On the turbulent prandtl number in a stably stratified atmospheric boundary layer

The results obtained from both atmospheric and laboratory measurements and from LES data show that, in the stably stratified flows of the atmospheric boundary layer, turbulent mixing occurs at gradient Richardson numbers Ri _g that significantly exceed one: the inverse turbulent Prandtl number Pr _t ⁻¹ decreases with an increase in the thermal flow stability. The decreasing trend of the inverse turbulent Ptandtl number is reproduced in a stably stratified atmospheric boundary layer in agreement with measurement data with the aid of an improved three-parameter turbulence model. In this model, a modified model that takes into account the effect of stratification in the expression for the time scale of the scalar field is used for the pressure-scalar correlation.     

The mixing efficiency and the processes of restratification in a stably stratified medium

The mechanism of the effect of a collapsing turbulent eddy on diapycnal transport in a stably stratified fluid is considered. It is shown that at small Richardson turbulent numbersRi ₀ the mixing efficiency increases asRi ₀, and at large numbers it decreases in proportion toRi ₀ ^–1/2 .Translated by Mikhail M. Trufanov. UDK 551.465.15.     

Evaluation of various vertical mixing parameterizations in a tropical Pacific Ocean GCM

Coupled general circulation models (GCMs) have had weak El Niño/Southern Oscillation variability that has been attributed to a diffuse thermocline in the modeled equatorial Pacific Ocean. Consequently, there have been many attempts to improve the thermocline by developing new or improved ocean vertical mixing schemes. This paper investigates the influence of gradient Richardson Number-based vertical mixing scheme profiles in a tropical Pacific Ocean GCM. It has been common for vertical mixing schemes to be assessed in tropical Pacific Ocean models that have a limited latitudinal domain bounded by zonal walls with sponge layers. However, recent work has shown that warm surface water can accumulate in these models and stop them from achieving the observed sharp equatorial thermocline. The present model employs a parameterized wall heat transport scheme that prevents warm surface water from accumulating. Thus we are able assess the influence of vertical mixing profiles in an ocean model that does not allow warm surface water to accumulate and influence the thermocline.In this paper we evaluate the equatorial performance of three different Richardson number (Ri)-based vertical mixing profiles: an integer power (IP) profile based on the observations of Peters, Gregg and Toole; a form of the Pacanowski and Philander profile modified to have low background mixing; and the Max Planck Institute profile. With the accumulation of warm surface water prevented, each of these profiles is able to achieve a sharp thermocline. When compared with observations, the IP profile achieves a better upwelling velocity distribution. We also examine the influence on equatorial performance of very high mixing coefficients at low Richardson number, and of low background mixing coefficients.     

Modeling the eddy transport of momentum and heat: Comparison with direct measurements in free atmosphere

Direct measurements of eddy diffusivities for momentum K _m and heat K _h by Doppler radar and by a radio acoustic sounding system in the upper troposphere and lower stratosphere were used to examine the applicability of three Reynolds-averaged Navier-Stokes (RANS) schemes of stratified turbulence in the environment: the E — ? turbulence scheme modified for stratified flows, the algebraic two-parameter E — ? Reynolds-stress scheme, and the three-parameter \(E - \varepsilon - \overline {\theta ^2 } \) turbulence scheme. All turbulence parameters-the turbulent kinetic energy (E), the dissipation rate (?), and vertical profiles of potential temperature (atmospheric stability) and mean wind velocity-were derived from direct measurements for all three turbulence schemes. It is shown that the profile of the vertical diffusivity of momentum (K _m ) obtained from the three-parameter RANS turbulence scheme agrees well with its directly measured analog. The profile of K _m calculated by the two-parameter turbulence schemes fits measurements rather qualitatively.     

Large eddy simulation of stratified mixing in two-dimensional dam-break problem in a rectangular enclosed domain

Mixing in both coastal and deep ocean emerges as one of the important processes that determines the transport of pollutants, sediments and biological species, as well as the details of the global thermohaline circulation. Both the observations, due to their lack in space and time resolution, and most coastal and general circulation models due to inadequate physics, can only provide partial information about oceanic mixing processes. A new class of nonhydrostatic models supplemented with physically based subgrid-scale (SGS) closures, or so-called large eddy simulation (LES), is put forth as another tool of investigation to complement observational and large-scale modeling efforts.However, SGS models have been developed primarily for homogeneous, isotropic flows. Here, four SGS models based on Smagorinsky eddy viscosity and diffusivity are tested for stratified flows in the context of 2D dam-break problem in a rectangular enclosed domain. This idealized testbed leads to a number of simplifications about the initial conditions, boundary conditions and geometry, while exhibiting the dynamically complex characteristics of stratified flows involving the interaction of shear-induced mixing and internal waves. Direct numerical simulations (DNS) at high resolutions are taken as benchmark solutions. Under-resolved simulations without SGS terms (so-called DNS¹) are used to quantify the impact of SGS stresses. The performance of LES is assessed by using the time evolution of the volume fraction of intermediate density water masses generated by mixing. The simulations are conducted using a nonhydrostatic high-order spectral element model Nek5000 developed to exhibit minimal numerical dissipation and dispersion errors, which is advantageous to quantify accurately the impact of SGS stresses.It is found that all tested SGS models lead to improved results with respect to those from DNS¹. Also, SGS models allow for simulations with coarse resolutions that blow up in DNS¹ due to lack of adequate dissipation where needed. The SGS model in which the vertical eddy diffusion is modulated via a function that depends on the Richardson number Ri shows the most faithful reproduction of mixed water masses at all resolutions tested.The sensitivity of the results to the tunable parameter of the SGS model, to changes in the Ri-dependent function and resolution of the turbulent overturning scales is shown.     

Validating a turbulence closure against estuarine microstructure measurements

A simple k–ε turbulence closure is introduced which has no stability functions but instead a Richardson number-dependent turbulent Prandtl number. Its free parameters are determined in a comparison with microstructure observations from a stratified and sheared tidal estuary and laboratory measurements. The closure is able to simulate observed turbulent dissipation rates (ε) and turbulent length scales (l_th) in regions of strong mean shear and small gradient Richardson number (R_g) to within factors of 2–3. It fails in regions of small shear and large R_g, presumably because of the dominance of internal wave-driven mixing. Additional simulations with a k–ε closure with stability functions taken from Canuto et al. [Canuto, V.M., Howard, A., Cheng, Y., Dubovikov, M.S., 2001. Ocean turbulence I: one-point closure model. Momentum and heat vertical diffusivities. J. Phys. Oceanogr. 31, 1413–1426] and with the closure of Baumert and Peters [Baumert, H., Peters, H., 2004. Turbulence closure, steady state, and collapse into waves. J. Phys. Oceanogr. 34, 505–512] show poor performance. Establishing a valid 1:1 comparison of simulated and observed ε and l_th requires nudging the model velocity and density toward observed values because free model integrations quickly diverge from the observations. Steady state gradient Richardson numbers are constrained to a range of 0.18–0.25, while flux Richardson numbers are constrained to the range of 0.1–0.22. The closure output is rather insensitive to such parameter variations. The simulations are sensitive, however, to the treatment of the observed velocity and density used to nudge the model. Good closure performance requires averaging the measured tidal flow over about an hour, a time scale for which conventional numerical models of estuarine circulations should be able to match observed shears. In the closure simulations the TKE balance stays close to a production–dissipation balance. The time rate of change and vertical diffusion of TKE are small, of the same order of magnitude, and vary in magnitude relative to each other systematically across the water column.     

Towards a mesoscale eddy closure

A turbulence closure for the effect of mesoscale eddies in non-eddy-resolving ocean models is proposed. The closure consists of a prognostic equation for the eddy kinetic energy (EKE) that is integrated as an additional model equation, and a diagnostic relation for an eddy length scale (L), which is given by the minimum of Rhines scale and Rossby radius. Combining EKE and L using a standard mixing length assumption gives a diffusivity (K), corresponding to the thickness diffusivity in the [Gent, P.R., McWilliams, J.C. 1990. Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr. 20, 150–155] parameterisation. Assuming downgradient mixing of potential vorticity with identical diffusivity shows how K is related to horizontal and vertical mixing processes in the horizontal momentum equation, and also enables us to parameterise the source of EKE related to eddy momentum fluxes.The mesoscale eddy closure is evaluated using synthetic data from two different eddy-resolving models covering the North Atlantic Ocean and the Southern Ocean, respectively. The diagnosis shows that the mixing length assumption together with the definition of eddy length scales is valid within certain limitations. Furthermore, implementation of the closure in non-eddy-resolving models of the North Atlantic and the Southern Ocean shows consistently that the closure has skill at reproducing the results of the eddy-resolving model versions in terms of EKE and K.     

Observations of turbulence under weakly and highly stratified conditions in the Ariake Sea

Observations of turbulence, stratification, and mean current were made using a microstructure profiler and an acoustic Doppler current profiler (ADCP) during four cruises at a central location in the Ariake Sea, under weakly and strongly stratified conditions. Continuous measurements of the dissipation rate of turbulent kinetic energy (TKE), ε, were made. These revealed that frictional bed turbulence with quarterdiurnal variation in the bottom boundary layer (BBL) was one of the most energetic sources of vertical mixing in the sea. Thickness of the BBL was strongly confined by the stable stratification. We investigate a relationship between the BBL height h and the Ozmidov scale. We present a systematic argument that describes the vertical structure and characteristic scales of velocity and turbulence inside the frictional BBL, where the stratification persisted. Considerable deviation of observed vertical shear from the law of the wall indicated a modification of turbulent scales by the stratification. Shear stress calculated from the velocity data using vertical integration of the equation of motion was found to decrease approximately linearly with height. The TKE production rate P, estimated using the shear stress, was highly correlated with the dissipation rate. The buoyancy contribution to TKE balance in the BBL was quantified in terms of the flux Richardson number R _f as R _f?=?0.12.     

Does the Ulleung eddy owe its existence to β and nonlinearities?

A nonlinear theory for the generation of the Ulleung Warm Eddy (UWE) is proposed. Using the nonlinear reduced gravity (shallow water) equations, it is shown analytically that the eddy is established in order to balance the northward momentum flux (i.e., the flow force) exerted by the separating western boundary current (WBC). In this scenario, the presence of β produces a southward (eddy) force balancing the northward momentum flux imparted by the separating East Korean Warm Current (EKWC).It is found that, for a high Rossby number EKWC (i.e., highly nonlinear current), the eddy radius is roughly 2R_d/ε^1/6 (here ε≡βR_d/f₀, where R_d is the Rossby radius), implying that the UWE has a scale larger than that of most eddies (R_d). This solution suggests that, in contrast to the familiar idea attributing the formation of eddies to instabilities (i.e., the breakdown of a known steady solution), the UWE is an integral part of the steady stable solution. The solution also suggests that a weak WBC does not produce an eddy (due to the absence of nonlinearity).A reduced gravity numerical model is used to further analyze the relationship between β, nonlinearity and the eddy formation. First, we show that a high Rossby number WBC which is forced to separate from the wall on an f plane does not produce an eddy near the separation. To balance the northward momentum force imparted by the nonlinear boundary current, the f plane system moves constantly offshore, producing a southward Coriolis force. We then show that, as β is introduced to the problem, an anticyclonic eddy is formed. The numerical balance of forces shows that, as suggested by the analytical reasoning, the southward force produced by the eddy balances the northward flow force imparted by the boundary current. We also found that the observed eddy scale in the Japan/East Sea agrees with the analytical estimate for a nonlinear current.     

Turbulent mixing in stably stratified flows of the environment: The current state of the problem (Review)

Specific features of the turbulent transfer of the momentum and heat in stably stratified geophysical flows, as well as possibilities for including them into RANS turbulence models, are analyzed. The momentum (but not heat) transfer by internal gravity waves under conditions of strong stability is, for example, one such feature. Laboratory data and measurements in the atmosphere fix a clear dropping trend of the inverse turbulent Prandtl number with an increasing gradient Richardson number, which must be reproduced by turbulence models. Ignoring this feature can cause a false diffusion of heat under conditions of strong stability and lead, in particular, to noticeable errors in calculations of the temperature in the atmospheric boundary layer. Therefore, models of turbulent transfer must include the effect of the action of buoyancy and internal gravity waves on turbulent flows of the momentum. Such a strategy of modeling the stratified turbulence is presented in the review by a concrete RANS model and original results obtained during the modeling of stratified flows in the environment. Semiempirical turbulence models used for calculations of complex turbulent flows in deep stratified bodies of water are also analyzed. This part of the review is based on the data of investigations within the framework of the large international scientific Comparative Analysis and Rationalization of Second-Moment Turbulence Models (CARTUM) project and other publications of leading specialists. The most economical and effective approach associated with modified two-parameter turbulence models is a real alternative to classical variants of these models. A class of test problems and laboratory and full-scale experiments used by the participants of the CARTUM project for the approbation of numerical models are considered.     

Turbulent structure in the upper layer of the western Equatorial Pacific Ocean

The Richardson number dependence of vertical eddy diffusion coefficients in the western Equatorial Pacific Ocean was examined on the basis of a Microstructure Profiler (MSP) observations during the cruise of Natsushima (JAPACS-89). The Richardson numberR _i was estimated by using the mean shear of velocity profile measured by an Acoustic Doppler Current Profiler (ADCP) with the vertical interval of 15 meters within one or two hours of the each MSP cast. The raw data plot of the vertical eddy diffusion coefficientK _p shows a large scatter with increasing tendency belowR _i =0.5. The relation between the mean vertical eddy diffusion coefficientK _p and the Richardson numberR _i , averaged over every 0.025 in theR _i , supports the model of Pacanowski and Philander (1981) in the range ofR _i >0.5, but coincides with the result of Peterset al. (1988) in the range ofR _i <0.5.     

Direct measurements of diapycnal mixing in a fjord reach—Puget Sound's Main Basin

In the spring of 1988, time series of microstructure and ADCP current profiles were collected at four locations in the North Main Basin of Puget Sound, Washington. Depth and time averages of diapycnal diffusivity at the four stations (1.8−67.0×10⁻⁴ m² s⁻¹) were one to three decades above typical open-ocean thermocline levels. The buoyancy frequency-squared N² was near open-ocean levels, but unlike the open-ocean where N²S², finescale shear-squared S² was three to six times N² over significant portions of the water column at two of the stations. The time and space mean of all measurements ( ) is close to inferred vertical eddy diffusivity from a primitive equation model for Puget Sound (K_z=3×10⁻³ m² s⁻¹) (J. Geophys. Res. 96 (1991) 16779). Large time and space variability of K_ρ was found, with differences of inter-station, depth–time means over one decade. A simple scaling argument using the observed K_ρ suggests significant exchange of mass between the layers of the subtidal flow over the basin's residence time. Additionally, measurements show that local mixing may be comparable to volume-weighted sill mixing in modifying the Main Basin's stratification. Both are contrary to the “advective reach” simplification of fjord dynamics. The mixing levels were dominated by the passage of a mid-depth, southward-flowing density intrusion and what we interpret as a strongly advected, non-linear internal tide. These mechanisms elevated profile-averaged K_ρ by more than 10 times background levels, with sustained patches of K_ρ≥1×10⁻² m² s⁻¹. Critical 8-m gradient Richardson numbers (Ri₈<0.25) matching regions of overturns (>20 m) and strong turbulence suggest that shear instabilities dominated the turbulence production, though there was support for double-diffusive convection in the warm core of the density intrusion.     

Mixing and biological production at eddy margins in the eastern Gulf of Alaska

We performed a multi-day shipboard experiment in June 2001 to test whether combining water from within an anticyclonic mesoscale eddy in the eastern Gulf of Alaska with water from outside could result in enhanced phytoplankton growth and to determine how mixing might influence planktonic assemblages. Initially, the eddy had lower standing stocks of algal pigments (chlorophyll a [chl a] and accessory pigments), nutrients, phytoplankton, and particulate organic carbon/nitrogen compared to waters outside of the eddy. The eddy possessed a greater diversity and abundance of coastal diatoms while the outside waters had a greater proportion of oceanic species, including the endemic pennate diatom, Nitzschia cylindroformis. After one week of incubation, rates of primary production were significantly higher in the mixed water compared to both the eddy and outside treatments. Pigment concentrations (except chl c₃, alloxanthin, and zeaxanthin) and the proportion of large diatoms (mainly Pseudo-nitzschia spp.) and heterotrophic dinoflagellates were greater in the mixed water than would be expected from the simple combination of inside and outside waters. Nutrient limitation (most likely by trace metals) appeared to be less severe in the mixed water. Chl a was enhanced in the mixed water, particularly when compared to the eddy water. The mixing of eddy and outside water masses stimulated primary production by ∼20%, but more importantly, the mixing resulted in a distinct planktonic assemblage. The biomass enrichment was short-lived, indicating that the maintenance of elevated chl a would require further mixing events in a physical setting that also permits an accumulation of biomass. We note that submesoscale processes, including the intensification of ageostrophic circulation that elicits strong vertical mixing in the presence of strain, might explain observed patterns of high phytoplankton standing stocks at the inner edges of Haida eddies in the field.     

Seasonal response of surface wind to SST perturbation in the Northern Hemisphere

The seasonal response of surface wind speed to sea surface temperature (SST) change in the Northern Hemisphere was investigated using 10 years (2002-2011) high-resolution satellite observations and reanalysis data. The results showed that correlation between surface wind speed perturbations and SST perturbations exhibits remarkable seasonal variation, with more positive correlation is stronger in the cold seasons than in the warm seasons. This seasonality in a positive correlation between SST and surface wind speed is attributable primarily to seasonal changes of oceanic and atmospheric background conditions in frontal regions. The mean SST gradient and the prevailing surface winds are strong in winter and weak in summer. Additionally, the eddy-induced response of surface wind speed is stronger in winter than in summer, although the locations and numbers of mesoscale eddies do not show obvious seasonal features. The response of surface wind speed is apparently due to stability and mixing within the marine atmospheric boundary layer (MABL), modulated by SST perturbations. In the cold seasons, the stronger positive (negative) SST perturbations are easier to increase (decrease) the MABL height and trigger (suppress) momentum vertical mixing, contributing to the positive correlation between SST and surface wind speed. In comparison, SST perturbations are relatively weak in the warm seasons, resulting in a weak response of surface wind speed to SST changes. This result holds for each individual region with energetic eddy activity in the Northern Hemisphere.     

On the onset of the instability of a three-dimensional jet in a stratified fluid

The onset of a three-dimensional jet flow in a stratified fluid is studied with the aid of a direct numerical simulation. An initially cylindrical jet with a Gaussian velocity profile is considered in a fluid with stable linear density stratification. The results indicate that, if an initial small perturbation of the velocity field has a wide spectrum, an exponential growth of the isolated quasi-two-dimensional mode occurs and its spectral maximum is shifted toward smaller wave numbers in comparison with the maximum of the helical mode of the instability of a nonstratified jet. The growth rate is proportional to Ri^0.5, where Ri is the global Richardson number. The onset of the instability leads to the formation of the flow’s vortex structure, which consists of a collection of different-polarity quasi-two-dimensional vortices located in a horizontal plane near the longitudinal axis of the jet. At sufficiently long times (Nt > 100, where N is the buoyancy frequency and t is time), the growth of instability reaches the saturation stage and further fluctuations in velocity and density decay under the effect of viscous diffusion. At this stage, the flow becomes self-similar and the time dependences of the transverse and vertical widths of the jet are consistent with the asymptotic behaviors of integral parameters of the flow that are observed experimentally in the far stratified wake. The results suggest that the onset of the instability of a quasitwo-dimensional mode can play the determining role in the dynamics of flow in the far stratified wake.     

Turbulent mixing in a stratified estuarine tidal channel: Hikapu Reach,Pelorus Sound,New Zealand

Channel constrictions within an estuary can influence overall estuary-sea exchange of salt or suspended/dissolved material. The exchange is modulated by turbulent mixing through its effect on density stratification. Here we quantify turbulent mixing in Hikapu Reach, an estuarine channel in the Marlborough Sounds, New Zealand. The focus is on a period of relatively low freshwater input but where density stratification still persists throughout the tidal cycle, although the strength of stratification and its vertical structure vary substantially. The density stratification increases through the ebb tide, and decreases through the flood tide. During the spring tides observed here, ebb tidal flow speeds reached 0.7?m?s^?1 and the buoyancy frequency squared was in the range 10^?5 to 10^?3?s^?2. Turbulence parameters were estimated using both shear microstructure and velocimeter-derived inertial dissipation which compared favourably. The rate of dissipation of turbulent kinetic energy reached 1?×?10^?6?m²?s^?3 late in the ebb tide, and estimates of the gradient Richardson number (the ratio of stability to shear) fell as low as 0.1 (i.e. unstable) although the results show that bottom-boundary driven turbulence can dominate for periods. The implication, based on scaling, is that the mixing within the channel does not homogenise the water column within a tidal cycle. Scaling, developed to characterise the tidal advection relative to the channel length, shows how riverine-driven buoyancy fluxes can pass through the tidal channel section and the stratification can remain partially intact.     

A pair of Langmuir cells in a laboratory tank (I) wind-only experiment

Three velocity components of subsurface flow, observed in a rectangular tank under the action of a constant wind speed, are measured systematically at mesh points distributed uniformly over a vertical cross-section of the tank. Measurements are carried out for two cases: 1) reference wind speedU _r =7.5 m/s and fetchF=10 m; and 2)U _r =10 m/s andF=25 m. A pair of Langmuir cells is observed for both cases; downwelling zones are found along both of the sidewalls and an upwelling zone in the centre of the tank. Near the water surface, the vertical momentum flux is dominated by the Reynolds stress resulting from small-scale turbulence, while over the entire cross-section except near the surface, the Reynolds stress due to the Langmuir cells dominates the vertical momentum flux. As the result of the occurrence of this Langmuir cells, the vertical momentum flux, which consists of both mean advection and small-scale turbulence, is markedly inhomogeneous in the spanwise direction; for example, the largest vertical flux of the order of the wind stress is observed in the downwelling zone near one sidewall, while at the centre of the tank, the vertical momentum flux occupies only 30% of the wind stress. This indicates that a pair of Langmuir cells plays more important role than small-scale turbulence in the mixing process in a greater part of the wind-wave tank.Address after April 1, 1992: Department of Civil Engineering, Hiroshima Institute of Technology, Miyake 2-1-1, Saeki-ku, Hiroshima 731-51, Japan.