Tables of Fourier Transforms and Fourier Transforms of Distributions

Oberhettinger, Fritz

doi:10.1007/978-3-642-74349-8

Cited by 99 publications

(52 citation statements)

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“…The relaxation time found using this formulation is | ln η| 2 and so is in agreement with that found by Craig & McClymont (1991;1993). Craig et al (2005) comment that it is surprising that fluid viscosity has been neglected in previous studies, and state that the leading terms in the viscous stress tensor actually dominate the plasma resistivity by many orders of magnitude for typical coronal plasmas (as emphasised by Hollweg 1986).…”

Section: Why Is This Area Of Study Interesting or Important?supporting

confidence: 86%

“…The first investigation that specifically made the extension away from polar symmetry was performed by Ofman (1992) and Ofman et al (1993), who performed nonlinear, resistive 2D MHD calculations of the reconnection in the stressed null point and obtained the | ln η| 2 scaling law numerically, as well as solving the linear dispersion relation (equation 15) for all azimuthally nonsymmetric perturbations (m > 0) analytically, in agreement with the results of Craig & McClymont (1991;1993) and Hassam (1992). However, the authors also stressed the significance of the choice of boundary condition, and the strong influence they have on the permissible solutions.…”

Section: Why Is This Area Of Study Interesting or Important?supporting

confidence: 58%

“…Again, this result was the first to indicate a relationship between Alfvén wave propagation and the location of the separatrices. An alternative approach to the study of MHD wave behaviour in the neighbourhood of a 2D magnetic null point was investigated in a series of papers by Craig and co-workers (Craig & McClymont 1991;1993;Craig & Watson 1992) in which the authors considered perturbations of the flux function Az. Here, the focus was on applications for magnetic reconnection, specifically to investigate if null points (viewed as weaknesses in the magnetic field) could collapse in response to imposed boundary motions.…”

Section: Why Is This Area Of Study Interesting or Important?mentioning

confidence: 99%

“…Hassam performed a similar derivation to that of equation (13) and recognised that equation (15) can be recast as a hypergeometric equation (e.g. Oberhettinger 1990) when m = 0:…”

Section: Why Is This Area Of Study Interesting or Important?mentioning

confidence: 99%

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Review Article: MHD Wave Propagation Near Coronal Null Points of Magnetic Fields

McLaughlin¹,

Hood²,

Moortel³

2010

Space Sci Rev

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We present a comprehensive review of MHD wave behaviour in the neighbourhood of coronal null points: locations where the magnetic field, and hence the local Alfvén speed, is zero. The behaviour of all three MHD wave modes, i.e. the Alfvén wave and the fast and slow magnetoacoustic waves, has been investigated in the neighbourhood of 2D, 2.5D and (to a certain extent) 3D magnetic null points, for a variety of assumptions, configurations and geometries. In general, it is found that the fast magnetoacoustic wave behaviour is dictated by the Alfvén-speed profile. In a β = 0 plasma, the fast wave is focused towards the null point by a refraction effect and all the wave energy, and thus current density, accumulates close to the null point. Thus, null points will be locations for preferential heating by fast waves. Independently, the Alfvén wave is found to propagate along magnetic fieldlines and is confined to the fieldlines it is generated on. As the wave approaches the null point, it spreads out due to the diverging fieldlines. Eventually, the Alfvén wave accumulates along the separatrices (in 2D) or along the spine or fan-plane (in 3D). Hence, Alfvén wave energy will be preferentially dissipated at these locations. It is clear that the magnetic field plays a fundamental role in the propagation and properties of MHD waves in the neighbourhood of coronal null points. This topic is a fundamental plasma process and results so far have also lead to critical insights into reconnection, mode-coupling, quasi-periodic pulsations and phase-mixing.

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Section: Why Is This Area Of Study Interesting or Important?supporting

confidence: 86%

Section: Why Is This Area Of Study Interesting or Important?supporting

confidence: 58%

Section: Why Is This Area Of Study Interesting or Important?mentioning

confidence: 99%

“…Hassam performed a similar derivation to that of equation (13) and recognised that equation (15) can be recast as a hypergeometric equation (e.g. Oberhettinger 1990) when m = 0:…”

Section: Why Is This Area Of Study Interesting or Important?mentioning

confidence: 99%

See 2 more Smart Citations

Review Article: MHD Wave Propagation Near Coronal Null Points of Magnetic Fields

McLaughlin¹,

Hood²,

Moortel³

2010

Space Sci Rev

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“…TWO-DIMENSIONAL NETWORKS Our approach for two-dimensional networks is completely different from the approach followed in the one-dimensional case. Note that the approach that leads to the tight upper bound for one-dimensional networks heavily depends on the existence of an explicit formula (15) for the determinant in (13) in the special case δ = 1. This allows to study the configuration of nodes that maximize this determinant, in terms of which we are able to upper bound the behavior of (9) for any random configuration.…”

Section: Proof Of Theorem 21: Part IImentioning

confidence: 99%

Scaling Laws for One- and Two-Dimensional Random Wireless Networks in the Low-Attenuation Regime

Özgür¹,

Lévêque

Preissmann

2007

IEEE Trans. Inform. Theory

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The capacity scaling of extended two-dimensional wireless networks is known in the high attenuation regime, i.e. when the power path loss exponent α is greater than 4. This has been accomplished by deriving information theoretic upper bounds for this regime that match the corresponding lower bounds. On the contrary, not much is known in the so-called low attenuation regime when 2 ≤ α ≤ 4. (For one-dimensional networks, the uncharacterized regime is 1 ≤ α ≤ 2.5.) The dichotomy is due to the fact that while communication is highly power-limited in the first case and power-based arguments suffice to get tight upper bounds, the study of the low attenuation regime requires a more precise analysis of the degrees of freedom involved. In this paper, we study the capacity scaling of extended wireless networks with an emphasis on the low attenuation regime and show that in the absence of small scale fading, the low attenuation regime does not behave significantly different from the high attenuation regime.

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Analytic Bases for Approximation

Talman

2007

Geometric Mechanics

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Tables of Fourier Transforms and Fourier Transforms of Distributions

Cited by 99 publications

References 0 publications

Review Article: MHD Wave Propagation Near Coronal Null Points of Magnetic Fields

Review Article: MHD Wave Propagation Near Coronal Null Points of Magnetic Fields

Scaling Laws for One- and Two-Dimensional Random Wireless Networks in the Low-Attenuation Regime

Analytic Bases for Approximation

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