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Energy Sources and Global Circuit
The development and maintenance of resonance phenomena in planetary cavities require suitable electromagnetic sources. On Earth, lightning is the most important excitation source for Schumann resonances; it has been extensively studied in relation with atmospheric electricity, weather phenomena, and hazard prevention. Most lightning discharges on Earth are produced by precipitating clouds that contain water in solid and liquid phases. The two major requirements for lightning generation are: (i) interaction between particles of different types or between particles of the same type, but with different thermodynamic properties; (ii) significant spatial separation of the oppositely charged particles by convection or gravitational forces. In terrestrial lightning, the cloud-to-ground discharges are the most studied because they are the strongest and easiest to detect. However, it is not certain this type of lightning events occurs in other planetary atmospheres. On Earth, lightning-like discharges produced by dust are also possible, though they release less energy per stroke. Electrical sparks, hundreds of meters long, can be produced via charge generation and separation in volcano eruptions (Rakov and Uman, 2003). Turbulent meteorological phenomena associated with dust storms are also closely related with discharging processes. Magnetohydrodynamic waves have also been proposed as a complementary source to lightning for the Earth’s Schumann resonance (Abbas, 1968).
However, unlike the unambiguous lightning contribution, the role played by other sources requires further investigations.
Though not exhaustive, Table 1 summarizes some of the most important works dealing with observations, models, and global features of the Schumann resonance. Several advances in Earth cavity characterization can be applied to other planetary environments; this is one of the objectives of this thesis (Simões et al., 2007a; Simões et al., 2007b; Simões et al., 2007c) [Papers 4, 9, and 10].
The 2D axisymmetric Approximation
A cavity consisting of concentric shells, where the medium properties are functions of radial distance only, can be solved in 2D axisymmetric configuration. This approach minimizes memory and time requirements and provides accurate solutions. There is no particular constraint regarding eigenfrequency analysis; the time-harmonic propagation studies require the utilization of a vertical dipole along the axis of symmetry (Simões and Hamelin, 2006) [Paper 8]. The eigenfrequency analysis does not require a very fine mesh, and the cavity is composed of about 5× 104 elements, which also provides a reasonable accuracy in the time harmonic propagation mode. Figure 4 shows a typical 2D axisymmetric geometry with a vertical dipole aligned with the axis of symmetry. Equations (08-10) can be simplified in 2D axisymmetric geometry (r, ϕ, z) when there is no variation with the angle ϕ. Considering E(r,z)=Er(r,z)êr+Eϕ(r,z)êϕ +Ez(r,z)êz and H(r,z)=Hr(r,z)êr+Hϕ(r,z)êϕ +Hz(r,z)êz, where (êr, êϕ , êz) represent the unit vectors, Equations (08-10) can be written 19 1 ∇ × μ and ∇ × ε iσ ω 2 ∇ × E( r,z ) − ε − E( r,z ) = 0 ωεo c iσ −1 ω 2 − ∇ × H( r,z ) μ H( r,z ) = 0 , − ωεo c (18).
where μ is the relative magnetic permeability. The eigenmode analysis uses Equations (08-10) written in the form 1 ∇ × ∇ × E( r,z ) − ΛS ε E( r,z ) = 0 (20) μ and 1 ∇ × ∇ × H( r,z ) − ΛS μ H( r,z ) = 0 , (21).
where ΛS denotes the eigenvalues, which are generally complex. In the case of TE and TM waves, the electric and magnetic field vectors can be simplified because TE ⇒ Er=Ez=Hϕ=0 and TM ⇒ Hr=Hz=Eϕ=0.
Eigenfrequency Analysis
The eigenmode solver uses the ARPACK package that is based on a variant of the Arnoldi algorithm and is usually called the implicitly restarted Arnoldi method (Zimmerman, 2006). Additional information can be found in the COMSOL Multiphysics user guide and model library documentation. For some combinations of the model and medium parameters, the eigenfrequency problem may not be linear, which means that the eigenvalue appears in the equations in a different way than the expected second-order polynomial form. Equation (25) with finite conductivity is a typical problem with a nonlinear solution. In this case, the equation is solved in several steps: an initial guess is made for the eigenvalue; the equation is solved and a new eigenvalue is found; the eigenvalue is updated and the new equation is solved; the cycle is repeated until eigenvalues converge. In general, this procedure converges rapidly unless the wave attenuation is significant. Whenever the medium is lossy, i.e. σ≠0, the eigenvalues are complex; the real and imaginary parts characterize the wave propagation and attenuation, respectively.
Time harmonic Propagation Analysis
The propagation mode solves stationary problems with the tools supplied by the UMFPACK package. The harmonic propagation code computes the frequency spectra, identifies the propagating eigenmodes, calculates the electric field over a wide altitude range and evaluates the influence of the source distribution on the propagation modes. The solver employs the unsymmetrical-pattern multifrontal method and the direct LU-factorization of the sparse matrix obtained by discretizing Equations (18-19) or (22-23).
The harmonic propagation approach is especially suited to the analysis of global features and to the study of the electromagnetic field distribution generated by sparse sources. For the sake of simplicity, we shall assume a localized electromagnetic stimulus. The source is a pulsating Hertz dipole approximated by two spheres, on the surface of which a uniform surface current density is imposed. The intensity of the current is not important provided its density is uniform and its frequency spectrum flat. The dipole size is small compared to the wavelength and the cavity size. The algorithm calculates the frequency response of the cavity, in the specified frequency range. It is also possible to use a monopole, with a specified field distribution, that minimizes meshing requirements.
Accuracy and Numerical Solvers
Meshing is an important step in the finite element method towards the solution of the numerical model. The selection of appropriate meshes minimizes memory needs, optimizes accuracy and, in the case of intricate geometries, may improve convergence efficiency. A free mesh consisting of triangular elements is chosen for the 2D axisymmetric geometry. A swept mesh structured over the angular direction is used for the eigenmode problem. The 3D geometry is meshed with tetrahedral elements with a resolution compatible with the hardware and software capabilities, i.e. nearly 106 elements and ~8×106 degrees of freedom. The models are run in a dual core, dual processor, 16 Gbytes RAM station. A comparison between the 2D and 3D results, when axial symmetry applies validates the model accuracy. Continuity conditions are imposed at the surface of the body unless the latter coincides with the inner boundary of the cavity.
The numerical algorithms have been validated by comparing the eigenfrequencies computed with the finite element model and those derived from Equation (11), taking Earth and Titan as examples. In the case of a thin lossless cavity, the same results are obtained with Equations (02), Equation (11), and the finite element method (Table 4, Test A). The analytical and numerical results are similar, as long as the medium is lossless and homogeneous, and the PEC boundary conditions apply (Table 4, Test B). However, the two approaches give different results when the medium is heterogeneous (Table 4, Test C), which illustrates the limited validity of the analytical approximation.
Table of contents :
1. Introduction .
2. Wave Propagation and Resonances in Ionospheric Cavities
2.1. The Resonant Cavity
2.1.1. Basic Description
2.1.2. General Formalism
2.2. Energy Sources and Global Circuit
3. Numerical Model for Ionospheric Planetary Cavities
3.1. Numerical Tool
3.1.1. The 2D axisymmetric Approximation
3.1.2. The 3D Model
3.1.3. Boundary and Continuity Conditions
3.1.4. Eigenfrequency Analysis
3.1.5. Time harmonic Propagation Analysis
3.1.6. Accuracy and Numerical Solvers
3.2. Cavity Parameterization
3.2.1. Parameter Description
3.2.2. Cavity Description
3.2.2.1. Venus
3.2.2.2. Earth
3.2.2.3. Mars
3.2.2.4. Jupiter
3.2.2.5. Io and Europa
3.2.2.6. Saturn
3.2.2.7. Uranus
3.2.2.8. Neptune
3.3. Results
4. Titan Electromagnetic Environment Characterization
4.1. The Cassini-Huygens Mission
4.2. The Permittivity, Waves and Altimetry Analyzer
4.2.1. Instrument Configuration
4.2.2. Relaxation Probe
4.2.3. Mutual Impedance Probe
4.2.4. Dipole Antenna
4.2.5. Acoustic Sensor
4.2.6. Radar
4.3. Experimental Results
4.3.1. Data Synopsis
4.3.2. Relaxation Data
4.3.3. Radar Data
4.3.4. Extremely Low Frequency Spectra
4.3.5. Very Low Frequency Spectra
4.3.6. Mutual Impedance Data
4.3.7. Lightning Data
4.3.8 Acoustic Spectra
4.4. PWA Data Analysis
5. Prospective Space Instrumentation and Missions
5.1. Planetary Surfaces Dielectric Properties Measurement
5.2. Dielectric Properties of Water, Ice, and Soils
5.2.1. Polarization Mechanisms and Effects
5.2.1.1. Polarization Theory
5.2.1.2. Frequency Effects
5.2.1.3. Temperature Effects
5.2.1.4. Composition Effects
5.2.2. Water Dielectric Properties
5.2.3. Ice Dielectric Properties
5.2.4. Water and Ice Dielectric Signatures in Soils
5.3. Subsurface Permittivity Probe
5.3.1. General Description
5.3.2. Electrodes
5.3.3. Electronics
5.3.4. Signal Processing
5.3.5. Operation
5.4. Laboratory Experiments
5.4.1. Experimental Setup and Instrument Calibration
5.4.2. Soil Analogues Testing
5.4.3. Results
5.4.3.1. Gravimetric Water Content
5.4.3.2. Stratigraphic Measurements
5.5. Probe Modelling
5.6. Future Missions
6. Conclusions
7. References
