Strong-flow gyrokinetics with a unified treatment of all lengthscales

Amil Y. Sharma and Ben F. McMillan

Centre for Fusion, Space and Astrophysics, University of Warwick, UK

1. Brief review of strong-flow gyrokinetics

2. Unified equations

3. Numerical implementation and simulations

4. Conclusions and future work

A brief summary of strong-flow gyrokinetics

Flows Strong-flow formulation k⊥ρt

Dimits, 2012 Weak - Arbitrary

Madsen, 2010 Strong Symplectic O(ε), O(1)

Miyato et al., 2009 Strong Hamiltonian O(ε)

Dimits, 20101 Strong Symplectic Arbitrary

1 We are further developing this theory via an alternative derivation and general geometry, EM

extension (McMillan & Sharma, 2016).

1 / 18

Weak-flow gyrokinetics (Dimits, 2012)

The weak-flow gyrokinetic ordering is

εw ∼ω

Ω∼ k‖ρt ∼

ρtLB∼ u

vt 1,

which cannot be applied to all modern tokamak plasmas (and space plasmas) ingeneral due to the presence of strong flows, for which

u

vt∼ 1.

2 / 18

Strong-flow gyrokinetics 1 (Madsen, 2010)

The electrostatic potential can be split into

φ = φ0 + φ1,

with φ0 a large, long-wavelength component, and φ1 a small, short-wavelengthcomponent.

However:

1. we have that tokamak turbulence exhibits interaction on all length scales;

2. this splitting introduces ambiguity into the energetically consistent computation ofa field equation via

∂L

∂φ= 0.

3 / 18

Strong-flow gyrokinetics 2 (Miyato et al., 2009)

By using a guiding-centre ordering,

k⊥ρt ∼ εgc 1,

qφ0T∼ ε−1gc ,

and including φ0 in the guiding-centre transform,

X = x − 1

Ωb ×

(v⊥ +

1

Bb ×∇φ0(X , t)

)+O

(ε2gc),

equations with the same form as those for weak flows in the k⊥ρt 1 limit can berecovered (although with the full Hamiltonian being more complex).

4 / 18

Strong-flow gyrokinetics 3 (McMillan & Sharma, 2016)The gyrokinetic ordering may be modified (Dimits, 2010) for strong-flows as

εs ∼ω

Ω∼ k‖ρt ∼

ρtLB∼ |∇× u|

Ω 1, (1)

where v is defined to be in a frame moving with a velocity u.

• The eddy turnover time is ∼ |∇× u|−1, which means that, if

|∇× u|Ω

∼ 1,

thenω

Ω∼ 1,

thus, Ordering (1) appears to be a minimum requirement for a gyrokinetic theory.

• It is compatible with the MHD ordering, and similar to the Hasegawa-Mimaordering.

5 / 18

Strong-flow gyrocentre Lagrangian

The electrostatic, slab Lagrangian up to first order is

Γ =[A(R) + v‖b+u

]· dR + µdθ −

(12v

2‖ + µB+1

2u2 + 〈φ〉

)dt,

u = B−1b ×∇〈φ〉,

where v‖ and µ are defined to be in a frame moving with velocity u.

As u appears outside the Hamiltonian, implicit time-dependence prevents applicationof standard direct numerical schemes.

6 / 18

Strong-flow Vlasov-Poisson systemThe Vlasov-Poisson system obtained as a whole, directly from the gyrocentreLagrangian is

f,t + Ri f,i = 0,

R = v‖b + u + B∗−1‖ b × u1,

v‖ = 0,

0 =

∫d6Zδ(R + ρ− r)[B∗‖ f + B−1∇ · f u1],

whereu = B−1b ×∇〈φ〉,

B∗‖ = b · (B + ∇× u),

u1 = (∂t + u ·∇)u

and we consider two-dimensional potential perturbations.7 / 18

Fluid correspondence

In the weak-flow and k⊥ρt ∼ ε limits, our Vlasov-Poisson system yields theHasegawa-Mima equation,

d

dt

(b ·∇× u − ln ne

)= 0,

whered

dt=

∂

∂t+ u ·∇

andne = n0e

φ.

8 / 18

Pull-back to original coordinates

For k⊥ρt ∼ ε, the actual location of the particle is

Weak-flow representation:

x = R + ρ + B−2∇φ(R) +O(ε2)

= R + ρ +O(ε).

Strong-flow representation:

x = R + ρ + B−2∇[φ(R)−〈φ〉] +O(ε2)

= R + ρ +O(ε2).

And the fields are evaluated on the gyroring,

x ′ = R + ρ.

9 / 18

Formal solution

The system Lagrangian is

L =

∫dZ0f (Z0)Lp

(Z (Z0, t), Z (Z0, t), φ

),

=

∫dZ0f (Z0)Lp

(Z (Z0, t), Z (Z0, t), φ(f , Zs)

).

• The formal solution is computationally intractable.

• Thus, we perform an iterative solution of the equations.

10 / 18

Numerical scheme

We choose to use the δf particle-in-cell (PIC) method, with

f = f0 + δf .

We evolve our markers using an iterative method:

1. Take an initial RK4 step by neglecting the terms involving u1;

2. Compute a cubic spline representation of R(t) and δf (t) on the interval[t, t + ∆t];

3. Compute u1 via R(t), δf (t)→ n(t)→ φ(t)→ u(t) and finite differences;

4. Take an RK4 step including all terms using this estimated value of u1.

5. Iterate to desired level of convergence.

Alternatively, a multistep or hybrid method could be considered.

11 / 18

Weak-flow code verification via the Kelvin-Helmholtz instability

We initialise

φ = A(sin kyy + 10−4 cos kxx).

Growth-rate spectra:simulations (points);semi-analytic (solid curve).

For comparison:simplified, three-wave analytic(dashed curve) assuming

|φk1 | |φk2 | ∼ |φk3 | |φkn |,

n 6= 1, 2, 3.

12 / 18

Blobs (motivation)

Plasma and Fusion Research: Review Articles Volume 4, 019 (2009)

to closed magnetic surfaces (ρ < 0) and scrape-off layer(0 < ρ < 1) and wall shadow regions (ρ > 1) with strongdamping of all dependent variables to account for transportalong field lines intersecting material walls. All profilesevolve self-consistently with order unity fluctuation levels.Time series of the plasma parameters at fixed radial posi-tions are recorded and analyzed similar to the experimentaldata [48–50].

For a large range of model parameters, the numeri-cal simulations show intermittent eruptions of plasma andheat into the scrape-off layer. An example of the structuresobserved in these simulations is presented in Fig. 5. Asso-ciated with the blob structures are large amplitude bursts in

Fig. 5 Motion of blob-like structures in the particle density fromtwo-dimensional turbulence simulations. The verticalline labeled ρ = 0 corresponds to the last closed fluxsurface while the line labeled ρ = 1 corresponds to thewall radius. The bottom panel follows a time 30/ωci af-ter the top panel, where ωci = eB/mi is the ion gyrationfrequency. The size of the simulation domain is 150 ρs

in the radial direction (horizontal axis) and 100 ρs in thepoloidal direction (vertical axis), where ρs = Cs/ωci.

the probe time series, as shown in Fig. 4. These were thefirst simulations to reproduce the salient experimental ob-servations and a detailed comparison with probe measure-ments on TCV showed excellent agreement for the tempo-ral correlations and statistical distribution of the particledensity and turbulence-driven transport in the scrape-offlayer [28–30].

Probe measurements in the TCV scrape-off layerdemonstrate that the fluctuations have universal proper-ties for a large variation in experimental control parame-ters [26–31]. This is clearly demonstrated in Figs. 6 and7, which show the rescaled conditional averages and prob-ability distribution functions of the particle density for ascan in line-averaged density (ne measured in 1019 m−3)and plasma current (Ip measured in kA). The probabil-ity distribution functions of the particle density signals arepositively skewed and flattened due to the many large-amplitude bursts in the time series. When appropriatelyrescaled, the conditional averages and distribution func-tions have similar shapes across both the density and cur-rent scans. This strongly suggests that the same physicalmechanism underlies the fluctuations in all these parameterregimes.

Fig. 6 Conditionally averaged particle density fluctuations inTCV scrape-off layer for a scan in line-averaged densityand plasma current.

019-4

(euro-fusion.org; Garcia, 2009)

13 / 18

Blobs (motivation)

chamber filled with argon gas at 105–104 torr. Since theplasma density is typically 2 1016 m3, the ionizationfraction is only about 1%, and there is a constant back-ground of neutrals even after breakdown.

The plasma is tracked by an array of 200 Langmuirprobes. The tip spacing is 7 cm horizontally and 7 cmvertically, with triple resolution (horizontally) near thecenter. The main Langmuir probe array is located at asingle toroidal angle, but other Langmuir probes are usedto verify the azimuthal symmetry of the blobs. The otherprobes (not shown in Fig. 1) include 3 vertical lines ofstainless-steel cylinders and a horizontal line of cylindri-cally shaped, heated tungsten filaments. These filamentsare used to measure the full I-V characteristic (analyzed bytaking into account finite sheath size, i.e., using ABRtheory [15]), and hence the electron temperature andplasma potential. Heating the filaments between dischargeseliminates important surface contamination effects, andprevents overestimation of the electron temperature (see,e.g., [16,17]).

We observe experimentally for the first time the mush-room blob shape, which has been seen in many simulations(e.g., [13,18]). This shape is displayed in Fig. 2, whichshows the propagation of a typical blob in poloidal crosssection. The time step between adjacent density plots is100 s, and the first plot occurs 25 s after the micro-

waves are turned off. The blob shape exhibits ‘‘wings,’’which develop about a blob length away from the creationregion. The right-hand part of Fig. 2 shows the floatingpotential with some overlaid density contours. The poten-tial is obtained by combining data from a vertical array anda horizontal array.

The propagation seen in Fig. 2 can be quantified and it isfound to depend on the neutral pressure in the chamber.This dependence is explored in separate plasma dischargescovering a range of neutral pressures. As Fig. 3 shows, wefind that the blob’s center-of-mass speed is inversely pro-portional to the neutral pressure. The speed measurement isbased on a time-of-flight calculation using density traces atmultiple probes. Also plotted is a line indicating that thesound speed (cs

Te=mi

p 2 103 m=s assuming 2 eV

electrons) is an upper bound on the blob velocity. However,the three low-pressure points that give evidence for thisbound are from blobs with different shape and very lowdensity.

To describe the blob propagation, we use the standardvorticity equation [19] derived from MHD, with the addi-tion of a neutral-collision term,

r min

B2

Dr?Dt

rkJk2

Bb rpr

min

B2 r?;

(1)

where ? and k are defined with respect to the magneticfield, D=Dt @=@t v r, b B=B, b rb is themagnetic curvature, is the ion-neutral collision fre-quency, and we have assumed v cs and jB=rBj jn=rnj jv=rvj. The vorticity is given by r v r2=B (where v r B=B2). Equation (1) may besimplified for our experimental geometry. We have purelytoroidal magnetic field B Be / 1=R, so that b eand eR=R. We then neglect rkJk, since the toroi-

FIG. 2 (color). Poloidal cross section of typical blob at 3different times (t 100 s), showing characteristic mush-room shape. The density is calculated from ion saturation cur-rent; its decrease is consistent with the expansion of the blob.The blob propagation is consistent with the vertical electric field,which is reflected in the potential structure at right. The overlaidEB velocity arrows show the velocity field of a vortex pair.

10−4

10−3

102

103

104

Neutral Pressure (torr)

Ave

rage

Blo

b Sp

eed

(m/s

)

1/Pn

cs

FIG. 3. Blob center-of-mass speed versus neutral pressure(Pn). The speed scales inversely with the pressure, but thisscaling appears to break down at low pressure. The error inspeed is approximated by the standard deviation of the inferredblob speed as it fluctuates in time.

PRL 101, 015003 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending4 JULY 2008

015003-2

S.I. Krasheninnikov / Physics Letters A 283 (2001) 368–370 369

Fig. 1.∇B plasma polarization and associatedE × B drift result inoutward motion of plasma blob in tokamak far scrape off layer.

2. Equations and estimates

The ∇B drift of charged particles in a tokamakmagnetic field results in plasma polarization and,correspondingly, theE × B plasma flow. This effectof E × B flow becomes rather strong in the SOLdue to effective “sheath resistivity” [11] when plasmacontact with diverter targets at some distance from theseparatrix where the effects of strong magnetic shearinduced by X-point [12] can be neglected. Unlike [11]we assume that perpendicular plasma motion is fastand neglect impact of parallel plasma flow.

For simplicity we assume that the SOL plasmatemperature,T , is constant. Then, for a small plasmaresistivity electrostatic potential,ϕ, caused by the∇Bdrift, is constant alongB and can be found from theequation for electric current

(1)∇ j⊥ + ∇||j|| = 0,

with j⊥ = c( B × ∇P)/B2, whereP = nT , n is theplasma density,c is the light speed. Integrating Eq. (1)along the field line and using boundary conditionsj|||target≈ entCs(eϕ/T ) at the targets (we assume that|eϕ|< T ) we find

(2)eϕ

T= ρi

2ntB

∫d∇nB · ( B × ∇n),

wherent is the plasma density near the targets,Cs =√T/M is the plasma sound speed,M is the ion mass,

e is the elementary charge,ρi is the ion gyro-radius,and the coordinate goes along the magnetic fieldline. Consider plasma blob of densitynb with parallellength b situated around midplane and neglect the

effects of magnetic shear from Eq. (2) we have

(3)eϕ

T= bρi

2Rnt

∂nb

∂y,

where we used∇nB = ex/R, R is the tokamak ma-jor radius, andx andy are the coordinates along majorradius and poloidal direction, respectively. Using ex-pression (3) to findE× B drift velocity we write blobplasma continuity equation in the form

∂nb

∂t+Cs ρ

2i

2

b

R

∂

∂x

[nb∂

∂y

(1

nt

∂nb

∂y

)]

− ∂

∂y

[nb∂

∂x

(1

nt

∂nb

∂y

)]= 0.

(4)

For the casent = ξnb , whereξ = const, we find thatseparable solution

(5)nb(t, x, y)= n(x)b (t, x)n(y)b (y),with n(y)b (y) ∝ exp(−(y/δ)2), whereδ is the poloidalscale length of the blob, transforms Eq. (4) to ballisticequation forn(x)b (t, x), (∂t + Vb∂x)n(x)b = 0, with

(6)Vb = Cs(ρi

δ

)2b

R

nb

nt,

which sets the velocity of a blob propagation in radialdirection. Notice that separable solution does not setradial scale of the blobs.

Even though separable solution is just an exampleit gives an estimate for radial velocity of the blobsand allows to estimate radial distance,∆b, at whichblob can travel at it’s life time,τb ≈ b/Cs , beforedisappearing due to parallel plasma flows. Taking, asan estimate,b ∼ qR, whereq is the safety factor,from Eq. (6) we find

(7)∆b ≈ Vbτb ≈ R(qρi

δ

)2nb

nt.

Analysis of radial motion of isolated blobs makessense only when∆b > δ, which gives

(8)δ < δmax=R(qρi

R

)2nb

nt

1/3

and limits number of plasma particles in a blob,Nb ≈nbbδ

2,

Since, their theoretical description has been fastlydeveloping [44–47]. For the evolution of blobs inthe SOL it is important to understand that theplasma in the SOL is not confined, but streamingoff along magnetic field lines toward the divertortarget plates. Plasma parameters thus change drasti-cally along as well as across magnetic field lines,opposite to the confined region. In the SOL flux sur-faces can still be constructed, but loose their impor-tance as on a SOL flux surface the plasma pressureis not constant. Assuming that plasma is ejected as ablob from the edge into the SOL over a finite paral-lel extend, in a poloidal window of unfavorable cur-vature, the blob will be a plasma cloud expandingalong magnetic field lines into vacuum while propa-gating radially across magnetic field lines. Fig. 3contrasts the blob evolution in the SOL with thedrift type dynamics in the edge region. With perpen-dicular velocities of plasma filaments (VBlob) of acouple of percent of the ion-sound speed cs [6], blobscan move radially across the SOL, which typically isa couple of centimeters wide (DSOL), before theyexpand to the divertor target plates which aremeters ðLkÞ away: Lk=Cs DSOL=V Blob.

The SOL does not honor the resistive MHDequilibrium existing in the closed field line plasmaregion. The Pfirsch–Schluter current system can,for example, not be closed and thus plasma proper-

ties are characterized by a balance between paralleland perpendicular transport. Once again the timeaveraged flow velocities do not reflect the redistribu-tion of energy and particles during intermittenttransport events. There is no useful separationbetween fluctuations and background in the SOLand fluctuations easily exceed the long time averagevalues which define the background. In the SOLfueling from the edge in interplay with losses tothe divertor determines the average profiles withoutany relation to an equilibrium.

The description and simulation of blobs in theSOL has made significant progress in the last fewyears. Initial models were restricted to the SOLand accounted for sheath dominated parallel lossesonly [48–50]. Recently, 2D simulations with the par-allel losses being due to parallel expansion of anoriginally poloidally localised structure and encom-passing a fueling edge region in addition to theSOL, have had tremendous success in reproducingdetailed properties of the SOL, as transport statis-tics. Predicting the SOL profiles of the Lausannebased TCV Tokamak [51,24] and modeling of theJET SOL profiles appears to be in reach [52].

With ever more detailed simulations available, itshould be noted that it becomes increasingly impor-tant to also model the experimental diagnostics inthe simulations. Fig. 4 shows an ESEL simulated

Fig. 3. Difference of drift wave dynamics on closed field lines (left panel) to situation of a blob in the SOL (right panel).

radial position [ρs]

polo

idal

pos

ition

[ρ

s]

150 200 250 300 350

50

100

150

radial position [ρs]

polo

idal

pos

ition

[ρ

s]

150 200 250 300 350

50

100

150

Fig. 4. Blob in the ESEL simulation for Alcator C-Mod (left panel) and inferred Da image of the same structure (right panel) (Figurecourtesy of O. Grulke).

V. Naulin / Journal of Nuclear Materials 363–365 (2007) 24–31 29

Since, their theoretical description has been fastlydeveloping [44–47]. For the evolution of blobs inthe SOL it is important to understand that theplasma in the SOL is not confined, but streamingoff along magnetic field lines toward the divertortarget plates. Plasma parameters thus change drasti-cally along as well as across magnetic field lines,opposite to the confined region. In the SOL flux sur-faces can still be constructed, but loose their impor-tance as on a SOL flux surface the plasma pressureis not constant. Assuming that plasma is ejected as ablob from the edge into the SOL over a finite paral-lel extend, in a poloidal window of unfavorable cur-vature, the blob will be a plasma cloud expandingalong magnetic field lines into vacuum while propa-gating radially across magnetic field lines. Fig. 3contrasts the blob evolution in the SOL with thedrift type dynamics in the edge region. With perpen-dicular velocities of plasma filaments (VBlob) of acouple of percent of the ion-sound speed cs [6], blobscan move radially across the SOL, which typically isa couple of centimeters wide (DSOL), before theyexpand to the divertor target plates which aremeters ðLkÞ away: Lk=Cs DSOL=V Blob.

The SOL does not honor the resistive MHDequilibrium existing in the closed field line plasmaregion. The Pfirsch–Schluter current system can,for example, not be closed and thus plasma proper-

ties are characterized by a balance between paralleland perpendicular transport. Once again the timeaveraged flow velocities do not reflect the redistribu-tion of energy and particles during intermittenttransport events. There is no useful separationbetween fluctuations and background in the SOLand fluctuations easily exceed the long time averagevalues which define the background. In the SOLfueling from the edge in interplay with losses tothe divertor determines the average profiles withoutany relation to an equilibrium.

The description and simulation of blobs in theSOL has made significant progress in the last fewyears. Initial models were restricted to the SOLand accounted for sheath dominated parallel lossesonly [48–50]. Recently, 2D simulations with the par-allel losses being due to parallel expansion of anoriginally poloidally localised structure and encom-passing a fueling edge region in addition to theSOL, have had tremendous success in reproducingdetailed properties of the SOL, as transport statis-tics. Predicting the SOL profiles of the Lausannebased TCV Tokamak [51,24] and modeling of theJET SOL profiles appears to be in reach [52].

With ever more detailed simulations available, itshould be noted that it becomes increasingly impor-tant to also model the experimental diagnostics inthe simulations. Fig. 4 shows an ESEL simulated

Fig. 3. Difference of drift wave dynamics on closed field lines (left panel) to situation of a blob in the SOL (right panel).

radial position [ρs]

polo

idal

pos

ition

[ρ

s]

150 200 250 300 350

50

100

150

radial position [ρs]

polo

idal

pos

ition

[ρ

s]

150 200 250 300 350

50

100

150

Fig. 4. Blob in the ESEL simulation for Alcator C-Mod (left panel) and inferred Da image of the same structure (right panel) (Figurecourtesy of O. Grulke).

V. Naulin / Journal of Nuclear Materials 363–365 (2007) 24–31 29

Figure 4.6: Blob propagation through the edge, last closed flux surface (LCFS)

and scrape-off layer (SOL), where cs is the sound speed (Naulin, 2007).

tonically increasing weak- and strong-flow parallel vorticity at the end-state of

the simulation. This is shown in Figures 4.9 and 4.10. For weak flows, the

distribution of parallel vorticity is symmetric. For strong flows, the magnitudes

of the positive values of the parallel vorticity peak more than those of the nega-

tive values, however, there is a greater spread in the negative than the positive

values of the parallel vorticity.

We consider a circularly symmetric shear layer in the next subsection.

4.1.1 Circular symmetry

We may examine the Kelvin-Helmholtz instability of a shear layer with circular

symmetry.

The simulations used 223 markers, Ni = 16 and ∆t = 1. The potential was

initialised to contain a circularly symmetric shear layer. An example initialisa-

tion is shown in Figures 4.11 and 4.12.

The circularly symmetric shear layer is Kelvin-Helmholtz unstable, as shown

in Figures 4.11 and 4.12. Once again, we have the symmetric, weak-flow evolu-

tion (4.1), as shown in Figures 4.13, 4.14, 4.15 and 4.16, and the asymmetric,

strong-flow evolution (4.2), as shown in Figures 4.17, 4.18, 4.19 and 4.20.

72

(Katz et al., 2008; Krasheninnikov, 2001; Naulin, 2007)

14 / 18

Blobs (model)

We simulate:

• Two-dimensional blobs;

• After blob formation and ∇B polarisation has transpired;

• Thus, we initialise a dipole potential.

15 / 18

Strong-flow blob propagation depends on b ·∇× u

16 / 18

Conclusions and future work

• A strong-flow gyrokinetic theory with a unified treatment of all length scales hasbeen developed and numerically implemented.

• The Vlasov-Poisson system is obtained as a whole, directly from our gyrocentreLagrangian, and has correspondence to fluid equations.

• We use an iterative numerical solution of our Vlasov-Poisson system.

• We see strong-flow symmetry-breaking that depends on the sign of the parallelvorticity.

• Code verification has been performed with basic slab instabilities.

Centrifugal and drift instability simulations are to be performed.

A nonlinear Poisson solver is to be developed.

17 / 18

Stand-alone Poisson solver

Features:

• Arbitrary-wavelength perturbations;

• Cubic B-spline finite-element discretisation;

• Slab and cylindrical geometries;

• Background density and temperature gradients;

• MPI parallelisation;

• Fortran source code;

• Based on the solver from the ORB5 code (Dominski et al., 2017).

Future features:

Extension to three dimensions;

Field-aligned geometry.

18 / 18

Strong-flow Kelvin-Helmholtz instability of a shear layer

Analytic:-ve vorticity (solid);+ve vorticity (dashed).

Simulations:-ve vorticity (dotted);+ve vorticity (dot-dash).