Sack–Schamel equation

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

${\displaystyle {\ddot {V}}+\partial _{\eta }\left[{\frac {1}{1-{\ddot {V}}}}\partial _{\eta }\left({\frac {1-{\ddot {V}}}{V}}\right)\right]=0.}$

${\displaystyle V(\eta ,t)}$ is therein the specific volume of the ion fluid, ${\displaystyle \eta }$ the Lagrangian mass variable and t the time (see the following text).

We treat, as an example, the plasma expansion into vacuum, i.e. a plasma that is confined initially in a half-space and is released at t=0 to occupy in course of time the second half.[2][3][4][5] The dynamics of such 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, ${\displaystyle \partial _{t}n+\partial _{x}(nv)=0}$ and ${\displaystyle \partial _{t}v+v\,\partial _{x}v=-\partial _{x}\varphi }$, respectively.

Both species are thereby coupled through the self-organized electric field ${\displaystyle E(x,t)=-\partial _{x}\varphi (x,t)}$, which satisfies Poisson's equation, ${\displaystyle \partial _{x}^{2}\varphi =e^{\varphi }-n}$. 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.[6][5] 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 ${\displaystyle (x_{*},t_{*})}$. Here, the quantity ${\displaystyle V:=1/n}$ 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 ${\displaystyle (x,t)}$ are replaced by Lagrange coordinates ${\displaystyle (\eta ,\tau )}$, and by so-called open b.c.s, which are formulated by differential equations of the first order.[6][5]

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

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

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

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 ${\displaystyle (x_{*},t_{*})}$ 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.[6] The solution can be found in equation (6.37) and reads for small ${\displaystyle \eta }$ and t

${\displaystyle V(\eta ,t)=at\left[1+{\frac {t}{2a}}-b\eta +c(\eta ^{2}-\Omega ^{2}t^{2})+d(\eta -\Omega t)^{2}(\eta +2\Omega t)+\cdots \right]}$

where ${\displaystyle a,b,c,d,\Omega }$ are constants and ${\displaystyle (\eta ,t)}$ stand for ${\displaystyle (\eta _{*}-\eta ,t_{*}-t)}$. The collapse is hence at ${\displaystyle (\eta ,t)=(0,0)}$. ${\displaystyle V(\eta ,t)}$ is V-shaped in ${\displaystyle \eta }$ and its minimum moves linearly with ${\displaystyle \eta =\Omega t}$ towards the zero point (see Fig. 7 of [6]). This means that the density n diverges at ${\displaystyle (\eta _{*},t_{*})}$ when we return to the original Lagrangian variables.

It is easily seen that the slope of the velocity, ${\displaystyle \partial _{x}v={\frac {1}{V}}\partial _{\eta }v}$, diverges as well when ${\displaystyle V\rightarrow 0}$. In the final collapse phase, the Sack–Schamel equation transits into the quasi-neutral scalar wave equation: ${\displaystyle {\ddot {V}}+\partial _{\eta }^{2}{\frac {1}{V}}=0}$ and the ion dynamics obeys Euler's simple wave equation: ${\displaystyle \partial _{t}v+v\,\partial _{x}v=0}$.[7]

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

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

The limitation of ${\displaystyle \gamma }$ 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 [6] or more explicitly Sect. 2.2: "Constraints on the electron dynamics".

These results are in two respects remarkable. The collapse, which could be resolved analytically by the Sack–Schamel equation, signalizes through its singularity the absence of real physics. A real plasma can continue in at least two ways. Either it enters into the kinetic collsionless Vlasov regime [8] and develops multi-streaming and folding effects in phase space [3] or it experiences dissipation (e.g. through Navier-Stokes viscosity in the momentum equation [6][5][8]) which controls furtheron the evolution in the subsequent phase. As a consequence the ion density peak saturates and continues its acceleration into vacuum maintaining its spiky nature.[6][5] This phenomenon of fast ion bunching being recognized by its spiky fast ion front has received immense attention in the recent past in several fields. High-energy ion jets are of importance and promising in applications such as in the laser-plasma interaction,[9][10][11][12] in the laser irradiation of solid targets, being also referred to as target normal sheath acceleration,[13][14][15][16] in future plasma based particle accelerators and radiation sources (e.g. for tumor therapy)[17] and in space plasmas.[18] Fast ion bunches are hence a relic of wave breaking that is analytically completely described by the Sack–Schamel equation. (For more details especially about the spiky nature of the fast ion front in case of dissipation see http://www.hans-schamel.de or the original papers [19][20]). An article in which the Sack-Schamel's wave breaking mechanism is mentioned as the origin of a peak ion front was published e.g. by Beck and Pantellini (2009).[21]

Finally, the notability of the Sack–Schamel equation is clarified through a recently published molecular dynamics simulation.[22] In the early phase of the plasma expansion a distinct ion peak could be observed, emphasizing the importance of the wave breaking scenario as predicted by the equation.

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