Book review

Under New Management

Port Growth and Emerging Coastal Management Programs A Washington Sea Grant Publication     

Book review

Onshore Planning for Offshore Oil—Lessons from Scotland by Pamela L. Baldwin and Malcolm F. Baldwin The Conservation Foundation, Washington, D.C., 1975     

A New Coastal Policy for South Africa

Treaties on Trial: The Continuing Controversy Over Northwest Indian Fishing Rights by Fay G. Cohen (Seattle: University of Washington Press, 1986), 229 pages.     

Book reviews

Coastal Resources Management: Beyond Market and Bureaucracy by Robert B. Ditton, John L. Seymour, and Gerald C. Swanson. Published by D. C. Heath and Company, Lexington, Massachusetts, 1977, 191 pp., 6 figures.

The Urban Sea: Long Island Sound by Lee E. Koppelman, Peter K. Weyl, M. Grant Gross, and DeWitt S. Davies. Published by Praeger Publishers, 1976. 209 pp., 81 figures, 42 tables.     

Book review

Coastal Management: Readings and Notes. Edited by Marc J. Hershman and James H. Feldmann. Institute for Marine Studies, Coastal Resources Program, University of Washington, Seattle (January 1979), 806 pp. $22.50 (paper bound).

Tobey L. Winters, Deepwater Ports in the United States: An Economic and Environmental Impact Study. N.Y. Praeger, 1977, 201 pages.     

Spatial aspects of shoreline management in Puget sound reconsidered

Abstract

Shoreline management in the state of Washington attempts to provide opportunities for both urban and rural interests to express themselves. Local governments have wide latitude in determining the spatial patterns which they deem desirable. The state provides general guidelines to insure that local governments adhere to the policy of the act. On shores designated as shores of state‐wide significance the state has reserved the right to protect state interests over local interests. Since the Washington State Shoreline Management Act generally covers only 200 ft inland from the ordinary high tide line, and each local government develops its own management program, it is doubtful that nodal or spread development on a regional level in Puget Sound can be meaningfully addressed under its provisions. There may, however, be evidence of nucleation or spread patterns at the local level.     

Non-intrusive polynomial chaos for efficient uncertainty analysis in parametric roll simulations

Monte Carlo analyses are generally considered the standard for uncertainty analysis. While accurate, these analyses can be expensive computationally. Recently, polynomial chaos has been proposed as an alternative approach to the estimation of uncertainty distributions (Hosder et al. A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations. In: 44th AIAA aerospace sciences meeting and exhibit, Reno, Nevada, 2006; Wu et al. Uncertainty analysis for parametric roll using non-intrusive polynomial chaos. In: Proceedings of the 12th international ship stability workshop, Washington, DC, USA, 2011). This approach works by representing the function as a series of orthogonal polynomials; the weights for which can be calculated via several methods. Previous studies have demonstrated the usefulness of this technique for comparatively simple systems such as parametric roll modeled by the Mathieu equation with normally distributed parameter values (Wu et al. Uncertainty analysis for parametric roll using non-intrusive polynomial chaos. In: Proceedings of the 12th international ship stability workshop, Washington, DC, USA, 2011). In the present work, a polynomial chaos method is applied to a nonlinear computational ship dynamics model with normally distributed input parameters. Test cases were selected where parametric roll was expected to potentially occur. The resulting probability distributions are compared to the results of a Monte Carlo analysis. In general, these results demonstrate good agreement between Monte Carlo simulation and polynomial chaos in the absence of capsize with significant computation gains found with polynomial chaos. Overall, we conclude that polynomial chaos is an effective tool for reducing simulation time costs when studying parametric roll, and potentially other ship dynamics phenomena, particularly in the absence of capsize-like bifurcations.     

Air–sea gas transfer velocity estimates from the Jason-1 and TOPEX altimeters: Prospects for a long-term global time series

Estimation of global and regional air–sea fluxes of climatically important gases is a key goal of current climate research programs. Gas transfer velocities needed to compute these fluxes can be estimated by combining altimeter-derived mean square slope with an empirical relation between transfer velocity and mean square slope derived from field measurements of gas fluxes and small-scale wave spectra [Frew, N.M., Bock, E.J., Schimpf, U., Hara, T., Hauβecker, H., Edson, J.B., McGillis, W.R., Nelson, R.K., McKenna, S.P., Uz, B.M., Jähne, B., 2004. Air–sea gas transfer: Its dependence on wind stress, small-scale roughness and surface films, J. Geophys. Res., 109, C08S17, doi: 10.1029/2003JC002131.]. We previously reported initial results from a dual-frequency (Ku- and C-band) altimeter algorithm [Glover, D.M., Frew, N.M., McCue, S.J., Bock, E.J., 2002. A Multi-year Time Series of Global Gas Transfer Velocity from the TOPEX Dual Frequency, Normalized Radar Backscatter Algorithm, In: Gas Transfer at Water Surfaces, editors: Donelan, M., Drennan, W., Saltzman, E., and Wanninkhof, R., Geophysical Monograph 127, American Geophysical Union, Washington, DC, 325–331.] for estimating the air–sea gas transfer velocity (k) from the mean square slope of short wind waves (40–100 rad/m) and derived a 6-year time series of global transfer velocities based on TOPEX observations. Since the launch of the follow-on altimeter Jason-1 in December 2001 and commencement of the TOPEX/Jason-1 Tandem Mission, we have extended this time series to 12 years, with improvements to the model parameters used in our algorithm and using the latest corrected data releases. The prospect of deriving multi-year and interdecadal time series of gas transfer velocity from TOPEX, Jason-1 and follow-on altimeter missions depends on precise intercalibration of the normalized backscatter. During the Tandem Mission collinear phase, both satellites followed identical orbits with a mere 73-s time separation. The resulting collocated, near-coincident normalized radar backscatter (σ°) data from both altimeters present a unique opportunity to intercalibrate the two instruments, compare derived fields of transfer velocity and estimate the precision of the algorithm. Initial results suggest that the monthly gas transfer velocity fields generated from the two altimeters are very similar. Comparison of along-track Ku-band and C-band σ° during the collinear phase indicates that observed discrepancies are due primarily to small offsets between TOPEX and Jason-1 σ°. The Jason-1 k values have an apparent bias of + 4% relative to TOPEX, while the precision estimated from the two observation sets is 5–7% and scales with k. The resultant long-term, global, mean k is 16 cm/h.     

Application of new parameterizations of gas transfer velocity and their impact on regional and global marine CO2 budgets

One of the dominant sources of uncertainty in the calculation of air–sea flux of carbon dioxide on a global scale originates from the various parameterizations of the gas transfer velocity, k, that are in use. Whilst it is undisputed that most of these parameterizations have shortcomings and neglect processes which influence air–sea gas exchange and do not scale with wind speed alone, there is no general agreement about their relative accuracy.The most widely used parameterizations are based on non-linear functions of wind speed and, to a lesser extent, on sea surface temperature and salinity. Processes such as surface film damping and whitecapping are known to have an effect on air–sea exchange. More recently published parameterizations use friction velocity, sea surface roughness, and significant wave height. These new parameters can account to some extent for processes such as film damping and whitecapping and could potentially explain the spread of wind-speed based transfer velocities published in the literature.We combine some of the principles of two recently published k parameterizations [Glover, D.M., Frew, N.M., McCue, S.J. and Bock, E.J., 2002. A multiyear time series of global gas transfer velocity from the TOPEX dual frequency, normalized radar backscatter algorithm. In: Donelan, M.A., Drennan, W.M., Saltzman, E.S., and Wanninkhof, R. (Eds.), Gas Transfer at Water Surfaces, Geophys. Monograph 127. AGU,Washington, DC, 325–331; Woolf, D.K., 2005. Parameterization of gas transfer velocities and sea-state dependent wave breaking. Tellus, 57B: 87–94] to calculate k as the sum of a linear function of total mean square slope of the sea surface and a wave breaking parameter. This separates contributions from direct and bubble-mediated gas transfer as suggested by Woolf [Woolf, D.K., 2005. Parameterization of gas transfer velocities and sea-state dependent wave breaking. Tellus, 57B: 87–94] and allows us to quantify contributions from these two processes independently.We then apply our parameterization to a monthly TOPEX altimeter gridded 1.5° × 1.5° data set and compare our results to transfer velocities calculated using the popular wind-based k parameterizations by Wanninkhof [Wanninkhof, R., 1992. Relationship between wind speed and gas exchange over the ocean. J. Geophys. Res., 97: 7373–7382.] and Wanninkhof and McGillis [Wanninkhof, R. and McGillis, W., 1999. A cubic relationship between air−sea CO2 exchange and wind speed. Geophys. Res. Lett., 26(13): 1889–1892]. We show that despite good agreement of the globally averaged transfer velocities, global and regional fluxes differ by up to 100%. These discrepancies are a result of different spatio-temporal distributions of the processes involved in the parameterizations of k, indicating the importance of wave field parameters and a need for further validation.     

Cooperation of Science and Management for Harmful Algal Blooms: Domoic Acid and the Washington Coast Razor Clam Fishery

Harmful algal blooms (HABs) may be increasing in frequency and intensity worldwide. Coastal economies suffer significant income losses when fisheries or beaches are closed to protect human health and subsistence fishing communities are at risk. Despite these hardships, managers must often conservatively close harvests across a wide area or for long periods, because they lack scientific information that would allow them to predict HAB events. The outer coast of Washington State has experienced several closures of the razor clam (Siliqua patula) fishery starting in 1991, due to domoic acid (DA) contamination caused by toxic blooms of the diatom Pseudo-nitzschia. Improved science-based management was needed to minimize the impact of DA on this fishery and the coastal communities that relied on it for income, tourism, and subsistence. The Olympic Region Harmful Algal Bloom (ORHAB) Partnership, comprised of state and tribal managers, scientists, and local stakeholders, evolved in response to this need; it has been successful in its mission. Here we examine ORHAB through the lens of the Institutional Analysis and Development framework, in order to identify key factors contributing to its success. The relevance of our findings for other ORHAB-like institutions in the Pacific Northwest and elsewhere is discussed.     

Asymmetry of Columbia River tidal plume fronts

Columbia River tidal plume dynamics can be explained in terms of two asymmetries related to plume-front depth and internal wave generation. These asymmetries may be an important factor contributing to the observed greater primary productivity and phytoplankton standing crop on the Washington shelf. The tidal plume (the most recent ebb outflow from the estuary) is initially supercritical with respect to the frontal internal Froude number F_R on strong ebbs. It is separated from the rotating plume bulge by a front, whose properties are very different under upwelling vs. downwelling conditions. Under summer upwelling conditions, tidal plume fronts are sharp and narrow (< 20–50 m wide) on their upwind or northern side and mark a transition from supercritical to subcritical flow for up to 12 h after high water. Such sharp fronts are a source of turbulent mixing, despite the strong stratification. Because the tidal plume may overlie newly upwelled waters, these fronts can mix nutrients into the plume. Symmetry would suggest that there should be a sharp front south of the estuary mouth under summer downwelling conditions. Instead, the downwelling tidal plume front is usually diffuse on its upstream side. Mixing is weaker, and the water masses immediately below are low in nutrients. There is also an upwelling–downwelling asymmetry in internal wave generation. During upwelling and weak wind conditions, plume fronts often generate trains of non-linear internal waves as they transition from a supercritical to a subcritical state. Under downwelling conditions, internal wave release is less common and the waves are less energetic. Furthermore, regardless of wind conditions, solition formation almost always begins on the south side of the plume so that the front “unzips” from south to north. This distinction is important, because these internal waves contribute to vertical mixing in the plume bulge and transport low-salinity water across the tidal plume into the plume bulge.F_R and plume depth are key parameters in distinguishing the upwelling and downwelling situations, and these two asymmetries can be explained in terms of potential vorticity conservation. The divergence of the tidal outflow after it leaves the estuary embeds relative vorticity in the emerging tidal plume water mass. This vorticity controls the transition of the tidal plume front to a subcritical state and consequently the timing and location of internal wave generation by plume fronts.     

The egg morphology of 7 Acartiidae species: a preliminary survey of the ootaxonomy of calanoids

This is the first contribution to acartiid taxonomy based on egg morphology. Eggs laid under laboratory conditions by maintaining female specimens of Acartia adriatica, A. clausi, A. italica, A. margalefi, A. tonsa, Paracartia latisetosa and Pteriacartia josephinae were studied by S.E.M. Most species produced eggs with intraspecific variation in their external morphology. In spite of such variation, however, it was possible to separate three species groups.     

Vibration reduction of main hulls using semiactive absorbers

A semiactive-type absorber for vibration reduction of main hull girders was investigated. The semiactive absorber system includes a moving mass, support springs, dynamic dampers, and a control system. Only a small electrical power supply is needed for control of the damper valve and the operation of the control system. In this paper, the dynamics of the ship's hull and the constraints of the semiactive absorber are described first. Then, a suboptimal operation law is derived based on the properties of the absorber and the theory of optimal vibration reduction. The numerical simulation results show that the semiactive absorber is more efficient in hull vibration reduction than the passive absorber during critical periodical excitation from the propeller. The vibration caused by multifrequency excitation can also be suppressed by the semiactive absorber. In terms of effectiveness, the semiactive absorber is almost as effective as the active absorber. In particular, the performance of the semiactive absorber is excellent in the reduction of high-frequency fluctuations.List of symbols C _h ⁽ⁱ⁾ damping matrices of the segmenti - C _sb structural damping coefficient of bending - C _ss structural damping coefficient of shear - C _v hydrodynamic damping coefficient - EI flexural rigidity - f _a force generated by the absorber - f _ad damper force of the semiactive absorber - f _ext total excitation force - F _ext ⁽ⁱ⁾ generalized load vector in segmenti - teÎ the identity matrix - J performance index - J _r rotatory moment of inertia - k _a stiffness coefficient of the absorber - K _h ⁽ⁱ⁾ stiffnes matrices of the segmenti - K _s A _s G _s shear rigidity - k _v hydrodynamic spring coefficient - l _k length of the segmentk - m _a mass of the absorber - M _ext total exciting moment - M _h ⁽ⁱ⁾ mass matrices of the segmenti - m _v mass moment of inertia - w _h deflection of the center line of the hull - W _h ⁽ⁱ⁾ vertical translation and shear slope of nodes in segmenti - ¯ w _d displacement of the absorber mass relative to the hull - ¯ w _a absolute displacement of the absorber mass - ¯ w (a, t) absolute upward displacement of the hull atx=a - slope deflection due to bending - slope deflection due to shear - Dirac delta function - _k ⁽ⁱ⁾ Kronecker delta function - _k distribution function - shape function vector     

Present-day biosiliceous sedimentation in the northwestern Ross Sea, Antarctica

Sedimentological and oceanographic inferences have been obtained for the NW Ross Sea using sedimentary ²¹⁰Pb as a tracer together with determinations of biogenic silica and organic carbon. ²¹⁰Pb chronologies give apparent accumulation rates ranging between 14 and 80 mg cm⁻² yr⁻¹ (0.02–0.12 cm yr⁻¹) in the shelf basins. Even if a profile of ²¹⁰Pb is present in sediments from the top of the banks, here sediment accumulation rate is practically null, and physical mixing is responsible for the downward transport of fine particles and associated ²¹⁰Pb. The accuracy of ²¹⁰Pb-derived accumulation rates is discussed with respect to ¹⁴C dates. The annual rate of biogenic accumulation from ²¹⁰Pb appears to be ca. 8 times higher than the value derived using radiocarbon. Bioturbation is probably responsible for the discrepancy but also temporal and spatial variations in opal accumulation play a key-role. Contrasting measured and expected inventories of ²¹⁰Pb, a residence time of about 50 years has been tentatively estimated for the water in the NW Ross Sea. Furthermore, the data suggest that the pattern of present-day biosiliceous sediment accumulation in the Ross Sea is mainly driven by biogenic silica production in the water column, the SW area being the most productive part of the Ross Sea, by high sediment accumulation rate which enhances the seabed preservation, and by hydrodynamics, which is so effective to resuspend fine biogenic particles from the topographic highs. Resuspended particles are then deposited onto the flanks. The material which accumulates in the central part of the basins derives basically from primary production and settling through the water column.     

Digging Deep: Managing Social and Policy Dimensions of Geoduck Aquaculture Conflict in Puget Sound,Washington

Shellfish aquaculture can result in conflicts among stakeholders who perceive impacts and tradeoffs regarding sense of place, aesthetic, recreational, economic, and ecological values. Pacific geoduck clams (Panopea generosa Gould 1850) are grown in intertidal plots using gear- and labor-intensive techniques that result in a high value export product. A confluence of issues has resulted in on-going social and legal tensions surrounding geoduck aquaculture in southern Puget Sound, Washington (WA), USA. Using interviews and document analysis, we explored stakeholder perspectives and policy issues related to geoduck aquaculture in southern Puget Sound. Twenty-three stakeholders were interviewed, including state agency employees, representatives of the aquaculture industry, nongovernmental organizations, landowners, a tribal member, and an academic. Nine state hearings board decisions on challenges to aquaculture permits were also analyzed. Stakeholders articulated a variety of perspectives regarding aesthetic, recreational, land-use, ecological, political, regulatory, and economic aspects of geoduck aquaculture activities. Hearings board cases addressed similar issues (aesthetic, ecological, and recreational), as well as challenges to restrictions on aquaculture. Potential strategies for managing this conflict include emphasizing best management practices, identifying and incorporating best available science, joint fact-finding approaches, and initiating and improving communication among all stakeholders.     

Evaluating transfer velocity-wind speed relationship using a long-term series of direct eddy correlation CO2 flux measurements

Long-term observations of the marine atmospheric boundary layer were performed by an eddy correlation system, which was set-up on a platform in the Baltic Sea. In this experiment the three-dimensional wind vector and the turbulent fluxes of momentum, sensible and latent heat and CO₂ were measured for one and a half years. Simultaneously the CO₂ partial pressure p_CO2 in surface water was measured by a submersible autonomous moored instrument for CO₂ at the platform in 7-m depth. The high-resolution eddy correlation measurements of the atmospheric CO₂ flux F_CO2, together with the measurements of the CO₂ partial pressure differences between air and sea Δp_CO2 led to a long-term data set which provided the possibility to investigate the parameterization of the CO₂ transfer velocity k as a function of 10-m wind speed u in a statistical manner. From half-hour mean CO₂ fluxes and CO₂ partial pressure differences, k was calculated using k = F_CO2 / (K₀Δp_CO2), with K₀ the CO₂ solubility. The half-hour mean data points, used for the determination of the k–u parameterization, show large scatter. However, assuming a linear, quadratic dependency the analysis yields: k₆₆₀ = 0.365u² + 0.46u (k at 20 °C and salinity 35 psu) with a correlation coefficient of r² = 0.81. The large scatter indicates that the kinetics of the air–sea CO₂ transfer velocity is not only a function of the wind speed alone, but might also be controlled by other environmental parameters and mechanisms, such as sea state and surface coverage with surfactants.     

Air–sea gas exchange at extreme wind speeds measured by autonomous oceanographic floats

Measurements of the air–sea fluxes of N₂ and O₂ were made in winds of 15–57 m s^− 1 beneath Hurricane Frances using two types of air-deployed neutrally buoyant and profiling underwater floats. Two “Lagrangian floats” measured O₂ and total gas tension (GT) in pre-storm and post-storm profiles and in the actively turbulent mixed layer during the storm. A single “EM-APEX float” profiled continuously from 30 to 200 m before, during and after the storm. All floats measured temperature and salinity. N₂ concentrations were computed from GT and O₂ after correcting for instrumental effects. Gas fluxes were computed by three methods. First, a one-dimensional mixed layer budget diagnosed the changes in mixed layer concentrations given the pre-storm profile and a time varying mixed layer depth. This model was calibrated using temperature and salinity data. The difference between the predicted mixed layer concentrations of O₂ and N₂ and those measured was attributed to air–sea gas fluxes F_BO and F_BN. Second, the covariance flux F_CO(z) = wO₂′(z) was computed, where w is the vertical motion of the water-following Lagrangian floats, O₂′ is a high-pass filtered O₂ concentration and (z) is an average over covariance pairs as a function of depth. The profile F_CO(z) was extrapolated to the surface to yield the surface O₂ flux F_CO(0). Third, a deficit of O₂ was found in the upper few meters of the ocean at the height of the storm. A flux F_SO, moving O₂ out of the ocean, was calculated by dividing this deficit by the residence time of the water in this layer, inferred from the Lagrangian floats. The three methods gave generally consistent results. At the highest winds, gas transfer is dominated by bubbles created by surface wave breaking, injected into the ocean by large-scale turbulent eddies and dissolving near 10-m depth. This conclusion is supported by observations of fluxes into the ocean despite its supersaturation; by the molar flux ratio F_BO/F_BN, which is closer to that of air rather than that appropriate for Schmidt number scaling; by O₂ increases at about 10-m depth along the water trajectories accompanied by a reduction in void fraction as measured by conductivity; and from the profile of F_CO(z), which peaks near 10 m instead of at the surface.At the highest winds O₂ and N₂ are injected into the ocean by bubbles dissolving at depth. This, plus entrainment of gas-rich water from below, supersaturates the mixed layer causing gas to flux out of the near-surface ocean. A net influx of gas results from the balance of these two competing processes. At lower speeds, the total gas fluxes, F_BO, F_BN and F_CO(0), are out of the ocean and downgradient.     

Distributions of oxygen and carbon stable isotopes and CFC-12 in the water masses of the Southern Ocean at 30°E from South Africa to Antarctica: results of the CIVA1 cruise

This study presents oceanic distributions of stable isotopes (δ¹⁸O of water and δ¹³C of ΣCO₂) and CFC-12 from samples collected during the CIVA1 cruise (February/March 1993), across the Southern Ocean, along a meridian section at 30°E, from South Africa (44°S) to Antarctica (70°S). The isotopic measurements show important variations between the subantarctic surface waters with low δ¹⁸O–high δ¹³C values and the antarctic surface waters with very low δ¹⁸O–low δ¹³C values. The surface distributions of δ¹³C values follow the major frontal oceanic structures; the vertical distribution shows the progressive upwelling from the subantarctic zone to the antarctic divergence of ¹³C-depleted CO₂ derived from remineralization of organic matter. Along the Antarctic continental shelf, between 2500 and 4000 m, a core of water with δ¹⁸O values close to −0.1‰ is associated with a relative maximum in CFC-12 concentration, although this core is not detected by its temperature and salinity parameters. This water mass, which corresponds to recently formed deep water, may originate from the eastward extension of the Weddell gyre or from bottom waters coming from the East and formed near Prydz Bay.     

Simple Lagrangian formulation of bubbly flow in a turbulent boundary layer (bubbly boundary layer flow)

For the theoretical consideration of a system for reducing skin friction, a mathematical model was derived to represent, in a two-phase field, the effect on skin friction of the injection of micro air bubbles into the turbulent boundary layer of a liquid stream. Based on the Lagrangian method, the equation of motion governing a single bubble was derived. The random motion of bubbles in a field initially devoid of bubbles was then traced in three dimensions to estimate void fraction distributions across sections of the flow channel, and to determine local bubble behavior. The liquid phase was modeled on the principle of mixing length. Assuming that the force exerted on the liquid phase was equal to the fluid drag generated by bubble slip, an equation was derived to express the reduction in turbulent shear stress. Corroborating experimental data were obtained from tests using a cavitation tunnel equipped with a slit in the ceiling from which bubbly water was injected. The measurement data provided qualitative substantiation of the trend shown by the calculated results with regard to the skin friction ratio between cases with and without bubble injection as function of the distance downstream from the point of bubble injection.List of symbols B law of wall constant - C _f local coefficient of skin friction - C _f0 local coefficient of skin friction in the absence of bubbles - d _b bubble diameter [m] - g acceleration of gravity [m/s²] - k ₁ k₄ proportional coefficient - k _L turbulent energy of the liquid phase [m²/s²] - L representative length [m] - l _b mean free path of a bubble [m] - m _A added mass of a single bubble [kg] - m _b mass of a single bubble [kg] - N _x ,N _y ,N _z force perpendicular to the wall or ceiling exerted on a bubble adhering to that wall or ceiling [N] - P absolute pressure [Pa] - Q _G rate of air supply [/min] - q _L ⁽ⁱ⁾ turbulent velocity at the ith time increment [m/s] - R> _ex Reynolds number defined by Eq. 32 - T _*L integral time scale of the liquid phase [s] - U velocity of the main stream [m/s] - ,¯v,¯w time-averaged velocity components [m/s] - u,v,w turbulent velocity components [m/s] - û _L ,v_L root mean square values of liquid phase turbulence components in thex- and y-directions [m/s] - V volume of a single bubble [m³] - X,Y,Z components of bubble displacement [m] - x _s ,y _s ,z _s coordinate of a random point on a sphere of unit diameter centered at the coordinate origin - root mean square of bubble displacement in they-direction in reference to the turbulent liquid phase velocity [m] - local void fraction - _m mean void fraction in a turbulent region - regular random number - R _v increment of the horizontal component of the force acting on a single bubble, defined by Eq. 22 [N] - t time increment [s] - ₁ reduction of turbulent stress [N/m²] - _L rate of liquid energy dissipation [m²/s³] - _m coefficient defined by Eq. 30 - law of wall constant in the turbulent region in absence of bubbles - ₁ law of wall constant in the turbulent region in presence of bubbles