Entrainment in deep-sea hydrothermal plumes: explaining the scatter of reported entrainment coefficients and quantifying the effects of multiple vents, background currents, and rotation

A Reynolds-averaged Navier-Stokes (RANS) case study of the Daxi hydrothermal field, Carlsberg Ridge (Lou et al., 2020) left a composite open question about turbulent entrainment in deep-sea hydrothermal plumes: (i) reported entrainment coefficients scatter widely across studies (about 0.10-0.16 for plume-like discharges and up to about 0.25 for bent plumes in cross-flow) without a systematic physical explanation; (ii) the effect of multiple interacting adjacent plumes on plume structure, neutral-buoyancy-layer thickness, and temperature-anomaly magnitude - hypothesized to explain why the simulated Daxi temperature anomaly was slightly lower than in-situ measurements - was uninvestigated, since the model used a single equivalent vent; and (iii) the influence of Coriolis forces and background currents on entrainment in the plume cap and on long-range dispersal of the neutrally buoyant plume was unassessed, the model being axisymmetric with linear stratification and no rotation or currents. Later literature has resolved parts of the periphery - generic two-plume coalescence dynamics (laboratory, CFD, and integral models), background-current effects at the neighboring Wocan-1 field, rotation effects in generic stratified plumes, and near-field entrainment variability from large-eddy simulation - but no work reconciles the coefficient scatter or closes the Daxi-specific question. The problem asks for a physically grounded, quantitative account of entrainment in real deep-sea hydrothermal plumes that explains the scatter across studies and extends validated single-vent axisymmetric modeling to multi-vent, rotating, current-swept conditions, tested against field observations.

Background

Deep-sea hydrothermal plumes form where buoyant fluid, heated in the oceanic crust, is expelled at the seafloor through 'black smoker' vents (mineral-laden fluid at roughly 300-400 degrees C) and rises as a turbulent buoyant plume through the stably stratified deep ocean. Entrainment - the turbulent mixing of ambient seawater into the plume - controls the dilution of heat and chemicals, the maximum rise height, and the formation of a neutrally buoyant layer at which the plume stops rising and spreads laterally for kilometers, dispersing geochemical signals used to locate undiscovered vent fields and to estimate hydrothermal heat flux. The classical theory of Morton, Taylor, and Turner (1956; the 'MTT' integral model) closes the vertically integrated conservation equations with an entrainment assumption: the inward entrainment velocity is a constant fraction of the mean plume velocity, written $u_e=\alpha w$ with entrainment coefficient $\alpha$ (about 0.1 for a pure buoyancy-driven plume, smaller for a momentum-dominated jet). In the MTT framework plume variables are averaged over the cross-section using 'top-hat' profiles (uniform across a defined plume radius), which matters because a top-hat $\alpha$ and a Gaussian-profile $\alpha$ differ by a convention-dependent factor - one of several reasons reported values vary. The second-order integral model of Wang and Law (2002) treats $\alpha$ as a function of the local Richardson number $Ri$ (the ratio of buoyancy to inertial forces), interpolating between $\alpha_j=0.0742$ for a pure jet and $\alpha_p=0.1178$ for a pure plume at $Ri_p=0.584$. Computational studies solve the Reynolds-averaged Navier-Stokes (RANS) equations, in which turbulence is represented by a closure model such as the k-epsilon family, or resolve the large turbulent eddies directly with large-eddy simulation (LES). The source paper is written in Chinese and cited in the international literature as Lou et al. 2020; its findings are as follows: a RANS model with the Realizable k-epsilon closure, an axisymmetric domain, a single equivalent circular vent (0.2 m diameter, 300 degrees C, 0.0314 m^3/s), and linear background stratification reproduced the Daxi plume; the plume has a three-layer structure, and the classical constant-entrainment assumption holds only below about 0.6 times the maximum rise height $Z_{\max}$, where the diagnosed $\alpha_e$ is nearly constant at about 0.104, falling rapidly to zero near 0.72 $Z_{\max}$ as lateral spreading in the neutrally buoyant layer and recirculation in the 'plume cap' (the overshooting, overturning region above the neutral-buoyancy level) take over. The model under-predicted the amplitude of the observed temperature anomaly, and the authors attributed this tentatively to interactions among the several adjacent active plumes that their single-vent model could not represent. On this basis the composite open question addressed here combines a gap the paper itself notes - why the entrainment coefficients reported across hydrothermal-plume studies scatter so widely, with differences in initial momentum at the vent (whether the release is 'forced', momentum-dominated, or 'lazy', buoyancy-dominated), vent geometry, and measurement or evaluation technique as suspected but unquantified causes - with the consequences of its acknowledged simplifications: how multiple interacting vents (represented in the model by a single equivalent vent), background currents, and Coriolis forces (the rotational effects of the Earth's spin, measured by the Rossby number, the ratio of inertial to Coriolis accelerations), all absent from the model, change the plume structure, the neutral-buoyancy layer, the temperature anomaly, and the long-range dispersal of the plume. The general turbulent-plume literature gives partial ingredients - laboratory and direct-numerical-simulation work shows $\alpha$ depends on the source Richardson number and height (Ezzamel, Salizzoni, and Hunt, 2015; van Reeuwijk et al., 2016) - and dedicated studies exist for rotating, stratified, cross-flow plumes (Lavelle, 1997; Fabregat Tomàs et al., 2016) and for coalescing pairs of plumes in stratified ambience (He et al., 2018; Lou et al., 2019), but these have not been combined into a reconciliation specific to hydrothermal plumes.

Problem Statement

Provide a systematic physical explanation for the scatter of entrainment coefficients (the dimensionless constants quantifying turbulent ingestion of ambient seawater into the rising plume) reported in deep-sea hydrothermal plume studies - roughly 0.10-0.16 for plume-like discharges and up to about 0.25 for bent plumes in background cross-flow - by quantitatively assessing the contribution of candidate causes, including differences in initial momentum flux at the vent, vent geometry, and the measurement or evaluation technique used to diagnose the coefficient (including top-hat versus Gaussian profile conventions, i.e. the two standard ways of averaging plume variables over the cross-section, whose coefficients differ by a fixed conversion factor); and, building on that explanation, determine how multiple interacting adjacent plumes, background currents, and Coriolis forces (the inertial effects of Earth's rotation) alter the plume structure, the thickness of the neutral buoyancy layer (the depth interval in which the density-matched plume stops rising and spreads laterally), the amplitude of the temperature anomaly, the entrainment dynamics in the plume cap region (the overshooting, overturning region above the neutral-buoyancy level), and the long-range dispersal of the neutrally buoyant plume. The overarching objective is a physically grounded entrainment parameterization for real deep-sea hydrothermal plumes that extends the validated single-vent, axisymmetric, quiescent-ambient modeling of the Daxi hydrothermal field (Carlsberg Ridge) to realistic configurations - multiple vents, ambient currents, rotation - and is tested against field observations, including the in-situ temperature-anomaly amplitude that the single-vent model under-predicted and that the source tentatively attributed to interactions among several adjacent active plumes. The multi-vent, background-current, and rotation components are distinct sub-questions, each addressing a quantitative determination of the plume properties it governs.

References

1. Lou Yingzhong, Han Xiqiu, He Zhiguo, Wang Yejian, Qiu Zhongyan (2020). Numerical modeling of hydrodynamic processes of deep-sea hydrothermal plumes: A case study on Daxi hydrothermal field, Carlsberg Ridge. Scientia Sinica Technologica 50(2): 194-208 (in Chinese with English abstract). DOI: 10.1360/sst-2019-0111. https://doi.org/10.1360/sst-2019-0111
2. Morton B. R., Taylor G. I., Turner J. S. (1956). Turbulent gravitational convection from maintained and instantaneous sources. Proceedings of the Royal Society of London. Series A 234(1196): 1-23. DOI: 10.1098/rspa.1956.0011. https://doi.org/10.1098/rspa.1956.0011
3. Wang Hongwei, Law Adrian Wing-Keung (2002). Second-order integral model for a round turbulent buoyant jet. Journal of Fluid Mechanics 459: 397-428. DOI: 10.1017/S0022112002008157. https://doi.org/10.1017/S0022112002008157
4. Lavelle J. W. (1997). Buoyancy-driven plumes in rotating, stratified cross flows: Plume dependence on rotation, turbulent mixing, and cross-flow strength. Journal of Geophysical Research: Oceans 102: 3405-3420. DOI: 10.1029/96JC03601. https://doi.org/10.1029/96JC03601
5. Jiang H., Breier J. A. (2014). Physical controls on mixing and transport within rising submarine hydrothermal plumes: A numerical simulation study. Deep Sea Research Part I: Oceanographic Research Papers 92: 41-55. DOI: 10.1016/j.dsr.2014.06.006. https://doi.org/10.1016/j.dsr.2014.06.006
6. He Zhiguo, Zhang Wei, Jiang Houshuo, Zhao Liang, Han Xiqiu (2018). Dynamic interaction and mixing of two turbulent forced plumes in linearly stratified ambience. Journal of Hydraulic Engineering 144(12): 04018072. DOI: 10.1061/(ASCE)HY.1943-7900.0001545. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001545
7. Lou Yingzhong, He Zhiguo, Jiang Houshuo, Han Xiqiu (2019). Numerical simulation of two coalescing turbulent forced plumes in linearly stratified fluids. Physics of Fluids 31(3): 037111. DOI: 10.1063/1.5087534. https://doi.org/10.1063/1.5087534
8. He Zhiguo, Lou Yingzhong (2019). Integral model for multiple forced plumes arranged around a circle in a linearly stratified environment. Physical Review Fluids 4(12): 123501. DOI: 10.1103/PhysRevFluids.4.123501. https://doi.org/10.1103/PhysRevFluids.4.123501
9. Fabregat Tomàs Alexandre, Poje Andrew C., Özgökmen Tamay M., Dewar William K. (2016). Effects of rotation on turbulent buoyant plumes in stratified environments. Journal of Geophysical Research: Oceans 121(8): 5397-5417. DOI: 10.1002/2016JC011737. https://doi.org/10.1002/2016JC011737
10. Fabregat A., Deremble B., Wienders N., Stroman A., Poje A., Ozgokmen T. M., Dewar W. K. (2017). Rotating 2D point source plume models with application to Deepwater Horizon. Ocean Modelling 119: 118-135. DOI: 10.1016/j.ocemod.2017.10.005. https://doi.org/10.1016/j.ocemod.2017.10.005
11. Lavelle J. W., Di Iorio D., Rona P. (2013). A turbulent convection model with an observational context for a deep-sea hydrothermal plume in a time-variable cross flow. Journal of Geophysical Research: Oceans 118(11): 6145-6160. DOI: 10.1002/2013JC009165. https://doi.org/10.1002/2013JC009165
12. Adams Ian, Di Iorio Daniela (2021). Turbulence properties of a deep-sea hydrothermal plume in a time-variable cross-flow: Field and model comparisons for Dante in the Main Endeavour Field. Journal of Geophysical Research: Oceans 126(9): e2020JC016638. DOI: 10.1029/2020JC016638. https://doi.org/10.1029/2020JC016638
13. Ezzamel A., Salizzoni P., Hunt G. R. (2015). Dynamical variability of axisymmetric buoyant plumes. Journal of Fluid Mechanics 765: 576-611. DOI: 10.1017/jfm.2014.694. https://doi.org/10.1017/jfm.2014.694
14. van Reeuwijk Maarten, Salizzoni Pietro, Hunt George R., Craske John (2016). Turbulent transport and entrainment in jets and plumes: A DNS study. Physical Review Fluids 1(7): 074301. DOI: 10.1103/PhysRevFluids.1.074301. https://doi.org/10.1103/PhysRevFluids.1.074301
15. Zhao Wei, Chen Sheng, Yang Junyi, Zhou Weichang (2023). Assessment of RANS turbulence models in prediction of the hydrothermal plume in the Longqi hydrothermal field. Applied Sciences 13(13): 7496. DOI: 10.3390/app13137496. https://doi.org/10.3390/app13137496
16. Wang Kanghao, Han Xiqiu, Wang Yejian, Cai Yiyang, Qiu Zhongyan, Zheng Xiaoquan (2023). Numerical simulation-based analysis of seafloor hydrothermal plumes: A case study of the Wocan-1 hydrothermal field, Carlsberg Ridge, Northwest Indian Ocean. Journal of Marine Science and Engineering 11(5): 1070. DOI: 10.3390/jmse11051070. https://doi.org/10.3390/jmse11051070
17. Zhou Weichang, Chen Sheng, Yang Junyi (2024). Dynamic characteristics of two-vent submarine hydrothermal plumes: A case study of Longqi hydrothermal field. Deep Sea Research Part I: Oceanographic Research Papers 204: 104225. DOI: 10.1016/j.dsr.2023.104225. https://doi.org/10.1016/j.dsr.2023.104225
18. Lemaréchal Cyprien, Roullet Guillaume, Gula Jonathan (2025). Hydrothermal plume near-field dynamics from LES and observations. Journal of Geophysical Research: Oceans 130(10): e2024JC022277. DOI: 10.1029/2024JC022277. https://doi.org/10.1029/2024JC022277
19. Lou Yingzhong, He Zhiguo, Han Xiqiu (2020). Transport and deposition patterns of particles laden by rising submarine hydrothermal plumes. Geophysical Research Letters 47(20): e2020GL089935. DOI: 10.1029/2020GL089935. https://doi.org/10.1029/2020GL089935
20. Chen YaNan, Lou YingZhong, He ZhiGuo, Wang YeJian, Qiu ZhongYan, Han XiQiu (2021). Flow fields and outputs of the hydrothermal plume at the Wocan-1 field based on an in-situ video. Scientia Sinica Technologica 52(11): 1705-1715 (in Chinese with English abstract). DOI: 10.1360/sst-2021-0041. https://doi.org/10.1360/sst-2021-0041
21. Chen Sheng, Lin Xin, Zhou Weichang, Yang Junyi (2026). A CFD-informed Gaussian model for deep-sea hydrothermal plume dispersion under near-bottom cross-flow. SSRN preprint (not peer reviewed). DOI: 10.2139/ssrn.7176861. https://doi.org/10.2139/ssrn.7176861