perturbation energy, extracting it from the background flow due to the shear. The Maxwell stress, $-b_rb_{\phi}$, does not play a role because of the smallness of the induced field in the low-$Rm$ regime. The nonlinear term, not directly tapping into the flow energy and therefore not changing the total perturbation energy, acts only to redistribute energy among different wavenumbers. In the quasi-steady saturated state, energy injection by the Reynolds stress balances dissipation losses. Since $d\Omega/dr<0$, this implies that the stress should be positive for the long-term sustenance of this state.

For the total angular momentum, $L=\int ldV$, we get $$\frac{dL}{dt}=G_i-G_o,$$ where $$G_{i,o}=-\frac{R_{i,o}^3}{Re}\int_0^{2\pi}
\int_0^{L_z}\frac{d}{dr}\left(\frac{u_{\phi}}{r}\right)|_{r=R_{i,o}}d\phi
dz$$ are the total viscous torques exerted, respectively, by the inner and outer cylinders on the flow. These torques also characterize angular momentum transport by the perturbations. Other – advection (i.e., Reynolds stress) and magnetic – torques vanish after volume integration due to the boundary conditions and do not contribute to the total angular momentum evolution. In the equilibrium (laminar) state, the torques at both cylinders have the same absolute value, $G_{lam}=-(2\pi L_z/Re) R_i^3d\Omega/dr|_{r=R_i}=-(2\pi L_z/Re)
R_o^3d\Omega/dr|_{r=R_o}=4\pi L_z(1-\mu)a^2/Re(1-a^2)(1-a)^2>0$, implying that the inner cylinder tries to increase the angular momentum of the basic flow, while the outer cylinder to 