Read beyond linear MRI growth

Open Xue-Ning Bai's Hall-effect Controlled Gas Dynamics in Protoplanetary Disks: II. Full 3D Simulations toward the Outer Disk, arXiv:1409.2511v1, submitted 8 September 2014. Use this numbered version: its PDF header also carries a later internal draft date. Read Section 2, especially Sections 2.1–2.3 on printed pages 2–6. The main figure here is Fig.2 on printed page 4, which is also PDF page 4 when counted from one.

Keep a second source for the net-flux comparison: Bai and James M. Stone, Local Study of Accretion Disks with a Strong Vertical Magnetic Field: Magnetorotational Instability and Disk Outflow, arXiv:1210.6661v2, 17 February 2013. We will use its definition in equation 7, Section 3.2.1, and Table 2 on printed/PDF page 10. These are selected readings of simulations, not a reproduction of either paper's calculations.

Identify what was changed in Fig.2

Start below Fig.1, in the setup paragraphs of Bai's Section 2.3. These are three-dimensional, unstratified, isothermal shearing-box simulations, with Hall and ambipolar terms. “Unstratified” means vertical gravity is omitted. The box has radial, orbital, and vertical sides of $4H$, $4H$, and $2H$, where $H=c_s/\Omega$, $c_s$ is sound speed, and $\Omega$ is the local angular frequency.

Now read Fig.2's caption before following any line. The background vertical field points opposite to rotation. The initial gas-to-vertical-magnetic-pressure ratio is $\beta_0=10^4$, and the Hall length is $l_H=0.1H$. The calculation first evolves with ideal MHD. At $t=60\Omega^{-1}$, the Hall and ambipolar terms are switched on. The later colored curves compare three ambipolar coupling numbers, $\mathrm{Am}=100,10,1$.

The initial condition after the switch is an already turbulent state. That matters to the question: this experiment asks whether that finite disturbance persists. A linear eigenvalue calculation asks what happens to an arbitrarily small disturbance of a specified background.

Read the axes and the parameter symbols

The horizontal axis of Fig.2 is time in $\Omega^{-1}$. One full orbit lasts $2\pi\Omega^{-1}$, so the switch occurs after about 9.55 orbits, not 60 orbits. The vertical axis is Maxwell stress on a logarithmic scale. In the paper's units, magnetic permeability, reference density, and sound speed are one; the plotted stress corresponds to $-\langle B_xB_y\rangle$ divided by the reference gas pressure $P_0=\rho_0c_s^2$. Equal vertical distances indicate equal multiplicative changes. The wiggles are time variability, not error bars.

Section 2.1 defines the useful coupling quantities:

Here $\gamma_i\rho_i$ is the neutral–ion collision frequency, $\omega_H$ is the Hall frequency, and $v_A$ is the Alfvén speed. Smaller $\mathrm{Am}$ means weaker neutral–ion coupling relative to the orbital clock in this parametrization. The paper defines the Hall length and $\chi$ as positive and states field polarity separately. Keep that sign information when comparing cases; field strength alone does not specify a Hall experiment.

After the switch, the red $\mathrm{Am}=100$ curve maintains appreciable stress; the green and blue curves for 10 and 1 decay strongly. The caption explicitly places this result in a configuration with no linearly unstable MRI mode. Read that as a result about the survival of an existing nonlinear state under the stated conditions, over the plotted interval ending at $300\Omega^{-1}$. It is not a claim that a quiescent disk would reach the same state from any tiny seed. The caption, the switch, and the initialization belong together. Bai 2014, Fig.2 and Section 2.3.

For a cross-check, look at Fig.1 just above it. Its horizontal coordinate combines field polarity and inverse Hall coupling; its vertical coordinate is $kv_{A0}/\Omega$, and its color gives linear growth rate in units of $\Omega$. The two panels hold $\mathrm{Am}$ fixed at 100 and 1. Color in this figure answers the small-disturbance question; the time curves below answer the persistence question. No curve digitization is needed to see why the questions differ.

Check the pressure normalization before comparing net flux

Turn to the 2013 paper's equation 7. Its vertically integrated stress parameter is

An overbar denotes a horizontal average; the table then reports time averages. This pressure denominator differs from a single height's stress. That paper studies vertically stratified, net-vertical-flux shearing boxes, so density structure and the vertical domain enter the comparison.

As a short numerical exercise, read the Maxwell and Reynolds columns of Table 2. For B4 their rounded entries add to $0.061+0.014=0.075$; for B2 they add to $0.92+0.12=1.04$. The corresponding initial $\beta_0$ values are $10^4$ and $10^2$. At fixed initial gas pressure, this change means ten times the vertical field strength, because magnetic pressure scales as field squared. It does not mean a hundred times the field. The table illustrates how the transport level can depend on imposed net flux. Section 3.2.1 also warns that the strong-field B2 integrated value depends on the box's vertical extent and global field conditions. Treat its rounded number as a result of that setup. Bai and Stone 2013, equation 7, Section 3.2.1 and Table 2.

Saturation means that nonlinear evolution has limited the growing disturbance and established a later statistical state. The linear equations continue their exponential growth. Obtaining the table's long-time $\alpha$ requires a nonlinear calculation of the later state.

Use the linear Lab for the question it can answer

The MRI Lab follows an ideal, incompressible, horizontally homogeneous channel. Its controls change shear $q$ and magnetic tension frequency $K=kv_A$; the displayed $K/\Omega$ uses a rotating reference with $\Omega=1$. Velocity varies as a sine in height, horizontal magnetic perturbations as a cosine, and vertical fluid velocity is zero. Its small-amplitude fields let you connect growth, stress, and energy supply by the shear.

Its stress is normalized by $\rho L^2\Omega_{\rm ref}^2$, where $L$ is the chosen height scale. The incompressible model supplies no thermal pressure from which to form the paper's $\alpha$. It also has no Hall polarity, ambipolar coupling, or saturation control. Reversing the entire background field leaves this ideal linear growth problem unchanged; that symmetry is different from the polarity comparison in Bai's nonideal model.

Try the linear comparison first: at fixed Keplerian shear, compare the reference tension, the fastest-growing tension, and a strong tension that gives oscillation. Explain the energy source in each case. Then return to the paper and write down the extra inputs needed to ask its question: a nonlinear initial state, gas pressure, magnetic polarity, and nonideal coupling. Those inputs explain the distance between the two calculations.

Questions to carry to another MRI paper

If the switch in Fig.2 were moved to the start, could its caption alone tell you the later outcome? No: that would change the initial-value experiment. A useful follow-up calculation would need to specify the seed and test whether different starting states approach the same late behavior.

If two papers report different stresses, first compare the averaging volume, pressure denominator, imposed net flux, and averaging time. Then check vertical gravity, boundaries, and nonideal terms. A convergence study should compare the same physical setup as resolution changes. The 2013 paper discusses unresolved saturation-convergence questions in Section 3.1.3; resolving a linear wavelength and converging a later statistical stress are distinct tests.

On your next reading, write five things beside the figure: the exact version and page; the initial state and parameter held fixed; the axes and normalization; the change that produced the comparison; and the one conclusion the plotted quantity supports. That makes a growth diagram, a decaying time series, and a saturated transport table much harder to confuse.

Sources

Selected reading: Bai arXiv1409.2511v1, Section2 and Figures1–2 (fixed PDF); Bai and Stone arXiv1210.6661v2, equation7, Sections3.1.3/3.2.1 and Table2 (fixed PDF).