## 2 Materials and Methods
### The HJA-N Model
See Table 7 for parameter definitions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Rate** & **Definition** & **Units** \\ \hline \(M_{{}_{i,S}}=W_{{}_{i,S}}m_{{}_{i,S}}\) & \begin{tabular}{c} Mobilization of available N from instream (\(i=Na\) \\ represents unavailable, \(i=Nu\) represents available) \\ \end{tabular} & kg N/m\({}^{2}\) \\ \(I_{{}_{Na,S}}=W_{{}_{Na,S}}i_{{}_{Na,S}}\lambda\) & \begin{tabular}{c} Immobilization of available \(N\) from instream \\ \end{tabular} & kg N/m\({}^{2}\) \\ \(Q_{{}_{i,S}}=Q_{{}_{H^{\prime},S}}\left(\dfrac{W_{{}_{i,S}}}{W_{{}_{H^{\prime}, H^{\prime}}}+W_{{}_{H^{\prime},S}}}\right)\) & \begin{tabular}{c} Discharge of available N from stream. (\(i=Na\) \\ represents unavailable. \(i=Nu\) represents available) \\ \end{tabular} & kg N/m\({}^{2}\) \\ \(BI_{N}\) & \
\begin{tabular}{c} Biological input of N to the stream \\ \end{tabular} & kg N/m\({}^{2}\) \\ \(P_{{}_{N,S}}=A_{{}_{f}}{{}^{-1}}p_{{}_{{}_{P}}}\) &
\begin{tabular}{c} Particulate export from the stream \\ \end{tabular} & kg N/m\({}^{2}\) \\ \hline \(\text{when}\,Q_{{}_{H^{\prime},S}}\,/\,q_{{}_{max}}>1\) & \(\,q_{{}_{f}}\)-\(\left(\left(1+\dfrac{\left(\hat{\mathfrak{h}}-Q_{{}_{H^{\prime},S}}/q_{{}_{max}} \right)}{k_{{}_{3}}}\right)\right)\), Else \(\,q_{{}_{f}}=1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Rate terms comprising instream-processing of N. See Table 7 for parameter definitions. \begin{table}
\begin{tabular}{c c c} \hline \hline \begin{table}
\begin{tabular}{c c c} \hline
**Term** & **Definition** & **Units** \\ \hline \(\eta_{\forall}=(1-\mathsf{w}f_{i})*(1-\eta_{\omega_{r}})\) & Allocation fraction of foliage where \(i=C\) & – \\  & for Carbon and \(i=N\) for Nitrogen & – \\ \(\eta_{\omega_{r}}=\mathsf{w}f_{i}*(1-\eta_{\omega_{r}})\) & Allocation fraction of wood where \(i=C\) for & – \\  & Carbon and \(i=N\) for Nitrogen & – \\ \(\eta_{\omega_{r}}=a_{i}\bigg{[}a_{2}+(\frac{PAR_{R}}{PAR})a_{3}\bigg{]}^{1}\) & Allocation fraction root C & – \\ \(\eta_{\omega_{r}}=a_{i}\bigg{[}a_{5}+(\frac{CN_{r}}{CN_{\omega_{r}}})a_{6} \bigg{]}^{1}\) & Allocation fraction root N & – \\  & C (\(i=C\)) or N (\(i=N\)) remaining allocated & – \\ \(\mathsf{w}f_{i}\) & to wood & – \\ \(\lambda\)=\(q_{0}^{T/10}\) & Temperature modifier & – \\ \(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}\) & Root allocation parameters (hyperbola) & – \\ \hline \end{tabular}
\end{table} 
Table 7: _Cont._ \begin{table}
\begin{tabular}{c c c} \hline
**Term** & **Definition** & **Units** \\ \hline \(u_{\max}\) & Max uptake rate & kg N/m\({}^{2}\) \\ \(u_{{}_{1/2}}\) & Uptake \(\
icefrac{{1}}{{2}}\) saturation & kg N/m\({}^{2}\) \\ \(\tau_{{}_{fl}},\tau_{{}_{wt}},\tau_{{}_{rt}}\) & Turnover rate constants & year\({}^{-1}\) \\ \(d_{{}_{fl}},d_{wd},d_{rd}\) & Decomposition rate constants & day\({}^{-1}\) \\ \(q_{{}_{10}}\) & Coef by which D increases each 10 C & kg/m\({}^{2}\) \\ T & temperature & C \\ \(f_{CO_{2}f^{\prime}},f_{CO_{2}w},f_{CO_{2}r}\) & Fraction of D as CO\({}_{2}\) & – \\ \(CN_{{}_{\eta^{\prime}}},CN_{{}_{\mu\
u}},CN_{{}_{\
u}}\) & CN ratios of detrital pools & – \\ \(CN_{{}_{\sigma^{\prime}}},CN_{{}_{\omega_{r}}},CN_{{}_{\omega_{r}}}\) & Critical CN, above which N is retained & – \\ \(r_{{}_{R}},r_{{}_{B}}\) & Soil respiration coefficient & year\({}^{-1}\) \\ \(\tau_{{}_{I,R}}\) & Transfer rate of particulates & year\({}^{-1}\) \\ \(CN_{{}_{s,s}},CN_{{}_{s,s}}\) & CN ratios of SOM pools & – \\ \(i_{{}_{1/2}}\) & Max immobilization rate & year\({}^{-1}\) \\ \(dn_{{}_{R}},dn_{{}_{R}},dn_{{}_{B}}\) & Denitrification rate constant & year\({}^{-1}\) \\ \(S_{{}_{Na},R}\), \(S_{{}_{Na},B}\), \(S_{{}_{Na},R}\), \(S_{{}_{Na},B}\), \(S_{{}_{Na},B}\), \(S_{{}_{Na},B}\) & Sorption rate constant & year\({}^{-1}\) \\ \(d_{{}_{Ni},R},d_{{}_{Ni},B}\) & Decomposition rate of dissolved unavailable N & day\({}^{-1}\) \\ \(m_{{}_{Na,IS}},m_{{}_{Na,IS}}\) & Instream mobilization rate constant & day\({}^{-1}\) \\ \(i_{{}_{Na},IS},i_{{}_{Na},IS}\) & Instream immobilization rate constant & day\({}^{-1}\) \\ \hline \end{tabular}
\end{table}