Y

from which we get k2 - 0, k| - Y fu as d ^<x>. (2.67)

So, for a fixed scale there is an upper limit for the unstable mode wavenumber and hence a lower limit for the possible wavelengths of the spatial patterns. Figure 2.13(b) illustrates a typical case for the Thomas (1975) system given by (2.8).

With all kinetics parameters fixed, each parameter pair (d,y) defines a unique parabola h(k2) in (2.62), which in turn specifies a set of unstable modes. We can thus consider the (d,y) plane to be divided into regions where specific modes or a group of modes are diffusively unstable. When there are several unstable modes, because of the form of the dispersion relation, such as in Figure 2.5(b), there is clearly a mode with the largest growth rate since there is a maximum Re X for some km say. From (2.23), the positive eigenvalue X+(k2) is given by

2X+(k2) = y(fu + gv) - k2(1 + d) + {[y(fu + gv) - k2(1 + d)]2 - 4h(k2)}1/2

which has a maximum for the wavenumber km given by

Figure 2.13. (a) Isolation of unstable modes (that is, h(k2) < 0 in (2.23)) by setting the diffusion ratio d = dc + e, 0 < e ^ 1 and varying the scale y for the Thomas (1975) kinetics (2.8) with a = 150, b = 100, a = 1.5, p = 13, K = 0.05, d = 27.03: the critical dc = 27.02. (b) The effect of increasing d with all other parameters fixed as in (a). As d the range of unstable modes is bounded by k2 = 0 and k2 = y fu.

Figure 2.13. (a) Isolation of unstable modes (that is, h(k2) < 0 in (2.23)) by setting the diffusion ratio d = dc + e, 0 < e ^ 1 and varying the scale y for the Thomas (1975) kinetics (2.8) with a = 150, b = 100, a = 1.5, p = 13, K = 0.05, d = 27.03: the critical dc = 27.02. (b) The effect of increasing d with all other parameters fixed as in (a). As d the range of unstable modes is bounded by k2 = 0 and k2 = y fu.

fv gu

As we have noted the prediction is that the fastest growing km -mode will be that which dominates and hence will be the mode which evolves into the steady state nonlinear pattern. This is only a reasonable prediction for the lower modes. The reason is that with the higher modes the interaction caused by the nonlinearities is more complex than when only the simpler modes are linearly unstable. Thus using (2.68) we can map the regions in (d ,y) space where a specific mode, and hence pattern, will evolve; see Arcuri and Murray (1986). Figures 2.14(a) and (b) show the mappings for the Thomas (1975)

Was this article helpful?

0 0

Post a comment