## Appendix A Basic Equation Describing Large-Scale Atmospheric Dynamics
### The \(\beta\)-Plane Model  
Widely used in geophysical fluid dynamics, \(\beta\)-plane approximation allows for latitudinal variability of the Coriolis parameter, which is very important for accurate modelling of atmospheric wave motions on spherical earth. If \(\varphi_{0}\) is a given latitude and \(f_{0}\) is the Coriolis parameter at \(\varphi_{0}\), then \(f=f_{0}+\beta y\), where \(\beta=(2\Omega/a_{0})\cos\varphi_{0}\). Here \(\Omega\) is the angular speed of the Earth's rotation, and \(a_{0}\) is the Earth's radius. Taking into account the \(\beta\)-plane approximation Equations (A1)-(A5) that describe the large-scale atmospheric dynamics can be transformed into the following two equations [21; 37]:  
\[\left(\frac{\partial}{\partial t}+u\frac{\partial}{\partial x}+v\frac{ \partial}{\partial y}\right)\psi^{2}\psi+\beta\frac{\partial\psi}{\partial x }=f_{0}\frac{\partial\omega}{\partial p}\] (A24)  
\[\left(\frac{\partial}{\partial t}+u\frac{\partial}{\partial x}+v\frac{ \partial}{\partial y}\right)\frac{\partial\psi}{\partial p}+\frac{\sigma_{0} }{f_{0}}\omega=0\] (A25)  
where \(\psi=\Phi/f_{0}\) is a geostrophic streamfunction. Note that the geostrophic wind velocity components, \(u\) and \(v\), can be expressed in terms of stream function as \(u=-\partial\psi/\partial y\) and \(v=-\partial\psi/\partial x\) respectively. The vertical boundary conditions are specified at the top (\(p=0\)) and the bottom (\(p=p_{o}\)) of the atmosphere and assume that the vertical velocity is equal to zero. Applying the Equation (A24) at the 250 and 750 hPa levels designated as \(1\) and \(2\), respectively, and the Equation (A25) at the 500 hPa level designated as \({}^{3}/2\), we obtain  
\[\left(\frac{\partial}{\partial t}+u_{1}\frac{\partial}{\partial x}+v_{1} \frac{\partial}{\partial y}\right)\
abla^{2}\psi_{1}+\beta\frac{\partial\psi _{1}}{\partial x}=\frac{f_{0}}{\Delta p}\omega\ _{{}^{3}/2}\] (A26)  
\[\left(\frac{\partial}{\partial t}+u_{2}\frac{\partial}{\partial x}+v_{1} \frac{\partial}{\partial y}\right)\
abla^{2}\psi_{2}+\beta\frac{\partial\psi _{2}}{\partial x}=-\frac{f_{0}}{\Delta p}\omega_{{}^{3}/2}\] (A27)  
\[\left(\frac{\partial}{\partial t}+u_{3}/2\frac{\partial}{\partial x}+v_{3}/2 \frac{\partial}{\partial y}\right)(\psi_{1}-\psi_{2})=\frac{\sigma_{0}\Delta p }{f_{0}}\omega_{{}^{3}/2}\] (A28)  
where \(\Delta p=500\) hPa. Thus, we have a system of three Equations (A26)-(A28) in three variables \(\psi_{1}\), \(\psi_{2}\), and \(\alpha_{3/2}\). To explore the instability of the basic zonal flow with respect to infinitesimal perturbations, these equations are linearized around the basic state (A6) assuming that  
\[\psi_{1}=-u_{1}y+\psi_{1}^{\prime}(x,t)\psi_{2}=-u_{2}y+\psi_{2}^{\prime}(x,t) \omega_{3/2}=\omega_{3/2}^{\prime}(x,t)\] (A29)  
Substituting (A29) in the Equations (A26)-(A28) and defining the following notations  
\[u_{m}=\frac{1}{2}(u_{1}+u_{2}),\ u_{T}=\frac{1}{2}(u_{1}-u_{2}),\ \psi_{m}=\frac{1}{2} \Big{(}\psi_{1}^{\prime}+\psi_{2}^{\prime}\Big{)},\ \psi_{T}=\frac{1}{2}\Big{(}\psi_{1}^{\prime}-\psi_{2}^{ \prime}\Big{)}\] (A30)  
after the elimination of the variable \(\omega_{3/2}\), yields the perturbation equations:  
\[\left(\frac{\partial}{\partial t}+u_{m}\frac{\partial}{\partial x}\right) \frac{\partial^{2}\psi_{m}}{\partial x^{2}}+\beta\frac{\partial\psi_{m}}{ \partial x}+u_{T}\frac{\partial}{\partial x}\bigg{(}\frac{\partial^{2}\psi_{ T}}{\partial x^{2}}\bigg{)}=0\] (A31)  
\[\left(\frac{\partial}{\partial t}+u_{m}\frac{\partial}{\partial x}\right) \left(\frac{\partial^{2}\psi_{T}}{\partial x^{2}}-2\mu^{2}\psi_{T}\right)+ \beta\frac{\partial\psi_{T}}{\partial x}+u_{T}\frac{\partial}{\partial x} \bigg{(}\frac{\partial^{2}\psi_{m}}{\partial x^{2}}+2\mu^{2}\psi_{m}\bigg{)}=0\] (A32)  
where \(\mu^{2}=f_{0}^{2}/\Big{[}\sigma_{0}(\Delta p)^{2}\Big{]}\). We shall seek a normal modes solution:  
\[\psi_{m}=\Psi_{m}e^{ik(x-ct)},\ \psi_{T}=\Psi_{T}e^{ik(x-ct)}\] (A33)  
where \(\Psi_{m}\) and \(\Psi_{T}\) are the perturbation amplitudes.