Numerical studies of two-fluid tearing and dynamo in a ... · Outline Introduction – MHD...

Numerical studies of two-fluid tearing and dynamo in a Numerical studies of two-fluid tearing and dynamo in a cylindrical pinchcylindrical pinch

Jacob King, Carl Sovinec, and Vladimir Mirnov

University of Wisconsin – Madison

APS-DPP November 2009

NIMROD Team Meeting Remix

OutlineOutline

● Introduction

– MHD simulations of RFPs.

– Two-fluid system of equations

– The NIMROD code

– Simulation configuration and tearing phase orientation● Linear cases

– Magnetic sturcture and Ion-electron flow decoupling

– Dynamo electric field● Nonlinear island saturation

– Island structure and flows

– Fluctation driven dynamo electric field and mean current modification

– Saturated island force balance● Multihelicity – the RFP dynamo

● Conclusions and future work

Outline RemixOutline Remix

● Introduction

– MHD simulations of RFPs.

– Two-fluid system of equations

– The NIMROD code

– Simulation configuration and tearing phase orientation● Linear cases

– Magnetic sturcture and Ion-electron flow decoupling

– Dynamo electric field● Nonlinear island saturation

– Island structure and flows

– Fluctation driven dynamo electric field and mean current modification

– Saturated island force balance● Multihelicity – the RFP dynamo

● Conclusions and future work

Quick review

A quick reminderA quick reminder

● Unless otherwise indicated we are studying a single helicity tearing mode after nonlinear saturation.

● These cases are for a force free paramagnetic pinch equilibrium – there are no drift effects.

● By two-fluid we indicate the inclusion of

– a generalized Ohms law,

– and ion gyroviscosity.

● Both effects are significant as the ion sound gyroradius, ρs, become large relative to scales of

interest.

● The magnetic field is now advected by the electrons, and Hall term contributions represent electron and ion flow decoupling.

E=−v i×BJ×Bne

−∇ pene

Jmene2

∂ J∂ t

−v e×B

gyro=mi pi4eB

[ b×W⋅ I3 b b− I3 b b⋅W×b ]

d i=c pi

s=csci

=2 d i =0.1 a=1 a=1 Pm=1S=5000+

In the standard picture of a tearing mode, flows advect the In the standard picture of a tearing mode, flows advect the reconnecting magnetic flux through the resistive layer.reconnecting magnetic flux through the resistive layer.

● This 'slab' picture will be modified in the work here by

● Cylindrical curvature

● Regularity conditions

● Asymmetric parallel current profile

● The generalized Ohm's law

● Ion gyroviscosity

● After projecting the (1,1) component, the eigenmode is represented by complex funtions of radius.

● For the linear cases all eigenfunction components can be multiplied by a complex constant; we use this to normalize the phase such that B

r is real.

● In a resistive MHD case, this also makes Vr and B

┴ purely imaginary, however this isn't so in the

two-fluid cases.

Linear analysisLinear analysis

The linear growth rate varies as a function of The linear growth rate varies as a function of ρρss..

● The small Lundquist number dispersion exhibits a single fluid regime (ρs ≤ 0.01), an intermediate

regime where ion gyroviscosity slows the ion flows and the growth, and an electron flow dominated regime (ρ

s ≥ 0.20).

S=5000 S=105

The dynamo electric field is created by the mhd term at small The dynamo electric field is created by the mhd term at small ρρss; ;

both the Hall and mhd dynamos contribute at intermediate both the Hall and mhd dynamos contribute at intermediate ρρss..

● The net effect of the sum of the linear dynamos in all cases is similar.

ρs=0.01 cold ions

ρs=0.01 ion gyroviscosity

ρs=0.05 cold ions


Nonlinear saturated island analysisNonlinear saturated island analysis

The simulations are nonlinearly advanced until well after the The simulations are nonlinearly advanced until well after the island saturates. (~1.5island saturates. (~1.5ττ

RR))

● Without ion gyroviscosity the island width is 0.35 and independent of ρ

s.

● The use of the generalized Ohm's law has no effect on the saturation amplitude.

● The saturated island width varies with gyroviscosity, as shown in the table:

ρs

0.01 0.02 0.05 0.10 0.20

island width

0.34 0.24 0.21

r/a

z/a

ρs

0.01 0.05 0.20

island width

0.34 0.24 0.21

● The saturated island is dominated by the (1,1) perturbation, higher harmonics remain small.

The gyroviscous case saturates at a smaller amplitude, but the The gyroviscous case saturates at a smaller amplitude, but the shape of the magnetic perturbation is largely unchanged.shape of the magnetic perturbation is largely unchanged.

ρs=0.01

cold ions

ρs=0.01

ion gyroviscosity

ρs=0.05

cold ions


The ratio of the saturated perturbed kinetic energy to magnetic The ratio of the saturated perturbed kinetic energy to magnetic energy is 10 times less than the linear cases.energy is 10 times less than the linear cases.

ρs=0.01 cold ions


ρs=0.05 cold ions


● v┴ ≈ v

e┴, except for the ρ

s=0.05 case with ion gyroviscosity

● V┴ ≈ ve┴, except for the

We now superimpose the electron flows onto the islandWe now superimpose the electron flows onto the island..


The nonlinear inflow exhibits the decoupling of the ion and The nonlinear inflow exhibits the decoupling of the ion and electron flows at large electron flows at large ρρ

ss with ion gyroviscosity. with ion gyroviscosity.

ρs=0.01 cold ions


ρs=0.05 cold ions


Adding to our sketch we see an outboard eddy flowing into Adding to our sketch we see an outboard eddy flowing into the magnetic island.the magnetic island.


● The asymmetry in the inflow is a result of the cylindrical geometry.

We can use this sketch to probe the origins of the dynamo We can use this sketch to probe the origins of the dynamo electric field.electric field.

● The product of the flows and the magnetic island produce a positive parallel fluctuation induced electric field at the the rational surface, and a negative dynamo electic field on the outboard separatrix.

● This will reduce the parallel current at the rational surface and increase it on the outboard of the separatrix.

The fluctuation driven dynamo electric field modifies the The fluctuation driven dynamo electric field modifies the mean current profile.mean current profile.

ρs=0.01

cold ions

ion gyroviscosityρ

s=0.01

ρs=0.05 cold ions


The dynamo electric field flattens the mean current profile and The dynamo electric field flattens the mean current profile and eliminates the source of the free energy.eliminates the source of the free energy.

● Why do the gyroviscous cases saturate at a smaller island amplitude?

ρs=0.05 gyro

ρs=0.05 cold ions

Island width = 0.35


Island width = 0.24

The island force balance is a tool to study the effect of ion The island force balance is a tool to study the effect of ion gyroviscosity in reducing the island saturation width.gyroviscosity in reducing the island saturation width.

● As the fluctuation induced dynamo electric field modifies the mean current and magnetic profiles, they induce secondary forces.

● The secondary forces balance the driving forces, slowing and eventually stopping the growth of the island. (Rutherford 1973, Arcis et. al. 2006, Militello et. al. 2006)

i v 1,1 v⋅∇ v 1,1=J eq×b1,1j 1,1×Beq−∇ p1,1−∇⋅ 1,1j0,0×b1,1 j1,1×b0,0

inertial effects driving forces viscosity secondary forces

F d=J eq×b1,1 j1,1×Beq−∇ p1,1

F s=j0,0× b1,1j 1,1×b0,0

The ion gyroviscous force opposes the driving forces.The ion gyroviscous force opposes the driving forces.

● Other terms such as advection and the isotropic viscosity are small and are not plotted.

ρs=0.01 cold ions


ρs=0.05 cold ions


The primary contribution to the gyroviscous force is a term The primary contribution to the gyroviscous force is a term with a simple analytic form.with a simple analytic form.

● In a slab geometry with a large z-oriented guide field, the equation for gyroviscosity can be reduced.

● In our cylindrical cases, this for nearly holds as the curvature, equilibrium flow, and equilibrium gradient effects remain small.

● As the Laplacian operator primarily does not mix the phases (curvature excluded) the ion flows out of phase (black lines) with the flows responsible for reconnection (red lines) create the ion gyroviscous force.

−∇⋅ gyro=−∇⋅mi pi4eB

[ b×W⋅ I3 b b− I3 b b⋅W×b ]≃pi

2ci

∇2v y ,−∇

2 v x ,0

∇ v

We are extending these results to a higher Lundquist numberWe are extending these results to a higher Lundquist number

● As S becomes large we expect the linear resistive layer to become small as

● Thus the two-fluid effects may be more significant at smaller d

i, or ρ

s, than the small S

cases.

● We see that at S=20,000 the same results from S=5000 hold: the ion gyroviscous force causes the ion and electron flows to separate and plays a role in the saturation.

ρs=0.05 cold ions

S=20,000


S=20,000

1S a

S−1 /5

Multihelicity CasesMultihelicity Cases

Cases with a realisitic aspect ratio and current drive produce Cases with a realisitic aspect ratio and current drive produce the reversed state characteristic of the RFP.the reversed state characteristic of the RFP.

● We examine the initial relaxation event, the spike in the field reversal parameter, as this event is a close to a sawtooth as this simulation has come.

● We will make plots at two time points, before (t = 3282), during (t = 3328) the relaxation event.

F=B z a

⟨B z ⟩=

B a⟨B z ⟩

F

ln E

mag

Θ~1.6 ρ

s=0.05 warm ions

S=5000R/a = 3P

m=1

m= 0m= 1

m= 2

m= 3

m=5m=4

t/τa

t/τa

Both the Hall and MHD dynamos flatten the current profile Both the Hall and MHD dynamos flatten the current profile and contribute to the reversal of the mean field.and contribute to the reversal of the mean field.

● The dynamos suppresses current in the core, and drives current near the edge.

● This process reduces the axial current in the core and thus the poloidal magnetic field.

● The poloidal current is increased, and thus the axial magnetic field is increased in the core, and reduced at the edge driving the reversal of the mean magnetic field.

t=3,282 (before) t=3,328 (during)

ConclusionsConclusions

● In hot, low guide field plasmas, two-fluid effects may play a role if ρs is large.

● When ρs is on the order of the resistive layer width, the electron fluid becomes decoupled from the

ions and able to quickly advect magnetic flux into the resistive layer, increasing the growth rate of the tearing mode.

● In the intermediate ρs regime, where the ion flow is advecting the magnetic flux but ion

gyroviscosity is not negligible, the ion gyroviscous force slows the ion motion, and growth of the mode.

● The saturation amplitude is not affected by the use of a generalized Ohm's law for the parameter regime studied.

● However, the ion gyroviscosity can produce a force opposing the driving forces and thereby decreasing the saturation amplitude at large and intermediate ρ

s.

● The Hall dynamo is making a significant contribution to the full RFP dynamo in the multihelicity case studied.

Future WorkFuture Work

● It is important to continue to extend the results presented here into the experimentally relevant high S regime.

● We plan to extending the multihelicity results to higher Lundquist number, and further analysis of the existing cases.

● We will also examine effects such as drift tearing with an equilibrium pressure gradient.

Extra slidesExtra slides

MHD simulations of reversed field pinches, RFPs, have MHD simulations of reversed field pinches, RFPs, have produced many important results, but the models to date are produced many important results, but the models to date are

too simple to capture all the physics.too simple to capture all the physics.

● Previous simulations have been able to reproduce many aspects of the RFP

– Fluctuation spectrum

– Nonlinear coupling

– RFP dynamo

– See Schnack, Caramana, and Nebel 1985, Ho and Craddock 1991, Cappello and Biskamp 1996.

● However, some effects are missing in these models such as

– Drift effects

– Ion-electron decoupling

– Fast reconnection

– Kinetic FLR effects (Svidzinkski, Li, Albright, and Bowers 2009)● We concentrate on the two-fluid physics exposed by including ion and sound gyroradius

effects.

By 'two-fluid' we indicate the use of a generalized Ohm's law By 'two-fluid' we indicate the use of a generalized Ohm's law and the inclusion of ion gyroviscosity in the MHD model. and the inclusion of ion gyroviscosity in the MHD model.

d ndtn∇⋅v i=Dn∇

2 n

J=n e v i−ne ve

mi nd vdt=−∇ pJ×B−∇⋅

E=−v i×BJ×Bne

−∇ pene

Jmene2

∂ J∂ t

T e=T i=T /2

−v e×B

= iso gyro

iso=mi nW

gyro=mi pi4eB

[ b×W⋅ I3 b b− I3 b b⋅W×b ]

W=∇ v∇ vT−23I ∇⋅v

Pm==0.1

==Dn

● The two-fluid model exhibits modified plasma behavior when the ion sound gyroradius, ρs,

becomes appreciable relative to scales of interest.

● The electron fluid advects the magnetic field, and on scales smaller than ρs, it is decoupled from the

ion fluid.

● The ion gyroviscosity coefficient scales as ωciρ

s

2.

● Small density, temperature, and isotropic viscosity diffusion are included.

32nd TdtnT ∇⋅v=∇ 2T

Our linear and nonlinear calculations are solved with the Our linear and nonlinear calculations are solved with the NIMROD code.NIMROD code.

● NIMROD (nimrodteam.org) is a nonlinear, 3D, initial value code. It can also be applied to the linearized system of equations.

● It starts with a small perturbation imposed on steady state, equilibrium background fields, and is then evolved in time.

● We can characterize our system in terms of three dimensionless parameters: the normalized ion skin depth, d

i/a, the plasma beta, β, and the Lundquist number, S.

● We normalized all lengths to the minor radius, a, and all times to the Alfvén time, τa.

● The ion skin depth, and thus the ion sound gyroradius, will be varied in the following calculations, and dimensionless parameters will be fixed unless otherwise noted.

● The cases presented here are for collisional tearing, where the frozen flux theorem is predominantly broken by the plasma resistivity.

=p0

B02/20

=0.1 d i=c pi

a=a 0mi n

B0

=1 S=ra=aB0

0

mi n=5000 s=

csci

=2 d i+

The two-fluid algorithm has been benchmarked with analytic The two-fluid algorithm has been benchmarked with analytic theory for slab tearing and interchange.theory for slab tearing and interchange.

● The analytic results in (a) are from Mirnov et. al. 2004, and the plasma β is the varied parameter.

● In figure (b) di is the varied parameter, and the analytic model is developed by Ramos et. al.

● See Jardin et. al. IAEA proceedings TH/P9-29 2008.

● The model used in these studies does not include ion gyroviscosity.

● The two-fluid algorithm including gyroviscosity has been benchmarked in P. Zhu et. al., (PRL 2008) for the interchange mode.

The cylindrical paramagnetic pinch model is used to study a The cylindrical paramagnetic pinch model is used to study a tearing mode in an RFP-like discharge.tearing mode in an RFP-like discharge.

● We choose a force-free profile (no drift effects) to isolate a purely current driven tearing mode, with a pinch parameter of Θ=1.38.

●

● For R/a=3, this profile exhibits a bath of unstable modes.

● We cut a cylinder with R/a=3 down by a sixth, to R/a=0.5 to concentrate on the dynamics of a single mode at the q=1 rational surface, r

s=0.34.

● Conducting wall and no slip boundary conditions are enforced at r=1.

● For B = 0.2 T, a = 0.5 m, and Te = 300 eV, ρ

s/a = 0.03.

J 0× B0=∇ p0=0

The magnetic components in phase with the MHD The magnetic components in phase with the MHD eigenfunction are largely unmodified by the two-fluid effects.eigenfunction are largely unmodified by the two-fluid effects.

● The amplitudes are normalized to the saturation amplitude of the MHD island for later comparison.

ρs=0.01 cold ions


ρs=0.05 cold ions


The ion x-point outflow is suppressed by the ion gyroviscosity The ion x-point outflow is suppressed by the ion gyroviscosity and the ion-electron fluids decouple at intermediate and the ion-electron fluids decouple at intermediate ρρ

ss..

ρs=0.01 cold ions


ρs=0.05 cold ions


The ion inflow is The ion inflow is suppressedsuppressed by the ion gyroviscosity and the by the ion gyroviscosity and the electron inflow has a large contribution from j at large electron inflow has a large contribution from j at large ρρ

ss..

● Linear inflow shown

ρs=0.01 cold ions


ρs=0.05 cold ions


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