SurfaceLayer Flux–Gradient Relationships over Inclined Terrain Derived from a Local Equilibrium, Turbulence Closure Model
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Abstract
Derivation of surfacelayer flux–gradient relationships from a localequilibrium, turbulenceclosure model for a forced flow over inclined terrain is presented. Results are shown as a generalization of Monin–Obukhov universal functions respesenting nondimensional wind and temperature gradients.
Keywords
Katabatic flows Slope effects Surface layer Surfacelayer similarity theory Turbulence closure1 Introduction
The Monin–Obukhov surfacelayer similarity theory (Obukhov 1946, 1971; Monin and Obukhov 1954; hereafter MOST) has been proven successful as a framework for establishing flux–gradient relationships in the atmospheric surface layer over flat and uniform terrain. Such relationships have great practical importance; for example, virtually all threedimensional models used in numerical weather prediction utilize bulk schemes for calculating surface fluxes based on MOST, as their vertical grid spacing is grossly insufficient for resolving explicitly the large variability of main flow variables near the ground.
Universal functions of MOST can either be derived from turbulence models or determined from field data (e.g. Zilitinkevich and Chalikov 1968; Businger et al. 1971). While the performance of early theoretical models (KEYPS, Kazanski and Monin 1956) was rather poor, the secondorder Reynoldsstress modelling provided solutions consistent with field data (e.g. Lewellen and Teske 1973; Mellor 1973; Prenosil 1979). Nevertheless, simpler empirical methods dominated. While admitting this fact, we would like to point out that the alternative theoretical approach offers advantages over the empirical one, such as a broader generality, selfconsistency, and a prospect for consistency with the turbulence parametrization as used in the numerical model.
Although MOST assumptions seem rather restrictive with respect to real conditions wide application of MOST in atmospheric modelling has been very succesful. Perhaps this success is owed to sufficient accuracy in estimating relatively large fluxes that control the dynamics of atmospheric flows. However, with grid spacing of numerical weather prediction models approaching 1 km, our focus shifts to microscale problems, particularly in complex terrain where slope effects associated with stable equilibrium come into play. Hence, questions arise: are the slope effects important for influencing the exchange between the atmosphere and the underlying surface? If so, under what circumstances do slope effects become important? And finally, what modifications to existing algorithms are necessary to account for these effects?
While the empirical approach to these questions seems to be a natural path, it may not be easy to establish proper relationships. First, two new relevant parameters appear—wind direction with respect to the slope and terrain inclination angle. Second, while MOST is primarily designed for externally driven flows (volume forces acting in the flow direction within the surface layer are neglected) it does not apply to densitydriven flows that develop over inclined terrain. It is then difficult to separate the densitydriven component of the flow along the surface from the sheardriven component that is handled by MOST. Although there is a bulk of experimental results, most of the data come from field studies of katabatic flows (e.g. Horst and Doran 1983; van den Broeke 1997; Oerlemans et al. 1999). One of the few exceptions is the study by Park and Park (2006). However, this study is focused on turbulence statistics rather than flux calculations, and is limited to unstable conditions. An alternative approach to derive the desired relationships from a Reynoldsstress model is taken here. Of particular interest to this study is the work presented by Denby (1999), who successfully simulated katabatic flows over glaciers and ice caps with a onedimensional, high resolution, secondorder turbulence closure model transformed into slope coordinates. While Denby’s work confirms the feasibility of a onedimensional framework to slope flows the demand for a high density vertical grid near the surface precludes its use in most of the present threedimensional numerical models.
One might wonder whether the approach taken here is feasible. Indeed, in katabatic flows the wind maximum can be found at heights as low as 1 m (Denby 1999). Also, densitydriven flows are not steady. Therefore, the presented method does not include a mechanism of density flow formation for modelling katabatic flows with insufficient vertical resolution. The underlying assumption for practical application is a possibility of separation of scales of motions so that the densitydriving mechanism can be handled in resolvable scales while the transfer described by the solutions presented here would be treated as a subgrid process.
2 The Model
The turbulence model adopted herein stems from the Mellor–Yamada Level2 model (Mellor 1973; Mellor and Yamada 1974, 1982) widely used in applications such as numerical weather prediction (e.g. Janjić 1990; Gerrity et al. 1994; Saito et al. 2007), mesoscale atmospheric research (e.g. Sušelj and Sood 2010; Foreman and Emeis 2012), ocean modelling (e.g. Blumberg and Mellor 1987; Kantha and Clayson 1994; Burchard and Bolding 2001), and global atmospheric models (e.g. Sirutis and Miyakoda 1990). Here, we follow a modification of this model by Nakanishi (2001); the model equations for horizontally homogeneous conditions over flat terrain in a conventional coordinate frame are listed in the Appendix 1.
2.1 Slope Coordinates
For sloping terrain, one might consider a twodimensional Cartesian coordinate system, with its \(XZ\) vertical plane containing the wind vector, the \(X\)axis pointing in the wind direction, parallel to the slope, and the \(Z\)axis perpendicular to the \(X\)axis. Although not precise, the \(X\) direction will be termed tangent, and the \(Z\) direction termed normal in the following. This setting provides an exact transformation for pure downslope or upslope flows; for a general case, it should be treated as a simplification whose validity needs to be established. The transformation rules are summarized in Appendix 2.
While the basic set of prognostic equations for secondorder statistical moments is systematically derived from conservation equations, the formulation of the turbulent masterlength scale \(\ell \) applied in closure hypotheses rests largely on heuristic arguments. Mellor and Yamada (1982) recognized this as the possibly largest outstanding weakness of this class of models. Numerous formulations of the turbulence masterlength scale \(\ell \) have been proposed in the past, mainly regarding stable stratification (e.g. Grisogono 2010). However, some limitations were found to be helpful in ensuring physical realizability in rapidly growing turbulence (e.g. Janjić 2001). Since the avoidance of branching threads seems to be justified by a preliminary character of this study, a simple walleffect parametrization, \(\ell =\kappa z\) has been chosen here, at the cost of a certain limitation of the model applicability range.

the gravitational force splits now into two components, one acting along the \(Z\) axis, and the other acting along the slope. Note a new term in the downslope heat flux budget Eq. 5.

the masterlength scale is proportional to the distance from the surface (or to its projection onto the \(Z\) axis), not to the height
2.2 Equilibrium Parameters and the TKE Budget
2.3 NonDimensional Form of Model Equations
2.4 Solution Procedure
3 Results
Qualitative analysis of the relations in downslope flows is more difficult, as the individual terms in the tangential heatflux budget equation, Eq. 5, are not of the same sign and partially cancel each other. While the sign of the result can hardly be predicted by this argumentation, it is evident that the tangential heat flux should have a smaller value than in an upslope flow under comparable forcing. This is corroborated by calculations presented in the bottom panel of Fig. 4. While the tangential component is still larger than the normal one, its role in the TKE budget is currently marginal; over steeper terrain, we now see a relative reduction of TKE now. Consequently, dimensionless temperature and windspeed gradients should assume smaller values than in upslope winds.
Figure 5 displays the value of the \(q_n^2 = q^2 / u_*^2\) function, that is the doubled TKE normalized by kinematic surface stress. One may note that the slope influences agree, at least in qualitative terms with the field results presented by Park and Park (2006) (compare their Fig. 12) for unstable stratification. This agreement may seem encouraging, although more extensive experimental evidence is necessary to validate the theoretical predictions presented here.
4 Summary and Conclusions
Perspectives of the inclusion of terrain inclination effects into the surfacelayer flux–gradient relationships have been theoretically investigated with a localequilibrium, secondorder turbulence model. The aim of the present study was twofold: first, to propose a methodological approach to the inclusion of slope effects into the MOST; second, to assess the possible outcome of such an extension, at least in qualitative terms.

the external forcing is sufficient to maintain homogeneity along the slope so that a singlecolumn framework may be applied;

further, the MOST assumptions are imposed, with the exception of the requirement that the underlying surface be horizontal;

consequently, the results can only be applicable to the region where gravitational/buoyancy forces can be neglected in the budget of the downslope momentum component. In other words, existence of a turbulent sublayer where the mean motion is primarily sheardriven and the turbulence is driven by both the shear and buoyancy is postulated;

the wall proximity effects are parametrized via the masterlength scale as dependent on the distance from the inclined surface, while the buoyancy effect act along the vertical;

as a result of the flow’s time evolution, a quasistationary regime is reached through a process of selfadaptation of the profiles and forcing, and the potential temperature gradient assumes a direction perpendicular to the surface;

the influence of the crosswind component of the topographical slope is neglected, i.e. the theory is strictly applicable to pure downslope or upslope winds.
Corresponding to the aforementioned goals, we have opted for a widely applied, relatively simple turbulence closure model, thereby postponing the issue of optimal model choice to subsequent studies. The results presented here should be seen as preliminary and interpreted in qualitative terms rather than quantitative.
Calculations show that the classic MOST profiles may be used with almost no modification under unstable equilibrium, with simple geometric adjustments to account for the direction of the surface stress and for scaling the distance to the surface. However, in stable conditions, a modification of the universal functions \(\varphi _m(\zeta )\) and \(\varphi _h(\zeta )\) is also necessary. Such a modification can be quite simple for downslope winds over gently or moderately sloping terrain. However, the model results for upslope flows over steep terrain point to a more complex behaviour that may involve realizability limitations of the model. A further clarification of this issue is necessary, with the use of alternative turbulence closures, in particular the newly developed models that avoid the limitation of the gradient Richardson number (Zilitinkevich et al 2007), largeeddy simulation studies and analyses of field data.
Notes
Acknowledgments
This study was sponsored by the statutory research fund provided by the Polish Ministry of Science and Higher Education.
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