the inner disk, indicating inward propagating waves. These waves are excited at the "front" of MRI turbulence which propagates radially outward through the disk over the course of the simulation. Although potentially a nuisance, these waves do not play a major role in angular momentum transport in our simulations.

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In contrast to the outer disk, where the value of C_S at a given radius is approximately constant in time once MRI turbulence has saturated, C_S varies by more than an order of magnitude in the star and inner disk over the course of a simulation. This implies a mechanism of angular momentum transport that is decoupled from MRI turbulence. As we shall soon demonstrate, this mechanism is transport of angular momentum by magnetosonic modes. Given this fact, we can directly link magnetosonic modes to the gap formation and deepening events in Figure 10, since these are simultaneous with the periods when C_S is large in the BL and star in Figure 12.

In addition to waves dominating angular momentum transport in the star and BL, we find that they also contribute significantly to angular momentum transport in the inner disk. Figure 13 shows C_S and Ω superimposed on the same plot for simulations M9d, e, and f at times t = 550, 250, and 1100, when the angular momentum current due to waves is high in each simulation. The dashed vertical line denotes the radius at which C_S = 0, and for an anomalous viscosity model, C_S = 0 and dΩ/dϖ = 0 occur at the same radius (Equation (16)). However, we see in Figure 13 that the radius at which C_S = 0 is significantly displaced into the disk compared to the radius at which dΩ/dϖ = 0. This implies that C_S contains a contribution due to waves in the disk, since C_S < 0 for outgoing waves up to the corotation radius in the disk, which is located at ϖ_cr ≈ 1.8 (m = 14 mode) in each of the panels in Figure 13.

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We conclude that when waves are excited to a high amplitude, they significantly modify the profile of C_S in the inner disk compared to what is expected for a turbulent viscosity model. In particular, if both magnetosonic modes and anomalous viscosity contribute appreciably to C_S, then the radius at which C_S = 0 in the disk is neither at the corotation radius of the mode, nor at the radius where dΩ/dϖ = 0, but somewhere in between.

We also mention that the variation of C_S with radius in the star and BL in each of the panels in Figure 13 is consistent with C_S(ϖ) for the lower branch (Figure 12 of Belyaev et al. 2013). Since the lower branch is directly observed in our simulations (Section 6), this provides further confirmation that angular momentum is transported by waves in the BL and star.

Having established that magnetosonic modes transport angular momentum in the BL and star, and also influence transport in the inner disk, we now demonstrate that the mechanism of angular transport by acoustic modes is essentially hydrodynamical in nature. This provides evidence for the argument presented in Section 6.1 that when β ≫ 1, the properties of acoustic modes are not significantly affected by the magnetic field, and the results of Belyaev et al. (2013) are valid.

To show that stresses in the BL are hydrodynamical, we split C_S into magnetic and nonmagnetic components (Equation (14)). Figure 14 shows C_{S, B} and C_{S, H} for each of simulations M9d, e, and f, and several things are immediately apparent. The first is that C_{S, B} is several times larger than C_{S, H} in the disk, which is consistent with the fact that the magnetic stress is several times the hydrodynamical stress for MRI turbulence (Sorathia et al. 2012). The second is that C_{S, B} vanishes going into the star and there is no bump in C_{S, B} within the BL. This means that magnetic stresses are not important for transport in the BL. This is consistent with the results of Pessah & Chan (2012), who found that in the shearing sheet approximation, the magnetic stress oscillates about zero for a rising rotation profile (dΩ/dϖ > 0), as in the BL. The third is that |C_{S, H}| ≫ |C_{S, B}| in the star and BL, which is definitive proof that a hydrodynamical mechanism of angular momentum transport operates there—namely acoustic modes weakly modified by a magnetic field.

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Belyaev et al. (2012) and Belyaev et al. (2013) found that in hydro simulations the BL settled to a radial width of 6–7 radial scale heights (h_s ≡ s²/g(ϖ_⋆)). Using their definition for the BL as the region in which

$$\begin{eqnarray} &&0.1 < \langle v_\phi (\varpi)\rangle /v_K(\varpi) < 0.9, \end{eqnarray} \tag{ 27 }$$

we confirm that after the initial gap opening event, the BL in our MHD simulations also settles down to a radial width of 6–7 radial scale heights (Figure 15). However, the BL undergoes widening events that are coincident with gap deepening events (Section 7.1) and the periods when C_S is large in the star and BL (Section 7.2). Stochastic widening events were also observed by Belyaev & Rafikov (2012) in 2D hydro simulations, and it appears that in both hydro and MHD cases these events are mediated by the action of shocks on the fluid in the inner disk.

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The fact that the BL widening events occur simultaneously with gap deepening events, suggests a common origin, implying that magnetosonic modes play a critical role in regulating the width of the BL. However, given the occurrence of stochastic widening events in the simulations, the width of the BL is time-variable and it is not possible to assign a single number to it. Nevertheless, what the simulations suggest is that the BL initially settles down to a steady state width of 6–7 radial scale heights and is subsequently widened, during periods when magnetosonic modes are excited to a high amplitude.