Lecture 19
The Magnetorotational Instability
Overview
This lecture is the classic local calculation behind the modern theory of angular-momentum
transport in accretion disks.
-
1.
- Rayleigh’s criterion is already an energy principle for axisymmetric interchange
in a rotating fluid.
-
2.
- The epicyclic frequency emerges directly from the cylindrical equations of motion,
so one can see exactly how centrifugal and Coriolis effects combine.
-
3.
- A weak vertical magnetic field adds a spring-like coupling between neighboring rings,
and the decisive gradient changes from \(\dd {}{R}(R^4\Omega ^2)\) to \(\dd {\Omega }{R}\).
Accretion disks appear around young stars, compact binaries, and black holes. The basic difficulty is easy
to state: gravity wants matter to move inward, but angular momentum conservation prevents that unless
angular momentum is transported outward. In cylindrical coordinates the specific angular momentum is
\[L(R)=R^2\Omega (R). \tag{19.1}\]
If transport were due only to molecular viscosity, the characteristic accretion time would be
\[\tau _\nu \sim \frac {R^2}{\nu }, \tag{19.2}\]
which is generally far too long. This is why the effective-viscosity parameterization \[\nu _{\rm eff}=\alpha c_s H \tag{19.3}\]
was historically so influential: it captured the phenomenology of transport long before the underlying
mechanism was understood.
Historical Perspective
The historical arc of the MRI is unusually clean.
Rayleigh’s work on rotating fluids showed that axisymmetric hydrodynamic stability is
controlled by the outward increase of specific angular momentum, a result summarized
beautifully in Chandrasekhar’s classic monograph Chandrasekhar (1961). Velikhov and
Chandrasekhar then discovered that an axial magnetic field changes this criterion in a
profound way Velikhov (1959); Chandrasekhar (1960). For several decades the subject
lived mainly inside hydromagnetic Couette flow. Balbus and Hawley’s 1991 paper then
reframed the same instability as the missing local transport mechanism in accretion
disks Balbus and Hawley (1991). In hindsight that reinterpretation is one of the classic
moments in astrophysical MHD: an old stability calculation suddenly became the answer
to a much larger physical problem.
19.1 Rayleigh’s Criterion as a Rotational Energy Principle
The ideal-MHD energy principle in Lecture 15 teaches a general lesson: stability can often be read from
the sign of a quadratic energy functional. Rayleigh’s criterion is the rotating-fluid version of that same
idea.
Ring interchange.
Consider an inviscid axisymmetric flow
\[\uvec _0 = R\Omega (R)\,\ephi . \tag{19.4}\]
Take two thin rings of equal mass \(m\) at radii \(R_1<R_2\). During an axisymmetric interchange each ring conserves
its own specific angular momentum because the motion is inviscid and axisymmetric. Let
\[L_1=L(R_1), \qquad L_2=L(R_2).\]
The initial rotational kinetic energy is \[K_{\rm i} = \frac {m}{2} \left ( \frac {L_1^2}{R_1^2} + \frac {L_2^2}{R_2^2} \right ), \tag{19.6}\]
while after interchange it is \[K_{\rm f} = \frac {m}{2} \left ( \frac {L_1^2}{R_2^2} + \frac {L_2^2}{R_1^2} \right ). \tag{19.7}\]
Therefore \[\begin{aligned}\Delta W_{\rm rot} &\equiv K_{\rm f}-K_{\rm i} \nonumber \\ &= \frac {m}{2} \left [ L_1^2\left (\frac {1}{R_2^2}-\frac {1}{R_1^2}\right ) + L_2^2\left (\frac {1}{R_1^2}-\frac {1}{R_2^2}\right ) \right ] \nonumber \\ &= \frac {m}{2} \left (L_1^2-L_2^2\right ) \left (\frac {1}{R_2^2}-\frac {1}{R_1^2}\right ).\end{aligned} \tag{19.8}\]
Since \(R_2>R_1\), the geometric factor \((R_2^{-2}-R_1^{-2})\) is negative. Stability to every such interchange therefore requires
\[L_2^2>L_1^2 \qquad \text {whenever}\qquad R_2>R_1,\]
or equivalently \[\boxed {\dd {}{R}\left (L^2\right )>0} \qquad \Longleftrightarrow \qquad \boxed {\dd {}{R}\left (R^4\Omega ^2\right )>0}. \tag{19.10}\]
This is Rayleigh’s criterion.
Direct derivation from the cylindrical equations of motion.
The same criterion appears directly in the cylindrical inertial terms. For axisymmetric motion, the radial
and azimuthal components of the convective derivative are
\[\begin{aligned}\left (\pp {}{t}+\uvec \cdot \grad \right )u_R &= \pp {u_R}{t} + u_R\pp {u_R}{R} + u_Z\pp {u_R}{Z} - \frac {u_\phi ^2}{R}, \\ \left (\pp {}{t}+\uvec \cdot \grad \right )u_\phi &= \pp {u_\phi }{t} + u_R\pp {u_\phi }{R} + u_Z\pp {u_\phi }{Z} + \frac {u_Ru_\phi }{R}.\end{aligned}\]
Now linearize about the rotating equilibrium (19.4). Write
\[u_R=u_{1R}, \qquad u_\phi =R\Omega (R)+u_{1\phi }, \qquad u_Z=u_{1Z},\]
and keep only linear terms. Using the equilibrium radial force balance to cancel the background gravity
and centrifugal force, the remaining inertial terms are \[\begin{aligned}\pp {u_{1R}}{t}-2\Omega u_{1\phi }&=0, \\ \pp {u_{1\phi }}{t}+ \left (2\Omega +R\pp {\Omega }{R}\right )u_{1R}&=0.\end{aligned} \tag{19.14}\]
Differentiate (19.14) with respect to time and use (19.15) to eliminate \(\pp {u_{1\phi }}{t}\):
\[\begin{aligned}\pp {^2 u_{1R}}{t^2} -2\Omega \pp {u_{1\phi }}{t} &=0, \nonumber \\ \pp {^2 u_{1R}}{t^2} +2\Omega \left (2\Omega +R\pp {\Omega }{R}\right )u_{1R} &=0.\end{aligned}\]
This identifies the epicyclic frequency,
\[\kappa ^2 \equiv 2\Omega \left (2\Omega +R\pp {\Omega }{R}\right ) = 4\Omega ^2+R\pp {\Omega ^2}{R} = \frac {1}{R^3}\dd {}{R}\left (R^4\Omega ^2\right ). \tag{19.17}\]
If we write \(u_{1R}=\pp {\xi _R}{t}\) and choose the integration constant appropriate to a parcel released from equilibrium, this
becomes \[\pp {^2 \xi _R}{t^2}+\kappa ^2\xi _R=0. \tag{19.18}\]
The corresponding quadratic energy is \[\mathcal {E}_{\rm epi} = \frac {1}{2}\left (\pp {\xi _R}{t}\right )^2 + \frac {1}{2}\kappa ^2\xi _R^2. \tag{19.19}\]
Thus Rayleigh’s criterion is literally an energy principle: \[\boxed {\mathcal {E}_{\rm epi}>0 \iff \kappa ^2>0.}\]
For a Keplerian disk, \(\Omega \propto R^{-3/2}\), so \(\kappa ^2=\Omega ^2>0\). Keplerian rotation is therefore hydrodynamically stable even though it contains
strong shear.
Caution
The Rayleigh calculation isolates the centrifugal part of the problem. More general
rotating fluids can also support acoustic, buoyant, and baroclinic responses, and those
may matter in other lectures. Here, however, the point is exactly that the purely
rotational piece already has the structure of an energy principle.
19.2 MRI from the Cylindrical Ideal-MHD Equations
We now return to the ideal-MHD equations of the introduction, especially the momentum equation (1.8),
the ideal Ohm law (4.8), and the induction equation (1.13). Rather than adopting shearing-sheet notation,
we keep the cylindrical equations visible so the centrifugal and epicyclic terms can be read off
directly.
Choice of equilibrium and perturbation.
Take the equilibrium flow and magnetic field to be
\[\uvec _0 = R\Omega (R)\,\ephi , \qquad \B _0 = B_0\,\ez . \tag{19.21}\]
We consider local, axisymmetric perturbations of the form \[\propto e^{ikZ-i\omega t}, \tag{19.22}\]
with \(\vect {k}=k\ez \). For the classic worked example we choose incompressible vertical modes. Then the pressure
perturbation enforces the constraint but does not determine the onset criterion, so the MRI mechanism
can be seen without additional algebra from sound waves.
Define Alfvén-speed units for the perturbed field,
\[\vect {b}\equiv \frac {\B _1}{\sqrt {\muo \rho _0}}, \qquad V_A\equiv \frac {B_0}{\sqrt {\muo \rho _0}}, \qquad \omega _A\equiv kV_A. \tag{19.23}\]
Linearized cylindrical momentum equations.
The same cylindrical linearization used above now gives
\[\begin{aligned}-i\omega u_R - 2\Omega u_\phi &= f_R, \\ -i\omega u_\phi + \left (2\Omega +R\pp {\Omega }{R}\right )u_R &= f_\phi ,\end{aligned} \tag{19.24}\]
where \(f_R\) and \(f_\phi \) are the perturbed Lorentz forces per unit mass. It is convenient to rewrite the azimuthal
coefficient using (19.17):
\[2\Omega +R\pp {\Omega }{R} = \frac {\kappa ^2}{2\Omega }.\]
Therefore \[-i\omega u_\phi + \frac {\kappa ^2}{2\Omega }u_R = f_\phi . \tag{19.27}\]
This is the point of doing the derivation in cylindrical coordinates: the epicyclic restoring force is visible
from the start, rather than inserted later by a change of variables.
Linearized induction equations.
From the induction equation with the equilibrium (19.21) and perturbations (19.22), the radial and
azimuthal components are
\[\begin{aligned}-i\omega b_R &= i\omega _A u_R, \\ -i\omega b_\phi &= -R\pp {\Omega }{R} b_R + i\omega _A u_\phi .\end{aligned} \tag{19.28}\]
The first term in (19.29) is the crucial shear-winding term: a perturbed radial field is wound into an
azimuthal field by the background differential rotation.
Lorentz force.
For these vertical modes the perturbed Lorentz force is simply magnetic tension along the background
field,
\[f_R = i\omega _A b_R, \qquad f_\phi = i\omega _A b_\phi . \tag{19.30}\]
Substituting (19.30) into (19.24) and (19.27) gives the four-equation system \[\begin{aligned}-i\omega u_R - 2\Omega u_\phi &= i\omega _A b_R, \\ -i\omega u_\phi + \frac {\kappa ^2}{2\Omega }u_R &= i\omega _A b_\phi , \\ -i\omega b_R &= i\omega _A u_R, \\ -i\omega b_\phi &= -R\pp {\Omega }{R} b_R + i\omega _A u_\phi .\end{aligned} \tag{19.31}\]
First elimination: remove \(b_R\).
From (19.33),
\[b_R = -\frac {\omega _A}{\omega }u_R. \tag{19.35}\]
Substitute this into (19.31): \[\begin{aligned}-i\omega u_R - 2\Omega u_\phi &= i\omega _A\left (-\frac {\omega _A}{\omega }u_R\right ) = -i\frac {\omega _A^2}{\omega }u_R, \nonumber \\ \omega u_R + 2i\Omega u_\phi &= \frac {\omega _A^2}{\omega }u_R, \nonumber \\ \left (\omega ^2-\omega _A^2\right )u_R + 2i\Omega \omega u_\phi &= 0.\end{aligned} \tag{19.36}\]
So the radial momentum equation becomes a balance between the epicyclic motion and a magnetic
spring.
Second elimination: remove \(b_\phi \).
Use (19.35) in (19.34):
\[\begin{aligned}-i\omega b_\phi &= -R\pp {\Omega }{R}\left (-\frac {\omega _A}{\omega }u_R\right ) + i\omega _A u_\phi \nonumber \\ &= \frac {R\omega _A}{\omega }\pp {\Omega }{R}u_R + i\omega _A u_\phi .\end{aligned}\]
Divide by \(-i\omega \):
\[\begin{aligned}b_\phi &= \frac {iR\omega _A}{\omega ^2}\pp {\Omega }{R}u_R - \frac {\omega _A}{\omega }u_\phi .\end{aligned} \tag{19.38}\]
Now multiply by \(i\omega _A\):
\[\begin{aligned}i\omega _A b_\phi &= i\omega _A\left ( \frac {iR\omega _A}{\omega ^2}\pp {\Omega }{R}u_R - \frac {\omega _A}{\omega }u_\phi \right ) \nonumber \\ &= - \frac {R\omega _A^2}{\omega ^2}\pp {\Omega }{R}u_R - i\frac {\omega _A^2}{\omega }u_\phi .\end{aligned} \tag{19.39}\]
Substitute this into (19.32) and multiply through by \(i\omega \):
\[\begin{aligned}\omega ^2 u_\phi + i\omega \frac {\kappa ^2}{2\Omega }u_R &= - i\omega \frac {R\omega _A^2}{\omega ^2}\pp {\Omega }{R}u_R + \omega _A^2 u_\phi , \nonumber \\ \left (\omega ^2-\omega _A^2\right )u_\phi + i\omega \left [ \frac {\kappa ^2}{2\Omega } + \frac {R\omega _A^2}{\omega ^2}\pp {\Omega }{R} \right ]u_R &=0.\end{aligned} \tag{19.40}\]
Dispersion relation.
Equation (19.36) gives
\[u_\phi = \frac {i\left (\omega _A^2-\omega ^2\right )}{2\Omega \omega }u_R. \tag{19.41}\]
Insert this into (19.40): \[\begin{aligned}0 &= \left (\omega ^2-\omega _A^2\right ) \frac {i\left (\omega _A^2-\omega ^2\right )}{2\Omega \omega }u_R + i\omega \left [ \frac {\kappa ^2}{2\Omega } + \frac {R\omega _A^2}{\omega ^2}\pp {\Omega }{R} \right ]u_R \nonumber \\ &= \frac {i u_R}{2\Omega \omega } \left [ - \left (\omega ^2-\omega _A^2\right )^2 + \kappa ^2\omega ^2 + 2\Omega R\pp {\Omega }{R}\omega _A^2 \right ].\end{aligned}\]
A nontrivial solution therefore requires
\[\left (\omega ^2-\omega _A^2\right )^2 - \kappa ^2\omega ^2 - 2\Omega R\pp {\Omega }{R}\omega _A^2 =0.\]
Expanding, \[\omega ^4 - \omega ^2\left (\kappa ^2+2\omega _A^2\right ) + \omega _A^2 \left ( \omega _A^2 + 2\Omega R\pp {\Omega }{R} \right ) =0.\]
Since \[2\Omega R\pp {\Omega }{R} = \dd {\Omega ^2}{\ln R},\]
the standard MRI dispersion relation is \[\boxed { \omega ^4 - \omega ^2\left (\kappa ^2+2\omega _A^2\right ) + \omega _A^2 \left ( \omega _A^2 + \dd {\Omega ^2}{\ln R} \right ) =0.} \tag{19.46}\]
This is the clean payoff of the cylindrical derivation: the term \(\kappa ^2\) comes directly from the hydrodynamic
inertial structure, while the term \(\dd {\Omega ^2}{\ln R}\) comes from winding the perturbed radial field into an azimuthal
field.
Hydrodynamic and magnetic limits.
If \(\omega _A=0\), then (19.46) reduces to
\[\omega ^2\left (\omega ^2-\kappa ^2\right )=0,\]
so the nontrivial branch is the epicyclic oscillation \(\omega ^2=\kappa ^2\). If the field is weak but nonzero, the instability
threshold is obtained by setting \(\omega ^2=0\): \[\omega _A^2 + \dd {\Omega ^2}{\ln R} < 0. \tag{19.48}\]
In the weak-field limit this becomes \[\boxed {\dd {\Omega }{R}<0.} \tag{19.49}\]
This is the famous MRI criterion. It should be contrasted with Rayleigh’s hydrodynamic criterion
(19.10). A Keplerian disk satisfies \(\kappa ^2>0\) and is therefore hydrodynamically stable, but because \(\dd {\Omega }{R}<0\), it is
magnetorotationally unstable.
Keplerian growth rate.
For a Keplerian disk,
\[\Omega \propto R^{-3/2}, \qquad \kappa ^2=\Omega ^2, \qquad \dd {\Omega ^2}{\ln R}=-3\Omega ^2. \tag{19.50}\]
Then (19.46) becomes \[\omega ^4 - \omega ^2\left (\Omega ^2+2\omega _A^2\right ) + \omega _A^2\left (\omega _A^2-3\Omega ^2\right )=0. \tag{19.51}\]
Treat this as a quadratic in \(\omega ^2\): \[\omega ^2_{\pm } = \frac {1}{2} \left [ \Omega ^2+2\omega _A^2 \pm \sqrt {\Omega ^4+16\Omega ^2\omega _A^2} \right ]. \tag{19.52}\]
The unstable branch is \(\omega _-^2<0\). Write \(\omega _-^2=-\gamma ^2\), so that \[\gamma ^2 = \frac {1}{2} \left [ \sqrt {\Omega ^4+16\Omega ^2\omega _A^2} - \Omega ^2 - 2\omega _A^2 \right ]. \tag{19.53}\]
Introduce the dimensionless parameter \[x \equiv \frac {\omega _A^2}{\Omega ^2}.\]
Then \[\frac {\gamma ^2}{\Omega ^2} = \frac {1}{2}\left [\sqrt {1+16x}-1-2x\right ]. \tag{19.55}\]
Differentiate with respect to \(x\): \[\dd {}{x}\left (\frac {\gamma ^2}{\Omega ^2}\right ) = \frac {4}{\sqrt {1+16x}} - 1.\]
The maximum occurs when \[\frac {4}{\sqrt {1+16x}} - 1 = 0 \qquad \Longrightarrow \qquad \sqrt {1+16x}=4,\]
so \[x = \frac {15}{16} \qquad \Longrightarrow \qquad \omega _A^2 = \frac {15}{16}\Omega ^2.\]
Substitute this back into (19.55): \[\frac {\gamma _{\max }^2}{\Omega ^2} = \frac {1}{2}\left (4-1-\frac {30}{16}\right ) = \frac {9}{16},\]
which gives the standard result \[\boxed {\gamma _{\max } = \frac {3}{4}\Omega .} \tag{19.60}\]
Caution
This worked example suppresses pressure by choosing incompressible vertical modes.
That is a feature, not a bug: it isolates the rotational and magnetic coupling with the
minimum algebra. More general compressible or stratified calculations certainly exist,
and they introduce additional branches, but the essential MRI criterion (19.49) is already
visible here.
Physical interpretation.
In pure hydrodynamics, a displaced ring executes epicyclic motion at frequency \(\kappa \). The magnetic field ties
neighboring rings together like a weak spring. When \(\dd {\Omega }{R}<0\), the inner ring rotates faster than the outer ring. A
small radial displacement therefore causes magnetic tension to transfer angular momentum
outward: the inner ring loses angular momentum and falls inward, while the outer ring gains
angular momentum and moves outward. The restoring epicyclic motion is then turned into a
positive-feedback mechanism. The field must be weak enough that the rings are not simply locked
together, but strong enough to communicate tension. That is precisely what the quartic (19.46)
says.
19.3 Historical and Experimental Perspective
From Couette flow to accretion disks.
Velikhov and Chandrasekhar formulated the problem in rotating-cylinder geometry, where the question
was whether an imposed axial magnetic field destabilizes a centrifugally stable Couette profile
Velikhov (1959); Chandrasekhar (1960). Balbus and Hawley’s insight was that the same local
mathematics applies to accretion disks Balbus and Hawley (1991). Once that point was made, the MRI
moved from being a specialized hydromagnetic stability problem to being one of the central ideas in
modern astrophysical MHD.
Opinion
Laboratory studies of MRI-like physics naturally focused on Taylor–Couette flow. Early
work associated with Lathrop and collaborators helped motivate the experimental
program, and later liquid-metal efforts at Princeton and Potsdam showed how delicately
the background shear and boundary conditions must be engineered before one can
isolate the desired instability Lathrop and Forest (2011). In the author’s admittedly
editorial view, these were valuable but rather specialized experiments: they were not
especially rich plasma experiments, and they were not designed to explore fully developed
turbulence either, since much of the work went into suppressing or controlling unwanted
hydrodynamic structure. Once the correct base flow had been prepared, Maxwell’s
equations had very little choice but to couple neighboring rings in the way the theory
predicts.
Why plasma experiments were different.
Plasma MRI experiments opened a broader door. Hall physics, global eigenmodes, line-tying, equilibrium
modification, and boundary-driven current systems allowed behaviors that were absent from the most
stripped-down liquid-metal realizations. In that sense the plasma experiments were not merely
confirmations of the textbook dispersion relation; they also exposed additional physics that is probably
relevant in real astrophysical systems Collins et al. (2012, 2014); Flanagan et al. (2020); Milhone
et al. (2021); Ebrahimi et al. (2011).
Takeaways
The MRI lecture is really three classic results nested inside one another.
-
1.
- Rayleigh’s criterion is already a rotational energy principle: \(\kappa ^2>0\) means the epicyclic
energy is positive.
-
2.
- The cylindrical equations of motion make the role of centrifugal and Coriolis forces
transparent.
-
3.
- Magnetic tension plus differential rotation changes the decisive gradient from \(\dd {}{R}(R^4\Omega ^2)\) to \(\dd {\Omega }{R}\),
making Keplerian disks unstable.
Bibliography
S. Chandrasekhar. Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford, 1961.
E.P. Velikhov. Stability of an ideally conducting liquid flowing between cylinders rotating in a
magnetic field. Soviet Physics JETP, 36:995, 1959.
S. Chandrasekhar. The stability of non-dissipative Couette flow in hydromagnetics. Proceedings
of the National Academy of Sciences, 46(2):253–257, 1960. doi:10.1073/pnas.46.2.253.
S.A. Balbus and J.F. Hawley. A powerful local shear instability in weakly magnetized disks: I.
linear analysis. The Astrophysical Journal, 376:214, 1991. doi:10.1086/170270.
Daniel P. Lathrop and Cary B. Forest. Magnetic dynamos in the lab. Physics Today, 64(7):
40–45, 2011. doi:10.1063/pt.3.1166.
C Collins, N Katz, J Wallace, J Jara-Almonte, I Reese, E Zweibel,
and C B Forest. Stirring unmagnetized plasma. Physical Review Letters, 108(11):115001, 2012.
doi:10.1103/physrevlett.108.115001.
C. Collins, M. Clark, C. M. Cooper, K. Flanagan, I. V. Khalzov, M. D. Nornberg,
B. Seidlitz, J. Wallace, and C. B. Forest. Taylor-couette flow of unmagnetized plasma. Physics
of Plasmas (1994-present), 21(4):042117, 2014. doi:10.1063/1.4872333.
K. Flanagan, J. Milhone, J. Egedal, D. Endrizzi, J. Olson, E. E. Peterson, R. Sassella, and
C. B. Forest. Weakly magnetized, hall dominated plasma couette flow. Physical Review Letters,
125(13):135001, 2020. doi:10.1103/physrevlett.125.135001.
J. Milhone, K. Flanagan, J. Egedal, D. Endrizzi, J. Olson, E. E. Peterson, J. C. Wright, and
C. B. Forest. Ion heating and flow driven by an instability found in plasma couette flow. Physical
Review Letters, 126(18):185002, 2021. doi:10.1103/physrevlett.126.185002.
F Ebrahimi, B Lefebvre, C B Forest, and A Bhattacharjee. Global hall-MHD simulations of
magnetorotational instability in a plasma couette flow experiment. Physics of Plasmas, 18(6):
062904, 2011. doi:10.1063/1.3598481.
Problems
Problem 19.1. Rayleigh criterion and epicyclic energy
Consider an inviscid axisymmetric rotating flow with angular velocity \(\Omega (R)\).
-
(a)
- Starting from the interchange argument, derive (19.10).
-
(b)
- Starting from the cylindrical equations (19.14)–(19.15), derive (19.17).
-
(c)
- Show that the radial displacement energy (19.19) is positive definite if and only if \(\kappa ^2>0\).
Problem 19.2. MRI quartic from the cylindrical equations
Take the four linearized equations (19.31)–(19.34).
-
(a)
- Eliminate \(b_R\) and derive (19.36).
-
(b)
- Eliminate \(b_\phi \) and derive (19.40).
-
(c)
- Complete the algebra leading to the dispersion relation (19.46).
-
(d)
- Identify explicitly which term in the derivation comes from winding of the perturbed radial field
by the background shear.
Problem 19.3. Keplerian MRI
For \(\Omega \propto R^{-3/2}\):
-
(a)
- verify (19.50);
-
(b)
- derive (19.51);
-
(c)
- solve for \(\omega _\pm ^2\) and derive the unstable branch (19.53);
-
(d)
- show that the fastest growth occurs at \(\omega _A^2=15\Omega ^2/16\) and that \(\gamma _{\max }=3\Omega /4\).