A theory for the scalar roughness and the scalar transfer coefficients over snow and sea ice

Although the bulk aerodynamic transfer coefficients for sensible (C _H ) and latent (C _E ) heat over snow and sea ice surfaces are necessary for accurately modeling the surface energy budget, they have been measured rarely. This paper, therefore, presents a theoretical model that predicts neutral-stability values of C _H and C _E as functions of the wind speed and a surface roughness parameter. The crux of the model is establishing the interfacial sublayer profiles of the scalars, temperature and water vapor, over aerodynamically smooth and rough surfaces on the basis of a surface-renewal model in which turbulent eddies continually scour the surface, transferring scalar contaminants across the interface by molecular diffusion. Matching these interfacial sublayer profiles with the semi-logarithmic inertial sublayer profiles yields the roughness lengths for temperature and water vapor. When coupled with a model for the drag coefficient over snow and sea ice based on actual measurements, these roughness lengths lead to the transfer coefficients. C _E is always a few percent larger than CH. Both decrease monotonically with increasing wind speed for speeds above 1 m s^–1, and both increase at all wind speeds as the surface gets rougher. Both, nevertheless, are almost always between 1.0 × 10^–3 and 1.5 × 10^–3.     

A simple unified theory for flow in the canopy and roughness sublayer

The mean flow profile within and above a tall canopy is well known to violate the standard boundary-layer flux–gradient relationships. Here we present a theory for the flow profile that is comprised of a canopy model coupled to a modified surface-layer model. The coupling between the two components and the modifications to the surface-layer profiles are formulated through the mixing layer analogy for the flow at a canopy top. This analogy provides an additional length scale—the vorticity thickness—upon which the flow just above the canopy, within the so-called roughness sublayer, depends. A natural form for the vertical profiles within the roughness sublayer follows that overcomes problems with many earlier forms in the literature. Predictions of the mean flow profiles are shown to match observations over a range of canopy types and stabilities. The unified theory predicts that key parameters, such as the displacement height and roughness length, have a significant dependence on the boundary-layer stability. Assuming one of these parameters a priori leads to the incorrect variation with stability of the others and incorrect predictions of the mean wind speed profile. The roughness sublayer has a greater impact on the mean wind speed in stable than unstable conditions. The presence of a roughness sublayer also allows the surface to exert a greater drag on the boundary layer for an equivalent value of the near-surface wind speed than would otherwise occur. This characteristic would alter predictions of the evolution of the boundary layer and surface states if included within numerical weather prediction models.     

Shear stress partitioning in large patches of roughness in the atmospheric inertial sublayer

Drag partition measurements were made in the atmospheric inertial sublayer for six roughness configurations made up of solid elements in staggered arrays of different roughness densities. The roughness was in the form of a patch within a large open area and in the shape of an equilateral triangle with 60 m long sides. Measurements were obtained of the total shear stress (τ) acting on the surfaces, the surface shear stress on the ground between the elements (τ_S) and the drag force on the elements for each roughness array. The measurements indicated that τ_S quickly reduced near the leading edge of the roughness compared with τ, and a τ_S minimum occurs at a normalized distance (x/h, where h is element height) of (downwind of the roughness leading edge is negative), then recovers to a relatively stable value. The location of the minimum appears to scale with element height and not roughness density. The force on the elements decreases exponentially with normalized downwind distance and this rate of change scales with the roughness density, with the rate of change increasing as roughness density increases. Average τ_S : τ values for the six roughness surfaces scale predictably as a function of roughness density and in accordance with a shear stress partitioning model. The shear stress partitioning model performed very well in predicting the amount of surface shear stress, given knowledge of the stated input parameters for these patches of roughness. As the shear stress partitioning relationship within the roughness appears to come into equilibrium faster for smaller roughness element sizes it would also appear the shear stress partitioning model can be applied with confidence for smaller patches of smaller roughness elements than those used in this experiment.     

Bulk Transfer Relations for the Roughness Sublayer

In the roughness sublayer (RSL), Monin–Obukhov surface layer similarity theory fails. This is problematic for atmospheric modelling applications over domains that include rough terrain such as forests or cities, since in these situations numerical models often have the lowest model level located within the RSL. Based on empirical RSL profile functions for momentum and scalar quantities, and scaling the height with the RSL height z _*, we derive a simple bulk transfer relation that accounts for RSL effects. To verify the validity of our approach, these relations are employed together with wind speed and temperature profiles measured over boreal forest during the BOREAS experimental campaign to estimate momentum and heat fluxes. It is demonstrated that, when compared with observed flux values, the inclusion of RSL effects in the transfer relations yields a considerable improvement in the estimated fluxes.     

Sublayer-Stanton numbers of heat and matter for aerodynamically smooth surfaces: basic considerations and evaluation

Summary Sublayer-Stanton numbers, B_i, of heat and matter for the interfacial sublayer over aerodynamically smooth surfaces determined for forced convective conditions by elementary and numerical integration are reviewed and evaluated. The results are based on Roth’s modified Heisenberg model for the spectral energy transfer in the equilibrium range under locally isotropic conditions and the approaches of Reichardt, Elser, Deissler, van Driest, Rannie, Sheppard, and Spalding for the normalized eddy diffusivity K_m/v. The results substantiate that with the exception of Sheppard’s K_m/v-approach all formulations are appropriate to provide sublayer-Stanton numbers with a sufficient degree of accuracy. From a theoretical point of view the K_m/v-relationships of Roth, Reichardt, van Driest, and Spalding are to be preferred, when a turbulent Prandtl number Pr_t = 1 is presumed. Since within the framework of mesoscale meteorological modelling numerical integration techniques would consume too much CPU-time because of the large number of near-wall grid points, a parameterization formula for the sublayer-Stanton number is proposed and evaluated. Compared to the B_i ⁻¹-results obtained by numerical integration, this kind of parameterization leads to a relative error of less than 5 percent for roughness Reynolds numbers, η_r, ranging from 30 to 600. Received January 2001 Revised November 5, 2001     

Flux–Profile Relationships Over a Fetch Limited Beech Forest

The influence of an internal boundary layer and a roughness sublayer on flux–profile relationships for momentum and sensible heat have been investigated for a closed beech forest canopy with limited fetch conditions. The influence was quantified by derivation of local scaling functions for sensible heat flux and momentum (_h and _m) and analysed as a function of atmospheric stability and fetch. For heat, the influences of the roughness sublayer and the internal boundary layer were in agreement with previous studies. For momentum, the strong vertical gradient of the flow just above the canopy top for some wind sectors led to an increase in _m, a feature that has not previously been observed. For a fetch of 500 m over the beech forest during neutral atmospheric conditions, there is no height range at the site where profiles can be expected to be logarithmic with respect to the local surface. The different influence of the roughness sublayer on _h and _m is reflected in the aerodynamic resistance for the site. The aerodynamic resistance for sensible heat is considerably smaller than the corresponding value for momentum.     

Heat and mass transfer to and from surfaces with dense vegetation or similar permeable roughness

A differential equation is obtained to describe the concentration of passive admixtures (water vapor, sensible heat, pollutants, CO₂, etc.) of turbulent flow inside a dense and uniform vegetational canopy. The profiles of eddy diffusivity, wind speed and shear stress are assumed to be exponential decay functions of depth below the top of the canopy. This equation is solved for the case of a vegetation with constant concentration of the admixture at the foliage surfaces. The solution is used to formulate bulk mass or heat transfer coefficients, which can be applied to practical problems involving surfaces covered with a vegetation or with similar porous or fibrous roughness elements. The results are shown to be consistent with experimental data presented by Chamberlain (1966), Garratt and Hicks (1973) and Garratt (1978). Calculations with the model illustrate that, as compared to its behavior over surfaces with bluff roughness elements, ln(z ₀/z _0c ) (wherez ₀ is the momentum roughness andz _0c , the scalar roughness) for permeable roughness elements is relatively insensitive tou _* and practically independent ofz ₀.     

Near Wall Flow over Urban-like Roughness

In this study, comprehensive measurements over a number of urban-type surfaces with the same area density of 25% have been performed in a wind tunnel. The experiments were conducted at a free stream velocity of 10 m s^-1 and the main instrumentation was 120 ° x-wire anemometry, but measurement accuracy was checked using laser Doppler anemometry.The results haveconfirmed the strong three-dimensionalityof the turbulent flow inthe roughness sublayer and the depths of the inertial sublayer (log-law region) and roughness sublayer for each surface have been determined. Spatial averaging has been used to remove the variability of the flow in the roughness sublayer due to individual obstacles and it is shown that the spatially averaged mean velocity in the inertial sublayer and roughness sublayer can,together, be described by a single log-law with a mean zero-plane displacement and roughness length for the surface, provided that the proper surface stress is known. The spatially averaged shear stresses in the inertial sublayer and roughness sublayer are compared with the surface stress deduced from form drag measurements on the roughness elements themselves.The dispersive stress arising from the spatial inhomogeneity in the mean flow profiles was deduced from the data and is shown to be negligible compared with the usual Reynolds stresses in the roughness sublayer. Comparisons have been made between a homogeneous (regular element array) surface and one consisting of random height elements of the same total volume. Although the upper limits of the inertial sublayer for both surfaces were almost identical at equivalent fetch, the roughness sublayer was much thicker for the random surface than for the uniform surface, the friction velocity and the roughness length were significantly larger and the `roughness efficiency' was greater. It is argued that the inertial sublayer may not exist at all in some of the more extreme rough urban areas. These results will provide fundamental information for modelling urban air quality and forecasting urban wind climates.     

Velocity and Surface Shear Stress Distributions Behind a Rough-to-Smooth Surface Transition: A Simple New Model

A simple new model is proposed to predict the distribution of wind velocity and surface shear stress downwind of a rough-to-smooth surface transition. The wind velocity is estimated as a weighted average between two limiting logarithmic profiles: the first log law, which is recovered above the internal boundary-layer height, corresponds to the upwind velocity profile; the second log law is adjusted to the downwind aerodynamic roughness and local surface shear stress, and it is recovered near the surface, in the equilibrium sublayer. The proposed non-linear form of the weighting factor is equal to ln(z/z ₀₁)/ln(δ _i /z ₀₁), where z, δ _i and z ₀₁ are the elevation of the prediction location, the internal boundary-layer height at that downwind distance, and the upwind surface roughness, respectively. Unlike other simple analytical models, the new model does not rely on the assumption of a constant or linear distribution for the turbulent shear stress within the internal boundary layer. The performance of the new model is tested with wind-tunnel measurements and also with the field data of Bradley. Compared with other existing analytical models, the proposed model shows improved predictions of both surface shear stress and velocity distributions at different positions downwind of the transition.     

A parameterization of the boundary layer of a tropical cyclone

In an earlier paper by one of the authors (Smith, 1968), a momentum integral method was developed to parameterize the gross constraint imposed by the surface boundary layer of a steady, axisymmetric, tropical cyclone on the meridional circulation within the vortex itself. Specifically, the method provides an effective means of estimating the radial variation of mean upflow/downflow induced by the boundary layer, compatible with a prescribed radial variation of azimuthal velocity just above the boundary layer,V _gr. However, it relies on a judicious choice of vertical profiles of radial and azimuthal velocity components within the boundary layer. An especially suitable set of profiles is discussed herein; these are Ekman-like profiles in which turbulent mixing is characterized by a vertically constant eddy diffusivityK _M , matched to a constant stress sublayer just above the sea surface. An attractive feature of the formulation is that a suitable value forK _M as a function of radius, which is extremely difficult to extract from observational data, can be calculated when the state of the sea surface, described by a roughness lengthZ ₀, is prescribed. Although observations ofZ ₀ at high wind speeds are not yet available, the effect of radial variations in sea surface roughness can be assessed and it is shown that these affect the upflow to a significant degree.     

The relative importance of ejections and sweeps to momentum transfer in the atmospheric boundary layer

Using an incomplete third-order cumulant expansion method (ICEM) and standard second-order closure principles, we show that the imbalance in the stress contribution of sweeps and ejections to momentum transfer (ΔS _o ) can be predicted from measured profiles of the Reynolds stress and the longitudinal velocity standard deviation for different boundary-layer regions. The ICEM approximation is independently verified using flume data, atmospheric surface layer measurements above grass and ice-sheet surfaces, and within the canopy sublayer of maturing Loblolly pine and alpine hardwood forests. The model skill for discriminating whether sweeps or ejections dominate momentum transfer (e.g. the sign of ΔS _o ) agrees well with wind-tunnel measurements in the outer and surface layers, and flume measurements within the canopy sublayer for both sparse and dense vegetation. The broader impact of this work is that the “genesis” of the imbalance in ΔS _o is primarily governed by how boundary conditions impact first and second moments.     

A Simple Method of Estimating Scalar Fluxes Over Forests

A simple aerodynamic-variance method is proposed to fill gaps in continuous CO₂ flux measurements in rainy conditions, when open-path analysers do not function. The method requires turbulent conditions (friction velocity greater than 0.1 ms^–1), and uses measurements of mean wind speed, and standard deviations of temperature and CO₂ concentration fluctuations to complement, and at times replace, eddy-covariance measurements of friction velocity, sensible heat flux and CO₂ flux. Friction velocity is estimated from the mean wind speed with a flux-gradient relationship modified for the roughness sublayer. Since normalised standard deviations do not follow Monin-Obukhov similarity theory in the roughness sublayer, a simple classification scheme according to the scalar turbulence scale was used. This scheme is shown to produce sensible heat and CO₂ flux estimates that are well correlated with the measured values.     

Critical Evaluation of Scalar Roughness Length Parametrizations Over a Melting Valley Glacier

We present a field investigation over a melting valley glacier on the Tibetan Plateau. In the ablation zone, aerodynamic roughness lengths (z _0M ) vary on the order of 10⁻⁴–10⁻² m, whose evolution corresponds to three melt phases with distinct surface cover and moisture exchange: snow (sublimation/evaporation), bare ice (deposition/condensation), and ice hummocks (sublimation/evaporation). Bowen-ratio similarity is validated in the stably stratified katabatic winds, which suggests a useful means for data quality check. A roughness sublayer is regarded as irrelevant to the present ablation season, because selected characteristics of scalar turbulence over smooth snow are quite similar to those over hummocky ice. We evaluate three parametrizations of the scalar roughness lengths (z _0T for temperature and z _0q for humidity), viz. key factors for the accurate estimation of sensible heat and latent heat fluxes using the bulk aerodynamic method. The first approach is based on surface-renewal models and has been widely applied in glaciated areas; the second has never received application over an ice/snow surface, despite its validity in (semi-)arid regions; the third, a derivative of the first, is proposed for use specifically over rough ice defined as z _0M > 10⁻³ m or so. This empirical z _0M threshold value is deemed of general relevance to glaciated areas (e.g. ice sheet/cap and valley/outlet glaciers), above which the first approach gives notably underestimated z _0T,q . The first and the third approaches tend to underestimate and overestimate turbulent heat/moisture exchange, respectively, frequently leading to relative errors higher than 30%. Comparatively, the second approach produces fairly low errors in energy flux estimates both in individual melt phases and over the whole ablation season; it thus emerges as a practically useful choice to parametrize z _0T,q in glaciated areas. Moreover, we find all three candidate parametrizations unable to predict diurnal variations in the excess resistances to humidity transfer, thus encouraging more efforts for improvement.     

Near-Wall Flow Development After A Step Change In Surface Roughness

An experimental study of the initial flow field downstream of a step change in surface roughness is presented. The roughness length of the downstream surface was approximately tenfold that of the upstream roughness and, unlike all previous studies, attention was concentrated on the roughness sublayer region beneath the inertial (log-law) region. The experiments were conducted at a boundary layer Reynolds number of about 6 × 10⁴ (based on layer thickness andfree-stream velocity) and around a longitudinal location where the (downstream) roughness length, z_o2, was about 1% of the boundary-layer thickness atthe roughness change point.The thickness of the roughness sublayer was found for the two roughness. It was observed that the vertical profiles of mean velocity and turbulence characteristics started to show similarity after about 160z₀₂ downstream of the roughness change. The presence of a shear stress overshoot is shown to depend strongly on the precise location (with respect to the roughness elements) at which the measurements are made and the thickness of the equilibrium layer is shown to be very sensitive to the way it is defined. It is demonstrated that the growing equilibrium layer has first to encompass the roughness sublayer before any thickness of inertial sublayer can be developed. It follows that, in somepractical cases, like flows across some urban environments, the latter(log-law) region may never exist at all.     

The flow in a turbulent boundary layer upstream of a change in surface roughness

Modification of a turbulent flow upstream of a change in surface roughness has been studied by means of a stream function-vorticity model.A flow reduction is found upstream of a step change in surface roughness when a fluid flows from a smooth onto a rough surface. Above that layer and above the region of flow reduction downstream of a smooth-rough transition, a flow acceleration is observed. Similar flow modification can be seen at a rough-smooth transition with the exception that flow reduction and flow acceleration are reversed. Within a fetch of –500 < x/z ₀< + 500 (z ₀ is the maximum roughness length, the roughness transition is located at x/z ₀ = 0), flow reduction (flow acceleration) upstream of a roughness transition is one order of magnitude smaller than the flow reduction (flow acceleration) downstream of a smooth-rough (rough-smooth) transition. The flow acceleration (flow reduction) above that layer is two orders of magnitude.The internal boundary layer (IBL) for horizontal mean velocity extends to roughly 300z ₀ upstream of a roughness transition, whereas the IBL for turbulent shear stress as well as the distortion of flow equilibrium extend almost twice as far. For the friction velocity, an undershooting (overshooting) with respect to upstream equilibrium is predicted which precedes overshooting (undershooting) over new equilibrium just behind a roughness transition.The flow modification over a finite fetch of modified roughness is weaker than over a corresponding fetch downstream of a single step change in roughness and the flow stays closer to upstream equilibrium. Even in front of the first roughness change of a finite fetch of modified roughness, a distortion of flow equilibrium due to the second, downwind roughness change can be observed.     

Interpretation of Crown Radiation Temperatures of a Dense Douglas fir Forest with Similarity Theory

Infrared crown radiation temperatures as observed over a dense Douglas fir forest are analyzed in the context of similarity theory and the concept of transport resistances. As such we obtain a rather high value of the roughness length for heat, which is about equal to the roughness length for momentum. This value can be explained by the more efficient transport of heat relative to momentum in the roughness sublayer of the forest. Correcting for this effect we arrive at the classic value for homogeneous terrain of about 0.1 times the roughness length for momentum. For unstable cases the presence of enhanced mixing of heat in the roughness sublayer leads to a modified integral stability function for the dimensionless potential temperature difference between the surface and the top of the roughness sublayer. The observations give some evidence for this different stability behaviour. The analysis suggests that during daytime the radiative surface temperature and the aerodynamic surface temperature are not significantly different when used to estimate fluxes. Daytime trunk space air temperature is satisfactory parameterized with the concept of gusts and with surface renewal analysis. As such it is related to the sensible heat flux and the storage heat flux. Night time radiation temperatures at times strongly deviate from the expected behaviour based on similarity theory and the roughness length for heat, suggesting that the concept of a single surface temperature is too simple for such cases.     

Momentum Transfer and Turbulent Kinetic Energy Budgets within a Dense Model Canopy

Second-order closure models for the canopy sublayer (CSL) employ aset of closure schemes developed for `free-air' flow equations andthen add extra terms to account for canopy related processes. Muchof the current research thrust in CSL closure has focused on thesecanopy modifications. Instead of offering new closure formulationshere, we propose a new mixing length model that accounts for basicenergetic modes within the CSL. Detailed flume experiments withcylindrical rods in dense arrays to represent a rigid canopy areconducted to test the closure model. We show that when this lengthscale model is combined with standard second-order closureschemes, first and second moments, triple velocity correlations,the mean turbulent kinetic energy dissipation rate, and the wakeproduction are all well reproduced within the CSL provided thedrag coefficient (C_D) is well parameterized. The maintheoretical novelty here is the analytical linkage betweengradient-diffusion closure schemes for the triple velocitycorrelation and non-local momentum transfer via cumulant expansionmethods. We showed that second-order closure models reproducereasonably well the relative importance of ejections and sweeps onmomentum transfer despite their local closure approximations.Hence, it is demonstrated that for simple canopy morphology (e.g.,cylindrical rods) with well-defined length scales, standard closureschemes can reproduce key flow statistics without much revision.When all these results are taken together, it appears that thepredictive skills of second-order closure models are not limitedby closure formulations; rather, they are limited by our abilityto independently connect the drag coefficient and the effectivemixing length to the canopy roughness density. With rapidadvancements in laser altimetry, the canopy roughness densitydistribution will become available for many terrestrialecosystems. Quantifying the sheltering effect, the homogeneity andisotropy of the drag coefficient, and more importantly, thecanonical mixing length, for such variable roughness density isstill lacking.     

The Parameterisation of Scalar Transfer over Rough Ice

We test a surface renewal model that is widely used over snow and ice surfaces to calculate the scalar roughness length (z _s ), one of the key parameters in the bulk aerodynamic method. For the first time, the model is tested against observations that cover a wide range of aerodynamic roughness lengths (z ₀). During the experiments, performed in the ablation areas of the Greenland ice sheet and the Vatnajökull ice cap in Iceland, the surface varied from smooth snow to very rough hummocky ice. Over relatively smooth snow and ice with z ₀ below a threshold value of approximately 10^?3 m, the model performs well and in accord with earlier studies. However, with growing hummock size, z ₀ increases well above the threshold and the bulk aerodynamic flux becomes significantly smaller than the eddy-correlation flux (e.g. for z ₀ = 0.01 m, the bulk aerodynamic flux is about 50% smaller). Apparently, the model severely underpredicts z _s over hummocky ice. We argue that the surface renewal model does not account for the deep inhomogeneous roughness sublayer (RSL) that is generated by the hummocks. As a consequence, the homogeneous substrate ice grain cover becomes more efficiently ‘ventilated’. Calculations with an alternative model that includes the RSL and was adapted for use over hummocky ice, qualitatively confirms our observations. We suggest that, whenever exceedance of the threshold occurs (z ₀  >  10^?3 m, i.e., an ice surface covered with at least 0.3-m high hummocks), the following relation should be used to calculate scalar roughness lengths, ln (z _s /z ₀)  =  1.5  ? 0.2 ln (Re _*)  ? 0.11(ln (Re _*))².     

Determination of Roughness Lengths for Heat and Momentum Over Boreal Forests

The roughness length for momentum (z_0m), zero-plane displacementheight (d), and roughness length for heat (z_0h) are importantparameters used to estimate land-atmosphere energy exchange. Although many different approaches have been developed to parameterizemomentum and heat transfer, existing parameterizations generally utilizehighly simplified representations of vegetation structure. Further, a mismatch exists between the treatments used for momentum and heat exchange and those used for radiative energy exchanges. In this paper, parameterizations are developed to estimate z_0m, d, and z_0h for forested regimes using information related to tree crown density and structure. The parameterizations provide realistic representationfor the vertical distribution of foliage within canopies, and include explicit treatment for the effects of the canopy roughness sublayer and leaf drag on momentum exchange. The proposed parameterizationsare able to realistically account for site-to-site differences in roughness lengths that arise from canopy structural properties.Comparisons between model predictions and field measurements show good agreement, suggesting that the proposed parameterizations capture the most important factors influencing turbulent exchange of momentumand heat over forests.