Observation of organized structure in turbulent flow within and above a forest canopy

Ramp patterns of temperature and humidity occur coherently at several levels within and above a deciduous forest as shown by data gathered with up to seven triaxial sonic anemometer/thermometers and three Lyman-alpha hygrometers at an experimental site in Ontario, Canada. The ramps appear most clearly in the middle and upper portion of the forest. Time/height cross-sections of scalar contours and velocity vectors, developed from both single events and ensemble averages of several events, portray details of the flow structures associated with the scalar ramps. Near the top of the forest they are composed of a weak ejecting motion transporting warm and/or moist air out of the forest followed by strong sweeps of cool and/or dry air penetrating into the canopy. The sweep is separated from the ejecting air by a sharp scalar microfront. At approximately twice the height of the forest, ejections and sweeps are of about equal strength.In the middle and upper parts of the canopy, sweeps conduct a large proportion of the overall transfer between the forest and the lower atmosphere, with a lesser contribution from ejections. Ejections become equally important aloft. During one 30-min run, identified structures were responsible for more than 75% of the total fluxes of heat and momentum at mid-canopy height. Near the canopy top, the transition from ejection of slow moving fluid to sweep bringing fast moving air from above is very rapid but, at both higher and lower levels, brief periods of upward momentum transfer occur at or immediately before the microfront.     

Three-Dimensional Scalar Microfront Systems in a Large-Eddy Simulation of Vegetation Canopy Flow

Large-eddy simulation is used to reproduce neutrallystratified airflow inside and immediately above a vegetation canopy. A passive scalaris released from the canopy and the evolution of scalar concentration above the canopyis studied. The most significant characteristic of the scalar concentration is the repeatedformation and dissipation of scalar microfronts, a phenomenon that has been observedin nature. These scalar microfronts consist of downstream-tilted regions of highscalar concentration gradients. Computer visualization tools and a conditional samplingand compositing technique are utilized to analyze these microfronts. Peaks in positivepressure perturbation exceeding an experimental threshold are found to be effectiveindicators of scalar microfronts. Convergence of the streamwise velocity componentand divergence of the cross-stream velocity component are observed in the immediatevicinity of scalar microfronts, which helps explain their relatively longlifetimes. Many of these three-dimensional features have been observedin previous field studies of canopy flow.     

Large-Eddy Simulation of Coherent Turbulence Structures Associated with Scalar Ramps Over Plant Canopies

Large-eddy simulations were performed of a neutrally-stratified turbulent flow within and above an ideal, horizontally- and vertically-homogeneous plant canopy. Three simulations were performed for shear-driven flows in small and large computational domains, and a pressure-driven flow in a small domain, to enable the nature of canopy turbulence unaffected by external conditions to be captured. The simulations reproduced quite realistic canopy turbulence characteristics, including typical ramp structures appearing in time traces of the scalar concentration near the canopy top. Then, the spatial structure of the organised turbulence that caused the scalar ramps was examined using conditional sampling of three-dimensional instantaneous fields, triggered by the occurrence of ramp structures. A wavelet transform was used for the detection of ramp structures in the time traces. The ensemble-averaged results illustrate that the scalar ramps are associated with the microfrontal structure in the scalar, the ejection-sweep structure in the streamwise and vertical velocities, a laterally divergent flow just around the ramp-detection point, and a positive, vertically-coherent pressure perturbation. These vertical structures were consistent with previous measurements made in fields or wind tunnels. However, the most striking feature is that the horizontal slice of the same structure revealed a streamwise-elongated region of high-speed streamwise velocity impacting on another elongated region of low-speed velocity. These elongated structures resemble the so-called streak structures that are commonly observed in near-wall shear layers. Since elongated structures of essentially similar spatial scales were observed in all of the runs, these streak structures appear to be inherent in near-canopy turbulence. Presumably, strong wind shear formed just above the canopy is involved in their formation. By synthesis of the ensemble-averaged and instantaneous results, the following processes were inferred for the development of scalar microfronts and their associated flow structures: (1) a distinct scalar microfront develops where a coherent downdraft associated with a high-speed streak penetrates into the region of a low-speed streak; (2) a stagnation in flow between two streaks of different velocities builds up a vertically-coherent high-pressure region there; (3) the pressure gradients around the high-pressure region work to reduce the longitudinal variations in streamwise velocity and to enhance the laterally-divergent flow and lifted updrafts downstream of the microfront; (4) as the coherent mother downdraft impinges on the canopy, canopy-scale eddies are formed near the canopy top in a similar manner as observed in conventional mixing-layer turbulence.     

Retrieval of turbulent pressure fluctuations at the ground surface beneath a forest

The relationship between surface pressure fluctuations and the velocity field associated with turbulent coherent structures is examined for flow within and above a deciduous forest. Measurements were taken with tower-mounted sonic anemometer/thermometers at six heights, Lyman-alpha humidiometers at three heights, and a pressure sensor at the forest floor. We find a strong, near-linear relationship between the mean square turbulent velocity and the standard deviation of the high-pass-filtered pressure fluctuations. Lagged cross-correlations between vertical velocity fluctuations and those of pressure show maximum correlations of ± 0.5 but with a phase offset. Examination of surface pressure during the passage of coherent structures, which are characterized by a transition from ejection to sweep, reveals a period of overpressure about 20 s in duration roughly centered on the time of passage of the scalar microfront at the top of the canopy. Pressure patterns associated with coherent structures appear to be largely responsible for the form of the correlations stated above.Pressure patterns calculated from an integrated Poisson equation, using observed velocity and temperature signals during coherent structures, match the main features of the observed pressure. Retrieval of the pressure fluctuations in this manner reveals that the mean wind shear/turbulence interaction term is dominant, but that important contributions arise from two other terms in the equation. Buoyancy effects are negligible. We show that the surface pressure signal is mainly created by the velocity field near the top of the forest, and present evidence to suggest that features of the sub-crown air movement result directly from this pressure field.     

Coherent eddies and temperature structure functions for three contrasting surfaces. Part I: Ramp model with finite microfront time

Air temperature time series within and above canopies reveal ramp patternsassociated with coherent eddies that are responsible for most of thevertical transport of sensible heat. Van Atta used a simple step-changeramp model to analyse the coherent part of air temperature structurefunctions. However, his ocean data, and our own measurements for aDouglas-fir forest, straw mulch, and bare soil, reveal that even withoutlinearization his model cannot account for the observed decrease of thecubic structure function for small time lag. We found that a ramp model inwhich the rapid change at the end of the ramp occurs in a finite microfronttime can describe this decrease very well, and predict at least relativemagnitudes of microfront times between different surfaces. Averagerecurrence time for ramps, determined by analysis of the cubic structurefunction with the new ramp model, agreed well with values determined usingthe Mexican Hat wavelet transform, except at lower levels within theforest. Ramp frequency above the forest and mulch scaled very well withwind speed at the canopy top divided by canopy height. Within the forest,ramp frequency did not vary systematically with height. This is inaccordance with the idea that large-scale canopy turbulence is mostlygenerated by instability of the mean canopy wind profile, similar to aplane mixing layer. The straw mulch and bare soil experiments uniquelyextend measurements of temperature structure functions and ramp frequencyto the smallest scales possible in the field.     

Turbulent Statistics of Neutrally Stratified Flow Within and Above a Sparse Forest from Large-Eddy Simulation and Field Observations

Turbulent statistics of neutrally stratified shear-driven flow within and above a sparse forest canopy are presented from a large-eddy simulation (LES) and compared with those from observations within and above a deciduous forest with similar height and foliage density. First- and second-order moments from the LES agree with observations quite well. Third-order moments from the LES have the same sign and similar vertical patterns as those from the observations, but the LES yields smaller magnitudes of such higher-order moments. Turbulent spectra and cospectra from the LES agree well with observations above the forest. However, at the highest frequencies, the LES spectra have steeper slopes than observations. Quadrant and conditional analyses of the LES resolved-scale flow fields also agree with observations. For example, both LES and observation find that sweeps are more important than ejections for the transport of momentum within the forest, while inward and outward interaction contributions are both small, except near the forest floor. The intermittency of the transport of momentum and scalar increases with depth into the forest. Finally, ramp structures in the time series of a passive scalar at multiple levels within and above the forest show similar features to those measured from field towers. Two-dimensional (height-time cross-section) contours of the passive scalar and wind vectors show sweeps and ejections, and the characteristics of the static pressure perturbation near the ground resemble those deduced from field tower-based measurements. In spite of the limited grid resolution (2 m × 2 m × 2 m) and domain size (192 m × 192 m × 60 m) used in this LES, we demonstrate that the LES is capable of resolving the most important characteristics of the turbulent flow within and above a forest canopy.     

Organised Motion and Radiative Perturbations in the Nocturnal Canopy Sublayer above an Even-Aged Pine Forest

Using time series measurements of velocity, carbon dioxideand water vapour concentration, and temperature collected justabove a 15 m tall even-aged pine forest, we quantify the roleof organized motion on scalar and momentum transport withinthe nocturnal canopy sublayer (CSL). We propose a frameworkin which the nocturnal CSL has two end-members, bothdominated by organised motion. These end-members representfully developed turbulent flows at near-neutral or slightly stablestratification and no turbulence for very stable stratification.Our analysis suggests that ramps dominate scalar transport fornear-neutral and slightly stable conditions, while linear canopywaves dominate the flow dynamics for very stable conditions.For intermediate stability, the turbulence is highly damped andoften dominated by fine scale motions. Co-spectral analysissuggests that ramps are the most efficient net scalar mass-transportingagent while linear canopy waves contribute little to net scalartransport between the canopy and atmosphere for averagingintervals that include complete wave cycles. However, canopywaves significantly contribute to the spectral properties of thescalar time series. Ramps are the most frequently occurringorganised motion in the nocturnal CSL for this site.Numerous night-time runs, however, resided between thesetwo end-members. Our analysis suggests that whenradiative perturbations are sufficient large (>20 W m^-2 innet radiation), the flow can switch from being highly dampedfine-scale turbulence to being organized with ramp-like properties. We also found that when ramps are already the dominant eddymotion in the nocturnal CSL, radiative perturbations have aminor impact on scalar transport. Finally, in agreement withprevious studies, we found that ramps and canopy waves havecomparable length scales of about 30–60 metres. Consequencesto night-time flux averaging are also discussed.     

Two-Point Correlation Analysis Of Neutrally Stratified Flow Within And Above A Forest From Large-Eddy Simulation

Two-point space-time correlations ofvelocities, a passive scalar and static pressure arecalculated using the resolvable flow fields computedby large-eddy simulation (LES) of neutrally stratifiedflow within and above a sparse forest. Zero-time-lagspatial auto-correlation contours in thestreamwise-vertical cross-section for longitudinal andlateral velocities and for a scalar are tilted fromthe vertical in the downstream direction, as istypical in near-wall sheared flow. On the other hand,auto-correlations of vertical velocity and of staticpressure are vertically coherent. Zero-time-lagspatial auto-correlations in the spanwise-verticalcross-section show no distinct tilt, and those forboth longitudinal and vertical velocities demonstratedistinct negative side lobes in the middle forest andabove, while longitudinal velocity in the subcrowntrunk space is laterally in-phase. Static pressureperturbations appear to be spatially coherent in thespanwise direction at all heights, especially insidethe forest. Near the forest floor, longitudinalvelocity is found to be in-phase with static pressureperturbation and to be closely linked to theinstantaneous streamwise pressure gradient, supportinga previous proposal that longitudinal velocity in thisregion is dominantly modulated by the pressurepatterns associated with the coherent sweep/ejectionevents. Near treetop height, a lack of linkage betweenthe pressure gradient and the local time derivative ofthe longitudinal velocity supports the hypothesis ofadvection dominating turbulent flow.The major phase characteristics of the two-pointcorrelations essentially remained the same from fourLES runs with different domain size and/or gridresolution. A larger LES domain yielded betteragreement with field observations in a real forest onboth the magnitudes of the correlations and thesingle-point integral time scales. A finer gridresolution in the LES led to a faster rate of decreaseof correlation with increasing separation in space ortime, as did the higher frequency fluctuations in theturbulent records from field measurements. Convectivevelocities estimated from the lagged two-pointauto-correlations of the calculated flow fields werecompared with similar calculations from wind-tunnelstudies. At the canopy top, estimates from thecorrelation analyses agree with the translationvelocity estimated from instantaneous snapshots of ascalar microfront using both LES and field data. Thistranslation velocity is somewhat higher than the localmean wind speed. Convective velocities estimated fromlagged correlations increase with height above thecanopy. It is suggested that an appropriate filteringprocedure may be necessary to reduce the effects ofsmall-scale random turbulence, as was reported in astudy over an orchard canopy. The mean longitudinalvelocity near the treetops is found to be moreappropriate than the local mean longitudinal velocityat each height to link single-point integral timescales with directly calculated spatial integralstreamwise length scales.     

Turbulent exchange above a pine forest II. Organized structures

Data from two 100-min runs on the turbulent atmospheric wind, temperature and humidity fields above a pine forest have been analysed using conditional sampling techniques. With the aid of the temperature time series, ramp events were identified and all fields were averaged in order to remove smaller scale turbulence and random low-frequency turbulence, and to map the organized structures revealed in this way. It is shown that the turbulent fluxes of momentum, heat and humidity, determined from these ramp-events, in fact constitute a large part of the total fluxes, about 90% during the actual events.On average, the duration of and the time between ramps were found to be about 35 s and 100 s, respectively. These periods were also found in the uw-, wT- and wq-cospectra, not only in the data analysed in this paper, but in the major part of all day-time data from this forest site (about 80 runs of 60 or 100 min). This indicates that the ramp events described arc a rather common phenomenon. The organized structures also have remarkable resemblance to near-wall structures observed in the laboratory.     

Joint probability analysis of momentum and heat fluxes at a deciduous forest

Turbulent transport processes for momentum and scalar quantities are examined by a joint probability distribution analysis using data observed within and above a deciduous forest. Characteristics of transport processes in the frequency domain were also analyzed using Tukey's procedure. The results confirm that sweep phenomena prevail within and at the top of a tall plant canopy and that downdrafts are more effective for vertical transport of momentum and scalar quantities. On the other hand, updrafts become more efficient for vertical transport in the daytime at levels about twice treetop height. The results show that within the forest, the sweep phenomenon prevails over a wide frequency range, while above the forest, prevalence of the ejection phenomenon is limited to low frequencies. It is again noted that the plant canopy plays an important role in the sweep-ejection cycle as well as in turbulent transport processes.     

Pseudo-wavelet analysis of turbulence patterns in three vegetation layers

Ramp patterns in scalar traces such as temperature are the signature of coherent structures. A pseudo-wavelet analysis technique was developed in which ideal saw-tooth patterns of varying size were used as basis functions and fitted to temperature and velocity data. Data recorded from three very different vegetation stands were examined in this study. It was found that the most probable structure duration for the forest canopy was in the range 35–40 s, for the orchard canopy it was 20–25 s and for the maize it was 15–20 s. When expressed in non-dimensional form, the structure duration probability distribution for the maize canopy was about a decade larger than for the forest canopy, with the orchard canopy intermediate. The mean eddy duration versus wind shear relation falls on a narrow band for all three canopies, indicating that wind shear at the canopy top is the determining factor for the scale of the coherent eddies. The inverse of duration and intermittency of coherent structures exhibits a tendency of independence from wind shear at higher wind shear values. Coherent structures transport heat in a more efficient way than do smaller scale, less coherent motions. In all the canopies, the heat flux fractions associated with coherent structures are at least 10% higher than the corresponding time fraction.     

Coherent eddies and temperature structure functions for three contrasting surfaces. Part II: Renewal model for sensible heat flux

Sensible heat, latent heat, and other scalar fluxes cannot be measuredwithin short dense canopies, e.g., straw mulches, with standard approachessuch as eddy correlation, Bowen ratio-energy balance, aerodynamic, andvariance methods. However, recently developed surface renewal models, thatare based on the fact that most of the turbulent transfer within and abovecanopies is associated with large-scale coherent eddies, which are evidentas ramp patterns in scalar time series, offer a feasible solution. Wepresent a new air renewal model that calculates sensible heat flux atdifferent heights within and above a canopy from the average cubictemperature structure function, sampled at a moderate rate, and measuredaverage friction velocity. The model is calibrated and tested with datameasured above and within a Douglas-fir forest and above a straw mulch andbare soil. We show that the model describes half-hour variations ofsensible heat flux very well, both within the canopy and roughnesssublayers and in the inertial sublayer, for stable and unstable atmosphericconditions. The combined empirical coefficient that appears in the modelhas an apparently universal value of about 0.4 for all surfaces andheights, which makes application of the model particularly simple. Themodel is used to predict daytime and nighttime sensible heat flux profileswithin the straw mulch and within a small bare opening in the mulch.     

Evidence of pressure-forced turbulent flow in a forest

Lagged cross-correlation analyses between streamwise velocity at several heights within and above a forest, and between streamwise velocity and surface pressure, provide evidence that turbulence in the sub-crown region of the forest is to a large extent driven by pressure perturbations. The analyses support earlier results based on examination of coherent structures observed in the same forest. The phase of the streamwise velocity signal exhibits an increasing delay with decreasing height, indicative of a downwind tilted structure, until the upper region of the forest is reached, at which point the effect is reversed. It is suggested that positive pressure perturbations ahead of advancing microfronts induce streamwise accelerations in the trunk space. This link between the pressure pattern and the wind field explains why velocity spectra in the trunk space are depleted in the higher frequencies, relative to levels above.     

An Analytical Model for the Two-Scalar Covariance Budget Inside a Uniform Dense Canopy

The two-scalar covariance budget is significant within the canopy sublayer (CSL) given its role in modelling scalar flux budgets using higher-order closure principles and in estimating the segregation ratio for chemically reactive species. Despite its importance, an explicit expression describing how the two-scalar covariance is modified by inhomogeneity in the flow statistics and in the vertical variation in scalar emission or uptake rates within the canopy volume remains elusive even for passive scalars. To progress on a narrower version of this problem, an analytical solution to the two-scalar covariance budget in the CSL is proposed for the most idealized flow conditions: a stationary and planar homogeneous flow inside a uniform and dense canopy with a constant leaf area density distribution. The foliage emission (or uptake) source strengths are assumed to vary exponentially with depth while the forest floor emission is represented as a scalar flux. The analytical solution is a superposition of a homogeneous part that describes how the two-scalar covariance at the canopy top is transported and dissipated within the canopy volume, and an inhomogeneous part governed by local production mechanisms of the two-scalar covariance. The homogeneous part is primarily described by the canopy adjustment length scale, and the attenuation coefficients of the turbulent kinetic energy and the mean velocity. Conditions for which the vertical variation of the two-scalar covariance is controlled by the rapid attenuation in the mean velocity and turbulent kinetic energy profiles, vis-à-vis the vertical variation of the scalar source strength, are explicitly established. This model also demonstrates how dissimilarity in the emissions from the ground, even for the extreme binary case with one scalar turned ‘on’ and the other scalar turned ‘off’, modifies the vertical variation of the two-scalar covariance within the CSL. To assess its applicability to field conditions, the analytical model predictions were compared with observations made at two different forest types—a sparse pine forest at the Hyytiälä SMEAR II-station (in Finland) and a dense alpine hardwood forest at Lavarone (in Italy). While the model assumptions do not represent the precise canopy morphology, attenuation properties of the turbulent kinetic energy and the mean velocity, observed mixing length, and scalar source attenuation properties for these two forest types, good agreement was found between measured and modelled two scalar covariances for multiple scalars and for the triple moments at the Hyytiälä site.     

Modelling Vegetation-Atmosphere Co2 Exchange By A Coupled Eulerian-Langrangian Approach

A Eulerian-Lagrangian canopy microclimate model wasdeveloped with the aim of discerning physical frombiophysical controls of CO₂ and H₂O fluxes. The model couples radiation attenuation with mass,energy, and momentum exchange at different canopylevels. A unique feature of the model is its abilityto combine higher order Eulerian closure approachesthat compute velocity statistics with Lagrangianscalar dispersion approaches within the canopy volume. Explicit accounting for within-canopy CO₂,H₂O, and heat storage is resolved by consideringnon-steadiness in mean scalar concentration andtemperature. A seven-day experiment was conducted inAugust 1998 to investigate whether the proposedmodel can reproduce temporal evolution of scalar(CO₂, H₂O and heat) fluxes, sources andsinks, and concentration profiles within and above auniform 15-year old pine forest. The modelreproduced well the measured depth-averaged canopy surfacetemperature, CO₂ and H₂O concentrationprofiles within the canopy volume, CO₂ storageflux, net radiation above the canopy, and heat andmass fluxes above the canopy, as well as the velocitystatistics near the canopy-atmosphere interface. Implications for scaling measured leaf-levelbiophysical functions to ecosystem scale are alsodiscussed.     

Turbulence structure in a deciduous forest

Three-dimensional wind velocity components were measured at two levels above and at six levels within a fully-leafed deciduous forest. Greatest shear occurs in the upper 20% of the canopy, where over 70% of the foliage is concentrated. The turbulence structure inside the canopy is characterized as non-Gaussian, intermittant and highly turbulent. This feature is supported by large turbulence intensities, skewness and kurtosis values and by the large infrequent sweeps and ejections that dominate tangential momentum transfer. Considerable day/night differences were observed in the vertical profiles of the mean streamwise wind velocity and turbulence intensities since the stability of the nocturnal boundary layer dampens turbulence above and within the canopy.     

Dispersion of a Passive Scalar Within and Above an Urban Street Network

The transport of a passive scalar from a continuous point-source release in an urban street network is studied using direct numerical simulation (DNS). Dispersion through the network is characterized by evaluating horizontal fluxes of scalar within and above the urban canopy and vertical exchange fluxes through the canopy top. The relative magnitude and balance of these fluxes are used to distinguish three different regions relative to the source location: a near-field region, a transition region and a far-field region. The partitioning of each of these fluxes into mean and turbulent parts is computed. It is shown that within the canopy the horizontal turbulent flux in the street network is small, whereas above the canopy it comprises a significant fraction of the total flux. Vertical fluxes through the canopy top are predominantly turbulent. The mean and turbulent fluxes are respectively parametrized in terms of an advection velocity and a detrainment velocity and the parametrization incorporated into a simple box-network model. The model treats the coupled dispersion problem within and above the street network in a unified way and predictions of mean concentrations compare well with the DNS data. This demonstrates the usefulness of the box-network approach for process studies and interpretation of results from more detailed numerical simulations.     

Influence of Nocturnal Low-level Jets on Eddy-covariance Fluxes over a Tall Forest Canopy

Observations of low-level jets (LLJs) at the Howland AmeriFlux site in the USA and the jet’s impact on nocturnal turbulent exchange and scalar fluxes over a tall forest canopy are discussed. Low-frequency motions and turbulent bursts characterize moderately strong LLJs, whereas low-frequency motions are suppressed during periods with strong LLJs and enhanced shear. An analysis based on the shear-sheltering hypothesis seeks to elucidate the effect of LLJs on flux measurements. In the absence of shear sheltering, large eddies penetrate the roughness sublayer causing enhanced mixing while during periods with shear sheltering, mixing is reduced. In the absence of the latter, ‘upside-down’ eddies are primarily responsible for the enhanced velocity variances, scalar and momentum fluxes. The integral length scales over the canopy are greater than the canopy height. The variance spectra and cospectra from the wavelet analysis indicate that large eddies (spatial scale greater than the low-level jet height) interact with active canopy-scale turbulence, contributing to counter-gradient scalar fluxes.     

A Numerical Study Of Nocturnal Wavelike Motion In Forests

In this paper, we use a two-dimensional eddy-resolved model to investigate the instability of a parallel shear flow in a stably stratified boundary layer whose lower domain is occupied by a canopy. The results support our contention that wave motion in the canopy is initiated by shear in an air layer near the treetops. Significant modification by the wave motion of the mean velocity and temperature fields is found even before the wave reaches saturation. The wave fluxes of momentum and heat are not constant with height. Downwind tilting braids are found at the finite amplitude stage of the wave growth and could persist after wave breaking; these downwind tilting structures are believed to be the same as the temperature microfronts reported in the literature. We also present an analysis of the velocity and temperature fields of an observed wave event in the time-height domain and show that the simulation has captured the broad features of the observation.     

Scalar Concentration Profiles in the Canopy and Roughness Sublayer

The mean flow and scalar concentration profiles within and above a tall canopy are well known to violate the standard boundary-layer flux-gradient relationships. We present a theory for the scalar concentration profile that is comprised of a canopy exchange 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 canopy top. This analogy provides an additional length scale—the vorticity thickness—upon which the profiles depend and a set of criteria that allows a reduction in the empiricism associated with earlier forms in the literature. Predictions of the mean scalar concentration profiles are shown to match observations over a wide range of diabatic stabilities for both potential temperature and water vapour.