Solar wind energy transfer and the asymmetric geospace

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Solar wind energy transfer and the asymmetric geospace

Paul Tenfjord

Dissertation for the degree of Philosophiae Doctor (PhD)

Department of Physics and Technology University of Bergen

April 2017

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Preface

This thesis including three published papers are submitted for the degree of philosophiae doctor (PhD) in physics at the Department of Physics and Technology, University of Bergen.

The thesis is divided into an introductory part and a part consisting of three papers published in international peer reviewed journals.

Paper I P. Tenfjord, N. Østgaard, Energy transfer and flow in the solar wind-magnetosphere-ionosphere system: A new coupling function,Journal of Geophysical Research, Vol. 118,

doi:10.1002/jgra.50545, 2013

Paper II P. Tenfjord, N. Østgaard, K. Snekvik, K. M. Laundal, J. P. Reis- tad, S. Haaland, and S. E. Milan, How the IMF By induces a By component in the closed magnetosphere and how it leads to asymmetric currents and convection patterns in the two hemispheres,Journal of Geophysical Research, Vol. 120,

doi:10.1002/2015JA021579, 2015

Paper III P. Tenfjord, N. Østgaard, R. Strangeway, S. E. Haaland, K.

Snekvik, K. M. Laundal, J. P. Reistad, S. E. Milan, Magneto- spheric response and reconfiguration times following IMF By re- versals,Journal of Geophysical Research,

doi:10.1002/2016JA023018, 2016

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iv Preface

During my PhD studies, I have also contributed to the following papers.

A J. P. Reistad, N. Østgaard, K. M. Laundal, S. Haaland, P.

Tenfjord, K. Snekvik, K. Oksavik, and S. E. Milan, Intensity asymmetries in the dusk sector of the poleward auroral oval due to IMF B_x, Journal of Geophysical Research, Vol. 119, doi:10.1002/2014JA020216, 2014

B S. Haaland, J. P. Reistad, P. Tenfjord, J. Gjerloev, L. Maes, J.

DeKeyser, R. Maggio, C. Anekallu, N. Dorville, Characteristics of the flank magnetopause: Cluster observations,Journal of Geo- physical Research, Vol. 119, doi:10.1002/2014JA020539, 2014 C N. Østgaard, J. P. Reistad, P. Tenfjord, K. M. Laundal, K.

Snekvik, S. E. Milan, S. Haaland, What are the mechanisms that produce auroral asymmetries in the conjugate hemispheres?, AGU Monograph Series, Auroral Dynamics and Space Weather, Editors: Yongliang Zhang and Larry Paxton, 2015

D K. M. Laundal, S. E. Haaland, N. Lehtinen, J. W. Gjerloev, N.

Østgaard, P. Tenfjord, J. P. Reistad, K. Snekvik, S. E. Milan, S.

Ohtani, B. J. Anderson, Birkeland current effects on high-latitude ground magnetic field perturbations, Geophysical Research Let- ters, doi:10.1029/2015GL064831, 2015

E N. Y. Ganushkina, M. W. Liemohn, S. Dubyagin, I. A. Daglis, I. Dandouras, D. L. De Zeeuw, Y. Ebihara, R. Ilie, R. Katus, M.

Kubyshkina, S. E. Milan, S. Ohtani, N. Østgaard, J. P. Reistad, P. Tenfjord, F. Toffoletto, S. Zaharia, O. Amariutei, Defining and resolving current systems in geospace,Annales Geophysicae, 33, 1369-1402, doi:10.5194/angeo-33-1369-2015, 2015

F K. M. Laundal, J. W. Gjerloev, N. Østgaard, J. P. Reistad, S. Haa- land, K. Snekvik, P. Tenfjord, S. E. Milan, The impact of sunlight on high-latitude equivalend currents,Journal of Geophysical Re- search, Vol. 121, doi:10.1002/2015ja022236, 2016

G J. P. Reistad, N. Østgaard, P. Tenfjord, K. M. Laundal, K.

Snekvik, S. Haaland, S. E. Milan, K. Oksavik, H. U. Frey, and A. Grocott, Dynamic effects of restoring footpoint symmetry on closed magnetic field-lines, Journal of Geophysical Research, Vol 121, doi:10.1002/2015JA022058, 2016

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H S. E. Haaland , B. Lybekk, L. Maes, K. M. Laundal, A. Pedersen, P. Tenfjord, A. Ohma, N. Østgaard, J. P. Reistad , K. Snekvik, North-south asymmetries in cold plaams density in tyje magnetosphere: Cluster observations,Journal of Geophysical Research, doi:10.1002/2016JA023404, 2016

I K. Snekvik, N. Østgaard, P. Tenfjord, J. P. Reistad , K. M. Laun- dal, S. E. Milan, S. E. Haaland, Dayside and nightside magnetic field responses at 800 km altitude to dayside reconnection,Journal of Geophysical Research, doi:10.1002/2016JA023177, 2017

J S. E. Milan , L. B. N. Clausen, J. C. Coxon, J. A. Carter. M.

T. Walach. K. Laundal, N. Østgaard, P. Tenfjord, J. Reistad, K. Snekvik, H. Korth, B. J. Anderson, Overview of solar wind- magnetosphere-ionosphere-atmosphere coupling and the genera- tion of magnetospheric currents,Space Sci Rev, doi:10.1007/s11214- 017-0333-0, 2017

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Acknowledgements

This thesis is the result of four years work as a PhD student. Professor Nikolai Østgaard has been my supervisor. I am very thankful for the opportunity to do my PhD at the Birkeland Centre for Space Science (BCSS) under his supervision. I could not have asked for a better supervisor. I have very much enjoyed working with him and I am grateful for everything I have learned, experienced and all the fun we have had. Karl Magnus Laundal is not very good in chess, but he has been an excellent supervisor. He always finds time to help me out, and is always eager to discuss new ideas.

I have been privileged to work with a number of very skilled people during the past years. The work would not nearly have been as fun if it had not been for my colleges at BCSS. We are an efficient group, helping each other to succeed. I have enjoyed the numerous discussions, as well as coffee breaks and lunch breaks together we have had. We have been fortunate to participate at conferences in Vienna, San Francisco and Snowmass, to name a few. These conferences are not only scientifically useful, but have also made it possible for our group to build a network of colleagues and friends from all over the world.

In 2015 I was awarded a Fulbright scholarship. Between August 2015 and June 2016 I worked with Robert Strangeway at the University of California Los Angeles.

A huge thank you goes to Bob for inviting me. The UCLA group, led by Prof. Chris Russell were great hosts and I am grateful for the hospitality they showed. Thank you all for making this both a scientifically productive and a pleasant stay!

Collaborating with Kristian Snekvik, Jone Reistad, Steve Milan and Stein Haaland has been very rewarding. Thank you all for your contributions. I also really appre- ciate Kristian, Jone, Anders Ohma and Arve Aksnes taking the time to proofread this thesis. Finally, thank you Ann Karene for proofreading the thesis, and for trying to be enthusiastic about the content.

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Chapter 1 Introduction

The motivation for this thesis has been twofold: Advance our understanding of: 1) the energy transport from the solar wind to the magnetosphere-ionosphere system and 2) the processes leading to the asymmetric geospace. An unresolved and fundamental problem in space science is to understand the dynamical behaviour of the asymmetric geospace. There are several factors that can lead to asymmetries: Orientation of the In- terplanetary Magnetic Field (IMF), different solar illumination of the ionosphere in the two hemispheres, and asymmetries in the Earth’s magnetic field [Østgaard et al., 2015].

In this thesis the focus will be on the IMF orientation, and especially on the effect of the IMFBy-component. The orientation of the IMF defines the magnetic reconnection geometry at the magnetopause and controls how energy is transferred across the magnetopause. Often not appreciated is the fact that the most common orientation of the IMF is to have a dominantBy component (see Fig. 1.1). Our objective has been to understand how and on what timescales the magnetosphere and ionosphere changes from a symmetric state to a asymmetric state (or vice versa) due to the IMFBy component.

The presence of an IMFBycomponent leads to a variety of asymmetric behaviour, such as an induced B_y in the magnetosphere, asymmetric release of magnetic stress, asymmetric convection patterns and asymmetric auroral displays between the two hemispheres. We have focused on a dynamical description, where special attention is given to the causal relationships between the different asymmetric effects, understanding how one process leads to the other. While our work on the energy transfer is not concerned with hemispherical differences directly, it describes how energy is transmitted and de- posited in the magnetosphere-ionosphere system. Understanding the coupling between the solar wind and the magnetosphere-ionosphere has been paramount to understanding how asymmetric forcing from the solar wind results in asymmetric behaviour in the coupled system.

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2 Introduction

Figure 1.1: Distribution of IMF clockangle between 1981 and 2016

In Paper I we develop a semi-empirical energy coupling function based on upstream solar wind parameters and what we consider to be the best estimates for energy dissipation processes. To quantify the energy input we used the method of dimensional analysis to relate solar wind parameters to the transfer of energy across the magnetopause. Through a cross-correlation analysis we parametrize the energy input during a wide range of conditions.

In Paper II, we study the effect of IMFBy and how it leads to an inducedBy component in the magnetosphere using the Lyon-Fedder-Mobarry (LFM) MHD model [Lyon et al., 2004]. Here, the IMF By controls the dayside reconnection and the geometry of the newly reconnected field lines such that the energy transport across the magnetopause becomes asymmetric between the northern and southern lobes. This asymmetric energy flow also produces asymmetric stress between the northern and southern hemispheric footpoints, which is released through Alfvèn waves and leads to asymmetric convection patterns. We also explain the process of restoring symmetry in which asymmetric azimuthal flows in the ionosphere is the manifestation of asymmetric stress release.

Paper III is a follow-up to study to explore whether the predicted timing for the IMF By to induce a localBy component is supported by observations. To study the dynamic effect of the IMFBy component we identified events where the IMFBy changed polarity from positive to negative, or vice versa. Using this database of polarity changes we used four GOES spacecraft to study the response and reconfiguration of the magnetic field at geosynchronous distances. A significant change in the localB_y component was seen within 10 minutes of the arrival of IMF By at the magnetopause (response time), while it took approximately 40 minutes for the magnetic field to completely reconfig- ure.

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3

This thesis highlights the use of magnetohydrodynamics (MHD) to understand how the equations of Newton and Maxwell can describe a wealth of physics, from what is seemingly a simple set of equations. The focus is to describe the underlying physics and assumptions made in the three published papers. We will mainly let the papers speak for themselves while the purpose of the introduction is to establish an understanding of the physical processes. The examples and illustrations are often highly idealized, while sufficient to describe the point at hand. In many topics the formal mathematical methodology is replaced by explaining the physical meaning of the method. There are often many different ways of finding and describing relations between physical quantities. This is often a source of confusion, as the different ways may have different physical meaning, even when they lead to the same answer. In light of this, the physical explanations presented in this thesis are based on first principle physics. A discussion regarding the fundamental variables and equations is presented in Chapter 2. Here, the discussion of what will be known as theE,jvsv,B-paradigm sets the premises for how we will qualitatively describe the physics in this thesis.

Chapter 2 starts out with a discussion of when a collection of particles can be regarded as a fluid. We avoid the formal mathematical treatment which can be found in any introductory space plasma text-book and we focus instead on the physical description and understanding. We discuss the physical meaning of the thermal pressure and its importance in the fluid-description. A thorough example of how thermal momentum is transported across a surface is presented. The motivation is that analogues can be made from this example when we later discuss energy and momentum convergence. Once the fluid-properties of the plasma has been established, we present the MHD theory.

This includes a brief discussion regarding the validity and assumptions made.

In Chapter 3 we describe the fundamental processes governing energy and mass flow in the solar wind - magnetosphere - ionosphere system. We describe this energy transport in terms of conservation laws and look at the convergence of energy flux in the atmosphere and ring current, emphasising how energy and momentum is transported and how it is converted into other forms. This section also includes a discussion on different but sometimes equivalent ways of describing energy conversion.

Chapter 4 is about asymmetries in geospace. Previous work on inter-hemispherical asymmetries are summarized, but the main focus is on the effect of IMF By in gener- ating asymmetries. Our focus is to describe how momentum is transmitted via Alfvèn waves, and describe how the ionospheric footpoints reach a force balance with the source region. We describe the mechanism of how IMF By can lead to asymmetric aurora from simple force balance arguments. Timescales describing the time it takes the external By to induce a local By component found empirically are compared to

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4 Introduction

those found in simulations. The last section is a discussion about the effect of IMF B_y during northward IMF B_z. We describe the evolution of field lines reconnecting at high-latitudes on the dayside when an IMF By is present, and how this leads to an induced By component. This section is part of a paper which is in preparation.

Chapter 5 contains a brief summary of Paper I, Paper II and Paper III.

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Chapter 2 Basic concepts in space plasma physics

A plasma is a hot ionized gas consisting of approximately equal numbers of positively charged ions and negatively charged electrons. Individual charged particles behaves differently compared to neutral particles. Neutral particles interact only when they collide, whereas charged particles can interact through long range electromagnetic forces.

That means that a charged particle is subject to the electromagnetic fields produced by other particles in the plasma. Another important distinction from an ordinary gas is the collision frequency. Space plasmas can be considered as nearly collisionless. In a high density neutral gas it is collisions that transmits information, and eventually makes the gas relax to equilibrium when external forces are removed. Consider for example moving your hand through the air. Neutral molecules collide with your hand, extracting linear momentum from the motion of your hand and adding it to the molecules. These molecules will, in turn, collide with air molecules in front of your hand. Information is then propagated to the air molecules ahead of the hand forcing them to move (assuming that the movement of your hand is slower than the speed which information can prop- agate, the speed of sound). In a collisionless plasma, the charged particle you hit with your hand would, intuitively, behave as a projectile, due to the absence of collisions.

However, we will see that in the collisionless plasma it is the long range electromagnetic forces that transmits information, and the magnetic field constrains the motion of the charged particles.

This chapter provides an introduction to the different levels of theory for a plasma.

As noted in the introduction we will focus on the physical understanding instead of the formal mathematical treatment that can be found in any introductory plasma physics text-book [e.g.Baumjohann and Treumann, 2012; Choudhuri, 1998; Friedberg, 1987;

Goldston and Rutherford, 1995]. In Section 2.1-2.3 we describe the different ways to describe a collection of charged particles. In the description of the kinetic approach, we focus on the averaging scheme, from which a collection of particles can be de-

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6 Basic concepts in space plasma physics

scribed as continuum, such that a tractable theory with macroscopic quantities can be used to study nearly all plasma phenomena of interest. The motivation is to appreci- ate more fully the physics content of MHD, and the subtleties involved. In Section 2.4 we introduce MHD, and emphasize the physical meaning of the plasma pressure and its importance in the MHD description. In addition, the force balance in ideal MHD is briefly discussed, as well as giving an introduction to numerical simulations. Sec. 2.5 introduces to the two contradicting paradigms known as thee E,j andv,Bparadigms, and explain why we use the latter. This discussion sets the premise for how we describe the large-scale dynamics throughout this thesis.

2.1 Single particle motion

There are different levels of looking at a collection of particles. The most direct level is considering the plasma as a collection of individual particles. In this description the state of the system is prescribed by the position and velocity coordinates of each particle. The dynamics of the system can be studied by Newton’s laws of motion. A particle with mass mand chargeqmoving with velocityvin an electromagnetic field, satisfies the equation of motion:

mdv

dt =q(E+v×B) (2.1)

where the first term on the r.h.s. is the Coloumb force and the second term is the Lorentz force. The combination represents the electric fieldE⁰ experienced by the particle. HereE⁰=E+v×Bis the electric field in the plasma rest frame. A particle with velocity v_⊥ (perpendicular to the magnetic field) in a constant and uniform magnetic field will move in a circle in a plane perpendicular to the magnetic field. The particle performs the circular motion around a central point called the guiding centre [Alfven, 1940]. If the particle also has a parallel velocity component, it will lead to a uniform translation of the circular trajectory, that is, the particle spirals in a helix about the line of force. This translational motion can now be regarded as the motion of the guiding centre itself. This is known as the guiding centre approximation. The guiding centre model can be considered as a set of rules prescribing how the guiding centre moves in a few particular situations [e.g Choudhuri, 1998]. In the single particle description the magnetic and electric fields are prescribed, and it does therefore not constitute a self-consistent theory. Also, since the number of particles is very large, it is not real- istic to solve the dynamical equations for each particle. It can nevertheless aid us to develop an intuition of the motion of charged particles in electric and magnetic fields.

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2.2 Kinetic Description 7

2.2 Kinetic Description

This next higher level of description offers a solution to describing the evolution of a large collection of particles. The state of the system is now described by a distribution function f(x,v,t). In order to describe the state of nparticles in a single distribution function we introduce what is known asphase space. This is a six-dimensional space where each point represents the state of one particle, e.g. particle 1 has positionx₁ = (x₁,y₁,z₁) and velocity v₁ = (u₁,v₁,w₁). For n=10, the phase space consists of 10 points. Fig. 2.1 shows the six-dimensional phase space with n=10. At time t₁ the state of the system is given by the position and velocity coordinates of the dots. At a later time the particle has moved to new coordinates represented by the six-dimensional trajectory.

(v_x,v_y,v_z)

(x₁,x₂,x₃)

(x1,v1,t1) t2

Figure 2.1: Six-dimensional phase space withn=10. Each particle represented by a single point. As time evolves the particle traces out a trajectory in phase space.

In order to describe the change from t₁ tot₂ we need an equation describing how f(x,v,t)changes in time. This equation is theVlasov equation, or equivalently collionless Boltzman equation (with subtle differences in the derivation of the two). Fig. 2.2 shows an example of a Maxwellian velocity distribution at a fixed position in space at some fixed time. Fig. 2.2a shows a velocity distribution which has zero mean velocity.

A gas or plasma left to itself will eventually relax into such a distribution. Consider for instance the gas in a room where there are no external forces (mechanical equilibrium).

Clearly, there is no flow of gas and the mean velocity of the gas is zero. The width of the Maxwellian distribution (or spread of the particle distribution in velocity space) is our connection to the concept of temperature. At some later time the gas is subjected to an external force, causing a flow of the gas in some direction. The velocity distribution will then have non-zero mean velocity (drifting Maxwellian), as showed in Fig. 2.2b.

The Vlasov equation describes how the distribution function evolves in time.

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8 Basic concepts in space plasma physics f(v)

v 0

f(v)

v v₀

a) b)

Figure 2.2: a) Maxwellian distribution with zero velocity (equilibrium), b) drifting Maxwellian distribution.

The derivation of Vlasov’s equation was first shown starting with the Liouville equation of statistical mechanics [Baumjohann and Treumann, 2012]. However, an equivalent approach is here briefly described starting with Klimantovich-Dupree, and averaging the equation over a specific volume. By understanding the dimensions of the integration we also understand the validity of this description, that is, under which conditions our collection of particles can be described through an average distribution function. This methodology which we will use is different from the usual formal procedure taking the ensemble average. Instead we present a simple statistical method, based on the assumption that one need not distinguish between particles that have approximately the same velocity, and are located at approximately the same place. The motivation is to highlight the physical limits of the averaging scheme.

The Klimantovich-Dupree equation is an exact microscopic description where it is necessary to know the position and velocity of every particle at all times. We emphasize that no averaging assumptions are needed to derive this equation, and it still contains information about all particles in the system. We will not go into details of its derivation, but instead explain the physical meaning. We can define the exact microscopic phase space density as:

fm(x,v,t) =

N

∑

i=1

δ[x−x_i(t)]δ[v−v_i(t)] (2.2) The Dirac delta function δ[x−x(t)] tells us that the space phase density is different from zero only at the position and velocity of the particle (at a timet). If we only have one particle (N = 1) at a time t, the delta function is zero everywhere except at the positionand velocity coordinates where the particle happens to be. A different way to visualize this is to consider the coordinate x and v as the Eulerian (fixed) coordinate and x(t) and v(t) as the Lagrangian (follow the particle) coordinates [Goldston and Rutherford, 1995]. Our particle traces out a trajectory in phase space given by x(t)

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andv(t). If we integrate over theEuleriancoordinatesxandvat some timet, we will find the exact number of particles. By integrating over only the Eulerian velocity space we can find the particle density, this represents finding the position of all the particles, regardless of their velocity:

nm(x,t) = Z

fm(x,v,t)dv (2.3)

The resulting microscopic density distribution is a spikey distribution, meaning that it is infinite at the particles positionsx=x_i(t)and zero elsewhere [Goldston and Ruther- ford, 1995]. The evolution for each particle, x_i(t),v_i(t) (Lagrangian coordinates) is determined by the forces acting on it. The forces acting on each particle is given in Eq. 2.1, which we can write in a different form:

mdv_i

dt =q(E_m(x_i(t),t) +v_i(t)×B_m(x_i(t),t)), dx_i/dt=v_i (2.4) Here,E_m and B_m represents the electric and magnetic fields determined from the microscopic version of Maxwell’s equation, respectively. Lets consider one of the particles in our system. The electromagnetic forces acting on this particle are generated by the motions of all the other particles in our volume. The electromagnetic fields generated by all the other particles are what dictates the motion of our particle. The electromagnetic forces depend both on the particles position (x_i(t)), which is a function of time, and also explicitly on time. These fields are obtained from the microscopic Maxwell’s equations:

∇·E_m=ρm/ε₀, ∇×E_m =∂B_m/∂t

∇·B_m=0, ∇×B_m =µ₀(J_m+ε₀∂E_m/∂t) (2.5) ρm(x,t) :=

∑

s

qs

Z

fm(x,v,t)dv J_m(x,t) :=

∑

s

q_s Z

vf_m(x,v,t)dv (2.6) Maxwell’s equation are related to the particles motion through the charge density and the current density (Eq. 2.6).

Eq. 2.2-2.6 provide a complete, exact, self-consistent and fully dynamical theory describing the exact motion of all the particles, the electric and magnetic fields they gen- erate, and the collective effect on other particles. The Klimantovich-Dupree equation is an alternative form, which is equivalent to the above, it is given as [c.f.Baumjohann and Treumann, 2012]:

∂f_m

∂t +v·∇_xfm+ q

m(E_m+v×B_m)·∇_vfm=0 (2.7)

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Solving these equations for all particles simultaneously (coupled equations), and self- consistently yields the detailed motion of all the particles. However, it is important to realize that this is still a microscopic solution, and its solution (in terms of very many particle trajectories) is of limited value.

Instead, we seek a tractable set of macroscopic quantities (such as velocity, magnetic field, pressure) as a function of position and time. Thus, we continue by describing a way to average Eq. 2.7 and the coupled equations. Clearly, when we average Eq. 2.7 we average over all the particle properties over all the particles in the system.

Our goal is now to find an expression for the average distribution function. As mentioned, due to the presence of the δ function in Eq. 2.7, the distribution is spikey.

The average distribution function is smooth and will indicate the number of particles over some length, which we will define below. Thus, in this averaging scheme we are reducing our knowledge about individual particles to gain a smooth average distribution function.

The question is then, which limits should the integral have in position space (∆V_x =∆x∆y∆z) and velocity space (∆V_v = ∆u∆v∆w)? Lets consider only one component in the position space: ∆x. The same arguments hold for the velocity space. The lower limit of the integration is determined by the fact that we need enough particles in each basic cell so as to provide a statistically well-defined mean. This is related to the mean spacing of particles: ∆xn^−1/3. This becomes our lower integration limit. The basic cells are shown in Fig. 2.2 as the squares in the zoomed-in box, where the arrows represent the mean quantity in the basic cell.

The upper limit must be small compared to the macroscopic properties of the quantity.

Clearly, if the upper limit is too large, then we would lose information about the macroscopic property (e.g. average density). Thus, we require that the upper limit should be small enough so that macroscopic properties do not vary significantly within the box.

These upper limit boxes are shown in Fig. 2.2 and illustrates that a curved line (representing some macroscopic property) can be considered as straight lines within the box. This upper limit is related to the Debye length, so that ∆x<λD, where λD is the Debye length. A plasma particle is subject to the electromagnetic fields produced by other particles in the plasma. A charged particle produces long-range electromagnetic fields, however its effect is usually screened off by particles with the opposite charge within the distance of the Debye length. Therefore, our upper limit preserves collective plasma responses within the Debye length scale, but at the same time is shielded from external fields. This is indeed a very special limit, but only if this special limit exists can we pass from single particle description to the distribution function description.

We refer toGoldston and Rutherford[1995] for the complete derivation and discussion of the Vlasov equation.

The average distribution function f(x,v,t) is then the number of particles in our

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small phase space box, divided by the volume of the box:

f(x,v,t):= lim

n^−1/3<∆x<λ_D

R

∆Vx

R

∆Vv f_m(x,v,t) R

∆Vx

R

∆Vv

(2.8) Where f(x,v,t) now represents the smoothed properties of the plasma for distances greater than the Debye-length (∆x≥λD). We are also left with a term describing the deviation from the average. This can be ignored if we assume that mutual interactions among the particles can be ignored, that is, the plasma is collionless (no Coulomb collisions). The magnetic and electric fields are also split into a smooth average and a discrete part.

v

x

dX dV

Figure 2.3: Illustration of special integration limits shown for some macroscopic quantity. The larger boxes represent the upper limit of the integration related to the Debye-length. This limit is small compared to the overall spatial extension of some quantity (curved line). The smaller basic cells shown in the zoomed-in part represent the lower limit, which must be large enough to have a sufficiently large number of points so that a well-defined mean can be calculated.

The smoothed averaged Klimantovich-Dupree equation (Eq. 2.7) describes the evolution of the average distribution functions response to the smoothed, average electric and magnetic fields in the plasma. This is the Vlasov equation (also known as the collisionless Boltzmann equation):

∂f

∂t +v·∇_xf + q

m(E+v×B)·∇_vf =0 (2.9) When coupled with the now macroscopic Maxwell equations (as given in Eq. 2.6 but

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now without the subscriptm):

∇·E = ρc/ε₀ (Gauss’ Law) (2.10a)

∇×E = ∂B/∂t (Faraday’s Law) (2.10b)

∇·B = 0 (Non-existence of monopole) (2.10c)

∇×B = µ₀(J+ε₀∂E/∂t) (Ampère-Maxwell Law) (2.10d)

The average charge and current densities are given as:

ρ_c(x,t) :=

∑

s

q_s Z

f(x,v,t)dv (2.11)

J(x,t) :=

∑

s

q_s Z

vf(x,v,t)dv (2.12)

(2.13) Eqs. 2.9-2.13 are the fundamental set of equations that provides a complete kinetic description of plasma.

2.3 Fluid Description

In the previous section we showed that the evolution of a system of particles can be described by the Vlasov equation. We now seek a macroscopic description of the collection of particles. A macroscopic solution differs from the description in the previous section where quantities depend on location, velocity and time. At the macroscopic level the quantities depend only on location and time. This is how we usually perceive the world. Instead of thinking of a river as a collection of particles, we consider it as a continuum where macroscopic forces such as gravity or wind can change the velocity or flow of water. The fluid description is formally derived by taking appropriate moments of the distribution function (Eq. 2.9). Any introductory text-book will show the formal derivation of the fluid moments and fluid moment equations [Baumjohann and Treumann, 2012;Choudhuri, 1998]. Here, we will only present the parts which are important for the physical understanding of the scheme. The main idea of this section is to show that it is sufficient to consider the collection of particles as a continuum. Thus, by identifying the fluid moments: density, velocity, temperature and magnetic field, a set of equations describing the evolution of these quantities is a sufficient description of the large-scale plasma. Before we present the conservation laws it is convenient to define the fluid moments. The velocity moments of the averaged smoothed distribution

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2.3 Fluid Description 13

function are:

n(x,t) :=

Z

f(x,v,t)dv (density [#/m³]) (2.14) v_b(x,t) := 1

n Z

vf(x,v,t)dv (flow velocity [m/s]) (2.15) Π(x,t) := m

Z

vvf(x,v,t)dv (momentum flux density tensor[N/m²])(2.16)

For simplicity, the species subscript s=e,i are omitted here and throughout most of this section. The higher-order moments (such as heat flux) are not shown. The density (Eq. 2.14) is simply the smoothed average of the microscopic distribution function (Eq. 2.2). The bulk flow, v_b, is the macroscopic flow velocity of the particle. The momentum flux density tensor can be recast into another form by defining the particles velocity,v, in terms of bulk flow,vb, and thermal velocity, w. Thenv=vb+w, where the thermal velocity of the particle is defined as the velocity of the particle relative to the mean. From this definition the first momentum is unchanged since the thermal motions are defined to have no mean (hwi=0). The second moment can then be expressed as

Π=P_{i j}+mnv_bv_b (2.17)

where P_{i j} is the pressure tensor measured in the rest frame of this species of particles.

Before we emphasize the meaning of the pressure tensor and the assumptions made in defining the fluid moments, we will define the fluid moment equations. The large-scale dynamics of plasma is described by the equations of hydrodynamics:

∂n

∂t +∇·(nv_b) = 0 particle conservation (2.18)

∂nmv_b

∂t +nm∇·(v_bv_b) = −∇·Pi j+F momentum conservation (2.19) The formal derivation of these equations found in any text-book [e.g.Baumjohann and Treumann, 2012] is through appropriate moments of the Vlasov equation (Eq. 2.9).

Instead we treat the familiar laws of hydrodynamics as an unavoidable consequence of the simplest physical principles [Parker, 2007, Ch.8].

2.3.1 Conservation laws and the distribution function

In what follows we connect the concept of the fluid moments to the hydrodynamic conservation laws. The familiar continuity equation (Eq. 2.18) is constructed on the basis that the time rate of change of the density is equal to the negative divergence of the flux density of that quantity. It tells us that there are no sources or sinks of particles.

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The density of particles can change only by flow across its boundaries. Consider a differential volume as shown in Fig. 2.4 (the shear forces will be discussed later). In the Sec. 2.3 we defined the average distribution function f(x,v), which is the relative probability of finding a particle with a given velocity vector v. The total differential number of particles inside our differential volume is therefore n dx= f(x,v)dvdx, wheredx=dx₁dx₂dx₃as shown in Fig. 2.4. The particles leaving the shaded surface S (x₁-direction) are carried across by the velocityv₁=dx₁/dt. In terms of the distribution function, the differentialnumber of particles carried across the surface per second is then f(x,v)dvdx/dt = v₁f(x,v)dvdx₂dx₃ = nhv₁idx₂dx₃, where hv₁i is the average velocity of the particles in the x₁ direction (according to Eq. 2.15: hvi= v_b). The particles carried into or out of the opposite surface have the same expression but with opposite sign. Thus, the rate of change of the number of particles in the cube, in terms of the flow of particles across each of the six sides can be expressed as:

∂n

∂tdx=−[nhv₁idx₂dx₃]_dx₁+ [nhv₁idx₂dx₃]_x₁₌₀+... (2.20)

dx₁ dx₂

dx₃

P21

P22

P₁₁

P33

P₂₃

x₁

x₂

x₃

S

Figure 2.4: Illustration of differential element of volume with sides parallel to the coordinate surfaces.

P_{i j} represents the pressure tensor, whereiis the direction of the force acting on the plane perpendicular to j. The normal stresses are defined wherei= j. The two shear stresses illustrated represent forces acting in thex₂direction on surface j=1 and j=3.

By dividing by dx we obtain Eq. 2.18 [c.f. Goldston and Rutherford, 1995, Ch.

6]. The reason for this thorough explanation of the mass conservation equation is to show that the momentum conservation equation is constructed from exactly the same principle.

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2.3 Fluid Description 15

2.3.2 Transport of bulk and thermal momentum

The objective of this section is to understand the physical meaning of the pressure tensor, and at the same time show that the conservation of momentum follows from the same physical principle as the mass conservation. First, let us ignore the external forceFin Eq. 2.19 and focus only on momentum density arising from particle motion carrying momentum with it. Eq. 2.19 can then be expressed as:

∂nmv_b

∂t + nm∇·(v_bv_b) +∇·Pi j

= ∂nmv_b

∂t +∇·(nmvv) =0

= ∂nmv_i

∂t + ∂

∂xj

(nv_imvj) =0 (2.21) where the last term on the left hand side of the second line is now expressed in terms of the momentum flux density tensor (Eq. 2.16). The third expression is the same as the second, but written in index notion so that the directions can be specified in this discussion. It has a form similar to the mass conservation equation. Consider now a momentum transport across the shaded surface S in Fig. 2.4, that is the flux (nv₁) through S (normal tox₁) of x₂ directed momentum (mv₂). Put in yet another way, we quantify the number of particles per unit area per second (flux: nv₁) passing through our shaded surface, times the momentum in thex₂ direction (momentum mv₂), carried by each particle. This is the meaning ofP₂₁ in Fig. 2.4, where 2is the direction of the force acting on the plane perpendicular to1. The approach is similar to that of above, we are considering the number of particles carried per second across the shaded surface (normal tox₁) which as before is f(x,v)dvdx/dt=v₁f(x,v)dvdx₂dx₃. However, now these particles are carryingx₂-directed momentum,mv₂(in the mass conservation they only carried particles). Thus, the amount of momentum in this direction carried (per second) across our shaded surface is: mv₂v₁f(x,v)dvdx₂dx₃. The total amount of mv₂ carried across the shaded surface (x₁-direction) is then^R mv₂v₁f dv, which can be written as: mnhv₂v₁i. This is the momentum gained or lost throughoneof the surfaces.

The total rate of change of the momentum in thex₂-direction (mv₂) is:

∂nmv_b2

∂t =− ∂

∂x₁(mnhv₂v₁i)− ∂

∂x₂(mnhv₂v₂i)− ∂

∂x₃(mnhv₂v₃i) (2.22) The left hand side rate of change of x₂-directed momentum (hv₂i= v_b2), is balanced by the transport ofx₂-directed momentum across the various surfaces. The first term on the right hand side corresponds to P₂₁ in Fig. 2.4, whileP₂₂ and P₂₃ represents the middle and last, respectively. Eq. 2.22 can be generalized to equal Eq. 2.21. We can now define pressure, which will turn out to be nothing else then a momentum flux. The

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momentum density tensor in Eq. 2.16 was split into one term related to the bulk flow (nmv_bv_b) and a pressure term. The pressure tensor is defined as:

Pi j:= mnh(v−v_b)(v−v_b)i

= mnhwwi (2.23)

wherewis the velocity of the particle relative to the mean (see Eq. 2.17). The objective of this derivation has been to define the familiar concept of pressure in its fundamental sense, which is a momentum flux. Pressure is simply the momentum flux density transported by the thermal motions (w). Pressure has nothing to do with collisions [Parker, 2007]. Now returning to our derivation of the momentum balance (Eq. 2.22). The flux in thex₁,x₂andx₃-directions ofx₂momentum can now be written in terms of bulk flow and thermal motions:

∂nmv_b2

∂t =−∂P₂₁

∂x₁ −∂P₂₂

∂x₂ −∂P₂₃

∂x₃ − ∂

∂x₁(mnv_b1v_b2)− ∂

∂x₂(mnv_b2v_b2)− ∂

∂x₃(mnv_b3v_b2) (2.24) Generalized, this becomes:

∂mnvi

∂t =−∂Pi j

∂xj

− ∂

∂xj

(mnv_iv_j) (2.25)

where v_b is replaced by v in the remaining text. In this section we have ignored any external forces acting on the system (Eq. 2.19). Any bulk force F, e.g. gravity or Maxwell stress, applied to the fluid has no direct influence on the thermal motions, and will therefore only act on the bulk flow [Parker, 2007]. This force can therefore simply be added to the right side of Eq. 2.25 which makes it equal to the momentum equation in Eq. 2.19.

With our understanding of the pressure tensor it is now appropriate to discuss how well-defined this quantity actually is. In our fluid treatment our relevant quantities are functions ofxandtonly. A simple Maxwellian is uniquely described by the density and temperature. In the case where the velocity distribution is Maxwellian the pressure tensor can be replaced with a scalar pressure, meaning that the pressure is isotropic. Not all velocity distributions are as simple as the isotropic Maxwellian, but the fluid form can still handle fairly complex situations, such as bi-Maxwellian with different tem- peratures along the magnetic field than across the field [Paschmann and Daly, 1998, Ch.6]. However, we cannot expect the pressure tensor to represent all the effects associated with the full distribution function of the particles. For instance the very complex features of f(x,v)might require the complete kinetic approach [see discussion inGold- ston and Rutherford, 1995, p.387].

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2.4 Magnetohydrodynamics 17

In ordinary fluid and gas dynamics collisions are so frequent that the Maxwellian distribution is maintained on all time-scales of interest. Hence in an ordinary fluid or gas, collisions play an important role in establishing the fluid behaviour. In a plasma it is the magnetic field that, loosely speaking, has a similar role as collision and maintains approximately a Maxwellian distribution. So that perpendicular to the field, particles are confined to the vicinity of a given field line executing nearly isotropic motion if their gyroradius is much smaller than the characteristic plasma dimension. Therefore the perpendicular motion is fluid-like, implying that the perpendicular components of the momentum equation provides an excellent description of plasma behaviour in both collionless and collision-dominated regimes [Friedberg, 1987]. These characteristic temporal and spatial scales are often related to the ion gyroperiod and ion gyroradius.

Thus, on scales larger than the ion gyroradius and on timescales longer than the inverse plasma frequency an initially Maxwellian distribution will remain Maxwellian. The perpendicular isotropy is maintained by gyration around the field line, and therefore a bi-Maxwellian is sufficient for most plasma phenomena.

2.4 Magnetohydrodynamics

The MHD description is by far the simplest description of plasmas, and sufficiently accurate to describe the majority of macroscopic plasma phenomena [Parker, 1996;

Vasyli¯unas, 2010b].

The single fluid MHD model is derived from the more complex two-fluid model, where the electrons and ions have their own governing equations. Single fluid MHD (henceforth denoted MHD only) provides a tractable set of equations, and we will see that much of the important two-fluid effects are retained through the generalized Ohm’s law.

In our discussion about the plasma pressure tensor in Sec. 2.3, we made the claim that for certain plasma phenomena, the fluid treatment is inadequate. This was due to the difficulty in describing the thermal distribution in terms of a pressure tensor that would have to include all the information about the thermal velocity distribution. This problem is related to what is known as the closure problem, to be discussed below.

MHD treats the plasma as a single fluid with mass density:

ρ=n_im_i+n_em_e ≈nm_i (2.26) wheren_i≈n_e≈ncomes from the assumption that charge-neutrality is satisfied, this is known as the quasi-neutral approximation. Note that this is not essential for the theory and we can allow a small non-vanishing charge imbalance. MHD usually refers to the case where the quasi-neutrality approximation is invoked. The "MHD equations" can

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be summarized as:

∂ ρ

∂t +∇·(ρv) = 0 mass continuity equation (2.27a) ρdv

dt =ρ∂v

∂t +ρ(v·∇)v = −∇p+j×B momentum equation (2.27b) d

dt

p

ρ^γ

= 0 equation of state (2.27c)

these equations must be supplemented by:

E−v×B = ηj+ 1

ne(J×B−∇·P_e) + m_e ne²

∂j

∂t +∇·(jv+vj)

generalized Ohm’s law (2.28a)

∂B

∂t = −∇×E induction equation (2.28b)

in addition to the remaining equations in Maxwell’s equations (Eq. 2.10).

Ohm’s law and the momentum and mass conservation equations can be obtained by various linear combination of the individual ion and electron equations (Eq. 2.18- 2.19). For example, the mass conservation represents the sum of the ion and electron mass conservation equations. While Ohm’s law represents the difference between the ion and electron momentum equations. Thus, it is essentially the momentum equation of the electrons in the frame of reference of the ions.

We will now discuss the assumptions made so far, and assumptions that could be made while still being able to describe the large-scale dynamics sufficiently. In the momentum equation (Eq. 2.27b) we have assumed the pressure to be Maxwellian (or at least nearly symmetric). If this is not the case then the more complete form of the pressure tensor must be used, which then replaces the scalar pressure by a pressure tensor.

We refer to Sec. 2.3.2 for the discussion about the physical meaning of the pressure tensor. Even if the scalar pressure is sufficient we still need an equation describing how the scalar pressure pvaries in time. The relation given in Eq. 2.27c is often described as the equation which closes our set of equations or the adiabatic energy equation.

This equation refers to an assumption of adiabatic behaviour of the plasma. In this adiabatic energy equation pV^γ =const.the quantityγ refers to how much the temperature of the plasma increases as it is compressed. It is assumed that compression is fast enough to be adiabatic, but still slow enough that the energy is collisionally ex- changed between three degrees of freedom (see for example Goldston and Rutherford [1995] p.91). We have thus chosen to close the set of equations with our adiabatic energy equation. However, we could have proceed further and derived the energy transfer equation, which would have described heat transport by convection and conduction as well as various heat sources and sinks. This would in turn require consideration of the

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flow of energy flux. Just as we understand the thermal pressure as the transport of thermal momentum flow, the "heat-flux tensor" represents the transport of thermal energy.

It is evident that the assumption of a scalar pressure simplifies the closure problem. If a tensor representation is required, then 9 evolutionary equations would be required instead of the single equation given in Eq. 2.27c. By representing the pressure as a tensor, and deriving the necessary next order conservation equations, more complicated thermal distributions can be described. For example the inclusion of the heat-flux tensor could describe an Maxwellian with a superposed population of hot particles.

Next we consider the terms on the right hand side of Ohm’s law separately.

Generalized Ohm’s law

The last terms in Eq. 2.28a, proportional to the electron mass represent the electron inertia. By neglecting the electron inertia we assume that the processes we are considering are sufficiently slow so that the electrons have time to reach their final state. This is also the same as considering the electrons to have zero mass [Vasyli¯unas, 1975].

The second and third terms represent the Hall effect and the effects of a divergence in the electron pressure tensor. In order to describe these term’s importance compared to the convection term we can perform an estimation procedure based on order-of- magnitude sizes. We refer to for example [Goossens, 2003] for a detailed discussion.

In essence the Hall term can be dropped if the ions remain magnetized. That is, if the ion gyroradius is much smaller compared to the scale-length of the fluid motion. The Hall term is an important term in both the ionosphere and in magnetic reconnection. In terms of reconnection, ions get demagnetized as they enter the ion diffusion region - here the ions are no longer frozen to the magnetic field. The condition for neglecting the electron pressure term also involves the gyroradius to fluid motion scale length ratio, with additional criteria on the thermal velocities and the magnitude of the magnetic field. The off-diagonal terms of the pressure tensor are believed to play a crucial role in the electron diffusion region in magnetic reconnection [Hesse et al., 2011].

The first term on the right hand side of Eq. 2.28a is a resistive term related to ion- electron collisions. If the above terms are neglected, but this term is kept, we obtain:

E−v×B=ηj (2.29)

which is often referred to as "simple Ohm’s law". "Resistive MHD" usually refers to the set of equations with this simple Ohm’s replacing the generalized Ohm’s law. The relative importance of this resistive terms is given by the dimensionless quantity called the Reynolds number. It is in astrophysical settings a very large number, meaning that the resistive term can be neglected. In laboratory plasma the Reynolds number is much smaller, due to higher plasma density and collision frequency. This Ohmic terms is

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often neglected under the assumption of infinite conductivity. In the earliest magnetic reconnection models the magnetic dissipation (from which the conversion of magnetic to mechanical energy follows) were based on the resistive MHD equations [Parker, 1957b].

2.4.1 Ideal MHD

The limit in which the electrical resistivity in the plasma is negligible and the pressure can be represented as a scalar is of particular interest to our "large-scale" description.

The fluid description is then described as ideal MHD, in regard to the "tying" of plasma to magnetic field lines.

In the typical large-scale low-frequency astrophysical plasma, the induction term (v×B) is comfortably larger than all the other terms on the right hand side of the generalized Ohm’s law. The agile electrons have a remarkable ability to transport electric charge, and neutralize any charge imbalance that might arise in the plasma [Parker, 2007, Chapter 1]. The plasma can therefore not support any significant electric fieldE⁰ in its own moving frame of reference. For a completely negligibleE⁰, Ohm’s law can be written as:

E−v×B=E⁰=0 (2.30)

The assumption that no electric field exists in the frame of reference of the moving plasma leads to the dynamical description of the plasma known as ideal MHD. These equations can only adequately describe the dynamics as long as the initial assumption E⁰=0 is valid. The set of MHD equation together with the ideal Ohm’s law constitutes the framework denoted "ideal MHD".

2.4.2 Force balance in ideal MHD

We now look at an important consequence of the momentum equation (Eq. 2.27b) when the system can be considered steady-state. In a single particle description of the plasma we study the motion of the charged particles in a prescribed magnetic and electric field. In this description, currents arise from charged particles that drift across magnetic field lines (without influencing the prescribed field). For instance in a curved and non- uniform magnetic field, the charged particles will undertake gradient and curvature drift motions, giving rise to a net current. The fluid description of these drift motions are different, the resulting current from such motions in a fluid description is replaced by a diamagnetic current. We will not go into detail in describing the fluid equivalent drift motions, but instead discuss magnetospheric currents from a force balance point of view. The fluid-description represents a self-consistent description, meaning that the presence and motion of charged particles will modify or deform the existing magnetic

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field. A fundamental result by Parker [1957a] showed how the sum of all the drift motions from all the particles adds up to the momentum equation (Eq. 2.27b). So that the MHD equations does indeed include the requirement that the plasma currents needed for force balance are consistent with those required to deform the magnetic field [Vasyli¯unas, 2010b]. In magnetohydrodynamic statics the momentum equation reduces to:

∇p=J×B (2.31)

stating that the gradient of the plasma pressure must balance the Lorentz force. Recall that plasma pressure is thermal momentum flux carried by the thermal motions. This force balance describes that in a region where more thermal momentum flows into some imaginary box then out of it, a J×B force must exist to provide the required force balance. The current arising is then given byJ_⊥= ^B×∇p

B² . This term is known as the diamagnetic current, which corresponds to a charge dependent perpendicular fluid drift. We will return to an example of how a current develops, in order to balance the plasma pressure, in Sec. 3.2.5.

2.4.3 Magnetosphere-Ionosphere coupling and force balance

In the ionosphere, collisions dominate and must be included in the momentum equation.

The plasma pressure in the ionosphere is small. Under this assumption the ionospheric momentum equation can be expressed as:

ρ d

dtv=J×B−ρ νin(v−u) (2.32) where u is the neutral atmosphere velocity, and νin is the ion-neutral collision frequency. We have assumed quasi-neutrality, neglected the electric collision terms and represented the ion velocity as the plasma velocity (v_i≈v). If we consider time-scales longer than the time it takes to reach a quasi-equilibrium, the force balance is simply:

J×B=ρ νin(v−u) (2.33)

As emphasized by Parker [1996]; Vasyli¯unas [2001] and Strangeway [2012], the plasma momentum equation does not contain any electric field term, and neither does the ionospheric momentum equation given by Eq. 2.32 [Song et al., 2005]. The Lorentz force in both the ionospheric momentum equation and in Eq. 2.27b are the common force. The effect of the Lorentz force in the ionosphere is to make ionospheric and magnetospheric flow patterns match. When the two forces are out of balance, a curl of the electric field exists (∇×(v×B)) - changing the magnetic field through the induction equation (Eq. 2.28b). This increases the magnetic shear (or the "bending" of the field line) until the J×B force matches the drag. The force balance in Eq. 2.33

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clearly shows that the horizontal current density in the ionosphere results from a stress balance with the frictional ionosphere drag arising primarily from ion-neutral collisions. The corresponding magnetospheric and ionospheric currents are coupled by Birkeland currents, which is why Birkeland currents are an important aspect of the magnetosphere-ionosphere coupling. In recent years, stress and flow are emphasized for the ionosphere as well as the magnetosphere, and the limitations of the quasi-steady equilibrium scheme is better appreciated [Vasyli¯unas, 2012].

2.4.4 MHD numerical simulations

MHD consists of a set of coupled equation, that must be solved simultaneously and self-consistently. These equations are too difficult to solve analytically for any realis- tic conditions, and one must use numerical computation. These global numerical MHD models, based on first principle physics, are the only self-consistent description that can span the enormous distances associated with large scale system science. Although most models are based on single-fluid MHD, they have been used to successfully simulate a variety of plasma phenomena and continues to provide a powerful tool for understanding how the global system evolves over time. Ideal MHD does not support diffusion, and in many models the physical resistive term in Ohm’s law is replaced with a numerical dissipation term that mimics the physical process. In this way, reconnection can be included by the inclusion of a local resistivity. More complex models also incorporates two-fluid effects such as the Hall term, and anisotropic plasma pressure. The inclusion of such terms drastically increases the computational time, and are usually important only in specific regions of the system (e.g. magnetopause, reconnection region).

The coupling between the magnetosphere and the ionosphere represents a boundary where the fully dynamic self-consistent MHD description is replaced with a static ionospheric description. The inner boundary of the MHD domain is placed usually at 3 RE from the Earth. High Alfvèn speeds in this region would impose strong limitations on global step size. Therefore, Birkeland currents (and precipitation parameters) are mapped to the ionosphere, and used to solve the ionospheric potential, which is then mapped back to the magnetosphere as boundary condition for the flow. The ionosphere is assumed steady-state and the electric potential can be solved relatively easily from the Poisson equation [Song et al., 2000]. It is commonly stated that MHD is not ad- equate to describe the physics within this region. It is of course true that ideal MHD breaks down in the ionosphere because of the high collision frequencies between the plasma and the neutrals. However, the need for including two-fluid effects in MHD does not justify rearranging the equations to a steady-state version (E,j) and claiming superiority [Parker, 2007]. Despite several caveats owing to the applied boundary conditions [Song et al., 2000], and numerical artefacts [Ridley et al., 2010], global MHD

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2.5 Contradicting paradigms 23

simulation of the magnetosphere continues to provide a powerful tool for expanding our understanding of the dynamical plasma phenomena in the solar wind-magnetosphere- ionosphere system.

2.5 Contradicting paradigms

Equation 2.27 and 2.28 does not require an evolutionary equation for the electric field.

The electric field appears in the induction equation, but is readily obtained via the generalized Ohm’s law. The current density is given by Ampere’s law, so that no evolutionary equation for the current density is needed. Electric fields and currents are secondary or derived quantities, they do not describe the dynamics of the system. MHD describe the physics from first principles, and a qualitative understanding of the equations is necessary to guide our intuition of physical processes. These equations also depend on the characteristics (spatial and temporal) of the medium, such that the question whether quantity A produces quantity B, or vice versa, depends on the characteristic scale sizes.

The causal relation is a non-issue in space plasma, but parallels are often drawn between electric circuit theory and plasma, which acts only to enhance the confusion. These different ways of regarding physics of the plasma is divided into two paradigms. The MHD description is called thev,B-paradigm, while the description treating the electric field and currents as the drivers of the system is called the E,j-paradigm. TheE,j description ignores all information about cause and effect, by neglecting time-derivatives.

This results in a theory that is neither dynamical, or self-consistent and will in several cases depict an unphysical pictures not in complience with first principles (several examples given byParker[1996]). The causality of the system, for example, doesvcause E or vice versa, or does j produceB or vice versa is often presented as a controversy in the literature, even though it is unambiguously resolved [Parker, 1996, 2007; Va- syli¯unas, 2001, 2005a, b]. The purpose of this discussion is that it sets the premise for how to quantitatively describe the large-scale dynamics of the system. Throughout this thesis we try to avoid any confusion by considering only the fundamental variables that determine the dynamics. The secondary variablesj andEare derived only if needed.

There are of course conditions where theE,j description is sufficient, but the limitations of the equations must be understood. The question whether the reduced set of equations is sufficient to understand the system science depends very much on the problem at hand. Many aspects of the magnetosphere can be described as a sequence of equilibrium or quasi-equilibrium states; The success of theE,j-paradigm in describing the average behaviour in the magnetosphere relies on the average behaviour being approximately steady-state. Regardless of its success in describing the steady-state magnetosphere, one is left with merely descriptions and not physical explanations. In this description, the plasma pressure and the Lorentz force are always in balance. Phys-

Solar wind energy transfer and the asymmetric geospace