A three-dimensional time-dependent convection model is used to describe circulation and property fields in rotating, stratified, and moving fluids near a point source of heat. The study context and, consequently, model scales are those of chronically discharging hydrothermal vent fields found at submarine ridge crests. Hydrothermal plumes having distinctive thermal and chemical anomalies have been observed to rise several hundreds of meters above the deep-sea floor before being advected away by background cross flows typically of magnitude 1-4 cm s. The model is used to study effects of rotation and indicate differences in plumes with respect to variation of subgrid-scale turbulence intensity and cross-flow strength. Counterrotating vorticity () couplets in all three coordinate directions develop in the lower plume stem at startup and follow the plume to the level of neutral buoyancy; for a nonrotational case ( = 0), patterns resemble those previously found for jets injected into homogenous cross flow. Ambient fluid entrainment into the convecting column is primarily from the upstream side, but deflection of background flow around both sides of the rising column is the root of the relative vorticity (_{z}) couplet in the lower plume. Turbulence intensity within the buoyant region of the plume and/or globally controls smoothness and temporal variability of distal nonbuoyant plume distributions, allowing or preventing oscillations of potential temperature, , for example, at background buoyancy frequency, N. Over the range of turbulent mixing studied, rise height of plumes did not change appreciably, but breadth of plumes, counterintuitively, increased for decreasing turbulent mixing strength. Increasing cross-flow strength, U, bends model plumes such that rise height U^{-0.4}. For the two largest values of cross flow, for which R, the ratio of maximum vertical velocity to U, took values of 2.8 and 1.0, plumes showed evidence of bifurcation.
Convection from near-point sources of buoyancy occurs at the deep-sea floor where hot hydrothermal fluids are steadily vented. All along ridge crest spreading centers, buoyant plumes rich in chemicals and particles ascend several hundred meters into the water column while being bent over and advected away by ambient currents [Baker et al., 1995]. In those benthic environments, cross-flow strength, fluid turbulence, planetary rotation, background stratification, and source buoyancy flux all combine to determine dispersion patterns of heat and other constituents emanating from hydrothermal vents. In this paper, a model of buoyant plumes in rotating, stratified cross flows appropriate to conditions of ridge crest hydrothermal plumes is described. The focus of results is on rotation, intensity of turbulent mixing, and cross-flow strength as factors that determine downstream plume attributes.
Hydrothermal plumes are one of many examples in the natural environment where a localized source of buoyancy causes materials to rise into an overlying fluid and forced circulation to develop. In all cases, turbulence and, in many cases, rotation and cross flow are important to the dynamics of ensuing plumes. Plumes from industrial stacks, terrestrial volcanoes, forest fires, and oceanic thermal and waste water discharges provide instances of buoyant plumes or buoyant jets occurring in a cross flow. A related problem in the industrial realm involves injection of one fluid as a jet into another as occurs, for example, with fuel injection into jet engines [e.g., Claus and Vanka, 1992]. Numerical model results for jets entering unstratified, nonrotating cross flows [Sykes et al., 1986] provide useful points of comparison for results here. Jets and buoyant jets are different from buoyant plumes in that jets have momentum at source points independent of forces of buoyancy caused by density anomalies. Buoyant plumes, which lack initial upward momentum, are reasonable idealizations for many hydrothermal discharges; initial upward momentum found at a typical hydrothermal vent orifice must give way to buoyancy forces within vertical distances very much smaller than plume rise height.
Model approaches to plumes of this kind are generally classified as integral or numerical. The seminal paper for the integral method is that of Morton et al. [1956], who closed one-dimensional vertical mass, momentum, and heat conservation equations by making fluid entrainment into the rising plume stem at each level proportional to plume upward velocity. The resulting set of ordinary differential equations can be readily solved to address maximum rise height and dilutions within the plume stem. On the other hand, integral models provide no information on distributions in the plume cap region or in regions exterior to the plume where recirculations develop. The integral approach has since been extended and heavily used to describe a variety of jets and plumes, including those rising into cross flows [e.g., Slawson and Csanady, 1967; Middleton and Thomson, 1986; Davidson, 1989]. The literature on jets, buoyant jets, and plumes in cross flows is extensive; reviews by List [1982], Hanna et al. [1982], and Weil [1988] are good sources of information on fundamentals of integral models.
Unlike integral models, more computationally taxing numerical models allow plume cap distributions and induced circulation surrounding the plume to be computed. No uniform entrainment coefficient need be assumed, though as in all fluid dynamical models a turbulent mixing closure assumption of some kind is required. Numerical models of plumes have tended to be two-dimensional [e.g., Lilly, 1962] or quasi-three-dimensional (3-D) [Golay, 1982; Zhang and Ghoniem, 1994] because of computational demands. On the other hand, models of thunderstorms [e.g., Klemp, 1987] and cumulus cloud formation [e.g., Smolarkiewicz and Clark, 1985] have shown the benefits of full 3-D approaches for some time, and those convection problems bear some commonality with point source plume problems. In some sense the oceanic convection problem is easier: source point is stationary, initial conditions are more easily prescribed, and fluid can be assumed to be incompressible.
Hydrothermal plumes result from episodic and chronic discharges of chemically anomalous, heated water at the seafloor along crustal spreading centers. Plumes are of two main types. Megaplumes appear to be the result of short-lived discharge events [Lavelle, 1995]. These plumes are kilometers in diameter and hundreds of meters in thickness and rise many hundreds to a thousand or more meters above the seafloor [e.g., Baker et al., 1995]. The plumes are characterized by thermal anomalies as large as ~0.3°C [e.g., Baker et al., 1995] and by their distinctive chemical signatures [e.g., Massoth et al., 1995]. The few observed have been associated with linear crustal ruptures brought on by episodic magmatic intrusions that rise from depth to the seafloor leaving telltale lava flows [Embley et al., 1995; Fox et al., 1995]. More commonly observed are hydrothermal plumes that result from the continuous release of heat and chemicals stripped from the underlying rock by the hot fluids. Fluxes from chronic sources individually are many orders of magnitude less than those that result in megaplumes, but in aggregate over a ridge crest segment and over a year's time, flux is comparable. Chronic discharge can emanate from sulfide chimneys, e.g., black and white smoker chimneys, more diffusely from sulfide mounds typically meters on a side [e.g., Delaney et al., 1992], or it can percolate from seafloor fissures. Heat release from single vents is estimated to range from several to tens of megawatts [e.g., Schultz et al., 1992] and from vent fields from several hundreds to several thousand megawatts [Baker et al., 1996]. Because heat release is continuous in these cases, background current flows play a major role in plume development.
Observations of effects on the thermal, chemical, and particulate environments resulting from chronic discharges at ridge crests are numerous and are summarized by Baker et al. [1995]. Plumes from chronic sources rise several hundred meters into the water column before reaching density equilibrium. Measured temperature anomalies in the nonbuoyant plume region are typically <0.05 °C [Baker et al., 1995]. Such plumes are also made distinctive by their particle [e.g., Feely et al., 1992], metal [e.g., Massoth et al., 1994], and gas content [e.g., Lupton and Craig, 1977; Mottl et al., 1995]. Transects made along the strike of ridge crests show that plumes from chronic sources can extend longitudinally for many tens of kilometers, as plumes from separate sources coalesce in background currents, which are often along ridge [Cannon et al., 1991]. The model in this paper is intended to address features of a plume from a single, somewhat broad chronic discharge site, e.g., a sulfide mound.
Until now, there have been no models for chronic discharge of hydrothermal fluids that could address the three-dimensional distributions of plume properties in the nonbuoyant region and connect them with source discharge. The challenge of dealing with open boundary conditions when background flows are superimposed on the convection has been one reason for this absence. But the need for a plume model incorporating background flow is clear. Observations typically show that background currents advect hydrothermal plumes away from their source points. Furthermore, without lateral advection, thermal equilibrium cannot be achieved in any fixed volume enclosing a chronically discharging hydrothermal vent.
The plan of the paper is to set out the design of the numerical model, show results for a reference case, and then look at variations in results for changes in single model parameters. The reference case is calculated for source buoyancy flux, cross flow, and background stratification like those expected for hydrothermal discharges on the Juan de Fuca Ridge (JDFR) in the northeast Pacific. The reference case is then compared with a nonrotational case, with results of two experiments in which turbulent mixing is reduced, and with results of two experiments where cross flow is doubled and quadrupled. The emphasis of this initial report is on the general character of model plumes rather than on specific details of their dynamics.
The model describes a thermal plume rising into a temperature- and salinity-stratified background environment marked by steady, vertically sheared cross flow. The fluid is incompressible and the response to the localized buoyancy source nonhydrostatic. Together with the equation of state for seawater [Fofonoff and Millard, 1983], the underlying model equations in the Boussinesq approximation are
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
(8) |
(9) |
(10) |
where t is time; consists of u, v, and vertical w (positive upward) components; p is pressure; is density; (1028.11 kg m) is a reference density; g (9.81 m s) is acceleration of gravity; is the rotation vector; and is the unit vector in the vertical direction. Potential temperature, , and salinity, S, depend on rates of discharge of heat Q_{}H, via Q_{} = Q_{H} /(C_{P}), and salt Q_{S}, where C_{P} (4200 J kg °C) is the specific heat of seawater. Equations (1)-(3) are statements of conservation for momentum, mass (given fluid incompressibility), and heat (or salt). Chemical tracers have conservation equations analogous to (3). Retention of all terms in the vertical force balance in (1) makes the model nonhydrostatic. The viscosity tensor A and diffusivity tensor K are time- and space-dependent; diffusive transport is down gradient.
Equations (4)-(10) constitute the turbulent mixing submodel. Horizontal and vertical components of mixing tensors A_{H or V} (and K_{H or V} via (10)) are each composed of two parts: an isotropic (A_{I}) mixing coefficient and either horizontal (A_{HMIN}) or vertical (A_{VMIN}) background terms (equations (4)-(5)). The isotropic mixing coefficient A_{I} (equation (6)) is made to depend on fluid shear (S_{ij}, here given in Cartesian form, (9)), on shear Richardson number (R_{I} (8)), on Prandtl number (P_{R} (10)), and on the turbulence length scale (l_{S}) and Smagorinsky (C_{S}) constants (equation (6)). If R_{I} were zero, (6) would reduce to a form originating with Smagorinsky [1963]; it stems from an assumed local balance between shear production and turbulence dissipation in the turbulence kinetic energy (TKE) equation. When buoyancy production or dissipation is added to the local TKE balance, the R_{I} dependent factor becomes part of the formulation [Lilly, 1962]. Advection and diffusion of TKE are generally found to represent small contributions to the TKE equation balance when explicitly evaluated, and they are here implicitly set to zero. Mixing terms have no explicit dependence on rotation. The reason is that in forming TKE, terms involving sum to zero.
Smagorinsky-Lilly mixing, with or without the R_{I} dependent factor, is in widespread use in atmospheric calculations [e.g., Clark and Farley, 1984; Mason, 1989]. Subgrid-scale mixing of this type is frequently used in large eddy simulations (LES) of both engineering and geophysical natures [e.g., Reynolds, 1990; Galperin and Orszag; 1993; Smagorinsky, 1993; Mason, 1994]. Following the nomenclature of Reynolds [1990], Wyngaard [1990] notes that models using this kind of turbulence closure should actually be typed very large eddy simulation (VLES). VLES implies that, although resolved scales include much of the eddy energy, model resolution is not high enough that smallest resolved scales fall within the subinertial range of turbulence where details of dissipation ought not to matter.
Background mixing terms (equations (4)-(5)), presumed to be small, have here been added to A_{I} to ensure a modest amount of mixing in regions without shear, e.g., away from the plume region, where A_{I} will be zero. Other closure models are in current use as well. Sykes et al. [1986] used a full TKE equation to determine q, the square root of twice TKE; turbulent mixing was then made proportional to the product of q and l_{S}. In both the present, Sykes et al.'s, and many other approaches, l_{S} has a size that is imposed rather than calculated. The model summarized by (1)-(10), but without cross flow, has been studied in the context of laboratory convection experiments as a way of investigating the value of the subgrid turbulence closure subcomponent [Lavelle and Smith, 1996].
For hydrothermal venting, convective flow is forced by a source of heat at interior points along the lower boundary of the region of interest. Steady background cross flow is forced by a constant horizontal pressure gradient, P. Without point source heating, the steady background velocity profile u_{BKG} = (u_{BKG}, v_{BKG}) results from an Ekman horizontal force balance (equation (1)):
- P/ - 2 × u_{BKG} + d/dz [A_{z} d(u_{BKG})/dz] = 0 | (11) |
and boundary conditions u_{BKG} = 0 (z = 0, bottom) and du_{BKG}/dz = 0 (z = h, top). With uniform P in the y direction, nonzero , and A_{z} profiles that increase toward the seafloor as described later, flow is oriented in the x direction except near the bed. Profiles u_{BKG} and v_{BKG }(e.g., Figure 1a) were used to initialize velocity fields and maintain upstream velocity boundary conditions thereafter. Without source heat (and/or salt), u_{BKG }and v_{BKG }are maintained numerically over time as solutions to (1)-(10) as they must be. In reference experiment 21 (Table 1), outside the boundary layer, u_{BKG} was 1.5 cm s, while v_{BKG} was nonzero only within that layer (Figure 1a). Mean currents of that magnitude are typical at ridge crest depths on the JDFR [e.g., Cannon et al., 1991].
Figure 1. Idealized environmental profiles for u, v, A_{Z}, , S, and . Profile shapes and magnitudes nominally represent conditions in the lowest 300 m of water column above the seafloor spreading center of the Juan de Fuca Ridge, northeast Pacific. The A_{Z} profile controlling the boundary layer thickness is hypothetical, but the shape and magnitude are generally consistent with A_{Z} values deduced from benthic ocean measurements at other sites.
Table 1. Parameter Values for the Convective Plume Experiments
Exp. No. | 2_{Z}, s |
C_{S} | A_{H
MIN}, cm s |
U, m s |
h,
m |
21 | 1.03 × 10 | 0.2 | 100 | 0.015 | 300 |
22 | 0 | 0.2 | 100 | 0.015 | 300 |
23 | 1.03 × 10 | 0 | 100 | 0.015 | 300 |
26 | 1.03 × 10 | 0.2 | 10 | 0.015 | 300 |
24 | 1.03 × 10 | 0.2 | 100 | 0.03 | 200 |
25 | 1.03 × 10 | 0.2 | 100 | 0.06 | 150 |
The vertical force balance without heating is hydrostatic: dP/dz = g_{BKG}, where _{BKG }is background density; w_{BKG} = 0. The variable p in (1) thus represents the sum of P(z), P(x,y), and a perturbation pressure p(x,y,z) caused by heating. In actual practice, hydrostatic terms are first subtracted from (1), with the consequence that in resulting equations, p represents the sum of P and p and becomes = ( - _{BKG }).
Background profiles (Figures 1a-1b) hinge on the form given A_{z}. To simplify the initialization process and make background profiles independent of boundary layer shear, of (4) was set to zero. Vertical turbulent mixing coefficients are thus fixed from the start of calculations. Because vertical mixing rates in the ocean are small in the interior (A_{ZMIN}) and increase (to A_{ZMAX}) as z 0 [e.g., Garrett, 1979], the profile for A (Figure 1a) was given the form
(12) |
where h_{B} determines the boundary layer thickness. The value of A_{z} declines to within 1% of A_{ZMIN} by a distance of 5 h_{B} of the seafloor (z = 0). Values of A_{ZMIN} = 1 cm s, A_{ZMAX} = 100 cm s, and a viscosity height scale length h_{B} = 10 m were taken as nominally representative of benthic ocean conditions.
Without heating, background profiles _{BKG} and S_{BKG} must be time-invariant and laterally uniform. Since w_{BKG} = 0, (3) in that case leads to
d/dz [K (d_{BKG}/dz)] = 0 | (13) |
or once integrated over depth interval [z,h],
K [d_{BKG}/dz] = F_{0} | (14) |
where F_{0 }is a constant the value of which is set by conditions at z = h. Over depth interval [0,h], profiles of K_{z} and _{BKG} are thus linked; where K grows large near the seafloor, d_{BKG}/dz must adjust appropriately. Equations (13)-(14) imply that to maintain a steady _{BKG} profile, the gradient flux of _{BKG} must be balanced by a vertical flux of opposite sign. Otherwise, background profiles would continually change, making more difficult the resolution of small temperature anomalies caused only by hydrothermal heating. Analogous arguments apply to S_{BKG}.
Above a boundary layer, background profiles of and S were taken to be linear: = (1 + ) and S = S_{0 }(1 + ) , where , S, , and were given the values 1.6339°C, 34.616, 1.8925 × 10 m, and -1.5751 × 10 m and where is height above the sea floor. These values are taken from linear least squares fits of hydrographic profile data from the JDFR over the 2100-2400 m depth range, profiles judged least affected by hydrothermal influences. Given those profiles and (14), F_{0} = _{0}K (z = h) and F_{0S} = _{2}SK (z = h). Since A(z = h) = 10 m s (equation (12)) and P_{R} = 1.0 (equation (10)), F_{0} = 3.1 × 10 °C m s and F_{0S} = 5.5 × 10 m s.
Integrating (4) with the A profile of (12) for P_{R} = 1.0 results in the background profile _{BKG} for 0 < z < h:
(15) |
having a well-mixed boundary layer (Figure 1b), the height (~50 m) of which is determined by h_{B}. In (15), K represents (K_{ZMAX} - K_{ZMIN}). A like equation can be written for S_{BKG}. For z >> h_{B}, the second term of (15) is negligible. As with velocity, _{BKG} and S_{BKG} profiles (Figure 1b) are used to initialize fields and provide upstream boundary profiles. Note in Figure 1 that where gradients of and S are large, velocity shear is small, and vice versa. This situation suppresses shear instabilities that might otherwise develop. It follows that = - 1000 has a similarly well-mixed boundary layer (Figure 1b). Buoyancy frequency, N, over depths 2100-2350 m is 7.9 × 10 s.
A resolved Ekman boundary layer serves these purposes. It allows for directional shear of flow that can influence plume dispersion. It reduces flow speed in the boundary layer, flow that can affect entrainment of fluid into the plume stem, particularly on the downstream side. Finally, a resolved boundary layer is particularly important for plumes the B of which is not great enough to cause the plumes to rise distinctly above the seafloor. Diffuse low-temperature sources of hydrothermal heat are instances of that situation [Trivett and Williams, 1994].
Equations (1)-(10) were integrated over a rectangular domain with boundaries open in x, cyclic in y, and fixed in z. Specifically, conditions were
Lateral inflow boundary, x = X_{1 }
u = u_{BKG} v = v_{BKG} w = 0 = _{BKG} S = S_{BKG}. |
(16) |
Lateral outflow boundary, x = X
u_{XXX} = v_{XX} = w_{XX} = _{X} = S_{X} = 0. | (17) |
Lower boundary, z = 0
u = v = w = 0 K_{z} [/z] = F_{0} K_{z} [S/z] = F_{0S}. |
(18) |
Upper boundary, z = h
u/z
= v/z
= w = 0 K_{z} [/z] = F_{0} K_{z} [S/z] = F_{0S}. |
(19) |
Front and back boundaries, y = Y and y = Y
(20) |
Boundary conditions on p in the x and z directions were taken in the manner of Harlow and Welch [1965]. Cyclic conditions were used in the y direction for all variables (u, v, w, , S, and p) for two reasons: so the Ekman boundary layer of the background flow would be uniform across the region of calculation and so plumes could pass though y boundaries as if transparent. Cross-flow strength_{ }and width of the solution domain were always large enough that the downstream plume never filled the domain side to side, though in some cases the plume passed through y boundaries. Boundary openness in the x direction was essential to allow passage of heat and tracers out of the domain, which if prevented would disallow any thermal equilibrium state from being reached. X direction boundary conditions are those suggested by Johansson [1993]. In these experiments, no absorbing layers [e.g., Clark and Farley, 1984] were used to damp waves.
As with background profiles, length scales, B, and background were chosen with regard to conditions for chronic hydrothermal plumes at the JDFR. Equations were solved in a Cartesian domain 640 × 320 × 300 m when U was 1.5 cm s, but domain height was reduced to 200 and then 150 m for increasingly larger U_{0 }(Table 1). Domain depth h was chosen for each U so that plumes penetrated to ~ 0.5 h. x, y, z, t were 5, 5, and 9.4 (6.2 or 4.7) m, and 15 s, respectively. N, N_{y}, N_{z}, and N were 128, 64, 32, and 2000 (or 5760).
The source region was 10 × 10 m and was centered at the horizontal location (x,y) = (-105, 0) m; source lateral dimension will hereafter be represented by D. Individual hydrothermal vents have smaller surface cross sections, but separate vents can be clustered in fields [Ginster et al., 1994], or the hydrothermal source may be diffuse, like the sulfide mounds described by Schultz et al. [1992]. The intention here was to model plumes not from a single vent, but from a small composite venting source or vent field. Resolving a single small vent and its regional plume is beyond present computational resources. Model source area was thus chosen to be representative of typical vent field and sulfide mound surface areas. Despite having an extended area, the source is nonetheless point-like in that rise height, h_{RISE}, is very much greater than D.
Source heat flux, Q, was set at 1.3 × 10 J s (13 MW), a value chosen to cause maximum plume rise of ~200 m under reference experiment conditions. Since buoyancy flux B = gQ//C_{P}, where is the thermal expansion coefficient (7.3 × 10°C), B was fixed at 2.1 × 10 m s. Buoyancy flux density was consequently 2.1 × 10 m s.
The value assigned Q_{H} is consistent with the wide range of heat fluxes measured for single vents when extrapolated to a small vent field. For 18 vents on the southern Juan de Fuca Ridge, Bemis et al. [1993] estimated an average heat flux of 0.1-3.1 MW per vent. Ginster et al. [1994] indicated an average heat output of 3.1 MW for single high-temperature vents on the southern JDFR and higher values on the ridge's northerly Endeavor segment, where hydrothermal plumes can reach 300 m above the seafloor. There their estimated heat flux density of 0.139 MW m would, over a source area of 10 × 10 m, give a Q very much like the 13 MW heat flux value used in these calculations. Schultz et al. [1992] estimated a heat flux of 58 MW from a 4 × 5 m sulfide mound at Endeavor.
Calculations were made for a site latitude of 45°N. With the nonvertical component of rotation set to zero, the rotation vector (, , ) was (0, 0, 5.14 × 10 s); an example of the effect of nonzero [Garwood et al., 1985] on a hydrothermal plume is given by Lavelle [1995]. Mixing parameters were typically A_{HMIN} = 10 m s and A_{VMIN} = 10^{-4 }m s. C_{S} was set to 0.2, and P_{R}, the ratio of turbulent viscosity to diffusivity, was given the value 1. Mixing length, l_{s}, was made equal to (xyz)^{1/3} [e.g., Reynolds, 1990]. Values for l_{S}, C_{S}, and P_{R} are not uniquely assignable, but values here are in ranges commonly used. Sensitivity of results to these parameters can be found, in part, in studies of axisymmetric plumes by Lavelle and Baker [1994] and of plumes from line segment sources by Lavelle [1995].
Time and length scales pertinent to the development of convective plumes from point and extended sources in rotating but otherwise quiescent background environments may be found in the recent work of Jones and Marshall [1993], Maxworthy and Narimousa [1994], and Speer and Marshall [1995]. For point sources and under stratified and depth-unlimited conditions, the external set of three variables, B, f, and N, allows two length scales to be derived. The first, well known from nonrotating conditions, is l_{N} = (B/N)^{1/4} [e.g., Turner, 1973]. By replacing N by f, a second length scale is uncovered: l_{f} = (B/f^{ 3})^{1/4} [e.g., Speer and Marshall, 1995] or l_{ROT} = (B/f^{ 3})^{½}, where B in the last equation is buoyancy flux density rather than total buoyancy flux [Maxworthy and Narimousa, 1994]. Atmospheric and laboratory observations for plumes rising in nonrotating environments without cross flow have established that maximum rise height, h_{RISE}, scales like l_{N}: h_{RISE} ~ 3.75 l_{N} [e.g., Hanna et al., 1982]. An example in which l_{ROT} was found to be relevant comes from rotating tank studies of Maxworthy and Narimousa [1994] on brine convection. They found that once a brine from a distributed (i.e., nonpoint) source reached a fluid depth of the order of 12.7 l_{ROT}, convection transitioned from a three-dimensional turbulent condition to one dominated by descending vortical columns. Length scales such as l and l_{ROT} help organize experimental, observational, and occasionally numerical [e.g., Speer and Marshall, 1995; Lavelle and Smith, 1996] results, but in all cases the coefficient of scaling must be determined from data.
Previous modeling work on hydrothermal megaplumes, which are caused by short-lived thermal releases associated with episodic tectonic events [e.g., Embley et al., 1995], have confirmed the utility of those scales in such cases. Speer and Marshall [1995] used l_{f} and the timescale for Coriolis effects to be important, f^{ -1}, to scale rotational velocities in the counterrotating vortices that should develop during megaplume formation. Lavelle and Baker [1994] and Speer and Marshall [1995] found that l scales rise height as it does in the nonrotating case under typical benthic ocean conditions, and Lavelle [1995] has shown that the timescale for megaplumes reaching h_{RISE} is ~ 4N. The fact that megaplumes form quickly permits the modeling assumption of a quiescent background.
In the case of chronically discharging hydrothermal plumes, the assumption of a quiescent background environment is no longer tenable. Furthermore, these more typical hydrothermal plumes are not products of ephemeral sources of heat. They do not produce bottom-detached lenses of water, the geostrophic adjustment of which Gill [1981] and McWilliams [1988] have studied. Consequently, one must not expect that such plumes are associated with a Burger number ~1 nor expect that the description of bent-over plumes should involve length scales l and l_{f}, as is made clear below.
In the case of continuous discharging plumes in a cross flow, the set of external variables from which length scales can be constructed is increased from three to four: the variable U must be added to the previous set. Generalized length scales that result from the expanded set are B^{(1+) }N^{ }U ^{(-3-4)} and B^{(1+) }f^{ } U^{(-3-4)}. When = -1, advective length scales L_{NADV} = U/N and L_{fADV} = U/f emerge. When = -3/4, l_{N} and l_{f} are recovered. But can take other values. Atmospheric and laboratory observations of plume in a cross flow show that l_{N} (i.e., = -1) is not an appropriate scaling for plume rise height, for example. Those data (and entrainment theory, e.g., Middleton [1986]) point to a value for that is more nearly -2/3, so that rise heights of plumes in cross flows scale like l_{CROSS} = [B/(UN)]^{1/3} [e.g, Hanna et al., 1982], not like [BN]^{1/4} as in cases without background flow. For plumes in a cross flow, it is only through observations that the coefficient of scaling and indeed even the value of can be found. Neither l or l_{f} have demonstrated roles in the scaling of cross-flow plume results.
Transverse width of the plume is an important measure of the likelihood of counterrotating vortex pair development seen in studies of plume formation in quiescent backgrounds [e.g., Speer, 1989]. But it is clear that the width of a plume transverse to the cross-flow direction cannot scale like L_{NADV,} L_{fADV,} l or l_{f}, except in the limit U 0: plumes having the same source B grow narrower with increasing U, but none of those four scales permit plumes with that U_{0 }dependence. The width must be inversely proportional to some power of cross-flow velocity, perhaps scaling as l_{CROSS}. The same arguments can be made that l and l_{f} play no role in setting plume height. For the cross-flow magnitudes studied in this paper, plume width above the source is always much less than l_{f}, the size at which one might expect the Coriolis force to cause counterrotating vortices in the manner previously predicted for plumes without background flow [Lavelle and Baker, 1994; Speer and Marshall, 1995]. Balancing vertical and horizontal mass flux at the top of the plume stem in simple calculations leads to that conclusion too. For the smallest U_{0 }studied in the paper, cross flow overwhelms upstream density-driven flow in the plume cap above the source. With U larger than the density-driven upstream velocity in the plume cap, the plume bends forward. With U and B typical of chronic hydrothermal discharge conditions, the cross-flow carries plume mass injected to the level of neutral buoyancy away at a rate that prevents a cross-flow dimension of the plume to approach l_{f} there.
An interesting transition to the axisymmetric plume case should occur in a sequence of plumes as U 0. Then one can expect the plume to first develop an anvil shape, with some upstream progression of the plume and some widening of the plume in the transverse direction. At even smaller U, the plume must begin to take on all the conditions of plumes described in earlier papers, including a counterrotating vortex pair. U, of course, drops out of the scaling arguments in this limit and l and l_{f} prevail. The study of that transition from cross flow to axisymmetric plumes has not been undertaken, nor have enough cross-flow experiments been run that the process of scaling confirmation analogous to those of Speer and Marshall [1995] or Lavelle and Smith [1996] is possible now.
Momentum equations were centered differenced in space in energy conservation form and leapfrogged in time. An Asselin [Asselin, 1972] filter having = 0.15 was applied each time step to control temporal mode splitting. Integrations involved solving a 3-D Poisson equation for p [Harlow and Welch, 1965] on a staggered grid each time step using direct method solvers HS3CRI and HS3CRT (R. Sweet, National Bureau of Standards, Boulder, Colorado, 1985). Transport equations were forward time differenced. Their diffusion terms were centered differenced in space, while advection terms were upstream differenced but corrected each time step for unwanted numerical diffusion using the procedure of Smolarkiewicz [1983] and Smolarkiewicz and Clark [1986]. Even in the presence of strong advection, upstream differencing with correction can maintain relatively sharp property gradients like those found at plume stem walls, while maintaining positive definiteness of calculated quantities. The last attribute proves useful in eliminating the occurrence of unphysically low property anomalies.
The accuracy of the model has been examined in the following ways. Requisite conservation of mass, momentum, heat, salt, and energy was carefully checked under a variety of forcing, as was symmetry (or antisymmetry) of all field variables under symmetric forcing. Conditions for the independence of results from model grid size were examined by Lavelle and Baker [1994] using the axisymmetric realization of the model. Results on sensitivity to the Smagorinsky coefficient and Prandtl number and on the model relationship of plume rise height to buoyancy flux B_{0} are found in the same place. Additional sensitivity experiments, specifically for rise height dependence on buoyancy flux and buoyancy frequency N, were conducted for a line segment hydrothermal source rising into a quiescent background environment [Lavelle, 1995]. That analysis demonstrated that the rise height of the model scales as B^{1/3 }in a quiescent background setting, as would be expected for that source configuration. Model results on the scaling of plume rise height with cross-flow strength are also encouraging. Atmospheric observations show plume rise height h_{RISE} depending on the cross-flow velocity U as U^{-}, where is ~1/3 [Hanna et al., 1982]. As demonstrated later, this model shows a rise height dependence on cross-flow velocity of U^{-0.4}, with a coefficient of proportionality of similar magnitude to atmospheric cases.
Additional comparisons of model results to laboratory and field observations have been made. The utility of the Smagorinksy-Lilly subgrid-scale turbulence formulation was examined [Lavelle and Smith, 1996] by comparing model with laboratory results [Fernando and Ching, 1993] for a brine convection in a rotating tank. The importance of self-generated turbulence during convection has long been recognized [e.g., Priestly, 1956], and those numerical experiments in conjunction with laboratory results have reconfirmed the view that time- and space-dependent turbulence closure is essential. Furthermore, this model, under conditions of cross flow in a nonrotating environment, produces flow patterns comparable to the numerical results of Sykes et al. [1986] and to experimental results they cite, as will be described more fully below. Finally, hydrographic profiles from this model for regions affected by chronic hydrothermal sources have the form and magnitude of perturbed hydrographic profiles observed in the field (J. W. Lavelle et al., Effects of deep ocean hydrothermal discharge on near-source hydrography: Surrogate field studies with a convective plume model, Pacific Marine Environmental Laboratory contribution 1735, National Oceanic and Atmospheric Administration, Seattle, Washington, 1996).
Model content vis-a-vis the simpler entrainment model of Morton et al. [1956] was examined by computing the effective entrainment coefficient from axisymmetric convective plume results [Lavelle and Baker, 1994]. Horizontal velocities at the stem wall were divided by the average vertical velocity at the same height in the stem, and results were profiled as a function of height from the buoyancy source. Entrainment profiles so derived showed the entrainment coefficient having a magnitude comparable to that customarily used with integral theory ( ~ 0.1). Contrary to the assumption of entrainment theory the entrainment coefficient was not constant with height, even becoming negative in the plume cap region [Lavelle and Baker, 1994]. The number of model checks and comparisons to measurements enumerated here demonstrates how well tested this convection model is.
Six model experiments are reported here. Discussion begins with a reference case followed by contrasting results of perturbed cases (Table 1). Perturbed cases refer to those at zero rotation rate ( = 0), null shear-dependent turbulent mixing intensity (C_{S} = 0, (6)), reduced horizontal background mixing (A_{HMIN} = 10 m s, (4)), and higher U of 3 and 6 cm s. The three velocity cases to be compared have R = w_{MAX} /U values of 6.6, 2.8, and 1.0, where w_{MAX} is the computed maximum upward velocity in the plume stem and U is background flow velocity. Buoyancy dominates plume development for larger R and cross flow dominates for smaller R. The parameter R is often used to classify situations of jets entering cross flows, in which case w_{MAX} represents jet exit velocity. Sykes et al. [1986] ran numerical experiments for jets that spanned the ratio 2-8. Laboratory experiments of Ernst et al. [1994] on buoyant jets in cross flows showed plume bifurcation over the range R = 2-6.
Calculated fields in three dimensions and in time include , p, , S, and A_{I}. This paper focuses primarily on temperature and velocity/vorticity fields near temporal equilibrium as a way to describe differences caused by rotation, turbulence, and cross-flow speed.
Figures 2 and 3 present cross sections through the reference experiment plume at 24 hours past plume startup. With the given advection speed (U = 1.5 cm s), the advection distance over that time period is 3 times the distance (425 m) between source and outflow boundary. By 24 hours, the plume is in equilibrium; heat flux through the outflow boundary is equal to the heat input at the source. Even as early as 8 hours, corresponding to an advection distance just past the outflow boundary, the plume is in equilibrium around the source and near equilibrium at outflow, with outflow heat flux already 80% of vent heat input.
Figure 2. Plume distributions for the reference case (experiment 21, Table 1) at t = 24 hours. The cross-stream direction is y and the along-stream direction is x. (a) on the plane y = 0; (b) on the plane y = 0; (c) on the plane z = 2280 m; (d) on the plane x = 320 m. All contours are in degrees Celsius.
Figure 3. Velocity, , and relative vorticity patterns for the reference case (experiment 21, Table 1) at t = 24 hours. (a) (shaded) and _{z} nondimensionalized by U/D (contoured) at z = 2350 m (z/D = 5); (b) (shaded) and u velocity (contoured) at z = 2280 (z/D = 12); (c) (shaded) and v velocity (contoured) at z = 2280 (z/D = 12). Velocity is in meters per second.
Contours of (Figure 2a) show the effect of hydrothermal heat release on the surrounding environment. Isotherms are drawn down into the source region. The inverted J-shaped isotherm and a plume stem bent ~13° with respect to the vertical evidence the effects of background flow. The well-defined stem has lateral gradients as large as 6.8 × 10 °C m. Relatively steep gradients such as these are not numerically easy to preserve; their occurrence in these calculations results from the use of the upstream corrected advection scheme of Smolarkiewicz and Clark [1986]. Downstream internal waves above the height of the convection column with wavelength of ~140-170 m are also evident in Figure 2a. These are associated with internal waves in u and w velocities with amplitudes of ~0.2 and 0.6 cm s, respectively. Lees waves in the atmosphere resulting from convective motion have been modeled by Hauf and Clark [1989] and are used to good advantage by glider pilots [Kuettner et al., 1987]. A possible explanation for their occurrence is that a convection column can act, in part, like a hill, forcing environmental flow over and around, but further numerical experimentation is required to be unequivocal about the cause in this setting.
The anomaly, = - _{BKG}, contoured in units of 0.005°C (Figure 2b), better shows the maximum rise of the plume to be 180 m. For the given combination of U and B, the plume overshoots the neutral density level. Yet flow is strong enough, with the given source buoyancy and stratification, that the overshoot is small, and strong enough that no limb of the plume appears upstream of the source. For fixed B, an upstream limb can be expected as U is reduced [e.g., Ernst et al., 1994], and full plume symmetry about the then-vertical convection axis must occur when U = 0 [e.g., Lavelle, 1995]. In all cases reported here, U is sufficiently large (1.5 cm/s) that neither upstream nor significant cross-stem plume growth occurs. Flows in and around the stem and plume cap region, which result from the superposition of background and convective flows, are also much different than flows expected in and around convective plumes rising into a quiescent background environment [e.g., Lavelle and Baker, 1994].
Vertical velocities in the stem reach maxima of 0.1 m s in this example. The plume overshoots the equilibrium level and causes a slight positive density anomaly in the region above the stem. Just downstream of the positive region, flow has a downward directed component with vertical velocities of as much as 0.03 m s. The local maximum of centered near x = 50 m (Figure 2b) is at the terminal end of this downward directed flow. Note that > 0.02 °C extends only several hundred meters downstream because of lateral dispersion. Since resolving 0.01 °C in field data is difficult, a single hydrothermal source of the given size in this stratification environment ought to be difficult to detect thermally beyond several hundred meters from the vent source.
A planar view of the same plume at 120 m above the bottom (z = 2280 m, Figure 2c) shows the small aspect ratio of the plume for the given cross-flow strength. At the outflow boundary, the plume (i.e., the 0.005°C isopleth) at this depth is ~100 m wide. Growth of the plume in the flow transverse direction is limited by the downstream transport of plume material. The local maximum centered at x = 75 m (Figure 2c) is the same local maximum evident in Figure 2b, the consequence of initial plume overshoot with subsequent downward advection. Computations without rotation show planar distributions with perfect symmetry about the y = 0 axis. The slight asymmetry of the pattern of Figure 2c is thus caused by rotation, a topic to be taken up more fully later. No undesirable outflow boundary layer is apparent in Figures 2a-2c, evidence that supports Johansson's [1993] prescription for boundary conditions.
At the outflow boundary (x = 320 m), the equilibrium plume shows a maximum core of 0.015°C (Figure 2d). The plume (i.e., 0.005°C isopleth) has maximum width of ~200 m, about half the width of the computational domain (320 m). Cyclic boundary conditions in the y direction allow transport through the side walls of the calculational region, but little -distribution contamination by adjoining cyclic domains is apparent. A wider computational domain width would be necessitated if smaller U were used. A test experiment was performed to examine the change in results due to quadrupling the domain width. In that case the y-direction resolution was coarsened to 10 m, but all other aspects of the calculations were left intact. Results were not significantly different from those shown here. In consideration of computational costs, most experiments were run with the 320-m domain width.
Circulation in the region of the plume stem and above is considerably different from that predicted for point source convection in otherwise quiescent environments [e.g., Lavelle and Baker, 1994]. In the high -gradient region near the source, contours (shaded, Figure 3a) are kidney- or horseshoe-shaped, as found numerically, for example, of plumes in nonrotating environments by Sykes et al. [1986]. The same authors cite numerous laboratory observations of the same effect. While much of the upstream fluid enters the stem, there is also some spatial acceleration of flow around the stem. Downstream of the stem there is a u-velocity minimum and beyond that is a reconvergence of the stem-separated flow. The flow patterns result in a counterrotating _{z} couplet (contours, Figure 3a) with a _{z} maximum on the right (referenced to the downstream direction) and a minimum on the left, at the downstream end of the anomalies. Such a counterrotating couplet at the downstream edge of a jet entering cross flow was noted by Turner [1960] and observed by Moussa et al. [1977], for example. Besides the asymmetry caused by rotation (Figure 3a), the _{z} pair is also much like the one found numerically by Sykes et al. [1986] for jets in nonrotating, unstratified cross flows. Not shown is the perturbation pressure (p') distribution, which has two local minima of comparable size located asymmetrically about y = 0 and slightly downstream of the isopleth tips.
The kidney-shaped pattern of extends from the seafloor to the levels of neutral buoyancy. At z = 2280 m, the pattern of in the stem region (shaded, Figures 3b and 3c) also has two lobes, the right lobe being larger. Asymmetry about y = 0 is again the consequence of Coriolis forces. Superimposed on the distributions are isopleths of u (Figure 3b) and v (Figure 3c). With background flow at 0.015 m s, Figure 3b shows a region of reduced u ahead of the convection column and a region of near-zero u some 20 m downstream. The v distribution (Figure 3c) shows maxima to both side of the column, but much higher v in the direction of positive y. Downstream, the signs of the two v lobes reverse to allow the reconvergence of the flow that was deflected to either side of the column. Though Figure 3 bears evidence that the Coriolis force does affect plume structure, the simple anticyclonic flow for the upper plume predicted when convection occurs in a quiescent background environment [Lavelle and Baker, 1994] no longer occurs. Additional differences in plume structure with and without rotation are discussed in the following section.
A nonrotational case was next examined. To isolate direct effects of rotation, the u profile of Figure 1a was taken as the along-stream background current, and cross-stream v was taken to be zero in experiment 22. Using (11), a P profile consistent with those velocity profiles but unique to the nonrotating case was determined. The resulting P was used to force ambient cross flow (Table 1). Using the P of experiment 21 when = 0 would have resulted in a u_{BKG} profile with a much thicker boundary layer.
The primary difference in caused by rotation is the absence of distributional symmetry about y = 0; rise height, overshoot, and magnitudes are otherwise similar in a general sense. As expected, u and v too are symmetrical about y = 0 in experiment 22 (Figures 4a and 4b) but not in experiment 21 (Figures 4c and 4d). To allow easier comparison of these results with those of Sykes et al. [1986], velocities and distances in Figure 4 have been non-dimensionalized by U and D.
Figure 4. Comparison of rotating (experiment 21) and nonrotating (experiment 22) cases. Velocity u nondimensionalized by U on the plane z = 2380 m (z/D = 2) when = 0 (Figure 4a) and when 0 (Figure 4c). Velocity v nondimensionalized by U on the plane z = 2380 m (z/D = 2) when = 0 (Figure 4b) and when 0 (Figure 4d). For = 0, on the plane x = -60 m (x/D = 4) (Figure 4e) and nondimensional u on the plane x = -60 m (x/D = 4) (Figure 4f).
Figures 4a ( = 0) and 4c ( 0) show u in the vicinity of the source at a vertical distance of z/D = 2, where z is distance from the seafloor. When = 0, u is nearly doubled (U/U = 1.9) on both sides of the rising plume as background flow, in part, sweeps around the ascending fluid column. The bulk of the upstream flow is undeflected; after entering the column, u momentum is displaced vertically, with the result that little of the u momentum entering upstream is found downstream at z levels where the stem is well-defined. The u velocity immediately downstream of the stem is just greater than zero, although flow of small size (u < 0) occurs in the boundary layer (z/D < 2) and at some sites above z/D = 10. Though the possibility of downstream reverse flow (u < 0) must depend on boundary layer thickness and strength of upward convection, the general result, that nearly all entrainment into the stem occurs from the upstream side of plumes under similar forcing conditions, is likely not to be significantly altered.
In the rotating case (Figure 4c), reverse flow (u < 0) occurs on the left side of the convection column (y > 0), while larger along-stream flow (u/U = 2.4) occurs on the right (y < 0). This asymmetry helps shape the distribution of, for example, Figure 3a. The region of flow affected is small. Consequently, field observations of u enhancement near a heat source may prove difficult, if only because of the small size of the region involved.
Magnitude of the transverse velocity, |v| , reaches ~0.6U in both cases (Figures 4b and 4d). For = 0, maximum |v| occurs nearly twice as far downstream as || maxima, which occur at the downstream edge of the stem. The v convergence (Figure 4b) results in u values again having magnitudes ~U within a distance of 5D downstream of x = 0. The effect of 0 on v (Figure 4d) is to skew distributions across the plane of symmetry (y = 0) so maximum |v| occurs slightly farther downstream than does maximum |-v|. Magnitudes of v at this z level are little changed by rotation.
For cases when R = 4 and R = 8, Sykes et al. [1986] provide distributions of a passive scalar and u on the cross section x/D = 4. While their study of jets involved neither background stratification nor a boundary layer, their results show horseshoe-shaped patterns for the scalar (as in Figure 4e) and u distributions with a low velocity core underlying a higher velocity high region (as in Figure 4f). Results here show larger vertical gradients above the core (Figure 4e) and maximum u velocities the distribution of which drapes less over the sides of the lower u core (Figure 4f) than it does in the results of Sykes et al. [1986]. Those differences undoubtedly reflect the presence of background stratification. In none of the panels of Figure 4 is the full domain of computation shown.
Relative vorticity distributions allow a comparison to the results of Sykes et al. [1986] as well. Nondimensionalized stream-wise vorticity in the source region at z/D = 2 appears as a counterrotating couplet (Figure 5a); even during plume development (t < 1 hour) the couplet at this level has the indicated strength and shape. Low p is found just downstream of extremal _{x} sites. If contours were superimposed, _{x} extrema would be seen to be just downstream of the center. When 0, the axis separating the two counterrotating vortices of the couplet is oriented clockwise of the x axis, but magnitudes are comparable to those of the = 0 case. Sykes et al. [1986] found distributions of that are similar in both length scale and intensity at this height. The similarity is not surprising, in that at z/D = 2 the background environment is well mixed (Figures 1a-1b) and by this time (t = 8.3 hours) conditions in the stem have long ago reached equilibrium. The pattern (Figure 5a) is primarily the consequence of the w/y contribution to _{x} and, for fixed x, error function-like distributions of w in the y direction across the plume stem; the other term contributing to _{x}, v/z, is less than 10% the size of w/y.
Figure 5. Relative vorticity, nondimensionalized by U/D, about the x axis, , when = 0 on the planes (a) z = 2380 (z/D = 2), (b) x = -60 m (x/D = 4), and (c) x = 0 (x/D = 10). (d) Relative vorticity, nondimensionalized by U/D, about the y axis, , on the plane y = 0.
Downstream, the distribution of _{x} grows in complexity. For example, on the plane x/D = 4, has two pairs of counterrotating cells (Figure 5b). The underlying velocity field is like that measured by Fearn and Weston [1974] for a jet entering a cross flow. It is the w distribution, similar to Figure 4e, once differentiated (i.e., w/y), much more than the distribution of v/x, that determines the form of . At this x location, _{x} again resembles that found by Sykes et al. [1986]. At x/D = 10, where negative w at the level of neutral buoyancy is a response to the initial plume overshoot (Figure 3a), is preponderantly negative on the right side and positive on the left (Figure 5c). No similar result could be expected for jets in homogenous flow because then no vertical overshoot is possible. Stratification also broadens and flattens the _{x} distribution at this distance.
In experiment 22 ( = 0), initially only is nonzero, and then only in the boundary layer, because background shear is at first unidirectional. Both and quickly develop, however, and is substantially changed as convectively forced flow develops. On the plane y = 0, for example, singlet becomes a couplet extending much higher into the water column (Figure 5d) with extremal values (6.5 units) that dwarf original magnitudes (-0.65 units). The distribution of in Figure 5d is also determined by the distribution of w across the stem region: w/x, the significant factor determining , is positive entering the stem on the upstream side and negative exiting the stem downstream. Sykes et al. [1986] and Klemp [1987], among others, have analyzed the growth of vorticity components as convection occurs, so a full discussion of that time development is unnecessary to repeat here. Klemp [1987] shows that when cross-stream vorticity is present, it is tilted and drawn up by buoyancy driven flow during thunderstorm development to initiate a _{z} couplet (e.g., Figure 3a). Schlesinger [1980] suggests that no initial shear is needed for all three components to develop, but tilting by advection is a primary means of growth for downstream components during storm development.
Dominance of _{x }and by one of the horizontal derivatives of w points to the certainty of development of both relative vorticity components starting at the time buoyancy is first generated because vertical velocity is created by buoyancy from startup. For example, without cross flow the distribution of w in the budding stem would be Gaussian in both lateral directions, and the first derivative of w would lead to counterrotating vorticity pairs in both x and y directions, i.e., _{x }and _{y}. Clearly no cross-stream or stream-wise vorticity is needed initially when buoyancy forcing is present to generate _{x }and . Just as clearly does _{z} production begin at the same time: as the convection column first deflects a fraction of the background flow to both sides, flow that subsequently converges downstream, distributions of u and v (Figures 4a-4d) are created that once differentiated lead to nonzero _{z}. Production of all components of must occur at any location where buoyancy has begun to disturb background flow.
To gauge sensitivity of results to subgrid-scale mixing, two additional experiments were performed. In the first, dependence of mixing on shear (equation (6)) was eliminated by setting C_{S} = 0, so only constant mixing coefficients controlled turbulent diffusion (experiment 23). Those background mixing coefficients had relatively small values A_{ZMIN} (10 m^{2 }s) and A_{HMIN} (10 m^{2 }s), as indicated earlier. In another experiment (experiment 26), C_{S} was left at 0.2 so that A_{I} (equation (6)) would be a significant factor in mixing in the stem region, where shears are larger, but the value of A_{HMIN} that controls lateral mixing outside that region was reduced by a factor of 10. In this case A_{HMIN} is smaller (10 m s) than even A_{ZMAX} (equation (12)). In reference experiment 21, A_{I} in the plume stem was typically 10-20 × 10 m^{2 }s. With a small value in experiment 26, A_{HMIN} had little influence on mixing in the stem region and much reduced influence beyond. Effects of stirring by nonlinear advection beyond the convection region are thus highlighted in experiment 26.
Results at t = 8.3 hours for the three cases are shown in Figure 6. In both panels, isopleths are provided as solid lines (experiment 23), dotted lines (experiment 26), or shaded regions (experiment 21). Both experiments with reduced turbulent mixing show anomalies with greater spatial variability. Three plumes along y = 0 (Figure 6a) show that there is no substantial difference in the equilibrium level of the plumes. Using height of maximum averaged over each section in the downstream interval 200 < x < 320 m as indicator, average rise heights were 136, 125, and 122 m for experiments 21, 23, and 26, respectively (Table 1). Thus reduced lateral mixing leads to only slightly smaller rise heights. On the other hand, when time development of the plumes is examined, the starting pulse of anomalous water rose to a height greater by 27 m in the case of smallest stem viscosity (experiment 23) compared to the case of largest viscosity (experiment 21). Earlier work by Lavelle and Baker [1994] for plumes without cross flow had shown higher rise heights with less stem mixing. Present results agree only for the initial interval of rise to the level of neutral buoyancy, but not in the longer term. The explanation must lie in differences in entrainment when cross flow is present. In the cross flow case, background flow is forced into the plume stem region on the upstream side, while without cross flow, entrainment is caused by convection alone and occurs omnidirectionally.
Figure 6. for experiments that differ only in the subgrid-scale parameterization (Table 1). Results are represented by dotted contours for experiment 26 (A_{HMIN} = 10 cm s), solid contours for experiment 23 (C_{S} = 0), and shaded contours for experiment 21, the reference case. (a) Cross sections for y = 0, (b) cross sections for z = 2280 m. All contours are in degrees Celsius at 8.3 hours.
When viewed on the horizontal plane z = 2280 m, plumes of experiments 23 and 26 show larger lateral downstream spread than in the reference experiment (Figure 6b). Counterintuitively, smaller mixing coefficients either locally in the stem (experiment 23) or globally (experiment 26) cause greater lateral dispersion. Using the 0.005°C isopleth to designate a plume edge, widths averaged over 200 < x < 320 m for the three experiments were 90, 188, and 155 m, respectively. In experiment 26, that same edge shows signs of wispiness, as if Helmholtz shear instabilities were occurring.
Statistics of _{z} within the downstream plume ( > 0.005°C, x > 100 m ) show experiment 26 having larger relative vorticity. In experiment 26, _{z} was more patchy downstream of the source. Mean values of |_{z}| for the three experiments were 1.4 ×, 1.2 ×, and 2.9 × 10 s, while the standard deviation of || for experiment 26 is twice as large as for the other experiments. This suggests that resolved stirring rather than unresolved mixing is more significant as a dispersion process in experiment 26 than in the others. Thus reduced turbulent mixing in the far field, i.e., smaller A_{HMIN}, allows stronger small-scale stirring, which in turn leads to more widespread plume dispersal.
Differences in velocities also are apparent with changes in turbulent mixing intensity. In the stem, w maxima are smallest with full mixing (experiment 21) and largest when C_{S} = 0 (experiment 23). Downstream negative w are 40% larger in experiment 23 than experiment 21. Times series show that differences are much more than just changes in magnitude. Animations of fields show source heat, while steadily discharged at the seafloor, being pulsed to higher levels when mixing is small (experiments 23 and 26) but not when mixing is higher (experiment 21). Time series sampled at a site 120 m above and 27.5 m downstream of the source (Figure 7) show, in comparison, that quickly grows to a value of 0.06°C in all three experiments as the plume front passes but thereafter they are quite different. For largest mixing (experiment 21), smoothly seeks an equilibrium level. In experiments 23 and 26, on the other hand, values oscillate with periods of ~1300 s. The w time series at this location shows similar frequency content. Buoyancy period, based on the linear region of the profile (Figure 1b), is 1265 s. The oscillation period from model results is only a coarse estimate because model data were sampled only every 300 s. Since the amplitude of oscillations is nearly 0.03°C in experiment 26 (Figure 7), field observations of very near to hydrothermal heat sources at intervals of 1 min or less might be able to distinguish different mixing coefficient regimes.
Figure 7. Time series of at a fixed point in the plume stem (x/D = 2.25, y = 0, z/D = 12). The dotted line represents the experiment of largest subgrid-scale mixing (experiment 21), while the other two have reduced background mixing (dashed line, experiment 26) or shear independent mixing (solid line, experiment 23).
Effects of cross-flow strength were examined by increasing U from 1.5 cm s in experiment 21, to 3 and 6 cm s in experiments 24 and 25, while all other conditions were held fixed. That sequence of three experiments has R = w_{MAX}/U_{o} values of 6.6, 2.8, and 1.0, respectively, where w_{MAX} is maximum upward stem velocity determined empirically from results of each experiment. This range of R is comparable to that examined by Sykes et al. [1986] and is approximately the range over which Ernst et al. [1994], in laboratory experiments, found significant changes in the character of buoyant jets in cross flows.
Plumes bend increasingly with increasing cross-flow strength, as expected (Figures 8a and 8d); in all panels of Figure 8 the dotted line represents the = 0.005°C isotherm of experiment 21. Rise heights, h_{RISE}, based on the location of maximum in vertical sections at the outflow boundary for each of the three experiments are 136, 97, and 77 m, respectively, the last value representing the higher of two maxima (Figure 8f). On the basis of those three values alone, a best fit of h_{RISE} to U gives h_{RISE} U^{-0.40}. Extensive atmospheric observations have led to the canonical form h_{RISE }= 2.6 [B/(UN)]^{1/3} for bent-over plumes in the stratified atmosphere [e.g., Hanna et al., 1982], where h_{RISE} refers to distance between source level and the vertical midpoint of the plume downstream of the source. Thus, for atmospheric cases, h_{RISE} U^{-0.33}. The similarity of U dependence for these model results and atmospheric data is encouraging, though the paucity of model realizations, the difficulty of defining rise height when the distribution has more than a single maximum (Figure 8f), and the difference in rise height definitions between this and the atmospheric case are all causes for caution.
Figure 8. Contours of in degrees Celsius at 8.3 hours for experiment 24 (U = 0.03 m s, Table 1) at (a) y = 0, (b) z = 2320 m, (c) x = 300 m, and for experiment 25 (U = 0.06 m s, Table 1) at (d) y = 0, (e) z = 2360 m, and (f) x = 300 m. The dotted contour in each panel represents the 0.005°C isopleth for the reference case (experiment 21).
Unexpectedly, plumes of experiments 24 and 25 have voids in the downstream distributions. For experiment 23 (R = 2.8) this occurs just downstream of the stem, but the branches merge again farther downstream (Figure 8a). In experiment 26 (R = 1.0), branching occurs farther from the stem region and extends to the outflow boundary (Figure 8d). No such voids were seen in experiment 21 (R = 6.6, Figure 2).
In neither of the two cases is branching simple. For R = 2.8, a section for z = 2320 m (Figure 8b) shows that the void does not extend laterally all the way across the plume. The core region of highest gradients has a more exaggerated kidney shape than in experiment 21, but only the right branch spawns material downstream at this level; the left branch is truncated. A sequence of horizontal sections shows the void to be tubular with the principal axis of the tube skewed from the vertical. The irregularly shaped tube cuts through the plume wall, here defined as the = 0.005°C isopleth, on the left-hand side below z = 2310 m, creating the left-side void seen in Figure 8b. Above z = 2310 m, the right-side wall of the plume is interrupted. In the x direction near the stem, is continuous on the right-hand side below z = 2310 m and continuous on the left-hand side above. Beyond x = ~100 m, the plume has no voids. At x = 300 m (Figure 8c) it is wider than high, with two local maxima. The rightmost maximum evidently buds from the lower right-hand limb of the kidney-shaped region, while the upper maximum buds from the higher-rising left-hand limb.
For the plume rising into the strongest cross flow (R = 1.0, experiment 25), the picture of a top-to-bottom bifurcation suggested by Figure 8d is also not complete. The planar view (Figure 8e) shows a plume with a strong right-hand limb and a stunted left-hand limb at z = 2360 m. A sequence of horizontal slices shows that the left-hand limb is favored below z = 2380 m and the right-hand limb is favored above. The left-hand limb is attached to the seafloor and extends to x = ~200 m before disappearing; the attachment is in part caused by the location of the source at the seafloor, but effects of the low vertical resolution of the boundary layer by the model cannot be discounted. The right-hand limb splits vertically but does not completely separate; the section at x = 300 m (Figure 8f) shows that the bifurcation in Figure 8d was apparent only; the two vertically aligned maxima are connected. The plume would be earmarked as distinctly bifurcated only if = 0.005°C were too small an anomaly to be observed. Thus an observational threshold can affect judgement as to whether a plume has bifurcated or not. This result should also serve warning that two maxima in a single vertical profile in a hydrothermal region may not mean that two venting sources, each with different B, are nearby.
Distributions of within these plumes might seem peculiar if it were not for field and laboratory observations that confirm that plumes from buoyant jets can bifurcate. Scorer [1959] noted the occurrence of plume bifurcation in ordinary chimney plumes. Observations of bifurcating industrial stack plumes are exemplified in the report of Fanaki [1975]. Volcanic plume bifurcations are summarized by Ernst et al. [1994]. Several laboratory experiments of Wu et al. [1988] on buoyant jets in unstratified flows led to vertical bifurcation of the kind seen in experiment 25 (Figure 8f), though in the laboratory it was a source configuration skewed with respect to flow direction rather than environmental rotation or shear in cross flow that broke plume symmetry about the y axis.
Mechanisms that cause jet or plume bifurcations are not completely understood, though observations have pinpointed some conditions under which bifurcation is likely to occur. In cases of jets entering unstratified flows, the ratio R has been used to classify results. Ernst et al. [1994] saw buoyant laboratory jets that clearly bifurcated when R fell in the range 2-6, but bifurcation was blurred or did not occur at higher or lower R values. In model plumes addressed here, bifurcation occurred but was not complete when R = 1.0 and 2.8, but did not occur when R = 6.6. Ernst et al. [1994] noted that sharp density interfaces, orientation of a jet orifice with respect to the flow, and latent heat release all can influence the occurrence of a bifurcation. In the case of convection during severe storms, Klemp and Wilhelmson [1978] showed that vertical shear of environmental winds and downdraft caused by precipitation are important to the storm splitting and divergence process. On the basis of results reported here, it is appropriate to add rotation, background stratification, and boundary layer shear to the list of possible factors affecting plume bifurcation. The large number of potential factors involved, however, will likely make the identification of conditions and causes leading to the bifurcation of hydrothermal plumes difficult.
Initial parameter sensitivity experiments made with a three-dimensional time-dependent hydrodynamical model of buoyancy-driven plumes in sheared, stratified cross flows have shown a number of effects on plumes caused by differences in rotation rates, turbulent mixing intensity, and cross-flow strength. While the context of the work here is for hydrothermal plumes rising several hundreds of meters into the benthic ocean as the result of chronic releases of magmatic heat from vents along submarine ridge crests, the model has considerable generality. Important features of the model include: inflow and outflow boundaries that allow passage of fluid, heat, and salt without the development of unrealistic along-stream boundary layers; bottom Ekman boundary layers for velocity, temperature, and salinity; time- and space-dependent turbulent mixing; and the use of an advection scheme for and S that maintains a well-defined plume stem with its accompanying large lateral property gradients.
The model shows that most of the cross-flowing fluid encountering the stem is entrained into it on the upstream side of the plume. In the parameter regime examined, little entrainment into the stem occurs on the lee side of the stem. The rising column of fluid also deflects some cross flow around it, thus acting in part like an obstruction. The result is vertical counterrotating vortices on each side of the plume stem, long identified in studies of plumes and jets as the _{z} couplet. The rising column of fluid also leads to internal waves downstream at or above the level of neutral buoyancy.
Distributions of velocities around the heat source have properties, in a general sense, like those earlier observed for jets injected into cross streams in nonrotating environments: vorticity couplets in all three coordinate directions develop in the plume stem, then follow the plume to its level of neutral buoyancy, and ultimately decline in strength with downstream distance. Rotation, as expected, breaks the cross-stream symmetry or antisymmetry of the distributions.
Intensity of turbulent mixing changes width and wispiness of plumes, with higher stem viscosity/diffusivity resulting in steadier plumes with smaller downstream lateral spread. Though the source is steady, turbulent mixing coefficients of reduced size allow oscillations in and w at the buoyancy frequency, which are suppressed when the strength of turbulent mixing is increased.
For fixed buoyancy and increasing cross-flow strength, model plumes encompass instances of plume bifurcation. When the ratio R of maximum upward velocity to cross-flow strength was 2.8, the plume had a columnar void just downstream of the plume stem but none at greater distance. When R = 1.0, bifurcation was incomplete but vertical sections at increasing distance from the stem showed vertically bimodal distributions. When R = 6.6, no plume bifurcation was observed. Initial experiments are too few in number to identify mechanisms that cause bifurcations, but results suggest that this model is a tool that can contribute to that understanding.
Acknowledgments. Support for this work comes from the NOAA VENTS program. Encouragement to pursue this modeling work by VENTS colleagues is appreciated. Contribution 1687 from NOAA/Pacific Marine Environmental Laboratory.
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