PvdZon 2012 - les08

Resonant absorption and wave heating 11-1

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 ]



Observations

of MHD waves in the solar atmosphere

•

for a long time: almost no observations of waves in corona

only AWs (from non-thermal broadening spectral lines) (Doschek et al. ’76, Zirker ’93)

⇒

scepticism for wave heating mechanisms

•

Recently

: abundant evidence for the presence of MHD waves in the corona

–

SMWs in coronal plumes (DeForest and Gurman ’98, Ofman et al. ’00, Cuntz and Suess ’01) and

in coronal loops (e.g. De Moortel et al. ’00, Nakariakov et al. ’00, Robbrecht et al. ’01)

–

FMWs in coronal loops (Aschwanden et al. ’99, Nakariakov et al. ’00, O’Shea et al.’01)

–

‘damped oscillations’ in coronal loops and prominences (reviews Engvold ’01, Oliver ’01)

⇒

observational evidence now so overwhelming that coronal seismology arises

•

identification of MHD waves is complicated by geometrical effects, spatial variation of the physical quantities, non-trivial boundary and initial conditions and nonlinear behavior

⇒

wave couplings occur

Resonant absorption and wave heating 11-27

 



Wave generation and propagation

•

studied since the late 1940’s (Biermann ’46, Schwarzschild ’48)

•

basic idea enriched with a leading role of the magnetic field

• source

= overshooting convective motions in the photosphere

–

power spectrum shows a maximum around

3000 μ

Hz (periods

≈

5 min)

–

can generate FMWs, SMWs, and AWs, BUT

⇒

SMWs steepen into shocks, dissipated in chromosphere

⇒

FMWs are reflected and/or refracted at the transition region

⇒

only AWs are able to reach the corona (Hollweg ’84)

• problem

(?)

: periods are too long for resonating with the shorter loops (Parker ’92)

P

_fund

=

2L

v

A



Wave generation and propagation

•

studied since the late 1940’s (Biermann ’46, Schwarzschild ’48)

•

basic idea enriched with a leading role of the magnetic field

• source

= overshooting convective motions in the photosphere

–

power spectrum shows a maximum around

3000 μ

Hz (periods

≈

220–500 sec)

–

can generate FMWs, SMWs, and AWs, BUT

v

A

=

√B_μρ0 0

≈ 1.5 − 2 × 10

6_m/s

40 × 10

6

_{≤ L ≤ 400 × 10}

6 _m



⇒ P

fund

=

2L

v

A

≈

40 − 530

sec

→

when ‘slow’: twisting & braiding

⇒

reconnection (‘DC’) (cf. Parker, ’72, ’88)

Resonant absorption and wave heating 11-29

 



Wave generation and propagation

Watch out:

this is NOT TRUE!

•

resonant modes of a finite system can be excited most easily!, e.g. by

–

a ‘stick-slip’ mechanism (cf. bow on a violin) (Goedbloed ’95)

–

Kelvin-Helmholtz instabilities (cf. many instruments)

–

even a simple short pulse (cf. a drum) (De Groof et al. ’01)

in fact,

it is impossible to avoid the excitation of the fundamental modes!!

⇒

any perturbation of the loops, even random perturbations, excites the fundamental modes very efficiently as proven by De Groof et al. (2002)

OTHER WAVE SOURCES:

•

non-steady

magnetic reconnection

–

substantial part of energy released as MHD waves (Hood ’02)

–

occur in different topologies and length scales (from flares to ‘nanoflares’ (Parker ’88))

⇒

solves two problems at once:

– transport to corona not necessary – higher frequencies

⇒

resonances

(Parker ’00)

Resonant absorption and wave heating 11-31

OTHER WAVE SOURCES:

•

background velocity shear effects

⇒

MHD waves are coupled!

⇒ non-modal analysis

revealed:

•

waves can transform into each other

•

FMWs and SMWs can exchange energy with the flow

•

. . .

⇒

needs to be quantified! -4 -2 0 2 4x10-3 0 50 100 150 200 (a) τ by -0.04 -0.02 0 0.02 0.04 0 50 100 150 200 (b) τ K *v - K *v x y y x -4 -2 0 2 4x10-3 0 50 100 150 200 (c) τ Vz 0 0.5 1 1.5 2x10-4 0 50 100 150 200 (d) τ E FMW→ AW + SMW transition (from Poedts et al. ’98)



Resonant dissipation and phase mixing

•

can AWs be

dissipated efficiently

in the solar corona?

•

dissipation time scale:

   

τ

_D

=

μ

l

2

η

+ ν

⇒

extremely long in solar corona

⇒

higher ‘effective’ resistivity or viscosity required (cf. Nakariakov et al., 1999)

AND/OR

small length scales

l

required

⇒ ‘great challenge’ (20 years ago)

:

find a heating mechanism that is

equally efficient

in both open and closed magnetic structures, i.e. for the wide variety of time and length scalesobserved in the corona, in order to explain the observed

uniform temperature

in the solar corona

Resonant dissipation and phase mixing 11-33

Presently:

coronal heating model requirements changed drastically!!

•

waves also generated in corona itself! (by reconnection, flow instabilities, . . . )

•

wave modes are coupled to each other and to (sheared) background flows

•

temperature is not so uniform in the corona!

Yohkoh (1996)

:

•

temperature higher in the loops (up to

7 × 10

6

K

)

Presently:

coronal heating model requirements changed drastically!!

•

waves also generated in corona itself! (by reconnection, flow instabilities, . . . )

•

wave modes are coupled to each other and to (sheared) background flows

•

temperature is not so uniform in the corona!

Yohkoh (1996)

: (Kano & Tsuneta ’96)

•

temperature higher in the loops (up to

7 × 10

6

K

)

•

temperature higher at tops of loops!

⇒

heating at tops (reconnection?)!

Resonant dissipation and phase mixing 11-35

Presently:

coronal heating model requirements changed drastically!!

•

waves also generated in corona itself! (by reconnection, flow instabilities, . . . )

•

wave modes are coupled to each other and to (sheared) background flows

•

temperature is not so uniform in the corona!

TRACE (2000)

: (Aschwanden ’00)

•

temperature uniform in loops

(Yohkoh results ascribed to ‘statistical uncertainties’ of broadband SXT filter)

⇒

more heating in low corona!

(reconnection(?) at base of loops!)

Presently:

coronal heating model requirements changed drastically!!

•

waves also generated in corona itself! (by reconnection, flow instabilities, . . . )

•

wave modes are coupled to each other and to (sheared) background flows

•

temperature is not so uniform in the corona!

HOWEVER

:

•

TRACE 2002

: temperature varies in loops!!

(Testa et al. ’02)

⇒

temperature higher at tops of loops!

(cfr. Yohkoh results)

Alternative configurations 11-37

Alternative configurations

B

₀

B

₀

B

₀

Y

X

Z

X

Z

X

Z

(a)

(b)

(c)

Y

Y



cfr. MHD models: astrophysical plasmas

model V model IV a θ a pl. line tying plasma line tying b pl. plasma line tying b θ c model VI φ

((a) Model IV: Closed coronal magnetic loop, line-tied at both ends; (b) Model V: Open coronal magnetic loop, line-tied at one end and flaring on the other; (c) Model VI: Stellar wind outflow

Alternative configurations 11-39

Foot point driving

•

two boundaries at

z

= 0

and

z

= L

• B0

= B

0

e

z,

ρ

0

= ρ

0

(x)

⇒ ω

d

= ω

A

(x

s

) = k

z

B

0

/



μ

₀

ρ

₀

(x

s

)

•

steady state:

ξ

_y

(x, z, t) = A(x) e

i(kzz+ωdt)

⇒

real driving frequency (non-ideal!)

•

simplification: consider perturbations polarized in

y

-direction!

⇒ k

y

B

1y

= 0

so that

k

⊥

= k

y

= 0

⇒

classic resonant absorption does not work!

(AWs and FMWs decoupled)

0 Z = L Z = 0 Z Y X

ω

_d A

ω

X X₀

B

•

cfr. the 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 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎝ ξ_x 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

⇒

visco-resistive dynamic equation for the AW mode:

∂

2

ξ

_y

∂t

2

−

(˜η + ν)

∂

2

∂x

2

− k

2 z



∂ξ

_y

∂t

+ ω

2 A

(x) ξ

y

=

F

(x)

e

i(k_zz−ω_dt)

Alternative configurations 11-41

⇒

visco-resistive dynamic equation for the AW mode:

∂

2

ξ

y

∂t

2

−

(˜η + ν)

∂

2

∂x

2

− k

2 z



∂ξ

y

∂t

+ ω

2 A

(x) ξ

y

=

F

(x)

e

i(k_zz−ω_dt)

with external harmonic driving term

F

with frequency

ω

d (steady state)

•

characteristic time and length scales

:

– Taylor expansion:

ω

_d2

− ω

_A2

= (x − x

s

)[

d

(ω

_d2

− ω

_A2

)

dx

]

x=xs – drop

k

_z2-terms (

|k

_z

| = |∂/∂z| |∂/∂x|

)

⇒ (x − x

s

) 2 ω

A

(x

s

)

dω

A

dx

(x

s

) ξ

y

−

iω

d

(˜η + ν)

∂

2

ξ

y

∂x

2

= F (x, z) e

iω_dt

⇒

dissipative term comparable with first term when

l

₀

∼ |

˜η + ν

2 [(ω

A

)



]

x=x_s

|

1/3 τdis=l

=⇒

02/(˜η+ν)

Phase mixing

•

‘realistic’ driver has broad band spectrum

⇒

each field line picks up its own frequency from the broad spectrum

⇒

solution

∼ ξ

y

(x, z, t) = A(x) e

i(kzz+ωA(x)t)

⇒ k

x,eff

= (ω

A

)



t

⇒

‘cascade’ of energy to small scales

=

phase mixing

ξ x ξ x x x x x ξ

⇒

essential ingredient of resonant wave heating

•

dissipation when   

k

_x,eff

=

_l1 0

⇒ τ

mix

=

1 l₀(ω_A)

∼ (˜η + ν)

−1/3

[(ω

A

)



]

−2/3

Phase mixing 11-43   

Phase mixing

•

does not need resonances

⇒

also occurs in coronal holes!

⇒

running waves:

∼ ξ

_y

(x, z, t) ∼ e

i(kz(x)z+ωdt)

⇒ k

x,eff

∼ k

z

z

•

dissipation when

k

x,eff

=

1

l

₀

⇒

   

z

mix

=

1

l

₀

k

_z

•

numerous studies

: linear, numerical and analytical

Heyvaerts & Priest ’83 (weak damping, strong phase mixing

Nocera et al. ’84; Browning & Priest ’84; Ireland ’96; Cally ’91; Ofman et al. ’95; Hood et al. ’97; Poedts et al. ’97; Nakariakov et al. ’97; Ruderman et al. ’98, ’99; etc. etc.

•

De Moortel (2000): included effects of stratification & field divergence

– total heat deposited not affected by stratification

– heat deposited higher unless phase mixing is strong (in strat. atmosph.) – efficiency depends on geometry, scale height, and wave amplitudes



Side-ways excitation of coronal loops

•

small length scales due to ‘resonances’

⇒

resonant ‘absorption’

•

time scale:

τ

D

=

μ

l

2

η

+ ν

l

= Δ ∼

η

+ ν

ω

_A



1 3

⇒ τ

RD

∼ μ(η + ν)

− 1 3

ω

 A −2 3 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 solution with artificial damping

⇒

more quantitative results require more realistic models including dissipation, nonlin-earity, geometry, density and magnetic field stratification, . . .

Alternative configurations and applications 11-45

 



Side-ways excitation of coronal loops

⇒ η = 0 ⇒

nearly resonant behavior



Side-ways excitation of coronal loops

⇒

efficiency

⇒

‘fractional absorption’,

f

a

=

dissipated energy total energy

Alternative configurations and applications 11-47

•

‘collective mode’ or ‘quasi-mode’ with

Re

(ω) = 0.191

⇒

yields

perfect coupling

, i.e.

100% absorption

(dissipation)

•

fast magnetosonic waves needed to bring energy into the loop, across the magnetic flux surfaces

⇒

efficient coupling to global modes

⇒

‘quasi-modes’ play crucial role in



Energetics

Time-averaged power (curve a), change of kinetic energy (curve b), change of potential plasma energy (curve c), Ohmic dissipation rate (curve d), and change of vacuum magnetic energy (curve e) versus number of driving periods for

ω

_d

= 0.205

and

η

= 10

−6.

Alternative configurations and applications 11-49

 



Foot point excitation of coronal loops

•

Strauss & Lawson (’89): effect of ‘anchoring’ (‘line-tying’) and foot point excitation

•

Goedbloed & Halberstadt (’94, ’95): – AWs and FMWs are coupled! – AW continuum:    

ω

A

(r) =

nπ

L

B

_z

(r)



μρ

(r)

⇒

studied rigorously: linear and nonlinear studies, numerical and analytical, steady state approach, eigenvalue approach and time evolution

(Poedts et al. ’89–’96; Ofman et al. ’95, ’96, ’98; Berghmans & Tirry ’96; Tirry & Berghmans, ’97; Beli ¨en et al., ’97; Berghmans & De Bruyne, ’96; Erd ´elyi et al., ’96, ’97; Beli ¨en, Martens, and Keppens,’99; De Groof et al., ’99, 2000; etc. etc.)



Foot point excitation of coronal loops

•

Poedts & Boynton (’96): 2.5D simulations, nonlinear

⇒

very efficient heating even without quasi-mode!

⇒

acceptable time scales

⇒

Poynting fluxes can compensate radiative and conductive losses

(Poedts & Boynton ’96) 0 2 0 4 0 6 0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 T ime a a a a a a b b b b b b c c c _c c c d d d d d d e e e e e a: Poynting flux b: total energy = c + d + e c: kinetic energy d: magnetic energy e: thermal energy

Alternative configurations and applications 11-51

 



Foot point excitation of coronal loops

•

Poedts & Boynton (’96): 2.5D simulations, nonlinear

⇒

very efficient heating even without quasi-mode!

⇒

acceptable time scales

⇒

Poynting fluxes can compensate radiative and conductive losses

•

Ofman et al. (’95, ’96): 3D simulations, nonlinear

⇒

3D Kelvin-Helmholtz like vortices at resonance layers

⇒

heating rate drops

•

coupling to chromosphere is important:

⇒

Berghmans & De Bruyne (’96): leakage

⇒

Ofman et al. (’98) : chromospheric evaporation

⇒

tuning/detuning

Beli ¨en, et al. (’96, ’99):

⇒

include loop expansion (

B)



⇒ βfeet

>

1, β

_top

≈ 0.01

⇒

include density stratification (

g)

(not both at the same time. . . )

Δ g g L e_z e_r e_θ

⇒

drastic effect on ideal MHD continua:

•

two times more continuum branches (degeneracies lifted)

•

resonant Alfv ´en frequencies: order of magnitude lower!

⇒

efficient generation of SMWs, input energy does not reach corona (only 30% AWs)

⇒

heating much less efficient (but: unrealistic monoperiodic driver!?)

Alternative configurations and applications 11-53





More realistic

radial

drivers

(De Groof & Goossens, 2000)

0.2 0.4 0.6 0.8 1 200 600 1000 1400 t30 x E_A 0.2 0.4 0.6 0.8 1 100 300 500 700 t20 x EA 0.2 0.4 0.6 0.8 1 20 60 100 140 t10 x EA 0.2 0.4 0.6 0.8 1 1000 2000 3000 4000 t50 x EA 0.2 0.4 0.6 0.8 1 500 1500 2500 3500 t40 x E_A

Energy stored in Alfv ´en modes (from De Groof et al., 2000)

•

linear, ideal MHD

•

random footpoint driver, radial

•

short loop

⇒

only 5 quasi-modes

•

k

y

= 0 ⇒

FMW & AW coupled

⇒

energy is transformed into AW en-ergy

⇒

global heating: entire loop vol-ume!

More realistic

azimuthal

drivers

(De Groof & Goossens, 2001) 0.2 0.4 0.6 0.8 1 X 0.25 0.5 0.75 1 Z -4 -2 0 2 4 0.2 0.4 0.6 0.8 X 0.2 0.4 0.6 0.8 1 X 0.25 0.5 0.75 1 Z -5 0 5 0.2 0.4 0.6 0.8 X

3D and contour plot of ξ_y at t = 20 and t= 50 after a random pulse train (k_y = 2)

•

random (broadband) driver polarized in

y

−

direction

Alternative configurations and applications 11-55

 



p

-mode absorption by sunspots

Observational data

•

large sunspots act as strong absorbers of

p-mode wave energy

(Braun, Duvall and Labonte ’87, ’88)

⇒

as much as 50% of the incident acoustic wave power can be lost

⇒

opens up the new avenue of

sunspot seismology

(Bogdan ’95)

Resonant absorption of p-modes

•

static cylindrically symmetric magnetic flux tube model, radial symmetry (Lou ’90)

• α

2: amplitude of the incoming wave,

α

1: amplitude of the outgoing wave

⇒

absorption coefficient:

α

≡

α

2

2

− α

12

α

2₂

Absorption coefficient α as a function of the wave number of the acoustic oscillation for a sunspot with a straight magnetic field and radiusR = 4.2 × 106m

Alternative configurations and applications 11-57

Absorption coefficient α as a function of the radius of a sunspot with a straight magnetic field for B_z(0) = 0.2 T, ω= 0.02rad s−1, K = 1 × 10−6m−1, and m= 1, 2, 3, 5



Applications in the magnetosphere

•

Stewart (1861!) observed ULF oscillations of magnetic fields on the surface of the Earth

⇒

caused by ULF waves (peri-ods of seconds to minutes)

⇒

standing AWs with fixed ends in the ionosphere

⇒

‘box’ model: straightened magnetic field lines (Kivelson and Southwood ’86)

⇒

resonant field line model with the solar wind as (sideways) external driver is very successful in explaining many important properties (Kivelson and Russell ’95)