Waveform and envelope field statistics for waves with stochastically driven amplitudes

Abstract

The statistically steady distributions P͑log E͒ and P e ͑log E e ͒ of waveform field E and envelope field E e are studied for time-varying waves with stochastically driven amplitudes. The waves are represented in one dimension ͑1D͒ by a single mode or superposition of multiple independent modes, whose amplitudes follow stochastic differential equations. Both distributions at low fields follow power laws: P͑log E͒ ϰ E p and P e ͑log E e ͒ ϰ E e q with distinct exponents p and q. Transitions in both distributions are found between the single-mode and multimode cases, with the distributions in the latter essentially independent of the number N ͑provided N Ն 2͒ of modes. For N Ն 2, p Ϸ +1.0, q Ϸ +2.0, and both distributions agree quantitatively with independent analytic predictions. Applications to Langmuir waves observed in Earth's polar cusp ionosphere show that both distributions for N Ն 2 agree quantitatively with the respective observations, suggesting that the Langmuir waves may be 1D and have a stochastic driver.

Electric field statistics and modulation characteristics of bursty Langmuir waves observed in the cusp

“…Cairns et al [2008] show analytically that the resulting distribution is a power law with index +1 for low values of the field. Li et al [2010] confirm this effect with numerical simulations.…”

“…Its detection confirms that the modulations arise from the combination of the narrowband large amplitude signals comprising the spectrum shown in Figure 3a. This subtle effect has also been observed in Langmuir waves in the electron and ion foreshocks [ Sigsbee et al , 2005] and in theoretical calculations [ Li et al , 2010]. Its detection depends on having waveform data, preferably significantly oversampled as in the TRICE observations.…”

“…These statistics provide key clues about the nature of the modulated waves. For example, recently Li et al [2010] made predictions of the wave statistics expected for superpositions of independent wave modes, which can be tested experimentally using TRICE data. Two types of statistical analyses of the wavefields are considered here: statistics of the absolute values of the wavefields and statistics of the envelopes of the fields.…”

“…Section 2 describes instrumentation and flight parameters. Section 3.1 presents spectral and waveform data of observed Langmuir wave packets with emphasis on wave modulation, and section 3.2 presents corresponding wave and envelope field statistics which, when combined with recent theoretical advances [ Li et al , 2010], provide an experimental test of the nature of the modulations. Section 4 discusses the implications of the measurements for understanding beam‐generated Langmuir waves in general and cusp waves in particular.…”

Statistics of electric fields' amplitudes in Langmuir turbulence: A numerical simulation study

“…The study of the statistics of electric fields' amplitudes became a more and more popular tool in the investigation of Langmuir wave turbulence in space plasmas. For instance, the statistics of electric fields' measurements were presented in cusp regions [ LaBelle et al , ], in the vicinity of the Earth electron foreshock [ Bale et al , ; Kellogg et al , ], and in the solar wind [ Li et al , ; Reid and Kontar , ]. Small‐amplitude waves present distributions related to linear instability properties, while the large‐amplitude waves' statistics is influenced by the presence of nonlinear processes.…”