Gravity currents with tailwaters in Boussinesq and non-Boussinesq systems: two-layer shallow-water dam-break solutions and Navier–Stokes simulations

We consider the dam-break initial stage of propagation of a gravity current of density $\rho _{c}$ released from a lock (reservoir) of height $h_0$ in a channel of height $H$ . The channel contains two-layer stratified fluid. One layer, called the “tailwater,” is of the same density as the current and is of thickness $h_T (< h_0)$ , and the other layer, called the “ambient,” is of different density $\rho _{a}$ . Both Boussinesq ( $\rho _{c}/\rho _{a}\approx 1$ ) and non-Boussinesq systems are investigated. By assuming a large Reynolds number, we can model the flow with the two-layer shallow-water approximation. Due to the presence of the tailwater, the “jump conditions” at the front of the current are different from the classical Benjamin formula, and in some circumstances (clarified in the paper) the interface of the current matches smoothly with the horizontal interface of the tailwater. Using the method of characteristics, analytical solutions are derived for various combinations of the governing parameters. To corroborate the results, two-dimensional direct numerical Navier–Stokes simulations are used, and comparisons for about 80 combinations of parameters in the Boussinesq and non-Boussinesq domains are performed. The agreement of speed and height of the current is very close. We conclude that the model yields self-contained and fairly accurate analytical solutions for the dam-break problem under consideration. The results provide reliable insights into the influence of the tailwater on the propagation of the gravity current, for both heavy-into-light and light-into-heavy motions. This is a significant extension of the classical gravity-current theory from the particular $h_T=0$ point to the $h_T > 0$ domain.     

The flow of high-Reynolds axisymmetric gravity currents of a stratified fluid into a stratified ambient: shallow-water and box model solutions

We consider the axisymmetric flow (in a full cylinder or a wedge) of high-Reynolds-number Boussinesq gravity currents and intrusions systems in which both the ambient and the propagating “current” are linearly stratified. The main focus is on a current of fixed volume released from a cylinder lock; the height ratio of the fluids H, and the stratification parameter of the ambient S, are quite general. We develop a one-layer shallow-water model. The internal stratification enters as a new dimensionless parameter, ${\sigma \in [0, 1]}$ . In general, the time-dependent motion is obtained by standard finite-difference solutions; a self-similar analytical solution exists for S?= 0. We show that, in general, the speed of propagation decreases when the internal stratification becomes more pronounced (σ increases). We also developed a box-model approximation, and show that the resulting radius of propagation is in good agreement with the more rigorous shallow-water prediction.     

Gravity currents with double stratification: a numerical and analytical investigation

We consider high-Reynolds-number Boussinesq gravity current and intrusion systems in which both the ambient and the propagating “current” are linearly stratified. The main focus is on a current of fixed volume released from a rectangular lock; the height ratio of the fluids $H$ , the stratification parameter of the ambient $S$ , and the internal stratification parameter of the current, $\sigma $ , are quite general. We perform two-dimensional Navier–Stokes simulation and compare the results with those of a previously-published one-layer shallow-water model. The results provide insights into the behavior of the system and enhance the confidence in the approximate model while also revealing its limitations. The qualitative predictions of the model are confirmed, in particular: (1) there is an initial “slumping” stage of propagation with constant speed $u_N$ , after which $u_N$ decays with time; (2) for fixed $H$ and $S$ , the increase of $\sigma $ causes a slower propagation of the current; (3) for some combinations of the parameters $H,S, \sigma $ the fluid released from the lock lacks initially (or runs out quickly of) buoyancy “driving power” in the horizontal direction, and does not propagate like a gravity current. There is also a fair quantitative agreement between the predictions of the model and the simulations concerning the spread of the current.     

A general solution of Benjamin-type gravity current in a channel of non-rectangular cross-section

We consider the steady-state propagation of a high-Reynolds-number gravity current in a horizontal channel along the horizontal coordinate x. The bottom and top of the channel are at z =?0, H, and the cross-section is given by the quite general form ?f ₁(z) ≤?y ≤?f ₂(z) for 0 ≤?z ≤?H, where f _1,2 are piecewise continuous functions and f ₁ +?f ₂ >?0 for ${z \in(0,H)}$ . The interface of the current is horizontal, the (maximum) thickness is h, its density is ρ _c . The reduced gravity g′ =?|ρ _c /ρ _a ? 1|g (where ${- g\hat{z}}$ is the gravity acceleration and ρ _a the density of the ambient) drives the current with speed U into the stationary ambient fluid. We show that the dimensionless Fr =?U/(g′ h)^1/2, the rate of energy dissipation (scaled with the rate of pressure work), and the dimensionless head-loss Δ/h, can be expressed by compact formulas which involve three integrals over the cross-section areas of the current and ambient. By some standard manipulations these integrals are simplified into quite simple line-integrals of the shape-function of the channel, f(z) =?f ₁(z) +?f ₂(z), and of z f(z). This theory applies to Boussinesq and non-Boussinesq currents of “heavy” (bottom) and “light” (top) type. The classical results of Benjamin (J Fluid Mech 31:209–248, 1968) for a rectangular channel are fully recovered. We also recover the Fr results of Marino and Thomas (J Fluid Eng 131(5):051201, 2009) for channels of shape y =?±b z ^α (where b, α are positive constants); in addition, we consider the energy dissipation of these flows. The results provide insights into the effect of the cross-section shape on the behavior of the steady-state current, in quite general cases, for both heavy-into-light and light-into-heavy fluid systems, Boussinesq and non-Boussinesq. In particular, we show that a very deep current displays ${Fr = \sqrt{2}}$ , and is dissipative; the value of Fr and rate of dissipation (absolute value) decrease when the thickness of the current increases. However, in general, energy considerations restrict the thickness of the current by a clear-cut condition of the form h/H ≤?a _max ?< 1.     

Numerical simulations of intrusive gravity currents interacting with a bottom-mounted obstacle in a continuously stratified ambient

In this study, the flow dynamics of intrusive gravity currents past a bottom-mounted obstacle were investigated using highly resolved numerical simulations. The propagation dynamics of a classic intrusive gravity current was first simulated in order to validate the numerical model with previous laboratory experiments. A bottom-mounted obstacle with a varying non-dimensional height of \(\tilde{D}=D/H\), where D is the obstacle height and H is the total flow depth, was then added to the problem in order to study the downstream flow pattern of the intrusive gravity current. For short obstacles, the intrusion re-established itself downstream without much distortion. However, for tall obstacles, the downstream flow was found to be a joint effect of horizontal advection, overshoot-springback phenomenon, and associated Kelvin-Helmholtz instabilities. Analysis of the numerical results show that the relationship between the downstream propagation speed and the obstacle height can be subdivided into three regimes: (1) a retarding regime (\(\tilde{D}\) \(\approx \) 0–0.3) where a 30 % increase in obstacle height leads to a 20 % reduction in propagation speed, simply due to the obstacle’s retarding effect; (2) an impounding regime (\(\tilde{D}\) \(\approx \) 0.3–0.6) where the additional 30 % increase in obstacle height only leads to a further (negligible) 5 % reduction in propagation speed, due to the accelerating effect of upstream impoundment and downstream enhanced mixing; and (3) a choking regime (\(\tilde{D}\) \(\approx \) 0.6–1.0) where the propagation speed is dramatically reduced due to the dominance of the obstacle’s blocking effect. The obstacle thickness was found to be irrelevant in determining the downstream propagation speed at least for the parameter range explored in this study. The present work highlights the significance of topographic effects in stratified flows with horizontal pressure forcing.     

Quantifying a significance of sediment particle size to hyporheic sedimentary oxygen demand with a permeable stream bed

A mechanistic model of sedimentary oxygen demand (SOD) for hyporheic flow is presented. The permeable sediment bed, e.g. sand or fine gravel, is considered with hydraulic conductivity in the range $0.1 < K < 20$  cm/s. Hyporheic pore water flow is induced by pressure fluctuations at the sediment/water interface due to near-bed turbulent coherent motions. A 2-D advection–diffusion equation is linked to the pore water flow model to simulate the effect of advection–dispersion driven by interstitial flow on oxygen transfer through the permeable sediment. Microbial oxygen uptake in the sediment is expressed as a function of the microbial growth rate, and is related to the sediment properties, i.e. the grain diameter $(d_{s})$ and porosity $(\phi )$ . The model describes the significance of sediment particle size to oxygen transfer through the sediment and microbial oxygen uptake: With increasing grain diameter $(d_{s})$ , the hydraulic conductivity $(K)$ increases so does the oxygen transfer rate, while particle surface area per volume (the available surface area for colonization by biofilms) decreases reducing the microbial oxygen uptake rate. Simulation results show that SOD increases as the hydraulic conductivity $(K)$ increases before a threshold has been reached. After that, SOD diminishes with the increment of the hydraulic conductivity $(K)$ .     

Initial mixing of inclined dense jet in perpendicular crossflow

A comprehensive experimental investigation for an inclined ( $60^{\circ }$ to vertical) dense jet in perpendicular crossflow—with a three-dimensional trajectory—is reported. The detailed tracer concentration field in the vertical cross-section of the bent-over jet is measured by the laser-induced fluorescence technique for a wide range of jet densimetric Froude number $Fr$ and ambient to jet velocity ratios $U_r$ . The jet trajectory and dilution determined from a large number of cross-sectional scalar fields are interpreted by the Lagrangian model over the entire range of jet-dominated to crossflow-dominated regimes. The mixing during the ascent phase of the dense jet resembles that of an advected jet or line puff and changes to a negatively buoyant thermal on descent. It is found that the mixing behavior is governed by a crossflow Froude number $\mathbf{F} = U_r Fr$ . For $\mathbf{F} < 0.8$ , the mixing is jet-dominated and governed by shear entrainment; significant detrainment occurs and the maximum height of rise $Z_{max}$ is under-predicted as in the case of a dense jet in stagnant fluid. While the jet trajectory in the horizontal momentum plane is well-predicted, the measurements indicate a greater rise and slower descent. For $\mathbf{F} \ge 0.8$ the dense jet becomes significantly bent-over during its ascent phase; the jet mixing is dominated by vortex entrainment. For $\mathbf{F} \ge 2$ , the detrainment ceases to have any effect on the jet behavior. The jet trajectory in both the horizontal momentum and buoyancy planes are well predicted by the model. Despite the under-prediction of terminal rise, the jet dilution at a large number of cross-sections covering the ascent and descent of the dense jet are well-predicted. Both the terminal rise and the initial dilution for the inclined jet in perpendicular crossflow are smaller than those of a corresponding vertical jet. Both the maximum terminal rise $Z_{max}$ and horizontal lateral penetration $Y_{max}$ follow a $\mathbf{F}^{-1/2}$ dependence in the crossflow-dominated regime. The initial dilution at terminal rise follows a $S \sim \mathbf{F}^{1/3}$ dependence.     

Data depth for the uniform distribution

Given a set $X$ of $k$ points and a point $z$ in the $n$ -dimensional euclidean space, the Tukey depth of $z$ with respect to $X$ , is defined as $m/k$ , where $m$ is the minimum integer such that $z$ is not in the convex hull of some set of $k-m$ points of $X$ . If $z$ belongs to the closed region $B$ delimited by an ellipsoid, define the continuous depth of $z$ with respect to $B$ as the quotient $V(z)/\text{ Vol }(B)$ , where $V(z)$ is the minimum volume of the intersection of $B$ with the halfspaces defined by any hyperplane passing through $z$ , and $\text{ Vol }(B)$ is the volume of $B$ . We consider $z$ a random variable and prove that, if $z$ is uniformly distributed in $B$ , the continuous depth of $z$ with respect to $B$ has expected value $1/2^{n+1}$ . This result implies that if $z$ and $X$ are uniformly distributed in $B$ , the expected value of Tukey depth of $z$ with respect to $X$ converges to $1/2^{n+1}$ as the number of points $k$ goes to infinity. These findings have applications in ecology, namely within the niche theory, where it is useful to explore and characterize the distribution of points inside species niche.     

Waves and turbulence in katabatic winds

The measurements taken during the Vertical Transport and Mixing Experiment (VTMX, October, 2000) on a northeastern slope of Salt Lake Valley, Utah, were used to calculate the statistics of velocity fluctuations in a katabatic gravity current in the absence of synoptic forcing. The data from ultrasonic anemometer-thermometers placed at elevations 4.5 and 13.9 m were used. The contributions of small-scale turbulence and waves were isolated by applying a high-pass digital (Elliptical) filter, whereupon the filtered quantities were identified as small-scale turbulence and the rest as internal gravity waves. Internal waves were found to play a role not only at canonical large gradient Richardson numbers $(\overline{\hbox {Ri}_\mathrm{g} } >1)$ , but sometimes at smaller values $(0.1 < \overline{\hbox {Ri}_\mathrm{g}}<1)$ , in contrast to typical observations in flat-terrain stable boundary layers. This may be attributed, at least partly, to (critical) internal waves on the slope, identified by Princevac et al. [1], which degenerate into turbulence and help maintain an active internal wave field. The applicability of both Monin-Obukhov (MO) similarity theory and local scaling to filtered and unfiltered data was tested by analyzing rms velocity fluctuations as a function of the stability parameter z/L, where L is the Obukhov length and z the height above the ground. For weaker stabilities, $\hbox {z/L}<1$ , the MO similarity and local scaling were valid for both filtered and unfiltered data. Conversely, when $\hbox {z/L}>1$ , the use of both scaling types is questionable, although filtered data showed a tendency to follow local scaling. A relationship between z/L and $\overline{\hbox {Ri}_\mathrm{g} }$ was identified. Eddy diffusivities of momentum $\hbox {K}_\mathrm{M}$ and heat $\hbox {K}_\mathrm{H}$ were dependent on wave activities, notably when $\overline{\hbox {Ri}_\mathrm{g} } > 1$ . The ratio $\hbox {K}_{\mathrm{H}}/\hbox {K}_{\mathrm{M}}$ dropped well below unity at high $\overline{\hbox {Ri}_\mathrm{g} }$ , in consonance with previous laboratory stratified shear layer measurements as well as other field observations.     

Aeolian erosion of storage piles yards: contribution of the surrounding areas

Dust emissions from stockpiles surfaces are often estimated applying mathematical models such as the widely used model proposed by the USEPA. It employs specific emission factors, which are based on the fluid flow patterns over the near surface. But, some of the emitted dust particles settle downstream the pile and can usually be re-emitted which creates a secondary source. The emission from the ground surface around a pile is actually not accounted for by the USEPA model but the method, based on the wind exposure and a reconstruction from different sources defined by the same wind exposure, is relevant. This work aims to quantify the contribution of dust re-emission from the areas surrounding the piles in the total emission of an open storage yard. Three angles of incidence of the incoming wind flow are investigated ( $30^{\circ }, 60^{\circ }$ and $90^{\circ }$ ). Results of friction velocity from numerical modelling of fluid dynamics were used in the USEPA model to determine dust emission. It was found that as the wind velocity increases, the contribution of particles re-emission from the ground area around the pile in the total emission also increases. The dust emission from the pile surface is higher for piles oriented $30^{\circ }$ to the wind direction. On the other hand, considering the ground area around the pile, the $60^{\circ }$ configuration is responsible for higher emission rates (up to 67 %). The global emissions assumed a minimum value for the piles oriented perpendicular to the wind direction for all wind velocity investigated.     

On the front conditions for gravity currents in channels of general cross-section

We consider the propagation of a high-Reynolds-number gravity current in a horizontal channel with general cross-section whose width is \(f(z), 0 \le z\le H\), and the gravity acceleration g acts in \(-z\) direction. (The classical rectangular cross-section is covered by the particular case \(f(z) =\) const.) We assume a two-layer system of homogeneous fluids of constant densities \(\rho _{c}\) (current, of height \(h < H \)) and smaller \(\rho _{a}\) (ambient, filling the remaining part of the channel). We focus attention on the calculation and assessment of the nose Froude-number condition \(Fr = U/(g' h)^{1/2}\); here U is the speed of propagation of the current and \(g' = (\rho _{c}/\rho _{a}-1) g\) is the reduced gravity. We first revisit the steady-state current, and derive compact insightful expressions of Fr and energy dissipation as a function of \(\varphi \) (\(=\) area fraction occupied by the current in the cross-section). We show that the head loss \(\delta _0\) on the stagnation line is formally a degree of freedom in the determination of \(Fr(\varphi )\), and we clarify the strong connections with the head loss \(\delta \) in the ambient fluid, and with the overall rate of dissipation \(\dot{{\mathcal{D}}}\). We demonstrate that the closure \(\delta _0 = 0\) [suggested by Benjamin (J Fluid Mech 31, 209–248, 1968) for the rectangular cross-section] produces in general the smallest Fr for a given \(\varphi \); the results are valid for a significant range \([0, \varphi _{\max }]\), in which the current is dissipative, except for the point \(\varphi _{\max }\) where \(\delta = \dot{{\mathcal{D}}} = 0\). We show that imposing the closure \(\delta = \dot{{\mathcal{D}}} = 0\), which corresponds to an energy-conserving or non-dissipative current, produces in general unacceptable restrictions of the range of validity, and large values of Fr; in particular, deep currents (\(\varphi < 0.3\) say) must be excluded because they are inherently dissipative. On the other hand, the compromise closure \(\delta (\varphi ) =\delta _0(\varphi )\) produces the simple \(Fr(\varphi ) = \sqrt{2}(1 - \varphi )\) formula whose values and dissipation properties are very close, and the range of validity is identical, to these obtained with Benjamin’s closure (moreover, we show that this corresponds to circulation-conservation solutions). The results are illustrated for practical cross-section geometries (rectangle, \(\Delta \) and \(\nabla \) triangle, circle, and the general power-law \(f(z) = b z ^\alpha \) (\(b>0, \alpha \ge 0, 0< z \le H\)). Next, we investigate the connection of the steady-state results with the time-dependent current, and show that in a lock-released current the rate of dissipation of the system is equal to, or larger than, that obtained for Fr corresponding to the conditions at the nose of the current. The results and insights of this study cover a wide range of cross-section geometry and apply to both Boussinesq and non-Boussinesq systems; they reveal a remarkable robustness of Fr as a function of \(\varphi \).     

Extension of the skin shear stress Li’s relationship to the flat bed

A proper estimation of the skin shear stress τ _s is necessary for a proper evaluation of sediment flux at the sediment–fluid interface. Several empirical formulas of the skin shear stress have been proposed in the literature for rippled bed as function of the factor form η/λ (η and λ are respectively the height and wavelength of the bedform). These formulas express that in the presence of bedform, τ _s is a partition of the total shear stress τ _b . In contrast, when the bottom is flat, τ _s is exactly equal to τ _b . Based on in situ measurements, Li (J Geophys Res 99:791–799, 1994) has proposed a new formula of τ _s depending on u _*/η (u _* is the friction velocity, ${u_{*}=\sqrt{\tau_{b}/\rho}}$ ), but not as a function of η/λ. This formulation appears physically more realistic, but does not cover the flat bottom range. The purpose of this short note is therefore the extension of the Li’s expression to this type of bottom.     

Moisture effects on eolian particle entrainment

In wind tunnel experiments, we study the effects of soil moisture on the threshold condition to entrain fine grain sand/silt into eolian flow and the near-bed concentration of airborne particles. To study the effect of particle shape on moisture bonding, we use two types of particles nearly equal in size: spherical glass beads $(d_{50} = 134\,\upmu \mathrm{m})$ and sieved quartz sand $(d_{50} = 139 \,\upmu \mathrm{m})$ . Both are poorly graded soils. We conducted these experiments at low moisture contents $({<}1\,\%)$ . We found that the spherical particles were more sensitive to changes in moisture than the sand, attributable to the large differences in specific surface area of the two particles. The larger specific surface area for sand is due to the surface roughness of the angular sand particle. Consequently, sand “stores” more moisture via surface adsorption, requiring higher soil moisture content to form liquid bridges between sand particles. Based on these findings, we extend the concept of a threshold moisture content, $w^{\prime }$ —originally proposed for clayey soils—to soils that lack any measureable clay content. This allows application of existing models developed for clayey soils that quantify the moisture effect on the threshold friction velocity to sand and silty soils (i.e., clay content $=$ 0). Additionally, we develop a model that quantifies the moisture effects on near-surface airborne particulate concentration, using experimental observations to determine the functional dependence on fluid and particle properties, including soil specific area. These models can be applied to numerical simulation of particulate plume formation and dispersion.     

On the effective hydraulic conductivity and macrodispersivity for density-dependent groundwater flow

In this paper, semi-analytical expressions of the effective hydraulic conductivity ( $K^{E})$ and macrodispersivity ( $\alpha ^{E})$ for 3D steady-state density-dependent groundwater flow are derived using a stationary spectral method. Based on the derived expressions, we present the dependence of $K^{E}$ and $\alpha ^{E}$ on the density of fluid under different dispersivity and spatial correlation scale of hydraulic conductivity. The results show that the horizontal $K^{E}$ and $\alpha ^{E}$ are not affected by density-induced flow. However, due to gravitational instability of the fluid induced by density contrasts, both vertical $K^{E}$ and $\alpha ^{E}$ are found to be reduced slightly when the density factor ( $\gamma $ ) is less than 0.01, whereas significant decreases occur when $\gamma $ exceeds 0.01. Of note, the variation of $K^{E}$ and $\alpha ^{E}$ is more significant when local dispersivity is small and the correlation scale of hydraulic conductivity is large.     

Ammonium regeneration and carbon utilization by marine bacteria grown on mixed substrates

We examined the impact of exposing natural populations of marine bacteria (from seawater collected near Woods Hole, Massachusetts, USA) to multiple nitrogen and carbon sources in a series of batch growth experiments conducted from 1989 through 1990. The substrate C:N ratio (C:N_s) was varied from 1.5:1 to 10:1 either with equal amounts of NH ₄ ⁺ and different amino acids or an amino acid mixture, all supplemented with glucose to maintain the C:N_s ratio equal to that of the respective amino acid, or with combinations of glucose and NH ₄ ⁺ alone. A common feature of the experiments involving amino acids was the concurrent uptake of NH ₄ ⁺ and amino acids that persisted as long as a readily assimilable carbon source (glucose in our case) was taken up. There was no net regeneration of NH ₄ ⁺ , even though catabolism of amino acids occurred. Regeneration of NH ₄ ⁺ was evident only after glucose was completely utilized, which usually occurred at the end of exponential growth. The contribution of¹⁵NH ₄ ⁺ to total nitrogen uptake by the end of exponential growth varied from ~60 to 80% when individual amino acids were present and down to ~24% when the amino acid mixture was added. These estimates are conservative because we did not account for possible isotope dilution effects resulting from amino acid catabolism. When NH ₄ ⁺ and glucose were the sole nitrogen and carbon sources, there was a stoichiometric balance between glucose and NH ₄ ⁺ uptake over a wide range of C:N_s ratios, leading to a constant bacterial biomass C:N ratio (C:N_B) of ~4.5:1. As a result NH ₄ ⁺ usage varied from 50% when the C:N_s ratio was 3.6:1, to 100% when the C:N_s ratio was 10:1. Gross growth efficiency varied from ~60% when NH ₄ ⁺ plus glucose were added alone or with the amino acid mixture, to 47% when the individual amino acids were used in place of the mixture. It is thus evident that actively growing bacteria will act as sinks for nitrogen when a carbon source that can be assimilated easily is available to balance NH ₄ ⁺ uptake, even when amino acids are available and are being co-metabolized.     

Passive scalar roughness lengths for atmospheric boundary layer flow over complex, fractal topographies

When modeling atmospheric boundary layer flow over rough landscapes, surface fluxes of flow quantities (momentum, temperature, etc.) can be described with equilibrium logarithmic law expressions, all of which require specification of a roughness length that is, physically, the elevation at which the flow quantity equals its surface value. In high Reynolds number flows, such as the atmospheric boundary layer, inertial forces associated with turbulent eddy motions are responsible for surface momentum fluxes (form, or pressure drag). Surface scalar fluxes, on the other hand, occur exclusively via diffusion in the immediate vicinity of the topography—the interfacial region—before being advected by turbulent eddy motions into the bulk of the flow. Owing to this difference in surface transfer mechanism, the passive scalar roughness length, $z_{0S}$ , is known to be less than the momentum roughness length, $z_0$ . In this work, classical relations are used to specify $z_{0S}$ during large-eddy simulation of atmospheric boundary layer flow over aerodynamically rough, synthetic, fractal topographies which exhibit power-law height energy spectrum, $E_h (k) \sim k^{\beta _s}$ , where $\beta _s$ is a (predefined) spectral exponent. These topographies are convenient since they resemble natural landscapes and $\beta _s$ can be varied to change the topography’s aerodynamic roughness (the study considers a suite of topographies with $-2.4 \le \beta _s \le -1.2$ , where $-2.4$ and $-1.2$ are the “most smooth” and “most rough” cases, respectively, corresponding with roughness Reynolds number, $Re_0 \approx 10$ and $300$ ). It is often assumed that $z_{0S}/z_{0} \approx 10^{-1}$ for all $Re_0$ . But results from this work show that the roughness length ratio, $z_{0S}/z_{0}$ , depends strongly on $Re_0$ , ranging between $10^{-3}$ and $10^{-1}$ .     

A simple and precise method for fitting a von Bertalanffy growth curve

The parameter K of the von Bertalanffy equation, as developed by Beverton and Holt (1957), is first estimated by the relation $$\log _e \left( {dL_t /dt} \right) = A - Kt$$ where dLt/dt denotes growth increments per a unit of age, t denotes age, and A is a constant. The K estimate is used to evaluate L∞; $$L_\infty = \left( {e^K \sum\limits_2^n {L_t - \sum\limits_1^{n - 1} {L_t } } } \right)/\left( {n - 1} \right)\left( {e^K - 1} \right)$$ The L∞ estimate is used to estimate t _o, and to obtain a better estimate for K; $$\log _e \left( {1 - L_t /L_\infty } \right) = - Kt + Kt_0 $$ The K estimate may be used to obtain another estimate for L∞. Solved examples show that a single iteration is sufficient to obtain fitted equations which are, on the average, as precise as equations fitted by the least squares method shown by Tomlinson and Abramson (1961). This new method can be used, with a slight modification, for the second equation given above, if growth data have unequal age intervals. The variance of K, t _o and log _e L∞ can be estimated by applying the simple methods used in the case of straight-line relationships.