Plasma 2 Lecture 15:

Introduction to Drift WavesAPPH E6102y

Columbia University

1

These physical mechanisms can be seen in gyrokinetic simulations and movies

Unstable bad-curvature side, eddies point out, direction of effective gravity

particles quickly move along field lines, so density perturbations are very extended along field lines, which twist to connect unstable to stable side

Stable side, smaller eddies

21

Outline• Goal

• Geometry

• Drifts

• (Fast) electron motion along B (“adiabatic electrons”)

• Drift motion across B

• Ion inertial currents (i.e. polarization drifts)

• What’s next: collisions, Landau damping, (low-frequency) EM: δB⊥

"A gyro-Landau-fluid transport model," R. E. Waltz, G. M. Staebler, W. Dorland, G. W. Hammett, M. Kotschenreuther, and J. A. Konings, Physics of Plasmas 4, 2482 (1997).

Jan Weiland, Collective Modes in Inhomogeneous Plasma: Kinetic and Advanced Fluid Theory, IOP Publishing, 2000.

2

Low Frequency Modes in a Magnetized PlasmaThese physical mechanisms can be seen in gyrokinetic simulations and movies

Unstable bad-curvature side, eddies point out, direction of effective gravity

particles quickly move along field lines, so density perturbations are very extended along field lines, which twist to connect unstable to stable side

Stable side, smaller eddies

21

3

Vorticity and MixingThese physical mechanisms can be seen in gyrokinetic simulations and movies

Unstable bad-curvature side, eddies point out, direction of effective gravity

particles quickly move along field lines, so density perturbations are very extended along field lines, which twist to connect unstable to stable side

Stable side, smaller eddies

21

4

Simple Drift Wave Description

In general the perturbed potential is determined byPoisson’s equation

π2f524pe~dni2dne!. (9)

In this self-consistent field problem dna

5dna„x,f(x,t)…. For structures that are large comparedto the electron Debye length lDe5(kBTe/4pNe

2)1/2;however, the fractional charge separation allowed by thePoisson equation is small. From Eq. (9) the fractionaldeviation of electron and ion density is

~dni2dne!

N5S

2kBTe

4pe2N

Dπ2Sef

kBTe

D

&SlDe

2

dx2D S

ef

kBTe

D!1. (10)

Thus the principle of quasineutrality applies. In thequasineutral regime the plasma potential adjusts itsvalue to make ni(f)5ne(f) to the first order inlDe

2 /dx2. The resulting small fractional charge separa-

tion is computed from Poisson’s equation using the po-tential rather than vice versa. All the drift-wave dynam-ics discussed in this review are in the regime ofquasineutrality where rQ5Saqana50 determines theevolution of the electrostatic potential and the plasmaeigenmodes. Here a is the species index for particles ofcharge qa and density na . In terms of the particle cur-rents ja the quasineutral dynamics is given by Saπ•ja

50. In plasma physics it is conventional to measure tem-perature in energy units kBTe!Te leaving free the k

symbol for a wave number k associated with the fluctua-tions. The discussion regarding Fig. 1 applies equallywell to each cell of a wave with dx.p/k . Figure 2 showsthe three-dimensional structure of the drift wave as ob-served, for example, in either a Q machine or the Co-lumbia Linear Machine (see Sec. II.B), with the m52mode dominant. We return to analyze the convection inFigs. 1 and 2 in Sec. C after discussing the general fea-tures of the fluctuations.

The condition for the electrons to establish theBoltzmann distribution in the drift-wave–ion acousticwave disturbances follows from the parallel electronforce balance equation. For fluctuations with the parallelvariation k i52p/l i sufficiently strong so that electronswith thermal velocity ve5(Te /me)1/2 move rapidly alongk i compared with time rate of change of the fields (v

,kive) the dominant terms in the electron fluid forcebalance equation

meneSdu

i

e

dtD 52eneE i2π ipe1

nemej i

e(11)

are the electric field E i52π if and the isothermal pres-sure gradient π ipe5Teπ ine . The temperature gradientis small compared with density gradient due to the fastelectron thermal flow associated with k ive.v . Thusmuch of the low-frequency drift-wave dynamics falls inthe regime where the Boltzmann description of the elec-tron response

dne5NFexpSef

Te

D21 Gapplies in the form

dne>

Nef

Te

. (12)

Here in the last step it is convenient to adopt an equalityin the sense of defining the linear, adiabatic electronmodel. The model requires that ef/Te!1 and k ive

@(ne

21v2)1/2. For example, both the Hasegawa-Mimaequation and the ideal ion temperature gradient (ITG)model rely on the adiabatic electron model defined byEq. (12). For the simple drift-wave instability, however,the adiabatic electron response defined by Eq. (12) isbroken by dissipation as we now discuss.

After dropping the electron inertia (me!0) the par-allel electron force balance becomes

E i1π ipe

ene

1b•~π•pe!

ene

5hji ,

where pe is the traceless momentum stress tensor thatincludes the electron viscosity and h is the electricalresistivity. Using collisional or neoclassicaltransport theory, the nonadiabatic contributions toEq. (12) can be computed. For the collisional regimene.k ive the electron temperature fluctuations dTe /Te

;(vne /ki

2ve

2)(dne /ne) and the resistivity contributionto dne are of the same order, both giving rise to a dne-f

FIG. 1. Drift-wave mechanism showing E3B convection in anonuniform, magnetized plasma.

FIG. 2. Three-dimensional configuration of the drift-wavefields in a cylinder.

738 W. Horton: Drift waves and transport

Rev. Mod. Phys., Vol. 71, No. 3, April 1999

long parallel

wavelength

5

How Fast is the Diamagnetic Drift?

In general the perturbed potential is determined byPoisson’s equation

π2f524pe~dni2dne!. (9)

In this self-consistent field problem dna

5dna„x,f(x,t)…. For structures that are large comparedto the electron Debye length lDe5(kBTe/4pNe

2)1/2;however, the fractional charge separation allowed by thePoisson equation is small. From Eq. (9) the fractionaldeviation of electron and ion density is

~dni2dne!

N5S

2kBTe

4pe2N

Dπ2Sef

kBTe

D

&SlDe

2

dx2D S

ef

kBTe

D!1. (10)

Thus the principle of quasineutrality applies. In thequasineutral regime the plasma potential adjusts itsvalue to make ni(f)5ne(f) to the first order inlDe

2 /dx2. The resulting small fractional charge separa-

tion is computed from Poisson’s equation using the po-tential rather than vice versa. All the drift-wave dynam-ics discussed in this review are in the regime ofquasineutrality where rQ5Saqana50 determines theevolution of the electrostatic potential and the plasmaeigenmodes. Here a is the species index for particles ofcharge qa and density na . In terms of the particle cur-rents ja the quasineutral dynamics is given by Saπ•ja

50. In plasma physics it is conventional to measure tem-perature in energy units kBTe!Te leaving free the k

symbol for a wave number k associated with the fluctua-tions. The discussion regarding Fig. 1 applies equallywell to each cell of a wave with dx.p/k . Figure 2 showsthe three-dimensional structure of the drift wave as ob-served, for example, in either a Q machine or the Co-lumbia Linear Machine (see Sec. II.B), with the m52mode dominant. We return to analyze the convection inFigs. 1 and 2 in Sec. C after discussing the general fea-tures of the fluctuations.

The condition for the electrons to establish theBoltzmann distribution in the drift-wave–ion acousticwave disturbances follows from the parallel electronforce balance equation. For fluctuations with the parallelvariation k i52p/l i sufficiently strong so that electronswith thermal velocity ve5(Te /me)1/2 move rapidly alongk i compared with time rate of change of the fields (v

,kive) the dominant terms in the electron fluid forcebalance equation

meneSdu

i

e

dtD 52eneE i2π ipe1

nemej i

e(11)

are the electric field E i52π if and the isothermal pres-sure gradient π ipe5Teπ ine . The temperature gradientis small compared with density gradient due to the fastelectron thermal flow associated with k ive.v . Thusmuch of the low-frequency drift-wave dynamics falls inthe regime where the Boltzmann description of the elec-tron response

dne5NFexpSef

Te

D21 Gapplies in the form

dne>

Nef

Te

. (12)

Here in the last step it is convenient to adopt an equalityin the sense of defining the linear, adiabatic electronmodel. The model requires that ef/Te!1 and k ive

@(ne

21v2)1/2. For example, both the Hasegawa-Mimaequation and the ideal ion temperature gradient (ITG)model rely on the adiabatic electron model defined byEq. (12). For the simple drift-wave instability, however,the adiabatic electron response defined by Eq. (12) isbroken by dissipation as we now discuss.

After dropping the electron inertia (me!0) the par-allel electron force balance becomes

E i1π ipe

ene

1b•~π•pe!

ene

5hji ,

where pe is the traceless momentum stress tensor thatincludes the electron viscosity and h is the electricalresistivity. Using collisional or neoclassicaltransport theory, the nonadiabatic contributions toEq. (12) can be computed. For the collisional regimene.k ive the electron temperature fluctuations dTe /Te

;(vne /ki

2ve

2)(dne /ne) and the resistivity contributionto dne are of the same order, both giving rise to a dne-f

FIG. 1. Drift-wave mechanism showing E3B convection in anonuniform, magnetized plasma.

FIG. 2. Three-dimensional configuration of the drift-wavefields in a cylinder.

738 W. Horton: Drift waves and transport

Rev. Mod. Phys., Vol. 71, No. 3, April 1999

long parallel

wavelength

6

Simple Drift Wave Description: Parallel Dynamics

In general the perturbed potential is determined byPoisson’s equation

π2f524pe~dni2dne!. (9)

In this self-consistent field problem dna

5dna„x,f(x,t)…. For structures that are large comparedto the electron Debye length lDe5(kBTe/4pNe

2)1/2;however, the fractional charge separation allowed by thePoisson equation is small. From Eq. (9) the fractionaldeviation of electron and ion density is

~dni2dne!

N5S

2kBTe

4pe2N

Dπ2Sef

kBTe

D

&SlDe

2

dx2D S

ef

kBTe

D!1. (10)

Thus the principle of quasineutrality applies. In thequasineutral regime the plasma potential adjusts itsvalue to make ni(f)5ne(f) to the first order inlDe

2 /dx2. The resulting small fractional charge separa-

tion is computed from Poisson’s equation using the po-tential rather than vice versa. All the drift-wave dynam-ics discussed in this review are in the regime ofquasineutrality where rQ5Saqana50 determines theevolution of the electrostatic potential and the plasmaeigenmodes. Here a is the species index for particles ofcharge qa and density na . In terms of the particle cur-rents ja the quasineutral dynamics is given by Saπ•ja

50. In plasma physics it is conventional to measure tem-perature in energy units kBTe!Te leaving free the k

symbol for a wave number k associated with the fluctua-tions. The discussion regarding Fig. 1 applies equallywell to each cell of a wave with dx.p/k . Figure 2 showsthe three-dimensional structure of the drift wave as ob-served, for example, in either a Q machine or the Co-lumbia Linear Machine (see Sec. II.B), with the m52mode dominant. We return to analyze the convection inFigs. 1 and 2 in Sec. C after discussing the general fea-tures of the fluctuations.

The condition for the electrons to establish theBoltzmann distribution in the drift-wave–ion acousticwave disturbances follows from the parallel electronforce balance equation. For fluctuations with the parallelvariation k i52p/l i sufficiently strong so that electronswith thermal velocity ve5(Te /me)1/2 move rapidly alongk i compared with time rate of change of the fields (v

,kive) the dominant terms in the electron fluid forcebalance equation

meneSdu

i

e

dtD 52eneE i2π ipe1

nemej i

e(11)

are the electric field E i52π if and the isothermal pres-sure gradient π ipe5Teπ ine . The temperature gradientis small compared with density gradient due to the fastelectron thermal flow associated with k ive.v . Thusmuch of the low-frequency drift-wave dynamics falls inthe regime where the Boltzmann description of the elec-tron response

dne5NFexpSef

Te

D21 Gapplies in the form

dne>

Nef

Te

. (12)

Here in the last step it is convenient to adopt an equalityin the sense of defining the linear, adiabatic electronmodel. The model requires that ef/Te!1 and k ive

@(ne

21v2)1/2. For example, both the Hasegawa-Mimaequation and the ideal ion temperature gradient (ITG)model rely on the adiabatic electron model defined byEq. (12). For the simple drift-wave instability, however,the adiabatic electron response defined by Eq. (12) isbroken by dissipation as we now discuss.

After dropping the electron inertia (me!0) the par-allel electron force balance becomes

E i1π ipe

ene

1b•~π•pe!

ene

5hji ,

where pe is the traceless momentum stress tensor thatincludes the electron viscosity and h is the electricalresistivity. Using collisional or neoclassicaltransport theory, the nonadiabatic contributions toEq. (12) can be computed. For the collisional regimene.k ive the electron temperature fluctuations dTe /Te

;(vne /ki

2ve

2)(dne /ne) and the resistivity contributionto dne are of the same order, both giving rise to a dne-f

FIG. 1. Drift-wave mechanism showing E3B convection in anonuniform, magnetized plasma.

FIG. 2. Three-dimensional configuration of the drift-wavefields in a cylinder.

738 W. Horton: Drift waves and transport

Rev. Mod. Phys., Vol. 71, No. 3, April 1999

long parallel

wavelength

7

Simple Drift Wave Description: Continuity

In general the perturbed potential is determined byPoisson’s equation

π2f524pe~dni2dne!. (9)

In this self-consistent field problem dna

5dna„x,f(x,t)…. For structures that are large comparedto the electron Debye length lDe5(kBTe/4pNe

2)1/2;however, the fractional charge separation allowed by thePoisson equation is small. From Eq. (9) the fractionaldeviation of electron and ion density is

~dni2dne!

N5S

2kBTe

4pe2N

Dπ2Sef

kBTe

D

&SlDe

2

dx2D S

ef

kBTe

D!1. (10)

Thus the principle of quasineutrality applies. In thequasineutral regime the plasma potential adjusts itsvalue to make ni(f)5ne(f) to the first order inlDe

2 /dx2. The resulting small fractional charge separa-

tion is computed from Poisson’s equation using the po-tential rather than vice versa. All the drift-wave dynam-ics discussed in this review are in the regime ofquasineutrality where rQ5Saqana50 determines theevolution of the electrostatic potential and the plasmaeigenmodes. Here a is the species index for particles ofcharge qa and density na . In terms of the particle cur-rents ja the quasineutral dynamics is given by Saπ•ja

50. In plasma physics it is conventional to measure tem-perature in energy units kBTe!Te leaving free the k

symbol for a wave number k associated with the fluctua-tions. The discussion regarding Fig. 1 applies equallywell to each cell of a wave with dx.p/k . Figure 2 showsthe three-dimensional structure of the drift wave as ob-served, for example, in either a Q machine or the Co-lumbia Linear Machine (see Sec. II.B), with the m52mode dominant. We return to analyze the convection inFigs. 1 and 2 in Sec. C after discussing the general fea-tures of the fluctuations.

The condition for the electrons to establish theBoltzmann distribution in the drift-wave–ion acousticwave disturbances follows from the parallel electronforce balance equation. For fluctuations with the parallelvariation k i52p/l i sufficiently strong so that electronswith thermal velocity ve5(Te /me)1/2 move rapidly alongk i compared with time rate of change of the fields (v

,kive) the dominant terms in the electron fluid forcebalance equation

meneSdu

i

e

dtD 52eneE i2π ipe1

nemej i

e(11)

are the electric field E i52π if and the isothermal pres-sure gradient π ipe5Teπ ine . The temperature gradientis small compared with density gradient due to the fastelectron thermal flow associated with k ive.v . Thusmuch of the low-frequency drift-wave dynamics falls inthe regime where the Boltzmann description of the elec-tron response

dne5NFexpSef

Te

D21 Gapplies in the form

dne>

Nef

Te

. (12)

Here in the last step it is convenient to adopt an equalityin the sense of defining the linear, adiabatic electronmodel. The model requires that ef/Te!1 and k ive

@(ne

21v2)1/2. For example, both the Hasegawa-Mimaequation and the ideal ion temperature gradient (ITG)model rely on the adiabatic electron model defined byEq. (12). For the simple drift-wave instability, however,the adiabatic electron response defined by Eq. (12) isbroken by dissipation as we now discuss.

After dropping the electron inertia (me!0) the par-allel electron force balance becomes

E i1π ipe

ene

1b•~π•pe!

ene

5hji ,

where pe is the traceless momentum stress tensor thatincludes the electron viscosity and h is the electricalresistivity. Using collisional or neoclassicaltransport theory, the nonadiabatic contributions toEq. (12) can be computed. For the collisional regimene.k ive the electron temperature fluctuations dTe /Te

;(vne /ki

2ve

2)(dne /ne) and the resistivity contributionto dne are of the same order, both giving rise to a dne-f

FIG. 1. Drift-wave mechanism showing E3B convection in anonuniform, magnetized plasma.

FIG. 2. Three-dimensional configuration of the drift-wavefields in a cylinder.

738 W. Horton: Drift waves and transport

Rev. Mod. Phys., Vol. 71, No. 3, April 1999

long parallel

wavelength

8

Basic “Drift Wave”

In general the perturbed potential is determined byPoisson’s equation

π2f524pe~dni2dne!. (9)

In this self-consistent field problem dna

5dna„x,f(x,t)…. For structures that are large comparedto the electron Debye length lDe5(kBTe/4pNe

2)1/2;however, the fractional charge separation allowed by thePoisson equation is small. From Eq. (9) the fractionaldeviation of electron and ion density is

~dni2dne!

N5S

2kBTe

4pe2N

Dπ2Sef

kBTe

D

&SlDe

2

dx2D S

ef

kBTe

D!1. (10)

Thus the principle of quasineutrality applies. In thequasineutral regime the plasma potential adjusts itsvalue to make ni(f)5ne(f) to the first order inlDe

2 /dx2. The resulting small fractional charge separa-

tion is computed from Poisson’s equation using the po-tential rather than vice versa. All the drift-wave dynam-ics discussed in this review are in the regime ofquasineutrality where rQ5Saqana50 determines theevolution of the electrostatic potential and the plasmaeigenmodes. Here a is the species index for particles ofcharge qa and density na . In terms of the particle cur-rents ja the quasineutral dynamics is given by Saπ•ja

50. In plasma physics it is conventional to measure tem-perature in energy units kBTe!Te leaving free the k

symbol for a wave number k associated with the fluctua-tions. The discussion regarding Fig. 1 applies equallywell to each cell of a wave with dx.p/k . Figure 2 showsthe three-dimensional structure of the drift wave as ob-served, for example, in either a Q machine or the Co-lumbia Linear Machine (see Sec. II.B), with the m52mode dominant. We return to analyze the convection inFigs. 1 and 2 in Sec. C after discussing the general fea-tures of the fluctuations.

The condition for the electrons to establish theBoltzmann distribution in the drift-wave–ion acousticwave disturbances follows from the parallel electronforce balance equation. For fluctuations with the parallelvariation k i52p/l i sufficiently strong so that electronswith thermal velocity ve5(Te /me)1/2 move rapidly alongk i compared with time rate of change of the fields (v

,kive) the dominant terms in the electron fluid forcebalance equation

meneSdu

i

e

dtD 52eneE i2π ipe1

nemej i

e(11)

are the electric field E i52π if and the isothermal pres-sure gradient π ipe5Teπ ine . The temperature gradientis small compared with density gradient due to the fastelectron thermal flow associated with k ive.v . Thusmuch of the low-frequency drift-wave dynamics falls inthe regime where the Boltzmann description of the elec-tron response

dne5NFexpSef

Te

D21 Gapplies in the form

dne>

Nef

Te

. (12)

Here in the last step it is convenient to adopt an equalityin the sense of defining the linear, adiabatic electronmodel. The model requires that ef/Te!1 and k ive

@(ne

21v2)1/2. For example, both the Hasegawa-Mimaequation and the ideal ion temperature gradient (ITG)model rely on the adiabatic electron model defined byEq. (12). For the simple drift-wave instability, however,the adiabatic electron response defined by Eq. (12) isbroken by dissipation as we now discuss.

After dropping the electron inertia (me!0) the par-allel electron force balance becomes

E i1π ipe

ene

1b•~π•pe!

ene

5hji ,

where pe is the traceless momentum stress tensor thatincludes the electron viscosity and h is the electricalresistivity. Using collisional or neoclassicaltransport theory, the nonadiabatic contributions toEq. (12) can be computed. For the collisional regimene.k ive the electron temperature fluctuations dTe /Te

;(vne /ki

2ve

2)(dne /ne) and the resistivity contributionto dne are of the same order, both giving rise to a dne-f

FIG. 1. Drift-wave mechanism showing E3B convection in anonuniform, magnetized plasma.

FIG. 2. Three-dimensional configuration of the drift-wavefields in a cylinder.

738 W. Horton: Drift waves and transport

Rev. Mod. Phys., Vol. 71, No. 3, April 1999

long parallel

wavelength

9

Ion Inertial Currents (Polarization Drift)

In general the perturbed potential is determined byPoisson’s equation

π2f524pe~dni2dne!. (9)

In this self-consistent field problem dna

5dna„x,f(x,t)…. For structures that are large comparedto the electron Debye length lDe5(kBTe/4pNe

2)1/2;however, the fractional charge separation allowed by thePoisson equation is small. From Eq. (9) the fractionaldeviation of electron and ion density is

~dni2dne!

N5S

2kBTe

4pe2N

Dπ2Sef

kBTe

D

&SlDe

2

dx2D S

ef

kBTe

D!1. (10)

Thus the principle of quasineutrality applies. In thequasineutral regime the plasma potential adjusts itsvalue to make ni(f)5ne(f) to the first order inlDe

2 /dx2. The resulting small fractional charge separa-

tion is computed from Poisson’s equation using the po-tential rather than vice versa. All the drift-wave dynam-ics discussed in this review are in the regime ofquasineutrality where rQ5Saqana50 determines theevolution of the electrostatic potential and the plasmaeigenmodes. Here a is the species index for particles ofcharge qa and density na . In terms of the particle cur-rents ja the quasineutral dynamics is given by Saπ•ja

50. In plasma physics it is conventional to measure tem-perature in energy units kBTe!Te leaving free the k

symbol for a wave number k associated with the fluctua-tions. The discussion regarding Fig. 1 applies equallywell to each cell of a wave with dx.p/k . Figure 2 showsthe three-dimensional structure of the drift wave as ob-served, for example, in either a Q machine or the Co-lumbia Linear Machine (see Sec. II.B), with the m52mode dominant. We return to analyze the convection inFigs. 1 and 2 in Sec. C after discussing the general fea-tures of the fluctuations.

The condition for the electrons to establish theBoltzmann distribution in the drift-wave–ion acousticwave disturbances follows from the parallel electronforce balance equation. For fluctuations with the parallelvariation k i52p/l i sufficiently strong so that electronswith thermal velocity ve5(Te /me)1/2 move rapidly alongk i compared with time rate of change of the fields (v

,kive) the dominant terms in the electron fluid forcebalance equation

meneSdu

i

e

dtD 52eneE i2π ipe1

nemej i

e(11)

are the electric field E i52π if and the isothermal pres-sure gradient π ipe5Teπ ine . The temperature gradientis small compared with density gradient due to the fastelectron thermal flow associated with k ive.v . Thusmuch of the low-frequency drift-wave dynamics falls inthe regime where the Boltzmann description of the elec-tron response

dne5NFexpSef

Te

D21 Gapplies in the form

dne>

Nef

Te

. (12)

Here in the last step it is convenient to adopt an equalityin the sense of defining the linear, adiabatic electronmodel. The model requires that ef/Te!1 and k ive

@(ne

21v2)1/2. For example, both the Hasegawa-Mimaequation and the ideal ion temperature gradient (ITG)model rely on the adiabatic electron model defined byEq. (12). For the simple drift-wave instability, however,the adiabatic electron response defined by Eq. (12) isbroken by dissipation as we now discuss.

After dropping the electron inertia (me!0) the par-allel electron force balance becomes

E i1π ipe

ene

1b•~π•pe!

ene

5hji ,

where pe is the traceless momentum stress tensor thatincludes the electron viscosity and h is the electricalresistivity. Using collisional or neoclassicaltransport theory, the nonadiabatic contributions toEq. (12) can be computed. For the collisional regimene.k ive the electron temperature fluctuations dTe /Te

;(vne /ki

2ve

2)(dne /ne) and the resistivity contributionto dne are of the same order, both giving rise to a dne-f

FIG. 1. Drift-wave mechanism showing E3B convection in anonuniform, magnetized plasma.

FIG. 2. Three-dimensional configuration of the drift-wavefields in a cylinder.

738 W. Horton: Drift waves and transport

Rev. Mod. Phys., Vol. 71, No. 3, April 1999

long parallel

wavelength

10

Ion Inertial Currents (Polarization Drift)

In general the perturbed potential is determined byPoisson’s equation

π2f524pe~dni2dne!. (9)

In this self-consistent field problem dna

5dna„x,f(x,t)…. For structures that are large comparedto the electron Debye length lDe5(kBTe/4pNe

2)1/2;however, the fractional charge separation allowed by thePoisson equation is small. From Eq. (9) the fractionaldeviation of electron and ion density is

~dni2dne!

N5S

2kBTe

4pe2N

Dπ2Sef

kBTe

D

&SlDe

2

dx2D S

ef

kBTe

D!1. (10)

Thus the principle of quasineutrality applies. In thequasineutral regime the plasma potential adjusts itsvalue to make ni(f)5ne(f) to the first order inlDe

2 /dx2. The resulting small fractional charge separa-

tion is computed from Poisson’s equation using the po-tential rather than vice versa. All the drift-wave dynam-ics discussed in this review are in the regime ofquasineutrality where rQ5Saqana50 determines theevolution of the electrostatic potential and the plasmaeigenmodes. Here a is the species index for particles ofcharge qa and density na . In terms of the particle cur-rents ja the quasineutral dynamics is given by Saπ•ja

50. In plasma physics it is conventional to measure tem-perature in energy units kBTe!Te leaving free the k

symbol for a wave number k associated with the fluctua-tions. The discussion regarding Fig. 1 applies equallywell to each cell of a wave with dx.p/k . Figure 2 showsthe three-dimensional structure of the drift wave as ob-served, for example, in either a Q machine or the Co-lumbia Linear Machine (see Sec. II.B), with the m52mode dominant. We return to analyze the convection inFigs. 1 and 2 in Sec. C after discussing the general fea-tures of the fluctuations.

The condition for the electrons to establish theBoltzmann distribution in the drift-wave–ion acousticwave disturbances follows from the parallel electronforce balance equation. For fluctuations with the parallelvariation k i52p/l i sufficiently strong so that electronswith thermal velocity ve5(Te /me)1/2 move rapidly alongk i compared with time rate of change of the fields (v

,kive) the dominant terms in the electron fluid forcebalance equation

meneSdu

i

e

dtD 52eneE i2π ipe1

nemej i

e(11)

are the electric field E i52π if and the isothermal pres-sure gradient π ipe5Teπ ine . The temperature gradientis small compared with density gradient due to the fastelectron thermal flow associated with k ive.v . Thusmuch of the low-frequency drift-wave dynamics falls inthe regime where the Boltzmann description of the elec-tron response

dne5NFexpSef

Te

D21 Gapplies in the form

dne>

Nef

Te

. (12)

Here in the last step it is convenient to adopt an equalityin the sense of defining the linear, adiabatic electronmodel. The model requires that ef/Te!1 and k ive

@(ne

21v2)1/2. For example, both the Hasegawa-Mimaequation and the ideal ion temperature gradient (ITG)model rely on the adiabatic electron model defined byEq. (12). For the simple drift-wave instability, however,the adiabatic electron response defined by Eq. (12) isbroken by dissipation as we now discuss.

After dropping the electron inertia (me!0) the par-allel electron force balance becomes

E i1π ipe

ene

1b•~π•pe!

ene

5hji ,

where pe is the traceless momentum stress tensor thatincludes the electron viscosity and h is the electricalresistivity. Using collisional or neoclassicaltransport theory, the nonadiabatic contributions toEq. (12) can be computed. For the collisional regimene.k ive the electron temperature fluctuations dTe /Te

;(vne /ki

2ve

2)(dne /ne) and the resistivity contributionto dne are of the same order, both giving rise to a dne-f

FIG. 1. Drift-wave mechanism showing E3B convection in anonuniform, magnetized plasma.

FIG. 2. Three-dimensional configuration of the drift-wavefields in a cylinder.

738 W. Horton: Drift waves and transport

Rev. Mod. Phys., Vol. 71, No. 3, April 1999

long parallel

wavelength

11

How Much Transport from Drift Waves?

In general the perturbed potential is determined byPoisson’s equation

π2f524pe~dni2dne!. (9)

In this self-consistent field problem dna

5dna„x,f(x,t)…. For structures that are large comparedto the electron Debye length lDe5(kBTe/4pNe

2)1/2;however, the fractional charge separation allowed by thePoisson equation is small. From Eq. (9) the fractionaldeviation of electron and ion density is

~dni2dne!

N5S

2kBTe

4pe2N

Dπ2Sef

kBTe

D

&SlDe

2

dx2D S

ef

kBTe

D!1. (10)

Thus the principle of quasineutrality applies. In thequasineutral regime the plasma potential adjusts itsvalue to make ni(f)5ne(f) to the first order inlDe

2 /dx2. The resulting small fractional charge separa-

tion is computed from Poisson’s equation using the po-tential rather than vice versa. All the drift-wave dynam-ics discussed in this review are in the regime ofquasineutrality where rQ5Saqana50 determines theevolution of the electrostatic potential and the plasmaeigenmodes. Here a is the species index for particles ofcharge qa and density na . In terms of the particle cur-rents ja the quasineutral dynamics is given by Saπ•ja

50. In plasma physics it is conventional to measure tem-perature in energy units kBTe!Te leaving free the k

symbol for a wave number k associated with the fluctua-tions. The discussion regarding Fig. 1 applies equallywell to each cell of a wave with dx.p/k . Figure 2 showsthe three-dimensional structure of the drift wave as ob-served, for example, in either a Q machine or the Co-lumbia Linear Machine (see Sec. II.B), with the m52mode dominant. We return to analyze the convection inFigs. 1 and 2 in Sec. C after discussing the general fea-tures of the fluctuations.

The condition for the electrons to establish theBoltzmann distribution in the drift-wave–ion acousticwave disturbances follows from the parallel electronforce balance equation. For fluctuations with the parallelvariation k i52p/l i sufficiently strong so that electronswith thermal velocity ve5(Te /me)1/2 move rapidly alongk i compared with time rate of change of the fields (v

,kive) the dominant terms in the electron fluid forcebalance equation

meneSdu

i

e

dtD 52eneE i2π ipe1

nemej i

e(11)

are the electric field E i52π if and the isothermal pres-sure gradient π ipe5Teπ ine . The temperature gradientis small compared with density gradient due to the fastelectron thermal flow associated with k ive.v . Thusmuch of the low-frequency drift-wave dynamics falls inthe regime where the Boltzmann description of the elec-tron response

dne5NFexpSef

Te

D21 Gapplies in the form

dne>

Nef

Te

. (12)

Here in the last step it is convenient to adopt an equalityin the sense of defining the linear, adiabatic electronmodel. The model requires that ef/Te!1 and k ive

@(ne

21v2)1/2. For example, both the Hasegawa-Mimaequation and the ideal ion temperature gradient (ITG)model rely on the adiabatic electron model defined byEq. (12). For the simple drift-wave instability, however,the adiabatic electron response defined by Eq. (12) isbroken by dissipation as we now discuss.

After dropping the electron inertia (me!0) the par-allel electron force balance becomes

E i1π ipe

ene

1b•~π•pe!

ene

5hji ,

where pe is the traceless momentum stress tensor thatincludes the electron viscosity and h is the electricalresistivity. Using collisional or neoclassicaltransport theory, the nonadiabatic contributions toEq. (12) can be computed. For the collisional regimene.k ive the electron temperature fluctuations dTe /Te

;(vne /ki

2ve

2)(dne /ne) and the resistivity contributionto dne are of the same order, both giving rise to a dne-f

FIG. 1. Drift-wave mechanism showing E3B convection in anonuniform, magnetized plasma.

FIG. 2. Three-dimensional configuration of the drift-wavefields in a cylinder.

738 W. Horton: Drift waves and transport

Rev. Mod. Phys., Vol. 71, No. 3, April 1999

long parallel

wavelength

12

Next: Drift Wave Instability and Transport

13

VOLUME 18,NUMBER 12 PHYSICAL REVIEW LETTERS 20 MARGH 1967

COLLISIONAL EFFECTS IN PLASMAS —DRIFT-WAVE EXPERIMENTS AND INTERPRETATION*

H. W. Hendel, g B. Coppi, f F. Perkins, and P. A. PolitzerPlasma Physics Laboratory, Princeton University, Princeton, New Jersey

(Received 25 January 1967)

We report the results of experiments on low-temperature alkali plasmas in strong magnet-ic fields and their interpretation in terms ofcollisional drift modes, in which diffusion overthe transverse wavelength, resulting from ion-ion collisions, ' plays an important role.Collisional drift modes' arise in the presence

of a density gradient perpendicular to the mag-netic fieM and result from the combined effectsof ion inertia, electron-ion collisions, andmean electron kinetic energy along the magnet-ic-field lines.Our experiments determine frequencies, am-

plitudes, and azimuthal mode numbers of steady-state drift waves as functions of magnetic fieldstrength and ion density. Their interpretationis based on a theory which includes the effectsof ion-ion collisions on the ion motion. Althoughthe observed relative wave amplitudes are notsmall, the linearized approximation is expect-ed and found to predict correctly frequenciesand the abrupt appearance of certain modesas functions of the various physical parameters(density, magnetic field, temperature, and ionmass).The principal experimental results are the

consistent observations of single-mode steady-state oscillations, which we have identified asdrift modes, and, at certain critical values ofthe magnetic-field strength, sudden changesof both the azimuthal mode number and frequen-cy' of the oscillations. We have explained theseresults by a theory, which predicts abrupt sta-bilization of a particular mode with decreasingmagnetic field as a result of increased diffusionover the transverse wavelength due to ion-ioncollisions. This type of diffusion, in fact, sup-presses the instability when the ion gyroradi-us reaches a critical size relative to the trans-verse wavelength of the mode.The experimental work has been performed

on the Princeton Q-1 device. 7 The plasma con-sists of ions produced by surface ionizationof cesium or potassium atomic beams incidenton hot tungsten plates located on both ends ofthe plasma column and of thermionic electronsemitted from the same plates. The alkali va-por pressure is cryogenically reduced to below snl/st+u1. Vn+u. Vn1= 0,1i& i (1c)

10 Torr and the residual pressure is main-tained at approximately 10 ' Torr, so that theplasma, is fully ionized. The plasma columnis 3 cm in diameter and 128 cm long. Plasma(center) densities ranged from 5x10" to 5x10"cm ', plasma (ionizer-plate) temperaturesfrom 2100 to 2900'K, and magnetic fields from2 to 6.5 kG. Electric fields are not applied tothe plasma, and the thermionic voltage betweenend plates is maintained below 5 mV. The wavesare detected as either ion-density or plasma-potential fluctuations with Langmuir probes.Special effort was directed towards conclusiveidentification of the drift wave, since it wasrecognized that a plasma rotation comparablewith the electron diamagnetic velocity is pres-ent in Q machines for certain ranges of plas-ma temperature and density. ' Our experimentsdiffer from most previous drift-wave work inthat the neutral beam was collimated to reducethe ion density at the edge of the ionizer platewhere the temperature gradient is large. Thisprocedure spatially separates the effects oftemperature and density gradients. ' The oscil-lations reported here are confined to the regionwhere only the density gradient is important.The experiments have shown that the relevant

modes are localized in the radial direction withthe amplitude maximum at approximately —,

' ofthe plasma radius. Then, for these localizedmodes we can carry out the theory simulatingcylindrical geometry by a one-dimensional "slab"model, with density gradient in the g directionand magnetic field in the z direction. We adoptthe electrostatic approximation E = -V'y validwhen p = 8mp/B «1, as is the case here (p & 10 8).The time-independent electric field (experimen-tally found to be constant in the region wheretemperature gradients are negligible) is ignoredbecause it only produces a Doppler shift in therelevant frequencies. We use the following lin-earized equations adopting a standard notation:

nm(su . )/st p, V 'u, = —v p—+(J xB/c), (la)lid x z lid i1—V (n KT —en' )—v .nm u =0,

II 1 e 1 ei e 1eIl

439

VOLUME 18, +UMBER 12 PHYSlCAL REVIEW LETTERS 20 MARCH 1967

sn /8t+u ~ Vn+u ~ Vn +nV g =0,lei es 1 )) le li(ld)

u = -Vy xBcB s+KTc(eB n) 'Vn x B, (le)le& 1 1

u. =ETc(+eBan) iVn &B.i, ePerturbed quantities are indicated by the sub-script 1. Ion motion and electron inertia alongthe lines of force have been ignored. 0 Theeffects of ion-ion diffusion across the field e n-ter through the coefficient p& which is givenby pi = —,'(nETvf/Qi k~ ) for our experimentalconditicns, "with Qi the ion gyrofrequency andhz a dimensionless coefficient tabulated in Ref.11. For simplicity we assume a WEB-typesolution y = (pi epx(ifk dxx+ikyy+ikiiz+i~t) andconsider modes such that kx»n ~(dn/dx). Thisimplies mode localization in. the x directionand will restrict us to modes with azimuthalmode number m ) 1, since those with m = 1,equivalent to an off-axis shift of the whole plas-ma column, are less localized and cannot besimulated in a plane geometry. Moreover, inthe experiments, k)i=m/L, where L is the lengthof the plasma column. By expressing J1& andJ1ti in terms of cp, one derives for Te = Tz thedispersion relation in its simplest form,

rk IKT v, .bmi

ydmv. 4e ei

0.5—o -- GROWTH

RATE0.4—

/ /I /

I

(0)2.0-

cAE

i.6—uCL

C3

—0.3Cl

UJ 0,2I—

UJ0.1

8- aLIJ

4—~UJ

0.0

(b)

3—

values of magnetic field only one single modeis dete cte d, but in the mode transition regionstwo separate modes are observed. The rapidrise of y =-Im(a) with increasing magneticfield, once the condition b&b~ is satisfied, isevident. The second point is that the maximumgrowth rate is found to correspond to frequen-cies Re(a) = —0.5k vd. In particular, we findthat for b(b, the growth rate y is maximizedfor the dimensionless parameter ZO =ALII KTx (mev&&kyvdb) slightly above unity, mebeing the electron mass. At this point, themagnitude of y is -0.2k&gd and is comparablewith the instability frequency. Figure 1(b) showsthat the observed frequencies are proportion-

iv bi..(u-k v — Ibu),

y d 2 /(2) UJ

CFLU

U

2 ~ m=20 Al=3

m=4—DRIFT FREQUENCY

where vd = -(1/n)(dn/dx)(cKT/eB) is the elec-tron diamagnetic velocity, 5 =-,'(kx +k ')aLis assumed to be smaller than unity, al = (2KT/M)imam/Qf is the ion Larmor radius, and ve&=2(M/ms)iI vlf, taking for v&f the definitionof Ref. 11. Here we have neglected terms oforder k)) ET(mevefkyv&) i in comparison with 1.This dispersion relation is a quadratic equa-tion whose roots are easily evaluated numer-ically, and reveals two important points. First,there exists a critical value of b given by bz=4k')'KT(meve;vlf) ' such that the linearizedgrowth rate is positive only for b &b~. In con-nection with this, we display in Fig. 1 someexperimental results for fixed neutral-beamflux and plasma temperature, i.e., for approx-imately constant ion density. Note that for most

00

I I I

2 3 4MAGNETIC FIELD, kG

FIG. 1. (a) Observed oscillation amplitudes are com-pared with theoretical growth rates as a function of mag-netic field strength for various azimuthal mode num-bers. The absolute value of the magnetic field strengthfor the theoretical (slab model) curves has been scaledby a factor of -1.5 to give a good fit to the data. Therelative amplitude is defined as the ratio of the maxi-mum density fluctuation to the central density. (b) Theoscillation frequency (after subtraction of the rotationalDoppler shift) is compared with the drift frequency vd=kyvd/2m as a function of the magnetic field strength.The drift frequency, which has an uncertainty of +0.5kc/sec, is computed from the experimental values ofk, T, snd n ~(dn/dx). The data, are for a potassiump asma, no ——3.5&&10 cm 3, T =2800'K.

440

VOLUME 18, +UMBER 12 PHYSlCAL REVIEW LETTERS 20 MARCH 1967

sn /8t+u ~ Vn+u ~ Vn +nV g =0,lei es 1 )) le li(ld)

u = -Vy xBcB s+KTc(eB n) 'Vn x B, (le)le& 1 1

u. =ETc(+eBan) iVn &B.i, ePerturbed quantities are indicated by the sub-script 1. Ion motion and electron inertia alongthe lines of force have been ignored. 0 Theeffects of ion-ion diffusion across the field e n-ter through the coefficient p& which is givenby pi = —,'(nETvf/Qi k~ ) for our experimentalconditicns, "with Qi the ion gyrofrequency andhz a dimensionless coefficient tabulated in Ref.11. For simplicity we assume a WEB-typesolution y = (pi epx(ifk dxx+ikyy+ikiiz+i~t) andconsider modes such that kx»n ~(dn/dx). Thisimplies mode localization in. the x directionand will restrict us to modes with azimuthalmode number m ) 1, since those with m = 1,equivalent to an off-axis shift of the whole plas-ma column, are less localized and cannot besimulated in a plane geometry. Moreover, inthe experiments, k)i=m/L, where L is the lengthof the plasma column. By expressing J1& andJ1ti in terms of cp, one derives for Te = Tz thedispersion relation in its simplest form,

rk IKT v, .bmi

ydmv. 4e ei

0.5—o -- GROWTH

RATE0.4—

/ /I /

I

(0)2.0-

cAE

i.6—uCL

C3

—0.3Cl

UJ 0,2I—

UJ0.1

8- aLIJ

4—~UJ

0.0

(b)

3—

values of magnetic field only one single modeis dete cte d, but in the mode transition regionstwo separate modes are observed. The rapidrise of y =-Im(a) with increasing magneticfield, once the condition b&b~ is satisfied, isevident. The second point is that the maximumgrowth rate is found to correspond to frequen-cies Re(a) = —0.5k vd. In particular, we findthat for b(b, the growth rate y is maximizedfor the dimensionless parameter ZO =ALII KTx (mev&&kyvdb) slightly above unity, mebeing the electron mass. At this point, themagnitude of y is -0.2k&gd and is comparablewith the instability frequency. Figure 1(b) showsthat the observed frequencies are proportion-

iv bi..(u-k v — Ibu),

y d 2 /(2) UJ

CFLU

U

2 ~ m=20 Al=3

m=4—DRIFT FREQUENCY

where vd = -(1/n)(dn/dx)(cKT/eB) is the elec-tron diamagnetic velocity, 5 =-,'(kx +k ')aLis assumed to be smaller than unity, al = (2KT/M)imam/Qf is the ion Larmor radius, and ve&=2(M/ms)iI vlf, taking for v&f the definitionof Ref. 11. Here we have neglected terms oforder k)) ET(mevefkyv&) i in comparison with 1.This dispersion relation is a quadratic equa-tion whose roots are easily evaluated numer-ically, and reveals two important points. First,there exists a critical value of b given by bz=4k')'KT(meve;vlf) ' such that the linearizedgrowth rate is positive only for b &b~. In con-nection with this, we display in Fig. 1 someexperimental results for fixed neutral-beamflux and plasma temperature, i.e., for approx-imately constant ion density. Note that for most

00

I I I

2 3 4MAGNETIC FIELD, kG

FIG. 1. (a) Observed oscillation amplitudes are com-pared with theoretical growth rates as a function of mag-netic field strength for various azimuthal mode num-bers. The absolute value of the magnetic field strengthfor the theoretical (slab model) curves has been scaledby a factor of -1.5 to give a good fit to the data. Therelative amplitude is defined as the ratio of the maxi-mum density fluctuation to the central density. (b) Theoscillation frequency (after subtraction of the rotationalDoppler shift) is compared with the drift frequency vd=kyvd/2m as a function of the magnetic field strength.The drift frequency, which has an uncertainty of +0.5kc/sec, is computed from the experimental values ofk, T, snd n ~(dn/dx). The data, are for a potassiump asma, no ——3.5&&10 cm 3, T =2800'K.

440

VOI.UMZ 18,NUMsZR 12 PHYSICAL RKVIKW LKTTKRS 20 MARCH 1967

al to, but less than, k vd. These considerationsindicate that the criterion" g =&~, which canbe written as

2 +k 2)l/2XB

k ( 4e4k ) (Z' " 1

B (M'c KTm v,v. .f kn M '8 8Z 2'E

1000—

500—

+ m =5 STABILIZATION

m=4X N=)

m=2

—=7xi8kg

( ~ '

describes the onset of the single modes. InFig. 2(a) we plot the experimental values atinstability onset of B/k& vs n'~' and find agree-ment between the experiments and this theoret-ical prediction, including the numerical coef-ficient. In addition, other measurements haveshown dependence on plasma temperature andion mass as predicted by the transition crite-rion, Eq. (3). ln Fig. 2(b), the radial extentof the oscillation is shown as a function of mag-netic field. The extent of the oscillation in theradial direction was found to increase for de-creasing values of ~, as expected from thefull analysis of the normal-mode equation andcorresponding to (a/Bg) -k&. The position ofthe amplitude maximum does not coincide withthe position of maximum density gradient andmay be determined by the radial dependenceof the growth rate, taking into account both thevariations of n and dn/dx.It is evident from our treatment that the lin-

earized approximation is inadequate to explainthe large experimental amplitudes (typicallyn, /no-ep/KT-20%) at which the growth of thewaves ceases. However, there are good argu-ments for predicting that the saturation stageis reached when the perturbed (ExB)c/B' driftvelocity becomes of the order of the diamagnet-ic velocity (in our case -2x10' cm/sec). Thispoint is confirmed by the experiment and, inaddition, it is shown [see Fig. 1(a)] that thepattern of the measured amplitudes followsclosely that of the calculated growth rates a,sa function of the magnetic field. If we consid-er this as an indication that at the saturationstage the amplitude is proportional to a pow-er of the growth rate, '4 we can associate theobserved frequency with that of the maximumgrowth rate. 'On the other hand, if we make use of a quasi-

linear approximation" in treating the problemand include higher order terms in the disper-sion relation (see Ref. 12), we observe thatat the saturation level the frequency of oscil-

00

I/2 I/2A, ', I05I I

2.0-X }r

0 =3

}(,t:m

I.O-

0.5—I on L o rior Rod ius

B, kQ

lation is l~i-k&vd. In this connection we noticethat more detailed measurements than thosegiven in Fig. 1(b) have shown &u/k&vd = —~.The observation of one single mode at a giv-

en magnetic field is explained, within the lim-its of the linearized theory, by the fact thatonly one mode has appreciable growth rate out-side the mode-transition regions.Several important considerations of general

nature arise from this work. First, the agree-ment of the theoretically predicted frequencieswith observations, coupled with the fact thatthe mode amplitudes maximize when 5, = 1,implies that the large growth rates given bythe linearized approximation y= 0.2k gd-Re(&u)

FIG. 2. (a) The ratio of magnetic field strength toperpendicular wave number is plotted versus the squareroot of the density for the stabilization points of sever-al modes. Theory I.Eq'. (3)] gives a proportionality fac-tor of 9.7 X 10 . (b) The measured radial (~z) and azi-muthal (~0} wavelengths of the perturbation are dis-played as a function of the magnetic field.14