MHD Waves in the Solar Atmosphere

MHD Waves in the Solar Atmosphere Ineke De Moortel School of Mathematics & Statistics University of St Andrews

Overview 1. Some Observational Examples 2. Propagating Transverse Waves 3. Global Standing Transverse Waves 4. Global Standing Transverse Waves - Seismology 5. Wave Heating (?) 6. Waves & Flows

Sausage Oscillations in Photospheric Pores Morton et al (2011) Intensity Pore size Ø Oscillations in Photospheric Magnetic Pores (ROSA 4170A line) Use Empirical Mode Decomposition to separate out timescales. Periods: 30 450 seconds Sausage modes: oscillations in pore size and intensity are 180 degrees out of phase. Wave energy: E = 108 ergs cm-2 s-1 à ~10% transmission coeff needed?

Transverse & Compressive Waves in the Chromosphere Morton et al (2012) Ø Concurrent observations of (on disk) compressible and incompressible wave modes (ROSA). Transverse motions fast kink wave Periodic changes in intensity & cross section fast MHD sausage mode Incompressible energy ~ 4300 ± 2700 W m -2 Compressible energy ~ 11700 ± 3800 W m -2 Assume 4-5% connected to corona Incompressible energy ~ 170 ± 110 W m -2 Compressible energy ~ 460 ± 150 W m -2 Intensity Width

Alfvénic Waves in the Chromosphere De Pontieu et al (2007, 2012) Swaying spicules everywhere (Hinode/SOT) Transverse motions ~ 500-1000km Periods ~ few minutes Chromospheric energy flux ~ 4-7 kw m -2 Coronal energy flux ~ 120 W m -2 (transmission coefficient ~ 3%) Sufficient to heat the Quiet Sun corona and/or drive the solar wind (~100 W m -2 )

Alfvén Waves in the Chromosphere Jess et al (2009) Ø Chromospheric bright point oscillations (SST) Periodic spectral line broadening; no intensity oscillations Interpreted as torsional Alfvén waves Chromosphere Chromospheric energy flux ~ 15,000 W m-2 1.6% of surface covered in Bright Points Global average ~ 240 W m-2 Transmission coefficient ~ 42% Coronal energy flux ~ 100 W m-2 Photosphere

Transverse Standing Waves in the Corona Aschwanden et al 2001

Propagating Transverse Waves in the Corona Alfvénic" motions everywhere (SDO/AIA) Amplitudes ~ 5-20 km/s Periods ~ 100 500 sec (lifetimes ~ 50-500 sec) Energy flux Quiet Sun & Coronal Holes ~ 100 200 W m -2 Active Region Loops ~ 100 W m -2 (2000 W m -2 needed) McIntosh et al 2011

Ubiquitous Propagating Transverse oscillations Tomczyk et al. (2007) and Tomczyk & McIntosh (2009)

Alfvén Waves in the Coronal Holes Ø Alfven waves in coronal holes (Hinode/EIS) Line widths from (double) Gaussian fits to EIS lines At larger heights, v nt < undamped waves à wave damping Wave energy at base of coronal hole: E = 6.7 ± 0.7 x10 5 ergs cm -2 s -1 Waves lose ~85% of energy by 1.44 R o Damping length L D =0.18 ± 0,04 R o and damping time = 68 ± 15 seconds Hahn & Savin 2013

Nobeyama Radioheliograph Observations Spectra at different parts of the loop P ~ 14 17 sec NoRH + Yohkoh periodicity stronger at loop apex Ø global standing mode Nakariakov et al (2003)

MHD Waves in a Uniform Medium Wave Driving forces Speed Compressible Energy Alfvén Magnetic tension v A no // to magnetic field Slow magnetoacoustic Plasma & magnetic pressure out of phase c T yes Essentially // to magnetic field Fast magnetoacoustic Plasma & magnetic pressure in phase c F yes Isotropic (fastest to magnetic field) Alfvén speed: v A 2 = B2 µρ & Sound speed: c s 2 = γ p ρ ~ T Fast speed:c F 2 = c s 2 + v A 2 Tube speed: c T 2 = c 2 2 s v A c 2 s + v & Kink speed:c 2 2 K = ρ v 2 0 A0 A 2 + ρ e v Ae ρ 0 + ρ e

Coronal Loops Corona does not appear uniform filled with loop-like structures. Model loops as fluxtubes support different types of MHD waves. Coronal fluxtubes usually modelled as cylindrical density structures in uniform magnetic field

MHD Waves in Fluxtube Sausage Wave B > 0, r > 0 Kink Wave r» 0 Torsional (m=0) Alfvén Wave r = 0 V q Slow (longitudinal) Wave B > 0, r < 0 V z

Propagating Transverse Waves

Alfvén(ic) Waves in the Corona Ubiquitous quasi-periodic fluctuations in velocity but no fluctuations in intensity Interpretation as propagating Alfvén waves based on high phase speeds (~ 1 Mm/s), field-aligned, and very small intensity fluctuations (incompressible) Tomczyk et al 2007

Observations of Ubiquitous Waves (COMP) De Pontieu et al 2007; Okamoto et al 2007; Cirtain et al 2007; McIntosh et al 2011 Tomczyk et al. 2007; Tomczyk & McIntosh 2009 Disparity between outward and inward wave power (even along closed loops) suggests significant amplitude decay in situ Energy insufficient to account for heating? F W = 10 100 erg cm 2 s 1 vs 3 10 5 erg cm 2 s 1 needed for Quiet Sun

3D Numerical Model Field is straight, uniform and in the z direction Density profile describes a cylindrical tube aligned with the z axis and defines 3 regions (core, shell, environment)

3D Numerical Simulations flux tube Pascoe et al 2010 3D Loop Model core shell Kink mode propagating along coronal loop Driver: models buffeting by solar surface motions Mode conversion to Alfvén wave Damping of original kink mode

3D Numerical Simulations uniform Pascoe et al 2010 3D Loop Model Driver: models buffeting by solar surface motions

Uniform Alfvén Mode Still Alfvén Mode Flux tube Kink Mode Alfvénic (Azimuthal m=1) Alfvén Mode

Coupled kink-alfvén (Alfvénic) Mode Kink Alfvén (m=1) Z=0 Z=200

Mode Coupling

Randomly structured loop/loop strands Propagating transverse waves couple efficiently to Alfvén waves when the medium is non-uniform. Ø This is seen as a decay of the driven wave fields. In an inhomogeneous corona, any general footpoint shaking motion deposits energy in Alfvén modes. Pascoe et al 2011

Mode Coupling - Wave Energy total shell core Wave energy becomes increasingly localised in tube boundary. Damping in qualitative agreement with CoMP observation Damping length ~ 750 Mm Footpoint seperation ~ 500 Mm Observations: outward power dominates in loops with footpoint separation > 300 Mm Pascoe et al 2010, 2011, 2012; Terradas et al 2010; Verth et al 2010; Soler et al 2011a,b,c

Gaussian Damping Profile A( z) 2 2 = A0 exp( -z / L g ) L g» 52Mm A( z) = A0 exp( -z / Ld ) Ld» 25Mm Gaussian damping most pronounced at low heights i.e. when damping is strong - Wide inhomogeneous layers - Weaker density contrasts Hood et al 2013; Pascoe et al 2013

Mode Coupling Frequency Filtering Harmonic Driver Broadband Driver t = C P a l r0 + re r - r 0 e Distance along Loop t ~ P & Ld = Vgt Þ Ld ~ P Ø Frequency dependence evident in CoMP data Time Goossens et al 1992,2002; Ruderman & Roberts 2002;Verth et al 2010; Terradas et al 2010; Pascoe et al 2010, 2012

Application to CoMP data? Exponential decay Gaussian decay Original CoMP Alfvenic wave observations do not allow us to distinguish the two profiles. Verth et al 2010; Pascoe et al 2015

What if there was more than one oscillating loop in the corona? Construct 10 randomly distributed loops with random density contrasts (2.5 4) Use the same footpoint driver in each loop random directions and periods (P= 13.5 16.5) How does LOS integration affect the visible density (intensity) structure and LOS velocity components De Moortel & Pascoe 2012

Horizontal Velocity Perturbations z=1 z=165 Randomly directed driver clearly visible at bottom boundary Loops have different density contrasts and driver periods: Mixture of azimuthal Alfvén waves and transverse (Alfvénic/kink) modes at higher height De Moortel & Pascoe 2012

LOS (Density) 2 integration Y Integration along x: only 4 loops visible X Integration along y: only 1 broad loop and possibly one very weak loop visible (appears nearly uniform!) Density contrasts strongly reduced Density contrasts derived through coronal seismology might not agree well with LOS integrated observed density contrasts De Moortel & Pascoe 2012

Integrated (weighted) LOS velocities (Doppler) De Moortel & Pascoe 2012

Integrated (weighted) LOS velocities (Doppler) Driven kink mode generally corresponds to bulk motion (periodic Doppler shifts) but cannot tell with which loops the oscillations are associated Doppler velocities much smaller than actual perturbations in domain y-los: strong oscillation which does not line up with a loop Alfvén wave (m=1) appears as bulk motion in some locations (but very low amplitude in this simulation) De Moortel & Pascoe 2012

LOS Energy vs Dark Energy? Compare kinetic energy integrated over 3D domain with kinetic energy derived from LOS velocities (and LOS densities) Only between 7 20% of energy visible Footpoint driver also contains magnetic energy Visible (kinetic) energy only 3 10% of total energy in 3D domain (kinetic + magnetic) De Moortel & Pascoe 2012

Observing Dark Energy Ø Apparent discrepancy between CoMP velocities (~0.5 km/s) and Hinode & SDO (~20 km/s) Monte Carlo model of Alfvenic waves based on threads ( elementary oscillating structures ) Vary number of threads and input wave amplitude determine v RMS and line broadening Real CoMP wave amplitudes estimated as 40-60 km/s Most of the CoMP non-thermal line broadening < LOS superposition of low frequency waves McIntosh & De Pontieu 2012

Global Standing Transverse Waves

wave amplitude Resonant Absorption & Phase Mixing Ionson 1978, Heyvaerts & Priest 1983 + Hollweg,, Goossesns, Sakurai Density Flux tube Transverse MHD wave Phase speed Alfvén wave speed Transverse MHD wave speed Alfvén wave Resonance between Alfvén & transverse MHD wave Pascoe et al 2010, Verth et al 2010, Terradas et al 2010, Pascoe et al 2011, 2013, 2015

Hinode & IRIS observations - Resonant Absorption Okamoto et al 2015, Antolin et al 2015 x-t diagram Hinode/SOT Ca II, 10,000 K) 104 K 105 K 1,000 10 500 5 0 0-500 -5-1,000 Doppler velocity from IRIS [km/s] Displacement [km] 200 s Motion of prominence plasma crossing the slit -10 0 100 200 300 400 Time [sec] IRIS/SJI (Si IV, 100,000) Heating: fading in cool line (104 K), subsequent appearance in hot line (105 K) POS motion out-of-phase with LOS velocity Thread-like structure

Hinode & IRIS observations - Resonant Absorption Okamoto et al 2015, Antolin et al 2015 Prominence thread - Mg II k intensity Displacement [km] 2,000 1,000 0-1,000-2,000 Numerical results Motion of prominence plasma crossing the slit Synthetic IRIS Doppler signal [km/s] 3D MHD transverse wave model: resonant absorption + Kelvin-Helmholtz instability Cross-section 0 200 400 600 800 Time [sec] Observational results Motion of prominence plasma crossing the slit Displacement [km] 1,000 500 0-500 -1,000 0 100 200 300 400 Time [sec] 10 5 0-5 -10 Doppler velocity from IRIS [km/s]

Observable signatures in the corona SDO/AIA Observations Nisticò+(2013), Anfinogentov+ (2013, 2015), Goddard+(2015) Coronal loop Apparently decay-less Strand-like structure Broad arrow-shaped Doppler maps Broad line widths at edges (Antolin+2016, submitted)

Global Standing Transverse Waves - Seismology

TRACE 14 Jul 1998 Aschwanden et al (1999); Nakariakov et al (1999); Ofman & Nakariakov (2001); Schrijver et al (2002); Aschwanden et al (2002); Verwichte et al (2004);.

Time Series 14 July 1998

Time Series 14 July 1998 P ~ 256 sec Choose cut perpendicular to loop axis Maximum intensity along cut = loop Track loop for each frame

Interpretation P=256 seconds and looplength L = 1.3 x 10 5 km Assume the observed wave is the basic harmonic : l= 2L The speed is given by: v obs l 2 L = = =1020 km/s P P Transversal loop oscillation Propagation speed > Alfvénspeed Ø (Standing) fast magneto-acoustic kink waves Excitation? Damping? Aschwanden et al (1999); Nakariakov et al (1999); Ofman & Nakariakov (2001); Schrijver et al (2002); Aschwanden et al (2002); Verwichte et al (2004);.

Magnetic Field Strength Try to use the observed oscillations to determine local plasma properties: Observed wave interpreted as fast kink wave: æ v obs» v A ç è1+ 2 / r r e 0 ö ø 1/ 2 = 1020 km/s From the definition of the Alfven speed: B 0 = 2 ( µr ) 1/ = ( µr ) 0 v A v obs 0 1/ 2 æ 2 ö ç 1 / è + re r0 ø -1/ 2 Assume r 0 = 1 x 10-14 1.5 x 10-15 g cm -3 and r e /r 0 = 0.1 è B 0 = 4 30 G Nakariakov & Ofman (2001)

Estimate of Uncertainty!!Nakariakov!&!Ofman!(2001):!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!δB (δl) 2 +(δp) 2 +(δρ 0 / 2) 2!!!where!!δL 10%,δP 3%!and!δρ 0 50%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! δb 30%

Kink Oscillations observed by EIS P=296 s c k = B µm m m p (n in + n ex ) / 2 L=390 Mm P=296 sec c K =v ph =2L/P=2600 ± 500 km/s n in = (9.8 ± 0.3) x 10 14 m -3 n ex = (8.1 ± 0.2) x 10 14 m -3 B=39 ± 8 G Van Doorsselaere et al 2008

STEREO Observations of Loop Oscillations Use 3D information to find loop geometry and correct plane of oscillation.» Appears parallel to loop!» L = 340 ± 15 Mm Mode of oscillation identified as horizontal kink mode Verwichte et al 2009

STEREO Loop Oscillations Loop Oscillation Properties P=630 ± 30 sec t=1000 ± 300 sec Displacement ampl = 4Mm P A =642±29 s v ph =2L/P=1100±100 km/s c K = 2 2 va = 1+ r / r µ ( r + e i 0 i re B ) Magnetic Field Estimate P B =627±32 s Density ratio estimated to be ~ 0.1 0.3 c K =v ph =1100 ± 100 km/s v A = 800 ± 80 km/s n i =(6.7±0.5) x 10 14 m -3 Ø B=11±2 G Verwichte et al 2009

Model Setup 3D Numerical domain with arcade magnetic field B ~ 1/r and b ~ 0.1 at loop apex Modified Epstein profile density r = r + ( r r e æ r - r ç è a 2 c 2 0 - e)sech sech Pressure pulse (d = 40 Mm): dp/p = 1000 De Moortel & Pascoe 2009; Pascoe & De Moortel 2014 ö ø p æ ç è y a ö ø p

Modelling Coronal Loop Oscillations Pressure pulse propagates through numerical domain Excitation of global, transverse oscillation of coronal loop

Global Kink Mode Impulsively generated fast wave: initial, aperiodic, transitory phase followed by MHD mode (damped harmonic oscillation). Global mode: amplitude of the oscillation is zero at the footpoints and has single maximum at the apex De Moortel & Pascoe 2009; Pascoe & De Moortel 2014

Loop Displacement

Kink Mode Period

Estimate of Magnetic Field Strength Ø Difference in predicted magnetic field strength of the order of 50% De Moortel & Pascoe 2009; Pascoe & De Moortel 2014

Estimate of Uncertainty!!Nakariakov!&!Ofman!(2001):!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!δB (δl) 2 +(δp) 2 +(δρ 0 / 2) 2!!!where!!δL 10%,δP 3%!and!δρ 0 50%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! δb 30%!!With!δP 45%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! δb 50% Ø Difference in predicted magnetic field strength of the order of 50%

Estimate of Magnetic Field Strength: 2014 Update, a secondary mode with a period determined by the loop length and external Alfven speed is also present. When a low number of oscillations is considered, these two periods can result in a single, non-resolved (broad) peak in the power spectrum, [ ]. [ ] the presence of this additional mode would lead to ambiguous seismological estimates of the magnetic field strength. Pascoe & De Moortel 2014

Wave Heating (?)

Wave Energy vs Wave Heating Recent observations suggest a substantial amount of wave power is available. It is not clear if/how/when these waves contribute to heating: Sufficient flux (right) heating (1) Damping Dissipation e.g. mode coupling (e.g. Lee & Roberts 1986) (2) Timing? (dissipation time?) Phase mixing and resonant absorption: Require a transverse density structure. Self-consistent? (3) Is this sustainable by wave heating? ρ(x) (4) Does the wave heating change the density profile?

Alfven Wave Phase Mixing Shear Alfven waves become quickly out of phase as they propagate along the field lines with large (perpendicular) gradients in the Alfven speed Small length scales are generated dissipation is enhanced. Eventually, all the wave energy is dissipated.

Alfven Wave Phase Mixing Shear Alfven waves become quickly out of phase as they propagate along the field lines with large (perpendicular) gradients in the Alfven speed Small length scales are generated dissipation is enhanced. Eventually, all the wave energy is dissipated. ~exp(-z 3 )

Coronal Heating - Resonant Absorption Ionson 1978, Goossens et al 1992, Ruderman & Roberts, 2002 t R = 2 a (ρ i / ρ e +1) π l (ρ i / ρ e 1) t W a = radius of loop l= width of the layer t w = period of the global mode

(2) Relevant time and spatial scales for wave energy transfer/heating

Coronal Parameters Cargill et al (2016)

(3) Can wave heating sustain the required density profiles? Consider steady state with a radiative loss function L(T)=cT a : d " ds κ T 5/2 T % $ ' = n 2 χt α H 0 # s & Scaling laws: T 7/2 ~H L 2, nl~t (7-2a)/4, n ~H (7-2a)/14 L -2a/7 Ø High density region requires more heating. Ø For both phasemixing and resonant absorption, heating is situated near the maximum density gradient! n(x) Required heating Wave heating Cargill et al (2016)

Assume Example Phasemixing Density Profiles Uniform background magnetic field (along z) A density gradient in x-direction: n(x) = n " 0 # 1+(f / 2)(1+ tanh{(x x 1 ) / x sc } $ % Perturbations b y and v y Damping due to phasemixing: v y (x,t) = v y (x,0)exp" # (t / t P (x)) 3 $ % Characteristic damping time: t P = 3 & 6 /( ν" # dω / dx$ ' % Ø t p of the order of thousands of seconds. 2 ) + * n = n m + n v (resistive + viscous damping coefficients) W = pqv A (x)/2l (q=1: global mode) Cargill et al (2016)

Example Phasemixing Density Profiles Damping due to phasemixing & scaling laws à density profile due to heating by phasemixing Assumed density profile Density profile from heating by phasemixing The imposed density profile is incompatible with the density implied by the wave heating. Cargill et al (2016)

Temporal Evolution no heating Consider ρ(x) as before, T constant à temporal evolution of density profile as loop cools n(x) Time (100s) Time T(x) Period of the global mode Damping time Max density gradient Relative location of resonance point High density region cools rapidly à location of the resonance point Cargill et al (2016)

Temporal Evolution no heating Consider r(x) and T(x) related through n L~T (7-2a)/4 n(x) T(x) Time Time Period of the global mode Damping time Max density gradient Relative location of resonance point Density profile still present à location of the resonance point remains the same Cargill et al (2016)

Temporal Evolution no heating Resonant absorption static equilibrium: Energy continues to enter fixed resonant layer of cooling loop Resonant absorption other initial state (constant T): Resonant layer moves à can wave damping still occur? Phase mixing: Density gradient decreases: damping becomes weaker/slower damping time ~ (density gradient) 2/3 To maintain fixed (background) density profile needed for wave heating mechanisms: Need dissipation time << draining time whereτ drain = n dn 1! $ 2kTγ # & = ~ 1000sec " dt % ( (γ 1)nRL (T)* ) + Phasemixing: need very large transport coefficients Resonant absorption: perhaps ok for short loops/strong B

Summary & References Observations: waves are present beyond doubt in a wide range of structures in all layers of the solar atmosphere. Seismology is highly dependent on the mode identification and the robustness (applicability) of the underlying model. Mode identification is not straightforward in highly structured and dynamic atmosphere. Optically thin nature of the corona adds to the confusion. Wave heating: has come full circle and is now back at the forefront of the coronal heating debate. Vast amount of literature! Damping Dissipation Ø Can heating be delivered in the right locations and on the right timescales by waves? Some review papers: Nakariakov & Verwichte (2005) Living Reviews in Solar Physics Space Science Review (2009) Space Science Review (2010/2011) De Moortel & Nakariakov (2012) + Parnell & De Moortel (2012) Arregui (2015)