Recent developments in “added-mass” planing theory

The results of several recent isolated investigations in planing theory are consolidated in this paper, together with new insights generated by a recent numerical solution of the vertically impacting wedge problem by Zhao and Faltinsen [(1992), Water entry of two-dimensional bodies. J. Fluid Mech. 246, 593–612]. As a result, in contrast to some earlier studies, it is found that the “wetted width” associated with the added mass is not that of the intersection of the wedge with the undisturbed water surface, but the wetted width of the splashed-up water, as originally proposed by Wagner [(1932), Uber Stoss-und Gleitvorgange an der Oberflache von Flussig-Keiten, Zeitschrift für Angewandte Mathematik und Mechanik, Band 12, Heft 4 (August)]. However, the splash-up ratio is not the value of (π/2–1) which he proposed, but a value which decreases with increasing deadrise, originally proposed in the late-1940s by Pierson (“Pierson's hypothesis” in the paper). For 30° deadrise, for example, Pierson's splash-up ratio is two-thirds that of Wagner's.The new equations are employed to determine the increase in the “added mass” of prismatic hull sections due to chine immersion, using experimental data. If m_o^′ is the added amss of the hull section whose chines are just wetted, Payne [(1988), Design of High-speed Boats. Volume 1: Planing. Fishergate, Inc., Annapolis, Maryland, U.S.A.] postulated that the increase in added mass due to a chine submergence (z_c) would be

where b is the chine beam and k is a constant which Payne [(1988), Design of High-speed Boats. Volume 1: Planing. Fishergate, Inc., Annapolis, Maryland, U.S.A.] gave as .The present analysis includes the “one-sided flow” correction introduced in Payne [(1990), Planing and impacting forces at large trim angels. Ocean Engng 17, 201–234]. Partly for that reason and partly because of the more precise analysis of the experimental data, the present paper revises the value to k = 2 for wetted length to beam ratios normally employed. For deadrise angles in excess of 40° and wetted keel to beam ratios in excess of 2.0, there is some evidence that k < 2.0.The revised theoretical formulation is compared with eight different sets of experimental data for flat plate and prismatic hull forms and is found to be in excellent agreement when the speed is high enough for “dynamic suction” (a loss of buoyancy at low speeds and low wetted lenghts) to be unimportant. This is true for “chines-dry” operation with deadrise angles up to 50° and chines-wet operation at length to beam ratios far in excess of the most extreme conventional practice.The research involved in performing this analysis led to the realization that different towing tanks measure different wetted chine lengths for the same hulls and test conditions. Some consistently measure more splash-up than “theory” (based on Pierson's splash-up hypothesis) predicts and others measure somewhat less than the theory. Some examples are given in Appendix B. The reason for this is not understood.     

The β-induced drift of separated boundary currents

Western boundary currents flow poleward from low latitudes until they ultimately separate from the coast and turn eastward into the ocean interior. The separation is mainly due to either: (i) the variation of the Coriolis parameter with latitude (β) which causes vanishing of the near-wall depth; (ii) vanishing wind stress curl over the ocean interior which forces zero meridional transport; or (iii) opposing currents that flow toward the equator and force the northward flowing currents to turn offshore (Agra and Nof, Deep Sea Research I, 40, 2259–2282). Here, we focus on the third kind of separated currents and show that, due to β, such separated currents migrate along the wall. A nonlinear “reduced gravity” one-and-a-half layer model is used to compute the desired migration speed. Solutions of the primitive equations are constructed analytically assuming that the translation rate is steady. It is found that the migration rate along the wall is given by βR_d² cosα/2 sinγ, where R_d is the Rossby radius, α an angle that measures the inclination of the joint offshore currents relative to the north, and γ is the angle between the axis of the joint offshore currents and the wall. The migration meridional component can be either northward or southward (depending on the inclination of the wall) but the zonal component is always westward. When the separated joint offshore flow is in the east-west direction (i.e. α = π/2 or 3π/2 so that the separated flow is zonal) no migration is taking place. It turns out that the above migration formula is so robust that it is also describes the migration rate in a two-and-a-half layer model where one current is allowed to, at least partially, dive under the other. For most separated currents the computed migration rate is a few centimeters per second.Possible application of this theory to the Confluence zone in the South Atlantic (where significant seasonal movement of the separation latitude has been observed) is discussed.     

A polar method for obtaining wave resonating quadruplets in finite depths

A polar method for obtaining wave resonating quadruplets {K₁, K₂, K₃, K₄} in the computation of nonlinear wave–wave interaction source term of the wave model is presented with results for both deep and finite water depths. The method first determines the end radial points of the locus equation for K₂, for each set of input wave vectors (K₁, K₃) on the symmetry. The locus of K₂ (and hence K₄) is then traced in the anti-clockwise direction starting with the maximum radial point on the line of symmetry. It is shown that when k₃>k₁, the number of points on the locus varies when the orientations of the input wave vectors are changed and reduces when the difference in the magnitude of the input wave vectors is increased. A significant advantage in this method is that the angular increment on the locus for K₂ can be kept constant.     

Planing and impacting plate forces at large trim angles

Added mass theory has been shown to give excellent agreement with experimental measurements on planing surfaces at normal planing angles [e.g. Payne, P.R. (1982, Ocean Engng9, 515–545; 1988, Design of High-Speed Boats, Volume 1: Planning. Fishergate. Inc., Annapolis, Maryland)] and to agree exactly with more complex conformal transformations where such a comparison is possible. But at large trim angles, it predicts non-transient pressures that are greater than the free-stream dynamic pressure and so cannot be correct. In this paper, I suggest that the reason is because, unlike a body or a wing in an infinite fluid, a planing plate only has fluid on one side—the “high pressure” side. So the fluid in contact with the plate travels more slowly as the plate trim angle (and therefore static pressure) increases. This results in lower added mass forces than Munk, M. (1924) The Aerodynamic Forces on Airship Hulls (NACA TR-184) and Jones, R. T. (1946) Properties of Low-Aspect-Ratio Pointed Wings at Speeds Below and Above the Speed of Sound (NACA TR-835) originally calculated for wings and other bodies in an infinite fluid.For simplicity of presentation, I have initially considered the example of a triangular (vertex forward) planning plate. This makes the integration of elemental force very simple and so the various points are made without much trouble. But the penalty is that there seem to be no experimental data for such a configuration; at least none that I have been able to discover. But at least the equations obtained in the limits of zero and infinite aspect ratio, small trim angles (τ) and τ = 90° all agree with established concepts and the variation of normal force with trim angle looks like what we would expect from our knowledge of how delta wings behave in air.I then employed the new equation to calculate the force on a rectangular planing surface at a trim angle τ, having a constant horizontal velocity u_o and a vertical impact velocity of ż. This happens to have been explored experimentally by Smiley, R. F. [(1951) An Experimental Study of Water Pressure-Distributions During Landings and Planing of a Heavily Loaded Rectangular Flat Plate Model (NACA TM 2453)] up to trim angles of τ = 45°, and so a comparison between theory and experiment is possible. The results of this comparison are encouraging, as is also a comparison with the large trim angle planing plate measurements of Shuford, C. L. [(1958) A Theoretical and Experimental Study of Planing Surfaces Including Effects of Cross Section and Plan Form (NACA Report)].As two practical applications, I first employed the new equations to calculate the “design pressures” needed to size the plating of a transom bow on a high-speed “Wavestrider” hull. The resulting pressures were significantly different to those obtained using semi-empirical design rules in the literature. Then I used the theory to critically review data obtained from tank tests of a SES bow section during water impact to identify how the “real world” of resilient deck plating diverged from the “model world” of extreme structural rigidity.     

A regeneration model to estimate addition fluxes and removal constants of trace metals in seawater

A model is presented for flux evaluation of trace metals in situ. For the practically feasible simplified model, mere estimation of metals in various dissolved states (e.g. Cr(III) and Cr(VI)) and in particulate phase, in a single vertical profile at least, together with temperature, salinity, O₂, titration alkalinity (TA), Ca²⁺, NO₃⁻, HPO₄²⁻ and H₄SiO₄ parameters enable evaluation of the amount of metal removed in situ, to a first approximation. From the evaluated net metal removed (M_R), estimation could be made for removal constants, i.e. removal times (t), residence times (τ) and removal rate constants (ψ), not only for the average ocean but also for the oxygen-minimum zone. Besides, M_R can be used to calculate t, τ and ψ of various species of the metal concerned, that are not yet available. The present model evaluates t, τ and ψ for total chromium from a single station, which are consistent with the reported values, and shows that chromium is removed with fast-settling particulate matter. This study also reveals that t, τ and ψ values of 0.001 years 7 years and 0.150 y⁻¹, respectively, for Cr(III) in the oxygen-minimum zone and 0.006 y, 211 y and 0.005 y⁻¹ in the marine environment, respectively.     

Hydrostatic data for cargo ships obtained with regression analysis

Starting from the regression analysis, equations are provided to determine the hydrostatic data for single-screw ships, fixing the block coefficient C_B and the principal dimensions L₀, B₀and T₀, with values T = (0÷1.5)T₀. This paper continues and develops a former one, in which the hydrostatic data were obtained in nondimensional form by means of graphs for tive parent hulls of the “Series 60”, with volume ₀ = 10,000 m³ at designed draught T₀.Numerical examples show the usefulness of the equations, both with respect of the direct calculations starting from the half-breadth values, and with respect to other methods.     

An approach for eliminating re-reflected waves

Theoretical and experimental studies were conducted to eliminate the re-reflected waves in a wave channel by installing a wavefilter in front of the wavemaker. A thin porous mesh is installed in front of the wavemaker.to serve as a wavefilter. A porous-effect parameter [Chwang, A. T. and Li, W. (1983), A piston-type porous wavemaker theory. J. Engng Math. 17, 301–313], G₀ = bω/μL₀, is employed to characterize the transmissability of the wavefilter. Theoretical relationships are established between the amplitude of progressive wave and G₀, and the distances between the wavefilter, the wavemaker and the test structure. The proper location for the wavefilter to eliminate re-reflected waves can be determined. Several experimental tests were conducted to verify the theory.Both theoretical and experimental studies show that re-reflected waves can be effectively eliminated by placing a wavefilter at a proper position between the wavemaker and the test structure, provided G₀ ≤ 1. For G₀ > 1 however, the wavefilter would fail.     

The speciation of Fe(II) and Fe(III) in natural waters

The interactions of Fe(II) and Fe(III) with the inorganic anions of natural waters have been examined using the specific interaction and ion pairing models. The specific interaction model as formulated by Pitzer is used to examine the interactions of the major components (Na⁺, Mg²⁺, Ca²⁺, K⁺, Sr²⁺, Cl⁻, SO₄⁻, HCO₃⁻, Br⁻, CO₃²⁻, B(OH)₄⁻, B(OH)₃ and CO₂) of seawater and the ion pairing model is used to account for the strong interaction of Fe(II) and Fe(III) with major and minor ligands (Cl⁻, SO₄²⁻, OH⁻, HCO₃⁻, CO₃²⁻ and HS⁻) in the waters. The model can be used to estimate the activity and speciation of iron in natural waters as a function of composition (major sea salts) and ionic strength (0 to 3 M). The measured stability constants (K_FeX^*) of Fe(II) and Fe(III) have been used to estimate the thermodynamic constants (K_FeX) and the activity coefficient of iron complexes (γ_FeX) with a number of inorganic ligands in NaClO₄ medium at various ionic strengths: In(K_FeX/γ_Feγ_X) = InK_FeX − In(γ_FeX) The activity coefficients for free ions (γ_Fe, γ_x) needed for this extrapolation have been estimated from the Pitzer equations. The activity coefficients of the ion pairs have been used to determine Pitzer parameters (B_FeX, B_FeX⁰, C_FeX^φ) for the iron complexes. These results make it possible to estimate the stability constants for the formation of Fe(II) and Fe(III) complexes over a wide range of ionic strengths and in different media. The model has been used to determine the solubility of Fe(III) in seawater as a function of pH. The results are in good agreement with the measurements of Byrne and Kester and Kuma et al. When the formation of Fe organic complexes is considered, the solubility of Fe(III) in seawater is increased by about 25%.     

Parameters of JONSWAP spectral model for surface gravity waves—II. Predictability from real data

Monte Carlo simulation of wave spectra was carried out to provide an assessment of JONSWAP spectral model and parameters. The simulation method is found to be satisfactory because (a) it excludes the spectral variability due to geophysical factors from the sampling errors in the spectral estimates and the statistical uncertainty in determining the model parameters; and (b) the simulated spectra can represent ideal spectral estimates where the sampling errors have been minimized by increasing the degrees of freedom of the spectra. The latter (b) allows both the magnitude of sampling errors to be evaluated and errors due to statistical uncertainty to be isolated. Thus, the stimulation study provides a useful error analysis to assess the JONSWAP spectral model and parameters. For instance, it is found from the results that the sampling errors could be as high as 20% while errors due to uncertainty in determining the model parameter could be as high as 17%. However, the overall errors may be reduced to the minimum of approx. 15% if the simulated spectra have 80 degrees of freedom and constant values of σ_a and σ_b i.e. σ_a = 0.07 and σ_b = 0.09. This implies that the maximum accuracy of 85% may be achieved in JONSWAP spectral model even though the α parameter has been underestimated by about 1.5%. The overestimated values of γ might come from the underestimated α and the biased φ_m estimator caused by the statistical uncertainty in the presence of a sharp spectral peak. Although the scale parameters (α and φ_m) exhibit smaller errors and variability than the shape parameters (ψ, σ_a and σ_b), they are more sensitive to the degrees of freedom of the spectra and their estimators are not better than the estimators of shape parameters. The simulation experiments have also shown that simulated spectra at 20–40 degrees of freedom contain a substantial amount of sampling errors. Therefore, the measured wave spectra at the same degrees of freedom (20–40) are not suitable and should not be used for evaluating the accuracy of any wave spectral model.     

On the origin and the spreading of the shallow Mediterranean water core in the Iberian basin

The origin and the spreading of the shallow Mediterranean water core (Ms) in the Iberian basin is discussed with a quasi-synoptic hydrographic data set enhanced by chlorofluoromethane (CFM) measurements. Its characteristic density level is found to be σ_t = 27.4. Characterized by high temperature and CFM values, Ms enters the Iberian basin in the region of Cape St Vincent between depths of 500–750 dbar. A heat anomaly of >11.8 × 10⁹ J m⁻² is chosen as the boundary between the presence of Ms and the background field. The core is found in a tongue-like shape as well as in separate isolated eddies of both cyclonic and anticyclonic circulation. Using the optimum multiparameter analysis (Tomczak and Large, 1989, Journal of Geophysical Research, 94, 16141–16149), the North Atlantic Central Water (NACW), which mixes with the Mediterranean outflow to form Ms, turned out to be in the mean 1°C warmer and 0.11 saltier than in regions with minor Mediterranean influence. This points to the Gulf of Cadiz as the origin of Ms, where the Mediterranean oufflows is in contact with NACW of the appropriate characteristics.     

Effect of major ion variations in the marine environment on the specific gravity-conductivity-chlorinity-salinity relationship

Partial equivalent conductances and partial equivalent volumes of the major constituents in seawater were used to evaluate the specific gravity-conductivity-chlorinity-salinity relationships in the marine environment. For example, in the open ocean, the relationships between Cl‰ and both S‰ and specific gravity are valid to within 0.014‰ and 0.014 σ_t, respectively. The relationships between conductivity and S‰ and specific gravity are valid to within 0.006‰ and 0.007 σ_t. In river diluted nearshore areas specific gravity anomalies inferred from Cl‰, can be as great as 0.06 σ_t and 0.04 σ_t when inferred from a conductivity ratio measurement.     

Modelling of natural and synthetic polyelectrolyte interactions in natural waters by using SIT, Pitzer and Ion Pairing approaches

In this paper SIT and Pitzer models are used for the first time to describe the interactions of natural and synthetic polyelectrolytes in natural waters. Measurements were made potentiometrically at 25 °C in single electrolyte media, such as Et₄NI and NaCl (for fulvic acid 0.1 < I /mol L^− 1 < 0.75), and in a multi-component medium simulating the composition of natural waters at a wide range of salinities (for fulvic and alginic acids: 5 < S < 45) with particular reference to sea water [Synthetic Sea Water for Equilibrium studies, SSWE]. In order to simplify calculations, SSWE was considered to be a “single salt” BA, with cation B and anion A representing all the major cations (Na⁺, K⁺, Mg²⁺, Ca²⁺) and anions (Cl⁻, SO₄²⁻) in natural sea water, respectively. The ion pair formation model was also applied to fulvate and alginate in artificial sea water by examining the interaction of polyanions with the single sea water cation. Results were compared with those obtained from previous speciation studies of synthetic polyelectrolytes (polyacrylic and polymethacrylic acids of different molecular weights). Results indicate that the SIT, Pitzer and Ion Pairing formation models used in studies of low molecular weight electrolytes may also be applied to polyelctrolytes with a few simple adjustments.     

Variability of C02 distributions and sea-air fluxes in the central and eastern equatorial Pacific during the 199–1994 El Nin˜o

As part of the U.S. JGOFS Program and the NOAA Ocean-Atmosphere Carbon Exchange Study (DACES), measurements of C02 partial pressure were made in the atmosphere and in the surface waters of the central and eastern equatorial Pacific during the boreal spring and autumn of 1992, the spring of 1993, and the spring and autumn of 1994. Surface-water pC0₂ data indicate significant diurnal, seasonal, and interannual variations. The largest variations were associated with the 1991–1994 ENSO event, which reached maximum intensity in the spring of 1992. The lower values of surface-water ΔpC0₂ observed during the 1991–1994 ENSO period were the result of the combined effects of both remotely and locally forced physical processes. The warm pool, which reached a maximum eastward extent in January-February of 1992, began in September of 1991 as a series of westerly wind events lasting about 30 days. Each wind event initiated an eastward propagating Kelvin wave which caused a deepening of the thermocline. By the end of January 1992 the thermocline was at its maximum depth, so that the upwelled water was warm and C0₂-depleted. In April of the same year, the local winds were weaker than normal, and the upwelling was from shallow depths. These changes resulted in a lower-than-normal C02 flux to the atmosphere. The results show that for the one-year period from the fall of 1991 until the fall of 1992, approximately 0.3 GtC were released to the atmosphere; 0.6 GtC were released in 1993, and 0.7 GtC in 1994, in good agreement with the model results of Ciais et al. [Science,269,1098–1102;J. Geophys. Res.,100, 5051–5070]. The net reduction of the ocean-atmosphere C02 flux during the 1991–1994 El Nifio was on the order of 0.8 – 1.2 GtC. Thus, the total amount of C02 sequestered in the equatorial oceans during the prolonged 1991–1994 El Nin˜o period was about 25% higher than the severe El Nin˜o of 1982–1983.     

On interaction between intermediate-depth long waves and deep-water short waves

The interaction between a unidirectional deep-water short-wave train and an intermediate water-depth long wave is studied. The steady solutions are derived up to third order in wave steepness, respectively, using two different approaches: a conventional perturbation method employing linear phase functions to describe both long- and short-wave phases and a phase modulation method using a modulational phase function to model the short-wave phase. The two results are shown to be identical for a parametric range χ₁ coth Kdχ₃ ≤ 0.5, where χ₃ is the short-to-long wavelength ratio, χ₁ and K are, respectively, the long-wave steepness and wavenumber, and d the water depth. When χ₁ coth Kd approaches χ₃, the conventional solution converges slowly and eventually diverges for χ₁ coth Kdχ₁. The slow convergence of the conventional solution results from the approximation of a modulated short-wave phase by a linear phase formulation. In addition to the increasing modulation of the short-wave phase, amplitude and wavenumber as the water depth decreases, it is found that the modulation of the short-wave intrinsic frequency and potential amplitude along the long-wave surface become significant. Previous results about virtually non-modulated short-wave intrinsic frequency and potential amplitude are only limited to the case of unidirectional wave modulation in very deep water.     

Formation of the salinity minimum in the Mixed Water Region between the Oyashio and Kuroshio Fronts

The salinity minimum frequently occurring in the Mixed Water Region between the Oyashio and Kuroshio Fronts seems to originate from the salinity minimum at the density of 26.8σ_θ called the North Pacific Intermediate Water. We examined water exchange of this region with the Oyashio and the Kuroshio Extension using mixing ratio R_K defined as (θ - θ_OY)/(θ_K - θ_OY) × 100, where θ_OY, θ_K, and θ represent potential temperature of the Oyashio and Kuroshio Waters and their mixture on the isopycnal surfaces, respectively. CTD data were obtained by repeated observation from January 1990 to May 1991. R_K increases southward from the Oyashio Front to the Kuroshio Front with the range of −20 to 120%. The gradient of R_K on the isopycnal surfaces is large around the Oyashio Front above the 26.8σ_θ surface, while it is large around the Kuroshio Front below it. This agrees with the average R_K in the Mixed Water Region decreasing greatly with the increase of density at densities less dense than 26.8σ_θ. We calculated thickness and volume transport of the Oyashio between the isopycnal surfaces near the coast of Hokkaido. They increase largely with density at densities less dense than 26.8σ_θ. It is supposed that the salinity minimum in the Mixed Water Region is the upper limit of the water largely influenced by the Oyashio Water. Its density could depend only on the density structure of the Oyashio.     

Spectrophotometric pH measurement in estuaries using thymol blue and m-cresol purple

The conditional acid dissociation constants (pK_a′) of two sulfonephthalein dyes, thymol blue (TB) and m-cresol purple (mCP), were assessed throughout the estuarine salinity range (0<S<40) using a tris/tris–HCl buffer and spectrophotometric measurement. The salinity dependence of the pK_a′ of both dyes was fitted to the equations (25 °C, total proton pH scale, mol kg soln⁻¹):

The estimated accuracy of pH measurements using these calculated pK_a′ values is considered to be comparable to that possible with careful use of a glass electrode (±0.01 pH unit) but spectrophotometric measurements in an estuary have the significant advantage that it is not necessary to calibrate an electrode at different salinities. pH was measured in an estuary over a tidal cycle with a precision of ±0.0005 pH unit at high (S>30) salinity, and ±0.002 pH unit at low (S<5) salinity. The pH increased rapidly in the lower salinity ranges (0<S<15) but less rapidly at higher salinities.     

Characterization of vertical mixing at a tidal-front on Georges Bank

Studies of mixing were done at the northern flank of Georges Bank in the summer and autumn of 1988. Two time-series of the evolution and intensity of microstructure were examined over a tidal period in the context of tidal forcing and the evolution of the density and velocity field at the site. From the CTD, ADCP and microstructure observations (EPSONDE) on Georges Bank, several interesting features of the mixing processes were found. High dissipation and diffusivity regions appear near the bottom of the Bank. Turbulence near the bottom is highest in intensity and reaches farthest from the bottom at peak tidal flow and diminishes in intensity and vertical extent as the flow decreases. The thickness of the bottom turbulent layer has its maximum value when the flow is strongest and the stratification is weakest. Characterization of the dissipation rate and turbulent diffusivities in respect to buoyancy frequency N, current shear S, Richardson Number R_i and ε/νN² was done. Dissipation and χ_T showed little dependence on shear or N² but decreased at larger R_i. χ_t was found to be higher in regions of higher N² and increased as ε/νN² increased. K_T, K and K_ν, were all highest near the bottom in excess of 10⁻²m²s⁻¹ and decreased towards the surface. There was little suggestion of a dependence of mixing efficiency on S², R_i or ε/νN², but some indication that Γ decreases with decreasing N².     

The reliability of the thermodynamic constants for the dissociation of carbonic acid in seawater

Laboratory measurements of all four CO₂ parameters [f_CO2 ( = fugacity of CO₂), pH, TCO₂ ( = total dissolved inorganic carbon), and TA ( = total alkalinity)] were made on the same sample of Gulf Stream seawater (S = 35) as a function of temperature (5–35 °C) and the ratio of TA/TCO₂ (X) (1.0–1.2). Overall the measurements were consistent to ±8 μ atm in f_CO2, ± 0.004 in pH, ± 3 μ mol kg⁻¹ in TCO₂, and ± 3 μ mol kg⁻¹ in TA with the thermodynamic constants of Goyet and Poisson (1989), Roy et al. (1993), and Millero (1995). Deviations between the measured pH, TCO₂, TA and those calculated from various input combinations increase with increasing X when the same constants are used. This trend in the deviations indicates that the uncertainties in pK₂ become important with increasing X (surface waters), but are negligible for samples with the lower X (deep waters). This trend is < 5 μ mol kg⁻¹ when the pK₂ values of Lee and Millero (1995) are used.The overall probable error of the calculated f_CO2 due to uncertainties in the accuracy of the parameters (pH, TCO₂, TA, pK₀, pk₁, and pK₂) is ± 1.2%, which is similar to the differences between the measured values and those calculated using the thermodynamic constants of Millero (1995).The calculated values of pK₁, (from f_CO2-TCO₂-TA) agree to within ± 0.004 compared to the results of Dickson and Millero (1987), Goyet and Poisson (1989), Roy et al. (1993), and Millero (1995) over the same experimental conditions. The calculated values of pK₂ (from pH-TCO₂-TA) are in good agreement (± 0.004) with the results of Lee and Millero (1995) and also in reasonable agreement (± 0.008) with the results of Goyet and Poisson (1989), Roy et al. (1993), and Millero (1995). The salinity dependence of our derived values of pK₁ and pK₂, (S = 35) can be estimated using the equations determined by Millero (1995).     

Ventilation of the North Pacific subtropical pycnocline and mode water formation

The annual subduction rate of the North Pacific was calculated based on isopycnally averaged hydrographic climatology (HydroBase), high-resolution winter mixed-layer climatology (NWMLC), and various wind stress climatologies from ship reports, numerical weather prediction products, and satellite products. The calculation was performed using Lagrangian coordinates in the same manner as in previous works, except a less smoothed oceanic climatology (HydroBase and NWMLC) was used instead of a World Ocean Atlas. Differences in the wind stress climatologies have very little effect on subduction rate estimates. The subduction rate census for density classes showed peaks corresponding to subtropical mode water (STMW), central mode water (CMW), and eastern subtropical mode water (ESTMW). The deeper mixed layer and the associated sharper mixed-layer fronts in the present climatology resulted in a larger lateral induction, which boosted the subduction rate, especially for the potential density anomaly (σ_θ) range of the lighter STMW (25.0 < σ_θ < 25.2 kg m⁻³) and lighter CMW (26.0 < σ_θ < 26.2 kg m⁻³), compared to previous estimates. The renewal time of permanent pycnocline water was estimated as the volume of water divided by the subduction rate for each σ_θ class: 2–4 years for ESTMW (24.5 < σ_θ < 25.2 kg m⁻³), 2 years for the lighter STMW (25.0 < σ_θ < 25.3 kg m⁻³), 5–9 years for the denser STMW (25.3 < σ_θ < 25.6 kg m⁻³), 10–20 years for the lighter CMW (26.0 < σ_θ < 26.2 kg m⁻³), 20–30 years for the middle CMW (26.2 < σ_θ < 26.3 kg m⁻³), and 60 years or longer for the denser CMW (26.3 < σ_θ < 26.6 kg m⁻³). A comparison of the water volume and subduction rate in potential temperature–salinity (θ–S) space indicated that the upper permanent pycnocline water (25.0 < σ_θ < 26.2 kg m⁻³) was directly maintained by nondiffusive subduction of winter surface water, including STMW and lighter CMW. The lower permanent pycnocline water (26.2 < σ_θ < 26.6 kg m⁻³) may be maintained through the subduction of fresher and colder water from the subarctic–subtropical transition region and subsequent mixing with saltier and warmer water. Diagnosis of the potential vorticity (PV) of the subducted water demonstrated that the low PV of STMW was mainly due to the large subduction rate, whereas that of both ESTMW and CMW was due mainly to the small density advection rate (cross-isopycnal flow). Additionally, a relatively large subduction rate probably contributes to the low PV of part of the lighter CMW (ESTMW) formed in the region around 38°N and 170°W (28°N and 145°W), which is characterized by a relatively thick winter mixed layer and an associated mixed-layer front, causing a large lateral induction rate.