Space weather is the branch of physics that studies the time varying phenomena in the solar system. The principal driver of space weather phenomena is the Sun, specifically its magnetic field variations and the solar wind. The effect of solar variations on the planetary environment are caused by the coupling between solar wind particles and the magnetic field produced by the Earth. This chapter gives a semi-quantitative treatment of various scientific ideas relevant to space weather research.
3.1. Space Plasma
Plasma, also known as the fourth state of matter due to its properties that differentiate it from the conventional gaseous state, is ubiquitous through-out the visible Universe. Plasma is a gas which is composed of roughly equal number of positive and negatively charged particles, a property known as charge quasi-neutrality. The term quasi-neutral is used because although the gas has almost equal amounts of positive and negative charges, the mixture is electromagnetically active. Due to incomplete charge shielding, long range electromagnetic fields play a big role in the dynamics of plasma.
In a quasi-neutral plasma, due to the presence of partial electric shielding the potential due to the charges now takes the well known Debye form
where r is the spatial distance with respect to the charge and is the permittivity of vacuum. The electric potential decays with the Debye length scale λ_d at which a balance between thermal vibrations which can disturb quasi-neutrality, and electrostatic forces due to charge separation, is achieved. The Debye length scale depends on the electron temperature and plasma density.
In equation (3.2) above, the Debye length scale is expressed in terms of the Boltzmann constant k_b, the electron temperature T_e, free space permittivity , and electron charge e. One can visualise the positively charged ions having a cloud of electrons shielding them at the distance of λ_d.
It is also possible to take into account the shielding effect of the ions. The effective Debye length is now expressed as an addition of two terms: one for electrons (equation (3.2)) and a similar term for the ions by replacing T_e for the ion temperature T_i (n_i ≈ n_e).
Plasma Parameter
Consider a Debye sphere of radius λd. This sphere contains N_e = 4/3 π λ^3_d n_e electrons. The plasma parameter g is defined as N^{−1}_e . Rewriting this, we can say:
The description of plasma used in many applications in space is applicable when g << 1. In this situation the Debye shielding is significant, and the quasi-neutral plasma obeys collective statistical behavior. The plasma parameter g also correlates with the collision frequency. The collisions in plasma increase with increasing density and decreasing temperature, and if g → 0 the plasma becomes nearly collisionless. The collisionless property helps in making simplifying assumptions about plasma dynamics and serves as the starting point for the adiabatic theory of plasma motions in the Earth's magnetosphere which will be discussed in section 3.3.1.
德拜长度
在准中性等离子体中，由于部分电屏蔽的存在，电荷产生的电势现在呈现为众所周知的德拜形式：
其中 r 是相对于电荷的空间距离， 是真空介电常数。电势随着德拜长度标度 λ_d 衰减，在这个长度标度上，能够扰乱准中性的热振动与电荷分离产生的静电力之间达到平衡。德拜长度标度取决于电子温度和等离子体密度。
在许多空间应用中使用的等离子体描述适用于 g << 1 的情况。在这种情况下，德拜屏蔽很重要，且准中性等离子体服从集体统计行为。等离子体参数 g 也与碰撞频率相关。等离子体中的碰撞随着密度的增加而增加，随着温度的降低而增加，如果 g -> 0，等离子体几乎变为无碰撞的。无碰撞特性。
3.2 Sun & the Solar Wind
The Sun is an almost perfectly spherical ball of plasma which is the the center of our solar system and the only source of light and energy for all living and meteorological processes on Earth. Apart from terrestrial weather, the Sun is also the primary driver of space weather which results from the interaction between the solar wind and planetary magnetospheres.
Figure 3.1 shows a cross section of the Sun with various layers. We give a brief description of them below.
Core:
The core of the Sun is the site for the thermonuclear fusion reactions which produce its energy. It extends from the center to about 20 − 25% of the solar radius [Garc´ıa et al., 2007]. It has a temperature close to 1.57 × 107 K and a density of 150 g/cm3 [Basu et al., 2009]. Nuclear fusion in the core takes place via the well known proton-proton chain (pp).
Radiative Zone:
The radiative zone extends from 25% to 70% of the solar radius. The nuclear reactions in the core are highly sensitive to temperature and pressure. In fact, they are almost shut off at the edge of the core. In the radiative zone, energy transfer takes place via photons (radiation) which bounce around nuclei until they reach the convective zone.
Convective Zone:
The convective zone lies between 70% of the solar radius to a point close to the solar surface. Density decreases dramatically going from the core to the radiative zone and subsequently the convective zone. In this region, the solar material behaves more like a fluid. Due to the temperature gradient which exists across it, the primary source of transport is here via convection.
Photosphere:
The photosphere is the visible 'surface' of the Sun, since the layers below it are all opaque to visible light. A layer of about 100 km thickness, the photosphere is also the region from where sunlight can freely escape into space. The photospheric surface has a number of features i.e. sunspots, granules and faculae. Sunspots (see section 3.2.3) are magnetic regions where the solar material has a lower temperature compared to its surroundings. Magnetic field lines are concentrated in sunspot regions, and the field strength in sunspots can often be thousands of times stronger than the on the Earth.
Chromosphere:
The Chromosphere extends for a distance of almost 5000 km after the photosphere. The chromosphere is known for the existence of features called spicules and prominences. The chromosphere has a red colour which is generally not visible due to the intense light given off by the photosphere but can be observed through a filter centered on the Hydrogen Hα spectral line.
Solar Transition Region:
A thin (100 km) region between the chromosphere and the solar corona where the temperature rises from about 8000 K to 5 × 105 K, the solar transition region might not be well defined at all altitudes; however its existence is evidenced by a bifurcation of the dynamics of the solar plasma. Below the transition region, the dynamics is dictated by gas pressure, fluid dynamics, and gravitation while above the region, the dynamics is dictated more by magnetic forces.
Corona:
An aura of plasma around the Sun that extends millions of kilometers into space, the corona can be observed during a total solar eclipse (figure 3.2) or with a coronagraph. The temperature of the corona is dramatically higher than the photosphere and chromosphere. The average temperature can range from 1 × 106 K to 2 × 106 K while in the hottest regions it can be as high as 2 × 107 K [Erd´elyi and Ballai, 2007]. Although the reason for this dramatic increase is still not well understood, there exist various explanations using concepts of magnetic reconnection [Russell, 2001, Erd´elyi and Ballai, 2007] and Alfv´en waves [Alfv´en, 1947]. There is a critical height in the corona, known as the source surface, below which the magnetic field controls the plasma completely. Above it the plasma carries the magnetic field with it into the interplanetary medium.
日冕是太阳周围数百万公里远的等离子体光环，可以在日全食（图3.2）期间或通过日冕仪观察到。日冕的温度远高于光球层和色球层。平均温度范围可以从1×10^6 K到2×10^6 K，而在最热的区域，温度可以高达2×10^7 K [Erd´elyi和Ballai，2007]。尽管这种急剧增加的原因尚不完全清楚，但存在各种解释，使用了磁重联[Russell, 2001, Erd´elyi和Ballai, 2007]和阿尔文波[Alfv´en, 1947]的概念。在日冕中，有一个称为源表面的关键高度，低于该高度，磁场完全控制等离子体。在其上方，等离子体将磁场带入行星际介质中。
3.2.2 Solar Wind & Heliospheric Magnetic Field
The idea that the Sun was ejecting charged particles outwards into space was first hinted at after the solar storm of 1859 by Richard Carrington [Cliver and Dietrich, 2013] and later by George FitzGerald [Meyer-Vernet, 2007]. Arthur Eddington, in a footnote of an article about the comet Morehouse in 1910, was the first to suggest the existence of the solar wind, without naming it so [Durham, 2006].
In the 1950s, studies of the anti-solar orientation of the ion tails of Halley's comet led to the theory of solar corpuscular emission [Biermann, 1952, 1957, 1951]. Parker [1958b, 1960, 1965] argued that the corona cannot remain in hydrostatic equilibrium and that supersonic expansion of the corona is responsible for the outward expulsion of charged particles, which the author referred to as the solar wind. Parker [1958b] also proposed a spiral model for the Heliospheric Magnetic Field (HMF) and suggested that the solar wind carried with it the solar magnetic field. The Parker model was further supported its ability to explain the effect of the HMF on the modulation of galactic cosmic rays and their measured intensities close to the Earth [Parker, 1958a]. In 1959 the Soviet spacecraft Luna 1 was the first to directly observe the solar wind and measure its strength [Harvey, 2007]. Subsequently, the Mariner 2 mission recorded properties of the positive ion component of the solar wind and confirmed the Parker spiral HMF model [Neugebauer and Snyder, 1966].
The structure of the HMF is central to explaining the formation and propagation of the solar wind. The HMF in steady state points radially outward and rotates with the Sun, producing an Archimedean spiral structure as postulated in Parker [1958b] and shown schematically in figure 3.3. Photospheric observations of the magnetic field (see Global Oscillation Network Group https://gong.nso.edu) are often extrapolated to compute approximations to the coronal HMF topology. There exist a number of techniques used to perform such extrapolations: potential field based methods such as Potential-Field Source Surface (PFSS) [Schatten et al., 1969, Altschuler and Newkirk, 1969], PFSS variants such as Potential-Field Current Sheet (PFCS) [Schatten, 1971], Current-Sheet Source Surface (CSSS) [Zhao and Hoeksema, 1995], and several others. Apart from potential based models, there exist more involved techniques based on Magnetohydrodynamics (MHD) such as Magnetohydrodynamics Around a Sphere (MAS) [Linker et al., 1999], ENLIL [Odstrˇcil et al., 1996, Odstrˇcil and Pizzo, 1999a,b, Odstrˇcil, 2003, Odstrˇcil et al., 2004] and EUHFORIA [Pomoell and Poedts, 2018].
The HMF can be seen as a combination of two components: the poloidal magnetic field and the toroidal magnetic field. The two fields often exchange energy between themselves over the course of several years in a cyclical phenomenon known as the solar cycle (section 3.2.3). Interested readers can read Owens and Forsyth [2013] for an in-depth review on the phenomena that drive the HMF.
The expansion of the coronal magnetic field leads to an eventual opening of field lines at the source surface (see figure 3.3) and the ejection of the solar wind. This hot plasma consists mostly of protons, electrons and a small number of helium and heavy ions. The solar wind spirals outwards in all directions, carrying with it the magnetic field. Close to the Earth's magnetosphere, this wind has a nominal speed of about 400 km s−1 while its high speed component has an average velocity of ∼ 700 km s−1 (figure 3.4).
Near Earth Measurements
The solar wind has the heliospheric magnetic field frozen in1 , and as it propagates in the interplanetary medium, it carries the solar magnetic field with it [Alfv´en, 1942, 1943]. Important solar wind quantities such as: 1. solar wind speed, 2. proton density, and 3. magnetic field strength are recorded at the well known L1 Lagrangian point where the gravitational fields of the Earth and the Sun approximately balance out.
3.2.2 太阳风与日球层磁场
1859年太阳风暴之后，理查德·卡林顿（Richard Carrington）首次暗示了太阳正在向外太空喷射带电粒子的观点[Cliver and Dietrich, 2013]，随后乔治·菲茨杰拉德（George FitzGerald）也提出了类似看法[Meyer-Vernet, 2007]。1910年，亚瑟·爱丁顿（Arthur Eddington）在一篇关于莫豪斯彗星（Morehouse comet）的文章的脚注中，首次提出了太阳风的存在，尽管当时并未使用这一名称[Durham, 2006]。
HMF的结构对于解释太阳风的形成和传播至关重要。在稳定状态下，HMF径向向外延伸并随太阳旋转，形成了帕克[1958b]所假设的阿基米德螺旋结构，如图3.3所示。通常，通过对光球层磁场（参见全球振荡网络组 NSO: Global Oscillation Network Group (GONG) ）的观测进行外推，可以计算日冕HMF拓扑结构的近似值。进行此类外推的技术有多种：基于势场的方法，如势场源表面（PFSS）[Schatten et al., 1969, Altschuler and Newkirk, 1969]；PFSS变体，如势场电流片（PFCS）[Schatten, 1971]、电流片源表面（CSSS）[Zhao and Hoeksema, 1995]等。除了基于势的模型外，还有基于磁流体动力学（MHD）的更复杂技术，如球体周围的磁流体动力学（MAS）[Linker et al., 1999]、ENLIL[Odstrˇcil et al., 1996, Odstrˇcil and Pizzo, 1999a,b, Odstrˇcil, 2003, Odstrˇcil et al., 2004]和EUHFORIA[Pomoell and Poedts, 2018]。
Sunspots are temporarily occurring regions on the Sun's photosphere that appear as dark spots. They are areas of magnetic field concentration where the field lines often 'puncture' the solar surface inhibiting convection and producing regions with lower temperatures than the surroundings. Sunspots generally last anywhere between a few days to a few months. They can occur in pairs or groups and can accompany other phenomena such as coronal loops, prominences, and reconnection events.
Since the 19th century the number of sunspots on the Sun's surface have been recorded as the sunspot number (SSN). Sunspots populations increase and decrease, thereby behaving as markers for solar activity levels. The cyclical behavior of sunspot populations is called the sunspot cycle or solar cycle (figure 3.5).
Figure 3.5 depicts how the area occupied by sunspots changes with solar latitude and time. During the start of a solar cycle (solar minimum), sunspots start appearing at higher latitudes. Over the course of the cycle, they move towards the equatorial regions and their number increases to some maximum (solar maximum). Towards the end, the number of sunspots diminishes and the entire cycle starts over. This repetitive behavior happens over approximately 11 years.
Because sunspots are magnetic phenomena, the solar cycle represents cyclical behavior of the HMF. During solar minimum, the poloidal component of the solar magnetic field is at its strongest and it is the closest it can get to a magnetic dipole configuration. Towards solar maximum, energy is transferred from the poloidal component to the toroidal component, resulting in complex field configurations which are evidenced by larger numbers of sunspot clusters.
The solar cycle also gives rise to variations in solar irradiance [Willson et al., 1981]. Between 1645 and 1715, very few sunspots were observed, a period known as the Maunder minimum. This coincided with lower than average temperatures in Europe, which was called the little ice age. Although the Maunder minimum was a period of lower solar irradiance, recent research [Owens et al., 2017] has demonstrated that this was neither the only factor nor the most significant in causing lower than average temperatures during the little ice age.
In chapter 7, the sunspot number data as well as the flux tube expansion factor (fS or FTE) and the magnetic field strength computed by the CSSS model will be used to create a input data set for building the dynamic time lag regression model proposed therein. Using the input parameters, the DTLR model provides an estimate for the near Earth solar wind speed as well as the propagation time. Measurements of the solar wind speed will also be used in chapters 4 and 5 as inputs to the Dst forecasting models applied therein.
The Earth's magnetosphere (figure 3.6) is a region surrounding the planet where its magnetic field dominates the interplanetary magnetic field. The Earth's magnetic field shields the atmosphere and terrestrial life from the impact of the solar wind.
As the solar wind approaches the Earth, it is slowed down and deflected by the Earth's magnetic field. Since the solar wind is supersonic when it arrives and slows down to subsonic levels, a shock wave is generated in the process (bow shock). Much of the solar wind kinetic energy is converted to thermal energy when it crosses the bow shock into the magnetosheath. The magnetosheath spans from the bow shock to the magnetopause. The magnetopause is the outer boundary of the Earth's magnetic shield. Its location is ∼ 10R_E (R_E = 6372 km, the radius of the Earth).
Earth's magnetic shielding is not perfect, and some particles manage to get trapped inside the cavity of the magnetosphere. This region of trapped plasma is known as the the van Allen radiation belts. Particles trapped in the radiation belts execute complex motions which can be approximately modelled using ideas from adiabatic theory and diffusion described in section 3.3.1 below. The plasmasphere is the inner region of the radiation belts which contains cold, dense plasma. The portion of the magnetosphere facing away from the Sun (called the nightside) is stretched out in a tail-like shape by the deflected solar wind, hence referred to as the magnetotail. The magnetotail has an approximate extent of up to 1000R_E.
3.3.1 Particle Motions & Adiabatic Theory
This section gives a quick introduction to the theory of charged particle motions in the magnetosphere. The reader may refer to Roederer [1970] for an in depth treatment of this subject. To understand the motions of charged particles in the magnetosphere, the role of electric and magnetic forces must be understood.
It is well known from classical electromagnetism that the force exerted on a particle with charge q by a magnetic field B and an electric field E is given by the well known Lorentz force (equation (3.3)).
The first component of equation (3.3) (qE) is either parallel or opposite to the local electric field depending on the charge of the particle. The second component qv × B involves a vector cross product so it is always perpendicular to the plane spanned by vectors v and B . In order to understand its effects, we can decompose the particle velocity in two components; v_{||} parallel to B and v⊥ perpendicular to B. If E = 0, then the particle executes a circular motion with properties shown in equation (3.4). Here ρ is the gyroradius and ω is the gyrofrequency or cyclotron frequency. In the case where , the trajectory is helical.
Apart from the gyro motion, there are some important drift forces that significantly influence particle motions.
• Electric field drift: If E has a component E_⊥ perpendicular to B , the electric field accelerates and decelerates the particle in the two hemispheres of the orbit. The orbit becomes a distorted circle, and the particle drifts in a direction perpendicular to the electric field with a velocity v_d = E × B/B^2 .
• Magnetic gradient drift: When the magnetic field varies in space (as is the case of the Earth), a gradient in the field strength in the direction perpendicular to B gives rise to a gradient drift velocity given by vg = 1/2 m v^2⊥ B × ∇B/aB^3.
• Magnetic curvature drift: If the magnetic field has a curvature, this creates an additional drift motion with velocity vc = mv{||} B ×(b ˆ·∇)b ˆ qB2 (b = B/B ).
The equations of motion for charged particles in the general case of spatially varying electric and magnetic fields do not admit closed-form solutions. The motions are generally complex and require lengthy numerical integrations to be resolved.
The guiding center approximation helps us to decompose particle motions into three periodic components (figure 3.7): 1. gyration around magnetic field lines, 2. bounce motions between magnetic north and south poles, and 3. equatorial drift of electrons and protons, each with its own time scale.
Adiabatic Invariants
When a physical system with periodic motion is varied slowly as compared to the time period of its periodicity, the transformation can be characterized as adiabatic. Formally speaking, for systems which are described by Hamiltonian dynamics, we can write the equations of motion in terms of the canonical position q, the canonical momentum p, external parameters θ, and the system's Hamiltonian H(q, p|θ)：
If the system shown in equation (3.5) executes a periodic motion in the q, p phase space, it admits an adiabatic invariant A given in equation (3.6).
The quantity A would remain approximately constant if the external parameters θ were varied adiabatically (i.e. if changes in θ happen over a time period much greater than the period of oscillation of the system).
Applying the idea of adiabatic invariance to charged particle motion in the magnetosphere, it is possible to associate one adiabatic invariant with each periodic motion i.e. gyromotion M, bounce J, and drift Φ (equation (3.7)).
The first invariant M is associated with the Larmor gyration - it is the magnetic moment of the current generated by the circular motion of the particle around the field line.
The second invariant J is associated with the bounce motion between the two magnetic mirrors near the north and south poles (the quantity s is an appropriately chosen arc length coordinate along the bounce trajectory). The bounce motion between the magnetic poles can be explained by the conservation of the particle energy and the first invariant M. Because field strength |B| increases near the poles, v⊥ also increases to conserve M; however, to conserve energy, v_{||} decreases until the particle can no longer move farther along the field line (and bounces back).
The third invariant Φ, associated with equatorial drift motion, is actually the magnetic flux through the barrel shape envelope of the particle drift. A particle's guiding magnetic field line can be identified by its radial position r and its longitude . The magnetic flux of the drift can then be computed by integrating over .
Associated with each adiabatic invariant is a timescale which determines how easily its conservation can be violated. The timescales for M, J, and Φ are the time periods of the gyromotion, bounce motion, and equatorial drift motion respectively. Since it takes much a longer time for the particles to complete a drift motion around the Earth as compared to bounce and gyromotion (in that order), the invariance of Φ is most easily violated - a fact which is used in the simplification of the Fokker-Planck diffusion system described below.
Plasma Diffusion
Because we consider populations of charged particles, it is natural to employ some kind of distribution based picture for magnetospheric plasma. The adiabatic invariants give us a phase space or coordinate system by which we can express quantities of interest.
The main quantity of interest in this case is the phase space density f(t,M, J, Φ) which is a function of time and three invariants. The phase space density tells us the number of particles in a particular region of the phase space, and at a particular point of time. Diffusion behavior arises when one or more of the invariants are violated, which can happen due to a number of reasons such as: 1. non-adiabatic variations of the magnetic field, 2. external forces, 3. interaction with electromagnetic waves, and 4. collisions with atmosphere/ionosphere. The plasma diffusion system [Schulz and Lanzerotti, 1974] can be written as a generalized Fokker-Planck system as shown in equation (3.10).
It is possible to simplify this system by considering the two main categories of diffusion: radial diffusion and pitch angle diffusion. Radial diffusion allows particles to move farther or closer to the Earth, and pitch angle diffusion moves the magnetic mirror points along the field lines.
Rewriting Φ ∝ 1/l, the third invariant can be expressed in terms of the drift shell l (larger value of l implies greater distance from the Earth). The radial diffusion system can be obtained from equation (3.10) by keeping M and J fixed, considering diffusion in ` (violation of Φ invariance), and by approximating pitch angle diffusion as a loss process [Walt, 1970, Roederer, 1970]. The resulting system is shown in equation (3.14).
The first term on the right hand side, , models diffusive phenomena in Φ but is expressed in the drift shell coordinate `. Pitch angle diffusion is approximated using a loss process λ(l, t)f, where λ(l, t) is the loss rate. As an alternative it is also possible to express the loss rate as a loss time scale τ (l, t) = 1/λ(l,t) , but in this thesis we will use the former convention.
The radial diffusion system in equation (6.1) is the starting point for chapter 6 where a surrogate model of the phase space density ˆf is built to perform Bayesian inference over the parameters of the diffusion coefficient κ and loss rate λ.
As was noted earlier, the solar wind is largely deflected by the Earth's magnetic field but some particles still leak into the magnetosphere. This particle injection is governed by the interaction between the magnetic field carried by the solar wind and the Earth's magnetic field, also known as solar wind - magnetosphere coupling. It plays an important role in determining space weather conditions in the Earth's vicinity.
Solar wind plasma gets trapped in the Earth's magnetic field at a rate that is modulated by the solar wind - magnetosphere coupling. The drift motions of charged particles in the magnetosphere as discussed in section 3.3 lead to many current systems. The prominent current systems (pictured in figure 3.6) are 1. the ring current, 2. field aligned current, 3. tail current, and 4. magnetopause current. These current systems induce magnetic fields that interact with the Earth's magnetic field and mutate it. Weakening of the Earth's magnetic field strength due to strong ring currents leads to geomagnetic storm conditions which can have adverse impacts on orbiting satellites and ground based infrastructure.
For the purposes of space weather monitoring and forecasting, the state of the magnetosphere and geomagnetic phenomena are often represented by proxies known as geomagnetic indices. Geomagnetic indices give us the ability to summarize the state of the magnetosphere in terse framework. They are often calculated by averaging several ground based measurements of magnetic fluctuations, generally at a cadence of a few hours.
Chapter 4 gives a brief introduction to the popular geomagnetic indices and formulates gaussian process models for producing probabilistic one hour ahead forecasts of the Dst index. In chapter 5, we augment the Dst model from chapter 4 with long short-term memory (LSTM) networks and obtain five hour ahead forecasts of the Dst.
is the permittivity of vacuum. The electric potential decays with the Debye length scale λ_d at which a balance between thermal vibrations which can disturb quasi-neutrality, and electrostatic forces due to charge separation, is achieved. The Debye length scale depends on the electron temperature and plasma density.
and the first invariant M. Because field strength |B| increases near the poles, v⊥ also increases to conserve M; however, to conserve energy, v_{||} decreases until the particle can no longer move farther along the field line (and bounces back).
, models diffusive phenomena in Φ but is expressed in the drift shell coordinate `. Pitch angle diffusion is approximated using a loss process λ(l, t)f, where λ(l, t) is the loss rate. As an alternative it is also possible to express the loss rate as a loss time scale τ (l, t) = 1/λ(l,t) , but in this thesis we will use the former convention.
