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Home , Magnetic helicity, Solar wind

Solar Turbulence

A Study of Corotating Interaction Regions at AU

Jerey A Tessein

DepartmentofPhysics

University of New Hampshire

Durham NH

May

Abstract

The is a sup ersonic ow of coming out of the and propagating

throughout the Because of the Suns rotation and the variability in the source

of the solar wind fast moving wind can crash into slow wind This results in an interface

that can rep eat as the Sun rotates known as a Corotating Interaction Region CIR where

the slippage of fast wind past slow wind is a source of turbulence Ihave taken CIR

events from the years and to analyze their turbulence parameters

and compared them to nonCIR ambient turbulence from years The CIR

turbulence diers from the ambient turbulence b ecause it is eectively a p ointsourceand

the hop e is to nd evidence of turbulent energy emanating from the interface This should

present some nonhomogeneities not seen by nonCIR turbulence We nd that there are

some dierences with the CIR turbulence but for the most part the CIR turbulence has the

same results as past turbulence studies This most likely means that the evolution of solar wind turbulence is faster than that of the source

Contents

Basics of the Solar Wind

Prop erties of the Solar Wind

Termination Sho ck

History of Solar Wind Measurements

Current Missions

Coronal Mass Ejections

Origin of the Solar Wind

Parker Solutions

Three Mathematical Approaches to the Solar Wind

Solar Wind Development

Bulk Solar Wind at AU

Fast And Slow Wind

Transients in the Solar Wind

Sho cks

Corotating Interaction Regions

Magnetic Clouds

Cosmic Rays

Plasma

Magnetohydro dynamics

Plasma

Fluctuations

Developmentofturbulence

Sp ectral Index

Geometry

Inertial Range

Dissipation Range

Observations

Dierences Between Fast and SlowWind

Smaller Scale Analyses

Tessein Hourly Analysis

Background

DataAnalysis

ACE s Merged Data i

ACE High Reolution MAG Data Analysis

Helios MAG Data

Conclusions

Corotating Interaction Regions at AU

Motivation

Data

Results

Discussion ii

Acknowledgements

First and foremost I would liketothankmy advisor Dr Charles W Smith without

whom none of this would b e p ossible I have come a long way from the rst year

ma jor I was several years ago Working for Dr Smith has op ened many do ors for me

including publications attending meetings and acceptance to graduate scho ol The career

path that lays ahead of me has b een made p ossible by his excellentmentoring

Iwould like to thank my fellowphysics ma jors who have b een excellent supp ort over the

last four years Instead of b eing in constant comp etition with one another my class has b een

more like a supp ortivecommunityofwhich b eing a memb er has b een a pleasant exp erience

Iwould also like to thank Dr Bernie Vasquez for helping out with some imp ortant things

Lastly I need to thank my family and friends They are go o d p eople and have always

b een there for me

Chapter

Basics of the Solar Wind

Prop erties of the Solar Wind

The solar wind is a sup ersonic gas propagating throughout interplanetary space It is an

extension of the solar driven by the dierence in gas pressure b etween the Sun

and space The vast ma jority of the constituent particles are and alpha particles



He also has an average abundance of bynumb er In addition isotop es

up to Iron have b een observed The solar wind has an average densityofabout



cm near is dep endent on solar activity and wind sp eed during times of

solar activity the density tends to spike Wind sp eed can vary from kms to kms

however nominal sp eed during times of reduced solar activity is kms to kms The



average temp erature is ab out K although at such an extremelylowdensity there is very

little heat capacity Like density the temp erature is related to solar activity

Just as with Earth the Sun has a mostly dip olar magnetic eld The pro cess that

generates the magnetic eld is called a dynamo It is susp ected that the magnetic eld is

generated by rotating charged uid in the interior The solar magnetic eld encompasses the

entire and is known as the Interplanetary hereafter referred to

as the IMF

The IMF is coupled to the solar wind and is carried out by it Lo cal electromagnetic

interactions are governed by the geometry of the IMF gyrate around the eld lines

The gyration has a characteristic radius given by

mv

r

jB j

Magnetosphere

When the solar wind reaches a with a magnetic eld like Earth it interacts Earths

magnetic eld generates a barrier around the planet known as the magnetosphere On the

day side of the magnetosphere and around part of the night side there exists an interface

known as the b ow sho ck This is the lo cation where the of the solar wind

equilibrates with the magnetic pressure of Earths magnetosphere This eect is similar to

arockinafastmoving river There is a stationary sho ckwave in the stream that causes

the water to slide around the ro ck In the case of the the sho ckdoesmove

dep ending on the pressure from solar wind for example a can compress the

bow sho ck closer to Earth Likewise during quiet solar wind Earths magnetic eld is

Figure Artists rendition of the heliosphere Image courtesy of NASA

able to push the b ow sho cksunward As the solar wind is diverted around the side of the

magnetosphere it forms a long narrowing b oundary on the night side of Earth called the

magnetotail

Termination Sho ck

The solar wind extends out to approximately AUAU is equivalent to the EarthSun

distance where it reaches the termination sho ck This is the p oint where pressure from the

solar wind balances with the Lo cal LISM When the the solar wind

reaches the heliopause it joins with the ow of the LISM The termination sho ck completely

surrounds the heliosphere and is approximately bullet shap ed The sho ck has b ecome an

ob ject of study in recentyears b ecause we are getting our rst in situ measurements of it

from Voyagers and An artists depiction of the entire heliosphere is shown in Figure

History of Solar Wind Measurements

Studies of solarterrestrial relations have a long history Before the space age there were

numerous ways to remotely measure the eects of solar activity on our planet

studies go back thousands of years and chemical analysis of tree rings is also an indicator

of the sunsp ot record sunsp ot numb er correlates to solar activity A lack of solar activity

from to called the has b een linked to unusually cold

Viking settlements in Greenland correpsond to what is known as the Medievel Warm Perio d

immediately preceeding the Maunder Minimum During the Maunder Minimum Vikings

abandoned Greenland p ossibly due to climate change In the twentieth century studies of

tails p ortended the existence of a solar wind Astronomers found that contain a

tail that always p oints antisunwardInadditiontoadusty tail a comet has a second

tail that contains ions This is known as a plasma tail The observation that the plasma

tail always p oints antisunward meant that there must b e a continuous solar wind During

the early s most astronomers thought that the solar wind only came in intermittent

bubbles It was referred to as solar corpuscular radiation In predicted

the existence of a continuous sup ersonic solar wind This was so on conrmed by in situ

measurements from early satellites This forms the basis for our current picture of the solar

wind

Current Missions

The arrival of the space age has greatly increased our knowledge of the solar wind In the

s the Helios spacecraft wentwithinAU of the Sun the closest of any spacecraft to

date A new mission Solar Prob e is scheduled to get much closer to the Sun around

The twoVoyager spacecraft launched in have reached the termination sho ck and are

returning the rst in situ measurements of the b oundary regions b etween the Heliosphere and

the LISM As of the IBEX mission is mapping the termination sho ckby measuring

neutral particles coming in from the LISM launched in did several p olar

orbits ab out the Sun revealing much ab out prop erties of the highlatitude solar wind The

spacecraft that will b e used for this thesis ACE Advanced Comp osition Explorer was

launched in and takes plasma measurements at the L p ointbetween the Sun and

Earth

Solar Cycle

ween stages of solar min The Sun go es through eleven year cycles of activity switching b et

imum and solar maximum This eect is thought to b e a result of the Suns dierential

rotation Because the Sun is not a rigid b o dy its equatorial regions rotate faster than the

p olar regions Because low latitudes on the Sun rotate faster than high latitudes the equator

has a rotational p erio d of ab out days and the p olar latitudes take ab out days to com

plete a rotation the magnetic eld can quickly b ecome twisted and mangled It is thought

that as the eld b ecomes more and more distorted there is an increase in solar activity This

is the rise toward solar maximumEventually when the eld b ecomes extremely distorted

it is able to reset itself and the pro cess b egins again At this p oint the Sun would b e re

turning to solar minimum The details of the resetting pro cess are mostly unknown During

the time that the magnetic eld is highly distorted the sunsp ot count b ecomes elevated

likewise when the Suns magnetic eld is more dip olar the sunsp ot countislow The last

solar maximum was around and as of the Sun is coming out of solar minimum

into the next maximum It is during the solar maximumsthatwe see the most solar ares

and CMEs see Section which can o ccur several times a week During solar minimum

these phenomena are only seen a few times a year and tend to b e smaller in magnitude

Coronal Mass Ejections

Sometimes magnetic energy on the Sun can cause a violent release of energyeventually

triggering a CME The CME b egins as a prominence A prominence

is a magnetic eld line coming o the surface of the Sun and going back in forming a lo op of

plasma emitting visible light The fo otprints of the prominence are areas of opp osite magnetic

p olarityAt some p oint the prominence builds up enough energy to release massive amounts

of matter This is thought to b e a result of a pro cess where magnetic

eld lines cross disconnect then reconnect with a new top ology releasing energyInatyp cial

CME a massive amount of matter is released from the Sun in one giant burst This release

of energy triggers extremely hot and fast solar wind and b ecuse of the sup ersonic sp eeds

a sho ck When the CME reaches Earth it interacts with the day side of the magnetopause

The interaction sends current around to the night side of the magnetosphere and up on

reaching it connects to the Up on reaching the ionosphere a

is triggered The visible result of geomagnetic storms are however there are other

eects as well A geomagnetic storm can cause current to run through p ower lines damaging

them and can throw o GPS signals and radio communication One little known impact of

geomagnetic storms is on racing pigeons Since the pigeons use the geomagnetic eld as a

guidance system they lose their sense of direction during a geomagnetic storm During solar

activity the Prediction Center in Boulder Colorado warns pigeon racers to

cancel anyevents that they have planned until the geomagnetic eld settles CMEs are also

often seen at the same time as solar ares which are an intense release of radiation from the

Sun This source of extreme solar activity is coronal holes A is a structure on

the Sun that acts as a source of hot fast solar wind It is not really a hole in the corona

rather a coronal hole is an area of lower density plasma in the solar atmosphere The coronal

holes tend to app ear during times of increased solar activity and are asso ciated with ares

and CMEs At high solar latitudes there are p ermanent coronal holes High sp eed solar

wind often coming from coronal holes V kms is funademantally dierent from

SW

low sp eed solar wind V kms High sp eed streams are hotter and more tenuous

SW

while low sp eed have a higher density and are not as hot

Origin of the Solar Wind

Parker Solutions

In his landmark publication Eugene Parker unveiled his hydro dynamic approachto

the solar wind In this publication Parker predicted that the hydrostatic mo del of the

corona the leading theory of the time was unphysical Parker applied Bernoullis equations

and came up with several branches of solutions to the dierential equations These

are shown in Figure Plotted on the xaxis is distance from the Sun while the yaxis is

solar wind sp eed The critical p oint in the middle is the sp eed of sound for Alfven waves

Solutions E and D are unphysical and solutions B and C do not matchobservations The

F branchisknown as the solar breeze Parker was able to determine that solution A is

the one representing the solar wind and that it is sup ersonic This led to the theory of the

hydro dynamic corona Parkers predictions were conrmed by the rst in situ observations

a few years later

Three Mathematical Approaches to the Solar Wind

The solar wind can now b e describ ed at dierentlevels of complexity The rst level of

abstraction is the purely kinetic particle approach In this approachwe lo ok at the solar

Figure The Parker solutions

wind as a collection of individual particles In principle every charged particle sees every

other charged particle ignoring Debye shielding for the moment This means that the total

force F on any particle i is the sum of all electric and magnetic forces from all other particles

i

X

F F F

i E ij M ij

j

where



F q q r

E ij i j

ij

is the electric force on a particle i from particle j and

F q v B

M ij i i j

where B is the magnetic eld from particle j and F is the magnetic force The energy

j M

velo citycharge mass etc of each particle is taken into account This metho d is the most

precise but can get very complicated

The second level of abstraction is the Vlasov Equation

f q v B f

v rf E

t m c v

where f is particle density in six dimensional phase space v is velo cityq ischarge and m is

mass To this we add Maxwells equations to evolve the electromagnetic elds This fo cuses

on an ensemble of particles and distinguishes each particle typ e

The third level of abstraction is magnetohydro dynamics MHD This approach treats

the solar wind as a magnetized uid MHD is describ ed by the induction equation

which can b e derived from Amp eres Law

E

rB j





c t

 E

In the nonrelativistic limit Hence

c t

rB j



rB

j E v B



j

E v B



Thenwe apply Faradays Law of Induction Let



B

rE

t

B j

r v B

t

B

rv B r r B

t

rrB rr B rrB

This brings us to the MHD Induction Equation

B



r v B r B

t

The rst term describ es the changes in the magnetic eld from plasma oscillations and is

called the convection term The second term is the resistive term In the innite conductivity

limit we can eliminate the second term This is the limit in which the frozenin

condition applies The frozen in condition states that as the plasma propagates and changes

its shap e the ux will not change This means that in the reference frame of a parcel of solar

wind the magnetic eld top ology is unchanging Tothiswe add the NavierStokes equation

which describ es the evolution of a uid Magnetic terms derived from the VlasovMaxwell

equations are added to the NavierStokes equation

Solar Wind Development

The Sun rotates with an average p erio d of ab out days recall that the rotation is dier

ential and dep endent on latitude This causes the magnetic eld coming o the Sun to b e

wound in the shap e of an archimedian spiral This also aects the solar wind parcels coming

o the Sun The eect is muchlikea water sprinkler releasing water while spinning The

water still moves radially but has a tangential comp onent to its motion In this same way

the p ower in the solar wind propagates radially This winding of the magnetic eld leads to

the shap e of what weknow as the Heliospheric Current Sheet HCS the neutral plane of

the Suns magnetic eld The shap e of the HCS is often describ ed as a ballerina skirt It

b ecomes more wavy and exhibits larger disturbances in amplitude during solar maximum

During solar minimum the shap e of the HCS b ecomes more simplied

All these prop erties aect the solar wind we measure at AU It is a magnetized uid

containing sho cks interface regions and turbulence All the asp ects of in terplanetary space

written ab out ab ovecontribute to the transient and turbulent prop erties of the solar wind

discussed in Chapter

Chapter

Bulk Solar Wind at AU

Fast And SlowWind

Weseetwodierenttyp es of solar wind at AU As mentioned earlier there is the hot

tenuous fast wind and the co oler more dense slow wind The following plots come from

acodeIworked on from into this co de is explained further in Chapter

Figures and show this dierence remarkably well Also shown is plotted against

P

sp eed is the ratio of plasma pressure to magnetic pressure for the protons If

P

the plasma is called warm and if the plasma is cold In the solar wind

P P

Figures and all show standard solar wind prop erties discussed

P

in Chapter These gures were a nice test to see if the co de was working The scatter in

each of these plots is exactly what one would exp ect from nominal solar wind

Transients in the Solar Wind

The solar wind can b e thought of as a continous uid p erio dically interrupted by what

is known as transientevents These transients disturb the solar wind and do not exhibit

nominal solar wind prop erties Indeed many solar wind studies exlcude transients from their

analyses

Figure Proton density plotted against proton sp eed

Figure Proton temp erature plotted against proton sp eed

Figure Plasma is the ratio of plasma pressure to magnetic pressure app ears to

P P

converge at higher V

SW

Figure SchematicofaCIR

Sho cks

One ma jor transient in the solar wind is a sho ck Sho cks have already b een mentioned in

the discussion of b owshocks Bowshocks are known as standing sho cks b ecause they are

stationary in the frame of a magnetosphere However sho cks do propagate A moving sho ck

is generated by a source moving faster than the signal sp eed of the medium In plasma physics

a sho ck is imp ortant b ecause it represents a b oundary b etween two dierent mediums The

sho cks that we are familiar with such as sonic b o oms are known as gas dynamic sho cks

Gas dynamic sho cks transfer imformation through particle collisions In space plasmas

the densityislow and collisions are very rare the mean free path is around AU This

results in collisionless sho cks Collisionless sho cks are able to transfer information through

electromagnetic interactions rather than collisions This is very similar to the concept of

magnetic pressure and tension in MHD Pressure and tension pro cesses normally thoughtto

o ccur with collisions o ccur without collisions b ecause the magnetic eld has the same eect

Behind the sho ck there is a transition back to the medium of the foresho ck the region ahead

of the sho ck If the transition is fast a reverse sho ck can o ccur

Corotating Interaction Regions

As the Sun rotates there is variability in the solar wind it emits This variabilityismost

pronounced at low latitudes As a result of the variability and rotation fast solar wind can

catch up to slow solar wind This results in an interface region and creates a transientin

the solar wind known as a Corotating Interaction Region CIR Figure shows a schematic

of a CIR

A CIR can easily b e identied by the discontinuities it creates in the solar wind At

the interface temp erature rises and abundance increases Radial

evolution of the CIR leads to a steep ening of the velo city gradientattheinterface if this

b ecomes suciently steep a sho ck will form Upstream of the interface is a spike in proton

density and temp erature A common feature of the fasttoslowinterface is an increase in

the MgO ratio

Flowvelo city is in opp osite directions always away from the interface As a result a sho ck

can propagate in b oth directions away from the interface This is known as a forwardreverse

sho ck pair When a CIR o ccurs there is a compression region at the interface where magnetic

eld lines come togther Magnetic structures do not merge together b ecause they are frozen

in to the solar wind and the eld lines do not cross

Because the solar variability that causes CIRs is mostly limited to to low latitudes on

the Sun CIRs are not seen as frequently at high solar latitude However higher latitude

CIRs can o ccur as a result of the tilt of the Suns dip ole which is around Gosling and

Pizzo found that at high latitudes only reverse sho cks were found The transition from

forwardreverse sho ck pairs to only reverse sho cks o ccurrs at a latitude roughly equivalent

to the tilt of the Suns magnetic dip ole and this tilt may b e the reason for the eect

Magnetic Clouds

Magnetic clouds were rst dened by Burlaga et al in and an example of one is

shown in Figure A is a conguration in the IMF in which there exists

higher than average magnetic eld strength and the eld vectors rotate parallel to a plane

Burlaga et al found that the frequency of o ccurence of these dep ends on the solar cycle

The radius of a typical magnetic cloud at AUisAU and is prop ortional to helio centric

distance Clouds tend to expand with a sp eed of around kms and also have translational

motion that is usually equivalent to the sp eed of the background solar wind An example of

magnetic cloud data is shown in Figure

Cosmic Ra ys

A is a high energy charged particle moving through the plasma It can come

from the Sun or from outside the heliosphere Galactic Cosmic Rays GCRs come from

outside the heliosphere and have enough energy to p enetrate the interstellar b ow sho ck

As a result they are often travelling near the sp eed of light These cosmic rays frequently

reach Earth and make a signicantcontribution to the natural background radiation we

exp erience When cosmic rays reach Earths atmosphere the particle will crash into it and

break apart This will generate new particles and gamma rays which will also interact with

the atmopshere The gamma rays and particles that nally reach the ground can b e traced

back to their parent particles

The cosmic rays describ ed ab ove are known as Galactic Cosmic Rays GCRs There are

also Anamolous Cosmic Rays ACRs These are high energy particles that orginated in the

heliosphere and interacted with the termination sho ck b efore returning to the inner helio

sphere Weknow that this happ ens b ecause they disrupt the nominal cosmic ray sp ectrum

Forbush showed that cosmic rays are mo dulated by solar activity Cosmic ray ux at Earth

increases and decreases with anticorrelation to the solar cycle The decrease during solar

maximum is known as the

Figure Magnetic cloud data

Figure Large scale schematic of a magnetic cloud

In addition to cosmic rays it is commonplace for neutral particles from the LISM to enter

the heliosphere In the of space there are essentially no barriers to these neutral

particles like there are for the ions

Plasma

The constituent particles of the solar wind are at suciently high energies that all particles

are disasso ciated from their molecular b onds This is the main feature of a plasma An

imp ortant asp ect of any plasma is the shielded p otential The shielded p otential eect

known as Debye Shielding is a result of a group of ions o ccupying the same area In the

frame of an in space electromagnetic interactions will result in that ion pulling toward it

ions of opp osite charge The opp ositely charged ions pulled in will eectively shield the ion

from other particles and reduce the eects of their forces to the p oint that electromagnetic

interactions are eliminated outside of the Debye Shield This spherical p otential formed

around the ion has a radius known as the Debye Length

Another asp ect of a plasma is plasma oscillations Because of the magnetic elds that

exist in the solar wind the ions are constantly in motion In a plasma there exist domains

that are dominated byonecharge or another This happ ens b ecause the scattering of charges

is not homogeneous An oscillation can form as a result of an ion drifting into an area that

is dominated by ions of the same charge The coulomb repulsion will force the ion out of

the domain However this ion will go to o far and enter a dierent domain of likecharge

The restoring force is the plasma trying to equilibrate the domains When this happ ens to

an ensemble of particles plasma oscillations result

Magnetohydro dynamics

One of the most useful ways to study a plasma is with Magnetohydro dynamics MHD Some

MHD concepts imp ortant to the study of plasmas are magnetic pressure magnetic tension

reconnection and ux tub es Magnetic pressure is a result of energy density in a magnetic

B

eld and is dened by the equation When magnetic eld lines b ecome compressed they





will eventually reacha point where they are so dense they will resist any further forcing An

example of magnetic pressure is in Earths magnetopause Magnetic pressure there resists

the ram pressure from the solar wind

Magnetic tension is the rigidity of magnetic eld lines When a magnetic eld line is

p erturb ed it will oscillate then return to where it was This eect is muchlike the plucking

of a guitar string A ux tub e is formed by the gyrokinetic rotation of particles As a particle

travels down a magnetic eld line it rotates in a helical path with a characteristic radius

An ensemble of identical particles with the same radius form a tub e In a publication

Borovsky states that interplanetary space is lled with ux tub es likea bowl of spaghetti

Plasma Waves

Central to the study of turbulence in the solar wind is plasma waves A plasma waveisany

typ e of disturbance that propagates through the plasma As a plasma propagates the

uctuating quantity can b e any prop erty of the solar wind but is often limited to magnetic

eld and particle velo city The plasma waves most imp ortant to solar wind study are Alfven

waves and magnetosonic waves

Figure Alfven waves The magnetic eld and velo city uctuate together while the

density remains constant

An Alfven wave propagates along a magnetic eld line It is similar to plucking a guitar

string and sending a disturbance along the string The restoring force is magnetic pressure

and tension When there is a transverse p erturbation on the eld line magnetic tension

restores the eld line to where it was In Figure we can see Alfven waves

Magnetosonic waves have the same restoring force as Alfven waves but propagate p er

p endicular to the eld lines These are longitudinal waves that compress the eld lines

increasing the density By compressing the eld lines togther a magnetic eld gradientis

formed and the eld lines return to where they where

Belcher Davis prop ose several theories for the origin of Alfven waves One p ossibility

is that they are pro duced by instabilities in the plasma These instabilities can arise from

expansion of the corona Another theory is that they are pro duced lo cally in the solar

wind This would b e a result of CIRs causing turbulence which is the origin of the waves

One problem with this theory is that CIRs are not common enough to explain howmany

Alfven Waves there are in the solar wind A third theory is that these waves are remnants

of pro cesses o ccurring in the and of the Sun A problem with

this theory is that Alfven waves are more energetic than theory would predict them to b e

through damping pro cesses

Another quantity that relates to solar wind uctuations and wa ves whichshows a sp eed

dep endent relationship is V B where denotes uctuations relative to the mean This is

p

where b est studied by putting the magnetic eld in Alfven units by dividing B by

is the mass density of the magnetized uid By diving the total energy we obtain V

c

B E This limits the value of V B so that it b ecomes a measure of the angle b etween the

velo cityvector and the magnetic eld vector It is shown in Figure part of the published

analysis covered in Chapter where is the normalized angle represents antiparallel

c

propagation relative to the mean eld represents p erp endicular and represents parallel

Since the interplanetary magnetic eld can b e oriented radially inward or radially outward

we can multiply by sg nB to obtain a pro duct that is p ositive for antiSunward

c R

propagation and negative for Sunward propagation Figure illustrates an interesting

Figure This plot shows the degree of correlation b etween velo city and magnetic eld in

the solar wind Most of the time wave propagation is away from the sun but this analysis

showthatsunward propagation can o ccur frequently

characteristic of V B which is the maximum for sg nB Recall that the

R c

core maximum at sg nB represents antiSunward propagation The smaller

R c

maximum at sg nB represents a p opulation of events dominated bySunward

R c

propagating waves

The plasma waves discussed in this chapter are what make up the large scale structure of

turbulence in the heliosphere As we will see in Chapter an understanding of these waves

forms an essential foundation for the study of turbulence and uctuations

Chapter

Fluctuations

Turbulence describ es a uid whose prop erties are nondeterministic The solar wind is an

ideal lab oratory for in situ turbulence studies b ecause it contains conditions unattainable in

a lab oratory setting In the solar wind the turbulence is driven by uctuations these could

b e uctuations in the magnetic eld velo city temp erature densityorany other plasma

parameter Understanding these uctuations can tell us how energy is tranferred in the

plasma

Development of turbulence

It has b een shown that the magnetic p ower in a turbulent plasma cascades through a

hierarchyofwaves Most of the p ower resides in the low frequency waves Damping of the

waves causes the energy to cascade from large wavelengths to smaller wavelengths As it

go es through this cascade energy is removed from the system This pro cess follows a p ower

law and the rate of energy dissipation is the sp ectral index often referred to as q In

Kolomogrov published the following formula



 

P C k

k k

where P is the p ower sp ectrum of the velo city uctuations C is a universal constant

k k

is the cascade rate and k is the wavenumber Even though this study predates in situ

observations the prediction has b een observed in the solar wind In turbulence there

exist eddies which are rotating vortices in the uid The time it takes for one rotation is the

where is is eddy turnover time Kolmogorov established the eddy turnover time as

v

the size of the and v is the sp eed This is a characteristic time scale for turbulence

In Kraichnan predicted that the critical timescale of the turbulence is the p erio d

of the Alfven waves rather than the eddy turnover time This led to a sp ectral index of

rather than the prediction of Kolmogorov describ ed by the following equation

 

E k Ab k



where is the cascade rate the rate of through the inertial range and k is

the wavenumb er

There are three regimes in the turbulent cascade at AU Frequencies Hz spatial



scales greater than km are considered to b e dominated by solar source dynamics

CMEs for example and are known as the energy containing range that drives interplanetary

turbulence Frequencies from Hz to Hz dene the inertial range In this range p ower

law sp ectra are observed and the dynamics are thought to b e describ ed by energy cascade

theories to Hz and probably higher is known as the dissipation range where the p ower

sp ectral index steep ens and dissipation of turbulentenergyinto heat is thought to o ccur

Figure shows these regimes

The transition from the inertial range to the dissipation range is likely caused byvarious

dynamics including ion cyclotron damping and results in the steep ening of the sp ectrum

Thus the frequency at which this change o ccurs is near the ion cyclotron frequency

Sp ectral Index

Early studies of plasma uctuations by Kolmogorov and Kraichnan form an imp ortant basis

for more mo dern theories involving the uctuations Coleman used magnetic eld data

from Mariner I I to p erform a sp ectral analysis of the plasma waves using actual data In

a subsequent pap er the turbulent cascade was shown with frequency plotted against

magnetic p ower and the sp ectral index was shown to b e

Belcher Davis computed the sp ectral index for the inertial range several years

later They found that it ranges b etween and and that the uctuations are not

isotropic This means that the sp ectral index can change as a function of the angle of the

mean magnetic eld with resp ect to the radial direction Further they found that the part

 

of the inertial range containing the frequencies Hz to Hz has a p ower law index

between and However the lower frequency end of the inertial range Hz to



Hz has a larger spread with indices b etween and

Geometry

Geometry refers to the orientation of the wavevectors The wavevectors parallel to the

mean magnetic eld are known as the slab comp onent and the p erp endicular wavevectors

are the D comp onent When studying anglular dep endence the angle refers to the

BV

angle b etween the wavevector and the mean eld In Matthaeus et al lo oked

at correlation functions for magnetic uctuations For this analysis they used months

of ISEE data with minute time resolution and analyzed spatial scales of to



km The result known as the Maltese Cross displayed in Figure shows that

most of the p ower resides parallel and p erp endicular to the mean eld and that intermediate

angles have signicantly reduced magnetic p ower

In Goldreich and Sridhar p ostulated that the sp ectral index has an angular

dep endence They stated that p erp endicular uctuations are dominated by the timescale

of wave propagation while parallel uctuations are dominated by the timescale of p erp en

dicular uctuations This is known as the critical balance assumption The critical balance

and assumption means that the sp ectral index for quasip erp endicular uctuations is

the sp ectral index for quasiparallel uctuations is Goldreich and Sridhar showed that

for intermediate angles the sp ectral index will evolvetoward one or the other As of

this theory is unproven but evidence is gathering for and against it

Bieber et al established a metho d for distinguishing slab turbulence and D turbu

lence They used a correlation matrix to compute where the p ower is Bieber et al assume a

simplied mo del in which there are only p erp endicular and parallel wavevectors By taking

Figure The Maltese Cross Result

the ratio of p ower in the p erp endicular wavevectors to the p ower in the parallel wavevec

tors they found which comp onentcontains more p ower The result of this analysis is that

of the p ower is in the p erp endicular comp onent Bieber et al used a second test called

the anisotropy test whichuses and nds that of the p ower is in the p erp edicular

BV

wavevectors By averaging the results of these two metho ds Bieb er concludes that ab out

of the p ower resides in the p erp endicular wavevectors

Inertial Range

In a lowlatitude study of the inertial range using Mariner data Coleman found a

slop e in the magnetic uctuations that is consistent with the prediction of Kolmogorov

The sp ectrum of the inertial range app ears to b e dierent for magnetic and velo city

uctuations as well Podesta et al found using Wind data that the velo city uctuations

in the solar wind have a sp ectral index of Another asp ect of the inertial range

sp ectrum is the the dep endence of the sp ectral index on wavevector orientation Recall

that the Critical Balance assumption p ostulated by Goldreich Sridhar separates wave

vector comp onents by dominating time scale Horbury found a similar result using a

elet metho d and highlatitude Ulysses data The frequency range of Horbury et al wav

is mHz to mHz Tessein et al conducted a similar study using lowlatitude data

from the Advanced Comp osition Explorer ACE in the frequency range mHz to mHz

with structure functions and found no dep endence Note that these two studies are

BV

lo oking at dierent scales with Horbury et al at the high frequency end of the inertial range

and Tessein et al closer to the middle This issue as well as the second order structure

function metho d is discussed further in Chapter

Dissipation Range

The dissipation range is the steep ening of the sp ectral cascade at small length scales Prior

to the onset of the dissipation range uid dynamics describ e the system The NavierStokes

equation represents these uid dynamics

vx t



v rv r r v

t

The MHD equations break down at the onset of dissipation b ecause they are only valid at

lengths greater than the ion intertial scale This scale isgiven by the following equation

ii

A

ii

ic

where is the Alfven sp eed and is the ion cyclotron frequency The ion inertial

A ic

scale represents the breakdown of single uid theory and the increasing imp ortance of ion

kinetics where the dissipated waves are converted into heat

The ma jor characteristic of the dissipation range is the steepness of its sp ectrum which

was explored by Leamon et al Leamon et al analyzed b oth the inertial and

dissipation ranges using ACE data They found a wellformed Kolmogorov sp ectrum in the

inertial range q The value of q found for the dissipation range was but

ranged b etween and Leamon also found the p ower in the dissipation range to b e

Figure The p ower sp ectrum Notice the frequency at which the slop e changes This

marks the transition from the inertial range to the dissipation range

slab and D This is a greatly increased slab fraction from the inertial range which

typically has values of around for the slab comp onent Smith et al investiagates why

the dissipation range sp ectral index varies more than that of the inertial range They

found that the steepness of the dissipation range is dep endent on the cascade rate in the

inertial range

Helios Observations

The Helios spacecraft in the late s p ermitted solar wind measurements close to the Sun

for the rst time Helios to ok measurements b etween AUandAU Bavassano et al

computed sp ectral indices from Helios observations and found only a slight radial evolution

Values found from this analysis range from to at AU and AU resp ectively

This indicates that the sp ectral index can change with helio centric distance however the

change is minimal

Figure Result of separation b etween fast and slowwind Slow wind left has more

power in the p erp endicular direction b ecause of the larger gradient in that direction Fast

wind right has a larger gradient in the parallel direction

Dierences Between Fast and SlowWind

In addition to the plots displayed in Chapter where the dierences b etween fast and slow

streams are shown there exist dierences in the uctuations as well Dasso et al

extend the two dimensional analysis of Matthaeus et al and separate the solar wind



into fast and slow subsets on a spatial scale of to km Any stream slower than

kms was dened as slow wind and faster than kms was dened as fast wind

Anything b etween and kms was excluded to b etter separate the two p opulations

This analysis used veyears of data from Advanced Comp osition Explorer ACE Using

correlation functions for large scale uctuations in the intertial range they found that fast

solar wind is dominated by quaiparallel uctuations and slow solar wind is dominated by

quasip erp endicular uctuations This result is shown in Figure

Smaller Scale Analyses

Two publications in by Smith et al and one in by Hamilton et al study

turbulence at smaller scales from the analyses discussed ab ove This high frequency regime

is unresolved by most plasma instruments including the one ab oard the ACE spacecraft

The rst pap er Smith et al fo cuses on the ratio of magnetic p ower in the comp onent

p erp endicular to the eld to the magnetic p ower in the parallel comp onent this is known as

B

the variance anisotropy The variance anisotropy dened as may b e a proxy for density

B

k

uctuations in the high frequency range b ecause the density seems to scale with magnetic

power in the parallel uctuations Earlier studies predict that the variance anisotropy should

b e correlated to proton b eta Smith et al found that variance anisotropyvaries

P



as Figure Earlier studies also predict a correlation b etween variance anosotropy

P

B

and uctuation level Smith et al also nd this correlation in b oth op en and closed

B



magnetic eld lines Figure Do es this mean that and uctuation level are correlated

P

Figure shows that these seemlingly unrelated quantities are prop ortional and includes

further relationships

Hamilton studied the variance anisotropy further in a publication this is de

Figure In all plots red represents magnetic clouds and black represents op en eld lines

a Variance anisotropy vs B B b vs B B This shows correlation b etween

 P 

seemingly unrelated quantities c N T vs B showing correlation b etween quantities not

P P

causally connected d T vs B e Plot of variance anisotropy vs cascade rate This

P

shows that variance anisotropy is not well correlated to the rate of energy cascade

Figure Computed magnetic variance anisotropyvs Red represents magnetic clouds

P

and black represents op en eld lines

Figure Distribution of magnetic variance anisotropy Black represents op en eld lines

and red represents magnetic clouds

scrib ed in more detail in Section Figure shows the distribution functions for

magnetic variance anisotropy Note that magnetic clouds have high anisotropy in the iner

tial range but not in the dissipation range Figures and show a scatter of variance

anisotropy in the inertial range vs dissipation range The inertial range anisotropy is greater

than dissipation range anisotropywhich tells us that the dissipation range is generally more

compressive recall that this is b ecause the density scales prop ortionally to the parallel mag

netic uctuations

Leamon et al showed for the rst time that there exists a strong correlation b etween

the in the dissipation range normally asso ciated with cyclotron damping

and the cross helicity in the inertial range normally asso ciated with propagation direction

Magnetic helicity is dened by the following equation

Ak B k

q

k

M

E k

B

Figure Scatter of variance anisotropy plotting inertial range against dissipation range

Black represents op en eld lines and red represents magnetic clouds Note that the dissipa

tion range is more compressive

Figure Same as Figure but subsetted for wind sp eed

where E k is the magnetic energy sp ectrum Cross helicityis

B

V B

c

 

V B

where represents cyclotron damping which means that Alfven waves are picked out while

r

r

represents everything b esides Alfven waves Their theory argues that

  r

M

r 

They obtain which leads to Figure shows Hamiltons cross helicity

r

M

analysis Hamilton nds that

c M

In traditional hydro dynamics the scale at which the dissipation range app ears is dep en

dent on the cascade rate of the inertial range Leamon et al showed that the app earance

of the dissipation range in the solar wind do es not follow predictions derived from uid vis

cosity and resisitivity However they did nd that the dissipation range dynamics are

in agreement with kinetic plasma theory Smith et al found that the sp ectral form of the

IMF dissipation also do es not adhere to the hydro dynamic predictions of Kolmogorov

and that the app earance of the dissipation range has no dep edence on the cascade rate

Figure However they found that the steepness of the dissipation range sp ectrum is

dep endent on the cascade rate Figure Given that that the uid approximation in the

solar wind breaks down at scales where the dissipation range b egins Smith et al conclude

that the onset of the dissipation range represents a coupling of a uid cascade approxima

tion and kinetic dissipation They also conclude that the traits of the inertial range have

nothing to do with the onset of dissipation and that traditional hydro dynamics is not a go o d

approximation for the dissipation range

Figure Comparison of h i to h i in the intertial range

c M

Figure Plot of cascade rate vs dissipation range sp ectral index q The sp ectral

dissip

form of the dissipation range is dep endent on the cascade rate

Figure The length scale of the sp ectral break V vs cascade rate Traditional

SW bf

hydro dynamics predicts that the cascade rate correlates to the onset of dissipation However

there is no correlation here

Chapter

Tessein Hourly Analysis

The Tessein Hourly Analysis is presented here as a precursor to the CIR studyItwas taken

up by the author in and published February The purp ose of the study was

to lo ok at distinctions in the dynamics b etween parallel and p erp endicular sp ectra in the

inertial range In addition welooked to resolve evidence of dierent sp ectral forms for the

p erp endicular uctuations and search for an anisotropic sp ectrum by fo cusing on p ower law

indices We did this by lo oking at howthepower lawindexchanges with as explained

BR

in Chapter and extending previous results to the current study

Background

MHD turbulence in the presence of a magnetic eld has two distinct timescales the eddy

 

turnover time k and the Alfven propagation p erio d k where V

nl A A

ku k V

k A

k

B



p

andkisthewavenumb er In traditional hydro dynamics is the Alfven sp eed V

A

there is only the eddy turnover time and this is what Kolomogorov used to obtain the

prediction The Goldreich Sridhar mo del attempts to separate these two timescales

and is shown pictorially in Figure This theory creates a separatrix which establishes

in what angles the dominant timescales applyItshows that nonlinear timescales dominate

quasip erp endicular uctuations while wavelike time variations dominate the quasiparallel

uctuations This results in a sp ectral index of at small angles of and and at

BV

angles near p erp endicular Intermediate angles will evolvebetween and This is

known as the critical balance assumption

Horbury et al studied this further using a wavelet analysis This study used high

latitude Ulysses data at AU on a high sp eed stream coming out of a p olar coronal hole

The wavelet analysis was used to compute the p ower sp ectra in the frequency range

mHz This range is the high frequency end of the inertial range The result they found

is consistent with the critical balance assumption All this leads up to the publication of

Tessein et al Tessein et al lo ok at this further using the second order structure

function metho d

Data Analysis

The data for this study comes from three distinct data sets The rst is a database consisting

of almost all of the ACE data from through It has s resolution and is analyzed k WA = Wnl A 3/2 kA ~ k||

WA > Wnl Wnl > WA Wnl > WA k||

Wnl > WA WA = Wnl

WA > Wnl

Figure The Goldreich Sridhar mo del Axes are the p erp endicular and parallel uctu

ations and the curved lines denote the b oundary b etween regions dominated by a particular

timescale

in the frequency range of to mHz the middle of the inertial range There are roughly

data p oints The second dataset used is that of Hamilton This dataset comes

from ACE as well and consists of hand picked intervals with intervals from

magnetic clouds from the years through solar maximum years This data has a

vectors magneotometer resolution and analyzes the frequency range to mHz high

freq end of the inertial range The third dataset comes from the Helios spacecraft during

a solar minimum p erio d This consists of hand picked intervals b etween and AU

This data has a vectors resolution and the frequency range analyzed is to mHz Note

that the Hamilton data set covers the same scales as Horbury

ACE s Merged Data

The ACE data consists of merged data from the ACE SWEPAM and MAG instruments

This data covers the time interval from dayoftoday of The vectors in the

data are in the RTN co ordinate system where R stands for Radial T stands for transverse

N stands for Normal and R T N The solar wind nominally ows in the R direction

unless it is deected by upstream ejecta The RT plane is the Suns equatorial plane and

the N direction is solar north The data is subsetted to one hour intervals and asso ciated

uncertainties are calculated The data then go es through a linear detrend and variances

with uncertainties are computed

At this p oint second order structure functions are obtained from p oints which

represents a lag of to s The structure function is dened by the following equivalence

Z x for parameter

D E



S L Z x Z x L



where hi denotes ensemble average and L is the lag This is done for each comp onent

and the diagonal elements of the resulting matrix are summed to obtain the magnetic trace

n

Resulting structure functions are t to the form S LAL where q n isthe



implied index of the p ower sp ectrum Then the same is done for the velo city and the results

of the structure functions are t to a p ower lawform

Figure shows distributions of computed structure function indices Recall that these

are related to the sp ectral index by the equivalence q n These plots displaythe

distribution of tted p ower law indices and how often each o ccurs B is plotted for having

n

very few current sheet crossings which can contaminate the samples WeincludeV because

n

changes in the R comp onent of solar wind velo cityover the course of an hour can dominate

the trace The N comp onent is not sub ject to as much large scale change and is useful for

this analysis We further subset the data into fast and slow wind as dened by Dasso et al

Wind sp eed in the subsets are based on wind sp eed averaged to a one hour interval

We see from the plots that the most probable value for n is which corresp onds to

q V is centered near which corresp onds to a sp ectral index This

n

magnetic eldsolar wind duality is consistent with the ndings of Po desta where the

magnetic eld sp ectral index agrees with the Kolmogorov sp ectrum and the

velo city sp ectral index agrees with the prediction of Kraichnan Also note that the

N comp onents of B and V are ab out the same as the trace which eliminates the need to

worry ab out current sheet crossing and large scale velo citychanges Also recall the predicted

sp ectral index of from Goldreich Sridhar and observed by Horbury We see

but it is very uncommon The discrepancy b etween Tessein et al and Horbury et al

mayhave to do with timescales

Figure shows computed uncertainties for the ts Uncertainties are less than and

well determined so go o d statistics have b een obtained Wehave computed errorweighted

means of the form

X X

 

hni n

i

i i

i i

where n is an individual t index and is the error asso ciated with the t We can then

i i

compute the variance of the asso ciated distribution

X X



 

h i n hni

i

stdev i

i i

The two sources of error are individual measurement error and the width of the distribution

We dene the errorofthemean as

X

  

N

er r stdev i

i

Errorofthemean is smaller than the plot symb ol b ecause each bin contains hundreds

of p oints and in some cases thousands Note that a sp ectral index of is more than one

standard deviation o the mean meaning that its o ccurrence is rare

Figure shows computed indices in the same manner as the ab ove analysis In this case

the computed p owerlaw indices are a function of and separated into bins where error

BR

bars represent standard deviation The plots show that there is no statistically signicant

t with the most probable dep endence The sp ectral index shown in this plot is consisten

BR

values from Figure and the standard deviations are comparable to the width of the

distributions in Figure This means that is consistently more than one standard

deviation from the mean for all values of

BR

Figure Analysis using ACE data from through with s data resolution

averaged to one hour intervals The top panel shows the distribution of tted p ower law

indices for the B tensor The second panel shows the distribution of tted p ower law indices

for the V tensor The third panel shows the distribution of tted p ower law indices for the

northsouth comp onentoftheB tensor The fourth panel shows the distribution of tted

power law indices for the northsouth comp onentofthe V tensor Dashed lines indicate n

and q and resp ectively

Figure The distribution of computed uncertainties in p ower law indices for each tted

interval

Figure Same analysis as Figure but tted p owerlaw indices are a function of

BR

Figure Same analysis as Figure but samples are limited to those hours when is

B

smaller than the prescrib ed angle

The amount of p oints in eachbinvaries The B distribution for

tr ace BR

contains estimates of n The computed hn i is while the computed standard

i i

deviation is and the errorofthemean is For the bin there are

BR

estimates of n The computed hn i while the computed standard deviation is

i i

and the errorofthemean is If the Goldreich Sridhar prediction is to b e viewed

as a partial theory that only holds under certain conditions then it is standard deviations

away from the mean However if it is a universal theory for MHD then its predictions are

standard deviations away from the mean

It has b een said that the wavelet analysis used by Horbury et al b etter resolves small

We can address the discrepancy with Horbury et al by limiting our analysis to small

BR

uctuations of instantaneous This is also helpful b ecause the Goldreich Sridhar

BR

prediction is based on the assumption that the mean eld is much larger than the uctuations

in the magnetic eld To do this we dene arctan B jB jwhereB is the

B r ms  r ms

ro otmeansquare variation of the comp onent of the magnetic eld p erp endicular to the

mean eld direction In this way b ecomes a measure of the variability of the eld

B

direction and intervals with b oth small and small can b e inferred to sample the

BR B A

dominated region of the sp ectrum without contamination from the dominated p ortion of

nl

the sp ectrum Figure shows the computed indices for values of less than and

B

These small angle bins tend to b e very dep opulated there are only nine intervals for

and Even though these bins do not pro duce go o d statistics there is no

BR B

Figure Means and variances for tted p ower law indices as a function of using the

BR

Hamilton database Top Mean indices for all op en eld lines black fast wind op en eld

lines red and slow wind op en eld lines green Bottom Mean indices for all magnetic

clouds

trend toward q The sp ectral index is still for the magnetic eld and for

velo city

ACE High Reolution MAG Data Analysis

This analysis comes from a data set known as the Hamilton database Recall that

the Hamilton database consists of intervals from solar minimum years The intervals

one to three hours in duration are handpicked to contain a broad range of solar maximum

conditions including distinct magnetic clouds In order to keep the data as pure as

p ossible there are no sho cks present in this database The sp ectra were t to two distinct

frequency ranges and only those with prop er p ower law form were retained The purp ose

of this database was to test the conclusions of various earlier pap ers The analysis found a

sp ectral index of for the intertial range and for the dissipation range

We apply the ab ove dataset to the analysis from Section The data used here has

a resolution of vectors s and is analyzing the high frequency end of the inertial range

Figure Same as top panel of Figure excp et using Helios magnetic eld data b etween

and AU during the years to in the frequency range mHz

mHz to mHz Note that the study of Horbury et al analyzed a similar frequency

range mHz to mHz

Figure shows the results of the studyThepower sp ectrum has b een computed over

the sp ecied frequency range and the indices separated into bins As b efore the error

bars display standard deviations errorsofthemean are roughly ten times smaller than this

Also the database is again subsetted for fast and slow wind and we add the magnetic clouds

subset app ears to b e consistentwith However and

BR BR

suer from a lack of data p oints Indeed is entirely

BR BR

absent of fast wind In spite of this we see in the angles that there is no trend

BR

toward if anything there is attening at these small angles

Helios MAGData

The motivation to use Helios data is to avoid contamination by upstream waves and extend

our analysis closer to the Sun where the turbulence is less evolved This database contains

interv als of IMF sp ectra from the Helios spacecraft during the years The

data ranges from to AU and the intervals are typically several hours We use the same

analysis here as was used in Section The frequency range analyzed here is to mHz

the middle of the inertial range Every eort has b een made to capture the broadest p ossible

range of solar wind conditions Figure shows the results of the analysis from the Helios

database We see from this plot that there is no sp ectral index Instead there app ears to

b e a attening of the sp ectrum asso ciated with radial mean eld conditions for

BR

This may b e related to the ndings of Bavassano et al that sp ectral index steep ens with

increasing helio centric distance Also it is interesting to note that the fast wind is atter

than the slow wind

Conclusions

It is unknown whythevelo city app ears to have the sp ectrum of Kraichnan while the

magnetic sp ectrum is that of Kolmogorov Roberts et al argues that the velo city

sp ectrum evolves to byAU but this dichotomy remains unexplained at AU

The study of Tessein et al has found no evidence of critical balance It do es not argue that

sp ectra are unobserved rather Figure shows that it is present but infrequent Tessein

et al and Horbury et al can b e made compatible if we p ostulate that dierent

inertial range scales have dierent prop erties As this thesis is b eing written studies of the

discrepancy are underway and will b e investigated during the summer of

Chapter

Corotating Interaction Regions at

AU

Motivation

The results of this analysis will b e compared with the turbulence parameters obtained from

the Hamilton database see Chapter The key dierence b etween this CIR study and

Hamilton et al is that the latter analyzes turbulence during solar maximum conditions that

were taken from nominally homogenous time p erio ds My database contains solar minimum

events and the CIR is exp ected to b e nonhomogeneous Since the data of Hamilton et al

comes from times of solar maximum we will b e able to lo ok at how the turbulence changes

throughout the solar cycle Also since the CIR is a source region for the generation of

turbulence and the database of Hamilton et al studied the prop erties of turbulence with

many distributed sources that failed to resolve source prop erties this study will enable us

to study the source prop erties and p otentially separate them from the state to whichthe

turbulence evolves If clear distinctions are seen then we can learn something ab out the

source dynamics that drive turbulence If few distinctions are seen then it may b e p ossible

that the turbulence evolves faster than the source or p ossesses prop erties in common with

the source In addition to comparing our results with results from the Hamilton database

wehave also compared these CIR turbulence prop erties to one another and found that they

are mostly identical to one another

Figure depicts an example of data used for this study Leading into daythe

spacecraft observes slow solar wind that is slowly decreasing in sp eed The lo cal IMF is

directed away from the Sun The densityisslowly rising and the temp erature is

steady while alpha particle density scales with proton density and IMF uctuations are low

At the start of day the wind sp eed rises density b egins to rise and IMF intensity rises

marking the start of the CIR Overaperiodof hours the wind sp eed rises from to

kms and the density p eaks The IMF intensit y p eaks with the density as the magnetic

eld is compressed b etween the converging ow By late in the day the spacecraft emerges

into a highsp eed wind that is hotter than its previous environment with an increased IMF

intensity marking the passage of the CIR

Figure One of the CIRs used for this study Note the standard feautres of a CIR spike

in magnetic eld and density and rise in wind sp eed and temp erature

Table Time intervals of CIRs used in this study

Year CIR interval days Total interval studied days

inclusive

inclusive

inclusive

inclusive

inclusive

Data

The data used for this study consists of veCIRevents from and Each CIR

is one to twodays long and we analyze the surrounding area such that the entire analyzed

interval encompasses four to vedays Table lists the time intervals studied The data

is broken down byeyeinto smaller intervals around one hour of similar activity Of this

data only intervals that pro duced well dened sp ectra were retained but the value of the

power law index was not a discriminating factor in the elimination of these intervals The

power sp ectra pro duced were manually split over two freqeuncy intervals the inertial range

and the dissipation range The frequency range around the sp ectral break was not analyzed

These two frequency intervals were t to separate p ower law forms and the p ower law indices

were recorded An example of this tting to separate frequencies is shown in Figure

Results

Figure shows two results immediately First the IMF uctuation level as measured

by the integrated sp ectrum in the top panel increases as the spacecraft passes through the

CIR Second the inertial range p ower law index remains throughout the interval so

that the rise in p ower is derived from a rising sp ectral form and not a lo cal enhancemen t

over a limited frequency interval as might b e seen upstream of sho cks The dissipation range

sp ectral index decreases in the fast wind when the IMF uctuation level is highest in keeping

with Smith et al The inertial range slab fraction p ercentage seems highest b efore the

CIR and lower in the fast wind b ehind This is an unexp ected result The inertial range

variance anisotropy A is mo dest throughout the interval There is no magnetic helicityin

the inertial range as exp ected and very little in the dissipation range

There are key dierences b etween the database used for this analysis and the one used

for the analyses we are comparing to Smith et al and Hamilton et al

The three previous pap ers use solar maximum data while the CIR analysis is during solar

minimum The main dierence we exp ect from the solar cycle is the fundamental dierence

in solar wind sp eed Note that the CIR has two distinct solar wind p opulations high sp eed

winds and low sp eed winds Figure This is b ecause of the structure of a CIR fast

wind crashing into low sp eed wind and an interface region marking the transition from one

to another The Hamilton et al wind sp eed distribution is not separated whichistypical

of solar maximum conditions Figure

Another dierence is geometry The three publications b eing compared to the CIR rep

resentambient turbulence This means that the turbulence do esnt come from one place in

Figure Summary of a CIR showing typical results of all ve CIRs studied Explanation

of panels from top to b ottom Magnetic p ower sp ectral index slab p ercentage anisotropy

and magnetic helicity marks the measured spatial scale for the onset of dissipation and

d

marks the measured Taylor scale not discussed in this pap er

t

Figure Distribution of solar wind sp eeds for the CIR analysis

Figure Distribution of solar wind sp eeds from Hamilton et al

Figure Before during and after the CIR Outward propagation b efore and after but a

mixture of b oth during the actual event

particular and is well develop ed byAU The CIR turbulence comes from a p oint source

the CIR itself and as a result might not b e isotropic and is exp ected to b e isolated within

the source region

Figure shows three panels from b efore during and after the CIR resp ectivelyEach

panel shows the energy cross helicity H and sign of the radial magnetic eld B prop erties

C R

which togther indicate the dominant propagation direction If H and B are of opp osite

C R

signs then dominantAlfven wave propagation is outward If they are of the same sign then

propagation is inward We see that propagation is outward as shown by Belcher Davis

b efore and after the CIR but there is a mixture of b oth during the actual event This is

something not seen in ambient turbulence and the only dierence we found b etween the CIR

turbulence and ambient turblence during this studyHowever this dierence is imp ortant

as it indicates lo cal generation of uctuation energy by means of the ow gradient Any

smaller scale turbulence within the region is exp ected to result from the turbulent casacde

of this newly injected energy It is imp ortant to note that Figure depicts larger scales

than those wenow examine

Another comparison we can makebetween CIR turbulence and ambient turbulence is the

correlation b etween cascade rate and sp ectral index Figure shows this distribution We

see that they are correlated in b oth CIR and am bient Figure turbulence

We pro ceed to present further comparisons b etween CIR turbulence and ambient turbu

lence that yielded similar results The Kolmogorov prediction that the spatial scale V

SW bf

is correlated to the rate of energy cascade has b een shown by Smith et al not to apply to

the solar wind We nd the same result in the CIR that there is no correlation b etween

these twovalues See Figure

Figure shows the panel that is analogous to Figure from Smith et al In each

of the plots wehave go o d agreement with the results of Smith et al In plot e where

Smith et al nds no correlation b etwen variance anisotropy and cascade rate we also nd

no correlation in the CIR

Variance anisotropywas lo oked at in this study in the same manner as in Smith et al

and Hamilton et al The distribution of variance anisotropy for the CIR turbulence

is shown in Figure This app ears to b e ab out the same as the ambient turbulence result

Figure In b oth gures the distinct p eak at low anisotropy is present

Figure is analogous to Figure from Hamilton et al Solid lines represent

Hamiltons trend lines for magnetic clouds and op en eld lines in the inertial range Dashed

lines represent Hamiltons dissipation range trend and the scatter represents CIR data Note

Figure Cascade rate vs sp ectral index in the dissipation range for CIR turbulence The

trend line is the t from Smith et al

Figure There is no correlation b etween cascade rate and characteristic sptatial scale

contrary to the predictions of traditional hydro dynamics This result found for the CIR is

the same as the result found for ambient turbulence

Figure In all plots trend lines representattotheanalysisofSmith et al b

There are two lines b ecause the analysis was subsetted for magnetic clouds and op en eld

lines a Variance anisotropyvsB B b vs B B This shows correlation b etween

 P 

seemingly unrelated quantities c N T vs B showing correlation b etween quantities not

P P

casually connected d T vs B e Plot of variance anisotropy vs cascade rate This

P

shows that variance anisotropy is not well correlated to the rate of energy cascade

Figure Variance anisotropy distribution function for CIR turbulence

that neither p opulation ts the trend lines p erfectly but the dissipation range is more similar

to Hamilton than the inertial range

In the ambient turbulence publications it was found from comparing variance anisotropy

in the inertial range to the dissipation range that the dissipation range is slightly more

compressive See Figure Figure shows that this is the same for CIR turbulence

Recall the discussion of the correlation b etween magnetic helicity in the dissipation range

and cross helicity in the inertial range See Section Hamilton et al found that

c M

Figure The result for CIR turbulence is shown in Figure The line is

and matches the CIR data very well

c M

Figure Variance anisotropy plotted against for the inertial and dissipation ranges

P

The trend line is the Smith et al result

Figure Correlation for variance anisotropybetween the inertial range and dissipation

range

Figure Correlation b etween magnetic helicity in the dissipation range vs cross helicity

in the inertial range for CIR turbulence with error bars the same result

c M

as Hamilton et al

Chapter

Discussion

Wehave found from the results of analyses discused in Chapters and that scale seems to

b e an imp ortant factor in the study of plasma waves in interplanetary turbulence Recall the

unresolved disagreementbetween Tessein et al and Horbury et al which seems to

have arisen from their analysis of dierent scales The matter has cropp ed up again in the

CIR study b ecause at small scales CIR turbulence is the same as ambient turbulence but

there is a clear dierence at large scales Wemay b e able to explain this in terms of eddy

lifetimes

We see from Figure that the eddy lifetime is shorter at smaller scales Since turbulence

evolves on the scale of an eddy lifetime a length scale that contains many lifetimes maybe

fully evolved while a scale with fewer lifetimes may not b e This means that small scales

are representative of universal trends while large scales show only lo cal prop erties This

makes sense in the case of the CIR b ecause we found common prop erties b etween the CIR

and ambient turbulence at small scales but dierences at larger scales As the energy is

cascading from large scale to small scale in the CIR the turbulence is evolving suchthatit

matches prop erties found in the ambient turblence

Figure Plot of p ower sp ectrum solid and eddy lifetime dashed The time for a parcel

to travel AU is marked by the arrow

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