First-Order Coupled Differential Equations for Waves in Inhomogeneous Warm Magnetoplasmas*

The first-order coupled differential equations for waves in temperate inhomogeneous magnetoplasmas are well known [1]. The extension of this method for the case of waves in inhomogeneous warm magnetoplasmas is presented here. In this form, they are well suited for solution on a digital computer by the Runge–Kutta method. The plasma is assumed to be neutral and in equilibrium, and the motion of the ions is neglected. Taking E ¯ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqi10b_1.jpg"/> , H ¯ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqi10b_2.jpg"/> , p ₀, N ₀, and T 0 u ¯ 0 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqi10b_3.jpg"/> as the stationary values of the plasma which are given functions of position, and E ¯ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqi10b_4.jpg"/> , H ¯ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqi10b_5.jpg"/> , p ₁, N ₁, T ₁, and u ¯ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqi10b_6.jpg"/> as the harmonic time-varying (e^iωt ) small components of the wave in the plasma, and defining the plasma parameters X r ¯ = e 2 N 0 r ¯ ω 2 ε m , Y r ¯ = e μ m ω H ¯ 0 r ¯ , U r ¯ = 1 − i ¯ ν r ¯ ω ¯ , δ r ¯ = a 2 r ¯ c 2 = μ ε γ K T 0 r ¯ m , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/equ10b_1.jpg"/> where a r ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqi10b_7.jpg"/> is the acoustic velocity in the electron gas, by taking a new set of dependent variables 10B.1 E ¯ = E ¯ 1 , H ¯ = μ ε 1 2 H ¯ 1 , u ¯ = ω m e u ¯ 1 , p = e ω m μ ε 1 2 p 1 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn10b_1.jpg"/> one obtains [2] for small-signal theory, where k ₀ = ω(με)^1/2 = ω/c: 10B.2a ∇ × E ¯ = − i k 0 H ¯ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn10b_2a.jpg"/> 10B.2b ∇ × H ¯ = i k 0 E ¯ − k 0 X u ¯ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn10b_2b.jpg"/> 10B.2c − i U X u ¯ = 1 k 0 ∇ p + X E ¯ + δ γ k 0 2 ∇ p 0 p 0 ∇ ⋅ E ¯ + X u ¯ × Y ¯ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn10b_2c.jpg"/> 10B.2d − ∇ ⋅ u ¯ = i k 0 δ X p + u ¯ ⋅ ∇ p 0 γ p 0 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn10b_2d.jpg"/>