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- Herausgeber: Wiley-VCH GmbH
- Kategorie: Lebensstil
- Sprache: Englisch
- Veröffentlichungsjahr: 2024
The book summarises the theory of magnetohydrodynamic waves in Earth's magnetosphere and provides an extension that allows for an accurate interpretation of data acquired by current and future satellite missions.
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
List of Figures
List of Tables
Author Biography
Preface
Acknowledgements
Acronyms
Symbols
Introductionintroduction
Note
1 Hydromagnetic Oscillations in Homogeneous Plasma
2 MHD Oscillations in 1D‐Inhomogeneous Model Magnetosphere
2.1 A Qualitative Picture of MHD Wave Propagation in a 1D‐Inhomogeneous Plasma
2.2 Model of a Smooth Transition Layer and Basic Equations for MHD Oscillations
2.3 FMS Wave Reflected from the Transition Layer in a Cold Plasma. Alfvén Resonance
2.4 Alfvén Resonance Excited by a Wave Impulse
2.5 Energy Balance in the Problem of an Incident FMS Wave Reflected from the Transition Layer Containing an Alfvén Resonance Point
2.6 FMS Wave Reflected from the Transition Layer in a ‘warm’ Plasma. Alfvén and Magnetosonic Resonances
2.7 Alfvén Resonance in Non‐ideal Plasma. Kinetic Alfvén Waves
2.8 FMS Waveguide
2.9 Waveguide for Quasilongitudinal Alfvén Waves
2.10 Waveguides for Kinetic Alfvén Waves in a ‘cold’ Plasma. Waveguide Mode Attenuation
2.11 Waveguide for Kinetic Alfvén and FMS Waves in a ‘warm’ Plasma. Waveguide Mode Resonance
2.12 Waveguides in Plasma Filaments
2.13 FMS Wave Passing Through a Tangential Discontinuity
2.14 Unstable MHD Shear Flows in the Presence/Absence of Boundary Walls
2.15 Geotail Instability Due to Shear Flow at the Magnetopause
2.16 Kelvin–Helmholtz Instability in the Geotail Low‐Latitude Boundary Layer
2.17 Cherenkov Radiation of the Fast Magnetoacoustic Waves
2.18 MHD Oscillation Field Penetrating from the Magnetosphere to Ground
Notes
3 MHD Oscillations in 2D‐Inhomogeneous Models
3.1 Resonance Between FMS and Kinetic Alfvén Waves in a Dipole‐Like Magnetosphere
3.2 Alfvén Resonance in a Dipole‐Like Magnetosphere
3.3 Resonant Alfvén Waves Excited in a Dipole‐Like Magnetosphere by Broadband Sources
3.4 Magnetosonic Resonance in a Dipole‐Like Magnetosphere
3.5 FMS Oscillations in a Dipole‐Like Magnetosphere
3.6 FMS Resonators in Earth's Magnetosphere
3.7 Monochromatic Transverse‐Small‐Scale Alfvén Waves with in a Dipole‐Like Magnetosphere
3.8 Electromagnetic Oscillations Induced at Earth Surface by Magnetospheric Standing High‐ Alfvén Waves
3.9 Linear Transformation of Standing High‐ Alfvén Waves Near the Toroidal Resonance Surface
3.10 Magnetospheric Resonator for Standing High‐ Alfvén Waves
3.11 High‐ Alfvén Waves Generated in the Magnetosphere by Stochastic Sources
3.12 Broadband Standing High‐ Alfvén Waves Generated by Correlated Sources
3.13 Model Equation to Determine the Transverse Structure of Standing Alfvén Waves in the Magnetosphere
3.14 Spatial Structure of Alfvén Oscillations Excited in the Magnetosphere by Localised Monochromatic Source
3.15 High‐ Alfvén Oscillations Generated in the Magnetosphere by Localised Pulse Sources
3.16 Ballooning Instability of Alfvén and SMS Oscillations on Field Lines Crossing the Current Sheet
3.17 Coupled Alfvén and SMS Oscillation Modes in the Geotail
Chapter 4: MHD Oscillations in 3D‐Inhomogeneous Models of the Magnetosphere
4.1 MHD Oscillation Properties in Non‐homogeneous Models of the Magnetosphere of Different Dimension
4.2 Coordinate System
4.3 Basic Equations
4.4 Qualitative Investigation of the Equation for Characteristics
4.5 Wave Singularity in the 3D‐Inhomogeneous Magnetosphere
Notes
Chapter 5: Conclusion
Appendixes
A Transverse Dispersion of MHD Waves in a ‘Cold’ and ‘Hot’ Plasma
B Deriving an Equation for MHD Oscillations in a 1D‐Inhomogeneous Moving Plasma
C A Model of the Spectral Function of the Solar Wind FMS Oscillations
D Stability of MHD Oscillations with , in the Shear Layer, for in a Boundless Medium
E Deriving Equations for Potentials and for MHD Waves in a ‘Warm’ Plasma, in a Curvilinear Orthogonal System of Coordinates
F WKB Solution of the Longitudinal Problem for FMS Waves Having Two Turning Points on the Field Line
G Integrals of Functions and Describing the Transverse Structure of Standing Alfvén Waves
H Parameters of the Polarisation Ellipse of Stochastic Oscillations
I Deriving Coefficients of the Differential Equation Based on the Given WKB Solution
J Strictly Deriving a Transverse Model Equation for Standing Alfvén Waves, for the Case
K Calculating Characteristics
L Calculating the Integral (3.390) Near the Characteristics and
M Calculating the Integral (3.388) Near the Characteristics and
N Determining the Shape of a Field Line from Given Components of Background Magnetic Field
O Defining Tri‐orthogonal System of Coordinates Related to Magnetic‐Field Lines
P Determining Metric Tensor Components in a Curvilinear Orthogonal System of Coordinates
Q Coefficients of the Equation for the Coupled Modes of MHD Oscillations
R Equation for MHD Oscillations in a Cylindrical Coordinate System
S Equality of the Alfvén Oscillation Specific Power Absorbed Near the Resonance Surface and the Density of Energy Carried Away by KAWs
References
Index
End User License Agreement
List of Tables
Chapter 1
Table 1.1 Amplitudes of wave field components for various MHD modes.
Chapter 2
Table 2.1 Averaged plasma and magnetic field parameters in the solar wind (
Table 2.2 Main parameters of the model medium at the geotail boundary.
Chapter 3
Table 3.1 Eigen frequencies ...
Table 3.2 Eigen frequencies (rad/s), of several first harmonics of the FMS...
Table 3.3 Eigen frequencies of several first harmonics of a FMS resonator ...
List of Illustrations
f10
Figure I.1 Basic structural elements of Earth's magnetosphere.
Chapter 1
Figure 1.1 Friedrichs diagrams for the Alfvén (black)...
Figure 1.2 Schematic of magnetic field oscillations (field lines), plasma pr...
Figure 1.3 Relative SMS decrement vs. non‐isothermality, , of homogeneous...
Figure 1.4 Qualitative behaviour of the function in the complex plane as...
Chapter 2
Figure 2.1 Model 1D‐inhomogeneous plasma (with density ) in a homogeneous m...
Figure 2.2 Parallel structure of electric (black lines) and magnetic (gr...
Figure 2.3 (a) Alfvén speed distribution along the ...
Figure 2.4 Spatial distribution of MHD oscillation ‐component (left‐hand ax...
Figure 2.5 The dependence of the absorption coefficient for FMS waves inci...
Figure 2.6 Evolution of an FMS wave packet incident onto a smooth transition...
Figure 2.7 Distributions of Alfvén speed ...
Figure 2.8 Derivative distribution in the problem of FMS wave incident on/...
Figure 2.9 Distribution of MHD oscillation magnetic field components (Re())...
Figure 2.10 Hodograph behaviour for resonant MHD oscillations in the neighbo...
Figure 2.11 Absorption coefficient dependence for FMS waves incident on the ...
Figure 2.12 Distribution of the real and imaginary parts of ...
Figure 2.13 Structure of MHD oscillations across magnetic shells when FMS wa...
Figure 2.14 Potential in (2.79) for quasi‐parallel MHD oscillations: (a) the...
Figure 2.15 The structure of waveguide modes in the potential decaying as ...
Figure 2.16 Graphic solution of dispersion Eqs. (2.122), (2.123). Curves I c...
Figure 2.17 Alfvén speed distribution in the dayside...
Figure 2.18 Schematic representation of a magnetospheric duct model in a cyl...
Figure 2.19 Model medium and coordinate system. Roman numbers indicate the f...
Figure 2.20 The ...
Figure 2.21 The characteristic form of the dependencies of the relative de...
Figure 2.22 Total energy transferred into the magnetosphere by the ‘geoeff...
Figure 2.23 Model medium and coordinate system: is the characteristic scal...
Figure 2.24 Growth rate (...
Figure 2.25 Growth rate () isoline distribution for MHD oscillations with
Figure 2.26 Growth rate distribution of MHD oscillations with for a shea...
Figure 2.27 Growth rate isoline distribution for MHD oscillations with f...
Figure 2.28 Growth rate isoline distribution for MHD oscillations with f...
Figure 2.29 Dependence of growth rate for MHD oscillations with for a sh...
Figure 2.30 Growth rate isoline distribution for MHD oscillations with f...
Figure 2.31 Growth rate isoline distribution for MHD oscillations with f...
Figure 2.32 (a) cylindrical model of the geotail wrapped around by the solar...
Figure 2.33 Spatial structure of monochromatic MHD waves with for two diff...
Figure 2.34 Alfvén and SMS wave speed distribution (light and dark gray ...
Figure 2.35 The Mach number dependence of the frequency , dark gray line)...
Figure 2.36 Radial structure of unstable oscillations in the geotail for the...
Figure 2.37 The Mach number, , dependence of the growth rate of the ‘global...
Figure 2.38 The dependence of the growth rate of unstable azimuthal harmon...
Figure 2.39 The low‐latitude boundary layer (LLBL): (a) schematic of the mag...
Figure 2.40 Radial distribution of Alfvén speed , sound speed , SMS speed
Figure 2.41 Radial structure of unstable MHD oscillations generated by the m...
Figure 2.42 dependence of the oscillation growth rate for the given azim...
Figure 2.43 Growth rate () isoline maps for the basic and first () azimu...
Figure 2.44 Dependence of the growth rate of unstable MHD oscillations on ...
Figure 2.45 The box model of the magnetosphere.
Figure 2.46 The spatial structure of the fast mode wave field generated by t...
Figure 2.47 The section of the wave field along the azimuthal coordinate (ex...
Figure 2.48 (a) typical height profiles of the conductivity tensor component...
Figure 2.49 Mutual positions of three systems of coordinates used in the pro...
Chapter 3
Figure 3.1 Toroidal and poloidal oscillations of field lines. The fundamenta...
Figure 3.2 Coordinate systems related to geomagnetic field lines in an axisy...
Figure 3.3 Equatorial dependence of Alfvén speed in the model under study ...
Figure 3.4 Eigenfrequency distribution for the first seven harmonics of stan...
Figure 3.5 The structure of monochromatic ( Hz) FMS waves across magnetic s...
Figure 3.6 Amplitude distributions of magnetic field components, , and ,...
Figure 3.7 Mean amplitude distribution for the field component of the two ...
Figure 3.8 Distribution of the component (thick lines) of the field of res...
Figure 3.9 Amplitude distribution in the meridional plane for the full field...
Figure 3.10 (a) FMS wave packet; (b–d) behaviour of the th harmonic of stan...
Figure 3.11 Standing Alfvén waves excited in the magnetosphere by a stochast...
Figure 3.12 Model axisymmetric magnetosphere and systems of coordinates: ()...
Figure 3.13 Distribution across magnetic shells of plasma equatorial velocit...
Figure 3.14 Parameter isolines in the meridional plane (). Coordinates ...
Figure 3.15 (a) Distribution across magnetic shells of the eigenfrequencies ...
Figure 3.16 (a) Structure along magnetic field lines of the first three harm...
Figure 3.17 Transverse structure of the magnetic field components for the fi...
Figure 3.18 Dependence of function on geomagnetic latitude and azimuthal...
Figure 3.19 Longitudinal structure of first four harmonics () of the FMS ...
Figure 3.20 Squared quasi‐classical wave‐vector component for the first fi...
Figure 3.21 The boundaries of transparent regions (in the meridional plane) ...
Figure 3.22 The boundaries of the transparent regions for FMS oscillation ha...
Figure 3.23 Model FMS resonator in the form of a rectangular box with ideall...
Figure 3.24 Global Alfvén speed distribution in Earth's magnetosphere.
Figure 3.25 Axisymmetric model magnetosphere with a plasma sheet.
Figure 3.26 Orthogonal system of dimensionless parabolic coordinates (), in...
Figure 3.27 Alfvén speed ( km/s) distribution isolines in the meridional p...
Figure 3.28 Distribution of the square of the wave‐vector WKB component fo...
Figure 3.29 Lines of constant phase (characteristic lines) in the transver...
Figure 3.30 Structure of the first two harmonics () of standing toroidal (d...
Figure 3.31 Poloidal and toroidal eigenfrequencies () vs. magnetic shel...
Figure 3.32 The polarisation splitting of the spectrum, equatorial splitti...
Figure 3.33 The group velocity components and for the main harmonic of s...
Figure 3.34 Schematic representation of the structure of standing Alfvén wav...
Figure 3.35 Penetration of the high‐ Alfvén wave field from the magnetosphe...
Figure 3.36 Distribution in of the quasiclassical wave vector squared . S...
Figure 3.37 Possible integration contours in integrals (3.242). Sectors with...
Figure 3.38 Spatial structure of Alfvén waves with across magnetic shells....
Figure 3.39 Schematic plots of functions in the dayside magnetosphere of E...
Figure 3.40 Schematic plots of spectral density of the components () of pe...
Figure 3.41 Schematic plots of spectral density of the components () of p...
Figure 3.42 Hodographs of monochromatic Alfvén oscillations with , at diffe...
Figure 3.43 Hodographs of non‐stationary Alfvén oscillations with excited ...
Figure 3.44 Field structure across magnetic shells of standing Alfvén waves ...
Figure 3.45 Spatial distribution over the transverse coordinate for the po...
Figure 3.46 Same as Figure 3.45, for wave with resonant type transverse stru...
Figure 3.47 Distribution of the amplitude of standing Alfvén waves excited b...
Figure 3.48 Calculated dependence of the splitting, between the toroidal a...
Figure 3.49 Calculated dependence of the eigenfrequencies of toroidal Alf...
Figure 3.50 Graphical solution of (3.398): (a) in the tolerance range (3.396...
Figure 3.51 Region occupied by Alfvén wave oscillations in the asymptotic re...
Figure 3.52 Amplitude distribution for a separate harmonic of standing Alfvé...
Figure 3.53 Amplitude distribution for a separate harmonic of standing Alfvé...
Figure 3.54 Amplitude distribution of the ‐component of the full field of s...
Figure 3.55 Behaviour of the ‐component of the full field of Alfvén oscilla...
Figure 3.56 Model axisymmetric magnetic field with elongated field lines for...
Figure 3.57 Shape of magnetic field lines calculated from (3.421), in a geot...
Figure 3.58 Distribution isolines for: (a) Alfvén speed (km/s); (b) sound ...
Figure 3.59 Graphical solution of dispersion equation (3.428). The solution ...
Figure 3.60 Eigen frequency distribution for first odd harmonics of azimut...
Figure 3.61 Distribution of parameter along the field line located on magn...
Figure 3.62 Distribution of the wavenumber squared, along different field li...
Figure 3.63 Eigen frequency distribution for the first five harmonics of s...
Figure 3.64 Eigen frequency distribution for the first five harmonics of s...
Figure 3.65 Longitudinal structure of the main harmonic () of unstable polo...
Figure 3.66 Distribution along the field line of the scalar potential of the...
Figure 3.67 Distribution along the field line of the electromagnetic field c...
Figure 3.68 Distribution across magnetic shells of the eigenfrequencies of t...
Figure 3.69 Parameter distribution along the field line, for the main harm...
Figure 3.70 Azimuth variations for saddle points () in (3.452), from to...
Figure 3.71 Coupled Alfvén and SMS mode structure across magnetic shells, ne...
Chapter 4
Figure 4.1 (a) Characteristics (energy flux lines) of transverse small‐scale...
Figure 4.2 The inclination of the channel with respect to the coordinate sur...
Figure 4.3 The structure transverses the channel of and electromagnetic ...
1
Figure A.1 Dependence for: (
a
) the ‘cold’ dispersion of Alfvén waves (). ...
Figure H.1 Distribution, in the () plane, of the mean square of the amplitu...
Figure L.1 Integration contour and the location of sectors of exponentiall...
Figure L.2 Integration contour for the integral (3.388) near the character...
Figure N.1 Length elements necessary for calculating the components of metri...
Figure N.2 Defining tri‐orthogonal system of coordinates. The coordinate c...
Guide
Cover
Table of Contents
Title Page
Copyright
List of Figures
List of Tables
Author Biography
Preface
Acknowledgements
Acronyms
Symbols
Introduction
Begin Reading
Appendixes
References
Index
End User License Agreement
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Magnetospheric MHD Oscillations
A Linear Theory
Anatoly Leonovich
Vitalii Mazur
Dmitri Klimushkin
Authors
Dr. Anatoly LeonovichInst. of Solar‐Terrestrial PhysicsLermontova st. 126aIrkutskRussia, 664033
Prof. Vitalii MazurInst. of Solar-Terrestrial PhysicsLermontova st. 126aIrkutskRussia, 664033
Dr. Dmitri KlimushkinInst. of Solar‐Terrestrial PhysicsLermontova st. 126aIrkutskRussia, 664033
All books published by WILEY‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No.: applied for
British Library Cataloguing‐in‐Publication DataA catalogue record for this book is available from the British Library.
Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.
© 2024 WILEY‐VCH GmbH, Boschstr. 12, 69469 Weinheim, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN: 978‐3‐527‐41430‐7ePDF ISBN: 978‐3‐527‐84574‐3ePub ISBN: 978‐3‐527‐84575‐0oBook ISBN: 978‐3‐527‐84573‐6
Cover Image: © MARK GARLICK/SCIENCE PHOTO LIBRARY/Getty Images
List of Figures
Figure 1.1
Friedrichs diagrams for the Alfvén (black), slow (light gray) and fast (dark gray) MHD modes for case when the Alfvén speed is greater than the speed of sound : (a) for the phase velocity and (b) for the group velocity . Here indices and mean projections on the direction parallel and perpendicular to the ambient magnetic field, respectively
Figure 1.2
Schematic of magnetic field oscillations (field lines), plasma pressure (shades of gray) and group velocity directions for Alfvén (a), fast magnetosonic (b) and slow magnetosonic (c) waves propagating in a homogeneous plasma
Figure 1.3
Relative SMS decrement vs. non‐isothermality, , of homogeneous plasma
Figure 1.4
Qualitative behaviour of the function in the complex plane as the argument varies from to
Figure 2.1
Model 1D‐inhomogeneous plasma (with density ) in a homogeneous magnetic field
Figure 2.2
Parallel structure of electric (black lines) and magnetic (gray lines) field of the MHD wave. Solid and dashed lines depict the fundamental () and second () harmonics, respectively
Figure 2.3
(a) Alfvén speed distribution along the axis in a 1D‐inhomogeneous model of the medium. (b) Dependence of the squared WKB component of the wave vector, , of MHD oscillations in a ‘cold’ plasma ( is the resonance point for Alfvén wave, is the turning point for FMS wave), (c) The structure of wave field components of MHD oscillations in a 1D‐inhomogeneous plasma
Figure 2.4
Spatial distribution of MHD oscillation ‐component (left‐hand axis) in the FMS wave which is incident to/reflected from the transition layer with an Alfvén resonance point. The gray line is the full oscillation field. WKB approximation: line 1 is FMS wave incident onto the transition layer, line 2 is FMS wave reflected from the transition layer. Oscillation hodographs for various points are shown above. The right‐hand axis line is Alfvén speed distribution
Figure 2.5
The dependence of the absorption coefficient for FMS waves incident onto the transition layer that are partially absorbed at the Alfvén resonance point. Distributions for various are shown. The black marginal curve, for , corresponds to an infinite layer with a linear profile. The dark gray dash lines show analytical distributions for two limiting cases: (1) and (2) , described by (2.21) and (2.23)
Figure 2.6
Evolution of an FMS wave packet incident onto a smooth transition layer containing an Alfvén resonance point, . The distribution of the Alfvén speed (right coordinate axis) and the component of the wave field (left coordinate axis) are shown. (a) Initial state of the unit‐wide wave packet. (b) Field structure at the moment the wave packet is reflected from the transition layer. (c) Wave packet structure upon reflection from the transition layer
Figure 2.7
Distributions of Alfvén speed , SMS wave speed (black lines, left‐hand axis) and the square of the WKB component of the wave vector, (gray lines, right‐hand axis), across the transition layer, in two limiting cases: (1) (solid gray line for ) the opaque region () is present for FMS waves, (2) (gray dashed line for ) the transparent region for SMS waves spreads from the resonance surface for SMS waves, , to infinity
Figure 2.8
Derivative distribution in the problem of FMS wave incident on/reflected from the transition layer with resonance surfaces for Alfvén () and SMS waves (). The gray line is the (numerically calculated) full oscillation field, (1) FMS wave incident on the transition layer, (2) FMS wave reflected from the transition layer (WKB approximation)
Figure 2.9
Distribution of MHD oscillation magnetic field components (Re()) in the problem of FMS wave incident on/reflected from the transition layer with resonance surfaces for Alfvén () and SMS waves ()
Figure 2.10
Hodograph behaviour for resonant MHD oscillations in the neighbourhood of resonance surfaces, and , for various decrements of Alfvén () and SMS oscillations. The curves and circles with hodograph rotation directions labelled 1,2 and 3, correspond to three values of SMS oscillation decrement: , and
Figure 2.11
Absorption coefficient dependence for FMS waves incident on the transition layer with two resonance surfaces: for Alfvén and SMS waves. The distributions are shown for the plasma layer (for waves with : curves for and curves for ) and the plasma layer (for : curves for and curves for )
Figure 2.12
Distribution of the real and imaginary parts of , a function of a real argument
Figure 2.13
Structure of MHD oscillations across magnetic shells when FMS waves are in resonance with kinetic Alfvén waves. The FMS wave amplitude decreases exponentially into the opaque region. In a ‘cold’ plasma (), a kinetic Alfvén wave is excited on the resonance magnetic shell and travels away leftwards, while in a ‘warm’ plasma (), an Alfvén wave travels away rightwards from the resonance shell
Figure 2.14
Potential in (2.79) for quasi‐parallel MHD oscillations: (a) the form of potential with Alfvén resonance points () for waveguide FMS oscillations (); (b) potential for waveguide‐travelling quasi‐parallel Alfvén waves ()
Figure 2.15
The structure of waveguide modes in the potential decaying as the related MHD waves escape. Top: waveguide FMS mode and related escaping kinetic Alfvén wave (A2) (see Figure A.1, (a) in Appendix A); bottom: waveguide mode of kinetic Alfvén waves that decays due to the FMS wave escaping from the waveguide
Figure 2.16
Graphic solution of dispersion Eqs. (2.122), (2.123). Curves I correspond to the right side of (2.122), and curves II to the right side of (2.123)
Figure 2.17
Alfvén speed distribution in the dayside magnetosphere (gray scale). distribution is shown in the meridional section inside the plasmapause and in the plasma filament (duct for MHD waves) in the outer magnetosphere. Smaller are shown by light gray. The white arrows indicate the direction of the FMS group velocity in the meridional section
Figure 2.18
Schematic representation of a magnetospheric duct model in a cylindric system of coordinates (): (a) direction of outer magnetic field and plasma density distribution in a plasma filament; (b) the form of the potential in (2.128)
Figure 2.19
Model medium and coordinate system. Roman numbers indicate the following regions: – solar wind, – magnetosphere. FMS waves are labelled as:
1
– incident on the magnetosphere,
2
– reflected from the magnetopause,
3
– penetrating into the magnetosphere
Figure 2.20
The dependence of function for fixed and . (a) – in the magnetosphere, the points labelled: 1 – , 2 – , 3 – . (b) – in the solar wind, the points labelled: 1 – , 2 – , 3 – , 4 – , 5 – , 6 –
Figure 2.21
The characteristic form of the dependencies of the relative densities of monochromatic FMS energy fluxes in the solar wind and in the magnetosphere, in the sector. Energy flux densities are labelled as: 1 – FMS wave incident to the magnetopause; 2 – wave reflected from the magnetopause; 3 – wave penetrating into the magnetosphere
Figure 2.22
Total energy transferred into the magnetosphere by the ‘geoeffective’ FMS flux over time s vs. the geotail growth rate index in the model Eq. (2.168)
Figure 2.23
Model medium and coordinate system: is the characteristic scale of the shear layer, is the location of possible boundaries in the form of rigid walls, , are the unperturbed speed and background magnetic field vectors, is the tangential wave vector of the oscillations
Figure 2.24
Growth rate () isoline distribution for MHD oscillations with generated by a shear flow in the form of a smooth transition layer in a boundless medium, for two values of : (
a
) , (b)
Figure 2.25
Growth rate () isoline distribution for MHD oscillations with generated by a shear flow in the form of a smooth transition layer in a boundless medium
Figure 2.26
Growth rate distribution of MHD oscillations with for a shear flow in the form of a tangential discontinuity bounded by a rigid wall on one side, for different values of parameters and : (
a
) , plots 1–5 refer to ; (
b
) , plots 1–5 refer to
Figure 2.27
Growth rate isoline distribution for MHD oscillations with for a shear flow with a smooth transition layer bounded by a rigid wall () on one side, for two different values of parameter : (a) – , (b) –
Figure 2.28
Growth rate isoline distribution for MHD oscillations with for a shear flow with a smooth transition layer bounded by a rigid wall on one side (). Thick lines correspond to the surface and radiative oscillation modes, thin lines to the oscillation mode reflected from the wall
Figure 2.29
Dependence of growth rate for MHD oscillations with for a shear flow in the form of a tangential discontinuity between two rigid walls, for different values of parameters and : (
a
) , plots
1
–
3
correspond to values ; (
b
) , plots
1
–
4
correspond to values
Figure 2.30
Growth rate isoline distribution for MHD oscillations with for a shear flow in the form of a smooth transition layer between two rigid walls () for two different values of parameter : (
a
) , (
b
)
Figure 2.31
Growth rate isoline distribution for MHD oscillations with for a shear flow in the form of a smooth transition layer bounded by rigid walls on both sides (). The thick lines correspond to the surface and radiative oscillation modes, and the thin lines to oscillation modes reflected from the walls
Figure 2.32
(a) cylindrical model of the geotail wrapped around by the solar wind plasma flow and a schematic structure of the unstable ”global mode” of the geotail. (b) Alfvén speed and SMS speed distributions in the geotail and the solar wind. On the resonance surfaces (points
2
and 3) and (points
1
and
4
), the parallel phase velocity of the monochromatic wave coincides with, respectively, the local SMS speed and the Alfvén speed
Figure 2.33
Spatial structure of monochromatic MHD waves with for two different values of the parallel phase speed : (a) oscillations with resonant surfaces for SMS waves in the geotail, , (b) oscillations with no resonance surfaces in the geotail
Figure 2.34
Alfvén and SMS wave speed distribution (light and dark gray lines; the vertical axis is to the right) inside and outside the geotail in the plasma cylinder model. Squared wave‐vector WKB component distribution over radius, (black lines; the vertical axis is to the left, the thick dashed lines correspond to ). Coordinates and correspond to resonance surfaces for Alfvén and SMS oscillations, are the turning points for magnetosonic waves. Numbers and shades of grey denote the transparent regions:
1
– for SMS waves,
2
– for FMS waves with , and
3
– for FMS waves with
Figure 2.35
The Mach number dependence of the frequency , dark gray line) and growth rate ( light gray lines) for unstable oscillations driven at the geotail boundary. (a) WKB solution for the model with a boundary in the form of a tangential discontinuity, where are the roots of the dispersion equation determining, in the WKB approximation, the FMS‐waveguide eigenfrequencies () in the geotail lobes. (b) solution for the model with a boundary in the form of a smooth transition layer of characteristic thickness for the same parameters as in panel (a)
Figure 2.36
Radial structure of unstable oscillations in the geotail for the azimuthal harmonic , normalised to the maximum value of : (a) oscillations close to the second harmonic, , of the eigenmodes propagating in the FMS waveguide in the geotail lobes (), (b) ‘global mode’ oscillations for small values of
Figure 2.37
The Mach number, , dependence of the growth rate of the ‘global’ modes in the geotail for the first azimuthal harmonics and
Figure 2.38
The dependence of the growth rate of unstable azimuthal harmonics and on the global mode frequency for various speeds of the solar wind flow around the magnetosphere:
1
– km/s,
2
– km/s,
3
– km/s,
4
– km/s
Figure 2.39
The low‐latitude boundary layer (LLBL): (a) schematic of the magnetic field lines and the electric current configuration in the geotail LLBL, (b) cylindrical model of the geotail enwrapped by a helical plasma flow. Here: I is magnetosphere, II is solar wind
Figure 2.40
Radial distribution of Alfvén speed , sound speed , SMS speed (left vertical axis) and the parameter (right vertical axis) in the equilibrium geotail model for km/s and . Here (see Fig. 2.39(b)):
I
– magnetosphere,
II
– solar wind
Figure 2.41
Radial structure of unstable MHD oscillations generated by the magnetopause shear flow: (a) – surface mode structure (1), radiative mode structure (2); (b) – structure of the first (1) and second (2) eigen‐mode harmonics of the FMS waveguide in the geomagnetic tail. On both panels: solid lines are , dotted lines are (in panel (a) for the surface mode only); black curves (3) are the squared wave number,
Figure 2.42
dependence of the oscillation growth rate for the given azimuthal wave number and the Mach number : (0) – for the surface mode, (1,2,3) – for three harmonics of the FMS waveguide in the geotail
Figure 2.43
Growth rate () isoline maps for the basic and first () azimuthal harmonics of MHD oscillations in the () plane in the longitudinal solar wind flows (left‐hand panels, helicity index ) and in the strongly twisted flows (right‐hand panels, ). The bold line depicts the boundary separating the areas of unstable () and stable () oscillations
Figure 2.44
Dependence of the growth rate of unstable MHD oscillations on their frequency (for azimuthal harmonics ) for different velocities of the solar wind flowing around the magnetosphere: km/s (curves 1,2,3). Solid lines depict the oscillations generated by a longitudinal flow (the helicity index ), dashed lines stand for the oscillations generated by a strongly twisted flow ()
Figure 2.45
The box model of the magnetosphere
Figure 2.46
The spatial structure of the fast mode wave field generated by the Cherenkov mechanism in the source reference frame (normalised amplitude). The parameters chosen in Eq. (2.264) are , . The Mach number on the magnetopause is chosen . The sum of the first 20 harmonics is depicted. The source is denoted by the circle on the top left, the grey line denotes the general reflection surface. The numbers denote different wave branches
Figure 2.47
The section of the wave field along the azimuthal coordinate (exact solution at ). The numbers denote wave branches, as in Figure 2.46
Figure 2.48
(a) typical height profiles of the conductivity tensor components and the Alfvén speed . Roman numerals indicate the following layers: I – ground with isotropic conductivity , II – atmosphere with conductivity , III – lower ionosphere with transverse Pedersen and Hall conductivities and parallel conductivity , IV – upper ionosphere, where , V – magnetosphere. (b) model of near‐Earth medium with a vertical magnetic field; Alfvén wave penetrating to Earth is shown schematically: 1 – incident wave from the magnetosphere, 2 – wave reflected from the ionosphere, 3 – field of the wave penetrating to ground
Figure 2.49
Mutual positions of three systems of coordinates used in the problem of MHD wave field penetrating from the magnetosphere to Earth, for a geospace model with inclined geomagnetic field: , and . Roman numerals denote the following layers: I – ground with conductivity , II – atmosphere with conductivity , III – lower ionosphere (E‐layer) and IV – upper ionosphere with anisotropic conductivities, V – magnetosphere
Figure 3.1
Toroidal and poloidal oscillations of field lines. The fundamental () and second () harmonics are shown
Figure 3.2
Coordinate systems related to geomagnetic field lines in an axisymmetric model magnetosphere: curvilinear orthogonal system of coordinates (), curvilinear non‐orthogonal system of coordinates ()
Figure 3.3
Equatorial dependence of Alfvén speed in the model under study and the corresponding dependence of the basic period of magnetospheric Alfvén eigen oscillations on the magnetic shell parameter
Figure 3.4
Eigenfrequency distribution for the first seven harmonics of standing toroidal Alfvén waves across magnetic shells in a dipole model magnetosphere. Horizontal dashed lines indicate the frequencies of magnetosonic waves incident on the magnetosphere from the solar wind. Vertical dashed lines are conditional boundaries of the transition layer of the magnetopause
Figure 3.5
The structure of monochromatic ( Hz) FMS waves across magnetic shells for two first longitudinal harmonics (), with (dashed lines) or without (solid lines) feedback from resonant Alfvén waves. The vertical dashed lines denote FMS turning points ( and ) and the magnetopause ()
Figure 3.6
Amplitude distributions of magnetic field components, , and , across magnetic shells for monochromatic ( Hz) magnetosonic oscillations: (a) inside the magnetosphere, (b) in the solar wind region. Numbers (1) and (2) in the panels correspond to the two first longitudinal harmonics of FMS oscillations ()
Figure 3.7
Mean amplitude distribution for the field component of the two first harmonics () of resonant Alfvén waves across magnetic shells. These harmonics are excited by a monochromatic ( Hz, ) magnetosonic waves with (lines 1,2) and (lines 3,4). The amplitude distribution of the first harmonic () is shown (line 1(10)) for comparison, as excited by magnetosonic wave with , which practically does not penetrate into the magnetosphere
Figure 3.8
Distribution of the component (thick lines) of the field of resonant Alfvén oscillations excited by monochromatic ( Hz) FMS waves incident on the magnetosphere. The numbers indicate the first four harmonics of resonant Alfvén waves (). The distribution of the parallel component (thick line) of the field of the second parallel harmonic () of the FMS wave is given for comparison
Figure 3.9
Amplitude distribution in the meridional plane for the full field of resonant Alfvén oscillations excited by monochromatic (frequency 0.01 Hz) magnetosonic wave in a dipole magnetosphere. Numbers 1, 2 denote the first () and the second () harmonics of standing Alfvén waves
Figure 3.10
(a) FMS wave packet; (b–d) behaviour of the th harmonic of standing Alfvén waves excited by FMS wave packet: (b) exciting FMS wave frequency is higher than the frequency of Alfvén eigen oscillations, ; (c) – standing Alfvén waves and FMS oscillations are in resonance; (d)
Figure 3.11
Standing Alfvén waves excited in the magnetosphere by a stochastic source of FMS oscillations. Top left – the spectrum of stochastic FMS oscillations, . Top right – eigenfrequency distribution for three first harmonics () of resonant Alfvén waves across magnetic shells ( – McIlwain parameter). Bottom – distribution, across magnetic shells, of the oscillation amplitude for the first three harmonics of the excited standing Alfvén waves and their envelope (dash‐dotted line)
Figure 3.12
Model axisymmetric magnetosphere and systems of coordinates: () is an orthogonal system of curvilinear coordinates related to magnetic field lines, () is a non‐orthogonal system of curvilinear coordinates. Dashed lines indicate the plasmapause () and magnetopause ()
Figure 3.13
Distribution across magnetic shells of plasma equatorial velocity (curves 1 and 2), Alfvén speed (curve 3 in the equatorial plane , curve 4 along radius at angle to the equator) and the basic harmonic period of magnetospheric standing Alfvén waves (curve 5)
Figure 3.14
Parameter isolines in the meridional plane (). Coordinates and are in Earth radius units
Figure 3.15
(a) Distribution across magnetic shells of the eigenfrequencies of the first three harmonics of standing SMS waves () and the SMS transit time along the field line. (b) Frequencies of the first three harmonics of standing toroidal Alfvén waves () and Alfvén speed transit time between magneto‐conjugated ionospheres,
Figure 3.16
(a) Structure along magnetic field lines of the first three harmonics of standing Alfvén waves () and SMS waves ( ) at magnetic shell . (b) Structure of the first three harmonics of standing SMS waves () at magnetic shell
Figure 3.17
Transverse structure of the magnetic field components for the first harmonic () of resonant SMS oscillations near the resonant shell : (a) standing SMS wave amplitude and phase distribution () near the equatorial surface; (b) oscillation amplitude and phase distributions near the ionosphere (longitudinal component turns zero)
Figure 3.18
Dependence of function on geomagnetic latitude and azimuthal wave number at magnetic shell . Curve 1 refers to , curve 2 to , curve 3 to , curve 4 to . Horizontal dashed lines are possible eigenvalues of parameter . Areas where labelled as I, II, III, IV (shown in grey) correspond to the four types of longitudinal structure of FMS eigen oscillations
Figure 3.19
Longitudinal structure of first four harmonics () of the FMS eigenmodes in the dipole‐like magnetosphere, at magnetic shell
Figure 3.20
Squared quasi‐classical wave‐vector component for the first five harmonics () of the FMS eigenmodes vs. the magnetic shell parameter : (a) distribution inside the magnetosphere and (b) distribution in the magnetic shell range , including the solar wind region
Figure 3.21
The boundaries of transparent regions (in the meridional plane) for the first four parallel harmonics of FMS oscillations (), in a dipole‐like model magnetosphere, for frequency and azimuthal wavenumber
Figure 3.22
The boundaries of the transparent regions for FMS oscillation harmonic (, ), for different frequencies: (1) , (2) , (3)
Figure 3.23
Model FMS resonator in the form of a rectangular box with ideally reflecting walls (‘box model’)
Figure 3.24
Global Alfvén speed distribution in Earth's magnetosphere
Figure 3.25
Axisymmetric model magnetosphere with a plasma sheet
Figure 3.26
Orthogonal system of dimensionless parabolic coordinates (), in the meridional section, including the axis. The focus, , is at Earth's centre. The surface coincides with the magnetopause. The semiaxes and correspond to the coordinate surfaces and , respectively
Figure 3.27
Alfvén speed ( km/s) distribution isolines in the meridional plane, in the parabolic model magnetosphere
Figure 3.28
Distribution of the square of the wave‐vector WKB component for Alfvén waves with , in the transverse coordinate
Figure 3.29
Lines of constant phase (characteristic lines) in the transverse section of an axisymmetric magnetosphere. Curves 1 correspond to , and curves 2 to . The circles are the transverse sections of the resonance surfaces: the inner circle is the poloidal (), the outer the toroidal surface ()
Figure 3.30
Structure of the first two harmonics () of standing toroidal (dark lines) and poloidal (light lines) Alfvén waves, along field‐lines, at magnetic shell
Figure 3.31
Poloidal and toroidal eigenfrequencies () vs. magnetic shell (the McIlwain) parameter
Figure 3.32
The polarisation splitting of the spectrum, equatorial splitting of resonance surfaces and the characteristic scale of the transverse inhomogeneity of the Alfvén speed for the basic harmonic of Alfvén oscillations () vs. magnetic shell parameter
Figure 3.33
The group velocity components and for the main harmonic of standing Alfvén waves vs. the radial coordinate in the equatorial plane (), the toroidal resonance surface for the waves is at magnetic shell
Figure 3.34
Schematic representation of the structure of standing Alfvén wave with in a dipole‐like magnetosphere. Function describes the wave structure along magnetic field lines, and across magnetic shells
Figure 3.35
Penetration of the high‐ Alfvén wave field from the magnetosphere to ground: (a) in the case, (b) in the case. Shown are the hodographs of oscillations in the plane (), at various points inside the transparent region () at the upper ionospheric boundary and at Earth surface
Figure 3.36
Distribution in of the quasiclassical wave vector squared . Slanting dashed lines show the asymptotics . The (a) case corresponds to , and the (b) case to
Figure 3.37
Possible integration contours in integrals (3.242). Sectors with exponentially growing asymptotics are in grey
Figure 3.38
Spatial structure of Alfvén waves with across magnetic shells. The thick (black) line is ‘large‐scale’ Alfvén wave, the thin line is kinetic Alfvén wave: (a) and (b)
Figure 3.39
Schematic plots of functions in the dayside magnetosphere of Earth. The McIlwain parameter is used as the coordinate.
1
– the transparent region is located between the poloidal and toroidal resonant surfaces;
2
– Alfvén resonator;
3
– the presence of two toroidal turning points makes Alfvén oscillations impossible
Figure 3.40
Schematic plots of spectral density of the components () of perturbed magnetic field of standing Alfvén waves for strong decay of oscillations ()
Figure 3.41
Schematic plots of spectral density of the components () of perturbed magnetic field of standing Alfvén waves for weak decay of oscillations ()
Figure 3.42
Hodographs of monochromatic Alfvén oscillations with , at different points in the transparent region between the resonance surfaces
Figure 3.43
Hodographs of non‐stationary Alfvén oscillations with excited at the observation point by a source of the ‘sudden pulse’ type, at different moments of time: 1 – – oscillations of the poloidal type (); 2 – – oscillations of the intermediate type (); 3 – – oscillations of the toroidal type ()
Figure 3.44
Field structure across magnetic shells of standing Alfvén waves with , for various values of parameter : (a) – a typical structure of resonant oscillations; (b) – structure of intermediate type; (c) – ‘travelling wave’‐type structure
Figure 3.45
Spatial distribution over the transverse coordinate for the poloidal () and toroidal () components of the full energy of standing Alfvén wave (in relative units) for ‘travelling wave’ type oscillations (, see Figure 3.44c) for various values of the dimensionless dissipation index : (a) ; (b) ; (c)
Figure 3.46
Same as Figure 3.45, for wave with resonant type transverse structure (, see Figure 3.44a): (a) – weak dissipation – toroidal type wave; (b) – moderate dissipation – intermediate type wave; (c) – strong dissipation – poloidal type wave
Figure 3.47
Distribution of the amplitude of standing Alfvén waves excited by strongly localised monochromatic sources over dimensionless transverse coordinates ( and ). Source location: (a) in the opaque region behind the toroidal surface, (b) at the toroidal surface, (c) at the poloidal surface, and (d) in the opaque region behind the poloidal surface
Figure 3.48
Calculated dependence of the splitting, between the toroidal and poloidal resonance magnetic shells as mapped onto the ionosphere on the magnetic shell parameter , for the first (, bold line, right‐hand axis) and two next harmonics (, left‐hand axis) of standing Alfvén waves
Figure 3.49
Calculated dependence of the eigenfrequencies of toroidal Alfvén oscillations on the magnetic shell parameter , for the first five eigen harmonics of standing Alfvén waves
Figure 3.50
Graphical solution of (3.398): (a) in the tolerance range (3.396), (b) in the tolerance range (3.397). The dark gray curves are the left‐hand sides (panel (b) the upper curve is for the ‘’ sign, the lower for ‘’), and straight horizontal black lines are the right‐hand sides of (3.398)
Figure 3.51
Region occupied by Alfvén wave oscillations in the asymptotic regime (‘butterfly wings’). Solid lines (1–6) are characteristic lines , dashed lines are characteristic lines . The Roman numerals I, II, III and IV indicate the domains of existence for the roots of (3.398) – and
Figure 3.52
Amplitude distribution for a separate harmonic of standing Alfvén wave, in the initial regime of oscillations, in the plane of dimensionless transverse coordinates (). Sectors with smallest amplitude are in shades of grey
Figure 3.53
Amplitude distribution for a separate harmonic of standing Alfvén wave, in the asymptotic regime of oscillations, in the plane of dimensionless transverse coordinates (). Sectors with smallest amplitude are in shades of grey
Figure 3.54
Amplitude distribution of the ‐component of the full field of standing Alfvén waves excited by a pulsed source near the ionosphere, in the horizontal plane (). Possible view of the Aureole‐3 satellite trajectory (dark gray line) in the MASSA experiment relative to the region occupied by the Alfvén oscillations
Figure 3.55
Behaviour of the ‐component of the full field of Alfvén oscillations onboard the satellite crossing the equatorial boundary of the ‘butterfly wing’, in the strong decay case. Variants (1–6) correspond to the satellite crossing the boundary at time moments (3.413)
Figure 3.56
Model axisymmetric magnetic field with elongated field lines formed by the vector sum of the dipole magnetic field and the field of the axisymmetric current sheet. The systems of coordinates used in the calculations: () – orthogonal and () non‐orthogonal curvilinear systems of coordinates, () – cylindric system of coordinates
Figure 3.57
Shape of magnetic field lines calculated from (3.421), in a geotail model with a thin current sheet
Figure 3.58
Distribution isolines for: (a) Alfvén speed (km/s); (b) sound speed (km/s), in the meridional plane, in the geotail model with a thick current sheet. Lines 1 in panel (b) indicate the meridional sections of the magnetic field‐line inflexion surfaces
Figure 3.59
Graphical solution of dispersion equation (3.428). The solution is determined by the points where the parabola crosses the straight lines corresponding to the right‐hand side of (3.428), for different ratios between the parameters. Straight line 1 corresponds to solutions for neutral poloidal Alfvén and azimuthally small‐scale SMS waves; straight line 2 to the neutral Alfvén and aperiodically unstable SMS wave; straight line 3 to neutral Alfvén and SMS waves, in a force‐free magnetic field
Figure 3.60
Eigen frequency distribution for first odd harmonics of azimuthally small‐scale standing Alfvén waves ( – solid thick lines) and Eigen frequency ( – solid thin lines) and growth rate ( – dashed lines) distribution for standing SMS waves, calculated in the local approximation from the dispersion equation (3.428): (a) for the geotail model with a thin current sheet and (b) for the model with a thick current sheet
Figure 3.61
Distribution of parameter along the field line located on magnetic shell , in models with a thick (thick line) and a thin (thin line) current sheet. in the equatorial plane
Figure 3.62
Distribution of the wavenumber squared, along different field lines ( – solid lines, – dashed line), for the main harmonic () of standing poloidal Alfvén waves, in the thin current sheet model. Neutrally stable oscillations on magnetic shells: (1) – , (2) – ; (3) – ; (4) – unstable () periodic oscillations on magnetic shell
Figure 3.63
Eigen frequency distribution for the first five harmonics of standing poloidal ( – thin solid lines, – dashed lines) and toroidal ( – thick solid lines) Alfvén waves across magnetic shells: (a) for the geotail model with a thin current sheet and (b) for the model with a thick current sheet
Figure 3.64
Eigen frequency distribution for the first five harmonics of standing azimuthally small‐scale SMS waves ( – solid lines, – dashed lines), across magnetic shells, in the model geotail with a thin current sheet
Figure 3.65
Longitudinal structure of the main harmonic () of unstable poloidal Alfvén waves (A) on magnetic shell and, for comparison, the structure of the SMS wave (SMS) of the same frequency (which fails to satisfy the boundary conditions on the ionosphere). Gaps in the Alfvén wave structure correspond to the neighbourhoods of turning points (where )
Figure 3.66
Distribution along the field line of the scalar potential of the electric field of the main () and first () harmonics of the coupled modes: (a, b) in the inner magnetosphere, at shell ; (c, d) in the current sheet region, at magnetic shell
Figure 3.67
Distribution along the field line of the electromagnetic field components () of the coupled modes main harmonic (): (a–c) in the inner magnetosphere, at shell ; (d–f) in the current sheet region, at magnetic shell
Figure 3.68
Distribution across magnetic shells of the eigenfrequencies of the main () and first () harmonics of toroidal Alfvén (thick dashed lines) and coupled (thick solid lines) modes of MHD oscillations, as well as their growth rates (thin solid lines) in the current sheet region. The transition layer region is shown in grey
Figure 3.69
Parameter distribution along the field line, for the main harmonic of coupled modes (), on magnetic shells (line 1) and (line 2)
Figure 3.70
Azimuth variations for saddle points () in (3.452), from to and integration contours for the solutions of (3.447) for
Figure 3.71
Coupled Alfvén and SMS mode structure across magnetic shells, near the resonance surface for poloidal Alfvén waves: (a) oscillations with a growth rate exceeding their decrement due to dissipation in the ionosphere (); (b) oscillations with a lower instability growth rate than the decrement ()
Figure 4.1
(a) Characteristics (energy flux lines) of transverse small‐scale Alfvén waves for a fixed sign of , in an axisymmetrical model magnetosphere. PT‐type characteristics only are present. (b) Characteristics in a 3D‐inhomogeneous model magnetosphere. The types of characteristics,
1– 6
, are described in the text. Asymmetry can be seen in the behaviour of the characteristics in sectors – and –. Both PT‐ (numbered
1– 4
) and TT‐type characteristics (numbered
5
and
6
) are present
Figure 4.2
The inclination of the channel with respect to the coordinate surfaces
Figure 4.3
The structure transverses the channel of and electromagnetic components of the wave in vicinity of the resonant point (corresponding ) for (a) and (b). The structure was calculated for value
