Shear dispersion in the surface layers of the sea
The kinetic energy density, k, and lifetime, t, of a turbulent eddy, volume V, are shown to be related to a characteristic length scale, 2, by: 13 =vk= c2/3e2/3 t-C 1/312/3 where c is the energy dissipation rate. A self similar cascade of discrete eddy sizes is derived, each size related to the next larger by: P 22 CC i+l i where C= 81'2. With some simple assumptions as to the turbulent production process the mean logarithmic velocity profile is derived. The relationship between the friction velocity and Reynolds stress is explained in terms of the large eddy intermittency, n. Below a critical free stream velocity, U', n is proportional to the free stream velocity. The dissipation rate, e, is then constant and given by: e U'3/N3L where N is the number of discrete eddy sizes in the boundary layer. When the boundary layer has reached the surface N- 11. The observed turbulent spectral characteristics are derived from the eddy equations without using dimensional reasoning and an explanation of the mechanism behind surface layer similarity scaling is proposed. An experiment was carried out in the North Sea to test the model predictions. Correlations showed that, except at slack water, the largest eddies were approximately cubic, occupying the whole flow depth, and were advected with the mean flow. Frequency spectra provided evidence that the cascade formulation was correct. The turbulent intermittency was proportional to the current speed and the value of c, calculated by several methods, was found to be constant with a value: C=0.3 cm2/s3 The characteristics of the largest eddies were isolated using a spectral cropping technique and plotted as a phase portrait of the turbulent strange attractor. This demonstrated that the boundary layer sat at preferred, discrete energy levels. The levels observed could be related to the discete cascade model. A computer code based on the model equations was tested against a series of large scale oil and dye releases in the North Sea. The observed intermittency, meandering, and dispersion were well simulated with the value of c given above.