introduction.tex

% Chapter 1: Introduction

\chapter{Introduction}
\label{sec:introduction}

Plasmas are fluid masses of charged particles, often formed when neutral atoms and mol\-e\-cules in a gas are ionized by external energy.  Instances of plasma include everything from fluorescent lightbulbs and flames to more exotic examples such as nuclear fusion reactor experiments, most stars, nebulas, and the interstellar medium.  In fact, about 99\% of visible matter in the known universe exists in the plasma state.  Of particular interest to this thesis is the plasma in the Earth's ionosphere.

Ionospheric plasma is partially ionized, meaning that only a fraction of the particles are ionized and the rest are neutral.  Investigation of the dynamics of plasmas therefore requires not only collisions between particles to be considered but also charge-charge interactions and external and internal electric and magnetic fields.  These dynamics can generate waves in the plasma, which cause density perturbations.  Similarly, variations in plasma temperature and electric and magnetic field strength can occur on a variety of scales.  This thesis focuses on mechanisms by which density variations in ionospheric plasma occur, often referred to as plasma structures.

\section{The Solar-Terrestrial Environment}
The solar-terrestrial environment begins with the Sun in the center of the solar system.  The Sun is composed of highly energized plasma that is gravitationally bound together.  The surface of the Sun is highly dynamic, populated with sunspots, which are dark and relatively cool regions with intense magnetic fields, as well as coronal holes, which are low-density areas characterized with a continuous outflow of plasma.  The intense magnetic fields associated with sunspots can break down releasing a large amount of energy and plasma, known as a solar flare.  Planet-sized masses of plasma known as coronal mass ejections (CMEs) can be ejected from the surface, which can be large enough to maintain an internal magnetic field as they travel through the solar system.  The frequency of sunspots and large outbursts of plasma changes with the 11-year-long solar cycle.  A large number of sunspots, high occurrence of fast and dense plasma outbursts, strong x-ray or extreme ultraviolet (EUV) flux, and a highly variable magnetic field indicate solar maximum.  Conversely, solar minimum is characterized by a low sunspot number and slow and relatively steady outflow of plasma.  This thesis examines factors that control irregularity production in the Earth's ionosphere, which can be strongly influenced by solar activity.

The Sun interacts with the rest of the solar system through solar radiation and the solar wind, a continuous outflow of plasma that carries the Sun's magnetic field with it.  This is shown in Figure \ref{fig:ste}.  The solar wind, represented by the blue arrows in Figure \ref{fig:ste}, typically has a density around 10 cm\(^{-3}\) and is traveling away from the Sun on the order of 400 km/s at the Earth's orbit.  However, like the Sun, the solar wind is highly dynamic and these parameters can change drastically over a period of minutes.  CMEs, for instance, create a substantial increase in density and are often preceded by a shock, a region of hot, compressed plasma caused by the CME moving at supersonic speeds.  In addition to the outflow of plasma, the solar wind also contains the Sun's magnetic field extended into the outer limits of the solar system, called the interplanetary magnetic field (IMF), orange lines in Figure \ref{fig:ste}.  Because the Sun has a short rotation period (about 27 days at the equator), the IMF gets twisted into a spiral, commonly called the Parker Spiral \citep{Parker1958,Kivelson1995}.  When the IMF reaches the Earth, the magnetic field is typically oriented at a \(45\deg\) inclination relative to the ecliptic, but this is highly variable depending on solar wind conditions.

\begin{figure}
	\centering
	\includegraphics[width=\textwidth,angle=0]{ste.pdf}
	\caption[Solar-terrestrial environment]{{\:}Solar-terrestrial environment\vspace{1mm} 
	
	Diagram of the solar-terrestrial environment.  The solar wind, blue arrows, contains both a steady stream of charged particles and the interplanetary magnetic field, orange.  The purple region represents the magnetosheath, the shock layer formed due to the supersonic plasma outflow from the Sun.  The bow shock is indicated by the green line.  The magnetopause encloses the magnetosphere dominated by the geomagnetic field and is shown by the black dashed line.  Closed magnetic field lines from the Earth are shown in grey, while open field lines are shown in light blue.  The ionosphere is shown in yellow above the Earth's surface (not to scale).}
	\label{fig:ste}
\end{figure}

The IMF interacts with the Earth's own magnetic field, and this interaction creates the magnetosphere.  Without the solar wind, the Earth's magnetosphere would be approximately in a dipole shape, but dynamic pressure from the solar wind causes the Sun-facing side to be compressed to 6--10 Earth radii and the opposite side to be stretched into an extended tail (hundreds of Earth radii).  Because the solar wind is typically a supersonic flow, a bow shock forms on the dayside of the magnetosphere, green line in Figure \ref{fig:ste}.  The heated and compressed solar wind plasma created by the bow shock is known as the magnetosheath, the purple region in Figure \ref{fig:ste}.  The magnetopause, black dashed line, is the actual boundary between solar wind plasma in the magnetosheath and magnetospheric plasma.  Magnetic field lines that are part of the closed magnetosphere are shown in grey in Figure \ref{fig:ste}.  However, usually the magnetosphere is not fully closed.  At the magnetopause, magnetic reconnection can occur between the IMF and the Earth's magnetic field creating magnetic field lines that directly connect the Sun to the Earth via the solar wind, commonly referred to as open field lines, light blue in Figure \ref{fig:ste}.  This reconnection occurs on last closed field line intersecting the magnetopause (highlighted in orange in Figure \ref{fig:ste}).  Open field lines allow highly energized solar particles to enter the magnetosphere and tend to occur around the Earth's polar caps, making these regions particularly interesting to study due to very complex behavior and coupled interactions between the Earth and the Sun.

Closer to the Earth's surface, the magnetosphere also interacts with the Earth's ionosphere, a layer of partially ionized plasma in the upper atmosphere, yellow in Figure \ref{fig:ste}.  The interaction between charged particles in the ionosphere with magnetic field lines couples the ionosphere, magnetosphere, and solar wind to create non-trivial dynamics.  Additionally,  neutral winds in the thermosphere can influence the ionosphere through viscous interactions.  Overall, the charged particle interactions in the ionosphere create a highly complex system, which is further complicated by coupling with both the thermosphere from below and the magnetosphere and solar wind from above.  

\section{The Earth's Ionosphere}
\label{sec:ionosphere}
The focus of this work is the Earth's ionosphere, a region of the upper atmosphere that extends approximately 50--1000 km in altitude where neutral gases have been partially ionized, resulting in free electrons and ions mixed with neutral particles.  Two competing processes occur in the ionosphere to create a peak in the electron density.  Solar illumination and particle precipitation ionize neutrals, and the density of ions increases as altitude decreases because there are more neutral molecules available to ionize.  However, ionizing radiation does not penetrate the atmosphere easily below a certain altitude, and ion recombination becomes a substantial factor, reducing the ion density.  The exact altitude where this density peak occurs varies with time of day, location, season, and solar cycle.  Representative model examples of electron density changes with altitude are shown in Figure \ref{fig:densprofile}, found from the International Reference Ionosphere (IRI) model run on January 1, 2007 at \(75\deg\) N, \(0\deg\) E, geographic (the IRI model is discussed further in Section \ref{sec:iri}).

\begin{figure}
	\centering
	\includegraphics[width=\textwidth]{densprofile.pdf}
	\caption[Ionosphere density profiles]{{\:}Ionosphere density profiles\vspace{1mm}
	
	Altitude profiles of electron density throughout the ionosphere found from the IRI model.  Typical extents of the \(D\), \(E\), and \(F\) regions are identified.}
	\label{fig:densprofile}
\end{figure}

Above 50 km, the primary neutral particles are atomic oxygen, O, molecular oxygen, O\(_2\), and molecular nitrogen, N\(_2\), with N\(_2\) dominating at low altitudes but O becoming predominate above 250 km \citep{Kelley2009}.  Common ions at these altitudes are therefore mostly composed of oxygen and nitrogen, O\(^+\), NO\(^+\), and O\(_2^+\), but hydrogen, H\(^+\), can contribute as well, particularly at higher altitudes \citep{Kelley2009}.  In the ionosphere, neutral densities are typically several orders of magnitude higher than ion densities, however even a small number of charged particles changes the behavior of the region dramatically.  Because there are so many different ion species, plasma density is usually discussed simply in terms of electron density.  Plasmas are assumed to be quasineutral, meaning that electron and ion densities are approximately the same so the plasma does not have an overall charge.  The electron density is equal to the sum of densities of all the ions in the plasma, or \(n = n_e = \sum n_i\).

\subsection{Ionospheric Regions}
\label{sec:ionosphere_regions}
The ionosphere is typically divided into three regions, Figure \ref{fig:densprofile}.  The \(D\) region is located between 80 and 90 km in altitude and typically only exists in the daytime when the ionosphere is sunlit.  The \(E\) region typically exists between 90 and 130 km with a density peak between 105 and 110 km, depending on factors such as time of day and season.  Above 130 km is considered to be the \(F\) region, which has a density peak around 250 km, Figure \ref{fig:densprofile} \citep{Kivelson1995}.  The \(F\) region can actually have two peaks under some daytime conditions, notated as the \(F1\) peak and the \(F2\) peak.  The peak heights and densities of each of these regions is highly variable both diurnally and seasonally.  In Figure \ref{fig:densprofile}, the daytime profile is shown with the solid line next to a nighttime profile shown with a dashed line.  Note that night electron densities are generally lower at all altitudes due to the lack of photoionization during the night time, however densities can be high in the cases of aurora or a sporadic \(E\) layer.  In addition, the \(F\)-region peak moves to higher altitudes at night.  The focus of this thesis is on the \(F\) region.

Because the concentrations of ions and neutral particles changes with altitude, each of these regions have distinct plasma physical properties that make them unique.  The main issue is whether the large-scale motion of a particle is controlled more by the magnetic field (the particle is magnetized) or by collisions with other particles (the particle is collisional).  This can be determined by comparing the collision frequency, \(\nu_\alpha\), to the gyrofrequency, \(\Omega_\alpha\), of a particular species, where \(\alpha\) refers to either electrons, \(e\), or the various ion species, \(i\).  If \(\Omega_\alpha \gg \nu_\alpha\), the particle completes many gyrations between each collision, so its motion is determined mostly by the magnetic field and it is considered to be magnetized.  If \(\nu_\alpha \gg \Omega_\alpha\), the particle collides with other particles much more often than it completes a gyration, so it is considered collisional.  The gyrofrequency is dependent on the mass (\(m_\alpha\)) and charge (\(q_\alpha\)) of the particle, and the strength of the magnetic field (\(B\)), \(\Omega_\alpha = q_\alpha B/m_\alpha\), so it is roughly constant throughout the ionosphere because the magnetic field strength does not change much throughout these altitudes.  Collision frequencies depend on the relative densities of different species within the plasma and can be calculated from a series of standard expressions \citep{Schunk1980,Schunk2009}.  In the \(E\) region, \(\Omega_e \gg \nu_e\), so electrons are magnetized.  However, the motion of ions is dominated by collisions between ions and neutrals (\(\nu_i > \Omega_i\)), so ions in the \(E\) region are considered collisional.  In the \(F\) region where the neutral density is much lower, both ions and electrons are magnetized, \(\Omega_\alpha \gg \nu_\alpha\).  These differences in how charged particles move impact the conductivity and plasma waves development, which is further discussed in Section \ref{sec:lit_instabilities}.

\subsection{Plasma Structuring in the Polar Ionosphere}
\label{sec:polar_structure}

This thesis is concerned with the polar ionosphere, which is a particularly interesting and dynamic region due to the presence of open field lines that are directly connected to the solar wind.  The main focus of this work is plasma density waves or structures, also known as irregularities.  Density irregularities in the polar cap can occur on scales ranging from thousands of kilometers to less than a centimeter \citep{Tsunoda1988}, and a summary of some of these irregularities is presented in Figure \ref{fig:polarirreg}.  

Large-scale plasma irregularities are typically considered to be any density structure on the scale of greater than 30 km \citep{Kelley2009}.  Some common examples are polar patches, polar holes, and sun-aligned arcs.  Polar patches are large density enhancements (usually at least twice the background plasma density) that travel across the polar cap with the background convection, Figure \ref{fig:polarirreg}a \citep{Weber1984,Valladares1994}.  Polar holes are plasma density depletions that typically occur in the \(F\) region slightly poleward of the auroral oval \citep{Brinton1978}.  Sun-aligned arcs are large auroral forms that stretch across the polar cap along the Sun-Earth line.  They are very narrow and often characterized by large and complex velocity shears on either side of the arc \citep{Valladares1991}.  Large-scale plasma structures and the density gradients that surround them often serve as a platform for smaller-scale structures to form as well.  Polar patches, for instance, are known to form large finger-like structures along their trailing edges, Figure \ref{fig:polarirreg}b \citep{Gondarenko2004b,Hosokawa2016}.

\begin{figure}
	\centering
	\includegraphics[height=0.75\textheight]{irregularity.pdf}
	\caption[Plasma irregularities in the polar ionosphere]{{\:}Plasma irregularities in the polar ionosphere\vspace{1mm}
	
	Plasma density irregularities occur in the polar cap at different scale sizes.  Polar patches, orange in Figures \ref{fig:polarirreg}a and \ref{fig:polarirreg}b, move with the background ionospheric convection, indicated by red arrows.  Smaller-scale structures often develop along the edges of polar patches where strong density gradients exist, Figure \ref{fig:polarirreg}b.  Intermediate-scale irregularities are primarily responsible for disrupting radio signals traveling through the ionosphere, Figure \ref{fig:polarirreg}c.  Field-aligned irregularities, FAIs, are density perturbations that form along the magnetic field lines, Figure \ref{fig:polarirreg}d.}
	\label{fig:polarirreg}
\end{figure}

Intermediate-scale plasma irregularities are those with sizes of 100 m -- 30 km \citep{Kelley2009}.  These are also referred to as scintillation-causing structures and are responsible for many of the adverse space weather effects that are observed on Earth.  One important example of a space weather effect is when ionospheric irregularities cause fluctuations of the phase and amplitude of the original radio signal, called radio scintillation, which affects the ability of satellites to communicate with ground receivers.  This can cause communication blackouts or introduce large errors into GPS/GNSS (Global Positioning System/Global Navigation Satellite System) calculations, affecting navigation.

Small-scale structuring is generally considered to be all irregularities less than 100 m \citep{Kelley2009}.  Although small-scale irregularities are not responsible for scintillation that directly impacts GPS/GNSS performance, they can be the easiest to study because of extensive datasets collected with a large number of ionospheric radars around the world that have been in operation for many decades (discussed further in Section \ref{sec:superdarn}).  Because electrons move far more easily along magnetic field lines than across them, density perturbations tend to be aligned with the magnetic field and are often known as field-aligned irregularities (FAI), Figure \ref{fig:polarirreg}d.  The irregularity classifications presented here are rough categories only and a high degree of coupling occurs between different scales, which makes plasma structures of all scales (large, intermediate, and small) important for plasma structuring processes.  For instance, large-scale structures such as polar patches may very well have internal intermediate- and/or small-scale structuring.  A turbulent cascade is often assumed to connect these different scales, but because various instruments or techniques usually only observe structuring on a particular scale, it is difficult to find direct evidence of this or how exactly it occurs \citep{Kintner1985,Tsunoda1985}.  A description of some of the particular instruments and models that are used in this thesis follows.

\section{Observational Techniques and Models}

\subsection{Coherent Scatter Radars}
\label{sec:csr}
In this thesis, we make extensive use of coherent scatter radars (CSRs), which have been used to study the ionosphere for the last half century.  The radar transmits a radio wave, which can scatter off plasma structures in the ionosphere, and then be detected by the receiving antennas.  The time difference between when the signal was transmitted and when it was detected determines the location of plasma irregularities in the ionosphere and the power, phase, Doppler shift, and Doppler width of the returned signal determines their characteristics.

There are three measurements typically made by CSR systems: backscatter power, line-of-sight (LoS) velocity (derived from the Doppler shift), and spectral width.  Backscatter power is the power of the returned signal that the radar's receivers detect.  Typically, a threshold is selected for the minimum signal-to-noise ratio (SNR) acceptable for a return to be considered an actual signal instead of background noise.  LoS velocity is the component of the total plasma irregularity velocity that is measured along the transmitted signal direction.  Because radars measure velocity through the Doppler shift of backscatter, only a component of the velocity vector can be measured \citep{Greenwald1985}.  Spectral width represents the broadening of the returned Doppler power spectrum and can be used to infer other characteristics of irregularities.  For instance, a large spectral width can indicate irregularities that have a short lifetime, possibly due to Bohm diffusion, or fluctuations in the plasma drift velocity due to turbulence \citep{Ponomarenko2006}.  

Radars can operate by transmitting pulses instead of a continuous signal.  This allows the distance between the radar and the target to be found (based on the time difference between when the pulse is transmitted and received).  The lower limit of the range gate size is determined by the length of a radar pulse,
\begin{equation}
	\label{eqn:range_gate}
	\Delta r = \frac{c\Delta t}{2}.
\end{equation}
Here, \(\Delta r\) is the size of a range gate, \(\Delta t\) is the pulse length, and \(c\) is the speed of light.  Longer pulses result in larger range gates, resulting in poorer spatial resolution.

When the radar transmits continuously, backscatter will be received, but because it is impossible to know when that signal was transmitted, the difference in time between transmission and receiving cannot be used to calculate how far away the backscatter volume is.  Conversely, if the radar only transmits a pulse of a certain length and then ``listens'' for its return, the time between transmission and return is known and the distance the pulse traveled can be calculated and therefore the location of the backscattering volume can be found \citep{Farley1972,Greenwald1983}.  However, using this simple single-pulse method, another pulse cannot be transmitted until the first pulse returns, limiting the temporal resolution that can be achieved.  This can be improved by using a multi-pulse scheme, which is further discussed below \citep{Farley1972,Greenwald1983,Greenwald1985}.

A short time between pulse returns is necessary to allow Doppler velocities to be calculated.  The maximum Doppler shifted frequency, \(\delta f\), that can be observed by a radar is the Nyquist frequency, \(f_n\), which is half the sampling frequency, \(f_s\),
\begin{equation}	
	\label{eqn:nyquist}
	\delta f < f_n = \frac{f_s}{2}.
\end{equation}
Radars operating in a single-pulse mode must have a very low sampling frequency, as described above, particularly when the extent over which echoes are expected is large and the pulse must travel a long distance.  This results in a low Nyquist frequency which severely limits the Doppler shift that can be measured and places an upper limit on the LoS velocity that can be calculated with this technique.  For an ionospheric radar observing targets a maximum of 2000 km away, it would take about 13 ms for a signal traveling at the speed of light to travel the extent of the target area and back to the radar, which corresponds to a sampling frequency of \(f_s = 75\) Hz.  The Nyquist frequency is then \(f_n = 37.5\) Hz, which corresponds to a maximum measurable Doppler velocity of \(\sim550\) m/s.  Flows in the ionosphere are known to exceed 1000--2000 m/s, so single-pulse sampling is insufficient for these purposes.

Instead of waiting for each individual signal to be received before transmitting the next, ionospheric radars typically employ a multi-pulse mode.  In a multi-pulse mode, the radar transmits a series of pulses with different time intervals or lags between them.  This can introduce complications when backscatter from different pulses at different ranges is received by the radar at the same time, referred to as cross-range interference, but these effects can be mitigated using correlation techniques \citep{Farley1972}.  The smallest lag between two pulses is known as the multi-pulse increment, \(\tau\).  All other lags are integer multiples of \(\tau\), which allows the calculation of the corresponding lags of the complex autocorrelation function (ACF).  The ACF is used to find the spectral characteristics of the backscatter, from which the LoS velocity, spectral width, and backscatter power can be obtained.  Overall, multi-pulse techniques are generally considered far more appropriate for ionospheric studies than single-pulse techniques \citep{Farley1972,Greenwald1983,Greenwald1985,Sulzer1993,Barthes1998,Ponomarenko2006}.

The first observations of the ionosphere by CSRs were made in the 1930s \citep{Eckersley1937,Harang1938}.  Over the next several decades, numerous ground-based radars were used to advance our understanding of the ionosphere  \citep{Hultqvist1964,Leadabrand1965,Unwin1972,Sahr1996}.  In the 1970s and 1980s, the Scandanavian Twin Auroral Radar Experiment (STARE) produced the first continuous large data-set, similar to how modern CSR systems are run.  STARE consisted of two very high frequency (VHF) radars with overlapping FoVs that were designed to measure FAIs in the \(E\) region of the ionosphere \citep{Greenwald1997}.  The advantage of two radars with overlapping FoVs was the ability to observe the same structures from two different directions.  Because each radar can only measure the LoS velocity of a structure, two simultaneous observations from different directions allows two different velocity vector components to be found, and hence the total velocity vector can be calculated.  This technique is still commonly used with CSR system.

\subsubsection{Super Dual Auroral Radar Network (SuperDARN)}
\label{sec:superdarn}
The primary instruments used within this body of work are radars within the Super Dual Auroral Radar Network (SuperDARN), a global network of high frequency (HF) CSRs that was designed to observe small-scale plasma structures in the ionosphere and map the global plasma convection.  The network currently consists of about 35 operational radars between the northern and southern hemispheres distributed at mid-, high-, and polar latitudes, Figure \ref{fig:superdarnmap}.  After the STARE experiment \citep{Greenwald1978}, the first HF ionospheric radar was built in Goose Bay, Canada in 1983, which would become the first of the SuperDARN radars \citep{Greenwald1985}.  

\begin{figure}
	\centering
	\includegraphics[width=\textwidth,angle=180]{SuperDARNmap.pdf}
	\caption[SuperDARN map]{{\:}SuperDARN map\vspace{1mm}
	
	Fields-of-view of all operational SuperDARN radars in both the northern (a) and southern (b) hemispheres.  Polar-latitude radars are shown shaded in blue.  The radars at Rankin Inlet (northern hemisphere) and McMurdo Station (southern hemisphere) are outlined in pink because these two instruments are particularly important to this body of work.  Lines of constant magnetic latitude are shown in yellow at \(\Lambda = 40\deg, 60\deg, 80\deg\) in the northern hemisphere and at \(\Lambda = -40\deg,-60\deg,-80\deg\) in the southern hemisphere.}
	\label{fig:superdarnmap}
\end{figure}

The most important condition for the observation of FAIs with CSR systems with colocated transmitters and receivers is the so-called perpendicularity condition, according to which the radar beam has to reach the scattering volume perpendicular to the local magnetic field direction.  At mid-latitudes, this is plausable through straight line propigation because the magnetic field is angled relative to the Earth's surface, so a radar with a suitably chosen elevation angle can produce a beam perpendicular to the local magnetic field.  However, at the polar cap, the magnetic field is close to vertical so it is necessary to use HF radars to study ionospheric structures because the beam must be refracted through a dense ionosphere to reach perpendicularity with the magnetic field.  VHF radar beams are not refracted enough in the polar cap and backscatter produced by FAIs will not return to the radar.  SuperDARN radars operate nominally between 8--20 MHz, which, according to the Bragg scatter condition, refers to decameter-scale plasma waves.  Each radar consists of 16 independent antennas that transmit a radio wave and receive backscatter from the ionosphere.  Beams are electronically steerable and most radars have between 16 and 24 beams in normal operation mode, each beam spanning \(3.25\deg\) in azimuth.  Each beam consists of 75--100 range gates, which are generally either 15 km, 30 km, or 45 km in length \citep{Chisham2007}.  SuperDARN radars were originally designed to use a 7-pulse ACF \citep{Farley1972,Greenwald1983,Greenwald1985}, however since 2011, most radars have begun using an 8-pulse sequence.   The multi-pulse increment, \(\tau\), is 2400 \(\mu\)s, which corresponds to a Nyquist frequency of about 200 Hz, small enough to measure plasma drift velocities up to 3000 m/s.  The spectral characteristics of backscatter are derived from the ACF using the FITACF algorithm \citep{Ponomarenko2006}.

One of the original purposes of SuperDARN was to produce maps of the plasma convection patterns at high latitudes using 2D velocity vectors.  Originally this was accomplished by considering two radars with overlapping FoVs.  If both received backscatter from the same scattering volume, two LoS velocities could be found and combined to find a 2D velocity vector \citep{Ruohoniemi1989}.  The modern method for creating convection maps involves taking all data recorded by all radars in a particular hemisphere and finding the best fit to a 2D ionospheric electrostatic potential using Legendre functions \citep{Ruohoniemi1998}.  For convection maps, only data from the \(F\) region is considered.  If there is not enough velocity data to confine the fit sufficiently, the measured data points are supplemented with a statistical model based on the IMF conditions \citep{Ruohoniemi1995,Ruohoniemi2005}.

Because convection maps require data covering as wide a range of magnetic local time sectors as possible, SuperDARN radars are continuously operational, usually in a common mode.  This creates a vast database of high-quality measurements of small-scale FAIs, which is very useful for other studies on irregularity occurrence and plasma structuring.  Data used in the studies presented here came primarily from the SuperDARN radars located at Rankin Inlet, Canada (RKN) and McMurdo Station, Antarctica (MCM).  These two radars have been outlined in pink in Figure \ref{fig:superdarnmap}.  MCM is the only polar radar in the southern hemisphere that has been operational for more than 5 years.  Although the FoVs of other radars cover parts of the polar cap, these parts refer to either \(E\) region ranges or the furthest of the \(F\) region ranges, for which there is rarely much backscatter.  There are currently three polar radars in the northern polar cap, but the FoV of RKN overlaps the viewing area of an important complementary instrument, the Resolute Bay Incoherent Scatter Radar, discussed in Section \ref{sec:isr}, which makes it particularly useful for multi-instrument studies of the ionosphere.

\subsection{Resolute Bay Incoherent Scatter Radar (RISR)}
\label{sec:isr}
Similar to CSRs, incoherent scatter radars (ISRs) are ground-based radars that are used to probe the ionosphere.  However, they use a fundamentally different technique to observe structuring, which allows them to measure bulk plasma parameters in contrast to small-scale irregularity characteristics.  While CSRs receive backscatter from relatively large, ``coherent'' density structures in the ionosphere, ISRs operate by radar scattering from waves due to the random thermal motion of electrons in the plasma.  The radar receivers then measure a spectrum of frequencies with different power \citep{Gordon1958}.

The scattering cross section for a volume of \(N\) electrons in the ionosphere is given by \(\sigma_n = N \sigma_e\) for radio waves with a wavelength small compared to the Debye length of the plasma, where \(\sigma_e\) is the scattering cross section of a single electron \citep{Gordon1958,Fejer1960}.  The relationship to the plasma Debye length is important because at scale sizes smaller than the Debye length, the plasma is not capable of organized motion, so the motion of the elections is only due to their own thermal energy and not larger-scale plasma waves.  The signal returned from this kind of scatter was postulated to have a very low power and a very broad spectral width, however after the first ISR was built, the spectrum was found to be much narrower than previously expected \citep{Evans1969}.  This can be attributed to the effect that the much slower ions have on free electrons in the plasma through ion acoustic waves \citep{Bowles1958}.  The practical results of this is that ISRs of much lower sensitivity can be used to study the ionosphere.  Very high sensitivity is only necessary for wavelengths less than the Debye length when true Thompson scatter occurs.

Analysis of the returned power spectrum reveals the electron density, electron and ion temperatures, and LoS ion velocity \citep{Evans1969,Rishbeth1985,Nicolls2007a}.  Using a model of chemical composition at different altitudes, the average ion mass can also be found.  In addition, 3D convection velocities and electric field vectors can be derived from LoS ion velocity measurements in some of the more modern ISRs by assuming that both \(\vec{E}\) and \(\vec{V}_E\) are constant across the radar's FoV \citep{Heinselman2008}.  More recently, a new algorithm has been developed that used the Lagrange method of undetermined multipliers to generate 2D convection velocity and electric field vectors without making this limiting assumption \citep{Nicolls2014}.  Each instrument must be calibrated with known electron density measurements to account for noise and system constants \citep{Nicolls2007a}.  This is done using the plasma frequency measured from either the plasma line of the ISR spectra during summer daytime period or ionosonde measurements for all other times \citep{Bahcivan2010,Themens2014}.  The plasma line refers to a discrete line in an ISR spectra that occurs at a Doppler shift approximately equal to the plasma frequency, from which the plasma density can be derived \citep[e.g.][Figure 1]{Evans1969}.

\begin{figure}
	\centering
	\includegraphics[width=\textwidth]{ISRmap.pdf}
	\caption[ISR map]{{\:}ISR map\vspace{1mm}
	
	FoVs of all ISRs in the northern polar region.  The individual radars shown are identified by their commonly used acronyms: Poker Flat (PFISR), Resolute Bay North (RISR-N), Resolute Bay Canada (RISR-C), Sondrestrom (SFJ), European Incoherent Scatter Scientific Association (EISCAT), and EISCAT Svalbard (ESR).  RISR-N, which is primarily used throughout this thesis, is shown in orange.  The FoV of the SuperDARN RKN radar is shown by the pink outline.  Lines of constant MLAT are shown at \(\Lambda=80\deg,60\deg\) in yellow.}
	\label{fig:isrmap}
\end{figure}

The first ISR was built in Arecibo, Puerto Rico in the 1950s \citep{Gordon1958}.  Since then, at least 8 other ISRs have been deployed around the world at equatorial, mid-, and high-latitudes.  A map of the ISR systems currently operational near the northern polar cap is shown in Figure \ref{fig:isrmap}.  The most recent advancement has been the development of Advanced Modular Incoherent Scatter Radar (AMISR) systems \citep{Heinselman2008}, of which there are currently three in use, one at the Poker Flat Rocket Range (PFISR), just north of Fairbanks, Alaska and two in Resolute Bay, Canada (RISR-N and RISR-C).  The advantage of these new systems is that they are electronically steerable, unlike older ISRs which consisted of a large dish that had to be physically moved to change the beam direction.  This allows beams to be transmitted in many different directions at a very high time cadence, giving the radar the capability to make measurements in multiple directions almost simultaneously \citep{Nicolls2007a,Nicolls2007b,Heinselman2008,Bahcivan2010}.  This is very important for creating 2D maps of ionospheric conditions and tracking how they change in time \citep{Semeter2009,Dahlgren2012a,Dahlgren2012b}.

AMISR systems typically have two types of pulses, a long pulse (LP) with 72-km range gates for \(F\)-region studies and an alternating code pulse (AC) with 4.5-km range gates for \(E\)-region studies.  The exact number and configuration of beams depends on the radar mode used and can be changed easily due to the system's electronic steering.  In the common WorldDay mode, there are typically 11 beams and data is collected at \(\sim\)1 min intervals, although this can change slightly between different renditions of the WorldDay mode.

One of the most important advantages of ISR systems over CSRs is the ability to directly measure electron density in the ionosphere.  However, because ISRs are much scarcer and typically only run a few days each month, data are not as widely available.  In addition, ISRs do not actually observe small-scale coherent plasma structures like the SuperDARN array does.  However, they can directly measure large-scale structures and image density variations on scales larger that 100 km (limited by the number of beams and range gate size).  For these reasons, it is useful to use a combination of CSR and ISR measurements in ionospheric structuring studies, whenever possible.  The primary ISR used in the present volume of work is the north face of the AMISR system at Resolute Bay, Canada, RISR-N, shown in orange in Figure \ref{fig:isrmap}.  The FoV of RISR-N overlaps that of the SuperDARN RKN radar, red outline in Figure \ref{fig:isrmap}, making it ideal for these types of comparison studies in the polar cap.  The two experimental studies that employ both the SuperDARN RKN and RISR-N systems are presented in Chapters \ref{sec:paper1} and \ref{sec:paper3}.

\subsection{International Reference Ionosphere (IRI) Model}
\label{sec:iri}
In addition to radars, models can be a useful tool for understanding the background conditions in the ionosphere, particular in regions with poor instrumental coverage.  The International Reference Ionosphere (IRI) is an empirical model of the Earth's ionosphere.  It was originally created in 1969 as a joint effort between the Committee on Space Research (COSPAR) and the International Union of Radio Science (URSI) and has been periodically updated since then \citep{Rawer1975,Rawer1978,Rawer1981,Bilitza1985,Bilitza1986,Bilitza1990,Bilitza1997,Bilitza2001,Bilitza2008}.  The current version is IRI-2012 \citep{Bilitza2014}, which is used throughout the work done here.

The IRI model produces outputs of electron density, temperatures of electrons, ions, and neutrals, ion composition (O\(^+\), H\(^+\), He\(^+\), O\(_2^+\), NO\(^+\), N\(^+\)), and total electron content (TEC).  The model requires the input of a location (latitude, longitude, and altitude) and time (year, date, and time).  Additionally, the model gives the option to choose between two \(F\) peak models and three bottomside thickness models.  There are a variety of other optional input parameters such as sunspot number, ionospheric index, and F10.7 radio flux which can help make the output parameters more accurate for a particular situation.  Additionally, the model is capable of producing profiles of any of the output parameters in either space or time.  This is particularly significant for height profiles, and allows the peak heights and densities of the \(E\) and \(F\) region to be calculated.

Although the IRI model is very useful for providing background plasma density conditions where no instruments are available to make measurements, it is important to bear in mind that it is a model and therefore only represents the average conditions.  Although it is fairly accurate at mid-latitudes \citep{Coisson2006,Bilitza2012}, it does not necessarily represent well variations within the polar cap \citep{Themens2014,Makarevich2015b}.  In particular, the height of the \(F\)-region peak tends to be underestimated in daytime and the bottomside thickness does not show realistic seasonal and diurnal variations \citep{Themens2014}.  

Despite this, the IRI model is still a very important tool for establishing a density profile at any time in any location in the ionosphere.  An example of an altitude profile of plasma density provided by the IRI model was shown in Figure \ref{fig:densprofile}.  The displayed profiles show how the electron density changes with altitude at local noon (marked Day in Figure \ref{fig:densprofile}) and midnight (Night) on January 1, 2007 at \(75\deg\) N, \(0\deg\) E, geographic.

One of the most important uses of the IRI model has been its integration into ray tracing simulations.  Throughout this work, it is valuable to determine the degree to which HF radar beams refract through a dense ionosphere.  This can be done through standard raytracing tools based on numerical solutions to the Hamiltonian raypath equations, as shown in Chapter \ref{sec:paper1} \citep{Haselgrove1963,Jones1975}, however these tools require a 2D density profile along the path that the radar beam travels.  It is possible to assume a Gaussian distribution or Chapman layer as a simple model of the \(F\)-region peak, but it is much more instructive to use output from the IRI model, as is done in this work, as it is sensitive to changes in latitude as well as seasonal and diurnal variations.

\section{A Brief Review of Plasma Structuring Theory and Observations in the Polar Cap}
Much work has been done over the past several decades to study the complex picture of plasma structuring in the polar cap, including theoretical analysis of plasma dispersion relations, simulations of plasma dynamics, and observations of plasma structuring using a variety of instruments.  In this section, a survey is given of some of these advancements focusing on large- and small-scale structuring processes in the polar cap.

\subsection{Polar Patch Formation}
\label{sec:lit_patches}
Large-scale structuring in the polar cap can take a variety of forms, Section \ref{sec:polar_structure}, but here the focus is on large regions of enhanced plasma density known as polar patches  \citep[e.g.][]{Weber1984,Weber1986,Buchau1983,Buchau1985}.  One of the primary interests of this thesis is the relationships between large-scale plasma structures and small-scale irregularity formation.  This structuring has long been observed around polar patches \citep{Weber1984,Milan2002b,Moen2012}.  Although small-scale structures have also been observed around sun-aligned arcs \citep[e.g.][]{Koustov2012}, these structures also tend to be more complicated and often are surrounded by intense shears and electric fields \citep{Safargaleev2000,Aikio2002,Kozlovsky2007}.  Additionally, polar patches occur relatively frequently, providing a good opportunity to study irregularity coupling at different scales \citep{Rodger1996}.  In theory, many of the results presented in this work pertaining to the strong gradients on the edges of patches could also easily be applied to the edges of polar holes, but to date, far less attention has been paid to polar holes in the literature, making it challenging to accurately put observations in context \citep{Makarevich2015b}.

The first observations of polar patches were made in the 1960s by \citet{Hill1963}.  Polar patches can range in size from 100 km up to 1500 km and can have peak densities as much as eight times that of the background ionospheric plasma \citep{Weber1986,Hosokawa2014}.  Polar patches are known to emerge on the dayside and then propagate with the background plasma convection across the polar cap to the nightside, where they either disintegrate or recombine with enhanced-density plasma in the auroral oval \citep{Weber1985,Weber1986}.

There have been at least 6 mechanisms proposed for how polar patches form on the dayside \citep{Crowley1996,Carlson2012}:
\begin{enumerate}
	\item Discrete changes in solar wind parameters, such as IMF, density, speed, and pressure \citep{Sojka1994}
	\item Average flow patterns varying in time \citep{Anderson1988}
	\item Transient magnetopause reconnection \citep{Lockwood1992b}
	\item Plasma production by cusp particle precipitation \citep{Rodger1994,Millward1999}
	\item Plasma flow jet channels that cut continuous stretches of plasma into segments 	\citep{Valladares1998}
	\item Alfv\'{e}n wave coupling \citep{Prikryl1999}.
\end{enumerate}
Although particle precipitation has been shown to create enhanced plasma density in the cusp region \citep{Rodger1994}, it is unreasonable for this mechanism to account for the largest density enhancements observed in polar patches.  These high densities must originate from the reservoir of solar illuminated plasma on the dayside.  Additionally, a mechanism is still required to separate patches from the cusp region.  High speed flow channels can ``cut'' segments off of a tongue of high-density plasma that has been pulled into the polar cap by convection \citep{Valladares1994,Valladares1998}.  Models have also shown that fast-changing large-scale convection patterns do produce density structures similar to patches \citep{Anderson1988}.  However, the mechanism that seems to be dominant for the majority of polar patches observed is transient magnetic reconnection \citep{Carlson2012}.

In order to discuss patch formation by transient magnetic reconnection, it is important to first understand the steady-state ionospheric convection patterns in the polar cap and how they relate to the magnetosphere \citep{Cowley1980}.  Figure \ref{fig:magnetosphere} is a schematic of how the IMF, magnetosphere, and ionospheric polar cap are all interconnected.  In general, magnetic field lines in the magnetosphere can either be closed (grey lines in Figure \ref{fig:magnetosphere}), meaning they connect only the regions near the north and south pole of the Earth, or open (light blue lines in Figure \ref{fig:magnetosphere}), when one of the ends of the field lines originates from the Sun such that the field line is connected directly to the IMF.  IMF lines that are not connected to the Earth are shown in orange in Figure \ref{fig:magnetosphere}.  The polar cap boundary is generally defined as the border between open and closed magnetic field lines, shown by the green line in Figure \ref{fig:magnetosphere}.

\begin{figure}
	\centering
	\includegraphics[width=\textwidth,angle=0]{magnetosphere.pdf}
	\caption[Magnetic reconnection footprint in the ionosphere]{{\:}Magnetic reconnection footprint in the ionosphere\vspace{1mm}
	
	Diagram of the interactions between the IMF, magnetosphere, and polar cap ionosphere.  Closed field lines in the Earth's magnetosphere are shown in grey, while open field lines are light blue.  The polar cap open-closed boundary is in green.  Magnetic field lines that are part of the IMF are shown in orange.  Magnetic reconnection occurs along the X-line (red) on the dayside of the magnetopause (dashed line).  The X-line can be mapped back along magnetic field lines (yellow) to the ionosphere, where its projection is the merging gap (pink dotted line).  During reconnection events, the open-closed boundary expands equatorwards as more open field lines are formed and there is plasma flow across the merging gap. Figure adapted from \citet{Cowley1991}.}
	\label{fig:magnetosphere}
\end{figure}

By Faraday's Law, the total electromotive force, \(\xi\) around the polar cap boundary is equal to the rate of change of magnetic flux, \(\Phi_B\) within the polar cap, Equation \ref{eqn:pc_imf} \citep{Lockwood1992a}.  In steady-state solutions, \(d\Phi_B/dt = 0\), so \(\xi=0\) by definition, however the polar cap is known to be a highly dynamic system, so changes in magnetic flux are expected
\begin{equation}
	\label{eqn:pc_imf}
	-\xi = \frac{d\Phi_B}{dt} = B\frac{dA_{pc}}{dt}.
\end{equation}
In the polar cap at ionospheric altitudes, the plasma is incompressible so the magnetic field line density is relatively constant and any changes in magnetic flux must be due to changes in the polar cap area, \(A_{pc}\) due to the polar cap boundary expanding or contracting \citep{Lockwood1992a}.  The polar cap boundary changes in response to magnetic reconnection between the magnetosphere and the IMF on both the day and night sides changing the amount of open magnetic flux.

Magnetic reconnection between the IMF and the Earth's magnetosphere occurs along the X-line on the dayside magnetopause (red line in Figure \ref{fig:magnetosphere}).  The X-line in the magnetopause can be mapped along the magnetic field lines (orange) to the ionosphere, where it is referred to as the merging gap (pink dotted line in Figure \ref{fig:magnetosphere}) and lies along the steady state polar cap boundary.  

During reconnection events, the polar cap boundary moves equatorwards as the open magnetic flux into the polar cap increases, Figure \ref{fig:magnetosphere} \citep{Cowley1991,Lockwood1992a}.  In addition, the potential along the X-line also gets mapped to the merging gap, assuming there is minimal drop in potential along the magnetic field lines \citep{Lockwood1992b}.  The combination of these two effects results in flux being transferred across the boundary at a rate equal to the applied voltage  along the X-line and hence plasma flow across the merging gap \citep{Lockwood1992b}, which will be examined further in Figure \ref{fig:patch_formation}.  

Figure \ref{fig:patch_formation} is a schematic of polar patches being formed through transient magnetopause reconnection.  In all panels, the dayside is the top of the figure and the green line shows the open-closed polar cap boundary, similar to Figure \ref{fig:magnetosphere}.  Plasma flow contours are shown in black and the orange region represents high density plasma from the dayside reservoir.  The reconnection burst starts at Figure \ref{fig:patch_formation}a and the increasing amount of open magnetic flux moves the dayside open-closed boundary equatorwards, as described above \citep{Cowley1991}.

As reconnection continues in Figure \ref{fig:patch_formation}b, the open-closed boundary continues to expand equatorwards as more open flux is produced, which excites a convection pattern in the polar cap \citep{Cowley1991}.  Simultaneously, this convection starts to transport dense dayside plasma polewards \citep{Lockwood1992b}.  As more open flux is produced, Figure \ref{fig:patch_formation}c, the strength of the plasma flow increases and a large blob of dense, dayside plasma is pulled into the polar cap \citep{Lockwood1992b}.

\begin{figure}
	\centering
	\includegraphics[width=\textwidth,angle=0]{patch_formation.pdf}
	\caption[Polar patch formation]{{\:}Polar patch formation\vspace{1mm}
	
	Polar patch formation through transient magnetopause reconnection.  The open-closed boundary is the green line.  Black contours represent plasma flow.  Dense daytime plasma is shown by the orange regions.  The burst of reconnection is occurring in Figures \ref{fig:patch_formation}a--\ref{fig:patch_formation}c, but stops in Figures \ref{fig:patch_formation}d--\ref{fig:patch_formation}f.  Figure adapted from \citet{Cowley1991}.}
	\label{fig:patch_formation}
\end{figure}

At Figure \ref{fig:patch_formation}d, the burst of reconnection stops and the open-closed boundary begins to relax polewards \citep{Cowley1991}.  As it moves, continuing convection begins to pull from lower-density plasma instead of the high-density dayside reservoir, separating the dense blob in the polar cap \citep{Lockwood1992b}.  Before the convection flow stops completely, Figure \ref{fig:patch_formation}e, the blob of dense plasma ``pinches off'' from the dayside plasma to become an independent patch \citep{Lockwood1992b}.  The exact mechanism by which this happens is not well understood, but it is thought to be related to small variations in the IMF \(B_y\) component shifting the convection pattern slightly \citep{Cowley1980,Lockwood1992b}.  By the time shown in Figure \ref{fig:patch_formation}f, the open-closed boundary has returned completely to its original position and there is a newly formed patch within the polar cap.  The process can now repeat to form another patch within the polar cap.

A series of reconnection bursts can create a line of patches all propagating across the polar cap.  Reconnection bursts typically last about 2 min with anywhere between 7 min and 25 min between bursts \citep{Foster1984,Etemadi1988,Lockwood1992b}.  However, observations have shown convection flow to be close to continuous for much longer periods.  This can be explained by each reconnection burst exciting flow for a much longer interval than the time between bursts.  In this way, a reconnection burst can excite plasma flow before the convection from the previous burst has stopped, and a series of overlapping reconnection events like this can create continuous plasma flow through the polar cap \citep{Cowley1991}. 

\subsection{Theory of Plasma Instabilities}
\label{sec:lit_instabilities}
Plasma structuring, especially at smaller scales, is often discussed in terms of the growth or damping of plasma waves, or instabilities.  As discussed previously in Section \ref{sec:ionosphere_regions}, the plasma characteristics change with altitude in the ionosphere, which causes a variety of different instability mechanisms to be operational depending on the region considered.  The Farley-Buneman instability (FBI), or the modified two-stream instability tends to be a major factor in the \(E\) region where ion inertial effects are strong \citep{Farley1963,Buneman1963}.  Also operational in the \(E\) region but more dominant at higher, \(F\)-region altitudes is the gradient-drift instability (GDI) \citep{Simon1963,Hoh1963,Linson1970}.  If a wave propagation vector has a component along the magnetic field line, the current-convective instability (CCI) can be operational even if GDI is not \citep{Hoh1960,Ossakow1979,Chaturvedi1981}.  Additional instability mechanisms emerge if shears are considered, when \(\nabla E \neq 0\), such as the Kelvin-Helmholtz instability (KHI) \citep{Kintner1977,DAngelo1965}.

In the polar \(F\)-region ionosphere, GDI is typically considered to be the dominant structuring process \citep{Weber1984,Cerisier1985,Basu1988,Tsunoda1988,Fukumoto2000}.  GDI is operational when high-density perturbations in the plasma move to regions of lower background density and low-density perturbations in the plasma move to regions of higher background density, such that the wave amplitude grows relative to background conditions.  As plasma drifts and carries irregularities with it, electrons and ions have slightly different velocities and create a perturbed electric field.  This perturbed electric field \(\vec{E}'\) combined with the background magnetic field causes the density perturbations to drift in the \(\vec{E}'\times\vec{B}\) direction.  If the drift causes high- and low-density perturbations to move as described above, the instability is operational and the perturbations grow.  If instead high-density perturbations drift into regions of higher background density and low-density perturbations drift into regions of lower background density, the wave is damped.  Because GDI is considered the primary structuring mechanism at \(F\)-region altitudes, it will be the focus of this work.  Other instabilities, notably FBI and CCI, have been briefly described previously, and details can be found in the accompanying references \citep{Farley1963,Buneman1963,Ossakow1979}.

Some of the first 1D investigations of the linear GDI growth rate found that it can be described by the following simple expression involving the plasma drift speed, \(V_E = |\vec{E}\times\vec{B}|/B^2\), and the gradient scale length, \(L^{-1} = |(\partial n/\partial x)/n|\) \citep{Simon1963,Hoh1963,Linson1970}
\begin{equation}
	\label{eqn:gdi_old}
	\gamma = \frac{V_E}{L}.
\end{equation}
This expression is technically only valid if the density gradient, \(\nabla n\), is in the same direction as the plasma drift velocity, \(\vec{V}_E\), and the wave is propagating perpendicular to both.  A more general expression that is used less often does consider an arbitrary wavevector, \(\vec{k}\), but still assumes density gradients parallel to the plasma drift \citep{Tsunoda1988}:
\begin{equation}
	\label{eqn:gdi_80s}
	\gamma = \frac{k_y}{k}\frac{\vec{k}\cdot\vec{V}_E}{kL}.
\end{equation}
An even more general consideration involves arbitrary gradient and plasma drift directions \citep{Keskinen1982,Makarevich2014c}.  In this more general case, the directional dependence of GDI has been shown to be significant, such that the growth rate can actually 
change signs (determining whether wave growth or damping occurs) with different wavevector and gradient directions.  In addition, these expressions have historically been developed for a particular altitudinal regime, usually in the \(F\) region.

Recently, a new expression has been introduced that considers any arbitrary wavevector, density gradient, and altitude within the ionosphere, which allows three electrostatic instabilities, FBI, GDI, and CCI, to be considered within a single dispersion relation \citep{Makarevich2016a}
\begin{equation}
	\label{eqn:gdi_Mak16}
	(H_i-H_e)\omega = (H_i\vec{V}_{e0}-H_e\vec{V}_{i0})\cdot\vec{k}+(C_i-C_e)H_eH_i.
\end{equation}
Here, \(\vec{k}\) is the wavevector, \(\omega\) is the wave frequency, and \(\vec{V}_{\alpha 0}\) is the background velocity of species \(\alpha\).  All other quantities in Equation \ref{eqn:gdi_Mak16} are not standard and described below.
\begin{equation}
\begin{split}
	H_\alpha = S_\alpha F_\alpha + D_\alpha^{-1} F_\parallel \quad\quad\quad\quad&\quad\quad\quad\quad 
	C_\alpha = \frac{T_\alpha}{m_\alpha \Omega_\alpha} \\
	F_\alpha = i k_\perp^2 D_\alpha + \vec{G}\cdot\vec{k}_\perp D_\alpha + \vec{G}\cdot\vec{k}\times\uvec{b} \quad\quad&\quad\quad\quad\quad
	F_\parallel = i k_\parallel^2 + \vec{G}\cdot\vec{k}_\parallel \\
	S_\alpha = \frac{1}{1+r_\alpha^2} \quad\quad\quad\quad\quad\quad&\quad
	D_\alpha = -\frac{i}{\Omega_\alpha}(\omega-\vec{k}\cdot\vec{V}_{\alpha 0})+r_\alpha
\end{split}
\end{equation}
Throughout this thesis, \(T_\alpha\) refer to the temperature of species \(\alpha\).  The unit vector \(\uvec{b}\) is in the direction of the magnetic field \(\vec{B}\) and \(\vec{k}_\perp\) is the component of the wavevector \(\vec{k}\) perpendicular to the magnetic field while \(\vec{k}_\parallel\) is the component parallel to it.  The gradient strength, \(\vec{G}\), is defined by \(\vec{G} = \nabla n/n\) such that the magnitude of the gradient strength is equivalent to the inverse gradient scale length, \(G = L^{-1}\).  The quantity \(r_\alpha\) is the ratio of the collision frequency to the gyrofrequency of a particular species, given by \(r_\alpha = \nu_\alpha/\Omega_\alpha\).  

By considering certain limiting cases, expressions for the instability growth rate in different regions can be derived from Equation \ref{eqn:gdi_Mak16}.  These expressions are summarized in Table \ref{tab:gdi_exps}.  In addition to the quantities previously described in relation to Equation \ref{eqn:gdi_Mak16}, the following definitions are useful for the expressions in Table \ref{tab:gdi_exps}.
\begin{equation}
\begin{split}
	\Psi = -D_iD_e \quad\quad\quad\quad\quad\quad
	\psi =& -r_ir_e \quad\quad\quad\quad
	\hat{\psi} = \psi\left(1+\frac{k_\parallel^2}{r_e^2 k_\perp^2}\right) \\
	b = -\frac{\vec{G}\cdot\vec{k}\times\uvec{b}}{k_\perp^2} \quad\quad\quad\quad\quad\quad
	y =& \frac{k_\parallel}{k_\perp} \quad\quad\quad\quad\quad
	R = \frac{\sigma_H}{\sigma_P} = \frac{D_i+D_e}{1+\Psi} \\
	\vec{V}_d = \vec{V}_{e0}-\vec{V}_{i0} = (r_i-r_e)&\left[s_es_i(1+\psi)\left(R\vec{V}_E-\frac{\vec{E}}{B}\right)\right]
\end{split}
\end{equation}

Most of the quantities defined above are simply for mathematic convenience, however \(\psi\) is also known as the anisotropy parameter and \(R\) is the ratio of the Hall conductivity, \(\sigma_H\), to the Pedersen conductivity, \(\sigma_P\), both of which are discussed further in Chapter \ref{sec:paper2}.  Additionally, \(\vec{V}_d\) is known as the differential plasma velocity and is the difference between the background electron and ion drift.  Furthermore, the standard isothermal ion-acoustic speed, \(C_s\), is used in the expressions in Table \ref{tab:gdi_exps}.  For the purposes of this thesis, the cold plasma limit is most relevant and can be rewritten using the definition of \(b\) \citep{Makarevich2014c,Makarevich2016a}
\begin{equation}
	\label{eqn:gdi}
	\gamma = \frac{1}{1+\psi}\left(\uvec{k}\cdot\uvec{b}\times\vec{G}\right)\uvec{k}\cdot\left(\frac{\vec{E}}{B}-R\vec{V}_E\right).
\end{equation}
Equation \ref{eqn:gdi} is applicable throughout the \(E\) and lower \(F\) regions.  

\begin{table}
	\centering
	\caption[Irregularity growth rates]{{\:}Irregularity growth rates\vspace{1mm}
	
	Irregularity growth rates in the ionosphere for three different limiting cases \citep{Makarevich2016a}.}
	\vspace{5pt}
	\renewcommand{\arraystretch}{1.5}
	\setlength{\tabcolsep}{15pt}
	\begin{tabular}{l l}
	\textbf{Limiting Case} & \textbf{Growth Rate, \(\gamma\)} \\
	\hline
	GDI/FBI in \(E\) Region & \(\gamma = \frac{1}{1+\hat{\psi}}\left[\frac{\hat{\psi}}{\nu_i}\left(\left(\frac{\vec{V}_d\cdot \vec{k}}{1+\hat{\psi}}\right)^2-C_s^2 k_\perp^2\right)+\frac{br_i\vec{V}_d\cdot \vec{k}}{1+\hat{\psi}}\right]\) \\
	GDI/CCI in \(F\) Region & \(\gamma = -\frac{b}{\psi+y^2}\left[\frac{\psi\vec{k}\cdot\vec{E}_{0\perp}+E_{0\parallel}k_\parallel}{B}\right]+\frac{C_s^2}{\Omega_i}\left[r_ek_\perp^2-\frac{k_\parallel^2}{r_i}\left(1+\frac{r_i^2}{\psi+y^2}\right)\right]\) \\
	Cold Plasma in \(F\) Region & \(\gamma = \frac{b}{1+\psi}\left(R\vec{V}_E-\frac{\vec{E}_{0\perp}}{B}\right)\cdot\vec{k}\) \\
	\end{tabular}
	\label{tab:gdi_exps}
\end{table}

From Equation \ref{eqn:gdi}, the growth rate exhibits a clear dependence on the propagation direction in the field-perpendicular plane \(\uvec{k}\) and from heretofore, we will refer to this dependence as an \textit{anisotropy}  or the \textit{directional dependence}.  Although linear GDI theory as described above predicts clear dependence on wavevector direction \citep{Makarevich2014c}, this theory is only valid in the long-wavelength limit, \(\vec{k}\cdot\vec{V}_E \ll \nu_\alpha\).  In the \(F\) region, the ion-neutral collision frequency, \(\nu_i\), is about 10 Hz and if the drift speed, \(V_E\), is about 1000 m/s, fluid theory only applies to wavelengths greater than 100 m, larger than the decameter-scale waves that are detected by SuperDARN.  This raises the important question of whether or not the same anisotropy that is predicted by linear GDI theory at larger scales can be expected in small-scale irregularities where other processes (not described by the linear fluid theory) may be important.  This issue has mostly been studied through numerical simulations due to the challenges associated with measuring a high-resolution 2D spectrum experimentally.

The first simulations of GDI were done in the 1970s; they have successfully reproduced structures developing on the trailing edge of large density gradients \citep{Zabusky1973,Doles1976,Scannapieco1976,Ossakow1975,Ossakow1977}.  Later, 2D simulations were expanded to include magnetosphere-ionosphere coupling, but it was found that this only had a substantial effect for very large-scale structures \citep{Keskinen1990}.  3D simulations of GDI were introduced surrounding a large plasma density enhancement \citep{Guzdar1998} and were later improved by including plasma dynamics parallel to the magnetic field and inertial effects so that secondary KHI and tertiary shear-driven instability processes could be modeled \citep{Gondarenko1999}.  These simulations consistently showed asymmetry between the leading and trailing edge of large-scale density structures and fluctuations in plasma density that were as much as 10\%--20\% of the background plasma \citep{Gondarenko2004a,Gondarenko2004b}.  Plasma structuring can occur on the leading edge of large-scale density enhancements if the convection velocity changed rapidly such that the trailing edge becomes the leading edge and vice versa, or if there are large velocity shears initially present \citep{Gondarenko2004a,Gondarenko2004b}.

For the most part, the results of nonlinear numerical simulations very closely match expectations from linear GDI theory.  In particular, it was found that the spatial power spectra are anisotropic with most of the power concentrated in the direction where linear GDI growth is most favorable \citep{Keskinen1981a,Keskinen1981b,Keskinen1982,Gondarenko2001,Gondarenko2004b}.  However, very large shears or stronger ion-inertial factors can cause isotropy \citep{Gondarenko2001,Gondarenko2006}.  The power spectra derived from these simulations are similar to those from experimental observations \citep{Baker1978,Kelley1979}, however it is computationally challenging to run simulations down to decameter scales and observations are generally limited to rocket and satellite techniques, which can only be used to calculate power spectra in one dimension \citep{Villain1986,Moen2012}.


\subsection{Observations of Polar Cap Structuring}
\label{sec:lit_observations}
A comprehensive review of polar patch observations has been provided by \citet{Crowley1996}.  In the below overview, we focus on the observations that are most relevant to this thesis.  Polar patches were first observed in the 1960s \citep{Hill1963}, but much of the work of characterizing them using optical and radio techniques was not accomplished until two decades later \citep{Weber1981,Weber1984,Weber1986,Buchau1983,Buchau1985}.  Polar patches are density enhancements of as much as \(10^6\) cm\(^{-3}\) that occur primarily in the polar \(F\) region \citep{Buchau1983}.  There were initially categorized as extending 800--1000 km horizontally \citep{Weber1984}, but more recent studies have identified patches as large as 1500 km \citep{Hosokawa2014}.  Patches tend to drift antisunwards at speeds of 500--1000 m/s, following background convection \citep{Buchau1983,Weber1984}.

Polar patch occurrence varies both diurnally and seasonally, with a diurnal peak at magnetic noon and a seasonal peak during equinoctial months \citep{Rodger1996}.  Additionally, there is a bias towards patches occurring when there is a negative IMF \(B_z\) component \citep{Buchau1983,Rodger1996}.  The frequency of patch occurrence is greatest when the cusp is slightly daywards of the terminator.  This creates a dark polar cap, so background plasma density is low and blobs of enhanced density are easily visible \citep{Coley1998}.  In addition, the cusp then provides a gateway of highly ionized sunlit plasma into the dark polar cap.  Patches convect from the dayside through the cusp/throat region towards the polar cap \citep{Kelly1984,Foster1984,Foster1985,Foster1993,Sojka1982,delaBeaujardiere1985}.

The theoretical mechanism for patch formation via transient magnetic reconnection has been described previously, Section \ref{sec:lit_patches}.  A variety of experimental evidence has been presented to support this mechanism \citep{Cowley1998,Carlson2002}.  Experiments run in the 1980s using the European Incoherent Scatter (EISCAT) radar facility in Troms\o, Norway showed that plasma flow occurred near noon shortly after southward IMF \(B_z\) arrives at the magnetopause \citep{Etemadi1988,Todd1988}.  These plasma flows are pulsed in the region of the polar cap boundary \citep{Lockwood1993a,Lockwood1993b}, as predicted by \citet{Cowley1991}.  Observations made by an all sky imaging photometer (ASIP) in Svalbard in 1984 were fully consistent with patch formation through transient magnetopause reconnection \citep{Carlson1996,Carlson2002} and periods where polar patches developed on the dayside have been found to correspond to flux transfer events (FTE), or periods of ``bursty'' magnetic reconnection \citep{Rodger1996,Walsh2014}.  \citet{Carlson2004} identified five signatures of transient magnetic reconnection and then examined a patch formation event for these signatures using the EISCAT Svalbard radar.  All five signatures were observed as expected, strengthening the idea that transient magnetopause reconnection is responsible for patch formation.  Furthermore, \citet{Carlson2006} identified and tracked a series of patches directly from the subauroral plasma reservoir and showed that the boundary moves equatorwards before relaxing poleward, one of the main predictions of the transient magnetopause reconnection mechanism \citep{Lockwood1992b}.  Similar efforts using TEC measurements from GPS receivers have confirmed that patches form from dense dayside plasma convecting into the polar cap and ``pinch off'' due to bursty reconnection events allowing low density plasma to isolate them \citep{Zhang2013,Walsh2014}.

Besides polar cap patch formation and evolution, much of the previous experimental effort focused on small-scale structuring processes near polar patches.  One of the key observations in this regard was that, over time, the trailing edge of a patch became steeper than the leading edge and irregularities developed on the trailing edge \citep{Weber1984}.  Backscatter power from HF radars also tended to be greater on the trailing edges of both moving polar patches \citep{Milan2002b} and sun-aligned arcs identified from ASIP data \citep{Koustov2012}, which is generally attributed to plasma being unstable to GDI on the trailing edge but stable on the leading edge of such large and moving density structures.  Contradicting the results of \citet{Weber1984}, studies of patches using all-sky airglow imagers (ASI) found that the density gradient on the leading edge of polar patches tends to be 2--3 times steeper than that on the trailing edge \citep{Hosokawa2016}.  Additionally, large finger-like structures on the trailing edge of patches have been identified by \citet{Hosokawa2016} that were tens to hundreds of kilometers in size and agreed with predictions made by GDI simulations \citep{Gondarenko2004b}.  \citet{Moen2012} made direct measurements of plasma density structuring using the ICI-2 sounding rocket.  This study provided evidence that decameter-scale plasma structuring (the same as observed by HF radars) had spawned from kilometer-scale density gradients (observed by ASIP and ASI methods) and structuring was greatest where GDI was operational according to estimates of the growth rate.

Recent experimental advancements have improved the ability to image density structures in the ionosphere, particularly using multi-instrument approaches.  \citet{Semeter2009} introduced a method by which the density pattern in a particular volume could be imaged in 3D using AMISR systems.  This technique was later used by \citet{Dahlgren2012a,Dahlgren2012b} to image a polar patch.  \citet{Dahlgren2012b} additionally used ASIPs and SuperDARN radars to observe the polar patch and found that backscatter tended to be observed more on the trailing edge of the patch.  This agrees well with many studies that have examined the asymmetry of small-scale plasma structuring surrounding plasma patches.  

Even though observations of plasma irregularities near polar patches present certain advantages, small-scale irregularities may be present in the polar cap regardless of polar patch presence.  Plasma irregularities at decameter scales are typically studied with HF coherent scatter radars, Section \ref{sec:csr}.  The factors that affect irregularity observation can roughly be divided into two groups: irregularity production and radar propagation.  Irregularity production factors are related to the existence of plasma irregularities in the ionosphere, while radar propagation factors are related to the ability to observe irregularities with a ground-based radar, Section \ref{sec:superdarn}.  

Recent radar observations suggested that plasma irregularity production in the polar cap may be much higher than at lower latitudes.  When background plasma conditions are favorable for radar propagation in the polar caps, FAIs are observed nearly continuously \citep{Bristow2011}, however, there are still definite diurnal, seasonal, and solar cycle trends similar to what is observed at lower, auroral latitudes \citep{Kane2012}.  There is an increase in echo occurrence at solar maximum, particularly in the midnight sector \citep{Milan1997,Koustov2004}.  Seasonally, high-latitude radars observe a peak in echo occurrence during equinoctial months \citep{Koustov2004}.  

Additionally, backscatter tends to occur at closer ranges in summer at midnight, but moves further away from the radar in other seasons at midday \citep{Milan1997}.  In general, echo occurrence is high during the day and low at night, but nighttime occurrence is directly proportional to the \(F\) peak density and daytime occurrence can be significantly reduced by a dense and conductive \(E\) layer \citep{Koustov2004,Kane2012}.  In the daytime, solar illumination causes the electron density of the \(E\) region to increase dramatically, making it highly conductive.  This conductive layer can provide a conduit by which the potential differences associated with FAIs in the \(F\) region can be shorted out, effectively reducing \(F\)-region plasma irregularities \citep{Vickrey1982}.  Both irregularity production and radar propagation factors probably contribute to these regular variations of echo occurrence, because while a dense, daytime ionosphere creates favorable radar propagation conditions, it can also contribute to \(E\)-region shorting or smoothing of gradients which surpress irregularity production \citep{Koustov2004}.  

The two most important production factors for irregularities produced through GDI are density gradients and convection velocity or, equivalently, convection electric fields, Section \ref{sec:lit_instabilities}.  Properly-oriented density gradients and electric fields are a necessary condition for GDI to be operational, Section \ref{sec:lit_instabilities}, and the stronger these are, the more likely the irregularity production rate will be high.  Experimental studies focused on the latter factor, with both backscatter power and occurrence showing an increase with increasing electric field strength \citep{Fukumoto2000,Danskin2002,Makarevich2014b}.

Amongst the propagation factors, arguably the primary one is the background plasma density and the associated amount of the radar beam refraction.  If the background density is too low, the beam is under refracted and is never perpendicular to the magnetic field, however if the density is too high, the beam is over refracted. Previous studies show that observation of small-scale FAIs is most favorable if the background density is between \(1\times 10^{11}\) m\(^{-3}\) and \(4\times 10^{11}\) m\(^{-3}\) \citep{Danskin2002,Makarevich2014b}.  If the \(D\) region is too dense, this can also in theory attenuate the signal and reduce the rate of returned backscatter, although observations have shown this is not a major factor in echo occurrence \citep{Danskin2002}.  

Generally, both production and propagation factors are important in determining when irregularities are observed by ground-based HF radars, so it is important to distinguish between the two to interpret data correctly.  Additionally, the relationship between plasma irregularities of different scales is important to understand how small-scale irregularities develop in the context of larger ionospheric dynamics.


\section{Motivation and Objectives}

The plasma in the polar cap ionosphere is highly structured in a non-trivial manner \citep{Tsunoda1988,Carlson2012}.  In addition to being of purely scientific interest, this structuring creates complications when receiving radio signals through the ionosphere.  Because the structuring is highly irregular and varies on many spatial and temporal scales, unpredictable wave refraction, phase shifts, and amplitude attenuation can occur, making radio signals difficult to detect.  This introduces problems in any system that involves ground-to-satellite, satellite-to-ground, or over-the-horizon communication, such as navigation or communication.  Because of the increasing interconnectedness of modern society, any interruptions or errors in these systems can have broad impacts on infrastructure, including commercial and defense interests, as well as everyday life.

The ionosphere is an important part of the highly coupled Sun-Earth environment.  The polar cap ionosphere essentially serves as the boundary conditions for waves propagating through the magnetosphere.  In addition, understanding plasma structuring on a range of scales is important for correctly interpreting backscatter from ionospheric radars.  In particular, one of the main purposes of SuperDARN is to create large-scale convection maps of both hemispheres including polar cap regions.  These convection maps have been used to give context of both the magnetosphere and thermosphere and are widely used in space physics research.  To map convection accurately, it is extremely important to understand when backscatter is observed and when it is absent, and why this is so.  Understanding plasma instability and wave growth mechanisms is very important for answering these questions.  Furthermore, models and simulations of ionospheric dynamics in the polar caps often incorporate plasma structuring details, such as instability growth mechanism.  Although GDI is often considered the dominant instability mechanism in the \(F\) region, it is helpful to understand what other mechanisms may be operational and how they are interconnected.

The aim of this work is to investigate small-scale plasma irregularity production in the \(F\)-region polar cap.  Both global and local control of irregularity production is studied, i.e. we consider global solar control through solar illumination and solar wind as well as much more local control by plasma density gradients and convection electric field.  The majority of observational data used in this thesis have been obtained by ground-based ionospheric radars, both CSR systems (Section \ref{sec:csr}) and ISR systems (Section \ref{sec:isr}), with particular emphasis on radars within the SuperDARN array, Section \ref{sec:superdarn}.  These datasets have also been complemented with information from models as well as satellite data when necessary.  Experimental results are considered in the context of a linear fluid theory of the gradient-drift instability.

The first objective of this body of work is to evaluate the extent of solar control over irregularity occurrence in the polar \(F\) region.  Although radar backscatter is the lower limit to plasma irregularity occurrence, examining the occurrence of radar backscatter in the context of both solar illumination (considered direct solar control) and IMF factors (considered indirect solar control) reveals physically relevant trends.  This gives a global or macrophysical view of where and when plasma irregularities are most likely to occur.

The second objective is to investigate asymmetry in the GDI growth rate surrounding large-scale density structures.  A model is used to identify where the growth rate is high and hense where plasma irregularities are likely to occur surrounding a polar patch and how they can be observed by a ground-based HF radar.  The model considers a large, elongated polar patch drifting in an arbitrary direction relative to a radar and assumes irregularity production is directly dependent on the linear GDI growth rate.  The model results are expected to contribute to more accurate interpretation of observations of irregularities near polar patches.

The third objective is to investigate directional dependencies of small-scale plasma irregularities produced through the gradient-drift instability operating in the linear regime.  Two-dimensional measurements of density gradients and electric fields are made such that the directionally-dependent GDI growth rate can be calculated, which is then compared with plasma irregularity power and occurrence in a particular location.  This gives insight into whether the production of small-scale irregularities is directionally dependent, similar to the linear GDI growth rate, or if the distribution with the propagation direction is isotropic due to other processes.

The outline of this thesis is as follows.  Chapter \ref{sec:introduction} provides an overview of the polar ionosphere and plasma structuring mechanisms at both large and small scales.  Chapter \ref{sec:paper1} presents an experimental investigation of decameter-scale plasma irregularities in the polar \(F\)-region and their solar control, with a particular focus on statistical analysis of data collected with a recently-deployed SuperDARN radar in the southern polar cap.  The same results were also presented in a recent journal article \citep{Lamarche2015}.  Chapter \ref{sec:paper2} presents a modeling study of the linear GDI growth rate of plasma irregularities near a large, elongated polar patch drifting in an arbitrary direction within the viewing area of an oblique-scanning radar such as SuperDARN.  These results have also been recently published \citep{Lamarche2016}.  Chapter \ref{sec:paper3} presents an experimental investigation of the directional dependence of small-scale plasma structures by considering simultaneous measurements of electric field and density gradients from an AMISR  system and FAIs observed by SuperDARN in the context of predictions made by linear GDI theory and the model results presented in Chapter \ref{sec:paper2}.  The same results have been recently published \citep{Lamarche2017}.  Chapter \ref{sec:conclusion} presents a summary of the most significant results of this body of work, their implications for interpretation of radar observations of plasma structures in the polar cap and suggestions for future research involving coherent and incoherent scatter radar facilities in the polar region.

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