Second-Order Nonlinear Processes in Warm Unmagnetized Plasmas

Nonlinear processes are commonly invoked to describe a wide range of phenomena in both space plasmas and laboratory plasmas, allowing wave energy in a particular mode to be transferred to different modes. Relevant processes include three-wave interactions and nonlinear wave-particle scattering. The strength of the wave coupling—and hence the nonlinear rates—for these processes is determined by the quadratic response tensor. The general expression for this tensor involves a number of velocity-space integrals of the velocity distribution function and denominators related to the Cerenkov resonance between waves and particles. Due to the difficulty of evaluating these integrals they are typically approximated by making assumptions about the phase speeds of the waves; for an unmagnetized Maxwellian plasma, the phase speeds of Langmuir and transverse waves are assumed to be much greater than the electron thermal speed, and the phase speed of ion-sound waves is assumed to be much less than the electron thermal speed. However, the ranges of validity for these approximations are unclear, and the resulting approximate quadratic response tensors and nonlinear rates may be inaccurate when modeling the nonlinear processes in particular space plasmas. Conversely, an exact expression for the quadratic response tensor of an unmagnetized Maxwellian plasma has been derived previously in terms of generalized plasma dispersion functions. This expression is valid for any phase speeds of the waves, but its length and complexity prevents its use in the calculation of nonlinear rates. What is lacking in the literature are more accurate explicit expressions for the quadratic response tensor that are appropriate for nonlinear rate calculations, from which also the accuracy of the typical approximations can be assessed. This thesis presents new, more accurate analytical expressions for the quadratic response tensor for various second-order processes in unmagnetized plasmas, and analytical and numerical calculations of the corresponding nonlinear rates. Comparisons are then made between these new nonlinear rates and the previous approximate rates, allowing the accuracy of the previous rates to be vii assessed and sharper bounds placed on their regimes of validity for the first time. The structure of the thesis is as follows. Chapter 1 presents an overview of the relevant theory for nonlinear processes, a discussion of the previous work on the topic, and an outline of the work presented in the following chapters. In Chapter 2 a first-order thermal correction to the cold-plasma quadratic response tensor is derived, which is valid for interactions between three waves with phase speeds greater than the electron thermal speed in an unmagnetized plasma with an arbitrary isotropic velocity distribution. From this a thermal correction to the rate of second harmonic plasma emission (via Langmuir-wave coalescence) is calculated, and its importance is assessed for various space physics contexts. Chapter 3 presents an exact evaluation of the longitudinal part of the quadratic response tensor (known as the quadratic longitudinal response function) for an unmagnetized Maxwellian plasma, which gives an exact description of the wave coupling in second-order processes involving electrostatic waves only. The expression we derive involves a number of generalized plasma dispersion functions. Errors in certain aspects of previous expressions for the generalized plasma dispersion functions are corrected, a new set of expressions are derived that converge more rapidly, and the accuracy of various approximations to these functions are assessed. Chapter 4 contains an exact evaluation of the rates of electrostatic decay and scattering off thermal ions using the exact quadratic longitudinal response function derived in Chapter 3, and comparisons are made between these exact rates and the previous approximate rates in various contexts. A summary of the results in this thesis and directions for future work are presented in Chapter 5.