The Sack–Schamel equation describes the nonlinear evolution of the cold ion fluid in a two-component plasma under the influence of a self-organized electric field. It is a partial differential equation of second order in time and space formulated in Lagrangian coordinates. The dynamics described by the equation take place on an ionic time scale, which allows electrons to be treated as if they were in equilibrium and described by an isothermal Boltzmann distribution. Supplemented by suitable boundary conditions, it describes the entire configuration space of possible events the ion fluid is capable of, both globally and locally.
The Sack–Schamel equation is in its simplest form, namely for isothermal electrons, given by
\ddotV+\partialη\left[
1 | |
1-\ddotV |
\partialη\left(
1-\ddotV | |
V |
\right)\right]=0.
V(η,t)
η
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.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,
\partialtn+\partialx(nv)=0
\partialtv+v\partialxv=-\partialx\varphi
Both species are thereby coupled through the self-organized electric field
E(x,t)=-\partialx\varphi(x,t)
\partial
2\varphi= | |
x |
e\varphi-n
Figs. 1a, 1b show an example of a typical evolution. 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*)
V:=1/n
This result is obtained by a Lagrange numerical scheme, in which the Euler coordinates
(x,t)
(η,\tau)
This transformation is provided by
η=η(x,t)
\tau=t
x | |
η(x,t)=\int | |
0 |
n(\tildex,t)d\tildex
x=x(η,\tau),t=\tau
x(η(x,t),\tau)=x
\partialxη\partialηx=1
\partialηx=
1 | |
\partialxη |
=
1 | |
n |
=V
(\partialt+v\partialx)η(x,t)=0
η
n
\partialxη
\partial\taux(η,\tau)=:
x(η, |
\tau)=v(η,\tau)
It immediately follows:
\ddotV=\partialη\ddotx=\partialη
v=\partial |
ηE
=-\partial | ||||
|
\partialη\varphi)
\partial | ||||
|
\partialη\varphi
η
\partialxη=n=
1 | |
V |
Replacing
\partialx
1 | |
V |
\partialη
\partialη\left(
1 | |
V |
\partialη\varphi\right)=Ve\varphi-1=-\ddotV
\varphi=ln\left( | 1-\ddotV |
V |
\right)
\varphi
\ddotV
\ddotV+\partialη\left[
1 | |
1-\ddotV |
\partialη\left(
1-\ddotV | |
V |
\right)\right]=0
V
(η,\tau)
V(η,\tau)
\tau
t
\ddotV
dx=
dx | |
dη |
dη=Vdη=Jdη.
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*)
η
V(η,t)=at\left[1+
t | |
2a |
-bη+c(η2-\Omega2t2)+d(η-\Omegat)2(η+2\Omegat)+ … \right]
where
a,b,c,d,\Omega
(η,t)
(η*-η,t*-t)
(η,t)=(0,0)
V(η,t)
η
η=\Omegat
(η*,t*)
It is easily seen that the slope of the velocity,
\partialxv=
1 | |
V |
\partialηv
V → 0
\ddotV+
2 | |
\partial | |
η |
1 | |
V |
=0
\partialtv+v\partialxv=0
A generalization is achieved by allowing different equations of state for the electrons. Assuming a polytropic equation of state,
pe\sim
\gamma | |
n | |
e |
pe\simneTe
Ten
1-\gamma | |
e |
=constant,
\gamma=1
\ddotV+\partialη\left[
\gamma | \left( | |
V |
1-\ddotV | |
V |
\right)\gamma-2\partialη\left(
1-\ddotV | |
V |
\right)\right]=0, 1\le\gamma\le2
The limitation of
\gamma
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 and develops multi-streaming and folding effects in phase space or it experiences dissipation (e.g. through Navier-Stokes viscosity in the momentum equation) 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. 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, in the laser irradiation of solid targets, being also referred to as target normal sheath acceleration, in future plasma based particle accelerators and radiation sources (e.g. for tumor therapy) and in space plasmas.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). 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).
Finally, the notability of the Sack–Schamel equation is clarified through a recently published molecular dynamics simulation. 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.