## 3 Results
### Influence of Climate Change on the Growth Rate of Unstable Waves
#### 3.2.2 Sensitivity Functions  
Sensitivity functions \(S_{\sigma}\) with respect to the static stability parameter as functions of zonal wavenumber \(k_{z}\) for the \"dry\" and \"moist\" \(f\)-plane and \(\beta\)-plane models are plotted in Figure 2. As the wavenumber increases, the absolute values of sensitivity functions \(S_{\sigma}\) grow exponentially meaning that the effect due to a change in the vertical stratification of the atmosphere substantially depends on the unstable wavelength. Short waves are more affected by changes in atmospheric static stability than longer waves. Atmospheric moisture also noticeably affects the sensitivity of unstable waves to the parameter of static stability (see Table 2). Results show that the \(\beta\)-plane model is more sensitive to the static stability then the \(f\)-plane model. Since sensitivity functions \(S_{\sigma}\) obtained from \(f\)-plane and \(\beta\)-plane \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Models** & \(\mathbf{L_{min}^{f}}\)**km** & \(\mathbf{L_{max}^{f}}\)**km** & \(\mathbf{L_{kcmax}^{f}}\)**km** & \(\mathbf{\chi_{k}}\)**day\({}^{-1}\)** & \(\mathbf{t^{d}}\)**day** \\ \hline \(f\)-plane models: & & & & & \\ “Dry” & 3.575 & - & 5340 & 0.59 & 1.18 \\ “Moist” & 2735 & - & 4090 & 0.77 & 0.90 \\ \hline \(\beta\)-plane models: & & & & & \\ “Dry” & 3085 & 5945 & 4130 & 0.45 & 1.54 \\ “Moist” & 2333 & 6015 & 3385 & 0.68 & 1.02 \\ \hline \end{tabular} Note: \(t^{f}\) is a doubling time.  
\end{table} 
Table 1: Characteristics of baroclinic instability obtained from the \(f\)-plane and \(\beta\)-plane models. models are negative for all unstable waves, the increase in static stability leads to the decrease in the growth rates of unstable modes. Sensitivity functions \(S_{\Lambda}\) and \(S_{uT}\) which represent the influence of MTG on the growth rate of unstable waves for the \"dry\" and \"moist\" \(f\)-plane and \(\beta\)-plane models are shown in Figure 3. From the comparison of graphs shown in Figures 1a and 3a, it follows that the graph of function \(S_{\Lambda}\) is similar to the graph of function \(\chi_{k}\). Therefore, in the \(f\)-plane model, the most unstable wave possesses the highest sensitivity to MTG, while in the \(\beta\)-plane model, the shorter the wave, the more sensitive it is to the horizontal temperature gradient, which is consistent with the results obtained from general circulation models of the atmosphere (e.g., [47; 87]). It can be shown that the sensitivity functions \(S_{\Lambda}\) and \(S_{uT}\), when considered over the same domain of wavenumbers, are related by the equation \(S_{\Lambda}=0.25\times S_{uT}\). Using this equation, we can compare the results obtained from both models. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Models** & \(S_{\sigma}\) & \(S_{\Lambda}\) & \(S_{uT}\) & \(S_{\sigma}^{R}\) & \(S_{uT}^{R}\) \\ \hline \(f\)-plane models: & & & & & \\ “Dry” & \(-1.71\) & \(0.0196\) & - & \(-0.50\) & - \\ “Moist” & \(-3.84\) & \(0.0256\) & - & \(-0.50\) & - \\ \hline \(\beta\)-plane models: & & & & & \\ “Dry” & \(-2.47\) & - & \(0.0864\) & \(-0.94\) & \(1.44\) \\ “Moist” & \(-4.34\) & - & \(0.1031\) & \(-0.64\) & \(1.14\) \\ \hline \end{tabular} Note: \(S_{\Lambda}^{R}=1\) for all unstable waves.  
\end{table} 
Table 2: Absolute and relative sensitivity functions for the most unstable wave obtained from the \(f\)-plane and \(\beta\)-plane models. Figure 2: Sensitivity functions \(S_{\sigma}\) with respect to the static stability parameter as functions of the zonal wavenumber \(k_{z}\) for the “dry” and “moist” (**a**) \(f\)-plane and (**b**) \(\beta\)-plane models. The relative importance of control parameters, the static stability parameter and MTG, in the development of baroclinic instability can be estimated using relative sensitivity functions displayed in Figure 4. The analysis of relative sensitivity functions shows that for each model (\"dry\" and \"moist\" \(f\)-plane and \(\beta\)-plane) there is a critical wavelength \(L_{c}\) that splits up the wave spectrum in two regions.