George L. Mellor

A Hierarchy of Turbulence Closure Models for Planetary Boundary Layers

Abstract Turbulence models centered on hypotheses by Rotta and Kolmogoroff are complex. In the present paper we consider systematic simplifications based on the observation that parameters governing the degree of anisotropy are small. Hopefully, we shall discern a level of complexity which is intuitively attractive and which optimizes computational speed and convenience without unduly sacrificing accuracy. Discussion is focused on density stratified flow due to temperature. However, other dependent variables—such as water vapor and droplet density—can be treated in analogous fashion. It is, in fact, the anticipation of additional physical complexity in modeling turbulent flow fields that partially motivates the interest in an organized process of analytical simplification. For the problem of a planetary boundary layer subject to a diurnally varying surface heat flux or surface temperature, three models of varying complexity have been integrated for 10 days. All of the models incorporate identical empirica...

The Pressure Gradient Conundrum of Sigma Coordinate Ocean Models

Abstract Much has been written of the error in computing the horizontal pressure gradient associated with sigma coordinates in ocean or atmospheric numerical models. There also exists the concept of “hydrostatic inconsistency” whereby, for a given horizontal resolution, increasing the vertical resolution may not be numerically convergent. In this paper, it is shown that the differencing scheme cited here, though conventional, is not hydrostatically inconsistent; the sigma coordinate, pressure gradient error decreases with the square of the vertical and horizontal grid size. Furthermore, it is shown that the pressure gradient error is advectively eliminated after a long time integration. At the other extreme, it is shown that diagnostic calculations of the North Atlantic Ocean using rather coarse resolution, and where the temperature and salinity and the pressure gradient error are held constant, do not exhibit significant differences when compared to a calculation where horizontal pressure gradients are c...

A Simulation of the Wangara Atmospheric Boundary Layer Data

Abstract Previously, the authors have studied a hierarchy of turbulent boundary layer models, all based on the same closure assumptions for the triple turbulence moments. The models differ in complexity by virtue of a systematic process of neglecting certain of the tendency and diffusion terms in the dynamic equations for the turbulent moments. Based on this work a Level 3 model was selected as one which apparently sacrificed little predictive accuracy, but which afforded considerable numerical simplification relative to the more complex Level 4 model. An earlier paper had demonstrated that the model produced similarity solutions in near agreement with surface, constant flux data. In this paper, simulators from the Level 3 model are compared with two days of Wangara atmospheric boundary layer data (Clarke et al., 1971). In this comparison, there is an easily identified error introduced by our inability to include advection of momentum in the calculation since these terms were not measured. Otherwise, the ...

The Three-Dimensional Current and Surface Wave Equations

Surface wave equations appropriate to three-dimensional ocean models apparently have not been presented in the literature. It is the intent of this paper to correct that deficiency. Thus, expressions for vertically dependent radiation stresses and a definition of the Doppler velocity for a vertically dependent current field are obtained. Other quantities such as vertically dependent surface pressure forcing are derived for inclusion in the momentum and wave energy equations. The equations include terms that represent the production of turbulence energy by currents and waves. These results are a necessary precursor for three-dimensional ocean models that handle surface waves together with wind- and buoyancy-driven currents. Although the third dimension has been added here, the analysis is based on the assumption that the depth dependence of wave motions is provided by linear theory, an assumption that is the basis of much of the wave literature.

EQUILIBRIUM TURBULENT BOUNDARY LAYERS

Empirical information is extracted from constant-pressure flows and, on this basis alone, the equations of motion are solved for flows where the pressure gradient parameter, β = δ*( dp / dx )/τ 0 is held constant. The experimental defect profiles of Clauser and the near-separating profile of Stratford are predicted quite well. The present work is an extension of the work of Clauser and Townsend in that a particular form for an effective or eddy viscosity is hypothesized. Here, however, a continuous, and analytically precise family of defect profiles are calculated for the entire range, −0·5 ≤ β ≤ ∞. The solutions span the whole profile with the exception of the viscous sublayer. A detailed consideration of the viscous sublayer and a comparative examination of various eddy viscosity hypotheses are included in a companion paper.

The Structure and Dynamics of the Ocean Surface Mixed Layer

Abstract The present paper describes a one-dimensional unsteady model of the ocean surface mixed layer. Themodel somewhat resembles the approach of Munk and Anderson in that the differential equations for meanvelocity and temperature are solved. The Richardson-number-dependent stability functions which enterthe model are significantly different, however, as is the fact that we are able to solve problems with realisticboundary conditions. Furthermore, all empirical constants have been determined from neutral turbulentflow experiments. Comparisons of prediction and data are favorable.

Analytic Prediction of the Properties of Stratified Planetary Surface Layers

Abstract By considering the complex of one-point, turbulent moment equations for velocity, pressure and temperature, it appears possible to predict some properties of diabatic, density-stratified planetary layers using empirical information obtained from laboratory turbulence data in the absence of density stratification. In this paper attention is focused on the near-surface, constant-flux layer. The results, like the empirical input, are simple and, hopefully, will be instructive and useful in the formulation of improved and possibly more complicated models in the future.

A Numerical Study of the Mediterranean Sea Circulation

Abstract A primitive equation ocean model that makes use of a curvilinear orthogonal grid and a sigma-coordinate system was used to simulate the Mediterranean Sea The model was forced with monthly climatological values of wind stress, heat, and salinity flux. With the help of the curvilinear horizontal grid, the larger scales of the entire Mediterranean Sea are modeled, and the topography around the narrow and shallow Straits of Gibraltar is also reasonably well represented. The resulting model inflow and outflow seems to mimic the real Mediterranean, often in considerable detail. Levantine Intermediate Water is formed in the Levantine Basin and exits through the Strait of Sicily and the Strait of Gibraltar. Deep-water formation processes are clearly represented by the model. The model results indicate that in the western Mediterranean the wind stress is very important in establishing the summer northward shift of the Atlantic inflow. Lateral boundary runoff, surface salinity, and heat fluxes are necessar...

Modeling Vertical and Horizontal Diffusivities with the Sigma Coordinate System

Abstract The use of diffusive terms in numerical ocean models is examined relative to different coordinate systems. The conventional model for horizontal diffusion is found to be incorrect when bottom topographical slopes are large. A new formulation is suggested which is simpler than the conventional formulation when transformed to a sigma coordinate system and makes it possible to model realistically both surface Ekman and bottom boundary layers.

Wave Breaking and Ocean Surface Layer Thermal Response

The effect of breaking waves on ocean surface temperatures and surface boundary layer deepening is investigated. The modification of the Mellor‐Yamada turbulence closure model by Craig and Banner and others to include surface wave breaking energetics reduces summertime surface temperatures when the surface layer is relatively shallow. The effect of the Charnock constant in the relevant drag coefficient relation is also studied.