## 3 Results
### Influence of Climate Change on the Growth Rate of Unstable Waves  
#### 3.2.1 Growth Rate of Unstable Baroclinic Waves  
It is expected that the static stability of the atmosphere and the temperature difference between the poles and equator will change under climate change. In mid-latitudes, changes in static stability and MTG will affect the development of baroclinic instability and, therefore, the atmospheric eddy meridional heat and moisture transport. In both models (\(f\)-plane and \(\beta\)-plane), the growth rate \(\chi_{k}\) of unstable baroclinic waves can be considered as the main characteristic of the development of baroclinic instability [21]. In these two models, the growth rate \(\chi_{k}\) is a function of two control parameters: static stability parameter \(\sigma_{0}\) and MTG. In the \(f\)-plane model, the MTG is expressed via the vertical wind shear \(\Lambda\) since the thermal wind equation (A7) relates \(\Lambda\) to MTG. In the \(\beta\)-plane model, the thermal wind serves as a characteristic of MTG. By varying parameters \(\sigma_{0}\), \(\Lambda\), and \(u_{T}\), we can estimate the climate change impact on \(\chi_{k}\) using sensitivity functions \(S_{\sigma}\), \(S_{\Lambda}\), and \(S_{ut}\) defined by Equations (13), (14), (16), and (17). These functions should be calculated around some reference values of static stability parameter and MTG (\(\sigma_{0}\), \(\Lambda\) and \(u_{T}\)), each pair of which corresponds to certain climate conditions (previous, current or future). For the current climate conditions, the reference values of static stability parameter and the vertical wind shear are taken to be \(\sigma_{0}^{*}=2{\cdot}10^{-6}\) m\({}^{2}\) Pa\({}^{-2}\) s\({}^{-2}\) and \(\Lambda^{ref}=30\) ms\({}^{-1}\) bar\({}^{-1}\)[21]. The value of \(\Lambda^{ref}=\) ms\({}^{-1}\) bar\({}^{-1}\) corresponds to \(u_{T}=7.5\) ms\({}^{-1}\), which is used in the \(\beta\)-plane model and serves as a characteristic of MTG. The reference temperature lapse rate is of 6.5 \({}^{\circ}\)C km\({}^{-1}\) which corresponds to \(\sigma_{0}^{ref}=2{\cdot}10^{-6}\) m\({}^{2}\) Pa\({}^{-2}\) s\({}^{-2}\). To take into consideration the influence of atmospheric moisture on the development of baroclinic instability, the effective static stability parameter \(\sigma_{0,eff}^{ref}\) was also calculated [45]: \(\sigma_{0,eff}^{ref}=1.16{\cdot}10^{-6}\) m\({}^{2}\) Pa\({}^{-2}\) s\({}^{-2}\). The latitude of interest is assumed to be \(\varphi_{0}=45\) N which gives \(f_{0}\approx 10^{-4}\) s\({}^{-1}\) and \(\beta=1.63{\cdot}10^{-11}\) m\({}^{-1}\) s\({}^{-1}\). The growth rate of the unstable waves as a function of zonal wavenumber \(k_{z}\) for the \"dry\" and \"moist\" \(f\)-plane and \(\beta\)-plane models are illustrated in Figure 1. The \(f\)-plane model has a short-wave cutoff. In other words, there is a critical value of wavelength \(L_{min}^{f}\) above which instability occurs. If wavelength \(L_{w}<L_{min}^{f}\) then the phase velocity is purely real and, therefore, this wave is stable, and its amplitude does not grow with time [36]. The value of \(L_{min}^{f}\) is obtained from the condition that the discriminant \(\eta^{2}-4(\eta{\rm coth}\eta-1)\) of the Equation (10) is equal to zero, which gives \(\eta=\eta_{c}=2.3994\) and, therefore, \(L_{min}^{f}\approx 3575\) km for the \"dry\" model, and \(L_{min}^{f}\approx 2735\) km for the \"moist\" model (see Table 1). Thus, under the influence of the atmospheric moisture, baroclinic unstable waves are shifted into the shortwave part of the spectrum. Figure 1: Growth rate \(\chi_{k}=kc_{i}\) as a function of zonal wavenumber \(k_{z}\) for the “dry” and “moist” (**a**) \(f\)-plane and (**b**) \(\beta\)-plane models, as described by Equations (10) and (15) respectively. Note that one of the disadvantages of \(f\)-plane model is that the long waves are unstable that contradicts, at least, the results of the numerical simulations with global models, which show that planetary-scale waves are practically stable.