Turbulent Buoyancy Flux Research Articles

Turbulence modification by stable stratification in channel flow

Direct numerical simulations of stably stratified, turbulent channel flow at low to moderate Reynolds number have been performed using large computational boxes and considering a wide range of stratification levels. For weak stratification or high Reynolds number, the turbulence is affected by buoyancy in the core of the channel, but the near-wall region differs little from the neutral case. With strong stratification, large laminar patches appear in the near-wall region and turbulent momentum and buoyancy fluxes vanish in the core of the channel. With increasing stratification, the near-wall streaks remain essentially unmodified, while large-scale global modes are damped. In the central region, internal gravity waves are dominant. In addition, there is an intermediate outer layer where the dynamics of the turbulent structures is governed by local fluxes. In this region, energy spectra collapse when using local Obukhov scaling.

Turbulent mixing in a strongly forced salt wedge estuary

Turbulent mixing of salt is examined in a shallow salt wedge estuary with strong fluvial and tidal forcing. A numerical model of the Merrimack River estuary is used to quantify turbulent stress, shear production, and buoyancy flux. Little mixing occurs during flood tides despite strong velocities because bottom boundary layer turbulence is dislocated from stratification elevated in the water column. During ebbs, bottom salinity fronts form at a series of bathymetric transitions. At the fronts, near‐bottom velocity and shear stress are low, but shear, stress, and buoyancy flux are elevated at the pycnocline. Internal shear layers provide the dominant source of mixing during the early ebb. Later in the ebb, the pycnocline broadens and moves down such that boundary layer turbulence dominates mixing. Mixing occurs primarily during ebbs, with internal shear mixing accounting for about 50% of the total buoyancy flux. Both the relative contribution of internal shear mixing and the mixing efficiency increase with discharge, with bulk mixing efficiencies between 0.02 and 0.07. Buoyancy fluxes in the estuary increase with discharge up to about 400 m3 s−1 above which a majority of the mixing occurs offshore. Observed buoyancy fluxes were more consistent with the k‐ɛ turbulence closure than the Mellor‐Yamada closure, and more total mixing occurred in the estuary with k‐ɛ. Calculated buoyancy fluxes were sensitive to horizontal grid resolution, as a lower resolution grid yielded less integrated buoyancy flux in the estuary and exported lower salinity water but likely had greater numerical mixing.

On the turbulent Prandtl number in homogeneous stably stratified turbulence

In this paper, we derive a general relationship for the turbulent Prandtl number Prt for homogeneous stably stratified turbulence from the turbulent kinetic energy and scalar variance equations. A formulation for the turbulent Prandtl number, Prt, is developed in terms of a mixing length scale LM and an overturning length scale LE, the ratio of the mechanical (turbulent kinetic energy) decay time scale TL to scalar decay time scale Tρ and the gradient Richardson number Ri. We show that our formulation for Prt is appropriate even for non-stationary (developing) stratified flows, since it does not include the reversible contributions in both the turbulent kinetic energy production and buoyancy fluxes that drive the time variations in the flow. Our analysis of direct numerical simulation (DNS) data of homogeneous sheared turbulence shows that the ratio LM/LE ≈ 1 for weakly stratified flows. We show that in the limit of zero stratification, the turbulent Prandtl number is equal to the inverse of the ratio of the mechanical time scale to the scalar time scale, TL/Tρ. We use the stably stratified DNS data of Shih et al. (J. Fluid Mech., vol. 412, 2000, pp. 1–20; J. Fluid Mech., vol. 525, 2005, pp. 193–214) to propose a new parameterization for Prt in terms of the gradient Richardson number Ri. The formulation presented here provides a general framework for calculating Prt that will be useful for turbulence closure schemes in numerical models.

The Zonal Dimension of the Indian Ocean Meridional Overturning Circulation

Abstract The three-dimensional structure of the meridional overturning circulation (MOC) in the deep Indian Ocean is investigated with an eddy-permitting ocean model. The amplitude of the modeled deep Indian Ocean MOC is 5.6 Sv (1 Sv ≡ 106 m3 s−1), a broadly realistic but somewhat weak overturning. Although the model parameterization of diapycnal mixing is inaccurate, the model’s short spinup allows the effective diapycnal velocity (the sum of model drift and the explicitly modeled diapycnal velocity) to resemble the true, real-ocean diapycnal velocity. For this reason, the model is able to recover the broad zonal asymmetry in the turbulent buoyancy flux that is suggested by observations. The model features a substantial deep, depth-reversing zonal circulation of nearly 50% of the MOC. The existence of this circulation, brought about by the zonally asymmetric distribution of diapycnal mixing, implies a much slower ventilation of the deep Indian Ocean (by a factor of 5–6) than would be in place without zonal interbasin exchanges. It is concluded that the zonal asymmetry in the distribution of diapycnal mixing must have a major impact on the deep Indian Ocean’s capacity to store and transform climatically significant physical and biogeochemical tracers.

Energy- and flux-budget (EFB) turbulence closure model for stably stratified flows. Part I: steady-state, homogeneous regimes

We propose a new turbulence closure model based on the budget equations for the key second moments: turbulent kinetic and potential energies: TKE and TPE (comprising the turbulent total energy: TTE = TKE + TPE) and vertical turbulent fluxes of momentum and buoyancy (proportional to potential temperature). Besides the concept of TTE, we take into account the non-gradient correction to the traditional buoyancy flux formulation. The proposed model grants the existence of turbulence at any gradient Richardson number, Ri. Instead of its critical value separating - as usually assumed - the turbulent and the laminar regimes, it reveals a transition interval, 0.1< Ri <1, which separates two regimes of essentially different nature but both turbulent: strong turbulence at Ri<<1; and weak turbulence, capable of transporting momentum but much less efficient in transporting heat, at Ri>1. Predictions from this model are consistent with available data from atmospheric and lab experiments, direct numerical simulation (DNS) and large-eddy simulation (LES).

A numerical study of the tidal circulation and buoyancy effects in a Scottish fjord: Loch Torridon

The tidal and buoyancy‐driven circulation in Loch Torridon, a Scottish fjord, is investigated using a three‐dimensional, hydrostatic, primitive equation model. The predictions from the model have been quantitatively compared to available data gathered between 1998 and 2001. The model performed well in reproducing the tidal characteristics and density evolution of the fjord over the 3‐month simulation period. Results show that tidal currents are dominated by the constraint imposed on the flow by the shallow sill separating the upper and middle basins: the flow is strongly accelerated during flood and ebb tides, and significant residual currents are generated that may influence exchange and residence times of the middle basin. A strong baroclinic response to the flow over the sill is predicted throughout the fjord, with a propagating linear M2 tide displacing isopycnals vertically by up to 32 m. At the sill the formation of stationary lee waves on both flood and ebb tides is predicted, with the wave generated during flood tide being notably larger as the growth of the ebb wave is inhibited by deep dense water. This allows more energy to transfer into the seaward propagating internal tide. As the tidal flow weakens, the lee waves propagate over the sill and upstream. Estimates of the total vertical turbulent buoyancy flux in the fjord are made, and the distribution of mixing over a tidal cycle is shown; vertical mixing is seen to be highly spatially and temporally variable and strongly linked to the presence and dissipation of lee waves near the sill.

Dense water formation in the Nordic Seas diagnosed from sea surface buoyancy fluxes

The relationship between the surface water mass transformation over the Nordic Seas and the large-scale diapycnal overturning flow through the same region was examined. The surface transformation was estimated from climatological and reanalyzed surface hydrography and air–sea buoyancy fluxes. Ocean currents and transports were assembled from climatological wind stresses and modern hydrographic data with reference level velocities for thermal wind calculations gathered from independent studies and adjusted in a linear inverse model. The data reveal a close match between the surface transformation and the magnitude of the large-scale diapycnal overturning. This result confirms an existing hypothesis that turbulent fluxes of buoyancy and nonlinearities in the equation of state are of secondary importance in high-latitude dense water production regions. Fields from a coupled numerical ocean–ice model also suggest that the seasonal cycle of freezing and melting within the Nordic Seas has a small net effect on the surface transformation, particularly at high densities ( < 10 % ) . Finally, the data confirm that the bulk of the dense water with eventually feeds the North Atlantic Ocean via the Greenland–Scotland Ridge is formed within the Nordic Seas. Most of this dense water production takes place east of the Mohn–Knipovich ridge system with particularly high production over the Norwegian and Lofoten basins.

Advanced insights into sources of vertical velocity in the ocean

Estimating vertical velocity in the oceanic upper layers is a key issue for understanding ocean dynamics and the transport of biogeochemical elements. This paper aims to identify the physical sources of vertical velocity associated with sub-mesoscale dynamics (fronts, eddies) and mixed-layer depth (MLD) structures, using (a) an ocean adaptation of the generalized Q-vector form of the ω-equation deduced from a primitive equation system which takes into account the turbulent buoyancy and momentum fluxes and (b) an application of this diagnostic method for an ocean simulation of the Programme Ocean Multidisciplinaire Meso Echelle (POMME) field experiment in the North-Eastern Atlantic. The approach indicates that w-sources can play a significant role in the ocean dynamics and strongly depend on the dynamical structure (anticyclonic eddy, front, MLD, etc.). Our results stress the important contribution of the ageostrophic forcing, even under quasi-geostrophic conditions. The turbulent w-forcing was split into two components associated with the spatial variability of (a) the buoyancy and momentum (Ekman pumping) surface fluxes and (b) the MLD. Process (b) represents the trapping of the buoyancy and momentum surface energy into the MLD structure and is identified as an atmosphere/oceanic mixed-layer coupling. The momentum-trapping process is 10 to 100 times stronger than the Ekman pumping and is at least 1,000 times stronger than the buoyancy w-sources. When this decomposition is applied to a filamentary mixed-layer structure simulated during the POMME experiment, we find that the associated vertical velocity is created by trapping the surface wind-stress energy into this structure and not by Ekman pumping.

Mixing within the interior of the Faeroe-Shetland Channel

Observations of the distribution of the density field, the dissipation rate of turbulent kinetic energy, e, the vertical diffusivity, Kz, and optical backscatter over a repeated cross-section of the FaeroeShetland Channel (FSC) are presented within the context of ocean mixing. Turbulence in the permanent pycnocline occurs in discrete vertical patches of thickness 20‐50 m coincident with layers of elevated vertical current shear magnitude, S 10 2 s 1 . The layers of elevated shear result from high-wavenumber internal waves which may break through shear instabilities as indicated by critical Richardson numbers, Ri 1. The internal waves are ubiquitous features of strong boundary currents flowing along continental shelves and are likely generated here by geostrophic adjustment of the pycnocline in response to lateral excursions of the boundary current over the Shetland slope or by the scattering of internal tide energy. The pycnocline as a whole exhibits Kz 10 4.5 m 2 s 1 , a factor of 3 larger than typical open ocean thermoclines but still implying a weak exchange of water mass properties across the density interface despite the confined and isolated patches of enhanced turbulence which elevate Kz by an order of magnitude. Both the boundary current and the deep interior are typified by weak stratification and locally high Kz 10 3 m 2 s 1 , but low turbulent buoyancy fluxes. The strongest observed turbulence and mixing in the channel, e O(10 7 Wk g 1 ) and Kz 10 3 m 2 s 1 respectively, is observed in the near-bed region over the Shetland slope and results from solibore propagation up the slope and the asymmetric response of the bottom boundary layer to the tidal currents. The overall contribution of the FSC to oceanic mixing would appear to be about half of the required canonical value of Kz 10 4 m 2 s 1 given the observed short duration and/or infrequent occurrence of the processes generating turbulence in the near-bed region, the spatially confined sporadic mixing patches in the pycnocline, and the low turbulent buoyancy fluxes in the interior and slope current.

Form Drag and Mixing Due to Tidal Flow past a Sharp Point

Barotropic tidal currents flowing over rough topography may be slowed by two bottom boundary–related processes: tangential stress of the bottom boundary layer, which is generally well represented by a quadratic drag law, and normal stress from bottom pressure, known as form drag. Form drag is rarely estimated from oceanic observations because it is difficult to measure the bottom pressure over a large spatial domain. The “external” and “internal” components of the form drag are associated, respectively, with sea surface and isopycnals deformations. This study presents model and observational estimates of the components of drag for Three Tree Point, a sloping ridge projecting 1 km into Puget Sound, Washington. Internal form drag was integrated from repeat microstructure sections and exceeded the net drag due to bottom friction by a factor of 10–50 during maximum flood. In observations and numerical simulations, form drag was produced by a lee wave, as well as by horizontal flow separation in the model. The external form drag was not measured, but in numerical simulations was found to be comparable to the internal form drag. Form drag appears to be the primary mechanism for extracting energy from the barotropic tide. Turbulent buoyancy flux is strongest near the ridge in both observations and model results.

Turbulent energy production and entrainment at a highly stratified estuarine front

Rates of turbulent kinetic energy (TKE) production and buoyancy flux in the region immediately seaward (∼1 km) of a highly stratified estuarine front at the mouth of the Fraser River (British Columbia, Canada) are calculated using a control volume approach. The calculations are based on field data obtained from shipboard instrumentation, specifically velocity data from a ship mounted acoustic Doppler current profiler (ADCP), and salinity data from a towed conductivity‐temperature‐depth (CTD) unit. The results allow for the calculation of vertical velocities in the water column, and the total vertical transport of salt and momentum. The vertical turbulent transport quantities (, ) can then be estimated as the difference between the total transport and the advective transport. Estimated production is on the order of 10−3 m2 s−3, yielding a value of ɛ(νN2)−1 on the order of 104. This rate of TKE production is at the upper limit of reported values for ocean and coastal environments. Flux Richardson numbers in this highly energetic system generally range from 0.15 to 0.2, with most mixing occurring at gradient Richardson numbers slightly less than . These values compare favorably with other values in the literature that are associated with turbulence observations from regimes characterized by scales several orders of magnitude smaller than are present in the Fraser River.

The front on the Northern Flank of Georges Bank in spring: 2. Cross‐frontal fluxes and mixing

Hydrographic observations made with an undulating vehicle carrying a CTD and concurrent shipboard ADCP velocity observations over a 12‐day period are combined to investigate vertical mixing and cross‐frontal fluxes on the Northern Flank of Georges Bank. The CTD density time series is analyzed to detect the presence of vertical overturns, the statistics of which are used to infer vertical mixing parameters. Vertical turbulent buoyancy fluxes are downward and most intense, reaching values of 5 × 10−7 W/kg, near the bottom at the edge of the bank and decrease both on‐ and off‐bank. Horizontal, cross‐bank buoyancy fluxes are partitioned into mean, tidal pumping, and nontidal eddy components and are computed as a function of cross‐isobath/vertical position by averaging in the along‐isobath direction. The tidal pumping component is dominant over most of the cross‐bank section with a peak value of ∼1 × 10−4 W/kg, directed off‐bank near the bank edge. A diagnosed tidal vertical velocity field is used with mean buoyancy gradients to compute the along‐isopycnal skew flux. The horizontal component of this skew flux has similar spatial structure and magnitude to that of the observed tidal pumping flux. The divergent component of the skew flux, at depths above the bottom boundary layer, appears to be convergent north of the bank edge and divergent at the bank edge, suggesting that tidally driven advective processes drive buoyant bank water downward and off‐bank at mid‐depth and force the upwelling of deep, dense water near the bottom at the bank edge.

Research Aircraft Observations of the Mean and Turbulent Structure of a Low-Level Jet Accompanying a Strong Storm*

Abstract A NOAA P-3 research aircraft collected measurements in a storm off of the U.S. West Coast that featured a low-level jet with wind speeds approaching 50 m s−1. These measurements have been used to document mean and turbulent boundary layer structures in the vicinity of the jet along a pair of transects (∼150 and ∼25 km off the coast). Large magnitudes of turbulent kinetic energy (TKE) and a lack of thermal wind balance were found in the region of the jet, indicating that mixing was an important process. The turbulent fluxes of buoyancy were generally negative, implying that the turbulence near the jet was generated by wind shear. The observed momentum fluxes associated with this wind shear were compared with a TKE-based parameterization for these fluxes. In certain locations, the momentum fluxes were up the vertical wind shear gradient, whereas the parameterization specifies that these fluxes should be down the gradient. Nevertheless, in general the observed and parameterized momentum fluxes agree...

The Efficiency of Mixing in Turbulent Patches: Inferences from Direct Simulations and Microstructure Observations

The time evolution of mixing in turbulent overturns is investigated using a combination of direct numerical simulations (DNS) and microstructure profiles obtained during two field experiments. The focus is on the flux coefficient Γ, the ratio of the turbulent buoyancy flux to the turbulent kinetic energy dissipation rate ϵ. In observational oceanography, a constant value Γ = 0.2 is often used to infer the buoyancy flux and the turbulent diffusivity from measured ϵ. In the simulations, the value of Γ changes by more than an order of magnitude over the life of a turbulent overturn, suggesting that the use of a constant value for Γ is an oversimplification. To account for the time dependence of Γ in the interpretation of ocean turbulence data, a way to assess the evolutionary stage at which a given turbulent event was sampled is required. The ratio of the Ozmidov scale LO to the Thorpe scale LT is found to increase monotonically with time in the simulated flows, and therefore may provide the needed time indicator. From the DNS results, a simple parameterization of Γ in terms of LO/LT is found. Applied to observational data, this parameterization leads to a 50%–60% increase in median estimates of turbulent diffusivity, suggesting a potential reassessment of turbulent diffusivity in weakly and intermittently turbulent regimes such as the ocean interior.

An experimental investigation of the vertical temperature structure of homogeneous stratified shear turbulence

The vertical temperature structure of homogeneous stratified shear turbulence is investigated using new rapid vertical temperature measurements in a thermally stratified wind tunnel. Six cases of gradient Richardson number, Rig = N 2 /(dU/dz) 2 , where N is the Brunt-Vaisala frequency (N 2 = (g/T)dT/dz), are studied, spanning a range 0.015 ≤ Ri g ≤ 0.5. Three- to five-hundred high-resolution temperature profiles are made for several streamwise stations for each case of Ri g . These measurements are supplemented with standard fixed-point, Eulerian measurements of streamwise and vertical velocity fluctuations and temperature fluctuations and with an eight-point vertical rake of temperature probes using standard hot-wire and cold-wire techniques. Vertical profiles uniquely enable the computation of available potential energy (APE), Thorpe scales (L Th ), and the diapycnal flux (o d ), as well as one-dimensional vertical wavenumber temperature spectra. These quantities are compared with Eulerian measurements of turbulent kinetic energy (KE), potential energy (PE), and buoyancy flux. It is found that the one-dimensional vertical wavenumber temperature spectrum contains more energy at smaller scales compared to the horizontal spectrum, owing in part to shear distortion, which leads to larger mean square vertical gradients of fluctuating temperature as compared to mean square horizontal gradients. The combination of shear and stratification, especially for cases where the turbulence decays with evolution, accelerates the evolution toward small-scale anisotropy compared to just shear or just stratification. It is found that in highly stratified cases, the diapycnal flux can persist after buoyancy flux has collapsed to negligible values, indicating enhanced heat transfer without turbulent mixing. For low Rig, large-scale vertical advection creates both high local temperature gradients and regions of static instability. Associated with the regions of instability is APE, which grows relative to KE for the least stratified cases. For high Ri g , the turbulence evolves to a wavelike state, containing some counter gradient fluxes and unstable patches. This wavelike state has higher heat flux efficiency than the more turbulent states owing to the low dissipation but relatively high diapycnal flux.

Statistics of the Richardson Number: Mixing Models and Finestructure

Abstract Parameterization of the dissipation rate of turbulent kinetic energy (ϵ) in terms of Richardson number (Ri = N2/S2) is examined for a variety in internal wave environments. Previous work with these data suggests a scaling of ϵ ∼ E2N2g(w) to within a factor of 2, where E represents a low-wavenumber shear spectral density, N the background buoyancy frequency, and g(w) a dependence upon average wave frequency content. The alternative Richardson-number-based parameterization of Kunze et al. is also shown to collapse the dissipation data to within a factor of 2. On the basis of a number of theoretical Richardson number probability distributions, however, the nominal (E, N) scaling of the Kunze et al. model is determined to be E2N3. The difference between the nominal (N3) and observed (N2) scaling is hypothesized to be an effect of turbulent momentum and buoyancy fluxes on the internal wave shear and strain profiles associated with the shear instability. For a statistically homogeneous subset of ...

On the turbulent fluxes of buoyancy, heat and moisture at the air‐sea interface at low wind speeds

Under low‐wind, unstable conditions over water, the Liu, Katsaros and Businger (LKB) estimate of the buoyancy flux into the atmosphere has the form ν1/3 ΔB4/3 g(ξ), where 22.7ξ is the ratio of the Monin‐Obukhov length to the calm‐water roughness length; ΔB is the buoyancy difference between air just above the ocean surface and the ambient air above, and ν is viscosity. An analytic expression for g(ξ) is obtained, for low winds; g(ξ) has a minimum at about ξ=40 (corresponding over tropical oceans to wind speeds of about 0.3–0.6 m s−1). For lower values of x the assumptions underlying the LKB algorithm are no longer valid. For a reasonable choice of dimensionless constants, g(ξ) varies only by 8% over 40&lt;ξ&lt;200, being just above the minimum. Thus the LKB flux estimates are nearly independent of wind speed at their lower limit of applicability; it seems reasonable to identify this LKB minimum flux with laboratory observations of fluxes at zero wind. These show that the buoyancy flux is given by C* Pr−⅔ ν1/3 ΔB4/3, where measured values of C* are about 0.14±0.01 (Golitsin and Grachev, 1986). With Pr=0.71 for air, this implies that g(0) = 0.19 — well within the allowable range of values for the minimum of g(ξ). However, the minimum in g(ξ) varies as (γAT)1/3, where γ is the dimensionless constant in the Dyer‐Hicks‐Businger stability profile and AT the constant in the scalar roughness length for low winds; this can be used to fix AT. In field applications, buoyant motions in the planetary boundary layer ensure that r.m.s. wind speeds wG just above the surface are generally greater than the wind speed at the minimum of g(ξ), even for dry convection — though this would not be true if the planetary boundary layer were typically 100 m thick, instead of 1000 m. This “gustiness” is allowed for by replacing mean wind speed U by (U2+wG2)½ in the bulk formulae.

Turbulence-Scale Condensation Parameterization

When a turbulence closure model is used for cloud simulations, the effect of turbulence-scale saturation must be taken into account in determining the turbulent buoyancy flux. Sommeria and Deardorff derived statistical relations through which the fractional saturation or cloudiness and the mean liquid water specific humidity can be calculated from the moments of conservative thermodynamic quantities, such as total water specific humidity and liquid water potential temperature. In deriving these relations, however, they had to rely on assumptions that were only empirically justified. Mellor showed, by a direct integration of the probability density function, that the assumptions which Sommeria and Deardorff used were not necessary for deriving those relations. But Mellor's direct integration method had to be carried out for each different relation, and can not be generalized to other derivations. In the present paper, the same relations are derived from theorems on Gaussian distributions by introducing a new quantity, qx, which converts the original two-parameter turbulence-scale saturation parameterization problem to a single-parameter problem. A systematic solution method is established based on these theorems. Quantities involving liquid water specific humidity can be obtained through this parameterization from those conservative variables for which predictive equations are available and for which Gaussian distributions are presumed.

On the Nature of Turbulence in a Stratified Fluid. Part I: The Energetics of Mixing

Abstract The definition of the flux Richardson number Rf is generalized to be the ratio of the turbulent buoyancy flux b to the net turbulent mechanical energy m available from all sources. For mechanically energized turbulence where turbulence kinetic energy is used to sustain an upward buoyancy flux (b > 0), it is shown the magnitude of Rf is quantitatively determined by the location of the event in the FrT,–ReT where FrT and –ReT are the local instantaneous overturn Froude number and Reynolds number. In this parameter space the value of Rf varies between 0 and 0.20 for a fluid with Prandtl number greater than one, and between 0 and 0.15 for a fluid with a Prandtl number less than one. For turbulence sustained by a negative buoyancy flux (b < 0), such as penetrative convection in a cooling surface layer, it is shown that the flux Richardson number Rf is a fraction of depth below the surface Rf−1 varies between 0.55 at the surface and –∞ towards the base of the surface layer where the buoyancy flux vanis...

Signatures of Doubly Diffusive Convection and Turbulence in an Intrusive Regime

Abstract Doubly diffusive convection and turbulence were measured from a submarine in a region where the mean gradients do not support doubly diffusive processes but intrusions provide favorable sites. The stability of the submarine-mounted CTD system enabled us to see the unique signature in the T-S relation. The “salt finger” type of doubly diffusive convection contributed substantially less to the buoyancy flux than the laboratory laws predicted and were not important when compared to the load turbulent buoyancy flux.