Turbulent Froude Number Research Articles

Stratification and temporal evolution of mixing regimes in diurnally heated river flows

Direct numerical simulations of stratified open channel flows subject to a varying surface heat flux are performed. The influence of the diurnal heating time on the spatial and temporal variation of mixing in the flow and the characteristics of the mean flow state are examined. The control parameters are the bulk stability parameter lambda_{B}, defined through the ratio of the channel height delta and a bulk Obukhov length scale mathscr{L}_{B}, and the diurnal time scale hat{t}, defined as the ratio of the heating time to an eddy turnover time. The Prandtl number Pr and Reynolds number Re_{tau} have values of 1 and 400. Simulations are performed over hat{t} = 1 to 24 and lambda_{B} = 0.6 to 26. Two key flow features are used to classify the flow regimes observed, namely the laminar layer depth (LLD) and stratified layer depth (SLD) where the LLD is defined as the depth from the free surface when the buoyancy Reynolds number Re_{B} approx 7 and the SLD is the depth from the free surface when the turbulent Froude number Fr approx 1. This study attempts to characterise how these length scales vary across the diel cycle. The LLD is a viscous length scale and a regime map of a viscous parameter, the bulk Obhukov Reynolds number Re_mathscr{L}, and hat{t} is presented to classify the LLD behaviour. A regime map of lambda _{B} and hat{t} is presented to classify the behaviour of the SLD. Three classifications for each layer depth behaviour within a diel cycle form the basis of the regime maps for this paper: a neutral flow where the LLD or SLD does not exist (denoted by NL and NS), a stratified flow where the LLD or SLD are diurnally varying (denoted as DL and DS) and a persistent layer of the LLD or SLD (denoted as PL and PS). The transition between the NL to DL is hat{t} propto Re_{mathscr{L}}^{4.5}, DL to PL is hat{t} propto Re_{mathscr{L}}^{- 0.5}, NS to DS is hat{t} propto lambda _{B}^{0} and DS to PS is hat{t} propto lambda _{B}^{1}. The regime maps may be used as a predictive tool to determine when suppressed mixing regimes occur in rivers. At each flow depth, the flow sweeps though a range of mixing states across the diel cycle. The local mixing efficiency are briefly assessed and found to scale well with the instantaneous Fr number according to the regimes proposed by Garanaik and Venayagamoorthy (J. Fluid Mech., vol. 867, 2019, pp. 323-333).

Mixing rates in stably stratified flows with respect to the turbulent froude number and turbulent scales

Mixing rates in stably stratified flows with respect to the turbulent froude number and turbulent scales

Revised mixing coefficient scaling for sheared stably stratified turbulence

We revisit and extend the turbulent Froude number ( $Fr_k$ ) scaling for the mixing coefficient ( $\varGamma$ ) introduced by Garanaik & Venayagamoorthy (GV) (J. Fluid Mech., vol. 867, 2019, pp. 323–333) by directly incorporating the effects of mean shear through the non-dimensional shear parameter $S_{\ast } = S k/\epsilon _k$ . For flows where the effects of mean shear are stronger than the background vertical stratification, we find $\varGamma \sim Fr_k^{-2} S_\ast ^{-1}$ for weakly stratified sheared turbulence and $\varGamma \sim Fr_k^{-1}S_\ast ^{-1}$ for moderately stratified sheared turbulence. The scaling procedure is inconclusive for strongly stratified sheared turbulence, but using two independent datasets of homogeneous, sheared, stably stratified turbulence, we empirically observe $\varGamma \sim Fr_k^{-0.5} S_\ast ^{-1}$ . Our revised scaling better collapses both datasets compared with the original GV scaling, and we note that the moderately stratified sheared regime is extremely narrow (or maybe even non-existent). We also apply our scaling to the time-varying open channel simulations of Issaev et al. (J. Fluid Mech., vol. 935, 2022) and observe $\varGamma \sim Fr_k^{-2}S_\ast ^{-1}$ for weakly stratified sheared turbulence, but we observe deviations from our revised scaling for moderate and strong stratifications due to time-varying mean shear and vertical transport. Finally, we apply our revised scaling to field measurements of Conry, Kit & Fernando (Environ. Fluid Mech., vol. 20, 2020, pp. 1177–1197) and observe $\varGamma \sim Fr_k^{-2} S_\ast ^{-1}$ . We emphasize that our revised scaling is applicable only for stably stratified, vertically sheared turbulence with weak spatio-temporal variations of the mean shear and stratification, and we expect different scaling to apply when additional effects such as depth-varying radiative heating/cooling are present or when the orientation of the mean shear relative to the gravity vector is modified (e.g. horizontal shear).

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).

Dynamics and energetics underlying mixing efficiency in homogeneous stably stratified turbulence

Global and regional ocean simulations rely on eddy viscosities and diffusivities to account for irreversible mixing of momentum and scalars due to unresolved scales of motion. These closures are often sensitive to the values of the mixing coefficient, whose shape has been well characterized in terms of turbulence parameters. In this paper, we connect this well-established shape of the mixing coefficient curve to the underlying physics of stably stratified turbulence as a function of the turbulent Froude number.

Scaling the Mixing Efficiency of Sediment‐Stratified Turbulence

AbstractThe flux Richardson number Rf, also called the mixing efficiency of stratified turbulence, is important in determining geophysical flow phenomena such as ocean circulation and air‐sea transports. Measuring Rf in the field is usually difficult, thus parameterization of Rf based on readily observed properties is essential. Here, estimates of Rf in a strongly turbulent, sediment‐stratified estuarine flow are obtained from measurements of covariance‐derived turbulent buoyancy fluxes (B) and spectrally fitted values of the dissipation rate of turbulent kinetic energy (ε). We test scalings for Rf in terms of the buoyancy Reynolds number (Reb), the gradient Richardson number (Ri), and turbulent Froude number (Frt). Neither the Reb‐based nor the Ri‐based scheme is able to describe the observed variations in Rf, but the Frt‐based parameterization works well. These findings support further use of the Frt ‐ based parameterization in turbulent oceanic and estuarine environments.

Parameterization of mixing in stratified open channel flow

The dynamics and parameterization of mixing in temporally evolving turbulent open-channel flow is investigated through direct numerical simulations as the flow transitions from an initially neutral state to stable stratification. We observe three distinctly different mixing regimes separated by transitional values of turbulent Froude number $Fr$ : a weakly stratified regime for $Fr >1$ ; an intermediate regime for $0.3< Fr<1$ ; and a saturated regime for $Fr<0.3$ . The mixing coefficient $\varGamma =B/\epsilon _K$ , (where $B$ is the buoyancy flux and $\epsilon _K$ is the dissipation rate of kinetic energy), is well predicted by the parameterization schemes of Maffioli et al. (J. Fluid Mech., vol. 794, 2016) and Garanaik & Venayagamoorthy (J. Fluid Mech., vol. 867, 2019, pp. 323–333) across all three regimes through instantaneous measurements of $Fr$ and the ratio $L_E/L_O$ , where $L_E$ and $L_O$ are the Ellison and Ozmidov length scales, respectively. The flux Richardson number $R_f = B/(B+\epsilon _K)$ shows linear dependence on the gradient Richardson number $Ri_g$ up to a transitional value of $Ri_g =0.25$ , past which it saturates again to a constant value independent of $Fr$ or $Ri_g$ . By examining the flow as a balance of inertial, shear and buoyancy forces, we derive physically based scaling relationships to demonstrate that $Ri_g \sim Fr^{-2}$ and $Ri_g \sim Fr^{-1}$ in the weakly and moderately stratified regimes and that $Ri_g$ becomes independent of $Fr$ in the saturated regime. Our results suggest that the $L_E/L_O \sim Fr^{-1}$ scaling of Garanaik & Venayagamoorthy (J. Fluid Mech., vol. 867, 2019, pp. 323–333) in the intermediate regime manifests due to the influence of mean shear. The differences in the relationships between $Fr$ and $L_E/L_O$ for non-sheared flows within this regime are discussed.

Diapycnal mixing efficiency in lock-exchange gravity currents

An experimental measure of irreversible mixing efficiency, $R\phantom{\rule{0}{0ex}}{i}_{f}^{*}$, for a lock-exchange gravity current is presented. The results show that mixing efficiency increases and is sustained in the presence of strong stratification provided the flow energy is high. Following this, various turbulence regimes are mapped based on turbulent Froude number and turbulent Reynolds number. Additionally, $R\phantom{\rule{0}{0ex}}{i}_{f}^{*}$ is shown to have a strong dependence on turbulence regime and molecular effects induced by Prandtl number.

Mixing in forced stratified turbulence and its dependence on large-scale forcing

We study direct numerical simulations of turbulence arising from the interaction of an initial background shear, a linear background stratification and an external body force. In each simulation the turbulence produced is spatially intermittent, with dissipation rates varying over orders of magnitude in the vertical. We focus analysis on the statistically quasi-steady states achieved by applying large-scale body forcing to the domain, and compare flows forced by internal gravity waves with those forced by vertically uniform vortical modes. By considering the turbulent energy budgets for each simulation, we find that the injection of potential energy from the wave forcing permits a reversal in the sign of the mean buoyancy flux. This change in the sign of the buoyancy flux is associated with large, convective density overturnings, which in turn lead to more efficient mixing in the wave-forced simulations. The inhomogeneous dissipation in each simulation allows us to investigate localised correlations between the kinetic and potential energy dissipation rates. These correlations lead us to the conclusion that an appropriate definition of an instantaneous mixing efficiency, (where and are the volume-averaged turbulent viscous dissipation rate and fluctuation density variance destruction rate respectively) in the wave-forced cases is independent of an appropriately defined local turbulent Froude number, consistent with scalings proposed for low Froude number stratified turbulence.

On the inference of the state of turbulence and mixing efficiency in stably stratified flows

Scaling arguments are presented to quantify the widely used diapycnal (irreversible) mixing coefficient $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D716}_{PE}/\unicode[STIX]{x1D716}$ in stratified flows as a function of the turbulent Froude number $Fr=\unicode[STIX]{x1D716}/Nk$. Here, $N$ is the buoyancy frequency, $k$ is the turbulent kinetic energy, $\unicode[STIX]{x1D716}$ is the rate of dissipation of turbulent kinetic energy and $\unicode[STIX]{x1D716}_{PE}$ is the rate of dissipation of turbulent potential energy. We show that for $Fr\gg 1$, $\unicode[STIX]{x1D6E4}\propto Fr^{-2}$, for $Fr\sim \mathit{O}(1)$, $\unicode[STIX]{x1D6E4}\propto Fr^{-1}$ and for $Fr\ll 1$, $\unicode[STIX]{x1D6E4}\propto Fr^{0}$. These scaling results are tested using high-resolution direct numerical simulation (DNS) data from three different studies and are found to hold reasonably well across a wide range of $Fr$ that encompasses weakly stratified to strongly stratified flow conditions. Given that the $Fr$ cannot be readily computed from direct field measurements, we propose a practical approach that can be used to infer the $Fr$ from readily measurable quantities in the field. Scaling analyses show that $Fr\propto (L_{T}/L_{O})^{-2}$ for $L_{T}/L_{O}>O(1)$, $Fr\propto (L_{T}/L_{O})^{-1}$ for $L_{T}/L_{O}\sim O(1)$, and $Fr\propto (L_{T}/L_{O})^{-2/3}$ for $L_{T}/L_{O}<O(1)$, where $L_{T}$ is the Thorpe length scale and $L_{O}$ is the Ozmidov length scale. These formulations are also tested with DNS data to highlight their validity. These novel findings could prove to be a significant breakthrough not only in providing a unifying (and practically useful) parameterization for the mixing efficiency in stably stratified turbulence but also for inferring the dynamic state of turbulence in geophysical flows.

Assessment of small-scale anisotropy in stably stratified turbulent flows using direct numerical simulations

In this study, the validity of the widely used Kolmogorov hypothesis of local isotropy of dissipative scales in stably stratified turbulent flows is investigated using high-resolution direct numerical simulations. Departure from isotropy is investigated for the first time in terms of the turbulent Froude number defined as Fr = ϵ/(NEk), where N is the buoyancy frequency, Ek is the turbulent kinetic energy, and ϵ is the rate of dissipation of turbulent kinetic energy. It is shown that for both ϵ and the rate of scalar (density) dissipation, ϵρ, the small-scale statistics are nearly isotropic in the weakly stratified turbulent regime (Fr > 1). For strongly stable flows (Fr<O(1)), the vertical gradients of the fluctuating velocity field as well as the scalar fluctuations overestimate, while horizontal gradients underestimate the true rates of dissipation computed directly using the three-dimensional fluctuating velocity and scalar fields. However, in practice, microstructure instruments are typically limited to one-dimensional measurements of the fluctuating fields and hence very likely to yield erroneous estimates of dissipation rates of turbulent kinetic energy and scalar variance. Thus, a new parameterization is proposed here that can be used to infer true dissipation rates from one-dimensional measured values in oceanic flows.

Self-similar mixing in stratified plane Couette flow for varying Prandtl number

We investigate fully developed turbulence in stratified plane Couette flows using direct numerical simulations similar to those reported by Deusebioet al.(J. Fluid Mech., vol. 781, 2015, pp. 298–329) expanding the range of Prandtl number$Pr$examined by two orders of magnitude from 0.7 up to 70. Significant effects of$Pr$on the heat and momentum fluxes across the channel gap and on the mean temperature and velocity profile are observed. These effects can be described through a mixing length model coupling Monin–Obukhov (M–O) similarity theory and van Driest damping functions. We then employ M–O theory to formulate similarity scalings for various flow diagnostics for the stratified turbulence in the gap interior. The midchannel gap gradient Richardson number$Ri_{g}$is determined by the length scale ratio$h/L$, where$h$is the half-channel gap depth and$L$is the Obukhov length scale. As$h/L$approaches very large values,$Ri_{g}$asymptotes to a maximum characteristic value of approximately 0.2. The buoyancy Reynolds number$Re_{b}\equiv \unicode[STIX]{x1D700}/(\unicode[STIX]{x1D708}N^{2})$, where$\unicode[STIX]{x1D700}$is the dissipation,$\unicode[STIX]{x1D708}$is the kinematic viscosity and$N$is the buoyancy frequency defined in terms of the local mean density gradient, scales linearly with the length scale ratio$L^{+}\equiv L/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$, where$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$is the near-wall viscous scale. The flux Richardson number$Ri_{f}\equiv -B/P$, where$B$is the buoyancy flux and$P$is the shear production, is found to be proportional to$Ri_{g}$. This then leads to a turbulent Prandtl number$Pr_{t}\equiv \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D705}_{t}$of order unity, where$\unicode[STIX]{x1D708}_{t}$and$\unicode[STIX]{x1D705}_{t}$are the turbulent viscosity and diffusivity respectively, which is consistent with Reynolds analogy. The turbulent Froude number$Fr_{h}\equiv \unicode[STIX]{x1D700}/(NU^{\prime 2})$, where$U^{\prime }$is a turbulent horizontal velocity scale, is found to vary like$Ri_{g}^{-1/2}$. All these scalings are consistent with our numerical data and appear to be independent of$Pr$. The classical Osborn model based on turbulent kinetic energy balance in statistically stationary stratified sheared turbulence (Osborn,J. Phys. Oceanogr., vol. 10, 1980, pp. 83–89), together with M–O scalings, results in a parameterization of$\unicode[STIX]{x1D705}_{t}/\unicode[STIX]{x1D708}\sim \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D708}\sim Re_{b}Ri_{g}/(1-Ri_{g})$. With this parameterization validated through direct numerical simulation data, we provide physical interpretations of these results in the context of M–O similarity theory. These results are also discussed and rationalized with respect to other parameterizations in the literature. This paper demonstrates the role of M–O similarity in setting the mixing efficiency of equilibrated constant-flux layers, and the effects of Prandtl number on mixing in wall-bounded stratified turbulent flows.

Local versus volume-integrated turbulence and mixing in breaking internal waves on slopes

Using direct numerical simulations (DNS), we explore local and volume-integrated measures of turbulence and mixing in breaking internal waves on slopes. We consider eight breaking wave cases with a range of normalized pycnocline thicknesses $k\unicode[STIX]{x1D6FF}$, where $k$ is the horizontal wavenumber and $\unicode[STIX]{x1D6FF}$ is the pycnocline thickness, but with similar incoming wave properties. The energetics of wave breaking is quantified in terms of local turbulent dissipation and irreversible mixing using the method of Scotti &amp; White (J. Fluid Mech., vol. 740, 2014, pp. 114–135). Local turbulent mixing efficiencies are calculated using the irreversible flux Richardson number $R_{f}^{\ast }$ and are found to be a function of the turbulent Froude number $Fr_{k}$. Volume-integrated measures of the turbulent mixing efficiency during wave breaking are also made, and are found to be functions of $k\unicode[STIX]{x1D6FF}$. The bulk turbulent mixing efficiency ranges from 0.25 to 0.37 and is maximized when $k\unicode[STIX]{x1D6FF}\approx 1$. In order to connect local and bulk mixing efficiency measures, the variation in the bulk turbulent mixing efficiency with $k\unicode[STIX]{x1D6FF}$ is related to the turbulent Froude number at which the maximum total mixing occurs over the course of the breaking event, $Fr_{k}^{max}$. We find that physically, $Fr_{k}^{max}$ is controlled by the vertical length scale of billows at the interface during wave breaking.

Mixing efficiency in stratified turbulence

We consider mixing of the density field in stratified turbulence and argue that, at sufficiently high Reynolds numbers, stationary turbulence will have a mixing efficiency and closely related mixing coefficient described solely by the turbulent Froude number$Fr={\it\epsilon}_{k}/(Nu^{2})$, where${\it\epsilon}_{k}$is the kinetic energy dissipation,$u$is a turbulent horizontal velocity scale and$N$is the Brunt–Väisälä frequency. For$Fr\gg 1$, in the limit of weakly stratified turbulence, we show through a simple scaling analysis that the mixing coefficient scales as${\it\Gamma}\propto Fr^{-2}$, where${\it\Gamma}={\it\epsilon}_{p}/{\it\epsilon}_{k}$and${\it\epsilon}_{p}$is the potential energy dissipation. In the opposite limit of strongly stratified turbulence with$Fr\ll 1$, we argue that${\it\Gamma}$should reach a constant value of order unity. We carry out direct numerical simulations of forced stratified turbulence across a range of$Fr$and confirm that at high$Fr$,${\it\Gamma}\propto Fr^{-2}$, while at low$Fr$it approaches a constant value close to${\it\Gamma}=0.33$. The parametrization of${\it\Gamma}$based on$Re_{b}$due to Shihet al.(J. Fluid Mech., vol. 525, 2005, pp. 193–214) can be reinterpreted in this light because the observed variation of${\it\Gamma}$in their study as well as in datasets from recent oceanic and atmospheric measurements occurs at a Froude number of order unity, close to the transition value$Fr=0.3$found in our simulations.

The quest for an unambiguous parameterization of mixing efficiency in stably stratified geophysical flows

AbstractEstimates of turbulent mixing in geophysical settings typically depend on the efficiency at which shear‐driven turbulence mixes density across isopycnals. To date, however, no unifying parameterization of diapycnal mixing efficiency exists due to the variability of natural flows and also due to certain ambiguities that arise from descriptions based on a single parameter. Here we highlight important ambiguities of some common single‐parameter schemes in the context of a multiparameter framework that considers the independent effects of shear, buoyancy, and viscosity. Parameterizations based on the gradient Richardson number (Ri), the turbulent Froude number (FrT), and the buoyancy Reynolds number (Reb) are considered. The diagnostic ability of these parameters is examined using published data from both direct numerical simulations and field observations.

Comparison of mixing efficiency and vertical diffusivity models from temperature microstructure

Due to the many difficulties associated with measuring buoyancy flux directly in the field, parameterizations for the flux involving eddy diffusivities are commonly used. These parameterizations are often cast in terms of a mixing efficiency (Γ), which itself is often assumed to be a constant throughout the water column (0.17–0.2). Alternatively, Γ can be calculated in terms of turbulence parameters which attempt to capture the varying mixing dynamics throughout the water column. In this paper three separate parameterizations are used to calculate the eddy diffusivities in a dynamically evolving stratified water column in the Gulf of Aqaba. Furthermore, we use two different approaches for calculating Γ by Ivey and Imberger (1991) and Shih et al. (2005) for these diffusivity parameterizations and compare the results obtained using each approach. Further work on the Shih et al. (2005) parameterization by D. Bouffard and L. Boegman (personal communication, 2011) has also been used in this analysis. While direct measurements of buoyancy flux are needed to validate the actual accuracy of the mixing efficiency parameterizations, we can make the following conclusions. First, we found that in some circumstances the eddy diffusivity models produce very different estimates from one another and the estimates are themselves sensitive to the choice of parameterization for Γ. Second, when the stratification is weak and temperature variance is small, parameterizations involving calculations of the Thorpe scale, χT as well as the turbulent Froude number and Reynolds numbers should be treated with great caution. Third, under the same circumstances, the Shih et al. (2005) approach for Γ applied to the κρ parameterization for vertical diffusivity seems to be the best vertical diffusivity (based on a comparison to κχ).

Parameterizing individual effects of shear and stratification on mixing in stably stratified shear flows

This study proposes new parameterizations of diapycnal mixing by reanalyzing the results of previous laboratory and numerical experiments on homogeneous stably stratified shear flows. Unlike previous studies that use either the turbulent Froude number Fr or gradient Richardson number Rig, this study parameterizes nondimensional momentum and buoyancy fluxes as functions of Fr and a turbulent shear number Sh, in order to quantify individual effects of shear and stratification. Turbulent momentum flux is found to depend linearly on Sh and to decrease monotonically with decreasing Fr. Turbulent buoyancy flux has a peak at moderate Fr. With increasing Sh, it decreases and increases at high and low Fr, respectively. The increase of Sh also cause relatively small but significant decreases of nondimensional turbulent properties, such as the nondimensional conversion rate of turbulent potential energy to background potential energy. The proposed parameterizations lie within the scatter of limited available field data. The parameterizations may be reduced to Rig‐based ones by incorporating the relationship between Rigand turbulence intensity observed in the field. Existing stability functions for two‐equation turbulent closure schemes are found to over‐predict mixing efficiency at low Fr.

The mixing efficiency of interfacial waves breaking at a ridge: 2. Local mixing processes

[1] The efficiency with which internal wave energy is converted irreversibly to diapycnal mixing in bathymetry-induced mixing events is of great importance to larger-scale ocean modeling efforts. High-resolution laboratory measurements are used to investigate the spatial and temporal variability of interfacial wave breaking events at a submerged bathymetric ridge. From high spatial resolution measurements, it appears the local efficiency can vary significantly. Parameterizations based on the turbulent Reynolds number and Froude number suggest that the local mixing efficiency within the overturning patch at the interface is 10–17% but the local efficiency is near zero within the relatively homogenous layers. When the local mixing efficiency is integrated over the entire event, the resulting overall efficiency is consistent with the result from Part 1 that the overall event efficiency is 3–8%. This spatial variation in the mixing efficiency reinforces the importance of dynamic mixing efficiency parameterizations based on local stratified turbulence parameters.

A three‐dimensional numerical simulation of spreading umbrella clouds

During explosive volcanic eruptions, an eruption column buoyantly rises as a turbulent plume, and an umbrella cloud spreads laterally as a gravity current at the neutral buoyancy level. The source conditions of explosive eruptions such as mass discharge rates of magma at vents have been estimated from the field observations (e.g., satellite images) on the height of the eruption columns and the spreading rate of umbrella clouds on the basis of the one‐dimensional (1‐D) plume model and the gravity current model. However, these simplified models contain empirical constants (entrainment coefficient of turbulent plume, k, and Froude number of gravity current, λ) that should be justified. We developed a time‐dependent three‐dimensional (3‐D) numerical model of eruption clouds to independently determine the values of these constants. The 3‐D model is designed to calculate quantitative features of turbulent mixing between an eruption cloud and the ambient air without any a priori empirical constants by applying a sufficiently fine grid size with a third‐order accuracy scheme. It has reproduced the fundamental features of eruption clouds including eruption columns, pyroclastic flows, coignimbrite ash clouds, and umbrella clouds. The altitudes of the spreading umbrella clouds in the 3‐D simulations are consistent with those of the neutral buoyancy level of eruption columns estimated by the 1‐D plume model with the entrainment coefficient k ∼ 0.1. The relationship between the volumetric flow rate and the spreading rate of the umbrella cloud in the 3‐D simulations is explained by the axisymmetrical gravity current model with λ ∼ 0.2 for eruption clouds in the tropical atmosphere and λ ∼ 0.1 for those in the midlatitude atmosphere. On the other hand, the total column height in the 3‐D simulations strongly oscillates even for a constant mass discharge rate, and its time average tends to be substantially greater than the column height estimated by the 1‐D plume model with k ∼ 0.1, particularly for large‐scale eruption clouds in the midlatitude atmosphere. We recommend a method to estimate the mass discharge rate from the observation of the column height or the spreading rate of umbrella cloud. The method and the accuracy of the 3‐D simulations are tested on the basis of the field data (e.g., the total height of the eruption column, the altitude and the spreading rate of the umbrella cloud, and the mass discharge rate) of the Pinatubo 1991 eruption.

Simulation of shear orientation effects on stably stratified homogeneous turbulence with RANS second-order modelling

The paper investigates the ability of a differential second-order turbulence closure to predict the effects of shear directionality on stably stratified homogeneous turbulence. The study does not aim to propose new turbulence models but rather aims to provide a detailed examination of the behaviour of the turbulence closure from an extensive set of available direct numerical simulations (DNS) of homogeneous and stably stratified flows that have been subjected to non-vertical shear. The study disclosed several previously unknown features about the predicted behaviour of turbulence subjected to a mean shear gradient that makes an arbitrary angle with the plane of the temperature gradient (which is aligned with the gravitational acceleration vector) for a wide range of Ri numbers. The turbulence closure comprises transport equations for all of the components of the Reynolds stress tensor, the heat flux vector and for the dissipation rates of kinetic energy and temperature fluctuations. Further, an anisotropic dissipation tensor is considered. The results show that the model captures the impact on the turbulence growth rate of increasing the horizontal shear. The model yields more energetic turbulence levels with increasing orientation angle, and it predicts accurately the “critical” inclination angle. One of the most important findings is that the model is capable of estimating the critical Ri for both the vertical and the horizontal shear cases when a properly calibrated buoyancy term is included in the ϵ-equation. For both cases, the model predicts the same turbulent Froude number for critical Richardson number, in agreement with the DNS data. Moreover, the present model is able to predict the strong increase in the mixing efficiency parameter from the vertical shear to the horizontal shear cases, which was not achieved in previous studies. The model is sensitive to buoyancy damping of vertical velocity fluctuations, which ultimately lead to the overall decay of all of the velocity components. The comparison of the present predictions with rapid distortion theory (RDT) analysis attests the importance of non-linear processes in the approach to “sudden suppression” of turbulence. However, some concerns remain in the behaviour of the model. A major source of uncertainty relates to the anisotropy of Reynolds stresses. It was found that the observed departure in the prediction of anisotropies is not simply related to the accuracy of the pressure–strain redistribution, but the anisotropic dissipation modelling has a role to play in stratified flows.