where K is the growth rate. On the right-hand side of the equation, the first term accounts for diffusion, while the second represents linear growth. A complicated algebraic solution led the authors to define a critical length (Lc) for the water mass, under which the population would decrease and above which it could increase:
It must be noted that this equation is analogous to that of the critical mass in a nuclear reactor. Associated with this critical length is a characteristic time (t) of the process, after which the critical length Lc becomes operative:
The above results are those given in the paper of Kierstead and Slobodkin. The same problem is now approached by means of dimensional analysis, which will allow one to compare the dimensional solution of Platt (1981) to the algebraic solution of Kierstead and Slobodkin. In order to approach the question from a dimensional point of view, the dimensions of variables in the problem must first be specified:
The only dimensions that are not immediately evident are those of D, but these can easily be found using the principle of dimensional homogeneity of theoretical equations.
The equation of Kierstead & Slobodkin involves three variables (c, t, x) and two constants (D, K). According to the general method developed in the previous ecological application, the variables are first transformed to dimensionless forms, through division by suitable characteristic values. Dimensionless variables C, T and X are defined using characteristic values c* , t* and x* :
Substitution of these values in the equation gives:
The next step is to make all terms in the equation dimensionless, by multiplying each one by x1 and dividing it by D, after eliminating from all terms the common constant c* :
The resulting equation thus contains three dimensionless variables (C, T and X) and two dimensionless products (in brackets).
Since the dimensions of the two products are , these may be transformed to isolate the
= , it follows that [x1] = [D] and thus [x*] = D
Using these relationships, the following proportionalities are obtained:
Dimensional analysis thus easily led to the same results as those obtained by Kierstead and Slobodkin (1953), reported above, except for the constant factors n and 8n2. This same example will be reconsidered in the next section (Ecological application 3.3a), where the two dimensionless products will be calculated directly.
Was this article helpful?