Toroidal quarter waves in the Earth’s magnetosphere: theoretical studies

Abstract

An analytic model has been developed for toroidal quarter wave (QW) oscillations in the Earth's magnetosphere using idealistic and highly asymmetric ionospheric boundary condition. The background magnetic field is dipolar and plasma density distribution is governed by a power law 1/r m where r is the geocentric distance of any point along the field line and m is the density index. The solution thus obtained has been compared with the numerical solutions. Earlier workers had developed the analytic model for trivial 1/r 6 (m = 6) type of plasma density distribution along the field line for which the period of the fundamental is twice that of corresponding half wave (i.e. toroidal oscillation in the symmetric ionospheric boundary). The present analytic model does reproduce this feature. In addition, it is seen that this ratio decreases for lower values of m. Moreover, for a particular value of m, this ratio shows a decreasing trend with increased harmonic number. The spatial characteristics of QW obtained from present analytic model are in excellent agreement with those computed numerically, thereby validating the model. The departure of frequency computed analytically from that obtained numerically is significant only for the fundamental and this departure reduces sharply with the increased harmonic number. It should be noted that there is no such departure for 1/r 6 type of plasma density distribution and spatial structures as well as frequency computed from the present analytic model match perfectly with those computed numerically.

The Alfvén mode gyrokinetic equation in finite‐pressure magnetospheric plasma

“…Note that these expressions for the toroidal and poloidal operators are identical with those derived in other papers although written in a different form [e.g., Leonovich and Mazur, 1993;Bulusu et al, 2014]. The operator…”

Role of finite ionospheric conductivity on toroidal field line oscillations in the Earth's magnetosphere—Analytic solutions

“…These modes are formed when the ionospheres have infinitely large or infinitesimally small symmetric conductivity at the conjugate ionospheres. These are ideal cases and are discussed in our earlier study [Sinha and Rajaram, 1997;Bulusu et al, 2014a]. However, for quarter waves, either Σ N P ≪ Σ A (R = −1) in northern conjugate point and Σ S P ≫ Σ A (R = +1) in southern conjugate point or Σ N P ≫ Σ A (R = +1) in northern conjugate point and Σ S P ≪ Σ A (R = −1) in southern conjugate point are satisfied.…”

“…A comprehensive analytic model discussing the rigid‐end, free‐end, and quarter wave toroidal modes is presented in our earlier papers [ Sinha and Rajaram , ; Bulusu et al , , ]. The main focus of our study was development of a theoretical model that can explain different types of standing Alfvén oscillations in different regimes of ionospheric conductivity.…”

“…However, the magnetic field has a finite component at the ionosphere. On the other hand, when the ionospheric conductivity is vanishingly small, A comprehensive analytic model discussing the rigid-end, free-end, and quarter wave toroidal modes is presented in our earlier papers [Sinha and Rajaram, 1997;Bulusu et al, 2014aBulusu et al, , 2015b. The main focus of our study was development of a theoretical model that can explain different types of standing Alfvén oscillations in different regimes of ionospheric conductivity.…”

“…These modes are most likely observed during June and December solstices [Allan and Knox, 1979a;Budnik et al, 1998;Obana et al, 2008;Bulusu et al, 2015a]. A detailed theoretical computation of quarter wave toroidal wave was discussed by Bulusu et al [2014a]. They formulated the field-aligned structures of electric and magnetic field associated with quarter wave.…”