Some consequences of the boundedness of scalar fluctuations

Values of the scalar field c(x,t), if initially bounded, will always be bounded by the limits set by the initial conditions. This observation permits the maximum variance ∼(c′2) to be computed as a function of the mean value c̄. It is argued that this maximum should be expected in the limit of infinite Schmidt numbers (zero scalar species diffusivity). This suggests that c′/ c̄ on the axis of turbulent jets, for example, may not tend to a constant, i.e., independent of x/d, in the limit of very large Schmidt numbers. It also underscores a difficulty with the k−1 scalar spectrum proposed by Batchelor [J. Fluid Mech. 5, 113 (1959)].