Study of the parametric oscillator driven by narrow-band noise to model the response of a fluid surface to time-dependent accelerations

Abstract

A stochastic formulation is introduced to study the large amplitude and high-frequency components of residual accelerations found in a typical microgravity environment (or g-jitter). The linear response of a fluid surface to such residual accelerations is discussed in detail. The analysis of the stability of a free fluid surface can be reduced in the underdamped limit to studying the equation of the parametric harmonic oscillator for each of the Fourier components of the surface displacement. A narrow-band noise is introduced to describe a realistic spectrum of accelerations, that interpolates between white noise and monochromatic noise. Analytic results for the stability of the second moments of the stochastic parametric oscillator are presented in the limits of low-frequency oscillations, and near the region of subharmonic parametric resonance. Based upon simple physical considerations, an explicit form of the stability boundary valid for arbitrary frequencies is proposed, which interpolates smoothly between the low frequency and the near resonance limits with no adjustable parameter, and extrapolates to higher frequencies. A second-order numerical algorithm has also been implemented to simulate the parametric stochastic oscillator driven with narrow-band noise. The simulations are in excellent agreement with our theoretical predictions for a very wide range of noise parameters. The validity of previous approximate theories for the particular case of Ornstein–Uhlenbeck noise is also checked numerically. Finally, the results obtained are applied to typical microgravity conditions to determine the characteristic wavelength for instability of a fluid surface as a function of the intensity of residual acceleration and its spectral width.