Majority theories of Langmuir circulations hitherto assume that the presence of surface waves is essential to their generation by a unidirectional wind. Without surface waves, the linear theory of instability predicts that a shearing current is neutrally stable. However, the growth of roll instability by distortion of the vortex lines is  a nonlinear mechanism. Accordingly, in this paper, we extend the nonlinear theory of instability, due to Orr [21] and others, so as to include unsteady flows in which the tangential stresses are prescribed on some portion of the boundary. The theory is applied to an accelerated surface current, and it is found that for any given spacing L of the rolls is a series of possible modes of instability, each member of the series having a corresponding number m of circulating cells in the vertical direction. The lowest mode, having just one cell in the vertical direction, is the most unstable. For each mode, there is a critical Reynolds number  for instability. As Reynolds numbers exceed this critical value, the rate of growth is a function of the wavenumber  and there exists one wavenumber for which the growth rate is maximum. Comparisons are made with laboratory experiments on the initial water motion generated by wind (Melville et al. [19]) in which the flow was observed to be always laminar during the critical stage of development. This ensures that the theory is applicable and a reasonable agreement is found with experimental data. In these experiments, surface waves were also observed, but the r.m.s. slopes were demonstrably too small for the waves to have a significant effect on the initial growth of the rolls. On the other hand, the rolls are affected by the growth of the waves at a later stage of their development.

Langmuir circulation, wind waves, hydrodynamic stability.