## 3 Models for Wind Turbine Wakes under Stable Conditions
### Turbulence Modeling  
In this paper, we apply the \(k-\varepsilon\) turbulence model to close Equations (12)-(14):  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho U_{i}k )=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}} \right)\frac{\partial k}{\partial x_{j}}\right]+\mathcal{P}+\mathcal{B}-\rho \varepsilon-S_{k,\text{ABL}} \tag{22}\]  
\[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{i}}( \rho U_{i}\varepsilon)=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{ \mu_{t}}{\sigma_{k}}\right)\frac{\partial\varepsilon}{\partial x_{j}}\right] +\left(C_{\varepsilon 1}\mathcal{P}-\rho C_{\varepsilon 2}\varepsilon+C_{ \varepsilon 3}\mathcal{B}\right)\frac{\varepsilon}{k}+S_{\varepsilon,\text{ wake}} \tag{23}\]  
where \(k\) is the turbulence kinetic energy (TKE), \(\varepsilon\) is the TKE dissipation, \(C_{\mu}=0.033\), \(\sigma_{k}=1.0,\sigma_{\varepsilon}=1.3\), and \(C_{\varepsilon 2}=1.92\). \(\mathcal{P}\) and \(\mathcal{B}\) are the TKE source production due to shear and buoyancy, respectively:  
\[\mathcal{P}\equiv-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\frac{\partial U _{j}}{\partial x_{i}}=\mu_{t}\left(\frac{\partial U_{i}}{\partial x_{j}}+\frac {\partial U_{j}}{\partial x_{i}}\right)\frac{\partial U_{j}}{\partial x_{i}} \tag{24}\]  
\[\mathcal{B}\equiv\frac{g_{i}}{\dot{\Theta}_{0}}\overline{u_{i}^{\prime}\theta ^{\prime}}=-\frac{g_{i}}{\dot{\Theta}_{0}}\frac{\mu_{t}}{Pr_{t}}\frac{\partial \Theta}{\partial x_{i}} \tag{25}\]  
where \(u_{i}^{\prime}\) is the fluctuation of the velocity component in \(x_{i}\) direction. \(S_{\varepsilon,\text{wake}}\) is a source term of TKE dissipation applied in the neighbor region of the turbine to correct the fast wake recovery of the standard \(k-\varepsilon\) model [30]:  
\[S_{\varepsilon,\text{wake}}=\rho C_{4\varepsilon}\frac{\mathcal{P}^{2}}{ \rho k} \tag{26}\]  
where \(C_{4\varepsilon}=0.37\). In this work, the methods of [PERSON] [15], the recently proposed model from [PERSON] et al. [16], and the indirect model that cannot solve the energy equation applied in [PERSON] et al. [14] are investigated, and the models are hereafter referred to as the AM, Laan, and El-Askary models, respectively. In the Laan model and its modified model (the proposed model), the TKE source term \(S_{\varepsilon,\text{ABL}}\) and the coefficient \(C_{\varepsilon 3}\) are functions of atmospheric stability and modeled to keep flow homogeneity under stable conditions. However, \(S_{\varepsilon,\text{ABL}}\) is absent in the AM model and the El-Askary model, and \(C_{\varepsilon 3}\) is set to \(-2.9\) in the AM model [15] and set to \(1\) in the El-Askary model [14]. In the El-Askary model, the energy Equation (14) is not solved, and the TKE production due to buoyancy \(\mathcal{B}\) is modeled as  
\[\mathcal{B}=-\mu_{t}\left(\frac{\partial u}{\partial z}\right)^{2}\frac{Ri}{ \phi_{m}} \tag{27}\]  
where  
\[Ri=\zeta\frac{0.74+4.7\zeta}{(1+4.7\zeta)^{2}},\quad\zeta>0 \tag{28}\]  
The Laan model is derived from homogeneous steady flows over flat terrain where the kinematic viscosity \(\mu\) can be ignored, and the \(k-\varepsilon\) model can be rewritten into normalized form by \(\kappa z/u_{*}^{3}\)  
\[\phi_{t,k}+\phi_{m}-\phi_{k}-\frac{\phi_{h}}{Pr_{t}\phi_{m}}\zeta-\phi_{S_{k} }=0 \tag{29}\]  
\[\phi_{t,\varepsilon}+\left(C_{\varepsilon 1}\phi_{m}-C_{\varepsilon 2}\phi_{ \varepsilon}-C_{\varepsilon 3}\frac{\phi_{h}}{Pr_{t}\phi_{m}}\zeta\right)\frac{ \varepsilon}{k}=0 \tag{30}\]  
in which \(\phi_{t,k}\equiv\frac{\kappa z}{\mu_{t}^{3}}\frac{\partial}{\partial z}\left( \frac{\mu_{t}}{\sigma_{k}}\frac{\partial k}{\partial z}\right)\) and \(\phi_{t,\varepsilon}\equiv\frac{\kappa z}{\rho u_{*}^{3}}\frac{\partial}{ \partial z}\left(\frac{\mu_{t}}{\sigma_{k}}\frac{\partial k}{\partial z}\right)\) are the normalized turbulent transport of \(k\) and \(\varepsilon\), \(\phi_{S_{k}}\equiv\frac{\kappa z}{u_{*}^{3}}S_{kwhich yields \(C_{\epsilon 1}=1.24\).