Lecture 18 Gravitational Interchange

← Previous Lecture Contents Next Lecture →

Lecture 18
Gravitational Interchange

Overview

This lecture is the gravitational prototype for several later stability problems. The same logic appears in three increasingly rich settings:
1.
Ordinary buoyancy: a stratified gas in gravity, leading to the Schwarzschild criterion and the Brunt–Väisälä frequency.
2.
Magnetic interchange: a stratified plasma supported in part by magnetic pressure, with $k_\parallel =0$ so that field lines are exchanged but not bent.
3.
Parker or undular instability: long-wavelength motion along the magnetic field, which allows plasma to drain from the crests of rising flux tubes and turns interchange into the prototype of ballooning.

The point is not merely astrophysical. These are the cleanest settings in which to learn how equilibrium profiles, compressibility, and field-line tension conspire to decide stability.

Historical Perspective

The intellectual line behind this lecture is beautifully continuous. Rayleigh’s work on convection and Schwarzschild’s criterion for stellar atmospheres gave the first modern stability criteria for a stratified fluid Rayleigh (1916); Schwarzschild (1906). Brunt and Väisälä identified the buoyancy oscillation frequency of a stably stratified atmosphere Väisälä (1925); Brunt (1927). Kruskal and Schwarzschild then translated the same physical idea into magnetized plasmas, showing how a magnetic field can support plasma against gravity and how that support can fail through interchange Kruskal and Schwarzschild (1954). Newcomb recast the problem in the language of the ideal-MHD energy principle Newcomb (1961). Parker’s later work on magnetic buoyancy in the solar interior made clear that one does not need literal “heavy fluid on light fluid” in the textbook sense; a magnetized atmosphere can become unstable because magnetic support changes the vertical stratification and because plasma can drain along field lines Parker (1955a,b). That is why this lecture naturally points forward to magnetic interchange, ballooning, and solar-flux-emergence problems.

18.1 Buoyancy Without Magnetic Field

We begin with a static atmosphere in a uniform gravitational field $\vect {g}=-g\,\ez $. Adding gravity to the momentum equation in Eq. (1.8) and setting $\uvec _0=\vect {0}$, $\B _0=\vect {0}$, the equilibrium condition is the usual hydrostatic balance

\frac {dp_0}{dz}=-\rho _0 g. \tag{18.1}

Parcel derivation of the Brunt–Väisälä frequency. Displace a fluid parcel upward by a small distance $\xi _z$. Pressure adjusts rapidly enough that the parcel stays in pressure balance with its surroundings, but the displacement is fast enough thermodynamically that the parcel evolves adiabatically. Thus

\left (\frac {p}{\rho ^\gamma }\right )_{\rm parcel}=\text {const}, \qquad p_{\rm parcel}(z+\xi _z)=p_0(z+\xi _z).

Expand the adiabatic relation to first order:

\begin{aligned}\frac {p_0(z+\xi _z)}{\rho _{\rm parcel}^\gamma (z+\xi _z)} &= \frac {p_0(z)}{\rho _0^\gamma (z)}, \nonumber \\ \frac {p_0+\xi _z p_0'}{\bigl (\rho _0+\rho _{\rm parcel,1}\bigr )^\gamma } &= \frac {p_0}{\rho _0^\gamma }, \nonumber \\ \left (1+\xi _z\frac {p_0'}{p_0}\right ) \left (1-\gamma \frac {\rho _{\rm parcel,1}}{\rho _0}\right ) &=1,\end{aligned}

so that

\rho _{\rm parcel,1} = \frac {\rho _0}{\gamma p_0}\,p_0'\,\xi _z. \tag{18.4}

The ambient density at the displaced position is instead

\rho _0(z+\xi _z)=\rho _0+\rho _0'\xi _z.

Hence the density contrast between parcel and environment is

\Delta \rho \equiv \rho _{\rm parcel}(z+\xi _z)-\rho _0(z+\xi _z) = \left (\frac {\rho _0}{\gamma p_0}p_0'-\rho _0'\right )\xi _z. \tag{18.6}

The buoyancy force per unit mass is $-g\Delta \rho /\rho _0$, so the parcel obeys

\ddot {\xi }_z = -g\left (\frac {1}{\gamma p_0}\frac {dp_0}{dz}-\frac {1}{\rho _0}\frac {d\rho _0}{dz}\right )\xi _z \equiv -N^2\xi _z. \tag{18.7}

Therefore

\boxed { N^2 = \frac {g}{\gamma }\frac {d}{dz}\ln \!\left (\frac {p_0}{\rho _0^\gamma }\right ) = g\left (\frac {1}{\gamma p_0}\frac {dp_0}{dz}-\frac {1}{\rho _0}\frac {d\rho _0}{dz}\right ). } \tag{18.8}

If $N^2>0$, the atmosphere supports stable buoyancy oscillations. If $N^2<0$, the displacement grows exponentially and the atmosphere overturns.

Tutorial

Entropy from the Sackur–Tetrode equation and why $p/\rho ^\gamma $ is an entropy proxy.
For a classical monatomic ideal gas, the Sackur–Tetrode equation gives the entropy as
$S = N k_B \left [ \ln \!\left ( \frac {V}{N} \left ( \frac {4\pi m U}{3 N h^2} \right )^{3/2} \right ) +\frac {5}{2} \right ],$ where $N$ is the particle number, $V$ the volume, $U$ the internal energy, and $m$ the particle mass.
For a monatomic ideal gas,
$U = \frac {3}{2} N k_B T, \qquad pV = N k_B T.$ Substituting $U=\tfrac 32 N k_B T$ into Sackur–Tetrode gives $\frac {S}{N k_B} = \ln \!\left (\frac {V}{N}\right ) +\frac {3}{2}\ln T +\text {const}.$ Now introduce the number density $n=N/V$, so that $V/N=1/n$. Then $\frac {S}{N k_B} = -\ln n + \frac {3}{2}\ln T + \text {const}.$ Using the ideal-gas law $p=n k_B T$, we may write $T = \frac {p}{n k_B},$ and therefore $\begin{aligned}\frac {S}{N k_B} &= -\ln n + \frac {3}{2}\ln \!\left (\frac {p}{n k_B}\right ) + \text {const} \nonumber \\ &= -\ln n + \frac {3}{2}\ln p - \frac {3}{2}\ln n + \text {const} \nonumber \\ &= \frac {3}{2}\ln p - \frac {5}{2}\ln n + \text {const}.\end{aligned}$
Since a monatomic gas has
$\gamma = \frac {5}{3}, \qquad \frac {1}{\gamma -1}=\frac {3}{2},$ this becomes $\frac {S}{N k_B} = \frac {1}{\gamma -1} \ln \!\left (\frac {p}{n^\gamma }\right ) +\text {const}.$ Finally, for a fixed species $n=\rho /m$, so $\frac {S}{N k_B} = \frac {1}{\gamma -1} \ln \!\left (\frac {p}{\rho ^\gamma }\right ) +\text {const},$ where the factor $m^\gamma $ has been absorbed into the additive constant.
Thus the entropy per particle is proportional to
$\boxed { \frac {S}{N} = \frac {k_B}{\gamma -1} \ln \!\left (\frac {p}{\rho ^\gamma }\right ) +\text {const} }$ and the specific entropy per unit mass is $\boxed { s = \frac {k_B}{m(\gamma -1)} \ln \!\left (\frac {p}{\rho ^\gamma }\right ) +\text {const}. }$
So $p/\rho ^\gamma $ is not literally the entropy; rather, it is a monotonic proxy for entropy:
$K \equiv \frac {p}{\rho ^\gamma }, \qquad s \propto \ln K.$ Because the logarithm is monotonic, comparing $K$ is equivalent to comparing entropy.
This is why buoyancy and interchange criteria are often written in terms of the gradient of $p/\rho ^\gamma $. For adiabatic motion of an ideal-gas fluid element,
$\frac {d}{dt}\left (\frac {p}{\rho ^\gamma }\right )=0 \qquad \Longleftrightarrow \qquad \frac {ds}{dt}=0.$ In other words, $p/\rho ^\gamma $ is the conserved adiabatic label carried by the fluid element.

Thus the Schwarzschild criterion can be viewed as a statement about the ordering of the entropy proxy $K=p/\rho ^\gamma $: a stratification is buoyantly stable when entropy increases upward.

Adiabatic lapse rate. For an ideal gas, $p_0\propto \rho _0 T_0$, so Eq. (18.8) can be rewritten as

\begin{aligned}N^2 &= \frac {g}{\gamma }\frac {d}{dz}\ln \!\left (p_0^{1-\gamma }T_0^\gamma \right ) \nonumber \\ &= \frac {g}{T_0}\left (\frac {dT_0}{dz}-\left .\frac {dT}{dz}\right |_{\rm ad}\right ),\end{aligned}

where

\left .\frac {dT}{dz}\right |_{\rm ad} = -(\gamma -1)\frac {g}{c_s^2}T_0, \qquad c_s^2=\frac {\gamma p_0}{\rho _0}. \tag{18.23}

Thus the atmosphere is convectively unstable when the temperature falls with height faster than the adiabatic lapse rate. For dry air near room temperature this gives the familiar value of roughly $10\,^{\circ }{\rm C}/{\rm km}$.

The same result from the energy principle. To connect directly to later lectures, it is worth deriving the Schwarzschild criterion from the ideal-MHD energy principle with $\B _0=\vect {0}$. For adiabatic perturbations in a static stratified atmosphere,

2\delta W = \int dV\, \left [ \gamma p_0(\divergence \vect {\xi })^2 +(\vect {\xi }\cdot \grad p_0)(\divergence \vect {\xi }) +(\vect {\xi }\cdot \vect {g})\bigl (\vect {\xi }\cdot \grad \rho _0+\rho _0\divergence \vect {\xi }\bigr ) \right ]. \tag{18.24}

For plane-parallel stratification,

\vect {\xi }\cdot \grad p_0=\xi _z\frac {dp_0}{dz}, \qquad \vect {\xi }\cdot \grad \rho _0=\xi _z\frac {d\rho _0}{dz}, \qquad \vect {\xi }\cdot \vect {g}=-g\xi _z.

Using hydrostatic balance, Eq. (18.1), this becomes

\begin{aligned}2\delta W &= \int dV\, \left [ \gamma p_0(\divergence \vect {\xi })^2 -2\rho _0 g\,\xi _z\divergence \vect {\xi } -g\frac {d\rho _0}{dz}\,\xi _z^2 \right ].\end{aligned}

Now complete the square:

\begin{aligned}2\delta W &= \int dV\, \Biggl [ \gamma p_0\left (\divergence \vect {\xi }-\frac {\rho _0 g}{\gamma p_0}\xi _z\right )^2 +\left (-g\frac {d\rho _0}{dz}-\frac {\rho _0^2g^2}{\gamma p_0}\right )\xi _z^2 \Biggr ] \nonumber \\ &= \int dV\, \Biggl [ \gamma p_0\left (\divergence \vect {\xi }-\frac {\rho _0 g}{\gamma p_0}\xi _z\right )^2 +\rho _0 N^2\xi _z^2 \Biggr ].\end{aligned} \tag{18.27}

The first term is positive definite and will always be stabilizing. Nature will find a way around this by having

\divergence \vect {\xi }=\frac {\rho _0 g}{\gamma p_0}\xi _z

to minimize the first term. IN this case, the sign of $\delta W$ is controlled by $N^2$. This is the energy-principle version of the Schwarzschild criterion.

18.2 Plasma supported by magnetic field against gravity

We can start now to consider how magnetic fields change this story by considering cold fluid that is supported from underneath by a magnetic field. Imagine a magnetic field which is decreasing vertically with a density that that is rising vertically.

Assume:

No gas pressure: $p_0 = 0$
Straight magnetic field $\vect {B}_0 = B_0(z)\vect {e}_x$
Gravity $\vect {g} = -g \vect {e}_z$

The equilibrium condition becomes

\frac {d}{dz}\frac {B_0^2(z) }{2\mu _0} = -\rho _0(z) g

The energy variation reduces to

\begin{aligned}2\delta W & = \int dV \left [ \frac {|\vect {Q}|^2}{\mu _0} +\vect {\xi }\cdot \vect {g}\,\nabla \cdot (\rho _0\vect {\xi }) \right ].\end{aligned}

Choose interchange-type perturbations:

\vect {k}\perp \vect {B}_0, \qquad k_\parallel = 0,

so that field lines are exchanged but not bent.

In this case

\vect {Q}_\perp = i k_\parallel B \vect {\xi }_\perp = 0;

the perturbed magnetic field is entirely compressional and parallel to $\vect {B}_0$, and can be minimized by careful choice:

\begin{aligned}\vect {Q} & = \nabla \times \bm {\xi } \times \bm {B}_0\\ & = -\bm {B} \nabla \cdot \bm {\xi }_\perp -\bm {\xi }_\perp \cdot \nabla \bm {B} \\ \left | \bm {Q}^2 \right | & = B_0^2 \left (\nabla \cdot \bm {\xi }_\perp + \bm \xi _\perp \cdot \frac {\nabla B_0}{B_0}\right )^2 \\ \mbox {Choose } & \Rightarrow \quad \nabla \cdot \vect {\xi }_\perp = - \vect {\xi }_\perp \cdot \nabla \ln B_0.\end{aligned}

to mimimize $\delta W$.

Substituting into $\delta W$ gives

\begin{aligned}2\delta W &= \int dV \, \cancel {\left |\vect {Q}^2\right |} + \vect {\xi }\cdot \vect {g} \, \left ( \vect {\xi }_\perp \cdot \nabla \rho _0 + \rho _0 \underbrace {\nabla \cdot \vect {\xi }_\perp }_{ - \vect {\xi }_\perp \cdot \nabla \ln B_0} \right ) \nonumber \\ &= -\int dV \, \rho _0 g \, \xi _z \left ( \vect {\xi }_\perp \cdot \nabla \ln \frac {\rho _0}{B} \right ) \nonumber \\ &= -\int dV \, \rho _0 g \, \frac {d}{dz} \left (\ln \frac {\rho _0}{B_0} \right ) \xi _z^2\end{aligned} \tag{18.37}

\[ \boxed { \text {Magnetic interchange instability if } \frac {d}{dz}\left (\frac {\rho _0}{B_0}\right ) > 0 } \]

Physically: heavy flux tubes above light ones are unstable Kruskal and Schwarzschild (1954).

18.3 Magnetic Buoyancy and Flute-Like Interchange

Now let a horizontal magnetic field help support the atmosphere,

\vect {B}_0=B_0(z)\,\vect {e}_y, \qquad \vect {g}=-g\,\ez ,

where $\vect {e}_y$ is the horizontal field direction. The static equilibrium condition is

\frac {d}{dz}\left (p_0+\frac {B_0^2}{2\mu _0}\right )=-\rho _0 g. \tag{18.39}

We consider flute or interchange perturbations with

k_\parallel =0, \qquad \vect {\xi }=\xi _x\,\vect {e}_x+\xi _z\,\ez , \tag{18.40}

so field lines are exchanged but not bent.

Figure 18.1: Kruskal–Schwarzschild gravitational interchange: plasma supported against gravity by magnetic pressure. The basic question is whether exchanging neighboring flux tubes lowers or raises the potential energy Kruskal and Schwarzschild (1954).

Field perturbation for interchange modes. Recall the linear perturbation of the magnetic field from Eq. (15.4),

\vect {Q} \equiv \B _1 = \curl (\vect {\xi }\times \B _0) = \cancel {(\B \cdot \grad )\vect {\xi }_\perp } - (\vect {\xi }_\perp \cdot \grad )\B - \B \,\divergence \vect {\xi }_\perp

reduces for $k_\parallel =0$ to a purely parallel perturbation,

\vect {Q} = -B_0\left (\divergence \vect {\xi }+\xi _z\frac {d\ln B_0}{dz}\right )\vect {e}_y. \tag{18.42}

such that the field lines remain straight. The equilibrium current is

\vect {J}_0 = \frac {\curl \vect {B}_0}{\mu _0} = -\frac {1}{\mu _0}\frac {dB_0}{dz}\,\vect {e}_x. \tag{18.43}

Energy principle for flute modes. The ideal-MHD potential energy including gravity is

2\delta W = \int dV\, \left [ \frac {|\vect {Q}|^2}{\mu _0} +\gamma p_0(\divergence \vect {\xi })^2 +(\vect {\xi }\cdot \grad p_0)(\divergence \vect {\xi }) -\vect {\xi }\cdot (\vect {J}_0\times \vect {Q}) +(\vect {\xi }\cdot \vect {g})\bigl (\vect {\xi }\cdot \grad \rho _0+\rho _0\divergence \vect {\xi }\bigr ) \right ]. \tag{18.44}

Substitute Eqs. (18.42) and (18.43). First,

\begin{aligned}\frac {|\vect {Q}|^2}{\mu _0} &= \frac {B_0^2}{\mu _0} \left (\divergence \vect {\xi }+\xi _z\frac {d\ln B_0}{dz}\right )^2, \nonumber \\ -\vect {\xi }\cdot (\vect {J}_0\times \vect {Q}) &= -\xi _z\frac {d}{dz}\left (\frac {B_0^2}{2\mu _0}\right ) \left (\divergence \vect {\xi }+\xi _z\frac {d\ln B_0}{dz}\right ).\end{aligned}

Therefore

\begin{aligned}2\delta W &= \int dV\, \Biggl [ \frac {B_0^2}{\mu _0} \left (\divergence \vect {\xi }+\xi _z\frac {d\ln B_0}{dz}\right )^2 +\gamma p_0(\divergence \vect {\xi })^2 +\xi _z\frac {dp_0}{dz}\,\divergence \vect {\xi } \nonumber \\ &\hspace {7em} -\xi _z\frac {d}{dz}\left (\frac {B_0^2}{2\mu _0}\right ) \left (\divergence \vect {\xi }+\xi _z\frac {d\ln B_0}{dz}\right ) -g\xi _z\left (\xi _z\frac {d\rho _0}{dz}+\rho _0\divergence \vect {\xi }\right ) \Biggr ].\end{aligned} \tag{18.46}

Now expand the magnetic square:

\begin{aligned}\frac {B_0^2}{\mu _0} \left (\divergence \vect {\xi }+\xi _z\frac {d\ln B_0}{dz}\right )^2 &= \frac {B_0^2}{\mu _0}(\divergence \vect {\xi })^2 +2\frac {B_0}{\mu _0}\frac {dB_0}{dz}\,\xi _z\divergence \vect {\xi } +\frac {1}{\mu _0}\left (\frac {dB_0}{dz}\right )^2\xi _z^2.\end{aligned}

The current term subtracts

\frac {B_0}{\mu _0}\frac {dB_0}{dz}\,\xi _z\divergence \vect {\xi } + \frac {1}{\mu _0}\left (\frac {dB_0}{dz}\right )^2\xi _z^2,

so one copy of the mixed term remains and the $\xi _z^2$ magnetic term cancels entirely. Hence

\begin{aligned}2\delta W &= \int dV\, \Biggl [ \left (\gamma p_0+\frac {B_0^2}{\mu _0}\right )(\divergence \vect {\xi })^2 +\left (\frac {dp_0}{dz}+\frac {B_0}{\mu _0}\frac {dB_0}{dz}-\rho _0 g\right )\xi _z\divergence \vect {\xi } -g\frac {d\rho _0}{dz}\,\xi _z^2 \Biggr ].\end{aligned}

Using equilibrium, Eq. (18.39), the mixed term becomes $-2\rho _0 g\,\xi _z\divergence \vect {\xi }$, and therefore

2\delta W = \int dV\, \Biggl [ \left (\gamma p_0+\frac {B_0^2}{\mu _0}\right )(\divergence \vect {\xi })^2 -2\rho _0 g\,\xi _z\divergence \vect {\xi } -g\frac {d\rho _0}{dz}\,\xi _z^2 \Biggr ]. \tag{18.50}

Define

c_s^2=\frac {\gamma p_0}{\rho _0}, \qquad v_A^2=\frac {B_0^2}{\mu _0\rho _0}.

Then Eq. (18.50) becomes

\begin{aligned}2\delta W &= \int dV\, \Biggl [ \rho _0(c_s^2+v_A^2)(\divergence \vect {\xi })^2 -2\rho _0 g\,\xi _z\divergence \vect {\xi } -g\frac {d\rho _0}{dz}\,\xi _z^2 \Biggr ].\end{aligned}

Complete the square once more:

\begin{aligned}2\delta W &= \int dV\, \Biggl [ \rho _0(c_s^2+v_A^2) \left (\divergence \vect {\xi }-\frac {g}{c_s^2+v_A^2}\xi _z\right )^2 +\left (-g\frac {d\rho _0}{dz}-\frac {\rho _0 g^2}{c_s^2+v_A^2}\right )\xi _z^2 \Biggr ].\end{aligned} \tag{18.53}

The coefficient of $\xi _z^2$ can be rewritten as

\begin{aligned}-g\frac {d\rho _0}{dz}-\frac {\rho _0 g^2}{c_s^2+v_A^2} &= \frac {g}{c_s^2+v_A^2} \left [ -(c_s^2+v_A^2)\frac {d\rho _0}{dz} +\frac {dp_0}{dz} +\frac {B_0}{\mu _0}\frac {dB_0}{dz} \right ] \nonumber \\ &= \frac {g\rho _0}{c_s^2+v_A^2} \left [ v_A^2\frac {d}{dz}\ln \!\left (\frac {B_0}{\rho _0}\right ) +\frac {c_s^2}{g}N^2 \right ].\end{aligned}

Thus

2\delta W = \int dV\, \rho _0 \left [ (c_s^2+v_A^2) \left (\divergence \vect {\xi }-\frac {g}{c_s^2+v_A^2}\xi _z\right )^2 +N_M^2\xi _z^2 \right ], \tag{18.55}

Now

\mbox {Choose } \qquad \divergence \vect {\xi }-\frac {g}{c_s^2+v_A^2}\xi _z \Rightarrow \quad \nabla \cdot \vect {\xi }_\perp = - \vect {\xi }_\perp \cdot \nabla \ln B_0.

to minimize compression so that

\boxed { N_M^2 \equiv \frac {g}{c_s^2+v_A^2} \left [ v_A^2\frac {d}{dz}\ln \!\left (\frac {B_0}{\rho _0}\right ) +\frac {c_s^2}{g}N^2 \right ]. } \tag{18.57}

This is the magnetic analogue of the Brunt–Väisälä frequency for flute modes. Stability requires $N_M^2>0$.

Cold-plasma limit. If $p_0\to 0$, then $c_s^2\to 0$ and Eq. (18.57) reduces to

N_M^2 \longrightarrow g\frac {d}{dz}\ln \!\left (\frac {B_0}{\rho _0}\right ).

Therefore a cold plasma supported by magnetic pressure is unstable when

\boxed { \frac {d}{dz}\left (\frac {\rho _0}{B_0}\right )>0. } \tag{18.59}

This is the Kruskal–Schwarzschild interchange criterion: the unstable ordering is the magnetic analogue of putting heavy fluid on top of light fluid. Here the relevant quantity is not density alone but mass per unit flux.

Finite-$\beta $ interpretation from a thin flux tube. Equation (18.57) can be understood directly from flux freezing. Along a moving element, the induction and continuity equations imply the familiar frozen-in relation from Eq. (4.7),

\frac {B}{\rho }=\text {const along a fluid element}. \tag{18.60}

Hence

\frac {B_1}{B_0}=\frac {\rho _1}{\rho _0}. \tag{18.61}

From this, one can derive that

\begin{aligned}\delta \rho & = \rho _0 \frac {\delta B^2/2 }{B_0^2} \\ & \frac {1}{v_A^2} \delta (B^2/2 \mu _0)\end{aligned} \tag{18.63}

Magnetic pressure as a $\gamma =2$ fluid. Equation (18.60) has an important thermodynamic reading. For these flute/interchange motions the field is simply carried with the mass, so along a moving element one has $B\propto \rho $. Therefore the magnetic pressure

p_B\equiv \frac {B^2}{2\mu _0} \propto \rho ^2, \qquad \Longrightarrow \qquad \frac {p_B}{\rho ^2}=\text {const along a fluid element}. \tag{18.64}

In that restricted sense the magnetic field behaves like a compressive medium with an effective adiabatic index $\gamma _B=2$. This is stiffer than a monatomic gas, for which $p\propto \rho ^{5/3}$: under compression the magnetic pressure rises faster, and under expansion it falls faster. The corresponding incremental stiffness is

\left .\frac {dp_B}{d\rho }\right |_{\text {frozen-flux}} = \frac {B^2}{\mu _0\rho } = v_A^2,

which is exactly why Eq. (18.63) contains the combination $c_s^2+v_A^2$. In other words, for these $k_\parallel =0$ buoyancy motions the field contributes to the restoring force like an extra pressure law with $\gamma =2$. One should not push the analogy too far: the field is not a true scalar gas because in general it also carries anisotropic tension. But for flute modes, where field lines are exchanged without being bent, this $\gamma =2$ picture captures the essential magnetic compressibility. In that sense, the quantity $p_B/\rho ^2$ plays much the same bookkeeping role for the field that $p/\rho ^\gamma $ plays for an adiabatic gas parcel.

Physical picture. For a tube displaced upward by $\xi _z$, pressure balance with the surrounding atmosphere at the new height gives

\begin{aligned}\delta \left ( p + \frac {B^2}{2 \mu _0} \right )_{\rm parcell} & = \xi _z\frac {d}{dz}\left (p_0+\frac {B_0^2}{2\mu _0}\right ) \nonumber \\ \left ( \pp {p}{\rho } + \pp {p_B} {\rho } \right ) \rho _1 & = -\rho _0 g\,\xi _z \nonumber \\ (c_s^2 + v_A^2) \rho _1 & = -\rho _0 g\,\xi _z \nonumber\end{aligned}

The surrounding atmosphere at the new height has density $\rho _0+\xi _z\rho _0'$, so the density contrast is

\Delta \rho = \rho _1-\xi _z\frac {d\rho _0}{dz}.

The vertical equation of motion is therefore

\begin{aligned}\ddot {\xi }_z & = -g\frac {\Delta \rho }{\rho _0} \nonumber \\ & =-\frac {g}{\rho _0} \left ( -\frac {\rho _0 g}{c_s^2+v_A^2}\,\xi _z -\frac {d\rho _0}{dz} \xi _z\right ) \nonumber \\ & =-\frac {g}{\rho _0} \left ( \frac {1}{c_s^2+v_A^2} \left ( \dd {p_0}{z} + \frac {d}{dz} \frac {B^2}{2 \mu _0} \right ) -\frac {d\rho _0}{dz} \right ) \xi _z \nonumber \\ & =- \frac {g}{c_s^2+v_A^2} \left ( \frac {1}{\rho _0} \left ( \dd {p_0}{z} + \frac {d}{dz} \frac {B^2}{2 \mu _0} \right ) - \frac {c_s^2 + v_A^2}{\rho _0} \frac {d\rho _0}{dz} \right ) \xi _z \nonumber \\ & =- \frac {g}{c_s^2+v_A^2} \left ( \frac {1}{\rho _0} \frac {d}{dz} \frac {B^2}{2 \mu _0} + \frac { v_A^2}{\rho _0}\frac {d\rho _0}{dz} + c_s^2 \left ( \frac {1}{\gamma p_0} \dd {p_0}{z} - \frac {1}{\rho _0} \frac {d\rho _0}{dz} \right ) \right ) \xi _z \nonumber \\ & =- \frac {g}{c_s^2+v_A^2} \left ( \frac {1}{\rho _0} \frac {d}{dz} \frac {B^2}{2 \mu _0} - \frac { v_A^2}{\rho _0}\frac {d\rho _0}{dz} + c_s^2 \left ( \frac {1}{\gamma p_0} \dd {p_0}{z} - \frac {1}{\rho _0} \frac {d\rho _0}{dz} \right ) \right ) \xi _z \nonumber \\ & =- \frac {g}{c_s^2+v_A^2} \left ( v_A^2 \frac {d}{dz} \ln \frac {B}{\rho _0} + c_s^2 N^2 \right ) \xi _z \nonumber \\ &= -N_M^2\xi _z,\end{aligned} \tag{18.67}

and one recovers exactly Eq. (18.57). The first piece of $N_M^2$ measures how magnetic support changes with height; the second is the ordinary Schwarzschild buoyancy term.

Caution

Flute versus Parker. For $k_\parallel =0$, field lines are exchanged but not bent, and the problem is controlled by the magnetic buoyancy frequency $N_M$. The Parker problem is qualitatively different because one allows a small but finite $k_\parallel $: field lines bend, plasma drains along them, and a new competition appears between buoyancy and line tension. That is the gravitational ancestor of ballooning theory.

18.4 The Parker Instability

Magnetic buoyancy plays a central role in connecting the deep-seated solar dynamo to the magnetic structures observed at the solar surface. In dynamo theory, as already discussed in Lecture 10, differential rotation stretches poloidal field into strong toroidal field in the solar interior through the $\Omega $–effect. Once the toroidal field becomes sufficiently strong, magnetic pressure partially replaces gas pressure within the field concentration, reducing the plasma density relative to the surrounding medium.

In a gravitationally stratified atmosphere this configuration becomes buoyant. As first described by Parker, undular perturbations of a horizontal magnetic field allow plasma to drain along the field from the tops of rising arches into adjacent troughs. This drainage enhances the density deficit at the crest of the loop, driving the exponential growth of buoyant magnetic structures. These rising loops are widely believed to be the progenitors of sunspot pairs and active regions emerging through the photosphere.

Magnetic buoyancy therefore provides the physical mechanism that transports magnetic flux from the dynamo region to the solar surface. The classical theoretical framework originates in Parker’s early work on magnetic buoyancy and solar dynamos Parker (1955b, 1966, 1979), together with the general stability theory of stratified magnetized plasmas developed by Newcomb and Chandrasekhar Newcomb (1961); Chandrasekhar (1961). Modern numerical work on thin flux tubes rising through the solar convection zone has shown that buoyant loops with strengths of order $10^4$–$10^5$ G reproduce many observed properties of active regions, including emergence latitudes and systematic tilts consistent with Joy’s law Fan et al. (1993); Caligari et al. (1995). Despite this success, several key questions remain unresolved. It is still debated where the toroidal field is generated and stored (tachocline versus distributed convection-zone dynamos), how strong the magnetic field must be to survive turbulent shredding during buoyant rise, and how rotation and convection combine with magnetic tension to produce the observed properties of emerging sunspot pairs. Magnetic buoyancy thus remains a crucial nonlinear link between solar dynamo theory and the surface manifestations of solar activity.

Basic Mechanism The essence of the Parker Instability can sussed out from the energy principle arguments from above but now adding a very long wavelength undular perturbation parallel to the magnetic field. Here we follow Kulsrud (2005). The idea is that the motion of a long thin flux tube is primarily governed by the perpendicular dynamics but that the plasma can slip along the magnetic field. We can use exactly the same machinery we have just developed that essentially minimized the stabilizing contribution of the parallel magnetic field to determine $\nabla \cdot \vect {\xi }_\perp $. We have done both zero pressure and finite pressure where only transverse motion was considered. Now all we need to do is to recognize that $\nabla \cdot \vect {\xi } = \nabla \cdot \vect {\xi }_\perp + ik_\parallel \xi _\parallel $ and be careful to only include the compression where appropriate (not for the magnetic term).

Now we need to include a stabilizing term into $k_\parallel ^2 B_0^2 \xi _\perp ^2$ as well as a buoyancy contribution that scales from a non-zero divergence of the parallel motion. This term is $i (\vect {g}\cdot \vect {\xi }_\perp ) \rho _0 k_\parallel \xi _\parallel = - i k_\parallel \xi _z g \rho _0 \xi _\parallel $. This term can always be destabilizing if the phase of $k_\parallel $ is properly chosen.

Parker with low pressure. ($c_s^2=0)$ or equivalently low $\beta =\frac {c_s^2}{v_A^2}$ case is easy. Recall that the condition chosen for minimizing the magnetic compression energy in Eq. (18.37) was

\begin{aligned}\nabla \cdot \vect {\xi }_\perp & = - \vect {\xi }_\perp \cdot \nabla \ln B \mbox { or equivalently}\\ i k_\perp \xi _\perp & = -\xi _z \frac {d}{dz} \ln B_0 \mbox { that leads directly to } \\ k_\perp ^2 \xi _\perp ^2 & = \frac {g^2}{v_A^4} \xi _z^2.\end{aligned}

The two additional terms needed to allow for parallel streaming and a finite $k_\parallel $ give

\begin{aligned}2\delta W &= \int dV \, \underbrace {k_\parallel ^2 \frac {B_0^2}{\mu _0} \xi _\perp ^2}_{\rm field\, line \,bending } - \rho _0 g \, \xi _z^2 \left ( \dd {}{z} \ln \frac {\rho _0}{B_0} \right ) + \underbrace {2 i \rho _0 g\xi _z k_\parallel \xi _\parallel }_{\rm parallel \, streaming} .\end{aligned}

The factor of two comes from the $\vect {\xi }_\parallel \cdot \vect {J}_0 \times \vect {Q}_\perp = i \rho _0 g \xi _z k_\parallel \xi _\parallel $ term. One can see there is now a competition between the additional buoyancy and the increased magnetic tension. Choosing $\xi _\parallel $ to be 90 degrees out of phase with $\xi _z$, ie. $\xi _\parallel = i A \xi _z $ with $\xi _z$ chosen to be real, the energy principle becomes

\begin{aligned}2\delta W &= \int dV \, \rho _0 g \left [ \frac {k_\parallel ^2}{k_\perp ^2} \frac {g}{v_A^2} - \, \left ( \dd {}{z} \ln \frac {\rho _0}{B_0} \right ) - 2 A k_\parallel \right ] \xi _z^2\end{aligned}

and for instability

k_\parallel A = k_\parallel \frac {|\xi _\parallel |}{|\xi _z|} > \frac {k_\parallel ^2}{k_\perp ^2} \frac {g}{v_A^2} - \, \left ( \dd {}{z} \ln \frac {\rho _0}{B_0} \right )

The first term can be made small by assuming $k_\parallel ^2 \ll k_\perp ^2 \frac { v_A^2}{g}\dd {}{z} \ln \frac {\rho _0}{B_0} $ and thus the system is unstable even with $\frac {d}{dz} \ln \frac {\rho _0}{B_0} < 0 $ when it would be otherwise stable to interchange.

The Solar Tachocline The base of the solar convection zone (near the tachocline at $r \approx 0.71R_\odot $) is a strongly high–$\beta $ plasma, meaning that gas pressure greatly exceeds magnetic pressure. Typical thermodynamic conditions there are $p \sim 10^{13}$–$10^{14}\,\mathrm {Pa}$ and $\rho \sim 0.2\,\mathrm {kg\,m^{-3}}$. Even if the solar dynamo produces strong toroidal magnetic fields of order $1$–$10\,\mathrm {T}$, the plasma beta

\beta = \frac {2\mu _0 p}{B^2} = \frac {c_s^2}{v_A^2}

remains extremely large, typically \[ \beta \sim 10^{5} - 10^{7}. \]

Thus the equilibrium structure of the plasma is primarily determined by gas pressure and gravity, while magnetic fields act as a relatively small perturbation embedded in the fluid.
In the absence of convective instability the entropy would be decreasing with height; convection ensues and enforces a temperature gradient at marginality–helioseismology together with solar models (including neutrinos) confirm this picture of a convection zone more or less controlled by unmagnetized convection.
Magnetic fields play a very weak role in the equilibrium, and self-organized convection enforces $N^2 \approx 0$.
In the interface region between the turbulent convection zone and the rigid "core" of the Sun where radiation controls heat transport (called the tachocline), the toroidal magnetic field is continuously being "wound up" by differential rotation (the $\omega $ effect mentioned already in Lecture 10.

This high–$\beta $ regime is precisely the one assumed in Parker’s theory of magnetic buoyancy Parker (1955a, 1979), where a modest reduction in gas pressure inside a magnetic flux tube produces a density deficit that allows the tube to rise through the stratified convection zone. The question Parker addressed is how a thin long tube of toroidal magentic flux generated by strong differential rotation at the tachoclines might break away, rise up through the turbulent convection zone without being shredded by the turbulence and then emerge as sun spots and contribute the $\alpha $ effect needed to close the dynamo feedback loop.

High $\beta $ Undular Modes. One can quickly see how much more complicated the energy principle becomes when the motion becomes fully three dimentional as pressure terms in $\delta W$ include "total" rather than just "perpendicular" compression. Adding these terms in can be done piecemeal from where we left off, and the tricky part is knowing where and when to add magnetic buoyancy forces.

\begin{aligned}2\delta W &= \int dV \, k_\parallel ^2 \frac {B_0^2}{\mu _0} \xi _\perp ^2 + \frac {g\rho _0}{ c_s^2 + v_A^2} \left ( v_A^2 \dd {}{z} \ln \frac {B_0}{\rho _0} + \frac {c_s^2}{g} N^2 \right ) \xi _z^2 + 2 i \rho _0 g\xi _z k_\parallel \xi _\parallel \\ & + \gamma p_0 \left [ (\nabla \cdot \bm {\xi })^2 - (\nabla \cdot \bm {\xi }_\perp )^2\right ] + (\bm {\xi }_\perp \cdot \nabla p_0)(\nabla \cdot \bm {\xi }_\parallel )\end{aligned}

and we can use

\begin{aligned}(\nabla \cdot \bm {\xi })^2 - (\nabla \cdot \bm {\xi }_\perp )^2 & = i k_\parallel \xi _\parallel \nabla \cdot \vect {\xi }_\perp ^* - i k_\parallel \xi _\parallel ^* \nabla \cdot \vect {\xi }_\perp + k_\parallel ^2 \xi _\parallel ^2 \\ \nabla \cdot \vect {\xi }_\parallel & = i k_\parallel \xi _\parallel \\ \nabla \cdot \vect {\xi }_\perp = \xi _z \frac {g}{( c_s^2 + v_A^2 )} & \rightarrow \xi _\perp ^2 = \frac {g^2}{k_\perp ^2 (c_s^2 + v_A^2)^2} \xi _z^2\end{aligned}

With these pieces

\begin{aligned}2\delta W &= \int dV \, k_\parallel ^2 \frac {B_0^2}{\mu _0} \xi _\perp ^2 + \frac {g\rho _0}{ c_s^2 + v_A^2} \left ( v_A^2 \dd {}{z} \ln \frac {B_0}{\rho _0} + \frac {c_s^2}{g} N^2 \right ) \xi _z^2 \nonumber \\ + 2 i \rho _0 g\xi _z k_\parallel \xi _\parallel \nonumber \\ & + \gamma p_0 \left ( i k_\parallel (\xi _\parallel \xi _z^* -\xi _\parallel ^* \xi _z) \frac {g}{( c_s^2 + v_A^2 )} + k_\parallel ^2 \xi _\parallel ^2\right ) + \xi _z \dd {p_0}{z} i k_\parallel \xi _\parallel \nonumber \\ &= \int dV \, k_\parallel ^2 \frac {B_0^2}{\mu _0} \xi _\perp ^2 + \gamma p_0 k_\parallel ^2 \xi _\parallel ^2 + \frac {g\rho _0}{ c_s^2 + v_A^2} \left ( v_A^2 \dd {}{z} \ln \frac {B_0}{\rho _0} + \frac {c_s^2}{g} N^2 \right ) \xi _z^2 \nonumber \\ & + i k_\parallel \left ( \left ( 2 \rho _0 g + \dd {p_0}{z} \right ) \xi _z \xi _\parallel + \rho _0 g \frac {c_s^2}{ c_s^2 + v_A^2 } (\xi _\parallel \xi _z^* -\xi _\parallel ^* \xi _z) \right ) \nonumber \\ &= \int dV \, \rho _0 k_\parallel ^2 (v_A^2 \xi _\perp ^2 + c_s^2 \xi _\parallel ^2) + \frac {g\rho _0}{ c_s^2 + v_A^2} \left ( v_A^2 \dd {}{z} \ln \frac {B_0}{\rho _0} + \frac {c_s^2}{g} N^2 \right ) \xi _z^2 \nonumber \\ & + i k_\parallel \left ( \left ( 2 \rho _0 g + \dd {p_0}{z} \right ) \xi _z \xi _\parallel + \rho _0 g \frac {c_s^2}{ c_s^2 + v_A^2 } (\xi _\parallel \xi _z^* -\xi _\parallel ^* \xi _z) \right ) \nonumber\end{aligned}

The additional quadratic term from the pressure changes the stability criteria. Nonetheless it is clear that the last line can be made negative and large by choosing a proper phase shift between $\xi _\parallel $ and $\xi _z$.

The big difference with the low beta case is that the parallel draining along the field line is now balanced by pressure. Newcomb Newcomb (1961) plowed through this form and showed that ultimate instability for "undular" modes was simply that

- \frac {d\rho }{dz}<\frac {\rho ^2 g}{\gamma p}

or equivalently

\frac {B}{\mu _0 } \frac {d B}{dz} < - \frac {\gamma p N^2}{g}

and for $N^2\approx 0$ this amounts to $\frac {dB}{dz} < 0$, ie that the magnetic field decrease with height. Little information is given about what such an instability would look like in Newcomb’s treatment.

Why long-wavelength motion along the field matters. In a flute mode the displacement has $k_\parallel =0$, so there is no field-line bending energy and no opportunity for mass to drain from the crest of a rising perturbation to neighboring troughs. Once we allow a small but finite $k_\parallel $, two new effects appear simultaneously:

1.: magnetic tension adds a stabilizing term of order $\rho _0 v_A^2 k_\parallel ^2|\xi _z|^2$,
2.: parallel motion lets plasma slip along the field, which can lighten the crest of a rising arch and thereby increase the buoyancy drive.

The Parker instability is the statement that for sufficiently long parallel wavelength, the second effect can beat the first.

Figure 18.2: The Parker or undular instability: a horizontal field bows upward, plasma drains from the crest to the valleys, and the crest becomes lighter still. This is the prototype of a ballooning mode in a gravitationally stratified atmosphere. The mode is localized across the field but long along it, so the crest can drain mass into the neighboring valleys.

A clean long-wavelength estimate. Take a horizontal flux tube whose crest is displaced upward by $\xi _z$. Let

H_B^{-1}\equiv -\frac {d\ln \rho _0}{dz}

be the density scale height of the background atmosphere and

H_p\equiv \frac {c_s^2}{g}

the gas-pressure scale height along the field line. In a long-wavelength undular mode, pressure equilibrates rapidly along the field, so the crest density of the displaced tube follows the gas-pressure law rather than the full magnetostatic law. To first order,

\rho _{\rm crest}\simeq \rho _0\left (1-\frac {\xi _z}{H_p}\right ), \qquad \rho _{\rm ext}\simeq \rho _0\left (1-\frac {\xi _z}{H_B}\right ).

Thus

\Delta \rho \equiv \rho _{\rm crest}-\rho _{\rm ext} = -\rho _0\left (\frac {1}{H_p}-\frac {1}{H_B}\right )\xi _z. \tag{18.85}

The buoyancy force density is $-g\Delta \rho $, while field-line bending supplies a restoring force density $-\rho _0 v_A^2k_\parallel ^2\xi _z$. Therefore

\ddot {\xi }_z = \left [ g\left (\frac {1}{H_p}-\frac {1}{H_B}\right )-v_A^2k_\parallel ^2 \right ]\xi _z. \tag{18.86}

The long-wavelength undular mode is unstable when

\boxed { k_\parallel ^2 < \frac {g}{v_A^2}\left (\frac {1}{H_p}-\frac {1}{H_B}\right ). } \tag{18.87}

Equivalently, only sufficiently long parallel wavelengths are unstable:

\lambda _\parallel > \lambda _c \equiv \frac {2\pi v_A}{\sqrt {g(H_p^{-1}-H_B^{-1})}}. \tag{18.88}

This estimate is deliberately simple, but it captures the central Parker idea: magnetic support changes the background scale height, while parallel drainage makes the crest obey the gas-pressure scale height instead.

Connection with the exact energy-principle criterion. The full Newcomb treatment is more careful than the estimate above, but it leads to the same lesson: long parallel wavelength and parallel drainage make the configuration easier to destabilize than a purely flute perturbation Newcomb (1961). In the high-$\beta $ limit the instability condition can be written in the compact form

\boxed { \frac {B_0}{\mu _0}\frac {dB_0}{dz} < -\frac {\gamma p_0}{g}N^2. } \tag{18.89}

If the atmosphere is close to adiabatic, so that $N^2\approx 0$, this reduces to the simple statement that a horizontal field which decreases with height is Parker unstable provided the parallel wavelength is long enough.

Solar context. This is why Parker instability sits so naturally beside the dynamo lecture. Differential rotation can wind poloidal flux into strong toroidal field in the solar interior, but that toroidal field is not useful observationally until it can rise. Magnetic buoyancy and Parker-type undular modes provide the classical route by which deep toroidal flux escapes magnetic burial and emerges toward the photosphere Parker (1955a,b, 1966, 1979). Modern thin-flux-tube calculations show that this basic picture remains remarkably robust: buoyant loops rising through a stratified rotating envelope reproduce many gross properties of active-region emergence Fan et al. (1993); Caligari et al. (1995).

Why this lecture matters for later stability theory. In a torus, true gravity is no longer essential. Magnetic curvature and pressure gradient combine to produce an effective buoyancy drive. In that sense, bad curvature in a tokamak or stellarator plays the role that gravity played here, while long parallel wavelength and localization along a field line produce the ballooning structure. The gravitational interchange problem is therefore the clean prototype for magnetic interchange and ballooning theory: first learn the buoyancy physics here, then replace gravity by curvature in the confinement geometry.

Takeaways

The Brunt–Väisälä frequency $N$ measures whether a stratified atmosphere restores or amplifies a vertical displacement.

In a magnetized atmosphere with $k_\parallel =0$, the relevant restoring quantity is the magnetic buoyancy frequency $N_M$, Eq. (18.57).

The cold-plasma interchange criterion depends on $\rho /B$, not on $\rho $ alone: mass per unit magnetic flux is the quantity that gets exchanged.

Allowing finite $k_\parallel $ changes the problem qualitatively. Parallel drainage can overcome line tension and produce Parker’s undular instability, the gravitational prototype of ballooning.

Bibliography

Lord Rayleigh. On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Philosophical Magazine, 32:529–546, 1916. doi:10.1080/14786441608635602.

K. Schwarzschild. On the equilibrium of the Sun’s atmosphere. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, pages 41–53, 1906.

V. Väisälä. Über die wirkung der windschwankungen auf die pilotballonaufstiege. Annales Academiae Scientiarum Fennicae, 24:1–41, 1925.

D. Brunt. The period of simple vertical oscillations in the atmosphere. Quarterly Journal of the Royal Meteorological Society, 53:30–32, 1927. doi:10.1002/qj.49705322103.

M. D. Kruskal and M. Schwarzschild. Some instabilities of a completely ionized plasma. Proceedings of the Royal Society of London A, 223:348–360, 1954. doi:10.1098/rspa.1954.0120.

W. A. Newcomb. Convective instability induced by gravity in a plasma with a frozen-in magnetic field. Physics of Fluids, 4:391–396, 1961. doi:10.1063/1.1706342.

Eugene N Parker. The formation of sunspots from the solar toroidal field. The Astrophysical Journal, 121:491, 1955a. doi:10.1086/146010.

E. N. Parker. Hydromagnetic dynamo models. Astrophysical Journal, 122:293–314, 1955b. doi:10.1086/146087.

E. N. Parker. The dynamical state of the interstellar gas and field. Astrophysical Journal, 145: 811–833, 1966. doi:10.1086/148828.

E. N. Parker. Cosmical Magnetic Fields: Their Origin and Activity. Oxford University Press, 1979.

S. Chandrasekhar. Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford, 1961.

Y. Fan, G. H. Fisher, and E. E. DeLuca. The rise of magnetic flux tubes in the solar convection zone. Astrophysical Journal, 405:390–401, 1993.

P. Caligari, F. Moreno-Insertis, and M. Schüssler. Emerging flux tubes in the solar convection zone. Astrophysical Journal, 441:886–902, 1995. doi:10.1086/175410.

Russell M. Kulsrud. Plasma Physics for Astrophysics. Princeton University Press, Princeton, NJ, 2005. ISBN 9780691120737.

Problems

Problem 18.1.

Starting from Eq. (18.24), complete the square explicitly and verify Eq. (18.27). Show directly that $\delta W\ge 0$ for all admissible perturbations if and only if $N^2\ge 0$.

Problem 18.2.

Take the magnetic buoyancy frequency in Eq. (18.57). Show that

(a): in the limit $v_A\to 0$ it reduces to the ordinary Brunt–Väisälä frequency;
(b): in the cold limit $c_s\to 0$ it reduces to the Kruskal–Schwarzschild interchange criterion, Eq. (18.59).

Problem 18.3.

Using the long-wavelength Parker estimate, Eq. (18.86), derive the critical wavelength $\lambda _c$ in Eq. (18.88). Then interpret physically why long parallel wavelength is destabilizing rather than stabilizing.