NSO Summer SchoolLecture 1:

Solar Wind Structure and Waves

Charles W. SmithSpace Science Center

University of New HampshireCharles.Smith@unh.edu

Good References:• Space Physics – May-Britt Kallenrode, Springer 2001.

(~ $70)

• Turbulence -- Uriel Frisch, Cambridge University Press 1995. (~ $35 to $55)

• Magnetohydrodynamic Turbulence – Dieter Biskamp, Cambridge 2003. (~ $110)

• The Solar Wind as a Turbulence Laboratory -- Bruno and Carbone, on line @ Living Reviews in Solar Physics (free)

Origin of the Solar Wind

Model drawn from many remote sensing methods:

Note rapid rise in temperature in so-called “transition region” around 2000 km about photosphere.

Source of heat is not known, although widely speculated upon.

Solar Wind Acceleration

Scatter plots of velocity as a function of distance for the emitted plasma puffs and solid line showing best fits to radial profile. These plots show the expected outflow that Solar Probe will encounter during its perihelion pass (Sheeley et al., 1997).

Heating the Solar Wind

1 AU ConditionsSolar wind speed, VSW, varies from 250 to 800 km/s.

(REarth = 6371 km, an 8 second transit at high wind speed.)

At least 2 transients have been observed at 2000 km/s at 1 AU!

Proton density varies, but averages about 10 p+/cm3.

We are unable to achieve these densities on the surface of the Earth.

Proton temperature averages about 105 K. (Vth ~ 50 km/s for ions.)

Magnetic clouds are colder (1/10’th), high speed streams are colder…

Magnetic field averages 6 to 8 nT and 45° from radial direction.

(BEarth = 5.7 x 104 nT at Washington DC.)

Can be just about anything at any time.

Have reached 55 nT during ACE era and << 1 nT.

Heavy ions (He++, He+, Fe?+, C?+, O?+, etc.) provide clues to origins.

Solar Wind Composition• About 96% is p+ and about 4% is He++.• The rest is:

Detailedcompositionmeasurementsattempt torefine physicsof acceleration.

Energy Output of the Sun

VSW ~ 300 to 800 km/s flowing away from sun

(averages ~1,000,000 miles/hr)

Stays ~ constant as it flows from the sun.

Tp ~ 4 x 104 to 2 x 105 K (Vpth ~ 50 km/s ~ Vs)

Np ~ 3 to 10 p+/cm3

Decreases ~ 1/R2 from sun.

|B| ~ 3 to 10 nT (1 nT = 10-5 G, BEarth = 0.5 G)

Decreases ~ 1/R from sun.

Solar output in light:

The solar constant is 135.3 mWatts/cm2 which corresponds to a radiant solar energy of ~1.4 GigaWatts/km2 (1.4 x 109 Watts/km2).

Multiplying by area of sphere 1 AU = 1.49 x 108 km in radius (4R2) gives solar output of light energy at 3.9 x 1026 Watts.

For comparison:

10 protons/cm3 traveling at 450 km/s has an energy flux density of…

(1/2) (MP NP VSW) VSW2 = 7.6 x 102 Watts/km2

Integrating over a 1 AU sphere, the sun’s energy production in the

solar wind is 2.1 x 1020 Watts.

Solar Activity 1992-1999

The Sun through the Eyes of SOHO

SOHO Project Scientist Team

Coronal mass ejections are plasma eruptions from the sun that are fast-moving.

They are predictable, but only approximately.

Some are self-contained magnetic structures called magnetic clouds that are observed to be cold at 1 AU.

Charge-state composition data strongly indicates they originate in hot electron environments.

Overexpansion is largely responsible for observed temperature at 1 AU.

In situ heating rate can be calculated from spectra.

These are Not Rare Events

Earth’s Magnetosphere

CME Interacting with Earth

Transient Arrival of

Bastille Day 2000

Over a week of extensive solar activity 3 shocks and associated ejecta arrived at Earth.

Each resulted in energetic particle enhancements to varying degrees.

Each resulted in magneto-spheric storm activity to varying degrees.

Shock Acceleration

Day 49, 1999 Upstream Waves“Classic” upstream wave spectrum for older, well-developed event (as expected).

Broad spectrum of upstream waves.

Not much polarization.

No real surprises here.

Upstream spectrum from prior to the onset of the SEP and ESP.

Power law form has no net polarization and there is a spectral break to form the dissipation range.

This is a typical undisturbed solar wind interval.

f5/3

Energetic Particle Populations Upstream of the Earth’s Bow Shock

Field-aligned beamSpecularly reflected ions

Gyrophase-bunched

Gyrotropic distributionIntermediate

distribution

Diffuse distribution

Quasi-Perpendicular

Quasi-Parallel

Diffuse ions

FAB

Ion

Fore

shoc

k B

ound

ary

B

GyrophaseBunched

I

Electrons

Earth

Proton Distribution Functions

Marsch, Phys. Inner Heliosphere 2, Chapter 8 (1991)

“Among the most typical features are…pronounced total temperature anisotropies with TP > TP|| and anisotropic cores in high-speed [winds] and… TP|| > TP and isotropic cores in low-speed [winds].”

Formalisms of Plasma Physics

• We can build a mathematical formalism for the theory of freely-interacting charged particles– Start with Maxwell’s equations governing E&M– Add the Boltzman equation from statistical

mechanics– Obtain the Maxwell-Vlasov equations of plasma

physics

• We can “simplify” this to a set of fluid equations– Magneto-fluid equations called the MHD equations– Generally applicable to low-frequency phenomena

Vlasov Equation

(Collisionless Boltzman’s Equation)

ss

ss

ss

s

ss

s

fdqt

fdqt

fcm

qf

t

tftc

tc

t

tct

t

t

vvxJ

vx

Bv

Evvx

ExJxB

BxE

xB

xE

),(

),(

0),,(

1),(

4),(

1),(

0),(

4),(

}

} Maxwell’s Equations governing Electro-Magnetic Fields

zyx

zyx

Charge density, currents, etc. are computed from the distribution of particles for all species

0BJB

BBVB

VV

VBBVVV

and 4

4

0or 0

4

1)(

22

2

c

c

t

t

Pt

MM

M

Let’s Do 5 Things Today:

1. Examine global structure of heliosphere

2. Examine properties of fluctuations

3. Compute wave solutions to MHD

4. Ask: Can waves heat the solar wind?• Nope! They can’t without something more.

5. Begin to re-examine properties of fluctuations

1. Global StructureWind speed is independent of heliocentric distance.

Density ~ R2

IMF wound in a spiral

|B| ~ R2 inside 2 AU

~ R1 outside 2 AU

VA = (B02/4NPmP)1/2

~ constant

P = eB0/mPc ~ |B|

1b. VSW is not Function of R

Richardson et al., GRL, 22, 325, 1995.

Density multiplied by R2.

1c. NP Varies as R2

2-2 R ~Density so R ~ surface

constant depth

depth x surface ~ Volume

Volumenumber / total~Density

A1

A2 =

A1 (R2/R1)2

1d. Adiabatic Expansion

s-Joules/kg 106.3 that so

2/33

4

5SWP

BSW

PPP

VTx

kV

m

R

T

dR

dT

4/3P

PP

R~T that so

0R

T

3

4

dR

dT

Vasquez et al., J. Geophys. Res., 112, A07101 (2007).

1e. Radial IMF

2

2

1RR

2R1R

3

R

R1B 2B so

AB AB

xd 0B

Flux of IMF through closed surfaces surrounding the sun is constant.

1f. Azimuthal IMF

RRS

2

0R0

1R

1RS

20R0

1RR0

20

220R0

S00

R000RR

RR

RR

/VBR RR

RBV B

0 /RV :const.RV

momentum,angular by determined is V

RV /RVRB B

RV BR R/RBRV B

R V rotation solar

BVR BVBVR

const. BVBVR

0 BVBVRR

1

0 BV

S-

S

R

Note error in Kallenrode book!

1g. IMF Intensity and Direction

2

SW2

20

022

RR

SWRS

2

0R0R

V

R1

R

RBBB |B| and R ~

B

B

/VBRB and R

RB B

S

Solar Wind Structure (Magnetic)Expanding solar wind draws sun’s magnetic field outward as sun rotates underneath to create a spiral interplanetary magnetic field (IMF).

IMF winding results from a “connect-the-dots” approach to plasma elements moving outward and sun rotating.

sin)/)(/(),,(

,0),,( ,)/(),,(

110

21

0

rrVrBrB

rBrrBrB

SWSR

RR

The Solar Wind Continues Flowing Outward, Away from the Sun

• Studied by Pioneers 10 & 11– Had enough power to function to 35 AU

• Studied by Voyagers 1 & 2– Still operating at ~100 AU– Recall crossing of termination region crossing

• To a lesser degree:– Cassini, Galileo, and other planetary missions

Solar Wind Properties

• At 1 AU:– VSW ~ 300-800 km/s

– Np ~ 10 p+/cm3

– Tp ~ 105 K

– |B| ~ 6 nT BR ~ 45

• At 100 AU:– VSW ~ 450 km/s

– Np ~ 103 p+/cm3

– Tp ~ 104 K

– |B| ~ 0.06 nT BR ~ 90

By 100 AU: Streams merge, gradients are smoothed, & few shocks remain,

Fluctuations are reduced, but persist,

Interstellar neutrals / pickup ions dominate pressure,

Thermal composition is unchanged.

2a. Properties of Fluctuations

• Low-frequency fluctuations tend to be:– Transverse to the mean magnetic field.

– Have low density compressions (/0~0.1).• Not generally well-correlated with fields.

– Correlated B and V fluctuations.• Indicative of outward propagation.

– Evolution of power consistent with WKB.– Reproducible power spectral form!

• Varying intensity.

Belcher & Davis, J. Geophys. Res., 76, 3534 (1971).

2b. Fluctuation Spectrum

Matthaeus et al., Phys. Res. Lett., 48, 1256 (1982).

The spectrum of magnetic fluctuations follows a predictable and reproducible form (power law with index ~ 5/3).

The magnetic helicity (polarization) is mixed.

2c. Fluctuation Spectrum

Podesta et al., J. Geophys. Res., 111, A10109, 2006.

M

0A

M

202

101

01

0110011

1

0110011

4

BV where

4

Bk

BBik4iBikBi

BVikVBikBVikBi

0Bik 0Vik

ikBBBBik4

1BBik

4

1Vi

M

MM

1

0BJB

BBVB

VV

VBBVVV

and 4

4

0or 0

4

1)(

22

2

c

c

t

t

Pt

MM

M

3a. Low-Frequency Waves

t-xki1

t-xki10

eV V

eBBB

Homework Problem:

• If you haven’t already, explore the MHD wave solutions that give Alfven and fast-mode waves in the fluid approximation.– Obtain dispersion relations, polarization,

fluctuation directions.

• If you haven’t already, review the kinetic theory descriptions of the same waves.– Obtain polarization and dissipation.

3b. Alfven and Fast-Mode Waves

• Alfven waves are:– Noncompressive– Transverse fluctuations– Right-hand polarized– Move energy parallel to

mean field– Decay via cyclotron

damping on ions– Decay via Landau

resonance (ions & electrons)

• Fast-Mode Waves:– Compressive for off-axis

propagation– Not transverse at off-axis

propagation– Left-hand polarized– Move energy across the

mean field for off-axis propagation

– Convert to whistler waves – Decay via cyclotron

damping on electrons– Decay via Landau

resonance (ions & electrons)

3c. So, Life is Good!

…or, is it?

4a. Can Waves Heat the Solar Wind?

In the range 0.3 < R < 1.0 AU, Helios observations demonstrate the following:

For VSW < 300 km/s, T ~ R -1.3 0.13

300 < VSW < 400 km/s, T ~ R -1.2 0.09

400 < VSW < 500 km/s, T ~ R -1.0 0.10

500 < VSW < 600 km/s, T ~ R -0.8 0.10

600 < VSW < 700 km/s, T ~ R -0.8 0.09

700 < VSW < 800 km/s, T ~ R -0.8 0.17

Why is this? It seems distinct from the high-latitude observations.

We can look to explain this through theory and link it to observations of the dissipation range and inferred spectral cascade rates.

Adiabatic expansion yields T ~ R-4/3.

Low speed wind expands without in situ heating!?

High speed wind is heated as it expands.

The slow wind continues to accelerate from 0.3 to 1 AU (due to in situ heating?), which complicated the analysis for years!

The temperature of all solar wind intervals 0.3 to 1 AU varies as R0.9!

All low-latitude winds are heated inside 1 AU!

Heating rate at 1 AU averages ~ 3 x 103 Joules/kg-s.

See Vasquez, J. Geophys. Res., 112, 2007.

4b. Can Waves Heat the Solar Wind?

Imagine that there is a spectrum of waves originating at the sun and propagating outward.

As the wave gets further from the sun, the mean magnetic field decreases and the cyclotron resonance shifts to lower frequencies (larger scales).

The spectrum of wave energy is slowly consumed from the high-frequency end.

Schwartz et al., JGR, 86, 541 (1981)

Homework Problem:

• Take the energy spectrum of solar wind fluctuations– Magnetic + Velocity

• Compute the rate of P 0 as R .• Compute the rate of energy input to the

solar wind by damping the static spectrum.• What is the local heating rate?• How long before the entire inertial range

energy is consumed?

Newborn Interstellar Pickup Ions

VSW

Interstellar neutral enter the heliosphere at small speeds, are ionized either by charged particle collision or photo-ionization, are “picked up” by the solar wind & IMF, and convected back to termination region.

In the process their circulation of the IMF provides “free energy” for exciting magnetic waves that can provide energy for heating the thermal plasma.

Waves Due to Pickup IonsTraditional plasma theory suggests that wave spectra due to pickup ions will reach LARGE enhancements in the background power levels (left, Lee & Ip, 1987).

They don’t! (below, Murphy et al., 1995)

Wave growth is slow and turbulent processes overwhelm the plasma kinetics of wave excitation (for once).

Pickup Ions Heat the Solar Wind

5a. Re-examine the Observations

• Are the fluctuations really waves at all?

• What is the orientation of k?– …and how did it get there?

• Is VB 0 indication of propagation?

• What happens when we add energy?– …where does it go?– …and how does it get there?

• What about that spectrum?

Summary• The heliosphere is vast, structured, and

containing a broad range of dynamics.– Filled with expanding, supersonic solar wind.– Transients and planetary interactions drive shocks

and particle acceleration.– Interstellar neutrals / ions provide new dynamics.

• Waves may explain (some of?) the fluctuations seen in the solar wind.– We will challenge this claim.

5b. Multi-D Correlation Fn.

Combining analyses of many intervals, adding correlation functions according to mean field direction, can lead to an understanding (statistically):

Matthaeus et al., J. Geophys. Res., 95, 20,673, 1990.

Bieber et al., J. Geophys. Res., 101, 2511, 1996 showed the 2D component dominates.

5c. Multi-D HC = VB

If there is a V and a B so that there is a VB, there must be sufficient energy to support VB.

Define C VB / EV+EB so that1 C +1

Milano et al., Phys. Rev. Lett., 93(15), 155005 (2004).

5d. Breaking Apart the Maltese Cross

Combining analyses of many intervals, adding correlation functions according to mean field direction, can lead to an understanding (statistically):

Matthaeus et al., J. Geophys. Res., 95, 20,673, 1990.

Bieber et al., J. Geophys. Res., 101, 2511, 1996 showed the 2D component dominates.

Dasso et al., ApJ, 635, L181-184 (2005).

Slow wind is 2D

Fast wind is 1D

5e. Decay of HC (Cross-Correlation)

HC VB / V2+B2

Strong correlations evident at 1 AU disappear by ~ 4 AU.

Roberts et al., J. Geophys. Res., 92(10), 11021, 1987.

6a. A New View• Coleman, Phys. Rev.

Lett., 17, 207 (1966)

Fluctuations in the wind and IMF are presumed to be waves, most likely Alfven waves, that are remnant signatures propagating out from the solar corona.

• Coleman, Astrophys. J., 153, 371 (1968)

Fluctuations arise in situ as result of large-scale interplanetary sources such as wind shear and evolve non-linearly in a manner analogous to traditional hydro-dynamic turbulence.

6b. A New View

0.5 Hz

If the inertial range is a pipeline, the dissipation range consumes the energy at the end of the process.

f -1 “energy containing range”

f -5/3

“inertial range”

f -3

“dissipation range”

Few hours

Mag

netic

Pow

er

6.1

re whe

3/53/2

K

Kk

C

kCP

= V3/L

This is the essence of hydrodynamic turbulence applied to MHD!

Inertial rangespectrum ~ -5/3

Spectral steepening with dissipation

Grant, Stewart, and Moilliet, J. Fluid Mech., 12, 241-268, 1962.

Spectral steepening with dissipation

Inertial range spectrum ~ 5/3

Ion Inertial Scale

6c. A New View• Complications:

– MHD is not hydrodynamics!– …but it contains hydrodynamics!– There are multiple time scales.– There are wave dynamics.

• The mean field provides a direction of special importance!– The spectra are not isotropic.

• Can we build a theory of MHD turbulence that brings ideas from hydrodynamics into the problem?

• Dissipation is NOT provided by the fluid equation!– Dissipation marks the breakdown of the fluid approximation.– Going to need kinetic physics for dissipation!?!

What is Turbulence?

It is the nonlinear evolution of the fluid away from its initial state and

toward a self-determined state.

Weak Turbulence Theory

• One view, popular in plasma theory, is that turbulence is just the 2nd-order interaction of linear waves.– Oppositely propagating low-frequency waves interact

to produce a “daughter” wave propagating obliquely in a 3rd direction.

• Requires that transfer rates be slow compared with wave periods.

• Traditionally give suspect predictions(in my humble opinion)

Hydrodynamic Turbulence:Laminar vs. Turbulent Flow

Interacting vortices lead to distortion, stretching, and destruction (spawning).

One Example of Turbulence

Contributed by Pablo Dimitric of the Bartol Research Inst.

Navier-Stokes Equation

0V

VνP)V(Vt

Vρ 2

M

Mass DensityFluid Pressure

Dissipation of Energy

Incompressibility

(constant mass)

Convective derivative (time variability following the flow)

Energy Conservation

VνP)V(Vt

Vρ V 2

M

t

E

t

V

2

1

t

VV

2

0

00

VVVVVV

VV2

1V

2

1V)V(VV

222

2

Energy Conserving

Wave Vector Dynamics

VνP)V(Vt

Vρ 2

M

k

xikkeVV(x) If

k2

kkkk

k2kk

M VνkikPVikVt

Vρ

21

21

Dissipation at large wave numbers

MHD Equations

The Navier-Stokes equation represents a system where the nonlinear terms move energy without changing the total energy and viscosity dissipates energy at the smallest scales.

Can the MHD equations do something comparable?

0 and 4

4

0or 0

4

1)(

22

2

BJB

BBVB

VV

VBBVVV

c

c

t

t

Pt

MM

M

zyx

zyx

Over the Next Two Days:

• In Tuesday’s seminar we will explore one very powerful formulation for measuring the rate of energy cascade in the spectrum.

• In Wednesday’s class we will continue to explore the concepts of MHD turbulence.

Extra Slides

Magnetohydrodynamics (MHD)

JB

BBVB

VV

VBBVVV

c

c

t

t

Pt

MM

M

44

0or 0

4

1)(

22

2

zyx

zyx

Spectral steepeningwith dissipation

Ion inertial scaleInertial range spectrum ~ 5/ 3

There is a mathematical formalism that (we believe) describes the low-frequency fluctuations in the solar wind.

It is derived from a combination of Maxwell’s equations for E&M and statistical mechanics.

It attempts to present a fluid description for a plasma without particle collisions.

These equations break down when spatial (temporal) scales become smaller (faster) than the orbit of a proton.

There is a formal description to plasmas under these conditions.

Waves vs. Turbulence and more…

• Can we understand reproducible observations in basic terms?

• What are the universal processes of space physics?

• What can we assume of unmeasured and partially measured physics?

• Can we move toward a predictive understanding that gets dynamics right, or just qualities in general?

…and Why?

• Because waves are a relatively static phenomenon.– The propagate, but are reluctant to give up

their energy.

• Turbulence is an inherently dynamic phenomenon.– Energy moves through the spectrum

continuously and defines EVERYTHING!

Temperature Gradients R < 1 AU

In the range 0.3 < R < 1.0 AU, Helios observations demonstrate the following:

For VSW < 300 km/s, T ~ R -1.3 0.13

300 < VSW < 400 km/s, T ~ R -1.2 0.09

400 < VSW < 500 km/s, T ~ R -1.0 0.10

500 < VSW < 600 km/s, T ~ R -0.8 0.10

600 < VSW < 700 km/s, T ~ R -0.8 0.09

700 < VSW < 800 km/s, T ~ R -0.8 0.17

Why is this? It seems distinct from the high-latitude observations.

We can look to explain this through theory and link it to observations of the dissipation range and inferred spectral cascade rates.

Adiabatic expansion yields T ~ R-4/3.

Low speed wind expands without in situ heating!?

High speed wind is heated as it expands.

Before There Were Spacecraft…

• There were comet tails.

• Something blows the tailsof comets away from sun.

• Light pressure cannot explain it.

• Sometimes the tailsbecome disconnected.

1 AU Conditions (Near Earth)Solar wind speed, VSW, varies from 250 to 800 km/s.

(REarth = 6371 km, an 8 second transit at high wind speed.)

At least 2 transients have been observed at 2000 km/s at 1 AU!

Proton density varies, but averages about 10 p+/cm3.

We are unable to achieve these densities on the surface of the Earth.

Proton temperature averages about 105 K. (Vth ~ 50 km/s for ions.)

Magnetic clouds are colder (1/10’th).

Interplanetary magnetic field averages 6 to 8 nT and 45° from radial direction.

(BEarth = 5.7 x 104 nT at Boulder, Colorado.)

IMF can be just about anything at any time.

Have reached 55 nT during ACE era and << 1 nT.

Heavy ions (He++, He+, Fe?+, C?+, O?+, etc.) provide clues to origins.

Solar Activity 1992-1999

The Sun through the Eyes of SOHO

SOHO Project Scientist Team

Solar Wind Variability

While we will see there is all sorts of variability from one hour to the next, there is also a systematic variability tied to the solar cycle.

This reflects the Sun’s changing state with high-speed wind sources moving to new latitudes and multiple wind sources interacting.

This variability propagates into the outer heliosphere, forms merged regions of plasma that alter the propagation of galactic cosmic rays Earthward, and effects the acceleration of energetic particles within the heliosphere.

Solar Energetic Particles

Solar flares on Bastille Day 2000 marked the eruption of coronal magnetic fields and associated plasma.

This ejection resulted in the acceleration of ions and electrons via mechanisms still under study.

ACE observed the energetic particles arrived at Earth several days ahead of the ejecta.

Orbiting spacecraft were damaged or destroyed.

Loss of electrical power was threatened.

Astronauts were endangered.

Analysis: Day 49 of 1999“Classic” upstream wave spectrum for older, well-developed event (as expected).

Broad spectrum of upstream waves.

Not much polarization.

No real surprises here.

Upstream spectrum from prior to the onset of the SEP.

Power law form has no net polarization and there is a spectral break to form the dissipation range.

This is a typical undisturbed solar wind interval.

f5/3

Low-Frequency Waves• Alfven waves:

– Transverse magnetic and velocity fluctuations– No density compressions k B0 / (4)

• Fast Mode waves:– Some field-aligned V and B fluctuations– Some compression for off-axis propagation k B0 / (4)

Solar Wind Flow Near Sun

Magnetohydrodynamics (MHD)

JB

BBVB

VV

VBBVVV

c

c

t

t

Pt

MM

M

44

0or 0

4

1)(

22

2

zyx

zyx

Spectral steepeningwith dissipation

Ion inertial scaleInertial range spectrum ~ 5/ 3

There is a mathematical formalism that (we believe) describes the low-frequency fluctuations in the solar wind.

It is derived from a combination of Maxwell’s equations for E&M and statistical mechanics.

It attempts to present a fluid description for a plasma without particle collisions.

These equations break down when spatial (temporal) scales become smaller (faster) than the orbit of a proton.

There is a formal description to plasmas under these conditions.

JB

BBVB

VV

VBBVVV

c

c

t

t

Pt

MM

M

44

0or 0

4

1)(

22

2

zyx

zyx

A Model to Think About:

• All that energy has to go someplace!– All energy goes into heat.

• Will MHD turbulence (in analogy with hydrodynamic turbulence) converts large-scale energy into small-scale energy?

• Will combination of processes (mostly kinetic physics) converts collective motions into heat?– Cyclotron damping, Landau damping, current sheet

formation, etc….