The models for root distribution proposed by Laio et al. (
2006), Preti et al. (
2010) and Schenk (
2008) are mainly intended to be used in situations where the vegetation uptake relies on water infiltrating into the soil (Tron et al.
2015). In riparian regions, ground water is the main source of nutrients and water for vegetation (Zeng et al.
2006), unlike in other situations, where the nutrient availability decreases with depth. The roots can concentrate in the top regions due to lack of oxygen resulting from high water table or can grow deep to reach the water table to exploit necessary nutrients and water. To model the effects of riparian vegetation on soil reinforcement, it is required to adopt a root distribution model, which takes into account the above-mentioned situations. Tron et al. (
2014) developed a stochastic analytical model for finding the vertical root distribution in ecosystems where rainfall infiltration is not the main source of plant water uptake, see Eqs.
1–
4:
$${\overline{\text{r}}}\left( {\text{z}} \right) = \frac{{2{\uptheta }\left( {\text{z}} \right){\text{k}}\left( {\text{z}} \right)}}{{{\uptheta }\left( {\text{z}} \right) + {\uptheta }\left( {\text{z}} \right){\text{K}}\left( {\text{z}} \right) + 1 - {\text{k}}\left( {\text{z}} \right)}}$$
(1)
$${\uptheta }\left( {\text{z}} \right) = \frac{{{\upbeta }\left( {\text{z}} \right)}}{{\upgamma }}$$
(2)
$${\text{k}}\left( {\text{z}} \right) = \left\{ {\begin{array}{*{20}c} {\left\{ {\begin{array}{*{20}c} {\frac{{\Gamma \left( {\frac{\lambda }{\eta },\frac{{{\text{h}}_{1} - {\text{z}} - {\text{L}}}}{{\upalpha }}} \right) - \Gamma \left( {\frac{\lambda }{\eta },\frac{{{\text{h}}_{1} - {\text{z}}}}{{\upalpha }}} \right)}}{{\Gamma \left( {\frac{\lambda }{\eta }} \right)}},} \\ { } \\ \end{array} } \right.} & { {\text{if}} - \infty < z < {\text{h}}_{1} - L} \\ {1 - \frac{{\Gamma \left( {\frac{\lambda }{\eta },\frac{{{\text{h}}_{1} - {\text{z}}}}{{\upalpha }}} \right)}}{{\Gamma \left( {\frac{\lambda }{\eta }} \right)}}} & {{\text{if}}\,{\text{h}}_{1} - L < z < {\text{h}}_{1} } \\ \end{array} } \right.$$
(3)
$$\alpha = \frac{\check{\alpha}}{{h_{2} }},$$
(4)
where
\(\stackrel{-}{r}\left(z\right)\) is the normalized root mass at depth z, and to obtain real root mass,
\(\stackrel{-}{r}\left(z\right)\) is multiplied by maximum root mass. At a certain depth from the water table, the root growth is enhanced and its range depends on the water table fluctuation.
\(k\left(z\right)\) is the probability that a depth
\(\mathrm{z}\) falls in the optimal root growth zone, the zone below the surface where the root growth is favoured. The range of the optimal root growth zone is represented by a root box of width
\(L\). The water table jumps, which are considered instantaneous at daily time scale, are taken into account using the mean depth of pulses
\(\check{\alpha}\).
\(\upbeta \left(z\right)\) represents the growth rate of roots and
\(\gamma\) represents decay rate of roots. λ is the mean rate of stochastic instantaneous rise of water level,
\({h}_{2}\) is the depth of water table at driest periods,
\({\eta}\) is the water level decrease in time, and
\({h}_{1}\) is depth of the root box.
According to this model, the roots concentrate on the upper layers if the variability of the water table is high, and deeper roots are found when the variability of the water table is less.