Introduction of a sub-grid hydrology in the ISBA land surface model

Abstract

In atmospheric models, the partitioning of precipitation between infiltration and runoff has a major influence on the terrestrial water budget, and thereby on the simulated weather or climate. River routing models are now available to convert the simulated runoff into river discharge, offering a good opportunity to validate land surface models at the regional scale. However, given the low resolution of global atmospheric models, the quality of the hydrological simulations is much dependent on various processes occurring on unresolved spatial scales. This paper focuses on the parameterization of sub-grid hydrological processes within the ISBA land surface model. Five off-line simulations are performed over the French Rhône river basin, including various sets of parameterizations related to the sub-grid variability of topography, precipitation, maximum infiltration capacity and land surface properties. Parallel experiments are conducted at a high (8 km by 8 km) and low (1° by 1°) resolution, in order to test the robustness of the simulated water budget. Additional simulations are performed using the whole package of sub-grid parameterizations plus an exponential profile with depth of saturated hydraulic conductivity, in order to investigate the interaction between the vertical soil physics and the horizontal heterogeneities. All simulations are validated against a dense network of gauging measurements, after the simulated runoff is converted into discharge using the MODCOU river routing model. Generally speaking, the new version of ISBA, with both the sub-grid hydrology and the modified hydraulic conductivity, shows a better simulation of river discharge, as well as a weaker sensitivity to model resolution. The positive impact of each individual sub-grid parameterization on the simulated discharges is more obvious at the low resolution, whereas the high-resolution simulations are more sensitive to the exponential profile with depth of saturated hydraulic conductivity.

Appendices

Appendix 1

1.1 Dunne surface runoff formulation

Here, the coupling between ISBA and TOPMODEL is briefly reviewed. More details can be found in Habets and Saulnier (2001) and Decharme et al. (2005). TOPMODEL describes generally the evolution of a water storage deficit near the soil surface. Therefore, the active layer chosen for the ISBA-TOPMODEL coupling is the root layer, and not the total soil column.

The Dunne surface runoff is then simply given by Q ^D_s =P _g × f _sat where f _sat is the saturated fraction of a grid cell. f _sat is inversely proportional to the mean water storage deficit, D _t (m), of a grid cell computed as follows:

$$0 \leq D_{\text{t}} = \left({w_{{\text{sat}}} - {\overline{w_{\text{2}} } }} \right) \times \overline {d_{\text{2}} } \leq d_{\text{0 }} $$

(10)

$$d_{\text{0}} = \left({w_{\operatorname{sat} } - w_{{\text{wilt}}} }\right) \times \overline {d_{\text{2}} } $$

(11)

where d ₀ (m) is the maximum local deficit, and$\overline{w_{\text{2}} }$ and $\overline {d_{\text{2}} }$ are the mean volumetric water contents and depth of the root zone according to the relative fraction of each land surface tiles within each grid cell. In other words, when $\overline {w_{\text{2}} }$ is below the wilting point, the mean water storage deficit is a maximum, D _t=d ₀, f _sat= 0 and no surface runoff occurs.

Appendix 2

1.1 Horton surface runoff formulation

As previously discussed in Sect. 3, the sub-grid variability in local rainfall, P _i , can be given by an exponential probability density distribution (Entekhabi and Eagleson 1989):

$$f\left({P_i } \right) = \frac{\mu }{{\overline P }}{\text{e}}^{ - \mu{{(P_i } \mathord{\left/ {\vphantom {{(P_i } {\overline P }}} \right.\kern-\nulldelimiterspace} {\overline P }})} $$

(12)

where$\overline P$ is the mean rainfall rate over the grid cell and μ is the fraction of the grid cell affected by precipitation reaching the surface. μ can be determined using the results of Fan et al. (1996), who showed an exponential relationship between the fractional coverage of precipitation and rainfall rate, based on their analyses of over 2 years radar observations and rain gauge measurements over the Arkansas-Red river basin in the southern plains of the USA. This relationship is:

$$\mu = {\text{1}} - {\text{e}}^{ - \beta \overline P } $$

(13)

where β is a parameter which depends on grid resolution, dx, according to the relationship given by Peters-Lidard et al. (1997):

$$\beta = {\text{0}}{\text{.2}} + {\text{0}}{\text{.5e}}^{ - 0.01dx}$$

(14)

In Fan et al. (1996), dx represents lengths of square grid cells ranging from 40 to 500 km. In consequence, the μ parameter here is equal to 1 at high resolution because the length of each grid cell is equal to 8 km. At low resolution, it is calculated according to Fan et al. (1996) (Eq. 13), where the β parameter depends, for each grid cell, on the root-square of the grid cell area in km² (Eq. 14).

The spatial heterogeneity of the local maximum infiltration rates, I _i , can also be approximated by an exponential probability density distribution (Yu 2000):

$$g\left({I_i } \right) = \frac{{\text{1}}}{{\overline I }}{\text{e}}^{- {{I_i } \mathord{\left/ {\vphantom {{I_i } {\overline I }}} \right.\kern-\nulldelimiterspace} {\overline I }}} $$

(15)

where $\overline I$ is the mean maximum infiltration rate over the grid cell. Distinction has been made between infiltration on unfrozen and frozen soil. The local maximum infiltration rates on unfrozen soil, I _unf, i, is given as in Chen and Kumar (2001) according to Abramopoulos et al. (1988) and Entekhabi and Eagleson (1989), and on frozen soil, I _f,i, by Johnsson and Lundin (1991). Generalization of these functions into the ISBA framework leads to:

$$I_{{\text{unf}},\,\,i} = k_{{\text{sat}},\,\,i} \left[ {\frac{{b\psi_{{\text{sat}}} }}{{\Delta z}}\left({\frac{{w_{\text{2}}}}{{w_{{\text{sat}}}^* }} - 1} \right) + 1} \right]$$

(16a)

$$I_{{\text{f}},\,i} = k_{{\text{sat}},\,i}^{{\text{ice}}}\left({\frac{{w_{\text{2}} }}{{w_{{\text{sat}}}^* }}}\right)^{{\text{2}}b + {\text{3}}} \times 10^{ - 6\frac{{w_{I2}}}{{w_{I2} + w_2 }}} $$

(16b)

where k _sat,i (ms⁻¹) is the local surface saturated hydraulic conductivity, ψ_sat (m) is the saturated soil water potential or air entry potential, Δz is a soil thickness of 0.1 m, and b is a dimensionless slope parameter (Brooks and Corey 1966; Clapp and Hornberger 1978). w ₁₂ is the layer-average volumetric ice content (m³  m⁻³) in the root zone and w ^*_sat is the soil porosity in presence of soil ice (Boone et al. 2000). k ^ice_sat,i is the local layer-average saturated hydraulic conductivity over a diagnostic reservoir, corresponding to the ice depth, d _ice, where the ice can be present. In this study, d _ice is equal to 50% of the rooting depth. This calibration gives the best results over the Rhône basin even if Johnsson and Lundin (1991) recommended a value of 0.2 m, which is used at the global scale. Finally, the fraction of the frozen soil in ISBA is given by $\delta _{\text{f}} = \min\left({{{w_{I2} \cdot d_2 } \mathord{\left/ {\vphantom {{w_{I2} \cdot d_2 } {w_{{\text{sat}}} \cdot d_{{\text{ice}}} }}} \right.\kern-\nulldelimiterspace} {(w_{{\text{sat}}} \cdot d_{{\text{ice}}})}},1} \right).$

According to Eq. 7, a new expression of the Horton runoff, Q ^H_s , is given by:

$$Q_{\text{s}}^{\text{H}} = \mu \left[ {\left({1 - \delta _{\text{f}} }\right)\int\limits_0^\infty {\int\limits_{I_{{\text{unf},}\,i} }^\infty{\left({P_i - I_{{\text{unf},}\,\,i} } \right)} f\left({P_i }\right)g\left({I_{{\text{unf},}\,\,i} } \right)\,{\text{d}}P_i\,{\text{d}}I_{{\text{unf},}\,\,i} + }\; \delta _{\text{f}}\int\limits_0^\infty {\int\limits_{I_{\operatorname{f}, \,i} }^\infty{\left({P_i - I_{{\text{f}},\,i} } \right)} f\left({P_i }\right)g\left({I_{{\text{f}},\,i} } \right)\,{\text{d}}P_i\,{\text{d}}I_{{\text{f}},\,i} } } \right]$$

(17)

where the first term corresponds to the Horton runoff on unfrozen soil and the second to the Horton runoff on frozen soil. Consequently and passing throughout several algebraic steps, the Horton runoff in ISBA is computed as follows:

$$Q_{\text{s}}^{\text{H}} = \left({1 - \delta _{\text{f}} }\right)\frac{{\overline P }}{{1 + \overline {I_{{\text{unf}}} }\frac{\mu }{{\overline P }}}} + \delta _{\text{f}} \frac{{\overline P}}{{1 + \overline {I_{\text{f}} } \frac{\mu }{{\overline P }}}}$$

(18)

where $\overline {I_{\text{f}} }$ and $\overline {I_{{\text{unf}}} }$ are the grid-average frozen and unfrozen maximum infiltration capacity, respectively. Finally, in the presence of snowmelt, S _m, any assumptions about its spatial variability can be made, and the Horton runoff is simply given by:

$$Q_{\text{s}}^{\text{H}} = \left({1 - \delta _{\text{f}} }\right){\text{max}}\left({{\text{0}},\,S_{\text{m}} - \overline{I_{{\text{unf}}} } } \right) + \delta _{\text{f}}{\text{max}}\left({{\text{0}},\,S_{\text{m}} - \overline {I_{\text{f}} }} \right)$$

(19)