Nonmodal Growth of the Magnetorotational Instability

We analyze the linear growth of the magnetorotational instability (MRI) in the short-time limit using nonmodal methods. Our findings are quite different from standard results, illustrating that shearing wave energy can grow at the maximum MRI rate $-d\mathrm{\Omega }/d\ln r$ for any choice of azimuthal and vertical wavelengths. In addition, by comparing the growth of shearing waves with static structures, we show that over short time scales shearing waves will always be dynamically more important than static structures in the ideal limit. By demonstrating that fast linear growth is possible at all wavelengths, these results suggest that nonmodal linear physics could play a fundamental role in MRI turbulence.

Figure 1

(a)–(c) Time evolution of the spatial structure of the radial magnetic field perturbation in $(r,\theta )$ that maximizes energy amplification at $t_M=10$. The global parameters are $m=2$, $k_z=15$, $U_0=1$, $B_{0z}=1/30$, $B_{az}=0$, $\overline{\nu }=\overline{\eta }=1/10000$. White and black regions show positive and negative values respectively. (d) Spatial structure of the most unstable eigenmode for the same parameters.

Figure 2

Energy amplification, $E_r(t)\equiv E(r=1,t)/E(r=1,0)$, of the global nonmodal solution (dashed line), the most unstable eigenmode (dotted line), and the solution of the local shearing wave equations [Eq. (4)] (solid red line). Parameters are the same as Fig. 1. The black dots illustrate the time frames shown in Fig. 1.