**1. Surface Runoff**

Surface runoff is one of the components in the surface boundary conditions. Its dynamics can be described as a water balance process of surface ponding (Kroes, J.G., 2008).

Where *h _{0}* is the storage height change of ponding,

*q*is the net precipitation flux,

_{prec}*q*is the flux from the soil surface to the ponding layer,

_{s }*q*is the irrigation flux,

_{ir}*q*is the evaporation flux.

_{evap}Surface runoff occurs when the storage height of ponding exceeds the critical depth *h _{0,crit}* limited by surrounding environment. The runoff flux

*q*is

_{runoff}Where *κ* is the resistance coefficient, *β *is the empirical parameter.

**2. Soil Evaporation & Crop Transpiration**

The ET approach used in STEMMUS model is one-step calculation of actual soil evaporation and potential transpiration by incorporating canopy minimum surface resistance and actual soil resistance into Penman-Montieth model. LAI is implicitly used to partition available energy into canopy and soil. Where *R*_{n}* ^{c}* and

*R*

_{n}*are the net radiation at the canopy surface and soil surface, respectively;*

^{s}*ρ*is the air density;

_{a}*c*is the specific heat capacity of air;

_{p}*r*and

_{a}^{c}*r*are the aerodynamic resistance for canopy surface and bared soil, respectively;

_{a}^{s}*r*is the minimum canopy surface resistance;

_{cmin}*r*is the soil surface resistance.

_{s}The net radiation reaching to the soil surface can be calculated using the Beer’s law relationship of the form

And the net radiation intercepted by the canopy surface is the residual part of total net radiation

The minimum canopy surface resistance *r _{cmin}* is given by

**3. Biomass Flow**

Some of the photosynthesis assimilation products are required to provide energy for maintaining the existing biostructures (refer as maintenance respiration), and some others are consumed in the conversion process of glucose molecules (refer as growth respiration), the remainder of the photosynthesis assimilation products are potentially available for the formation of the plant biomass.

In the model, total biomass is partitioned into the different plant organs (leaves, stems, storage organs and roots) according to the characteristics of plant growth development, which simplified as fixed distribution coefficients as a function of the development stage. The effect of environmental factors, such as temperature, water and nutrient status, are considered by applying the modification factors to these distribution coefficients.

### 3.1. Crop growth module (biomass flow)

When the incoming radiation reaches the canopy, a part of the radiation is reflected by the canopy. The reflection coefficient of the canopy depends on solar elevation, leaf angle distribution, leaf transmission and reflection properties. For a green canopy with a random spherical leaf angle distribution, the reflection coefficient can be presented by Goudriaan(1977).

The rest part of radiation is potentially available for absorption by the canopy. Following the beer’s law, radiation fluxes decreases exponentially with increasing leaf area within the canopy:

Where *f _{0}* is the radiation flux at the top of the canopy,

*L*is the cumulative leaf area index from the top of the canopy,

*f*is the net radiation flux at depth with leaf area index of L, and

_{L}*k*is the extinction coefficient, which is different for the direct and diffuse radiation fluxes as they attenuate light at a different rate.

On its way through the leaves, a part of the direct flux is scattered (i.e. reflected or transmitted). Hence, the direct flux inside the canopy turns into a scattered, diffused component and a direct component. Both are treated separately in the model.

Since the absorption of radiation flux is related to the transmission, the decline of the radiation flux should be a measure for its absorption. The rate of absorption at a depth *L* in the canopy is obtained by taking the derivative of Equation with respect to leaf area index in the canopy.

Where *f _{aL}* refers to the absorbed radiation flux.

Based on the radiation transfer process within the canopy, the diffuse radiation flux absorbed by the sunshade leaf stem from the incoming solar radiation and scattered radiation of direct flux when transmits through the leaves. The sunlit leaf area receives both diffuse and direct radiation flux.

### 3.2. Instantaneous assimilation rate per leaf layer and scale up

Plant leaves transform the intercepted radiation flux into photosynthesis assimilation products through rather a complex physiological process. On the basis of experimental analysis, the assimilation-light response curve seems practical and reliable for modeling use when well calibrated. Following (Spitters, 1986), the assimilation rate of the shaded leaves is written by

However, it is more reasonable to take the variation in leaf angle into account for the sunlit leaves. The direct flux absorbed by the leaves is perpendicular to the direct beam.

Where, solar elevation *β* is the direct irradiance on a horizontal plane.

The assimilation rate of sunlit leaf area can be given by

The average assimilation rate over a canopy layer is the sum of sunlit and shaded leaves weighted by their proportion of that layer.

With the fraction of sunlit leaf area *P _{sl}* equals to the fraction of the direct component of the direct flux reaching that canopy layer L.

The scale up of instantaneous assimilation rate per leaf layer to daily gross assimilation of the canopy is achieved by using the Gaussian integration method. For specific procedure, see (Goudriaan, 1986, Spitters, 1986).