A magnetic reconnection primer
Posted: Tue Dec 15, 2020 12:07 am
This is a rather long post presenting a simple introduction to the topic of magnetic reconnection in plasmas (hereafter reconnection). There are people on this forum who are intensely hostile to the idea that reconnection can explain anything about the natural world – hostile to the point of intellectual incoherence. It has always been and remains a puzzle to me why anyone would take that attitude, unless it is because the idea threatens what they believe to be the full and unique explanation of the causes of the coronal arcade or the temperature in the corona, a belief which they seem to hold with religious fervour. I haven’t seen any cogent argument against the science associated with reconnection, although I have seen a lot of rhetoric.
The comments that I have seen about the topic also lead me to think that some people on this forum don’t really know what reconnection is, and, if so, this post is for you. This is not meant to be argumentative or confrontational and it is meant to impart information. All I ask is that you read it with an open mind, and, at least for a few minutes, set aside the oft-repeated claim on this forum that magnetic reconnection is pseudoscience, and the product of professional scientists’ incompetence. If at the end of reading it you have objections or questions that are grounded in scientific considerations, that’s fine and we should discuss them, but I will not respond to rhetoricposts abusing the intelligence and integrity of plasma physicists or astrophysicists.
Let’s start. We can all agree, I hope, that the science of plasma physics is critical to the understanding of many astrophysical phenomena. People never tire of pointing out that the Universe is 99% or some greater percentage plasma. Well reconnection is simply a branch of plasma physics which seeks to explore what happens in plasmas when the topology of the magnetic field changes.
“…when the topology of the magnetic field changes” – what does that mean? Magnetic fields are vector fields – they have both direction and magnitude (strength). The field can be visualised by lines of magnetic force on a diagram or map where the direction of the line at any point is in the direction of the magnetic field and the lateral spacing between the lines is proportional to the magnitude of the field at that point. You can draw such lines for any continuous vector field. Now, of course those lines don’t physically exist; they are merely a means of visualising the magnetic field. The magnetic field lines are smooth and continuous in any uniform medium because the direction of the field changes smoothly.
Any magnetic field can be divided into domains where all the lines (the magnetic field) start and end at the same two opposite poles (or go to infinity). For a single isolated dipole, like a bar magnet, there is only one domain – all the lines start and end at the same two poles (the N and S poles of the single bar magnet). However, in more complicated systems with multiple poles there are multiple domains. For example, for two bar magnets side by side both with north poles up, there are two domains, one associated with one magnet and one with the other, with a region of zero magnetic field between them – the separator. If you imagine turning one of those magnets through 180 degrees so that one magnet remains N-up and the other is now S-up, then there are now four domains – one above and one below the two magnets where the magnet field joins the N-pole of one magnet to the S-pole of the other, and two domains on either side where the field joins the N and S-poles of the same magnet. There are now two separators dividing the total field into four domains which has the shape of an X and the magnetic field is zero along the separators. The topology of the magnetic field has changed between the arrangements. In going from the first to the second arrangement, you can keep track of what happens to any individual field line (ie the local direction of the magnetic field), and it is clear that some field lines have to disconnect from one pole and reconnect to another pole (because where there were originally two domains, there are now four). Field lines merge and break in order for the topology to change. Whenever field lines merge or break apart, this is reconnection. (Remember field lines are not physical – they are means of visualising the field and so breaking and reconnecting field lines are just a means of visualising how the topology is changing.)
So, the question then arises: what happens in a plasma when the topology of the magnetic field changes? The kind of astrophysical plasmas we are interested in, plasmas in the solar photosphere, chromosphere and corona, plasmas associated with aurorae, plasmas in star forming filaments and accretion discs, are all very dynamic. In other words, there are plasma flows which result in currents which in turn create magnetic fields and these are constantly changing. One consequence of a changing magnetic field is the creation of an electric field in a loop round the changing magnetic flux, which we call induction. In a plasma, this induced electric field accelerates the electrons and ions changing the current and thus the magnetic field again, and round and round we go.
If that seems complicated enough, we have barely begun with the complications, because a plasma is an ionised gas which also obeys the gas laws, for example such things as the relationships between pressure, temperature and density, the thermodynamic properties, and the mechanical properties such as viscosity. Moreover, the electrostatic properties of the plasma also matter, so that in addition to the usual pressure, there is electrostatic pressure arising from the fact that the gas is made up of charged particles moving independently. Also, in partly ionised plasmas, some of the gas is neutral and some ionised, and this affects the fluid dynamics. The degree of ionisation can change as a consequence of changing temperature within the process. Everything is highly interconnected and interactive. In order to understand the behaviour of the plasma, it is not enough to apply only electromagnetic theory (that is just to solve for Maxwell’s equations or as some people on this forum would have it, to solve for Faraday’s equation only). The currents are flowing because the ions and electrons are flowing, but that also involves mass flow, temperature changes and all the other physical processes I mentioned above. Just to take one example, if the flows result in an increased electron temperature, then that itself will affect the flow of electrons which will affect the current and so on.
If all this seems terribly complicated, that’s because it is. But it’s physicists’ jobs to describe complicated things. To begin to formulate a theory of plasma dynamics, who should come riding along on a great white charger, but Hannes Alfven, with a set of four linked equations to combine the theory of electromagnetism with the theory of fluid dynamics and to lay the groundwork for the combined theory of magnetohydrodynamics. Alfven’s original MHD made some simplifying assumptions, amongst which were that the plasma should obey Maxwellian statistics, timescales should be longer than the ion gyration time, distances longer than the gyration radius (ie the theory holds for slowly and continuously changing plasmas) and the resistivity should be low enough to ignore or zero.
In fact, in regions where reconnection is occurring in astrophysical plasmas, none of these conditions hold, mainly because the smaller scale and higher speed effects owing to current sheets, magnetic turbulence and wave propagation are highly significant, and so ideal MHD is inapplicable. The theory of magnetic reconnection in plasmas is therefore a branch and an extension of ideal MHD and is itself a kind of MHD.
Because the behaviour of dynamic plasmas is so complex, there is no single over-arching magnetic reconnection theory. Instead, several theories have been developed over the years which apply to different types of plasmas undergoing different sorts of topological change, and which take account of more and finer details as the theories have been developed and refined. Theories can be organised into 2D, 2.5D and full 3D geometries. In 2D models all of the flows, fields and regions of reconnection lie in a plane which simplifies things at the expense of accuracy, in 2.5D models some effects are out of plane such as current sheets, and 3D models are, well, fully 3D. Famous theories include Parker-Sweet for slow, collisional reconnection, and Petschek for fast, collision-free reconnection; these are both 2D theories. 3D theories have had a great deal of attention in the last 15 to 20 years and the most refined models adopt a full two fluid approach, where the electrons and ions are treated as two separate but interacting fluids. Various models also take account of pressure, electrical and magnetic turbulence, Alfven wave propagation and a neutral sheath between magnetic domains less thick than the ion gyro-radius. Collision-free non-Maxwellian conditions are important and are being studied. The importance of Hall effects (voltages perpendicular to the current flow in the presence of a perpendicular magnetic field, j x B) is predicted by two-fluid models. Finally, the non-zero resistivity of the plasma, known as the Spitzer resistivity, turns out to be important in many regimes, and is itself a function of electron temperature and other plasma dynamics.
There is little point in my writing down the theoretical equations for each of these cases as that would add little to the purpose of this post, and people would have to be familiar with vector calculus to understand them. In any case very few analytical solutions are available, and the set of describing equations are mainly used as the basis for numerical models. Those models show that events in plasma with reconnection can lead to very high and short term transfer of energy from the magnetic field as the stress in the lead-up to the topology change (think of it like an over-centre mechanism) is released to the electrical, kinetic and thermal energy of the plasma.
This is all very well, but it is important to keep in mind that theoretical models need to be grounded in empirical work. There are two sources for empirical studies, natural phenomena themselves such as solar flares, studies by satellite missions such as YOKOH, SOHO, TRACE, Hinode etc, where the observations of the geometry and UV, soft and hard X-rays confirms that the reconnection models are progressing on the right lines, and laboratory experiments. With regard to the latter, the problem of scaling arises, in that many of the naturally occurring processes occur on linear scales and at temperatures, densities, pressures and magnetic field strengths that are difficult or impossible to reach in the lab. Nevertheless, there is a full literature of work on reconnection experiments (not all directly relevant to astrophysical plasmas), exploring many aspects of the process. There are still a number of fundamental aspects of reconnection that are not fully understood. The reasons for the gaps are a combination of the extreme interconnected complexities of the process leading to the difficulties in accurately modelling the behaviour, and the difficulties in creating accurately scaled experiments and making the relevant measurements in the lab. So, it’s an ongoing study, but one which is essential to better understand dynamic plasma behaviour in any number of astrophysical environments.
There is a huge literature on all of this covering theory, observations of naturally occurring astrophysical events, and laboratory experiments and I can give anyone who is interested references to multiple papers, some of which are, admittedly rather indigestible for the lay person. This post is, of course, a superficial and simplified description of the field, but I hope it gives those who have heard the term “magnetic reconnection”, but who didn’t know what it means, or those who thought they knew and now find that they didn’t, a better idea of the science and a reason for why the phenomenon is studied.
The comments that I have seen about the topic also lead me to think that some people on this forum don’t really know what reconnection is, and, if so, this post is for you. This is not meant to be argumentative or confrontational and it is meant to impart information. All I ask is that you read it with an open mind, and, at least for a few minutes, set aside the oft-repeated claim on this forum that magnetic reconnection is pseudoscience, and the product of professional scientists’ incompetence. If at the end of reading it you have objections or questions that are grounded in scientific considerations, that’s fine and we should discuss them, but I will not respond to rhetoricposts abusing the intelligence and integrity of plasma physicists or astrophysicists.
Let’s start. We can all agree, I hope, that the science of plasma physics is critical to the understanding of many astrophysical phenomena. People never tire of pointing out that the Universe is 99% or some greater percentage plasma. Well reconnection is simply a branch of plasma physics which seeks to explore what happens in plasmas when the topology of the magnetic field changes.
“…when the topology of the magnetic field changes” – what does that mean? Magnetic fields are vector fields – they have both direction and magnitude (strength). The field can be visualised by lines of magnetic force on a diagram or map where the direction of the line at any point is in the direction of the magnetic field and the lateral spacing between the lines is proportional to the magnitude of the field at that point. You can draw such lines for any continuous vector field. Now, of course those lines don’t physically exist; they are merely a means of visualising the magnetic field. The magnetic field lines are smooth and continuous in any uniform medium because the direction of the field changes smoothly.
Any magnetic field can be divided into domains where all the lines (the magnetic field) start and end at the same two opposite poles (or go to infinity). For a single isolated dipole, like a bar magnet, there is only one domain – all the lines start and end at the same two poles (the N and S poles of the single bar magnet). However, in more complicated systems with multiple poles there are multiple domains. For example, for two bar magnets side by side both with north poles up, there are two domains, one associated with one magnet and one with the other, with a region of zero magnetic field between them – the separator. If you imagine turning one of those magnets through 180 degrees so that one magnet remains N-up and the other is now S-up, then there are now four domains – one above and one below the two magnets where the magnet field joins the N-pole of one magnet to the S-pole of the other, and two domains on either side where the field joins the N and S-poles of the same magnet. There are now two separators dividing the total field into four domains which has the shape of an X and the magnetic field is zero along the separators. The topology of the magnetic field has changed between the arrangements. In going from the first to the second arrangement, you can keep track of what happens to any individual field line (ie the local direction of the magnetic field), and it is clear that some field lines have to disconnect from one pole and reconnect to another pole (because where there were originally two domains, there are now four). Field lines merge and break in order for the topology to change. Whenever field lines merge or break apart, this is reconnection. (Remember field lines are not physical – they are means of visualising the field and so breaking and reconnecting field lines are just a means of visualising how the topology is changing.)
So, the question then arises: what happens in a plasma when the topology of the magnetic field changes? The kind of astrophysical plasmas we are interested in, plasmas in the solar photosphere, chromosphere and corona, plasmas associated with aurorae, plasmas in star forming filaments and accretion discs, are all very dynamic. In other words, there are plasma flows which result in currents which in turn create magnetic fields and these are constantly changing. One consequence of a changing magnetic field is the creation of an electric field in a loop round the changing magnetic flux, which we call induction. In a plasma, this induced electric field accelerates the electrons and ions changing the current and thus the magnetic field again, and round and round we go.
If that seems complicated enough, we have barely begun with the complications, because a plasma is an ionised gas which also obeys the gas laws, for example such things as the relationships between pressure, temperature and density, the thermodynamic properties, and the mechanical properties such as viscosity. Moreover, the electrostatic properties of the plasma also matter, so that in addition to the usual pressure, there is electrostatic pressure arising from the fact that the gas is made up of charged particles moving independently. Also, in partly ionised plasmas, some of the gas is neutral and some ionised, and this affects the fluid dynamics. The degree of ionisation can change as a consequence of changing temperature within the process. Everything is highly interconnected and interactive. In order to understand the behaviour of the plasma, it is not enough to apply only electromagnetic theory (that is just to solve for Maxwell’s equations or as some people on this forum would have it, to solve for Faraday’s equation only). The currents are flowing because the ions and electrons are flowing, but that also involves mass flow, temperature changes and all the other physical processes I mentioned above. Just to take one example, if the flows result in an increased electron temperature, then that itself will affect the flow of electrons which will affect the current and so on.
If all this seems terribly complicated, that’s because it is. But it’s physicists’ jobs to describe complicated things. To begin to formulate a theory of plasma dynamics, who should come riding along on a great white charger, but Hannes Alfven, with a set of four linked equations to combine the theory of electromagnetism with the theory of fluid dynamics and to lay the groundwork for the combined theory of magnetohydrodynamics. Alfven’s original MHD made some simplifying assumptions, amongst which were that the plasma should obey Maxwellian statistics, timescales should be longer than the ion gyration time, distances longer than the gyration radius (ie the theory holds for slowly and continuously changing plasmas) and the resistivity should be low enough to ignore or zero.
In fact, in regions where reconnection is occurring in astrophysical plasmas, none of these conditions hold, mainly because the smaller scale and higher speed effects owing to current sheets, magnetic turbulence and wave propagation are highly significant, and so ideal MHD is inapplicable. The theory of magnetic reconnection in plasmas is therefore a branch and an extension of ideal MHD and is itself a kind of MHD.
Because the behaviour of dynamic plasmas is so complex, there is no single over-arching magnetic reconnection theory. Instead, several theories have been developed over the years which apply to different types of plasmas undergoing different sorts of topological change, and which take account of more and finer details as the theories have been developed and refined. Theories can be organised into 2D, 2.5D and full 3D geometries. In 2D models all of the flows, fields and regions of reconnection lie in a plane which simplifies things at the expense of accuracy, in 2.5D models some effects are out of plane such as current sheets, and 3D models are, well, fully 3D. Famous theories include Parker-Sweet for slow, collisional reconnection, and Petschek for fast, collision-free reconnection; these are both 2D theories. 3D theories have had a great deal of attention in the last 15 to 20 years and the most refined models adopt a full two fluid approach, where the electrons and ions are treated as two separate but interacting fluids. Various models also take account of pressure, electrical and magnetic turbulence, Alfven wave propagation and a neutral sheath between magnetic domains less thick than the ion gyro-radius. Collision-free non-Maxwellian conditions are important and are being studied. The importance of Hall effects (voltages perpendicular to the current flow in the presence of a perpendicular magnetic field, j x B) is predicted by two-fluid models. Finally, the non-zero resistivity of the plasma, known as the Spitzer resistivity, turns out to be important in many regimes, and is itself a function of electron temperature and other plasma dynamics.
There is little point in my writing down the theoretical equations for each of these cases as that would add little to the purpose of this post, and people would have to be familiar with vector calculus to understand them. In any case very few analytical solutions are available, and the set of describing equations are mainly used as the basis for numerical models. Those models show that events in plasma with reconnection can lead to very high and short term transfer of energy from the magnetic field as the stress in the lead-up to the topology change (think of it like an over-centre mechanism) is released to the electrical, kinetic and thermal energy of the plasma.
This is all very well, but it is important to keep in mind that theoretical models need to be grounded in empirical work. There are two sources for empirical studies, natural phenomena themselves such as solar flares, studies by satellite missions such as YOKOH, SOHO, TRACE, Hinode etc, where the observations of the geometry and UV, soft and hard X-rays confirms that the reconnection models are progressing on the right lines, and laboratory experiments. With regard to the latter, the problem of scaling arises, in that many of the naturally occurring processes occur on linear scales and at temperatures, densities, pressures and magnetic field strengths that are difficult or impossible to reach in the lab. Nevertheless, there is a full literature of work on reconnection experiments (not all directly relevant to astrophysical plasmas), exploring many aspects of the process. There are still a number of fundamental aspects of reconnection that are not fully understood. The reasons for the gaps are a combination of the extreme interconnected complexities of the process leading to the difficulties in accurately modelling the behaviour, and the difficulties in creating accurately scaled experiments and making the relevant measurements in the lab. So, it’s an ongoing study, but one which is essential to better understand dynamic plasma behaviour in any number of astrophysical environments.
There is a huge literature on all of this covering theory, observations of naturally occurring astrophysical events, and laboratory experiments and I can give anyone who is interested references to multiple papers, some of which are, admittedly rather indigestible for the lay person. This post is, of course, a superficial and simplified description of the field, but I hope it gives those who have heard the term “magnetic reconnection”, but who didn’t know what it means, or those who thought they knew and now find that they didn’t, a better idea of the science and a reason for why the phenomenon is studied.
