.. _ecosystem:

Ecosystems impacts
=====================

Within the European domain and East Asia domains, `GAINS`_ quantifies the deposition of total nitrogen (N) as the sum of oxidized (:math:`NO_x`) and reduced (:math:`NH_y`) nitrogen compounds, as well as deposition of sulfur, using linear transfer coefficients from the `EMEP CTM`_ For quantifying ecosystems risks from acidification and eutrophication, the `GAINS`_ models employs the critical loads approach, with Critical Loads (CL) being defined to characterise the vulnerability of (parts of) an ecosystem in terms of deposition. If the CL of pollutant X at a given location is smaller than the deposition of X at that location, it is said that the CL is *exceeded* and the difference is called *exceedance*.  

Eutrophication
------------------

For a critical load of eutrophication :math:`CL_{eut}N` (also called CL of nutrient N), the exceedance :math:`Ex_{eut}N` for a deposition :math:`N_{dep}` is calculated as:

.. math::
   :nowrap:
   :label: ex_eut_n

   \begin{equation} 
   Ex_{eut}(N_{dep}) = \max\{0, N_{dep} - CL_{eut}N\} = 
   \begin{cases}
       N_{dep} - CL_{eut}N & \text{if } N_{dep} > CL_{eut}N \\
       0 & \text{if } N_{dep} \leq CL_{eut}N \tag{1}
   \end{cases}
   \end{equation} 

where:

============================   ==============================================================================================================================
:math:`N_{dep}, CL_{eut}N`     N deposition and Critical Load of N for eutrophication respectively 
:math:`Ex_{eut}(N_{dep})`      Exceedance of N eutrophication Critical Load for a given N deposition :math:`N_{dep}`
============================   ==============================================================================================================================


In case exceedances are negative, they are simply set to zero.




Acidification
----------------

For acidification impacts, there is no unique critical load of S and N acidity. Instead, acidity CL are defined by a trapezoidal-shaped critical load function (CLF), defined by the quantities :math:`CL_{max}S`, :math:`CL_{min}N`, and :math:`CL_{max}N`. Thus, the exceedance is defined as the sum of the N- and S-deposition reductions needed to reach the closest point on the CLF, with zero exceedance for deposition pairs (:math:`N_{dep}`, :math:`S_{dep}`) lying below the CLF (see below Figure **1**).

.. _fig-acidity_CL:
.. figure:: /_static/acidity_cl.png
   :align: center
   

**Figure 1** Critical Load Function (CLF) for acidifying Sulfur and Nitrogen defined by :math:`CL_{max}S`, :math:`CL_{min}N`, and :math:`CL_{max}N` (thick black line). The grey-shaded area below the CLF (region 0) defines deposition pairs (:math:`N_{dep}`, :math:`S_{dep}`) for which there is non-exceedance. The points *E1* to *E4* refer to deposition pairs in four different regions and the way the exceedance (:math:`N_{ex}`, :math:`S_{ex}`) is computed for a point in Region 2. 

Given the above Figure **1**, the exceedance of N and S acidity CLs is obtained as follows: 

.. math::
   :nowrap:
   :label: ex_acid
    
   \begin{equation} 
   Ex_{ac}(N_{dep},S_{dep}) = 
   \begin{cases}
       0 & \text{if } (N_{dep},S_{dep}) \in \text{Region } 0  \\
       N_{dep} - CL_{max}N + S_{dep} & \text{if } (N_{dep},S_{dep}) \in \text{Region } 1 \\
       N_{dep} - N_0 + S_{dep} - S_0 = (N_{ex},S_{ex})  & \text{if } (N_{dep},S_{dep}) \in \text{Region } 2 \\
       N_{dep} - CL_{min}N + S_{dep} - CL_{max}S & \text{if } (N_{dep},S_{dep}) \in \text{Region } 3 \\
       S_{dep} - CL_{max}S & \text{if } (N_{dep},S_{dep}) \in \text{Region } 4 \tag{2}
   \end{cases}
   \end{equation}

where:

================================================================    ===========================================================================================================================================
:math:`N_0, S_0, N_{dep},S_{dep},CL_{max}S,CL_{min}N,CL_{max}N`     N and S deposition for point Z2, N and S deposition, maximum Critical Load of S, and minimum and maximum Critical Load of N respectively 
:math:`Ex_{ac}(N_{dep},S_{dep})`                                    Exceedance of N and S acidity Critical Load for a given deposition pair (:math:`N_{dep}`, :math:`S_{dep}`)
================================================================    ===========================================================================================================================================

To obtain a single exceedance value for a given grid cell or an entire region, `GAINS`_ computes the *Average Accumulated Exceedance* (AAE), defined as follows:

.. math:: 
   :nowrap:
   :label: aae 
   
   \begin{align}
   AAE = \frac {\sum_{j=1}^n A_j Ex_j} {\sum_{j=1}^n A_j} \tag{3}
   \end{align}

where:

====================    =============================================================================================================================
*j, n*                  Ecosystem, number of ecosystems respectively 
:math:`A_j`             Area of ecosystem j
:math:`Ex_j`            Exceedance of N or S Critical Load for ecosystem A_j
AAE                     Average Accumulated Exceedance of N or S Critical Load
====================    =============================================================================================================================

The AAE is thus expressed as the area-weighted exceedance of the CLs for the individual ecosystems. To remain consistent with the latter definition, :math:`Ex_j` needs to be set to zero if the CL is not exceeded.

.. _`GAINS`: http://gains.iiasa.ac.at/models/
.. _`EMEP CTM`: https://github.com/metno/emep-ctm