# Evaluating the Effectiveness of Natural Carbon Sinks Through a Temperature-Dependent Model
## 2 Materials and Methods
### The Top-Down Carbon Sink Model from the Continuity Equation
#### 2.1.2 The Extended Linear Sink Model  
It is generally known and a consequence of Henry's Law that the gas exchange between the sea and the atmosphere depends on the sea surface temperature. Increased CO\({}_{2}\) emissions from the oceans are, therefore, expected as the temperature rises, like in a glass of beer on a warm day. It is also possible that absorptions are diminished as temperature rises. Similarly, the decay of biological matter and organisms is related to [PERSON]n't [PERSON]'s rule, which states an increase in decay rate and, thus, natural emissions with an increase in temperature. Of course, the sustainable availability of decayable substances depends on photosynthesis. But photosynthesis also scales with sunlight hours and temperature up to 30 \({}^{\circ}\)C besides scaling with CO\({}_{2}\) concentration and thus CO\({}_{2}\) fertilization. CO\({}_{2}\) fertilization dominates the greening of the earth, which has been more than 30% since 1900 [22]. So, it is logical that absorption dominantly scales with CO\({}_{2}\) concentration, while biological decay scales with temperature. When both are correlated as they have been for the last 65 years, a balanced growth of both is expected. These considerations motivate the introduction of temperature as a model parameter in the description of the effective sink capacity. This model extension has been first introduced in a previous publication [10]. But here, the model is refined by applying it to monthly data in conjunction with an optimization of the time lag between temperature and CO\({}_{2}\) concentration change. The extended sink model has two components. Absorptions are proportional to the CO\({}_{2}\) concentration preceding by \(lagC\) time units, with an offset \(C^{0}\), while natural emissions are proportional to temperature preceding by \(lagT\) time units, with an offset \(T^{0}\). The introduction of a potential time lag between temperature and change of CO\({}_{2}\) concentration is motivated by the fact that decay processes take time, typically a few months. The existence of a time lag in the relation between temperature change and CO\({}_{2}\) concentration change has also been observed in another investigation [23]. [PERSON]'s publication has been criticized because of the suppression of the long-term trend in both CO\({}_{2}\) concentration and temperature [24]. CO\({}_{2}\) concentration or temperature is not manipulated in such a way here, but their actually measured values are used in the regression model. Following the logic of Equation (13), the extended sink model becomes  
\[\hat{S}_{i}(C,T) = a\cdot\left(\sum_{k=12i}^{12i+11}\frac{1}{12}C_{k-lagC}^{month}- C^{0}\right)+b\cdot\left(\sum_{k=12i}^{12i+11}\frac{1}{12}T_{k-lagT}^{month}-T^{0}\right) \tag{14}\] \[= a\cdot\sum_{k=12i}^{12i+11}\frac{1}{12}C_{k-lagC}^{month}+b\cdot \sum_{k=12i}^{12i+11}\frac{1}{12}T_{k-lagT}^{model}+c \tag{15}\]  
with \(c=-a\cdot C^{0}-b\cdot T^{0}\) capturing all offsets of concentration or temperature measurements. With the special choice  
\[\hat{C}^{0}=\frac{-c}{a} \tag{16}\]  
(see Equation (22) in [10]), which implies \(T^{0}=0\), the extended sink model takes a simplified form:  
\[\hat{S}_{i}(C,T)=a\cdot\left(\sum_{k=12i}^{12i+11}\frac{1}{12}C_{k-lagC}^{month }-\hat{C}^{0}\right)+b\cdot\sum_{k=12i}^{12i+11}\frac{1}{12}T_{k-lagT}^{month} \tag{17}\]  
It must be noted that \(\hat{C}^{0}\) here is not the estimated preindustrial concentration, but the estimated hypothetical equilibrium concentration for zero anthropogenic emissions and the temperature anomaly \(T=0\). This means that with an expected negative value of \(b\), there is a reduction in the observed sink effect with rising temperature, but this is overcompensated by the increasing sink effect through the rising concentration as long as the temperature and concentration are correlated, as they have been over the last 60 years.