Dissipation and Kinetic Physics of Astrophysical Plasma Turbulence

NSF PRAC project #1614664

Vadim RoytershteynSpace Science Institute

Blue Waters Symposium, Sunriver, OR, June 5, 2018

Stanislav Boldyrev, University of Wisconsin, MadsionGian Luca Delzanno, LANLYuri Omelchenko, Space Science InstituteNikolai Pogorelov, University of Alabama, HuntsvilleHeli Hietala, UCLAChris Chen, Queen Mary University, LondonJohn Podesta, Space Science InstituteAaron Roberts, NASA GoddardWilliam Matthaeus University of DelawareHoma Karimabadi, CureMetrix, Inc

Acknowledgements

Funding: NASA, NSF

Collaborators:

The Problem(s)

Plasma Turbulence is a Ubiquitous Phenomenon

Fusion : magnetically confined, inertially confined, hybrid

Solar corona

Solar wind, planetary magnetospheres

Heliosphere, interstellar Medium

Jets, accretion disks, other astrophysical objects

Armstrong et al., 1995

Local Interstellar Medium

3

rsta.royalsocietypublishing.orgPhil.Trans.R.Soc.A373:20140155

.........................................................

10 −6 10−5 10−4 10−3 10−2 10−1 1 10

10−6

10−4

10−2

1

102

104

106

spacecraft frequency (Hz)

ACE MFI (58 days)ACE MFI (51 h)Cluster FGM + STAFF−SC (70 min)

−1.00 ± 0.04

−2.73 ± 0.01

−1.65 ± 0.01

tran

sitio

n re

gion

inertial range sub-ion rangef –1 range

trac

e po

wer

spe

ctra

l den

sity

(nT

2H

z–1)

lc ~ 106 km re ~ 1 km ri ~102 km

Figure 1. Typical trace power spectral density of the magnetic field fluctuations of a βi ∼ O(1) plasma in the ecliptic solarwind at 1 AU. Dashed lines indicate ordinary least-squares fits, with the corresponding spectral exponents and their fit errorsindicated. This spectrum represents an aggregate of intervals with each smaller interval being containedwithin the subsequentlarger interval—hence the higher frequencies of this spectrum are not representative of the interval describing the lowerfrequencies. At the largest scales is a 58 day interval [2007/01/01 00.00–2007/02/28 00.00 UT] from the MFI instrument onboard the ACE spacecraft, illustrating the large-scale forcing range (the so-called f−1 range). The inertial range is computedfrom a shorter 51 h interval [2007/01/29 21.00–2007/02/01 00.00 UT] also from the same instrument. Both these datasets are at1 Hz cadence, so they just begin to touch the beginning of the sub-ion range. The kinetic scale spectrum in the sub-ion scalerange is given by magnetometer data from the FGM and STAFF-SC instruments on the Cluster multi-spacecraft mission, fromspacecraft 4, while it was in the ambient solar wind [2007/01/30 00.10-01.10 UT] and operating in burst mode with a cadenceof 450 Hz—the two signals from both of these instruments have been merged as in [6]. The vertical dashed lines indicate thethree length scalesmentioned above:λc the correlation length,ρi the ion gyro-radius andρe the electron gyro-radius. (Onlineversion in colour.)

(a) Brief phenomenology of the energy cascadeWe ask the reader to turn their attention to figure 1, which shows a canonical power spectraldensity at 1 AU in the solar wind. We have chosen the power spectral density as it is not only thefocus of many, if not most, studies of turbulence, but also serves as a simple map to illustrate thescales of interest in the phenomena. It is also reflective—being the Fourier transform pair—of thetwo-point field correlation, another obsession of generations of turbulence researchers. Owing tothe extremely high speed of the solar wind, faster than most temporal dynamics in the system, wecan invoke the ‘Taylor frozen-in flow’ hypothesis to relate temporal scales to spatial scales (see [7]for caveats to this). Thus, although the abscissa shows a temporal scale of spacecraft frequency,for most of this spectrum (in the inertial range and above) it can be viewed as a proxy for spatialscales—some of which are marked at the top of the figure. In particular, we have highlighted fourdistinct regions of interest demarcated by three important length scales:

— The f −1 range. At these very small frequencies—corresponding to temporal scales overmany days—what we are actually measuring is the temporal variability of the source ofthe solar wind: the Sun and its solar atmosphere. Near the top of this range, we havethe first of our important length scales: the correlation length λc. Below this scale (higher

on February 27, 2017http://rsta.royalsocietypublishing.org/Downloaded from Focus of This Project: Turbulence in Solar Wind & Magnetosphere

• Turbulence is of interest because of:• Local energy input (e.g. to explain famously anomalous temperature profiles)• Transport of energetic particles (solar energetic particles, cosmic rays, etc)

• Solar Wind is the best accessible example of astrophysical (=large scale) plasma turbulence

Kiyani et al., 2015

Parker Solar Probe (NASA)

Solar Orbiter (ESA)

Two Upcoming Missions Are Expected to Revolutionize Understanding of the Sun, the Solar Wind and the Solar Corona

PSP will provide in-situ measurements as close as 9.8 R

We need theory & numerical tools to make predictions and Internet the data

10-3

10-2

10-1

100

101

102

10-2 10-1 100

ScaleSize

(km)

Heliospheric Distance (AU)

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Kinetic-Alfven regime

Inertial kinetic-Alfven regime

RsParker Solar

ProbeSolar

Orbiter 1 AU10-3

10-2

10-1

100

101

10-2 10-1 100

β s

Heliospheric Distance (AU)

ratio of plasma pressure to magnetic pressure evolution of plasma microscales

The Nature of Kinetic Processes Is Expected to Change Closer to the Sun

Supplemental Material for “Nature of Kinetic Scale Turbulence in the Earth’s Magnetosheath”

When the wave propagation is oblique, k⊥ ≫ k∥, theAlfven wave transforms into the kinetic Alfven wave forscales k⊥ρi > 1. The situation is different in the cases wheneither the ion or electron plasma beta are small. In the regimeof extremely small electron plasma beta βe ≪ me/mi, andβi ≪ 1, the Alfven wave transforms into the inertial Alfvenwave [e.g., 1]. In this case the electron thermal velocity ismuch smaller than the Alfven velocity and the electrons donot adjust instantaneously to the electric field acting alongthe magnetic field lines. Here we consider a different regimeme/mi ≪ βe ≪ 1, where the standard inertial Alfven the-ory does not apply. In this case, the strongly oblique kineticAlfven wave transforms into the new mode which we termthe inertial kinetic Alfven wave (IKAW). Here we derive theIKAW, together with the corresponding nonlinear equations.

The Inertial Kinetic Alfven Wave

The dispersion relation for the standard kinetic Alfvenwave, derived under the assumption ω ≪ kzvth,e, is

ω2 =k2zk

2v2Aρ2i (1 + Te/Ti)

2 + βi(1 + Te/Ti)≈

k2zk2v2Aρ

2i

2 + βi, (1)

where in the last expression we use the approximationTe/Ti ≪ 1 [e.g., 2]. It can be seen, however, that the fre-quency of this mode becomes larger than kzvth,e for

k2ρ2e > βi (2 + βi) (Te/Ti)2 . (2)

For Te/Ti ≪ 1 this happens at scales much larger than theelectron gyroscale.

In the frequency range kzvth,e ≪ ω ≪ kvth,i, the kineticAlfven wave transforms into the new mode, the inertial kineticAlfven wave (IKAW). To derive this mode, we consider a col-lisionless plasma with βi <∼ 1, and concentrate on the sub-ionscales k⊥ρi ≫ 1. The wave modes are found from the equa-tion DijEj = 0, where Dij = k2δij − kikj − (ω2/c2)ϵij . Inthis equation, ϵij is the plasma dielectric tensor, and Ej is theelectric field. Under the additional assumption Te ≪ Ti, thecomponents of the tensor Dij have the form

Dxx ≈ k2z −ω2

c2ω2pe

Ω2e

!

1 +2

βik2⊥d2i

"

, (3)

Dxy = −Dyx ≈ −iωω2

pe

Ωec2, (4)

Dxz = Dzx ≈ −kzk⊥, (5)

Dyy ≈ k2, (6)

Dzz ≈ k2⊥ +ω2pe

c2, (7)

Dyz = Dzy ≈ 0. (8)

Here we assume the coordinate frame where the uniform mag-netic field is in the z direction, and the wave vector has thecomponents (k⊥, 0, kz).

The dispersion relation for the inertial kinetic Alfven waveobtained from this equation is

ω2 =k2zk

2⊥d

4eΩ

2e

(1 + k2⊥d2e) (1 + 2/βi + k2⊥d

2e). (9)

When the electron inertial corrections are small, k2⊥d2e ≪ 1,

the IKAW relation Eq. (9) is similar to the dispersion relationof the standard kinetic Alfven wave Eq. (1), although in a dif-ferent phase space region ω ≫ kzvth,e. In the limiting case,k2⊥d

2e ≫ 1 + 2/βi, the IKAW frequency is approximated by

ω2 = Ω2

ek2

z/k2

⊥. (10)

The IKAW exists in the phase space region ω2 ≪ k2v2th,i.This condition, together with the dispersion relation Eq. (9),allows one to estimate the required obliquity. At k2⊥d

2e ∼ 1 we

obtain k2z/k2⊥ ≪ me/mi, and at k2⊥ρ

2e ∼ 1 we get k2z/k

2⊥ ≪

(me/mi)(Ti/Te). The IKAW is different from the previouslystudied whistler wave that exists in the opposite limit ω2 ≫k2v2th,i [e.g., 3, 4].

One finds from the kinetic derivation that the density fluc-tuations associated with the IKAW are

!

δn

n0

"2

=4

β2i

!

1 +1 + 2/βi + k2⊥d

2e

1 + k2⊥d2e

"−1 !

δB

B0

"2

, (11)

and that the density fluctuations are anti-correlated with thefluctuations of the magnetic field strength,

δBz

B0

= −βi2

δn

n0

. (12)

From Eqs. (11) and (12) one can derive the expression for themagnetic compressibility of the IKAW,

δB2z

δB2⊥

=1 + k2⊥d

2e

1 + 2/βi + k2⊥d2e

. (13)

Nonlinear Equations

We consider very oblique propagation, and use the orderingk∥/k⊥ ∼ δB/B0 ∼ δn/n0 ∼ ω/Ωe ≪ 1. The starting equa-tions are the electron continuity and momentum equations. Inthe electron continuity equation,

∂n

∂t+∇⊥ · (nv⊥) +∇∥(nv∥) = 0, (14)

we express the electron field-parallel velocity through the par-allel electron current, v∥ = −J∥/(ne), where the current is,in turn, expressed through the z-component of the magnetic

Supplemental Material for “Nature of Kinetic Scale Turbulence in the Earth’s Magnetosheath”

When the wave propagation is oblique, k⊥ ≫ k∥, theAlfven wave transforms into the kinetic Alfven wave forscales k⊥ρi > 1. The situation is different in the cases wheneither the ion or electron plasma beta are small. In the regimeof extremely small electron plasma beta βe ≪ me/mi, andβi ≪ 1, the Alfven wave transforms into the inertial Alfvenwave [e.g., 1]. In this case the electron thermal velocity ismuch smaller than the Alfven velocity and the electrons donot adjust instantaneously to the electric field acting alongthe magnetic field lines. Here we consider a different regimeme/mi ≪ βe ≪ 1, where the standard inertial Alfven the-ory does not apply. In this case, the strongly oblique kineticAlfven wave transforms into the new mode which we termthe inertial kinetic Alfven wave (IKAW). Here we derive theIKAW, together with the corresponding nonlinear equations.

The Inertial Kinetic Alfven Wave

The dispersion relation for the standard kinetic Alfvenwave, derived under the assumption ω ≪ kzvth,e, is

ω2 =k2zk

2v2Aρ2i (1 + Te/Ti)

2 + βi(1 + Te/Ti)≈

k2zk2v2Aρ

2i

2 + βi, (1)

where in the last expression we use the approximationTe/Ti ≪ 1 [e.g., 2]. It can be seen, however, that the fre-quency of this mode becomes larger than kzvth,e for

k2ρ2e > βi (2 + βi) (Te/Ti)2 . (2)

For Te/Ti ≪ 1 this happens at scales much larger than theelectron gyroscale.

In the frequency range kzvth,e ≪ ω ≪ kvth,i, the kineticAlfven wave transforms into the new mode, the inertial kineticAlfven wave (IKAW). To derive this mode, we consider a col-lisionless plasma with βi <∼ 1, and concentrate on the sub-ionscales k⊥ρi ≫ 1. The wave modes are found from the equa-tion DijEj = 0, where Dij = k2δij − kikj − (ω2/c2)ϵij . Inthis equation, ϵij is the plasma dielectric tensor, and Ej is theelectric field. Under the additional assumption Te ≪ Ti, thecomponents of the tensor Dij have the form

Dxx ≈ k2z −ω2

c2ω2pe

Ω2e

!

1 +2

βik2⊥d2i

"

, (3)

Dxy = −Dyx ≈ −iωω2

pe

Ωec2, (4)

Dxz = Dzx ≈ −kzk⊥, (5)

Dyy ≈ k2, (6)

Dzz ≈ k2⊥ +ω2pe

c2, (7)

Dyz = Dzy ≈ 0. (8)

Here we assume the coordinate frame where the uniform mag-netic field is in the z direction, and the wave vector has thecomponents (k⊥, 0, kz).

The dispersion relation for the inertial kinetic Alfven waveobtained from this equation is

ω2 =k2zk

2⊥d

4eΩ

2e

(1 + k2⊥d2e) (1 + 2/βi + k2⊥d

2e). (9)

When the electron inertial corrections are small, k2⊥d2e ≪ 1,

the IKAW relation Eq. (9) is similar to the dispersion relationof the standard kinetic Alfven wave Eq. (1), although in a dif-ferent phase space region ω ≫ kzvth,e. In the limiting case,k2⊥d

2e ≫ 1 + 2/βi, the IKAW frequency is approximated by

ω2 = Ω2

ek2

z/k2

⊥. (10)

The IKAW exists in the phase space region ω2 ≪ k2v2th,i.This condition, together with the dispersion relation Eq. (9),allows one to estimate the required obliquity. At k2⊥d

2e ∼ 1 we

obtain k2z/k2⊥ ≪ me/mi, and at k2⊥ρ

2e ∼ 1 we get k2z/k

2⊥ ≪

(me/mi)(Ti/Te). The IKAW is different from the previouslystudied whistler wave that exists in the opposite limit ω2 ≫k2v2th,i [e.g., 3, 4].

One finds from the kinetic derivation that the density fluc-tuations associated with the IKAW are

!

δn

n0

"2

=4

β2i

!

1 +1 + 2/βi + k2⊥d

2e

1 + k2⊥d2e

"−1 !

δB

B0

"2

, (11)

and that the density fluctuations are anti-correlated with thefluctuations of the magnetic field strength,

δBz

B0

= −βi2

δn

n0

. (12)

From Eqs. (11) and (12) one can derive the expression for themagnetic compressibility of the IKAW,

δB2z

δB2⊥

=1 + k2⊥d

2e

1 + 2/βi + k2⊥d2e

. (13)

Nonlinear Equations

We consider very oblique propagation, and use the orderingk∥/k⊥ ∼ δB/B0 ∼ δn/n0 ∼ ω/Ωe ≪ 1. The starting equa-tions are the electron continuity and momentum equations. Inthe electron continuity equation,

∂n

∂t+∇⊥ · (nv⊥) +∇∥(nv∥) = 0, (14)

we express the electron field-parallel velocity through the par-allel electron current, v∥ = −J∥/(ne), where the current is,in turn, expressed through the z-component of the magnetic

Chen & Boldyrev, 2017 Passot et al., 2017,18

s = 8nsTs/B2

<latexit sha1_base64="0fgftMuOepIsxxSrT3DaRwoSM14=">AAACE3icbZDNSgMxFIUz9a/Wv1GX3QSL4KrOFMG6EIpuXFbo2EJnHDJppg1NMkOSEUrpwpfwFdzq3pW49QHc+iSm7Sy09UDg49x7uTcnShlV2nG+rMLK6tr6RnGztLW9s7tn7x/cqSSTmHg4YYnsREgRRgXxNNWMdFJJEI8YaUfD62m9/UCkoolo6VFKAo76gsYUI22s0C77EdEoVPAS1qGfUigMt0J1enVfC+2KU3Vmgsvg5lABuZqh/e33EpxxIjRmSKmu66Q6GCOpKWZkUvIzRVKEh6hPugYF4kQF49knJvDYOD0YJ9I8oeHM/T0xRlypEY9MJ0d6oBZrU/PfWsQXNuu4HoypSDNNBJ4vjjMGdQKnCcEelQRrNjKAsKTmdogHSCKsTY4lE4q7GMEyeLXqRdW5Pas0Gnk6RVAGR+AEuOAcNMANaAIPYPAInsELeLWerDfr3fqYtxasfOYQ/JH1+QPTsZyr</latexit><latexit sha1_base64="0fgftMuOepIsxxSrT3DaRwoSM14=">AAACE3icbZDNSgMxFIUz9a/Wv1GX3QSL4KrOFMG6EIpuXFbo2EJnHDJppg1NMkOSEUrpwpfwFdzq3pW49QHc+iSm7Sy09UDg49x7uTcnShlV2nG+rMLK6tr6RnGztLW9s7tn7x/cqSSTmHg4YYnsREgRRgXxNNWMdFJJEI8YaUfD62m9/UCkoolo6VFKAo76gsYUI22s0C77EdEoVPAS1qGfUigMt0J1enVfC+2KU3Vmgsvg5lABuZqh/e33EpxxIjRmSKmu66Q6GCOpKWZkUvIzRVKEh6hPugYF4kQF49knJvDYOD0YJ9I8oeHM/T0xRlypEY9MJ0d6oBZrU/PfWsQXNuu4HoypSDNNBJ4vjjMGdQKnCcEelQRrNjKAsKTmdogHSCKsTY4lE4q7GMEyeLXqRdW5Pas0Gnk6RVAGR+AEuOAcNMANaAIPYPAInsELeLWerDfr3fqYtxasfOYQ/JH1+QPTsZyr</latexit><latexit sha1_base64="0fgftMuOepIsxxSrT3DaRwoSM14=">AAACE3icbZDNSgMxFIUz9a/Wv1GX3QSL4KrOFMG6EIpuXFbo2EJnHDJppg1NMkOSEUrpwpfwFdzq3pW49QHc+iSm7Sy09UDg49x7uTcnShlV2nG+rMLK6tr6RnGztLW9s7tn7x/cqSSTmHg4YYnsREgRRgXxNNWMdFJJEI8YaUfD62m9/UCkoolo6VFKAo76gsYUI22s0C77EdEoVPAS1qGfUigMt0J1enVfC+2KU3Vmgsvg5lABuZqh/e33EpxxIjRmSKmu66Q6GCOpKWZkUvIzRVKEh6hPugYF4kQF49knJvDYOD0YJ9I8oeHM/T0xRlypEY9MJ0d6oBZrU/PfWsQXNuu4HoypSDNNBJ4vjjMGdQKnCcEelQRrNjKAsKTmdogHSCKsTY4lE4q7GMEyeLXqRdW5Pas0Gnk6RVAGR+AEuOAcNMANaAIPYPAInsELeLWerDfr3fqYtxasfOYQ/JH1+QPTsZyr</latexit>

• Short-wavelength turbulence at 1 AU is dominated by KAW • KAW turbulence is relatively well studied • as β↓, the nature of fluctuations changes (iKAW) • iKAW is not well understood • separation of scales increases as β↓

Methods

A Variety of Models & Approximations Are Used in Plasma Physics to Tackle Different Scales

smaller scales

L: system size, energy injection scale,

correlation scaleion kinetic

scales

electron kinetic scales

debye lengthcollisional scale

(collisional)collisional

scale (c-less)

Magnetohydrodynamic approximation (MHD):

incompressible, fully compressible, kinetic MHD.. Hall MHD

multi-fluid multi-moments modelshybrid kinetic

…Landau FluidGyrokineticFully kinetic

…E +1

cv B = 0

@tv + v ·rv =1

cj B

@tB = crE rB =4

cjm

odel

com

plex

ity

In many situations, cross-scale coupling play a role an important role global dynamics. Full understanding of global evolution may require multi-scale, multi-physics models

@tfs + v ·rfs +qsms

E +

1

cv B

·rvfs = Cfs, fs0 , . . .

+ Maxwell’s equations

“First-Principle” description ofweakly coupled plasmas:

Our Go-To Model: Fully Kinetic and Hybrid PIC Simulations

@fs@t

+ v ·rfs +qsms

E +

1

cv B

·rvfs =

X

s0

Cfs, fs0

~up to 1010 cells~up to 4x1012 particles~120 TB of memory~107 CPU-HRS (~103 CPU-YRS)

Hybrid simulationskinetic ions + fluid electronscodes: H3D, HYPERES

Fully kinetic simulationsAll species kineticcode: VPIC

rB =4

cj +

1

c

@E

@t

1

c

@B

@t= rE

r ·E = 4

r ·B = 0

~up to 1.7x1010 cells~up to 2x1012 particles~130 TB of memory

Particle-In-Cell (PIC) is a very efficient, but relatively inaccurate method for solving full Vlasov-Maxwell system. Major limitations: noise and the need to resolve ALL kinetic spatial

and temporal scales (in explicit methods) . Typical 3D simulation takes up a significant portion of a good modern supercomputer for ~10 days => BW10

pexpensivesampling

p

t=t1x

efficientsampling

x

vv

Goal : high-accuracy, implicit, fully kinetic simulation Problem : 6D! Sample phase space with 100 points in each direction=1012

unknowns per species Solution: Efficient (spectral) Discretization of the Velocity Space

11

We Investigate Several Related Problems on BW.Focus Today: SpectralPlasmaSolver: a Simulation Tool for Fluid-Kinetic Coupling

@tfs + v ·rfs +qsms

E +

1

cv B

·rvfs = Cfs, fs0 , . . .

Accordingly, the dimensionless Vlasov equation reads:

@fs

@t+ v ·r

x

fs

+qs

e

!cs

!pe

(E+ v B) ·rv

fs

= 0, (1)

where s labels the plasma species (s = e, i for electrons and ions), the cyclotron frequency is!cs

= qs

B0/ms

and E (B) is the electric (magnetic) field. Equation (1) is coupled to Maxwell’sequations via the electromagnetic field:

@B

@t= rE,

@E

@t= rB !

pe

!ce

Z +1

1vf

i

dv Z +1

1vf

e

dv

, (2)

r ·E =!pe

!ce

Z +1

1fi

dv Z +1

1fe

dv

, r ·B = 0. (3)

We expand the particle distribution function in Hermite basis functions,

fs

=N

n

1X

n=0

N

m

1X

m=0

N

p

1X

p=0

Cs

n,m,p

(x, t) n

(sx

) m

(sy

) p

(sz

) (4)

(where Nn

, Nm

and Np

are the total number of Hermite modes in the vx

, vy

and vz

directions, respectively), to obtain a set of non-linear partial di↵erential equations (PDEs) forthe coecients of the expansion. We consider the asymmetrically-weigthed (AW) basis function,

n

(x) = (2nn!)1/2Hn

(x)ex

2, where H

n

is the Hermite polynomial of the n-th order [23]. The

argument of the basis functions is defined as s

=v

us

/↵s

, where us

and ↵s

are (constant)

shift and scaling parameters that have to be supplied by the user and = x, y, z. We furtherdecompose the spatial part of the particle distribution function into Fourier basis functions,

Cs

n,m,p

(x, t) =

N

x

/2X

k

x

=N

x

/2

N

y

/2X

k

y

=N

y

/2

N

z

/2X

k

z

=N

z

/2

Ck

x

,k

y

,k

z

,s

n,m,p

(t) exp

2i

kx

x

Lx

+ky

y

Ly

+kz

z

Lz

, (5)

where Nx, y, z

+ 1 is the total number of Fourier modes in each spatial direction.Using the orthogonality properties of the Hermite and Fourier basis functions, the Vlasov-

Maxwell equations can be rewritten as a set of ordinary di↵erential equations (ODEs) [20]:

dCk

x

,k

y

,k

z

,s

n,m,p

dt= 2ik

x

Lx

↵s

x

rn+ 1

2C

k

x

,k

y

,k

z

,s

n+1,m,p

+

rn

2C

k

x

,k

y

,k

z

,s

n1,m,p

+usx

↵s

x

Ck

x

,k

y

,k

z

,s

n,m,p

!

2iky

Ly

↵s

y

rm+ 1

2C

k

x

,k

y

,k

z

,s

n,m+1,p +

rm

2C

k

x

,k

y

,k

z

,s

n,m1,p +usy

↵s

y

Ck

x

,k

y

,k

z

,s

n,m,p

!

2ikz

Lz

↵s

z

rp+ 1

2C

k

x

,k

y

,k

z

,s

n,m,p+1 +

rp

2C

k

x

,k

y

,k

z

,s

n,m,p1 +usz

↵s

z

Ck

x

,k

y

,k

z

,s

n,m,p

!+

qs

e

!cs

!pe

(p2n

↵s

x

[Ex

Cn1,m,p

]k

x

,k

y

,k

z

+

p2m

↵s

y

[Ey

Cn,m1,p]

k

x

,k

y

,k

z

+

p2p

↵s

z

[Ez

Cn,m,p1]

k

x

,k

y

,k

z

)+

qs

e

!cs

!pe

[Bx

p

mp

↵s

z

↵s

y

↵s

y

↵s

z

Cs

n,m1,p1 +pm(p+ 1)

↵s

z

↵s

y

Cs

n,m1,p+1p

(m+ 1)p↵s

y

↵s

z

Cs

n,m+1,p1 +p2m

usz

↵s

y

Cs

n,m1,p p

2pusy

↵s

z

Cs

n,m,p1

]k

x

,k

y

,k

z

+

Accordingly, the dimensionless Vlasov equation reads:

@fs

@t+ v ·r

x

fs

+qs

e

!cs

!pe

(E+ v B) ·rv

fs

= 0, (1)

where s labels the plasma species (s = e, i for electrons and ions), the cyclotron frequency is!cs

= qs

B0/ms

and E (B) is the electric (magnetic) field. Equation (1) is coupled to Maxwell’sequations via the electromagnetic field:

@B

@t= rE,

@E

@t= rB !

pe

!ce

Z +1

1vf

i

dv Z +1

1vf

e

dv

, (2)

r ·E =!pe

!ce

Z +1

1fi

dv Z +1

1fe

dv

, r ·B = 0. (3)

We expand the particle distribution function in Hermite basis functions,

fs

=N

n

1X

n=0

N

m

1X

m=0

N

p

1X

p=0

Cs

n,m,p

(x, t) n

(sx

) m

(sy

) p

(sz

) (4)

(where Nn

, Nm

and Np

are the total number of Hermite modes in the vx

, vy

and vz

directions, respectively), to obtain a set of non-linear partial di↵erential equations (PDEs) forthe coecients of the expansion. We consider the asymmetrically-weigthed (AW) basis function,

n

(x) = (2nn!)1/2Hn

(x)ex

2, where H

n

is the Hermite polynomial of the n-th order [23]. The

argument of the basis functions is defined as s

=v

us

/↵s

, where us

and ↵s

are (constant)

shift and scaling parameters that have to be supplied by the user and = x, y, z. We furtherdecompose the spatial part of the particle distribution function into Fourier basis functions,

Cs

n,m,p

(x, t) =

N

x

/2X

k

x

=N

x

/2

N

y

/2X

k

y

=N

y

/2

N

z

/2X

k

z

=N

z

/2

Ck

x

,k

y

,k

z

,s

n,m,p

(t) exp

2i

kx

x

Lx

+ky

y

Ly

+kz

z

Lz

, (5)

where Nx, y, z

+ 1 is the total number of Fourier modes in each spatial direction.Using the orthogonality properties of the Hermite and Fourier basis functions, the Vlasov-

Maxwell equations can be rewritten as a set of ordinary di↵erential equations (ODEs) [20]:

dCk

x

,k

y

,k

z

,s

n,m,p

dt= 2ik

x

Lx

↵s

x

rn+ 1

2C

k

x

,k

y

,k

z

,s

n+1,m,p

+

rn

2C

k

x

,k

y

,k

z

,s

n1,m,p

+usx

↵s

x

Ck

x

,k

y

,k

z

,s

n,m,p

!

2iky

Ly

↵s

y

rm+ 1

2C

k

x

,k

y

,k

z

,s

n,m+1,p +

rm

2C

k

x

,k

y

,k

z

,s

n,m1,p +usy

↵s

y

Ck

x

,k

y

,k

z

,s

n,m,p

!

2ikz

Lz

↵s

z

rp+ 1

2C

k

x

,k

y

,k

z

,s

n,m,p+1 +

rp

2C

k

x

,k

y

,k

z

,s

n,m,p1 +usz

↵s

z

Ck

x

,k

y

,k

z

,s

n,m,p

!+

qs

e

!cs

!pe

(p2n

↵s

x

[Ex

Cn1,m,p

]k

x

,k

y

,k

z

+

p2m

↵s

y

[Ey

Cn,m1,p]

k

x

,k

y

,k

z

+

p2p

↵s

z

[Ez

Cn,m,p1]

k

x

,k

y

,k

z

)+

qs

e

!cs

!pe

[Bx

p

mp

↵s

z

↵s

y

↵s

y

↵s

z

Cs

n,m1,p1 +p

m(p+ 1)↵s

z

↵s

y

Cs

n,m1,p+1p

(m+ 1)p↵s

y

↵s

z

Cs

n,m+1,p1 +p2m

usz

↵s

y

Cs

n,m1,p p

2pusy

↵s

z

Cs

n,m,p1

]k

x

,k

y

,k

z

+

Maxweliian = 1 coefficient per direction; More coefficients = more complex distributions = more kinetic physics; Adaptivity = fluid-kinetic coupling

Grant and Feix, 1967; Armstrong et al., 1970; Vencels et al., 2015; Loureiro et al., 2013; Camporeale et al. 2016; Delzanno et al., 2015; Vencels et al., 2016;

Roytershteyn & Delzanno 2018;

An Implementation: FORTRAN90, PETSC + 2DECOMP&FFT + FFTW/MKL

Roytershteyn and Delzanno Spectral approach to kinetic simulations

for the Vlasov equation, and99

dBk

x

,k

y

,k

z

dt= 2i"

k

L

Ek

x

,k

y

,k

z

,

dEk

x

,k

y

,k

z

dt= 2i"

k

L

Bk

x

,k

y

,k

z

!pe

ce

X

s=e, i

qs

e↵s

x

↵s

y

↵s

z

↵s

p2

Ck

x

,k

y

,k

z

,s

a,b,c

+ us

Ck

x

,k

y

,k

z

,s

0,0,0

(8)

for Maxwell’s equations (written in index notation with summation over repeated indices implied). Here100"

is the Levi-Civita tensor and (a, b, c) = (1, 0, 0), (0, 1, 0), (0, 0, 1) for = x, y, z, respectively.101Equations (7) are defined for n 2 [0, N

n

1], m 2 [0, Nm

1] and p 2 [0, Np

1] and the convolution102operator is defined as103

[H Cn,m,p

]

k

x

,k

y

,k

z

=

N

x

/2X

k

0x

=N

x

/2

N

y

/2X

k

0y

=N

y

/2

N

z

/2X

k

0z

=N

z

/2

Hk

x

k

0x

,k

y

k

0y

,k

z

k

0zC

k

0x

,k

0y

,k

0z

,s

n,m,p

, (9)

where H is an arbitrary function. Note that Eq. (7) couples a given n,m, p mode with successive104modes of the hierarchy and it is necessary to specify how the system of equations is closed: We use105C

nN

n

,mN

m

,pN

p

,s

k

x

,k

y

,k

z

= 0. In order to address potential problems arising from the filamentation typical of106collisionless plasmas, we add a collisional term on the right hand side of Eq. (7)107

C[Ck

x

,k

y

,k

z

,s

n,m,p

] =

n(n 1)(n 2)

(Nn

1)(Nn

2)(Nn

3)

+

m(m 1)(m 2)

(Nm

1)(Nm

2)(Nm

3)

+

p(p 1)(p 2)

(Np

1)(Np

2)(Np

3)

C

k

x

,k

y

,k

z

,s

n,m,p

, (10)

that can be used to damp higher-order Hermite modes as necessary. The specific form of Eq. (10) is intended108to leave the first three modes of the Hermite expansion, which have connections to density, momentum and109energy of the system, untouched.110

Equations (7) and (8) can be cast in matrix form as111

dC

dt= L1C+N (C,F) ,

dF

dt= L2C+ L3F, (11)

where C represent the vector of Ck

x

,k

y

,k

z

,s

n,m,p

coefficients and F contains the electric and magnetic field.112Furthermore, L1,2,3 represent the linear part of Eqs. (7) and (8) (advection in the case of Vlasov equations)113while N (C,F) represents the nonlinear part with the convolutions.114

System (11) is discretized in time with a fully-implicit Crank-Nicolson scheme. In residual form, we have115

(R1

C

+1,F+1= C

+1 C

thL1C

+1/2+N

C

+1/2,F+1/2i

= 0

R2C

+1,F+1= F

+1 F

tL2C+1/2 tL3F

+1/2= 0

() R(X) = 0,

(12)

Frontiers 5

Roytershteyn and Delzanno Spectral approach to kinetic simulations

for the Vlasov equation, and99

dBk

x

,k

y

,k

z

dt= 2i"

k

L

Ek

x

,k

y

,k

z

,

dEk

x

,k

y

,k

z

dt= 2i"

k

L

Bk

x

,k

y

,k

z

!pe

ce

X

s=e, i

qs

e↵s

x

↵s

y

↵s

z

↵s

p2

Ck

x

,k

y

,k

z

,s

a,b,c

+ us

Ck

x

,k

y

,k

z

,s

0,0,0

(8)

for Maxwell’s equations (written in index notation with summation over repeated indices implied). Here100"

is the Levi-Civita tensor and (a, b, c) = (1, 0, 0), (0, 1, 0), (0, 0, 1) for = x, y, z, respectively.101Equations (7) are defined for n 2 [0, N

n

1], m 2 [0, Nm

1] and p 2 [0, Np

1] and the convolution102operator is defined as103

[H Cn,m,p

]

k

x

,k

y

,k

z

=

N

x

/2X

k

0x

=N

x

/2

N

y

/2X

k

0y

=N

y

/2

N

z

/2X

k

0z

=N

z

/2

Hk

x

k

0x

,k

y

k

0y

,k

z

k

0zC

k

0x

,k

0y

,k

0z

,s

n,m,p

, (9)

where H is an arbitrary function. Note that Eq. (7) couples a given n,m, p mode with successive104modes of the hierarchy and it is necessary to specify how the system of equations is closed: We use105C

nN

n

,mN

m

,pN

p

,s

k

x

,k

y

,k

z

= 0. In order to address potential problems arising from the filamentation typical of106collisionless plasmas, we add a collisional term on the right hand side of Eq. (7)107

C[Ck

x

,k

y

,k

z

,s

n,m,p

] =

n(n 1)(n 2)

(Nn

1)(Nn

2)(Nn

3)

+

m(m 1)(m 2)

(Nm

1)(Nm

2)(Nm

3)

+

p(p 1)(p 2)

(Np

1)(Np

2)(Np

3)

C

k

x

,k

y

,k

z

,s

n,m,p

, (10)

that can be used to damp higher-order Hermite modes as necessary. The specific form of Eq. (10) is intended108to leave the first three modes of the Hermite expansion, which have connections to density, momentum and109energy of the system, untouched.110

Equations (7) and (8) can be cast in matrix form as111

dC

dt= L1C+N (C,F) ,

dF

dt= L2C+ L3F, (11)

where C represent the vector of Ck

x

,k

y

,k

z

,s

n,m,p

coefficients and F contains the electric and magnetic field.112Furthermore, L1,2,3 represent the linear part of Eqs. (7) and (8) (advection in the case of Vlasov equations)113while N (C,F) represents the nonlinear part with the convolutions.114

System (11) is discretized in time with a fully-implicit Crank-Nicolson scheme. In residual form, we have115

(R1

C

+1,F+1= C

+1 C

thL1C

+1/2+N

C

+1/2,F+1/2i

= 0

R2C

+1,F+1= F

+1 F

tL2C+1/2 tL3F

+1/2= 0

() R(X) = 0,

(12)

Frontiers 5

10-2

10-1

100

101

102 103 104 105

time,s

MPI ranks

101

102

102 103 104tim

e,s

MPI ranks

single convolution complete cycle

communication starts dominating

The present version decomposes the solution space in 2/6 dimensions. There is more parallelism to explore!

(Some) Results on Turbulence

14

SPS: A Small Number of Hermite Modes Can Reproduce Collisionless Damping with Reasonable Accuracy.

-10-7

-10-6

-10-5

-10-4

-10-3

-10-2

-10-1

-100

-10110-2 10-1 100 101

γ/Ωci

kde

Vlasov, 1

Vlasov, 2

SPS, NH=4

SPS, NH=5

SPS, NH=6

10-3

10-2

10-1

100

101

10-2 10-1 100 101

ω/Ω

ci

kde

analytic

Vlasov, 1

Vlasov, 2

SPS, NH=4

SPS, NH=5

SPS, NH=6

= 89 e = 0.04Ti/Te = 10<latexit sha1_base64="Ic/e7MaItIDUp22en//eIu3Aip8=">AAACJXicbZBNS0JBFIbn2pfZl9WyzZAELcruDSFdCFKblgaagtcuc8ejDs79YObcQMRf0Z/oL7StfasIWgX9ksaPRVkvDLw85xzOmdePpdBo2x9Waml5ZXUtvZ7Z2Nza3snu7t3qKFEc6jySkWr6TIMUIdRRoIRmrIAFvoSGP7ia1Bv3oLSIwhoOY2gHrBeKruAMDfKypy72AVm5WLpzuVDcPaGub4AHtEztvF0woOaJs5oHZcf2sjnDpqJ/jTM3OTJX1ct+uZ2IJwGEyCXTuuXYMbZHTKHgEsYZN9EQMz5gPWgZG7IAdHs0/daYHhnSod1ImRcindKfEyMWaD0MfNMZMOzrxdoE/lvzg4XN2C22RyKME4SQzxZ3E0kxopPMaEco4CiHxjCuhLmd8j5TjKNJNmNCcRYj+Gvq5/lS3r4p5CqX83TS5IAckmPikAtSIdekSuqEkwfyRJ7Ji/VovVpv1vusNWXNZ/bJL1mf35/4oq0=</latexit><latexit sha1_base64="Ic/e7MaItIDUp22en//eIu3Aip8=">AAACJXicbZBNS0JBFIbn2pfZl9WyzZAELcruDSFdCFKblgaagtcuc8ejDs79YObcQMRf0Z/oL7StfasIWgX9ksaPRVkvDLw85xzOmdePpdBo2x9Waml5ZXUtvZ7Z2Nza3snu7t3qKFEc6jySkWr6TIMUIdRRoIRmrIAFvoSGP7ia1Bv3oLSIwhoOY2gHrBeKruAMDfKypy72AVm5WLpzuVDcPaGub4AHtEztvF0woOaJs5oHZcf2sjnDpqJ/jTM3OTJX1ct+uZ2IJwGEyCXTuuXYMbZHTKHgEsYZN9EQMz5gPWgZG7IAdHs0/daYHhnSod1ImRcindKfEyMWaD0MfNMZMOzrxdoE/lvzg4XN2C22RyKME4SQzxZ3E0kxopPMaEco4CiHxjCuhLmd8j5TjKNJNmNCcRYj+Gvq5/lS3r4p5CqX83TS5IAckmPikAtSIdekSuqEkwfyRJ7Ji/VovVpv1vusNWXNZ/bJL1mf35/4oq0=</latexit><latexit sha1_base64="Ic/e7MaItIDUp22en//eIu3Aip8=">AAACJXicbZBNS0JBFIbn2pfZl9WyzZAELcruDSFdCFKblgaagtcuc8ejDs79YObcQMRf0Z/oL7StfasIWgX9ksaPRVkvDLw85xzOmdePpdBo2x9Waml5ZXUtvZ7Z2Nza3snu7t3qKFEc6jySkWr6TIMUIdRRoIRmrIAFvoSGP7ia1Bv3oLSIwhoOY2gHrBeKruAMDfKypy72AVm5WLpzuVDcPaGub4AHtEztvF0woOaJs5oHZcf2sjnDpqJ/jTM3OTJX1ct+uZ2IJwGEyCXTuuXYMbZHTKHgEsYZN9EQMz5gPWgZG7IAdHs0/daYHhnSod1ImRcindKfEyMWaD0MfNMZMOzrxdoE/lvzg4XN2C22RyKME4SQzxZ3E0kxopPMaEco4CiHxjCuhLmd8j5TjKNJNmNCcRYj+Gvq5/lS3r4p5CqX83TS5IAckmPikAtSIdekSuqEkwfyRJ7Ji/VovVpv1vusNWXNZ/bJL1mf35/4oq0=</latexit>

• Collisionless damping is a very subtle physical phenomenon (2010 Fields Medal) • For some parameters, the method can describe it with just a few modes

“exact” solution

SPS simulations

10-10

10-8

10-6

10-4

S Bk-3.8

0

0.4

0.8

C||

100

102

10-1 100 101

Ce

kde

New vs Old (SPS vs PIC): Decaying Turbulence Test

Roytershteyn & Delzanno, 2018

reference solution (ignore overall constant)

SPS simulations

spectrum of magnetic fluctuations

magnetic compressibility - a “fingerprint” of the fluctuations

compressibility - another quantity revealing the nature of fluctuations

SPS Simulations of Decaying Turbulence in Low-Beta Plasma

10-11

10-8

10-5

10-2

k-11/3

k-2.8

k-5/3

0

0.4

0.8

261014

10-1 100 101

k?de<latexit sha1_base64="BXqykzSi4Wht29ie/ieN5zVrbhI=">AAACAnicbVBNS8NAFNzUr1q/qh69LBbBU0lEUG8FLx4rGFtoQthsXtqlu8myuxFK6NG/4FXvnsSrf8Srv8Rtm4NWBx4MM+8xj4klZ9q47qdTW1ldW9+obza2tnd295r7B/c6LxQFn+Y8V/2YaOAsA98ww6EvFRARc+jF4+uZ33sApVme3ZmJhFCQYcZSRomxUjCOAglKlkkE06jZctvuHPgv8SrSQhW6UfMrSHJaCMgM5UTrgedKE5ZEGUY5TBtBoUESOiZDGFiaEQE6LOc/T/GJVRKc5spOZvBc/XlREqH1RMR2UxAz0sveTPzXi8VSskkvw5JlsjCQ0UVwWnBscjwrBCdMATV8YgmhitnfMR0RRaixtTVsKd5yBX+Jf9a+aru3561Op2qnjo7QMTpFHrpAHXSDushHFEn0hJ7Ri/PovDpvzvtiteZUN4foF5yPb1UvmGY=</latexit><latexit sha1_base64="BXqykzSi4Wht29ie/ieN5zVrbhI=">AAACAnicbVBNS8NAFNzUr1q/qh69LBbBU0lEUG8FLx4rGFtoQthsXtqlu8myuxFK6NG/4FXvnsSrf8Srv8Rtm4NWBx4MM+8xj4klZ9q47qdTW1ldW9+obza2tnd295r7B/c6LxQFn+Y8V/2YaOAsA98ww6EvFRARc+jF4+uZ33sApVme3ZmJhFCQYcZSRomxUjCOAglKlkkE06jZctvuHPgv8SrSQhW6UfMrSHJaCMgM5UTrgedKE5ZEGUY5TBtBoUESOiZDGFiaEQE6LOc/T/GJVRKc5spOZvBc/XlREqH1RMR2UxAz0sveTPzXi8VSskkvw5JlsjCQ0UVwWnBscjwrBCdMATV8YgmhitnfMR0RRaixtTVsKd5yBX+Jf9a+aru3561Op2qnjo7QMTpFHrpAHXSDushHFEn0hJ7Ri/PovDpvzvtiteZUN4foF5yPb1UvmGY=</latexit><latexit sha1_base64="BXqykzSi4Wht29ie/ieN5zVrbhI=">AAACAnicbVBNS8NAFNzUr1q/qh69LBbBU0lEUG8FLx4rGFtoQthsXtqlu8myuxFK6NG/4FXvnsSrf8Srv8Rtm4NWBx4MM+8xj4klZ9q47qdTW1ldW9+obza2tnd295r7B/c6LxQFn+Y8V/2YaOAsA98ww6EvFRARc+jF4+uZ33sApVme3ZmJhFCQYcZSRomxUjCOAglKlkkE06jZctvuHPgv8SrSQhW6UfMrSHJaCMgM5UTrgedKE5ZEGUY5TBtBoUESOiZDGFiaEQE6LOc/T/GJVRKc5spOZvBc/XlREqH1RMR2UxAz0sveTPzXi8VSskkvw5JlsjCQ0UVwWnBscjwrBCdMATV8YgmhitnfMR0RRaixtTVsKd5yBX+Jf9a+aru3561Op2qnjo7QMTpFHrpAHXSDushHFEn0hJ7Ri/PovDpvzvtiteZUN4foF5yPb1UvmGY=</latexit>

Ce,i

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Ck

<latexit sha1_base64="55RqCIzDq/sHqlyJL9nt5yPnZag=">AAAB+nicbVBNS8NAFHypX7V+VT16WSyCp5KIoN4KvXisYGyhDWWz3bRLdzdhdyOE2L/gVe+exKt/xqu/xE2bg7YOPBhm3mMeEyacaeO6X05lbX1jc6u6XdvZ3ds/qB8ePeg4VYT6JOax6oVYU84k9Q0znPYSRbEIOe2G03bhdx+p0iyW9yZLaCDwWLKIEWwKqT0cPA3rDbfpzoFWiVeSBpToDOvfg1FMUkGlIRxr3ffcxAQ5VoYRTme1QappgskUj2nfUokF1UE+/3WGzqwyQlGs7EiD5urvixwLrTMR2k2BzUQve4X4rxeKpWQTXQc5k0lqqCSL4CjlyMSoKAKNmKLE8MwSTBSzvyMywQoTY+uq2VK85QpWiX/RvGm6d5eNVqtspwoncArn4MEVtOAWOuADgQk8wwu8OjPnzXl3PharFae8OYY/cD5/AIyLlJM=</latexit><latexit sha1_base64="55RqCIzDq/sHqlyJL9nt5yPnZag=">AAAB+nicbVBNS8NAFHypX7V+VT16WSyCp5KIoN4KvXisYGyhDWWz3bRLdzdhdyOE2L/gVe+exKt/xqu/xE2bg7YOPBhm3mMeEyacaeO6X05lbX1jc6u6XdvZ3ds/qB8ePeg4VYT6JOax6oVYU84k9Q0znPYSRbEIOe2G03bhdx+p0iyW9yZLaCDwWLKIEWwKqT0cPA3rDbfpzoFWiVeSBpToDOvfg1FMUkGlIRxr3ffcxAQ5VoYRTme1QappgskUj2nfUokF1UE+/3WGzqwyQlGs7EiD5urvixwLrTMR2k2BzUQve4X4rxeKpWQTXQc5k0lqqCSL4CjlyMSoKAKNmKLE8MwSTBSzvyMywQoTY+uq2VK85QpWiX/RvGm6d5eNVqtspwoncArn4MEVtOAWOuADgQk8wwu8OjPnzXl3PharFae8OYY/cD5/AIyLlJM=</latexit><latexit sha1_base64="55RqCIzDq/sHqlyJL9nt5yPnZag=">AAAB+nicbVBNS8NAFHypX7V+VT16WSyCp5KIoN4KvXisYGyhDWWz3bRLdzdhdyOE2L/gVe+exKt/xqu/xE2bg7YOPBhm3mMeEyacaeO6X05lbX1jc6u6XdvZ3ds/qB8ePeg4VYT6JOax6oVYU84k9Q0znPYSRbEIOe2G03bhdx+p0iyW9yZLaCDwWLKIEWwKqT0cPA3rDbfpzoFWiVeSBpToDOvfg1FMUkGlIRxr3ffcxAQ5VoYRTme1QappgskUj2nfUokF1UE+/3WGzqwyQlGs7EiD5urvixwLrTMR2k2BzUQve4X4rxeKpWQTXQc5k0lqqCSL4CjlyMSoKAKNmKLE8MwSTBSzvyMywQoTY+uq2VK85QpWiX/RvGm6d5eNVqtspwoncArn4MEVtOAWOuADgQk8wwu8OjPnzXl3PharFae8OYY/cD5/AIyLlJM=</latexit>

Ce

Ci

Cave

SESB

S <latexit sha1_base64="bSxOzVXiibPXJltCrfUdEqQ9744=">AAAB93icbVBNS8NAFHypX7V+VT16WSyCp5KIoN4KXjy2aLTQhrLZvrRLN5uwuxFq6C/wqndP4tWf49Vf4rbNQVsHHgwz7zGPCVPBtXHdL6e0srq2vlHerGxt7+zuVfcP7nWSKYY+S0Si2iHVKLhE33AjsJ0qpHEo8CEcXU/9h0dUmifyzoxTDGI6kDzijBortW571Zpbd2cgy8QrSA0KNHvV724/YVmM0jBBte54bmqCnCrDmcBJpZtpTCkb0QF2LJU0Rh3ks0cn5MQqfRIlyo40ZKb+vshprPU4Du1mTM1QL3pT8V8vjBeSTXQZ5FymmUHJ5sFRJohJyLQF0ucKmRFjSyhT3P5O2JAqyoztqmJL8RYrWCb+Wf2q7rbOa41G0U4ZjuAYTsGDC2jADTTBBwYIz/ACr86T8+a8Ox/z1ZJT3BzCHzifP0tGk04=</latexit><latexit sha1_base64="bSxOzVXiibPXJltCrfUdEqQ9744=">AAAB93icbVBNS8NAFHypX7V+VT16WSyCp5KIoN4KXjy2aLTQhrLZvrRLN5uwuxFq6C/wqndP4tWf49Vf4rbNQVsHHgwz7zGPCVPBtXHdL6e0srq2vlHerGxt7+zuVfcP7nWSKYY+S0Si2iHVKLhE33AjsJ0qpHEo8CEcXU/9h0dUmifyzoxTDGI6kDzijBortW571Zpbd2cgy8QrSA0KNHvV724/YVmM0jBBte54bmqCnCrDmcBJpZtpTCkb0QF2LJU0Rh3ks0cn5MQqfRIlyo40ZKb+vshprPU4Du1mTM1QL3pT8V8vjBeSTXQZ5FymmUHJ5sFRJohJyLQF0ucKmRFjSyhT3P5O2JAqyoztqmJL8RYrWCb+Wf2q7rbOa41G0U4ZjuAYTsGDC2jADTTBBwYIz/ACr86T8+a8Ox/z1ZJT3BzCHzifP0tGk04=</latexit><latexit sha1_base64="bSxOzVXiibPXJltCrfUdEqQ9744=">AAAB93icbVBNS8NAFHypX7V+VT16WSyCp5KIoN4KXjy2aLTQhrLZvrRLN5uwuxFq6C/wqndP4tWf49Vf4rbNQVsHHgwz7zGPCVPBtXHdL6e0srq2vlHerGxt7+zuVfcP7nWSKYY+S0Si2iHVKLhE33AjsJ0qpHEo8CEcXU/9h0dUmifyzoxTDGI6kDzijBortW571Zpbd2cgy8QrSA0KNHvV724/YVmM0jBBte54bmqCnCrDmcBJpZtpTCkb0QF2LJU0Rh3ks0cn5MQqfRIlyo40ZKb+vshprPU4Du1mTM1QL3pT8V8vjBeSTXQZ5FymmUHJ5sFRJohJyLQF0ucKmRFjSyhT3P5O2JAqyoztqmJL8RYrWCb+Wf2q7rbOa41G0U4ZjuAYTsGDC2jADTTBBwYIz/ACr86T8+a8Ox/z1ZJT3BzCHzifP0tGk04=</latexit>

Roytershteyn et al., 2018

17

x/de

y/d e

2510 251

025

1

0x/de 2510 x/de

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Simulations Will Allow us to Address Key Questions, Such as Energy Dissipation, Structure Formation, etc

• 2D simulation with a traditional PIC method; • the goal is to do this in 3D using new tools • multiple scale ranges • high-frequency fluctuations resembling iKAW • particle acceleration at structures

2D PIC; βe=0.04, Ti/Te=10,mi/me=100; 25x25 c/ωpi; 4000 particles/cell/species

Summary

1.Understanding of plasma turbulence is a grand challenge problem.

2.We are using Blue Waters to study some aspects of this problem, namely kinetic effects associated with turbulence dissipation.

3.New methods are emerging for efficient solution of kinetic problems

4.18 Months on BW. Many exciting results.

1 paper published

2 under review 5 paper in various stage of preparation (from “submitting next week” to just

starting)

data for many more