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Title: Simulation of energy-dependent electron diffusion processes in the Earth's outer radiation belt

Abstract

The radial and local diffusion processes induced by various plasma waves govern the highly energetic electron dynamics in the Earth's radiation belts, causing distinct characteristics in electron distributions at various energies. In this study, we present our simulation results of the energetic electron evolution during a geomagnetic storm using the University of California, Los Angeles 3-D diffusion code. Following the plasma sheet electron injections, the electrons at different energy bands detected by the Magnetic Electron Ion Spectrometer (MagEIS) and Relativistic Electron Proton Telescope (REPT) instruments on board the Van Allen Probes exhibit a rapid enhancement followed by a slow diffusive movement in differential energy fluxes, and the radial extent to which electrons can penetrate into depends on energy with closer penetration toward the Earth at lower energies than higher energies. We incorporate radial diffusion, local acceleration, and loss processes due to whistler mode wave observations to perform a 3-D diffusion simulation. Here, our simulation results demonstrate that chorus waves cause electron flux increase by more than 1 order of magnitude during the first 18 h, and the subsequent radial extents of the energetic electrons during the storm recovery phase are determined by the coupled radial diffusion and the pitch anglemore » scattering by EMIC waves and plasmaspheric hiss. The radial diffusion caused by ULF waves and local plasma wave scattering are energy dependent, which lead to the observed electron flux variations with energy dependences. Lastly, this study suggests that plasma wave distributions in the inner magnetosphere are crucial for the energy-dependent intrusions of several hundred keV to several MeV electrons.« less

Authors:
ORCiD logo [1];  [1];  [1];  [1];  [2]; ORCiD logo [3];  [4];  [4];  [4];  [3];  [5];  [6];  [7];  [7];  [8]
  1. Univ. of California, Los Angeles, CA (United States). Dept. of Atmospheric and Oceanic Sciences
  2. Univ. of California, Los Angeles, CA (United States). Dept. of Atmospheric and Oceanic Sciences; Univ. of California, Los Angeles, CA (United States). Inst. of Geophysics and Planetary Physics/Earth, Planetary and Space Sciences
  3. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  4. Univ. of Iowa, Iowa City, IA (United States). Dept. of Physics and Astronomy
  5. Univ. of New Hampshire, Durham, NH (United States). Inst. for the Study of Earth, Oceans, and Space
  6. Univ. of Colorado, Boulder, CO (United States). Lab. for Atmospheric and Space Physics
  7. Aerospace Corporation, Los Angeles, CA (United States)
  8. Univ. of California, Los Angeles, CA (United States). Inst. of Geophysics and Planetary Physics/Earth, Planetary and Space Sciences
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
National Aeronautics and Space Administration (NASA); National Science Foundation (NSF); USDOE
OSTI Identifier:
1402607
Report Number(s):
LA-UR-16-22083
Journal ID: ISSN 2169-9380; TRN: US1703005
Grant/Contract Number:  
AC52-06NA25396; 967399; 921647; NAS5-01072; FA9550-15-1-0158; NNX15AI96G; NNX15AF61G; NNX11AR64G; NNX13AI61G; NNX14AI18G; 1405054; 1451911
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Geophysical Research. Space Physics
Additional Journal Information:
Journal Volume: 121; Journal Issue: 5; Journal ID: ISSN 2169-9380
Publisher:
American Geophysical Union
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 58 GEOSCIENCES; Heliospheric and Magnetospheric Physics; energy-dependent diffusion; radiation belt simulation; electron acceleration and loss; radial diffusion

Citation Formats

Ma, Q., Li, W., Thorne, R. M., Nishimura, Y., Zhang, X. -J., Reeves, Geoffrey D., Kletzing, C. A., Kurth, W. S., Hospodarsky, G. B., Henderson, M. G., Spence, H. E., Baker, D. N., Blake, J. B., Fennell, J. F., and Angelopoulos, V.. Simulation of energy-dependent electron diffusion processes in the Earth's outer radiation belt. United States: N. p., 2016. Web. doi:10.1002/2016JA022507.
Ma, Q., Li, W., Thorne, R. M., Nishimura, Y., Zhang, X. -J., Reeves, Geoffrey D., Kletzing, C. A., Kurth, W. S., Hospodarsky, G. B., Henderson, M. G., Spence, H. E., Baker, D. N., Blake, J. B., Fennell, J. F., & Angelopoulos, V.. Simulation of energy-dependent electron diffusion processes in the Earth's outer radiation belt. United States. https://doi.org/10.1002/2016JA022507
Ma, Q., Li, W., Thorne, R. M., Nishimura, Y., Zhang, X. -J., Reeves, Geoffrey D., Kletzing, C. A., Kurth, W. S., Hospodarsky, G. B., Henderson, M. G., Spence, H. E., Baker, D. N., Blake, J. B., Fennell, J. F., and Angelopoulos, V.. Thu . "Simulation of energy-dependent electron diffusion processes in the Earth's outer radiation belt". United States. https://doi.org/10.1002/2016JA022507. https://www.osti.gov/servlets/purl/1402607.
@article{osti_1402607,
title = {Simulation of energy-dependent electron diffusion processes in the Earth's outer radiation belt},
author = {Ma, Q. and Li, W. and Thorne, R. M. and Nishimura, Y. and Zhang, X. -J. and Reeves, Geoffrey D. and Kletzing, C. A. and Kurth, W. S. and Hospodarsky, G. B. and Henderson, M. G. and Spence, H. E. and Baker, D. N. and Blake, J. B. and Fennell, J. F. and Angelopoulos, V.},
abstractNote = {The radial and local diffusion processes induced by various plasma waves govern the highly energetic electron dynamics in the Earth's radiation belts, causing distinct characteristics in electron distributions at various energies. In this study, we present our simulation results of the energetic electron evolution during a geomagnetic storm using the University of California, Los Angeles 3-D diffusion code. Following the plasma sheet electron injections, the electrons at different energy bands detected by the Magnetic Electron Ion Spectrometer (MagEIS) and Relativistic Electron Proton Telescope (REPT) instruments on board the Van Allen Probes exhibit a rapid enhancement followed by a slow diffusive movement in differential energy fluxes, and the radial extent to which electrons can penetrate into depends on energy with closer penetration toward the Earth at lower energies than higher energies. We incorporate radial diffusion, local acceleration, and loss processes due to whistler mode wave observations to perform a 3-D diffusion simulation. Here, our simulation results demonstrate that chorus waves cause electron flux increase by more than 1 order of magnitude during the first 18 h, and the subsequent radial extents of the energetic electrons during the storm recovery phase are determined by the coupled radial diffusion and the pitch angle scattering by EMIC waves and plasmaspheric hiss. The radial diffusion caused by ULF waves and local plasma wave scattering are energy dependent, which lead to the observed electron flux variations with energy dependences. Lastly, this study suggests that plasma wave distributions in the inner magnetosphere are crucial for the energy-dependent intrusions of several hundred keV to several MeV electrons.},
doi = {10.1002/2016JA022507},
journal = {Journal of Geophysical Research. Space Physics},
number = 5,
volume = 121,
place = {United States},
year = {2016},
month = {4}
}

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Works referenced in this record:

The THEMIS Mission
journal, April 2008


Electron densities inferred from plasma wave spectra obtained by the Waves instrument on Van Allen Probes: Van Allen Probes Electron Densities
journal, February 2015

  • Kurth, W. S.; De Pascuale, S.; Faden, J. B.
  • Journal of Geophysical Research: Space Physics, Vol. 120, Issue 2
  • DOI: 10.1002/2014JA020857

Radiation belt electron acceleration by chorus waves during the 17 March 2013 storm: ELECTRON ACCELERATION BY CHORUS WAVES
journal, June 2014

  • Li, W.; Thorne, R. M.; Ma, Q.
  • Journal of Geophysical Research: Space Physics, Vol. 119, Issue 6
  • DOI: 10.1002/2014JA019945

Analytic expressions for ULF wave radiation belt radial diffusion coefficients: Analytic Radial Diffusion Coefficients
journal, March 2014

  • Ozeke, Louis G.; Mann, Ian R.; Murphy, Kyle R.
  • Journal of Geophysical Research: Space Physics, Vol. 119, Issue 3
  • DOI: 10.1002/2013JA019204

Electron lifetimes from narrowband wave-particle interactions within the plasmasphere: Electron lifetimes from narrowband waves
journal, November 2014

  • Ripoll, J. -F.; Albert, J. M.; Cunningham, G. S.
  • Journal of Geophysical Research: Space Physics, Vol. 119, Issue 11
  • DOI: 10.1002/2014JA020217

Event-specific chorus wave and electron seed population models in DREAM3D using the Van Allen Probes: Tu et al.: Event-specific chorus and seed electrons
journal, March 2014

  • Tu, Weichao; Cunningham, G. S.; Chen, Y.
  • Geophysical Research Letters, Vol. 41, Issue 5
  • DOI: 10.1002/2013GL058819

First Results of the THEMIS Search Coil Magnetometers
journal, June 2008


Modeling the dynamics of the inner magnetosphere during strong geomagnetic storms
journal, January 2005


THEMIS observations of electromagnetic ion cyclotron wave occurrence: Dependence on AE, SYMH, and solar wind dynamic pressure: THEMIS EMIC WAVE DISTRIBUTION
journal, October 2012

  • Usanova, M. E.; Mann, I. R.; Bortnik, J.
  • Journal of Geophysical Research: Space Physics, Vol. 117, Issue A10
  • DOI: 10.1029/2012JA018049

Magnetic field power spectra and magnetic radial diffusion coefficients using CRRES magnetometer data
journal, February 2015

  • Ali, Ashar F.; Elkington, Scot R.; Tu, Weichao
  • Journal of Geophysical Research: Space Physics, Vol. 120, Issue 2
  • DOI: 10.1002/2014JA020419

Energy-dependent dynamics of keV to MeV electrons in the inner zone, outer zone, and slot regions: ENERGY-DEPENDENT RADIATION BELT DYNAMICS
journal, January 2016

  • Reeves, Geoffrey D.; Friedel, Reiner H. W.; Larsen, Brian A.
  • Journal of Geophysical Research: Space Physics, Vol. 121, Issue 1
  • DOI: 10.1002/2015JA021569

Radial diffusion analysis of outer radiation belt electrons during the October 9, 1990, magnetic storm
journal, January 2000

  • Brautigam, D. H.; Albert, J. M.
  • Journal of Geophysical Research: Space Physics, Vol. 105, Issue A1
  • DOI: 10.1029/1999JA900344

The Magnetic Electron Ion Spectrometer (MagEIS) Instruments Aboard the Radiation Belt Storm Probes (RBSP) Spacecraft
journal, June 2013

  • Blake, J. B.; Carranza, P. A.; Claudepierre, S. G.
  • Space Science Reviews, Vol. 179, Issue 1-4
  • DOI: 10.1007/s11214-013-9991-8

The Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) on RBSP
journal, June 2013


Evolution of electron fluxes in the outer radiation belt computed with the VERB code: VERB CODE
journal, November 2009

  • Shprits, Yuri Y.; Subbotin, Dmitri; Ni, Binbin
  • Journal of Geophysical Research: Space Physics, Vol. 114, Issue A11
  • DOI: 10.1029/2008JA013784

Modeling inward diffusion and slow decay of energetic electrons in the Earth's outer radiation belt: Modeling gradual diffusion processes
journal, February 2015

  • Ma, Q.; Li, W.; Thorne, R. M.
  • Geophysical Research Letters, Vol. 42, Issue 4
  • DOI: 10.1002/2014GL062977

ULF wave derived radiation belt radial diffusion coefficients: ULF WAVE DIFFUSION COEFFICIENTS
journal, April 2012

  • Ozeke, Louis G.; Mann, Ian R.; Murphy, Kyle R.
  • Journal of Geophysical Research: Space Physics, Vol. 117, Issue A4
  • DOI: 10.1029/2011JA017463

New conjunctive CubeSat and balloon measurements to quantify rapid energetic electron precipitation: QUANTIFYING RAPID E - PRECIPITATION LOSS
journal, November 2013

  • Blum, L. W.; Schiller, Q.; Li, X.
  • Geophysical Research Letters, Vol. 40, Issue 22
  • DOI: 10.1002/2013GL058546

Global distributions of suprathermal electrons observed on THEMIS and potential mechanisms for access into the plasmasphere: ELECTRON ACCESS INTO THE PLASMASPHERE
journal, December 2010

  • Li, W.; Thorne, R. M.; Bortnik, J.
  • Journal of Geophysical Research: Space Physics, Vol. 115, Issue A12
  • DOI: 10.1029/2010JA015687

Resonant scattering of plasma sheet electrons leading to diffuse auroral precipitation: 2. Evaluation for whistler mode chorus waves: DIFFUSE AURORAL SCATTERING BY CHORUS
journal, April 2011

  • Ni, Binbin; Thorne, Richard M.; Meredith, Nigel P.
  • Journal of Geophysical Research: Space Physics, Vol. 116, Issue A4
  • DOI: 10.1029/2010JA016233

A novel technique to construct the global distribution of whistler mode chorus wave intensity using low-altitude POES electron data: A novel technique for global wave model
journal, July 2014

  • Ni, Binbin; Li, Wen; Thorne, Richard M.
  • Journal of Geophysical Research: Space Physics, Vol. 119, Issue 7
  • DOI: 10.1002/2014JA019935

Diffusion Processes Influenced by Conjugate-Point Wave Phenomena
journal, July 1968


Radiation belt dynamics: The importance of wave-particle interactions: FRONTIER
journal, November 2010

  • Thorne, Richard Mansergh
  • Geophysical Research Letters, Vol. 37, Issue 22
  • DOI: 10.1029/2010GL044990

Rapid local acceleration of relativistic radiation-belt electrons by magnetospheric chorus
journal, December 2013


Pitch angle and energy diffusion coefficients from resonant interactions with ion–cyclotron and whistler waves
journal, December 1974


The occurrence and wave properties of H + ‐, He + ‐, and O + ‐band EMIC waves observed by the Van Allen Probes
journal, September 2015

  • Saikin, A. A.; Zhang, J. ‐C.; Allen, R. C.
  • Journal of Geophysical Research: Space Physics, Vol. 120, Issue 9
  • DOI: 10.1002/2015JA021358

Wave acceleration of electrons in the Van Allen radiation belts
journal, September 2005

  • Horne, Richard B.; Thorne, Richard M.; Shprits, Yuri Y.
  • Nature, Vol. 437, Issue 7056
  • DOI: 10.1038/nature03939

Substorm dependence of plasmaspheric hiss
journal, January 2004


Potential waves for relativistic electron scattering and stochastic acceleration during magnetic storms
journal, August 1998

  • Horne, Richard B.; Thorne, Richard M.
  • Geophysical Research Letters, Vol. 25, Issue 15
  • DOI: 10.1029/98GL01002

Electron Acceleration in the Heart of the Van Allen Radiation Belts
journal, July 2013


Velocity Space Diffusion from Weak Plasma Turbulence in a Magnetic Field
journal, January 1966


Relativistic electron pitch-angle scattering by electromagnetic ion cyclotron waves during geomagnetic storms
journal, January 2003


Electron losses from the radiation belts caused by EMIC waves
journal, November 2014

  • Kersten, Tobias; Horne, Richard B.; Glauert, Sarah A.
  • Journal of Geophysical Research: Space Physics, Vol. 119, Issue 11
  • DOI: 10.1002/2014JA020366

Geomagnetically trapped electrons from cosmic ray albedo neutrons
journal, December 1961

  • Lenchek, A. M.; Singer, S. F.; Wentworth, R. C.
  • Journal of Geophysical Research, Vol. 66, Issue 12
  • DOI: 10.1029/JZ066i012p04027

Science Objectives and Rationale for the Radiation Belt Storm Probes Mission
journal, September 2012


Resonant scattering of plasma sheet electrons by whistler-mode chorus: Contribution to diffuse auroral precipitation
journal, January 2008

  • Ni, Binbin; Thorne, Richard M.; Shprits, Yuri Y.
  • Geophysical Research Letters, Vol. 35, Issue 11
  • DOI: 10.1029/2008GL034032

Radial diffusion of ionized helium and protons: A probe for magnetospheric dynamics
journal, April 1972


An empirical plasmasphere and trough density model: CRRES observations
journal, November 2001

  • Sheeley, B. W.; Moldwin, M. B.; Rassoul, H. K.
  • Journal of Geophysical Research: Space Physics, Vol. 106, Issue A11
  • DOI: 10.1029/2000JA000286

Modeling radiation belt radial diffusion in ULF wave fields: 2. Estimating rates of radial diffusion using combined MHD and particle codes: ESTIMATE RADIAL DIFFUSION RATE
journal, June 2010

  • Huang, Chia-Lin; Spence, Harlan E.; Hudson, Mary K.
  • Journal of Geophysical Research: Space Physics, Vol. 115, Issue A6
  • DOI: 10.1029/2009JA014918

Modeling radiation belt electron dynamics during GEM challenge intervals with the DREAM3D diffusion model: MODELING RADIATION BELT WITH DREAM3D
journal, October 2013

  • Tu, Weichao; Cunningham, G. S.; Chen, Y.
  • Journal of Geophysical Research: Space Physics, Vol. 118, Issue 10
  • DOI: 10.1002/jgra.50560

Comparative study of outer-zone relativistic electrons observed by Akebono and CRRES
journal, January 2005


Calculation of pitch angle and energy diffusion coefficients with the PADIE code: PADIE DIFFUSION CODE
journal, April 2005

  • Glauert, Sarah A.; Horne, Richard B.
  • Journal of Geophysical Research: Space Physics, Vol. 110, Issue A4
  • DOI: 10.1029/2004JA010851

The Electric Field Instrument (EFI) for THEMIS
journal, December 2008


Multidimensional quasi-linear diffusion of radiation belt electrons: MULTIDIMENSIONAL DIFFUSION
journal, July 2005

  • Albert, J. M.; Young, S. L.
  • Geophysical Research Letters, Vol. 32, Issue 14
  • DOI: 10.1029/2005GL023191

Slot region electron loss timescales due to plasmaspheric hiss and lightning-generated whistlers: SLOT REGION ELECTRON LOSS TIMESCALES
journal, August 2007

  • Meredith, Nigel P.; Horne, Richard B.; Glauert, Sarah A.
  • Journal of Geophysical Research: Space Physics, Vol. 112, Issue A8
  • DOI: 10.1029/2007JA012413

The THEMIS ESA Plasma Instrument and In-flight Calibration
journal, October 2008


Statistics of whistler mode waves in the outer radiation belt: Cluster STAFF-SA measurements: WHISTLER WAVES STATISTICS
journal, June 2013

  • Agapitov, Oleksiy; Artemyev, Anton; Krasnoselskikh, Vladimir
  • Journal of Geophysical Research: Space Physics, Vol. 118, Issue 6
  • DOI: 10.1002/jgra.50312

General relations for resonant particle diffusion in pitch angle and energy
journal, August 1974


The effects of geomagnetic storms on electrons in Earth's radiation belts: RADIATION BELT ELECTRON RESPONSES TO STORMS
journal, November 2015

  • Turner, D. L.; O'Brien, T. P.; Fennell, J. F.
  • Geophysical Research Letters, Vol. 42, Issue 21
  • DOI: 10.1002/2015GL064747

Relativistic bounce-averaged quasilinear diffusion equation for low-frequency electromagnetic fluctuations
journal, November 2001

  • Brizard, Alain J.; Chan, Anthony A.
  • Physics of Plasmas, Vol. 8, Issue 11
  • DOI: 10.1063/1.1408623

Acceleration and loss of relativistic electrons during geomagnetic storms: ACCELERATION AND LOSS OF RELATIVISTIC ELECTRONS
journal, May 2003

  • Reeves, G. D.; McAdams, K. L.; Friedel, R. H. W.
  • Geophysical Research Letters, Vol. 30, Issue 10
  • DOI: 10.1029/2002GL016513

Precipitation of radiation belt electrons by EMIC waves, observed from ground and space
journal, January 2008

  • Miyoshi, Y.; Sakaguchi, K.; Shiokawa, K.
  • Geophysical Research Letters, Vol. 35, Issue 23
  • DOI: 10.1029/2008GL035727

Three-dimensional simulations of outer radiation belt electron dynamics including cross-diffusion terms: 3D SIMULATIONS INCLUDING CROSS TERMS
journal, May 2010

  • Xiao, Fuliang; Su, Zhenpeng; Zheng, Huinan
  • Journal of Geophysical Research: Space Physics, Vol. 115, Issue A5
  • DOI: 10.1029/2009JA014541

Dynamic evolution of energetic outer zone electrons due to wave-particle interactions during storms: EVOLUTION OF ENERGETIC ELECTRONS
journal, October 2007

  • Li, W.; Shprits, Y. Y.; Thorne, R. M.
  • Journal of Geophysical Research: Space Physics, Vol. 112, Issue A10
  • DOI: 10.1029/2007JA012368

Resonant scattering and resultant pitch angle evolution of relativistic electrons by plasmaspheric hiss: RELATIVISTIC ELECTRON SCATTERING BY HISS
journal, December 2013

  • Ni, Binbin; Bortnik, Jacob; Thorne, Richard M.
  • Journal of Geophysical Research: Space Physics, Vol. 118, Issue 12
  • DOI: 10.1002/2013JA019260

Global distribution of wave amplitudes and wave normal angles of chorus waves using THEMIS wave observations: CHORUS WAVE DISTRIBUTION ON THEMIS
journal, December 2011

  • Li, W.; Bortnik, J.; Thorne, R. M.
  • Journal of Geophysical Research: Space Physics, Vol. 116, Issue A12
  • DOI: 10.1029/2011JA017035

Global morphology and spectral properties of EMIC waves derived from CRRES observations
journal, July 2014

  • Meredith, Nigel P.; Horne, Richard B.; Kersten, Tobias
  • Journal of Geophysical Research: Space Physics, Vol. 119, Issue 7
  • DOI: 10.1002/2014JA020064

A background correction algorithm for Van Allen Probes MagEIS electron flux measurements: MAGEIS BACKGROUND CORRECTION ALGORITHM
journal, July 2015

  • Claudepierre, S. G.; O'Brien, T. P.; Blake, J. B.
  • Journal of Geophysical Research: Space Physics, Vol. 120, Issue 7
  • DOI: 10.1002/2015JA021171

Resonant scattering of energetic electrons by unusual low-frequency hiss
journal, March 2014

  • Ni, Binbin; Li, Wen; Thorne, Richard M.
  • Geophysical Research Letters, Vol. 41, Issue 6
  • DOI: 10.1002/2014GL059389

Works referencing / citing this record:

Long-term evolution of electron distribution function due to nonlinear resonant interaction with whistler mode waves
journal, April 2018

  • Artemyev, Anton V.; Neishtadt, Anatoly I.; Vasiliev, Alexei A.
  • Journal of Plasma Physics, Vol. 84, Issue 2
  • DOI: 10.1017/s0022377818000260

Parallel Acceleration of Suprathermal Electrons Caused by Whistler‐Mode Hiss Waves
journal, November 2019

  • Li, Jinxing; Ma, Qianli; Bortnik, Jacob
  • Geophysical Research Letters, Vol. 46, Issue 22
  • DOI: 10.1029/2019gl085562

Equatorial Magnetosonic Waves Observed by Cluster Satellites: The Chirikov Resonance Overlap Criterion
journal, April 2019

  • Aryan, Homayon; Walker, Simon N.; Balikhin, Michael A.
  • Journal of Geophysical Research: Space Physics, Vol. 124, Issue 4
  • DOI: 10.1029/2019ja026680

Characterization and Evolution of Radiation Belt Electron Energy Spectra Based on the Van Allen Probes Measurements
journal, June 2019

  • Zhao, H.; Johnston, W. R.; Baker, D. N.
  • Journal of Geophysical Research: Space Physics, Vol. 124, Issue 6
  • DOI: 10.1029/2019ja026697

Explaining the apparent impenetrable barrier to ultra-relativistic electrons in the outer Van Allen belt
journal, May 2018


From solar sneezing to killer electrons: outer radiation belt response to solar eruptions
journal, May 2019

  • Daglis, Ioannis A.; Katsavrias, Christos; Georgiou, Marina
  • Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 377, Issue 2148
  • DOI: 10.1098/rsta.2018.0097