Brian Henderson-Sellers

New formulation of eddy diffusion thermocline models

One-dimensional thermal stratification models (for lakes, reservoirs, oceans, fjords), based on eddy diffusion concepts to represent vertical mixing of heat, have not found extensive application due to the lack of an adequate analytical representation of the neutral eddy diffusion coefficient, KH~. In this paper such an analytic expression is developed and shown to be in good agreement with various pieces of empirical evidence. The temporal changes of temperature in a freshwater lake or reservoir or in the sea can be simulated by models of varying degrees of complexity,’ although in many circumstances a one-dimensional heat transfer model may be applicable. In this latter case the calculation can be undertaken using either an eddy diffusion approach or a mixed layer representation: two approaches which should be seen as complementary rather than competitive.’ In the former approach (discussed here) difficulties have arisen in the past in the analytical specification of both (the vertical component of) the eddy diffusion coefficient under neutral conditions, KH~, and the appropriate modification for nonneutral conditions. Of the several possible analytical formulations for KH,,, the most acceptable is found to be related to the vertical profile of the horizontal current. The ‘typicality’ of any specific velocity profile, however, remains in some doubt. For example, in addition to the log profile or modified log profile,3 Ekman layer theories give rise to decaying exponential profiles.4 Alternatively, Csanady,’ in his discussion of a developing oceanic shear layer, utilized a profile based upon the complementary error function; whereas McPhee’8 analysis expressed the dimensionless complex velocity u in terms of a series of terms, one of which contains a logarithmic function. In this paper the problem of specifying the current is obviated by the development of a new analytical model, which is shown to be in good agreement both with previous empirically based models and field observations and which, when applied to the specific example of the freshwater lake environment, is found to give accurate simulations. (It should be noted that this derived value of the vertical component of the eddy diffusion coefficient is also required

Sensitivity of thermal stratification models to changing boundary conditions

Abstract Simulations of the thermal structure of lakes and other water bodies rely on the input of a large body of data, mostly meterological, as boundary conditions to the model. Here, perturbations to a range of meteorological and limnological variables are imposed in order to assess for which of these many variables errors in the input data could be problematical. The results suggest that the most important parameters are generally air temperature and cloud cover. In specific situations other parameters may also be important: for example, changes in wind speed in low-wind-speed situations; changes in extinction coefficient (a measure of water turbidity) in lakes which may be classed as oligotrophic or mesotrophic, but not those which are eutrophic. Parameters to which the lake seems relatively insensitive are solar radiation (the lake appears to have little “memory” in turbulent transfer (assuming its vertical and temporal variation is included relatively realistically).

The zone of flow establishment for plumes with significant buoyancy

Abstract The self-similar assumption used in jet and plume models is only valid for distances of greater than about six stack diameters downstream, in the zone of established flow (ZEF). The ‘Gaussian’ profile, observed at the beginning of the ZEF, must be related to source ‘top hat’ parameter values. However, previously used formulae are shown here to be approximations, being valid only for non-buoyant sources (‘pure jets’). Extensions to sources of significant buoyancy are described in terms of the densimetric Froude number, based on recently published experimental work.

The behaviour of marginally buoyant plumes in an urban environment

Abstract A numerical model for a forced plume is described which takes into account the effects of turbulence. This full three-phase model of an urban plume permits the tracking of effluents for long distances downwind of the stack. Turbulence within the urban environment is induced both by non-planar natural topography and by the variety of land-surface types present (e.g. parkland, high-rise housing blicks).

Development and application of U.S.E.D. : A hydroclimate lake stratification model

Abstract Two model types are currently in use to model the thermal stratification cycle in lakes and reservoirs: the eddy diffusion and the mixed layer (or integral energy) approaches. Here the former is analysed and developments are proposed to remove the empiricisms previously implicit in these models. These discussions permit the reformulation of K H 0 independently of current shear, together with an expression for R i . The deduced formulae are in good agreement with observations. The newly formulated model (the University of Salford eddy diffusion model, U.S.E.D.) is subsequently used in simulations of lakes and reservoirs at different latitudes which are found to be in good agreement with observations without requiring inter-site calibration.

Plume rise modelling: The effects of including a wind shear and a variable surface roughness

Abstract Using a dynamic plume rise and dispersion numerical model, the importance of accurate specification of the wind profile has been investigated. Plume rise, plume dispersion and the corresponding value of the maximum ground level concentration (Cmax) and its distance of occurrence (xmax) are found to be sensitive to the wind profile, largely influenced by the value chosen for the surface roughness parameter. It would appear from these numerical experimental results that improved accuracy can be gained by incorpotaring a logarithmic or function of distance downstream; although it is suggested that the primary copcern should be the accurate specification of the values of the surface roughness close to the stack when the effluent plume is in its initial stages of rise.

Verification of the plume rise/dispersion model USPR: plume rise for single stack emissions

Abstract Despite proliferation of the use of air pollution models for regulatory application, major discrepancies still occur between models and also between models and observations, especially when oversimplistic models are used. The problem of predicting plume rise (and subsequently ground level concentrations) from a single source is evaluated here in terms of an integral plume rise and dispersion model (USPR) which encompasses both bouyant rise and turbulent spreading; thus avoiding the problems of the concatenation of separate plume rise and dispersion models. The wide range of validity of the USPR model is demonstrated is terms of plume rise by comparison with the highly buoyant GCOS and Kincaid plumes as well as with dense effluents. It is also shown to be in agreement with Briggs two-thirds law when the restrictions applicable to the latter model are imposed.

A comparison of the formulation for eddy diffusion in two one-dimensional stratification models

Abstract Two models for thermal stratification based on turbulent diffusion concepts are analysed and compared. The models by Henderson-Sellers; and by McCormick and Scavia, are shown to be equivalent at large values of the Richardson number, R i . At small R i , the simpler model reverts to specification of the turbulent diffusion as a constant value. This simplification is also demonstrated to be a realistic approximation only at low wind speeds and for deep lakes. By comparison of these model types, a (previously empirically defined by McCormick and Scavia) parameter β is related conceptually to the lake depth, H .

The dependence of surface velocity in water bodies on wind velocity and latitude

Abstract In analysis of the hydrodynamics of large and small water bodies, a relation is frequently required between surface velocity, U s , and wind speed, U . Here a theory is developed to explain the observed relationship between surface velocity in a water body and both wind speed and latitude.

Gaussian Plume Model

A second set of models are described in the air pollution literature: the so-called “Gaussian Plume Models” which describe the lateral dispersion of effluent emitted with zero buoyancy and momentum (i.e. there is no plume rise). This approach develops the initial work of Sutton 123, 124 in terms of Fickian diffusion and is still in use today (e.g. Melli and Runca 125) — see e.g. Pasquill and Smith 126.