## 3 Results  
### Qualitative Analysis  
Extratropical transient cyclonic eddies are wave phenomena, which are formed and evolved in response of the baroclinic instability of atmospheric quasi-zonal westerly flows drawing energy from the MAPE that arises from the MTG. As shown in the previous section, the meridional moisture flux depends on the wavelength of unstable wave which characterizes the size of cyclones. Numerous studies (e.g., [27; 28; 29; 30; 31; 82; 81; 83; 84; 81; 82]) suggest that over the past few decades the length-scale, frequency, and intensity of extratropical cyclones have changed, and a poleward shift of storm tracks and eddy driven jet streams have been observed. Apart from that, the projected changes in the climate system will, to a certain extent, affect the genesis and development of large-scale eddies in the mid-latitude atmosphere [83]. The impact of global warming on the horizontal sizes of transient eddies and their activity (the number of cyclones generated annually) can be qualitatively examined using a simplified theoretical framework (e.g., [84]). In a baroclinic atmosphere, the characteristic horizontal length scale of eddies is determined by the first internal Rossby radius of deformation \(L_{R}\)[21]:  
\[L_{R}=\frac{NH}{f}=\frac{1}{f}\sqrt{\frac{T}{g}(\Gamma_{d}-\Gamma)} \tag{20}\]  
where \(H\) is the height of homogeneous atmosphere. In the mid-latitudes, typical values for \(H\), \(N\), and \(f\) are respectively 8 km, \(10^{-2}\) s\({}^{-1}\), and \(1.1\times 10^{-4}\) s\({}^{-1}\)[21], consequently \(L_{R}\approx 850\) km. The impact of global warming on \(L_{R}\) can be estimated by the sensitivity coefficient \(dL_{R}/dT\). Using logarithmic differentiation to differentiate both sides of Equation (20) with respect to \(T\), we obtain  
\[\frac{1}{L_{R}}\frac{dL_{R}}{dT}=\frac{1}{2}\bigg{[}\frac{1}{T}-\frac{d\Gamma }{dT}(\Gamma_{d}-\Gamma)^{-1}\bigg{]} \tag{21}\]  
Equation (21) shows that in baroclinic atmosphere the influence of temperature growth on \(L_{R}\) is not so clear. The dry adiabatic lapse rate \(\Gamma_{d}\) is a constant 9.8 \({}^{\circ}\)C km\({}^{-1}\), while the typical value for \(\Gamma\) is about 6.5 \({}^{\circ}\)C km\({}^{-1}\). Since \(\Gamma_{d}>\Gamma\) and \(1/T\) are small, depending on the behavior of \(\Gamma\) under global warming, there could be two possible contrasting scenarios, \(dL_{R}/dT>0\) and \(dL_{R}/dT<0\). Therefore, in order to get a more clear picture of the global warming impact on \(L_{R}\), it is useful to explore these two scenarios. To begin, let us consider partial sensitivity coefficients:  
\[\frac{1}{L_{R}}\frac{\partial L_{R}}{\partial T}=\frac{1}{2T} \tag{22}\]  
\[\frac{1}{L_{R}}\frac{\partial L_{R}}{\partial\Gamma}=-\frac{1}{2(\Gamma_{d}- \Gamma)} \tag{23}\]  
From Equation (22), it follows that a small change \(\Delta T\) in temperature causes a fractional change \(\Delta L_{R}/L_{R}\) in Rossby radius of deformation of  
\[\frac{\Delta L_{R}}{L_{R}}\approx\frac{1}{2T}\Delta T \tag{24}\]  
Taking the annual global mean surface temperature \(T_{sfc}\approx 288\) K as a reference temperature, we find that \((1/L_{R})(\partial L_{R}/\partial T)\) is about 0.002 K\({}^{-1}\) which means that to increase the temperature by 1 K leads only to the 0.2% increase in characteristic length of large-scale eddies. Similarly, we can estimate the effect of small changes in the temperature lapse rate \(\Delta\Gamma\) on a fractional change \(\Delta L_{R}/L_{R}\):  
\[\frac{\Delta L_{R}}{L_{R}}\approx-\frac{1}{2(\Gamma_{d}-\Gamma)}\Delta\Gamma \tag{25}\]  
The characteristic value of \(\Gamma\) in the mid-latitude troposphere is 6.5 K/km, hence the partial sensitivity \((1/L_{R})(\partial L_{R}/\partial\Gamma)\) is about \(-0.15\) kmK\({}^{-1}\), meaning that to increase the lapse rate by 1 K/km (i.e., reducing the atmospheric static stability) results in the 15% decrease in characteristic length of large-scale eddies.