General MeetingMadison, August 4-6, 2004

Plans and Progressof

Magnetic Helicity Conservation and Transport

H. Ji

for participants of

the Center for Magnetic Self-organization

Outline• Introduction

– Magnetic helicity

– Physics issues

• Goals, plans and progress– The basics

– The role

– The applicabilities

• Summary: relations to other topics

Introduction: Magnetic Helicity

• Magnetic helicity is a global topological quantity which measures linkage (knottedness and twistedness) of magnetic field lines.

• Gauge invariant when B is tangential to the surface of V, which is simply-connected.

• K is conserved in ideal MHD plasmas.

€

K = (r A ⋅

r B )dV∫

Physics Issues

• Ambiguities in gauge-invariant definitions and their physical interpretations still exist in realistic geometries.– Non-uniqueness of gauge-invariant definitions

– Practical formulation

• Are they really conserved in reality?

• How does the helicity conservation regulate dynamo process?– The helicity conservation is a fundamental assumption in the

theories of magnetic self-organization (relaxation).

Physics Issues (Cont’d)

• Can the concept for helicity and relaxation be extended to 2-fluid plasmas?– Other helicities and possible new relaxed states?

• How does the helicity related to solar dynamo, flares, and CMEs (coronal mass ejections)?– “Reversed S” tends to appear in

the northern hemisphere while “S” tends to appear in the southern hemisphere.

Major Goals

I. Study basic properties of magnetic helicity in realistic geometries and its extensions to 2-fluid plasmas.

II. Determine role of conservation and transport of magnetic and kinetic helicities during magnetic self-organization processes.

III. Assess applicabilities of physics understandings gained in achieving Goals I&II to astrophysical circumstances, such as solar dynamo activity and accretion disk dynamics.

Tasks for Goal I - The Basics

• Revisit the concept of relative helicity (short)

• Consider helicity concept in 2-fluid plasmas (short-mid)

• Evaluate conservation of magnetic and kinetic helicities during 2-fluid dynamo simulations (short-mid)

• Evaluate conservation (and transport) of magnetic and kinetic helicities during self-organization events in MST (short-mid)

• Evaluate conservation of magnetic and kinetic helicities during merging processes in MRX and SSX (short-mid)

Gauge-invariant Definition

• Gauge-invariant definition when magnetic field intercepts with boundary and/or the volume is doubly-connected– Concept of relative helicity which subtracts off unknown linkage

outside of V (Berger & Field ‘84, Finn & Anderson ‘84)

– Non-uniqueness of gauge-invariant definitions identified (Blackman & Ji, ‘03).

• Practical formulation exist for spherical shell or slab geometry (Berger ‘85), but not for toroidal/cylinderical shell– Important for a sub-volume of a torus, accretion disks

€

Krel = (A + AV ) ⋅(B − BV )dV∫

Helicities in 2-fluid Plasmas

• Self-helicities (e.g. Steinhauer & Ishida ‘98):

• Electron-helicity reduces to magnetic helicity when size is larger than electron skin depth (c/wpe).

• Ion-helicity is equivalent to helicity of the generalized vorticity (e.g. Mahajan & Yoshida ’98):

• Discussions by C. Hegna (‘98)

€

Kα = Pα ⋅Ωα dV∫

€

Pα = mα Vα + qα A /c

€

Qα =∇ × Pα

canonical momentum: =e,i:

€

G = (A + V) ⋅(B +∇ × V)dV∫

• Helicity change:

• Energy change:

• Ratio of changes:

• In reality, is small in large systems

€

dK

dt= −2(E ⋅B) ⋅2δ ⋅2L ⋅2πR

€

dW

dt= −(E ⋅ j) ⋅2δ ⋅2L ⋅2πR

€

W

K

dK

dW= 2

δ

a

Bt

Bp

€

jt =Bp

μ0δ,W

K=

1

μ0a

⎛

⎝ ⎜

⎞

⎠ ⎟ a: plasma size

Magnetic Helicity Change Relative to Energy: an Estimate due to Reconnection

€

W

K

dK

dW

Test Helicity Conservation in Simulation

(Horiuchi & Sato, ‘88)

• MHD simulation:– Tested for magnetic helicity in some cases

• 2-fluid simulations :– NIMROD ready in a few months– DEBS later

Test Helicity Conservation in MST

External info only(MST ‘95)

Internal + external info (MST ‘03)

• Magnetic helicity:

• Progress in measuring velocity profiles (CHERS)• Neutral beam injection to drive flow

Test Helicity Conservation during Merging

Data from TS-3 (Ono et al., ‘96)

Test Helicity Conservation in SSX and MRX

• SSX:– Data available for evaluation of magnetic helicity– New velocity measurements in progress

• MRX:– Machine upgraded for merging– Plan to drive flow by biasing and/or neutral beam injection

Tasks for Goal II - The Role

• Investigate role of helicity flux for large -effect in RFP and compare with computations (short-mid)

• Evaluate 2-fluid relaxation theories and explore new relaxation phenomena experimentally (mid-long)

• Evaluate role of helicity conservation in cascades where wave anisotropy supercedes invariant constraints on cascades (mid)

– P. Terry’s talk

– E. Blackman’s talk

Quenching of the Dynamo Effect

• The -effect is a key ingredient of dynamo action.• Recent theories and simulations using periodic or

closed boundaries predict the quenching of -effect due to large Rm or B0.

(Cattaneo & Hughes ‘96)

• Large -effect accompanies an large outward helicity flux.

• Importance of open boundaries?

€

=−∇˜ φ ⋅ ˜ B

Β 02

+∇ ˜ P e ⋅ ˜ B

enΒ 02

−

∂ ˜ A

∂t⋅ ˜ B

Β 02

−η ˜ j ⋅ ˜ B

Β 02

smalllarge

The Dynamo Effect is Large in the Lab

large small

• Magnetic field generated in the convection zone– Polarity reverses every 11 years

• Has open boundaries, with helicity flux outward to corona– “S” and “reversed S” loops

• Anything common between lab and sun?

The Dynamo Effect is Also Large in the Sun

• Flow of magnetic energy– Generated in the convection zone– Transported to the corona,

reversing polarity every 11 years– Effectively dissipated by flares

• Flow of magnetic helicity– Generated in the convection zone,

with opposite signs each hemisphere– Transported to the corona, with no

polarity reversal– Almost no dissipation by flares– Removed effectively only by CME

==> Inevitability of CME

Magnetic Helicity and CME (Flux Rope)

(Zhang & Low ‘04)

2-fluid Self-organization or Relaxation

• New relaxed states can be readily found using additional conserved quantities in 2-fluid. An example is given below.

• Energy Magnetic helicity Generalized (or ion) helicity

• Minimizing W while keeping K and G constant:€

G = (A + V) ⋅(B +∇ × V)dV∫

€

K = A ⋅BdV∫

€

W = (B2 + V2)dV∫

€

δW + λ1δK + λ 2δG = 0

(Mahajan & Yoshida ‘98)

€

V −∇ × B = λ1B

V = −λ 2(B +∇ × V)

€

λ2∇ ×∇ × B + (λ1λ 2 +1)∇ × B + (λ1 + λ 2)B = 0

2-fluid Relaxation (Cont’d)

• When λ20, then V0 and

• When λ10 and λ2large, then V is large: €

∇2B = −λ12B

€

Bθ ∝ J1(λ1r)

BZ ∝ J0(λ1r)€

λ2∇ ×∇ × B + (λ1λ 2 +1)∇ × B + (λ1 + λ 2)B = 0

Taylor solution; paramagnetic

Scale arbitrary, decided by λ1

€

∇2B = B

€

Bθ ∝ I1(r)

BZ ∝ I0(r) Scale on order of c/pi

diamagnetic

Shock front, diamagnetic cavity, small s FRC, etc

Preliminary Indications From TS-3

Plan to drive large flows by biasing and/or NBI to explore new relaxation phenomena in MRX/MST

Tasks for Goal III - Application to Astrophysics

• Investigate solar dynamo using a comprehensive computational model (long)

– F. Cattaneo’s dynamo talk

• Relate above helicity concepts to solar dynamo and to computation (long)

– E. Blackman’s talk

• Understand helicity/poynting flux injection in disk-jet system

– Related to coronal MRI

Summary: Relations to Other Topics

• Dynamo:– Conservation and transport of helicity may regulate dynamo

• Reconnection:– Helicity is dissipated at the reconnection region

• Magnetic turbulence:– Conservation of helicity is linked to the inverse cascade

• Angular momentum transport:– Properties of kinetic helicity and cross helicity could be related to

(angular) momentum transport?

Links with Astrophysical Plasmas

• Physics of fast solar/disk dynamo:– Role of helicity conservation and transport

• Magnetic turbulence:– Inverse cascade due to helicity constraint

• High-beta magnetic structures (cavity, hole)– A relaxed state of 2-fluid plasma with significant flow