magnetic nozzle and plasma detachment...

In collaboration with an experimental team:R.D. Bengtson and J. Meyers (UT Austin)D.G. Chavers, C.C. Dobson, and J.E. Jones (MSFC)B.M. Schuettpelz (University of Alabama, Huntsville)C. Deline (University of Michigan, Ann Arbour)

Research supported in part by Ad Astra Rocket Company

Boris Breizman, Mikhail Tushentsov, and Alex ArefievInstitute for Fusion Studies, UT Austin

Magnetic Nozzleand Plasma Detachment Scenario

November 12-16, 2007, Orlando, Florida 49th APS-DPP Meeting

TALK OUTLINE

• Plasma detachment issue

• MHD detachment concept and theory (motivated by VASIMR project)

• Numerical model for steady-state magnetic nozzle

• Code testing and numerical results

• Experiment (DDEX, Marshall Space Flight Center)

• Summary

1APS - DPP 2007

PLASMA DETACHMENT PROBLEM

Plasma-based propulsion systems generate thrust by ejecting directed plasma flow.

• A strong magnetic field is used to guide the plasma.

• The ejected plasma must break free from the spacecraft to produce thrust.

There are two scenarios for plasma detachment:

• Detachment from the magnetic field(by breaking the frozen-in constraint viarecombination or some other mechanism).

• Detachment with the magnetic field(by stretching the field lines along the flowdue to plasma plasma current).

2

nozzle wall

plasma current

field lines stretchedby the plasma

vacuum field lines

APS - DPP 2007

MHD DETACHMENT CONCEPT

•Magnetic energy decreases downstream faster than plasma kinetic energy.

•An initially sub-Alfvénic flow becomes super-Alfvénic downstream.

•Plasma flow can stretch the magnetic field lines after drops below . B

28!

m

in

iV

�

22

0.6 0.8 1 1.2 1.40

1

2

3

4

.

2 28B S! "

#

2

1

2

i im nV

S!

�

SS (nozzle cross-section)

E.N.Parker, Astrophys. J. 128, 664 (1958)E.B.Hooper, Journal of Propulsion and Power 9, 757 (1993)

3

Magnetic flux (BS) = const

Flow velocity (V||) = const

Plasma flux (niV

||S) = const

Conserved quantities:

APS - DPP 2007

MAGNETIC NOZZLE LAYOUT

Nozzle endNozzle wall

Sub-Alfvénicflow

Transition to super-Alfvénic flow

Super-Alfvénicflow

Plasma-vacuuminterface

Key requirements to magnetic nozzle:

• Provide a smooth transition from sub- to super-Alfvénic plasma flow.

• Ensure efficient plasma detachment.

4APS - DPP 2007

The main part of the flow remains unperturbed after detachment!The main part of the flow remains unperturbed after detachment!

( )

0

,r z!

!

0z r

0r r

Plasma-vacuuminterface

Inward Alfvén-wave

Rarefaction wave

Unperturbedsuper-Alfvénic flow

CharacteristicNozzle end

Nozzle wall

PLASMA DENSITY PROFILE IN THE PLUME

A. Arefiev and B. Breizman, Phys. Plasmas 12, 043504 (2005)

5APS - DPP 2007

HIGHLY SUPER-ALFVÉNIC PLUME (V/VA>>1)

A. Arefiev and B. Breizman, Phys. Plasmas 12, 043504 (2005)

6

• Rarefaction wave stays at the edge of the plume when V /V

A>> 1 / !

0

B(r, z) =

B0

zmin

2

z2

0 < r < rC

B0

9

zmin

2

z2

V2

VA

2

r

z!"

0! 2

VA

V1!

z*

z

#

$%

&

'(

)

*++

,

-..

2

1!z

*

z

#

$%

&

'(

!2

rC< r < r

PV

0 r > rPV

/

0

111

2

111

rC= z!

01"

1

!0

V

VA

1"z

*

z

#

$%&

'()

*++

,

-..

rPV= z!

01+

2

!0

V

VA

1"z

*

z

#

$%&

'()

*++

,

-..

Rarefaction wave

Unperturbed main flow

θ0 zmin z*

Conical flow without vacuum gap between plasma and nozzle wall

rC! inner front of the rarefaction wave

rPV! outer front (plasma-vacuum interface)

APS - DPP 2007

GOVERNING EQUATIONS

• Collisionless MHD with cold electrons and anisotropic ion pressure (from ICRH)

•Steady-state axisymmetric flow flow along magnetic field lines

1

r

!

!r(r"V

r) +

!

!z("V

z) = 0

" Vr

!Vr

!r+V

z

!Vr

!z

#

$%&

'(= )

!

!rp*+

B2

2µ0

#

$%

&

'( + B

r

!

!r+ B

z

!

!z

#

$%&

'(B

r

µ0

+p*

B2

Br

#

$%

&

'(

" Vr

!Vz

!r+V

z

!Vz

!z

#

$%&

'(= )

!

!zp*+

B2

2µ0

#

$%

&

'( + B

r

!

!r+ B

z

!

!z

#

$%&

'(B

z

µ0

+p*

B2

Bz

#

$%

&

'(

!Bz

!z+

1

r

!

!r(rB

r) = 0

continuity equation

equationsof motion

magnetic flux conservation

7

vr

vz

||v

magnetic field linev!

miV!

2

2B= µ = const

miV

||

2

2+ µB = const

p!= µB

"

mi

; p||= 0

Vr

Vz

=B

r

Bz

APS - DPP 2007

PARAXIAL APPROXIMATION

!

!z+

Vr

Vz

!

!r

"

#$

%

&' (V

z( ) = )(V

z

1

r

!

!rr

Vr

Vz

"

#$

%

&'

!

!z+

Vr

Vz

!

!r

"

#$

%

&'V

r= )

1

(Vz

!

!r

Bz

2

2µ0

+ µBz

(

mi

"

#$

%

&'

!

!z+

Vr

Vz

!

!r

"

#$

%

&'V

z=µB

z

miV

z

1

r

!

!rr

Vr

Vz

"

#$

%

&'

!

!z+

Vr

Vz

!

!r

"

#$

%

&' B

z= )B

z

1

r

!

!rr

Vr

Vz

"

#$

%

&'

!

!z+

Vr

Vz

!

!r

"

#$

%

&' µ = 0

2 2

0 0

ˆ

2 2

z

z

i

B BBm

!µ

µ µ= +

Boundary condition at the plasma-vacuum interface (pressure balance)

B̂ ! external magnetic field at the plasma boundary

• Axial scale-length is much greater than radial scale-length

• Efficient detachment implies that radial velocity and radial magnetic field are small

Vr

Vz

=B

r

Bz

<< 1

8

We use Largangian radial coordinate toaccommodate this boundary condition.

APS - DPP 2007

LAGRANGIAN SOLVER

• Method of lines (ODE with adaptive step) (along flow lines)• Staggered grid second order FD scheme (radial direction)• Mapping to the fixed Eulerian grid

• Natural tracking of the moving plasma-vacuum interface

Zv

ZB

1/ 2jr+

3/ 2jr+

jr

1jr+

vR

rr

k!

1k!

+

9

• Non-uniform profile inputs• Treatment of vacuum gap between plasma and nozzle walls

Steps beyond analytic model

!

r(r0,! ) = r

0+

Vr

Vz

!0

!

" d! '

z(r0,! ) = !

#

#!=#

#z+

Vr

Vz

#

#r

Transformation to Lagrangian radial coordinate

• Written in MATLAB®

• Runs on single workstation

• Magnetic field at the plasma-vacuum boundary can be precalculated for slowly diverging flow

APS - DPP 2007

NOZZLE CODE VERIFICATION

10

Incoming flow parameters

Ion energy: !||= 250 eV

Plasma density: n = 5.0 "1014 cm-3

Vacuum field linein the presence of plasma

Analytic solution

Numerical solution

Nozzle end

B=const contours

Nozzle wall

The code accurately reproducesthe rarefaction wave at the plasma edge

Magnetic field at the plasma boundary

Rarefaction wave front

APS - DPP 2007

SUB- TO SUPER-ALFVÉNIC TRANSITION

Vacuum field lines(solenoid)

V V

A= 1

11

Alfvén Mach number

R (meter) R (meter)

Den

sity

(101

9 m-3

)




Thrust = 114 N

Power = 400 kW ( Ar)

APS - DPP 2007

2µB m

iV

�

2 = 1

Vacuum field lines

Thrust = 16 N

Power = 193 kW ( Ar)

!||=

miV

||

2

2

!"=

miV"

2

2= µB

PLASMA FLOW WITH ION GYROMOTION

12


Ion gyroenergy: !"= 100 eV

Axial energy: !||= 10 eV

Plasma density: n = 5.0 #1013 cm-3

Plasma radius: Rp= 10 cm

APS - DPP 2007

NOZZLE EFFICIENCY

• There are two factors that affect flow directivity:

• Definition of nozzle efficiency (momentum efficiency):

out

z

Momentum in

z

P

P! "

&

&

out

zP&

in

zP&

divergence of the nozzle itself;radial expansion of the plasma plume at its edge.

= axial momentum flux in the outgoingflow

= axial momentum flux in the incomingflow

13

!Momentum

="V

z

22#rdr$

"VVz2#rdr$

!Power

=

"Vz

2

2V

z2#rdr$

"V2

2V

z2#rdr$

Power efficiency of the nozzle: 1! "

Power( ) # 2 1! "

Momentum( )

APS - DPP 2007

NOZZLE EFFICIENCY CALCULATION

14









APS - DPP 2007

DETACHMENT DEMONSTRATION EXPERIMENT

D. Chavers et al., “Status of Magnetic Nozzle and Plasma DetachmentExperiment”, CP813, Space Technology and Applications InternationalForum, p. 465 – 473, AIP 2006

• Washer-stack plasma gun (300 kW) • Plasma density ~ 1019 m-3

• Magnetic field ~ 0.1 Tesla• 3 ms pulse at high power levels

15

Z

R

DDEX facility at MSFC

APS - DPP 2007

MAGNETIC FIELD IN DDEX FOR NORMAL SETUP

16

Horizontal slice Vertical slice

APS - DPP 2007

REVERSED CURRENT CONFIGURATION IN DDEX

17

Horizontal slice Vertical slice

APS - DPP 2007

PLASMA DENSITY MEASUREMENTS IN DDEX

18

Normal setup Reversed current

Upstream interferometer Upstream interferometer

Downstream interferometersDownstream interferometers

APS - DPP 2007

SIMULATION OF PLASMA FLOW IN DDEX

Vacuum field lines

V V

A= 1

19

nmax

= 0.53 !1011

cm-3

FWHM = 1.1 m

Alfvén Mach number



Maximum plasma density: n = 1.0 "1013 cm-3

Gaussian profile: FWHM = 9.3 cm @ 0.47 m

Downstream density profile

Experiment: Simulation:

nmax

= 1.0 !1011

cm-3

FWHM = 0.8 m

Magnetic field at the plasma boundary

APS - DPP 2007

REPRODUCTION OF PLASMA PROFILE IN DDEX

20

Den

sity

(101

3 cm

-3)

•15GHz Interferometer•Langmuir probe@ z=1.87 meters

Noz

zle

Effic

ienc

y

z (meter)



Maximum plasma density: n = 1.0 "1013 cm-3

Gaussian profile: FWHM = 9.3 cm @ 0.47 m

Expected downstream density for plasma flow along vacuum field lines

APS - DPP 2007

SUMMARY

• A properly shaped paraxial magnetic nozzle provides smooth transition from sub- to super-Alfvénic plasma flow and efficient detachment.

• Magnetic nozzle can simultaneously convert ion gyro-motion into axial flow to benefit from the ICRH power deposition.

• The MHD detachment concept is particularly relevant to high-power thrusters (VASIMR).

• The developed steady-state Lagrangian code enables broad parameter scan in detachment modeling with modest computational requirements (single workstation).

• Simulation results match the DDEX experimental data within the confidence range.

21APS - DPP 2007

magnetic nozzle and plasma detachment...

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