The MHD equations
So far we have applied the arguments of classical fluid dynamics to obtain a closed set of equations for the plasma fluid variables but, except for the introduction of Joule heating, we have taken almost no account of the fact that a plasma is a conducting fluid. This we do now by specifying the force per unit mass F. Except in astrophysical contexts, where gravity is an important influence on the motion of the plasma, electromagnetic forces are dominant. For a fluid element with charge density q and current density jwe then have
ρF= qE+ j× B(3.32)
where the fields Eand Bare determined by Maxwell’s equations (2.2)–(2.5). Equations (2.6) and (2.7) for q and jare not suitable in a fluid model. However, our first objective is to obtain a macroscopic description of the plasma in which the fields are those induced by the plasma motion. Thus, we now introduce the basic assumption of MHD that the fields vary on the same time and length scales as the plasma variables. If the frequency and wavenumber of the fields are ω and k respectively, we have ωτH ~1 and kLH ~1, where τH and LH are the hydrodynamic time and length scales. A dimensional analysis then shows that both the electrostatic force qEand displacement current ε0μ0∂E/∂t may be neglected in the non-relativistic approximation ω/k << c. Consequently, (3.32) becomes
ρF= j× B(3.33)
and (2.3) is replaced by Ampere’s law
j= (1 / μ0) grad x B(3.34)
Now, Poisson’s equation (2.4) is redundant (except for determining q) and just one further equation for jis required to close the set. Here we run into the main problem with a one-fluid model. Clearly, a current exists only if the ions and electrons have distinct flow velocities and so, at least to this extent, we are forced to recognize that we have two fluids rather than one. For the moment we side-step this difficulty by following usual practice in MHD and adopting Ohm’s law
j= σ(E+ u× B) (3.35)
as the extra equation for j. The usual argument for this particular form of Ohm’s law is that in the non-relativistic approximation the electric field in the frame of a fluid element moving with velocity uis (E+ u× B). However, this argument is over-simplified, unless uis constant so that the frame is inertial, and later, when we discuss the applicability of the MHD equations, we shall see that the assumption of a scalar conductivity in magnetized plasmas is rarely justified. The status of (3.35) should be regarded, therefore, as that of a ‘model’ equation, adopted for mathematical simplicity.
This closes the set of equations for the variables ρ, u, P, T, E, Band jbut before listing them it is useful to reduce the set by eliminating some of the variables. Although in electrodynamics it is customary to think of the magnetic field being generated by the current, in MHD we regard Ampere’s law (3.34) as determining jin terms of B. Then Ohm’s law (3.35) becomes
E= (1/σμ0) grad × B− u× B(3.36)
so determining E. Finally, substituting (3.36) in (2.2), treating σ as a constant, and using (2.5), we get the induction equation for B
∂B/∂t = (1/σμ0) grad2B+grad× (u× B) (3.37)
Since we have eliminated jand E, this is now the only equation we need add to the set derived at the end of the last section for the fluid variables.
Elements of Quantum Mechanics. History Max Planck laid the foundations of quantum physics in 1900 when he assumed that light of frequency v was emitted and absorbed in packets called photons or quanta, with energy given by E = hv (1) Planck made this bold hypothesis to explain the observed spectral- emitting power of black body cavities at various temperatures. Planck's constant h can be evaluated by confronting his theory with the experimental data. This constant is basic to quantum physics; in fact, letting it approach zero is a formal device for testing quantum equations to see whether they reduce, as they should, to classical equations under conditions for which quantum effects are not important. From 1900 to the early 1920's, the structure of what is now called "the old quantum theory" developed. A high spot was Bohr's theory of the hydrogen atom (B13). Implicit in this is the important notion that an atomic system can exist in a discrete set of stationary states of total energy El, E2 , … , En , and that light is emitted or absorbed when the system changes from one of these states to another. The frequency of the light is found from Bohr's frequency condition ωnm = (En—Em)/h (3) which combines Eq. 2 and the conservation of energy principle. In spite of the impressive successes of the old quantum theory, there were some points of disagreement with experiment. Also, there were a number of measurable quantities for which the theory did not seem able to make predictions. It was slowly realized that this theory had not made a clean enough break with classical physics. In 1926 Schrodinger (S26) developed an improved quantum theory called wave mechanics; it is based on the notion that matter can be described in terms of waves. In the same year Born, Heisenberg, and Jordan (B26), following up an idea formulated by Heisenberg the previous year (H25) developed another form of quantum theory called matrix mechanics; this is based on a feeling that the "raw facts" of atomic physics, e.g., the spectral frequencies, should be built solidly into the theory in a central fashion and that doing so is more important than hanging the theory on a particular model. This theory is particle-oriented, in contrast to the wave orientation of Schrbdinger’s theory. Although the two theories seem quite different, they lead to identical results. In 1948 Feynman, then of Cornell University, invented still a third form of quantum mechanics (F48). The operationally inclined reader will not spend much time wondering which of these equivalent theories is "true." All three theories represent a fresh start rather than an attempt to patch up classical physics. This seems wise since classical theory is a special case of quantum theory, rather than the converse. Neither wave nor matrix mechanics took into account the special theory of relativity. Dirac remedied this defect in 1928 with his relatiuistic quantum mechanics (D28). With its aid he was able to predict the existence of the positron before it was discovered in the cosmic rays by Anderson (A33). Dirac's theory also predicts the spin of the electron in a natural way. Uhlenbeck and Goudsmit (U25) deduced empirically from studies of atomic spectra that the electron must have a spin of Yz. This does not appear without some "forcing" in the non-relativistic quantum theories, however. Although relativistic quantum mechanics (or quantum electro- dynamics) involved some rather arbitrary and disturbing mathematical procedures, it did seem, up to 1947, to be in complete accord with experi- ment when applied to systems, such as atoms, involving electrons and radiation. In that year however, Lamb and Retherford of Columbia University (La47) discovered a small but definite discrepancy between theory and experiment in their careful study of the fine structure of the hydrogen spectrum. This spurred on a reinvestigation of the foundations of quantum electrodynamics. In the hands of Kramers, Schwinger, Bethe, Dyson, Tomonaga, and others, it became possible to reformulate the theory so as to account quantitatively for the results of the Lamb-Retherford experiment. Though cumbersome in its present form, modern quantum electrodynamics now seems to give precise answers when applied to atomic problems. It contains what Dyson (D52) calls a built-in miracle, in that certain expressions, whose presence in the final result would be embarrassing, always conveniently cancel, even though it is not clear why they do. Theoreticians are trying now to reformulate quantum electrodynamics with the hope of gaining new insights.
Дата добавления: 2015-09-21; просмотров: 623;