# Climate Model

[DRAFT]. Last updated 2016-03-23.

## Introduction

In Heureka, the effect of global warming on forest growth can be accounted for based on results from a process-based model called BIOMASS. BIOMASS was developed by McMurtrie et. al (1990)[1] and has been modified and validated for Swedish conditions in a number of studies [2] [3] [4] [5]. BIOMASS is computationally demanding to run and not suitable to include directly in Heureka’s growth simulator. Therefore BIOMASS has been run for different parts of the country, forest conditions and climate scenarios, and the results have been used to construct an approximation model of BIOMASS.

## How the model works

A special approach has also been developed to adapt the response of the growth models used in Heureka. The principal idea is to augment or shift the growth function when climate changes. This is done by changing tree ages; the size to age relation of trees is a critical variable in all growth functions used in Heureka. In the system, we differ between actual tree age and biological tree age. The biological age is the one subject to adjustment.

The climate response model affects the following variables directly:

• The biological age or basal area depending on user setting
• Site index (SIS)
• Vegetation index
• Temperature sum

#### Main calculation steps

1. The selected climate scenario has information on relative growth difference (α, see below) between an unchanged climate and a changed climate. This is called the growht response.
2. At a given year t, Heureka calculates the growth from year t to t+5, for an unchanged climate (ΔGempir).
3. Calculate Gclim = α · ΔGempir = The expected growth when there is a climate change.
4. A new growth calculation is done, with climate-adjusted values for site index and vegetation index (see below)
5. Depending on whether tree ages should be adjusted or not (see control table Climate Model, one of the following is applied:
By using binary search, all tree ages are adjusted with a certain factor c, until the obtained growth equals Gclim (within a certain tolerance).
The growth projection length is extended (or shortened if the climate model predicts decreased growth due to drought) until the expected growth is obtained. Non-linear interpolation is used to handle that this will unlikely coincide with an exact five-year interval. All tree attributes in at time t+5 (diameter, height etc) except tree ages are set to the values obtained in the extended (or shortened period).

## Calculation of growth response (changed growth due to climate change)

#### Step 1: Uncorrected growth response ($\alpha_{BIOMASS})$

A "climate scenario" file imported to Heureka is not an actually scenario but a table of coeffiicients used to calculate growth modification factors that are then used in the Heureka "climate model".

For a given prediction unit that has soil moisture class j, the growth correction factor is calculated as function of the parameters a, b and c, which depend on scenario, geographic locations, tree species and soil moisture code.

$\alpha_{BIOMASS}(s) = a(s,j) LAI(s)^2 + b(s,j) LAI(s) + c(s,j)$

where

αBIOMASS(i,j) = Growth correction factor for species j according to model BIOMASS, and
LAI(s) = Leaf area index for species s (m2 leaf area / m2 forest floor area)

Since leaf area is not calculated in Heureka, foliage biomass is used instead, and multiplied with a conversion factor (SLA) to get LAI.

$\displaystyle LAI(s) = \frac{fbm(s) SLA(s)}{f(s)}$

where
fbm(s) = Foliage biomass (kg/m2, dry matter / forest floor area), and
SLA(s) = Conversion factor for biomass foliage to leaf area for species s (m2 projected one-side leaf area / kg dry matter),
f(s) = Species proportion of species s in the stand, used as indicator for how much of the forest floor area is occupied by species s. The division with the species distribution is done because the divisor (forest floor area) should only include the forest floor occupied by the subject trees.

#### Step 2: Corrected response (β)

$\alpha_{BIOMASS}$ reflects "optimal" stand conditions and should therefore be modified. The response modifier is a linear function of vegetation index.

$\displaystyle \beta(s) = c_0 +(VIX-VIX_{min}(s))\frac{1-c_0(s)}{VIX_{max}(s)-VIX_{min}(s)}$

where

β(s) = Response modifier for species s, and
VIX = Vegetation index, and

c0(s) = Intercept for species s (see control table), and
VIXmin = Minimum vegetation index for species s, and
VIXmax = Maximum vegetation index for species s.

The modified response for each species s is:

$\alpha_{c1}(s)=1 + \beta(s)(\alpha_{BIOMASS}(s)-1)$

#### Step 3: Carbon dioxide response

Step 1 and 2 can either included the combined effect or water, temperature and carbon dioxide, or add the carbon dioxide component in a third step. In that case, other coefficients are used in step 1. The carbon dioxide response modifier is calculated with the same type of formula as α(s), but with other coefficients.

$\displaystyle \gamma(s) = d(s,j) LAI(s)^2 + e(s,j) LAI(s) + f(s,j)$

This value is then multiplied with the calculated growth response.

$\alpha_{c2}(s)= \gamma(s) \alpha_{c1}(s)$

## Temperature sum change

The change in temperature over time caused by a changing climate is expressed as an average temperature sum change factor (λ) for five years, and is part of the climate scenario input data.

## Site index change

The site index (sis) change from one five-year period to the next is calculated as

$\Delta sis = \lambda (21.9947 \Delta TS - 2.1384 \Delta TS2)$

where

$\lambda$ = Temperature change factor for a five-year period
$TS_{t} = \lambda \cdot TS_{t-1}$ (temperature sum in °C)
$\Delta TS$ = Change in temperature sum (1000 °C) between two five-year periods: $0.001\ (TS_{t} - TS_{t-1})$) where t is period index, and
$\Delta TS2$ = Change between squared temperature sums (in 1000 °C) from period t-1 to period t: $(0.001 TS_{t})^2 - (0.001 TS_{t-1})^2$

## Vegetation index change

The vegetation index is a mapping of the categorial variable vegetation type code, to a nominal variable. The vegetation index is used as one of the explanatory in the Whole-stand growth model (see page 39 in the "Growth modelling in Heureka" document).

The climate model modifies the vegetation index. The vegetation index increases when the climate model predicts increased growth. The vegetation index is updated in each projection period as a linear function of ΔTS2:

$\Delta VIX = 1.24 \cdot \Delta TS2 \cdot \frac{periodLength}{5}$

## Growth model

#### Case 1: Whole-stand growth model

In Heureka a stand-level growth model is used (as default) to calculate basal area growth per ha on each plot, in combination with a single-tree growth model that distributes the growth to individual tree objects. Mean age is used as one of the explanatory variables. Different species are not differentiated, and it is not possible to make an adjustment for each species separately. Therefore, the response for all species together is weighted proportionally to the species proportions:

$\displaystyle \alpha_{tot} = \sum_{s=1}^{m} f(s) \alpha_{c2}(s)$

where

m = number of species, and
f(p) = volume proportion of species s

#### Case 2: Single-tree growth model

Single-tree or other growth model that used species-wise ages as function parameter. In this case, the growth adjustment is differentiated with respect to species (binary search can be applied to each species group separately when adjusting tree ages), using indexed \alpha_{c2}(s) instead of alpha_{tot}.

## Input data

With the installation of Heureka, there are three climate model scenarios available:

MPI 4.5: Based on Max Planck Institute MPI-ESM model using radiation scenario RCP 4.5, which assumes that radiative forcing stabilises at 4.5 W/m² before the year 2100. See also RCP4.5 at SMHI's homepage.
MPI 8.5: Based on Max Planck Institute MPI-ESM model using radiation scenario RCP 4.5, which assumes that radiative forcing stabilises at 8.5 W/m² before the year 2100.. See also RCP8.5 at SMHI's homepage
ECHAMS_A1B: Based on Max Planck Institute climate model ECHAM using emission scenario SRES A1B.