## Appendix A Basic Equation Describing Large-Scale Atmospheric Dynamics
### The f-Plane Model  
In general, the life cycle of baroclinic unstable waves has three main stages: the initial stage, in which unstable waves grow exponentially; the mature stage, in which the waves reach saturation and transform into large-scale eddies (cyclones and anticyclones); and the decay stage, in which eddies dissipate. To explore the initial stage, theoretical models usually employ linearized dynamics equations whereby the instability problem is examined as an eigenvalue problem. To isolate the baroclinic mechanism of instability in pure form, we will exclude from consideration the \(y\)-dependence of atmospheric zonal flow. First, we consider the \(f\)-plane baroclinic instability model [36] in which the Coriolis parameter \(f\) is set to a constant value, i.e., \(f=f_{0}=2\Omega\sin\varphi_{0}\), where \(\Omega=7.29\times 10^{-5}\) s\({}^{-1}\) is the angular speed of rotation of the earth and \(\varphi_{0}\) is the reference latitude (\(\varphi_{0}\)-\(60^{0}\) N). Then the eigenvalue problem, with some additional simplifications discussed below, can be solved analytically using the perturbation method. This technique requires representing each dependent variable \(\psi(x,y,p,t)\) as the sum of a reference (basic) state \(\overline{\psi}(p)\) and the infinitesimal (sufficiently small) perturbation \(\psi^{\prime}(x,p,t)\) such that \(\psi^{\prime}\ll\psi\):  
\[\psi(x,y,p,t)=\overline{\psi}(p)+\psi^{\prime}(x,y,p,t),\;\psi=(u,\;v,\;\omega, \;T,\;\Phi)\]  
Substituting the above expression into Equations (A1)-(A5) and neglecting the second-order terms, we obtain the following linear set of differential equations for two-dimensional perturbations:  
\[\frac{\partial u^{\prime}}{\partial t}+\overline{u}\frac{\partial u^{\prime} }{\partial x}+\alpha^{\prime}\frac{\partial\overline{u}}{\partial p}-f_{0}v ^{\prime}+\frac{\partial\Phi^{\prime}}{\partial x}=0\] (A9)  
\[\frac{\partial v^{\prime}}{\partial t}+\overline{u}\frac{\partial v^{\prime} }{\partial x}+f_{0}u^{\prime}=0\] (A10)  
\[\frac{\partial u^{\prime}}{\partial x}+\frac{\partial\omega^{\prime}}{ \partial p}=0\] (A11)  
\[\frac{\partial}{\partial t}\bigg{(}\frac{\partial\Phi^{\prime}}{\partial p} \bigg{)}+\overline{u}\frac{\partial}{\partial x}\bigg{(}\frac{\partial\Phi^ {\prime}}{\partial p}\bigg{)}-f_{0}v^{\prime}\frac{\partial\overline{u}}{ \partial p}+\sigma_{0}\alpha^{\prime}=0\] (A12)  
where \(\sigma_{0}\) is the static stability parameter (for simplicity taken constant):  
\[\sigma_{0}=\frac{R^{2}\overline{T}}{gp^{2}}(\Gamma_{d}-\Gamma)\] (A13)  
To define a baroclinic instability problem that results in a unique solution, we must specify boundary conditions for the Equations (A9)-(A12). The linear equations are complemented by periodic boundary conditions in \(x\)-direction for all model variables, and the rigid lids at the top (\(p=0\)) and bottom (\(p=p_{0}=10^{5}\) Pa):  
\[\omega^{\prime}=0\;\text{at}\;p=0\;\text{and}\;p=p_{0}\] (A14)  
We will look for plane-wave solutions of the form:  
\[\psi^{\prime}(x,p,t)=\hat{\psi}(p)e^{ik(x-ct)}\] (A15)  
where the amplitude of perturbation \(\hat{\psi}\) is a function of \(p\) only, \(k\) is a zonal wavenumber, and \(c\) is a complex phase velocity of perturbations.