The application of Markov theory to runs of high waves is examined using extensive computer simulations of random Jonswap 3.3 seas. Results show direct application when account is taken of the variation of the Markov coefficient with threshold level defining high waves. The variation is shown to be consistent with theoretical results from Markov and envelope descriptions of wave runs.
Markov theory of wave runs in random seas involves the basic assumption that statistical correlation exists only with immediately preceding waves. The theory has been discussed previously by Kimura (1980), Longuet-Higgins (1984), Dawson et al. (1996) and Dawson (1997), among others. High waves in the theory are considered to be those with heights at or above a specified threshold level. The probability P+ that a high wave is followed by a high wave and the probability P_ that a low wave is followed by a low wave are expressible as (Dawson et al., 1996)
P+=I-C(1-P), P_ =I-CP (1)
where P denotes the Rayleigh probability and C denotes the Markov coefficient. The transition probabilities from high to low waves and low to high waves are given by the differences 1 - P+ and 1 - P_, respectively. Values of the probabilities from equation (1) Mso allow additional calculations of run statistics such as the average run length of high waves, the average number of waves between the beginnings of such runs, and the probability of a run length of high waves of specified number. The Rayleigh probability in equation (1) measures the fraction of high waves expected to equal or exceed a specified threshold level, and the Markov coefficient measures the statistical correlation existing for a given sea and specified threshold level.