Effect of atmospheric stability on the growth of surface gravity waves

In this paper we study the effect of atmospheric stability on the growth of surface gravity waves. To that end we numerically solved the Taylor-Goldstein equation for wind profiles which deviate from a logarithmic form because stratification affects the turbulent momentum transport. Using Charnock's relation for the roughness height z₀ of the wind profile, it is argued that the growth rate of the wave depends on the dimensionless phase velocity c/u_* (where u_* is the friction velocity) and a measure of the effect of atmospheric stability, namely the dimensionless Obukhov length gL/u_*², whereas it only depends weakly on gz_t/u_*² (where z_t is the roughness height of the temperature profile). Remarkably for a given value of u_*/c, the growth rate is larger for a stable stratification (L > 0) than for an unstable one (L < 0). We explain why this is the case. If, on the other hand, one considers the growth rate as a function of c/U₁₀ (where U₁₀ is the windspeed at 10 m), the situation reverses for c/U₁₀ < 1. For practical application in wave prediction models, we propose a new parameterization of the growth rate of the waves which is an improvement of the Snyder et al. (1981) proposal because the effect of stability is taken into account.