ALBERT

Description: An effort is made to understand turbulence in fluid systems like the oceans and atmosphere in which the Richardson number is generally large. Toward this end, a theory is developed for turbulent flow over a flat plate which is moved and cooled in such a way as to produce constant vertical fluxes of momentum and heat. The theory indicates that in a co-ordinate system fixed in the plate the mean velocity increases linearly with height z above a turbulent boundary layer and the mean density decreases as z3, so that the Richardson number is large far from the plate. Near the plate, the results reduce to those of Monin & Obukhov. The curvature of the density profile is essential in the formulation of the theory. When the curvature is negative, a volume of fluid, thoroughly mixed by turbulence, will tend to flatten out at a new level well above the original centre of mass, thereby transporting heat downward. When the curvature is positive a mixed volume of fluid will tend to fall a similar distance, again transporting heat downward. A well-mixed volume of fluid will also tend to rise when the density profile is linear, but this rise is negligible on the basis of the Boussinesq approximation. The interchange of fluid of different, mean horizontal speeds in the formation of the turbulent patch transfers momentum. As the mixing in the patch destroys the mean velocity shear locally, kinetic energy is transferred from mean motion to disturbed motion. The turbulence can arise in spite of the high Richardson number because the precise variations of mean density and mean velocity mentioned above permit wave energy to propagate from the turbulent boundary layer to the whole region above the plate. At the levels of reflexion, where the amplitudes become large, wave-breaking and turbulence will tend to develop. The relationship between the curvature of the density profile and the transfer of heat suggests that the density gradient near the level of a point of inflexion of the density curve (in general cases of stratified, shearing flow) will increase locally as time goes on. There will also be a tendency to increase the shear through the action of local wave stresses. If this results in a progressive reduction in Richardson number, an ultimate outbreak of Kelvin–Helmholtz instability will occur. The resulting sporadic turbulence will transfer heat (and momentum) through the level of the inflexion point. This mechanism for the appearance of regions of low Richardson number is offered as a possible explanation for the formation of the surfaces of strong density and velocity differences observed in the oceans and atmosphere, and for the turbulence that appears on these surfaces. © 1970, Cambridge University Press. All rights reserved.

Description: Some experiments are described in which steady-state shearing flows are developed in stratified brine solutions contained in a cyclically continuous tank of rectangular cross-section. Over the range of overall Richardson numbers studied, the results suggest that whenever turbulent layers are present on either side of a region of fluid with a gravitationally stable density gradient, they cause erosion of this region to occur. The erosion leads to the formation of two homogeneous layers separated by a thin layer of strong density and velocity gradients. The gradient Richardson number, computed by using the velocity and density gradients in this transition layer, tends to have a value of order one. If we define an overall Richardson number Ri* by averaging the velocity and density gradients over the entire depth of fluid in the tank, we find that the non-dimensional buoyancy flux, Q, is functionally related to Ri* by an equation of the form Q = C1(Ri*)⊟1 where C1 is a constant, approximately, and Ri* ranges in value between one and thirty. To check the effect of a large variation of the molecular diffusivity coefficient on flow conditions, we ran a limited number of experiments with thermally stratified fluid. Over a restricted range, 1·0 ≪ Ri* ≪ 5·0, velocity profiles very similar to those measured in the brine-stratified experiments at like values of Ri* were obtained. This suggests that the coefficient of molecular diffusion is not an important parameter in either type of experiment. Other experiments, made in the same apparatus, describe the entrainment by a turbulent, homogeneous layer of an initially quiescent layer of fluid with a linear density gradient. The depth of the turbulent layer, D, increases with time, t, according to the relation. [formula omited] This result is consistent with that found by Kato & Phillips (1969), although the turbulent layer in the present experiment is generated in a different manner. © 1971, Cambridge University Press. All rights reserved.

Description: This paper describes a laboratory experiment designed to compare measurements with published theoretical ideas of the mixed-layer growth of a two-layer system in which the turbulence is induced by an oscillating grid. Experimental results show excellent agreement with an earlier theory by one of us (Long), in which the mixed-layer depth D* measured from a virtual origin is given by $D_{*}sim V_0^{-frac{7}{11}}K^{frac{9}{11}}t^{frac{2}{11}}$, where K is action, t is time and V0 is a characteristic velocity of the problem. The experiments also verify Long's theoretical entrainment relation E = α2Ri−7/4, where E is the entrainment coefficient and $Ri = D^3_{*}Delta b/K^2$, and Δb is the buoyancy difference between the two layers. The interfacial-layer thickness was observed to be proportional to the depth of the mixed layer, as also predicted by Long. After a certain depth, the entrainment law tends to deviate from Long's theory. The deviation may be due to wall effects.

Description: Experiments were performed to investigate some aspects of turbulence in rotating and non-rotating fluid systems where the turbulence was induced by a horizontal grid oscillating vertically. An earlier theory by the second author made use of a planar source of energy, which appeared to be similar to the energy source of the grid, in determining the characteristics of the turbulence at points some distance away. The simplicity of the theory was in the parameterization of the grid ‘action’ by a single quantity K, with dimensions and characteristics of eddy viscosity. The experimental results provide additional confirmation of the theory in the non-rotating case, and indicate the usefulness of the idealized energy source in the rotating case. In the latter, we measured the propagation of the front separating disturbed and undisturbed fluid, moving along the axis of rotation. The thickness d(t) of the disturbed region increases at first as (Kt)1/2 as in a non-rotating fluid, until the Rossby number K/Ωd2k becomes of order unity. Beyond this the disturbances are wavelike and rotationally dominated, and the thickness now increases linearly with time, yielding a speed of propagation for the front proportional to the wave speed (KΩ)1/2. Finally, the disturbances reach the bottom and the vessel is in statistical steady state. Then a region of thickness dk independent of time is found, and it contains motion that resembles ordinary, three-dimensional turbulence. dk ~ (K/Ω)1/2 is analogous to the depth of the turbulent Ekman layer H ~ (K/Ω)1/2, where K is taken as an eddy viscosity. McEwan constructed a similar rotating experiment, although with a different energy source, and observed vortices parallel to the axis of rotation, provided that the Rossby number was less than a critical value. Our observations and theory indicate that the disappearance of the vortices corresponds to h 〈 dk, where h is the total depth of the fluid. At that point, the whole tank is filled with three-dimensional turbulence. © 1983, Cambridge University Press. All rights reserved.

Description: A useful parametrical model for the vertical structure of the pressure field induced by wind blowing over a field of surface gravity waves is proposed. The model is a linear expansion in a set of empirical orthogonal functions, derived from a set of 110 complex pressure profiles computed according to the theory of Miles (1957), and provides a compact, quantitative description of those profiles. The model has been used as an element in the analysis of a body of experimental data on wave-induced atmospheric pressure fluctuations obtained by Snyder et al. (1980). © 1980, Cambridge University Press. All rights reserved.

Description: A joint experiment to study microscale fluctuations of atmospheric pressure above surface gravity waves was conducted in the Bight of Abaco, Bahamas, during November and December 1974. Field hardware included a three-dimensional array of six wave sensors and seven air-pressure sensors, one of which was mounted on a wave follower. The primary objectives of the study were to resolve differences in previous field measurements by Dobson (1971), Elliott (1972b) and Snyder (1974), and to estimate the vertical profile of wave-induced pressure and the corresponding input of energy and momentum to the wave field. Analysis of a pre-experiment intercalibration of instruments and of 30 h of field data partially removes the discrepancy between the previous measurements of the wave-induced component of the pressure and gives a consistent picture of the profile of this pressure over a limited range of dimensionless height and wind speed. Over this range the pressure decays approximately exponentially without change of phase; the decay is slightly less steep than predicted by potential theory. The corresponding momentum transfer is positive for wind speeds exceeding the phase speed. Extrapolation of present results to higher frequencies suggests that the total transfer is a significant fraction of the wind stress (0.1 to 1.0, depending on dimensionless fetch). Analysis of the turbulent component of the atmospheric pressure shows that the ‘intrinsic’ downwind coherence scale is typically an order-of-magnitude greater than the crosswind scale, consistent with a ‘frozen’ turbulence hypothesis. These and earlier data of Priestley (1965) and Elliott (1972c) suggest a horizontally isotropic ‘intrinsic ’turbulent pressure spectrum which decays as k-v where k is the (horizontal) wave-number and v is typically —2 to — 3; estimates of this spectrum are computed for the present data. The implications of these findings for Phillips’ (1957) theory of wave growth are examined. © 1981, Cambridge University Press. All rights reserved.

Description: Based on theoretical analysis and laboratory data, we proposed a unified two-parameter wave spectral model as [Formula omitted] with β and m as functions of the internal parameter, the significant slope η of the wave field which is defined as [Formula omitted] is the mean squared surface elevation, and λ0, n0 are the wavelength and frequency of the waves at the spectral peak. This spectral model is independent of local wind. Because the spectral model depends only on internal parameters, it contains information about fluid-dynamical processes. For example, it maintains a variable bandwidth as a function of the significant slope which measures the nonlinearity of the wave field. And it also contains the exact total energy of the true spectrum. Comparisons of this spectral model with the JONSWAP model and field data show excellent agreements. Thus we established an alternative approach for spectral models. Future research efforts should concentrate on relating the internal parameters to the external environmental variables. © 1981, Cambridge University Press. All rights reserved.

Description: The paper is a study of experimental data in the light of new theories of turbulence recently developed by the first author for a number of problems including flow in a pipe, boundary layer at zero incidence, atmospheric boundary layer, turbulent convection and distribution of energy in wavenumber space in decaying, isotropic turbulence. In each of these, a basic element is a ‘mesolayer’ or ‘mesoregion’ in physical space or wavenumber space which is absent in earlier theories and which intrudes between the inner and outer regions preventing the overlap assumed in the derivation of the classical results, e.g. the logarithmic profile in shear flow. The new and old theories differ both in principle and in the final results: the new ideas replace rather than modify or extend the older ones. The main purpose of this paper is to bring together accumulated evidence concerning the mesolayer theories. We believe that this evidence provides overwhelming support for the existence of the mesolayer and for its pervasive importance in problems of turbulence. © 1981, Cambridge University Press. All rights reserved.

Description: Laboratory experiments were conducted to measure the surface elevation probability density function and associated statistical properties for a wind-generated wave field. The laboratory data together with some limited field data were compared. It is found that the skewness of the surface elevation distribution is proportional to the significant slope of the wave field, §, and all the laboratory and field data are best fitted by with § defined as ([formula omitted], where ζ is the surface elevation, and λ0 is the wavelength of the energy-containing waves. The value of K3 under strong wind could reach unity. Even under these highly non-Gaussian conditions, the distribution can be approximated by a four-term Gram-Charlier expansion. The approximation does not converge uniformly, however. More terms will make the approximation worse. © 1980, Cambridge University Press. All rights reserved.