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Lecture 21
FLR Stabilization: Roberts–Taylor Extended MHD

Overview

Why this lecture matters. This is the first place in the notes where we go beyond the resistive Ohm law (1.9) and the ideal limit (4.8) on purpose. The payoff is analytic: one can see, in a completely worked fluid derivation, exactly how Hall physics and ion gyroviscosity regularize the short-wavelength interchange mode. In the Roberts–Taylor ordering the result is

\[\omega ^2-\left (\frac {gk_\perp }{\Omega _i}+\nu _0\eta k_\perp \right )\omega +\frac {g}{L_n}=0, \tag{21.1}\]
with \(\eta \equiv L_n^{-1}\) and \[ \nu _0\eta k_\perp =\frac {\Omega _i}{2}\frac {1}{k_\perp L_n}(k_\perp \rho _i)^2. \] The Hall piece and the FLR piece both enter linearly in \(\omega \); the gravitational drive remains \(g/L_n\).

The Roberts–Taylor problem is a classic because the algebra is still manageable while all of the extended-MHD ideas are already present: Hall physics, ion inertia, gyroviscosity, electrostatic closure, and a non-solenoidal flow that replaces the ideal-MHD condition \(\divergence \uvec =0\). It is also a perfect bridge lecture. The previous lectures developed the single-fluid equations and frozen-in flux. Here we deliberately keep the terms that ideal MHD throws away and watch the interchange spectrum change.

Historical Perspective

The early 1960s were intensely focused on interchange stability and on whether finite-Larmor-radius effects could tame bad-curvature drive Rosenbluth et al. (1962); Rosenbluth and Simon (1965). Roberts and Taylor showed that this stabilization can be recovered from a fluid model provided one retains both of the ingredients that simpler one-fluid reductions tend to discard: the two-fluid correction in Ohm’s law and the nonscalar ion stress tensor Roberts and Taylor (1962). I have benefited enormously from Dalton Schnack’s notes Schnack (2005) for unpacking the terse PRL derivation.

The modern vocabulary may be drift-kinetic, gyrofluid, or gyrokinetic, but the intellectual core is already here. Bad curvature is still the organizing instability for confinement physics; one simply meets it later dressed as interchange, ballooning, ITG, or TEM turbulence. Roberts–Taylor therefore deserves to be worked through carefully, not just quoted. Indeed, this, together with Rosenbluth’s kinetic FLR treatment is the progenitor of modern day gyrokinetics.

21.1 Geometry, equilibrium, and notation

We use the slab geometry \[ \B _0=B_0\,\vect {e}_z, \qquad \vect {g}=-g\,\vect {e}_x, \qquad \dd {\rho _0}{x}=\eta \rho _0, \qquad \eta \equiv \frac {1}{L_n}>0, \qquad \uvec _1 = V_x\,\vect {e}_x+V_y\,\vect {e}_y. \] The equilibrium total pressure must satisfy

\[\dd {p_{T0}}{x}=-\rho _0 g, \qquad p_{T0}\equiv p_0+\frac {B_0^2}{2\muo }. \tag{21.2}\]
If \(B_0\) is constant, (21.2) reduces to the familiar hydrostatic balance \(\dd {p_0}{x}=-\rho _0 g\).

The perturbations are taken to be local normal modes,

\[\tilde f(x,y,t)=\hat f\,e^{i\omega t+iky}, \qquad k\equiv k_y=k_\perp , \tag{21.3}\]
and the local ordering is
\[\eta \ll k, \tag{21.4}\]
so perturbation amplitudes are treated as slowly varying in \(x\) while equilibrium coefficients may vary through \(\rho _0(x)\) and \(p_0(x)\).

SVG-Viewer needed.

A useful notation map is

This note

RT/Schnack

Meaning

\(\Omega _i\)

\(\Omega \)

ion gyrofrequency

\(\nu _0\)

\(2\nu \)

gyroviscous coefficient

\(L_n^{-1}=\eta \)

\(\eta \)

density-gradient scale

\(k_\perp \)

\(k\)

local transverse wavenumber

\(\rho _i^2\)

\(a^2\)

ion gyroradius squared

Caution

Two sign conventions are in play. First, Roberts and Taylor use the opposite signs for \(g\) and the equilibrium density gradient. Second, our earlier momentum equation (1.8) was written with the continuum-stress convention \(+\divergence \tens {\Pi }\). In this lecture we follow the plasma-transport convention and write the gyroviscous correction as a pressure tensor entering through \(-\divergence \tens {\Pi }^{\mathrm {gv}}\). The two conventions are related by

\[\tens {\Pi }_{\text {book}}=-\tens {\Pi }^{\mathrm {gv}}. \tag{21.5}\]
That sign map is worth keeping in mind for homework and for the Roberts–Taylor FLR lecture.

21.2 The clever step: rewrite generalized Ohm’s law with the momentum equation

Start from the collisionless, zero-resistivity, negligible-electron-inertia generalized Ohm law,

\[\E +\uvec \times \B +\frac {1}{ne}\grad p_e-\frac {1}{ne}\J \times \B =0. \tag{21.6}\]
Relative to the resistive Ohm law (1.9), the new term is the Hall force \((\J \times \B )/(ne)\), and relative to the fully reduced ideal law (4.8) we are also keeping electron-pressure effects. Compare also the more general Braginskii form in (5.10).

The one-fluid momentum equation is

\[\rho \frac {D\uvec }{Dt}=\J \times \B -\grad p+\rho \vect {g}-\divergence \tens {\Pi }^{\mathrm {gv}}, \qquad p=p_i+p_e. \tag{21.7}\]
Therefore
\[\J \times \B =\rho \frac {D\uvec }{Dt}+\grad p-\rho \vect {g}+\divergence \tens {\Pi }^{\mathrm {gv}}. \tag{21.8}\]
Substituting (21.8) into (21.6) gives
\[\boxed {\; \E =-\uvec \times \B +\frac {M}{e}\frac {D\uvec }{Dt}+\frac {1}{ne}\grad p_i-\frac {M}{e}\vect {g}+\frac {1}{ne}\divergence \tens {\Pi }^{\mathrm {gv}} \;} \tag{21.9}\]
with \(\rho =nM\).

This is the key reorganization. The Hall term and the electron-pressure term have been traded for a sum of physically transparent ion-fluid pieces. If we now cross (21.9) with \(\B /B^2\), the perpendicular flow can be read as

\[\uvec _\perp =\underbrace {\frac {\E \times \B }{B^2}}_{\text {common }E\times B} +\underbrace {\frac {1}{\Omega _i}\vect {b}\times \frac {D\uvec }{Dt}}_{\text {polarization}} +\underbrace {\frac {\vect {b}\times \grad p_i}{enB}}_{\text {ion diamagnetic}} -\underbrace {\frac {1}{\Omega _i}\vect {b}\times \vect {g}}_{\text {gravity drift}} +\underbrace {\frac {\vect {b}\times (\divergence \tens {\Pi }^{\mathrm {gv}})}{enB}}_{\text {gyroviscous drift}}. \tag{21.10}\]

Caution

Where did the diamagnetic drift go? It is still present: it sits explicitly in (21.10) as \(\vect {b}\times \grad p_i/(enB)\). In the barotropic, isothermal local benchmark its curl vanishes at leading order, so it drops out of the final scalar closure. But the derivation should not skip it, because it is part of the extended-MHD bookkeeping and it reappears directly in the separate ion/electron derivation below.

21.3 Electrostatic closure and the loss of \(\divergence \uvec =0\)

In the ideal-flux-freezing lecture, the ideal Ohm law (4.8) led directly to frozen flux (4.13). Here we take a different reduced limit: low \(\beta \), electrostatic perturbations with negligible magnetic perturbation but finite Hall and gyroviscous corrections. The dynamics are organized around the total pressure

\[p_T\equiv p+\frac {B^2}{2\muo }, \tag{21.11}\]
and the electrostatic assumption
\[\curl \E =0. \tag{21.12}\]
Taking the curl of (21.9) gives
\[-\curl (\uvec \times \B ) +\frac {M}{e}\curl \left [\frac {D\uvec }{Dt}+\frac {C_s^2}{\rho }\grad \rho +\frac {1}{\rho }\divergence \tens {\Pi }^{\mathrm {gv}}\right ]=0, \tag{21.13}\]
where barotropic ions have been used so that \(\grad p_i=C_s^2\grad \rho \).

For the present slab geometry, \(\B =B\,\vect {e}_z\) and \((\B \cdot \grad )\uvec =0\), so

\[\divergence \uvec +\frac {1}{\Omega _i}\vect {e}_z\cdot \curl \left [\frac {D\uvec }{Dt}+\frac {C_s^2}{\rho }\grad \rho +\frac {1}{\rho }\divergence \tens {\Pi }^{\mathrm {gv}}\right ]=0. \tag{21.14}\]
For the isothermal benchmark, \(C_s^2\) is constant, and the pressure-gradient curl vanishes explicitly because \[ \curl \left (\frac {C_s^2}{\rho }\grad \rho \right ) =C_s^2\,\grad \left (\frac {1}{\rho }\right )\times \grad \rho =0. \] Likewise, \[ \curl \left (\frac {1}{\rho }\divergence \tens {\Pi }^{\mathrm {gv}}\right ) =\grad \left (\frac {1}{\rho }\right )\times \left (\divergence \tens {\Pi }^{\mathrm {gv}}\right ) +\frac {1}{\rho }\curl \left (\divergence \tens {\Pi }^{\mathrm {gv}}\right ). \] For the local slab benchmark the second term does not contribute at the order retained, so Schnack’s closure reduces to
\[\boxed {\; \divergence \uvec +\frac {1}{\Omega _i}\vect {e}_z\cdot \curl \frac {D\uvec }{Dt} -\frac {1}{\Omega _i\rho ^2}\vect {e}_z\cdot \Big (\grad \rho \times (\divergence \tens {\Pi }^{\mathrm {gv}})\Big )=0 \;} \tag{21.15}\]
which is the extended-MHD substitute for the ideal-MHD incompressibility condition.

In the ideal-MHD limit \(\Omega _i\to \infty \) and \(\tens {\Pi }^{\mathrm {gv}}\to 0\), so (21.15) collapses to \(\divergence \uvec =0\). With Hall and FLR effects retained, electrostatic perturbations are no longer forced to be solenoidal.

21.4 Gyroviscosity in slab geometry: from Braginskii to Roberts–Taylor

The Braginskii lecture defined the symmetric traceless strain tensor in (5.25). In the present slab geometry,

\[\begin{aligned}W_{xx}&=2\pp {V_x}{x}-\frac {2}{3}\divergence \uvec , \\ W_{yy}&=2\pp {V_y}{y}-\frac {2}{3}\divergence \uvec , \\ W_{xy}&=\pp {V_x}{y}+\pp {V_y}{x}.\end{aligned} \tag{21.16}\]

In the strongly magnetized, collisionless limit relevant here, the Braginskii tensor collapses to its gyroviscous part with coefficient \(\eta _3=\rho \nu _0\). The nonzero perpendicular components are therefore

\[\Pi _{xx}^{\mathrm {gv}}=-\Pi _{yy}^{\mathrm {gv}}=-\rho \nu _0\left (\pp {V_y}{x}+\pp {V_x}{y}\right ), \qquad \Pi _{xy}^{\mathrm {gv}}=\Pi _{yx}^{\mathrm {gv}}=\rho \nu _0\left (\pp {V_x}{x}-\pp {V_y}{y}\right ), \tag{21.19}\]
with
\[\nu _0=\frac {a^2\Omega _i}{2}=\frac {\rho _i^2\Omega _i}{2}, \qquad a\equiv \rho _i, \qquad \rho \nu _0\sim \frac {p_i}{\Omega _i}. \tag{21.20}\]
Although \(\nu _0\) has the dimensions of a kinematic viscosity, the associated force is nondissipative: it is an FLR stress, not collisional damping.

Now insert the local mode (21.3). To keep the algebra visible, it is worth evaluating the divergence component by component. First,

\[\begin{aligned}(\divergence \tens {\Pi }^{\mathrm {gv}})_x &=\pp {\Pi _{xx}^{\mathrm {gv}}}{x}+\pp {\Pi _{yx}^{\mathrm {gv}}}{y} \\ &=\pp {}{x}\left [-\rho _0\nu _0\left (\pp {V_y}{x}+ikV_x\right )\right ] +ik\left [\rho _0\nu _0\left (\pp {V_x}{x}-ikV_y\right )\right ].\end{aligned}\]

Under the local ordering (21.4) we neglect \(\pp {V_x}{x}\) and \(\pp {V_y}{x}\) in the perturbation amplitudes but retain the equilibrium variation of \(\rho _0\nu _0\). Therefore

\[(\divergence \tens {\Pi }^{\mathrm {gv}})_x=-(\rho _0\nu _0)'ikV_x+\rho _0\nu _0k^2V_y. \tag{21.21}\]
Likewise,
\[\begin{aligned}(\divergence \tens {\Pi }^{\mathrm {gv}})_y &=\pp {\Pi _{xy}^{\mathrm {gv}}}{x}+\pp {\Pi _{yy}^{\mathrm {gv}}}{y} \\ &=\pp {}{x}\left [\rho _0\nu _0\left (\pp {V_x}{x}-ikV_y\right )\right ] +ik\left [\rho _0\nu _0\left (\pp {V_y}{x}+ikV_x\right )\right ] \\ &=-(\rho _0\nu _0)'ikV_y-\rho _0\nu _0k^2V_x.\end{aligned}\]

Hence

\[(\divergence \tens {\Pi }^{\mathrm {gv}})_y=-(\rho _0\nu _0)'ikV_y-\rho _0\nu _0k^2V_x. \tag{21.22}\]
If \(\nu _0\) is treated as constant and the equilibrium variation is carried entirely by the density, then \((\rho _0\nu _0)'=\eta \rho _0\nu _0\), and (21.21)–(21.22) become
\[\begin{aligned}(\divergence \tens {\Pi }^{\mathrm {gv}})_x&=-\nu _0\eta \rho _0 ikV_x+\rho _0\nu _0k^2V_y, \\ (\divergence \tens {\Pi }^{\mathrm {gv}})_y&=-\nu _0\eta \rho _0 ikV_y-\rho _0\nu _0k^2V_x.\end{aligned} \tag{21.23}\]

Note

Important point. The equilibrium-gradient piece \((\rho _0\nu _0)'\propto \eta \rho _0\nu _0\) is not an optional detail. It is exactly this piece that becomes the FLR stabilization term in the final quadratic. Dropping it would remove the effect one is trying to calculate.

21.5 Linearized extended-MHD system

The benchmark is obtained by linearizing three equations: continuity (1.7), one-fluid force balance (21.7), and the non-solenoidal closure (21.15).

From continuity, \[ \pp {\rho _1}{t}+\uvec _1\cdot \grad \rho _0+\rho _0\divergence \uvec _1=0. \] Using \(\grad \rho _0=\eta \rho _0\,\vect {e}_x\), \(\uvec _1=V_x\,\vect {e}_x+V_y\,\vect {e}_y\), and the local approximation \(\divergence \uvec _1\simeq ikV_y\), one finds

\[i\omega \rho _1+\eta \rho _0V_x+ik\rho _0V_y=0. \tag{21.25}\]

The \(x\)-component of the momentum equation is \[ \rho _0 i\omega V_x=-\pp {p_{T1}}{x}-g\rho _1-(\divergence \tens {\Pi }^{\mathrm {gv}})_x. \] In the local electrostatic treatment the \(x\)-variation of \(p_{T1}\) is neglected, so using (21.23) gives

\[\frac {g}{\rho _0}\rho _1+(i\omega -\nu _0\eta ik)V_x+\nu _0k^2V_y=0. \tag{21.26}\]
Similarly, the \(y\)-component reads \[ \rho _0 i\omega V_y=-ikp_{T1}-(\divergence \tens {\Pi }^{\mathrm {gv}})_y, \] so that
\[-\nu _0k^2V_x+(i\omega -\nu _0\eta ik)V_y+\frac {ik}{\rho _0}p_T=0. \tag{21.27}\]

Now linearize the closure (21.15). The three terms are

\[\begin{aligned}\divergence \uvec _1&\simeq ikV_y, \\ \frac {1}{\Omega _i}\vect {e}_z\cdot \curl \frac {D\uvec _1}{Dt} &=\frac {1}{\Omega _i}\left [\pp {}{x}(i\omega V_y)-\pp {}{y}(i\omega V_x)\right ] \simeq \frac {\omega k}{\Omega _i}V_x, \\ -\frac {1}{\Omega _i\rho _0^2}\vect {e}_z\cdot \Big (\grad \rho _0\times (\divergence \tens {\Pi }^{\mathrm {gv}})\Big ) &=-\frac {\eta }{\Omega _i\rho _0}(\divergence \tens {\Pi }^{\mathrm {gv}})_y \\ &=\frac {\nu _0\eta }{\Omega _i}k^2V_x+\frac {\nu _0\eta ^2}{\Omega _i}ikV_y.\end{aligned}\]

Adding them yields

\[\left (\frac {\omega k}{\Omega _i}+\frac {\nu _0\eta }{\Omega _i}k^2\right )V_x +\left (1+\frac {\eta ^2\nu _0}{\Omega _i}\right )ikV_y=0. \tag{21.28}\]
Equation (21.28) is the linearized form of the extended-MHD closure. Equations (21.25)–(21.28) are already the complete local Roberts–Taylor benchmark.

For later use, it is convenient to note the three simple limits obtained by turning terms on and off:

\[\begin{aligned}\text {ideal MHD:}&& \omega ^2+g\eta &=0, \\ \text {gyroviscosity only:}&& \omega ^2-\nu _0\eta k\,\omega +g\eta &=0, \\ \text {two-fluid only:}&& \omega ^2-\frac {gk}{\Omega _i}\,\omega +g\eta &=0.\end{aligned} \tag{21.29}\]

The full result is obtained next.

21.6 Deriving the full local dispersion relation step by step

Step 1: eliminate \(V_y\) with the closure. From (21.28), define

\[S\equiv \frac {\omega k}{\Omega _i}+\frac {\nu _0\eta }{\Omega _i}k^2, \qquad D\equiv 1+\frac {\eta ^2\nu _0}{\Omega _i}, \tag{21.32}\]
so that
\[ikV_y=-\frac {S}{D}V_x, \qquad V_y=\frac {iS}{kD}V_x. \tag{21.33}\]

Step 2: eliminate \(\rho _1\) with continuity. Using (21.33) in (21.25),

\[\frac {\rho _1}{\rho _0}=-\frac {\eta V_x+ikV_y}{i\omega } =\frac {i}{\omega }\left (\eta -\frac {S}{D}\right )V_x. \tag{21.34}\]

Step 3: insert into the \(x\)-momentum equation. Substituting (21.33) and (21.34) into (21.26) gives \[ \frac {g}{\rho _0}\left [\rho _0\frac {i}{\omega }\left (\eta -\frac {S}{D}\right )V_x\right ] +(i\omega -\nu _0\eta ik)V_x +\nu _0k^2\left (\frac {iS}{kD}V_x\right )=0. \] Dividing by \(iV_x\) gives the compact intermediate form

\[g\left (\eta -\frac {S}{D}\right )+\omega (\omega -\nu _0\eta k)+\frac {\omega \nu _0 kS}{D}=0. \tag{21.35}\]
Now substitute \[ S=\frac {k}{\Omega _i}(\omega +\nu _0\eta k), \qquad D=1+\frac {\eta ^2\nu _0}{\Omega _i}. \] Multiplying (21.35) by \(D\) and grouping like powers of \(\omega \) gives
\[\begin{aligned}0={}&g(\eta D-S)+D\omega (\omega -\nu _0\eta k)+\omega \nu _0kS \\ ={}&\left [1+\frac {\nu _0}{\Omega _i}(\eta ^2+k^2)\right ]\omega ^2 \\ &-\left \{\frac {gk}{\Omega _i}+\nu _0\eta k\left [1+\frac {\nu _0}{\Omega _i}(\eta ^2-k^2)\right ]\right \}\omega \\ &+g\eta \left [1+\frac {\nu _0}{\Omega _i}(\eta ^2-k^2)\right ].\end{aligned}\]

Hence the local exact quadratic is

\[\boxed {\; \left [1+\frac {\nu _0}{\Omega _i}(\eta ^2+k^2)\right ]\omega ^2 -\left \{\frac {gk}{\Omega _i}+\nu _0\eta k\left [1+\frac {\nu _0}{\Omega _i}(\eta ^2-k^2)\right ]\right \}\omega +g\eta \left [1+\frac {\nu _0}{\Omega _i}(\eta ^2-k^2)\right ]=0 \;} \tag{21.36}\]
which is Schnack’s local exact quadratic.

Step 4: reduce to the Roberts–Taylor ordering. Roberts and Taylor then use the short-wavelength FLR ordering

\[\frac {\nu _0k^2}{\Omega _i}=\frac {(k\rho _i)^2}{2}\ll 1, \qquad \eta ^2\ll k^2. \tag{21.37}\]
Keeping the lowest nontrivial order in these small quantities reduces (21.36) to
\[\boxed {\; \omega ^2-\left (\frac {gk}{\Omega _i}+\nu _0\eta k\right )\omega +g\eta =0 \;} \tag{21.38}\]
with solution
\[2\omega =\frac {gk}{\Omega _i}+\nu _0\eta k \pm \sqrt {\left (\frac {gk}{\Omega _i}+\nu _0\eta k\right )^2-4g\eta }. \tag{21.39}\]
The instability is stabilized when the discriminant becomes nonnegative, i.e.
\[k^2>k_{\text {EMHD}}^2 =\frac {4g\eta }{\left (\dfrac {g}{\Omega _i}+\nu _0\eta \right )^2}. \tag{21.40}\]

Model Quadratic Stability threshold
ideal MHD \(\omega ^2+g\eta =0\) none
2-fluid only \(\omega ^2-\dfrac {gk}{\Omega _i}\omega +g\eta =0\) \(k^2>4\eta \Omega _i^2/g\)
FLR only \(\omega ^2-\nu _0\eta k\,\omega +g\eta =0\) \(k^2>4g/(\nu _0^2\eta )\)
full ext-MHD (21.38) (21.40)

21.7 Writing the FLR term in \(k_\perp \rho _i\) form

Because

\[\nu _0=\frac {\rho _i^2\Omega _i}{2}, \tag{21.41}\]
the FLR coefficient in (21.38) is
\[\nu _0\eta k_\perp =\frac {\rho _i^2\Omega _i}{2}\,\eta k_\perp =\frac {\Omega _i}{2}\frac {1}{k_\perp L_n}(k_\perp \rho _i)^2. \tag{21.42}\]
Thus the ordered quadratic can be written directly as
\[\boxed {\; \omega ^2- \left [ \frac {gk_\perp }{\Omega _i} +\frac {\Omega _i}{2}\frac {1}{k_\perp L_n}(k_\perp \rho _i)^2 \right ]\omega +\frac {g}{L_n}=0 \;} \tag{21.43}\]
for this slab geometry, where \(k_\perp =k_y\).

A convenient frequency notation is

\[\omega _g^2\equiv \frac {g}{L_n}, \qquad \omega _H\equiv \frac {gk_\perp }{\Omega _i}, \qquad \omega _\pi \equiv \nu _0\eta k_\perp , \tag{21.44}\]
so that the stability condition becomes
\[(\omega _H+\omega _\pi )^2>4\omega _g^2. \tag{21.45}\]

Note

Where exactly does the FLR correction live? In this fluid benchmark it appears as the linear-in-\(\omega \) coefficient \(\omega _\pi =\nu _0\eta k_\perp \), not as a direct additive correction to the gravitational drive \(g/L_n\). The stabilizing term comes from the equilibrium-gradient part of the gyroviscous force and then feeds into the non-solenoidal closure.

It is also useful to note the Hall–FLR cross-term estimate

\[(\nu _0\eta k_\perp )\left (\frac {gk_\perp }{\Omega _i}\right ) =\frac {g}{L_n}\frac {(k_\perp \rho _i)^2}{2}, \tag{21.46}\]
which is smaller than the gravitational drive by the ordered quantity \((k_\perp \rho _i)^2/2\ll 1\). This is why the Roberts–Taylor reduction keeps Hall and FLR terms in the coefficient of \(\omega \) while discarding higher-order cross-couplings elsewhere.

21.8 Separate ion/electron derivation and physical interpretation

One of Schnack’s nicest additions is to return to separate ion and electron equations. Ignoring electron inertia, the equilibrium drifts are

\[V_{xi0}=0, \qquad V_{yi0}=\frac {g}{\Omega _i}+\frac {C_{Si}^2}{\Omega _iL_n}-\frac {E_{x0}}{B}, \qquad V_{xe0}=0, \qquad V_{ye0}=-\frac {E_{x0}}{B}. \tag{21.47}\]
This is already a useful bookkeeping result:

Thus the separate-fluid derivation makes the diamagnetic piece explicit even before linearization.

After linearization, quasi-neutrality, and the same low-\(\beta \) ordering used above, the perturbed electron \(E\times B\) drift satisfies

\[V_E\equiv \frac {E_y}{B} =\frac {W+W_{e0}}{W+W_{i0}}\left (v_{xi}+i\alpha v_{yi}\right ), \tag{21.48}\]
where \(W=\omega /\Omega _i\), \(\alpha =kL_n\), and \(W_{\alpha 0}=kV_{y\alpha 0}/\Omega _i\). Eliminating the remaining variables gives
\[\alpha (W+W_{i0})(W+W_{e0})+W_{i0}-W_{e0}=0. \tag{21.49}\]
In dimensional form this becomes
\[\omega ^2 +k\left (\frac {g}{\Omega _i}-2\frac {E_{x0}}{B}\right )\omega -k^2\frac {E_{x0}}{B}\left (\frac {g}{\Omega _i}-\frac {E_{x0}}{B}\right ) +\frac {g}{L_n}=0. \tag{21.50}\]
Two standard frame choices are then
\[\begin{aligned}E_{x0}=0 \quad \text {(stationary electrons)} &\Longrightarrow \omega ^2+\frac {gk}{\Omega _i}\omega +\frac {g}{L_n}=0, \\ E_{x0}=\frac {gB}{\Omega _i} \quad \text {(stationary ions)} &\Longrightarrow \omega ^2-\frac {gk}{\Omega _i}\omega +\frac {g}{L_n}=0.\end{aligned} \tag{21.51}\]

The sign of the linear \(\omega \)-coefficient changes because it is a frame effect. The stability threshold does not.

Note

Physical picture. In ideal MHD, electrostatic perturbations together with (4.8) force \(\divergence \uvec =0\), so ions and electrons share a common \(E\times B\) motion. In the two-fluid problem the ion and electron drifts differ. Quasi-neutrality then requires just enough compression or expansion of the ion fluid to prevent charge separation. That non-solenoidal ion motion is what the closure equation captures. Gyroviscosity adds a second ion drift channel, and that extra FLR drift shifts the linear \(\omega \)-coefficient in the stabilizing direction.

Two cautions are worth recording.

Equilibrium force balance matters. The benchmark is not just about linearization. The equilibrium must be thermodynamically and mechanically consistent with (21.2); otherwise the coefficient multiplying the FLR term is altered and one can manufacture spurious cancellations.

Higher-order ion heat-stress corrections are subleading here. Schnack shows that collisionless ion heat-stress corrections modify the result only at higher order in the same short-wavelength expansion. The leading Roberts–Taylor stabilization mechanism is already isolated by the Hall term plus gyroviscosity.

21.9 Comparison with the original Roberts–Taylor PRL

Roberts and Taylor write the gyroviscous coefficient as \[ \nu _{\mathrm {RT}}=\frac {a^2\Omega _i}{4}, \qquad \nu _0=2\nu _{\mathrm {RT}}, \] and their one-fluid quadratic is

\[F_0(\omega )=\omega ^2+\left (2\nu _{\mathrm {RT}}\eta k+\frac {gk}{\Omega _i}\right )\omega +g\eta =0. \tag{21.53}\]
This differs from (21.38) only by sign conventions for \(g\) and the equilibrium density gradient, together with the factor-of-two notation change from \(\nu _{\mathrm {RT}}\) to \(\nu _0\).

21.10 Experimental and modern perspective

Roberts–Taylor is a slab calculation, but the lesson survives far beyond the slab. Whenever ideal MHD predicts that interchange or bad-curvature modes grow faster and faster with \(k_\perp \), one should immediately ask what kinetic or extended-fluid physics cuts off that trend. In many magnetized plasma experiments the answer is some combination of ion gyroradius physics, diamagnetic drifts, parallel dynamics, and finite collisionality. This lecture isolates the cleanest possible fluid version of that logic.

That is why this benchmark keeps showing up in lecture notes and reduced models. It is not because laboratory plasmas are literally the Roberts–Taylor slab, but because the calculation teaches where the FLR correction lives, how it enters a closure, and why the shortest wavelengths are often the first place ideal MHD fails.

Takeaways
  • The Roberts–Taylor problem is the cleanest analytic example of finite-Larmor-radius stabilization in an extended-MHD fluid model.
  • The Hall term and the gyroviscous term both modify the coefficient of \(\omega \) in the dispersion relation; the gravitational drive remains \(g/L_n\).
  • The equilibrium-gradient piece of \(\rho _0\nu _0\) is essential. Dropping it drops the FLR stabilization effect.
  • The non-solenoidal closure (21.15) is the extended-MHD replacement for the ideal-MHD condition \(\divergence \uvec =0\).

Bibliography

    M.N. Rosenbluth, N.A. Krall, and N. Rostoker. Finite larmor radius stabilization of "weakly" unstable confined plasmas. Nuclear Fusion: 1962 Supplement, Part 1, page 143, 1962.

    Marshall N Rosenbluth and Albert Simon. Finite larmor radius equations with nonuniform electric fields and velocities. Physics of Fluids, 8(7):1300, 1965. doi:10.1063/1.1761402.

    K. V. Roberts and J. B. Taylor. Magnetohydrodynamic equations for finite larmor radius. Physical Review Letters, 8(5):197–198, 1962. doi:10.1103/physrevlett.8.197.

    Dalton D. Schnack. Gravitational instability as a test case for extended mhd computations. Technical Report UW-CPTC 12-3, Center for Plasma Theory and Computation, University of Wisconsin–Madison, Madison, Wisconsin, September 2005. URL https://cptc.wisc.edu/wp-content/uploads/sites/327/2017/09/UW-CPTC_12-3.pdf.

Problems

Problem 21.1. Gyroviscous Algebra in the Roberts–Taylor Slab

Starting from the gyroviscous tensor (21.19):

(a)
Using the mode ansatz (21.3), derive (21.21) and (21.22) step by step. Be explicit about which \(x\)-derivatives are neglected and which equilibrium-gradient terms are retained.
(b)
Show that if one incorrectly sets \((\rho _0\nu _0)'=0\), the FLR stabilization term in (21.38) disappears.
(c)
Explain why the gyroviscous force is nondissipative even though it is written as a divergence of a stress tensor.

Problem 21.2. From the Exact Local Quadratic to the Roberts–Taylor Limit

Starting from the exact local dispersion relation (21.36):

(a)
Apply the ordering (21.37) and recover the reduced quadratic (21.38).
(b)
Derive the stabilization threshold (21.40) directly from the discriminant.
(c)
Rewrite the FLR term in \(k_\perp \rho _i\) form and recover (21.42).
(d)
Compare the ideal-MHD, Hall-only, and FLR-only limits (21.29)–(21.31). Which term is responsible for the short-wavelength cutoff in each case?