NIMROD Simulations ofReconnection in MRX and SSX

Nicholas A. Murphy & Carl R. Sovinec

University of Wisconsin-Madison

NIMROD Simulations of Reconnection in MRX and SSX – p.1/29

Outline

Local vs. global effects in reconnection

Two-fluid global simulations of reconnection in MRX,including

Null-helicity cases showing the well-knownquadrupole fieldCo-helicity cases showing a tilt in the current sheetCounter-helicity cases with a radial shift in thecurrent sheet position

Comparisons between a global MRX simulation and asimulation of reconnection in a simplified domain

Resistive MHD simulations of spheromak formation andmerging in SSX with predicted Ion DopplerSpectroscopy measurements

NIMROD Simulations of Reconnection in MRX and SSX – p.2/29

Motivation

Most two-fluid studies of reconnection focus on local

reconnection physics (e.g. Birn et al., 2001) and most global

studies use resistive MHD (e.g. Lukin et al., 2001)

However, the reconnection layer can communicate with the

global magnetic field through MHD and non-MHD effects

Changes in the global magnetic field topology can in turn affect

the reconnection rate

NIMROD has the capability to perform two-fluid global

simulations in a realistic geometry and compare them to local

simulations

The choice of MRX and SSX as global cases allows

benchmarks to aid the simulation effort and provides

computational support to the experimentsNIMROD Simulations of Reconnection in MRX and SSX – p.3/29

Two-fluid Reconnection

The electrons and ions decouple, allowing the electrons to pull

in the magnetic field to a much smaller diffusion region

The signature is a quadrupole out-of-plane magnetic fieldNIMROD Simulations of Reconnection in MRX and SSX – p.4/29

Local vs. Global Effects

Magnetic reconnection is influenced by global effects(length scales, driving forces, geometry) and localeffects (wave-particle interactions, collisionality, Halleffects)

The reconnection layer communicates with the globalmagnetic field through the physics of dispersive wavessuch as whistlers and kinetic Alfvén waves in addition toshear Alfvén waves

This communication might regulate the reconnectionprocess to ensure that reconnection occurs as fast as itis driven

A strategy to gauge local vs. global effects is to take thefields from a global simulation to set up a localsimulation and look for differences in the evolution ofthe plasma NIMROD Simulations of Reconnection in MRX and SSX – p.5/29

Introduction to NIMROD

NIMROD (Non-Ideal Magnetohydrodynamics withRotation, Open Discussion; nimrodteam.org) is anextended MHD code designed to study fusion plasmas

The finite element representation of fields in thepolodial (R,Z) plane allows great geometric flexibility

Basis functions are polynomials of arbitrary degreeMesh packing is used to get high resolution in thereconnection layer

Finite Fourier series in the toroidal (φ) direction

This research is taking advantage of a recent two-fluidimplementation

NIMROD Simulations of Reconnection in MRX and SSX – p.6/29

NIMROD’s Non-Ideal Hall MHD Model

NIMROD solves the equations of extended MHD cast in a single fluidform. The relations for E, Π, and qα determine which model is solved.

∂B

∂t= −∇×

(

ηJ − V × B +1

neJ× B −

1

ne∇pe

)

µ0J = ∇× B

∇ · B = 0

ρ

(

∂V

∂t+ V · ∇V

)

= J× B −∇p −∇ · Π

∂n

∂t+ ∇ · (nV) = ∇ · D∇n

n

γ − 1

(

∂Tα

∂t+ Vα · ∇Tα

)

= −pα∇ · Vα −∇ · qα + Qα

This model is used in simulations of MRX and SSX to study theinterplay between local and global effects in the reconnectionprocess.

NIMROD Simulations of Reconnection in MRX and SSX – p.7/29

The Magnetic Reconnection Experiment

The Magnetic Reconnection Experiment (MRX) is located at

the Princeton Plasma Physics Laboratory and is designed to

study controlled axisymmetric magnetic reconnection.

Plasma parameters: T ∼ 15 eV, B ∼ 200 G, S ∼ 250 − 1000,

and n ∼ 1014 cm−3.

MRX is a good candidate for computational study because the

spatial scale separation is small and plasma parameters are

not too extreme NIMROD Simulations of Reconnection in MRX and SSX – p.8/29

MRX Modes of Operation

By changing the currents in the flux cores, two distinct modes ofreconnection can be induced in MRX (Yamada et al. 1997). OurNIMROD simulations of MRX are investigating both of these modesof operation.

NIMROD Simulations of Reconnection in MRX and SSX – p.9/29

Simulating MRX provides a chance to study global effects on reconnection

Simulations of MRX are axisymmetric with a Lundquist numbersimilar to the experiment (S ∼ 500 − 1000) and with an isothermalequation of state (usually ∼ 5 − 15 eV)

Applied voltage on the flux core surfaces drives reconnection andcan induce a toroidal field

Two-fluid effects that are seen include

The development of a quadrupole field

A tilt of the current sheet when a guide field is present

A radial shift of the current sheet during counter-helicity merging

In order to simplify analysis and avoid the complications of toroidicity,simulations sometimes use a ‘linear’ MRX

NIMROD Simulations of Reconnection in MRX and SSX – p.10/29

Finite Element Grid for MRX

MRX has a nontrivial geometry which requires significant

modification of NIMROD’s preprocessing routines.NIMROD Simulations of Reconnection in MRX and SSX – p.11/29

The well-known quadrupole is present in two-fluid simulations

Two-fluid push reconnection simulations in MRX’s geometry show the development ofthe quadrupole field signature of whistler-mediated reconnection. Here, S ∼ 750,BR ∼ 60 G and c/ωpi ∼ 4 cm.

NIMROD Simulations of Reconnection in MRX and SSX – p.12/29

There is significant electron outflow on scales well belowc/ωpi

The electron outflow is significantly more concentrated and at

higher speeds than the corresponding ion outflows

NIMROD Simulations of Reconnection in MRX and SSX – p.13/29

The current sheet tilts during two-fluid co-helicity reconnection

Introducing a guide field during a two-fluid simulation breaks

the symmetry. (Jφ contours and Bpol vectors)NIMROD Simulations of Reconnection in MRX and SSX – p.14/29

The tilt is associated with a localized vertical Hall electric field

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Electron flowCurrentHall Electric FieldImposed Magnetic Field

In two-fluid null-helicity reconnection, there is a large inward-pointing current densitycorresponding to the localized electron outflow

A guide field imposed on this setup would produce a vertical Hall J × B/ne electricfield, typically on scales below c/ωpi

This Lorentz force on the electrons is represented by the Hall electric field above andtips the flow and reconnection pattern

See also A. Frank et al. (2006) for experimental resultsNIMROD Simulations of Reconnection in MRX and SSX – p.15/29

A shift in position occurs during two-fluid counter-helicit y merging

An opposite shift is seen when the toroidal field directions are reversed (currently, Bφis out-of-plane for Z < 0)

The electrons in the current sheet pull the magnetic field with them, producing this shift

See also Inomoto et al. (2006) for experimental results and simulations by E. BelovaNIMROD Simulations of Reconnection in MRX and SSX – p.16/29

Pressure is higher on the side from which the current sheet was shifted

The uneven pressure distribution is an example of how the local reconnection processcan influence its own boundary conditions

We are working to compare simulation results to the pressure measurementspublished in Inomoto et al. (2006)

NIMROD Simulations of Reconnection in MRX and SSX – p.17/29

Reconnection in a Simplified Domain

To gauge the importance of global effects, two-fluidresults from a geometrically linear MRX case arecompared to a simulation of reconnection in a simplifieddomain

The simulation is in a 20 cm by 50 cm box

Voltage is applied with a half-sine position dependencewith a length of 15 cm with infinite wires setting up theinitial poloidal field

The simulations have the same physical parameters(e.g. resistivity, viscosity, density) and are matched bychoosing simulations with approximately the samereconnection electric field

Ample room downstream prevents the buildup ofmaterial in the outflow region

NIMROD Simulations of Reconnection in MRX and SSX – p.18/29

Comparing the Quadrupole Field

The shape and strength of the quadrupole field aresimilar for MRX (left) and the simplified reconnectioncase (right)

Maximum contours are about ±20 GNIMROD Simulations of Reconnection in MRX and SSX – p.19/29

Comparing the Electron Velocity

Contours of the outflow component of electron velocity(VR and VX , respectively) are similar for the two cases

The peak electron velocity is about 50 km s−1

(VA ∼ 30 − 50 km s−1)

NIMROD Simulations of Reconnection in MRX and SSX – p.20/29

Comparing Current Density

The two cases also show similar out-of-plane current densitycontours when the reconnection rate is matched

More work will be done to show the importance of global effects indifferent regimes

NIMROD Simulations of Reconnection in MRX and SSX – p.21/29

Swarthmore Spheromak Experiment

SSX (see Cothran et al. 2003) is devoted to studyingspheromak formation and merging, magneticreconnection, particle acceleration, and stability ofcompact toroids

The measurement of a quadrupole field has beenreported (Matthaeus et al. 2005)

Current research involves using ion Dopplerspectroscopy to investigate bi-directional outflowssimilar to line-of-sight averaged observations in thecorona

Research will likely soon investigate the formation andstability of a compact toroid in an oblate flux conserver

Plasma parameters: n ∼ 1015 cm−3, T ∼ 25 eV, andS ∼ 1000

NIMROD Simulations of Reconnection in MRX and SSX – p.22/29

Experimental Setup of SSX

Two spheromaks are formed by plasma guns

The spheromaks merge and reconnect at the midplaneof the flux conserver

The resulting compact toroid tilts

This setup is described by Cothran et al. (2003)NIMROD Simulations of Reconnection in MRX and SSX – p.23/29

Simulating SSX allows the study of dynamical time-dependent reconnection

Present simulations are axisymmetric and use resistive MHD with aLundquist number of 500 − 1000 and a temperature of T ∼ 5 − 15 eV

Voltage is applied at the top and bottom boundaries of the domain toforce spheromak merging

The flux conserver shape is faithful to the experiment, but the sharpcorners are rounded and the inner electrode is not included toprevent numerical problems

The density in the flux conserver is typically 5-15% of the peakdensity in the plasma gun

To reduce the need for a large number density diffusivity, an artificial

nonlinear diffusivity proportional to(

∆tV·∇n

n

)2is used to prevent

numerical problems in regions of large flow into a region of lowdensity

NIMROD Simulations of Reconnection in MRX and SSX – p.24/29

Poloidal flux contours show spheromak formation/merging inSSX

NIMROD Simulations of Reconnection in MRX and SSX – p.25/29

Pressure contours show a buildup of density nearR = 0

This buildup reduces reconnection outflow towards the central axisNIMROD Simulations of Reconnection in MRX and SSX – p.26/29

Simulated Ion Doppler Spectroscopy

Double peaked profiles are apparent, but at lower outflow speeds than the 40 km s−1

seen in the experiment

The buildup of density near R = 0 results in more emission from inward outflow asmerging commences (left)

After merging is well underway, the density buildup pushes out, leading to largeoutward velocities (right)

Mach probe investigations in the experiment and different mass loading techniques inthe simulation should shed more light on the outflow pattern in SSX

NIMROD Simulations of Reconnection in MRX and SSX – p.27/29

An oblate flux conserver looks promising for compact toroid formation

Spheromaks have ample time to break off from the plasma guns before merging into aCT using an oblate geometry based on SSX’s proposed upgrade

NIMROD Simulations of Reconnection in MRX and SSX – p.28/29

Conclusions

Several two-fluid results are seen in global simulations of MRX

The tilt of the current sheet during two-fluid co-helicityreconnection is associated with a localized vertical component ofthe Hall electric field

There is an asymmetric pressure buildup associated with theradial shift in position during counter-helicity reconnection

Direct comparisons of MRX to local simulations of reconnection showonly minor differences when toroidicity is ignored

Simulations of SSX show spheromak formation and merging

Predicted IDS measurements suggest that additionalobservations will be needed to understand outflow structure inSSX

Compact Toroid formation through spheromak merging in anoblate flux conserver looks viable for SSX’s proposed upgrade

NIMROD Simulations of Reconnection in MRX and SSX – p.29/29

OutlineMotivationTwo-fluid ReconnectionLocal vs. Global EffectsIntroduction to NIMRODsmall {NIMROD's Non-Ideal Hall MHD Model}The Magnetic Reconnection Experimentlarge {MRX Modes of Operation}scriptsize {Simulating MRX provides a chance to study global effects on reconnection}small {Finite Element Grid for MRX}small {The well-known quadrupole is present in two-fluid simulations}small {There is significant electron outflow on scales well below $c/omega _{pi}$}small {The current sheet tilts during two-fluid co-helicity reconnection}small {The tilt is associated with a localized vertical Hall electric field}small {A shift in position occurs during two-fluid counter-helicity merging}scriptsize {Pressure is higher on the side from which the current sheet was shifted}Reconnection in a Simplified Domainsmall {Comparing the Quadrupole Field}small {Comparing the Electron Velocity}small {Comparing Current Density}Swarthmore Spheromak Experimentsmall {Experimental Setup of SSX}�ootnotesize {Simulating SSX allows the study of dynamical time-dependent reconnection}small {Poloidal flux contours show spheromak formation/merging in SSX}small {Pressure contours show a buildup of density near $R=0$}small {Simulated Ion Doppler Spectroscopy}scriptsize {An oblate flux conserver looks promising for compact toroid formation}Conclusions