The acoustic scintillation method is applied to the investigation and monitoring of a vigorous hydrothermal plume from Dante within the Main Endeavour vent field (MEF) in the Endeavour Ridge segment. A 40 day time series of the plume's vertical velocity and temperature fluctuations provides a unique opportunity to study deep sea plume dynamics in a tidally varying horizontal cross flow. An integral plume model that takes into account ambient stratification and horizontal cross flows is established from the conservation equations of mass, momentum and density deficit. Using a linear additive entrainment velocity in the model (E = αUm + βU⊥) that is a function of both the plume relative axial velocity (Um) and the relative ambient flow perpendicular to the plume (U⊥) gives consistent results to the experimental data, suggesting entrainment coefficients α = 0.1 and β = 0.6. Also from the integral model, the plume height in a horizontal cross flow (Ua) is shown to scale as 1.8B1/3Ua−1/3N−2/3 for 0.01 ≤ Ua ≤ 0.1 m/s where B is the initial buoyancy transport and N is the ambient stratification, both of which are assumed constant.
 Geothermally heated fluids exit the seafloor at hydrothermal vents, which occur mostly at spreading centers located along Mid-Ocean Ridges. High temperature hydrothermal vents discharge mineral-rich water into the ambient environment. When the hot plume meets the cold ambient seawater, the dissolved minerals in the plume (mainly metal sulfides and anhydrite) precipitate as suspended particles. Depending on the relative composition of the particles, the plumes may look black, gray or white. Hydrothermal plumes emanating from high temperature focused vents can impose a significant impact on the local deep ocean environment. The heat and chemicals carried by the plumes support an abundant biosphere around the hydrothermal fields, and hydrothermal plumes can also help disperse organisms and chemicals within and beyond the local biological community [Roth and Dymond, 1989; Jannasch, 1995; Marsh et al., 2000]. In addition to the biological effect, hydrothermal plumes can also affect the local current and fluid structure and may even induce unique patterns of circulation around the vent field [Stommel, 1982; Thomson et al., 2003, 2005].
 The Endeavour Ridge segment has been under intensive study since the early 1980s. The ridge is like an elongated volcano with an axial valley that is 1 km wide and 10 km long with an average depth of 100 m (see Figure 1a). Within the axial valley there are 5 major vent fields (Mothra to the south, Main Endeavour, High Rise, Salty Dawg and Sasquatch to the north) [Delaney et al., 1992; Kelley et al., 2001]. Among the five vent fields, the Main Endeavour vent field (MEF) has approximately 21 venting edifices [Kelley et al., 2001], among which Dante is the largest and the most active. Dante is located in the northern part of the MEF. Recent measurements in 2008 show a height of 25 m with approximately ten high temperature black smokers on the top of Dante that have temperatures ranging from 325–338°C.
 Through mixing between hydrothermal plumes and ambient seawater, a significant amount of heat and geochemical constituents (from high temperature focused vents and diffuse flow fields) are transferred into the deep ocean from the sub-seafloor lithosphere (see reviews by Elderfield and Schultz  and Baker et al. ). Measurements of plumes are also important for the development of realistic models of hydrothermal plumes in the ocean (see for example three-dimensional modeling by Lavelle  and one-dimensional integral models by Carazzo et al.  and Speer and Rona ). A review linking buoyant hydrothermal venting to subseafloor processes and to interactions with the ambient ocean is summarized in Di Iorio et al.  and shows the importance of model/data comparisons. Here we describe a novel acoustic forward scatter technique for obtaining a long time series of the upward plume velocity to estimate the heat flux from a large sulfide structure in the MEF and to understand its interaction with ambient oscillatory currents using a one dimensional integral model.
 Temporal variability, especially tidal oscillations, have been observed in many hydrothermal systems for both diffuse flows [Little et al., 1988; Pruis and Johnson, 2004] and high temperature ‘black smokers’ [Fujioka et al., 1997; Kinoshita et al., 1998; Larson et al., 2007; Crone et al., 2010], which implies that there is a significant interaction between tides and hydrothermal systems. Previous studies attribute the tidal signals in temperature, chemistry and flow velocity of venting fluids to the bottom currents, tidal-loading effect and underlying geological events [Crone and Wilcock, 2005; Larson et al., 2007; Crone et al., 2010]. Given the unique characteristics of a ‘black smoker’ (high temperature, strong caustic effect), it is hard to get long-term measurements of the plume's temperature variability and flow velocity directly and simultaneously that would give a statistically reliable result representing the output of a venting edifice with multiple sources. The acoustic method described here integrates through the plume giving path-averaged measurements.
 To investigate the temporal variability of hydrothermal physical properties within the water column (temperature variability and vertical flow), we use an autonomous acoustic scintillation instrument described by Di Iorio et al.  and Xu and Di Iorio . Acoustic scintillation is a phenomenon in which the pattern of the modulation of the acoustic signal is evolving constantly due to the turbulence within the medium. By measuring the acoustic fluctuations passing through an isotropic and homogeneous turbulent medium, properties of the medium can be recovered through an inverse approach. Acoustic scintillation is a well-proven technology and has been applied in many geophysical flows during the past 30 years [Clifford and Farmer, 1983; Farmer and Clifford, 1986; Farmer et al., 1987; Di Iorio, 1994; Di Iorio and Farmer, 1994, 1998].
 When applied to the investigation of hydrothermal vents, path-averaged measurements of plume properties such as temperature variability and vertical flow are obtained. Acoustic scintillation is non-intrusive, since the transmitter and receiver are moored outside the plume during the measurement. Because of its path-averaged and non-intrusive characteristics, acoustic scintillation offers an alternative approach to the long-term monitoring of the integrated hydrothermal plume emanating from high temperature focused vents. Thus the major objectives for this paper are: (1) determine the relationship between the tidal current oscillations and the vertical velocity and temperature variability of the plume; (2) estimate the mound heat flux from Dante using the radially averaged vertical velocity measured by acoustic scintillation; (3) establish an integral model to simulate a hydrothermal plume's behavior in an environment with tidally varying horizontal cross flows and ambient stratification and use the model to quantify the entrainment velocity.
Section 2 is an introduction to the experimental approach including an introduction to the acoustic scintillation system and its deployment at the MEF. Section 3 gives a time series of the plume's vertical velocity and temperature fluctuations above Dante and an estimate of the total heat flux. The horizontal flow in the MEF is summarized in section 4 and the relationship between the hydrothermal plume and the horizontal cross flow within the axial valley is shown. In section 5 an integral model, based on an entrainment velocity that is dependent on flows along and perpendicular the plume's axis, is established to examine the behavior of a hydrothermal plume under a significant horizontal cross-flow. A discussion of model results is in section 6 and a summary of significant findings is in section 7.
2. Acoustic Scintillation System
 The self-contained (battery operated and internally logging) acoustic scintillation system was initially built in the early 1990s where it was first used to investigate hydrothermal vent flow from Hulk within the Main Endeavour Field in 1991 [Di Iorio et al., 2005]. Since then, the receiver was completely rebuilt incorporating digital signal processing boards, increased memory and a faster central processing unit (CPU). The deployment of the transmitter (TX) and receiver (RX) relative to Dante is shown in Figure 1b and was placed using the remotely operated vehicle (ROV) Jason II in 2007. The separation of the transmitter and receiver mooring was approximately 91 m and aligned along the axial valley of the Endeavour ridge segment. To ensure insonification of the hydrothermal plume at 20 m above the Dante structure, the transducers were designed to be horizontally omnidirectional and have a vertical beam width of 10 deg. Two transducers with a vertical separation of 15.5 cm make up the transmitter and receiver arrays. The vertically separated transmitters and receivers create both parallel acoustic rays (two transmitters, two receivers) and diverging rays (one transmitter, two receivers) as shown in Figure 2a.
 The center frequency of the transducers is 307.2 kHz with a bandwidth of 30 kHz. The transmitter uses a pulsed monochromatic sinusoidal signal with a pulse width of 0.1 ms. The pulses are separated by 100 ms giving a pulse repetition rate of 10 Hz. The delay between the signals from the two transmitters is 25 ms. The parameters of the system are summarized in Table 1. In order to obtain a long-term time series, the instrument was programmed in burst sampling mode with 15 min of data every hour.
Table 1. Parameters of the Acoustic Scintillation System
Path length (L), orientation
91 m, 19°T
Mean soundspeed (2155 m)
Fresnel scale ( )
 The in-phase (I) and quadrature (Q) components are obtained by sampling the received signal 1/4 carrier cycle apart every two cycles (150 kHz digitization rate) [Xu and Di Iorio, 2011]. The amplitude of the received pulse as a function of travel time is then calculated as . A window of 64 samples (0.43 ms) is used to capture the digitized pulses. The maximum amplitude above a noise threshold within each receiver window is detected and saved into the receiver's memory along with two adjacent samples on both sides of the maximum. In this way, each data ping contains 5 consecutive amplitude samples (as a function of travel time) with the maximum in the middle. A quadratic fit on the five data points is then performed to pinpoint the amplitude ( ) of the received signal. A despiking routine was then developed in order to eliminate abrupt amplitude changes that can result from system noise or lost tracking of the received signal. The log-amplitude fluctuation χ is then calculated from the despiked amplitudes as where is the mean amplitude averaged over 15 minutes of data (9000 data pings).
3. Vertical Velocity and Temperature Variability
 Acoustic waves propagating through a hydrothermal plume (as depicted in Figure 2a) are dominated by the refractive index fluctuations (η) within the medium that are a result of temperature fluctuations as discussed by Xu and Di Iorio . Using space-time coherence methods and calculating the variance of the log-amplitude fluctuations (σχ2), the flow perpendicular to the acoustic propagation direction and the variance of the temperature fluctuations (σT2) are obtained respectively.
 If the two line-of-sight acoustic rays between the transmitter(s) and receivers are close enough, the fluctuation pattern observed at the downstream receiver (R1) will be nearly identical to that of the upstream receiver (R2), except it will be shifted in time by a lag τ. This is because the turbulent eddies are embedded in the mean flow and their evolution during the time they cross the two line-of-sights is small (‘frozen turbulence’ assumption). The time lag τ can be measured by analyzing the space-time cross-covariance function of the received log-amplitude signals. This function, assuming weak scattering, is given by Clifford and Farmer  and is modified by Xu and Di Iorio  for path dependent refractive index fluctuations:
It is maximum when |ζ − Wτ| = 0 assuming Taylor's ‘frozen turbulence’ hypothesis [Tennekes and Lumley, 1972]. In this equation integration is over all refractive index wave numbers κ and over the acoustic path length from the transmitter at x = 0 to the receiver at x = L aligned along the axial axis and along the dominant horizontal flows; W is the mean vertical velocity of the plume; ζ is the vertical separation of the two acoustic rays shown in Figure 2a, ζ = z for parallel rays, ζ = zx/L is the separation between two diverging rays along the path and z = 15.5 cm; k is the acoustic wave number; J0 is the zero order Bessel function. The path dependent three-dimensional isotropic and homogeneous refractive index spectral density is [Xu and Di Iorio, 2011],
which corresponds to axisymmetric buoyant plumes with a centerline refractive index fluctuation of and bη ∼ 5 m is the e-folding distance from the centerline at 20 m above the Dante structure.
 The log-amplitude variance is then defined as
and the variance of the temperature fluctuations along the centerline, assuming isotropic and homogenous turbulence, is then
 Sample correlation functions for parallel and diverging rays are shown in Figure 2b from which the time lag τ can be calculated from the delay to the peak. The maximum time-lagged correlation function for diverging paths is always greater than that for parallel paths (for this highly turbulent environment, diverging rays may have an advantage over parallel rays due to the reduced vertical separation in the center of the path). Thus the time delay for diverging rays is half that for parallel rays because of this reduced vertical separation. The path averaged vertical velocity of the plume is then calculated using diverging paths and W = z/2τ.
 The path averaged vertical velocity and the temperature fluctuations of the hydrothermal plume of Dante (measured at 20 m above the orifice) are shown in Figure 3. Each 15 min burst of data each hour was broken up into three 5 min intervals from which the cross correlation and variance was derived. A quadratic fit to the cross correlation peak then gives the delay τ from which the vertical velocity is calculated. The time averaged vertical velocity and temperature variance are thus 0.14 m/s and 0.03 (°C)2 respectively. Tidal current oscillations can be observed in these results and the velocity range relative to the mean velocity is practically 100%. A variance preserving power spectral density for the vertical velocity is shown in Figure 4 and oscillations are forced at the dominant M2 tidal frequency. The 4-day oscillation observed by Cannon and Thomson  in the current meter measurements around the Endeavour segment is also identified though not as significant.
Figure 5 shows an expanded section with tidal heights superimposed. The tidal height is derived from a regional tidal model for the west coast based on the tidal inversion solutions developed by Egbert et al.  and Egbert and Erofeeva  and available online at http://volkov.oce.orst.edu/tides/otis.html. The tidal periodicity (dominated by the M2 tidal harmonic) is observed in the both vertical velocity and the temperature variance measurement. A maximum in the measurements occurs when water is moving from high level to low level (during ebbing tide) and reaches a minimum when water is moving from low level to high level (during flooding tide). This phenomenon suggests a certain interaction between the tidal currents and the vertical velocity of the hydrothermal plume.
 From the time averaged vertical velocity (W = 0.14 m/s) and mean temperature (T = 2.4°C, obtained by ROV Jason II while hovering in the plume at 20 m above the orifice), the heat flux can be approximated,
where the ambient temperature at 20 m above the orifice is Ta = 1.9°C and ρcp = 4 × 106J/(m3°C). The radius of the plume at 20 m above the orifice is approximately R = 7 m. The heat transport of the plume from Dante is then estimated as
4. Horizontal Flow Within the Axial Valley
 The tidal characteristics observed in the acoustic scintillation measurements indicate a significant interaction between horizontal flow and the hydrothermal plume. During the past two decades, currents within the axial valley have been studied and documented extensively [Thomson et al., 1990; Allen and Thomson, 1993; Thomson et al., 2003, 2005; Berdeal et al., 2006]. The sulfide structures within the MEF vary in size from 5–20 m tall and the steep talis wall on the western side causes significant spatial variations in the flow 0–20 m from the base of the sulfide mounds. As no current meters were deployed during the scintillation measurement we make use of current meter data collected by R. Thomson (Institute of Ocean Sciences) in 2001, as it is located in the central axial valley to the northeast of the MEF at a depth of 2145 m (19 m from the seafloor); see Figure 1a for the current meter mooring location. Three months of data with a sampling interval of 20 min was collected.
 Tidal current oscillations are prevalent within the axial valley. According to Allen and Thomson  and Thomson et al. , the oscillatory currents are nearly rectilinear above the valley but become amplified and clockwise rotary as they approach the valley's crests. Tidal current oscillations are then attenuated within the valley and the M2 oscillation is the most dominant frequency. Figure 6a shows the tidal current oscillations extracted using the tidal harmonic analysis program T-Tide developed by Pawlowicz et al. . From a principal component analysis the major axis is aligned at 2.3°T and shows that the tidal current oscillations within the central valley are nearly meridional. From the T-Tide results, the M2 tidal current accounts for nearly 45 percent of the total variance in the 2001 data.
 According to Thomson et al.  and Allen and Thomson , the residual flow within the axial valley is near steady and convergent. At the southern and central parts of the valley, the residual flow is predominantly northward and strongest (5 cm/s) while at the northern part of the valley, the residual flow becomes southward and weakest (1 cm/s). The convergent flow within the axial valley is believed to be induced by turbulent entrainment of the hydrothermal plumes [Thomson et al., 2003, 2005]. The northward residual flow extends to the central valley due to the shallow topographic saddle at the northern end of the valley and more intensive hydrothermal activity within the southern and central valley [Thomson et al., 2003]. Figure 6b shows a stick plot of the daily averaged residual flow within the central valley using 36 hour low-pass filtered data. The flow is northward with an average velocity of 3.12 cm/s and rarely reverses to the south.
 Hourly tidal currents within the axial valley during the acoustic scintillation deployment in 2007 can be extrapolated from the 2001 current meter data using the T-Tide harmonic analysis program. The total horizontal flow aligned along the acoustic axis is thus estimated by adding an along axis residual flow of 3.12 cm/s. Note that we have no way of knowing how the residual flow varies in time during our measurements and so a first order approximation is to assume constant residual flow. Figure 7 shows that the horizontal flow reaches a maximum during flooding tide and is small during the ebbing tide; dotted horizontal lines represent one standard deviation from the residual flow of 3.12 cm/s (shown as a dashed line).
Figure 8 shows an expanded comparison of the vertical velocity and temperature fluctuations with the estimated horizontal flow. A significant negative correlation (r ∼ −0.55 and −0.53 respectively) is observed. The plume's vertical velocity reaches a minimum when the horizontal flow is maximal and vice versa. The temperature fluctuations σT2 also reaches a minimum when the horizontal flow is maximal and vice versa. From this comparison, it is hypothesized that the plume's interaction with the horizontal cross flow within the Main Endeavour Field causes enhanced entrainment. According to the results of laboratory experiments carried out by Fan , entrainment can be enhanced by the ambient horizontal cross-flow for a turbulent buoyant plume thus affecting the rise speed and dilution.
5. Integral Model
 Deep sea hydrothermal vents are sources for turbulent buoyant jets and plumes. Driven by the gravitational buoyancy force, hydrothermal plumes can rise up to hundreds of meters above the orifice [McDuff, 1995]. Ambient ocean water is entrained into the plume through its ascent, which makes the plume diluted and cooled. Finally, an equilibrium level is reached when the density deficit between the plume and ambient environment is offset by the entrainment. The plume overshoots the equilibrium level owing to its remaining momentum and then falls back and spreads laterally at its neutral buoyancy level [Turner, 1986].
 The integral model used to describe the time-averaged plume behavior was first developed by Morton et al.  (MTT model). The MTT model is derived from the conservation equations for volume, momentum and density deficit. The essence of the MTT model is Taylor's entrainment hypothesis, which states that the velocity of the inflow of diluting water across the edge of the turbulent shear flow is proportional to a characteristic velocity in the jet/plume at the level of the inflow. The proportionality constant (α) is defined as the entrainment coefficient. In the case of a hydrothermal plume emanating into a quiescent environment (without horizontal cross-flows), the characteristic velocity is taken as the plume's axial velocity.
 The effect of the horizontal cross flow, however, is not incorporated into the MTT model. Due to the existence of the horizontal cross-flow, the trajectory of the plume will bend toward the downstream direction of the flow because of both the low-pressure wake-like region established behind the plume and the entrainment of horizontal momentum from the cross-flow [Fan, 1967]. In addition to the horizontal cross-flow, stratification of the ambient environment also has an important influence on the plume's behavior (a hydrothermal plume will eventually reach its terminal height where neutral buoyancy is reached in a stratified environment). Therefore, an extension to the MTT model is developed to make it suitable for modeling the rise of a hydrothermal plume in a stratified medium under significant horizontal cross-flows.
5.1. Scaling and Dimensional Analysis
 In general, the behavior of a hydrothermal plume can be represented as a buoyant jet or plume in a stratified medium with a spatially uniform horizontal cross-flow that varies with the tides. The hydrothermal plume is affected by both the initial momentum and buoyancy transport and is thus more of a momentum driven jet or a buoyancy driven plume depending on the distance from the orifice. Therefore the mean properties of a hydrothermal plume should be functions of the initial volume transport Q, initial specific momentum transport M, and the initial specific buoyancy transport B, which are defined by
where D is the diameter of the vent's orifice, W is the plume's exit velocity, ΔT = T0 − Ta0 is temperature anomaly at the orifice with Ta0 the ambient temperature at the depth of the orifice (z = 0) and αt ∼ 10−4°C−1, is the thermal expansion coefficient for water Turner and Campbell .
 Dante has approximately 10 major high temperature vents [Delaney et al., 1992] that were sampled in 2007 and 2008 from which the plumes coalesce to form a single integrated plume at several meters above the edifice (based on video and acoustic images as shown in Figure 2a). The average temperature measured using Alvin's high temperature probe in these black smoker vents is T0 = 325.7°C. The total cross sectional area of all the orifices sampled is then used to represent the base cross-section of the integrated plume using a radius 0.20 m (D = 0.40 m) and the average exit velocity of all black smoker vents sampled is W ∼ 0.3 m/s [Germanovich et al., 2009].
 The ambient stratification is calculated by a linear regression on conductivity-temperature-depth (CTD) profiles taken outside the Main Endeavour vent field. The vertical structure of ambient temperature, salinity and density (as shown in Figure 9), are then approximated as
where z is the height above the edifice of Dante (at a depth of 2175 m). Since dρa/dz = −1.247 × 10−4 kg/m4, then the buoyancy frequency is
It is assumed that changes in the ambient stratification due to mixing of the plume will be small and so N is assumed constant.
 Substituting T0, Ta0, D and W into equations (7) to (9) gives Q = 0.038 m3/s M = 0.011 m4/s2 and B = 0.012 m4/s3. Based on these constant initial conditions, several characteristic length scales are derived following Fischer et al. :
 1. LQ = Q/M1/2 = 0.35 m: for z ≫ LQ, the turbulence along the plume's axis reaches stationary decay; for z ≪ LQ the flow establishment zone is where the turbulence is at a non-stationary state due to the existence of large scale eddies and the mean properties of the buoyant jet/plume are influenced by the geometry of the orifice.
 2. LM = M3/4/B1/2 = 0.32 m: for z ≫ LM (plume zone), a purely buoyancy driven plume exists; for z ≪ LM (jet zone), a purely momentum driven jet exists.
 3. zB = B/U3 = 24 m, U = 0.08 m/s: for z ≫ zB (bent over zone), the plume is bent horizontally under the effect of the horizontal cross-flow; for z ≪ zB (vertical zone), the horizontal current has not yet imposed an appreciable effect on the plume and the plume still maintains its vertical shape.
 4. zmax = 3.8B1/4/N3/4 = 208 m: the terminal height of the plume in a stratified quiescent environment.
 The terminal height zmax may not be an accurate estimation due to the omission of the ambient horizontal cross-flow which can reduce the terminal height of a plume [Rona et al., 2006]. Similarly, zB may be an overestimate due to the omission of ambient stratification in a cross flow. With the existence of ambient stratification, a hydrothermal plume will continue entraining ambient seawater and carry it upward against the ambient stratification. In such a way, the plume will slow down and bend over more quickly than in an environment with a homogeneous density distribution. In summary, at z = 20 m above the orifice (where the acoustic scintillation measurement was conducted), we can assume that the plume becomes purely buoyancy driven (z ≫ LM) and can be influenced by the maximal horizontal flows (z ∼ zB) while the turbulence within the plume reaches stationary decay (z ≫ LQ).
5.2. Conservation Equations
 The integral model is developed following Fan , and is based on the conservation equations of mass, momentum, dissolved tracers and density deficit. The assumptions are: 1) within the range of variation, the density of the fluid is assumed to be a linear function of salinity and temperature; 2) the flow is fully turbulent and molecular transport can be neglected (molecular viscosity and diffusion are neglected). This assumption is valid as the Reynolds number (WL/ν) is ∼3 × 104 at the orifice, for a typical high temperature focused vent (with diameter 6 cm and exit velocity of 0.5 m/s); 3) axial turbulent transport is small compared with axial advective transport. According to Papanicolaou and List , turbulence adds approximately 16% to the specific momentum transport; 4) the mean properties of the plume are axisymmetric. This assumption may not be valid as 3D models of plumes in cross flows show significant asymmetry in the cross section [Lavelle, 1997; Di Iorio et al., 2012]; 5) the velocity and density deficit profiles are Gaussian functions at all cross sections normal to the plume's axis within the region z ≫ LQ (self-similarity). Again this assumption may not be valid when cross flows exist; 6) a modified Taylor entrainment hypothesis is applied, which relates the inflow velocity at the edge of the plume to the axial velocity within the plume and to the ambient cross-flow; 7) the plume is in steady state because the risetime (the time taken by the plume to reach its terminal height) is much shorter than the dominant tidal period.
 A schematic plot of a hydrothermal plume in a uniform cross-flow with linearly varying stratification is shown in Figure 10. A curvilinear coordinate system is used where s is the axial distance above the orifice along the plume's trajectory and r is the radial distance away from the axis in a perpendicular cross-section. In addition, Us is the axial velocity of the plume, Ua is the ambient horizontal cross-flow, U0 is the plume's exit velocity and Up(s) is the plume's residual axial velocity (Up = Us − Ua cos θ). We expect that the vertical shear in the ambient currents will be small because the model measurements start at 25 m above the base of Dante. The diameter of the orifice is D and FD is the drag force imposed by the horizontal flow on the plume and acts perpendicular to the plume's axis.
 The conservation of mass for the plume, transformed into cylindrical coordinates is
where the over bar denotes Reynolds averaged plume properties that are dependent on r and z. Multiplying by 2πr and integrating along the plume's radial cross section gives
Applying Leibniz integral rule gives
The second term on the left goes to zero because the mean vertical velocity ( ) evaluated at r = b(z) is at the edge of the plume where ambient water starts, and thus . Equation (16) is further transformed into a curvilinear coordinate system shown in Figure 10 using the following geometric relationships,
The vertical distance z can be replaced by an axial distance s and can be replaced by axial velocity Us giving
in which the left hand side is the axial gradient of the mass transport passing the cross-section perpendicular to the plume's axis. The right hand side represents the entrainment from the ambient environment where represents the horizontal mass flux through the perimeter of the plume toward the center and will be linked to an entrainment velocity (E) to be defined in section 5.3.
 The conservation of vertical momentum (per unit volume) as defined by the Navier-Stokes equation in cylindrical coordinates is
where the terms on the right hand side are respectively the pressure force in the z direction per unit volume induced by a drag force acting normal to the edge of plume by the ambient flow (see Figure 10) and the buoyancy force. Multiplying by 2πr and integrating along the plume's radial cross section gives the conservation of vertical momentum transport,
where the drag force is at the perimeter of the plume and acts over a thickness Δz of the plume. Further simplification as before leads to
The left hand side is the change of the transport of vertical momentum along the axial direction. The first term on the right hand side is the drag force imposed by the horizontal cross-flow, and the second term is the buoyancy force.
 The Navier-Stokes equation for the time rate of change of horizontal momentum (per unit volume) in cylindrical coordinates is
where the term on the right hand side is a pressure force in the x direction induced by a drag force acting normal to the edge of plume by the ambient flow per unit volume (see Figure 10). Following the same method as before this equation reduces to
where the left hand side is the change of the axial transport of horizontal momentum along the axial direction. On the right hand side, the first term describes the transport of ambient horizontal momentum into the plume per unit height since is the horizontal momentum flux on the boundary of the plume. This term will also be linked to an entrainment velocity (E) described in section 5.3. The last term is the drag force acting on the plume from the ambient ocean.
Equation (20) introduces a density anomaly between the ambient fluid and the plume, and so another equation is needed to define how this variable changes. Using the advection-diffusion equation, the conservation of a tracer C which is for example potential temperature (T) or salinity (S), in cylindrical coordinates gives
Integrating and transforming into curvilinear coordinates gives
The left hand side of equation (24) is the change of concentration transport along axial direction while the right hand side describes the entrainment of the tracer from the ambient environment into the plume.
 An equation for the transport of dissolved tracer concentration anomalies is obtained by first multiplying equation (14) by the ambient dissolved tracer concentration Ca (which is independent of r)
Integrating and transforming into curvilinear coordinates gives
which is the conservation equation for density deficit. This equation states that any changes along the plume axis are due to axial advection.
 Thus, the conservation equations for mass, vertical and horizontal momentum, as well as density deficit are summarized in equations (17), (20), (22) and (31) respectively.
 The conservation equations can be simplified based on the ‘self-similarity’ assumption which states that the radial distribution profiles of the plume's mean properties all have similar Gaussian distributions within the region where z ≫ LQ (above the flow establishment zone). As LQ is small we make the assumption that the flow is fully developed at the source and so the radial profiles can be written as
where Um and ηm are the center axis (r = 0) mean residual properties of the plume and are functions of z; bu is defined where the axial velocity Um decreases by a factor of 1/e so that the radius of the plume is defined as following Papanicolaou and List . Based on the results of laboratory experiments generalized by Papanicolaou and List , the radius of the dissolved tracer concentrations is greater than that of the axial velocity and the ratio of proportionality is 1.2. The density deficit is assumed to have an identical distribution profile to that of the tracer concentrations.
where α and β are the entrainment coefficients in the tangential and normal directions respectively.
 According to Fan , the drag force FD is perpendicular to the plume's axis acting on the edge of the plume at a radius (see Figure 10). It is defined as
where CD is the drag coefficient typically of order 1, the area over which the force acts is A = 2πb(s)Δz, and the velocity component (Ua sin θ) is perpendicular to the plume's axis on the boundary.
 Substituting these parameterizations into the conservation equations reduces the problem from a set of partial differential equations to a set of ordinary differential equations that can be solved numerically for: mass,
and density deficit,
In deriving the equations above, the radius of the plume in the integral limit is assumed to be . Thus the conservation equations for mass (36), vertical momentum (37), horizontal momentum (38) and density deficit (39) together with the geometric equations
form 6 equations with 6 unknown variables as a function of s: [x, z, bu, Um, θ, ηm].
5.4. Numerical Solutions
 In order to solve the simplified set of conservation equations, they need to be transformed into forms suitable for numerical calculation. By expanding the derivatives on the left hand sides, the equations are transformed into the following matrix equation,
in which bi, ui, λi and κi (i = 1:4) are the coefficients of the derivatives of the unknown variables bu, Um, θ, and ηm respectively, and Ri are the corresponding right hand sides of the conservation equations. Thus
A final form suitable for numerical calculation is solved symbolically using MAPLE. The MATLAB routine ode45 is used to solve the differential equations with an absolute error of 10−9 and with an interval of 0.01 m for the axial distance s.
5.5. Comparison with Experimental Data
 The initial conditions at z = 0 are set to
The initial velocity Um(0) represents the average velocity measured locally at black smokers on Dante using independent rotary devices developed by Germanovich et al.  and the source radius ( m) represents the cumulative size for all the vigorous black smokers that were sampled on Dante. According to Bischoff and Rosenbauer , the density of seawater with 3.2% NaCl at pressure 220 bars and temperature 350°C is 667 kg/m3, which is assumed to be the density of the plume at the orifice (ρ0). The initial conditions thus correspond to a Gaussian mound heat flux of 14 MW when integrated over the area. The model is forced with a horizontal cross-flow varying from 0 to 0.1 m/s, which reflects the range of the current oscillations shown in Figure 7.
 The plume's radially averaged vertical velocity is computed at z = 20 m above the orifice,
so that it can be compared to the acoustic path averaged measurement of vertical velocity. Results are plotted as a function of the horizontal cross-flow in Figure 11a for entrainment coefficients α = 0.1 and β = 0.6 and drag coefficient (CD = 1.7), which give the best fit to the data after multiple model runs were carried out with varying coefficients. These values were also in the range proposed by Fan  and Hoult and Weil . The acoustically derived vertical velocity (which represents a spatial average through the plume) is also plotted against the extrapolated horizontal currents. The comparison between the measurements and the integral model are quite remarkable although some of the variance in our acoustic measurement is probably due to incorrect estimations of the oscillatory flow. Future acoustic measurements of the plume vertical velocity will need to be made with simultaneous current meter measurements of the horizontal flow to confirm these results.
6. Discussion of Results
 Plume characteristics such as the rise height as a function of the horizontal cross flow is shown in Figure 11b with comparison to scaled distances summarized by Turner . For a quiescent environment (no ambient cross flows), plumes will rise under their buoyancy until they reach their height of neutral buoyancy governed by the background stratification N. The scaling for this case goes as 3.8B1/4N−3/4 where the constant 3.8 is determined empirically. By measuring the plume height and background stratification this equation can be used to quantify source conditions. Given the initial conditions used to force the integral model we see that this scaling is consistent with the model output. When there is a cross flow the plume begins to bend and is cooled by enhanced entrainment and thus the vertical velocity decreases and hence the rise height will decrease. At a flow of 0.06 m/s the rise height of the plume, where the vertical velocity goes to zero, is approximately 100 m compared to 200 m when there is no cross flow (see Figure 11c). As noted by Fischer et al.  and Middleton  the rise height of plumes in cross flows scales as B1/3Ua−1/3N−2/3 and the scaling coefficient that fits with our integral model result is 1.8 rather than the suggested 3.8 coefficient. For constant initial conditions and stratification, the range of validity for this scaling is from 0.01 ≤ Ua ≤ 0.1 m/s.
 The radial averaged vertical velocity (〈W〉) as a function of height is compared for two different horizontal cross flows (U = 0, 0.06 m/s) in Figure 11c. The radially averaged vertical velocity decreases exponentially through the plume's ascent due to the loss of buoyancy while mixing with entrained ambient ocean water and finally reaches zero at roughly 200 m above the orifice during slack and at 100 m during a strong cross flow. This level can be regarded as the terminal height of the plume where the vertical momentum goes to zero. The vertical variation indicates that the horizontal cross-flow has a major effect on the vertical velocity. Superimposed is the radially averaged theoretical scaling of (B/z)1/3 using the empirical coefficient of 3.8 determined by Papanicolaou and List  (the factor of 0.5981 is derived from an analytical solution to the radially averaged Gaussian function in equation (45)). Using this scaling at 20 m above the top of Dante, the radially averaged vertical velocity is 0.19 m/s, which compares very well to our acoustic measurement of 0.14 m/s. Because of stratification this scaling starts to depart from the integral model result at approximately 50 m above the top of Dante.
 The radius of the hydrothermal plume at 20 m above the orifice as measured by acoustic imaging shown in Figure 2, is approximately 7 m. This would correspond to a 1/e radial distance of bu = 5 m. According to Figure 11d, the plume radial distance (bu) increases linearly following the theoretical expansion rate of for zero ambient cross flow. This equation is obtained by solving equation (36) to first order (i.e. neglecting terms with ηm and writing Um ∝ z−1/3, E = αUm and bu = γz, where γ is the expansion rate). When cross flows exist the expansion occurs at a much greater rate ranging from 5.5 to 10 m at 20 m above Dante.
 This research documents for the first time a long time series measurement of vertical velocity and temperature fluctuations from deep sea hydrothermal vent plumes using a novel acoustic forward scatter method referred to as acoustical scintillation analysis. The advantages of this acoustical method are that the vertical velocity is obtained directly without any geometrical correction and that a path averaged result is obtained which is advantageous for monitoring hydrothermal volume and heat flux for a major edifice over the long term. Using a mean vertical velocity and an approximate cross sectional area of the plume at 20 m above the Dante structure, a heat flux of 43 MW is estimated. This value compares well to the range of estimates obtained by Bemis et al.  and Ginster and Mottl  for sulfide structures in the MEF.
 Our measurements also show that the variations in vertical velocity are much greater than the velocity variations expected from tidal loading as modeled by Crone and Wilcock  and thus entrainment of ambient waters in the plume is the dominant cause for the variability. Our integral modeling results and comparison with data show that entrainment of ambient water into a buoyantly driven hydrothermal plume is enhanced under the influence of ambient tidal current oscillations. When the horizontal flow is weak (during the ebbing tide), less ambient ocean water is entrained into the plume. In such a case, the vertical velocity of the plume is larger and the temperature is hotter thus temperature fluctuations will increase (as was shown in Figure 8). When the horizontal flow is strong (during the flooding tide), more ambient ocean water is entrained into the plume. In such a case, the vertical velocity of the plume is slower and the temperature is cooler (more like ambient conditions) resulting in reduced temperature fluctuations. Consequently, the vertical velocity and temperature fluctuations of the plume show significant tidal variations that are out-of-phase with the ambient current oscillations.
 According to the modeling result that best describes our data, several conclusions are reached and generalized as follows: 1) entrainment coefficients that best fit the model to the measurements are α = 0.1 and β = 0.6 with a drag coefficient of CD = 1.7; 2) the rise height of the plume scales with the horizontal flow as Ua−1/3, 0.01 < Ua < 0.1 m/s while buoyancy and stratification are kept constant and the best fit scaling coefficient is 1.8; 3) the expansion rate of the plume is increased from ∼ 0.2 m/m as a result of increased entrainment by the horizontal cross flow.
 This research was funded by the National Science Foundation CAREER grant OCE0449578 and any opinions, findings, or conclusions expressed here are those of the authors and do not necessarily reflect the views of the funding agency. Many thanks go to the captain and crew of the RV Atlantis for careful deployment of instrumentation. Special thanks to the pilots and crew of the DSV Alvin and ROV Jason II for positioning the acoustic scintillation transmitter and receiver arrays. Trent Moore (SkIO) was a key partner in carrying out all the measurements described here and we are grateful for his enthusiastic participation. Thanks to L. Germanovich (GaTech) for many helpful discussions. Two anonymous reviewers and the Associate Editor Edward Baker provided helpful comments that improved the paper considerably.