Linear refraction of waves on inhomogeneous current is known to provoke extreme waves. We investigate the effect of nonlinearity on this phenomenon, with respect to the variation of significant wave height, kurtosis and occurrence of freak waves. Monte Carlo simulations are performed employing a modified nonlinear Schrödinger equation that includes the effects of a prescribed non-potential current. We recommend that freak waves should be defined by a local criterion according to the wave distribution at each location of constant current, not by a global criterion that is either averaged over, or insensitive to, inhomogeneities of the current. Nonlinearity can reduce the modulation of significant wave height. Depending on the configuration of current and waves, the kurtosis and probability of freak waves can either grow or decrease when the wave height increases due to linear refraction. At the centre of an opposing current jet where waves are known to become large, we find that freak waves should be more rare than in the open ocean away from currents. The largest amount of freak waves on an opposing current jet is found at the jet sides where the significant wave height is small.
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