The essential role of an integrated land surface model is to physically predict the land–atmosphere interactions by resolving the transfer of energy, water, and trace gases (Katul et al., 2012; Liang et al., 1994; Sellers et al., 1997). Such land–atmospheric interactions are strongly modulated by the partitioning of solar energy at the land surface (Chen and Dudhia, 2001; McCumber and Pielke, 1981; Yang and Wang, 2014) which can be considered through the surface energy balance (SEB) equation (Oke, 1988): Q∗-ΔQS=QH+QE, where Q∗, ΔQS, QH, and QE are the net all-wave radiation, net storage, turbulent sensible, and latent heat fluxes, respectively. Equation (1) distinguishes the available energy at the land surface (left-hand side) from the heat transfer through turbulent transport (right-hand side).

The turbulent and radiative fluxes (Q∗, QH, and QE) are more readily measured using standard techniques (e.g. eddy-covariance instruments, radiometry) than ΔQS (Offerle et al., 2005; Pauwels and Daly, 2016; Roberts et al., 2006; Wang, 2012). For ΔQS the net energy stored or released by changes in sensible heat within the canopy air layer, roughness elements (e.g. vegetation, buildings in an urban environment), and the ground all have to be considered. The volume of interest extends from the top of the roughness sub-layer to the depth in the ground where the daily averaged vertical net heat conduction is zero (see Fig. 2 in Masson et al., 2002); this presents very significant challenges of spatial sampling.

Knowledge of ΔQS is crucial to a wide range of processes and applications: from modelling turbulent heat transfer and boundary layer development to predicting soil thermal fields. In rural sites, or simple bare soil sites, the flux may be a small fraction of the net all-wave radiation (Oliphant et al., 2004). However, in areas where there is more mass, such as cities, the term becomes much more significant (Kotthaus and Grimmond, 2014a) and a key element of the SEB and well-known effects such as the urban heat island.

In urban systems a wide range of techniques have been used to estimate ΔQS (Grimmond et al., 1991; Roberts et al., 2006). These include the following: a.

Heat conduction approach: the weighted average of heat flows through all urban materials and surfaces by solving heat conduction equations – e.g. buildings, streets, vegetated lands (Offerle et al., 2005; Wang et al., 2012; Yang et al., 2014).

OHM is a cornerstone in the urban land surface models SUEWS (Surface Urban Energy And Water Balance Scheme; Järvi et al., 2011, 2014; Ward et al., 2016) and LUMPS (Local-Scale Urban Meteorological Parameterization Scheme; Grimmond and Oke, 2002), and plays an essential role in determining the initial energy partitioning at each time step of the models' simulations. Previous modelling studies (Arnfield and Grimmond, 1998; Meyn and Oke, 2009) have led to better understanding of the OHM coefficients. Solution of the one-dimensional advection–diffusion equation of coupled heat and liquid water transport by Gao et al. (2003, 2008) was used to explore the physical relation of OHM coefficients a1 and a2 to the phase lag between ΔQS and Q∗. However, insight into a3 remain unclear (Sun et al., 2013).

In this paper, the solutions of the one-dimensional advection–diffusion equation of coupled heat and liquid water transport (Gao et al., 2003, 2008) are employed with the SEB (Eq. 1) to investigate more fully the three OHM coefficients, the outcomes of which lead to development of the Analytical Objective Hysteresis Model (AnOHM) (Sect. 2). The Monte Carlo-based subset simulation (Au and Beck, 2001) approach is then used to undertake a sensitivity analysis of AnOHM to surface properties and hydrometeorological conditions (Sect. 3). An offline evaluation of AnOHM's performance for five sites with different land covers (Sect. 4) provides evidence that this is an alternative approach to obtain OHM coefficients. Given that this allows applications across a much wider range of environments and meteorological conditions, we conclude that AnOHM has important implications for land surface modelling (urban and non-urban).

Given the complex dependence of AnOHM coefficients on surface properties and meteorological forcing (Sect. 2.3), the impacts of these coefficients are assessed further by a sensitivity analysis.

In this section, the actual ability of AnOHM to determine the storage heat flux relative to observations is evaluated using 30 min observations from five sites of different land use/covers (Table 2). The measurements include turbulent sensible and latent fluxes, along with incoming and outgoing shortwave and longwave radiation and basic meteorological variables (see Kotthaus and Grimmond, 2014a, b; Klazura et al., 2006; Coulter et al., 2006; Goulden et al., 2006; Scott et al., 2009; Luo et al., 2007, for details). Anthropogenic heat flux QF at the urban site (i.e. UK-Ldn) is estimated using the GreaterQF model (Iamarino et al., 2011); the heat storage flux ΔQS is thus estimated as the modified residual of urban energy balance as ΔQS=Q∗+0.75QF-1.2QH+QE (Kotthaus and Grimmond, 2014a, b), which is then used in this evaluation. A similar approach for estimating ΔQS (i.e. residual of surface energy balance, ΔQS=Q∗+QF-QH+QE) is applied at the other (non-urban) sites but with QF=0.

AnOHM is first calibrated with observations under sunny conditions, when the assumptions of AnOHM are best satisfied (i.e. diurnal cycles of K↓ and Ta follow sinusoidal forms), to obtain surface properties required by AnOHM (Table 3). As the Bowen ratio β varies daily and monthly (Kotthaus and Grimmond, 2014a, b), β is either determined as the daily value if available, or based on the observation-based monthly climatology (Table 3). The seasonality in albedo α is accounted for also by using its monthly climatology (Table 3). AnOHM is driven by atmospheric forcing (i.e. K↓, Ta, and U) and/or their derived scales (AK, K‾↓, AT, T‾a, and τ) to generate the OHM coefficients (i.e. a1, a2, and a3, see Fig. 5), from which the net heat storage flux ΔQS can be predicted (Fig. 6) using the observed Q∗ with Eq. (2).

To examine the seasonality of the OHM coefficients, rather than the daily variations in hydrometeorological forcing, LOESS (LOcally wEighted Scatter-plot Smoother; Cleveland and Devlin, 1988) curves are obtained to filter out day-to-day variations in the OHM coefficients (see Appendix B for a direct comparison of these coefficients by different modelling and observational regression approaches). Intra-annual variations are found in all the three OHM coefficients (Fig. 5), indicating the strong impact of seasonality of meteorological conditions. These controls, as indicated by Eqs. (23)–(25/27), are complex and will vary with local conditions. For instance, comparison of OHM coefficients between the AnOHM predictions (LOESS fitted solid lines in Fig. 5) and observations at an asphalt road site in Alland, Austria, reported in Anandakumar (1999) (empty squares in Fig. 5) demonstrates differences in a1 (Fig. 5a) and a2 (Fig. 5b) but general similarity in a3 (Fig. 5c). Compared to a1 and a2, it is noteworthy that, in addition to the S results (see Fig. 4) given the more explicit mechanism by which the atmospheric conditions moderate a3 (see Eqs. 25 and 27), such seasonality in a3 is predicted by AnOHM, and evident in the observations (Fig. 5c, also Ward et al., 2013). Larger K‾↓ in warm seasons (May–September) will lead to smaller a3 (Eqs. 25, 27) and vice versa.

The AnOHM simulated and observed ΔQS agree well at the five different land cover sites, with RMSE values of ∼ 30 W m-2. For comparison purposes, it is noted that the urban land surface model comparison (Best and Grimmond, 2015; Grimmond et al., 2011) found ΔQS to be the most poorly represented among all the SEB components with the best RMSE values of 53 W m-2 (Lipson et al., 2017). Although the much smaller ΔQS RMSE obtained by AnOHM uses a prescribed Bowen ratio in the offline evaluation, such improvement indicates the ability of AnOHM to simulate a more consistent ΔQS with observations. Compared with OHM predictions (orange lines in Fig. 6), AnOHM (blue lines in Fig. 6) better reproduces the seasonality in ΔQS but gives larger bias at two sites with natural land covers (i.e. US-SRM and US-SO4). This can be attributed to the overestimates of nocturnal ΔQS by AnOHM. Overall, the evaluation demonstrates good performance of AnOHM in predicting the long-term ΔQS with clear seasonality reproduced across a wide range of surface types.

In this study, the Analytical Objective Hysteresis Model (AnOHM) is developed to obtain OHM coefficients across a wide range of surface and meteorological conditions and to improve physical understanding of the interactions between ΔQS and Q∗. The sensitivity of AnOHM to surface properties and hydrometeorological conditions is analysed through Monte Carlo-based subset simulations (Au and Beck, 2001). The results highlight the importance of the albedo, the Bowen ratio, and the bulk transfer coefficient, and the importance of solar radiation and wind speed in regulating the heat storage. The importance of albedo in modulating the heat storage was also found by Wang et al. (2011), who also used the same subset simulation approach with the single-layer urban canopy model (SLUCM; for details see Kusaka et al., 2001). This demonstrates the consistency in heat storage modelling between AnOHM and SLUCM. From the sensitivity results, variations in OHM coefficients of a similar size may arise from either surface property parameters or hydrometeorological forcing that are associated with the same physical processes (see bulk transfer coefficient Ch in Fig. 2 and wind speed U in Fig. 4). This supports the ability of AnOHM in representing physical processes. An offline evaluation of AnOHM using flux observations from five sites with different land covers demonstrates its ability to predict the intra-annual dynamics of OHM coefficients and shows good agreement between simulated and observed storage heat fluxes. In particular, the seasonality in the OHM coefficient a3 observed in a previous study (Anandakumar, 1999) is well predicted by AnOHM.

The limitations of AnOHM are important to consider. First, given the assumption that the incoming solar radiation K↓ and air temperature Ta diurnal cycles are sinusoidal, optimal performance of AnOHM occurs under clear-sky conditions. The current parameterizations of K↓ and Ta within AnOHM only consider the harmonics of principal frequencies for formulation simplicity. More frequencies may potentially resolve more realistic diurnal variations in K↓ and Ta. As the reflected part of L↓ (i.e. 1-εsL↓) is assumed negligible, and similar emissivity values are assumed for sky and land surface (i.e. εs≈εa≈ε), the outgoing longwave radiation is underestimated. These simplifications greatly facilitate the AnOHM formulation without qualitatively changing the final results as the sensitivity analyses (see the minimal S values for ε in Fig. 2) demonstrate. The inclusion of water flux density W equips AnOHM with an ability to investigate the hydrological impacts of the underlying surface on land–atmosphere interactions. However, estimation of W remains challenging (Wang, 2014) and the resulting uncertainty in the final results warrants caution in conducting simulations over land covers with strong soil moisture dynamics (e.g. grassland with high soil moisture under clear-sky condition). Despite these limitations, AnOHM does permit improved modelling of the surface energy balance through its physically based parameterization scheme for storage heat flux ΔQS. Compared to OHM, AnOHM has the benefit of allowing ΔQS to be simulated for land covers for which coefficients are not available and to allow for seasonal variability to be accounted for. As AnOHM shares similar hydrometeorological forcing inputs (i.e. K↓, Ta and U) to other land surface models (LSMs), it can potentially be used within in LSMs to estimate ΔQS, or if turbulent fluxes are included to be a complete LSM. The overall improvements from adopting AnOHM in modelling land surface processes will be presented in forthcoming work in the SUEWS–AnOHM framework.