# In-flight Wind Field Identification and Prediction of Parafoil Systems
## 4 Wind Field Prediction  
### Average Wind Field Prediction Model  
Average wind speed over a given altitude band varies with the change of altitudes. The influence of internal viscous friction on the acceleration motion is little in the atmosphere motion process, which can be ignored. The atmospheric dynamics equations [29] in the local vertical, local horizontal coordinates is built as Equations (14)-(17). The atmosphere motion equation  
\[\left\{ \begin{array}{l}\frac{\mathrm{d}u}{\mathrm{d}t}=-\frac{1}{\rho}\frac {\partial p}{\partial x}+2\Omega v\sin\lambda-2\Omega w\cos\lambda\\ \frac{\mathrm{d}v}{\mathrm{d}t}=-\frac{1}{\rho}\frac{\partial p}{\partial y} -2\Omega u\sin\lambda\\ \frac{\mathrm{d}w}{\mathrm{d}t}=-\frac{1}{\rho}\frac{\partial p}{\partial z} -g+2\Omega wu\cos\lambda\end{array} \right. \tag{14}\]  
where \(u\) and \(v\) denote the components of horizontal velocity along \(x\) and \(y\) axis, respectively. \(w\) denotes the vertical velocity along \(z\) axis. \(\rho\) denotes the density of air. \(p\) is the ideal gas pressure. \(\Omega\) denotes the angular velocity (with respect to the earth, \(\Omega=15^{\mathrm{o}}/\mathrm{s}=7.29\times 10^{-5}\) rad/s). \(\lambda\) denotes the latitude. And \(f=2\Omega\sin\lambda\) denotes Coriolis parameter. The atmosphere state equation  
\[P=\rho RT \tag{15}\]  
where \(T\) denotes temperature, and \(R\) is the gas constant. The continuous equation  
\[\frac{\mathrm{d}\rho}{\mathrm{d}t}+\rho\
abla\vec{V}=0 \tag{16}\]  
where \(\vec{V}\) denotes the three-dimensional wind velocity vector. The thermodynamic equation  
\[\frac{\mathrm{d}\varsigma}{\mathrm{d}t}=F_{\varsigma} \tag{17}\]  
where \(\varsigma\) denotes potential temperature. For large-scale atmospheric motion systems, the vertical velocity during atmospheric motion is much smaller than the horizontal velocity and the inertial force generated is much smaller than the generated Coriolis force term [30]. The vertical velocity and inertia force can be ignored, and only the horizontal component is taken into account. We obtain  
\[\left\{ \begin{array}{l}-\frac{1}{\rho}\frac{\partial p}{\partial x}+2\Omega v \sin\lambda=0\\ -\frac{1}{\rho}\frac{\partial p}{\partial y}-2\Omega u\sin\lambda=0\end{array} \right. \tag{18}\]  
Equation (18) shows that the atmosphere reaches to geostrophic equilibrium in large scale motion, which means that Coriolis force and pressure gradient force are at equilibrium state. Plugging in \(f=2\Omega\sin\chi\) into Equation (18) yields to  
\[\left\{ \begin{array}{l}u_{g}=-\frac{1}{f\rho}\frac{\partial p}{\partial y} \\ v_{g}=\frac{1}{f\rho}\frac{\partial p}{\partial x}\end{array} \right. \tag{19}\]  
where \(u_{g}\) and \(v_{g}\) denote two components of geostrophic wind along \(x\) and \(y\) direction, respectively. Though the geostrophic wind calculated by Equation (19) is not the actual wind, they are very approximate beyond the area of \(\lambda<\pm 15\). When calculating the wind in a free atmosphere, the geostrophic wind can be applied to approximate the actual wind field. In the friction layer, the pressure gradient force, the turbulent friction, and the Coriolis force are similar [31]. Hence, they cannot be ignored. In the atmospheric boundary layer, the three forces reach to non-geostrophic wind balances  
\[\left\{ \begin{array}{l}-\frac{1}{\rho}\frac{\partial p}{\partial x}+2\Omega v \sin\lambda+F_{x}=0\\ -\frac{1}{\rho}\frac{\partial p}{\partial y}-2\Omega u\sin\lambda+F_{y}=0\end{array} \right. \tag{20}\]  
The turbulent friction in Equation (20) is calculated by  
\[\left\{ \begin{array}{l}F_{x}=\frac{\partial}{\partial x}(K_{m}\frac{ \partial u}{\partial z})=K_{m}\frac{\partial^{2}u}{\partial z^{2}}\\ F_{y}=\frac{\partial}{\partial z}(K_{m}\frac{\partial v}{\partial z})=K_{m} \frac{\partial^{2}v}{\partial z^{2}}\end{array} \right. \tag{21}\]  
where \(K_{m}\) is the turbulent viscosity coefficient.