Vertical Velocity Scale Research Articles

Self-similar solution for laminar bubbly flow evolving from a vertical plate

The development of a bubble plume from a vertical gas-evolving electrode is driven by buoyancy and hydrodynamic bubble dispersion. This canonical fluid mechanics problem is relevant for both thermal and electrochemical processes. We adopt a mixture model formulation for the two-phase flow, considering variable density (beyond Boussinesq), viscosity and hydrodynamic bubble dispersion. Introducing a new change of coordinates, inspired by the Lees–Dorodnitsyn transformation, we obtain a new self-similar solution for the laminar boundary layer equations. The results predict a wall gas fraction and gas plume thickness that increase with height to the power of 1/5 before asymptotically reaching unity and scaling with height to the power 2/5, respectively. The vertical velocity scales with height to the power of 3/5. Our analysis shows that self-similarity is only possible if gas conservation is entirely formulated in terms of the gas specific volume instead of the gas fraction.

Numerical validation of scaling laws for stratified turbulence

Recent theoretical progress using multiscale asymptotic analysis has revealed various possible regimes of stratified turbulence. Notably, buoyancy transport can either be dominated by advection or diffusion, depending on the effective Péclet number of the flow. Two types of asymptotic models have been proposed, which yield measurably different predictions for the characteristic vertical velocity and length scale of the turbulent eddies in both diffusive and non-diffusive regimes. The first, termed a ‘single-scale model’, is designed to describe flow structures having large horizontal and small vertical scales, while the second, termed a ‘multiscale model’, additionally incorporates flow features with small horizontal scales, and reduces to the single-scale model in their absence. By comparing predicted vertical velocity scaling laws with direct numerical simulation data, we show that the multiscale model correctly captures the properties of strongly stratified turbulence within regions dominated by small-scale isotropic motions, whose volume fraction decreases as the stratification increases. Meanwhile its single-scale reduction accurately describes the more orderly, layer-like, quiescent flow outside those regions.

Mixed Convection in an Idealized Coastal Urban Environment With Momentum and Thermal Surface Heterogeneities

AbstractCoastal marine heatwaves (MHWs) modulate coastal climate through ocean‐land‐atmosphere interactions, but little is known about how coastal MHWs interact with coastal cities and modify urban thermal environment. In this study, a representative urban coastal environment under MHWs is simplified to a mixed convection problem. Fourteen large‐eddy simulations (LESs) are conducted to investigate how coastal cities interact with MHWs. We consider the simulations by simple urban roughness setup (Set A) as well as explicit urban roughness representation (Set B). Besides, different MHW intensities, synoptic wind speeds, surface fluxes of urban and sea patches are considered. Results suggest that increasing MHW intensity alters streamwise potential temperature gradient and vertical velocity direction. The magnitude of vertical velocity and urban heat island (UHI) intensity decrease with increasing synoptic wind speed. Changing urban or sea surface heat flux also leads to important differences in flow and temperature fields. Comparison between Set A and B reveals a significant increase of vertical velocity magnitude and UHI intensity. To further understand this phenomenon, a canopy layer UHI model is proposed to show the relationship between UHI intensity and urban canopy, thermal heterogeneity and mean advection. The effect of urban canopy is considered in terms of an additional vertical velocity scale that facilitates heat transport from the heated surface and therefore increases UHI intensity. The model can well explain the trend of the simulated results and implies that overlooking the effect of urban canopy underestimates canopy UHI in urban coastal environment.

Dependency of vertical velocity variance on meteorological conditions in the convective boundary layer

Abstract. Measurements of vertical velocity from vertically pointing Doppler lidars are used to derive the profiles of normalized vertical velocity variance. Observations were taken during the FESSTVaL (Field Experiment on Submesoscale Spatio-Temporal Variability in Lindenberg) campaign during the warm seasons of 2020 and 2021. Normalized by the square of the convective velocity scale, the average vertical velocity variance profile follows the universal profile of Lenschow et al. (1980). However, daily profiles still show a high day-to-day variability. We found that moisture transport and the content of moisture in the boundary layer could explain the remaining variability of the normalized vertical velocity variance. The magnitude of the normalized vertical velocity variance is highest on clear-sky days and decreases as the absolute humidity increases and surface latent heat flux decreases on cloud-topped days. This suggests that moisture content and moisture transport are limiting factors for the intensity of turbulence in the convective boundary layer. We also found that the intensity of turbulence decreases with an increase in the boundary layer cloud fraction during FESSTVaL, while the latent heating in the cloud layer was not a relevant source of turbulence in this case. We conclude that a new vertical velocity scale has to be defined that would take into account the moist processes in the convective boundary layer.

Nonprecipitating Shallow Cumulus Convection Is Intrinsically Unstable to Length Scale Growth

Abstract Condensation in cumulus clouds plays a key role in structuring the mean, nonprecipitating trade wind boundary layer. Here, we summarize how this role also explains the spontaneous growth of mesoscale [>O(10) km] fluctuations in clouds and moisture around the mean state in a minimal-physics, large-eddy simulation of the undisturbed period during BOMEX on a large [O(100) km] domain. Small, spatial anomalies in condensation in cumulus clouds, which form on top of small moisture fluctuations, power circulations that transport moisture, but not heat, from dry to moist regions, and thus reinforce the condensation anomaly. We frame this positive feedback as a linear instability in mesoscale moisture fluctuations, whose time scale depends only on (i) a vertical velocity scale and (ii) the mean environment’s vertical structure. In our minimal-physics setting, we show both ingredients are provided by the shallow cumulus convection itself: it is intrinsically unstable to length scale growth. The upshot is that energy released by clouds at kilometer scales may play a more profound and direct role in shaping the mesoscale trade wind environment than is generally appreciated, motivating further research into the mechanism’s relevance.

Some new insights for inferring diapycnal (irreversible) diffusivity in stably stratified turbulence

New insight for inferring diapycnal diffusivity in stably stratified turbulent flows is obtained based on physical scaling arguments and tested using high-resolution direct numerical simulation data. It is shown that the irreversible diapycnal diffusivity can be decomposed into a diapycnal length scale that represents an inner scale of turbulence and a diapycnal velocity scale. Furthermore, it is shown that the diapycnal length scale and velocity scale can be related to the measurable Ellison length scale (LE) that represents outer scale of turbulence and vertical turbulent velocity scale (w′) through a turbulent Froude number scaling analysis. The turbulent Froude number is defined as Fr=ε/Nk, where ε is the rate of dissipation of turbulent kinetic energy, N is the buoyancy frequency, and k is the turbulent kinetic energy. The scaling analysis suggests that the diapycnal diffusivity Kρ∼w′LE in the weakly stratified regime (Fr >1) and Kρ∼(w′LE)×Fr for the strongly stratified regime (Fr <1).

Wildfire smoke-plume rise: a simple energy balance parameterization

Abstract. The buoyant rise and the resultant vertical distribution of wildfire smoke in the atmosphere have a strong influence on downwind pollutant concentrations at the surface. The amount of smoke injected vs. height is a key input into chemical transport models and smoke modelling frameworks. Due to scarcity of model evaluation data as well as the inherent complexity of wildfire plume dynamics, smoke injection height predictions have large uncertainties. In this work we use the coupled fire–atmosphere model WRF-SFIRE configured in large-eddy simulation (LES) mode to develop a synthetic plume dataset. Using this numerical data, we demonstrate that crosswind integrated smoke injection height for a fire of arbitrary shape and intensity can be modelled with a simple energy balance. We introduce two forms of updraft velocity scales that exhibit a linear dimensionless relationship with the plume vertical penetration distance through daytime convective boundary layers. Lastly, we use LES and prescribed burn data to constrain and evaluate the model. Our results suggest that the proposed simple parameterization of mean plume rise as a function of vertical velocity scale offers reasonable accuracy (30 m errors) at little computational cost.

Unified Entrainment and Detrainment Closures for Extended Eddy-Diffusivity Mass-Flux Schemes.

We demonstrate that an extended eddy‐diffusivity mass‐flux (EDMF) scheme can be used as a unified parameterization of subgrid‐scale turbulence and convection across a range of dynamical regimes, from dry convective boundary layers, through shallow convection, to deep convection. Central to achieving this unified representation of subgrid‐scale motions are entrainment and detrainment closures. We model entrainment and detrainment rates as a combination of turbulent and dynamical processes. Turbulent entrainment/detrainment is represented as downgradient diffusion between plumes and their environment. Dynamical entrainment/detrainment is proportional to a ratio of a relative buoyancy of a plume and a vertical velocity scale, that is modulated by heuristic nondimensional functions which represent their relative magnitudes and the enhanced detrainment due to evaporation from clouds in drier environment. We first evaluate the closures off‐line against entrainment and detrainment rates diagnosed from large eddy simulations (LESs) in which tracers are used to identify plumes, their turbulent environment, and mass and tracer exchanges between them. The LES are of canonical test cases of a dry convective boundary layer, shallow convection, and deep convection, thus spanning a broad rangeof regimes. We then compare the LES with the full EDMF scheme, including the new closures, in a single‐column model (SCM). The results show good agreement between the SCM and LES in quantities that are key for climate models, including thermodynamic profiles, cloud liquid water profiles, and profiles of higher moments of turbulent statistics. The SCM also captures well the diurnal cycle of convection and the onset of precipitation.

Inner, meso, and outer scales in a differentially heated vertical channel

Reynolds shear stress and Reynolds normal stress in a differentially heated vertical channel were shown in a previous paper to scale by a mixed scale of the friction velocity uτ and the maximum mean vertical velocity Umax. This scale was empirically determined; this work presents a physical understanding of the new scaling. A new dimensional analysis is developed that involves a simple modification of the traditional inner–outer scaling. The functional dependencies in the new dimensional analysis are determined using direct numerical simulation (DNS) data. The DNS data confirm the new physical reasoning and reveal, as follows: (i) The velocity fluctuation variance at the centerline scales with an outer scale, which is the mixed scale uτUmax. (ii) The location of the maximum mean vertical velocity scales as an Obukhov-style length scale, which, in turn, scales as the geometric mean of the inner length scale and the outer length scale. (iii) The temperature fluctuation variance at the centerline scales with an outer temperature scale. (iv) The temperature fluctuation variance peaks in the inner layer, and its peak value scales with an inner temperature scale. The new outer scales obtained from the dimensional analysis are shown to be consistent with the properties of the mean momentum balance equation.

Influence of Relative Humidity, Mixed-Layer Height, and Mesoscale Vertical-Velocity Variations on Column and Surface Aerosol Characteristics Over an Urban Region

Investigations into the influence of variations in relative humidity, mixed-layer height, and mesoscale vertical velocity on column and surface aerosol characteristics over urban regions are quite rare. Here we report on a comprehensive investigation that was conducted over Ahmedabad, an urban location in western India, during December 2006. In this campaign columnar and surface aerosol characteristics were measured and compared to relative humidity, mixed-layer height, and mesoscale vertical velocity to examine their influence on urban aerosol characteristics. The 500-nm aerosol optical depth was found to be approximately 0.8 between 24 and 26 December while on a cleaner day such as 7 and 18 December the aerosol optical depth was 0.1. Aerosol optical depths based on Moderate Resolution Imaging Spectroradiometer (MODIS) level-2 and level-3 data, and on in situ Sun photometer measurements, show good agreement. The scattering coefficient ( $$\beta _{\mathrm{sca}}$$ ) and absorption coefficient ( $$\beta _{\mathrm{abs}}$$ ) increased by a factor of 5–10 on 26 December compared to 7 December (a normal day). This latter date was characterized by a clear atmosphere, a lower mixed-layer height ( $$\approx $$ 1650 m), positive vertical velocity and higher aerosol scale height (>3 km), while 26 December was marked with hazy and smoky conditions, a larger mixed-layer height ( $$\approx $$ 2500 m), negative vertical velocity and smaller aerosol scale height ( $$\approx $$ 1 km). These atmospheric conditions lead to lower and higher values of surface and columnar aerosol concentrations on 7 and 26 December respectively. A measure of spectral aerosol absorption, $$\alpha '_{\mathrm{abs}}$$ > 2, indicating the dominance of carbonaceous aerosols from biomass/biofuel emissions (open biomass burning), is rather surprising as fossil-fuel emissions that produce strongly light absorbing carbonaceous particles usually dominate urban regions. The single-scattering albedo on both days is 0.56 and 0.67 respectively, while monthly mean hemispheric backscatter fraction b and asymmetry parameter g values are 0.16 and 0.53 respectively. Higher b and lower g values on 7 December, and lower b and higher g values on 26 December, provide the relative scale of variation in the amount of sub-micron aerosols that dominate on a normal/clear day vis-a-vis a smoky/perturbed day. Lower single-scattering albedo indicates the dominance of absorbing aerosols above Ahmedabad, and higher b and lower g values suggest the abundance of fine mode particles in aerosol size distribution. The in-depth results serve as representative inputs to modelling the column and surface characteristics, and the resultant radiative forcing of urban aerosols influenced by variations in relative humidity, mixed-layer height, and mesoscale vertical velocity.

An unambiguous definition of the Froude number for lee waves in the deep ocean

There is a long-standing debate in the literature of stratified flows over topography concerning the correct dimensionless number to refer to as a Froude number. Common definitions using external quantities of the flow include $U/(ND)$, $U/(Nh_{0})$, and $Uk/N$, where $U$ and $N$ are, respectively, scales for the background velocity and buoyancy frequency, $D$ is the depth, and $h_{0}$ and $k^{-1}$ are, respectively, height and width scales of the topography. It is also possible to define an internal Froude number $Fr_{\unicode[STIX]{x1D6FF}}=u_{0}/\sqrt{g^{\prime }\unicode[STIX]{x1D6FF}}$, where $u_{0}$, $g^{\prime }$, and $\unicode[STIX]{x1D6FF}$ are, respectively, the characteristic velocity, reduced gravity, and vertical length scale of the perturbation above the topography. For the case of hydrostatic lee waves in a deep ocean, both $U/(ND)$ and $Uk/N$ are insignificantly small, rendering the dimensionless number $Nh_{0}/U$ the only relevant dynamical parameter. However, although it appears to be an inverse Froude number, such an interpretation is incorrect. By non-dimensionalizing the stratified Euler equations describing the flow of an infinitely deep fluid over topography, we show that $Nh_{0}/U$ is in fact the square of the internal Froude number because it can identically be written in terms of the inner variables, $Fr_{\unicode[STIX]{x1D6FF}}^{2}=Nh_{0}/U=u_{0}^{2}/(g^{\prime }\unicode[STIX]{x1D6FF})$. Our scaling also identifies $Nh_{0}/U$ as the ratio of the vertical velocity scale within the lee wave to the group velocity of the lee wave, which we term the vertical Froude number, $Fr_{vert}=Nh_{0}/U=w_{0}/c_{g}$. To encapsulate such behaviour, we suggest referring to $Nh_{0}/U$ as the lee-wave Froude number, $Fr_{lee}$.

Comment on “Using an ADCP to Estimate Turbulent Kinetic Energy Dissipation Rate in Sheltered Coastal Waters”

AbstractGreene et al. revisit the suggestion that the turbulent kinetic energy dissipation rate could be estimated through a “large-eddy estimate,” employing acoustic measurements of velocity fields associated with the largest energy-containing scales of ocean turbulence. While the large-eddy estimate as originally proposed used vertical velocity and a vertical eddy length scale, Greene et al. chose instead to substitute a horizontal length scale for the latter. This comment argues that combining a horizontal scale for length with a vertical velocity scale produces a large-eddy estimate of the dissipation rate that is accurate only if the energy-containing eddies are isotropic, and that this condition is highly unlikely in naturally occurring ocean turbulence, subject as it is to influences of stratification, vertical shear, and/or the presence of horizontal boundaries. The problem is documented using data from a large-eddy simulation of Langmuir supercells.

Penetrative convection in slender containers

An experimental study was conducted to investigate the penetration of a convective mixed layer into an overlying stably (solutally) stratified layer contained in a narrow, tall vessel when the fluid is subjected to a destabilizing heat flux from below. The interest was the evolution of the bottom mixed-layer height (\(h\)) with time (\(t\)) in the presence of side-wall effects, but without the formation of conventional double-diffusive layers. The side-wall effects are expected at small mixed-layer aspect ratios, \(\varGamma_{h} = (W/h)\), where \(W\) is the container width. This case has not been studied hitherto, although there are important environmental and industrial applications. The mixed-layer growth laws for low aspect ratio convection were formulated by assuming a balance between the vertical kinetic energy flux at the interface and the rate of change of potential energy of the fluid system due to turbulent entrainment. The effects of sidewalls were considered using similarity arguments, by taking characteristic rms velocities to be a function of \(\varGamma_{h}\), in addition to buoyancy flux (\(q_{0}\)) and \(h\). In all stages of evolution, the similarity variables \(\xi = h/W\) and \(t^{\prime } = Nt/A\), where \(A = N^{3} W^{2} /4q_{0}\) and \(N\) is the buoyancy frequency, scaled the mixed-layer evolution data remarkably well. Significant wall effects were noted when \(\varGamma_{h} < 1\), and for this case the interfacial vertical turbulent velocity and length scales were identified via scaling arguments and experimental data.

Increasing vertical mixing to reduce Southern Ocean deep convection in NEMO3.4

Abstract. Most CMIP5 (Coupled Model Intercomparison Project Phase 5) models unrealistically form Antarctic Bottom Water by open ocean deep convection in the Weddell and Ross seas. To identify the mechanisms triggering Southern Ocean deep convection in models, we perform sensitivity experiments on the ocean model NEMO3.4 forced by prescribed atmospheric fluxes. We vary the vertical velocity scale of the Langmuir turbulence, the fraction of turbulent kinetic energy transferred below the mixed layer, and the background diffusivity and run short simulations from 1980. All experiments exhibit deep convection in the Riiser-Larsen Sea in 1987; the origin is a positive sea ice anomaly in 1985, causing a shallow anomaly in mixed layer depth, hence anomalously warm surface waters and subsequent polynya opening. Modifying the vertical mixing impacts both the climatological state and the associated surface anomalies. The experiments with enhanced mixing exhibit colder surface waters and reduced deep convection. The experiments with decreased mixing give warmer surface waters, open larger polynyas causing more saline surface waters and have deep convection across the Weddell Sea until the simulations end. Extended experiments reveal an increase in the Drake Passage transport of 4 Sv each year deep convection occurs, leading to an unrealistically large transport at the end of the simulation. North Atlantic deep convection is not significantly affected by the changes in mixing parameters. As new climate model overflow parameterisations are developed to form Antarctic Bottom Water more realistically, we argue that models would benefit from stopping Southern Ocean deep convection, for example by increasing their vertical mixing.

Flow characteristics within different configurations of submerged flexible vegetation

Summary The effects of three configurations (aligned, staggered, and columnar) of submerged flexible vegetation on flow structure are investigated experimentally in the laboratory. Time-averaged flow velocity and turbulence behavior are evaluated at different positions in each configuration by using a 3D acoustic Doppler velocimeter (ADV). According to the hydrodynamic regimes in experimental results, the vertical distribution of streamwise velocity can be separated into three layers—the upper non-vegetated layer, middle vegetation layer, and lower sheath layer. This three-layer model, which is associated with different logarithmic equations, can be applied to describe the vertical distribution of streamwise velocity. The local maximum velocity within vegetation occurs at the sheath section of a plant clump (0.10–0.15 vegetation height ( H v )) where the frontal width is minimal. Turbulent intensities in the streamwise ( u rms ) and spanwise ( v rms ) directions peak at the sheath section and at the approximate top of the canopy (0.9 – 1.2 H v ). The maximum Reynolds stresses exist at roughly 0.9 – 1.2 H v , which may be migrated vertically as the frontal width of a plant clump is increased. This high frontal width also increases streamwise velocity above vegetation, leading to increase variations in Reynolds stresses around the canopy top. On the vertical turbulent velocity scale ( w rms ), the vortices above a still canopy rotate faster than those above a waving canopy. Therefore, the experimental results demonstrate that the flow field can vary significantly at the sheath section and at the top of a plant clump due to altered flow pass. These analytical findings will likely prove useful when designing ecological habitats and preventing riverbed erosion.

Vertical Water Velocities from Underwater Gliders

Abstract The underwater glider is set to become an important platform for oceanographers to gather data within oceans. Gliders are usually equipped with a conductivity–temperature–depth (CTD) sensor, but a wide range of other sensors have been fitted to gliders. In the present work, the authors aim at measuring the vertical water velocity. The vertical water velocity is obtained by subtracting the vertical glider velocity relative to the water from the vertical glider velocity relative to the water surface. The latter is obtained from the pressure sensor. For the former, a quasi-static model of planar glider flight is developed. The model requires three calibration parameters, the (parasite) drag coefficient, glider volume (at atmospheric pressure), and hull compressibility, which are found by minimizing a cost function based on the variance of the calculated vertical water velocity. Vertical water velocities have been calculated from data gathered in the northwestern Mediterranean during the Gulf of Lions experiment, winter 2008. Although no direct comparison could be made with water velocities from an independent measurement technique, the authors show that, for two different heat loss regimes (≈0 and ≈400 W m−2), the calculated vertical velocity scales are comparable with those expected for internal waves and active open ocean convection, respectively. High noise levels resulting from the pressure sensor require the water velocity time series to be low-pass filtered with a cutoff period of 80 s. The absolute accuracy of the vertical water velocity is estimated at ±4 mm s−1.

Parameterization of the roughness length over the sea in forced and free convection

In the condition of free convection, the Charnock relation is inadequate. In this paper we extend the Charnock relation to include the effect of free convection on the roughness length. As a result, the singularity in the Monin–Obukhov similarity theory can be avoided. This paper shows two approaches to derive the roughness length formula in the forced and free convections. The first approach is based on the mixing length theory and the use of the observational data of the vertical velocity variance. We introduce a new vertical velocity scale based on the vertical velocity variance; this velocity variance is well behaved in the atmospheric boundary layer and easy to obtain from field experiments. The second approach is based on the theoretical framework of Sykes et al. (Q R Met Soc, 119: 409–421). From that framework, we develop a theory to obtain the roughness length formula. The results of these two approaches are in agreement with each other. In the past, a multiplication factor associated with free convection was considered to be a constant. This paper shows that this multiplication factor is, in fact, also dependent on the depth of the mixing height. In previous studies, experimental works were often conducted without taking into account the depth of the mixing height. Not taking into account the mixing height in the estimation of the roughness length in free convection would result in an inaccurate estimate of the roughness length and hence the drag coefficient.

A New Bulk Shallow-Cumulus Model and Implications for Penetrative Entrainment Feedback on Updraft Buoyancy

Abstract A new refinement of Albrecht et al.’s bulk model for shallow-cumulus convection is presented. It is used to illuminate fundamental aspects of oceanic shallow-cumulus boundary layer structure, including updraft buoyancy and vertical velocity scales, cumulus mass flux, temperature and humidity profiles, and trade-inversion height. The new model, like Albrecht et al.’s, includes a subcloud mixed layer and a cumulus layer with linear gradients of conserved thermodynamic variables, topped by a sharp capping inversion. Albrecht et al.’s model was not mathematically well posed, leading to an inconsistency between its heat and moisture balances at the inversion base. The new bulk model resolves this problem by diagnosing, rather than prognosing, the cumulus-layer gradients and introducing a penetrative entrainment closure to determine the growth of the cumulus layer into the overlying free troposphere. It uses more realistic assumptions about lateral cumulus entrainment and detrainment and a simplified sub-cloud-layer entrainment closure. When applied to a Barbados Oceanographic and Meteorological Experiment (BOMEX) trade-cumulus case, the steady state and the transient response of the bulk model to a sudden increase in sea surface temperature both compare favorably to a large-eddy simulation (LES). The new closure shows that penetrative entrainment constantly adjusts the cumulus-layer temperature and moisture profiles to keep the updraft buoyancy and vertical velocity small in inverse proportion to a penetrative entrainment efficiency A, estimated to be 5 from LES. Energy-balance arguments provide a skillful prediction of the steady-state bulk-model trade-inversion height. They also show that penetrative entrainment is driven by destabilization of the cumulus layer by radiative cooling and Lagrangian surface temperature increases.

Large-Eddy Simulation of Langmuir Turbulence in Pure Wind Seas

Abstract The scaling of turbulent kinetic energy (TKE) and its vertical component (VKE) in the upper ocean boundary layer, forced by realistic wind stress and surface waves including the effects of Langmuir circulations, is investigated using large-eddy simulations (LESs). The interaction of waves and turbulence is modeled by the Craik–Leibovich vortex force. Horizontally uniform surface stress τ0 and Stokes drift profiles uS(z) are specified from the 10-m wind speed U10 and the surface wave age CP/U10, where CP is the spectral peak phase speed, using an empirical surface wave spectra and an associated wave age–dependent neutral drag coefficient CD. Wave-breaking effects are not otherwise included. Mixed layer depths HML vary from 30 to 120 m, with 0.6 ≤ CP/U10 ≤ 1.2 and 8 m s−1 &amp;lt; U10 &amp;lt; 70 m s−1, thereby addressing most possible oceanic conditions where TKE production is dominated by wind and wave forcing. The mixed layer–averaged “bulk” VKE 〈w2〉/u*2 is equally sensitive to the nondimensional Stokes e-folding depth D*S/HML and to the turbulent Langmuir number Lat = u*/US, where u* = |τ0|/ρw in water density ρw and US = |uS|z=0. Use of a D*S scale-equivalent monochromatic wave does not accurately reproduce the results using a full-surface wave spectrum with the same e-folding depth. The bulk VKE for both monochromatic and broadband spectra is accurately predicted using a surface layer (SL) Langmuir number LaSL = u*/〈uS〉SL, where 〈uS〉SL is the average Stokes drift in a surface layer 0 &amp;gt; z &amp;gt; − 0.2HML relative to that near the bottom of the mixed layer. In the wave-dominated limit LaSL → 0, turbulent vertical velocity scales as wrms ∼ u*La−2/3SL. The mean profile (z) of VKE is characterized by a subsurface peak, the depth of which increases with D*S/HML to a maximum near 0.22HML as its relative magnitude /〈w2〉 decreases. Modestly accurate scalings for these variations are presented. The magnitude of the crosswind velocity convergence scales differently from VKE. These results predict that for pure wind seas and HML ≅ 50 m, 〈w2〉/u*2 varies from less than 1 for young waves at U10 = 10 m s−1 to about 2 for mature seas at winds greater than U10 = 30 m s−1. Preliminary comparisons with Lagrangian float data account for invariance in 〈w2〉/u*2 measurements as resulting from an inverse relationship between U10 and CP/U10 in observed regimes.

Unsteady natural convection on an evenly heated vertical plate for Prandtl numberPr&lt;1

The transient behavior of the natural convection boundary-layer flow adjacent to a vertical plate heated with a uniform flux in a quiescent homogeneous ambient fluid with Prandtl number Pr<1 is investigated by scaling analysis and direct numerical simulation (DNS). The flow is characterized by a startup stage, a short transitional stage and a steady state. The flow is parametrized by the thermal and velocity boundary-layer thickness scales, the vertical velocity scale, the time scale for the boundary layer to reach the steady state and the plate temperature scale. Scaling analysis is used to obtain laws relating these quantities to the flow governing parameters, the Rayleigh number Ra, the Prandtl number, and the Boussinesq number Bo=RaPr which is a much more important control parameter than Ra for small Pr fluids. A series of DNS with selected values of Ra and Pr in the ranges of 10(6)< or =R< or =10(10) and 0.01< or =Pr < or =0.5 are used to validate the scaling laws and obtain scaling constants.