Diffusive Scaling meaning

Dear community,

I actually understand the general concept of working under conditions where the diffusive scales predomine over the convective scale.

Also I assume that:
Having a model which needs tMax time steps to reach converge. If I change my grid step (for example refining a level my grid, which means multiply by aprox. 4 the total number of nodes) this leads that I will now need more timesteps to reach the same numerical solution(, actually, 2 times more which is the square root of the times that I have increase my nodes).

having all this in mind, my concern is on what does the expression “work under diffusive scaling” or “diffusive scaling is necessary for the convergence of this model”.