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Coronal Heating: Basic 1-Fluid Plasma Equations I’m going to begin by writing down some basic one-fluid plasma equations that provide a framework for thinking about coronal heating processes. To derive these equations (which I’m not doing here), we could first write down the equations for electrons and each ion species and then sum them to get one-fluid plasma equations. Since the gas may be partially ionized, as it is in the lower transition region and upper chromosphere, I’m going to include source and sink terms in the mass conservation equation, since in the partially ionized region, the loss of the ionized gas through recombination processes and its production through ionization processes may not be negligible. An alternative approach to this is to write down the equations for each species (electrons, ions, neutrals) separately, and that’s the way we actually address the chromosphere-corona problem, but for what I want to illustrate here, that is too cumbersome. For now, let’s assume we’re dealing with a hydrogen plasma (composed of electrons, protons, and hydrogen atoms). Then the variables we’ll be working with are as follows: n n p n e number density u u p flow velocity p p p pe n kB Tp Te pressure q heat flux density J n u p ue electric current density E electric field B magnetic field e n p ne electric charge density Here, T p and Te are the proton and electron temperatures, kB is the Boltzmann constant, and e is the magnitude of the electron charge. Although I am using the term “magnetic field” for B , which is standard terminology in our area of research, many authors (such as Jackson or Panofsky and Phillips) use the term “magnetic induction” for B , reserving the term magnetic field for H . Now I’ll write down the basic equations, beginning with the mass conservation equation, where we represent the particle mass by m m p . Recall that I’m going to include a source/sink term in this equation, and this implies the existence of related terms in the momentum and energy equations, which I am not going to include at the present time. -1- n m n m u S t (1) We will write the momentum and energy equations in this same general “conservation” form. In the case of (1) this conservation form can be described in words by saying that the time rate of change of mass in an infinitesimal volume ( nm t ) is given by the rate at which mass is flowing into that volume ( n m u ) plus the net rate at which mass is being produced in that volume ( S ). The same sort of verbal representation can be given to the following momentum and energy equations. In the momentum equation, I am going to write the pressure term as a force (i.e., the pressure gradient force), which means I’m treating it as a momentum “source” (equivalent to the mass source, S , in (1)). We could equally well treat it as a contributor to the momentum flux density, just as we could the viscosity. (Together, the pressure gradient term and the viscous term make up the divergence of the pressure tensor, which can be thought of as the divergence of one part of the momentum flux density—the other part being the momentum flux density associated with bulk flow.) Our momentum equation thus takes the form GM n m u n m u u p π J B n m 2 eˆ r FR Ff t r (2) where G is the gravitation constant, M the solar mass, π is the viscous stress tensor, FR is the radiative force (not generally important for us), and Ff is the frictional force (owing to the presence of hydrogen atoms). In principle, there is also a thermal force term, although this is not as important for ion-neutral interactions as it is for ion-ion interactions in a multi-species description. As noted earlier, there should also be a source term associated with the mass source term, but we are not including that here. Finally, in the context of the earlier discussion, we could have used a pressure tensor ( P ), in which case P p π would appear as part of the divergence term on the left side of (2), instead of as a source term (i.e., a force) on the right side. When the momentum equation (2) is derived, the electron and ion momentum equations are added to each other. If we subtract these two equations instead, we obtain what is called the generalized Ohm’s law: me n e2 J 1 1 1 uJ Ju J J E u B η J JB pe ne ne ne t (3) where me is the electron mass, and η is the electrical resistivity tensor. Only the first three terms on the right side of (3) are included in the ordinary Ohm’s law for a plasma, and for a perfectly conducting plasma, only the first two terms. The terms on the left side of (3) are the inertial terms and must be considered when rapid time variations or sharp spatial gradients are present. The so-called Hall term is the fourth term on the right side. Note that the resistivity can often be treated as a scalar (rather than a tensor); the tensor nature generally must be considered -2- in a weakly ionized plasma, such as the E-region of the terrestrial ionosphere, and perhaps in parts of the solar chromosphere (particularly, near the temperature minimum). Now we turn to energy balance, which in conservation form can be written 1 p 2 12 n m u 2 u p u u π q 2nmu t 1 1 (4) GM J E u n m 2 eˆ r FR u uH Ff r Here, is the adiabatic index (i.e., the ratio of specific heats), which for three degrees of freedom (as in our case) is 5 3 . Except for the J E term, the source terms on the right side of (4) describe the change in energy of our plasma owing to its direct acceleration (or deceleration). The J E term arises both from such an acceleration or deceleration (through the action of the J B force) and from heating or cooling, but in both cases, it is associated with the exchange of energy between the electromagnetic field and the plasma. To understand this energy exchange issue we must also consider Maxwell’s equations (of course, from a mathematical perspective, we must consider them anyway, because with E and B as unknowns, we find ourselves with six more unknowns than we have equations). In SI units, Maxwell’s equations can be written B 0 J 0 E t (5) B t (6) 0 E (7) B 0 (8) E Here, 0 is the electric permittivity of free space, and 0 is the magnetic permeability of free space. Equations (7) and (8) are usually considered as initial conditions, and the sense of this can be seen by taking the divergence of (5) and (6), which yields J J 0 E 0 t t (9) B 0 t (10) -3- Equation (10) tells us that B does not vary with time, so that if it initially vanishes, then it always vanishes. Similarly, (9) (which incidentally can also be derived by subtracting the electron and ion continuity equations) describes the time evolution of E or alternatively the charge density. If we wish to follow the time evolution of the charge density, we can do this by using (9), but it is not necessary to do this in order to solve our system of equations. Speaking of our system of equations, we now have the 17 unknowns n , u , p , q , J , E , and B , and the 14 equations (1)-(6) (if we choose to follow the time evolution of , this becomes 18 unknowns and 15 equations, including (9)). [Actually, if we were to include π as an unknown, that would give us nine more unknowns, but for now at least, we’ll ignore the effects of viscosity. If we were to include them, then we would need a closure relation equivalent to the one we are about to specify for q .] The additional three equations we need can be provided by specifying a closure relation for q (closure in this case refers to closing the moment equations, the first three of which we have written as (1), (2), and (4)), and for this purpose we will use (for now) the classical expression for the heat flux: q T 5 2 T (11) where the thermal conduction coefficient, , is a constant. Of course, (11) doesn’t quite solve our problem, because we have introduced the new variable, T . To take care of this, we use the ideal gas law to specify T in terms of n and p : p 2 n kB T (12) where, for the time being, we are taking T Te Tp . Now, let’s return to the subject of the J E term in (4), which couples the electromagnetic energy and the fluid energy. If we take the scalar product of (5) and E and then make use of (6), we can show that J E 12 B 2 0 12 0 E 2 E B 0 t (13) So the J E term is the negative of the sum of the time rate of change of the electromagnetic energy density and the divergence of the Poynting vector, which measures the electromagnetic energy flux density. If we had wanted to, we could have written (4) as conservation equation for the total (fluid and field) energy by substituting (13) into (4) and moving the electromagnetic terms to the left side of the equation. Sometimes, this is a more useful form of the energy equation, but it is important to keep in mind that the two forms are completely equivalent and simply provide different perspectives. Actually, people who forget about the equivalence of these two perspectives tend to get into arguments about things like whether or not it is currents that heat the corona. Finally, let’s return to the subject of heating versus direct acceleration. The direct acceleration component of the J E term is just the work done by the J B force, which is -4- u J B J u B . So the heating component of the J E term, which we will label QEM is given by QEM J E u J B J E u B (14) One could substitute for E u B using the generalized Ohm’s law (3) to obtain a general form for the electromagnetic heating term, but we will just note that in the commonly considered case where Ohm’s law is written E u B J our heating term takes the familiar form associated with resistive heating: QEM J 2 (15) I suppose that people who claim that all coronal heating is associated with currents have this term in mind. Of course, as we can see from (14), if the heating involves a conversion of electromagnetic energy into fluid energy, then it can always be said to be associated with electric currents, even if the resistivity vanishes. Anyway, I’ll stop for now. Obviously the above comments raise a variety of issues we’ll have to deal with, and I hope what I have written begins to provide a framework we can use for thinking about these issues. -5-