Scaling arguments are presented to quantify the widely used diapycnal (irreversible) mixing coefficient
$\unicodeSTIX{x1D6E4}=\unicodeSTIX{x1D716}_{PE}/\unicodeSTIX{x1D716}$
in stratified flows as a ...function of the turbulent Froude number
$Fr=\unicodeSTIX{x1D716}/Nk$
. Here,
$N$
is the buoyancy frequency,
$k$
is the turbulent kinetic energy,
$\unicodeSTIX{x1D716}$
is the rate of dissipation of turbulent kinetic energy and
$\unicodeSTIX{x1D716}_{PE}$
is the rate of dissipation of turbulent potential energy. We show that for
$Fr\gg 1$
,
$\unicodeSTIX{x1D6E4}\propto Fr^{-2}$
, for
$Fr\sim \mathit{O}(1)$
,
$\unicodeSTIX{x1D6E4}\propto Fr^{-1}$
and for
$Fr\ll 1$
,
$\unicodeSTIX{x1D6E4}\propto Fr^{0}$
. 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.
The flux Richardson number
$R_{f}$
(often referred to as the mixing efficiency) is a widely used parameter in stably stratified turbulence which is intended to provide a measure of the amount of ...turbulent kinetic energy
$k$
that is irreversibly converted to background potential energy (which is by definition the minimum potential energy that a stratified fluid can attain that is not available for conversion back to kinetic energy) due to turbulent mixing. The flux Richardson number is traditionally defined as the ratio of the buoyancy flux
$B$
to the production rate of turbulent kinetic energy
$P$
. An alternative generalized definition for
$R_{f}$
was proposed by Ivey & Imberger (J. Phys. Oceanogr., vol. 21, 1991, pp. 650–658), where the non-local transport terms as well as unsteady contributions were included as additional sources to the production rate of
$k$
. While this definition precludes the need to assume that turbulence is statistically stationary, it does not properly account for countergradient fluxes that are typically present in more strongly stratified flows. Hence, a third definition that more rigorously accounts for only the irreversible conversions of energy has been defined, where only the irreversible fluxes of buoyancy and production (i.e. the dissipation rates of
$k$
and turbulent potential energy (
$E_{PE}^{\prime }$
)) are used. For stationary homogeneous shear flows, all of the three definitions produce identical results. However, because stationary and/or homogeneous flows are typically not found in realistic geophysical situations, clarification of the differences/similarities between these various definitions of
$R_{f}$
is imperative. This is especially true given the critical role
$R_{f}$
plays in inferring turbulent momentum and heat fluxes using indirect methods, as is commonly done in oceanography, and for turbulence closure parameterizations. To this end, a careful analysis of two existing direct numerical simulation (DNS) datasets of stably stratified homogeneous shear and channel flows was undertaken in the present study to compare and contrast these various definitions. We find that all three definitions are approximately equivalent when the gradient Richardson number
$Ri_{g}\leqslant 1/4$
. Here,
$Ri_{g}=N^{2}/S^{2}$
, where
$N$
is the buoyancy frequency and
$S$
is the mean shear rate, provides a measure of the stability of the flow. However, when
$Ri_{g}>1/4$
, significant differences are noticeable between the various definitions. In addition, the irreversible formulation of
$R_{f}$
based on the dissipation rates of
$k$
and
$E_{PE}^{\prime }$
is the only definition that is free from oscillations at higher gradient Richardson numbers. Both the traditional definition and the generalized definition of
$R_{f}$
exhibit significant oscillations due to the persistence of linear internal wave motions and countergradient fluxes that result in reversible exchanges between
$k$
and
$E_{PE}^{\prime }$
. Finally, we present a simple parameterization for the irreversible flux Richardson number
$R_{f}^{\ast }$
based on
$Ri_{g}$
that produces excellent agreement with the DNS results for
$R_{f}^{\ast }$
.
The time-averaged flow dynamics of a suspended cylindrical canopy patch with a bulk diameter of
$D$
is investigated using large-eddy simulations (LES). The patch consists of
$N_{c}$
constituent solid ...circular cylinders of height
$h$
and diameter
$d$
, mimicking patchy vegetation suspended in deep water (
$H/h\gg 1$
, where
$H$
is the total flow depth). After validation against published data, LES of a uniform incident flow impinging on the canopy patch was conducted to study the effects of canopy density (
$0.16\leqslant \unicodeSTIX{x1D719}=N_{c}(d/D)^{2}\leqslant 1$
, by varying
$N_{c}$
) and bulk aspect ratio (
$0.25\leqslant AR=h/D\leqslant 1$
, by varying
$h$
) on the near-wake structure and adjustment of flow pathways. The relationships between patch geometry, local flow bleeding (three-dimensional redistribution of flow entering the patch) and global flow diversion (streamwise redistribution of upstream undisturbed flow) are identified. An increase in either
$\unicodeSTIX{x1D719}$
or
$AR$
decreases/increases/increases bleeding velocities through the patch surface area along the streamwise/lateral/vertical directions, respectively. However, a volumetric flux budget shows that a larger
$AR$
causes a smaller proportion of the flow rate entering the patch to bleed out vertically. The global flow diversion is found to be determined by both the patch geometrical dimensions and the local bleeding which modifies the sizes of the patch-scale near wake. While loss of flow penetrating the patch increases monotonically with increasing
$\unicodeSTIX{x1D719}$
, its partition into flow diversion around and beneath the patch shows a non-monotonic dependence. The spatial extents of the wake, the flow-diversion dynamics and the bulk drag coefficients of the patch jointly reveal the fundamental differences of flow responses between suspended porous patches and their solid counterparts.
The structure and propagation of lock-release bottom gravity currents in a linearly stratified ambient with the presence of a submerged canopy are investigated for the first time using large-eddy ...simulations. The canopy density (i.e. the solid volume fraction), the strength of ambient stratification and the canopy height are varied to study their respective effects on the gravity current. Both denser canopies and stronger ambient stratification tend to switch the horizontal boundary along which the current propagates from the channel bed towards the canopy top (i.e. the through-to-over flow transition). It is found that the dilution of the current density is enhanced by denser canopies but is weakened by stronger ambient stratification. The non-monotonic relationship between front velocity and canopy density proposed by Zhou et al. (J. Fluid Mech., vol. 831, 2017, pp. 394–417) in homogeneous environments is also observed in stratified environments. However, as the ambient stratification is strengthened, the present study shows a shift of the turning point (beyond which increasing canopy density leads to faster current propagation) towards sparser canopies, accompanied by a more pronounced recovery of the front velocity. This is the combined action of three stratification-induced mechanisms: the promotion of through-to-over flow transition (less canopy drag), the upward displacement of current nose in a stably stratified water column (more buoyancy gain) and the weakening of current dilution (less buoyancy loss). Under stronger ambient stratification, the propagation of gravity currents shows a lower sensitivity to the retarding effect of the submerged canopy.
Estimates 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.
Key Points
We highlight important ambiguities of some common single‐parameter schemes
The findings will be useful for modeling geophysical flows
Abstract
Oceanic density overturns are commonly used to parameterize the dissipation rate of turbulent kinetic energy. This method assumes a linear scaling between the Thorpe length scale
L
T
and the ...Ozmidov length scale
L
O
. Historic evidence supporting
L
T
~
L
O
has been shown for relatively weak shear-driven turbulence of the thermocline; however, little support for the method exists in regions of turbulence driven by the convective collapse of topographically influenced overturns that are large by open-ocean standards. This study presents a direct comparison of
L
T
and
L
O
, using vertical profiles of temperature and microstructure shear collected in the Luzon Strait—a site characterized by topographically influenced overturns up to
O
(100) m in scale. The comparison is also done for open-ocean sites in the Brazil basin and North Atlantic where overturns are generally smaller and due to different processes. A key result is that
L
T
/
L
O
increases with overturn size in a fashion similar to that observed in numerical studies of Kelvin–Helmholtz (K–H) instabilities for all sites but is most clear in data from the Luzon Strait. Resultant bias in parameterized dissipation is mitigated by ensemble averaging; however, a positive bias appears when instantaneous observations are depth and time integrated. For a series of profiles taken during a spring tidal period in the Luzon Strait, the integrated value is nearly an order of magnitude larger than that based on the microstructure observations. Physical arguments supporting
L
T
~
L
O
are revisited, and conceptual regimes explaining the relationship between
L
T
/
L
O
and a nondimensional overturn size
are proposed. In a companion paper, Scotti obtains similar conclusions from energetics arguments and simulations.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
Most commonly used models for turbulent mixing in the ocean rely on a background stratification against which turbulence must work to stir the fluid. While this background stratification is typically ...well defined in idealized numerical models, it is more difficult to capture in observations. Here, a potential discrepancy in ocean mixing estimates due to the chosen calculation of the background stratification is explored using direct numerical simulation data of breaking internal waves on slopes. Two different methods for computing the buoyancy frequency
$N$
, one based on a three-dimensionally sorted density field (often used in numerical models) and the other based on locally sorted vertical density profiles (often used in the field), are used to quantify the effect of
$N$
on turbulence quantities. It is shown that how
$N$
is calculated changes not only the flux Richardson number
$R_{f}$
, which is often used to parameterize turbulent mixing, but also the turbulence activity number or the Gibson number
$Gi$
, leading to potential errors in estimates of the mixing efficiency using
$Gi$
-based parameterizations.
The propagation of full-depth lock-exchange bottom gravity currents past a submerged array of circular cylinders is investigated using laboratory experiments and large eddy simulations. Firstly, to ...investigate the front velocity of gravity currents across the whole range of array density
$\unicodeSTIX{x1D719}$
(i.e. the volume fraction of solids), the array is densified from a flat bed (
$\unicodeSTIX{x1D719}=0$
) towards a solid slab (
$\unicodeSTIX{x1D719}=1$
) under a particular submergence ratio
$H/h$
, where
$H$
is the flow depth and
$h$
is the array height. The time-averaged front velocity in the slumping phase of the gravity current is found to first decrease and then increase with increasing
$\unicodeSTIX{x1D719}$
. Next, a new geometrical framework consisting of a streamwise array density
$\unicodeSTIX{x1D707}_{x}=d/s_{x}$
and a spanwise array density
$\unicodeSTIX{x1D707}_{y}=d/s_{y}$
is proposed to account for organized but non-equidistant arrays (
$\unicodeSTIX{x1D707}_{x}\neq \unicodeSTIX{x1D707}_{y}$
), where
$s_{x}$
and
$s_{y}$
are the streamwise and spanwise cylinder spacings, respectively, and
$d$
is the cylinder diameter. It is argued that this two-dimensional parameter space can provide a more quantitative and unambiguous description of the current–array interaction compared with the array density given by
$\unicodeSTIX{x1D719}=(\unicodeSTIX{x03C0}/4)\unicodeSTIX{x1D707}_{x}\unicodeSTIX{x1D707}_{y}$
. Both in-line and staggered arrays are investigated. Four dynamically different flow regimes are identified: (i) through-flow propagating in the array interior subject to individual cylinder wakes (
$\unicodeSTIX{x1D707}_{x}$
: small for in-line array and arbitrary for staggered array;
$\unicodeSTIX{x1D707}_{y}$
: small); (ii) over-flow propagating on the top of the array subject to vertical convective instability (
$\unicodeSTIX{x1D707}_{x}$
: large;
$\unicodeSTIX{x1D707}_{y}$
: large); (iii) plunging-flow climbing sparse close-to-impermeable rows of cylinders with minor streamwise intrusion (
$\unicodeSTIX{x1D707}_{x}$
: small;
$\unicodeSTIX{x1D707}_{y}$
: large); and (iv) skimming-flow channelized by an in-line array into several subcurrents with strong wake sheltering (
$\unicodeSTIX{x1D707}_{x}$
: large;
$\unicodeSTIX{x1D707}_{y}$
: small). The most remarkable difference between in-line and staggered arrays is the non-existence of skimming-flow in the latter due to the flow interruption by the offset rows. Our analysis reveals that as
$\unicodeSTIX{x1D719}$
increases, the change of flow regime from through-flow towards over- or skimming-flow is responsible for increasing the gravity current front velocity.
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.
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