Reply to Greg Holloway

Abstract

HOLLOWAY (1993, hereafter GH) states that YAMAZAKI and KAMYKOWSKI (1991, hereafter YK) used a random walk model inconsistent with the continuity condition of an incompressible fluid. Because a spatial dependence of diffusivity induces an advectionlike effect, YK's formulation results in unmixed state from a completely mixed state. Based on his mathematical reasoning GH is correct. However, one must be careful in making comments on an immature subject since it may put constraints on creative thinking. We argue that GH's comments are not appropriate for the following reasons: (1) GH does not consider the bottom line of our work; (2) GH does not mention the theories of stochastic differential equations. As we clearly stated in YK, our original intent was to investigate the spatial dependence of the diffusivity effect from a Lagrangian point of view because previous studies traditionally were based on Eulerian modelling and almost always used a constant diffusivity coefficient. We did not propose our model as a perfect solution to bio-physical coupling problems. Also, we did not claim that the way the initial cluster of particles dispersed into the resulting trajectories or distributions (not aggregations as stated by GH) necessarily was right. Instead, we merely hoped that our model would serve as a reference point for future development. GH criticizes a minor inconsistency in our Lagrangian random walk model and does not consider how much this mathematical problem alters the final conclusions. Although GH claims that our formalism is incorrect, we disagree with his reasoning. In YK, we avoided the details of random walk theories in order to focus the application on the bio-physical coupling problem. We now mention why equation (2) of YK, namely,