## 2 Materials and Methods
### Numerical Simulation Method
#### 2.3.3 Numerical Simulation of Working Conditions  
Buoyancy frequency and buoyancy flux are the key factors affecting the movement of hydrothermal plumes. Taking the environmental reference condition (case2) as the control group, environmental factors such as the initial velocity of the vent orifice fluid, the temperature of the vent orifice fluid and the temperature gradient of the environment are changed to observe the influence of those on the dynamic characteristics of hydrothermal plume. To evaluate the ability of different models to simulate hydrothermal plumes, the rke model and the sst model are selected for comparative analysis, and 14 groups of numerical simulation conditions are set. See Table 5 for specific simulation parameter settings. In addition, temperature gradient initialization is used in the simulation of each set of conditions.  
In order to investigate the process of hydrothermal plume from eruption to quasi-steady state, the calculation time of the simulation is set as follows. The total simulation time of each condition is 4t*, where t* is the buoyancy time scale, defined as t* = 2\(\pi\)/N. We use a second-order implicit time integration scheme in our simulations. An implicit scheme is unconditionally stable with respect to time step size. At the same time, a time step of 0.5 s is selected so that the calculation in a single time step can reach convergence within 20 iterations (that is, the total residual of each physical quantity in two successive iterations is less than 10 \(\times\) 10\({}^{-3}\), and the total residual of energy is less than 10 \(\times\) 10\({}^{-6}\)). In addition, because the simulated physical time is short relative to the rotation period of the Earth, the  
Figure 6: Characteristic parameters of hydrothermal plume under different grid conditions.  
effect of the Coriolis force is ignored. When the simulation time is 3t*, the curves of each variable tend to be stable. Therefore, readers can judge the convergence speed and stability (that is, the asymptotic stability of numerical calculations) by observing the convergence of the iterative residual in each working condition.  
In this paper, the computational fluid dynamics software ANSYS Fluent (version 2020 R2) based on the finite volume method was used to numerically solve the above turbulence models and the specific settings for hydrothermal plume modeling. In order to reduce the numerical dissipation and ensure the rationality of the numerical model results, the following settings were employed in this paper. The highly accurate third-order MUSCL (monotone upstream-centered schemes for conservation laws) scheme was used for spatial discretization for all the convection terms in the momentum, energy and turbulence equations. The second-order accurate central-differencing scheme was used for spatial discretization for the diffusion terms. We chose the PRESTOI (Pressure Staggering Option) scheme as the discretization method of pressure. The PISO (pressure implicit with splitting of operators) scheme was adopted in the pressure-velocity coupling scheme. Second-order implicit scheme was used for temporal discretization.