## 2 Data and Methodology
### WRF Setup and Experiment  
#### 2.2.1 Equations of WRF  
In this paper, WRF (ARW) version 4.1.2 is employed, and a terrain following vertical coordinates is adopted, which is expressed as  
\[\eta=\frac{p_{\rm h}-p_{\rm ht}}{\mu}, \tag{1}\]  
where \(p_{\rm h}\) is the static part of the air pressure, \(\mu=p_{\rm hs}-p_{\rm ht}\), and \(p_{\rm hs}\) and \(p_{\rm ht}\) are the air pressure at the model surface and upper boundary, respectively. Considering the horizontal pressure gradient error, the new variables are defined as the deviation from the reference state, and the atmospheric motion variables are expressed as follows  
\[\left. \begin{array}{l}p=\overline{p}(\overline{z})+p^{\prime}\\ \phi=\overline{\phi}(\overline{z})+\phi^{\prime}\\ \alpha=\overline{\alpha}(\overline{z})+\alpha^{\prime}\\ \mu_{\rm d}=\overline{\mu}_{\rm d}(x,y)+\mu^{\prime}_{\rm d}\end{array} \right\}, \tag{2}\]  
where \(p\) is the corrected air pressure, \(\phi\) is the geopotential height, \(\phi=gz\), \(g\) is the gravitational acceleration, and \(z\) is the height of the position. \(\alpha\) is the derivative of density, which is a non-conservative variable, and is a function of the vertical profile \((x,y,\eta)\), \(\mu_{d}=p_{\rm dhs}-p_{\rm dht}\), d represents dry air, and \(p^{\prime},\phi^{\prime},\alpha^{\prime},\mu^{\prime}_{\rm d}\) are the original values of those variables without considering the horizontal pressure gradient error. In the calculation process, the perturbation form of momentum equations of the WRF model can be written as:  
\[ \begin{array}{l}\frac{\partial U}{\partial t}+m_{x}\left(\frac{\partial Uu} {\partial x}+\frac{\partial Vu}{\partial y}\right)+\frac{\partial\Omega u}{ \partial\eta}+\left(\mu_{d}\alpha\frac{\partial p^{\prime}}{\partial x}+\mu_ {d}\alpha^{\prime}\frac{\partial\overline{p}}{\partial x}\right)+\\ \left(\alpha/\alpha_{d}\right)\left(\mu_{d}\frac{\partial\phi^{\prime}}{ \partial x}+\frac{\partial p^{\prime}}{\partial\overline{\alpha}}\frac{ \partial\phi}{\partial x}-\mu^{\prime}_{d}\frac{\partial\phi}{\partial x} \right)=F_{U},\end{array}  \tag{3}\]  
\[ \begin{array}{l}\frac{\partial V}{\partial t}+m_{y}\left(\frac{\partial Uv} {\partial x}+\frac{\partial Vv}{\partial y}\right)+\left(m_{y}/m_{x}\right) \frac{\partial\Omega v}{\partial\eta}+\left(\mu_{d}\alpha\frac{\partial p^{ \prime}}{\partial y}+\mu_{d}\alpha^{\prime}\frac{\partial\overline{p}}{ \partial y}\right)+\\ \left(\alpha/\alpha_{d}\right)\left(\mu_{d}\frac{\partial\phi^{\prime}}{ \partial y}+\frac{\partial\phi}{\partial y}-\mu^{\prime}_{d}\frac{\partial \phi}{\partial y}\right)=F_{V},\end{array}  \tag{4}\]  
\[ \begin{array}{l}\frac{\partial W}{\partial t}+\left(m_{x}m_{y}/m_{y}\right) \left(\frac{\partial Uw}{\partial x}+\frac{\partial Vw}{\partial y}\right)+ \frac{\partial\Omega w}{\partial\eta}\\ -\frac{g}{m_{y}}\left(\alpha/\alpha_{d}\right)\left(\frac{\partial p^{\prime}} {\partial\eta}-\overline{\mu}_{d}(q_{\rm p}+q_{\rm c}+q_{\rm r})\right)+\frac{ g}{m_{y}}\mu^{\prime}_{d}=F_{w},\end{array}  \tag{5}\]  
And the potential temperature equation can be expressed as:  
\[\frac{\partial\Theta}{\partial t}+m_{x}m_{y}\left(\frac{\partial U\theta}{ \partial x}+\frac{\partial V\theta}{\partial y}\right)+m_{y}\frac{\partial \Omega\theta}{\partial\eta}=F_{\Theta}, \tag{6}\]  
where \(U\), \(V\), and \(W\) are the momentum components controlling the three directions of the equation of atmospheric motion on the projection plane, respectively. \(\Omega\) is the momentum of the vertical velocity field, \(\Theta\) is the momentum of the potential temperature field, \(u\), \(v\), and \(w\) are velocity components in three directions, respectively, mx and my are magnification factors for map projections in \(x\) and \(y\) directions, respectively, \(\left(m_{x},m_{y}\right)=\frac{\left(\Delta x,\Delta y\right)}{\left(\Delta x, \Delta y_{\rm c}\right)}\), and \(F_{U},F_{V},F_{W},F_{\Theta}\) are forced terms caused by model physical processes, turbulent mixing, spherical projection, and Earth rotation, respectively.