Waveform and envelope field statistics for waves with stochastically driven amplitudes

The statistically steady distributions P(log E) and Pe(log Ee) of waveform field E and envelope field Ee are studied for time-varying waves with stochastically driven amplitudes. The waves are represented in one dimension (1D) by a single mode or superposition of multiple independent modes, whose amplitudes follow stochastic differential equations. Both distributions at low fields follow power laws: P(log E)∝Ep and Pe(log Ee)∝Eeq 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≥2) of modes. For N≥2, p≅+1.0, q≅+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≥2 agree quantitatively with the respective observations, suggesting that the Langmuir waves may be 1D and have a stochastic driver.