On the similarity of the characteristics of turbulence in an unstable boundary layer

The statistics of turbulence, such as the standard deviation of fluctuating velocities, in an unstable atmospheric boundary layer are assumed to be characterized by the combination of three specific lengths, Monin-Obukhov length L, observation height z and the height of mixing layer h. Unlike Monin-Obukhov similarity, even near the ground the effect of h is taken into account. According to observation, the length scale of the vertical velocity is proportional to z at least near the ground, but the lateral component depends mostly on h alone. The length scale of the longitudinal component depends on z and h.     

An Improvement of the Louis Scheme for the Surface Layer in an Atmospheric Modelling System

The Louis scheme for calculating the vertical eddy fluxes within the atmospheric surface layer is improved by broadening the original assumptions. In our approach, the momentum and heat transfer roughness lengths (z₀ and z_T respectively) can be different, and z₀ need not be negligibly small compared with the lowest height (z) in modelling. For these conditions, we choose more consistent wind and potential temperature profile forms, then derive new algorithms for calculating fluxes. Improvement is demonstrated for a wide range of z/L (L is the Obukhov length), z/z₀ and z₀ z_T, by comparing these fluxes with those derived from a theoretical surface-layer model. The improved algorithms can be used in atmospheric modelling systems for more varied surfaces and a wide range of atmospheric stability.     

Models of eddy viscosity for numerical simulation of horizontally inhomogeneous,neutral surface-layer flow

Modification of a turbulent flow due to a change from a smooth to a rough surface has been studied by means of a stream function-vorticity model. Results of four models of eddy viscosity (or turbulent exchange coefficient) K _mhave been compared. The models are: (1) K _m = l²S, where l is the mixing length and S is the deformation of mean flow; (2) K _m E/S, which is based on the assumption that turbulent momentum flux is proportional to turbulent kinetic energy E; (3) K _m lE^1/2, the so called Prandtl-Kolmogoroff approach; and (4) K _m E²/, the E — closure, where is the dissipation of turbulent kinetic energy.It is found that net-production, i.e., the difference of production and dissipation of turbulent kinetic energy counteracts the influence of mean shear on turbulent shear stress and diminishes turbulent shear stress. The reduction of mixing-length, being predicted by Model 4 only, adds to this attenuation. As a consequence, in Models 2 and 4, loss of horizontal mean momentum is concentrated close to the ground, which results in an inflexion point in the logarithmic, vertical profile of horizontal mean velocity. By contrast, in Models 1 and 3, modification of turbulent shear stress reaches larger heights causing deeper internal boundary layers. Concerning the existence of an inflexion point in U(lnz), the depth of the internal boundary layer for mean velocity, and the modification of bottom shear stress, Model 4 comes closest to experimental data.A remarkable difference of Models 1, 2, 3 and Model 4 is that only Model 4 predicts a very slow relaxation of eddy viscosity which can be attributed to the reduction of mixing-length.     

Zero-plane displacement and roughness length for tall vegetation,derived from a simple mass conservation hypothesis

A method for the determination of the zero-plane displacement, d, and roughness length, z ₀, for tall vegetation is described. A new relationship between d and z ₀ is developed by imposing the condition of mass conservation on the logarithmic wind profile. Further, d and z ₀ can be evaluated directly if independent measurements of friction velocity are available in addition to wind profile measurements. The proposed method takes into account the existence of a transition layer immediately above the vegetation where the logarithmic wind profile law is not valid. Only one level of wind speed measurements is necessary within the inertial sub-layer.The method is applied to wind profile and eddy correlation measurements taken in and above an 18.5 m pine forest to yield d = 12.7 m and z ₀ = 1.28 m. The choice of height for the upper level of measurement and problems with measuring canopy flow are discussed.Work carried out while on leave at the Institute of Hydrology.     

Regional roughness of the landes forest and surface shear stress under neutral conditions

Mean wind velocity profiles were measured by means of radio-windsondes over the Landes region in southwestern France, which consists primarily of pine forests with scattered villages and clearings with various crops. Analysis of neutral profiles indicated the existence of a logarithmic layer between approximately z – d ₀ = 67(±18)z ₀ and 128(+-32)z ₀ (z is the height above the ground, z ₀ the surface roughness and d ₀ the displacement height). The upper limit can also be given as z – d ₀ = 0.33 (±0.18)h, where h is the height of the bottom of the inversion. The profiles showed that the surface roughness of this terrain is around 1.2 m and the displacement height 6.0 m. Shear stresses derived from the profiles were in good agreement with those obtained just above the forest canopy at a nearby location with the eddy correlation method by a team from the Institute of Hydrology (Wallingford, England).     

Effects of Wall Heating on Flow Characteristics in a Street Canyon

We develop a large-eddy simulation (LES) model based on a meteorological numerical model for a real scale street-canyon flow with rough building facets heated by a given temperature. The model is applied to a canyon with the aspect ratio of unity for two idealized heating scenarios: (1) the roof and the entire upstream wall are heated, named as ‘assisting cases’, and (2) the roof and the entire downstream wall are heated, named as ‘opposing cases’. These facets were heated up to 15 K above the air temperature. A wall function for temperature is proposed for a rough facet with an assumption that the thermal roughness length, z _0T, is much smaller than the aerodynamic roughness length, z ₀. It is demonstrated that the sensible heat flux and canyon-air temperature are significantly influenced by the near-facet process that is parametrized by z _0T as the primary factor; other processes such as in-canyon mixing and roof-level exchange are secondary. This new finding strongly suggests that it is vital to choose an appropriate value of z _0T in a numerical simulation of street-canyon flows with the facet-air exchange processes of heat or any scalar. The finding also raises an awareness of the demand for carefully designed laboratory or field experiments of quantifying z _0T values for various urban surfaces. For the opposing cases, an unsteady penetrating narrow updraft zone appears occasionally along the heated wall and this feature is consistent field observations. The unique result indicates the superior capability of LES. The results of this study can be used to guide the parametrization of turbulent processes inside the urban canopy layer.     

Coherent eddies and turbulence in vegetation canopies: The mixing-layer analogy

This paper argues that the active turbulence and coherent motions near the top of a vegetation canopy are patterned on a plane mixing layer, because of instabilities associated with the characteristic strong inflection in the mean velocity profile. Mixing-layer turbulence, formed around the inflectional mean velocity profile which develops between two coflowing streams of different velocities, differs in several ways from turbulence in a surface layer. Through these differences, the mixing-layer analogy provides an explanation for many of the observed distinctive features of canopy turbulence. These include: (a) ratios between components of the Reynolds stress tensor; (b) the ratio K _H/K _M of the eddy diffusivities for heat and momentum; (c) the relative roles of ejections and sweeps; (d) the behaviour of the turbulent energy balance, particularly the major role of turbulent transport; and (e) the behaviour of the turbulent length scales of the active coherent motions (the dominant eddies responsible for vertical transfer near the top of the canopy). It is predicted that these length scales are controlled by the shear length scale % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa% aaleaacaWGtbaabeaakiabg2da9iaadwfacaGGOaGaamiAaiaacMca% caGGVaGabmyvayaafaGaaiikaiaadIgacaGGPaaaaa!3FD0!\[L_S = U(h)/U'(h)\] (where h is canopy height, U(z) is mean velocity as a function of height z, and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyvayaafa% Gaeyypa0JaaeizaiaadwfacaGGVaGaaeizaiaadQhaaaa!3C32!\[U' = {\rm{d}}U/{\rm{d}}z\]). In particular, the streamwise spacing of the dominant canopy eddies is _x = mL _s, with m = 8.1. These predictions are tested against many sets of field and wind-tunnel data. We propose a picture of canopy turbulence in which eddies associated with inflectional instabilities are modulated by larger-scale, inactive turbulence, which is quasi-horizontal on the scale of the canopy.     

Turbulence Structure of the Unstable Atmospheric Surface Layer and Transition to the Outer Layer

We present a new model of the structure of turbulence in the unstable atmospheric surface layer, and of the structural transition between this and the outer layer. The archetypal element of wall-bounded shear turbulence is the Theodorsen ejection amplifier (TEA) structure, in which an initial ejection of air from near the ground into an ideal laminar and logarithmic flow induces vortical motion about a hairpin-shaped core, which then creates a second ejection that is similar to, but larger than, the first. A series of TEA structures form a TEA cascade. In real turbulent flows TEA structures occur in distorted forms as TEA-like (TEAL) structures. Distortion terminates many TEAL cascades and only the best-formed TEAL structures initiate new cycles. In an extended log layer the resulting shear turbulence is a complex, self-organizing, dissipative system exhibiting self-similar behaviour under inner scaling. Spectral results show that this structure is insensitive to instability. This is contrary to the fundamental hypothesis of Monin--Obukhov similarity theory. All TEAL cascades terminate at the top of the surface layer where they encounter, and are severely distorted by, powerful eddies of similar size from the outer layer. These eddies are products of the breakdown of the large eddies produced by buoyancy in the outer layer. When the outer layer is much deeper than the surface layer the interacting eddies are from the inertial subrange of the outer Richardson cascade. The scale height of the surface layer, z _s, is then found by matching the powers delivered to the creation of emerging TEAL structures to the power passing down the Richardson cascade in the outer layer. It is z _s = u _* ³ /k_s, where u _* is friction velocity, k is the von Kármán constant and _s is the rate of dissipation of turbulence kinetic energy in the outer layer immediately above the surface layer. This height is comparable to the Obukhov length in the fully convective boundary layer. Aircraft and tower observations confirm a strong qualitative change in the structure of the turbulence at about that height. The tallest eddies within the surface layer have height z _s, so z _s is a new basis parameter for similarity models of the surface layer.     

Turbulent dispersion in the atmospheric surface layer

By non-dimensionalizing a trajectory-simulation (TS) model of turbulent dispersion, it is shown that the dimensionless concentration z ₀cu_*/kQ (cu _*/kQ) due to a continuous line (area) source of strength Q in the atmospheric surface layer depends only on z/z ₀, x/z ₀, z ₀/L and z _s/z₀, where z _s is the source height. The TS model is used to tabulate concentration profiles due to ground-level line and area sources. Concentration profiles generated by the TS model for elevated sources are shown to be inconsistent with the Reciprocal Theorems of Smith (1957) and it is suggested that this is because the flux-mean gradient closure scheme inherent in the Reciprocal Theorem is invalid for an elevated source.     

A Case Study Of An On-Ice Air Flow Over The Arctic Marginal Sea-Ice Zone

A case study of warm air advection over the Arctic marginalsea-ice zone is presented, based on aircraft observations with direct flux measurements carriedout in early spring, 1998. A shallow atmospheric boundary layer (ABL) was observed, which wasgradually cooling with distance downwind of the ice edge. This process was mainly connected with astrong stable stratification and downward turbulent heat fluxes of about 10–20 W m^-2, but wasalso due to radiative cooling. Two mesoscale models, one hydrostatic and the other non-hydrostatic,having different turbulence closures, were applied. Despite these fundamental differences betweenthe models, the results of both agreed well with the observed data. Various closure assumptions had amore crucial influence on the results than the differences between the models.Such an assumption was, for example,the parameterization of the surface roughness for momentum (z₀) and heat (z_T). This stronglyaffected the wind and temperature fields not only close to the surface but also within and abovethe temperature inversion layer. The best results were achieved using a formulation for z₀ that took intoaccount the form drag effect of sea-ice ridges together withz_T = 0.1z₀. The stability within theelevated inversion strongly depended on the minimum eddy diffusivity K_min. A simple ad hocparameterization seems applicable, where K_min is calculated as 0.005 timesthe neutral eddy diffusivity. Although the longwave radiative cooling was largest within the ABL, theapplication of a radiation scheme was less important there than above the ABL. This was related to theinteraction of the turbulent and radiative fluxes. To reproduce the strong inversion, it wasnecessary to use vertical and horizontal resolutions higher than those applied in most regional andlarge-scale atmospheric models.     

The impact of the PBL scheme and the vertical distribution of model layers on simulations of Alpine foehn

Summary This paper investigates the influence of the planetary boundary-layer (PBL) parameterization and the vertical distribution of model layers on simulations of an Alpine foehn case that was observed during the Mesoscale Alpine Programme (MAP) in autumn 1999. The study is based on the PSU/NCAR MM5 modelling system and combines five different PBL schemes with three model layer settings, which mainly differ in the height above ground of the lowest model level (z ₁). Specifically, z ₁ takes values of about 7 m, 22 m and 36 m, and the experiments with z ₁ = 7 m are set up such that the second model level is located at z = 36 m. To assess if the different model setups have a systematic impact on the model performance, the simulation results are compared against wind lidar, radiosonde and surface measurements gathered along the Austrian Wipp Valley. Moreover, the dependence of the simulated wind and temperature fields at a given height (36 m above ground) on z ₁ is examined for several different regions. Our validation results show that at least over the Wipp Valley, the dependence of the model skill on z ₁ tends to be larger and more systematic than the impact of the PBL scheme. The agreement of the simulated wind field with observations tends to benefit from moving the lowest model layer closer to the ground, which appears to be related to the dependence of lee-side flow separation on z ₁. However, the simulated 2 m-temperatures are closest to observations for the intermediate z ₁ of 22 m. This is mainly related to the fact that the simulated low-level temperatures decrease systematically with decreasing z ₁ for all PBL schemes, turning a positive bias at z ₁ = 36 m into a negative bias at z ₁ = 7 m. The systematic z ₁-dependence is also observed for the temperatures at a fixed height of 36 m, indicating a deficiency in the self-consistency of the model results that is not related to a specific PBL formulation. Possible reasons for this deficiency are discussed in the paper. On the other hand, a systematic z ₁-dependence of the 36-m wind speed is encountered only for one out of the five PBL schemes. This turns out to be related to an unrealistic profile of the vertical mixing coefficient. Correspondence: Günther Z?ngl, Meteorologisches Institut der Universitat München, 80333 München, Germany     

Formulation of the thermal internal boundary layer in a mesoscale model

Mesoscale models using a non-local K-scheme for parameterization of boundary-layer processes require an estimate of the planetary boundary layer (PBL) height z _i at all times. In this paper, two-dimensional sea-breeze experiments are carried out to evaluate three different formulations for the advective contribution in the z _i prognostic equation of Deardorff (1974).Poor representation of the thermal internal boundary layer in the sea breeze is obtained when z _i is advected by the wind at level z _i . However, significantly better results are produced if the mean PBL wind is used for the advecting velocity, or if z _i is determined simply by checking for the first sufficiently stable layer above the ground.A Lagrangian particle model is used to demonstrate the effect of each formulation on plume dispersion by the sea breeze.     

Aerodynamic roughness of vegetated surfaces

Available experimental results indicate that as the density of roughness elements over a horizontally homogeneous surface is varied, the roughness length, z ₀, varies in a manner that exhibits a maximum at intermediate density values. In an attempt to explain this behaviour, the available analytical solutions for the wind profile inside dense homogeneous canopies were reviewed. The review indicated that the variation of z ₀ with density depends on the interrelationship between the leaf density, a, and the mixing length, l. In view of this finding, a numerical model was devised based on a simple rule for constructing mixing-length profiles in the canopy. The rule states that the actual value of l is the maximum possible under the two constraints: l l _i and ¦dl/dz¦ k, where k is the von Karman constant and the intrinsic mixing length, l _i, is a function of the local internal structure of the canopy. The model which ensures a smooth transition from dense to thin canopy, was used to reproduce the observed maximum of z ₀. The model is also capable of handling vertically non-homogeneous canopies.     

On the effects of vertical eddy viscosity on rossby and eady waves in the ocean

The effects of vertical eddy viscosity on simple mesoscale waves in the ocean are studied. The decay of Rossby waves is investigated by one-dimensional depth-dependent linear stability problems which are derived for the interior non-viscous or viscous quasigeostrophic flow using parameterizations of the top and bottom boundary layers corresponding to Ekman suction, no-stress and bottom-stress boundary conditions.The non-slip condition at the bottom yielding an O(E_v^1/2)-Ekman layer causes very short damping times for the 0th Rossby mode. This suggests that this boundary condition is not suitable for mesoscale wave studies, because a Rossby wave fit for the MODE eddy can be done satisfactorily without any damping. Reasonable results for damping times of Rossby waves are obtained by prescribing the bottom stress, resulting from the constant-stress layer at the bottom, and the free-slip condition at the surface. The growth rates of Eady waves are reexamined using this bottom-stress condition.Vertical viscosity in the interior of the ocean, e.g. internal wave induced viscosity, may have a significant influence on the dynamics of the mesoscale motions, comparable to that of the boundary layers in some cases. The results are compatible with the sparse observations available.     

Intermittent wind close to the ground within a grass canopy

Wind speed was measured at a height of 1 cm above the ground and at several other heights in and above a canopy of tall fescue grass (Festuca arundinacea) using single hot-wire and triple hot-film anemometers. The plant area density in the canopy was concentrated close to the ground, with 75% of the plant area standing belowz=15 cm, wherez is height above the ground. The frequency distributions of horizontal wind speeds,s, were sharply skewed towards positive values at all measurement heights, but were most highly skewed near the ground where the coefficient of skewness ranged from 1.6 to 2.9. Above mid-canopy height, the frequency distribution ofs was described reasonably well by a Gumbel extreme value distribution. Average wind speed,S, decreased exponentially with depth into the canopy with an exponential scale length of abouth/2.8, whereh is the height of the canopy. Atz=1 cm, the value ofS was about 11% of the surface-layeru _*. The standard deviation of the fluctuations of the vertical and horizontal components of the wind speed also decreased exponentially with depth inside the canopy with a scale length of abouth/2.5.Inside the canopy, the Eulerian integral time scales for the vertical ( _w ) and horizontal ( _u ) components of wind speed were about 0.1 s and 1.0 s, respectively, and were approximately constant with height. Above the canopy, these time scales increased sharply and, atz=2.25h, _w and _u were approximately 1.0 and 3.0s, respectively. Turbulence length scales in the vertical and downwind directions, _u and _w ·U, respectively, were approximately 1 cm for heights between 1 to 10 cm above the ground inside the canopy, while atz=2.25h, they were about 55 cm and 277 cm. Relatively quiescent periods (lulls) in the air close to the ground were interrupted frequently by gusts. The frequency of occurrence of gusts appears to be correlated with the value of the local shear near the top of the canopy.     

A model of airflow above changes in surface heat flux,temperature and roughness for neutral and unstable conditions

A numerical model of airflow in the lowest 50–100 m of the atmosphere above changes in surface roughness and temperature or heat flux has been developed based on boundary layer approximations, the Businger-Dyer hypotheses for the non-dimensional wind shear and heat flux and a mixing length hypothesis.Results have been obtained for several situations, in particular, airflow with neutral upstream conditions encountering a step change in surface temperature or heat flux with no roughness change. In these cases large increases in shear stress at the outer edge of the internal boundary layer are predicted. The case of unstable upstream flow encountering a step change to zero heat flux is also considered.Two situations that may be encountered near the shores of the Great Lakes are considered.Notation B Businger-Dyer constant (= 16.0) in form for _M, _H - c _p Specific heat at constant pressure - g Acceleration due to gravity - H Upward vertical heat flux - H ₀ , H ₁ Surface heat fluxes for x < 0, x 0 - k von Kármán's constant ( = 0.4) - l Mixing length - L Monin-Obukhov length - L ₀ Upstream value of L - m Ratio of roughness lengths (= z ₁/z ₀) - RL ^* Non-dimensional parameter, see Equations (20, 22 and 24) - RL ₁ ^* Same as RL ^* but with z ₁ scaling (= mRL ^*) - T Scaled temperature - T ₀ (z) Upstream temperature profile - u ₀, u ₁(x) Surface friction velocities for x < 0, x 0 - U, W Horizontal and vertical mean velocities - U ₀ (z) Upstream velocity profile - x, z Horizontal and vertical coordinates - z _i Local roughness length     

An Alternative Eddy-Viscosity Model for the Horizontally Uniform Atmospheric Boundary Layer

This paper explores the utility of specifying the eddy viscosity for the horizontally uniform boundary layer as the product of the variance of vertical velocity and an empirical time scale τ _w , as opposed to the more usual formulation where k is the turbulent kinetic energy (TKE), λ _k is a length scale and α is a dimensionless coefficient. Simulations were compared with the observations on Day 33 of the Wangara experiment, and with a plausible specification of τ _w (or λ _k ) each model simulated convective boundary-layer development reasonably well, although the closure produced a more realistic width for the entrainment layer. Under the light winds of Day 33, and with the onset of evening cooling, an excessively shallow and strongly-stratified nocturnal inversion developed, and limited its own further deepening. Boundary-layer models that neglect radiative heat transport and parametrize convective transport by eddy viscosity closure are prone to this runaway (unstable) feedback when forced by a negative (i.e. downward) surface flux of sensible heat.     

An Improved Ekman Layer Approximation for Smooth Eddy Diffusivity Profiles

The Ekman boundary-layer model is extended analytically for a gradually varying eddy diffusivity K(z) ≥ 0, z ≥ 0. A solution for the Ekman layer is provided having similar structure to the constant-K case; that is, exponentially decaying sine functions for the two horizontal wind components. The analytical asymptotic solution compares well with its numerical counterpart for various K(z). The result can be useful in theoretical studies such as Ekman pumping, for efficient estimation of the Ekman layer profiles in various analyses with near-neutral stratifications, or for a rapid initialization of mesoscale models.     

The Apsley and Castro Limited-Length-Scale $${{k-\varepsilon}}$$ Model Revisited for Improved Performance in the Atmospheric Surface Layer

The limited-length-scale k-e{k-\varepsilon} model proposed by Apsley and Castro for the atmospheric boundary layer (Boundary-Layer Meteorol 83(1):75–98, 1997) is revisited with special attention given to its predictions in the constant-stress surface layer. The original model proposes a modification to the length-scale-governing e{\varepsilon} equation that ensures consistency with surface-layer scaling in the limit of small ℓ _m/ℓ _max (where ℓ _m is the mixing length and ℓ _max its maximum) and yet imposes a limit on ℓ _m as ℓ _m/ℓ _max approaches one. However, within the equilibrium surface layer and for moderate values of z/ℓ _max, the predicted profiles of velocity, mixing length, and dissipation rate using the Apsley and Castro model do not coincide with analytical solutions. In view of this, a general e{\varepsilon} transport equation is derived herein in terms of an arbitrary desired mixing-length expression that ensures exact agreement with corresponding analytical solutions for both neutral and stable stability. From this result, a new expression for C_e3{C_{\varepsilon3}} can be inferred that shows this coefficient tends to a constant only for limiting values of z/L; and, furthermore, that the values of C_e3{C_{\varepsilon3}} for z/L → 0 and z/L →∞ differ by a factor of exactly two.     

A comparison of turbulence statistics in the surface layer over plant canopies with those over several other surfaces

Turbulence statistics, including higher order moments, in the surface layer over plant canopies were compared with those observed over several different surfaces, using a nondimensional height (z – d)/z ₀: The values of (z – d)/z ₀extend over a very wide range from 10 over plant canopies to 10⁷ over the ocean. Several properties such as intensities of turbulence and skewness factors show a remarkable height-dependency in the air layer below (z – d)/z ₀ = 10², which is supposed to be much influenced by the underlying surface. In that layer, some peculiar phenomena, such as a downward energy transport and positive flux of shear stress, are frequently observed.