Appendix H — Solution procedure process-based root water uptake models
The equation systems for both the dW and dJvL concepts are solved via a double, nested iteration procedure. First, two estimates for \(h_{\text L}\) are searched for such that
\[ F(h_{\text L}) = U(h_{\text L}) - T_{\text{act}}(h_{\text L}) \tag{H.1}\]
results in a positive and a negative value for the function \(F\). Then the final value for \(h_{\text L}\) can be found such that \(F\) = 0 (or less than some small convergence criterion). The function \(U(h_{\text L})\) is the solution of the inner iteration loop in which the set of \(N\) equations with \(N\) unknowns is solved. For the concept of dW the elements of \(U\) are given by
\[ \begin{array}{l} {u_1} = {Q_1}\left( {{x_1} - {h_{\text{L}}}} \right) - {S_1}\left( {{M_{{\text{s,}}1}} - {M_{{\text{0,}}1}}} \right) = 0\\ {u_2} = {Q_2}\left( {{x_2} - {h_{\text{L}}}} \right) - {S_2}\left( {{M_{{\text{s,}}2}} - {M_{{\text{0,}}2}}} \right) = 0\\ ...\\ {u_{\text{N}}} = {Q_{\text{N}}}\left( {{x_{\text{N}}} - {h_{\text{L}}}} \right) - {S_{\text{N}}}\left( {{M_{{\text{s,N}}}} - {M_{{\text{0,N}}}}} \right) = 0 \end{array} \tag{H.2}\]
where \(x\) = \(h_0\) and \(Q\) = \(\Delta z L_{\text {rv}}K_r\) (d-1) and \(S\) = \(\Delta z \rho_{\text {dW}}\) (cm-1). In matrix notation this is given by \(\mathbf U(\mathbf x)\) = 0.
For the concept of dJvL the elements of \(U\) are given by \[ \begin{array}{l} {u_1} = {h_{\text{L}}} - {x_1} + S{S_1}\left( {{M_{{\text{s,1}}}} - {M_{{\text{0,}}1}}} \right) + \frac{{{T_{\text{a}}}}}{{{L_1}}} = 0\\ {u_2} = {h_{\text{L}}} - {x_2} + S{S_2}\left( {{M_{{\text{s,}}2}} - {M_{{\text{0,}}2}}} \right) + \frac{{{T_{\text{a}}}}}{{{L_1}}} = 0\\ ...\\ {u_{\text{N}}} = {h_{\text{L}}} - {x_{\text{N}}} + S{S_{\text{N}}}\left( {{M_{{\text{s,N}}}} - {M_{{\text{0,N}}}}} \right) + \frac{{{T_{\text{a}}}}}{{{L_1}}} = 0 \end{array} \tag{H.3}\]
where \(x\) = \(h_0\) and \(SS\) = \(\phi_{\text {dJvL}}\) (d cm-1). In matrix notation this is given by \(\mathbf U(\mathbf x)\) = 0.
So, for both concepts the solutions are found in a similar way.
Following a Newton-Raphson method (Press et al. 1992) the problem \(\mathbf U(\mathbf x)\) = 0 is rewritten based on a Taylor expansion as
\[ {\bf{U}}\left( {{\bf{x}} + \delta {\bf{x}}} \right) = {\bf{U}}\left( {\bf{x}} \right) + {\bf{J}} \cdot \delta {\bf{x}} + O\left( {\delta {{\bf{x}}^2}} \right) \tag{H.4}\] Disregarding the second-order term on the right-hand side and requiring that \({\bf{U}}\left( {{\bf{x}} + \delta {\bf{x}}} \right)\) = 0, the problem reduces to
\[ {\bf{J}} \cdot \delta {\bf{x}} = - {\bf{U}} \tag{H.5}\] This matrix problem can be solved and the estimate for \(\bf{x}\) can be updated as
\[ {{\bf{x}}_{\text{new}}} = {{\bf{x}}_{\text{old}}} + \delta {\bf{x}} \tag{H.6}\]
This continues iteratively until \(\delta {\bf{x}} \rightarrow 0\), or the sum of all \(|\delta {x}|\) is less than a convergence criterion. The Jacobian \(\bf J\) contains the derivatives \({\text d}u/{\text d}x\). For the concept of dW this is given as (note: \({\text d}M/{\text d}h = K\))
\[ {\bf{J}} = \left| {\begin{array}{*{20}{c}} {{Q_1} + {S_1}{K_1}}&0&{...}&0\\ 0&{{Q_2} + {S_2}{K_2}}&{...}&0\\ {...}&{...}&{...}&{...}\\ 0&0&{...}&{{Q_{\text{N}}} + {S_{\text{N}}}{K_{\text{N}}}} \end{array}} \right| \tag{H.7}\]
Similarly, for the concept of dJvL this is given by
\[ {\bf{J}} = \left| {\begin{array}{*{20}{c}} { - 1 - S{S_1}{K_1}}&0&{...}&0\\ 0&{ - 1 - S{S_2}{K_2}}&{...}&0\\ {...}&{...}&{...}&{...}\\ 0&0&{...}&{ - 1 - S{S_{\text{N}}}{K_{\text{N}}}} \end{array}} \right| \tag{H.8}\]
Since \(\bf J\) only contains non-zero entries on the main diagonal, its inverse can be obtained directly. All diagonal elements of \({\bf{J}}^{ - 1}\) are equal to the reciprocal of the diagonal elements of \(\bf J\). Thus, Equation H.5 can be solved directly: \(\delta {\bf{x}} = - {{\bf{J}}^{ - 1}} \cdot {\bf{U}}\).