## 3 Results and Discussion
### Feature Height and Entrainment Coefficient
#### 3.3.2 Entrainment Coefficient (\(\alpha_{e}\))  
The entrainment strength of hydrothermal plume is measured by entrainment coefficient. According to analyzing the calculation method of the entrainment coefficient, the calculation formula is defined as [34; 38; 39]  
\[\alpha_{e}(z)=\frac{dM(z)}{dz}\cdot\frac{1}{2\pi\rho_{0}\cdot w\cdot r}. \tag{19}\]  
Figure 17a shows that the entrainment coefficients of the rke model and the sst model have a similar variation trend. When z/Z\({}_{\text{max}}\) is greater than 0.3, the entrainment coefficient increases rapidly from 0 to a larger value. When z/Z\({}_{\text{max}}\) is 0.03 to 0.60, the entrainment coefficient is stable within 0.10-0.11. When z/Z\({}_{\text{max}}\) is 0.60 to 0.75, the entrainment coefficient decreases rapidly from a stable value to 0. When z/Z\({}_{\text{max}}\) is less than 0.5, the average entrainment coefficients calculated by the rke model and the sst model are analyzed. The result of the rke model is 0.10760, which is slightly higher than that of the sst model (the sst model result is 0.10695). Figure 17b shows the difference between the entrainment coefficient calculated by the rke model and the sst model under environmental reference conditions. We take the average entrainment coefficient \(\alpha_{e}\) is 0.104 of [PERSON] et al. as a reference and compare the entrainment coefficient calculated by the rke model and the sst model [35]. The calculation error of rke_case2 is 2.71%, and that of sst_case2 is 1.94%. In terms of the prediction of the entrainment coefficient of hydrothermal plume, the resulting curve of the sst model tends to be gentle, while the velocity change and enrolling effect of the hydrothermal plume in the plume stem area are relatively intense.  
Figure 17: Vertical profile distribution of entrainment coefficient of hydrothermal plume ([PERSON] et al. 2020 [35]). **(a)** The distribution of entrainment coefficient for all working conditions calculated based on the definition; **(b)** Comparison of results of different models based on definition calculations (case2); **(c)** Entrainment coefficient comparison of rke model based on Fr method (case2); **(d)** Entrainment coefficient comparison of sst model based on Fr method (case2). These vertical dotted lines are the average entrainment coefficient for hydrothermal plumes below 0.5 Zmax height.  
[PERSON] et al. introduced the Richardson number (Ri) to calculate the entrainment coefficient [35]. However, when z/Z\({}_{\rm max}\) is greater than 0.6, there is a slight deviation in the results of this calculation method. To better evaluate the accuracy of the entrainment coefficients of the rke model and the sst model, the Froude number (\(Fr\)) is introduced to improve the calculation method of entrainment coefficient [40].  
\[Fr=\left[\frac{w_{0}^{2}\cdot\rho_{a}}{gd(\rho_{a}-\rho_{0})}\right]^{1/2}, \tag{20}\]  
\[\alpha_{Fr}=\alpha_{j}-\big{(}\alpha_{j}-\alpha_{p}\big{)}\left(\frac{Fr_{p}} {Fr}\right)^{2}. \tag{21}\]  
[PERSON] et al. determined that the entrainment coefficient of pure jet \(\alpha_{\rm i}\) is 0.0742 and the entrainment coefficient of pure plume \(\alpha_{\rm p}\) is 0.1178 [41]. According to the above equation, we calculate Froude number \(Fr_{p}\) is 4.71879. Figure 17c,d show good self-consistency between \(\alpha_{\rm Fr}\) and \(\alpha_{\rm e}\). It shows that the entrainment coefficients obtained by the two methods in this paper are correct. The normal range of entrainment coefficient is 0.10\(\sim\)0.16 [42; 43; 44; 45]. The results show that these two models can effectively reproduce the enrolling process of hydrothermal plume.