Electrostatic solitary wave

The discovery of solitary waves in general is attributed to John Scott Russell in 1834,{{Cite journal |date=April 1845 |title=Proceedings of the Central Committee of the British Archaeological Association |url=http://dx.doi.org/10.1080/00681288.1845.11886760 |journal=Journal of the British Archaeological Association |volume=1 |issue=1 |pages=43–67 |doi=10.1080/00681288.1845.11886760 |issn=0068-1288|url-access=subscription }} with their first mathematical conceptualization being finalized in 1871 by Joseph BoussinesqBoussinesq, J. (1871). "Théorie de l'intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire". C. R. Acad. Sci. Paris. 72. (and later refined and popularized by Lord Rayleigh in 1876{{Cite journal |date=April 1876 |title=XXXII. On waves |url=http://dx.doi.org/10.1080/14786447608639037 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=1 |issue=4 |pages=257–279 |doi=10.1080/14786447608639037 |issn=1941-5982|url-access=subscription }}). However, these observations and solutions were for oscillations of a physical medium (usually water), and not describing the behavior of non-particle waves (including electromagnetic waves). For solitary waves outside of media, which ESWs are classified as{{Ref|a|a}}, the first major framework was likely developed by Louis de Broglie in 1927,{{Cite journal |last=De Broglie |first=Louis |date=1952 |title=La physique quantique restera-t-elle indéterministe ? |url=http://dx.doi.org/10.3406/rhs.1952.2967 |journal=Revue d'histoire des sciences et de leurs applications |volume=5 |issue=4 |pages=289–311 |doi=10.3406/rhs.1952.2967 |issn=0048-7996|url-access=subscription }} though his work on the subject was temporarily abandoned and was not completed until the 1950s.

Electrostatic structures were first observed near Earth's polar cusp by Donald Gurnett and Louis A. Frank using data from the Hawkeye 1 satellite in 1978.{{Cite journal |last1=Gurnett |first1=D. A. |last2=Frank |first2=L. A. |date=April 1978 |title=Plasma waves in the polar cusp: Observations from Hawkeye 1 |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/JA083iA04p01447 |journal=Journal of Geophysical Research: Space Physics |language=en |volume=83 |issue=A4 |pages=1447–1462 |bibcode=1978JGR....83.1447G |doi=10.1029/JA083iA04p01447 |issn=0148-0227|url-access=subscription }} However, it is Michael Temerin, William Lotko, Forrest Mozer, and Keith Cerny{{ref|b|b}} who are credited with the first observation of electrostatic solitary waves in Earth's magnetosphere in 1982.{{Cite journal |last1=Temerin |first1=M. |last2=Cerny |first2=K. |last3=Lotko |first3=W. |last4=Mozer |first4=F. S. |date=1982-04-26 |title=Observations of Double Layers and Solitary Waves in the Auroral Plasma |url=https://link.aps.org/doi/10.1103/PhysRevLett.48.1175 |journal=Physical Review Letters |language=en |volume=48 |issue=17 |pages=1175–1179 |bibcode=1982PhRvL..48.1175T |doi=10.1103/PhysRevLett.48.1175 |issn=0031-9007|url-access=subscription }} Since then, a wide variety of magnetospheric satellites have observed and documented ESWs, allowing for analysis of them and the surrounding plasma conditions.{{Cite journal |last1=Graham |first1=D. B. |last2=Khotyaintsev |first2=Yu. V. |last3=Vaivads |first3=A. |last4=André |first4=M. |date=April 2016 |title=Electrostatic solitary waves and electrostatic waves at the magnetopause |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1002/2015JA021527 |journal=Journal of Geophysical Research: Space Physics |language=en |volume=121 |issue=4 |pages=3069–3092 |doi=10.1002/2015JA021527 |bibcode=2016JGRA..121.3069G |issn=2169-9380}}{{Cite journal |last1=Matsumoto |first1=H. |last2=Kojima |first2=H. |last3=Miyatake |first3=T. |last4=Omura |first4=Y. |last5=Okada |first5=M. |last6=Nagano |first6=I. |last7=Tsutsui |first7=M. |date=1994-12-15 |title=Electrostatic solitary waves (ESW) in the magnetotail: BEN wave forms observed by GEOTAIL |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/94GL01284 |journal=Geophysical Research Letters |language=en |volume=21 |issue=25 |pages=2915–2918 |doi=10.1029/94GL01284 |issn=0094-8276|url-access=subscription }}{{Cite journal |last1=Fu |first1=H. S. |last2=Chen |first2=F. |last3=Chen |first3=Z. Z. |last4=Xu |first4=Y. |last5=Wang |first5=Z. |last6=Liu |first6=Y. Y. |last7=Liu |first7=C. M. |last8=Khotyaintsev |first8=Y. V. |last9=Ergun |first9=R. E. |last10=Giles |first10=B. L. |last11=Burch |first11=J. L. |date=2020-03-03 |title=First Measurements of Electrons and Waves inside an Electrostatic Solitary Wave |url=https://link.aps.org/doi/10.1103/PhysRevLett.124.095101 |journal=Physical Review Letters |language=en |volume=124 |issue=9 |page=095101 |doi=10.1103/PhysRevLett.124.095101 |pmid=32202894 |issn=0031-9007|url-access=subscription }}

Electrostatic solitary waves, by their nature, are a phenomenon occurring in the electric field of a plasma. As such, ESWs are technically detectable by any instrument that can measure changes to the electric field during a sufficiently short time window. However, since a given plasma's electric field can vary widely depending on the properties of the plasma and since ESWs occur in short time windows, detection of ESWs can require additional screening of the data in addition to the measurement of the electric field itself. One solution to this obstacle for detecting ESWs, implemented by NASA's Magnetospheric Multiscale Mission (MMS), is to use a digital signal processor to analyze the electric field data and isolate short-duration spikes as a candidate for an ESW.{{Cite journal |last1=Ergun |first1=R. E. |last2=Tucker |first2=S. |last3=Westfall |first3=J. |last4=Goodrich |first4=K. A. |last5=Malaspina |first5=D. M. |last6=Summers |first6=D. |last7=Wallace |first7=J. |last8=Karlsson |first8=M. |last9=Mack |first9=J. |last10=Brennan |first10=N. |last11=Pyke |first11=B. |last12=Withnell |first12=P. |last13=Torbert |first13=R. |last14=Macri |first14=J. |last15=Rau |first15=D. |date=December 2, 2014 |title=The Axial Double Probe and Fields Signal Processing for the MMS Mission |journal=Space Science Reviews |volume=199 |issue=1–4 |pages=167–188 |doi=10.1007/s11214-014-0115-x |issn=0038-6308|doi-access=free }} Though the following detection algorithm is specific to MMS, other ESW-detecting algorithms function on similar principles.{{Cite journal |last1=Mozer |first1=F. S. |last2=Ergun |first2=R. |last3=Temerin |first3=M. |last4=Cattell |first4=C. |last5=Dombeck |first5=J. |last6=Wygant |first6=J. |date=1997-08-18 |title=New Features of Time Domain Electric-Field Structures in the Auroral Acceleration Region |url=https://link.aps.org/doi/10.1103/PhysRevLett.79.1281 |journal=Physical Review Letters |language=en |volume=79 |issue=7 |pages=1281–1284 |doi=10.1103/PhysRevLett.79.1281 |issn=0031-9007|url-access=subscription }}{{Citation |last1=Gurnett |first1=D. A. |title=The Wide-Band Plasma Wave Investigation |date=1997 |work=The Cluster and Phoenix Missions |pages=195–208 |editor-last=Escoubet |editor-first=C. P. |url=http://link.springer.com/10.1007/978-94-011-5666-0_8 |access-date=2024-11-06 |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-94-011-5666-0_8 |isbn=978-94-010-6389-0 |last2=Huff |first2=R. L. |last3=Kirchner |first3=D. L. |editor2-last=Russell |editor2-first=C. T. |editor3-last=Schmidt |editor3-first=R.|url-access=subscription }}{{Cite journal |last1=Gustafsson |first1=G. |last2=BostrÖM |first2=R. |last3=Holback |first3=B. |last4=Holmgren |first4=G. |last5=Lundgren |first5=A. |last6=Stasiewicz |first6=K. |last7=ÅHLÉN |first7=L. |last8=Mozer |first8=F. S. |last9=Pankow |first9=D. |last10=Harvey |first10=P. |last11=Berg |first11=P. |last12=Ulrich |first12=R. |last13=Pedersen |first13=A. |last14=Schmidt |first14=R. |last15=Butler |first15=A. |date=1997-01-01 |title=THE ELECTRIC FIELD AND WAVE EXPERIMENT FOR THE CLUSTER MISSION |url=https://link.springer.com/article/10.1023/A:1004975108657 |journal=Space Science Reviews |language=en |volume=79 |issue=1 |pages=137–156 |doi=10.1023/A:1004975108657 |issn=1572-9672|url-access=subscription }}

To detect an ESW, the data from a device measuring the electric field is sent to the digital signal processor. This data is analyzed across a short time window (in the case of MMS, 1 millisecond), taking both the average electric field magnitude and the largest electric field magnitude during that time window. If the peak field strength exceeds some multiple of the average field strength (4 times the field strength in MMS), then the time window is considered to contain an ESW. After this occurs, the ESW can be associated with the peak electric field strength and categorized accordingly. These algorithms vary in success at detection, since both the time window and detection multiplier are chosen by scientists based on the parameters they wish to detect. As such, these algorithms often have false positives and false negatives.

One of the primary physical consequences of ESWs is their creation of electron phase-space holes, a type of structure which prevents low velocity electrons from remaining close to the source of the ESW.{{Cite journal |last1=Muschietti |first1=L. |last2=Ergun |first2=R. E. |last3=Roth |first3=I. |last4=Carlson |first4=C. W. |date=1999-04-15 |title=Phase-space electron holes along magnetic field lines |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/1999GL900207 |journal=Geophysical Research Letters |volume=26 |issue=8 |pages=1093–1096 |bibcode=1999GeoRL..26.1093M |doi=10.1029/1999GL900207 |issn=0094-8276}} These phase-space holes, like the ESWs themselves, can travel stably through the surrounding plasma. Since most plasmas are overall electrically neutral, these phase-space holes often end up behaving as a positive pseudoparticle.

In general, in order to form an electron phase-space hole, the electric potential energy associated with the ESW's potential needs to exceed the kinetic energy of electrons in the plasma (behavior analogous to potential hills). Research has shown that one possible set of situations where this occurs naturally are kinetic instabilities.{{Cite journal |last=Buneman |first=O. |date=1963-04-01 |title=Excitation of Field Aligned Sound Waves by Electron Streams |url=https://link.aps.org/doi/10.1103/PhysRevLett.10.285 |journal=Physical Review Letters |volume=10 |issue=7 |pages=285–287 |bibcode=1963PhRvL..10..285B |doi=10.1103/PhysRevLett.10.285 |issn=0031-9007|url-access=subscription }} One observed example of this is the increased occurrence of these holes near Earth's bow shock and magnetopause, where the incoming solar wind collides with Earth's magnetosphere to produce large amounts of turbulence in the plasma.{{Cite journal |last1=Hansel |first1=P. J. |last2=Wilder |first2=F. D. |last3=Malaspina |first3=D. M. |last4=Ergun |first4=R. E. |last5=Ahmadi |first5=N. |last6=Holmes |first6=J. C. |last7=Goodrich |first7=K. A. |last8=Fuselier |first8=S. |last9=Giles |first9=B. |last10=Russell |first10=C. T. |last11=Torbert |first11=R. |last12=Strangeway |first12=R. |last13=Khotyaintsev |first13=Y. |last14=Lindqvist |first14=P. -A. |last15=Burch |first15=J. |date=December 2021 |title=Mapping MMS Observations of Solitary Waves in Earth's Magnetic Field |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2021JA029389 |journal=Journal of Geophysical Research: Space Physics |volume=126 |issue=12 |bibcode=2021JGRA..12629389H |doi=10.1029/2021JA029389 |issn=2169-9380}}

The definition of an ESW is broad enough that, on occasion, research distinguishes between different types:

• Ion-acoustic solitary waves: A type of ESW that occurs when the electric potential that causes the ESW produces an ion acoustic wave.{{Cite journal |last=Ikezi |first=H. |date=1973-10-01 |title=Experiments on ion-acoustic solitary waves |url=https://doi.org/10.1063/1.1694194 |journal=The Physics of Fluids |volume=16 |issue=10 |pages=1668–1675 |doi=10.1063/1.1694194 |issn=0031-9171}}
• Electron-acoustic solitary waves: A type of ESW that produces an acoustic wave associated with electrons. These tend to be substantially faster and higher frequency than ion-acoustic solitary waves.{{Cite journal |last1=Lakhina |first1=Gurbax Singh |last2=Singh |first2=Satyavir |last3=Rubia |first3=Rajith |last4=Devanandhan |first4=Selvaraj |date=2021-10-21 |title=Electrostatic Solitary Structures in Space Plasmas: Soliton Perspective |journal=Plasma |volume=4 |issue=4 |pages=681–731 |doi=10.3390/plasma4040035 |doi-access=free |issn=2571-6182}}
• Supersolitary waves: A type of ESW whose electric potential include pulses on even smaller time scales than the ESW itself.{{Cite journal |last1=Varghese |first1=Steffy Sara |last2=Singh |first2=Kuldeep |last3=Kourakis |first3=Ioannis |date=2024-06-01 |title=Electrostatic supersolitary waves: A challenging paradigm in nonlinear plasma science and beyond – State of the art and overview of recent results |url=https://linkinghub.elsevier.com/retrieve/pii/S277282852400013X |journal=Fundamental Plasma Physics |volume=10 |pages=100048 |doi=10.1016/j.fpp.2024.100048 |issn=2772-8285|doi-access=free }}

• Soliton
• Interplanetary magnetic field
• Solar wind
• Electric potential
• Turbulence
• Time domain electromagnetics

:a.{{Note|a}}An ESW itself is strictly an electromagnetic phenomenon, and as such is technically non-dependent on media. However, this technicality should be observed with caution. Nearly all conditions that give rise to an ESW are theorized to be dependent on the plasma medium they reside in.

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:b.{{note|b}}Though the identity of the other 3 co-authors is known for certain, the career of K. Cerny after the publishing of their paper is poorly documented. The first name, date, school, and major associated with graduation heavily suggest that Keith Cerny is the K. Cerny credited on the paper, but this is (as-of-yet) unconfirmed.

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