A single-layer conceptual SEB-SL model was developed by to study the impact of the wind speed on the near-surface temperature gradient. In the presence of trees or other tall vegetation, however, the introduction of a second layer becomes necessary. Here, such a model is developed. It is composed of the surface, a “canopy” layer where the air is in thermodynamic equilibrium with the vegetation, and an overlying air layer (Fig. ). The effects of the canopy on the long-wave radiative and turbulent fluxes are then taken into account. In the following, the equations and notations draw on the one-layer model of . The surface emissivity is assumed to be equal to 1, which is a good approximation for snow-covered surfaces. The short-wave radiation and the latent heat fluxes are neglected in Arctic wintertime conditions.

The surface energy balance equation in this system can be written as (Appendix ) 1-(1-ϵc)Qi-Λs(Ta-Tg)+ΔTcsρCpCD,cUc+4σTa3+Λs+ΔTac(1-ϵc)4σTa3+Λs=0, where ΔTcs=Tc-Ts is the difference between the surface temperature (Ts) and the air temperature at canopy height (Tc). ΔTac=Ta-Tc is the difference between the canopy temperature and the air temperature at height za, corresponding to the top of the surface layer (Ta). Tg is the ground temperature (Fig. ). Qi=-LWd+σTa4 (where LWd is the downwards long-wave flux) is termed the isothermal net radiation: indeed, it is equal to the net long-wave flux if Ts=Ta . Λs=λsds, where λs is the snow conductivity and ds the snow depth. ρ is the air density, Cp the heat capacity of the air and Uc the wind speed at height zc. For the time being, Uc is very roughly estimated to be proportional to Ua: for example, Uc=0.25⋅Ua. ϵc is the canopy emissivity, and CD,c is the turbulent diffusion coefficient for the canopy-to-surface heat exchange.

A second equation can be obtained by considering the canopy energy balance (Appendix ): 2-Qi-Λs(Ta-Tg)+ΔTcs(1-ϵc)4σTa3+Λs+ΔTac4σTa3+Λs+ρCpCD,aUa=0, where Ua is the wind speed at height za, and CD,a is the turbulent diffusion coefficient for the air–canopy heat exchange. This is very similar to the one-layer model of , except that the energy source term (here, the first line of Eq. ) has an additional term, which is proportional to ΔTcs. The difficulty in solving Eqs. () and () to obtain ΔTcs and ΔTac is that the turbulent diffusion coefficients depend on the stability: CD,c=κ2[log⁡(zaz0s)-ψ(ζ)+ψ(z0sL)]23CD,a=κ2[log⁡(za-dz0c)-ψ(ζ)+ψ(z0cL)]2, where κ=0.4 is the van Kármán constant, and z0s and z0c are the roughness lengths of snow or of the canopy, respectively. Here, d is the displacement height due to the presence of the canopy. The snow and canopy momentum roughness length z0m,s/c is assumed to be equal to the heat roughness length z0h,s/c, both referred to as z0,s/c. ζ=z/L is the Obukhov parameter, with L being the Monin–Obukhov length. ψ is the integral stability function, which tends to 0 when ζ≈0 and tends to infinity with increasing ζ. The turbulent diffusion coefficients therefore tend to κ2/log⁡(zz0)2 at weak stability and 0 at strong stability. Many different expressions of ψ are found in the literature . Usually, these are classified as “short tail” (i.e. with a very sharp decrease so that CD quickly drops to 0 at increasing stability) or “long tail” (i.e. the transition is smoother so that some turbulent sensible heat flux is maintained for longer). There are also other ways to estimate the below-canopy turbulent diffusion coefficient CD,c, for example by assuming an exponential wind profile in the canopy as in . However, Eq. () is the simplest expression and is utilised for illustrative purposes.

The first insights into the behaviour of ΔTcs and ΔTac can be gained by studying the asymptotic cases: the weakly and strongly stable limits, defined by Ua→∞ and Ua→0, respectively, while keeping ΔTas>0. Here only the case where ϵc=1 (corresponding to an opaque canopy) is considered. In this situation, and if turbulence is completely collapsed (i.e. CD,a=CD,c=0), Eqs. () and () lead to the following values for the temperature gradients: ΔTac=Qi1+Λs/(4σTa3)+Λs(Ta-Tg)4σTa3+2Λs4ΔTcs=-ΛsQi/(4σTa3)+Λs(Ta-Tg)4σTa3+2Λs, and therefore, 5ΔTas=ΔTac+ΔTcs=Qi+2Λs(Ta-Tg)4σTa3+2Λs.

The total temperature gradient ΔTas (between the air and the surface) in Eq. () is very similar to the one-layer case but with an “equivalent snow conductivity” that is twice the real value.

It can also be noted that ΔTac will usually be positive for typical Arctic winter values of the different parameters (Table ) because radiation is the dominant process, unless the snow cover is very thin. For example, in a very cold, high-synoptic-pressure situation with Qi=70 W m-2, Ta=-20 °C and Tg=0 °C, ΔTac will be positive unless the snow depth is less than 7 cm. Similarly, in a warmer, cloudier situation with Qi=30 W m-2, Ta=-10 °C and Tg=0 °C, ΔTac will be positive unless the snow depth is less than 8 cm. For the same reason, ΔTcs will usually be negative. In short, the very stable case is characterised by a temperature decrease from the surface to the canopy and an increase from the canopy to the overlying air. This is illustrated in Fig.  (continuous line, corresponding to Ua=0.5 m s-1).

This is contradictory to the idea that CD,c collapses to 0 because in the presence of a negative temperature gradient, buoyancy effects may generate turbulence without significant mechanical shear. Indeed, solving Eqs. () and () numerically for Uc=Ua=0.001 m s-1 (using appropriate schemes for calculating the turbulent diffusion coefficients such as those described in Sect. ) shows that CD,cUc maintains a value of around 0.0017 m s-1. Therefore, while the surface layer as a whole may be considered strongly stable (because Ta-Ts is very large), this may not be the case for the canopy layer. This is in agreement with , who found that the canopy layer may remain unstable even when the air aloft is very stably stratified.

Next, the turbulence is solved iteratively using the Ameriflux stability function (Sect. ) over a complete range of Ua values. This makes it possible to study the behaviour of the inversion outside of the weakly and strongly stable regimes and for different values of the canopy emissivity. The result of this estimation is shown in Fig. a, b for three values of ϵc. ΔTac exhibits the same S shape as in the one-layer conceptual model of . On the other hand, ΔTcs is larger in the weakly stable regime than in the strongly stable regime (where it is negative, in coherence with the above discussion), and its shape is more dependent on values of the canopy emissivity. For ϵc=1, ΔTcs appears to tend to 0 at large values of Ua while keeping negative values. On the other hand, for ϵc=0.5 and 0.1, it turns positive before decreasing with increasing wind speeds, therefore reaching a maximum somewhere between 2 and 6 m s-1. In sum, the total temperature difference (ΔTas) decreases more slowly than in the one-layer model, even when accounting for the long-tailed function chosen. The transition between the two modes (weak winds associated with strong inversions and high winds associated with weak inversions) is much more gradual in this two-layer conceptual model than the characteristic marked inverted S shape reported in the one-layer conceptual model of . This phenomenon corresponds to observations over a surface covered by trees that attenuate the transition.

This behaviour can be understood in the following way. At low wind speeds, the dominant process influencing the canopy layer is radiation: it emits more than it receives and therefore loses its heat to both the surface and the air above. As a result, it is colder than both. Although some turbulence remains due to buoyancy, this is not enough to compensate for the radiative heat loss. At high wind speeds, on the other hand, the whole SL is well mixed. Turbulence is the dominant process, linking the canopy layer to both the surface and the air above and maintaining their temperatures closely.

Starting from the strongly stable state, a small increase in wind speed will lead to increased turbulence mixing and a positive heat flow from the air above to the canopy. The canopy temperature will therefore increase. If the canopy has a strong emissivity, this increase in temperature will lead to increased radiation downwards to the surface and a corresponding increase in surface temperature. The canopy and surface will therefore warm at relatively the same pace (Fig. , right). On the other hand, if the emissivity of the canopy is low, it will not as easily convert its increased temperature into radiation. Its temperature will therefore increase rapidly without contributing to warming the surface, leading to a high ΔTcs (Fig. , left). As the wind speed continues to increase, the canopy temperature will eventually be more or less equal to the temperature of the air above, and the surface continues to warm, thus leading to a decrease in ΔTcs.

The Ameriflux PRR site is located in the Poker Flat Research Range (65°07′24.4′′ N, 147°29′15.2′′ W), around 30 km away from Fairbanks (interior Alaska). It has been operating since 2010, when it was established as part of the JAMSTEC-IARC Collaboration Study (JICS) , and its data are made available online on the Ameriflux website (https://ameriflux.lbl.gov/sites/siteinfo/US-Prr, last access: 22 April 2024). Its 17 m measurement tower is implanted in a black spruce forest with sparsely distributed and short trees (Fig. ). The tree density, as measured in 2010, was 3967 trees ha-1, and the average tree height was 2.44 m: the tallest tree was 6.4 m, but 75 % of trees were shorter than 3 m . The leaf area index (LAI) was 0.73 . Both of these values are much smaller than those found in Noah-MP for evergreen forests (Table ).

The Ameriflux PRR measurements that are used here are summarised in Table . These include wind speeds and temperatures at eight different heights (from 1.5 to 16 m), as well as turbulent and radiative fluxes. The surface temperature was calculated from the radiative flux measurements at 1.3 m, assuming a snow surface emissivity of 0.99. As this study focuses on the clear-sky surface layer in wintertime conditions, the data were curated accordingly. Only time points in the months of November–March with snow depth greater than 10 cm were kept. As no measurements of the cloud cover are available at the PRR site, clear-sky instants were defined as those with net long-wave radiation less than -30 W m-2. Indeed, as is typical in high-latitude sites, the net long-wave flux (Rn) distribution at the PRR site was bimodal; the low-Rn mode was considered to correspond to the absence of clouds and the high-Rn mode to their presence. As the PRR site is located slightly below the Arctic circle, there is still some solar radiation at the surface in the wintertime. In order to simplify the analysis, only time points with downwelling short-wave radiation of less than 30 W m-2 were kept; as the snow albedo is very high, this corresponds to a net short-wave flux of less than 5 W m-2 and therefore to negligible short-wave impact. Lastly, measurements with a latent heat flux greater than 5 W m-2 in absolute value were discarded.

The average emissivity of the canopy layer ϵc=fvegϵv can be calculated from above- and below-canopy radiation measurements: 6LWd,bc-LWu=(1-fvegϵv)LWd-LWu,bc. As shown in Fig. c, this gives a best estimation of ϵc≈0.15. This is in accordance with the Noah-MP calculation of fveg and ϵv as a function of LAI. Indeed, measured LAI at the PRR site is 0.73, yielding fveg≈0.3 (Eq. ) and ϵv≈0.5.

The value of z0m,c can also be calculated from the sonic anemometer data. Indeed, at weak stability (ζ≪1), the wind profile is approximately logarithmic: Ua≈u*κlog⁡z-dz0m,c. Therefore, d and z0m,c can be determined through a linear regression of eκUa/u* against z when the data are restricted to values of ζ<10-2. Here, z0m,c was found to be 0.39 m with d=1.4±1.4 m (Fig. b), which makes sense for a forest environment with short trees. The integral stability function ψ can also be determined from the data (assuming, as is often done, that it is the same for momentum and heat). The aim of this determination is to reproduce the measurements over the zone of transition, which is approximately between ζ=0.1 and ζ=1. For higher values (i.e. z/L>3), the specific values of ψ are less problematic because in this range the turbulent heat flux will tend to collapse anyway. In order to determine ψ, Ψ=-ψ(ζ)+ψz0m,cL=Uau*κ-log⁡z-dz0m,c is plotted as a function of ζ (Fig. a). Because z0m,c/L is negligible compared to ζ, Ψ is approximately equal to -ψ(ζ) at the first order. Here, we found that measured ψ is more long tailed than the Businger–Dyer or WRF (Eq. ) functions. The intermediate zone (ζ between 0.1 and 1) exhibits a marked difference between the Businger-Dyer and WRF functions and the measurements. In fact, when plotted on a log–log scale, it became apparent that Ψ was proportional to ζ at low values of ζ and proportional to ζ2 at high values of ζ. This differs from the often-used z-less scaling, which implies that ψ must be proportional to ζ at least up to ζ≈0.1 as is the case for the WRF and Businger–Dyer functions. Measurement error may explain some of the difference at ζ<0.1 as Ψ is very small in these conditions.

For our purposes, we determined an analytical expression of ψ to best fit to the Ψ measurements and left aside the question of the z-less scaling. The following expression for ψ was therefore considered: ψ(ζ)=-a⋅ζr(ζ)7r(ζ)=0.75⋅arctan⁡(b⋅ζ-c)2π+1.25. This was chosen because arctan⁡(x) tends to ±π2 when x tends to ±∞, with a smooth transition around x=0. r(ζ), which therefore tends to 0.5 at low values of ζ and 2 at high values of ζ, similar to observations. The b and c coefficients must be chosen so that the timing and speed of the transition between the ζ and ζ2 asymptotes match the observations. The function Ψ was therefore fitted for the different heights (z=7.5, 9, 11, 13 and 16 m), yielding coefficients which varied between the ranges [4.5,5.5], [-1,2] and [10,40] for a, b and c, respectively. Plotting the function with these different parameters revealed little difference in behaviour over this range. Values of a=5, b=20 and c=0.1 were found to give a good fit to the observations (Fig. a). This expression of ψ was termed the PRR stability function.

The Ameriflux PRR site characteristics are summarised in Table . Although the location of the PRR site is given as evergreen needleleaf forest by the MODerate resolution Imaging Spectroradiometer (MODIS) land use categories as well as by the Ameriflux website, its canopy height and turbulent and radiative characteristics are actually the most similar to a wooded or mixed tundra.

The average temperature profile (and its difference from the temperature at 16 m) is shown in Fig. a. The impact of wind speed on the surface layer temperature profile is clear. To highlight the importance of the wind speed on the regime stability, data are gathered in two groups according to their wind speed values: Ua< 2 m s-1 or Ua> 4 m s-1. This distinction is only used in Fig.  for illustrative purpose. This separation is based on the distribution of the bulk Richardson number Rb at 16 m (Fig. b): 65% of the data with Ua>4 m s-1 indeed have an Rb≤0.25, and 99.2% of the data with Ua<2 m s-1 have an Rb>0.25. The two modes are therefore clearly separated with a negligible overlap.

For Ua<2 m s-1, the temperature decreases rapidly all the way down to the surface, and the Richardson number is overwhelmingly greater than 0.25 (Fig. b), which is the traditionally cited limit value beyond which turbulence collapses. However, while the turbulent sensible heat flux has a low mean value of 4 W m-2, its distribution remains quite spread out, with 5th and 95th percentiles of -12 and 30 W m-2, respectively (Fig. c). This indicates that there is some remaining turbulence.

For Ua>4 m s-1, the temperature gradient is very weak (approximately 0.5 °C) down to 1.5 m, with a strong temperature gradient remaining in the last metres. The top to bottom temperature difference is nevertheless smaller than for Ua<2 m s-1, leading to Rb values that are smaller than 0.25. Accordingly, the turbulent sensible heat flux is much larger than for the lower wind speeds: its mean is 32 W m-2, with 90 % of values between 11 and 60 W m-2. The fact that both Rb and the turbulent sensible heat flux have clearly distinct distributions for wind speeds greater than 4 and lower than 2 m s-1 suggests that a threshold wind speed for sustainable turbulence probably occurs in this range. It should further be noted that while only the bulk Richardson number at 16 m is calculated here, the distributions are similar at other altitudes higher than 6 m. The impact of the radiative input (Qi) is also clear in Fig. a. All the observed values have been gathered in two groups that do not overlap and are delimited by their values of the isothermal net radiation Qi. Thresholds of 50 and 60 W m-2 hence provide a clear view of the impact of Qi on the temperature profiles as a function of the wind speed. The average profiles corresponding to values of Qi>60 W m-2 exhibit a larger temperature gradient than those corresponding to values lower than 50 W m-2, especially at low wind speeds. This is in accordance with Sect. : greater radiative cooling leads to a larger SBI.

The relationship between the average air-to-surface temperature difference and wind speed is shown in Fig. c. ΔTas=Ta-Ts decreases with Ua, reaching a minimum for Ua>5 m s-1, and there is a clear distinction between the averages corresponding to Qi lower than 50 and greater than 60 W m-2, respectively. ΔTas can further be broken down into ΔTcs=T1.5m-Ts (Fig. b) and ΔTac=Ta-T1.5m (Fig. a). ΔTac exhibits a very clear S shape, collapsing to less than 1 K at wind speeds higher than 4 m s-1. ΔTcs, on the other hand, is maximum around 3 m s-1 for both ranges of Qi. These behaviours are reminiscent of the two-layer model (Sect. ): the main difference here is that ΔTcs remains positive instead of decreasing to negative values at low wind speeds.

Examination of the temperature profiles and gradients in relation to wind speed at the PRR site therefore suggests that a two-layer model may be able to reproduce the temperature gradients, with the temperature at 1.5 m being a proxy for the canopy temperature. The observations are compared to the models in more detail in Sect. .

Within the WRF code, the SEB-SL model is split into two parts. First, the surface layer module calculates the turbulent diffusion coefficient. Then, the land surface model uses the turbulent diffusion coefficient to solve the SEB and determine the surface temperature. Many different schemes are available for each module. In this work, we have chosen to focus on the Noah land surface model (Noah-LSM) , used in conjunction with the Mellor–Yamada–Janjić (MYJ) surface layer scheme , and the Noah multiphysics scheme , which is nominally a land surface model but actually functions as an entire SEB-SL model, as it calculates its own turbulent diffusion coefficients. In the rest of this article, the Noah-LSM and MYJ combination is termed Noah-MYJ.

In this section, two one-layer and two two-layer models are compared (Table ). The one-layer models include the original Noah-MYJ (oMYJ), presented in Sect. , and a modified version of this scheme (mMYJ). The two-layer models include the original Noah-MP (oMP) and a modified version of Noah-MP (mMP).

The guiding principle for the modifications to both original models was to improve the modelled dependency of the temperature inversion on the wind speed, in particular the transition between the two regimes. This included removing the imposed maximum on ζ so that a truly stable regime is allowed to develop. The stability function was also modified to a more long-tail formulation (Sect. ): this makes the transition more gradual and avoids the non-monotonicity associated with Eq. () at ζ>1. Furthermore, a displacement height is added in the mMYJ model.

Modifications implemented in mMP included forcing the vegetation and canopy air temperature to be equal so that the energy balance for the vegetated part is as described in Sect. . The canopy-to-ground turbulent diffusion coefficient was also calculated as in the MYJ surface layer instead of using Eq. (). Lastly, a constant coefficient K=5×10-4 m s-1 was added to CD,aUa. This effectively imposes a lower limit on the turbulent diffusion coefficient in a gradual way, without having to force a minimum which would create a discontinuity. At wind speeds greater than 3 m s-1, this constant coefficient is negligible compared to the calculated value of CD,aUa. It should be noted that in effect, the original Noah-MP also imposes such a limit through indirect methods (for example, by imposing a minimum value of 1 m s-1 for Ua or a maximum value of 1 for ζ). The reasons for imposing a lower limit on the turbulence are explored further in Sect. .

In order to evaluate the models “offline”, the oMYJ and oMP models were extracted from the WRF framework and recoded in Python in a minimal form; i.e. only the parts relating to the surface temperature calculation were kept. In particular, all latent heat flux calculations were ignored; the snow conductivity was assumed to be constant, equal to 0.3 W m-1 K-1, and snow depth and ground temperature were used as input variables rather than being calculated. mMYJ and mMP were similarly coded in Python.

Input parameters to all four models are set to correspond to the characteristics of the PRR site as determined in Sect. : i.e. d=1.4 m, z0m,c=0.4 m and z0m,s=0.002 m. For oMP and mMP, fveg was determined from the LAI using Eq. (). For the two modified versions, the stability function used is the one determined from the Poker Flat Research Range measurements (Sect. ).

Here, the top of the surface layer was considered to be the top of the measurement tower; i.e. za=16 m. The five input variables to the Python models are the measured air temperature at 16 m, wind speed at 16 m, downwards long-wave flux above the canopy, snow depth and ground temperature. The output is the surface temperature (and canopy temperature for the two-layer models). Running the models over the entirety of the curated PRR dataset yielded 5412 modelled values of surface temperature, which were then compared to the corresponding 5412 measured values of Ts. The results are analysed in Sect. .

The outputs of the one-layer models (oMYJ and mMYJ) are shown in blue in Fig. c. Both tend to have similar values to the observations for low wind speeds, although oMYJ does not reach a constant regime because ζ is limited to values of 1. Because this limit is removed in mMYJ, it better reproduces two regimes separated by a transition; this transition is however more gradual because the PRR stability function was used. Both models, however, predict too small values of ΔTas at high wind speeds compared to the observations.

The two-layer models, on the other hand, both show a much more gradual decrease in ΔTas with Ua. Indeed, the decrease is so gradual in the output of oMP that it is not possible to discern two distinct regimes – even though the stability function used is Businger–Dyer, which is very short tailed (see Fig. a, black lines compared to the grey line which corresponds to the PRR stability function). One reason for this is that many limits are placed to maintain turbulence: u* cannot become larger than 0.07 m s-1; ζ must remain smaller than 1; and when the wind speed is used for calculating the turbulent diffusion coefficient, it takes a minimum value of 1 m s-1 (this is only the case within the surface layer modules so that WRF still outputs wind speed values less than 1 m s-1). Although it is true that some turbulence is always maintained, as shown by the measurements at the PRR site (Fig. ), the result is that the Noah-MP model outputs too low ΔTas values at very low wind speeds. oMP also does not reproduce the individual behaviour of ΔTcs and ΔTac: its calculated ΔTac does not have an S shape as a function of wind speed, and ΔTcs exhibits no maximum.

The behaviour of mMP is more satisfactory. ΔTac shows a clear transition between a low-wind-speed, high-gradient state and high-wind-speed state where the gradient is close to 0. ΔTcs has a maximum between 3 and 5 m s-1 (depending on the value of fveg). ΔTas, finally, is close to the observed value in both the high- and the low-wind-speed limits. Two things must be noted here: first, that values of ΔTcs remain positive at low wind speeds because, as noted in Sect. , a constant K equal to 5×10-4 m s-1 has been added to CD,aUa. Similar to the limits imposed in oMP, this serves to maintain a certain level of turbulence and avoid the collapse of the turbulent sensible heat flux. Without this, ΔTcs would decrease much more strongly, as described in Sect. . Adding a constant, as opposed to imposing a maximum value, is a more gradual method which does not distort the shape of the transition. The constant value is chosen to best represent the observations and should be discussed in regard to other datasets.

Secondly, two versions of mMP are shown in Fig. . The first corresponds to fveg=0.3 and ϵv=0.5, which are the values which would be calculated by WRF from a LAI of 0.73 according to Eq. (). The second corresponds to fveg=1 and ϵv=0.15. The results are substantially different, especially concerning the canopy temperature (and therefore ΔTcs and ΔTac). Indeed, as outlined in Sect. , the canopy tends to become colder than the surface for higher values of ϵv, and this is the case for the simulation with fveg=0.3. Furthermore, the transition wind speed (for ΔTac) and wind speed at maximum ΔTac are shifted to lower values for the simulation with fveg=1. These two sets of values both correspond to ϵc=0.15 and therefore to the same radiative flux balance. However, the difference in outcome suggests that due to the turbulent fluxes, separating a bare fraction from a vegetated fraction is not equivalent to considering only one layer but with lower emissivity. The simulation with fveg=1 seems to perform better. One possible explanation is linked to the size of the eddies transporting heat. If they are of similar size to the typical distance between the trees, the turbulent transport of heat would not necessarily behave differently over a bare fraction than over a vegetated fraction. Instead, all turbulent transport would occur in an averaged manner. Indeed, the turbulent characteristics calculated in Sect.  are likely representative of this average, depending on the instrument footprint. At the PRR site, average tree distance is approximately 2.2 m, assuming that the trees are homogeneously distributed (which seems reasonable from site photos, Fig. ). It is, however, not clear whether this is a robust feature of the two-layer models. Indeed, oMP does not appear to perform better when fveg is set to 1 and ϵc to 0.15: its calculated values of ΔTas decrease more rapidly with wind speed and therefore remain smaller than the measured temperature difference over the whole wind speed range (not shown). In the following, the values of fveg=0.3 and ϵv=0.5 are used for both two-layer models.

The models are then run over all the PRR measurement points. Compared to the above analysis, this makes it possible to evaluate their behaviour for a wide variety of input values. Overall, all models capture some of the variability in ΔTas, probably due to the influence of the downwards radiative fluxes. It is clear however that the one-layer models always underestimate ΔTas when the measured ΔTas is the lowest, which corresponds to conditions of high wind speeds (Fig. ). oMYJ also underestimates ΔTas when the measured ΔTas is very high, but this effect has been corrected in mMYJ by allowing for a stronger decrease in turbulence. The root mean square error (RMSE) of mMYJ is therefore approximately 2.8 K as opposed to 3.4 K for the original MYJ scheme.

The two-layer models both perform better than the one-layer models, supporting the idea that they are more adapted for use in a forest environment. The original Noah-MP model cannot reproduce strong values of ΔTas because of excessive forced turbulence; mMP fares better in that regard. Its RMSE is slightly better (2.2 instead of 2.3 K). Note that running mMP with fveg=1 and ϵv=0.15 leads to an RMSE of 2.1 K. To gain insight into the reliability of the model, all available observations are selected and binned according to their wind speed Ua values in intervals of width of 0.5 m s-1. This eliminates assumptions regarding input parameters such as the net radiation at the surface. Results are shown in Fig. .

It clearly shows that MYJ, whether in its original (oMYJ) or its modified (mMYJ) versions, reproduces a too sharp transition due to the fact that it only considers a single layer and strongly differs from the observations when Ua values become larger than 2.5 m s-1. mMYJ is in better agreement with the observations when the wind speed is weaker because the modelled ΔTas values are obtained in a constant regime and enhanced by 2 K. This is the consequence of removing the limitation of ζ values to 1.

Regarding the two-layer models, oMP slightly underestimates the strength of the inversion for small values of the wind speed Ua, even though the results are not too far from the error bars: the interquartile intervals barely overlap with those of the observed values. On the other hand, it appears obvious that it is actually due to compensation errors in the two layers taken individually: ΔTac is overestimated, while ΔTcs is underpredicted.

The two versions of mMP by far provide the best results compared to the observations, especially when fveg=1 (Fig. ). It captures the dependency of the two individual layers (atmosphere–canopy and canopy–surface) on the wind speed well. A key result is that the transition between the two modes is much more gradual in two-layer models. Over a forest area or in multi-layer models, the S shape, reported in one-layer models, becomes blurred: this smoother transition between the two modes can be ascribed to the presence of trees, which attenuates the S shape. This is the reason why two-layer models perform better.