Ion acoustic wave

In plasma physics, an ion acoustic wave is one type of longitudinal oscillation of the ions and electrons in a plasma, much like acoustic waves traveling in neutral gas. However, because the waves propagate through positively charged ions, ion acoustic waves can interact with their electromagnetic fields, as well as simple collisions. In plasmas, ion acoustic waves are frequently referred to as acoustic waves or even just sound waves. They commonly govern the evolution of mass density, for instance due to pressure gradients, on time scales longer than the frequency corresponding to the relevant length scale. Ion acoustic waves can occur in an unmagnetized plasma or in a magnetized plasma parallel to the magnetic field. For a single ion species plasma and in the long wavelength limit, the waves are dispersionless ( ω = v s k {displaystyle omega =v_{s}k} ) with a speed given by (see derivation below) γ e T e ⟨ Z i 2 m i v s 2 − γ i T i ⟩ = ⟨ Z i ⟩ ( 1 + γ e k 2 λ D e 2 ) {displaystyle gamma _{e}T_{e}leftlangle {Z_{i}^{2} over m_{i}v_{s}^{2}-gamma _{i}T_{i}} ight angle =langle Z_{i} angle (1+gamma _{e}k^{2}lambda _{De}^{2})} .    (dispgen) In plasma physics, an ion acoustic wave is one type of longitudinal oscillation of the ions and electrons in a plasma, much like acoustic waves traveling in neutral gas. However, because the waves propagate through positively charged ions, ion acoustic waves can interact with their electromagnetic fields, as well as simple collisions. In plasmas, ion acoustic waves are frequently referred to as acoustic waves or even just sound waves. They commonly govern the evolution of mass density, for instance due to pressure gradients, on time scales longer than the frequency corresponding to the relevant length scale. Ion acoustic waves can occur in an unmagnetized plasma or in a magnetized plasma parallel to the magnetic field. For a single ion species plasma and in the long wavelength limit, the waves are dispersionless ( ω = v s k {displaystyle omega =v_{s}k} ) with a speed given by (see derivation below) where K B {displaystyle K_{B}} is Boltzmann's constant, M {displaystyle M} is the mass of the ion, Z {displaystyle Z} is its charge, T e {displaystyle T_{e}} is the temperature of the electrons and T i {displaystyle T_{i}} is the temperature of the ions. Normally γe is taken to be unity, on the grounds that the thermal conductivity of electrons is large enough to keep them isothermal on the time scale of ion acoustic waves, and γi is taken to be 3, corresponding to one-dimensional motion. In collisionless plasmas, the electrons are often much hotter than the ions, in which case the second term in the numerator can be ignored. We derive the ion acoustic wave dispersion relation for a linearized fluid description of a plasma with electrons and N { extstyle N} ion species. We write each quantity as X = X 0 + δ ⋅ X 1 {displaystyle X=X_{0}+delta cdot X_{1}} where subscript 0 denotes the 'zero-order' constant equilibrium value, and 1 denotes the first-order perturbation. δ {displaystyle delta } is an ordering parameter for linearization, and has the physical value 1. To linearize, we balance all terms in each equation of the same order in δ {displaystyle delta } . The terms involving only subscript-0 quantities are all order δ 0 {displaystyle delta ^{0}} and must balance, and terms with one subscript-1 quantity are all order δ 1 {displaystyle delta ^{1}} and balance. We treat the electric field as order-1 ( E → 0 = 0 {displaystyle {vec {E}}_{0}=0} ) and neglect magnetic fields, Each species s {displaystyle s} is described by mass m s {displaystyle m_{s}} , charge q s = Z s e {displaystyle q_{s}=Z_{s}e} , number density n s {displaystyle n_{s}} , flow velocity u → s {displaystyle {vec {u}}_{s}} , and pressure p s {displaystyle p_{s}} . We assume the pressure perturbations for each species are a Polytropic process, namely p s 1 = γ s T s 0 n s 1 {displaystyle p_{s1}=gamma _{s}T_{s0}n_{s1}} for species s {displaystyle s} . To justify this assumption and determine the value of γ s {displaystyle gamma _{s}} , one must use a kinetic treatment that solves for the species distribution functions in velocity space. The polytropic assumption essentially replaces the energy equation.