The cardinal temperatures Tb, To and Tm in this analysis have been kept the same as , which were chosen based on the literature review in . As in , there was the assumption of no dependence of thermal time on the photoperiod.

The thermal times were calculated using the available Mead data for the sowing date, the date at which 50 % of the plants had emerged

Emergence dates for 2001–2003 were estimated by the site investigator based on weather.

The carbon partitioning parameters αi, βi were tuned to observations of the biomass of green leaves, yellow leaves, stem and reproductive parts of maize. The ratio of carbon to biomass in each part of the plant was assumed to be the same and constant in time. The Cleaf pool in the model contains green leaves only (since Cleaf is directly linked to LAI and photosynthesis) and the Charv pool consists of both the reproductive parts of the plants and the yellow leaves. Stem carbon in the model is split between the Cstem and Cresv pools. The biomass observations were linearly interpolated to get a daily time series and then differentiated with respect to time. Ratios of these rates were then plotted as a function of DVI (Fig. ). Using these plots alongside the function for root carbon from (since there were no direct measurements of root biomass available from the Mead sites), new, tuned values for αi, βi were found. These tuned parameters (dashed lines) show an improvement in the proportion of the increase in above-ground carbon that goes to the green leaves (Fig. , top) and the proportion of the increase in above-ground carbon that goes to the stem (Fig. , middle) for DVI < 0.8 as compared to the parameters used in (solid line). However, note that, even after the tuning, the proportion of the increase in above-ground carbon that goes to the green leaves does not drop off sharply enough for DVI > 0.8 compared to the observations. The tuned partition fractions are shown more clearly in Fig.  (colours), together with the functions given in (the αi, βi in were fitted to these functions with minor adjustments as a result of global runs). It was not possible to fit proot accurately to the expression from for approximately a DVI of 1.0 to 1.4 given the constraints above. In addition, in reality, water stress can also increase the fraction of NPP going to the roots (see discussion in, e.g., and ), but this effect is not taken into account in JULES-crop. However, we do not see a notable difference between the irrigated sites US-Ne1 and US-Ne2 (blue and green lines respectively) and rainfed site US-Ne3 (red lines) in Fig. .

The stem biomass observations were used to tune the value for the stem reserve remobilisation constant τ. The relation governing the stem reserve remobilisation can be rearranged to 1-MstemMstemmax⁡=τ1-0.9dmax⁡, where Mstem is stem biomass (including reserves), Mstemmax⁡ is the maximum value of Mstem in that site in that year and dmax⁡ is the day since Mstemmax⁡ occurred.

Therefore, plotting 1-MstemMstemmax⁡ against 1-0.9dmax⁡ should give a straight line with gradient τ. Using the assumption that the day with maximum stem biomass was approximately the same day as the day with the maximum stem biomass measurement, a straight line was fitted to the observations and an approximate value of τ=0.12 was obtained. However, as can be seen in Fig.  (which displays both the new, tuned value τ=0.12 (black, dashed line) and the value used in of τ=0.35 (black, solid line), which was obtained from ), this parametrisation does not capture the large spread in the observations (blue, green and red lines). The uncertainty this introduces into the model is not critical, since there are no strong feedbacks involved (unlike, for example, uncertainty in specific leaf area (SLA) just after emergence), but it will affect the outputted yield.

The observations of green leaf biomass and above-ground biomass were used to tune the senescence parameters µ, ν and DVIsen. The above-ground biomass measurements were combined with the partition fractions from Sect. , the carbon to biomass ratios from Sect.  and the senescence parametrisation from Eq. () to get a time series for green leaf biomass (Fig. , centre and right plots, black lines), normalised to the maximum value in each year. This could then be compared to the normalised observed time series for green leaf biomass (Fig. , left, coloured lines). It is clear that, if the parametrisation from is used (Fig. , centre plot, solid black lines), senescence starts late and then progresses too abruptly as compared to the observations. However, with the new parametrisation (with the new free parameters µ, ν and DVIsen), it is possible to get a much better fit to the observations (Fig. , right plot, dashed black lines). Note that this tuning partially compensates for the bias in the proportion of carbon going to the leaves between DVI 0.8 and 1.0 in Fig.  (top). If this bias was not present, senescence could start more gradually, which would enable a better fit to leaf carbon at around DVI = 1.75. Also, the tuned lines underestimate the leaf biomass at around DVI = 1.75, which will help to compensate for the model being unable to capture the drop in photosynthetic capacity in the green maize leaves towards the end of the season.

Stem biomass measurements up until the maximum in each year and the corresponding crop height measurements from the Mead FLUXNET sites were used to fit the allometric constants κ and λ, through rearranging Eq. () to h=κ′Mstemλ where κ′=κ1-τλ. For consistency, it is important that the τ used in this expression is the same value as the τ used in Eq. (). Figure  shows the observations (points), along with the fit using parameters from (solid black line, λ=0.4, κ′=3.06) and a tuned fit (dashed black line, λ=0.38, κ′=3.43).

The allometric constants γ and δ relating specific leaf area to DVI (Eq. ) are tuned using Fig. , which plots SLA observations against DVI (points), the tuned fit (dashed line) and the parameters used in (solid line). The crop in the model is very sensitive to SLA for low values of DVI because of the feedback between leaf area index and leaf carbon. The model lines in Fig.  have the steepest gradient for low values of DVI, where there is also a greater spread of observations.

The observations of carbon fraction of the green leaf biomass (canopy mean) against day after sowing is shown in Fig. . The mean of all of these observation together is 0.43, although there are possible indications of a slight downward trend in each site in each year with time, which would indicate that this value might be sensitive to the dates on which the carbon is measured and all three sites in 2001 have lower values. In these runs we have used fC,leaf=fC,stem=0.439. For comparison, gave the carbon fraction of leaves and stems for non-leguminous and no-rice crops as 0.459 and 0.494 respectively, and used fC,leaf=fC,stem=0.5.

Assuming that, near emergence, approximately half of the plant carbon is above ground (Fig. ), values for the parameters governing initialisation of Cinit=8.0×10-4 and DVIinit=0.1 can be derived from the above-ground biomass measurements plotted in Fig.  and the carbon to biomass ratios. Since there are no measurements below DVI = 0.1, and the model is very sensitive to these parameters, we do not attempt to set a DVIinit below 0.1 and extrapolate. Note also that the initial value of carbon is very sensitive to the thermal time for emergence. Figure  also shows that the value Cinit=1.0×10-2, which was used in to initialise the crop at DVIinit=0.0, is too high to be consistent with the above-ground biomass observations.

As discussed above, the Charv pool in JULES contains both the reproductive parts of the maize crop (kernel, cob, husk, ear shank and silk) and the yellow leaf carbon and the proportion of this carbon pool that contributed to yield carbon fyield is set by the user. The value of fyield can be derived using the latest observations in each season of the biomass of the reproductive part of the crop, the proportion of this reproductive biomass which is composed of kernels, and the yellow leaf biomass. The yield fraction is then calculated as the kernel fraction of the sum of the reproductive part of the crop and the yellow leaves, leading to an approximate value of fyield=0.74 (Fig. ). This assumes that there is no significant change in fyield between the last measurement of the season and the harvest and also that the carbon fraction of the biomass in the both the reproductive parts and the yellow leaves is the same. Typically, an accurate value of fyield is not important in impact studies, since this constant can be incorporated into a yield gap parameter.

The JULES default C4 grass settings for the PAR leaf scattering coefficient ωPAR=0.17 and the PAR leaf reflection coefficient αrefl,PAR=0.1 were used (these are very similar to the values quoted in for live maize leaves: ωPAR=0.175, αrefl,PAR=0.105) as well as a spherical angle distribution. These are the same parameter values and options that were used in to model maize. The soil albedo was set to 0.133, which was the value from the nearest grid box in the ancillary used in the HadGEM2-ES model , which was used in the global runs.

The canopy clumping factor was tuned by comparing the fraction of incident PAR absorbed by the canopy (fraction of absorbed PAR (FAPAR)), using absorbed and incident PAR observations and interpolated LAI observations, to the model FAPAR, using observed diffuse radiation fraction and interpolated LAI observations up until flowering. The python package pySellersTwoStream (see code availability section) was used to calculate the model FAPAR since it is able to reproduce the results of the JULES radiation scheme exactly but can be called directly from our (python) analysis scripts, without the need for extra JULES runs for each combination of parameters tested.

Absorbed PAR through the canopy in the model closely follows a exponential decay function. Calculating FAPAR involves integrating this exponential decay over the canopy; Fig.  (centre row) shows the resulting FAPAR distribution against total LAI for a uniform canopy (canopy clumping factor a=1). For mostly direct radiation (diffuse radiation fraction 0.2–0.3), the rate of decay with layer LAI in the model shows a clear dependence on the zenith angle (Fig. , centre right), whereas for mostly diffuse radiation (diffuse radiation fraction 0.8–0.9), this zenith angle dependence is greatly reduced (Fig. , centre left). While the observations (Fig. , top row) also show a strong zenith angle dependence as the fraction of diffuse radiation decreases, the observations are, in general, consistent with a much lower effective decay constant (in particular, the model FAPAR values are higher than the observations at intermediate LAI values ∼ 2). The observed FAPAR values also have a much larger scatter than seen in the model FAPAR.

Decreasing the canopy clumping factor is equivalent to decreasing the effective decay constant in the model. Figure  shows the value of the clumping factor that would be needed to reproduce each FAPAR observation, given the observed LAI and diffuse radiation fraction. While there is a large spread in clumping values derived in this way, these results appear to indicate that a clumping factor between 0.5 and 0.8 would be consistent with the majority of the observations. In this study, we therefore set a=0.65. Figure  (bottom row) shows that using this clumping factor value to calculate model FAPAR gives a better fit to the observations, particularly for the intermediate LAI values.

Erectile, vertical and horizontal leaf angle distributions (for a uniform canopy) were also investigated, but the spherical distribution gave the best fit to the FAPAR observations.

The FAPAR observations can not be used to tune the model once green leaf area index has started to drop significantly, as the observations include PAR absorbed by any part of the plant, whereas the JULES canopy scheme models the PAR absorbed by photosynthesising leaves only. Whether the model canopy scheme needs to be extended to include the shading of green leaves by yellow leaves and other non-root biomass depends on the distribution of the remaining green leaves through the canopy (essentially, the model is roughly assuming that all the green LAI is at the top of the plant and so does not get shaded by other plant material). Different approaches have been used in the literature. For example, modelled maize assuming that green and dead leaves are evenly distributed throughout the canopy, whereas showed that “maximum leaf photosynthesis in a senescencing crop declines with time. The oldest leaves in the base of the canopy are affected first”.

In the literature, the photosynthetic capacity of maize leaves (per leaf area) declines with age and the older leaves are lower in the canopy . As discussed in Sect. , change in photosynthetic capacity through the canopy can be modelled in JULES by a non-zero knl, which we assume is due to change in nitrogen per unit leaf area through the canopy.

The nitrogen per unit leaf area as a function of layer LAI at anthesis (60 days after sowing) in for the highest nitrogen availability level (150 kg N ha-1; residual soil nitrate 31 kg ha-1) was consistent with a knl of approximately 0.07. Since this is low, in this study, the variation of nitrogen per unit leaf area through the canopy is neglected; i.e. knl=0.0. The inclusion of a non-zero knl would have the effect of increasing GPP, as the plant would be able to make more efficient use of the incoming radiation.

In this study, trait-based physiology was switched off (i.e. l_trait_phys=F). However, the same results could be obtained by switching trait-based physiology on and choosing values for the new parameters that are equivalent to the ones used here.

Figure shows the observations of the nitrogen mass per unit carbon mass (left) and per unit leaf area (right) averaged over the canopy. In both plots, nitrogen rapidly decreases with time at the beginning and end of the season, which cannot be captured by JULES. The inclusion of a non-zero knl would also not solve this problem, as this would simply increase the nitrogen per leaf area mid-season, as can be seen in Fig.  for knl=0.2.

In this study, the temperature dependence of Vcmax is fixed by fitting Eq. () to the expression given in (Fig. ). The default JULES C4 grass parametrisation of Vcmax is more sharply peaked, has its maximum at a higher temperature and is more asymmetrical. Also plotted is the expression for the temperature dependence for maize Vcmax from . modelled Vcmax for maize at the Mead site and fit the results with MaizeGro, using the default temperature dependence, which gives a peak at approximately 33 ∘C. verified this relation by successfully fitting the model to results from modern maize cultivars from , and , which all show the peak in Vcmax at approximately the same temperature. related the normalisation of Vcmax to the leaf nitrogen per biomass; for example, at 30 g N kg-1 at the V14 growth stage, maximum assimilation at 25 ∘C was 37 µmol m-2 s-1. The temperature dependence of maize at high temperatures was looked at in more detail in , which included an investigation into the dependence on the rate of temperature change. The experiment with the more gradual temperature change in corresponds well to the high temperature dependence of the expression.

The canopy average Vcmax,norm was tuned using the value of Vcmax at 25 ∘C at 340 vppm CO2 at a specific leaf weight of 450 kg h-1, which is the canopy average at DVI = 1 (for maize cv Pioneer) from . nl0 was set to the approximate value of the observations in Fig.  (left) at DVI = 1, which then constrains ne (since ne=Vcmax,norm/nl0 when knl=0). The quantum efficiency α was set to the value from of 0.055 µmol C m-2 s-1 (µmol photons m-2 s-1)-1 for maize, which was quoted for temperatures lower than 45 ∘C (above this temperature, it drops sharply – an effect which is not reproduced in JULES). This is consistent with values in the literature (e.g. , and references therein) and consistent with the fitted values of α from . The value of α for maize is not dependent on leaf age or position . This method of tuning the JULES parameters has assumed that the two limiting rates are predominantly Wc and Wlight, not We.

Note, however, that the photosynthesis light response curve in has an exponential dependence on the absorbed radiation, which causes the shape to vary slightly from the non-rectangular hyperbolae used in JULES (with hard-wired values of curvature from ), leading to lower values of photosynthesis below approximately 1500 µmol photons m-2 s-1.

The parameters involved in calculating the leaf internal carbon dioxide partial pressure, Δqcrit and f0 (in Eq. ), were not expected to strongly limit the results since this current study focusses on carbon fluxes rather than water fluxes, the runs are irrigated and the rate We is not expected to be limiting. Δqcrit was left at its default C4 grass value (as in ) and f0 was set to 0.4 (consistent with the range of maize measurements quoted in ).

Values for μrl and μsl (from Eq. ) were obtained for maize from of μrl=0.39 and μsl=0.43 (note that this assumes one constant value for the nitrogen per carbon in leaves over the crop season and τ=0.12).

Fixing the value for the dark respiration coefficient fdr (used in Eq. ) is complicated by the inclusion in the code of inhibition of leaf respiration in the light. Also, demonstrated that the dark respiration in darkness decreases as the time the leaf has been in darkness increases. This complicates the use of the light response curves for fitting this parameter, since this means that parameters measured during the day will not necessarily correspond to those needed in JULES for modelling the average dark respiration over a 24 h period. Using values for the maximum rate of leaf photosynthesis at 450 kg biomass per hectare and maintenance respiration at 25 ∘C for maize gives fdr24 h=0.0081 over the course of 24 h. Even with a correction for inhibition of dark respiration in the light, this is inconsistent with the spread of fitted values of dark respiration to maximum assimilation to light response curves measured at the site between 10:00 and 14:00 local time, presented in (leaf is exposed to ambient light pre-measurements), which are much higher, unless the dark respiration derived from the light curves is assumed to have a contribution from what JULES considers the “growth respiration”. In general, the dark respiration coefficient estimated from light response curves for maize appears to be higher than the value derived from the maintenance respiration measurement in (e.g. ; ), which is consistent with there being a component from growth respiration. In our JULES runs, we will use fdr derived from the maintenance respiration observation in , corrected assuming that in the day of measurement 50 % of leaves experienced inhibition of the dark respiration by light; i.e. fdr is set to 0.0081/0.85=0.0095 (this assumption was later tested, and found to be accurate to within 2 %).

gave a growth respiration coefficient of 0.22, 0.18, 0.19 and 0.18 for maize leaves, stem, roots and cob/grain respectively. These values can not be used directly in JULES since, as described earlier, the growth respiration coefficient in JULES is a constant for each carbon pool. Here, we set rg to 0.25 for every PFT, as in the JULES Global Land (GL4.0) configuration (note, however, that this approximation of a constant rg for each plant carbon pool would break down for other crops e.g. soybean).

It is also worth noting that found that the maximum assimilation rate had a much stronger relationship with leaf nitrogen than the leaf dark respiration rate. In addition, shows a strong dependence in dark respiration in maize over time (using fits to light response curves), which can not be captured in JULES: at degree day 220 (roughly where the leaf area reaches a maximum), it is approximately twice as high at degree day 50. As we have discussed, maintenance respiration and Vcmax co-vary in JULES, but the growth respiration is linked to net primary productivity, which increases in the crop up until approximately anthesis. Therefore, the total leaf respiration in the model will vary in time, and will have a different dependence on time to Vcmax. However, the issues we have already identified with the modelling of the evolution of Vcmax over time will impact the accuracy of the modelling of the maintenance component of the leaf respiration over time. Leaf dark respiration rates also differ between different maize hybrids . There is therefore a large uncertainty in the parameter fdr and the overall determination of growth respiration.

Plots of modelled GPP (blue) against observed GPP (green) are shown in Fig.  for years in which irrigated maize was grown at the Mead FLUXNET sites US-Ne1 and US-Ne2. While the overall shape of the plots is good, it is clear that GPP in the model is significantly overestimated after the mid-season peak in observed GPP (corresponding to where LAI declines as the crop leaves senesce). As discussed earlier, the model Vcmax at a certain temperature stays constant, whereas in reality it would decline over the crop season. Implementing this decline into JULES would result in a much closer fit between the model GPP and observed GPP.

To a lesser extent, there also appears to be an overestimation of GPP in the model before senescence. This was investigated in more detail by comparing plots of FLUXNET GPP against observed Absorbed Photosynthetically Active Radiation (APAR) with plots of model GPP against APAR, for hourly measurements before the crop reaches DVI = 1, for LAI bins of size 1. Figure  shows the LAI bin 3.5 to 4.5. There is a clustering of points due to the hourly resolution of the data, which is most clearly seen in the model output. Hours with high diffuse radiation fractions (red) are similar in both the FLUXNET data and the model output, although the scatter in the FLUXNET data is higher, as expected from the plots of observed FAPAR (Fig. ). For lower diffuse radiation fractions in the model, GPP decreases due to a combination of the effect of sunflecks and an increase in the effective decay constant of absorbed PAR through the canopy at the beginning and end of the day. Even when the scatter in the FAPAR observations is taken into account, the decrease in GPP for lower diffuse radiation fractions does not appear to be as large in the model as in the GPP observations, and this is the source of the overestimation of GPP we saw in the model output in Fig.  before the onset of senescence.

This effect was investigated further by considering the dependence on air temperature and vapour pressure deficit in the FLUXNET GPP data. As expected, the lower temperature points (Fig. , top left) and lower vapour pressure deficit (VPD) points (Fig. , top right) are clustered at low values of APAR. However, there does not seem to be a dependence on temperature or VPD at a constant APAR across the range of GPP observations.

Soil moisture stress is a factor that we have neglected in our runs (since we have assumed perfect irrigation), which could, if implemented, reduce GPP when the soil moisture is low. However, as Fig.  shows for soil moisture content at a depth of 10 cm (bottom left) and 25 cm (bottom right), at higher APAR values, points below a threshold of 30 % appear to be distributed evenly across the range of GPP observations for a constant APAR.

Including a decrease in leaf nitrogen concentration through the canopy (while keeping the total amount of nitrogen constant) would have the effect of making the light use of the plant more efficient, which would increase model GPP still further. Decreasing Vcmax,norm would have the effect of decreasing model GPP at higher APAR values, but this would not solve the issue at mid-range APAR points ∼800 µmol photons (m2ground)-1s-1 and would also worsen the fit of the points with high diffuse radiation fractions.

It is therefore difficult to see a clear way in which the model parameter settings or processes should be improved. It would be possible to improve the validation against observations by decreasing α or changing the curvature parameter in the non-rectangular hyperbola implemented for light response within JULES (currently hard wired) but it is difficult to justify this theoretically.

The results from the model runs without the crop model can also be used to test the parametrisation of respiration.

Using a number of assumptions, the measurements from Mead can be used to get an approximate value for leaf maintenance respiration. First, approximate values for NPP were obtained by linearly interpolating the Mead above-ground biomass measurements to get a daily time series, and then differentiating. The fraction of NPP directed to the roots at each DVI was calculated from the expression for maize in (plotted in Fig. ) and then used to obtain the total NPP. Combining these NPP values with the GPP observations and assuming a value for the growth respiration coefficient of rg=0.25 and summing over the crop season leads to an estimation of the plant maintenance respiration Rpm. It is necessary to sum over the whole season, since the NPP and GPP calculated in this way appear to be slightly out of step with each other, and this effect dominates the daily time series of derived maintenance respiration.

The interpolated carbon pool observations were used to calculate the factor 1+μrlCrootCleaf+μslCstemCleaf that converts between the leaf maintenance respiration and the total plant maintenance respiration. Note that the stem carbon observations had to be corrected using τ to get Cstem. This factor was used to convert the leaf maintenance respiration outputted by the model to the total plant maintenance respiration.

Figure shows the Rpm derived from observed GPP against Rpm/fdr derived from the outputted model leaf maintenance respiration. The x axis therefore is independent of fdr, which can be obtained from the gradient. Data from 2010 is not included (since the crop was damaged by hail). Both the default JULES C4 grass fdr (solid line) and the fdr used in our maize configuration (dashed line) are shown. It can clearly be seen that the new maize fdr is a better fit than the default C4 grass value. While there are many model and parameter assumptions (rg, μrl, μsl, β=1, Croot, τ) that have gone into this plot, this is still an important consistency check of our parameters.