Statistical theories of Langmuir turbulence. I. Direct-interaction-approximation responses

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Abstract
A statistical theory of Langmuir turbulence is developed for the equations of Zakharov in the approximation formally equivalent to the direct-interaction approximation (DIA). At this level the theory is realizable, conserves the mean values of the three invariants of the Zakharov equations, contains the modulational instability, and goes over continuously to weak-turbulence theory in the proper limit. This paper concentrates on the properties of the electrostatic and ion-density response functions which arise in the theory; a new continued-fraction representation is given which is numerically more efficient than direct integration of the DIA equations. These response functions contain a new nonlinear dispersion relation for the modulational instability of a fluctuating Langmuir spectrum. Comparison is made between the continued-fraction representation, direct integration of the DIA, and a Monte Carlo simulation. The noninvariance of the DIA to random gauge transformation is discussed as well as the saturation of the modulational instability by spectral broadening.