136 



Fishery Bulletin 97(1), 1999 



30'N 



28'N 



26'N 



24'N 



22'N 



20-N 



30'N 



28"N 



26"N 



24'N 



- n 22"N 



20'N 



180' 



izyw 



174*W 



17rW 



168'W 



165"W 



162'W 



159'W 



156'W 



Figure 4 



Sea surface height and geostrophic current estimateci from a 10-day TOPEX/POSEIDON cycle, 16-26 August 

 1993. Only currents 8 cm/sec or larger are plotted with arrows. Solid circles from lower right (east) to upper left 

 (west), denote Oahu, Necker, Maro, and Midway Islands, respectively, 



X and V = the larval position in degrees of longi- 

 tude and latitude respectively; 

 u and I' = longitudinal and latitudinal compo- 

 nents of velocity in degrees/day; 

 e = a normal (mean 0, standard deviation 



1) random variate; and 

 D - the eddy diffusion rate in units of degree"/ 

 day ( 1 m-/sec = 7 x 10"*' degrees/day). 



The first term in brackets in each equation is advec- 

 tive displacement and the second term, diffusional 

 displacement. The cosine function in the longitudi- 

 nal movement equation corrects for the fact that dis- 

 tance per degree of longitude decreases from the 

 equator to the poles. 



Values of i/ and v at particular locations and times 

 were obtained by linear interpolation from the 

 precalculated 0.5 degree by 10-day grid of;/ and v val- 

 ues computed from the T-P data. The time step was set 

 to one day (A^=l) with 365 iterations for one year of 

 simulation. The 1-day time step was used to match the 

 spatial resolution of the velocity field, but simulations 

 showed that the choice of time step was robust up to 

 about two weeks. Larvae dispersed in the model be- 

 cause a different diffusional displacement (a different 

 e) was chosen for each larva at each time step. Once 

 larvae were even slightly separated, dispersal was fur- 

 ther advanced because the larvae experienced differ- 

 ent advective displacements as u and i' varied spatially. 



Simulations 



Although lobsters presumably spawn on all banks 

 in the Hawaiian Archipelago, for simplicity, the simu- 



lations released larvae from only four banks: Mid- 

 way and Oahu (chosen as representative of the north- 

 western and southeastern ends of the archipelago) 

 and Necker and Maro (representative of the main 

 fishing banks [Fig. 1] ). Each simulation released 5000 

 larvae at the beginning of the month from each of 

 the four banks for the four months of the summer 

 spawning season and the two months of the winter 

 spawning season from 1993 to 1995. The simulations 

 tracked the positions of all released larvae for 365 

 days; no larval mortality was assumed. For example, 

 Figure 5 shows the distribution of simulated larvae 

 180 and 365 days after release at Maro Reef in July 

 1995. The distribution showed patchiness caused by 

 underlying oceanographic features and expanded 

 spatially over time. This simulation was based on 

 an eddy diffusion rate of 1000 m"/sec, and the same 

 simulation, but with the diffusion rate reduced to 

 100 m-/sec, showed a striking increase in patchiness 

 and the same general dispersal (Fig. 6). The eddy 

 diffusion rate for the NWHI is not known, but a range 

 from 100 m-/sec to 1000 m-/sec has been proposed 

 for another large archipelago, the Great Barrier Reef 

 (Gabric and Parslow, 1994). Because the level of dif- 

 fusion rate impacts larval patchiness and because 

 we had a measure of actual larvae patchiness from 

 numerous larval tows throughout the archipelago 

 (Polovina and Moffitt, 1995), we decided the appro- 

 priate level for eddy diffusion rate by comparing the 

 frequency distribution of larvae sampled in actual 

 lai-val tows with the frequency distribution of larvae 

 from line transects and the simulated spatial pat- 

 terns for different eddy diffusion-rate values. The 

 frequency distribution from transects with an eddy 



