Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 57 / 3 / c

Let $a=v_A^2{k}^2/{\Omega }^2$. The growing root of the ideal magnetorotational dispersion relation isIts derivative is $y^{\prime }(a)=-1+4/\sqrt{1+16a}$, so the continuum maximum occurs at $a=15/16$, with $s=3\Omega /4$. For the bounded vertical modes of a sech-squared magnetized disk,Modes $n=1,\dots ,5$ are unstable; $n\ge 6$ have $a_n\ge 3$. The function $y$ increases up to $15/16$ and decreases thereafter, so the only candidates for the discrete maximum are $n=2$, with $a_2=1/2$, and $n=3$, with $a_3=1$. Their values are $y(a_2)=1/2$ and $y(a_3)=(\sqrt{17}-3)/2>1/2$. HenceThe associated velocity profile is $F_3=[5{\tanh }^3(z/H)-3\tanh (z/H)]/2$. Discrete vertical quantization makes its growth slightly smaller than the continuum maximum. This is the fastest discrete mode of a stratified magnetorotational instability.