Add Plant N Demand and N Limiting

docs/model-structure.md

@@ -62,6 +62,7 @@ When LITTER_POOL=0:
 - Carbon fluxes that would go to the litter pool ($F^C_\text{litter}$) are routed directly to the soil carbon pool
 - All decomposition occurs in the soil pool
 - This affects carbon routing from harvest events, organic matter additions, plant senescence, and other processes involving $F^C_\text{litter}$
+- All nitrogen cycle modeling is off (that is, NITROGEN_CYCLE=1 requires LITTER_POOL=1)
 
 ### Maximum Photosynthetic Rate
 
@@ -101,6 +102,8 @@ The total adjusted gross primary production (GPP) is the product of potential GP
 
 The water stress factor $D_{\text{water,}A}$ is defined in equation \ref{eq:A16} as the ratio of actual to potential transpiration, and therefore couples GPP to transpiration by reducing GPP.
 
+Note that nitrogen limitation can further reduce GPP; see [Nitrogen Limitation](#nitrogen-limitation).
+
 ### Plant Growth
 
 $$
@@ -112,10 +115,11 @@ Net primary productivity  $(\text{NPP})$ is the total carbon gain of plant bioma
 To make explicit what contributes to autotrophic respiration, we decompose $R_A$ into maintenance and optional growth components:
 
 $$
-R_A = R_\text{leaf} + R_\text{wood} + R_\text{root} +\ R_\text{growth} \tag{1a}\label{eq:ra_components}
+R_A = R_\text{leaf} + R_\text{wood} + R_\text{fine_root} + R_\text{coarse_root} +\ R_\text{growth} \tag{1a}\label{eq:ra_components}
 $$
 
-Here, $R_\text{leaf}$ and $R_\text{wood}$ are maintenance respiration terms (Eqs. \ref{eq:A18a}, \ref{eq:A19}); $R_\text{root}$ denotes root maintenance respiration; and $R_\text{growth}$ is an optional growth respiration term. Because these components are part of $R_A$, their costs are subtracted from GPP before calculating NPP and before allocating NPP to plant pools.
+Here, $R_\text{leaf}$ and $R_\text{wood}$ are maintenance respiration terms (Eqs. \ref{eq:A18a}, \ref{eq:A19}); 
+$R_\text{fine_root}$ and $R_\text{coarse_root}$ denote root maintenance respiration; and $R_\text{growth}$ is an optional growth respiration term. Because these components are part of $R_A$, their costs are subtracted from GPP before calculating NPP and before allocating NPP to plant pools.
 
 Note that $\alpha_i$ are specified input parameters and $\sum_i{\alpha_i} = 1$.
 
@@ -129,18 +133,30 @@ $$
 
 Summing \ref{eq:Z3} over all plant pools shows that NPP is partitioned into biomass growth, litter production, and removed harvest.
 
+**TODO:** do we need to explain the five-day averaging of NPP here, or is sufficient in the Wood Carbon section below?
+
 ### Plant Death
 
 Plant death is implemented as a harvest event with the fraction of biomass transferred to litter, $f_{\text{harvest,transfer,}i}$ set to 1.
 
 ### Wood Carbon
 
+SIPNET uses a five-day averaged NPP when allocating gained carbon to plant growth. To implement this, the adjusted GPP 
+is added to the wood carbon pool as a storage mechanism, and all allocations from the averaged NPP are deducted from that pool. 
+We can represent this storage of carbon conceptually as:
+$$ 
+NPP_\text{storage} = (GPP - R_a) - \overline{NPP}_\text{alloc}
 $$
-\frac{dC_\text{wood}}{dt} = \alpha_\text{wood}\cdot\text{NPP} - F^C_\text{litter,wood} \tag{Braswell A1}\label{eq:A1}
-$$
+where $\overline{NPP}_\text{alloc}$ is the sum of the carbon allocated to the biomass pools as growth. Note that we do not explicitly track this storage term.
 
-Change in plant wood carbon  $(C_W)$ over time is determined by the fraction of net primary productivity allocated to wood, and wood litter production  $(F^C_\text{litter,wood})$.
+Thus, changes to wood carbon over time are determined by:
+
+$$
+\frac{dC_\text{wood}}{dt} = NPP_\text{storage} + \alpha_\text{wood} \cdot \overline{\text{NPP}} - F^C_\text{litter,wood}
+\tag{Braswell A1, modified}\label{eq:A1}
+$$
 
+where $\alpha_\text{wood}\cdot\overline{\text{NPP}}$ represents the amount of carbon allocated to growth and $(F^C_\text{litter,wood})$ is the wood litter production.
 
 ### Leaf Carbon
 
@@ -150,6 +166,22 @@ $$
 
 The change in plant leaf carbon  $(C_\text{leaf})$ over time is given by the balance of leaf production  $(L)$ and leaf litter production  $(F^C_\text{litter,leaf})$.
 
+**TODO:** explain $L$ in terms of $\alpha_\text{leaf}\cdot \overline{NPP}$ and leaf on/leaf off mechanics.
+
+### Root Carbon
+
+Both fine and coarse root carbon are treated similarly. Change in carbon for these pools is determined by these equations, applied separately to fine and coarse roots:
+
+$$
+\frac{dC_\text{i}}{dt} = \alpha_\text{i} \cdot \overline{NPP} - F^C_\text{i,root loss,}
+$$
+
+for $i \in \{\text{fine root}, \text{coarse root}\}$, where $F^C_\text{i,root loss}$ is determined by:
+$$
+F^C_\text{i,root loss} = k_\text{i,turnover} \cdot C_\text{i}
+$$
+and $k_\text{i,turnover}$ is the root turnover rate.
+
 ### Leaf Maintenance Respiration
 
 $$
@@ -427,26 +459,41 @@ Because we expect $N_2O$ emissions will be dominated by fertilizer N inputs, we
 A new fixed parameter $K_\text{vol}$ will represent the proportion of $N_\text{min}$ that is volatilized as $N_2O$ per day.
 
 $$
-F^N_\mathrm{vol} = K_\text{vol} \cdot N_\text{min} \cdot D_{\text{temp}} \cdot D_{\text{water}R_H} \tag{17}\label{eq:n_vol}
+F^N_\mathrm{vol} = K_\text{vol} \cdot N_\text{min} \cdot D_{\text{temp}} \cdot D_{\text{water}R_H}
+\tag{17}\label{eq:n_vol}
 $$
 
 ### Nitrogen Leaching $F^N_\text{leach}$
 
 $$
-F^N_\text{leach} = N_\text{min} \cdot F^W_{drainage} \cdot f_{N leach} \tag{18}\label{eq:n_leach}
+F^N_\text{leach} = N_\text{min} \cdot F^W_{drainage} \cdot f_{N leach} 
+\tag{18}\label{eq:n_leach}
 $$
 
 Where $f^N_\text{leach}$ is the fraction of $N_{min}$ in soil that is available to be leached, $F^W_{drainage}$ is drainage.
 
-### $\frak{Nitrogen \ Fixation \ F^N_\text{fix}}$
+### Plant Nitrogen Demand  $F^{N}_{\text{demand}}$
+
+Plant N demand is the amount of N required to support plant growth. This is calculated as the sum of changes in plant N pools:
+
+$$
+F^N_\text{demand}=\frac{dN_\text{plant}}{dt} = \sum_{i} \frac{dN_{\text{plant,}i}}{dt} 
+\tag{19}\label{eq:plant_n_demand}
+$$
+
+$$\small i \in \{\text{leaf, wood, fine root, coarse root}\}$$
+
+Each term in the sum is calculated according to equation \ref{eq:plant_n}. Total plant N demand $F^N_\text{demand}$ is then partitioned between fixation and soil N uptake using equations \ref{eq:n_fix_demand} and \ref{eq:n_uptake_demand}.
+
+### Nitrogen Fixation and Uptake $F^N_\text{fix}, F^N_\text{uptake}$
 
 For N-fixing plants, symbiotic nitrogen fixation is represented as supplying a fraction of plant nitrogen demand, and is inhibited by high soil mineral N. Plant N demand is defined in Eq. \ref{eq:plant_n_demand}.
 
 The fraction of plant N demand met by biological N fixation is defined as:
 
 $$
 f_\text{fix} = f_{\text{fix,max}} \cdot D_{N_\text{min}}
-\tag{19}\label{eq:f_fix}
+\tag{20}\label{eq:f_fix}
 $$
 
 where:
@@ -458,7 +505,7 @@ We use a simple down-regulation function with increasing soil mineral N:
 
 $$
 D_{N_\text{min}} = \frac{{K_N}}{{K_N} + N_\text{min}}
-\tag{19a}\label{eq:n_fix_supp_demand}
+\tag{21}\label{eq:n_fix_supp_demand}
 $$
 
 where $N_\text{min}$ is the soil mineral N pool (g N m$^{-2}$) and $K_N$ is the amount of mineral N at which fixation is reduced by half (g N m$^{-2}$).
@@ -467,50 +514,48 @@ Nitrogen fixation and soil N uptake are then partitioned from total plant N dema
 
 $$
 F^N_\text{fix} = f_\text{fix} \cdot F^N_\text{demand}
-\tag{19b}\label{eq:n_fix_demand}
+\tag{22a}\label{eq:n_fix_demand}
 $$
 
 $$
 F^N_\text{uptake} = (1 - f_\text{fix}) \cdot F^N_\text{demand}
-\tag{19c}\label{eq:n_uptake_demand}
+\tag{22b}\label{eq:n_uptake_demand}
 $$
 
 Fixed N ($F^N_\text{fix}$) is added directly to the plant N pool via Eq. \ref{eq:plant_n}, while $F^N_\text{uptake}$ is removed from the soil mineral N pool in Eq. \ref{eq:mineral_n_dndt}. If the available soil mineral N is insufficient to supply $F^N_\text{uptake}$, then actual uptake is capped at $N_\text{min}$ and any residual unmet demand contributes to nitrogen limitation as described in Eq. \ref{eq:n_limit}.
 
 We do not consider free-living nonsymbiotic N fixation, which is approximately two orders of magnitude smaller (less than 2 kg N ha$^{-1}$ yr$^{-1}$, Cleveland et al. 1999) than crop N demand and typical N fertilization rates.
 
-### $\mathfrak{Plant\ Nitrogen\ Demand\ and\ Uptake\ (F^{N}_{\text{uptake}})}$, $F^{N}_{\text{demand}}$
+### Nitrogen Limitation
 
-Plant N demand is the amount of N required to support plant growth. This is calculated as the sum of changes in plant N pools:
+What happens when plant N demand exceeds available N? This is N limitation, a challenging process to represent in biogeochemical models.
+
+The initial approach to representing N limitation in SIPNET will be simple, and the primary motivation for implementing this is to avoid mass imbalance. First we will identify the presence of nitrogen limitation with an indicator variable:
+First we calculate the demand based on our 5-day time-averaged NPP, pool allocation parameters, and C:N ratio:
 
 $$
-F^N_\text{demand}=\frac{dN_\text{plant}}{dt} = \sum_{i} \frac{dN_{\text{plant,}i}}{dt} \tag{20}\label{eq:plant_n_demand}
+N_\text{demand,est} = \overline{NPP} \cdot \sum_i{(\alpha_i \cdot CN_i)} 
+\tag{23}\label{eq:n_demand_est}
 $$
-
 $$\small i \in \{\text{leaf, wood, fine root, coarse root}\}$$
 
-Each term in the sum is calculated according to equation \ref{eq:plant_n}. Total plant N demand $F^N_\text{demand}$ is then partitioned between fixation and soil N uptake using equations \ref{eq:n_fix_demand} and \ref{eq:n_uptake_demand}.
-
-#### $\frak{Nitrogen \ Limitation \ Indicator \ Function \mathfrak{I_{\text{N limit}}}}$
-
-What happens when plant N demand exceeds available N? This is N limitation, a challenging process to represent in biogeochemical models.
-
-The initial approach to representing N limitation in SIPNET will be simple, and the primary motivation for implementing this is to avoid mass imbalance. First we will identify the presence of nitrogen limitation with an indicator variable:
+Note that this differs from Eq. \ref{eq:plant_n_demand} as this term is the expected growth, based solely on NPP.
 
+Next we compute the nitrogen deficit and convert back to carbon (in order to determine an appropriate scaling factor for GPP):
 $$
-I_{\text{N limit}} = \begin{cases}
-1, & \text{if } \frac{dN_\text{plant}}{dt} \leq N_{\text{min}} \\
-0, & \text{if } \frac{dN_\text{plant}}{dt} > N_{\text{min}}
-\end{cases} \tag{21}\label{eq:n_limit}
+\begin{array}{lcr}
+\Delta_N = N_\text{demand,est} - N_\text{min} \\
+\Delta_C = \frac{\Delta_N}{\sum_i{\alpha_I \cdot CN_i}} 
+\end{array}
+\tag{24}
 $$
-
-When $I=0$, SIPNET will throw a warning and increase autotrophic respiration to $R_A=GPP$ to stop plant growth and associated N uptake:
+Next we define a nitrogen dependency function $D_N$ as a function of estimated demand and current mineral N, scaled to GPP:
 
 $$
-R_A = \max(R_A, I_{\text{N limit}} \cdot GPP) \tag{22}\label{eq:n_limit_ra}
+D_N = \min(1, \frac{\text{GPP} - \Delta_C}{\text{GPP}})
 $$
 
-This will effectively stop plant growth and N uptake when there there is insufficient N.
+Last, we update GPP from Eq. \ref{eq:A17} by multiplying by this factor.
 
 We do expect N limitation to occur, including in vineyards and woodlands, but we assume that effect of nitrogen limitation on plant growth will have a relatively smaller impact on GHG budgets at the county and state scales. This is because nitrogen limitation should be rare in California's intensively managed croplands because the cost of N fertilzer is low compared to the impact of N limitation on crop yield.