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The Sack-Schamel equation is a partial differential equation of second order in time and space. It is formulated in Lagrangian coordinates and describes physically the nonlinear evolution of the cold ion fluid in a two-component plasma under the influence of a self-organized electric field. The dynamics takes place on the ionic time scale. Hence electrons can be treated in equilibrium and be described e.g. by an isothermal Boltzmann distribution for the density. Supplemented by suitable boundary conditions it describes the entire spectrum of possible events the ion fluid is capable of, both globally and locally in space-time. Its most spectacular application is the 1-D expansion of a plasma into vacuum, that is initially confined in a half-space, and the subsequent occurrence of an ion density collapse locally in space-time (spiky ion front) .

The equation

The Sack-Schamel equation is in its simplest form, namely for isothermal electrons, given by

${\ddot {V}}+\partial _{\eta }{\Bigl [}{\frac {1}{1-{\ddot {V}}}}\partial _{\eta }{\Bigl (}{\frac {1-{\ddot {V}}}{V}}{\Bigr )}{\Bigr ]}=0$

Derivation and application

The dynamics of a two-component plasma, consisting of isothermal Botzmann-like electrons and a cold ion fluid, is governed by the ion equations of continuity and momentum, $\partial _{t}n+\partial _{x}(nv)=0$ and $\partial _{t}v+v\partial _{x}v=-\partial _{x}\phi$, respectively.

Both species are thereby coupled through the self-organized electric field $E(x,t)=-\partial _{x}\phi (x,t)$, which satisfies Poisson's equation, $\partial _{x}^{2}\phi =e^{\phi }-n$. Supplemented by suitable initial and boundary conditions (b.c.s), they represent a self-consistent, intrinsically closed set of equations that represent the laminar ion flow in its full pattern on the ion time scale.

Fig. 1a Plasma expansion into vacuum, Fig. 1b A small section of the density front.

Figs. 1a, 1b show an example of a typical evolution, the expansion of a plasma which is initially enclosed in a half-space and is released at $t=0$ . Fig. 1a shows the ion density in x-space for different discrete times, Fig. 1b a small section of the density front.

Most notable is the appearance of a spiky ion front associated with the collapse of density at a certain point in space-time $(x_{*},t_{*})$. Here, the quantity $V:=1/n$ becomes zero. This event is known as "wave breaking" by analogy with a similar phenomenon that occurs with water waves approaching a beach.

This result is obtained by a Lagrange numerical scheme, in which the Euler coordinates $(x,t)$ are replaced by Lagrange coordinates $(\eta ,\tau )$, and by so-called open b.c.s, which are formulated by differential equations of the first order .

This transformation is provided by $\eta =\eta (x,t)$, $\tau =t$, where $\eta (x,t)=\int _{0}^{x}n({\tilde {x}},t)d{\tilde {x}}$ is the Lagrangian mass variable. The inverse transformation is given by $x=x(\eta ,\tau ),t=\tau$ and it holds the identity: $x(\eta (x,t),\tau )=x$. With this identity we get through an x-derivation $\partial _{x}\eta \partial _{\eta }x=1$ or $\partial _{\eta }x={\frac {1}{\partial _{x}\eta }}={\frac {1}{n}}=V$. In the second step the definition of the mass variable was used which is constant along the trajectory of a fluid element: $(\partial _{t}+v\partial _{x})\eta (x,t)=0$. This follows from the definition of $\eta$, from the continuity equation and from the replacement of $n$ by $\partial _{x}\eta$. Hence $\partial _{\tau }x(\eta ,\tau )=:{\dot {x}}(\eta ,\tau )=v(\eta ,\tau )$. The velocity of a fluid element coincides with the local fluid velocity.

It immediately follows: ${\ddot {V}}=\partial _{\eta }{\ddot {x}}=\partial _{\eta }{\dot {v}}=\partial _{\eta }E=-\partial _{\eta }({\frac {1}{V}}\partial _{\eta }\phi )$ where the momentum equation has been used as well as $\partial _{x}\phi ={\frac {1}{V}}\partial _{\eta }\phi$, which follows from the definition of $\eta$ and from $\partial _{x}\eta =n={\frac {1}{V}}$.

Replacing $\partial _{x}$ by ${\frac {1}{V}}\partial _{\eta }$ we get from Poisson's equation: $\partial _{\eta }\left({\frac {1}{V}}\partial _{\eta }\phi \right)=Ve^{\phi }-1=-{\ddot {V}}$. Hence $\phi =\ln \left({\frac {1-{\ddot {V}}}{V}}\right)$. Finally, replacing $\phi$ in the ${\ddot {V}}$ expression we get the desired equation:${\ddot {V}}+\partial _{\eta }\left[{\frac {1}{1-{\ddot {V}}}}\partial _{\eta }\left({\frac {1-{\ddot {V}}}{V}}\right)\right]=0$. Here $V$ is a function of $(\eta ,\tau )$: $V(\eta ,\tau )$ and for convenience we may replace $\tau$ by $t$. Further details on this transition from one to the other coordinate system can be found in . Note its unusual character because of the implicit occurrence of ${\ddot {V}}$. Physically V represents the specific volume. It is equivalent with the Jacobian J of the transformation from Eulerian to Lagrangian coordinates since it holds $dx={\frac {dx}{d\eta }}d\eta =Vd\eta =Jd\eta$.

Wave breaking solution

An analytical, global solution of the Sack-Schamel equation is generally not available. The same holds for the plasma expansion problem. This means that the data $(x_{*},t_{*})$ for the collapse cannot be predicted, but have to be taken from the numerical solution. Nonetheless, it is possible, locally in space and time, to obtain a solution to the equation. This is presented in detail in Sect.6 "Theory of bunching and wave breaking in ion dynamics" of . The solution can be found in equation (6.37) and reads for small $\eta$ and t
$V(\eta ,t)=at\left[1+{\frac {t}{2a}}-b\eta +c(\eta ^{2}-\Omega ^{2}t^{2})+d(\eta -\Omega t)^{2}(\eta +2\Omega t)+...\right]$
where a,b,c,d,$\Omega$ are constants and $(\eta ,t)$ stand for $(\eta _{*}-\eta ,t_{*}-t)$. The collapse is hence at $(\eta ,t)=(0,0)$. $V(\eta ,t)$ is V-shaped in $\eta$ and its minimum moves linearly with $\eta =\Omega t$ towards the zero point (see Fig. 7 of ). This means that the density n diverges at $(\eta _{*},t_{*})$ when we return to the original Lagrangian variables.
It is easily seen that the slope of the velocity, $\partial _{x}v={\frac {1}{V}}\partial _{\eta }v$, diverges as well when $V\rightarrow 0$. In the final collapse phase, the Sack-Schamel equation transits into the quasi-neutral scalar wave equation: ${\ddot {V}}+\partial _{\eta }^{2}{\frac {1}{V}}=0$ and the ion dynamics obeys Euler's simple wave equation: $\partial _{t}v+v\partial _{x}v=0$ .

Generalization

A generalization is achieved by allowing different equations of state for the electrons. Assuming a polytropic equation of state, $p_{e}\sim n_{e}^{\gamma }$ or with $p_{e}\sim n_{e}T_{e}$: $T_{e}n_{e}^{1-\gamma }$ = constant, where $\gamma =1$ refers to isothermal electrons, we get (see again Sect. 6 of ):

${\ddot {V}}+\partial _{\eta }\left[{\frac {\gamma }{V}}\left({\frac {1-{\ddot {V}}}{V}}\right)^{\gamma -2}\partial _{\eta }\left({\frac {1-{\ddot {V}}}{V}}\right)\right]=0$, $\qquad 1\leq \gamma \leq 2$

The limitation of $\gamma$ results from the demand that at infinity the electron density should vanish (for the expansion into vacuum problem). For more details, see Sect. 2: "The plasma expansion model" of , or more explicitly Sect. 2.2: "Constraints on the electron dynamics".