Directional Shear in the Nocturnal Atmospheric Surface Layer

Stratified nocturnal flow above and within a small valley of approximately 12-m depth and a few hundred metres width is examined as a case study, based on a network of 20 sonic anemometers and a central 20-m tower with eight levels of sonic anemometers. Several regimes of stratified flow over gentle topography are conceptually defined for organizing the data analysis and comparing with the existing literature. In our case study, a marginal cold pool forms within the shallow valley in the early evening but yields to larger ambient wind speeds after a few hours, corresponding to stratified terrain-following flow where the flow outside the valley descends to the valley floor. The terrain-following flow lasts about 10 h and then undergoes transition to an intermittent marginal cold pool towards the end of the night when the larger-scale flow collapses. During this 10-h period, the stratified terrain-following flow is characterized by a three-layer structure, consisting of a thin surface boundary layer of a few metres depth on the valley floor, a deeper boundary layer corresponding to the larger-scale flow, and an intermediate transition layer with significant wind-directional shear and possible advection of lee turbulence that is generated even for the gentle topography of our study. The flow in the valley is often modulated by oscillations with a typical period of 10 min. Cold events with smaller turbulent intensity and duration of tens of minutes move through the observational domain throughout the terrain-following period. One of these events is examined in detail.     

Intermittent Bursting of Turbulence in a Stable Boundary Layer with Low-level Jet

The atmospheric stable boundary layer (SBL) with a low-level jet is simulated experimentally using a thermally stratified wind tunnel. The turbulence structure and flow characteristics are investigated by simultaneous measurements of velocity and temperature fluctuations and by flow visualization. Attention is focused on the effect of strong wind shear due to a low-level jet on stratified boundary layers with strong stability. Occasional bursting of turbulence in the lower portion of the boundary layer can be found in the SBL with strong stability. This bursting originates aloft away from the surface and transports fluid with relatively low velocity and temperature upward and fluid with relatively high velocity and temperature downward. Furthermore, the relationship between the occurrence of turbulence bursting and the local gradient Richardson number (Ri) is investigated. The Ri becomes larger than the critical Ri, Ri_cr = 0.25, in quiescent periods. On the other hand, the Ri number becomes smaller than Ri_cr during bursting events.     

Large-Eddy Simulation Of The Stably Stratified Planetary Boundary Layer

In this work, we study the characteristics of a stably stratifiedatmospheric boundary layer using large-eddy simulation (LES).In order to simulate the stable planetary boundary layer, wedeveloped a modified version of the two-part subgrid-scalemodel of Sullivan et al. This improved version of themodel is used to simulate a highly cooled yet fairly windy stableboundary layer with a surface heat flux of(W)_o = -0.05 m K s^-1and a geostrophic wind speed of U_g = 15 m s^-1.Flow visualization and evaluation of the turbulencestatistics from this case reveal the development ofa continuously turbulent boundary layer with small-scalestructures. The stability of the boundary layercoupled with the presence of a strong capping inversionresults in the development of a dominant gravity wave atthe top of the stable boundary layer that appears to be relatedto the most unstable wave predicted by the Taylor–Goldsteinequation. As a result of the decay of turbulence aloft,a strong-low level jet forms above the boundary layer.The time dependent behaviour of the jet is compared with Blackadar'sinertial oscillation analysis.     

Early Warning Signals for Regime Transition in the Stable Boundary Layer: A Model Study

The evening transition is investigated in an idealized model for the nocturnal boundary layer. From earlier studies it is known that the nocturnal boundary layer may manifest itself in two distinct regimes, depending on the ambient synoptic conditions: strong-wind or overcast conditions typically lead to weakly stable, turbulent nights; clear-sky and weak-wind conditions, on the other hand, lead to very stable, weakly turbulent conditions. Previously, the dynamical behaviour near the transition between these regimes was investigated in an idealized setting, relying on Monin–Obukhov (MO) similarity to describe turbulent transport. Here, we investigate a similar set-up, using direct numerical simulation; in contrast to MO-based models, this type of simulation does not need to rely on turbulence closure assumptions. We show that previous predictions are verified, but now independent of turbulence parametrizations. Also, it appears that a regime shift to the very stable state is signaled in advance by specific changes in the dynamics of the turbulent boundary layer. Here, we show how these changes may be used to infer a quantitative estimate of the transition point from the weakly stable boundary layer to the very stable boundary layer. In addition, it is shown that the idealized, nocturnal boundary-layer system shares important similarities with generic non-linear dynamical systems that exhibit critical transitions. Therefore, the presence of other, generic early warning signals is tested as well. Indeed, indications are found that such signals are present in stably stratified turbulent flows.     

TURBULENCE STRUCTURE IN A STRATIFIED BOUNDARY LAYER UNDER STABLE CONDITIONS

Turbulence structure in stably stratified boundary layers isexperimentally investigated by using a thermally stratified wind tunnel. Astably stratified flow is created by heating the wind tunnel airflow to atemperature of about 50 °C and by cooling the test-section floor to asurface temperature of about 3 °C. In order to study the effect ofbuoyancy on turbulent boundary layers for a wide range of stability, thevelocity and temperature fluctuations are measured simultaneously at adownwind position of 23.5 m from the tunnel entrance, where the boundarylayer is fully developed. The Reynolds number, Re, ranges from 3.14× 104 to 1.27 × 105, and the bulk Richardson number, Ri,ranges from 0 to 1.33. Stable stratification rapidly suppresses thefluctuations of streamwise velocity and temperature as well as the verticalvelocity fluctuation. Momentum and heat fluxes are also significantlydecreased with increasing stability and become nearly zero in the lowest partof the boundary layer with strong stability. The vertical profiles ofturbulence quantities exhibit different behaviour in three distinct stabilityregimes, the neutral flows, the stratified flows with weak stability(Ri = 0.12, 0.20) and those with strong stability (Ri= 0.39,0.47, 1.33). Of these, the two regimes of stratified flows clearly showdifferent vertical profiles of the local gradient Richardson number Ri,separated by the critical Richardson number Ri cr of about 0.25. Moreover,turbulence quantities in stable conditions are well correlated with Ri.     

Turbulence Structure of Stable Boundary Layers with a Near-Linear Temperature Profile

By using a thermally stratified wind tunnel, we have successfullysimulated stably stratified boundary layers (SBL), in which the meantemperature increases upward almost linearly. We have investigated the flow structure and the effects of near-linearstable stratification on the transfer of momentum and heat. Thevertical profiles of turbulence quantities exhibit different behaviour in two distinct stability regimes of the SBLflows with weak and strong stability. For weak stability cases, theturbulent transfer of momentum and heat is basically similar to that for neutral turbulent boundary layers, although it is weakenedwith increasing stability. For strong stability cases, on the other hand,the time-mean transfer is almost zero over the whole boundary-layer depth.However, the instantaneous turbulent transfer frequently occurs in bothgradient and counter-gradient directions in the lower part of the boundary layer. This is due to the Kelvin–Helmholtz (K–H) shear instability and therolling up and breaking of K–H waves. Moreover, the internal gravity wavesare observed in the middle and upper parts of all stable boundary layers.     

A model of turbulence spectra in the atmospheric surface layer

The spectral equations of turbulent kinetic energy and temperature variance have been solved by using Onsager's energy cascade model and by extending Onsager's model to closure of terms that embody the interaction of turbulent and mean flow.The spectral model yields the following results: In a stably stratified shear flow, the peak wave numbers of the spectra of energy and temperature variance shift toward larger wave numbers as stability increases. In an unstably stratified flow, the peak wave numbers of energy spectra move toward smaller wave numbers as instability increases, whereas the opposite trend is observed for the peak wave numbers of temperature variance spectra. Hence, the peak wave numbers of temperature spectra show a discontinuity at the transition from stable to unstable stratification. At near neutral stratification, both spectra reveal a bimodal structure.The universal functions of the Monin-Obukhov similarity theory are predicted to behave as _m ~ _H ~ (- Z/L)^-1/3 in an extremely unstable stratification and as _m ~ _H ~ z/L in an extremely stable stratification. For a stably stratified flow, a constant turbulent Prandtl number is expected.     

Stable Boundary-Layer Scaling Regimes: The Sheba Data

Turbulent and mean meteorological data collected at five levels on a 20-m tower over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are analyzed to examine different regimes of the stable boundary layer (SBL). Eleven months of measurements during SHEBA cover a wide range of stability conditions, from the weakly unstable regime to very stable stratification. Scaling arguments and our analysis show that the SBL can be classified into four major regimes: (i) surface-layer scaling regime (weakly stable case), (ii) transition regime, (iii) turbulent Ekman layer, and (iv) intermittently turbulent Ekman layer (supercritical stable regime). These four regimes may be considered as the basic states of the traditional SBL. Sometimes these regimes, especially the last two, can be markedly perturbed by gravity waves, detached elevated turbulence (‘upside down SBL’), and inertial oscillations. Traditional Monin–Obukhov similarity theory works well in the weakly stable regime. In the transition regime, Businger–Dyer formulations work if scaling variables are re-defined in terms of local fluxes, although stability function estimates expressed in these terms include more scatter compared to the surface-layer scaling. As stability increases, the near-surface turbulence is affected by the turning effects of the Coriolis force (the turbulent Ekman layer). In this regime, the surface layer, where the turbulence is continuous, may be very shallow (< 5 m). Turbulent transfer near the critical Richardson number is characterized by small but still significant heat flux and negligible stress. The supercritical stable regime, where the Richardson number exceeds a critical value, is associated with collapsed turbulence and the strong influence of the earth’s rotation even near the surface. In the limit of very strong stability, the stress is no longer a primary scaling parameter.     

Water-tank experiment on the thermal circulation induced by the bottom heating in an asymmetric valley

Water tank experiments were carried out to investigate the thermal convection due to the bottom heating in an asymmetrical valley under neutral and stably stratified approach flows with the Particle Image Velometry (PIV) visualization technique. In the neutral stratification approach flow, the ascending draft induced by bottom heating is mainly located in the center of the valley in calm ambient wind. However~with ambient wind flow, the thermal convection is shifted leeward, and the descending draft is located on the leeward side of the valley, while the ascending draft is located on the windward side. The descending draft is minorly turbulent and organized, while the ascending draft is highly turbulent. With the increase of the towing speed, the descending and ascending drafts induced by the mechanical elevation begin to play a more dominant role in the valley flow, while the role of the thermal convection in the valley airflow becomes limited. In the stable stratification approach flow, the thermal convection is limited by the stable stratification and no distinct circulation is formed in calm ambient wind. With ambient wind, agravity wave appears in the upper layer in the valley. With the increase of the ambient wind speed, a gravity wave plays an important role in the valley flow, and the location and intensity of the thermal convection are also modulated by the gravity internal waves. The thermal convection has difficulty penetrating the upper stable layer. Its exchange is limited between the air in the upper layer and that in the lower layer in the valley, and it is adverse to the diffusion of pollutants in the valley.     

Wind-Tunnel Study Of Atmospheric Stable Boundary Layers Over A Rough Surface

Wind-tunnel simulations of theatmospheric stable boundary layer (SBL) developedover a rough surface were conducted by using athermally stratified wind tunnel at the Research Institutefor Applied Mechanics (RIAM), Kyushu University. Thepresent experiment is a continuation of the workcarried out in a wind tunnel at Colorado StateUniversity (CSU), where the SBL flows were developed over asmooth surface. Stably stratified flows were createdby heating the wind-tunnel airflow to a temperature ofabout 40–50°and by cooling the test-section floor toa temperature of about 10°. To simulate therough surface, a chain roughness was placed over thetest-section floor. We have investigated the buoyancyeffect on the turbulent boundary layer developed overthis rough surface for a wide range of stability,particularly focusing on the turbulence structure andtransport process in the very stable boundary layer.The present experimental results broadly confirm theresults obtained in the CSU experiment with the smoothsurface, and emphasizes the following features: thevertical profiles of turbulence statistics exhibitdifferent behaviour in two distinct stability regimes with weak and strong stability,corresponding to the difference in the verticalprofiles of the local Richardson number. The tworegimes are separated by the critical Richardsonnumber. The magnitudes in turbulence intensities andturbulent fluxes for the weak stability regime aremuch greater than those of the CSU experiments becauseof the greater surface roughness. For the very stableboundary layer, the turbulent fluxes of momentum andheat tend to vanish and wave-like motions due to theKelvin–Helmholtz instability and the rolling up andbreaking of those waves can be observed. Furthermore,the appearance of internal gravity waves is suggestedfrom cross-spectrum analyses.     

Flow splitting at the bifurcation between two valleys: idealized simulations in comparison with Mesoscale Alpine Programme observations

Summary This paper investigates the characteristics of channelled airflow in the vicinity of a junction of three idealized valleys (one valley carrying the incoming flow and two tributaries carrying the outflow), using a two-dimensional single-layer shallow water model. Particular attention is given to the flow splitting occurring at the junction. Nondimensionalized, the model depends on the valley geometry, the Reynolds number, which is related to the eddy viscosity, and on the difference of the hydrostatic pressure imposed at the exit of the tributaries. At the spatial scale considered in this study, the Rossby number relating the inertial and Coriolis forces is always larger than 1, implying that the effect of earth rotation can be neglected to a first approximation. The analysis of the flow structure within the three valleys as well as the calculation of the split ratio (fraction of the air flow diverted into one of the two downstream valleys with respect to the total mass flux in the upstream valley) show that (i) the flow pattern depends strongly on the Reynolds number while the split ratio is comparatively insensitive; (ii) the valley geometry and the difference between the upstream and downstream hydrostatic pressures affect the flow pattern, the location of the split point and the split ratio; (iii) the relative contribution of flow deflection by the sidewalls and the blocking/splitting mechanism differs between the settings of a “Y-shape” valley and a “T-shape” valley. Quantitative comparison of the present results with numerical simulations of realistic cases and with observations collected in the region of the Rhine and Seez valleys (Switzerland) (“Y-shape” valley) and in the region of the Inn and Wipp valleys (Austria) (“T-shape” valley) during the Mesoscale Alpine Programme (MAP) field experiment shows good agreement provided that the normalized valley depth NΔH/U_u significantly exceeds 1, i.e., when “flow around” is expected. A structural disagreement between the idealized simulations and the observed wind field is found only when NΔH/U_u ≃ 1, that is, in the “flow over” regime. This shows that the dimensionless valley depth is indeed a good indicator for flow splitting, implying that the stratification is a key player in reality.     

The structure of the stably stratified internal boundary layer in offshore flow over the sea

Observations obtained mainly from a research aircraft are presented of the mean and turbulent structure of the stably stratified internal boundary layer (IBL) over the sea formed by warm air advection from land to sea. The potential temperature and humidity fields reveal the vertical extent of the IBL, for fetches out to several hundred of kilometres, geostrophic winds of 20–25 m s^–1, and potential temperature differences between undisturbed continental air and the sea surface of 7 to 17 K. The dependence of IBL depth on these external parameters is discussed in the context of the numerical results of Garratt (1987), and some discrepancies are noted.Wind observations show the development of a low-level wind maximum (wind component normal to the coast) and rotation of the wind to smaller cross-isobar flow angles. Potential temperature () profiles within the IBL reveal quite a different structure to that found in the nocturnal boundary layer (NBL) over land. Over the sea, profiles have large positive curvature with vertical gradients increasing monotonically with height; this reflects the dominance of turbulent cooling within the layer. The behaviour is consistent with known behaviour in the NBL over land where curvature becomes negative (vertical gradients of decreasing with height) as radiative cooling becomes dominant.Turbulent properties are discussed in terms of non-dimensional quantities, normalised by the surface friction velocity, as functions of normalised height using the IBL depth. Vertical profiles of these and the normalised wavelength of the spectral maximum agree well with known results for the stable boundary layer over land (Caughey et al., 1979).     

Impact of the Diurnal Cycle of the Atmospheric Boundary Layer on Wind-Turbine Wakes: A Numerical Modelling Study

The wake characteristics of a wind turbine for different regimes occurring throughout the diurnal cycle are investigated systematically by means of large-eddy simulation. Idealized diurnal cycle simulations of the atmospheric boundary layer are performed with the geophysical flow solver EULAG over both homogeneous and heterogeneous terrain. Under homogeneous conditions, the diurnal cycle significantly affects the low-level wind shear and atmospheric turbulence. A strong vertical wind shear and veering with height occur in the nocturnal stable boundary layer and in the morning boundary layer, whereas atmospheric turbulence is much larger in the convective boundary layer and in the evening boundary layer. The increased shear under heterogeneous conditions changes these wind characteristics, counteracting the formation of the night-time Ekman spiral. The convective, stable, evening, and morning regimes of the atmospheric boundary layer over a homogeneous surface as well as the convective and stable regimes over a heterogeneous surface are used to study the flow in a wind-turbine wake. Synchronized turbulent inflow data from the idealized atmospheric boundary-layer simulations with periodic horizontal boundary conditions are applied to the wind-turbine simulations with open streamwise boundary conditions. The resulting wake is strongly influenced by the stability of the atmosphere. In both cases, the flow in the wake recovers more rapidly under convective conditions during the day than under stable conditions at night. The simulated wakes produced for the night-time situation completely differ between heterogeneous and homogeneous surface conditions. The wake characteristics of the transitional periods are influenced by the flow regime prior to the transition. Furthermore, there are different wake deflections over the height of the rotor, which reflect the incoming wind direction.     

The vertical structure of summertime local winds in the Wright Valley,Antarctica

Intensive observations of summertime up- and down-valley winds in a dry valley utilising airsondes, pilot balloons and a monostatic acoustic sounder are described. Both circulations show a distinctive layered vertical wind and temperature structure. Westerly down-valley flow is typically neutral and is characterised by strong surface winds overlain by light variable winds extending to an inversion between 2000 and 4000m in depth. Above this inversion, gradient winds prevail. This structure is similar to that of downslope winds observed elsewhere. The thermally-induced up-valley easterly flow is shown to be extremely well-developed in terms of its strength, depth and persistence. The strong surface easterly may reach 800 m in depth and usually undercuts the warmer westerly. The boundary between the two regimes is marked by an inversion. During easterly flow a surface-based, super-adiabatic layer of 100–200 m in depth is evident and is associated with weak convective activity. An intriguing aspect of the wind regime is the interaction between the easterly and westerly circulations in the valley. These are separated at the surface by a frontal zone which migrates up and down the valley. Further observational and modelling studies are recommended.     

Global Intermittency and Collapsing Turbulence in the Stratified Planetary Boundary Layer

Direct numerical simulation of the turbulent Ekman layer over a smooth wall is used to investigate bulk properties of a planetary boundary layer under stable stratification. Our simplified configuration depends on two non-dimensional parameters: a Richardson number characterizing the stratification and a Reynolds number characterizing the turbulence scale separation. This simplified configuration is sufficient to reproduce global intermittency, a turbulence collapse, and the decoupling of the surface from the outer region of the boundary layer. Global intermittency appears even in the absence of local perturbations at the surface; the only requirement is that large-scale structures several times wider than the boundary-layer height have enough space to develop. Analysis of the mean velocity, turbulence kinetic energy, and external intermittency is used to investigate the large-scale structures and corresponding differences between stably stratified Ekman flow and channel flow. Both configurations show a similar transition to the turbulence collapse, overshoot of turbulence kinetic energy, and spectral properties. Differences in the outer region resulting from the rotation of the system lead, however, to the generation of enstrophy in the non-turbulent patches of the Ekman flow. The coefficient of the stability correction function from Monin–Obukhov similarity theory is estimated as \(\beta \approx 5.7\) in agreement with atmospheric observations, theoretical considerations, and results from stably stratified channel flows. Our results demonstrate the applicability of this set-up to atmospheric problems despite the intermediate Reynolds number achieved in our simulations.     

A Hierarchy of Energy- and Flux-Budget (EFB) Turbulence Closure Models for Stably-Stratified Geophysical Flows

Here we advance the physical background of the energy- and flux-budget turbulence closures based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy, and a new relaxation equation for the turbulent dissipation time scale. The closure is designed for stratified geophysical flows from neutral to very stable and accounts for the Earth’s rotation. In accordance with modern experimental evidence, the closure implies the maintaining of turbulence by the velocity shear at any gradient Richardson number Ri, and distinguishes between the two principally different regimes: “strong turbulence” at ${Ri \ll 1}$ typical of boundary-layer flows and characterized by the practically constant turbulent Prandtl number Pr _T; and “weak turbulence” at Ri > 1 typical of the free atmosphere or deep ocean, where Pr _T asymptotically linearly increases with increasing Ri (which implies very strong suppression of the heat transfer compared to the momentum transfer). For use in different applications, the closure is formulated at different levels of complexity, from the local algebraic model relevant to the steady-state regime of turbulence to a hierarchy of non-local closures including simpler down-gradient models, presented in terms of the eddy viscosity and eddy conductivity, and a general non-gradient model based on prognostic equations for all the basic parameters of turbulence including turbulent fluxes.     

Some aspects of determining the stable boundary layer depth from sodar data

The question of estimating the height of the stable boundary layer (SBL) based on digitalized vertical profiles of sodar signal intensity has been re-examined. A simple one-dimensional numerical boundary-layer model is used to compute vertical profiles of the temperature structure parameterC _T ² . It is shown that especially at the beginning of the night (when stratification is weak) one can not expect any significant profile structure in the upper part of the SBL if its depth is determined in terms of common turbulent height scales. From this it is concluded that the SBL-height will be underestimated early in the night when derived from the maximum gradient in the signal intensity profiles. Later in the night in contrast, the computations often show elevated maxima or even zones with reduced, and above them enhanced, vertical gradients ofC _T ² , from which a SBL-height can be deduced that compares well with other common height scales. The computed profiles ofC _T ² are shown to be in qualitative agreement with observed profiles of sodar signal intensity for several analysed cases from the HAPEX-MOBILHY experiment.Comparing different SBL-depth scales with sodar observations, it is demonstrated that most of them are often closely related to a sodar-derived SBL-height only during certain phases of the night. Thus the sodar-SBL-height can, after a transition period, be perhaps associated with the lower turbulent layer of the growing surface inversion during the first part and with the height of the low-level wind maximum during the second part of the night.     

In the Core of the Night-Effects of Intermittent Mixing on a Horizontally Heterogeneous Surface

Intermittent breakdowns that accompany wind gusts at the surface are responsible for a large fraction of the turbulent exchange between the surface and the upper boundary layer in the core of clear nights. Vertical and horizontal structure of the breakdowns are investigated using data from a network of 26 stations in an area of 30 km × 30 km. Surface heterogeneity in the area includes complex terrain with different types of land cover. We treat the fine-scale landscape structure near sensors (sheltering) as a separate component of heterogeneity. These features have important consequences on the spatial distribution of mean variables and surface fluxes. We found that breakdowns connect the surface layer to a higher level (level H_C). Weak wind gusts below a threshold (approximately 1.5 m s^-1) mix the air down to the colder ground, cooling the surface layer. On the other hand, wind gusts above this threshold promote mixing with upper levels, warming the surface layer. The spatial maximum of surface temperature over the network can be used as an estimate of the temperature at H_C, allowing vertical gradients and stability to be approximated. Minimum temperature is a function of topography and sheltering. Appreciable surface fluxes at night occur primarily at high, open locations, and can be large enough there to influencearea-averaged values. Surface-fluxparameterizations currently used in mesoscale models were tested first by estimating fluxes at each station and aggregating, and then by formingarea-averages before estimating fluxes. Results show that these formulations underestimate the average surface fluxes over a region for most of the nights.     

A numerical study of boundary-layer dynamics in a mountain valley

A higher order closure model is applied to simulate the dynamics in an area with a deep valley characterized by complex terrain in the southwestern US. The simulation results show generally good agreement with measured profiles at two locations within the valley. Both the measurements and the simulations indicate that the flow dynamics in the area are highly influenced by the topography and meandering of the valley, and can be resolved only by the full three-dimensional model code. The wind veering simulated over the range of the topographic elevations is often larger than 100 deg and in some cases as large as 180 deg, as a consequence of topographic forcing. In the case of an infinitely long valley, as is assumed in two-dimensional test simulations, a strong low-level jet occurs within the valley during stable conditions. The jet is mainly a consequence of the Coriolis effect. However, the jet development is significantly reduced due to asymmetric effects of the actual topography treated in the three-dimensional simulations. Tests with the two-dimensional nonhydrostatic version of the model show significant wave responses for a stable stratified flow over the valley. The structure resembles nonlinear mesoscale lee waves, which are intrinsically nonhydrostatic. However, considering the three-dimensional nature of this valley system, a better understanding and verification of the nonhydrostatic effects requires both a three-dimensional nonhydrostatic numerical model and an observational data set which is fully representative in all three dimensions.List of symbols (unless otherwise defined in the text) B ₁ closure constant - f Coriolis parameter - g acceleration of gravity - K _M ,K _H ,K _R turbulent exchange coefficients for momentum, heat and moisture - k von Karman constant - L Monin-Obukhov length - q ² twice the turbulent kinetic energy - R specific humidity - s height of the model top - T _g ground surface temperature - t time - U, V horizontal components of wind - U _g ,V _g geostrophic wind components - u, w perturbation components ofU andW wind components - u _* friction velocity - W vertical wind component in the terrain-following coordinates - x, y horizontal coordinates - Z actual height above sea level - z actual height above ground - z ₀ roughness length - z _g terrain height - z _i depth of the convective boundary layer - ₁ closure constant - coefficient of thermal expansion - height in the terrain-following coordinate - master length scale in the turbulent parameterization - scaled pressure (Exner function) - potential temperature - _m normalized vertical wind shear