## Materials and Methods
### Moisture Content and Transport in the Global Atmosphere
Note that  
\[W=\frac{1}{g}\int_{0}^{p_{0}}qdp \tag{2}\]  
\[\mathbf{Q}=\int_{0}^{p_{0}}q\mathbf{V}\frac{dp}{g}=\underbrace{\mathbf{i}_{ \lambda}\int_{0}^{p_{0}}qu\frac{dp}{g}}_{\text{Zonal component }Q_{\lambda}}+ \underbrace{\mathbf{i}_{\varphi}\int_{0}^{p_{0}}q\varphi\frac{dp}{g}}_{ \text{Meridional component }Q_{\varphi}} \tag{3}\]  
In this study, the quantity of interest is the mean vertically integrated meridional moisture flux (MMF) \(\overline{Q}_{\varphi}\), which can be decomposed into three components [73]:  
\[\underbrace{\{[\overline{q}\overline{v}]\}}_{\text{MMF}}=\underbrace{\{[ \overline{q}][\overline{v}]\}}_{\text{MMC}}+\underbrace{\{[\overline{q} ^{\ast}\overline{v}^{\ast}]\}}_{\text{SE}}+\underbrace{\{[\overline{q^{ \ast}v^{\ast}}]\}}_{\text{TE}} \tag{4}\]  
The first term on the right-hand side of this equation (MMC) is the contribution from mean meridional circulation, the second term (SE) is from stationary eddies (planetary waves), and the third term (TE) is from transient eddies (extratropical cyclones) [67; 68]. Numerous studies involving the evaluation of the right-hand side of Equation (4) with observations datasets and CGCM-simulated data (Coupled atmosphere-ocean general circulation model) have shown that transient eddies dominate meridional moisture transport at mid-latitudes: \(\sim\)80% between 30\({}^{\circ}\) and 40\({}^{\circ}\) north and south latitudes (e.g., [20; 40; 41; 68; 73; 74; 75; 76; 77]). One can easily show that for the plane wave solutions (A15), the TE meridional moisture flux is exponentially dependent on the growth rate \(\chi_{k}=kc_{i}\) of unstable modes [78]. Let \(L_{w}\) be the wavelength of a baroclinic unstable wave and \(k=2\pi/L_{w}\) its wavenumber. Then  
\[q^{\prime}(x,p,t)=\Re e\Big{(}\hat{q}(p)e^{ik(x-ct)}\Big{)},\ v^{\prime}(x,p,t )=\Re e\Big{(}\hat{v}(p)e^{ik(x-ct)}\Big{)} \tag{5}\]  
or  
\[q^{\prime}(x,p,t)=\big{|}\hat{q}(p)\big{|}\cos\!\big{[}k(x-c_{r}t)+\theta_{q}( p)\big{]}e^{kc_{i}t} \tag{6}\]  
\[v^{\prime}(x,p,t)=\big{|}\hat{v}(p)\big{|}\cos\!\big{[}k(x-c_{r}t)+\theta_{v}( p)\big{]}e^{kc_{i}t} \tag{7}\]  
where \(\big{|}\hat{q}(p)\big{|}\) and \(\big{|}\hat{v}(p)\big{|}\) are the absolute values and \(\theta_{q}(p)\) and \(\theta_{v}(p)\) are the arguments of the complex values \(\hat{q}(p)\) and \(\hat{v}(p)\) respectively, \(c_{r}=\Re e(c),\ c_{i}=\Im m(c)\). The meridional equator-to-pole moisture flux averaged over one period of the unstable wave with wavelength \(L_{w}\) is as follows [21]:  
\[\overline{q^{\prime}v^{\prime}}^{x}=\frac{1}{L_{w}}\int_{0}^{L_{w}}q^{\prime} (x,p,t)v^{\prime}(x,p,t)dx \tag{8}\]  
We can find the flux \(\overline{q^{\prime}v^{\prime}}^{x}\) by substituting (6) and (7) into (8) and assuming the periodic boundary conditions along the \(x\) axis:  
Considering baroclinic instability on the \(\beta\)-plane [37] whereby the Coriolis parameter, \(f\), is set to vary linearly in the \(y\)-direction, we derived the following expression for the growth rate of unstable waves (see Equation (A35) in the Appendix A.3):  
\[\chi_{k}=\frac{\sqrt{\left|\beta^{2}\mu^{4}+u_{T}^{2}k^{4}(k^{4}-4\mu^{4}) \right|}}{k(k^{2}+2\mu^{2})} \tag{15}\]  
Here \(\mu^{2}=f_{0}^{2}/\left[\sigma_{0}(\Delta p)^{2}\right]\) and \(u_{T}=(u_{1}-u_{2})/2\), where \(u_{1}\) and \(u_{2}\) are the wind speeds at 250 hPa and 750 hPa pressure levels respectively.