Topics in Fusion and Plasma Studies

Topics in Fusion and Plasma Studies

459.666A 004

Part II. Plasma Turbulence and Turbulent Transport

T.S. Hahm

Department of Nuclear Engineering Seoul National University

Fall 2012

References:

• J. Wesson, Tokamaks

• R. J. Goldston and P. H. Rutherford, Introduction to Plasma Physics Topics:

• Microinstabilities in tokamaks

• Anomalous transport

Fluid Approach Kinetic approach

−Linear −Linear

−Nonlinear −Nonlinear

0. Magnetic confinement (Sept. 3)

⇒ T_i(r), T_e(r), n(r) are radially inhomogeneous.

• ∇T_i,∇T_e,∇n act as an expansion free energy to drive

⇒ any physical system tends to have homogeneous physical quantities

⇒various waves unstable

⇒instabilities, with various wavelengths and frequencies

• Each one has its own characteristics and conditions for excitation.

• It’s unlikely to suppress (make stable) all instabilities simultaneously.

This is the reason why tokamak transport rate is higher than prediction based on Coulomb collisions (in toroidal geometry, called “neoclassical theory”), even in the absence of large scale MHD instabilities.

1. Generic features of tokamak microturbulence

• “δn” density fluctuations are commonly observed from all tokamaks.

• There’s a general trend that:

– Whenδn%,χe, D% andτE &.

– Whenδn&,χe, D& andτE %.

(χ_e: electron thermal diffusivity, D: particle diffusivity, τ_E: energy confinement) δnis easier to measure than other parameters.

|e|δφ/T_e measured frequently at edge using Langmuir probe (cf. at core Heavy Ion Beam Probe, but expensive diagnostic).

δT_e/T_esometimes measured at core and edge (Electron Cyclotron Emission, spec- troscopy).

If one constructs a contour of n_e(r) (level curve), we can see the evolution of

“turbulent eddys”. This is a self-organized nonlinear structure which originates from specific instabilities with wave-like characters.

Wavelengths of instabilities (≈eddy size):

λr'2π/k_r λθ'2π/k_θ

If “∆t” large ⇒coherent structure (not called turbulent structure).

From measurements (microwave scattering, beam emission spectroscopy):

*Until ’90, using microwave scattering one could see only high-k part.

(Beam emission spectroscopy: R. Fonck ’93.)

Frequency spectrum of fluctuations: δnk,ω ∼e^{−iωt+ik·x} If a fluctuation satisfies a linear dispersion relationω=ω(k):

But experiments show that: S(ω) has a broad peak (frequency broadening).

“∆ω” significant ⇒ we should face nonlinearity.

Note Weak turbulent theory is nice since it assumes small higher order effects so that one just add higher terms after 0th and 1st order terms. But now we should consider strong turbulence theory.

2. Examples of Basic Microinstabilities (Sept. 5)

Consider a uniform magnetic field B = B₀z, nonuniform density profileˆ n₀ = n₀(x), periodic system in y. Let’s consider uniform temperatures for simplicity.

In this simple geometry, any perturbed quantities can be Fourier-decomposed in

y and zdirections. E.g.,

δφ(x, y, z) =X

k,ω

δφ_k,ω(x) expi(k_yy+k_zz−ωt) δn(x, y, z) =X

k,ω

δn_k,ω(x) expi(k_yy+k_zz−ωt)

We’ll pursuing a local theory at first (at one point in x).

2.1. Electron Drift Wave

Electron drift wave was discussed seriously in theoretical community, since it can be driven only by density gradient and can be easily unstable.

Let’s search for an “electrostatic” (i.e. ∇ ×E= 0⇒E=∇φ) wave with a phase velocity satisfying v_{T i} ω/k_z v_{T e}. Herev_{T e} =p

T_e/m_e, v_{T i} =p

T_i/M_i, k_z = k_k ≡B·k/|B|.

A.Electron response

Since electrons move fast (can cover the system size during one wave period!

kkvT e ω), we can consider them in a thermal equilibrium in the presence of electrostatic fluctuationδφ.

Maxwell-Boltzmann Statistics

⇒fe(E)∝exp (−E/T_e) = exp (−(m_ev²− |e|δφ)/Te)

⇒n_e =R

d³vf_e(E) =n_e0exp (|e|δφ/T_e) : Boltzmann relation

“δne=ne−ne0 =ne0

h

1 +^|e|δφ_T

e +O _|e|δφ

Te

2

−1 i

=ne0|e|δφ Te ” δn_e/n_e0 =|e|δφ/T_e : Electrons obey Boltzmann response.

This Boltzmann response is also called the adiabatic response. “Adiabatic” here refers to a slow time variation of a wave.

From a fluid description;

mene

d

dtue=−n_e|e|

E+1

cue×B

− ∇p_e Linearize (assumingue0= 0):

m_en_e ∂

∂tδu_e =−n_e|e|

δE+1

cδu_e×B

− ∇δp_e Takeb·and ignore electron inertia;m_e →0 (recall v_{T e}=p

T_e/m_e very fast)

⇒ |e|n₀∇_kδφ−T_e∇_kδn_e= 0

δne/ne0 =|e|δφ/Te (1)

(∇_k=b· ∇=∂/∂z, we assumed uniformT_e ⇒δp_e=T_eδn_e) B. Ion Response

Ions which satisfyω/k_k v_{T i}, we further assume “cold ions”, i.e. k⊥ρ_i1 (ignore FLR effect), but T_i T_e ⇒ k⊥ρ_s∼1 . Hereρ_i =v_{T i}/Ω_ci, Ω_ci≡ |e|B₀/M_ic,ρ_s =c_s/Ω_ci=p

T_e/T_iρ_i,c_s=p T_e/M_i. k⊥ = ky in this simple system. In tokamak, Ohmically heated, or ECR heated plasma satisfyTi Te, but here just for simplicity.

In this situation, most of ions move slowly enough. From the wave’s point of view,

they more or less move together like a fluid.

m_en_ed

dtu_i=−n_i|e|

E+1

cu_i×B

− ∇p_i Linearize and drop∇p_i (cold ions!) (withu_i0 = 0)

⇒M_in_i ∂

∂tδu_i =n_i|e|

δE+1

cδu_i×B

(2) alongB⇒

M_in₀ ∂

∂tδu_ik=n₀|e|δE_k=−n₀|e| ∇_kδφ (3) acrossB⇒

we can solve Equation (2) via iteration knowing (or assuming) ω/Ω_ci1.

(We are dealing with “low frequency” microinstabilities.) (ω/Ω_ci∼ω/^|e|B_M

ic ∝ |M_i/e| 1) 1st order : RHS=0.

δE+1

cδu⁽¹⁾_i ×B= 0

⇒δu⁽¹⁾_⊥ =δu_E = cb× ∇δφ

B (x-direction here) 2nd order :

M_in₀ ∂

∂tδu_E=n₀|e|1

cδu⁽²⁾_⊥ ×B (4)

δu⁽²⁾_⊥ =δupolarization drift= Mic²

|e|B²

∂

∂tE⊥=−Mic²

|e|B²

∂

∂t∇_⊥δφ (5) (1)⇒

δne

n0

= |e|δφ Te

(3), (4), (5) with _∂t^∂n_i+∇ · n_iu_i

= 0 ⇒ linearize to get

∂

∂tδni+δu_E· ∇n₀+n0∇ ·δu_pol+n0∇_kδu_ik = 0

Here you can check ∇ ·δu_E and other contributions are even smaller or vanish.

Fourier decompose, i.e. ∼exp (kyy+kzz−ωt) ⇒

∂

∂t → −iω, ∇_k →ikk, ∇_⊥→ik⊥,but∇n_=−ˆxn

Ln

HereL_nk⊥1, i.e. (system size)(⊥wave size), and thereforeδu_pol· ∇n₀ term is dropped in the linearized continuity equation.

⇒ δn_i n0

=ω∗e

ω +k_k²c²_s

ω² −k_⊥²ρ²_s|e|δφ Te

Here the 1st term on RHS is fromδu_E· ∇n₀, and the 2nd term and the 3rd term are from∇_kδu_ik and∇ ·δu_pol.

For long enough wavelengthλλ_De, the Poisson equation can be approximated by the quasi-neutrality equation : δn_e=δn_i

⇒ we obtain the linear dispersion relation for the electron drift wave.

1 +k²_⊥ρ²_s−ω∗e

ω −k_k²c²_s ω² = 0 Hereω∗e=k_yL^ρs

nc_s=k_yv∗eis electron diamagnetic frequency, wherev∗eis electron diamagnetic drift velocity.

(cf. the notationvde could be confused with∇B or curvature drift.)

2.2. Electron Drift Wave in Uniform Magnetic Field (Sept. 10) Linear dispersion relation:

1 +k²_⊥ρ²_s−ω∗e

ω −k_k²c²_s ω² = 0

Here, k_⊥² = k_x² +k²_y, B = B₀zˆ and the diamagnetic drift frequency is ω∗e ≡ (kyρs/Ln)cs=kyv∗e. We assumed

δφ(x, y, z) =X

k,ω

δφ_k,ω(x) expi(k_yy+k_zz−ωt)

and used the WKB approximation,

δφ_k,ω(x) =δφˆ(x) expi Z x

k_x(x)dx,

which is valid for k_xL_n1. Here, the eikonal actor e^iRxkx(x)dxcaptures the fast variation inx and δφˆ(x) andk_x(x) are slowly varying inx. To the lowest order in the 1/kxLn expansion, there is only a local value in kx in the linear dispersion relation.

Let’s calculate the particle flux in thex direction carried by a drift wave.

Γptl=hδn_eδvxi

Here,h. . .iis an ensemble average, or a long time average. Practically, it’s replaced by an average over ignorable coordinate(s) (i.e., direction of symmetry).

In this simple slab geometry, bothyandzare ignorable coordinates. (In tokamak geometry, only the toroidal angle “φ” is an ignorable coordinate.) TheE×Bdrift is

δv_x= cδEy

B² B_z =−c B

∂

∂yδφ and the density fluctuation is

δn= |e|δφ T_e n0

Therefore,

Γ_ptl=hδnδv_xi=−c|e|

BT_e

δφ ∂

∂yδφ

=− c 2B

|e|

T_en0

∂

∂y(δφ)²

= 0 with

h. . .i= 1 L_y

I _Ly

0

dy(. . .)⇐ 1 2π

I

dθ(. . .) rdθ=dy.

Therefore, electron drift waves (not instabilities) cannot carry any particle flux.

The underlying reason is thatδvxandδneareπ/2 out of phase (becauseδne∝δφ).

Also note that there’s no instability in this simple limit. Electron drift wave just wobbles without growing in almplitude or driving a net particle flux. For an instability and net particle flux, there should be a phase-shift betweenδne and δφ.

Note We can controll only the initial condition of macroscopic quantities (likeI_p, B, etc). Thus we hope an ensemble average can describe plasma phenomena. But in nonlinear phenomena, like in chaotic system, slightly different initial condition causes very different result. ⇒ ansemble average method cannot be used.

2.3. Electron Drift Instability in Uniform Magnetic Field

Linear dispersion relation:

1−iδ_k,ω+k²_⊥ρ²_s−ω∗e

ω − k_k²c²_s ω² = 0 δn_e

n₀ = (1−iδ_k,ω)|e|δφ T_e δ_k,ω =δ_k,ω(k, ω), |δ_k,ω| 1

where the minus sign comes from the inverse-dissipation of the electron drift wave (i.e., wave gain something) andδk,ω is the corresponding phase-shift.

⇒ =(ω)∝δ_k,ωω∗e for (k⊥ρ_s)²1 and k_k²c²_s/ω² 1.

Then what mechanism then will lead to the inverse dissipation?

Electron drift instabilities are classified according to the specific inverse dissipation mechanism which destabilizes the electron drift wave.

• Collisionless (electron) drift instability (“universal instability” because only requiresn₀(x),B₀zband electron Landau damping) γ >0

Due to Inverse Landau damping of (passing) electronsω/k_k∼v_k: these can be discussed in the context of an uniform magnetic field.

• Collisional (resistive) drift instability γ <0 (’78∼’79)

Due to Magnetic shear induced damping of drift waves (via ion Landau damping): this requires introducing of sheared magnetic field (for simple model ofBp,B=B0[ˆz+ (x/Ls)ˆy]).

• Trapped Electron(-driven electron instability) Mode (TEM) γ >0

This requires treating magnetically trapped electrons (banana orbits) in toroidal geometry (resonance with precession of banana orbitsω∼k_φv_prec).

Furthermore,∇B and curvature drift of ions will couple neighboring poloidal har- monics of drift waves and render magnetic shear induced damping of drift waves ineffective.

2.4. Inverse Landau Damping in 1-D Plasma (uniform n0,B, near-Maxwellian electron distributionfe)

The net result is that the wave will lose energy (damping) by accelerating more particles than deccelerating. ⇒ “surfing” the wave.

When can we get inverse Landau damping? With a Bump On (the distribution) Tail (∂f_e/∂v >0). ⇒ B.O.T. instability.

More particles losing energy than gaining energy from the wave. This is inverse Landau damping.

For collisionless (universal) electron drift instability, fe(x, v)∝n0(x)e^−v2/2vTe2

⇒ how on Earth can we get the inverse Landau damping?

It comes from the nonuniform densityn0(x).

2.5. Drift-kinetic Equation

The drift kinetic equation is a simplification of the Vlasov equation for ω

Ωce

1, k⊥ρ_e1 ρ_e

λ⊥

1

, 1 k⊥Ln

1 λ⊥

Ln

1

, ρ_e Ln

1 in a strong magnetic field. (λ⊥ ∼1cm,Ln∼10²cm∼a,ρe≪1mm)

• Electrons gyrate very very fast: Ω_ce ∼10GHz

• while drift waves oscillate slowly: ω∼∆ω ∼100kHz fe(x, y, z, vx, vy, vz, t)→fe,gc xgc, ygc, zgc,

ξ, µ, v_k, t Gyrophase (ξ) angle dependance can be eliminated (not simply ignored).

For ions,k_rρ_i ∼1 will modify the situation⇒ gyrokinetic equation.

2.5.1. Electron Drift-kinetic Equation (Sept. 12) fe(x, y, z, vx, vy, vz, t)→fe,gc xgc, v⊥, vk, t

for uniformB. (xgc: position of guiding center, no gyro-angle dependance) Let’s consider a Maxwellianfgc,0

fgc,0=n0(x) m

2πT_e 3/2

exp

−mv² 2T_e

For an application for drift-waves, the existence of diamagnetic drift flow was es- sential : v∗eyˆ= (ρ_s/L_n)c_sy.ˆ

Homework 1 Discuss how we can treat drift wave with a Maxwellian equilibrium distributionf_gc,0which is symmetric inv_y(Hint: Maxwellian for “guiding center”).

Consider both electron and ion contributions.

2.5.2. Derivation of the Drift-kinetic Equation

The total number of electron guiding centers in a 6-D phase-space volumeV is N_e=

Z

f_ed³xd³v= Z

f_edV Here,f_e is the guiding-center distribution function.

In effect, for guiding centers in an uniform magnetic field, dV =d³x_gc2πv⊥dv⊥dv_k

The 2π factor is there because the gyro-dependance has been eliminated, and the system has a cylindrical symmetry inv-space, so (vx, vy, vz)⇒ vk, v⊥

. Conservation of the number of guiding centers :

0 = d dtNe=

Z

∂fe

∂t

dV + Z

S

feU·dS

= Z

∂f_e

∂t +∇ ·(f_eU)

dV

where the 2nd term is the flux out of the volumeV through the phase-space surface S which bounds volumeV. Since V is arbitrary, the integrand is zero:

∂f_e

∂t +∇ ·(f_eU) = ∂f_e

∂t +∇_x·( ˙xf_e) +∇_v·( ˙vf_e) = 0

with

∇_v·( ˙vf_e) = 1 v⊥

∂

∂v⊥

(v⊥v˙⊥f_e) + ∂

∂v_k

v_k˙f_e

˙ x_gc = d

dtx_gc=v_kzˆ+vE×B

Here,

vE×B = cδE×B

B² = cb× ∇δφ B (Note that∇ ·vE×B = 0 for uniformB: incompressible)

There’s no∇B and curvature drift, since we are assuming uniform B.

Then, how aboutv_pol?

⇒ actually, in the modern derivation of drift-kinetic equation, vpol can be elimi- nated when eliminating gyrophase angle dependency. (In ancient years, pioneers of drift-kinetic equation prefered to putv_pol term in their derivation.) Physically, we are considering just electron andvpol∝m, sovpol is neglected.

˙

v⊥= 0 (sinceB is uniform, the magnetic moment ˙µ=_v˙2

⊥

2B

= ^v_B^⊥v˙⊥= 0).

This is the first adiabatic invariant which is a constant for ωΩ_ce,ρ_e L_B≡

∂|B|

∂x

−1

=∞ for uniform magnetic field.

For ˙v, we only need to consider

˙

v_k = dv_k dt = |e|

m_eδE_k= |e|

m_e

∂

∂zδφ

Therefore, the electron drift-kinetic equation with electrostatic fluctuation in an uniform magnetic field is given by

∂fe

∂t +

v_kzˆ+cb× ∇δφ B

· ∇f_e+ |e|

m_e

∂

∂zδφ∂fe

∂v_k = 0 Let’s linearize it around a maxellianf0: f =f0+δfe

• 0th order,

∂

∂t+v_k ∂

∂z

f₀= 0 satisfied.

• 1st order,

∂

∂tδf_e+v_k ∂

∂zδf_e+cb× ∇δφ

B · ∇ f₀+ δf_e

+ |e|

me

∂

∂zδφ ∂

∂vk

f₀+ δf_e

= 0 TheE×Bnonlinearityand thevelocity-space nonlinearity are neglected.

Therefore, the linear kinetic electron response is given by “δf_e”.

Sinceδφ, δne, δfe· · · ∝e^{−iωt+ik·x}, the linear (unperturbed) propagator is

−i ω−k_kz

δfe=−cˆz B ×yˆ ∂

∂yδφ·xˆ∂f0

∂x −i|e|

m_ek_kδφ∂f0

∂v_k

= +c

Bik_yδφ∂n₀

∂x f₀ n₀ −i|e|

m_ek_kδφ

− vk

v_{T e}²

f₀ (Note thatf₀ only depends on xin real space.)

The 1st term is the relaxation of expansion free energy in configuration space, and the 2nd is a “heating term”. For non Maxwellian (e.g. B.O.T.), this corresponds to the relaxation of velocity space free energy.

Therefore, withk_k ≡k_z,

δfe = k_yv∗e−k_kv_k ω−k_kv_k

|e|δφ me

f0=

1− ω−ω∗e

ω−k_kv_k

|e|δφ Te

f0

δne= Z

d³vδfe= |e|δφ T_e −

Z d³v

ω−ω∗e

ω−k_kv_k

f0

|e|δφ T_e Defining 1-D distribution functionF0 vk

dvk =f0 x, vk

2πv⊥dv⊥dvk, the 2nd term contains

Z

d³v f₀ ω−kkvk

= Z ∞

−∞

dv_k F0 vk

ω−kkvk

with

F₀ v_k

=n₀ m

2πTe

3/2

exp −mv²_k 2Te

!

How should we treat an apparent singularity at v_k =ω/k_k?

Following Landau’s prescription, we’ll evaluate the integral as ifω had a real and imaginary part ω ⇒ ω+iε ≡ω+i0⁺ (|ε| 1, ε >0). Therefore, in a complex vk-plane, we’ll chose a contour which passes below the pole atvk =ω/kk.

(This is true for k_k >0 which is a usual convention.)

Z ∞

−∞

dz f(z)

z−i0⁺ = Pr Z ∞

−∞

dzf(z)

z +iπf(0) 1

z−i0⁺ = Pr 1

z

+iπδ(z) (Pr: principal value)

Physical interpretation: sinceδφ∝e^−iωt, forω =ω+iε,δφ∝e^−iωte^εt. Fort→ −∞, this is extremely small.

Justifying linearization: on the other hand, t → ∞ is irrelevant since we face nonlinearities at finite time (present even ifδne/ne∼10⁻²).

2.6. Electron Response to Drift-wave Fluctuation,

Including Contribution from Wave-particle Interaction (Sept. 17) Contribution from the pole atv_z =ω/k_k according to Landau’s prescription:

• Resonant part:

Res Z ∞

−∞

Fe0(vz)dvz

ω−k_kvz

=− iπ kk

Fe0(vz)

=−iπ 2

1/2 ne0

k_k

vT e

exp − ω² 2k_k²v²_{T e}

!

' −iπ 2

1/2 n_e0 k_k

v_{T e}

• Principal value of the integral (from the rest of the real axis):

Pr Z ∞

−∞

Fe0(vz)dvz

ω−k_kv_z = Pr Z ∞

0

Fe0(vz)dvz

ω−k_kv_z + Pr Z 0

−∞

Fe0(vz)dvz

ω−k_kv_z

= Pr Z ∞

−∞

2ω

ω²−k²_kv_z²dvz ∼ O ne0ω k_k²v_{T e}²

!

which is negligible.

Therefore,

δne

n_e0 = |e|δφ T_e

"

1+i

π

2

1/2 ω−ω∗e

k_k v_{T e}

#

where the first term corresponds to the adiabatic (Boltzmann) response and the second term to thenon-adiabatic response (ω is linked to the gradient in velocity space andω∗e to the gradient in configuration space).

Now, we can discuss the collisionless drift instability (26.34).

1+k²_⊥ρ²_s−ω∗e

ω − k_k²c²_s

ω² =−iπ 2

1/2 ω−ω∗e

kk

vT e

In most cases,k_⊥²ρ²_s1 and k_kcsω

⇒ we can solve Eq. (26.34) perturbatively, withω'ω∗e. The small corrections to this are

ω' ω∗e

1−k²_⊥ρ²_s 'ω∗e 1−k²_⊥ρ²_s

wherek²_⊥ρ²_s proceeds from the ion polarization drift with Mime,

<(ω)'ω∗e 1−k_⊥²ρ²_s

+k_k²c²_s ω∗e

γ ω∗e

= =(ω) ω∗e

= π

2

1/2 ω∗e

k_k

vT e

k²_⊥ρ²_s− k_k²c²_s ω_∗e²

!

For an instability, downward shift ofω belowω∗eis required. In this simple limit, k_⊥²ρ²_s k_k²c²_s

ω²_∗e Note For wave-particle interaction to be important,

• The plasma needs to be collisionless enough R∞

−∞

Fe0(vz)dvz

ω−k_kvz−iνe

• <(ω) =(ω)

Otherwise the instability is “reactive” (i.e. can be obtained from fluid description).

2.7. Physical meaning of i ^π₂1/2 “ω−ω_∗e⁰⁰

|^kk|^{vT e} term

Recall the expression forδf_e from the drift-kinetic equation (26.20):

−i ω−k_kv_z

δf_e=−cδE_y B0

∂

∂xF_e0+ e me

δE_z ∂

∂vz

F_e0

where the1st termis proportional to ^∂n_∂x⁰ and associated withω∗e, and the 2 term is proportional tok_kvz = k_kvz−ω

+ωafter substracting the adiabatic response.

Note that

δv_x= cδE_y

B₀ =−ick_yδφ B₀ dδv_z

dt =−eδE_z me

=iekkδφ me

SinceδE_y/δE_z =k_y/k_z>0,δv_x and _dt^dδv_y are 180^◦ out of phase!

• For thefirst term:

– δvx >0⇒ _dt^dδvz <0, electrons lose energy – δv_x <0⇒ _dt^dδv_z >0, electrons gain energy

– Since _∂x^∂ n₀(x)<0, there are more electrons which lose energy to drift wave⇒destabilizing! (∝ω∗e)

• For the second term, since _∂v^∂

zFe0 < 0, the wave-particle response ends up heating electrons, i.e. drift waves gives energy to particles ⇒ stabilizing!

(∝ω) In summary,

• k_⊥²ρ²_s ⇒ω&, destabilizing

• k_k²c²_s ⇒ω %, stabilizing And we have

ω < ω∗e ⇒ k_⊥²ρ²_s > k²_kc²_s ω_∗e² Since

ω∗e= ρ²_s L²_nk_y²c²_s the RHS becomes

k_k k_y

2

L²_n ρ²_s Since k_⊥²ρ²_s .1 andL_n/ρ_s 1, we need k_k/k_y2

1 to have a drift instability, i.e. λ_k λ⊥.

2.8. Effect of Electron Temperature Gradient on Drift Instability (Sept. 19)

Now, we consider an equilibrium Maxwellian distribution F_e0 =n_e0(x)

m 2πT_e(x)

3/2

exp

− m_e 2T_e(x)

v_⊥² +v²_k

(with T_e=T_e0)

Then the expansion free energy is related to

∂

∂xF_e0 =F_e0 ∂

∂xlnF_e0 = Fe0

n_e0 dne0

dx −Fe0

T_e dTe

dx 3

2 − v² 2v_{T e}²

= F_e0 ne0

dn_e0 dx

1−η_e

3 2− v²

2v²_{T e}

wherev_{T e}² =Te/me and η_e≡ Ln

L_Te = 1 T_e

dTe

dx 1

n_e0 dne0

dx −1

, L_Te ≡ − 1 T_e

dTe

dx Note ηe → ∞for a flat density and finite LTe.

We can repeat the same calculation we did with an additional term related toη_e. δfe= |e|δφ

T_e Fe0− 1 ω−k_zv_k

ω−ω∗e

1−ηe

3 2 − v²

2v_{T e}²

|e|δφ T_e Fe0

δn_e n_e0 = 1

n_e0 Z

d³vδf_e=. . .

= |e|δφ

T_e −|e|δφ T_e

Z ∞

−∞

F_e0 v_k dv_k ω−k_zv_k

ω−ω∗e

1−ηe

3

2 −1− v² 2v_{T e}²

where the−1comes from the integrationR∞

0 dv⊥v⊥(. . .)h

3/2−

v_⊥² +v_k²

/2v_{T e}² i . Here, F_e0 is a one-dimensional Maxwellian distribution. Evaluating a resonant contribution from a pole atv_k =ω/k_k as before,

δne

ne0

= |e|δφ Te

"

1 +i π

2

1/2ω−ω∗e(1−ηe/2)

kk

#

Note ^dT_dx^e does not affect ion dynamics!

Therefore, the linear dispersion relation is 1 +k²_⊥ρ²_s−ω∗e

ω − k_k²c²_s

ω² =iπ 2

1/2ω−ω∗e(1−η_e/2)

kk

vT e

Evaluating=(ω) for ω/ω∗e'1−k_⊥²ρ²_s+k_k²c²_s/ω_∗e² (η_e does not affect<(ω)), we getγ(ω):

γ ω∗e

≡π 2

1/2 ω∗e

k_k v_{T e}

"

k²_⊥ρ²_s−k_k²c²_s ω_∗e² − ηe

2

#

as in Eq. (26.46) with typo for the 1/2 factor forηe.

In this case,∇T_e is a stabilizing influence on drift instability!

Here, we are only considering a specific (particular) example, in whichω/k_k v_{T e} (i.e. vkresonantvT e) andω'ω∗e.

δn_e∼Res (Z ∞

−∞

dv_k 1 ω−k_kv_k

ne0

Te^1/2(x) (. . .)

(

ω−ω∗e

"

1−η_e 1 2−

v_k² 2v_{T e}

!#))

whereFe0 v_k

∝ne0/Te^1/2(x) and v²_k/2v²_{T e}∼v_kresonant² /2v²_{T e}1.

In this limit, ne0(x)/Te^1/2(x) is the effective density which characterizes the ex- pansion free energy!

Of course, in other examples with different frequency ordering, the electron tem- perature gradient can destabilize a wave and excite an instability. e.g., for Elec- tron Temperature Gradient (ETG) instability (in a fluid limit: ω/k_k &v_{T e}) or Ion Temperature Gradient (ITG) instability driven by∇T_i (ω/k_k &v_{T i}).

2.9. Effect of Electron Current on Drift Instability

Now, we consider

F_e0 =n_e0(x) m

2πT_e 3/2

exp

−m_e 2T_e

v²_⊥+ v_k−u_e02

i.e. 1-D shifted Maxwellian. Here we ignore∇T_eeffect for simplicity, andT_e=T_e0. (The electron current isj_k=− |e|n_e0u_e0=− |e|2πR∞

0 dv⊥v⊥R∞

−∞dv_kF_e0.) Then, in the electron drift equation,

∂

∂v_kF_e0 =−vk−ue0

v_{T e}² F_e0 We again repeat a similar calculation forδn_e.

δf_e= |e|δφ

T_e F_e0−ω−ω∗e−kkue0

ω−k_kv_k

|e|δφ T_e F_e0 Do 2πR∞

0 dv⊥v⊥R∞

−∞dv_k, δne

n_e0 = |e|δφ

T_e − |e|δφ

T_e ω−ω∗e−k_kue0

Z ∞

−∞

F_e0 v_k ω−k_kv_kdv_k Taking the resonant contribution from the integral, we obtain

δn_e ne0

= |e|δφ Te

"

1 +iπ 2

1/2 ω−ω∗e−kkue0

kk

vT e

#

forue0 vT e, since exp h

− vkresonant−ue0

2

/2v²_{T e} i

'1.

The linear dispersion equation becomes 1 +k²_⊥ρ²_s−ω∗e

ω −k_k²c²_s

ω² =−iπ 2

1/2 ω−ω∗e−kkue0

k_k v_{T e}

Note the effect ω−k_ku_e0 can be regarded as a Doppler shifted frequency in the frame moving withue0.

γ ω∗e

'π 2

1/2 ω∗e

k_k

v_{T e} k_⊥²ρ²_s−k²_kc²_s

ω²_∗e +k_ku_e0 ω∗e

!

With our sign convention (k_k>0, ue0 >0), ue0 is destabilizing.

Homework 2 Problem 26.3 and 26.4 of Goldston and Rutherford.

3. Ion Temperature Gradient Instability (Sept. 24)

High central ion temperature is required for magnetic nuclear fusion→ ∇T_i exists and ion heat transport is a topic of primary interest.

In uniform B, ∇T_i alone can excite ITG instability by driving the ion acoustic wave unstable. ∇ninfluences conditions for excitation and growth rate. This can happen even with Boltzmann (adiabatic) electron response (i.e. ωk_kv_{T e}).

Let’s take Ti0(x) = T0(x), Ti = T0 +δTi, B = B0]ˆz. Now, we generalize ion dynamics (still assumingω k_kv_{T i} andk²_⊥ρ²_i 1→fluid description is justified).

Linearize

⇒ ∂

∂tδn+δu_E· ∇n₀+n₀∇_kδu_k+n₀

∇_⊥·δu_pol= 0 (6) We’ll relax cold ion assumption (T_eT_i) which was used before for electron drift wave. ⇒ T_i ∼T_e, but this leads tok_⊥²ρ²_s 1 (ρ_s=c_s/Ω_ci,c_s=p

T_e/M_i).

M_i ∂

∂tδu_k =− |e| ∇_kδφ− 1

n₀∇_kδp_i (7)

and

∂

∂tδp_i+δu_E · ∇p₀+ Γn₀∇_kδu_k = 0 (8) Here, “Γ” is an “adiabatic” exponent which appears in the equation of state:

p

ρ^Γ = const

Here the term adiabatic implies “thermal insulation” in thermodynamics.

E.g. for a sound wave in a gas, period of vibration

!

relaxation time for volume element of gas to exchange energy with the rest of fluid

through heat flow

!

As before, we assumed perturbed quantities∝exp (−iωt+ik·x)

→ _∂t^∂ ⇒ −iω,∇_k ⇒ik_k when they are operated on perturbed quantities.

Let

∂

∂xTi(x) x=x0

=− 1 LT i

Ti(x) x=x0

ω∗e= kyρs

Ln

cs>0 pi=n0Ti ω∗pi=−kyρs

L_pi v_{T i}<0

⇒ 1 Lpi

= 1 LT i

+ 1 Ln

, τ ≡ T_e Ti

ω∗T i =−k_yρ_s LT i

v_{T i}<0

⇒ with a proper normalization, Equations (6), (7) and (8) become

−iωδn n0

+iω∗e

|e|δφ Te

+ikkcs

δu_k cs

= 0

−iωδuk

cs

+ik_kc_s|e|δφ Te

+ik_kc_sT_i Te

δp_i p0

= 0

−iωδpi

p₀ +iω∗pi

τ

|e|δφ

T_e +ik_kcs

δu_k c_s = 0

⇒ withδn/n0=|e|δφ/Te,





−i(ω−ω∗e) ikkcs 0 ik_kc_s −iω ik_kc_s¹_τ

−i^ω_τ^pi ikkcsΓ −iω









δn/n0

δu_k/c_s δpi/po



= 0 Determinant [3×3] = 0⇒ Cubic algebraic equation for “ω”

1−ω∗e

ω −k_k²c²_s ω²

1 +Γ

τ −ω∗pi

ω

= 0 (9)

Note that asτ → ∞,LT i → ∞; we recover 1− ω∗e

ω −k²_kc²_s ω² = 0

(Since se assumed k_⊥²ρ²_s 1, there’s no k⊥ term.) Now, we consider more inter- esting case with strong gradient, 1/LT i %, 1/L_n%. Then,ω∗e,|ω∗pi| % and the 2nd and 5th terms of Equation (9) become dominant.

ω∗e

ω ' k²_kc²_s ω²

ω∗pi

ω =−ω∗e

ω k²_kc²_s

ω²

1 +ηi

τ

Hereη_i≡L_n/L_{T i}.

ω²=−1 +η_i τ

k_k²c²_s ⇒ γ =1 +η_i τ

1/2

k_kc_s (10) (This dissipation relation is valid forη_i1,k_k very small, 1/L_n relatively high.)

⇒Purely growing (reactive) instability! (without a help from wave-particle reso- nance) i.e. <(ω)'0,=(ω)>0.

↔ Electron drift instability which is “resonant”. Let’s check the validity regime.

|ω| k_kv_{T i}⇒1 +ηi

τ 1/2

1 But

ω∗e

|ω| 1⇒k_kc_s1 +η_i τ

1/2

k_yρ_s Ln

c_s (⇒“k_k should be very small⁰⁰) Equation (10) is the most pessimistic estimation of ITG linear growth rate.

So far we are addressing only the “local” stability or instability. To address more realistic spatial variation of perturbation, we need to consider non local theory (δφ∼δφ(x) expˆ i(k_yy+k_zz)).

3.1. Sound Wave

“Who derived the sound wave velocity at first?”

⇒ I. Newton first derived the equation of sound wave even before measuring it.

• In plasma,ω² =k_k²c²_s

• In gas,v²_ph=ω²/k²_k = Γp0/ρ0 (Newton’s answer was for Γ = 1)

Newton used Boyle’s law in the derivation (Γ = 1, pV = const. i.e., isothermal) rather than a proper equation of state p/ρ^Γ= const.

3.2. Ion Temperature Gradient Instability in Uniform Magnetic Field (Sept. 26)

We obtained

ω² k²_k =−

1 +ηi

τ

c²_s

when (1 +ηi)/τ 1,kk very small, ω/kk vT i. The dispersion relation clearly shows the “sound-wave-like” character of ITG instability.

Let’s review a sound wave in a gas:

ρ0

∂

∂tδu_k =− ∂

∂zδp (11)

∂

∂tδp=−Γp₀ ∂

∂zδuk (12)

The determinant of this [2×2] system is ω²

k²_k = Γp0

ρ0

Note that the linearized continuity equation _∂t^∂δp+ρ0 ∂

∂zδu_k = 0 andp/ρΓ = const yields Equation (12).

Let’s check how this is related to the compressibility.

κ_s=−V⁻¹ ∂V

∂p

s

is the adiabatic compressibility (in thermal insulation or at constant entropy “s”).

pV^Γ=p0V₀^Γ⇒p=p0V₀^ΓV^−Γ ∂p

∂V

s

=−Γp₀V₀^ΓV^−Γ−1 ⇒V ∂p

∂V

s

=−Γp₀ Therefore,

κs≡ −V⁻¹ ∂V

∂p

s

=

−V ∂p

∂V

s

−1

= (Γp0)⁻¹

so that

ω kk

2

= (ρ₀κ_s)⁻¹

From this, we can interpret that the effective “compressibility” of the ITG insta- bility is a “negative”.

Instability mechanism: P % when V &in a gas, but the opposite happens here!

Previous example: the 2nd and 5th terms of Equation (9) are dominant.

Now, if Ln→ ∞(flat density): ω∗e/ω &0.

Therefore, |ω_∗e/ω| 1 so that the 1st term should balance the 5th term. Also, we need|ω_{∗T i}/ω| 1 and k_k²c²_s/ω² 1 to make this dominant balance justified.

1 = k²_kc²_s ω²

ω∗T i

ω =−k_k²c²_s|ω_{∗T i}| ω³ This cubic equation has 3 roots.

Now the ITG is no longer purely growing. <(ω)<0 in the ω∗i direction.

ω =e^i2πN/3|ω∗T i|^1/3 kkcs

2/3

withN = 0,1,2.

Actually, for a tokamak core plasma, this limit is usually more relevant than another limiting case,ω∼i[(1 +ηi)/τ]^1/2kkcs.

ITG has a hybrid character of adrift waveand asound wave. Its frequency is also of opposite sign to the electron drift wave.

Now, back to (k²_⊥ρ²_i 1):

1−ω∗e

ω −k²_kc²_s ω²

1 +Γ

τ −ω∗pi

ω

= 0

We have ω∗e ∝ 1/Ln, ω∗pi ∝ 1/Lpi, 1/Lpi = 1/Ln + 1/LT i and (neglecting 1/L_n) ω∗pi ⇒ ω∗T i. Also, for generality, Γ/τ =O(1). Therefore, if 1/L_{T i} → 0, τ ≡T_e/T_i→ ∞,ω∗T i →0 and we recover the electron dispersion relation.

Recall the evolution equations determining dynamics:

∂

∂tpi+δuE · ∇p₀+Γp0∇ ·δuk = 0 M_i ∂

∂tδu_k=−e∇_kδφ− 1 n0

∇_kδp_i

∂

∂tδn+δuE· ∇n₀+n0∇ ·δuk = 0 withsoundand drift wave contributions and δn/n₀ =|e|δφ/T_e.

In these determining equations, the terms of sound wave and drift wave co-exist.

⇒ In this sense, we can say that drift wave and sound wavea are “coupled”.

What happens if 1/LT i →0?

Even for very weak ITG (1/L_{T i} → 0), it remains unstable! which is unphysical (does not match with experimental results). We need to re-examine the approxi- mations we’ve used. In particular,ω/kk vT i will break down as |ω| →0, so we need to consider ion kinetic effects, including ion Landau damping.

Inclusion os ion Landau damping using ion drift kinetic equation (similar to what we did for electrons for the electron drift wave) leads to a conclusion that ITG mode is unstable forηi ≡Ln/LT i >2 (for (k⊥ρi)² 1).

The condition _L¹

T i(1−η_i/2) is similar to the free energy relaxation in the electron drift wave.

Question when does the conditionη_i>2 become unphysical? (exam)

4. Basic Plasma Physics for Microinstabilities (Oct. 18)

4.1. Review of Single Particle Motion in a Strong Magnetic Field

4.1.1. Adiabatic Invariant

When magnetic field varies in space smoothly (i.e. ρ_i L_B≡ |∇lnB|⁻¹), we can identify approximate constants of motion.

“Adiabatic” in here means slow variation in time and space. This is well illustrated from the point of view of Quantum Mechanics (QM).

Let’s consider a Simple Harmonic Oscillator (SHO): Schr¨odinger Equation is Hψ(x) =

− ~² 2m

∂²

∂x² +1

2mω₀²x²

ψ(x) =Eψ(x) Here the eigenvalues are

E =~ω₀ N +1

2

whereN is quantum number (N = 0,1,2, . . .).

Suppose that potential well is changed very (very) slowly in time (τ 1/ω₀).

In this adiabatic process, what remains constant is “N” (eigenstate is preserved).

Energy (eigenvalue) changes in time. “N” is an example of adiabatic invariant.

N =E/ω0 : classical limit

4.1.2. Adiabatic Invariant in Classical Limit

In Classical Mechanics (CM), the Hamiltonian of SHO is given by H(p, q) = p²

2m +1

2mω²₀q² =E

Adiabatic invariant in CM is coupled to the conservation of the volume in the phase space for appropriate action-angle variables.

I = I

dq p This is also called “Action Invariant”.

For SHO the action invariant is I =πLqLp =π

s

2mE 2E

mω₀² = 2πE ω₀

Thus except for a numerical factor 2π, we recoverN =E/ω₀ from SHO in QM.

(The useful formulaN =E/ω₀ represents the “Duality of Wave and Particles”.) This illustration of geometric meaning of action invariant can be extended to quasi-periodic motion (recallτ 1/ω₀).

4.1.3. Gyromotion in Slowly Varying Magnetic Field

Consider the gyrating motion of charged particles in slowly varying magnetic field in time (1/ω) and space (LB).

For this gyration, the corresponding action invariant is the magnetic moment (the 1st adiabatic invariant).

• Energy corresponding to gyration: E⊥=mv_⊥²/2

• Frequency corresponding to gyration: Ωc=eB/mc

⇒µ∝ 1 2mv²_⊥/

eB mc

∝v²_⊥ 2B

: not exact constant What is the error or precision of the statement?

What is the expansion parameter or smallness parameter in this motion?

If_ω≡ω/Ω_c1 and _B ≡ρ_i/L_B1,

the adiabatic invariant is good up to arbitrary order!

Error =O

exp (−const

_B ), exp (−const _ω )

4.1.4. Charged Particle Motion in Inhomogeneous Magnetic Field

E = 1

2mv_⊥² +1

2mv²_k =µB+1 2mv_k² Ifµis a constant,

E =µB(x) +1 2mv_k²

Mathematically,µB(x) acts as “potential” energy for parallel motion: V_eff(x).

Whether a particle passes or be trapped depend onE and µ.

Example Cosmic ray in spcae: abundance of high energy cosmic ray

Fermi explained this using particle trapping in [Phys. Rev. 75, 1169 (1949)].

1. Particles get trapped in magnetic field 2. Magnetic field changes

3. Particles are accelerated

⇒ “Magnetic mirror”

In tokamaks;

|B|=B = R₀

R B∼=B₀ 1− r

R0

cosθ

(R=R₀+rcosθ)