Turbulence of the Solar Wind

Home , Aphelion (software), TRACE

Studies of the Solar Wind Using the ACE and Helios Spacecraft

Bejamin T MacBride

DepartmentofPhysics

University of New Hampshire

Durham NH

May

Abstract

The solar wind is a sup ersonic ow of plasma emanating from the sun and traveling through

the interplanetary medium to the outermost reaches of the heliosphere The solar wind

exp eriences in situ magnetohydro dynamic MHD turbulence as it expands from its source

This thesis has two comp onents with the common theme of turbulence one examines the

heating from the turbulent cascade at AU and the other studies the spatial evolution of

the interplanetary sp ectrum of solar wind turbulence

Mo dels based on turbulence provide enough energy to account for the heating of the solar

wind relative to an adiabatic expansion that has b een observed bytheVoyager spacecraft

In this work we employ Kolmogorovs expression for the rate of energy cascade p er unit

mass based on the thirdorder structure function of velo city uctuations in hydro dynamic

turbulence The solar wind however is MHD and its turbulence is anisotropic Politano and

Pouquet have derived an expression valid in anisotropic situations for the cascade

rate based on the third moment of uctuations in MHD turbulence We p erform an analysis

of the third moment derived byPolitano and Pouquet and use years of combined magnetic

eld and plasma measurements from the Advanced Comp osition Explorer to compute the

cascade rate of the turbulence in the solar wind We nd the energy cascade inferred from

traditional metho ds using the p ower sp ectrum of uctuations is higher than what is shown

from the lo cal temp erature gradient of the solar wind and that the thirdorder structure

function formalism provides an accurate prediction of the energy cascade

By analyzing the small scale prop erties of the turbulence it is p ossible to come to an

understanding of the dynamics of the energy as it moves through the inertial range and is

dissipated to heat the background plasma We construct a database with intervals of

Helios observations at spacecraft frame frequencies ranging from Hz We

nd the variance anisotropy scales with b oth plasma b eta and the amplitude of the p ower

sp ectrum of magnetic uctuations We also nd the energy to b e nearly equally distributed

between eldaligned and p erp endicular wavevectors These results are derived in the region

between and AU and show no dep endence on helio centric distance This conrms

that the results of Leamon et al Smith et al and Hamilton et al in the

nearearth atmosphere at AU hold true inside the orbit of Mercury which implies the

turbulence of the solar wind is fully develop ed by the time it reaches AU

Contents

Intro duction

Solar Wind Acceleration

Structured Solar Wind

Solar Cycle

Spacecraft Missions

Solar Wind Structure and Fluctuations

Parkers Mo del of the Solar Wind

Interplanetary Magnetic Field

Plasma Waves in the Solar Wind

WaveVectors in the Solar Wind

Heating of the Solar Wind

Solar Wind Turbulence

Interplanetary Sp ectrum

EnergyContaining Range

Inertial Range

Dissipation Range

Published Pap ers

Third Moment of Fluctuations in the Solar Wind

Introduction

Theory

Structure Function Formalism

Analysis

AllData

High and LowWindSpeed

Minority Comp onents

Yearly Analysis

Summary

Detrending and Stationarity

Co de Testing

Turbulence of the Inner Heliosphere

Anisotropic Turbulence in the Solar Wind

Helios Analysis

Magnetic Variance of the Inner Heliosphere

Spatial Evolution of the Power Sp ectral Index i

WaveVector Anisotropy of the Inner Heliosphere

Conclusion

Parting Remarks ii

Acknowledgements

This work would not have b een p ossible without the help of a supp orting cast First and

foremost to the work presented in this thesis is the help provided by Professor Chuck Smith

Professor Smith has p ositively inuenced my professional and academic career in a wayI

didnt think to b e p ossible He showed innite patience and p edagogical abilities as he to ok

me from the level of having zero programming exp erience and a rudimentary understanding

of outer space to the p oint where I was able to presentmywork to and b ecome engaged in

conversations with leading space physicists at professional conferences I owe a great deal of

any success I maystumble up on to the years of supp ort and guidance Professor Smith has

given me

As an educational institution the University of New Hampshire has b een very kind to

me From the rst day I arrived on campus the Physics Department has always treated

me with a great deal of kindness and resp ect and I have found the academic environment

in Demeritt Hall to b e intellectually engaging and stimulating I have b een treated to the

great teachings of Professors Calarco Chupp Echt Harp er Holtrop Meredith and Ryan

and I will take all that they havetaughtmeinto myyears of graduate scho ol I would also

like to thank my classmates MikeAntosh Don Carlson Jason Carrol Jason Conwell Kevin

Go din Andrew Gustafson and Will Morrison for helping me to endure the homework sets

and exams that accompany a degree in physics

Iwould like to thank Professor Miriam Forman of the State Univ ersityofNewYork at

Stonybro ok for her continual advice and guidance on solar wind turbulence theory Professor

Bernie Vasquez from the Space Science Center at UNH was also very helpful to me with the

insightheprovided on the heating and acceleration of the solar wind

On a p ersonal note I would like to thank the lady I love Hannah Varn for b eing my

companion in life You never cease to amaze me with all that you do with your life

Chapter

Intro duction

The solar wind is a plasma that is an expansion of the suns corona through the heliosphere at

a sup ersonic rate untilitreaches the termination sho ck where it slows down to a subsonic ow

as it interacts with the heliopause heliosheath and the lo cal interstellar medium There exist

fundamental questions that to this day remain unanswered by all the spacecraft observation

and research in the eld of solar wind studies These questions are where do es the solar

wind come from and what is the source of its acceleration The questions are so basic and

simple yet the answers haveproven to b e elusively complex

The rst predictions for the existence of a solar wind were made by Ludwig Biermann

Biermanns prediction of a solar wind came from his study of comet tails At the time

of his research the tails of comets were b elieved to b e a result of electromagnetic radiation

pressure alone Biermann observed long cometary tails that werent always p ointed exactly

radially away from the sun He argued that these long tails could not b e pro duced by radia

tion pressure alone This radiation pressure must b e coupled with another massive particle

ux originating at the sun He b elieved the lighter dust tails in comets are pro duced by

electromagnetic radiation but that the more massive tail region is pro duced by corpuscular

radiation

In Chapman develop ed a static subsonic mo del of the solar wind but this mo del

predicted ion densities and temp eratures at the vicinity of earth that were to o high Eugene

Parker argued against the static mo del of the solar wind and provided a mo del

that predicts a sup ersonic ow of plasma extending throughout the heliosphere His mo del

assumes the pressure gradient in the solar atmosphere is such that particles are accelerated

as they stream outwards from the sun until they reach a sup ersonic ow sp eed Parkers

sup ersonic solar wind prediction has b een validated by in situ observations of the solar wind

Parkers pap er Dynamics of the Interplanetary Gas and Magnetic Fields was

groundbreaking and is the foundation for mo dern solar wind research

This pap er is broken into Chapters whichgiv e a description of solar wind prop erties

and a particular fo cus on the turbulence of the solar wind

Solar Wind Acceleration

The core of the sun is a nuclear reactor in whichhydrogen nuclei are comp ounded together

in the pro cess known as nuclear fusion to create helium and excess heat energy The heat

energy pro duced in the core makes its way rst through the radiative zone of the solar

interior by the pro cess of diusive radiation As the heat expands towards the edges of

the radiative zone it co ols down and it b ecomes more ecient to transfer heat by means of

convection and so convective cells develop and transfer heat through the convectivezone

Ab ove the convection zone lies the layer of the sun we refer to as the surface of the sun or the

photosphere The photosphere is the layer of the sun that emits light and is the reason we

refer to this region as the solar surface The photosphere has a temp erature of approximately

K Ab ove the photosphere is a region that has b een lab eled the chromosphere When

the bright photosphere is blotted out by either a solar eclipse or with a coronagraph the

chromosphere can b e seen as the redcolored region just ab ove the photosphere The red color

gives the chromosphere its name chroma Greek for color The regions of the sun inside

the chromosphere demonstrate a decrease in temp erature with increasing distance from the

core of the sun However the chromosphere which is further from the core of the sun than

the photosphere has a higher average temp erature than the photosphere The corona is the

the solar atmosphere and lies ab ovethechromosphere This is the outermost layer of the

sun and the solar wind is an extension of the corona into the interplanetary medium Like

the chromosphere the corona can only b e seen during a solar eclipse or with sp ecial optical

instruments b ecause of its dimness in comparison to the photosphere The corona is even

K hotter than the chromosphere and exhibits an average temp erature of approximately

more than a factor of times as hot as the photosphere

The temp erature of the sun is the highest at its core and it decreases steadily as the heat

is transp orted away from the core until it reaches the photosphere Between the photosphere

and the corona the temp erature of the sun increases from approximately six thousand to one

million Kelvin There are several theories for the source of this heating but to this day none

has b een proven The heat energy at the base of the corona is transferred to kinetic energy

as the solar wind accelerates The solar wind is accelerated to typical velo cities ranging from

kms see Sections and

Structured Solar Wind

Observations show the solar wind exhibits twodistinctow sp eeds fast and slow The

solar wind is highly structured at solar minimum typically with high wind sources at high

latitudes and low wind sources at low latitudes The degree of order and coherence of the

solar wind is strongly correlated with the solar cycle see Section The slowvelo city

mo de is characterized by a sp eed range of kms and the fast mo de usually has

sp eeds b etween from kms The origin and unique prop erties and characteristics

of the twotyp es of wind have led physicists to treat them as two separate entities

Studies haveshown that the fast solar wind originates from coronal holes Coronal

holes are asso ciated with op en magnetic eld lines that allow the solar wind to ow at a faster

rate into the heliosphere The fast solar wind has an average proton density of approximately

three protons p er cubic centimeter at AU and has approximately a four p ercent helium

abundance The typical proton temp erature in the fast stream wind is Kandthe

electron temp erature is K

The slow solar wind is b elieved to originate from multiple regions in the corona Some

of these sources include the b oundaries b etween coronal holes and helmet streamers plasma

sheets near the cusps of streamers small coronal holes and active regions Helmet

streamers in the corona are regions of closed magnetic eld lines that form lo ops on the

surface of the sun The average proton density of the slow wind at a lo cation of AU from

the sun is approximately protons p er cubic centimeter The slow solar wind is

more aected by the solar cycle than the fast solar wind During solar minimum conditions

the slow wind usually originates near the heliomagnetic equator and has a variable helium

abundance The electrons in the slow wind haveaverage temp eratures of K and the

protons in the slow solar wind haveanaverage temp erature of K which is nearly

a factor of ten lower than the protons in the fast solar wind

The lo cation of the critical p oint discussed in Section in relation to where the solar

wind receives the ma jority of its heating is an imp ortant factor in determining the nature of

the solar wind In fact it determines whether the owisfastorslow If the ma jorityof

the heating o ccurs b elow the critical p oint when the solar wind is still subsonic and near the

sun then the plasma expands and the increase in energy provided by the heating is balanced

by the mass ux and density increase as the wind draws in more particles and thus more

mass from the corona and this limits the overall sp eed of the ow On the other hand if

most of the heating o ccurs ab ove the critical p oint once the ow is already sup ersonic and

beyond the inuence of its origin at the corona the increase in energy that is provided by

the heating is dep osited directly to the kinetic energy of the particles in the plasma there

is no increase in mass ux or density and the sp eed of the wind is increased Thus the fast

wind is heated ab ove and the slow wind is heated b elow the critical p oint This assessment

shows that the height of the critical p oint is dierentforvarious lo cations on the sun

The high sp eed wind originating from coronal holes has a lower critical p oint and lower

density since it receives most of its heating when it is already sup ersonic The low sp eed

wind originating near helmet streamers has a higher critical p oint and higher density since

it receives most of its heating when it is still subsonic

The fast and slo w solar wind are the twovarieties of the ambient or quasistationary

solar wind There also exist a variety of transient structures that interrupt the ambient

background plasma ow The frequency at which transientevents o ccur on the sun dep ends

on the stage of the solar cycle Solar maximum conditions pro duce more transientevents

than p erio ds of lower solar activityTransientevents in the solar wind include solar ares

and coronal mass ejections It is imp ortant to note that transient structures exist in the

solar wind and they must b e considered when describing the dynamimcs of the solar wind

Solar Cycle

The sun reverses its p olarityevery eleven years during a p erio d known as solar maximum

After approximately years the sun makes the full cycle from one p olarity to the next

and then back to the original p olarity This year cycle is known as the solar cycle which

consists of solar maximum and solar minimum conditions During solar maximumthere

are many active regions on the sun which lead to a higher pro duction of solar ares and

coronal mass ejections During solar maximum conditions the numb er of sunsp ots visible on

the surface of the sun also increases Sunsp ots are co oler regions that corresp ond to regions

of strong magnetic eld lines In b etween solar maximum conditions the sun settles into

solar minimum During solar minimum there is less solar activity and therefore there are

fewer sunsp ots and the solar wind is less variable The sun is also highly structured during

solar minimum with high sp eed solar wind streams emerging from the p olar regions and low

sp eed streams originating from lower latitudes The magnetic eld of the sun is also more

um conditions structured and predictable during solar minim

The theory that describ es the cause of the suns magnetic eld is known as solar dynamo

theory Although this theory describ es the origin of the magnetic elds it provides no

Figure The numb er of observed sunsp ots plotted as a function of time Image courtesy

of NOAA NGDC Solar Data Services

information as to what causes the shift in p olarityevery eleven years The cause of the

solar cycle is an area of active research Sunsp ots have b een do cumented through the years

and so there exists a record of solar maxima and minima Within the database of sunsp ot

activity there exists a p erio d of approximately seventyyears where hardly any sunsp ots were

observed This ep o ch has b een lab eled the Maunder Minimum and it spans the years

to AD Figure displays the sunsp ot numb er as a function of time from

The numb er of sunsp ots visible on the surface of the sun isnt the only thing that changes

with the solar cycle Along with an increase in sunsp ots solar maximum also brings an

increase in overall magnetic eld strength and a decrease in the overall velo city of the solar

wind Solar activity is a measure of the magnetic activity of the sun Solar maximum is a

p erio d that exhibits a lot of solar activity and this corresp onds to an increase in magnetic

eld activity and strength One of the regions on the sun from which the slow wind is thought

to originate is active regions Solar maximum contains more active regions than normal and

this means that more sources of slow solar wind exist As a result of a higher number

of sources of slow solar wind during solar maximumtheoverall sp eed of the solar wind

decreases Figure demonstrates the eect of the solar cycle on the solar wind velo city

and magnetic eld There is a strong correlation b etween sunsp ot numb er and magnetic

eld strength and the solar wind velo city app ears to b e anticorrelated to the other two

quantities

Spacecraft Missions

The Helios mission consists of two spacecraft launched in a joint eort b etween the National

Aeronautics and Space Administration NASA and the Federal Republic of Germany

Helios and were launched on Decemb er and January resp ectively The

aim of the Helios mission was to make in situ observations of the solar wind in the inner

heliosphere The Helios spacecraft traveled highly elliptical helio centric orbits that had a

p erihelion of AU for Helios and AU for Helios These spacecraft were the

rst satellites to b e sent closer to the sun than the innermost planet Mercury The orbital

p erio ds of Helios and were and days resp ectively The Helios spacecraft had

twelve exp eriments on b oard including plasma instruments magnetometers and cosmic ray

instruments The ma jor contribution of the Helios mission is that these spacecraft provided

the rst in situ measurements of the interplanetary medium in the inner heliosphere and to

this day they hold the record for the closest approach to the sun

Figure top Solar wind velo city as a function of time middle sunsp ot counts over

time and b ottom magnetic eld strength at AU plotted as a function of time image

courtesy of Burlaga

The Voyager mission consists of two identical spacecraft Voyager and Voyager Voy

ager was launched on Septemb er and Voyager was launched August

The initial goal of the Voyager mission was to explore Jupiter and Saturn The two

Voyager spacecraft are still op erational and in ighttothisday and they have done far more

than simply explore two of the giant planets The two spacecraft did fulll their original

exp ectations of exploring Jupiter and Saturn and provided us with new information ab out

the planets including discoveries of activevolcano es on Jupiters mo on Io and a b etter un

derstanding of Saturns rings The spacecraft had such success that their missions have

b een continuously extended and they are still traveling outwards from the sun into the outer

heliosphere and approaching the lo cal interstellar medium In fact it is accepted in the space

science community that Voyager crossed the termination sho ckorwas crossed by the ter

mination sho ck on Decemb er at a distance of AU from the sun Voyager

yager is traveling at an angle of approximately degrees ab ove the ecliptic plane and Vo

istraveling at an angle of ab out degrees relative to the ecliptic Figure is an artists

rendition of the twoVoyager spacecraft and their tra jectories through the heliosphere and

beyond

The Voyager missions initially had functional instruments on b oard to take readings

of the solar wind and planetary data including optical photographs Today there are ve

investigator teams that are still supp orted The Voyager missions continue to provide us

with new information regarding the outer heliosphere and the termination sho ck The two

spacecraft are p owered by the decay of Plutonium and it is b elieved that the two missions

will lose the p ower required to provide information around the year Until then we

can only hop e that they will continue to function and that p erhaps when Voyager crosses

the termination sho ckwe will learn even more from a second crossing

Ulysses was launched on Octob er as a joint eort b etween the Europ ean Space

Agency ESA and NASA The motivation for Ulysses was to explore the heliographic lat

Figure An artists rendition of the heliosphere and the lo cation of Voyager and

as they travel through the outer heliosphere and p erhaps on into the interstellar medium

Image courtesy of Feimer

itudinal dep endence of the solar wind and the structure and dynamics of the corona In

order to explore dierent heliographic latitudes Ulysses must traverse a p olar orbit around

the sun A p olar orbit takes Ulysses out of the ecliptic plane and such an orbit requires

more energy than an orbit in the ecliptic Scientists and engineers designed a tra jectory

that would make use of Jupiters gravitational energy in the form of a gravity assist to place

Ulysses into a p olar orbit Ulysses used a gravity assist in and it has b een orbiting

the sun in a p olar orbit since that day The intersection of Ulysses orbital plane with the

suns equatorial plane is approximately degrees the p erihelion of the orbit is AU from

the sun the aphelion is AU and its orbital p erio d is years

With its p olar orbit Ulysses has provided us with novel discoveries It has conrmed

the theory that the p olar regions havecharacteristically high solar wind velo cities on the

order of kms and are nearly uniform over the p oles Ulysses has demonstrated a

clear dierence b etween the slow and fast solar wind temp eratures and ion comp ositions

The radial comp onent of the suns magnetic eld has b een shown to b e indep endentofthe

helio centric latitude Ulysses has shown that turbulence in the solar wind is slower to evolve

in the p olar regions than it is in the ecliptic region It has also shown a latitudinal gradient

of cosmic rays that is less than exp ected

The Solar and Heliospheric Observatory SOHO was launched on Decemb er

with the primary goal of studying the sun from its deep core to the outer corona and the

solar wind SOHO was launched as a co op erative eort b etween ESA and NASA with

atwoyear lifetime that has b een extended several times to its current length of years

where it will b e able to observethesunover one complete solar cycle SOHO orbits the L

Lagrangian p oint and always faces the sun providing us with an uninterrupted view of the

sun

SOHO was launched in order to provide answers to three fundamental questions The

rst is an inquisition into the structure and dynamics of the interior of the sun where SOHO

will use helioseismology a practice that makes use of the elastic prop erties of the sun and

allows for the propagation of waves By analyzing the waves that propagate through the sun

SOHO intends to gain a b etter understanding of the internal structure and dynamics of the

sun The second question SOHO hop es to answer is why do es the corona exist and howisit

heated to a temp erature over times as hot as the photosphere The sp ectrometers and

imaging telescop es on SOHO will provide information that will help to answer this question

The third question SOHO was made to answer is the fundamental question I stated earlier

in this thesis That is where do es the solar wind come from and what is the source of its

acceleration

The Advanced Comp osition Explorer ACE was launched August with the

primary ob jective of measuring the energetic particles from the sun heliosphere and the

galaxy ACE contains eight instruments for measuring comp osition and a magnetometer

to measure magnetic eld strength ACE like SOHO also orbits the L Lagrangian p oint

where it is able to take samples of data outside the inuence of the earths atmosphere and

magnetic eld There is enough fuel ab oard ACE for it to remain on its L orbit through

Due to the fact that ACE has so many instruments on b oard it is able to provide a wide

range of information ab out solar wind prop erties The mission has four main ob jectives

The rst is to provide detailed information regarding elemental and isotopic abundances of

nuclei in the solar wind ACE is able to study nuclei ranging from Hydrogen Z to Nickel

Z A second task of ACE is to provide information on the origin and evolution of the

elements in our solar system It is theorized that the solar system was not homogeneous when

it was formed and also that the galaxy itself is not homogeneous ACE will b e able to provide

elemental and isotopic measurements of solar material as well as that from galactic cosmic

rays and from the lo cal interstellar medium It is able to do so b ecause it has instruments

on b oard capable of measuring energies ranging from solar wind energies on the order of

keVnucleon to galactic cosmic ray energies on the order of MeVnucleon The third

task is to study the acceleration of the solar wind and the formation of the corona ACE will

compare samples of coronal and photospheric abundances as well as comparing features of

solar energetic particles and the solar wind The fourth ma jor ob jectiveofACE is to study

particle acceleration and transp ort in nature It is still a ma jor ob jective in heliospheric

studies to determine the source of heating and acceleration of the solar wind ACE hop es

to provide insightinto the form of acceleration in nature by isolating solar are coronal

sho ck and interplanetary sho ck acceleration mo dels with charge mass and sp ectral data

The dynamics of a satellite interacting with the sun and earth can b e represented in what

is called the restricted threeb o dy problem This problem is aimed at nding lo cations

ab out the earth and sun at which there is no net force acting on the satellite It turns out

that ve such lo cations exist The ve lo cations are the ve Lagrangian p oints L through L

named after JosephLouis Lagrange L through L are collinear In the sunearth system

L lies b etween the earth and sun L is on the night side of the earth and L is on the

opp osite side of the sun from the earth L and L lie ab oveandbelow the sunearth line L

through L are saddle p oints and so they are unstable p oints of equilibrium The slightest

deviation from these p oints leads to a gravitational force that will pull the satellite towards

the sun or the earth The fourth and fth Lagrangian p oints are stable regions of balanced

force b etween the sun and earth The satellites ACE and SOHO are b oth orbiting the rst

Lagrangian p oint L in a halo orbit The status of these two satellites must b e checked

regularly so as to keep them within the force free region of the unstable L p oint

Chapter

Solar Wind Structure and

Fluctuations

Parkers Mo del of the Solar Wind

In E N Parker wrote a pap er that forever changed our p erception of the solar wind

In his mo del Parker did not deal with the issue of coronal heating He assumed that a

certain temp erature level was maintained it had b een determined from absorption lines and

the charge states that were deduced from these lines and he built his mo del up on a corona

that was suciently hot Parkers mo del sets the sun in a vacuum such that the pressure

go es to zero as the radial distance from the sun go es to innity This mo del implies that the

atmosphere of the sun cannot remain in static equilibrium but that the solar atmosphere

expands into space and is replaced by particles from the inner atmosphere Therefore a

continual outow or streaming of particles from the sun is predicted byParkers mo del

This corpuscular radiation or stream of particles has b een lab eled the solar wind

Parkers assumptions lead to the dierential equation

V dV dr GM

v V v r r

s s

where V is the solar wind velo city v is the velo city of sound r is the radial distance

SW s

from the center of the sun G is the gravitational constant and M is the mass of the sun

There are four solutions to this dierential equation but the solution that ts the conditions

present in the solar wind is

V V r r

SW c

ln ln

v v r r

s c

GM m

where r is the critical p oint where the solar wind b ecomes sup ersonic Parker

explains the transition from subsonic to sup ersonic ow in the solar wind with an analogy to

aLaval nozzle A Laval nozzle is a tub e that has a pinch in the middle to form an hour glass

shap e Gas ows through the nozzle and is accelerated from subsonic to sup ersonic sp eeds

as it passes through the throat of the nozzle Parker relates the throat of the nozzle to the

force of gravity acting on the solar wind In the instance of the solar wind the velo cityis

subsonic until it reaches the critical p oint and the ow b ecomes sup ersonic

Figure provides a graph of theoretical predictions from Parkers mo del of the solar

wind velo city as a function of radial distance from the sun for various coronal temp eratures

Figure Solar wind velo city plotted against the radial distance from the sun as predicted

byParkers solar wind mo del describ ed by equation Image courtesy of Kal lenrode

At the earths orbital distance from the sun the predicted solar wind velo city ranges from

around kms to kms corresp onding to coronal temp eratures of K

In situ observations of the solar wind taken by the Soviet spacecraft Lunik and were

sp oradic and inconsistent but the American spacecraft Mariner provided consistent

readings and a conrmation that the solar wind ows at sup ersonic sp eeds as predicted by

Parkers mo del of the solar wind The term sup ersonic isnt technically correct When

using the term sup ersonic I really mean sup erAlfvenic Alfven waves see Section are

the waves that propagate in interplanetary space and the solar wind b ecomes sup ersonic

when the wind travels at a rate faster than the waves within it propagate the Alfven

sp eed Therefore when I use the term sup ersonic I am actually referring to sup erAlfvenic

The results of Advanced Comp osition Explorer observations are shown in Figure The

solar wind V is measured to owbetween kms to kms whichiswell within

the predictions made byParkers solar wind mo del These observations show a general

anticorrelation b etween the solar wind velo city and density This topic of the densityand

the velo city of the solar wind was desrib ed in Section

In his pap er Parker also provided a theoretical prediction of the interplanetary

magnetic eld Parkers magnetic eld mo del is based up on the following assumptions

The solar wind ows radially outward from the sun at a constantvelo city V Thesun

rotates at an angular sp eed that is found by dividing by the solar rotation p erio d of

approximately days The solar wind is azimuthally symmetric ab out the rotation axis of

the sun and the interplanetary magnetic eld is frozen into the solar wind and has its feet

anchored at the sun

The Parker prediction for the magnetic eld in the equatorial plane of the sun is an

Archimedes spiral Figure is taken from Parkers pap er and demonstrates the

orientation of the interplanetary magnetic eld in the ecliptic plane This gure extends

Figure Advanced Comp osition Explorer solar wind observations for one week The fth

panel from the top displays the solar wind velo city in units of kms Image courtesy of the

ACE MAG team

Figure The Archimedes spiral of the orientation of the interplanetary magnetic eld

predicted byParkers mo del of the solar wind Image courtesy of Parker

the eld to just b eyond the orbit of Mars but the theory predicts an extension of this eld

into the outer heliosphere until this structured nature is disrupted at the termination sho ck

where the solar wind is slowed to subsonic sp eeds as it meets the interstellar medium

In his mo del of the interplanetary magnetic eld Parker describ ed the strength of the

large scale magnetic eld as the square ro ot of the sum of the squares of each comp onent

Parkers equation for the strength of the magnetic eld is a function of source strength

heliographic latitude radial distance from the sun and solar wind velo city

Many spacecraft have carried magnetometers and in Section I describ ed that the

twoVoyager spacecraft are traveling through the outer heliosphere Figure shows the

theoretical predictions of Parkers magnetic eld strength plotted on the same grid as the

observed magnetic eld strength measured byVoyager It is apparent from the plot that

Parkers theoretical prediction is in very close agreement with empirical observations

Interplanetary Magnetic Field

Solar dynamo theory explains the creation of the solar magnetic eld I will not discuss solar

dynamo theory in this pap er but I will discuss the suns magnetic eld and its presence in

the solar wind The solar cycle see Section aects the strength and p olarityofthe

suns magnetic eld The p olarity reverses every eleven years and after eacheleven year

halfcycle during solar maximum is when the p olarity reverses During solar maximum

conditions the p olarity of the eld isnt wellb ehaved or structured at all In b etween

transition stages during solar minimum the magnetic eld of the sun is highly structured

as rst prop osed byParker in his pap er is still used to day The Parker spiral that w

to describ e the orientation of the interplanetary magnetic eld The continual stream of

plasma from the solar source combined with the approximate day rotation p erio d of

the sun pro duce an Archimedes spiral that originates at the sun and extends out into the

interplanetary medium

At the lo cation of earth one astronomical unit from the sun the magnetic eld is usually

Figure Magnetic eld strength as a function of time predicted byParker solid line and

observed byVoyager circles Image courtesy of Burlaga

oriented at a degree angle from the radial direction At the b eginning of solar minimum

conditions in situ observations taken at earth normally showinward outward oriented

magnetic eld for the rst half of the solar rotation and outward inward oriented magnetic

eld for the second half of the solar rotation A solar rotation as seen by an observer at

earth has duration of approximately days and so eachorientation will span ab out

days These time p erio ds of uniform magnetic eld orientation are known as sectors

The start of solar minimum therefore contains a two sector structure During the middle

of solar minimum observations haveshown that a four sector structure exists and solar

maximum since it is so variable and unpredictable usually contains no such magnetic eld

structure The four sector structure has alternating intervals of eld orientation and each

sector has equal duration of approximately days each

The magnetic sector pattern that is observed in the solar wind is an extension of the

suns magnetic eld into the solar wind The two sector pattern corresp onds to a magnetic

dip ole structure at the sun the four sector pattern is a result of a quadrup ole structure

and the lack of a sector pattern during solar maximum reects the lack of a structure in

the suns magnetic eld at this time The extension of the heliomagnetic equator into the

heliosphere is known as the heliospheric current sheet and it is a plane of neutral magnetic

orientation Figure is a mo del of what the heliospheric current sheet mightlooklike

This current sheet is not a p erfectly level plane but it varies its shap e as a function of the

o sector structure is observed and the sun b ehaves as a magnetic solar cycle When the tw

dip ole the heliospheric current sheet is a fairly level surface but the four sector pattern and

the magnetic quadrup ole b ehavior of the sun distorts the current sheet and during solar

maximum conditions the structure and orientation of the current sheet is highly variable

and unpredictable

When describing the solar wind it is imp ortant to discuss some of the global prop erties it

Figure A mo del of the heliospheric current sheet Image courtesy of Burlaga

Table Global Prop erties of the Solar Wind Data courtesy of Schwenn

Common Physical Parameters

Parameter Solar Wind Values Terrestrial Values

Flow Sp eed kms average Sp eed of Sound

Density particlescm At Sea Level

Temp erature K protons Ro om

electrons no free electrons

Magnetic Field Tesla

exhibits The solar wind is a currentconducting magnetic eldcarrying sup ersonicowing

plasma Table gives a summary of some of the key features shown by the solar wind

and puts these values into p ersp ectiveby comparing them to some more familiar terrestrial

values

Plasma Waves in the Solar Wind

The solar wind is a plasma and so the waves that propagate through the solar wind are

plasma waves Since the density of the solar wind is so low protonscm the mean free

path the mean distance a particle travels b efore colliding with another particle is approxi

mately one astronomical unit Instead of the normal physical collisions b etween particles

that is seen in a hydro dynamic gas particles within a magnetohydro dynamic MHD uid

such as plasma normally interact via plasma waves Plasma waves carry energy and mo

mentum in the solar wind and are b elieved to b e resp onsible for much of the heating and

acceleration of the solar wind

Multiple typ es of plasma waves exist and the propagation velo city of these waves is

dep endent up on several factors including the sp eed of lightc magnetic eld strength

B particle numb er densityN and the particle temp erature T The subscript s

s s

refers to the sp ecies of the plasma wave where s e corresp onds to electrons and s i

corresp onds to ions Plasma waves are usually describ ed as a function of frequencyand

some common frequencies include the electron cyclotron frequency f eB m

ce e

where e is the elementary charge and m is the mass of the electron the ion cyclotron

frequency f m m f e where m is the mass of the ion the electron plasma frequency

ci e i c i

m where is the p ermittivity of free space and the ion plasma f N e

e pe e

Figure Observations of the magnetic eld strength and velo city in three comp onents

and the magnitude of the magnetic eld and densityprovided by Mariner Image courtesy

of Belcher and Davis

frequency f m m f

pi e i pe

The most imp ortant reference velo city in plasmas is the Alfven sp eed V B

A m

where is the p ermeability of free space and is the mass density The Alfven sp eed is

the sp eed at whichAlfven waves propagate through a plasma An Alfven wave is a transverse

wave that is the uctuation of the magnetic eld If one pictures the magnetic eld as a taut

string the Alfven wave is analogous to the transverse wave that travels along the string if

the string is plucked The solar wind has b oth massive particles in the form of a plasma and

a magnetic eld in the form of the IMF When the magnetic eld is disturb ed waves are

created in the plasma by the restoring force of the magnetic eld Suchwaves are kno wn

as Alfven waves Other imp ortant sp eeds in the solar wind are the electron sound sp eed

V kT m where is the adiabatic compression factor and k is the Boltzmann

e e e

constant and the ion sound sp eed V kT m

s i i

Three of the plasma waves are MHD waves the fast mo de slow mo de and Alfven waves

The sound sp eed V corresp onds to the electron sound sp eed for electron waves and to the

ion sound sp eed for ion waves Alfven waves were discovered by Alfven and were shown

to exist in the solar wind by Unti and Neugebauer and Belcher et al The fast mo de

wave has higher characteristic velo cities than Alfven mo de wave which in turn has a higher

characteristic v elo city than the slowmodewave

Figure provides evidence for the existence of Alfven waves in the solar wind The plot

contains velo city and magnetic eld observations sup erimp osed on the same plot and the

values along each direction b b andb are stacked on top of one another The b ottom plot

r t n

contains the magnitude of the magnetic eld and the density of the solar wind The magnetic

eld and velo city are measured in the RTN co ordinate system This co ordinate system has its

origin at the sun and the R comp onent is the radial direction from the sun to the spacecraft

making measurements If is the unit vector that p oints in the direction of the suns spin

axis the T comp onentoftheRTN co ordinate system is in the direction of the vector resulting

from RandN is the unit vector in the direction of R T to complete the righthanded

co ordinate system The uctuations of magnetic eld strength and propagation velo cityin

each comp onentshown in Figure are Alfven waves It is clear that these uctuations are

Alfven waves b ecause there exists a variation of each individual spatial comp onentofthe

solar wind but no variation in the overall magnitude or density of the solar wind Alfven

waves are transverse waves not compressivewaves and the characteristics demonstrated in

Figure indicate that the observed uctuations are Alfven waves

WaveVectors in the Solar Wind

As stated in Section Unti and Neugebauer and Belcher et al demonstrated

the existence of Alfven waves in the solar wind Twoyears after their initial pap er on

the sub ject Belcher and Davis showed uctuations in velo city and the magnetic eld to

b e primarily transverse to the mean magnetic eld It is worth noting that these studies

made use of low frequency observations This work is often interpreted in terms of the

minimum variance direction In the case of uctuations transverse to the magnetic eld

the minimum variance direction is along the eld In the early studies of waves in the

ve solar wind it was assumed that the minimum variance direction was the same as the wa

vector orientation A predominant mo del used at the time for describing uctuations of

the interplanetary magnetic eld is that all uctuations arise from noninteracting outward

propagating Alfven waves that move in the direction parallel to the mean magnetic eld

With this description and the assumption that the minimum variance direction is equivalent

to the wavevector orientation all wavevectors and minimum variance directions would b e

along the mean magnetic eld However wavevectors may b e oriented b oth p erp endicular

and parallel to the mean eld while the minimum variance direction is parallel to the mean

eld The minimum variance direction should not b e equated with wavevector orientations

The issue of wavevector orientation was addressed by Matthaeus et al using a

correlation function of the magnetic eld The result of the analysis has come to b e known

as the Maltese cross This Maltese cross demonstrates that there is a strong wavevector

orientation in b oth the onedimensional eldaligned comp onentandthetwodimensional

plane p erp endicular to the mean magnetic eld but not muchinbetween Although the

assumption that the wavevectors are a comp osite of strictly parallel and stricly p erp endicular

orientations is a simplication of the true situation in the solar wind it is a useful mo del that

p ermits the extraction of the amount of energy contained in the two comp onents Building

up on the results of the Maltese cross Dasso et al compute magnetic and velo city

correlation functions using solar wind observations provided bytheACE spacecraft but

they p erform the analysis on the basis of wind sp eed Dening the fast solar wind to b e

intervals with an average wind sp eed greater than kms and the slow solar wind to b e

intervals with an average wind sp eed less than kms they nd the fast wind to b e largely

p opulated bywavevectors aligned with the magnetic eld and the slow wind to b e p opulated

bywavevectors p erp endicular to the magnetic eld

Motivated by the implications of the Maltese cross Bieber et al derive a metho d

for determining the relative amount of energy contained in the wavevectors parallel and

p erp endicular to the magnetic eld see Section for an analytical description of the

Bieb er metho d This derivation relies up on the simplication that wavevectors in the solar

wind are oriented either p erfectly parallel or p erp endicular to the mean magnetic eld Again

we note that this case is an oversimplication but allows a distinction to b e made b etween

the two comp onents that are clearly demonstrated to exist by Matthaeus et al Bieber

et al apply the test for determining the amount of energy in the parallel and p erp endicular

comp onents to observations of the solar wind made by the Helios spacecraft They nd

that of the energy is contained in the two dimensional comp onent p erp endicular to

the mean magnetic eld These results are in stark contrast with the assumption that

uctuations of the solar wind are caused primarily by noninteracting Alfven waves with

wavevectors aligned with the magnetic eld It is evident that there are other causes for

solar wind uctuations The nonlinear interactions asso ciated with magnetohydro dynamic

turbulence is one such p ossible cause

Heating of the Solar Wind

Observations from the Helios spacecraft taken inside AU demonstrate isotropic velo city

distribution functions for thermal ions The proton temp erature in the fast wind is

higher for wavevectors p erp endicular to the magnetic eld than for vectors aligned with the

mean eld T T The opp osite is true in the slow wind Cyclotron resonance increases

the temp erature of wavevectors p erp endicular to the magnetic eld Landau resonance is

a phenomenon that aects charged particles in the solar wind traveling in the longitudinal

direction Therefore the temp erature of wavevectors parallel to the mean eld is increased

by Landau resonance

Although the temp erature prole in wavevectors p erp endicular and parallel to the mag

netic eld in the solar wind is an interesting topic what is more relevant to this pap er is

the radial variation of the temp erature with increasing distance from the sun If the solar

wind were to expand adiabatically the temp erature as a function of distance r from the

sun would b e prop ortional to r In situ measurements of the solar wind taken by Helios

and Voyager have shown that the temp erature of the solar wind decreases slower

than an adiabatic expansion would predict By AU the solar wind is hotter and by

AU the solar wind is hotter than the adiabatic prediction The departure from an

adiabatic co oling pro cess indicates the presence of a source of heating in the solar wind

Zhou and Matthaeus Zanketal and Matthaeus et al have shown that

a turbulent transp ort mo del based on the large energycontaining scales of magnetohydro

dynamic MHD turbulence can provide the right amount of energy cascade to resupply

the dissipation pro cess in a continuous fashion and thereby account for the heating of the

solar wind Their mo del do es not assume any particular dissipation pro cess but assumes

that the inertial range is energyconserving so that the energy cascade rate in the inertial

range will b e equal to the heating rate in the dissipation range This mo del do es assume the

uctuations are twodimensional and that the cascade of energy to the dissipation range is

twodimensional in nature Section demonstrates the existence and in some cases the

dominance of D uctuations in the solar wind whichvalidates the assumption of a D

cascade Smith et al demonstrate that a turbulent transp ort mo del together with

the Isenberg et al analysis of the pickup ion pro cess can account for thermal proton

energies to AU where their analysis stops for reasons of data availability

Chapter

Solar Wind Turbulence

Interplanetary Sp ectrum

The mo del of solar wind turbulence I employ is based on the p ower sp ectrum of uctuations

depicted in Figure The turbulent energy is generated at the low frequencies and large

length scales of the energy containing range The energy is then fed through the energy

conserving intermediate scales of the inertial range The turbulent energy is then converted

to heat energy at the small scales of the dissipation range

EnergyContaining Range

The energycontaining range is characterized bya f form when lo oking at p ower sp ectra of

the interplanetary magnetic eld The q power law index is typically found at frequen

cies b elowT where T hours is the average correlation time of the interplanetary

c c

magnetic eld Matthaeus and Goldstein study the f noise in the interplanetary

magnetic eld They explain that the source of the interplanetary uctuations at these low

frequencies is a pro duct of magnetic structures on the surface of the sun The fact that the

energy containing range o ccurs at frequencies lower than T means that uctuations on this

time scale which are greater than the correlation time are uncorrelated events that have

their source at the sun There have b een turbulent transp ort mo dels prop osed to accountfor

the nonadiabatic expansion of the solar wind These mo dels rely on the presence

of energy in the large scales of the energycontaining range to drive turbulence in the smaller

scales where the energy is ultimately dissipated to heat the background plasma As stated

in Section Smith et al and Isenbergetal demonstrate that these mo dels of

turbulent transp ort and the added heating provided bypickup ions in the outer heliosphere

account for the amount of energy required to pro duce the heating of the solar wind that is

observed Thus the connection b etween the energy contained in the large scale uctuations

in the solar wind and the heating of the background plasma is successfully made

Inertial Range

In b etween the large scale uctuations in which the energy is contained and the small scale

uctuations at which the uid energy is dissipated as heat lies the inertial range The inertial

range is a selfsimilar energy conserving pip eline through which energy travels b efore it is

Figure A model power sp ectrum of interplanetary turbulence displaying the energy

containing inertial and dissipation ranges

dissipated as heat energy in the dissipation range In this range of frequencies p ower sp ectra

of the uctuations of the interplanetary magnetic eld are characterized bya power law index

q AN Kolmogorov made signicantcontributions to our understanding of intertial

range prop erties in hydro dynamic turbulentows In a theory that predicted the energy

cascade rate in the solar wind Kolmogorov prop osed a power law index in the inertial

range of homogeneous turbulence One reason that wehave b een able to successfully adapt

the hydro dynamic mo del of turbulence for solar wind studies is b ecause this same f

form is observed in magnetic eld uctuations However Podesta et al demonstrate a

f form in the p ower sp ectra of velo city uctuations This q sp ectral index is in

agreement with the predictions of Kraichnan Kraichnan p ostulates a f form of the

power sp ectra for b oth velo city and magnetic eld uctuations in a MHD uid Leamon et

al Podesta et al and Hamilton et al demonstrate that the p ower sp ectra

of magnetic uctuations in the solar wind exhibit a f sp ectral form Some of the work

presented in this thesis is done with an assumed Kolmogorovpower lawbehavior with the

power law index q This assumption is well supp orted by observations of the p ower

law index in the inertial range which are closer in agreement with the Kolmogorov q

than the Kraichnan q see Table and Figure

In the same year that Kolmogorov describ ed the rate of energy cascade through the

inertial range as a function of the p ower sp ectrum of velo city uctuations he also derived a

more fundamental metho d of determining the cascade rate based on the thirdorder structure

function The rst relation was primarily a result of unit analysis whereas the structure

function metho d is derivable from the nonlinear terms of the NavierStokes equations For

a more detailed discussion of the two metho ds see Section The nal twochapters of this

thesis analyze two asp ects of solar wind turbulence in the inertial range Chapter studies

the cascade of energy in the solar wind at AU using measurements of the lowfrequency

end of the inertial range Chapter studies turbulent prop erties at various lo cations in the

inner heliosphere at higher frequencies to determine whether there is a spatial evolution of

the turbulence as the solar wind expands from to AU

Dissipation Range

The physical pro cesses taking place within the small scales of the dissipation range are not

well known but spacecraft have made in situ observations of this region to understand some

of its turbulent prop erties The onset of the dissipation range is usually found at several

tenths of a Hz and typically has an average f power law form in the magnetic

eld This q index is an average value and p ower sp ectra are known to

have b oth steep er and shallower p ower law indices In fact Leamon et al demonstrate

that the sp ectral index of the dissipation range is dep endent on the temp erature of the

background plasma the intervals with higher temp eratures have steep er p ower law indices

indicating a higher rate of energy dissipation Smith et al measure b oth the energy

cascade rate in the inertial range and the p ower sp ectral index in the dissipation range

and nd the twoquantities to b e directly related They show that higher energy cascade

rates in the inertial range lead to steep er p ower law indices in the dissipation range At

the length scales of the dissipation range the energy that travels through the inertial range

from the energycontaining range is dissipated to heat the background plasma Although the

uid approximation used to describ e the turbulence in the solar wind is very successful in

describing many observed prop erties at the small scales in volved with the dissipation range

there is a breakdown of the single uid mo del It is likely that in the dissipation range there

exists a transition from uid turbulence to the dominance of kinetic physics in the solar

wind

Published Pap ers

Up to this p ointmy thesis has b een a description and explanation of various prop erties

of the solar wind that can b e understo o d on the basis of previous research that has b een

done in the eld The following twochapters are the result of original research that I have

beenapartofduringmy time here at the University of New Hampshire Chapter is a

mo died version of the journal article The Turbulent Cascade at AU Energy Transfer

and the ThirdOrder Scaling for MHD by BM MacBride CW Smith and MA Forman

which is b eing submitted to The Astrophysical Journal I am the primary author of this

pap er and the pap er has had writing and theoretical input from coauthors CW Smith

and MA Forman Chapter is the rst draft written only bymyself of a journal article

InertialRange Anisotropies in the Solar Wind from to AU Helios Observations by

BM MacBride CW Smith and BJ Vasquez which will b e submitted to the Journal of

Geophysical Research this summer

Chapter

Third Moment of Fluctuations in the

Solar Wind

Intro duction

In Section we discuss the need for a source of heating in the solar wind as a result of a

departure from an adiabatic expansion We also refer to a turbulent transp ort mo del that

provides the prop er amount of energy to account for the heating of the solar wind Using

the mo del of Chapter where turbulent energy is generated at the large scales of the inertial

range fed through the intermediate scales of the inertial range and converted to heat energy

at the small scales of the dissipation range we seek to determine the cascade rate for energy

within the inertial range of interplanetary uctuations The energy driven by the largescale

structures in the energy containing range must pass through the inertial range It is dicult

to measure the energy content of the largest scales Moreover conrmation of the turbulent

transp ort theories requires that the energy b e tracked through the intermediate scales if we

are to b elieve that a sp ectral cascade in the manner analogous to hydro dynamic turbulence

exists and is resp onsible for the heating of the solar wind

In the sections that followwe outline a multidimensional extension of structure function

expressions derived to p ermit direct measurement of the energy cascade at inertial range

scales We go on to apply these expressions to Advanced Comp osition Explorer ACE

measurements at AU We nd that it is p ossible to determine the energy cascade rate and

that it matches the exp ected value required to explain in situ heating at AU Wealsond

that fast winds demonstrate slightly higher heating rates than slow winds In the case of

slowwindintervals we nd clear evidence that the wind is evolving toward an equipartition

between inward and outward propagating uctuations by a stronger damping of the dominant

outward propagating comp onent Howev er we nd evidence that the fast wind is evolving to

enhance the dominance of the outward propagating comp onentby a stronger damping of the

inward propagating comp onent This suggests that highsp eed winds at greater helio centric

distances should continue to exhibit a high degree of crosscorrelation b etween magnetic and

velo city uctuations

Theory

Kolmogorovs landmark conclusions of provided two indep endent and complementary

expressions describing the cascade of energy through the turbulenthydro dynamic HD

sp ectrum Kolmogorov predicted that the omnidirectional inertial range sp ectrum for

fully develop ed stationary and isotropic hydro dynamic turbulence would followapower law

form

P k C k

where P k isthepower integrated over wavenumb ers with magnitude from jk j to jk dk j

C is the socalled Kolmogorov constant and is the rate of energy cascade through

the sp ectrum Although the ab ove expression was derived to explain HD turbulence it seems

reasonable to apply it to MHD turbulence in case it should yield go o d comparisons with the

infered heating rate Leamon et al demonstrate this using the measured sp ectrum of

magnetic uctuations and correcting for the dierence b etween the omnidirectional sp ectrum

referred to in eq and the reduced sp ectra that are measured with single spacecraft in

the solar wind We can rewrite the Leamon et al result to use wavenumb er sp ectra

P jk jC jk j

P r r

We add the subscript P above since the expression yields an estimate for based on the

power sp ectrum alone

One added complication of MHD over HD turbulence is the existance of the Elsasser

variables

Z V B

and the fact that these variables intro duce a symmetry to the MHD equations Noting

that energy levels of Z can dier Matthaeus and Zhou Zhou and Matthaeus and

Marsch derive a generalized Kolmogorov phenomenology that takes into accountthe

power sp ectra of Z to write eq

h i

C P jk j P jk j k

r r

P K r

where P are the intensities of the p ower sp ectra for Z and Z resp ectively and the total

This intro duces a basic asymmetry for solar wind energy cascade rate is

P P P

turbulence studies since it is long recognized that outward propagating uctuations dominate

energetically o ver inward propagating uctuations at AU The relation b etween Z

inout

and Z is such that when the interplanetary magnetic eld is oriented radially away

inout

from the sun B Z Z and when the magnetic eld is oriented radially toward

outin

the sun B Z Z where B is the radial comp onentoftheinterplanetary

R R

magnetic eld IMF Using this relation we can write

h i

inout

inout outin

C P jk j P jk j k

r r

P K r

if the sp ectral index of the p ower sp ectrum is in go o d agreement with the prediction

T in out

As with the ab ove relation we can write of Kolmogorov

P P P

Structure Function Formalism

In the same year Kolmogorov derived a more general and analytically sup erior expression

based on the thirdorder structure function that do es not require a k inertial range

sp ectrum and which is applicable throughout the inertial range

h i

S L V x L V x

k k

jLj

where L is the separation vector V is the comp onentofthevelo city uctuation parallel to

the separation vector V V LL where L jLj h i denotes ensemble average and

is the rate of energy cascade dened for the hydro dynamic system

Politano and Pouquet have extended the Kolmogorov concept to address the

turbulent cascade within a magnetohydro dynamic MHD system based on the Els asser

variables Their expression for isotropic geometry can b e applied to the sup ersonic owof

the solar wind according to

h i

ISO

Z L Z L D L

R i

D L

where Z L Z x L Z x is the dierence b etween Z measured at twopoints separated

by a distance L the equations for and corresp ond to the cascade rate for the Z and

Z comp onents D is the dimensionality assumed for spherically isotropic turbulence

and subscript R denotes the Radial comp onent When the energy cascade is p ositiveand

energy moves from large to small scales cascade rates are synonymous with dissipation rates

All lagged separations are along the solar wind ow directione and the sign conforms to

the convention that p ositive distances corresp ond to negative time lags in the data set

h i

ISO

D Z Z

R i

The total energy dissipation rate p er unit mass isgiven by

Matthaeus et al demonstrate that the IMF uctuations at AU and at larger

spatial scales than those studied here are a comp osite of eldaligned and p erp endicular

wavevectors Dasso et al used this same ACE data set to show that the eldaligned

and p erp endicular wavevectors are spatially separated with eldaligned wavevectors in the

fast wind and p erp endicular wavevectors in the slow wind Both Matthaeus et al and Dasso

et al showpower conned to parallel and p erp endicular directions preferentially Therefore

wewould liketohave a form ulation of the ab ove theory that can address sp ecically the

eldaligned and p erp endicular wavevectors Wedothisby rst rotating the data into the

mean eld co ordinates Belcher and Davis and Bieber et al where

e e e je e j

x R B R B

e e e

y z x

e e

z B

If we adopt the D MHD geometry where the magnetic and velo city uctuations are

allowed to have all three comp onents but the wavevectors are conned to the D plane

p erp endicular to the mean magnetic eld wemayfollowthePolitano and Pouquet formalism

for this reduced set of wavevectors and write

i h

Z Z D

i y

V sin

where V is the average solar wind sp eed Z is the Elsasser uctuation measured along the

directione and is the angle b etween the mean eld direction and the owvelo city

y BR

For MHD geometries where the wavevector is along the mean eld wemay similarly

write

h i

D Z Z

z i

V cos

where Z is the measure of uctuations parallel to the mean eld

Analysis

This pro ject uses ACE data from through and applies the ab ove expressions

of Kolmogorovs law and its isotropic D and D MHD analogues to the solar wind

The merged MAG and SWEPAM data has a resolution of seconds Data is stored in solar

rotation les and in our analysis we break eachday solar rotation into fourteen twoday

subintervals Eachtwodayinterval is treated as an indep endent set for whichwe compute

the ab ove cascade rates The expression for the cascade rate involves the lag vector L and in

order to combine statistics from dierentintervals with dierent wind sp eeds it is necessary

to use a common spatial grid To solve this issue weinterp olate the results from eachtwoday

subinterval onto a common length grid that is based on a s data resolution and a kms

ow sp eed The resulting interp olated structure function estimates derived from eac htwo

dayinterval are then summed and the ensemble average is obtained by dividing the sum by

the total numb er of estimates at eachlagvalue In the case of one and twodimensional

MHD geometries we rotate the Elsasser variables into meaneld co ordinates so it

is necessary to havea welldened mean eld direction Theuseoftwoday subintervals

allows for this sp ecication We also use the mean density from each dayinterval This

work exp erimented with various additional data selection and pro cessing mechanisms that

are explained in the app endix but we ultimately decided not to use them

From the p ersp ective of solar wind physics the distinction b etween Elsasser variables Z

and Z which describ e propagation parallel or antiparallel to the mean magnetic eld can

b e made more signicantby considering the direction of the mean eld In the solar wind

the key distinction is outward vs inward propagation with outward propagation generally

dominating the data at AU We use the mean radial orientation of the magnetic eld

inout

in a twodayinterval and the conversion b etween Z and Z to rewrite eq

h i

outin inout inout

ISO

Z Z D

R i

inout

h i

inout inout

D outin

D Z Z

inout

V sin

h i

inout inout

D outin

D Z Z

inout

V cos

With these mo dications in the analysis and binning eachtwoday estimate of the structure

functions accordinglywe are able to separately determine the cascade rate for uctuations

traveling toward and a way from the sun

All Data

Figure displays the results of applying the ab ove formalism to seven years of ACE data

The analysis describ ed here forms the basis for all that follows in this pap er The trace of

the second order structure function for the Elsasser variables

Z t Z t S

i i

can then b e used to estimate the trace of the auto correlation matrix

R hZ t Z ti

i i

under the assumption of homogeneity

R B S

rms

For data intervals of this size with a maximum lag b ordering on the correlation time it is

p ossible to estimate B from the value of S at maximum lag The trace of the p ow er

rms

sp ectrum can then b e computed from R in the usual manner Figure top right shows

the resulting p ower sp ectra for outward propagation blue and inward propagation red

with the solid black line giving the average or total p ower We can use equation to

estimate the average energy cascade rate at AU according to the computed average p ower

sp ectrum A b est t p ower law index for the total MHD p ower sp ectrum in Figure is

q whichis from the Kolmogorov prediction for isotropic turbulence

Table lists the b estt p ower law indexes for all sp ectra shown in this pap er In every

out in

instance examined here the p ower law index of the sp ectra of Z Z andthetotalarecon

sistent with Fitting of eq for this same sp ectrum yields Jouleskgs

which is greater than the observed radial dep endence of the thermal proton temp erature will

allow Moreover this heating rate will result in the thermal proton temp erature increasing

with helio centric distance

Figure shows that outward propagating uctuations are energetically dominant in this

sample as exp ected This asymmetry b etween outward and inward propagation suggests

that we use the generalized Kolmogorov phenomenology of eq Applying this formalism

out in

to Figure yields Jouleskgs and Jouleskgs and the total

P P

Figure Lifetime analysis using ACE data from Top panels show S and

resulting p ower sp ectra for outward blue and inward red propagating MHD Solid black

line gives average MHD Dashed black line gives hydro dyamic value Leftright shows D

and for topdown isotropic D D and DD sum

Table Computed p ower law indexex from p ower sp ectra

q Stand Dev

Figure MHD Total MHD Outward MHD Inward

Fig

T out in

cascade rate Jouleskgs Note that we retain the subscript

P P P

P b ecause these estimates are derived exclusively from the p ower sp ectra All estimates

for derived from the p ower sp ectra alone are listed in Table Again this estimate for

the total heating rate is to o large Note that neither of these three expressions is capable of

accounting for a negativecascaderateonany comp onent of the turbulence

ISO

The plot D derived from eq displays the results of applying the structure

ISO

function equations and to seven years of ACE data and to its right D

is the result of the division by the time lag The rate of dissipation p er unit mass for

outward blue and inward red propagating waves the total rate of dissipation solid

black and the hydro dynamic estimate dashed black can all b e derived from this plot

The outward traveling waves are more aggressively damp ed than those traveling inward

D derived from eq is an analysis of the D and hydro dynamic expressions and

D is the rate of turbulent dissipation p er unit mass assuming a D geometry with

k vectors p erp endicular to the mean eld direction Outward owing disturbances blue

again pro duce a higher rate of dissipation than their inward owing counterparts red The

solid black line again shows the total heating rate while the dashed black line is derived from

the hydro dynamic approximation eq which is the same for all panels in this gure

D D

D shows the D geometry result according to eq and D is the ensuing

plot for the rate of energy cascade along the meaneld direction The same color scheme

is employed and it is worthytonotethereversal of roles b etween the rate of dissipation

corresp onding to outward and inwardtraveling waves as the inwardmoving disturbances

nowhave a higher asso ciated rate of dissipation

Table Energy cascade rates computed from p ower sp ectra

Stand Dev Jouleskgs

out in T

Figure

P P P

Fig

Table Computed energy cascade rates for all D in Figure

Stand Dev Jouleskgs

Geometry MHD Outward MHD Total MHD Inward Hydro dynamic

ISO

Politano and Pouquet left the dimension of the turbulence as an op en variable

although they did derive the isotropic case as an example Our eq and

assumed one and two dimensional geometries in acknowledgementofthe Matthaeus

et al Bieber et al and Dasso et al analyses The third order structure function

for the comp osite DD geometry

D D D D

D D D

D D

then allows us to compute D which is the calculated rate of energy dissipation

p er unit mass for this comp osite geometry Figure shows the result of this analysis

where outward propagating waves blue again dominate over those traveling inward red

It is interesting to note that this result agrees very well with the isotropic calculation A

tabulation of the calculated rates of energy dissipation for all geometries and directions can

b e found in Table

The ab ove treatment of the solar wind has aws in that it assumes a direction of k for

each of the D and D geometry calculations The analysis do es not reliably separate parallel

and p erp endicular wavevectors thereby risking double counting and misiden tication To

solve these issues we adapt the treatment used by Dasso et al which employs a division of

the solar wind into high and low sp eed categories With this separation we can nowapply

eq and reliably

High and Low Wind Sp eed

In our analysis the classication of the slow solar wind is a twodayinterval for whichthe

average ow sp eed is b elow kms a classication consistentwith Dasso et al The

results of Dasso et al demonstrate a dominantkvector p erp endicular to the mean eld in

the low sp eed solar wind For this reason we treat the low sp eed wind with an assumed D

geometry The top panel of Figure shows the trace of the second order structure function

from which the trace of the p ower sp ectrum PSD is computed The same color scheme is

employed here where the blue line denotes the p ower asso ciated with outward propagation

the red line is the p ower of inward propagation and the solid black line gives the total p ower

contained within the sp ectrum From the p ower sp ectrum we can extract an estimate for

the rate of energy dissipation p er unit mass Jouleskgs which is again to o

large and would result in thermal proton temp eratures increasing with helio centric distance

Using the generalized Kolmogorov phenomenology of eq we get a total heating rate of

Jouleskgs which is marginally in agreement with exp ectations

Figure Secondorder structure function S for V kms Power sp ectral density

PSD computed from S D computed for D comp onent p erp endicular to mean eld

Resulting energy cascade rate computed from the D formalism

Figure Secondorder structure function S for V kms Power sp ectral density

PSD computed from S D computed for D comp onent parallel to mean eld Resulting

energy cascade rate computed from the D formalism

D is the resulting plot from the application of eqn to lowspeedintervals of

data The b ottom panel of gure shows the cascade rate p er unit mass of the slow solar

wind as a function of lag The outwardtraveling uctuations are again transfering energy

to small scales more strongly than uctuations traveling toward the sun The resulting

total for the energy cascade rate derived from structure functions for the D geometry

is Jouleskgs whichisvery consistent with the lo cal gradient in proton

temp erature

The classication of the fast solar wind is anyinterval with an average sp eed ab ove

kms Dasso et al show a highly onedimensional turbulence geometry oriented along

the mean magnetic eld direction for high sp eed streams As a result we apply eqn to

the fast solar wind Figure is a result of the high wind sp eed selection pro cess The top

two panels show the trace of the second order structure function S and the ensuing p ower

sp ectrum from whichwe extract a heating rate of and Jouleskg

s b oth of whicharemuch to o large

D D

D is the cal D is an application of eqn to the fast solar wind

culated cascade rate for the fast solar wind Figure applies these formalisms to the

fast wind Here the inward traveling uctuations show a stronger cascade to small scales

than those uctuations propagating away from the sun with the ma jority comp onentof

outward propagating uctuations exp eriencing negative cascade rates or sp ectral transfer to

larger length scales away from the dissipation mechanism This may b e a form of selective

decay or dynamic alignment acting on the eldaligned wavevectors The resulting total

energy cascade rate to small scales derived from structure functions for the D geometry is

Jouleskgs whichisagainvery consistent with the lo cal gradient of proton

temp eratures Table displays the computed cascade rates for slow and fast solar wind

derived from the structure function metho ds

Minority Comp onents

The ab ove analysis of the turbulence in high and lowsp eed streams fo cused on the ma jority

comp onents k vectors along the mean eld direction in the fast wind and k vectors p erp en

dicular to the mean eld in the slow wind Wewillnow consider the minority comp onentin

b oth the high and lowsp eed solar wind by selecting intervals in the correct range of wind

sp eeds with mean elds suitably oriented to display a single comp onent of the turbulence

Figure uses twodayintervals with an av erage owspeedbelow kms and

By restricting data intervals to those with nearly radial mean elds we pref

erentially observe the eldaligned k vectors while the p erp endicular k vectors are Doppler

shifted to zero frequency The top two panels of gure show the trace of the second

order structure function and the resultantpower sp ectrum which yields heating rates of

Jouleskgs which is marginally which is again to o large and

acceptable if it were the total cascade rate for the turbulence However the latter value

seems high for the cascade of the minority geometry alone

D is a result of applying the D formalism of eq to the slow solar wind

with radial mean elds The division by lag results in D and is a measure of the

turbulent cascade rate of the minority comp onent in the slow solar wind The dominant

outwardtraveling uctuations exhibit a stronger cascade to small scales than uctuations

traveling inward until the larger lag v alues at which p oint the inwardmoving uctuations

pro duce a higher cascade rate The total heating rate of Jouleskgs seems entirely

Figure Same as Figure but computed from data subset where V kms and

plausible

Figure is an analysis of the minority comp onent in the fast solar wind The gure

is comp osed of intervals with an average ow sp eed ab ove kms and a magnetic eld

orientation The top two panels show the trace of the second order

structure function and the p ower sp ectrum We extract dissipation rates of

Jouleskgs from the p ower sp ectra and b oth values are to o large The and

power contained in the outward and inward mo des are nearly indistinguishable

With the line of sight conned to the plane p erp endicular to the mean eld direction we

examine the twodimensional geometry describ ed by eq The lower two panels show

D D

derived from eq The heating rate the D geometry results D and D

of the outwardtraveling uctuations is only slightly larger than that for inwardpropagating

uctuations seemingly in keeping with the equalityofthepower sp ectra of the twocom

p onents The total heating rate of Jouleskgs is high but in this instance may

reect a transient pulse as the system evolves toward twodimensionality Ta

ble displays the results of the energy cascade rate of the minority comp onent in the fast

and slow solar wind as derived from structure function metho ds

Figure Yearly analysis of D using isotropic formalism Color convention holds

Yearly Analysis

Figure displays the results of applying eqn to eachyear of data from through

This is the isotropic analysis which app ears to give a go o d assessment of the average

heating rate if one disregards the geometry issues Table lists the computed cascade

rates for Figure and these same average values are shown in Figure Uncertainties

are plotted in Figure but are often smaller than the symbol used We nd that the

outwardpropagating comp onent is energetically dominantevery year and that this comp o

nent p ossesses a more aggressive direct cascade to small scales than the minorityinward

propagating comp onent No clear evidence of a solar cycle eect is seen Some years

and show strongly asymmetric cascade rates b etween the two comp onents but the

total heating rate remains somewhere b etween Jouleskgs This is the correct

and exp ected range of values and the source of the extreme asymmetry some years is not

well understo o d The hydro dynamic expression eq given by the dashed line in Figure

consistently yields lower estimates for the cascade rate at Jouleskgs which are

themselves not unreasonable estimates

Table Computed energy cascade rates for high and low wind sp eeds and their minority

comp onents in Figures and

Stand Dev Jouleskgs

Figure MHD Outward MHD Total MHD Inward Hydro dynamic

Fig

Figure Summary of heating rates shown in Figure and listed in Table Color

convention holds

Table Computed yearly energy cascade rates in Figure

Stand Dev Jouleskgs

Year MHD Outward MHD Total MHD Inward Hydro dynamic

Summary

With growing evidence that a turbulent cascade provides the energy to heat the background

plasma wehave turned to the thirdorder structure function formalism of Politano and

Pouquet in an eort to accurately resolve without heuristic assumption the rate

of energy cascade through the inertial range Wehave used the formalism of Politano and

Pouquet to express structure function forms applicable to hybrid turbulent geometries where

energy is distributed over wavevectors b oth parallel and p erp endicular to the mean magnetic

eld and we gather our statistics according to inward and outward propagating uctuations

Wehaveshown that the ab ove formalism works remarkably well and predicts a total energy

cascade that is consistent with the lo cal gradient of the proton temp erature Wehave

shown that most often the energetically dominant outward propagating comp onent exhibits

a stronger cascade to small scales than the minorityinward propagating comp onent which

seems consistent with observations b eyond AU The exception to this is seen in eld

aligned wavevectors within highsp eed winds

What is most signicant in this work apart from the demonstration that the inertial

range cascade provides a measurable dynamic to bring V B is the unication of

implied heating rates provided by largescale transp ort theory with that required

of dissipation at the small scales of the dissipation range and the heating rate inferred from

the lo cal gradient in proton temp erature This analysis accurately and ob jectively

resolves the rate of energy cascade through the inertial range It tracks the energy injected by

the largescale transients that are mo delled by transp ort theory and measures the passage of

energy though the intermediate scales of the inertial range as it cascades to the small scales

for dissipation As with Vasquez et al wehaveshown that the heuristic theories of

cascade rates based on p ower sp ectrum levels fails in this regard by consistently predicting

cascade rates that are to o large

Detrending and Stationarity

There is a philosophical distinction in the data that is dicult to address ob jectively Are

transients and various disturbances of a solar source part of the turbulence we wish to study

or are they p ossible sources that are separate from the homogeneous turbulence concept

This question is made more comp elling when one realized that a kms ramp in solar

wind sp eed over the course of hours which is a common and rather b enign observation is

comparable to the signal we extract in the analysis The general rotation of the IMF within

a magnetic cloud has a similar eect If these represent shears in the ow then they are

sources of the turbulence and should b e included in this analysis If they are converging

ows or ideal rotations without lo cal gradients then it is less clear what their role is lo cally

in the generation of turbulence

The fo cus of the study is to analyze the in situ dynamics of the turbulence and not

the largescale structures that drive the turbulence and so we attempted to lter through

the intervals and only use those that pass our tests We made attempts to remove such

signals from the data with a combination of detrending and stationarity testing We

found no signicant or consistentchange in the results except in the general decrease of

statistical signicance that comes with a reduction in the ensemble size For this reason

wecho ose not to employ such metho ds here and present the results of the simpler analyses

that use all data sub ject to selection criteria listed in the text Under this assumption all

transients contribute to the interplanetary turbulence and with enough distinct contributions

the ensemble includes all p ossible observations This seems to b e b oth the most reliable

approach and philosophically the most appropriate since at AU the suns activity remains

a ma jor source of energy driving the turbulence

Co de Testing

Because the results for the k cascade are unexp ected and strongly indicate that the ma

jority comp onent of outward propagating uctuations are enhanced through the cascade of

the minorityinwardpropagating comp onent we should describ e how the co de was tested

Ordinarily one would test co des of this typ e by building a synthetic data set with known

prop erties that closely resembles the statistics of the measured variables In this case it

is very dicult to build synthetic data with known and nontrivial S prop erties that still

closely mimic other asp ects of interplanetary data S hV B i etc After manyat

tempts we realized that this was not necessary to testing the co de we develop ed here and

that all that is needed is to create a time series of known prop erties and predictable con

clusions We did this by rst creating a time series for magnetic comp onents B i i and

velo city comp onents V i i with Eachcomponentof B has the same value

and likewise each comp onentof V is the same These time series b ear little resemblence

i to reality but they do lead to predictable and testable conclusions They yield Z

h i

ISO

and Z i so that S L L L L S D and L L L

h i

ISO

L L Similar results can b e obtained for the D and D formalisms D L

This p ermits endtoend testing of the co des with a high degree of validation Later tests

can relax asp ects of test conditions as warranted

Chapter

Turbulence of the Inner Heliosphere

Anisotropic Turbulence in the Solar Wind

Up to this p oint this thesis has dealt with prop erties and measurements in the inertial range

of the solar wind Recalling the mo del of energy transfer in the solar wind discussed in

Chapter we can trace the cascade of energy from large scales of the energycontaining

range through the intermediate scales of the energyconserving inertial range into the small

scales of the dissipation range where the uid energy is converted to heat Chapter provides

a quantitative description of the rate at which the energy cascades through the inertial range

We fo cus here on the results of Leamon et al Smithetal and Hamilton

et al who studied prop erties of solar wind turbulence at AU in the high frequency

end of the inertial range Smith et al study the magnetic variance anisotropy the ratio

of magnetic uctuations p erp endicular to the mean magnetic eld to uctuations parallel

to the mean eld and nd it scales with b oth proton b eta and the amplitude of the p ower

sp ectrum In their analysis they also plot proton b eta against the amplitude of the p ower

sp ectrum of magnetic eld oscillations and nd them to b e correlated which leads to the

result that magnetic variance anisotropy scales with b oth quantities Proton b eta is a unitless

quantity dened as the ratio of thermal to magnetic energy densityandifthevariation of

the magnetic eld is related to the proton temp erature it will b e dicult to discern

whether the anisotropy arises due to the variation of the proton b eta or the amplitude of

the p ower sp ectrum

Leamon et al and Hamilton et al study the wavevector anisotropy in the solar wind

by applying the Bieber et al metho d This metho d of analysis is based on the ratio of

power sp ectra in the x and ydirections in mean eld co ordinates as dened by Belcher and

Davis wherez Bj B j is the unit vector in the direction of the mean magnetic eld

y z R jz R j where R is the radial direction and is assumed to b e the direction

of the solar wind ow andx y zThewavevector anisotropy gives an estimate of the

relative energy contained in the uctuation vectors p erp endicular and parallel to the mean

magnetic eld Bieber et al determined that approximately of the energy is contained

in the D plane p erp endicular to the magnetic eld Leamon et al apply the same

analysis to observations in the inertial range at AU and nd of the energy to reside

in the p erp endicular comp onent Hamilton et al use the same analysis in b oth the inertial

and dissipation ranges at AU but with an additional wind sp eed selection criteria Their

results demonstrate a nearly equal division of energy b etween wavevectors p erp endicular

and parallel to the mean eld with no dep endence of wavevector anisotropy on wind sp eed

Figure Helio centric distance for Helios during years when data for this study was

obtained

Figure Number of intervals selected for study in increments of AU

In this analysis we extend the metho ds that have b een used previously at AUtothe

inner heliosphere with the Helios spacecraft We analyze the magnetic variance and wave

vector anisotropies as a function of b oth wind sp eed and helio centric distance We attempt

to determine whether there is an observed spatial evolution of the these turbulent prop erties

as a function of helio centric distance and whether there is a distinction b etween high and

low wind sp eeds

Helios Analysis

In the analysis we use Helios spacecraft magnetic eld data from late through

with second resolution The Helios spacecraft traversed an orbit with a p erihelion of

AU and an aphelion of AU so the data spans this range as well Figure displays the

helio centric distance of the Helios spacecraft from until We use over hand

Figure Distribution function of events according to wind sp eed

selected stationary intervals ranging from four to six hours in length to create magnetic p ower

sp ectra bythe Blackman and Tukey metho d with auto correlation functions Intervals

with a dened p owerlaw index through the inertial range were kept and intervals without a

powerlaw index were discarded Intervals were chosen on the basis of providing a wide range

of prop erties to analyze In particular we made sure to include intervals spanning the inner

heliosphere down to AU and to cho ose intervalswithbothahighandlowaverage solar

wind velo city Figure displays the number of intervals chosen as a function of distance

from the sun in increments of AU and Figure is a probability distribution function

of events selected as a function of average solar wind sp eed These two plots demonstrate

a full coverage of the inner heliosphere down to AU and a range of wind sp eeds from

kms

In order to b e able to select intervals based on stationarity wind sp eed and helio centric

distance I rst needed to convert raw data les into data plots This pro cess required

that I learn a programming language with plotting capabilities I have used the scientic

computing languages of FORTRAN and C but neither of these have the ability to create

gures To solve this problem I learned enough commands in the In teractive Data Language

IDL to adapt co des written by RJ Leamon for the study of ACE measurements so they

would b e able to plot Helios spacecraft observations Once the Helios measurements were

plotted I could then cho ose the intervals and create magnetic p ower sp ectra From the

power sp ectra we analyze twointervals in the inertial range The frequency ranges weused

are from to Hz and from to Hz whichwe refer to as

the low and high frequency intervals resp ectively The two parameters of the solar wind

turbulence we examine are the magnetic eld variance and wavevector anisotropies

Magnetic Variance of the Inner Heliosphere

The magnetic eld variance anisotropy is the ratio of magnetic uctuations p erp endicular

to the mean magnetic eld to the uctuations parallel to the mean eld As stated in the

intro duction Smith et al found variance anisotropy to scale with b oth proton b eta

and the amplitude of the sp ectrum of magnetic uctuations BB This result was

the pro duct of an analysis based on measurements taken with the ACE spacecraft at AU

and we seek to extend this analysis to the inner heliosphere as near to the sun as AU

Figure Scatter plot of variance anisotropy for high frequency range side axis vs low

frequency range b ottom axis using all data

Figure Scatter plot of variance anisotropy side axis for low frequency range vs thermal

proton b ottom axis for all data

with the Helios spacecraft In our analysis we measure the magnetic variance anisotropy

in b oth the high and low frequency intervals Figure is a comparison of the variance

anisotropy measured in the high frequency range plotted against the anisotropy measured

for low frequencies The dashed line shows equality and is provided as a reference This

gure displays a direct relationship b etween the two frequency ranges with p erhaps a slightly

stronger anisotropy in the low range at high values This direct relationship implies that the

anisotropy of the turbulence do es not change as it travels through the inertial range The

higher anisotropyvalues in the low frequency range for strongly anisotropic events implies

that such o ccurences b ecome less anisotropic at higher frequencies

The magnetic variance anisotropy of the low frequency range is plotted against in

Figure This plot is created using all of the intervals we analyzed with no selection based

on helio centric distance or wind sp eed As in the case of Smith et al we again nd the

variance anisotropy to scale with Figure plots the magnetic variance anisotropyof

the low frequency range against BB The top panel of the gure uses all intervals

Figure top Scatter plot of variance anisotropy for low frequency range side axis vs

integrated magnetic p ower normalized to mean eld intensity b ottom axis for all data

b ottom Same for helio centric distance range AU

from the Helios data base while the b ottom panel uses only the intervals in the AU

range Both results show a scaling of the variance anisotropywith BB This is again in

accordance with the Smith et al results at AU In fact the trend line shown in Figures

and is the same line used in the analysis at AU These results indicate that the magnetic

variance anisotropy exp eriences no spatial evolution b etween and AU

Similar to the results of Smith et alwe again nd a dep endence of the magnetic variance

anisotropy on b oth the proton b eta and the amplitude of the p ower sp ectrum of magnetic

uctuations Prior to the analysis wethought taking samples from varying helio centric

distances might break the correlation b etween the dep endence on b oth and BB to

determine which quantity is fundamental to the variance anisotropy but this did not happ en

Variance anisotropy scales with b oth quantities all the waydown to AU Figure plots

the thermal energy of the protons in the solar wind against the amplitude of the p ower

sp ectra and they are shown to b e correlated This correlation is what leads the magnetic

variance anisotropy to scale with b oth parameters in Figure

Spatial Evolution of the Power Sp ectral Index

The intervals included in this analysis were chosen on the basis of having a w elldened

power law index That is on a loglog plot of the p ower sp ectrum of magnetic uctuations

as a function of frequencyallintervals are easily t with a straight line within the inertial

range The p ower lawindexq is the slop e of this straight line In the turbulence describ ed

by Kolmogorov the inertial range is characterized bya power law index but in our

analysis we do not require intervals to havea power law index so long as the index is

welldened The p ower law index of the intervals therefore exp eriences muchvariation

Bavassano et al demonstrate a dep endence of the p ower law index in the inertial range

Figure Scatter plot of proton thermal energy side axis vs integrated magnetic uctu

ation amplitude b ottom axis for all data

of solar wind uctuations on the helio centric distance They show a steepining of the p ower

sp ectra with increasing distance from the sun Figure plots the sp ectral index as a

function of helio centric distance in an attempt to conrm the results of Bavassano et al

This gure contains results from b oth the high and low frequency intervals and analyzes all

of the data in the top panel high wind sp eed streams in the middle panel and low sp eed

streams in the b ottom panel Although the errors are rather large to rst order the sp ectral

index of all data seems to show the same dep endence on helio centric distance as was shown

by Bavassano et al The circular data p ointattheAUvalue in all three panels is a result

drawn from Hamilton et al This plot is a source of concern and casts some doubt

on the results that follow The low frequency sp ectral indices hover around the q

mark whichischaracteristic of Kolmogorov inertial range turbulence The high frequency

sp ectral indices however are much steep er around the q mark The selfsimilar

inertial range should not exp erience such a drastic change of p ower law index b etween two

frequencies that lie securely within it This result has made it necessary to run additional

tests of the programs used in the analysis to ensure these results are physically signicant

and not a synthetic pro duct of the co des

WaveVector Anisotropy of the Inner Heliosphere

As stated in the intro duction the test to nd the orientation of the wavevectors in solar

wind turbulence was develop ed by Bieber et al and its rst application showed of

the energy exists in the D comp onent p erp endicular to the mean magnetic eld Hamilton

et al employed this same test and found a nearly equal mixture of wavevectors parallel and

p erp endicular to the mean magnetic eld Furthermore Hamilton et al selected events based

on the average ow sp eed of the solar wind for eachinterval and found no distinction in wave

vector orientation b etween the intervals with high sp eed V kms and low sp eed

V kms streams The metho d for determining the energy in the p erp endicular

Figure Means and variances of p ower law indexes for low frequency range squares and

high frequency range triangles computed from all data top high sp eed wind intervals

middle and low sp eed winds b ottom

and parallel directions relies on the ratio of the p ower in the two comp onents p erp endicular

to the mean eld

jq j

r k k

y jq j

jq j jq j

k r k

jq j

and k is the angle b etween the mean magnetic where k

s BR

V cos V sin

SW BR SW BR

eld and the ow direction of the solar wind is the spacecraft frame frequency r is the

energy in b oth the eld aligned and p erp endicular comp onents and r is the fraction

of total energy in the eldaligned comp onent This explanation of the Bieb er analysis is

courtesy of Smith et al

Here we employ the Bieb er test to examine the dep endence of wavevector anisotropyon

not only wind sp eed as demonstrated by Hamilton et al but also on helio centric distance By

using data from the Helios spacecraft wehave the opp ortunity to compare the p ercentage of

energy contained in the wavevectors p erp endicular and parallel to the mean eld at various

lo cations in the inner heliosphere By comparing results at dierent helio centric distances

it is p ossible to determine whether the wavevector anisotropyevolves as the solar wind

expands from its solar source Figure displays the ratio of p ower in the two comp onents

p erp endicular to the mean eld as a function of the angle b etween the mean eld and the

solar wind ow direction The minimum t of this plot gives an estimate of the

p ercentage of energy contained in the eldaligned wavevector All of the data ranging

from helio centric distances of to AUwas used to create this gure with no selection

based on wind sp eed The top panel displays the results of the analysis in the low frequency

Figure Distribution of computed ratio P P binned by for all low frequency top

yy xx BR

solution solid line and panel and high frequency b ottom panel data Minimum

r r

curves dashed lines are shown

intervals and the b ottom panel displays the results from the high frequency intervals These

results are dep endent on the sp ectral index and will change if Figure changes The

dubious values of the sp ectral index lie in the high frequency range and so it would b e

the b ottom panel of Figure that is likely to change if the sp ectral indices also change

As the plots stand however it app ears that the low frequency interval contains a stronger

p ercentage of eldaligned wavevectors than the high frequency interval This implies that

as the turbulent energy moves through the inertial range it b ecomes less eldaligned as the

energy is transferred to the p erp endicular D comp onent

Figure displays the fraction of energy contained in the wavevectors aligned with

the mean magnetic eld as a function of helio centric distance The top panel of the gure

displays results using all intervals the middle panel selects for events with an average wind

sp eed greater than kms and the b ottom panel includes only intervals with a wind sp eed

b elow kms Although the variance of the results is large vertical lines the p ercentage

of energy contained in the wavevectors aligned with the mean magnetic eld for b oth high

frequency triangles and low frequency squares intervals app ears to b e indep endentof

lo cation in the inner heliosphere The indep endence of helio centric distance is particularly

true in the top panel of Figure The fast wind results seem to show a slight progression

towards turbulence that is more highly aligned with the mean magnetic eld as the distance

from the sun increases but the slow wind results show no real correlation with helio centric

distance The high wind sp eed streams seem to show no more p ercentage of energy contained

in the wavevector aligned with the mean magnetic eld than the low wind sp eed streams

This is contrary to the results of Dasso et al who showed the turbulence in the fast

solar wind to b e strongly onedimensional in the direction parallel to the mean magnetic

eld and the slow solar wind to b e strongly twodimensional in the plane p erp endicular to

the mean magnetic eld Similar to the results of this analysis Hamilton et al also show

Figure Means and variances of slab fraction for all wind sp eeds top high wind

sp eeds middle and low wind sp eeds b ottom for low frequency intervals squares and

high frequency intervals triangles

no dep endence of wavevector anisotropy on the solar wind sp eed Hamilton et al provide

a resolution to these seemingly contradictory results The resolution lies in the dierence

of frequencies used for the two analyses Hamilton et al show that f Hz represents

uctuations that are to o shortlived to b e of solar origin and this holds down to the scop e

of the present analysis at AU All of our results arise from f Hz whereas the

work of Dasso et al contains data from f Hz Hamilton et al suggest that the two

dimensional turbulence in slow wind streams and the one dimensional turbulence in the fast

wind streams found by Dasso et al could b e a remnant of the solar source whichdoesnot

app ear at the high frequencies used in this study

Conclusion

In this analysis we extend the ndings of Leamon et al Smith et al and Hamilton

et al to AU We extend the previous results based on observations made bythe

ACE and WIND spacecraft to observations made by Helios in an attempt to determine

whether the prop erties of solar wind turbulence exp erience a spatial evolution b etween

andAU Wehave conrmed the results of Smith et al made at AU that the magnetic

variance anisotropy scales with b oth proton b eta and the amplitude of the p ower sp ectrum

Similar to their results wewere unable to break the correlation b etween the proton b eta

and the amplitude of the p ower sp ectrum These results hold down to AU the closest

to the sun Helios travels Hamilton et al found the wavevector anisotropyatAUinthe

high frequency end of the inertial range to b e indep endent of solar wind sp eed We also

analyze the high frequency end of the inertial range but this analysis is not restricted to

measurements made at AU instead it includes measurements made in the range of to

AU We also nd the wavevector anisotropy to b e indep endent of solar wind sp eed In

addition we nd that there is no dep endence of the wavevector anisotropy on helio centric

distance These results of the magnetic variance and wavevector anisotropies that were

made using data from the Helios spacecraft demonstrate that the turbulence measured at

AU is nearly identical to the turbulence at AU This implies that the turbulence in the

solar wind is fully develop ed by the time it reaches AU

Chapter

Parting Remarks

When I b egan my academic career at the University of New Hampshire I had not the faintest

idea of what the solar wind was Over three years later I am completing an honors thesis on

the topic Turbulence of the Solar Wind The journey has not b een easy but it has b een

enjoyable and I have learned a lot Through the researchI have done under the guidance

of Professor Charles Smith I have learned several new programming languages I have b een

intro duced to structure functions and turbulence theoryIhave had the opp ortunityto

presentmywork at professional conferences around the country and when I leave New

Hampshire for graduate scho ol I will have written at least one and p ossibly three pap ers

for publication in professional refereed journals In Chapter of this thesis I outlined and

explained some of the basic prop erties of the sun and the solar wind Chapter is an

extension of these prop erties to a sub discipline of solar wind studies that is concerned with

the turbulent prop erties of the solar wind One of the topics discussed in this chapter is

the observed departure of the solar wind temp erature as a function of helio centric distance

from the temp erature predicted by an adiabatic expansion This nonadiabatic expansion

implies the presence of a heating source in the solar wind Mo dels based on the turbulent

transp ort of energy provide a sucientamount of energy to account for the observed heating

of the solar wind This mo del shows that turbulence is the source of heating in the solar

wind Chapter presented a mo del of magnetohydro dynamic turbulence that I followas

an explanation of the turbulence in the solar wind Energy is generated at large scales it

then cascades through the energyconserving intermediate scales of the inertial range and is

eventually dissipated as heat in the small scales of the dissipation range Chapter to ok a

closer lo ok at the inertial range and in particular examined the energy cascade rate inferred

from p ower sp ectra and the cascade rate calculated with the thirdorder structure function of

magnetohydro dynamic uctuations We show that the cascade rate pro duced by the p ower

sp ectral metho ds are to o high They are so high that they imply the solar wind heats up as

it expands from AU The heating rate computed from the thirdorder structure function

provides a more reasonable estimate of energy cascade Chapter uses the high frequency

end of the inertial range to analyze the spatial ev olution of the magnetic variance and wave

vector anisotropies We nd solar wind turbulence measured by these twoquantities to b e

nearly identical at and AU The implication of these results is that the turbulence of

the solar wind is fully evolved by the time it reaches AU

Bibliography

Alfven H Existence of electromagnetichydro dynamicwaves Nature

Balogh A Solar Wind Ulysses in Encyclopedia of Astronomy and Astrophysics

Bavassano B M DobrowolnyGFanfoni F Mariani and N F Ness Radial evolution

of p ower sp ectra of interplanetary Alfvenic turbulence J Geophys Res

Belcher J W L Davis and E J Smith Largeamplitude Alfven waves in the inter

planetary medium Mariner J Geophys Res

Belcher J W and L Davis Largeamplitude Alfven waves in the interplanetary

medium J Geophys Res

Bieb er J W W Wanner and W H Matthaeus Dominanttwodimensional solar wind

turbulence with implications for cosmic ray transp ort J Geophys Res

Biermann L Kometenschweife und solare Korpuskularstrahlung Z Astrophys

Blackman R B and J W Tukey The Measurement of Power SpectraDover Mineola

N Y

Burlaga L F Solar wind Magnetic eld Encyclopedia of Astronomy and Astrophysics

Burlaga L F N F Ness A H Acuna R P Lepping J E P ConnerneyEC

Stone Crossing the termination sho ckinto the heliosheath Magnetic elds Science

Chapman S Note on the solar corona and the terrestrial ionosphere Smithsonian

Centr Astrophys

Cranmer S R Why is the fast solar wind fast and the slow solar wind slow Solar

Wind SOHO ConferenceProceedings

Dasso S L J Milano W H Matthaeus and C W Smith AnisotropyinFast and

Slow Solar Wind Fluctuations Astrophys J Lett LL

Elsasser W M The hydromagnetic equations Phys Rev

Europ ean Space Agency and the National Aeronautics and Space Administration

SOHO Online June httpsohowwwnascomnasagov

Feimer W Voyager at AU National Aeronautics and Space Administration Novem

b er

Fowles G R and G L Cassiday Analytical Mechanics Belmont CA Thomson

Bro oksCole

Freeman J W Estimates of solar wind heating inside AU Geophys Res Lett

Ghosh S W H Matthaeus D A Rob erts and M L Goldstein The evolution of slab

uctuations in the presence of pressurebalanced magnetic structures and velo city shears

J Geophys Res a

Ghosh S W H Matthaeus D A Rob erts and M L Goldstein Waves structures

and the app earance of twocomp onent turbulence in the solar wind J Geophys Res

Gurnett D A Solar Wind Plasma Waves Encyclopedia of Astronomy and Astro

physics

Hamilton K C W Smith B J Vasquez and R J Leamon J Geophys Res sub

mitted

Hundhausen A J Coronal Expansion and Solar Wind Berlin Germany Springer

Isenberg P A C W Smith and W H Matthaeus Turbulent heating of the distant

solar wind byinterstellar pickup protons Astrophy J

Jet Propulsion Lab oratory UlyssesMission Overview Online January

httpulyssesjplnasagov

Jet Propulsion Lab oratory Voyager Mission Online August

httpvoyagerjplnasagov

Krieger A S A F Timothy and E C Ro elof A coronal hole and its identication

as the source of a high velo city solar wind wtream Solar Phys

Kolmogorov A N The lo cal structure of turbulence in incompressible viscous uid for

very large Reynolds numb ers Dokl Akad Nauk SSSR a Reprinted in

Proc R Soc London A

Kolmogorov A N Energy dissipation in lo cally isotropic turbulence Dokl Akad Nauk

SSSR b Reprinted in Proc R Soc London A

Kraichnan R H Inertial range of hydromagnetic turbulence Phys Fluids

Leamon R J C W Smith N F Ness W H Matthaeus and H K Wong Observa

tional constraints on the dynamics of the interplanetary magnetic eld dissipation range

J Geophys Res

Leamon R J C W Smith N F Ness and H K Wong Dissipation range dynamics

Kinetic Alfven waves and the imp ortance of J Geophys Res

Marsch E Kinetic Physics of the Solar Wind Plasma in Physics of the Inner Helio

sphere vol edited byRSchwenn and E Marsch p SpringerVerlag Berlin

Marsch E Turbulence in the solar wind in Reviews in Modern Astronomy vol

edited by G Klare p SpringerVerlag New York

Matthaeus W H and M L Goldstein Stationarity of magnetohydro dynamic uctua

tions in the solar wind J Geophys Res

Matthaeus W H M L Goldstein and J H King J Geophys Res

Matthaeus W H and M L Goldstein Lowfrequency f noise in the interplanetary

magnetic eld Phys Rev Lett

Matthaeus W H and Y Zhou Extended inertial range phenomenology of magneto

hydro dynamic turbulence Phys Fluids B

Matthaeus W H M L Goldstein and D A Rob erts Evidence for the presence of

quasitwodimensional nearly incompressible uctuations in the solar wind J Geophys

Res

Matthaeus W H C W Smith and S Oughton Dynamical age of solar wind turbulence

in the outer heliosphere J Geophys Res A

Matthaeus W H G P Zank C W Smith and S Oughton Turbulence spatial

transp ort and heating of the solar wind Phys Rev Lett

McComas D J S J Bame PBarker W C Feldman J L Phillips PRileyand

J W Griee Solar wind electron proton alpha monitor SWEPAM for the Advanced

Comp osition Explorer Space ScienceRev a

McComas D J et al Ulysses return to the slow solar wind Geophys Res Lett

Neugebauer M and C W Snyder Mariner observations of the solar wind J Geo

phys Res

Parker E N Dynamics of the interplanetary gas and magnetic elds Astrophys J

Parker E N Interplanetary Dynamical Processes WileyInterscienceNewYork

Parks G K Physics of Space Plasmas Boulder Colorado Westview Press

Plasma Physics of the Local CosmosWashington DC The National Academies Press

Po desta J J D A Rob erts and M L Goldstein Power sp ectrum of small

scale turbulentvelo city uctuations in the solar wind J Geophys Res A

doi JA

Politano H and A Pouquet von KarmanHowarth equation for magnetohydro dynam

ics and its consequences on thirdorder longitudinal structure and correlation functions

Phys Rev E RR a

Politano H and A Pouquet Dynamical length scales for turbulent magnetized ow

Geophys Res Lett b

Porsche H Heliosmission Mission ob jectives mission verication selected results

SSE ConferenceProceedings

Richardson J D K I Paularena A J Lazarus and J W Belcher Radial evolution

of the solar wind from IMP to Voyager Geophys Res Lett

Rob erts D A L W Klein M L Goldstein and W H Matthaeus The nature and

evolution of magnetohydro dynamic uctuations in the solar wind Voyager observations

J Geophys Res

Russell C T R A Mewaldt and T T von Rosenvinge The Advanced Composition

Explorer Mission Boston Kluwer Academic Publishers

Schwenn R Solar Wind Global Prop erties Encyclopedia of Astronomy and Astro

physics

Smith C W W H Matthaeus and N F Ness Measurement of the dissipation range

sp ectrum of magnetic uctuations in the solar wind with applications to the diusion of

cosmic rays Proc st Int Conf Cosmic Ray Conf

Smith C W M H Acuna L F Burlaga J LHeureux N F Ness and J Scheifele

The ACE magnetic eld exp eriment Space Sci Rev

Smith C W W H Matthaeus G P Zank N F Ness S Oughton and J D

Richardson Heating of the lowlatitude solar wind by dissipation of turbulent magnetic

uctuations J Geophys Res

Smith C W PAIsenb erg W H Matthaeus and J D Richardson Turbulent

heating of the solar wind by newb orn interstellar pickup protons Astrophys J

Smith C W B J Vasquez K Hamilton Interplanetary magnetic uc

tuation anisotropy in the inertial range J Geophys R es A

doi JA b

Smith C W K Hamilton B J Vasquez and R J Leamon Dep endence of the

dissipation range sp ectrum of interplanetary magnetic uctuations on the rate of energy

cascade Astrophys J Lett

Smith C W B J Vasquez K Hamilton B T MacBride J A Tessein M A Forman

R J Leamon Turbulence sp ectrum of interplanetary magnetic uctuations and the rate

of energy cascade IGPP Conference Pro ceedings

Stone E C The Advanced Comp osition Explorer The Advanced Composition Explorer

Mission Boston Kluwer Academic Publishers

Totten T L J W Freeman and S Arya An empirical determination of the p olytropic

index for the freestreaming solar wind using Helios data J Geophys Res

Unti T W J and M Neugebauer Alfven waves in the solar wind Phys Fluids

Vasquez B J CW Smith K Hamilton B T MacBride and R J Leamon Evalua

tion of the turbulent energy cascade rates from the upp er inertial range in the solar wind

at AU J Geophys Res in press

Verma M K D A Rob erts and M L Goldstein Turbulent heating and temp erature

evolution in the solar wind J Geophys Res

Wilcox J M The interplanetary magnetic eld Solar origin and interplanetary eects

Space Sci Rev

Zank G P W H Matthaeus and C W Smith Evolution of turbulent magnetic

uctuation p ower with heliospheric distance J Geophys Res

Zhou Y and W H Matthaeus Mo dels of inertial range sp ectra of interplanetary

magnetohydro dynamic turbulence J Geophys Res