PvdZon 2012 - les07

Chapter 11: Resonant absorption and wave heating

 

Overview

•

Ideal MHD theory of resonant absorption

Analytical solution of a simple model problem [ book: Sec. 11.1.1 ]

The role of singularities [ book: Sec. 11.1.2 ]

Absorption versus dissipation [ book: Sec. 11.1.3 ]

•

Heating and wave damping in tokamaks and coronal magnetic loops

Tokamaks [ book: Sec. 11.2.1 ]

Coronal loops and arcades [ book: Sec. 11.2.2 ]

Numerical analysis [ book: Sec. 11.2.3 ]

•

Alternative excitation mechanisms

Foot point driving [ book: Sec. 11.3.1 ]

Phase mixing [ book: Sec. 11.3.2 ]





Motivation

•

plasma

WAVES and INSTABILITIES

play an important role. . .

–

in the dynamics of plasma perturbations

–

in energy conversion and transport

–

in the heating & acceleration of plasma

•

characteristics (

ν

,

λ

, amplitude. . . ) are determined by the ambient plasma

⇒

can be exploited as a diagnostic tool for plasma parameters, e.g.

–

wave generation, propagation, and dissipation in a confined plasma

⇒

helioseismology (e.g. Gough ’83)

MHD spectroscopy (e.g. Goedbloed et al. ’93)

–

interaction of external waves with (magnetic) plasma structures

⇒

sunspot seismology (e.g. Thomas et al. ’82, Bogdan ’91)







Resonant ‘absorption’ and phase mixing

⇒

involves direct or indirect excitation of

continuum waves

– singular behavior – small length scales

⇒

efficient dissipation

⇒

small length scales due to inhomo-geneity of the plasma

(cf. fusion research:

Tataronis & Grossmann (’73): incompressible Chen & Hasegawa (’74): shear, compressible)

driv

ω

A

ω

a

r

_s

0

r

continuum

incident wave

Resonant AW heating in 1D cylinder

•

first applied to coronal loops by Ionson (’78) and to sunspots by Lou (’90)

Re(

ε

)

x

ε

2

ε

1

plasma

vacuum

(1)

(2)

(3)

(4)

antenna

x

₀

c

x

₁

x

₂

=

0

a

Simplified (slab) set-up (cf. Chen & Hasegawa ’74), here (x) ≡ ρ(ω2− ω_A2(x))

•

semi-infinite plasma slab (

x <

0

) next to a vacuum (

x >

0

)

•

unidirectional magnetic (choose

z

-axis along field)

•

equilibrium quantities constant in (1) and

x

-dependent for

x

₁

≤ x ≤ x

₂ (

≡ 0

)







cfr. MHD models: laboratory plasmas

model I model II (*) model III

ϕ n wall plasma a a n ϕ b n ϕ c b c coil vacuum vacuum plasma pl-vac wall vac. / plasma ( * ) plasma pl-vac

(a) Model I: Plasma surrounded by a wall; (b) Model II (*): Plasma isolated from the wall by a vacuum (*: or another plasma); (c) Model III: Plasma excited by currents in external coils

•

equilibrium only

x

-dependent:

ξ(x, y, z; t) = ξ(x) e

i(kyy+kzz−ωt) where

ω

= ω

d

+ iν

, with

0 ≤ ν  1

(cf. later)

•

ignore gravity + low-

β

approximation

⇒

SMWs dropped

• B

uniform (

z

-dir.)

⇒ k

_

= f = k

z and

k

⊥

= g = k

y constants

⇒

wave or spectral equation for a plane slab

⎛ ⎜ ⎜ ⎜ ⎜ ⎝ d dx(γp+ B 2₎ d dx − f 2_B2 d dxg(γp + B 2₎ d dxf γp − g(γp+ B2) d dx −g 2_{(γp+ B}2_{) − f}2_B2 _−gfγp − fγp d dx −f g γp −f 2_γp ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎝ ξ η ζ ⎞ ⎟ ⎟ ⎟ ⎠ = −ρω2 ⎛ ⎜ ⎝ ξ η ζ ⎞ ⎟ ⎠ simplifies drastically:

g

= g(x) ≡ −ie

_⊥

· ∇ = G/B

= k

y

,

G

≡ k

y

B

z

− k

z

B

y

= k

y

B

z

f

= f(x) ≡ −ie

_

· ∇ = F/B

= k

z

,

F

≡ k

y

B

y

+ k

z

B

z

= k

z

B

z

•

equilibrium only

x

-dependent:

ξ(x, y, z; t) = ξ(x) e

i(kyy+kzz−ωt) where

ω

= ω

d

+ iν

, with

0 ≤ ν  1

(cf. later)

•

ignore gravity + low-

β

approximation

⇒

SMWs dropped

• B

uniform

⇒ k

_

= f = k

z and

k

⊥

= g = k

y constants

⇒

_⎛

⎜

⎜

⎜

⎝

ρω

2

+

d

dx

B

2

d

dx

− k

2 

B

2

d

dx

k

⊥

B

2

−

k

_⊥

B

2

d

dx

ρω

2

− k

2 0

B

2

⎞

⎟

⎟

⎟

⎠

⎛

⎝

ξ

x

iξ

_⊥

⎞

⎠ =

⎛

⎝

0

0

⎞

⎠

where

k

₀2

≡ k

_⊥2

+ k

_2

⇒

k

_⊥

= 0

: AWs and FMWs are coupled!

⇒

vital for resonant absorption!

⇒

d

dx

(

ρω

2

− k

_2

B

2

ρω

2

− k

2 0

B

2

B

2

dξ

x

dx

) + (ρω

2

_{− k}

2 

B

2

) ξ

x

= 0 ,

2nd-order ODE, special (simple) form of HL eq.

•

focus on nearly perpendicular propagation, i.e.

k

_

 k

_⊥

≈ k

and very strong coupling (although

ω

_A2

= k

_2

v

_A2

 ω

_{f 0}2

≈ k

2

v

_A2)

⇒

d

2

ξ

x

dx

2

+

1



d

dx

dξ

x

dx

− k

2 ⊥

ξ

x

= 0

with



(x) ≡ ρ(ω

2

− ω

_A2

(x))

⇒

Chen & Hasegawa (’74):

Re[(x)] = ⎧ ⎨ ⎩ ₁ (x < x₁) ₁ + 2 − 1 x₂ − x₁ (x − x1) (x1 ≤ x ≤ x2) ,

⇒

choose

ω

_d

: ∃x

₀

:



(x

₀

) = 0

Re(ε) x ε2 ε1 plasma vacuum (1) (2) (3) (4) antenna x₀ c x₁ x₂=0 a

Plasma solution

region (1)

: simple Helmholtz equation (since

d

dx

= 0

)

•  = 

1

⇒

ξ

x(1)

= A

₁

e

k⊥(x+a)

+ B

₁

e

−k⊥(x+a) (

A

₁

, B

₁ : from BCs)

region (2)

:

•

use normalized variable

X

≡

k

⊥



d/dx

=

ak

_⊥





₂

− 

₁

⇒

0th-order modified Bessel eq.:

d

2

_ξ

x

dX

2

+

1

X

dξ

_x

dX

− ξ

x

= 0

⇒ ξ

(2)

x

= A

1

(A

2

I

0

(X) + B

2

K

0

(X))

(constants such that BCs simple)

•

BC: solution finite as

x

→ −∞ ⇒ B

₁

= 0

•

BC at

x

= x

₁

(= −a)

: both

ξ

x and

ξ

_x continuous

A₂I₀(X₁) + B₂K₀(X₁) = 1 ⇒ A₂ = X₁[K₀(X₁) + K₁(X₁)] A₂I₁(X₁) − B₂K₁(X₁) = 1 ⇒ B₂ = −X₁[I₀(X₁) − I₁(X₁)]

Vacuum solution

•

simpler equations, viz.

∇ × ˆQ = 0

and

∇ · ˆQ = 0

⇒

Q

ˆ

y

=

k

_⊥

k

_

Q

ˆ

z

,

Q

ˆ

x

=

−i

k

_

∂ ˆ

Q

z

∂x

,

∂

2

Q

ˆ

z

∂x

2

= k

2 0

Q

ˆ

z (Helmholtz)

⇒ ˆ

Q

(3)_z

= A

₃

e

−k0(x−c)

+ B

₃

e

k0(x−c) and

Q

ˆ

(4)_z

= A

₄

e

−k0(x−c)

+ B

₄

e

k0(x−c) Re(ε) x ε2 ε1 plasma vacuum (1) (2) (3) (4) antenna x0 c x₁ x2=0 a

Vacuum solution

•

simpler equations, viz.

∇ × ˆQ = 0

and

∇ · ˆQ = 0

⇒

Q

ˆ

y

=

k

_⊥

k

_

Q

ˆ

z

,

Q

ˆ

x

=

−i

k

_

∂ ˆ

Q

z

∂x

,

∂

2

Q

ˆ

z

∂x

2

= k

2 0

Q

ˆ

z (Helmholtz)

⇒ ˆ

Q

(3)_z

= A

₃

e

−k0(x−c)

+ B

₃

e

k0(x−c) and

Q

ˆ

(4)_z

= A

₄

e

−k0(x−c)

+ B

₄

e

k0(x−c)

•

BC: solution finite as

x

→ +∞ ⇒ B

₄

= 0

•

BC at

x

= x

a: surface current

j

_c

= J



e

i(kyy+kzz−ωt)

⇒ [[ ˆ

Q

z

]]

c

= −J

_y but

[[ ˆ

Q

x

]]

c

= 0

⇒ A

3

= A

4

+

1₂

J

y and

B

3

=

1₂

J

y

⇒

Q

ˆ

(3)z

= A



₄

e

−k



0(x−c)



‘induced’

− J

 y

cosh(k

0

(x − c))







‘driven’

Linking the plasma and vacuum solutions

•

already 6 BCs

⇒

only

A

₄ and

A

₁ left

⇒

two more BCs required for unique solution of this driven problem!

•

BCs at

x

= x

₂

(= 0)

: continuity of

Q

ˆ

x and total pressure (i.e.

Q

ˆ

z)

⇒ k



Bξ

x(2)





x₂

= −

1

k

_

∂ ˆ

Q

(3)z

∂x







x₂ and

B

k

_⊥2

B

2

− 

∂ξ

x(2)

∂x







x₂

= ˆ

Q

(3)_z





x₂

⇒

A₄ = [C cosh(k₀c) − D sinh(k₀c)]A₁, A₁ = −J_y e −k₀c C − D, with C ≡ (k_2/k₀)B[A₂I₀(X₂) + B₂K₀(X₂)], and D ≡ k⊥B2 k_⊥2B2 −  2[A2 I₁(X₂) − B₂K₁(X₂)]

Some remarks. . .

•

Chen & Hasegawa (’74) conveniently avoided the singularity at

x

₀ where

ω

d

= ω

A

(x

₀

)

by considering a complex

ω

= ω

d

+ iν

⇒ ν =

‘artificial damping’ (

⇔

real damping!)

⇒

resonant ‘absorption’ (solution

∼ e

νt)

•

solution in terms of modified Bessel functions!

⇒

what happened to the logarithmic singularity??

⇒

close to

x

₀ :



 1

and thus

X

 1

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 x ~ 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 0 . 0 5 0 . 0 9 0 . 1 4 x ~ 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 x ~ 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 x ~ Re( ) Im( ) Re( ) Im( ) ξ ξ ξ ξ x x y y

⇒

Analytic solutions ξx(2) and ξ_⊥(2)(= ξy) in region (2) (−a ≤ x ≤ 0) as a function of the shifted coordinate

˜x ≡ x/a + 1 for a = B = 1, ωd = k = J_y∗ = 1, and ˜x₀ = 0.5; δ = 0.001, k⊥ = 5

Energy ‘absorption’ rate

•

remark: complex notation

⇒

time average of

Ae

iωt and

Be

iωt is 1₂

AB

∗

•

energy ‘absorption rate’ (in steady state) derived from Poynting vector

S =

1₂

E

∗

× Q

:

dW

dt

= L

y

L

z Re

[S(x

2

) − S(x

1

)] · e

x

with

L

y and

L

z the size of the plasma in the

y

−

and

z

−

direction

• E = iω ξ × B

⇒ E



= 0,

and

E

⊥

= −iω B ξ

x

⇒

dW

dt

=

ω

d

2

L

y

L

z Im



BQ

_

ξ

_x∗



x_x2 1

,

=

ω

d

2

L

y

L

z Im



B

2



k

_⊥2

B

2

− 

dξ

_x

dx

ξ

∗ x



x₂ x₁

0 2 4 6 8 10 1 1.2 1.4 1.6 1.8 2 ωd dW/dt

Scan of the average energy absorption rate dW/dtin the steady state versus the driving frequency for _{1 ≤ ωd}≤ 2 for δ = 0.001, a = 0.005, c = 0.1, k_{ = 1}, k_{⊥ = 10}, ρ_{1 = 1}, ρ_{2 = 0} and J y = 1. ‘quasi-mode’

Re ω = ±



ρ

₁

ω

_A12

+ ρ

₂

ω

_A22

ρ

₁

+ ρ

₂

ρ

₂

= 0

=⇒

Re ω =

√

2 ω

_A1







Ideal quasi-modes (cf. Chap. 10)

x₁ x₂ x x 0 ω2 A2 ω2 A1

⇒

x x₁ x₂ ξ x 0

step discontinuity of Alfv ´en frequency surface mode

-a 0 a x ω2 A2 ω2 A1

⇒

• x x • n = 0 n = -1 n = 0 n = 1 n = 0 x x

smoothing the discontinuity damped (!) ‘quasi-mode’







Different kinds of quasi-modes

0 2.5 5 7.5 10 12.5 15 0 2.5 5 7.5 10 12.5 15 17.5 20 ω k_z

Eigenfrequencies of the first three fast eigenmodes (grey lines) with upper and lower bound of the Alfv ´en continuum (black lines) as functions of k_z (for _{L = a = 1}, and k_{y = 0}). [From: De Groof et al. [?]]

⇒

increasing

k

z results in ever more FMWs ‘swallowed’ by the continuum

⇒

for

k

y

= 0

the FMWs couple to AWs and become quasi-modes

Resonant

dissipation

•

visco-resistive MHD equivalent of previous Eqs. reads (for hot plasmas)

⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ρω2 + d dx B 2 d dx − k 2 B2 − iωρ(˜η + ν) d2 dx2 d dxk⊥B 2 −k⊥B2 d dx ρω 2 _{− k}2_B2 _{− iωρ(˜η + ν)} d2 dx2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎝ ξx iξ_⊥ ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ 0 0 ⎞ ⎟ ⎠

where the (scalar) viscosity

ν

and the magnetic diffusivity

˜η ≡ η/μ

₀

= η/(4π×10

−7

)

, are only retained in terms with derivatives in the

x

-direction

⇒

remove the singularity (like ‘artificial’ damping)

⇒

nearly-singular behavior

⇒

dissipative effects cause a real mono-periodically driven plasma to reach a stationary state after a finite time





Remark on resistive solution

•

Kappraff & Tataronis (’77): similar planar set-up, resistive solution:

ξ

_⊥

(t) ∼

=



N

(ω

d

)

t

exp(−iω

d

t

)

(for

t < t

h)

N

(ω

d

) t

h

exp(−iω

d

t

)

(for

t > t

h)

,

with

N

(ω

d

)

a constant (independent of

η

), and with

t

h

=



24μ

0

ω

_d2

η



_1/3



2B

 1

(x

0

)

B

−

ρ



(x

₀

)

ρ



_−2/3

.







Remark on resistive energetics

•

Maxwell’s law + Faraday’s law:

−∇ · (E

∗

× Q) =

j

₁

· E

∗

+ Q ·

∂_∂tQ∗

•

electrical energy:

j

₁

· E

∗

= η|j

₁

|

2

+ v

∗

· j

₁

× B

₀

•

equation of motion

⇒

‘mechanical energy equation’:

ρ

₀

v

∗

·

∂

v

∂t

= −v

∗

_{· ∇p}

1

+ v

∗

· j

0

× Q + v

∗

· j

1

× B

0

.

⇒

combined and integrated over the plasma volume

V

: resistive energy balance

−

1 2



V

∇ · (E

∗

× Q) dV =

1 2



V

ρ

₀

v

∗

·

∂

v

∂t

dV







˙ K

+

1 2



V

v

∗

· ∇p

1

− v

∗

· j

0

× Q + Q ·

∂

Q

∗

∂t

dV







˙ W_p

+

1 2



V

η

|j

₁

|

2

dV







˙ D

.







Remark on resistive solution (cont.)

•

LHS is related to power emitted by the antenna: LHS

=

+

1₂



ˆ V

∇ · (ˆE

∗

× ˆ

Q) = −

1 2



ˆ V

ˆj

 c

· ˆE

∗

d ˆ

V







Pant

−

1 2



ˆ V

ˆQ ·

∂ ˆ

Q

∗

∂t

d ˆ

V







˙ Wv

(the inflow of electromagnetic energy in the plasma equals the outflow of electromag-netic energy out of the vacuum region and the power emitted by the antenna (

P

_ant) minus the rate of change of the vacuum magnetic energy (

W

˙

v))

•

combining the previous two equations yields

P

_ant

= ˙K + ˙W

_p

+ ˙

W

_v

+ ˙D

•

it has been shown that:



V

ηJ

₁2

dx

=

dW

  

Applications in tokamaks

supplementary HEATING:

•

theory from early ’70s (Tataronis & Gross-mann (’73); Chen & Hasegawa (’74))

•

many experiments (TCV!):

⇒

very efficient!

⇒

but: mainly edge heating

⇒

too efficient for 2-resonance heating!

DAMPING of gap modes

(discrete and global modes)

•

destabilized by

α

-particles

⇒ α

-particles lost by particle-wave resonance (was demonstrated in TFTR and DIII-D)

⇒

experiments showed a higher threshold than predicted from theory (clear indication of damping

∼ 0.5 − 1 %

)

Continuum branches and gap modes for n = −3, m = 2, 3, 4, 5, 6

⇒

damping of gap modes may be caused by coupling to continuum modes corresponding to continuum branches overlying the gaps

Convergence study of the relative damping versus plasma resistivity (3.4%)