Last changes:

- April 2015, March 2017, January 2018: Failure of Landau theory
- August, October 2018: Cnoidal electron and ion holes: the basis of intermittent plasma turbulence
- June 2019: Comment on "Symmetry in electron and ion dispersion"
- February 2020: Cnoidal electron holes exhibiting intrinsically substructures; update of stationary linear van Kampen & Landau modes; Ultra slow electron holes in subcritical plasmas
- May 2020: Diversity of hole structures due to non-perturbative trapping scenarios
- June 2020: Novel electron holes of Gaussian type due to second order, non-perturbative electron trapping and the general loss of identifiability of hole structures in experiments
- Sept. 2020: Two-parametric, mathematically undisclosed solitary electron holes and their evolution equation
- Oct. 2021: Pattern formation in Vlasov-Poisson plasmas beyond Landau, as caused by the continuous spectra of electron and ion hole equilibria
- January 2023: Pattern formation in Vlasov–Poisson plasmas beyond Landau caused by the continuous spectra of electron and ion hole equilibria

Hans Schamel received his doctorate in physics in 1970 from the Ludwig-Maximilians-Universität München

in Germany with a theoretical investigation of self-consistent, shock-like structures in collisionless plasmas.

From 1969-1974 he was employed at the Max-Planck-Institut für Physik and Astrophysik in München

and worked as a research scientist at the University of California, Los Angeles, USA in 1974-1975.

In 1975 he joined the Ruhr-Universität in Bochum, Germany, as an Assistant Professor, where he received

the Habilitation degree in Theoretical Physics in 1978 and was appointed extra-ordinary professor in 1984.

After a half-year employment at JET, the European Nuclear Fusion Center in Culham, England in 1986 he

was appointed as professor for Theoretical Physics at the Universität Bayreuth, Germany, from which he

retired in 2004.

Hans Schamel's theoretical research work is in the physics of plasmas, particle accelerators and fluid dynamics

with focus on structure formation, transport and turbulence. Some highlights of his investigations are

- Laser-plasma interaction [1]
- plasma expansion into vacuum [2,3]
- cross-field transport [4]
- ponderomotive effects [5]
- solitary waves on coasting beams in synchrotrons [6]
- phase space holes & intermittent plasma turbulence [7-15]
- nonlinear nature of electrostatic shielding [16]
- diode dynamics [17-19]
- magnetosonic wave collapse [20]
- Coulomb relaxation of anisotropic distributions [21]

PREVIOUS REVIEW and KEY ARTICLES

[1] Ch. Sack and H. Schamel, SUNION - An Algorithm for One-dimensional Laser-Plasma Interaction,

J. Comp. Phys. **53**(1984)395-428.

[2] Ch. Sack and H. Schamel, Plasma expansion into vacuum - a hydrodynamic approach, Phys. Reports

[3] Hans Schamel, Lagrangian fluid description with simple applications in compressible plasma and gas dynamics,

Phys. Reports **392**(2004)279-319.

[4] G. Hübner and H. Schamel, Enhanced cross-field transport in cylindrical, anisotropic magneto-plasmas in

the very long mean-field-path regime,

Transport Theory and Stat. Phys.**23**(1994)971-1000.

[5] U. Wolf and H. Schamel, Wake-field generation by the ponderomotive memory effect,Transport Theory and Stat. Phys.

Phys. Rev.E **56**(1997)4656-4664.

[6] Hans Schamel, Theory of Solitary Holes in Coasting Beams,

Phys. Rev. Lett. **79**(1997)2811-2814.

[7] Hans Schamel, Stationary Solitary, Snoidal and Sinusoidal Ion Acoustic Waves,

Plasma Phys. **14**(1972)905-924.

[8] Hans Schamel, Theory of Electron Holes,

Physica Scripta. **20**(1979)336-342.

[9] H. Schamel and S. Bujarbarua, Theory of finite-amplitude electron and ion holes,

J. Plasma Phys. **25**(1981)515-529.

[10] H. Schamel and S. Bujarbarua, Analytical double layers,

Phys. Fluids **26**(1983)190- 193.

[11] Hans Schamel, Kinetic Theory of Phase Space Vortices and Double layers,

Physica Scripta. Vol. T2/1(1982)228-237.

Phys. Reports **140**(1986)161-191.

[13] J. Korn and H. Schamel, Electron holes and their role in the dynamics of current-carrying weakly collisional plasmas

Part 1. Immobile ions, Part 2. Mobile ions,

J. Plasma Phys. **56**(1996)307-337, 339-359.

[14] Hans Schamel, Hole equilibria in Vlasov-Poisson systems: A challenge to wave theories of ideal plasmas,

Phys. Plasmas **7**(2000)4831-4844.

Phys. Reports **415**(2005)261-359.

[16] N. Das and H. Schamel, Nonlinear shielding of planar test charge in one-dimensionalVlasov-Poisson plasmas,

J. Plasma Phys. 71(2005) 769-784.

[17] A. Ya. Ender, H. Kolinsky, V. I. Kuznetsov, and H. Schamel, Collective diode dynamics: an analytical approach, Phys. Reports **328**(2000)1-72.

[18] P. V. Akimov and H. Schamel, Space-charge-limited current in electron diodes under the influence of collisions,

J. Appl. Phys**92**(2002)1690-1698.

J. Appl. Phys

[19] P. V. Akimov, H. Schamel, A. Ya. Ender, and V. I. Kuznetsov, Switching as a dynamical process in electron diodes,

J. Appl. Phys**93**(2003)1246-1256.

J. Appl. Phys

[20] N. Chakrabarti, C. Maity, and H. Schamel, Non-stationary Magnetosonic Wave Dynamics in Plasmas Exhibiting

Collapse,

Phys. Rev. E**88**(2013)023102.

Phys. Rev. E

[21] H. Schamel, H. Hamnén, D. F. Düchs, and T. E. Stringer, Nonlinear analysis of Coulomb relaxation of anisotropic distributions,

Phys. Fluids B 1(1989)76-86.

MORE RECENT ARTICLES (A PICK)

Phys. Plasmas **19**(2012)020501

[R2] Hans Schamel, Particle trapping:A key requisite of structure formation and stability of Vlasov-Poisson plasmas,

Phys. Plasmas **22**(2015)042301

[R3] Hans Schamel, Debraj Mandal, and Devendra Sharma, On the nonlinear trapping nature of undamped, coherent structures

in collisionless plasmas and its impact on stability,

Phys. Plasmas **24**(2017)032109

[R4] Debraj Mandal, Devendra Sharma, and Hans Schamel, Electron holes instability as a primordial step towards sustained

intermittent turbulence in linearly subcritical plasmas,

New J. Phys. **20**(2018)073004

[R5] H. Schamel, D. Mandal, and D. Sharma, Evidence of a new class of cnoidal electron holes exhibiting intrinsic

substructures, its impact on linear (and nonlinear) Vlasov theories and role in anomalous transport,

Phys. Scr. **95**(2020)055601

[R6] H. Schamel, D. Mandal, and D. Sharma, Diversity of solitary electron holes operating with non-perturbative trapping,

Phys. Plasmas** 27**(2020)062302

[R7] H. Schamel, Novel electron holes of Gaussian type due to second order, non-perturbative trapping and the general loss of identifiability of hole structures in experiments,

Phys. Lett. A** 384**(2020)126752

[R8] H. Schamel, Two-Parametric, Mathematically Undisclosed Solitary Electron Holes and Their Evolution equation,

Plasma** 3**(2020)166

[R9] H. Schamel, Pattern formation in Vlasov-Poisson plasmas beyond Landau caused by
the continuous spectra of

electron and ion hole equilibria,

arXiv:2110.01433v2 [physics.plasm-ph]

- Plasma expansion fast ions, trapped electrons
- Ponderomotive effects
- Phase space holes: short, long
- Magnetoacoustic wave collapse: short, long
- Failure of Landau theory

- Intermittent plasma turbulence

APPLICATIONS:

~ intense laser irradiation of matter and atomic clusters[1-3]

~ plasma based acceleration of ions [1-3]

~ matter generation in the early phase of our expanding universe [3]

REVIEW ARTICLES:

[1] Ch.Sack and H.Schamel, Sunion - an algorithm for one-dimensional

laser-plasma interaction, J. Comp. Phys. 53(1984)395-428

[2] Ch.Sack and H.Schamel, Plasma expansion into vacuum -

a hydrodynamic approach, Phys. Reports 156(1987)311-395

[3] H.Schamel, Lagrangian fluid description with simple applications in

compressible plasma and gas dynamics,

Phys.Reports 392(2004)279 - 319 and references therein

BASIC EQUATIONS:

~ cold ion fluid equations without and with Navier-Stokes viscosity term

~ polytropic electron equation of state

~ Poisson's equation

at t=0: semi-infinite plasma with a step-like, but continuous ion density

NEW RESULTS:

a) inviscid case

~ discovery of ion density collapse in the laser-plasma interaction [1]

~ explanation of density collapse during plasma expansion as an ion

wave breaking phenomenon resulting from wave steepening [2,3]

Keywords: simple wave analysis, derivation and solution of nonlinear

scalar wave equation in Lagrangian fluid description-> **Sack-Schamel equation**:

(see also https://en.wikipedia.org/wiki/ Sack-Schamel equation)

V = 1 / n , the specific volume.

b) viscid case

to allow long term simulations: inclusion of an ion viscosity term

accounting for anomalous plasma transport

(keywords: plasma turbulence due to multiple ion streaming after the

collapse event, coarse-grained ion Vlasov distribution function)

~ supersonic propagation of a peaked ion density front, the peak

being a remnant of the density collapse [2,3]

~ essentially 3 phases of evolution can be distinguished :

i) initial phase up to ion density bunching

ii) a stable peaked ion density front propagating with constant

velocity supersonically towards vacuum

iii) decay of the leading density hump due to debunching and

further acceleration of the resultant tiny ion front up to a

value determined by self-similar theory [2,3]

NUMERICAL SUBTLETIES :

~ exact numerical treatment of ion dynamics by a Lagrangian code [1,2]

~ need for a Poisson solver of high resolution and fast convergence of

the iteration procedure [1,2]

~ tri-diagonal coefficient matrix for the discretized evolution equations

with open (i.e. differential) boundary conditions [1,2]

b) viscid case

to allow long term simulations: inclusion of an ion viscosity term

accounting for anomalous plasma transport

(keywords: plasma turbulence due to multiple ion streaming after the

collapse event, coarse-grained ion Vlasov distribution function)

~ supersonic propagation of a peaked ion density front, the peak

being a remnant of the density collapse [2,3]

~ essentially 3 phases of evolution can be distinguished :

i) initial phase up to ion density bunching

ii) a stable peaked ion density front propagating with constant

velocity supersonically towards vacuum

iii) decay of the leading density hump due to debunching and

further acceleration of the resultant tiny ion front up to a

value determined by self-similar theory [2,3]

NUMERICAL SUBTLETIES :

~ exact numerical treatment of ion dynamics by a Lagrangian code [1,2]

~ need for a Poisson solver of high resolution and fast convergence of

the iteration procedure [1,2]

~ tri-diagonal coefficient matrix for the discretized evolution equations

with open (i.e. differential) boundary conditions [1,2]

APPLICATIONS

~laser-plasma (matter) interaction by laser pulses

~laser wake-field generation and acceleration of charged particles

~wave heating of plasmas

The fast-time-averaged electron distribution function as a solution of the kinetic diﬀusion

equation in the presence of a hf electric wave ﬁeld is found to be given by

E(x, t) representing the slowly varying electric ﬁeld envelope such as a hump localized in x and t.

Three ponderomotive eﬀects can be distinguished:

- cavity formation by pp in the exponential,
- fake heating by pp in the denominator of the argument in the exponential, and
- wake ﬁeld generation by pmt manifested by the emission of streamers, jets and heatﬂuxes.

Notice that the latter eﬀect weakens cavity formation.

Space-time plot of ponderomotive potential (assumed):

Space-time plot of ponderomotive density (derived):

[1] H.Schamel, Ponderomotive Effects in a Plasma, Phys.Rev.Lett. 42 (1979) 1339-1341

[2] H.Schamel and Ch.Sack, Existence of a time-dependent heat flux-related

ponderomotive effect, Phys.Fluids 23 (1980) 1532-1545

[3] H.Schamel and Ch.Sack, A new ponderomotive effect in wave heating,

Proc.2nd Joint Grenoble-Varenna Int.Symp.on Heating in Toroidal Plasmas,

Como 1980, Vol.II 1123-1128 ,

published by Commission of the European Communities EUR 7424 EN

[4] U.Wolf and H.Schamel, Wake-field generation by the ponderomotive

memory effect, Phys.Rev.E 56 (1997) 4656-4664

The reason of failure in two lines:

Coherent equilibria of VP plasmas necessarily imply trapping and hence exclude van Kampen modes, which on the other hand are a central issue for linear Landau solutions.

For more details see

- H.Schamel, Phys. Plasmas 22, 042301 (2015)
- H.Schamel, D.Mandal, D.Sharma, Phys. Plasmas 24, 032109 (2017)
- H.Schamel, N. Das, and P. Borah, Phys. Lett. A 382, 168 (2018)
- H.Schamel, Pattern formation in Vlasov-Poisson plasmas beyond Landau caused by the continuous spectrum of electron and ion holes, https://arxiv.org/pdf/2110.01433v2.pdf or ReviewSoElHo-Nov.pdf

A key issue in electrostatically driven **intermittent plasma turbulence** is **particle trapping**, which is associated with its **patches of coherency** in electron and ion phase space. The most plausible description of these coherent substructures is thereby provided by the spectrum of **Privileged Cnoidal Electron & Ion Holes,** which are **non-BGK** equilibrium solutions of the governing Vlasov-Poisson system. These modes, being extinguished by smooth (i.e. regular) **trapped particle distributions**, are challenging standard wave theory by superseding **Landau**'s linear wave approach (e.g. with respect to the onset of structure formation).

This underlying **nonlinear** picture of **structure** formation is further** strengthened** by the following new publications in 2018, 2019, 2020:

- H. Schamel, Unconditionally marginal stability of harmonic electron hole equilibria in current-driven plasmas,

Phys. Plasmas**25**, 062115 (2018) - D. Mandal, D. Sharma and H. Schamel, Electron hole instability as a primordial step towards sustained intermittent turbulence in linearly subcritical plasmas,

New J. Phys.**20**(2018) 073004 - N. Das, P. Borah and H. Schamel, Free energy and spatial periodicity of generalized cnoidal ion holes,

Phys. Lett. A**382**(2018) 2693 - P. Borah, N. Das and H. Schamel, The wavenumber of privileged cnoidal electron and ion holes - a nonlinearly nontrivial parameter,

Phys. Plasmas**25**(2018) 094506 - H.Schamel, Comment on "Symmetry in electron and ion dispersion in 1D Vlasov-Poisson plasma" [Phys. Plasmas 25,112102(2018)], Phys.Plasmas 26 (2019) 064701
- H.Schamel, D. Mandal, and D. Sharma, Evidence of a new class of cnoidal electron holes exhibiting intrinsic substructures, its impact on linear (and nonlinear) Vlasov theories and role in anomalous transport, Phys. Scr. 95 (2020) 055601
- H.Schamel, D. Mandal, and D. Sharma, Diversity of solitary electron holes operating with non-perturbative trapping, Phys. Plasmas 27 (2020) 062302
- H. Schamel, Novel electron holes of Gaussian type due to second order, non-perturbative trapping and the general loss of identifiability of hole structures in experiments,Phys. Lett. A
**384**(2020)126752 - H. Schamel, Two-Parametric, Mathematically Undisclosed Solitary Electron Holes and Their Evolution equation,Plasma
**3**(2020)166 - H. Schamel, Pattern formation in Vlasov-Poisson plasmas beyond Landau caused by the continuous spectra of electron and ion hole equilibria, arXiv:2110.01433v2, (2022)

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