## 3 Results and Discussion
### Feature Height and Entrainment Coefficient
#### 3.3.1 Feature Height (\(\mathrm{Z_{max}}\), \(\mathrm{Z_{neutral}}\))
As can be seen from Equation (18), when the temperature \(T_{bottom}\) is 1.74 \({}^{\circ}\)C, the seabed density \(\rho_{bottom}\) is 1038 kg m\({}^{-3}\) and the thermal expansion coefficient \(\beta_{bottom}\) is \(8\times 10^{-5}\) K\({}^{-1}\). \[B_{asymp}=g\beta_{bottom}(T_{exit}-T_{bottom})Q_{exit}, \tag{17}\]  
\[\beta_{bottom}=\frac{1}{V}\bigg{(}\frac{\partial V}{\partial T}\bigg{)}_{P,S} =-\frac{1}{\rho}\bigg{(}\frac{\partial\rho}{\partial T}\bigg{)}_{P,S}. \tag{18}\]  
The empirical coefficient of the plume theoretical model was determined by many experiments, and its value was generally recommended to be 3.76 [9]. Figure 16a shows the linear fitting of the maximum plume rising height with the buoyancy flux and buoyancy frequency. Figure 16b shows the linear fitting of the neutrally buoyant plume height, the buoyancy flux and the buoyancy frequency of the hydro-thermal plume. Table 7 shows the scaling law of feature height obtained according to the working conditions of the rke model and the sst model. The MTT theoretical model predicts the result based on an ideal point source, while the numerical simulation predicts the result based on a certain vent radius. The results obtained by using the progressive buoyancy flux are closer to the results of the MTT theoretical model than those obtained by using the initial buoyancy flux. [PERSON] et al. proposed that the empirical coefficient of the maximum plume rising height was 3.56 through the laboratory experiment of plume, which was 5.3% higher than the empirical value of the theoretical model [12]. When the initial buoyancy flux is adopted, the scaling \begin{table}
\begin{tabular}{c c} \hline \hline
**Scaling Law of \(\mathbf{Z_{\text{max}}}\)** & **Scaling Law of \(\mathbf{Z_{\text{neutral}}}\)** \\ \hline \(Z_{\text{max,rke}}=2.68\) (B\({}_{\text{exit}}\) N\({}^{-3}\))\({}^{1/4}\) & \(Z_{\text{neutral,rke}}=1.95\) (B\({}_{\text{exit}}\) N\({}^{-3}\))\({}^{1/4}\) \\ \(Z_{\text{max,rke}}=4.59\) (B\({}_{\text{asymp}}\) N\({}^{-3}\))\({}^{1/4}\) & \(Z_{\text{neutral,rke}}=3.34\) (B\({}_{\text{asymp}}\) N\({}^{-3}\))\({}^{1/4}\) \\ \(Z_{\text{max,sst}}=2.69\) (B\({}_{\text{exit}}\) N\({}^{-3}\))\({}^{1/4}\) & \(Z_{\text{neutral,sst}}=1.97\) (B\({}_{\text{exit}}\) N\({}^{-3}\))\({}^{1/4}\) \\ \(Z_{\text{max,sst}}=4.61\) (B\({}_{\text{asymp}}\) N\({}^{-3}\))\({}^{1/4}\) & \(Z_{\text{neutral,sst}}=3.39\) (B\({}_{\text{asymp}}\) N\({}^{-3}\))\({}^{1/4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Scaling law of maximum plume rising height and neutrally buoyant plume height. Figure 16: Regression results of the height of hydrothermal plume rise ([PERSON] et al. 2020 [35]). (**a**) The fitting result of the maximum plume rising height; (**b**) The fitting result of the neutrally buoyant plume height. Dotted lines of the same color are obtained by fitting corresponding scatter points [9]. law coefficients of neutrally buoyant plume height and maximum plume rising height of rke model and sst model are like the results of [PERSON] et al. [35]. Their fitting stability R\({}^{2}\) is greater than 0.99. When the progressive buoyancy flux is used, the deviation of the fitting coefficient between the rke model and theoretical model is smaller than that between the sst model and the theoretical model. The heights calculated by the rke model and the sst model are not much different. The rke model and the sst model have equivalent prediction effects on the characteristic height of hydrothermal plume, and they can reproduce the macroscopic kinetic process of hydrothermal plume well.