Sack–Schamel equation
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.
[1] 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.
[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,
Both species are thereby coupled through the self-organized electric field
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.
1a shows the ion density in x-space for different discrete times, Fig.
Most notable is the appearance of a spiky ion front associated with the collapse of density at a certain point in space-time
This result is obtained by a Lagrange numerical scheme, in which the Euler coordinates
, and by so-called open b.c.s, which are formulated by differential equations of the first order.
In the second step the definition of the mass variable was used which is constant along the trajectory of a fluid element:
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
It is equivalent with the Jacobian J of the transformation from Eulerian to Lagrangian coordinates since it holds
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.
and the ion dynamics obeys Euler's simple wave equation:
[7] A generalization is achieved by allowing different equations of state for the electrons.
results from the demand that at infinity the electron density should vanish (for the expansion into vacuum problem).
The collapse, which could be resolved analytically by the Sack–Schamel equation, signalizes through its singularity the absence of real physics.
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.
