Lecture9: Plasma Physics 1 - Columbia...

Lecture9:

Plasma Physics 1APPH E6101x

Columbia University

Last Lecture• Force balance (equilibrium) in a magnetized plasma

• Z-pinch

• θ-pinch• screw-pinch (straight tokamak)

• Grad-Shafranov Equation

- Conservation principles in magnetized plasma (“frozen-in” and conservation of particles/flux tubes)

- Alfvén waves (without plasma pressure)

MHD

116 5 Fluid Models

nmi∂ui

∂t= ne(E + ui × B) − ∇ pi + nmig + nνeime(ue − ui)

nme∂ue

∂t= −ne(E + ue × B) − ∇ pe + nmeg + nνeime(ui − ue) . (5.37)

The momentum exchange between the electron and ion fluid is described by a colli-sion frequency νei and the mean exchanged momentum per volume, nme(ue − ui).

Instead of solving the pair of fluid equations, it is useful to transform these equa-tions into a set of new variables that describe the mean mass motion vm and therelative motion ∝ j of the two fluids. This approach is similar to splitting a two-particle problem into center-of-mass motion and relative motion. The mean massmotion is described by

ρm∂vm

∂t= j × B − ∇ p + ρmg (5.38)

with the mass density ρm = n(mi + me), total pressure p = pe + pi and the meanmass velocity

vm = (miui + meue)

me + mi. (5.39)

Note that now the Lorentz force j×B acts on the total current density. Moreover, themass motion is not affected by the friction between electron and ion fluid becauseit does not change the total momentum, but leads only to redistribution betweenelectron and ion fluid.

When ∂vm/∂t = 0, (5.38) defines the static equilibria of a magnetized plasma,which are defined by the force balance

0 = j × B − ∇ p + ρmg . (5.40)

This framework is called magnetohydrostatics. In the next two paragraphs we willdiscuss two simple applications of this concept.

5.2.1 Isobaric Surfaces

Let us shortly return to the problem of toroidal confinement. Neglecting gravita-tional forces as small compared to the magnetic forces, we define the magnetohy-drostatic equilibrium by

j × B = ∇ p . (5.41)

By taking the dot product with B on the both sides of the equation, the dot productvanishes yielding 0 = B · ∇ p, i.e., B and ∇ p are perpendicular to each other.

5.3 Magnetohydrodynamics 121

5.3.1 The Generalized Ohm’s Law

A dynamic equation for the spatio-temporal evolution of the current results from themomentum (5.37) after multiplying the ion equation by me and the electron equationby mi, and subtracting the equations:

nmime∂

∂t(ui − ue) = ne(me + mi)E + ne(meui + miue) × B

−me∇ pi + mi∇ pe + n(me + mi)νeime(ue − ui) . (5.53)

This equation can be simplified by neglecting me in the sum of the masses. Themixed term

meui + miue = miui + meue + mi(ue − ui) + me(ui − ue) (5.54)

= 1nρmvm − (mi − me)

1ne

j (5.55)

can be decomposed into contributions from mass motion and current density, whichresults in

mime

e∂j∂t

= eρm

!E + vm × B − νeime

ne2 j"

−mij × B − me∇ pi + mi∇ pe . (5.56)

As long as we are interested in slowly varying phenomena, we can set ∂j/∂t = 0 andneglect terms of the order of me/mi. In this way we obtain the generalized Ohm’slaw

E + vm × B = ηj + 1ne

(j × B − ∇ pe) . (5.57)

Here, η = νeime/ne2 is the plasma resistivity that arises from Coulomb collisionsbetween electrons and ions. The l.h.s. of (5.57) is the correct electric field in themoving reference frame. This electric field balances the voltage drop η j by theresistivity, the contribution from the Hall effect j×B/(ne), and the electron pressureterm −∇ pe/ne.

5.3.2 Diffusion of a Magnetic Field

As an application of the generalized Ohm’s law, we consider a plasma that is movingat a velocity vm, and at an arbitrary angle to the magnetic field direction. We startfrom

E + vm × B = ηj = η

µ0∇ × B , (5.58)

5.1 The Two-Fluid Model 111

In the last step, we have Taylor-expanded the particle flux and retained onlythe differential change of the flux. Dividing by !V = A!x and taking the limit!V → 0 gives

∂n∂t

+ ∂(n ux )

∂x= 0 . (5.7)

This result can easily be generalized to a three-dimensional flow pattern, whichresults in the continuity equation

∂n∂t

+ ∇ · (nu) = 0 . (5.8)

This balance equation describes the conservation of the number of particles in theflow. When particles are generated or annihilated inside the cell, say by ionizationor recombination, the zero on the right hand side is replaced by a net production rateS (see Sect. 4.2.3).

The continuity equation can be easily generalized to an equation for the conser-vation of charge by introducing the charge density ρ = !

α nαqα and the currentdensity j = !

α nαqαuα

∂ρ

∂t+ ∇ · j = 0 . (5.9)

5.1.4 Momentum Transport

The net force in the balance of the considered cell is a result of the sum of all forcesacting on the particles within the cell plus the export and import of momentumby particles that leave and enter the cell. The starting point of our calculation isNewton’s equation for the force acting on a single particle

mdvdt

= q(E + v × B) . (5.10)

Here, d/dt is the derivative calculated at the position of the point-like particle.The correct momentum balance for a many-particle system can be obtained by mul-tiplying (5.10) with the density n. However, in an inhomogeneous flow, the timederivative has to be calculated according to the rules of hydrodynamic flow

dudt

= ∂u∂t

+ ∂u∂x

dxdt

+ ∂u∂y

dydt

+ ∂u∂z

dzdt

. (5.11)

The vector (dx/dt, dy/dt, dz/dt) is just the velocity u of the cell. This leads tothe compact notation

dudt

= ∂u∂t

+ (u · ∇)u , (5.12)

plus magnetostatics

http://www.sandia.gov/z-machine/

Modelling of edge localised modes and edge localised mode control

G. T. A. Huijsmans,1 C. S. Chang,2 N. Ferraro,3 L. Sugiyama,4 F. Waelbroeck,5 X. Q. Xu,6

A. Loarte,1 and S. Futatani71ITER Organization, Route de Vinon sur Verdon, 13067 Saint Paul Lez Durance, France2Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543, USA3General Atomics, P.O. Box 85608, San Diego, California 92186-5608, USA4Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139-4307, USA5Institute for Fusion Studies, University of Texas at Austin, Austin, Texas 78712, USA6Lawrence Livermore National Laboratory, Livermore, California 94551, USA7Ecole Centrale de Lyon, 69130 !Ecully, Lyon, France

(Received 6 August 2014; accepted 3 December 2014; published online 7 January 2015)

Edge Localised Modes (ELMs) in ITER Q¼ 10 H-mode plasmas are likely to lead to large tran-sient heat loads to the divertor. To avoid an ELM induced reduction of the divertor lifetime, thelarge ELM energy losses need to be controlled. In ITER, ELM control is foreseen using magneticfield perturbations created by in-vessel coils and the injection of small D2 pellets. ITER plasmasare characterised by low collisionality at a high density (high fraction of the Greenwald densitylimit). These parameters cannot simultaneously be achieved in current experiments. Therefore,the extrapolation of the ELM properties and the requirements for ELM control in ITER relies onthe development of validated physics models and numerical simulations. In this paper, wedescribe the modelling of ELMs and ELM control methods in ITER. The aim of this paper is nota complete review on the subject of ELM and ELM control modelling but rather to describe thecurrent status and discuss open issues. [http://dx.doi.org/10.1063/1.4905231]

I. INTRODUCTION

The ITER scenario to obtain an energy amplificationQ¼ 10 is based on the well-established ELMy H-mode. Inthis scenario about 20–40% of the energy losses across theplasma boundary are due to Edge Localised Modes (ELMs).ELMs have been identified as MHD instabilities destabilisedby the large pressure gradients and the associated bootstrapcurrent in the H-mode pedestal. Each ELM can expel a sig-nificant fraction of the plasma energy and density on a typi-cal time scale of a few hundred microseconds. Extrapolatingthe ELM energy loss from current experiments to ITER,based on the observed scaling with collisionality,1 leads tothe prediction of very large losses per ELM of up to 10% ofthe plasma thermal energy. The time scale for the energyflux to the divertor has been found to scale well with the par-allel propagation time,2 s//"L/cs, where L is the connectionlength, cs the local sound speed. This time scale is expectedto be in the range of 250–500 ls, similar to the time scale incurrent machines. As a consequence of the large thermalenergy content of ITER plasmas, the large fraction of theenergy lost per ELM and the time scales, the expectedenergy flux to the divertor will be significantly larger inITER compared to present experiments. Natural ELMs inITER are likely to lead to enhanced erosion of the divertor.In order to avoid a reduction of the lifetime of the divertor,the amplitude of ELMs in ITER will need to be controlled tobe below 0.7 MJ per ELM.3

The control of ELMs can be achieved either by anincrease in the ELM frequency or by a complete stabilisationof ELMs. The total ELM energy losses appear to be largely

independent of the ELM frequency, an increase in the fre-quency leads to smaller losses per ELM and a reduction inthe peak heat loads. An increase in ELM frequency can beachieved by providing an external trigger for the ELM, like,for example, the injection of frozen hydrogenic pellets4 orthe application of fast variations (“kicks”) vertical positionof the plasma.5 A reduction of the ELM energy loss to0.7 MJ in ITER will require an increase of the ELM fre-quency by a factor of 30 over the natural ELM frequency.The estimate of the natural ELM frequency in ITER is basedon a power through the separatrix of "100 MW of which30% is assumed to be lost through ELMs. The largest naturalELM size in ITER3 is assumed to be "20 MJ. Complete sta-bilisation of ELMs has been obtained in DIII-D by applyingan external magnetic field perturbation.6 In other tokamaks,the application of external magnetic perturbations has beenseen to lead to an increase of the ELM frequency.7–11 InITER, there will be two systems for ELM control, a high fre-quency pellet injector system12 and a set of 27 magnetic fieldcoils inside the vacuum vessel.13 Vertical kicks may be anoption for ELM control in ITER at plasma currents up to10 MA, but this method will not be available at the largerplasma currents needed for Q¼ 10 operation.3

The expected energy loss per ELM in ITER is based onthe extrapolation from current experiments of the scaling ofthe ELM size with collisionality. There is, however, nounderlying physics model that would explain or justify thecollisionality as the relevant parameter. In fact, a reductionof the ELM size with increasing density (relative to theGreenwald density NGW ½1020m$3% ¼ I½MA%=pa½m%2) has alsobeen observed1 albeit with a larger scatter in the data. ITER

1070-664X/2015/22(2)/021805/19/$30.00 22, 021805-1

PHYSICS OF PLASMAS 22, 021805 (2015)

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occur around an X-point outside, but near the plasmaboundary.

A homoclinic tangle is limited to the separatrix surfaceand the interior region. Open field lines outside the separa-trix may shift slightly in response but do not split. Thenonlinear simulations show that the field lines can extendwell beyond the original plasma edge, especially very closeto the X-point legs, as expected for a tangle. Where theextended tangles intersect the divertor, secondary strike

points are formed leading to multiple stripes in the powerdeposition profile during the ELM. Figure 4 shows aPoincare plot of the magnetic field structure during an ELMsimulation in an ITER Q¼ 10 scenario.60

With the formation of filaments and magnetic tangles,one can identity different loss mechanisms for the thermalenergy during the ELM. The ejection of filaments, due to theconvective motion of a ballooning mode corresponds to con-vective energy losses of density and energy. The losses fromthe filaments, losses once outside plasma, are both in paralleldirection along the filament and due to the radial movementof the filament into the first wall. The magnetic tangles andthe stochasticity cause a direct connection of field lines frominside the plasma to the divertor (or the first wall). This leadsto large parallel conductive losses causing mostly a reductionin the pedestal temperature in the simulations. The relativeimportance of these two loss channels may account for theexperimental classification of ELMs into so-called convec-tive and conductive ELMs.62 Convective ELMs, character-ised by significant density losses and small temperaturechanges, are small amplitude ELMs at high density wherethe parallel conduction is low. Conductive ELMs, with largeenergy losses and density losses comparable to the convec-tive ELMs, occur at low density. The minimum ELM densityloss (for both convective and conductive ELMs) would thenbe determined by the ejected filaments. If the conductedlosses are indeed related to the magnetic tangles, one wouldexpect a correlation with the amplitude (and duration) of themagnetic perturbation of the ELM. This remains to be veri-fied experimentally.

C. MHD models

The BOUTþþ code has been applied to investigate theinfluence on the linear stability of the pedestal and the ELMcrash of different MHD models from 3-field model, whichevolves vorticity, pressure and vector potential31,38,46,61 to

FIG. 3. More and less localised filament formation during ELM simulationsof DIII-D (density perturbation, left), NSTX (density perturbation, middle)[Reprinted with permission from Sugiyama et al., Phys. Scr. 86, 058205(2012). Copyright 2014 IOP Publishing] and AUG (perturbed poloidal flux,right) [Hoelzl et al., Phys. Plasmas 19, 082505 (2012). Copyright 2014 AIP]plasmas.

FIG. 4. A Poincare plot of the magnetic field, showing the magnetic tanglesat the time of the maximum magnetic perturbation during a simulated ELMin ITER [Reprinted with permission from Huijsmans et al., Nucl. Fusion 53,123023 (2013). Copyright 2014 IAEA].

021805-6 Huijsmans et al. Phys. Plasmas 22, 021805 (2015)


localisation is also found in ELM simulations in AUG50,56

using the JOREK code (see Fig. 3(c)).

F. Discussion

One of the important questions for extrapolation toITER is what determines the amplitude of the ELM energylosses. Assuming that one cannot significantly cross an idealMHD stability boundary, the ELM losses are determined bythe state to which the pedestal relaxes during the ELM.Thus, the question is: what determines this state immediatelyafter an ELM. Despite the importance of this issue, there hasbeen little emphasis on this topic in recent studies bothexperimentally and from simulations. A return to a margin-ally stable state after the ELM crash would lead to too smallELMs. The ELM energy loss may be enhanced when theELM itself changes the marginal stability boundary as itoccurs. It has been proposed in Ref. 34 that the reduction inplasma rotation due to an ELM may lower the marginal sta-bility limit from the rotation stabilised higher limit to thezero rotation lower limit. This could be especially relevant inspherical tokamaks with a large rotation. The locking of therotation or the reduction in rotation shear may also decreasethe efficiency of the stabilisation of the turbulence in the H-mode pedestal, leading to an increased turbulent transport.

The formation of magnetic tangles due to the magnetic per-turbation of the ballooning mode leads to a direct connectionfrom inside the original separatrix to the divertor. The result-ing ELM energy losses depend on the parallel conductionand the lifetime of the tangles. The formation of anavalanche-like process may also contribute to increasedenergy losses. In this case, the flattening of the pressure inthe pedestal leads to large gradients further inwards, destabil-ising additional MHD instabilities increasing the flattenedregion by the ELMs and moving the place where large gra-dients occur further inwards. This behaviour is sometimesobserved in simulations, leading to very large ELM affectedareas. However, this is mostly observed in case of high resis-tivity simulations, raising the question whether this is rele-vant also for realistic values of the resistivity.

For more realistic simulation of ELMs, it is likely to benecessary to include more divertor physics in the non-linearMHD codes. This includes, for example, the divertor bound-ary conditions. In steady state, the standard Bohm boundaryconditions link the convective and conductive fluxes at thetarget through a constant sheath transmission factors, cSH.PIC simulations76 in 1D show that during a transient ELMphase, cSH can vary significantly in time. However, recentcomparisons of 1D fluid simulations with PIC and Vlasovsolutions indicate that the error in the parallel electron andion heat fluxes at the divertor target due to the assumption ofa constant cSH in the fluid approach is relatively small.77 Inaddition to the divertor boundary conditions, the divertor re-gime may also have a strong influence on the ELM drivenheat loads. In the long term, predictions for ELMs in ITERwill need to include a description of the high recycling, par-tially detached divertor that is expected between ELMs inITER.

As mentioned above, typical non-linear MHD ELM sim-ulations start from an MHD unstable initial state and onlyconsider a single ELM. This approach does not allow predic-tions for the amplitude of the ELM energy and particlelosses, as these tend to depend on how unstable the initialstate was chosen to be. Simulations of multiple full ELMcycles will be necessary such that the ELMs do not dependon the choice of the initial state. From simulations of multi-ple ELMs is in simplified geometry,40,43 it appears that thediamagnetic terms in the MHD model are essential obtain acycling regime with repeated ELM crashes. A similar con-clusion was reached from non-linear MHD simulations ofsawteeth where obtaining repeated fast crashes requires aminimum threshold in the diamagnetic stabilisation.78

An alternative concept to explain the crash during theELM cycle, based on the phase coherence time, was pro-posed in Ref. 79. The interaction of many unstable peeling-ballooning modes destroys the phase coherence of the lin-ear instability, not giving enough time for the modes togrow to large amplitude and leading to a background,peeling-ballooning, turbulence. The formation of discreteELM-like crashes requires a long enough “phase coherencetime,” i.e., coherence between the pressure and potential(ExB flow) so that a mode can grow from the turbulence tolarge event. This concept leads to a non-linear criterion forthe ELM onset.

FIG. 5. Comparison of visible camera image of an ELM in MAST (right)with the predicted image from non-linear MHD ELM simulation [Reprintedwith permission from Pamela et al., Plasma Phys. Controlled Fusion 55,095001(2013). Copyright 2014 IOP Publishing].

021805-8 Huijsmans et al. Phys. Plasmas 22, 021805 (2015)


This Lecture- Force balance (equilibrium) in a magnetized plasma

- Z-pinch

- θ-pinch- screw-pinch (straight tokamak)

- Grad-Shafranov Equation

• Conservation principles in magnetized plasma (“frozen-in” and conservation of particles/flux tubes)

• Alfvén waves (without plasma pressure)

MHD

116 5 Fluid Models

nmi∂ui

∂t= ne(E + ui × B) − ∇ pi + nmig + nνeime(ue − ui)

nme∂ue

∂t= −ne(E + ue × B) − ∇ pe + nmeg + nνeime(ui − ue) . (5.37)

The momentum exchange between the electron and ion fluid is described by a colli-sion frequency νei and the mean exchanged momentum per volume, nme(ue − ui).

Instead of solving the pair of fluid equations, it is useful to transform these equa-tions into a set of new variables that describe the mean mass motion vm and therelative motion ∝ j of the two fluids. This approach is similar to splitting a two-particle problem into center-of-mass motion and relative motion. The mean massmotion is described by

ρm∂vm

∂t= j × B − ∇ p + ρmg (5.38)

with the mass density ρm = n(mi + me), total pressure p = pe + pi and the meanmass velocity

vm = (miui + meue)

me + mi. (5.39)

Note that now the Lorentz force j×B acts on the total current density. Moreover, themass motion is not affected by the friction between electron and ion fluid becauseit does not change the total momentum, but leads only to redistribution betweenelectron and ion fluid.

When ∂vm/∂t = 0, (5.38) defines the static equilibria of a magnetized plasma,which are defined by the force balance

0 = j × B − ∇ p + ρmg . (5.40)

This framework is called magnetohydrostatics. In the next two paragraphs we willdiscuss two simple applications of this concept.

5.2.1 Isobaric Surfaces

Let us shortly return to the problem of toroidal confinement. Neglecting gravita-tional forces as small compared to the magnetic forces, we define the magnetohy-drostatic equilibrium by

j × B = ∇ p . (5.41)

By taking the dot product with B on the both sides of the equation, the dot productvanishes yielding 0 = B · ∇ p, i.e., B and ∇ p are perpendicular to each other.


5.3.1 The Generalized Ohm’s Law

A dynamic equation for the spatio-temporal evolution of the current results from themomentum (5.37) after multiplying the ion equation by me and the electron equationby mi, and subtracting the equations:

nmime∂

∂t(ui − ue) = ne(me + mi)E + ne(meui + miue) × B

−me∇ pi + mi∇ pe + n(me + mi)νeime(ue − ui) . (5.53)

This equation can be simplified by neglecting me in the sum of the masses. Themixed term

meui + miue = miui + meue + mi(ue − ui) + me(ui − ue) (5.54)

= 1nρmvm − (mi − me)

1ne

j (5.55)

can be decomposed into contributions from mass motion and current density, whichresults in

mime

e∂j∂t

= eρm

!E + vm × B − νeime

ne2 j"

−mij × B − me∇ pi + mi∇ pe . (5.56)

As long as we are interested in slowly varying phenomena, we can set ∂j/∂t = 0 andneglect terms of the order of me/mi. In this way we obtain the generalized Ohm’slaw

E + vm × B = ηj + 1ne

(j × B − ∇ pe) . (5.57)

Here, η = νeime/ne2 is the plasma resistivity that arises from Coulomb collisionsbetween electrons and ions. The l.h.s. of (5.57) is the correct electric field in themoving reference frame. This electric field balances the voltage drop η j by theresistivity, the contribution from the Hall effect j×B/(ne), and the electron pressureterm −∇ pe/ne.

5.3.2 Diffusion of a Magnetic Field

As an application of the generalized Ohm’s law, we consider a plasma that is movingat a velocity vm, and at an arbitrary angle to the magnetic field direction. We startfrom

E + vm × B = ηj = η

µ0∇ × B , (5.58)

5.1 The Two-Fluid Model 111

In the last step, we have Taylor-expanded the particle flux and retained onlythe differential change of the flux. Dividing by !V = A!x and taking the limit!V → 0 gives

∂n∂t

+ ∂(n ux )

∂x= 0 . (5.7)

This result can easily be generalized to a three-dimensional flow pattern, whichresults in the continuity equation

∂n∂t

+ ∇ · (nu) = 0 . (5.8)

This balance equation describes the conservation of the number of particles in theflow. When particles are generated or annihilated inside the cell, say by ionizationor recombination, the zero on the right hand side is replaced by a net production rateS (see Sect. 4.2.3).

The continuity equation can be easily generalized to an equation for the conser-vation of charge by introducing the charge density ρ = !

α nαqα and the currentdensity j = !

α nαqαuα

∂ρ

∂t+ ∇ · j = 0 . (5.9)

5.1.4 Momentum Transport

The net force in the balance of the considered cell is a result of the sum of all forcesacting on the particles within the cell plus the export and import of momentumby particles that leave and enter the cell. The starting point of our calculation isNewton’s equation for the force acting on a single particle

mdvdt

= q(E + v × B) . (5.10)

Here, d/dt is the derivative calculated at the position of the point-like particle.The correct momentum balance for a many-particle system can be obtained by mul-tiplying (5.10) with the density n. However, in an inhomogeneous flow, the timederivative has to be calculated according to the rules of hydrodynamic flow

dudt

= ∂u∂t

+ ∂u∂x

dxdt

+ ∂u∂y

dydt

+ ∂u∂z

dzdt

. (5.11)

The vector (dx/dt, dy/dt, dz/dt) is just the velocity u of the cell. This leads tothe compact notation

dudt

= ∂u∂t

+ (u · ∇)u , (5.12)

plus magnetostatics

“Frozen-in” FluxThe plasma moves along with the magnetic field

orThe plasma within flux tubes remains invariant


5.3.3 The Frozen-in Magnetic Flux

The conductivity of a hot plasma is many times larger than that of metals. Therefore,we can describe hot laboratory plasmas or astrophysical plasmas by the concept ofinfinitely large conductivity (zero resistivity). This theory is named ideal magneto-hydrodynamics and is reached for Rm → ∞. From (5.60), this limit gives us therelationship

∂B∂t

= ∇ × (vm × B) . (5.65)

Using the identity

∇ × (vm × B) = (B · ∇)vm − (vm · ∇)B + vm (∇ · B)! "# $=0

−B(∇ · vm) (5.66)

and the continuity (5.8) in the form

∇ · vm = − 1ρm

%∂ρm

∂t+ (vm · ∇)ρm

&= − 1

ρm

dρm

dt(5.67)

we obtain the relation

dBdt

= (B · ∇)vm + Bρm

dρm

dt. (5.68)

We can further use the identity

ddt

%Bρm

&= 1ρm

dBdt

− Bρ2

m

dρm

dt, (5.69)

which results in the theorem of Truesdell [80]

ddt

%Bρm

&=%

Bρm

· ∇&

vm . (5.70)

The quantity B/ρm can be considered as the number of field lines per unit massof the plasma. When the mass flow is strictly perpendicular to the magnetic field,the r.h.s. vanishes. Hence, B/ρm becomes a conserved quantity. This means that themass motion can only occur together with the magnetic field. In other words, themagnetic flux is frozen into the plasma. When the mass motion has a field-alignedcomponent, the r.h.s. describes the (B/ρm)-weighted rate of change of the mass flowvelocity along the field line. This slipping along the field line for inhomogeneousflows allows a change of B/ρm.




∂B∂t

= ∇ × (vm × B) . (5.65)

Using the identity

∇ × (vm × B) = (B · ∇)vm − (vm · ∇)B + vm (∇ · B)! "# $=0

−B(∇ · vm) (5.66)


∇ · vm = − 1ρm

%∂ρm

∂t+ (vm · ∇)ρm

&= − 1

ρm

dρm

dt(5.67)


dBdt

= (B · ∇)vm + Bρm

dρm

dt. (5.68)


ddt

%Bρm

&= 1ρm

dBdt

− Bρ2

m

dρm

dt, (5.69)


ddt

%Bρm

&=%

Bρm

· ∇&

vm . (5.70)





∂B∂t

= ∇ × (vm × B) . (5.65)

Using the identity

∇ × (vm × B) = (B · ∇)vm − (vm · ∇)B + vm (∇ · B)! "# $=0

−B(∇ · vm) (5.66)


∇ · vm = − 1ρm

%∂ρm

∂t+ (vm · ∇)ρm

&= − 1

ρm

dρm

dt(5.67)


dBdt

= (B · ∇)vm + Bρm

dρm

dt. (5.68)


ddt

%Bρm

&= 1ρm

dBdt

− Bρ2

m

dρm

dt, (5.69)


ddt

%Bρm

&=%

Bρm

· ∇&

vm . (5.70)





∂B∂t

= ∇ × (vm × B) . (5.65)

Using the identity

∇ × (vm × B) = (B · ∇)vm − (vm · ∇)B + vm (∇ · B)! "# $=0

−B(∇ · vm) (5.66)


∇ · vm = − 1ρm

%∂ρm

∂t+ (vm · ∇)ρm

&= − 1

ρm

dρm

dt(5.67)


dBdt

= (B · ∇)vm + Bρm

dρm

dt. (5.68)


ddt

%Bρm

&= 1ρm

dBdt

− Bρ2

m

dρm

dt, (5.69)


ddt

%Bρm

&=%

Bρm

· ∇&

vm . (5.70)





∂B∂t

= ∇ × (vm × B) . (5.65)

Using the identity

∇ × (vm × B) = (B · ∇)vm − (vm · ∇)B + vm (∇ · B)! "# $=0

−B(∇ · vm) (5.66)


∇ · vm = − 1ρm

%∂ρm

∂t+ (vm · ∇)ρm

&= − 1

ρm

dρm

dt(5.67)


dBdt

= (B · ∇)vm + Bρm

dρm

dt. (5.68)


ddt

%Bρm

&= 1ρm

dBdt

− Bρ2

m

dρm

dt, (5.69)


ddt

%Bρm

&=%

Bρm

· ∇&

vm . (5.70)


(Ohm’s Law & Faraday’s Law)

“Frozen-in” Flux




∂B∂t

= ∇ × (vm × B) . (5.65)

Using the identity

∇ × (vm × B) = (B · ∇)vm − (vm · ∇)B + vm (∇ · B)! "# $=0

−B(∇ · vm) (5.66)


∇ · vm = − 1ρm

%∂ρm

∂t+ (vm · ∇)ρm

&= − 1

ρm

dρm

dt(5.67)


dBdt

= (B · ∇)vm + Bρm

dρm

dt. (5.68)


ddt

%Bρm

&= 1ρm

dBdt

− Bρ2

m

dρm

dt, (5.69)


ddt

%Bρm

&=%

Bρm

· ∇&

vm . (5.70)


Alfvén Waves5.3 Magnetohydrodynamics 127

Fig. 5.11 The deformedfield-line pattern of atransverse Alfvén wavepropagating along themagnetic field B0

B0

B1B

z

vphase

Although the Alfvén speed contains the term B20/µ0, it is misleading to associate

this with the magnetic pressure. The transverse wave, like any shear wave, conservesthe volume between the field lines and there is no compression of the plasma or ofthe bundle of magnetic field lines. Hence, the wave must be driven by a differentmechanism.

The force, which a magnetic field exerts on a certain volume, is obtained byintegrating the Maxwell stress tensor [74] over the surface of that volume

Fα =∮

Sαβ dAβ (5.84)

and the magnetic part of the stress tensor is defined as

Sαβ = µ0

2

⎛

⎝−B2

0 0 00 −B2

0 00 0 +B2

0

⎞

⎠ = −pmag

⎛

⎝1 0 00 1 00 0 1

⎞

⎠+

⎛

⎝0 0 00 0 00 0 2pmag

⎞

⎠ . (5.85)

The Maxwell stress can be decomposed into an isotropic magnetic pressure pmagand an opposing tension of magnitude T = 2pmag along the field line. Then, thepropagation velocity can be rewritten as

vA =(

T

ρm

)1/2

, (5.86)

which shows the similarity of the Alfvén wave mechanism to a plucked string. Here,the tension T of the magnetic field line replaces the mechanical tension, whichrestores the string to its resting position. This was already discussed for the curva-ture force in Sect. 5.2.2. Moreover, the inertia of the string is replaced by the massdensity ρm of the plasma. This means that the mass remains attached to the field lineas predicted by the concept of frozen-in flux.

5.3.5.2 The Compressional Alfvén Wave

For completeness, it should be mentioned at the end that there is a different typeof Alfvén wave, which propagates across the magnetic field direction and involvescompression of the magnetic field (see Fig 5.12). When effects from gas pressure

128 5 Fluid Models

Fig. 5.12 CompressionalAlfvén wave propagatingacross the magnetic fieldlines. Note the bunching ofthe field lines that can beinterpreted as thesuperposition of a parallelpertubing field B1

x

z

vphase

B1

B0

can be neglected (pkin ≪ pmag) the phase velocity becomes again vφ = vA, but thistime the analogy with a sound wave, cs = (γ p/ρ)1/2, is justified. In this situation,the magnetic pressure takes the role of gas pressure and γ = 2 reflects that thereare two degrees of freedom corresponding to the two directions perpendicular tothe magnetic field. When the kinetic pressure cannot be neglected, the compres-sional Alfvén wave becomes a magnetosonic wave, whose propagation speed isdetermined by vϕ = (v2

A + c2s )

1/2.

5.3.6 Application: The Parker Spiral

The solar wind is a highly conducting medium. Therefore, the magnetic field isfrozen into the mass flow of the expanding plasma. In Sect. 1.2.3 we had seen thatthe rotation of the Sun shapes the mass flow into an Archimedian spiral, as shownin Fig. 1.5, which is named the Parker spiral in honor of Eugene Parker who firstdescribed this structure by MHD [20]. Here, we will now consider the consequencesfor the interplanetary magnetic field.

If the Sun did not rotate, the solar wind would simply expand in flux tubes formedby radial magnetic field lines, see Fig. 5.13. Because of flux conservation in thisspherical geometry, the mass density would decrease as (r/r⊙)−2. In the same way,

Fig. 5.13 Cartoon of ahypothetical purely radialmagnetic flux tube for anon-rotating Sun and the fluxtubes forming the Parkerspiral

Bhðr; z;xÞ ¼4pr0 j0

c

ð1

0

J1ðkr0Þk

J1ðkrÞeikjj ðkÞzdk; (8)

in which the confinement magnetic field is in the z direction.In this expression the relationship between k and k\ is medi-ated by the dispersion relation which in practice may requirethe inclusion of collisional effects.

For the case of the inertial wave19 the spatial structurepredicted by Eq. (8) is governed by the ratio p ¼ d

r0. For large

values of p (narrow current channels) the theory predicts thatthe parallel wave current lies within a cone, which emanatesfrom the edge of the exciter disk. Further away from thesource the cone spreads at an angle h given by

tan h ¼ r $ r0

z¼

ffiffiffiffiffiffime

M

rxxci

# $1$ x

xci

# $2" #1

2

: (9)

Outside of the cone the wave current is zero, and the wavemagnetic field decays as 1/r. The wave current is confinedwithin the cone, which expands radially with distancealong the magnetic field away from the current source. Farfrom the source the wave magnetic field in the center ofthe cone vanishes. The magnetic field is in some sensesimilar to that of a current carrying wire; however, herethe “plasma wire” expands along its axis. Unlike the fieldof a wire the pattern exhibits a radial diffraction pattern farfrom the source.

The behavior in the kinetic regime does not exhibit thecharacteristics of the cone given in Eq. (8). Instead the waveis confined to a maximum angle of propagation given by18

tan hmax ¼xxci

2

332

ffiffiffiffiffibe

p; (10)

where be is the electron beta. The magnetic field patternexhibits radial oscillations within hmax which evolves into a1/r behavior at large radial positions.

Figure 1 shows the spatial pattern in an x-z plane fromEq. (5), here the magnetic field is in the z direction and thecenter of the current filament is located at x¼ 0.

II. EARLY ALFVEN WAVE EXPERIMENTS

There were no laboratory plasma sources adequate toperform Alfven wave measurements at the time of their pre-diction. The first theory was for a conducting fluid in whichthe particle’s gyroradii are assumed to be zero, and the con-ductivity assumed to be infinite. The initial experimentswere done in liquid metals. Lundquist20 used mercury, whichis a liquid at room temperature. Waves on the surface ofstirred magnetized column were observed on the surfacewith a mirror. Lehnert21 used sodium that is 34 times moreconductive than mercury but has to be heated (Tmelt¼ 98 %C)to the liquid state. The experimental apparatus is shown inFig. 2. The sodium column was “twisted” in the azimuthaldirection by the stirrer. At higher conductivity the magneticfield is frozen into the fluid, and the shear (or torsional)

FIG. 1. (Color) Theoretical patterns of one component, By, of the Alfven wave in the kinetic and inertial regimes. The waves propagate from left to right.

055501-3 Many faces of shear Alfven waves Phys. Plasmas 18, 055501 (2011)

This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:128.59.145.17 On: Tue, 22 Oct 2013 16:32:09

Low-Frequency MHD Dynamics


from a cylindrical wire-cage that vaporizes into a plasma and implodes under themagnetic pressure from several ten MA current.1

5.3.5 Application: Alfvén Waves

The concept of frozen-in magnetic flux can be best demonstrated by studying low-frequency waves of a magnetized plasma. Such MHD waves were predicted byAlfvén [84, 85] and were first generated in liquid metals [86, 87] before they weredemonstrated in magnetized plasmas, e.g., [88–90].

5.3.5.1 The Shear Alfvén Wave

We know from introductory mechanics courses that simple harmonic waves of afluctuating quantity, e.g., the gas density n in a sound wave, are described by asecond-order differential equation of the type

!∂2n∂t2 − v2 ∂

2n∂z2

"= 0 . (5.74)

The quantity v is the phase velocity of the wave and wave solutions have the form

n(x, t) = n sin[k(x ± vt)] . (5.75)

Here, k = 2π/λ is the wavenumber. The ± sign indicates that waves can propagatein ±x direction because the wave equation only depends on v2.

The starting point of our calculation is the ideal MHD (η = 0), which ensuresthat an internal magnetic field cannot leave the plasma by magnetic diffusion. Themomentum equation is written in the simplified form

ρm∂vm

∂t= j × B (5.76)

and the evolution of the magnetic field is given by (5.65) and (5.66) as

∂B∂t

= (B · ∇)vm − (vm · ∇)B − B(∇ · vm) . (5.77)

Since we are not interested in sound waves of the plasma, the additional assumptionof an incompressible flow ∇ ·vm = 0 is made, which gives ρm = const. Sound waveswill be discussed separately in Sect. 6.5.3. The linear wave analysis assumes that the

1 For recent experiments, see Sandia National Lab’s website http://zpinch.sandia.gov/







!∂2n∂t2 − v2 ∂

2n∂z2

"= 0 . (5.74)


n(x, t) = n sin[k(x ± vt)] . (5.75)



ρm∂vm

∂t= j × B (5.76)


∂B∂t

= (B · ∇)vm − (vm · ∇)B − B(∇ · vm) . (5.77)



5.2 Magnetohydrostatics 117

Fig. 5.6 Nested magneticsurfaces in a tokamak. Eachsurface is spanned by a set ofmagnetic field lines andcurrent stream lines. Theforce j × B points inward,balancing the pressuregradient

j

B

The same is true for j and ∇ p. Therefore, the vectors B and j must lie in a planeof constant pressure. The magnetic field lines and the current streamlines span amagnetic surface, which is also an isobaric surface. Figure 5.6 shows that the forcej × B is directed inward and balances the pressure force.

5.2.2 Magnetic Pressure

The relationship between current density and magnetic induction follows fromAmpere’s law

∇ × B = µ0j , (5.42)

which yields

j × B = 1µ0

(∇ × B) × B = − 1µ0

B × (∇ × B) . (5.43)

This expression can be evaluated by using the vector identity for arbitrary vectors aand b,

a × (∇ × b) = (∇b) · ac − (a · ∇)b . (5.44)

Then, we obtain a tensor ∇B with components (∇B)i j = ∂B j/∂xi . The symbol acmeans that a is held constant in the differentiation by the ∇-operator on its left side.Finally, we can use (∇B) · B = (1/2)∇(B · B) and obtain

j × B = − 12µ0

∇(B2) + 1µ0

(B · ∇)B . (5.45)

In the term (B · ∇)B we recognize the analogy to the convective derivative in fluidmotion discussed in Sect. (5.1.4). Here, the derivative describes the change of B(regarding magnitude and orientation) along a field line. Combining (5.41) and(5.45), we obtain a pressure balance

∇(p + pmag) = (B · ∇)Bµ0

, (5.46)

Linearize

126 5 Fluid Models

magnetic field and the mass velocity can be decomposed into a homogeneous andstationary equilibrium (subscript 0) and a wavelike perturbation (subscript 1).

B = B0 + B1

vm = v0 + v1 . (5.78)

The magnetic field B0 = (0, 0, B0) defines the z-direction and the plasma isassumed to be at rest, v0 = 0. Then the perturbed quantities are described by

ρm∂v1

∂t= 1

µ0(∇ × B1) × B0 (5.79)

∂B1

∂t= (B0 · ∇)v1 . (5.80)

Here, we have dropped the second-order term (vm · ∇)B. We now seek for perpen-dicular pertubations of the magnetic field B1 = (B1x , 0, 0), which describe a localtransverse displacement of a field line in x-direction. For calculating the stream-ing velocity, we must decompose the double vector product (∇ × B1) × B0 =(B0 · ∇)B1 − (∇B1) · B0. Since B1 has only an x-component, and B0 is ori-ented in z-direction, the expression (∇B1) · B0 vanishes. Likewise, (B0 · ∇)B1 =B0(∂B1x/∂z)ex and the acceleration of the mass is also in x-direction. In this way,we obtain the coupled set of equations

ρm∂v1x

∂t= B0

µ0

∂B1x

∂z∂B1x

∂t= B0

∂v1x

∂z, (5.81)

which can be combined into a wave equation for either of the perturbed quantities

!∂2

∂t2 − v2A∂2

∂z2

"v1x = 0

!∂2

∂t2 − v2A∂2

∂z2

"B1x = 0 . (5.82)

This equation describes a transverse wave that propagates along the magnetic fieldline (see Fig. 5.11). This is the shear-Alfvén wave. A different shear-wave in a solidis described in Fig. 10.37b. The propagation velocity vA is the Alfvén speed

vA =#

B20

µ0ρm

$1/2

. (5.83)

126 5 Fluid Models

magnetic field and the mass velocity can be decomposed into a homogeneous andstationary equilibrium (subscript 0) and a wavelike perturbation (subscript 1).

B = B0 + B1

vm = v0 + v1 . (5.78)

The magnetic field B0 = (0, 0, B0) defines the z-direction and the plasma isassumed to be at rest, v0 = 0. Then the perturbed quantities are described by

ρm∂v1

∂t= 1

µ0(∇ × B1) × B0 (5.79)

∂B1

∂t= (B0 · ∇)v1 . (5.80)

Here, we have dropped the second-order term (vm · ∇)B. We now seek for perpen-dicular pertubations of the magnetic field B1 = (B1x , 0, 0), which describe a localtransverse displacement of a field line in x-direction. For calculating the stream-ing velocity, we must decompose the double vector product (∇ × B1) × B0 =(B0 · ∇)B1 − (∇B1) · B0. Since B1 has only an x-component, and B0 is ori-ented in z-direction, the expression (∇B1) · B0 vanishes. Likewise, (B0 · ∇)B1 =B0(∂B1x/∂z)ex and the acceleration of the mass is also in x-direction. In this way,we obtain the coupled set of equations

ρm∂v1x

∂t= B0

µ0

∂B1x

∂z∂B1x

∂t= B0

∂v1x

∂z, (5.81)

which can be combined into a wave equation for either of the perturbed quantities

!∂2

∂t2 − v2A∂2

∂z2

"v1x = 0

!∂2

∂t2 − v2A∂2

∂z2

"B1x = 0 . (5.82)

This equation describes a transverse wave that propagates along the magnetic fieldline (see Fig. 5.11). This is the shear-Alfvén wave. A different shear-wave in a solidis described in Fig. 10.37b. The propagation velocity vA is the Alfvén speed

vA =#

B20

µ0ρm

$1/2

. (5.83)



B0

B1B

z

vphase




Fα =∮

Sαβ dAβ (5.84)


Sαβ = µ0

2

⎛

⎝−B2

0 0 00 −B2

0 00 0 +B2

0

⎞

⎠ = −pmag

⎛

⎝1 0 00 1 00 0 1

⎞

⎠+

⎛

⎝0 0 00 0 00 0 2pmag

⎞

⎠ . (5.85)


vA =(

T

ρm

)1/2

, (5.86)




Linear Plane Waves

Shear Alfvén Waves5.3 Magnetohydrodynamics 127


B0

B1B

z

vphase




Fα =∮

Sαβ dAβ (5.84)


Sαβ = µ0

2

⎛

⎝−B2

0 0 00 −B2

0 00 0 +B2

0

⎞

⎠ = −pmag

⎛

⎝1 0 00 1 00 0 1

⎞

⎠+

⎛

⎝0 0 00 0 00 0 2pmag

⎞

⎠ . (5.85)


vA =(

T

ρm

)1/2

, (5.86)




Compressional Alfvén Waves







!∂2n∂t2 − v2 ∂

2n∂z2

"= 0 . (5.74)


n(x, t) = n sin[k(x ± vt)] . (5.75)



ρm∂vm

∂t= j × B (5.76)


∂B∂t

= (B · ∇)vm − (vm · ∇)B − B(∇ · vm) . (5.77)









!∂2n∂t2 − v2 ∂

2n∂z2

"= 0 . (5.74)


n(x, t) = n sin[k(x ± vt)] . (5.75)



ρm∂vm

∂t= j × B (5.76)


∂B∂t

= (B · ∇)vm − (vm · ∇)B − B(∇ · vm) . (5.77)



5.2 Magnetohydrostatics 117

Fig. 5.6 Nested magneticsurfaces in a tokamak. Eachsurface is spanned by a set ofmagnetic field lines andcurrent stream lines. Theforce j × B points inward,balancing the pressuregradient

j

B

The same is true for j and ∇ p. Therefore, the vectors B and j must lie in a planeof constant pressure. The magnetic field lines and the current streamlines span amagnetic surface, which is also an isobaric surface. Figure 5.6 shows that the forcej × B is directed inward and balances the pressure force.

5.2.2 Magnetic Pressure

The relationship between current density and magnetic induction follows fromAmpere’s law

∇ × B = µ0j , (5.42)

which yields

j × B = 1µ0

(∇ × B) × B = − 1µ0

B × (∇ × B) . (5.43)

This expression can be evaluated by using the vector identity for arbitrary vectors aand b,

a × (∇ × b) = (∇b) · ac − (a · ∇)b . (5.44)

Then, we obtain a tensor ∇B with components (∇B)i j = ∂B j/∂xi . The symbol acmeans that a is held constant in the differentiation by the ∇-operator on its left side.Finally, we can use (∇B) · B = (1/2)∇(B · B) and obtain

j × B = − 12µ0

∇(B2) + 1µ0

(B · ∇)B . (5.45)

In the term (B · ∇)B we recognize the analogy to the convective derivative in fluidmotion discussed in Sect. (5.1.4). Here, the derivative describes the change of B(regarding magnitude and orientation) along a field line. Combining (5.41) and(5.45), we obtain a pressure balance

∇(p + pmag) = (B · ∇)Bµ0

, (5.46)

128 5 Fluid Models

Fig. 5.12 CompressionalAlfvén wave propagatingacross the magnetic fieldlines. Note the bunching ofthe field lines that can beinterpreted as thesuperposition of a parallelpertubing field B1

x

z

vphase

B1

B0

can be neglected (pkin ≪ pmag) the phase velocity becomes again vφ = vA, but thistime the analogy with a sound wave, cs = (γ p/ρ)1/2, is justified. In this situation,the magnetic pressure takes the role of gas pressure and γ = 2 reflects that thereare two degrees of freedom corresponding to the two directions perpendicular tothe magnetic field. When the kinetic pressure cannot be neglected, the compres-sional Alfvén wave becomes a magnetosonic wave, whose propagation speed isdetermined by vϕ = (v2

A + c2s )

1/2.

5.3.6 Application: The Parker Spiral

The solar wind is a highly conducting medium. Therefore, the magnetic field isfrozen into the mass flow of the expanding plasma. In Sect. 1.2.3 we had seen thatthe rotation of the Sun shapes the mass flow into an Archimedian spiral, as shownin Fig. 1.5, which is named the Parker spiral in honor of Eugene Parker who firstdescribed this structure by MHD [20]. Here, we will now consider the consequencesfor the interplanetary magnetic field.

If the Sun did not rotate, the solar wind would simply expand in flux tubes formedby radial magnetic field lines, see Fig. 5.13. Because of flux conservation in thisspherical geometry, the mass density would decrease as (r/r⊙)−2. In the same way,

Fig. 5.13 Cartoon of ahypothetical purely radialmagnetic flux tube for anon-rotating Sun and the fluxtubes forming the Parkerspiral

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