Late Minoan Jar ca. 1450-1400 bce

TURBIDITY CURRENTS ON STEEP SLOPES: APPLICATION OF AN AVALANCHE-TYPE NUMERIC MODEL FOR OCEAN THERMAL ENERGY CONVERSION DESIGN


A. T. DENGLER, JR. and P. WILDE*


Marine Sciences Group, Department of Paleontology, University of California, Berkeley, California 94720
*Present Contact Address:
Pangloss Foundation, 1735 Highland Place #28, Berkeley, California 94709
Revised and Modified for the WWW: 9 December 2001 from Ocean Engineering, Vol. 14, No 5, pp. 409-433, 1987.

Abstract


To minimize cold water pipe lengths, the most favorable land or fixed platform based Ocean Thermal Energy Conversions (OTEC) sites have subbottom slopes greater than 5. Observations at OTEC sites in Hawaii indicate that turbidity currents of an impulsive or episodic nature can occur with frontal speeds of several meters per second. Such speeds and the attendant potential for sediment transport and abrasion along routes containing OTEC installations indicate that the pertinent features of these flows are an important design criteria for OTEC or any other steep-slope marine installation. To satisfy this need, models of oceanic turbidity flows and similar flows have been examined. The model that addresses OTEC steep-slope conditions most succinctly was developed originally by Hopfinger and Tochon-Danguy (1977) for snow avalanches on land. This two-dimensional avalanche model is used to estimate the speed and growth characteristics of potential turbidity currents downslope for various postulated marine conditions of initial flow density, height, volume, and length at slopes from 5 to 60. The areas of additional research required to increase reliability of the analyses are in the initiation and initial development of a turbidity plume, the mechanisms of sediment entrainment to and loss from the plume, and three-dimensional in addition to two-dimensional studies.

INTRODUCTION

This research is designed to evaluate the characteristics of turbidity/suspension currents that can occur at locations of interest for Ocean Thermal Energy Conversion (OTEC) facilities. Turbidity/suspension currents are a sub-class of density current where the driving force is due to density differences caused by suspended solids (clays, muds, etc.). Turbidity/suspension currents generally are natural phenomena and episodic in occurrence. The potential for turbidity/suspension currents at sites of interest for OTEC is recognized from the study of Normark et al. (1982) that examined sediments and bathymetry at the OTEC 40MW Pilot Plant site off Kahe Point, Oahu, Hawaii, and the analyses of Dengler et al. (1984a,b) that reported on the actual observations, at the same site, of a turbidity current that appeared to have been initiated by associated effects of a passing hurricane, Iwa. The immediate concern for OTEC is that bottom mounted installations (pipelines, cables, foundations, moorings), could be damaged or destroyed by turbidity/suspension current events. OTEC systems have two properties that predispose them toward locations with steep slopes and therefore toward risk of turbidity/suspension currents. The first is that many of the islands that are within the area where the ocean thermal resource is sufficient for OTEC power generation are of volcanic origin and exhibit steep slopes in the surrounding bathymetry (Chase et al., 1986). The second is that the length and, to a large degree, the cost of the pipeline necessary to reach the cold water resource are less when steeper bathymetry can be followed (Lewis et al., 1985). The importance of knowledge of turbidity/suspension currents for design has been recognized in the design of the pipe system for the 41-MW OTEC Pilot Plant at Kahe Point, Oahu, Hawaii (OTC, 1984). The rough estimates of speed, direction and sediment concentration developed by Dengler et al. (1984a,b) were used by R. J. Brown, Inc. to provide limits on the conditions that the pipeline would need to survive after installation. The Steep-Slope Numeric Program is designed to provide the engineers with estimates of the characteristics of turbidity currents that might be expected at OTEC sites. The approach taken is to identify, from published literature and experts in field, the present state-of-art turbidity/suspension current numeric models. Appropriate test candidates can then be selected from the identified models. The criteria for selection are: boundary conditions sufficiently flexible to encompass potential OTEC; parameters (slopes, volume concentration of sediment, Froude numbers, Reynolds numbers, Richardson numbers, thickness of flow, entrainment coefficient, sediment density, velocity) appropriate for ranges of OTEC variables; separation of dependent and independent variables possible for variables of importance for OTEC design (drag coefficients, flow thickness, bed shear stress); formulae that can be numerically coded; and scalability of model results to OTEC time and space scales. Ranges of available parameters most appropriate for OTEC applications are defined for use as independent variables in the model simulations. The simulations are run over defined ranges of parameters and with specified boundary conditions to obtain the responses of dependent variables. Finally, the results (plots and tabulated values of dependent variables vs independent variables) are interpreted for use in evaluating designs for OTEC structures.

DESCRIPTION OF TURBIDITY CURRENTS

Definitions of turbidity and density currents

Turbidity currents are a subclass of the broader category of density currents. Density currents also are termed gravity currents. A density current "is the flow of one fluid within another caused by the density difference between the fluids. The difference in specific weight that provides the driving force may be due to dissolved or suspended material or to temperature differences" (Simpson, 1982). Turbidity currents exist when the density difference is due to suspended sediment and, of course, the fluids involved are both water.

Examples of density currents Density currents occur in many forms in nature. Examples of density currents in lakes and seas are: cold river inflows at the warmer surface of lakes and reservoirs; saline outflow of evaporative basins into less saline bodies, such as the outflow of the Mediterranean basin into the Atlantic through the Straits of Gibraltar; and, the formation of cold dense water on polar shelves that then sinks to form deep ocean water. At the land-atmosphere boundary, density currents include several types of catastrophic phenomena: avalanches; mudslides; nuees ardentes (gas and ash flows) Turbidity currents on steep slopes and lahars (mud and melted snow and ice flows) associated with volcanoes. In the atmosphere, density current phenomean include: catabatic drainage winds from plateau (such as the "bora", "mistral", and "Santa Ana" winds; sea breeze flows; thunderstorm base flows (wind shear); and, manmade smokestack discharges).

Marine observations of turbidity currents Dengler et al. (1984a) observed that "Current measurements on coastal slopes and in submarine canyons have resulted in an increasing number of observations of low- speed turbidity currents. Although current or wave event speeds as high as 750 cm/sec, calculated from cable breaks following the 1929 Grand Banks earthquake, appear possible (Shepard, 1963), actual measurements in submarine canyons show lower water current speeds of 50-70 cm/sec to be more common (Shepard et al., 1977; Shepard and Marshall, 1978). Current speeds as high as 190 cm/sec have been observed (Inman et al., 1976), and speeds of 200-250 cm/sec have been estimated from submersible observation (Keller and Shepard, 1978). All but one of the submarine canyons where turbidity currents have been observed are at the mouths of rivers where large quantities of sediment are discharged. The turbidity current speeds are sufficient to suspend and transport sediment (Middleton and Hampton, 1976; Miller et al., 1977), and sediment has been deposited on the current sensors (Shepard et al., 1977; Shepard et al., 1975; Lambert et al., 1976)."

Initiation of marine turbidity currents A number of phenomena have been implicated in the initiation of turbidity currents. Dengler et al., (1984a) observed that "The initiation of low-speed turbidity currents have been linked to surface waves (Shepard et al., 19-/5; Reimnitz, 1971), winds (Inman et al., 1976; Shepard and Marshall, 1973), and storm surges in rivers (Lambert et al., 1976; Gennesseaux et al., 1971) that resuspend or destabilize accumulated sediment. The sediment and water then form a suspension that cascades down the canyon slopes."

Description of an observed turbidity current Limited observations of real turbidity currents make it difficult to develop a physical sense of the phenomenon. Most observations are a-postiori examinations of the results of turbidity currents; for example, estimation of current velocity from grain size (Wilde, 1965). The most extensive in-situ observations are of a turbidity current that occurred at a site of interest for OTEC applications, from Dengler et al. (1984b). "On November 23, 1982, during the passage of hurricane Iwa, current sensor moorings in place along the proposed pipe-line route for the Ocean Thermal Energy Conversion (OTEC) Pilot Plant at Kahe Point, Oahu, Hawaii, moved downslope during a sequence of slump and/ or turbidity current events [Fig 1]. Moorings were initially situated between 100 and 760 m water depth within a 4-km wide re-entrant of the submarine volcanic slope. Sensors recorded total depth changes, during at least four events, of as much as 220 m, implying downslope movement of as much as 2.4 km. The downslope movement occurred as a series of episodes that could be traced sequentially through the series of moorings. Episodes of downslope movement were associated with a rapid increase in near-bottom current speed of up to 220 cm/sec. Temperature of 2-4C above ambient indicates that the sources for the water in the downslope current are several hundred meters above the initial position of each sensor. At least four specific events, which we interpret at turbidity currents, can be recognized in a 2 h period. The arrival times at successive sensors indicate a downslope speed of 300 cm/sec for the events. Communication cables in water depths of 1100 - 2000 m exhibited breaks or damage concurrent with the turbidity flow."

SURVEY OF MODELS

A number of theoretical and experimental models have been developed to address the phenomenon of density currents. Simpson (1982) offers a general review of research performed in the area of gravity currents in laboratory, atmosphere, and ocean environments. Of most interest for OTEC situations are examinations of gravity currents on slopes and of suspension flows. These studies can be divided into two regimes; shallow slopes from 0 to 5, and steep slopes from 5 to 90. A simplifying constraint on work in this field is that all of the available studies address two-dimensional currents. Therefore, the present analysis is similarly constrained. Behavior of density currents on shallow slopes has been subject of several experimental analyses. Middleton (1966 a, b, c) studied the advance of the head of a Turbidity currents on steep slopes saline gravity current down a gently sloping (less than 5) two-dimensional channel, building on Keulegan's (1957, 1958) lock exchange work on saline gravity surges in freshwater channels. Komar (1977) extended these results theoretically to two- dimensional turbidity currents on low slopes. The findings of these investigators have been used in an a-posteriori study by Bowen et al. (1984) to determine the nature of an earlier turbidity current from channel morphology and sediment thickness and grain size in the Navy Submarine Fan off the California coast. The work on gravity currents on steep slopes can be divided further into three classes; continuous plumes, starting plumes, and thermal plumes or finite surges. Ellison and Turner (1959) studied the behavior of a continuous two-dimensional plume on slopes varying from 12 to 90. They evaluated the entrainment of ambient fluid into the plume and found entrainment to be dependent upon the velocity of the layer and the Richardson number, determined from the buoyancy flux and the dimensions of the plume. Tochon-Danguy and Hopfinger (1975) performed further experiments with continuous plumes, measuring frontal velocity and characteristics of the head, and included the possibility of the flow entraining additional material in its path. Frontal velocity increased with increasing inflow rate and was roughly independent of slope, and rate of head growth increased with increasing inflow rate and slope. Experiments with starting plumes from continuous sources were performed by Britter and Simpson (1978) on level slopes and by Britter and Linden (1980) on inclines. They found that frontal velocity was proportional to the cube root of the buoyancy flux at the source. Hay (1983) was able to reproduce theoretically these experimentally determined frontal velocities. Thermal plumes, or instantaneously released flows, have been addressed in two studies. The first, by Hopfinger and Tochon-Danguy (1977), in a principally theoretical examination, studied speeds and growth of a thermal plume with the potential for entrainment of additional dense material by the plume from its path. The second, by Beghin et al. (1981), is an experimental analysis of frontal velocities and rates of growth of two-dimensional plumes on steep slopes. They find that, in a thermal plume without entrainment of material in its path, the frontal velocity decreases with distance from the source, due to entrainment of ambient fluid and enlargement of the thermal. At low slopes, less than 5, a thermal plume becomes increasingly like a continuous gravity plume. Conversely, when the continuous source of a starting plume is removed, the plume development with time will correspond increasingly to a thermal.

SELECTED MODEL

For OTEC interests, there is concern with steep-slopes of greater than 5, and the type of gravity flow anticipated and observed corresponds to the thermal plume. Also, we can anticipate the possibility of entraining sediment into the plume from along the path of the plume. For these reasons, we have selected the Avalanche model outlined by Hopfinger and Tochon-Danguy (1977) for examination. As noted above, their model exhibits the appropriate set of features.

General description The Avalanche model treats the flow of a finite volume of dense fluid down a slope. The flow may entrain additional dense material from a layer along its path. The speed and dimensions of the plume will vary with distance from the source, amount of dense material initially at the source and entrained from the path, and the slope. Speed increases with increasing dense material in the plume and slightly with the slope, and decreases with distance downslope. Size increases with source size, distance downslope, and the slope itself. The amount of material entrained along the path is a constant fraction of the material available on the path. Throughout its development, the theoretical plume maintains a similar profile of approximately an half-ellipse, although in reality, we can anticipate a wake developing behind the plume, especially when the plume is a turbidity current and the density difference is due to sediment (Beghin et al., 1981).

Assumptions and limitations In order to keep the phenomenon of turbidity currents tractable with this straightforward technique, a number of assumptions must be incorporated into the Avalanche model. The first assumption requires the previously stated two-dimensional simplification. This assumption is necessary due to the status of work in this field, and it will be reasonable whenever the flow is constrained to a channel or canyon. When flow is unconstrained by lateral boundaries, the violation of this assumption will have uncertain effects. A second assumption is that the density of the plume is approximately equal to the density of the ambient fluid. This second assumption is reasonable and we do not expect it to be violated for turbidity currents (Bagnold, 1962). A third assumption is of a simplified entrainment process for sediment along the path of the turbidity current. In the model the amount of sediment entrained is proportional to the thickness of the available sediment layer, and must be only a small fraction of the sediment load in the plume. In practice, the entrained sediment should be related to the particle-size distribution of the sediment and the speed, size, and duration of the turbidity current and should be limited to the available sediment layer. Correction of this simplified sediment entrainment process is an appropriate topic for further research. A fourth assumption is that there is no loss of sediment from the turbidity either to a wake or through settling. In the model, the potential for loss of sediment may be adjusted into the along path entrainment, but an accurate treatment of these losses is as yet an unaddressed problem. For the steep slopes likely to be encountered in OTEC development, the no-sediment loss assumption is reasonable. A fifth assumption is that the plume is fully developed from its inception, with the appropriate dimensions, characteristics and velocity. Finally, the model applies to slopes greater than 5. At less than 5, drag at the bottom of the turbidity current becomes relatively more important, and the thermal plume nature of the turbidity current grades into a continuous starting plume structure as modeled by Britter and Simpson (1978).

Fig 2 plume description
Figure 2 shows a reference sketch of an inclined thermal and the associated variables (Hopfinger and Tochon-Danguy, 1977). The following is a list of variables used in the model.
Variables

A a HL = volume per unit width A0 a (H0)(L0) C = experimentally determined constant C1 = experimentally determined constant C2 = theoretically determined constant E = experimentally determined entrainment constant g = acceleration due to gravity H = height normal to direction of travel H0 = initial height hn = depth of sediment on path L = length along direction of travel L0 = initial length ra = ambient density rm = mean density of plume rn = density of sediment in path r0 = initial density of plume Drm =rm-ra Drn = rn-ra Dr0 = r0-ra S = shape factor Q = slope angle U = frontal speed U0 = initial speed of turbidity V0 = initial volume of sediment available for turbidity X = direction of travel x = distance travelled from virtual origin xi = initial position Z = direction normal to direction of travel z = distance normal to direction of travel. Algorithms The equations governing the model are listed below: The experimentally determined fluid entrainment factor is E = 3(10-3) (5+Q). The height of the plume as a function of distance traveled is H = E(x). The cross sectional area of the plume is A = SH^2 where shape factor, given that the plume is an half-ellipse, is S = (1/4)P. (4) The initial height of the plume is Ho = (A0/S)1/2 the initial position of the plume is xo = H0E, x0 >0, and the initial cross section of the plume is Ao = Vo(Drn)/(Dr0), AO > or = V0 . The frontal speed is then U = C[(g(1/rnE) (sin Q) (C2Drnhn + Dr0A0/x0]1/2, x >0. Substituting Equations (5), (6) and (7) into Equation (8), and setting hn = 0, Uo = C[g(1/rm)(sin Q)^1/2[V0DrnDr0]1/4 C is experimentally determined as C=2 and C2 assumed to be, C2 = 1 for cases where sediment in the path will be eroded and C2= 0 otherwise. For turbidity currents, rm. in Equations (8) and (9) may be approximated as 1 with only slight loss of accuracy. Guide to use of algorithms For the level of prediction available from this model, we can specify several steps to follow to obtain the impressionistic estimate of the turbidity current that could occur at a given site. Turbidity currents on steep slopes

	
(1)	From surveys of the site, estimate bottom slope, Q, volume of sediment per unit 
width, V,, available for initiation of turbidity currents, and depth of sediment, hn, 
available for entrainment along the path of the turbidity.
(2)	Calculate, from V0, the possible ranges of initial size, A0, density, r0, and A0r0, using Equation (7), or Fig. A1.
(3)	From Q, calculate E, using Equation (1).
(4)	From range of A0 estimate corresponding H0 using Equation (5), or Fig. A2.
(5)	From H0,, estimate x0 using Equation (6), or Fig. A4.
(6)	Calculate U0 using Equation (9), or Figs A5 and A7.
(7)	Using an independent method and estimates for H0, and U0, evaluate whether 
sediment in path of turbidity will erode.
(8)	If sediment will not erode, calculate U(x,A0,Q0,Q), setting C2 = 0, using Equation (8), or Appendix, Figs A8-A14.
(9)	If sediment will erode, calculate U(hn,Q), for x large, (1/x) approachs 0, using Equation 
(8), or Fig. A6.
(10)	Calculate H(x) from Equation (2), Fig. A3.
Following these steps, we will obtain estimates for speed and size of the turbidity current verses distance downslope, for different slopes, initial sizes, and depths of entrainable sediment in the path. Future work Several possibilities present themselves for direct extension of this analysis. The most straightforward possibility is to incorporate the work of Beghin et al. (1981) to allow a relaxation of the assumption that the turbidity current thermal is initially in motion at the appropriate speed. A pair of recent studies (Pantin, 1979; Parker, 1982) have addressed part of the problem of the ability of turbidity currents to erode and entrain sediments along the path, and thereby to sustain motion. Analysis of these studies may allow for more realistic evaluation of bed entrainment and results might be obtainable in a near-term (1 - 2 yr) research effort. A pair of more distant research goals also present themselves. The first goal would be to understand the mechanisms for loss of sediment from the turbidity current. The two principal mechanism for loss are: (1) sedimentation of grains within and at the edges of the turbidity current and (2) loss of plume body to a wake behind the thermal plume. The second research effort would be to create a body of theoretical and experimental results using three-dimensional analyses.

>Acknowledgements
The authors would like to thank Dr L. F. Lewis of the Department of Energy and Mr B. Shelpuk of Solar Energy Research Laboratory for their encouragement and support. M. Krup did her usual outstanding job in preparing the figures and preparing the manuscript for publication. This work was sponsored by the Department of Energy, Solar Energy Research Laboratory contract #XX-5-05000-1 and is contribution MSG-87-001 of the Marine Sciences Group.of the University of California, Berkeley.

APPENDIX A-AVALANCHE MODEL CALCULATIONS

This model is designed to calculate the speed and growth of a turbidity current with distance downslope. It allows for additional entrainment of sediment in the path and treats the current as a 2-d thermal pulse. The model assumes: ambient density and plume density are approximately equal, entrained sediment is a small fraction of plume sediment, entrained sediment is a constant fraction of available sediment, slope is greater than 5 degrees so that bottom drag can be ignored, there is no loss of entrained sediment. The following figures with tabular data give a sample set of calculations for a realistic marine case. The list of variables are defined as given in the main text. For a detailed analysis and discussion of the mathematical basis of the model, the reader is referred to Hopfinger and Tochon-Danguy (1977, pp. 343-356). Values for parameters C = 2 C2 = 1 g = 10 m/sec2 ra = 1 kg/m3 rn = 2.7 kg/m3 Drn = 1.7 kg/m3 S = 0.78539 H = Ex A = SH2 E = 310-3(5+S) U = C[g(1/rmE)(sin Q)((C2Drnhn) + (Dr0A0/x))]0.5 Initial steps (also see in main text Guide to use of Algorithms): (1). Determine initial amount of sediment available = V0. (2). Estimate range of densities r0of the plume. (3). Calculate initial volume per unit length, A0 where: DrnV0 = Dr0A0 .

Computational Figures and Tables

Fig. A1.  Initial volume per unit width:  A0 as a function of V0 for various r0
Fig. A2.  Initial height of plume H0 as a function of V0 for various r0
Fig. A3.  Height of plume H, as a function of distance traveled x, for various slopes Q
Fig. A4.  Initial position x0 as a function of slope Q for various initial heights, H0
Fig. A5.  Initial speed U0 as a function of slope Q for various A0Dr0/x0
Fig. A6.  Limit speed U as a function of slope Q for various Drnhn
Fig. A7.  Initial speed U0 as a function of slope Q, for various V0.  r0 
     and V0 proportional to (Dr)0.25
Fig A8.  Speed U as a function of distance x from virtual origin for various V0
     with hn = 0, Q = 5degs., and E = 0.03
Fig A9.  Speed U as a function of distance x from virtual origin for various V0
     with hn = 0, Q = 10degs., and E = 0.045
Fig A10. Speed U as a function of distance x from virtual origin for various V0
     with hn = 0, Q = 15degs., and E = 0.06
Fig A11. Speed U as a function of distance x from virtual origin for various V0
     with hn = 0, Q = 20degs., and E = 0.075
Fig A12. Speed U as a function of distance x from virtual origin for various V0
     with hn = 0, Q = 45degs., and E = 0.15
Fig A13. Speed U as a function of distance x from virtual origin for various V0
     with hn = 0, Q = 60degs., and E = 0.195
Fig A14. Speed U as a function of distance x from virtual origin for various V0
     with hn = 0, Q = 90degs., and E = 0.285

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