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Constraints on the characteristics of the hydrothermal release resulting in
a megaplume can be derived from models of turbulent buoyancy-driven convective
fluids [e.g.,
*Morton et al.*, 1956; *Turner*,
1973]. Models of this kind have already been employed for hydrothermal plumes
[*Converse
et al.*, 1984; *Middleton
and Thompson*, 1986; *Little
et al.*, 1987]. Such models propose that upward convection of the hydrothermal
plume is forced by the density deficiency of the hot hydrothermal fluid with
respect to the surrounding water. The deficiency in density is diminished over
distance from the source by entrainment of ambient water into the plume. Eventual
complete quenching of the density deficiency in a stably stratified environment
limits the height of rise of the plume. The observed height of rise can be used
to infer the value of the heat flux at the vent. In the model presented below
we specifically apply these concepts to the megaplume observations. The computations
are based on an idealized plume source configuration and plume ascent, however,
so the model results must be taken as suggestive rather than definitive.

The model assumes that the turbulent entrainment of ambient fluid is proportional
to upward plume velocity, that profiles across the plume are similar at all
heights (here top-hat profiles), and that the fluid is incompressible. Previous
applications of such models to hydrothermal systems [*Converse
et al.*, 1984; *Middleton
and Thompson*, 1986; *Little
et al.*, 1987] have addressed plumes arising from individual vents and
have thus been confined to axisymmetric geometry, but for reasons to be indicated
below, we choose to model a linear source and planar plume. Because of the near-radial
symmetry of the observed plume cap, however, the source length must be small
with respect to the diameter of the plume cap. End effects of a finite length
source are not part of the model. On the basis of the development of *Morton
et al*. [1956], the conservation equations for a two-dimensional plume
are as follows: for mass,

(2)

for momentum,

(3)

for heat,

(4)

and for salinity,

(5)

where *b* is the plume width (e.g., the distance across the linear source
fissure), *w* is the upward velocity, *z* is the vertical coordinate (positive
upward), is the entrainment coefficient, *g* is the
acceleration of gravity, is the density in the plume, *S*
and *T* are the salinity and temperature, respectively, *C _{p}* is the specific heat of salt water at constant
pressure, and

(6)

Notice that the model employs an equation conserving heat rather than temperature
because *C _{p}* (as does ρ) depends nonlinearly on

Because of the assumption of incompressibility, potential temperatures and density referenced to the source depth (~2300 m) were used throughout. For *T* < 40°C and *S* < 42‰, densities at 230 bars were taken from the formula for density of seawater given by *Gill*
[1982]; densities at *T* > 40°C were interpolated from tables given by *Potter and Brown* [1977] and *Burnham et al*. [1969]. The specific heat of seawater at 230 bars varies from
4100 J kg^{−1} °K^{−1} to 6600 J kg^{−1} °K^{−1}
over the temperature range 200°C to 350°C [*Bischoff and Rosenbauer*, 1985]. Below 200°C, *C _{p}* = 4100 J kg

The model equations were consequently integrated using initial values of *T*_{0}
= 350°C and *S*_{0} = 34.63‰, the ambient salinity at 2300 m, for a range of mass fluxes. In all calculations,
the coefficient of entrainment of ambient water into the buoyant plume was given a value of α = 0.1 based on laboratory experiments.
For example, for two-dimensional plumes a range for α of 0.055 to 0.11 has been suggested [*Kostsovinos and List*, 1977; *Wright and Wallace*, 1979], the lower value for a pure jet and the upper value
for a pure plume. For axisymmetric plumes, α values of 0.09 to 0.11 have been employed [*Turner*, 1973; *Middleton and Thompson*, 1986].

The axisymmetric analogs of (equations (2)–(6)) were evaluated first. To fulfill the observed megaplume 1 rise of 1025 m, vent diameters of approximately 4 m (vent exit velocity of 5 m/s) to 9 m (exit velocity of 1 m/s) would be required. This result did not change when the initial temperature of the vented fluids was varied over the range 325°–375°C because the nearly counterbalanced dependencies of density and specific heat on temperature makes the initial heat flux virtually constant. We judged such a spatially restricted source of fluids to be unlikely and turned instead to a model in which the source was a linear one (equations (2)–(6)), a geometric analog to seafloor fissures.

Equations (2)–(6) were consequently integrated under varying conditions of vent width and fluid exit velocities. The calculated heights of rise were found to depend on the heat flux at the vent (Figure 8) just as in the case of buoyancy-conserving, linearly stratified regimes [*Turner*, 1973]. To force a plume to the observed megaplume 1 rise height of ~1025 m requires a source heat flux of approximately 3.5 × 10^{8} J/s per meter of vent fissure (Figure 8). The associated volume flux per unit length of the vent is 0.22 m^{3} m^{−1} s^{−1}.

**Figure 8. Relationship between plume rise height and the vent heat flux per meter of
vent fissure for a linear plume with an entrainment coefficient of 0.1. The megaplume 1
rise height required a heat flux of ~3.5 × 10 ^{8} J m^{−1} s^{−1}.**

The total excess heat in the megaplume, *Q _{t}* , is the product of the length of the line source, ℓ, the duration of venting, τ, and the excess heat flux per unit length of source,

(7)

where the reference temperature is the ambient temperature at height *h*. Given a field-measured value of *Q _{t}* and a
model-estimated value for

Because there are no heat sources within the plume, *H _{h}* must be the sum of

(8)

where *T _{e}*(

The model constraint on τℓ thus takes the following form:

(9)

For megaplume 1, *Q _{t}* is estimated to be 6.7 × 10

Using a value of 4000 m d for ℓτ, the total volume and heat released during the megaplume 1 event were 7.6 × 10^{7} m^{3} and 1.2 × 10^{17} J, respectively. (Both of these figures are about twice the preliminary estimates of *Baker et al.* [1987].) The comparative source strength for a megaplume 2-sized event is 1900 m d with a total volume and heat content of 3.6 × 10^{7} m^{3} and 5.8 × 10^{16} J, respectively. From considerations of the megaplume 1 geometry and suspended particle population presented earlier, reasonable bounds on the duration of
the causative venting event are 2–20 days. Consequently, from (9) with *h* = 700 m, the length of a venting fissure would be 2000–200 m.

Profiles of properties in the plume over the interval 1–1000 m from the source are shown in Figure 9 for the case of megaplumelike venting with an exit velocity of 2.5 m/s. The plume width remains relatively narrow for the first several hundred meters. Beyond that it rapidly enlarges. This is consistent with the observations that show the plume cap to begin several hundred meters from the seafloor. On the basis of laboratory studies of linear plumes in stratified environments, the thickness of the plume cap ought to be about one half the maximum height of the plume [*Wallace and Wright*, 1984]. In the case of these observations that would mean a plume cap thickness of ~500 m, not unlike the observations of ~700 m.

**Figure 9. Vertical profiles of temperature ( T), excess salinity (ΔS), plume width (b), and vertical velocity (w)
for a linear plume with an exit velocity of 2.5 m/s.**

Upward velocity in the plume (Figure 9) remains greater than 0.8 m/s to 700 m above bottom, after which it rapidly declines toward zero. A rough estimate of the rise time to maximum height is thus 20–30 min. These calculated velocities indicate that the plume would have no difficulty in lifting even the largest grains we observed in the filtered samples because their settling velocities are ~2 × 10^{−3}
m/s. Temperature in the plume (Figure 9) falls off rapidly with height, decreasing to ~30°C at 10 m and ~6°C at 100 m. Excess salinity rises to ~0.1‰ near the top of the plume, reflecting the transport of salt in the plume from the more saline water at depth.

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