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Edited by: Miguel A. C. Teixeira, University of Reading, UK

Reviewed by: Peter F. Sheridan, Met Office, UK; Lev Ingel, Institute of Experimental Meteorology, SPA “Typhoon”, Russia

*Correspondence: Ivan Güttler

This article was submitted to Atmospheric Science, a section of the journal Frontiers in Earth Science

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The Prandtl model succinctly combines the 1D stationary boundary-layer dynamics and thermodynamics of simple anabatic and katabatic flows over uniformly inclined surfaces. It assumes a balance between the along-the-slope buoyancy component and adiabatic warming/cooling, and the turbulent mixing of momentum and heat. In this study, energetics of the Prandtl model is addressed in terms of the total energy (

Katabatic and anabatic winds are downslope and upslope flows that form when a density difference between the air near the slope and the nearby atmosphere develops at the same height. This type of flow is often observed in regions of complex orography and substantially affects the weather and climate in these regions (e.g., Poulos and Zhong,

In the model of Prandtl (

We limit ourselves only to the linear and weakly nonlinear solution of the (extended) Prandtl model. A detailed description of the extended Prandtl model is presented in Grisogono et al. (2015; their Section 2). The new term that extends the original Prandtl model is presumably weak and regulated by the nonlinearity parameter ε. Our approach is relatively simple and general, and may be applied to solutions of Prandtl-type models that include 3D effects (e.g., Burkholder et al.,

This study is independent but based on the work of Grisogono et al. (

The goal of this study is to evaluate an ensemble of linear and weakly nonlinear solutions of the (extended) Prandtl model for katabatic and anabatic flows, and to examine the model energetics related to these solutions. In order to explore the sensitivity of our results to several model assumptions, we present a set of solutions where three governing parameters are perturbed: (1) the turbulent Prandtl number

The structure of the paper is as follows. In Section Methodology, we present the governing equations of our model and define the ensemble of solutions. In Section Results, the solutions of the (extended) Prandtl model are described, with a specific focus on the variability in the ensemble of solutions and impacts on the model energetics. Some specific differences between the nonlinear and linear solutions, as well as the limitations of our extended Prandtl model are discussed in Section Discussion. The paper is finalized in Section Summary and Conclusions, where the summary and outlook are presented.

We first present the governing equations of the Prandtl model and develop a simple, basic energy framework where wind and potential temperature are linked with the concepts of kinetic, potential and total energy. The full description of the system would include the energy components of not only the mean slope flow, but of the background atmosphere and the turbulent part of the slope flow, and their interactions. We limit our analysis only to the part of the slope flow described by the Prandtl model, i.e., the mean slope flow with relatively large eddy diffusivity and conductivity; hence, the model may emulate a simple turbulent slope flow (Defant,

Potential temperature and wind can be decomposed into _{r} = θ_{0} + γ_{0} is the surface potential temperature in a statically stable background atmosphere (e.g., Zardi and Serafin, _{r} = 0. Next,

The governing equations of the Prandtl model, including the weak nonlinearity extension (without invoking the steady-state assumption for the moment) are:

Four types of steady-state solutions of Equations (1) and (2) are analyzed in Grisogono et al. (_{LIN} and _{LIN}, and (2) the nonlinear solution with the constant turbulent diffusivity profile θ_{NOLIN} and _{NOLIN}. Initial results concerning the vertical variability of _{LIN} and _{LIN} [for the nonlinear solutions please refer to Grisogono et al. (

Following, e.g., Grisogono et al. (_{LIN}(_{LIN}(_{0}^{1/2}, _{P} = ^{1/2}_{P} can be interpreted as a characteristic depth of the Prandtl layer), σ = [^{1/2})]^{1/2} (σ can be interpreted as a characteristic inverse length scale), ^{2} = γ_{o} is background buoyancy frequency squared, and _{LIN}(

In the case of linear and stationary flow, Equation (3) simplifies to:

We evaluate the sensitivity of our solutions to the slope angle α, the value of the Prandtl number _{o} = 273.2 K, ^{2}/s (3.0 m^{2}/s) in katabatic (anabatic) flow.

(a) Linear case

The vertical profiles of θ_{LIN} and _{LIN} for katabatic flow are shown in Figure _{LIN} increasing in the first 30 m above the slope. At the same time, _{LIN} starts from the no-slip condition at the sloped surface, attains a local maximum (i.e., a low-level jet is formed at the height _{j}) and slows progressively upwards. The corresponding vertical profiles of kinetic _{LIN} profile leads to a corresponding kinetic energy profile with its maximum in the first 15 m. We proceed next with the evaluation of the sensitivity of the ensemble of solutions for the katabatic flow described by the linear model.

_{j} _{REF} from the solutions when

The following three heights are of interest to us:

The height of the low-level jet _{j}. This is the maximum of _{j} = π/4 _{P} in the linear solutions, i.e., it increases with increasing _{LIN} is _{j} in Equation 5) as is confirmed in Figure

The depth of the stable (in the anabatic case, unstable) layer. At the top of the stable layer _{P}. It also increases with increasing

The level where _{j}_{P} cos^{−1}([1/(1 + ^{1∕2}); i.e., it also increases with increasing _{LIN}. More details about this measure are presented in Supplementary Materials

(b) Nonlinear case

The deviation of the nonlinear from the linear solution for katabatic flow is presented in Figure _{NOLIN} and _{NOLIN} profiles are equivalent to θ_{LIN} and _{LIN}, and their corresponding

_{REF} are based on the solutions when

We also examine the sensitivity of the nonlinear solution to the choices of _{NOLIN} (and the corresponding maximum _{LIN} is constant; cf. Figures

In this subsection we present a general overview of anabatic flow solutions from the linear and weakly nonlinear Prandtl model. The main difference when compared to katabatic flow is the existence of the surface temperature surplus that induces the anabatic flow (now +6 K; cf. Defant, ^{2}/s to ^{2}/s and the increase of the nonlinearity parameter from ε = 0.005 to ε = 0.03, as explained in Grisogono et al. (_{P}_{Anabatic}_{Katabatic} ~ _{Anabatic}_{Katabatic}^{1/2}. With this choice of ε, perturbations to the linear solution are present, but the general structure of the solution does not change. Although anabatic upslope winds are generally deeper than typical katabatic flows, in our comparisons the same amplitude of potential temperature deviations at the surface is set so that the same potential energy of the slope flow

(a) Linear case

The vertical profiles of the upslope wind _{LIN}, potential temperature deviations θ_{LIN},

(b) Nonlinear case

The nonlinear solution of anabatic flow is described in this subsection. When compared to its katabatic counterpart, the vertical profiles of along-the-slope wind speed and potential temperature have the same general structure and this is also the case for kinetic, potential and total energy of the nonlinear vs. linear solution. However, all three energy components (

The sensitivity of the selected height measures to

Common to all previous solutions, while the maximum in _{p}cos^{−1}[1/(1+^{1∕2}] and this height is usually between _{j} and 2 _{j}. It is related to the corresponding gradient Richardson number, which compares the vertical gradients of

_{REF}. For presentation purposes, the lines in

The potential energy maximum (

At the same time, the kinetic energy maximum (_{j}) increases when _{j} and _{max}, show a similar sensitivity to ε. Grisogono et al. (_{j} and _{max} decrease in the nonlinear katabatic solution, and an _{j} and _{max} increase in the nonlinear anabatic solution.

The amplitude of

The diffusion maximum (

In the case of nonlinear solutions for anabatic and katabatic flow, the additional interaction term is present. Both the amplitude and height of the interaction term maximum ^{−3} J/kg/s to ~4·10^{−3} J/kg/s in katabatic flow, while it is negative and varies from ~ −0.22 J/kg/s to ~ -0.04 J/kg/s in anabatic flow. Also, by examining the maximum of the triple product in _{j} (this result can be derived by using the linear solutions to find numerically the local maximum of the triple product; a more precise estimation would include the use of the nonlinear solutions _{NOLIN} and θ_{NOLIN}). This means that _{j}, which is one of the new results of this study.

_{REF} _{REF} _{REF}

The last quantity examined in this subsection is the tendency of total energy ∂

In this section, we briefly discuss some of the results where references to LES studies and the issue of the non-stationarity present in our weakly nonlinear solutions are addressed.

The reduction of _{j} with an increasing slope is a well-known feature of katabatic flows (in both LES results and the Prandtl model, see e.g., Grisogono and Axelsen, _{LIN} is

Conceptually, there are no crucial differences (besides the vertical extent) in

Another important difference between the linear and nonlinear katabatic (and anabatic) solutions is the nonzero ∂

Lastly, the question is how the results of this study are comparable to the real atmosphere. While high-resolution observations over long gentle slopes and specific background atmospheric conditions are hard to acquire, we estimate

In this study, we have evaluated the energetics of the linear and weakly nonlinear solutions of the (extended) Prandtl model from Grisogono et al. (

The nonlinearity effect induced small to moderate variations in the total energy

We have limited our analysis to the energy terms and prognostic total energy equation of the mean slope flow only. It is shown that the strongest interaction between the θ- and _{j}, with _{j} = π/4 _{p}, i.e., about half the height of the low-level jet. Moreover, kinetic energy dominates over potential energy above _{p}cos^{−1}[1/(1 + ^{1∕2}], which is typically between _{j} and 2 _{j}. Thus, this is the sublayer where dynamic instabilities might occur. It is directly, although nonlinearly, related to the corresponding gradient Richardson number, which compares the differential change of potential energy vs. kinetic energy of the flow. This number falls significantly below 1 in that sublayer. However, the height where

A more complete energy framework would include an estimation of the potential and kinetic energy contributions from the basic state, turbulence and possibly mesoscale components (e.g., waves) in the system. Since there is still no satisfactory approach that would include the effects of sub-grid slope flows in the form of parameterizations in mesoscale and large-scale weather and climate models, greater effort should be made in order to increase the applicability of these types of models in complex orography regions (e.g., Bornemann et al.,

Finally, the results of our simple small-ensemble exercise may be compared with observations (where care is needed to ensure high-resolution measurements in order to correctly estimate the first and second vertical derivatives in the total energy equation). A second approach to an independent evaluation of our analytical model includes the construction of the total energy budget from an ensemble of LES simulations (e.g., Grisogono and Axelsen,

All authors listed, have made substantial, direct and intellectual contribution to the work, and approved it for publication.

IG was supported by the Croatian Science Foundation, CARE project, No. 2831; IM, ŽV, and BG were supported by the Grant Agency of the Czech Science Foundation under the GAČR project 14-12892S and by the Croatian Science Foundation, CATURBO project, No. 09/151.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors wish to thank the reviewers and the editor for providing detailed and constructive comments that have substantially improved the initial versions of the paper.

The Supplementary Material for this article can be found online at: