Mortality

Tree mortality rate (mort) is assumed to have a fixed component not related to stand density ( \( mort\_{natural}\)), e.g. caused by diseases, and a stand density related component ( \( mort\_{dens}\)). The latter only applies if trees are growing across a threshold of tree canopy coverage. Then, the number of trees is reduced to this threshold.

\[ mort = max(mort\_{dens}, mort\_{natural}) \]

\[ mort\_{natural} = \frac {MORTNORM}{365} \]

\[ if (relcover > 1) mort\_{dens} = 1.0 - (MORTCROWD \cdot \frac {cdr ^ {2}}{cdr\_{pot} ^ {2}}) \]

\[ relcover = \frac {(dbh \cdot cdr) ^ {2} \cdot PI \cdot 0.25 \cdot tree\_{number} }{10000 \cdot MORTCROWD} \]

with

dbh: diameter at breast height (1.3m) (m)
cdr: ratio between dbh and crown diameter
\( cdr\_{pot}\): ratio between dbh and potential crown diameter under unconstrained conditions
tree_number: number of trees per hectare
relcover: relation between actual tree cover and potential coverage with no constraints
PI: constant (3.146)
MORTNORM, MORTCROWD: parameters for annual intrinsic mortality rate and maximum tree coverage, respectively

Dimensional growth

Concept

The dimensions of a tree (height, crown base height, rooting depth, stem diameters and sapwood area) are not derived from empirical size relations but follow from the volume of living wood (sapwood). The sapwood of an average tree of a cohort is described by a set of simple geometric bodies that are all linked by one common quantity, the sapwood area \( A_s \) [m2 tree-1], i.e. the conducting cross-section of the stem (pipe model):

Compartment	Shape	Sapwood volume [m3 tree-1]
trunk (below crown)	cylinder	\( V_{trunk} = A_s \cdot h_{trunk} \)
stem axis in the crown	cone	\( V_{crown} = A_s \cdot h_{crown} / 3 \)
branches	-	\( V_{bra} = FBRA \cdot V_{crown} \)
tap root	cone	\( V_{croot} = A_s \cdot h_{depth} / 3 \)
other coarse roots	-	\( V_{crt} = FCRT \cdot V_{croot} \)

with \( h_{trunk} \) the height of the crown base, \( h_{crown} \) the crown length, \( h_{depth} \) the rooting depth, and the species parameters FBRA (branch to crown-axis volume ratio) and FCRT (coarse root to tap root volume ratio). The total sapwood volume of a tree is therefore

\[ V_{sap} = A_s \cdot L_{eff}, \qquad L_{eff} = h_{trunk} + \frac{h_{crown}}{3} (1 + FBRA) + \frac{h_{depth}}{3} (1 + FCRT) \]

where \( L_{eff} \) is the effective length of the sapwood body. The sapwood area that a tree needs is defined by its leaf area and the species-specific sapwood to foliage area ratio QSF (Huber value).

Driving variables

The function does not calculate any carbon fluxes itself. All dimensional changes are driven by two daily fluxes that must be provided by the calling physiology module:

\( \Delta C_{sap} \): carbon allocated to sapwood growth during the day [kgC m-2 d-1]
\( S_{sap} \): sapwood senescence during the day [kgDW m-2 d-1], i.e. living wood that dies (becomes corewood or branch/coarse root litter)

Both are converted into sapwood volumes per tree using the wood density DSAP [kgDW dm-3], the carbon content of dry matter (CCDM) and the number of trees per hectare \( N \):

\[ \Delta V_{sap} = \frac{\Delta C_{sap} / CCDM}{DSAP \cdot 1000} \cdot \frac{10^4}{N}, \qquad \Delta A_{s,loss} = \frac{S_{sap} / (DSAP \cdot 1000) \cdot 10^4 / N}{L_{eff}} \]

Note

Tree dimensions change only through these two fluxes. If a physiology module passes zero (or does not compute them), height, diameters, crown and rooting depth stay constant regardless of biomass changes. The fluxes have to be daily totals because the function is called once per day, independent of the time resolution of the physiology module. Currently they are provided as:

PSIM: d_dcSap + dcSap and d_sSap + sSap. PSIM calls this function in the last time step of the day, before the SubdailyFinalizer has added this step to the daily sums d_*. In daily mode the whole day is computed in this last step, so d_* is still zero and dcSap / sSap hold the daily values.
PnET: d_dcSap and the daily sapwood to corewood transfer (mSapLoss_vt)

The geometric volumes are only consistent with the sapwood biomass of the physiology module if every change of the sapwood pool (growth, senescence, but also e.g. re-initialisations) is passed to this function. Changes of the biomass pools themselves (sapwood, corewood, litter) are not done here but in the physiology module.

Step 1: sapwood senescence

The sapwood area is reduced by \( \Delta A_{s,loss} \). The crown is assumed to keep its shape, so that the crown length decreases in proportion to the diameter of the remaining sapwood, while the tree height is unchanged:

\[ h_{crown}' = h_{crown} \cdot \frac{d_s'}{d_s}, \qquad h_{trunk}' = h_{top} - h_{crown}', \qquad d_s = \sqrt{4 A_s / \pi} \]

Thus, sapwood loss raises the crown base (self-pruning): the lower part of the former crown becomes trunk. The volumes of all compartments are recalculated accordingly; the rooting depth is not reduced.

Step 2: sapwood growth

The new sapwood volume \( \Delta V_{sap} \) can be used either to thicken the tree (increase of \( A_s \)) or to elongate the crown and roots (increase of height and depth). Two extreme cases are calculated:

horizontal growth: the volume increases the sapwood area only, the heights stay (nearly) constant, which results in a low height:diameter ratio \( hd_{h} \);
vertical growth: the sapwood area stays constant and the volume elongates the crown (and tap root, limited by soil depth, the species-specific maximum root extension rate FRT_DGR_MAX and the depth/height preference QDRH), which results in a high height:diameter ratio \( hd_{v} \).

The realised partitioning is chosen such that the tree approaches its target height:diameter ratio \( hd^{*} \), which depends on tree size and on stand density (competition factor, see Height-diameter ratio):

\[ f = \frac{\min(hd^{*}, hd_{v}) - hd_{h}}{hd_{v} - hd_{h}}, \qquad \Delta A_s = \Delta A_{s,v} + (\Delta A_{s,h} - \Delta A_{s,v}) \cdot \min(1, 1 - f) \]

Trees in dense stands (high \( hd^{*} \)) therefore invest more into height, solitary trees more into diameter. With the new sapwood area, the volume increment is distributed to trunk, tap root and coarse roots, and crown and branches (remainder), and the new dimensions follow from the geometric relations above:

\[ h_{crown} = 3 V_{crown} / A_s, \qquad h_{depth} = 3 V_{croot} / A_s, \qquad h_{trunk} = \max(h_{trunk}, V_{trunk} / A_s), \qquad h_{top} = h_{trunk} + h_{crown} \]

The effective stem diameter increases by the increment of the sapwood diameter \( \Delta d_s \).

Derived properties

After the volume update, the following properties are derived:

diameter at breast height: zero below 1.3 m, derived from the sapwood diameter during the transition period (tree height between 1.3 m and 1.3 m + HREF), and equal to the effective stem diameter for taller trees;
diameter at ground level, increased by the sapwood diameter increment;
crown diameter ratio (Vegetation Libraries), branch fraction (branch wood relative to aboveground wood), sapwood to foliage area ratio (QSF), and the ground area covered by the cohort, optionally reduced by crown clumping of overlapping cohorts.

Parameters

_p	tree cohort
_dcSap_day	carbon allocated to sapwood during the day [kgC m-2 d-1]
_sSap_day	sapwood senescence during the day [kgDW m-2 d-1]
_competition_method	consider stand density in the target height:diameter ratio
_clumping_factor	degree of crown clumping between cohorts [-]

Sapwood:Foliage relation

Assuming a relation between sapwood area and foliage area (Sapwood to foliage area ration), which is calculated empirically in dependence on tree height, the sapwood biomass necessary to supply the maximum foliage in the crown can be determined assuming a cone shape decline within the canopy and throughout the rooting zone. This optimum sapwood biomass (mSap_opt) is then devided by current foliage (and bud) masses in order to gain a sapwood/foliage biomass ratio.

\[ qsm = \frac { mSap\_{opt} }{ mFol + mBud } \]

\[ mSap\_{opt} = 1000 \cdot DSAP \cdot mSap\_{demand} \cdot (height\_{base} + \frac { rooting\_{depth} } { 3.0 } + \frac { height - height\_{base} } { 3.0 } ) \]

\[ mSap\_{demand} = lai\_{max} \cdot qsfa \]

with

mFol, mBud: biomass of foliage and structural reserves (buds) (kg m-2)
height_base: crown base height (m)
rooting_depth: rooting depth (m)
lai_pot: potential leaf area calculated from foliage and bud mass weighted by specific leaf area, derived from canopy length and species-specific parameters (SLAMAX, SLAMIN)
qsfa: relation between sapwood area and foliage area using QSF_P1, and QSF_P2 (sap-fol-ratio)
DSAP: species-specific sapwood density (kg dm3)

The height at crown base is calculated EITHER from a parameterized relationship to tree height (= option 'height, described under 'PSIM') using the species-specific parameters HREF (tree height at which crown base height starts to increase) and CB (relation between crown base height and top
height at a mature state) or based on crown diameter (crown-length) using the species specific parameters (CL_P1 and CL_P2).

Biomass and leaf area distribution

Foliage biomass throughout the foliated canopy and fine root biomass throughout the rooted soil are empirically distributed using the same function that depends on distribution length (either canopy depth or rooting depth) with different parameters [31], veglibs_biomass_distribution.

Leaf area throughout the foliated canopy follows foliage biomass distribution but is weighted by specific leaf area per canopy layer. The latter varies linearly between a predefined minimum (at the top of the canopy) and maximum value (SLAMIN, SLAMAX).

Forest disturbances

Defoliate
Regrowth
Throwing
Thinning
Harvest