tokens: so that we could compare the value of the dimensionless growth rate directly to the predicted value. However, there does not appear to be an accurate way to measure this directly from the simulation data. Future simulations that insert a growing mode with a single, well-defined value for k would be desirable. In the dynamic simulations we present here, it is likely that there is significant power at a range of wavenumbers. In this case, we can expect growth to be slower initially, until sufficient power is in the fastest-growing modes. For the mode to fit in the disk (recall that we cannot bend the disk on scales ≲ H ), we must have Hk ≲ 1 (Ogilvie 2000 ), so we can expect the fastest-growing modes to appear with Hk less than, but not much less than, unity. However, to be consistent, we report the values of that can be measured from the simulation data. For simulation A with α = 0.03, H / R = 0.02, and θ = 30°, we find that the warp amplitude in the inner disk regions rapidly grows to greater than the critical value for R ≲ 15 R g (which for this viscosity is ψ c ≈ 0.5). This inner region breaks into multiple rings, starting from the inner parts and ending at ≈ 30 R g . Outside of this region, ψ remains below the critical value at all times, and no breaks are observed in this region. For this simulation, we analyze two clear breaks. The first break occurs between R ≈ 12–18 R g , with a peak in the warp amplitude at R p ≈ 15 R g while the warp amplitude is growing from ψ 0 = 2.5 to ψ 1 = 4. The second break forms an initial peak in the warp amplitude at R ≈ 20 R g , and by the time the peak warp amplitude is ≈ 2, the peak has settled at a radius of R p ≈ 27 R g . For the first break, we measure Δ t ≈ 1500 (in units of GM / c 3 ), and with the peak at R p ≈ 15 R g we have Ω ≈ 0.02. This yields χ ≈ 0.015. For the second peak, we have Δ t ≈ 4000, R p ≈ 27 R g , Ω ≈ 8 × 10 −3 , and therefore χ ≈ 0.015. This means that, when scaled to the local dynamical time, the growth rate was approximately the same in both cases, as expected. In this simulation, we expect the dimensionless growth rate to be in the range of 0.2–0.5 (see Figures 2 and 3 ). If , this would imply k −1 ≈ (3–5) H , which seems reasonable, but we also note that it is likely that the total level of viscosity in the simulation is slightly higher than α = 0.03 (see discussion in Section 2 ) and thus we would expect the growth rate to be reduced somewhat (i.e., if α ≈ 0.05, then the predicted dimensionless growth rate implies k −1 ≈ (2–3) H ). For simulation B, with α = 0.03, H / R = 0.02, and θ = 60°, we analyze two breaks in the disk, with the first occurring at R p ≈ 24 R g and the second beginning at R ≈ 30 R g and settling into a strong peak at R p ≈ 40 R g . For the first break, we measure Δ t ≈ 1650, and with the peak at R p ≈ 24 R g we have Ω ≈ 0.01. This yields χ ≈ 0.03. For the second peak, we have Δ t ≈ 5400, R p ≈ 40 R g , Ω ≈ 4 × 10 −3 , and therefore χ ≈ 0.02. As before, when
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