Carazzo, Kaminski and Tait (2008) showed that a Richardson-number-dependent (variable) entrainment closure raises the heat fluxes inferred for the TAG submarine hydrothermal field from observed plume-rise heights by a factor of about three relative to constant-entrainment estimates, but their submarine plume model computed density from a linearized equation of state with separate constant coefficients for temperature and salinity, explicitly leaving the effect of salt variations at high temperature unknown. The open problem is to quantify how the nonlinear, pressure-, temperature- and salinity-dependent thermodynamics of hot NaCl-seawater — including vent-fluid salinity anomalies produced by phase separation — modify plume buoyancy, neutral-buoyancy height, and the height-to-heat-flux inversion for high-temperature submarine hydrothermal plumes, and to bound the bias of the linearized formulation used in existing inversions, including whether the factor-of-three entrainment-driven revision of the TAG heat fluxes survives.

Background

Submarine hydrothermal plumes rise from 'black smoker' vents — seafloor hot springs at mid-ocean ridges that discharge mineral-laden seawater circulated through the hot crust, at temperatures up to about 400 degrees C — into the cold (about 2 degrees C), stably stratified deep ocean. Because the vent heat flux is extremely hard to measure directly at depth, a standard practice is to invert an easily observed quantity — the height at which the turbulent plume stops rising and spreads laterally at neutral buoyancy — for the source heat output, using an integral ('one-dimensional') plume model of the type introduced by Morton, Taylor and Turner (1956): the plume is described by cross-sectionally averaged ('top-hat') values of velocity, radius, and density deficit, and turbulent mixing with ambient water is parameterized by an entrainment coefficient αe\alpha_e that sets an inflow velocity proportional to the plume's mean vertical velocity. Speer and Rona (1989) and then Rudnicki and Elderfield (1992) adapted this framework to the TAG (Trans-Atlantic Geotraverse) hydrothermal field on the Mid-Atlantic Ridge, carrying the plume's temperature and salinity and computing its density from a linearized equation of state with separate, constant coefficients for temperature and salinity. Building on their setup, Carazzo, Kaminski and Tait (2008) showed that replacing the constant entrainment coefficient by a closure in which αe\alpha_e varies with the local Richardson number Ri=g′R/U2\mathrm{Ri} = g' R / U^2 (the ratio of buoyancy to inertia, with g′g' the reduced gravity, RR the plume radius, and UU its mean velocity) roughly triples the TAG heat fluxes inferred from observed rise heights, a revision with direct consequences for ridge-crest heat budgets. Integral to any such inversion is the equation of state (EOS), the relation ρ(T,S,P)\rho(T, S, P) giving seawater density from temperature, salinity, and pressure. At deep-ocean conditions the standard linearization ρ≈ρe−α(T−Te)+β(S−Se)\rho \approx \rho_e - \alpha (T - T_e) + \beta (S - S_e) — with thermal expansion coefficient α\alpha, haline contraction coefficient β\beta, and ambient reference values TeT_e, SeS_e — is adequate, because turbulent entrainment dilutes the plume to near-ambient temperature within the first tens of meters. The source fluid itself, however, sits at 300–400 degrees C and 20–40 MPa, where seawater thermodynamics are strongly nonlinear and approach the critical region: α\alpha and β\beta vary by orders of magnitude, the heat capacity peaks, and the linearization has no validity. Two further facts sharpen the question. First, vent-fluid salinity commonly deviates from ambient seawater because phase separation (boiling under pressure) in the crust produces both salt-depleted and salt-enriched (briny) end-member fluids. Second, mixing of hot saline fluid with cold seawater can, for sufficiently saline sources, produce mixtures denser than the ambient fluid — 'buoyancy reversal', analyzed by Turner and Campbell (1987) — while McDougall (1990) showed that with the nonlinear EOS a hot smoker's source buoyancy flux is not conserved: the far-field plume is governed by the conserved heat and salt contents weighted by the expansion and contraction coefficients at ambient conditions (the 'asymptotic buoyancy flux'). Carazzo et al. (2008) nevertheless carried out their TAG inversion with the linearized EOS of their Appendix B (Eq. B5), stating that the effect of salt variations at high temperature is unknown and therefore not accounted for — leaving unknown how much the inferred heat fluxes, already steeply sensitive to plume height through an approximately fourth-power relation, are biased by the density treatment. Quantitative thermodynamics for exactly this regime became available with the saline-thermal-fluid equations of Sun, Feistel, Koch and Markoe (2008), valid to 374 degrees C and 40 g/kg salinity, and the H2O–NaCl correlation formulae of Driesner and Heinrich (2007), so the question is no longer limited by data; it is limited by the absence of any analysis propagating these effects through a plume-rise inversion.

Problem Statement

The problem, as left open by Carazzo, Kaminski and Tait (2008) in their submarine hydrothermal application, is to determine quantitatively the effect of salt variations at high temperature on the buoyancy and rise of submarine hydrothermal plumes and on the vent heat fluxes inferred from them. Concretely: (1) Quantify how the plume's buoyancy flux, neutral-buoyancy height, and maximum rise height differ between the linearized density formulation ρ=1.041548−2.13×10−4(T−Te)+7.5×10−4(S−Se)\rho = 1.041548 - 2.13\times10^{-4}(T-T_e) + 7.5\times10^{-4}(S-S_e) used in their TAG inversion (Appendix B, Eq. B5) and the nonlinear, coupled temperature–salinity–pressure thermodynamics of hot saline seawater, conserving heat and salt separately, for high-temperature sources (of order 300–400 °C at ridge-axis pressures), including the near-critical regime where the thermal expansion and haline contraction coefficients vary strongly. (2) Quantify the effect of vent-fluid salinity anomalies produced by phase separation — both salt-depleted and salt-enriched end-members — on the buoyancy and rise of such plumes, including the conditions under which buoyancy reversal occurs when evaluated with a realistic equation of state. (3) Propagate these equation-of-state effects through the height-to-heat-flux inversion — with its steep, roughly fourth-power sensitivity of inferred flux to rise height — to establish how much the linearized treatment biases existing estimates, in particular whether the factor-of-three entrainment-driven revision of the TAG heat fluxes survives.

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