Profile derived fluxes above inhomogeneous terrain : a numerical approach

In Chapter 1 the goals of the present study were presented. These goals are (i) the estimation and analysis of the errors introduced in the standard flux determination methods when they are applied above non-homogeneous terrain(ii) providing simple techniques for estimating these errors, using a minimum number of data concerning sensor location,surrounding terrain(s) etc.These goals suggested that a direct treatment of the flux profile methods above a non-homogeneous terrain, with the aidof a second order model, was feasible. In the course of this study, however, it turned out that a direct treatment was notimmediately possible. First of all, the performance of the model had to be analyzed in order to tackle the posed problem. Thisanalysis was needed for the correct interpretation and apprecia tion of the results of the flux-profile methods above non-homogeneous terrain. Hence, the conclusions in this final chapter can be separated in two groups: (i) those which concern the model used in the present study, and (ii) those which concern the application of the model in order to study the flux-profile methods.In Chapter 2 the first simplifications were introduced. These simplifications referred to the initial conditions as well asthe upper boundary conditions of the numerical model. E.g. the present model is restricted to the atmospheric surface layer,which in its initial state is supposed to be in equilibrium with the surface. Also, the upper boundary value of every variable(except W) remains fixed after the change in surface conditions. This led to the conclusion that the model is only applicable toa limited downstream distance (x max ). As soon as the flow perturbations, induced by the new surface, reach the highest grid level (z max ) the fixed upper boundary values start to contaminate the solution. The deterioration of the solution subsequent-ly diffuses downwards as the downstream distance increases beyond x max . To avoid this problem the model should be extended to comprise the whole of the atmospheric boundary layer. This has already been accomplished (e.g. Wyngaard, 1975).In Chapter 3 we concluded that second order models are superior to first order models, mainly when the second moments (fluxes and other (co)variances) are considered. However, the disadvantage of second order models is the difficulty related to the modeling of the third order terms and the pressure terms. Sometimes, mean strain and buoyancy terms have to be incorporated in the approximation of the 3rd order and pressure terms, in order to get physically realistic results. At present, there is no general agreement on the modeling of these terms. Usually the "engineering approach" is used, where model constants are tuned to obtain the desired results. The third order models which have recently appeared, merely shift the problem to the modeling of the fourth order terms. Obviously, much has still to be learned here.The above conclusions hold for any 2nd order model. The present model was examined more closely in Chapter 4. It was found that flaws in the upper boundary conditions may be responsible for the deviation of the dimensionless gradients of U, θand Q from the expected universal φ-curves in diabatic equilibrium conditions. Efforts to overcome these imperfections were only partly successful. A comparison of the budgets of several 2nd moment equations with experimental results showed that the present model reproduces the equilibrium equation budgets quite accurately. A more serious problem was encountered in the statement of the lower boundary conditions downstream of the step change in surface characteristics. It was found that the condition of a constant surface relative humidity after the surface transition causes an unphysical solution for the change from a cool and humid terrain to a hot and dry one. A possible solution for this problem has been presented. Another problem, possibly related to the lower boundary conditions, emerged when the experimental results of e.g. Lang et al. (1983) became available. It appears that the model in its present state is not able to reproduce the inequality of K h and K w in the IAL, found in the experiment. This problem remains to be solved. In the same chapter the model was modified and extended in order to be able to treat rough-to-smooth transitions. It was then possible to demonstrate that, when the ASL is subjected to a temporary change of the lower boundary conditions, the solution will eventually approach its original equilibrium.The above results necessitated the analysis of the results of the application of the standard flux-profile methods on the equilibrium (or initial) profiles, generated with the initialisation part (part I) of the model. Chapter 5 is devoted to this subject. In Chapter 5 the so-called "second order flux-profile relations" were derived from the modeled 2nd order equations. This was done in order to be able to relate (in Chapter 6) the distribution of every term of the modeled 2nd order equations to the errors produced when applying the standard flux-profile relations in inhomogeneous conditions. We demonstrated that the 2nd order flux-profile relations are identical with the standard flux-profile relations in homogeneous and neutral conditions. When the ASL above homogeneous terrain has a diabatic stratification, the interpretation of the structure of the 2nd order flux-profile relations is rather difficult. It was shown that buoyancy effects enter these relations in two ways:(i) through the generation of an additional term in the relations, and(ii) through the modification of the factor in the relation which could be identified with the eddy diffusivity in neutral conditions.It was found that the aerodynamic method yields fluxes which deviate considerably from the fluxes generated with the model.This is attributed to the disagreement between the dimensionless profile of the wind shear (∂U/∂z) and the expected φ m -curve.This was discussed above and in Chapter 4. The Bowen ratio method is not sensible for the exact definition of φ h and φ w as long as these functions are equal. In Chapter 5 it was found that the dimensionless equilibrium profiles ∂θ/∂z and ∂Q/∂z are exactly equal. Hence, the application of the Bowen ratio method to the equilibrium profiles yields sensible- and latent heat fluxes which agree very well (within 5%) with the fluxes generated with the model.In Chapter 6, both the Bowen ratio method and the aerodynamic method were applied to the relaxation profiles of U, θand Qafter the surface change. Both methods yield heat fluxes in the IAL which are larger than the heat fluxes calculated with themodel (λE MD , H MD ). The difference of the Bowen ratio heat fluxes (λE BR , H BR ) and the model fluxes within the IAL is less than 10% if the region just downstream of the surface change (x < 1 m.) is disregarded. This difference decreases as the downstream distance increases. The analysis of the various terms of the wq-equation downstream of the surface change, showed that the two advection terms () and the modeled turbulent transport term are responsible for the differences mentioned. Of these three terms the horizontal advection term and the turbulent transport term nearly cancel each other. Hence, the relatively small vertical advection term is important in this respect. Above the IAL the difference in transport of water vapour and heat partly compensates the deviation caused by the three above mentioned terms. The results of the Bowen ratio method also indicate that the determination of the value of the heat fluxes at the surface is quite accurate (within 10%) even when this method is applied just above the IAL (z < 1.5 δ).As was mentioned earlier (Chapter 4) the present model is not capable to reproduce the φ m -curve exactly. Because the application of the aerodynamic method critically depends on the shape of this curve, the analysis of the aerodynamic method is rather complicated and uncertain. Nevertheless, it was found that the difference of the heat fluxes produced by the aerodynamic method (λE AE , H AE ) and the model fluxes within the IAL is larger than the difference of λE BR (H BR ) and λE MD (H MD ). This is attributed to the more stringent conditions which the aerodynamic method has to satisfy. At x = x max the IBL reaches the upper grid level. Beyond x max the upper boundary conditions contaminate the solution. Especially the aerodynamic method is susceptible to these errors, and a proper analysis cannot be made for these large downstream distances.The final conclusion is that the present state of the second order model used in this study is subject to improvement, both in the modeling of the pressure terms and in the lower boundary conditions. Until a substantial improvement has been achieved, the main merit of this (and similar) models lies primarily in the qualitative prediction of the structure of the ASL after a change in surface conditions. Used in this way, it is an excellent tool for the understanding of the various processes which take place above an inhomogeneous terrain. This possibility of the application of a 2nd order model has been explored in this thesis for one specific purpose: the analysis of fluxprofile relations in inhomogeneous conditions. The application of this model for quantitative purposes turned out to be quite hazardous. This is the sole reason why the second goal of this study has not been achieved. This application awaits the progress in the modeling of the higher order- and pressure terms. The problems concerned with the definition of the lower boundary conditions must be approached experimentally. The performance of an experiment, designed for the precise measurement of the processes which take place near the lower boundary, logically is the next step to the ultimate solution of this problem.