The standard κ-ϵ equations and other turbulence models are evaluated with respect to their applicability in swirling, recirculating flows. The turbulence models are formulated on the basis of two separate viewpoints. The first perspective assumes that an isotropic eddy viscosity and the modified Boussinesq hypothesis adequately describe the stress distributions, and that the source of predictive error is a consequence of the modeled terms in the κ-ϵ equations. Both stabilizing and destabilizing Richardson number corrections are incorporated to investigate this line of reasoning. A second viewpoint proposes that the eddy viscosity approach is inherently inadequate and that a redistribution of the stress magnitudes is necessary. Investigation of higher-order closure is pursued on the level of an algebraic stress closure. Various turbulence model predictions are compared with experimental data from a variety of isothermal, confined studies. Supportive swirl comparisons are also performed for a laminar flow case, as well as reacting flow cases. Parallel predictions or contributions from other sources are also consulted where appropriate. Predictive accuracy was found to be a partial function of inlet boundary conditions and numerical diffusion. Despite prediction sensitivity to inlet conditions and numerics, the data comparisons delineate the relative advantages and disadvantages of the various modifications. Possible research avenues in the area of computational modeling of strongly swirling, recirculating flows are reviewed and discussed.

Bubbles, Drops, and Particles

If model parameterizations of unresolved physics, such as the variety of upper ocean mixing processes, are to hold over the large range of time and space scales of importance to climate, they must be strongly physically based. Observations, theories, and models of oceanic vertical mixing are surveyed. Two distinct regimes are identified: ocean mixing in the boundary layer near the surface under a variety of surface forcing conditions (stabilizing, destabilizing, and wind driven), and mixing in the ocean interior due to internal waves, shear instability, and double diffusion (arising from the different molecular diffusion rates of heat and salt). Mixing schemes commonly applied to the upper ocean are shown not to contain some potentially important boundary layer physics. Therefore a new parameterization of oceanic boundary layer mixing is developed to accommodate some of this physics. It includes a scheme for determining the boundary layer depth h, where the turbulent contribution to the vertical shear of a bulk Richardson number is parameterized. Expressions for diffusivity and nonlocal transport throughout the boundary layer are given. The diffusivity is formulated to agree with similarity theory of turbulence in the surface layer and is subject to the conditions that both it and its vertical gradient match the interior values at h. This nonlocal “K profile parameterization” (KPP) is then verified and compared to alternatives, including its atmospheric counterparts. Its most important feature is shown to be the capability of the boundary layer to penetrate well into a stable thermocline in both convective and wind-driven situations. The diffusivities of the aforementioned three interior mixing processes are modeled as constants, functions of a gradient Richardson number (a measure of the relative importance of stratification to destabilizing shear), and functions of the double-diffusion density ratio, Rρ. Oceanic simulations of convective penetration, wind deepening, and diurnal cycling are used to determine appropriate values for various model parameters as weak functions of vertical resolution. Annual cycle simulations at ocean weather station Papa for 1961 and 1969–1974 are used to test the complete suite of parameterizations. Model and observed temperatures at all depths are shown to agree very well into September, after which systematic advective cooling in the ocean produces expected differences. It is argued that this cooling and a steady salt advection into the model are needed to balance the net annual surface heating and freshwater input. With these advections, good multiyear simulations of temperature and salinity can be achieved. These results and KPP simulations of the diurnal cycle at the Long-Term Upper Ocean Study (LOTUS) site are compared with the results of other models. It is demonstrated that the KPP model exchanges properties between the mixed layer and thermocline in a manner consistent with observations, and at least as well or better than alternatives.

Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization

Communities of microscopic plant life, or phytoplankton, dominate the Earth's aquatic ecosystems. This important new book by Colin Reynolds covers the adaptations, physiology and population dynamics of phytoplankton communities in lakes and rivers and oceans. It provides basic information on composition, morphology and physiology of the main phyletic groups represented in marine and freshwater systems and in addition reviews recent advances in community ecology, developing an appreciation of assembly processes, co-existence and competition, disturbance and diversity. Although focussed on one group of organisms, the book develops many concepts relevant to ecology in the broadest sense, and as such will appeal to graduate students and researchers in ecology, limnology and oceanography.

/pdf/the-ecology-of-phytoplankton-4ev70n3jsq.pdf

The Ecology of Phytoplankton

Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind. Most of the new material presented relates to the initial-value problem for the equation u$\_{t}$ + u$\_{x}$ + uu$\_{x}$ - u$\_{xxt}$ = 0, (a) whose solution u(x,t) is considered in a class of real nonperiodic functions defined for -$\infty $ < x < $\infty $, t $\geq $ 0. As an approximation derived for moderately long waves of small but finite amplitude in particular physical systems, this equation has the same formal justification as the Korteweg-de Vries equation u$\_{t}$ + u$\_{x}$ + uu$\_{x}$ + u$\_{xxx}$ = 0, (b) with which (a) is to be compared in various ways. It is contended that (a) is in important respects the preferable model, obviating certain problematical aspects of (b) and generally having more expedient mathematical properties. The paper divides into two parts where respectively the emphasis is on descriptive and on rigorous mathematics. In section 2 the origins and immediate properties of equations (a) and (b) are discussed in general terms, and the comparative shortcomings of (b) are reviewed. In the remainder of the paper (section section 3, 4) - which can be read independently of the preceding discussion - an exact theory of (a) is developed. In section 3 3 the existence of classical solutions is proved; and following our main result, theorem 1, several extensions and sidelights are presented. In section 4 solutions are shown to be unique, to depend continuously on their initial values, and also to depend continuously on forcing functions added to the right-hand side of (a). Thus the initial-value problem is confirmed to be classically well set in the Hadamard sense. In appendix 1 a generalization of (a) is considered, in which dispersive effects within a wide class are represented by an abstract pseudo-differential operator. The physical origins of such an equation are explained in the style of section 2, two examples are given deriving from definite physical problems, and an existence theory is outlined. In appendix 2 a technical fact used in section 3 is established.

Model Equations for Long Waves in Nonlinear Dispersive Systems

The term soliton has recently been coined to describe a pulselike nonlinear wave (solitary wave) which emerges from a collision with a similar pulse having unchanged shape and speed. To date at least seven distinct wave systems, representing a wide range of applications in applied science, have been found to exhibit such solutions. This review paper covers the current status of soliton research, paying particular attention to the very important "inverse method" whereby the initial value problem for a nonlinear wave system can be solved exactly through a succession of linear calculations.

The soliton: A new concept in applied science

The term 'vortex breakdown', as used in the reported investigation, refers to a disturbance characterized by the formation of an internal stagnation point on the vortex axis, followed by reversed flow in a region of limited axial extent. Two forms of vortex breakdown, which predominate, are shown in photographs. One form is called 'near-axisymmetric' (sometimes 'axisymmetric'), and the other is called 'spiral'. A survey is presented of work published since the 1972 review by Hall. Most experimental data taken since Hall's review have been in tubes, and the survey deals primarily with such cases. It is found that the assumption of axial-symmetry has produced useful results. The classification of flows as supercritical or subcritical, a step that assumes symmetry, has proved universally useful. Experiments show that vortex breakdown is always preceded by an upstream supercritical flow and followed by a subcritical wake. However, a comparison between experiments and attempts at prediction is less than encouraging. For a satisfactory understanding of the structure of vortex breakdown it is apparently necessary to take into account also aspects of asymmetry.

The structure of vortex breakdown

A realistic theoretical model of steady Langmuir circulations is constructed. Vorticity in the wind direction is generated by the Stokes drift of the gravity-wave field acting upon spanwise vorticity deriving from the wind-driven current. We believe that the steady Langmuir circulations represent a balance between this generating mechanism and turbulent dissipation.Nonlinear equations governing the motion are derived under fairly general conditions. Analytical and numerical solutions are sought for the case of a directional wave spectrum consisting of a single pair of gravity waves propagating at equal and opposite angles to the wind direction. Also, a statistical analysis, based on linearized equations, is developed for more general directional wave spectra. This yields an estimate of the average spacing of windrows associated with Langmuir circulations. The latter analysis is applied to a particular example with simple properties, and produces an expected windrow spacing of rather more than twice the length of the dominant gravity waves.The relevance of our model is assessed with reference to known observational features, and the evidence supporting its applicability is promising.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112076001420

A rational model for Langmuir circulations

Analyse theorique et experimentale des conditions de stabilite et d'eclatement de tourbillons isoles incompressibles pour des valeurs elevees du nombre de Reynolds. On rappelle des methodes d'experimentation en laboratoire qui ont ete utilisees. On decrit les deux types principaux d'eclatements de tourbillons. On discute de l'importance de la symetrie par rapport a l'axe. On donne un bref apercu de la theorie de l'onde transcritique

Vortex stability and breakdown - Survey and extension

When the wind blows at modest speeds over natural bodies of water, numerous streaks or slicks nearly parallel to the wind direction may appear on the surface. This form of surface streakiness is commonplace, and under favorable conditions it is readily apparent to the casual observer. The streaks result from the collection of floating sub­ stances-seaweed, foam from breaking waves, marine organisms, or organic films-into long narrow bands. Flotsam makes the bands visible directly, and compressed films make them visible by the damping of capillary waves, thereby giving the bands a smoother appearance. Naturalists and seafarers often note color variations of the sea due to minute marine organisms. Bainbridge (1957) cited many old descriptions of long narrow "bands," "streaks," or "lanes" including several by Darwin in 1839 during the voyage of the Beagle. James Thomson (1862) described observations of streaks made jointly with his brother, Lord Kelvin, in a paper that also indicated increased abundances of marine life below the streaks. The first connection between the wind and streak directions, among the authors cited by Bainbridge, was made by Collingwood (1868): "if a moderate breeze were blowing and the sea

The form and Dynamics of Langmuir Circulations

Flow visualization studies and laser Doppler anemometer measurements on swirling water flows reveal six distinct types of very large amplitude disturbance modes of the vortex core. Three, ’’axisymmetric’’ and spiral vortex breakdowns, and the ’’double helix,’’ have been described by others. A definite order of evolution in parameter space (Reynolds number and circulations) occurs, and is described. Puzzling responses of the axisymmetric and spiral vortex breakdowns to imposed flow transients reported previously are confirmed here, and are traced to the shedding of starting and stopping vortices from swirl vanes. Conclusions bearing upon the validity of some theories of vortex breakdown are possible from the data.

Sidney Leibovich

Papers

The structure of vortex breakdown

A rational model for Langmuir circulations

Vortex stability and breakdown - Survey and extension

The form and Dynamics of Langmuir Circulations

Disrupted states of vortex flow and vortex breakdown