Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to...
Transcript of Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to...
![Page 1: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/1.jpg)
Progress in Vlasov-Fokker-Planck simulations of laser-
plasma interactions C. P. Ridgers, M. W. Sherlock, R. J. Kingham,
A.Thomas, R. Evans Imperial College London
![Page 2: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/2.jpg)
Outline
Part 1 – simulations of long-pulse laser-plasma interactions. (1) The interplay of magnetic field dynamics and
non-locality. (2) Magnetic field generation
Part 2 – Vlasov-Fokker-Planck simulations of short-pulse laser-plasma interactions relevant to fast ignition. The importance of rear-surface effects.
![Page 3: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/3.jpg)
Background – inertial fusion
HiPER
• High Power Exper. Research facility
• Proposed European academic facility, with 200--400 kJ
• Site: TBD… UK ?
• 1.8 MJ in UV based at LLNL
• Partially working now… Ignition in 2011
NIF - National Ignition Facility
![Page 4: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/4.jpg)
Indirect drive ICF1,2
1Lindl J.D. et al, Physics of Plasmas, 11, 339, 2004 2Glenzer S.H. et al, Physics of Plasmas, 6, 2117, 1999
Gradients in Te, ne large enough so that classical diffusive transport (e.g. Spitzer) not valid.
electron density ∇n
∇T
Bθ
electron temperature
B-field capsule
![Page 5: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/5.jpg)
Indirect drive ICF1,2
1Lindl J.D. et al, Physics of Plasmas, 11, 339, 2004 2Glenzer S.H. et al, Physics of Plasmas, 6, 2117, 1999
Gradients in Te, ne large enough so that classical diffusive transport (e.g. Spitzer) not valid.
electron density ∇n
∇T
Bθ
electron temperature
B-field capsule
1) Does the heat-flow obey classical theory? (non-locality)
![Page 6: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/6.jpg)
Indirect drive ICF1,2
1Lindl J.D. et al, Physics of Plasmas, 11, 339, 2004 2Glenzer S.H. et al, Physics of Plasmas, 6, 2117, 1999
Gradients in Te, ne large enough so that classical diffusive transport (e.g. Spitzer) not valid.
electron density ∇n
∇T
Bθ
electron temperature
B-field capsule
B-field
1) Does the heat-flow obey classical theory? (non-locality)
2) Interplay between non-locality and B-fields.
![Page 7: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/7.jpg)
Indirect drive ICF1,2
1) Does the heat-flow obey classical theory? (non-locality)
2) Interplay between non-locality and B-fields.
First part of talk will examine these three points:
![Page 8: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/8.jpg)
Modelling long-pulse LPI
N-body equations of motion
Vlasov-Fokker-Planck equation ‘Distribution
function’ picture
Particle-in-cell PIC MHD
Distribution function is Maxwellian
![Page 9: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/9.jpg)
The Vlasov-Fokker-Planck equation
The VFP equation is the appropriate kinetic equation for a plasma:
Distribution function - density of particles in 6D phase space
![Page 10: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/10.jpg)
The Vlasov-Fokker-Planck equation The VFP equation is the appropriate kinetic equation for a plasma:
VFP equation is continuity is 6D phase space. Compare to:
![Page 11: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/11.jpg)
Modelling long-pulse LPI
Vlasov-Fokker-Planck equation
MHD
Distribution function is Maxwellian
![Page 12: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/12.jpg)
Modelling long-pulse LPI
Distribution function is Maxwellian: Vlasov-Fokker-Planck equation
MHD
Distribution function is Maxwellian
![Page 13: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/13.jpg)
Modelling long-pulse LPI
Vlasov-Fokker-Planck equation
MHD
Distribution function is Maxwellian
MHD requires closure - classical heat flow equation:
Distribution function is Maxwellian:
![Page 14: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/14.jpg)
Modelling long-pulse LPI
Vlasov-Fokker-Planck equation
MHD
Distribution function is Maxwellian X
X MHD requires closure - classical heat flow equation:
Distribution function is Maxwellian:
VFP - Distribution function is NOT Maxwellian
![Page 15: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/15.jpg)
What causes the break-down of classical transport?
Non-Locality: Scale-length of physical variables becomes comparable to mean-free-path
T Mean free path
x A highly non-local situation
![Page 16: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/16.jpg)
What causes the break-down of classical transport?
Non-Locality: Scale-length of physical variables becomes comparable to mean-free-path
T Mean free path
x A highly non-local situation
Non-locality causes distribution to move away from Maxwellian
![Page 17: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/17.jpg)
What causes the break-down of classical transport?
T Mean free path
x A highly non-local situation
f
fM
DISTRIBUTION FUNCTION
Non-Locality: Scale-length of physical variables becomes comparable to mean-free-path
![Page 18: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/18.jpg)
What causes the break-down of classical transport?
Effect on B-field on non-locality?
Non-Locality: Scale-length of physical variables becomes comparable to mean-free-path
T Mean free path
x A highly non-local situation
Larmor radius
![Page 19: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/19.jpg)
• Solves the VFP Equation (and Maxwell’s equations) with hydrodynamic ion flow
• Expands distribution function in spherical harmonics so can only deal with weakly anisotropic distributions
• Implicit finite-differencing - very robust + large Δt (e.g. ~ps for Δx~1µm vs 3fs)
• Valid when collisions important (unlike PIC)
• 2D (3V) VFP code with self consistent Bz-fields
IMPACT - Implicit Magnetised Plasma And Collisional Transport5
5Kingham R.J. & Bell A.R., Journal of Computational Physics 194, 1, 2004
![Page 20: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/20.jpg)
What’s new?
FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields
Ideal for studying interaction between non-local transport and magnetic fields
IMPACT - Implicit Magnetised Plasma And Collisional Transport5
5Kingham R.J. & Bell A.R., Journal of Computational Physics 194, 1, 2004
![Page 21: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/21.jpg)
Indirect drive ICF1,2
1) Does the heat-flow obey classical theory? (non-locality)
2) Interplay between non-locality and B-fields.
First part of talk will examine these three points:
![Page 22: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/22.jpg)
B-Field and Non-Locality: Experimental Measurement6
N2 gas jet 1µm, 1J, 1ns laser beam
0.5µm, 1J, 200ps probe beam - Thomson scattering
B-field
Gas jet: Initial temperature - 20eV Number density- 1.5x1019cm-3
Laser: Energy - 100J Spot size - 100µm Intensity - 1x1015W/cm2 Wavelength - 1.053µm Pulse duration - 1ns
Magnetic field: Non-local case - 0T Local case - 12T
Strong B-field expected to “localize” transport
6Froula D.H. et al, Physical Review Letters, 98, 135001, 2007
![Page 23: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/23.jpg)
IMPOSE A RANGE OF FIELDS B=0-12T
Using IMPACT with MHD to model magnetized transport experiments
Simulation geometry:
![Page 24: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/24.jpg)
Simulation results - pre-heat
Transverse Space (µm)
LAS
ER
Elec
tron
Tem
pera
ture
(eV)
EXPERIMENT
TEMPERATURE PROFILE ACROSS LASER BEAM
![Page 25: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/25.jpg)
Simulation results - pre-heat7
Transverse Space (µm)
LAS
ER
Elec
tron
Tem
pera
ture
(eV)
EXPERIMENT IMPACT SIMULATION (After 440ps)
7Ridgers C.P., Kingham R.J. & Thomas A.G.R., Phys Rev Lett, 100, 075003, 2008
B=0T
B=12T
TEMPERATURE PROFILE ACROSS LASER BEAM
![Page 26: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/26.jpg)
Simulation results - pre-heat7
We do see less pre-heat in un-magnetized case - but is it non-local?
Transverse Space (µm)
LAS
ER
Elec
tron
Tem
pera
ture
(eV)
EXPERIMENT IMPACT SIMULATION (After 440ps)
7Ridgers C.P., Kingham R.J. & Thomas A.G.R., Phys Rev Lett, 100, 075003, 2008
B=0T
B=12T
TEMPERATURE PROFILE ACROSS LASER BEAM
![Page 27: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/27.jpg)
Simulation results - is it non-local?7
7Ridgers C.P., Kingham R.J. & Thomas A.G.R., Phys Rev Lett, 100, 075003, 2008
RADIAL HEAT FLOW
B=0T
B=12T
![Page 28: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/28.jpg)
Simulation results - is it non-local?7
Braginskii heat flow:
q from instantaneous temperature profile.
RADIAL HEAT FLOW
12T field suppresses non-locality
B=0T
B=12T
7Ridgers C.P., Kingham R.J. & Thomas A.G.R., Phys Rev Lett, 100, 075003, 2008
![Page 29: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/29.jpg)
Indirect drive ICF1,2
1) Does the heat-flow obey classical theory? (non-locality)
2) Interplay between non-locality and B-fields.
First part of talk will examine these three points:
A: Non-locality causes classical theory to break down BUT transport can be relocalised by a strong B-field, in which case classical theory applies.
![Page 30: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/30.jpg)
Indirect drive ICF1,2
1) Does the heat-flow obey classical theory? (non-locality)
2) Interplay between non-locality and B-fields.
First part of talk will examine these three points:
![Page 31: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/31.jpg)
The Nernst effect8,9
First moment of VFP equation gives Ohm’s law:
8Nishiguchi A. et al, Physics of Fluids, 28, 3683, 1985
9Haines M.G., Plasma Physics and Controlled Fusion, 28, 1705, 1986
![Page 32: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/32.jpg)
The Nernst effect8,9
First moment of VFP equation gives Ohm’s law:
Put this into Faraday’s law (and simplify!):
8Nishiguchi A. et al, Physics of Fluids, 28, 3683, 1985
9Haines M.G., Plasma Physics and Controlled Fusion, 28, 1705, 1986
![Page 33: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/33.jpg)
The Nernst effect8,9
First moment of VFP equation gives Ohm’s law:
Put this into Faraday’s law (and simplify!):
8Nishiguchi A. et al, Physics of Fluids, 28, 3683, 1985
9Haines M.G., Plasma Physics and Controlled Fusion, 28, 1705, 1986
![Page 34: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/34.jpg)
The Nernst effect8,9
First moment of VFP equation gives Ohm’s law:
Frozen-in flow Resistive diffusion Nernst effect
Put this into Faraday’s law (and simplify!):
8Nishiguchi A. et al, Physics of Fluids, 28, 3683, 1985
9Haines M.G., Plasma Physics and Controlled Fusion, 28, 1705, 1986
![Page 35: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/35.jpg)
The Nernst effect8,9
First moment of VFP equation gives Ohm’s law:
Put this into Faraday’s law (and simplify!):
8Nishiguchi A. et al, Physics of Fluids, 28, 3683, 1985
9Haines M.G., Plasma Physics and Controlled Fusion, 28, 1705, 1986
Nernst effect
![Page 36: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/36.jpg)
B=2T
B=12T
Nernst Advection dominates over Frozen-in-Flow7
Solid lines - full B-field calculation
Dashed lines - frozen-in flow only
B-FIELD ACROSS BEAM
![Page 37: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/37.jpg)
B=2T
B=12T
Nernst Advection dominates over Frozen-in-Flow7
Solid lines - full B-field calculation
Dashed lines - frozen-in flow only
B-FIELD ACROSS BEAM
![Page 38: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/38.jpg)
Evolution of non-locality
Degree of non-locality increases with time
B=2T
B=4T
VS TIME
![Page 39: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/39.jpg)
Indirect drive ICF1,2
1) Does the heat-flow obey classical theory? (non-locality)
2) Interplay between non-locality and B-fields.
First part of talk will examine these three points:
A: Enhanced B-field advection, caused by the Nernst effect, causes B to drop so that even if it were initially sufficient to localise transport, non-locality can re-emerge in time
![Page 40: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/40.jpg)
Summary - non-locality & B-fields
• Magnetic field advection can be dominated by the Nernst effect which therefore should be included.
• Enhanced B-field advection can lead to the re-emergence of non-locality, even if the initial B-field suppresses it.
• MAGNETIC FIELD EVOLUTION AND NON-LOCALITY COUPLED
![Page 41: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/41.jpg)
Can these effects be measured experimentally?
The Nernst effect leads to cooling in the laser heated region when the laser is still on at full power
B=2T
B=12T
B=4T
B=2T
B=12T
B=4T
TEMPERATURE VS TIME VS TIME
![Page 42: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/42.jpg)
Nernst cooling
Equilibrium: = Cooling by thermal conduction
Heating from laser
![Page 43: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/43.jpg)
Nernst cooling
Equilibrium: = Cooling by thermal conduction
Thermal conduction rate controlled by Hall parameter.
Heating from laser
![Page 44: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/44.jpg)
Nernst cooling
Equilibrium: = Heating from laser
Cooling by thermal conduction
Thermal conduction rate controlled by Hall parameter.
Thermal conductivity
![Page 45: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/45.jpg)
Nernst cooling
Equilibrium: = Cooling by thermal conduction
Thermal conduction rate controlled by Hall parameter.
Thermal conductivity
Nernst drives out B-field faster than central hot-spot drives out plasma hydrodynamically, Hall parameter decreases
Heating from laser
![Page 46: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/46.jpg)
Nernst cooling
Equilibrium: = Heating from laser
Cooling by thermal conduction
Thermal conduction rate controlled by Hall parameter.
Thermal conductivity
Conductivity increases
Nernst drives out B-field faster than central hot-spot drives out plasma hydrodynamically, Hall parameter decreases
![Page 47: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/47.jpg)
Nernst cooling
Equilibrium: = Heating from laser
Cooling by thermal conduction
Thermal conduction rate controlled by Hall parameter.
Thermal conductivity
Conductivity increases
Nernst drives out B-field faster than central hot-spot drives out plasma hydrodynamically, Hall parameter decreases
Central temperature decreses – Nernst cooling
![Page 48: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/48.jpg)
Nernst cooling
Equilibrium: = Heating from laser
Cooling by thermal conduction
Thermal conduction rate controlled by Hall parameter.
Thermal conductivity
Nernst drives out B-field faster than central hot-spot drives out plasma hydrodynamically, Hall parameter decreases
Conductivity increases
Central temperature decreses – Nernst cooling
![Page 49: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/49.jpg)
Can these effects be measured experimentally?
B=2T
B=12T
B=4T
B=2T
B=12T
B=4T
TEMPERATURE VS TIME VS TIME
![Page 50: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/50.jpg)
Nernst cooling
Equilibrium: = Heating from laser
Cooling by thermal conduction
Thermal conduction rate controlled by Hall parameter.
Frozen-in flow – ratio B/ne constant – Hall parameter constant
COOLING OF CENTRAL LASER-HEATED REGION DOES NOT OCCUR
![Page 51: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/51.jpg)
Cooling can’t happen for frozen-in-flow
TEMPERATURE VS TIME VS TIME
B=2T
B=12T
B=4T
B=2T
B=12T B=4T
![Page 52: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/52.jpg)
Outline
Part 1 – simulations of long-pulse laser-plasma interactions. (1) The interplay of magnetic field dynamics and
non-locality. (2) Magnetic field generation
Part 2 – Vlasov-Fokker-Planck simulations of short-pulse laser-plasma interactions relevant to fast ignition. The importance of rear-surface effects.
![Page 53: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/53.jpg)
Short-pulse modelling
Fast ignition13 – decouple compression and heating phase.
Long-pulse laser compresses target
Short-pulse laser interacts with cone and generates fast electrons, whose absorption in the core heats target like a spark-plug
13M. Tabak et. al., Physics of Plasmas, 1, 1626 (1971)
![Page 54: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/54.jpg)
Modelling long-pulse LPI
N-body equations of motion
Vlasov-Fokker-Planck equation ‘Distribution
function’ picture
Particle-in-cell PIC MHD
Distribution function is Maxwellian
![Page 55: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/55.jpg)
Modelling long-pulse LPI
N-body equations of motion
Vlasov-Fokker-Planck equation ‘Distribution
function’ picture
Particle-in-cell PIC
VFP handles collisions much better than PIC. PIC prone to noise
![Page 56: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/56.jpg)
• Solves the VFP Equation (and Maxwell’s equations) without expanding distribution function
• Implicit EM solve but explicit momentum space solve limits Δt to 3fs
• Can deal with very anisotropic distribcutions - beams
FIDO14
14Sherlock M.W. submitted to PRL
• Runs in hybrid mode or full VFP mode
• 2D (3V) VFP code with self consistent Ex, Ey and Bz-fields
![Page 57: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/57.jpg)
Short-pulse modelling
Usual model:
Fast electrons treated kinetically
Background electrons treated as fluid particle
source
jf
![Page 58: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/58.jpg)
‘Standard’ hybrid mode
Standard hybrid usually uses reflective boundary conditions
particle source
![Page 59: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/59.jpg)
‘Standard’ hybrid mode
Standard hybrid usually uses reflective boundary conditions
particle source
Ey
Justified by large sheath field at rear surface – reflects particles. But ….
![Page 60: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/60.jpg)
‘Standard’ hybrid mode
Standard hybrid usually uses reflective boundary conditions
particle source
Ey
Justified by large sheath field at rear surface – reflects particles. But ….
… only if there is no magnetic field
![Page 61: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/61.jpg)
‘Improved’ hybrid mode
e
e
particle source
Ey
Improved hybrid:
B-fields lead to more complicated motion of electrons at rear-surface.
Bz
![Page 62: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/62.jpg)
‘Improved’ hybrid mode
e
e
particle source
Ey
Improved hybrid:
B-fields lead to more complicated motion of electrons at rear-surface.
Bz
I will simulate rear-surface effects
![Page 63: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/63.jpg)
‘Improved’ hybrid mode
What’s new?
FIRST continuum VFP hybrid simulation of fast-electron transport in solids to include target edge effects
e
e
particle source
Ey
Bz
![Page 64: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/64.jpg)
Solid-target simulations
Background electron number density ne=7.8x1023cm-3
BACKGROUND PROPERTIES:
BACKGROUND ELECRON NUMBER DENSITY
Ionic charge – Z=13 (Aluminium)
Initial temperature Te=50eV (somewhat arbitrary)
![Page 65: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/65.jpg)
Solid-target simulations
INJECTED ELECTRON DENSITY (after 200fs)
LASER AND ELECTRON INJECTION:
Laser intensity I=5x1019Wcm-2
Absorption fraction η=0.3
Max injection angle θ=28o
Injection region 8µm x 1µm using square mask
![Page 66: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/66.jpg)
Solid-target simulations
Fast electron number density
![Page 67: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/67.jpg)
Solid-target simulations
Electric field
![Page 68: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/68.jpg)
Solid-target simulations
Magnetic field
![Page 69: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/69.jpg)
Particle-tracking Particles injected at 350keV affected by E fields ONLY
E = 350keV
![Page 70: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/70.jpg)
Particle-tracking Particles injected at 350keV affected by E and B fields
E = 350keV
![Page 71: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/71.jpg)
Particle-tracking Particles injected at 1.8MeV affected by E and B fields
E = 1.8MeV
![Page 72: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/72.jpg)
Experiments?
Proposals for experiments:
(1) Is filamentation inside the target real? – use tracer layers and rear surface measurements to see where it happens
(2) Measure rear surface B and look at effects of scale-length?
![Page 73: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/73.jpg)
Summary
• Realistic inclusion of rear surface effects is important for the explanation of experiments.
• Large fields develop at the rear surface of the target and dominate the behaviour of electrons as they exit the target (and head towards experimental diagnositics)
• Large magnetic fields develop, necessitating a two dimensional treatment of the rear surface
![Page 74: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/74.jpg)
Conclusions
• Kinetic modelling gives useful insight into both long and short-pulse laser-plasma interactions
• Magnetic field dynamics can only be modelled correctly by using the appropriate Ohm’s law…
• …In long-pulse gas-jet interactions this means the inclusion of the Nernst effect…
• …In short-pulse solid-target interactions this means including edge effects by blending the Ohm’s law with a full solution of Maxwell’s equations
![Page 75: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/75.jpg)
![Page 76: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/76.jpg)
B=2T
B=12T
Nernst Advection dominates over Frozen-in-Flow
Dimensionless ratio quantifying Nernst vs. Frozen-In-Flow…
B-FIELD ACROSS BEAM
![Page 77: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/77.jpg)
B=2T
B=12T
Nernst Advection dominates over Frozen-in-Flow
Dimensionless ratio quantifying Nernst vs. Frozen-In-Flow…
In sims.
B-FIELD ACROSS BEAM
![Page 78: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/78.jpg)
B=2T
B=12T
Nernst Advection dominates over Frozen-in-Flow
Dimensionless ratio quantifying Nernst vs. Frozen-In-Flow…
Importance of Nernst is reduced with larger ωτ
In sims.
B-FIELD ACROSS BEAM
![Page 79: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/79.jpg)
The Nernst effect8,9
8Nishiguchi A. et al, Physics of Fluids, 28, 3683, 1985
9Haines M.G., Plasma Physics and Controlled Fusion, 28, 1705, 1986
![Page 80: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/80.jpg)
The Nernst effect8,9
ONLY see Nernst if collision frequency depends on velocity
Hotter (faster) particles carry B-field further
Magnetic field ‘frozen’ to heat flow 8Nishiguchi A. et al, Physics of Fluids, 28, 3683, 1985
9Haines M.G., Plasma Physics and Controlled Fusion, 28, 1705, 1986
![Page 81: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/81.jpg)
Non-Local Nernst10
Calculate dB/dt from:
10T.H. Kho & M.G. Haines, Physics of Fluids, Vol. 29, No. 8, 1986
B=2T B=12T
The Nernst effect is modified by non-locality
dB/dt
Solid lines – E from VFP code
![Page 82: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/82.jpg)
Non-Local Nernst10
10T.H. Kho & M.G. Haines, Physics of Fluids, Vol. 29, No. 8, 1986
B=2T B=12T
The Nernst effect is modified by non-locality
dB/dt Calculate dB/dt from:
Solid lines – E from VFP code
Dashed lines – E from classical Ohm’s law
Nernst not classical
![Page 83: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/83.jpg)
Particle-tracking Particles injected at 350keV affected by E fields ONLY
E = 1.8MeV
![Page 84: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/84.jpg)
Reflective boundary conditions
Fast electron number density
![Page 85: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/85.jpg)
‘Improved’ hybrid mode
e
e
particle source
Ey
Bz Need to solve Maxwell’s equations and Ohm’s law.
VFP code must supply current
![Page 86: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/86.jpg)
‘Improved’ hybrid mode
e
e
particle source
Ey
Bz Need to solve Maxwell’s equations and Ohm’s law.
Standard Ohm’s law:
![Page 87: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/87.jpg)
‘Improved’ hybrid mode
e
e
particle source
Ey
Bz Need to solve Maxwell’s equations and Ohm’s law.
BUT... This doesn’t work for very low density
Standard Ohm’s law:
![Page 88: Progress in Vlasov-Fokker- Planck simulations of laser ... · FIRST 2D VFP code to be able to simulate long-pulse laser-plasma interactions with self-consistent magnetic fields Ideal](https://reader034.fdocuments.in/reader034/viewer/2022042412/5f2b2ace3c0a6168ab4bf92d/html5/thumbnails/88.jpg)
‘Improved’ hybrid mode
e
e
particle source
Ey
Bz Need to solve Maxwell’s equations and Ohm’s law.
Include more terms and ensure realistic resistivity in vacuum
Standard Ohm’s law: