Three-Disk Accretion Model

• Three-Disk Model is a global 3D MHD framework that decomposes boundary-layer accretion into three radial zones with distinct transport physics.
• It delineates the outer MRI-active disk, intermediate transition layer, and inner boundary layer where different mechanisms (turbulence, spiral shocks, Maxwell stress) dominate angular momentum redistribution.
• The model emphasizes vertical processes, magnetic flux accumulation, and episodic mass ejection, enhancing our understanding of complex protostellar accretion dynamics.

Searching arXiv for the cited paper and closely related boundary-layer accretion work. The Three-Disk Model is a global three-dimensional magnetohydrodynamic description of boundary-layer accretion onto a low-mass protostar in which the star–disk system separates into three radial zones with distinct transport physics: an outer MRI-active disk, an intermediate transition layer, and an inner boundary layer adjacent to the stellar surface (Takasao et al., 19 Mar 2025). In this usage, “three-disk” does not denote three detached astrophysical disks, but three dynamically differentiated disk structures that together mediate mass inflow, angular-momentum redistribution, magnetic-flux accumulation, and episodic mass ejection in a magnetized star–disk system.

1. Zonal decomposition of the accretion flow

The model partitions the flow according to the radial coordinate measured in units of the stellar radius $R_1$. The outer zone, at $R \gtrsim 3$–$4\,R_1$, is the MRI-active disk. The intermediate zone, at $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$, is the transition layer. The innermost zone, at $R_1 \lesssim R \lesssim 1.2\,R_1$, is the boundary layer where the gas decelerates from Keplerian rotation to the stellar spin, which is zero in the reference simulation (Takasao et al., 19 Mar 2025).

Zone	Radial extent	Dominant transport
MRI-active disk	$R \gtrsim 3$–$4\,R_1$	MRI turbulence, coronal accretion, disk winds, jets
Transition layer	$\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$	Maxwell stress, spiral-shock Reynolds stress, vertical loss
Boundary layer	$R_1 \lesssim R \lesssim 1.2\,R_1$	Shear-amplified Maxwell stress, jets, subdominant hydrodynamic torque

In the MRI-active disk, magneto-rotational instability drives turbulence in the midplane, coronal or surface accretion drags vertical fields inward, and disk winds and jets extract angular momentum vertically. Reynolds stress associated with spiral shocks is subdominant in this region. In the transition layer, spiral shocks launched from the magnetized stellar surface propagate outward and deposit angular momentum, MRI remains active but is suppressed toward the inner edge, and both Maxwell and Reynolds stresses contribute comparably. In the boundary layer, MRI is stable because $\partial\Omega/\partial R>0$, so angular-momentum removal relies mainly on Maxwell stress from sheared disk-origin and stellar fields, together with vertical transport in jets, while hydrodynamic torque is subdominant but non-negligible (Takasao et al., 19 Mar 2025).

This decomposition is significant because it replaces a single transport mechanism with a radially stratified picture. The model therefore treats boundary-layer accretion not as a narrow local phenomenon alone, but as the inner continuation of a global MHD system in which the dominant stress channel changes with radius.

2. Governing equations and dynamical closure

The model solves the resistive MHD equations in conservation form in Cartesian coordinates. In cgs units,

$R \gtrsim 3$0

$R \gtrsim 3$1

$R \gtrsim 3$2

$R \gtrsim 3$3

with

$R \gtrsim 3$4

Here $R \gtrsim 3$5 is an artificial diffusivity and $R \gtrsim 3$6 is a $R \gtrsim 3$7-cooling term (Takasao et al., 19 Mar 2025).

The MRI stability criterion is stated in local form as $R \gtrsim 3$8, with maximum local growth rate $R \gtrsim 3$9. This immediately distinguishes the outer and inner regions: the MRI-active disk satisfies the instability condition, whereas the boundary layer, with $4\,R_1$0, is MRI-stable. A plausible implication is that the Three-Disk Model is fundamentally a model of transport-mode transition: turbulence-mediated angular-momentum extraction in the outer disk gives way to shock- and shear-mediated transport nearer the star.

The use of a global 3D MHD framework is central. The simulation resolves not only the accretion stream, but also the atmosphere, coronal inflow, wind and jet channels, and the magnetic coupling between disk and convective protostar. This suggests that local shearing-box or purely radial closures would miss transport pathways that are explicitly vertical or star-coupled.

3. Angular-momentum fluxes and accretion diagnostics

Angular-momentum transport is decomposed into hydrodynamic and magnetic contributions using the poloidal vectors $4\,R_1$1 and $4\,R_1$2. The radial flux of $4\,R_1$3-angular momentum per unit area is

$4\,R_1$4

Equivalent stress components are

$4\,R_1$5

with $4\,R_1$6 (Takasao et al., 19 Mar 2025).

The net accretion rate within $4\,R_1$7 is defined by

$4\,R_1$8

Under steady state and axisymmetry,

$4\,R_1$9

$\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$0

where $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$1. The corresponding predicted accretion rate is decomposed as

$\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$2

In the simulation these channels combine to yield a nearly constant $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$3 in code units, corresponding to $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$4, over $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$5 (Takasao et al., 19 Mar 2025).

This transport accounting is one of the model’s defining features. Rather than reducing accretion to a single effective viscosity, it isolates four mechanisms: Reynolds stress, Maxwell stress, vertical stress, and mass ejection. In the MRI-active disk, Maxwell stress from MRI turbulence dominates; in the transition layer, Maxwell and Reynolds stresses are comparable; in the boundary layer, Maxwell stress and jet-mediated angular-momentum loss dominate. This suggests that the usual scalar $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$6 parameter is informative but insufficient on its own unless it is tied to the underlying stress decomposition.

4. Stress scalings, radial structure, and magnetic topology

The model defines an effective dimensionless stress

$\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$7

Typical values are zone-dependent. In the MRI-active disk, $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$8, while the midplane plasma beta is initially $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$9, dropping to $R_1 \lesssim R \lesssim 1.2\,R_1$0 in the atmosphere. In the transition layer, $R_1 \lesssim R \lesssim 1.2\,R_1$1–$R_1 \lesssim R \lesssim 1.2\,R_1$2 from combined Reynolds and Maxwell stresses, and $R_1 \lesssim R \lesssim 1.2\,R_1$3 peaks at a few $R_1 \lesssim R \lesssim 1.2\,R_1$4 near $R_1 \lesssim R \lesssim 1.2\,R_1$5. In the boundary layer, Maxwell-dominated stress yields local $R_1 \lesssim R \lesssim 1.2\,R_1$6–$R_1 \lesssim R \lesssim 1.2\,R_1$7 (Takasao et al., 19 Mar 2025).

The magnetic-field scalings emphasize flux accumulation and shear amplification: $R_1 \lesssim R \lesssim 1.2\,R_1$8 in the disk midplane, and

$R_1 \lesssim R \lesssim 1.2\,R_1$9

so that $R \gtrsim 3$0. These relations summarize how inward flux advection and rotational shear restructure the field geometry toward the star.

Equatorial radial profiles at $R \gtrsim 3$1 mark the three zones sharply. For $R \gtrsim 3$2, the density obeys $R \gtrsim 3$3; it flattens, and can even decline, over $R \gtrsim 3$4; then it rises sharply inside $R \gtrsim 3$5. The temperature follows $R \gtrsim 3$6 in the disk and connects to a hot stellar photosphere with $R \gtrsim 3$7 at $R \gtrsim 3$8. The rotation profile is nearly Keplerian, $R \gtrsim 3$9, for $4\,R_1$0, but has $4\,R_1$1 in the boundary layer. Meanwhile $4\,R_1$2 peaks near $4\,R_1$3 and $4\,R_1$4 is largest in the boundary layer, where $4\,R_1$5 (Takasao et al., 19 Mar 2025).

The stellar magnetic topology is not imposed as a static boundary condition alone. Poloidal vertical fields are advected inward by coronal accretion and accumulate at the poles, with the surface magnetic flux growing to $4\,R_1$6. Differential rotation wraps $4\,R_1$7 into $4\,R_1$8 around the stellar equator and the boundary layer, and a convective dynamo in the subsurface layer $4\,R_1$9 regenerates small-scale fields that produce starspot-like surface concentrations of kG strength. Magnetic reconnection in polar current sheets and equatorial spot concentrations occurs at a local rate

$\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$0

driving explosive jets and episodic flux escape through Parker buoyancy and reconnection (Takasao et al., 19 Mar 2025).

5. Physical implications for protostellar accretion

The model attributes a central dynamical role to spiral shocks generated by the interaction between the rotating disk gas and magnetic concentrations on the protostellar surface. These shocks propagate into the transition layer and contribute substantially to angular-momentum redistribution there, even though they are subdominant in the MRI-active outer disk and only secondary in the boundary layer. This establishes a direct causal link between stellar magnetoconvection and disk accretion dynamics (Takasao et al., 19 Mar 2025).

The simulation also identifies several vertically mediated processes that are not reducible to midplane transport. Coronal accretion drags vertical magnetic flux inward, disk winds and jets remove angular momentum, and explosive reconnection occurs in both the protostar and the disk atmosphere. In addition, decretion flows appear in the disk midplane; the study notes that these may be important for the radial transport of refractory materials, such as Calcium-Aluminium-rich Inclusions precursor gas, to the outer disk (Takasao et al., 19 Mar 2025).

More broadly, the Three-Disk Model argues that boundary-layer accretion onto a weakly magnetized convective protostar is a coupled star–disk problem in which angular momentum is redistributed by a changing mixture of MRI turbulence, spiral shocks, Maxwell stress, and vertical outflows. A plausible implication is that observational or theoretical descriptions that isolate only one of these channels risk misidentifying the dominant transport agent in particular radial zones.

6. Limitations and terminological scope

The model has several stated caveats. The finest resolution is $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$1, which is marginal for resolving sonic instability or the scale height of the narrow boundary layer, so the true $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$2 and shock structure may differ at higher resolution. The Cartesian grid biases convective cells into an $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$3 pattern, likely exaggerating a four-armed spot distribution and the associated four-fold spiral shocks. The protostar is initialized as non-rotating; finite stellar spin would weaken shock strength and modify boundary-layer Maxwell stress. The $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$4-cooling prescription relaxes to a reference temperature and does not capture full radiative transfer, while limited outer-disk mass flux, top and bottom inflow caps of $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$5, and artificial diffusivity may affect wind, jet, and reconnection dynamics (Takasao et al., 19 Mar 2025).

The expression “three-disk” is also used in several unrelated domains. In open quantum chaos, the three-disk scattering system consists of three identical hard disks of radius $\sim 1.2\,R_1 \lesssim R \lesssim 3\,R_1$6 at the vertices of an equilateral triangle, and serves as a paradigmatic model of fully chaotic scattering and fractal Weyl scaling (Schmidt et al., 2023). In soft active matter, a three-disk microswimmer is formed by three rigid thin disks linked by two arms in a supported membrane, with mean speed controlled by a geometric factor exhibiting three asymptotic regimes (Ota et al., 2018). In Galactic dynamics, a Milky Way mass model represents each disk subcomponent by a superposition of three Miyamoto–Nagai disks, yielding a fully analytical gravitational potential suited to orbit calculations (Barros et al., 2016). Related but terminologically distinct accretion frameworks include three-regime RMHD black-hole flows—supercritical, standard thin, and radiatively inefficient (Ohsuga et al., 2011)—and a three-hybrid-flow black-hole model combining a Keplerian disk, ADAF, and a third low-angular-momentum component (Kumar et al., 2022).

Within protostellar boundary-layer accretion, however, the Three-Disk Model specifically denotes the global 3D MHD decomposition into MRI-active disk, transition layer, and boundary layer, together with the corresponding zonal transition in transport physics (Takasao et al., 19 Mar 2025).