scholarly journals Role of overturns in optimal mixing in stratified mixing layers

2017 ◽  
Vol 826 ◽  
pp. 522-552 ◽  
Author(s):  
A. Mashayek ◽  
C. P. Caulfield ◽  
W. R. Peltier
Keyword(s):  
Numerical Simulations ◽  
Ocean Circulation ◽  
Turbulent Mixing ◽  
Isotropic Turbulence ◽  
Life Cycles ◽  
Shear Instability ◽  
Mixing Efficiency ◽  
Small Scale ◽  
Mean Flow

Turbulent mixing plays a major role in enabling the large-scale ocean circulation. The accuracy of mixing rates estimated from observations depends on our understanding of basic fluid mechanical processes underlying the nature of turbulence in a stratified fluid. Several of the key assumptions made in conventional mixing parameterizations have been increasingly scrutinized in recent years, primarily on the basis of adequately high resolution numerical simulations. We add to this evidence by compiling results from a suite of numerical simulations of the turbulence generated through stratified shear instability processes. We study the inherently intermittent and time-dependent nature of wave-induced turbulent life cycles and more specifically the tight coupling between inherently anisotropic scales upon which small-scale isotropic turbulence grows. The anisotropic scales stir and stretch fluid filaments enhancing irreversible diffusive mixing at smaller scales. We show that the characteristics of turbulent mixing depend on the relative time evolution of the Ozmidov length scale $L_{O}$ compared to the so-called Thorpe overturning scale $L_{T}$ which represents the scale containing available potential energy upon which turbulence feeds and grows. We find that when $L_{T}\sim L_{O}$, the mixing is most active and efficient since stirring by the largest overturns becomes ‘optimal’ in the sense that it is not suppressed by ambient stratification. We argue that the high mixing efficiency associated with this phase, along with observations of $L_{O}/L_{T}\sim 1$ in oceanic turbulent patches, together point to the potential for systematically underestimating mixing in the ocean if the role of overturns is neglected. This neglect, arising through the assumption of a clear separation of scales between the background mean flow and small-scale quasi-isotropic turbulence, leads to the exclusion of an highly efficient mixing phase from conventional parameterizations of the vertical transport of density. Such an exclusion may well be significant if the mechanism of shear-induced turbulence is assumed to be representative of at least some turbulent events in the ocean. While our results are based upon simulations of shear instability, we show that they are potentially more generic by making direct comparisons with $L_{T}-L_{O}$ data from ocean and lake observations which represent a much wider range of turbulence-inducing physical processes.

Time-dependent, non-monotonic mixing in stratified turbulent shear flows: implications for oceanographic estimates of buoyancy flux

2013 ◽  
Vol 736 ◽  
pp. 570-593 ◽  
Author(s):  
A. Mashayek ◽  
C. P. Caulfield ◽  
W. R. Peltier
Keyword(s):  
Reynolds Number ◽  
Dissipation Rate ◽  
Isotropic Turbulence ◽  
Buoyancy Flux ◽  
Shear Instability ◽  
Turbulent Shear ◽  
Mixing Efficiency ◽  
Small Scale ◽  
Stratified Turbulence ◽  
Flow Evolution

AbstractWe employ direct numerical simulation to investigate the efficiency of diapycnal mixing by shear-induced turbulence in stably stratified free shear layers for flows with bulk Richardson numbers in the range $0. 12\leq R{i}_{0} \leq 0. 2$ and Reynolds number $Re= 6000$. We show that mixing efficiency depends non-monotonically upon $R{i}_{0} $, peaking in the range 0.14–0.16, which coincides closely with the range in which both the buoyancy flux and the dissipation rate are maximum. By detailed analyses of the energetics of flow evolution and the underlying dynamics, we show that the existence of high mixing efficiency in the range $0. 14\lt R{i}_{0} \lt 0. 16$ is due to the emergence of a large number of small-scale instabilities which do not exist at lower Richardson numbers and are stabilized at high Richardson numbers. As discussed in Mashayek & Peltier (J. Fluid Mech., vol. 725, 2013, pp. 216–261), the existence of such a well-populated ‘zoo’ of secondary instabilities at intermediate Richardson numbers and the subsequent high mixing efficiency is realized only if the Reynolds number is higher than a critical value which is generally higher than that achievable in laboratory settings, as well as that which was achieved in the majority of previous numerical studies of shear-induced stratified turbulence. We furthermore show that the primary assumptions upon which the widely employed Osborn (J. Phys. Oceanogr. vol. 10, 1980, pp. 83–89) formula is based, as well as its counterparts and derivatives, which relate buoyancy flux to dissipation rate through a (constant) flux coefficient ($\Gamma $), fail at higher Richardson numbers provided that the Reynolds number is sufficiently high. Specifically, we show that the assumptions of fully developed, stationary, and isotropic turbulence all break down at high Richardson numbers. We show that the breakdown of these assumptions occurs most prominently at Richardson numbers above that corresponding to the maximum mixing efficiency, a fact that highlights the importance of the non-monotonicity of the dependence of mixing efficiency upon Richardson number, which we establish to be characteristic of stratified shear-induced turbulence. At high $R{i}_{0} $, the lifecycle of the turbulence is composed of a rapidly growing phase followed by a phase of rapid decay. Throughout the lifecycle, there is considerable exchange of energy between the small-scale turbulence and larger coherent structures which survive the various stages of flow evolution. Since shear instability is one of the most prominent mechanisms for turbulent dissipation of energy at scales below hundreds of metres and at various depths of the ocean, our results have important implications for the inference of turbulent diffusivities on the basis of microstructure measurements in the oceanic environment.

scholarly journals Goldilocks mixing in oceanic shear-induced turbulent overturns

2021 ◽  
Vol 928 ◽  
Author(s):  
A. Mashayek ◽  
C.P. Caulfield ◽  
M.H. Alford
Keyword(s):  
Ocean Circulation ◽  
Turbulent Mixing ◽  
Intermediate Phase ◽  
Turbulent Flux ◽  
Geographical Location ◽  
Small Scale ◽  
Supporting Evidence ◽  
Stratified Turbulence ◽  
Wide Range ◽  
Variable Flux

We present a new, simple and physically motivated parameterization, based on the ratio of Thorpe and Ozmidov scales, for the irreversible turbulent flux coefficient $\varGamma _{\mathcal {M}}= {\mathcal {M}}/\epsilon$ , i.e. the ratio of the irreversible rate ${\mathcal {M}}$ at which the background potential energy increases in a stratified flow due to macroscopic motions to the dissipation rate of turbulent kinetic energy $\epsilon$ . Our parameterization covers all three key phases (crucially, in time) of a shear-induced stratified turbulence life cycle: the initial, ‘hot’ growing phase, the intermediate energetically forced phase and the final ‘cold’ fossilization decaying phase. Covering all three phases allows us to highlight the importance of the intermediate one, to which we refer as the ‘Goldilocks’ phase due to its apparently optimal (and so neither too hot nor too cold, but just right) balance, in which energy transfer from background shear to the turbulent mixing is most efficient. The value of $\varGamma _{\mathcal {M}}$ is close to 1/3 during this phase, which we demonstrate appears to be related to an adjustment towards a critical or marginal Richardson number for sustained turbulence ${\sim }0.2\text {--}0.25$ . Importantly, although buoyancy effects are still significant at leading order for the turbulent dynamics during this intermediate phase, the marginal balance in the flow ensures that the turbulent mixing of the (density) scalar is nevertheless effectively ‘locked’ to the turbulent mixing of momentum. We present supporting evidence for our parameterization through comparison with six oceanographic datasets that span various turbulence generation regimes and a wide range of geographical location and depth. Using these observations, we highlight the significance of parameterizing an inherently variable flux coefficient for capturing the turbulent flux associated with rare energetic, yet fundamentally shear-driven (and so not strongly stratified) overturns that make a disproportionate contribution to the total mixing. We also highlight the importance of representation of young turbulent patches in the parameterization for connecting the small scale physics to larger scale applications of mixing such as ocean circulation and tracer budgets. Shear-induced turbulence is therefore central to irreversible mixing in the world's oceans, apparently even close to the seafloor, and it is critically important to appreciate the inherent time dependence and evolution of mixing events: history matters to mixing.

scholarly journals Dynamics of Orographically Triggered Banded Convection in Sheared Moist Orographic Flows

2007 ◽  
Vol 64 (10) ◽  
pp. 3542-3561 ◽  
Author(s):  
Oliver Fuhrer ◽  
Christoph Schär
Keyword(s):  
Small Amplitude ◽  
Three Dimensional ◽  
Horizontal Resolution ◽  
Small Scale ◽  
Mean Flow ◽  
Topographic Roughness ◽  
Wide Range ◽  
Orographic Convection ◽  
Wind Profiles

Abstract Shallow orographic convection embedded in an unstable cap cloud can organize into convective bands. Previous research has highlighted the important role of small-amplitude topographic variations in triggering and organizing banded convection. Here, the underlying dynamical mechanisms are systematically investigated by conducting three-dimensional simulations of moist flows past a two-dimensional mountain ridge using a cloud-resolving numerical model. Most simulations address a sheared environment to account for the observed wind profiles. Results confirm that small-amplitude topographic variations can enhance the development of embedded convection and anchor quasi-stationary convective bands to a fixed location in space. The resulting precipitation patterns exhibit tremendous spatial variability, since regions receiving heavy rainfall can be only kilometers away from regions receiving little or no rain. In addition, the presence of banded convection has important repercussions on the area-mean precipitation amounts. For the experimental setup here, the gravity wave response to small-amplitude topographic variations close to the upstream edge of the cap cloud (which is forced by the larger-scale topography) is found to be the dominant triggering mechanism. Small-scale variations in the underlying topography are found to force the location and spacing of convective bands over a wide range of scales. Further, a self-sufficient mode of unsteady banded convection is investigated that does not dependent on external perturbations and is able to propagate against the mean flow. Finally, the sensitivity of model simulations of banded convection with respect to horizontal computational resolution is investigated. Consistent with predictions from a linear stability analysis, convective bands of increasingly smaller scales are favored as the horizontal resolution is increased. However, small-amplitude topographic roughness is found to trigger banded convection and to control the spacing and location of the resulting bands. Thereby, the robustness of numerical simulations with respect to an increase in horizontal resolution is increased in the presence of topographic variations.

scholarly journals Vortex-ring-induced stratified mixing: mixing model

2017 ◽  
Vol 837 ◽  
pp. 129-146 ◽  
Author(s):  
Jason Olsthoorn ◽  
Stuart B. Dalziel
Keyword(s):  
Vortex Ring ◽  
Turbulent Mixing ◽  
Quantitative Agreement ◽  
Vortex Rings ◽  
Mixing Efficiency ◽  
Mean Flow ◽  
Density Profiles ◽  
One Dimensional ◽  
Physical Experiments ◽  
Dynamical Features

The study of vortex-ring-induced mixing has been significant for understanding stratified turbulent mixing in the absence of a mean flow. Renewed interest in this topic has prompted the development of a one-dimensional model for the evolution of a stratified system in the context of isolated mixing events. This model is compared to numerical simulations and physical experiments of vortex rings interacting with a stratification. Qualitative agreement between the evolution of the density profiles is observed, along with close quantitative agreement of the mixing efficiency. This model highlights the key dynamical features of such isolated mixing events.

Identifying the tangle of vortex tubes in homogeneous isotropic turbulence

2019 ◽  
Vol 874 ◽  
pp. 952-978 ◽  
Author(s):  
Shiying Xiong ◽  
Yue Yang
Keyword(s):  
Isotropic Turbulence ◽  
Time Instant ◽  
Small Scale ◽  
Homogeneous Isotropic Turbulence ◽  
Complex Flow ◽  
Time Step ◽  
Vortex Sheets ◽  
Helical Geometry ◽  
Vortex Tubes

We extend the vortex-surface field (VSF), whose isosurface is a vortex surface consisting of vortex lines, to identify vortex tubes and sheets in homogeneous isotropic turbulence. The VSF at a time instant is constructed by solving a pseudo-transport equation. This equation is convected by a given instantaneous vorticity obtained from direct numerical simulation. In each pseudo-time step, we develop a novel local optimization algorithm to minimize a hybrid VSF constraint, balancing the accuracy and smoothness of VSF solutions. This key improvement makes the numerical construction of VSFs feasible for arbitrarily complex flow fields, as a general flow diagnostic tool. In the visualization of VSF isosurfaces in decaying homogeneous isotropic turbulence, the initial curved vortex sheets first evolve into vortex tubes, and then the vortex tubes are stretched and tangled, constituting a complex network. Some vortex tubes exhibit helical geometry, which suggests the important role of vortex twisting in the generation of small-scale structures in energy cascade.

Parameterizing gravity waves in the DYNAMICO-Saturn Global Climate Model to understand Saturn's equatorial oscillation

2020 ◽  
Author(s):  
Deborah Bardet ◽  
Aymeric Spiga ◽  
Sandrine Guerlet ◽  
Ehouarn Millour ◽  
François Lott
Keyword(s):  
Gravity Waves ◽  
Climate Model ◽  
Global Climate ◽  
Global Climate Model ◽  
Life Cycles ◽  
Small Scale ◽  
Mean Flow ◽  
Time Step ◽  
Maximum Value ◽  
The Mean

<p>To address questions about the driving mechanisms of Saturn's equatorial oscillation, our team at the Laboratoire de Météorologie Dynamique built the DYNAMICO-Saturn Global Climate Model to study tropospheric dynamics, tropospheric waves activity (Spiga et al. 2020) and equatorial stratospheric dynamics (Bardet et al. 2020) of Saturn. Previous studies (Guerlet et al. 2014, Spiga et al. 2020, Cabanes et al. 2020) have shown that our model produces consistent thermal structure and seasonal variability compared to Cassini CIRS measurements, mid-latitude eddy-driven tropospheric eastward and westward jets commensurate to those observed and following the zonostrophic regime, and planetary-scale waves such as Rossby-gravity (Yanai), Rossby and Kelvin waves in the tropical channel. Extending the model top toward the upper stratosphere allowed our model to produce an almost semi-annual equatorial oscillation with opposite eastward and westward phases. Associated temperature anomalies have a similar behavior than the Cassini/CIRS observations, but the amplitude of the temperature oscillation is twice smaller than the observed one. The absence of sub-grid-scale waves in the model produces an imbalance in eastward- and westward-wave forcing on the mean flow and could be an explanation to the irregularity in both the oscillating period and the downward rate propagation of the resolved Saturn equatorial oscillation.</p> <p>To explore the impact of those small-scale waves on the spontaneous equatorial oscillation emerging in the DYNAMICO-Saturn GCM (Bardet et al. 2020), we add a sub-grid-scale non-orographic gravity waves drag parameterization in our model.<br />This parameterization is directly adapted from the stochastic terrestrial model of Lott et al. (2012). This formalism represents a broadband gravity wave spectrum, using the superposition of a large statistical set of monochromatic waves. As the time scale of the life cycles of gravity waves is much longer than the time step of our GCM, our parametrization can launch a few waves whose characteristics are randomly chosen at each time step. This stochastic gravity waves drag parameterization is applied in DYNAMICO-Saturn on all points of the horizontal grid.</p> <p>A key parameter used in the non-orographic gravity waves drag parameterization is the maximum value of the Eliassen-Palm flux. The Eliassen Palm flux represents the momentum carried by waves that could be transferred to the mean flow. This value has never been measured in Saturn's atmosphere and it represents an important degree of freedom in the parameterization of gravity waves.</p> <p>We performed several test simulations, lasting two Saturn years whose initial state is derived from Bardet et al (2020), with an horizontal resolution of 1/2° in longitude/latitude and a vertical resolution ranging between 3 bar to 1 μbar. For these test simulations, the maximum value of the Eliassen-Palm fulx is set to 10<sup>-6</sup>, 10<sup>-5</sup>, 10<sup>-4</sup> and 10<sup>-3</sup> kg m<sup>-1</sup> s<sup>-2</sup>. </p> <p>Preliminary results show that the appropriate value of our main parameter is between 10<sup>-5</sup> and 10<sup>-4</sup> kg m<sup>-1</sup> s<sup>-2</sup>. Eliassen-Palm flux value of 10<sup>-3</sup> kg m<sup>-1</sup> s<sup>-2</sup> demonstrates a too large impact: the equatorial oscillation is entirely vanished is this configuration. The simulation using the value of 10<sup>-6</sup> kg m<sup>-1</sup> s<sup>-2</sup> is equivalent to the control simulation without the gravity waves drag parameterization.  </p> <p>The next step is to test other parameters, as phase velocity of the gravity waves, horizontal wavenumber, to understand how gravity waves impact the equatorial oscillation.</p>

On the vorticity transport due to dissipating or breaking waves in shallow-water flow

2000 ◽  
Vol 407 ◽  
pp. 235-263 ◽  
Author(s):  
OLIVER BÜHLER
Keyword(s):  
Numerical Simulations ◽  
Shallow Water ◽  
Gravity Waves ◽  
Large Scale ◽  
Bottom Topography ◽  
Breaking Waves ◽  
Small Scale ◽  
Practical Consideration ◽  
Mean Flow ◽  
Vorticity Transport

Theoretical and numerical results are presented on the transport of vorticity (or potential vorticity) due to dissipating gravity waves in a shallow-water system with background rotation and bottom topography. The results are obtained under the assumption that the flow can be decomposed into small-scale gravity waves and a large-scale mean flow. The particle-following formalism of ‘generalized Lagrangian-mean’ theory is then used to derive an ‘effective mean force’ that captures the vorticity transport due to the dissipating waves. This can be achieved without neglecting other, non-dissipative, effects which is an important practical consideration. It is then shown that the effective mean force obeys the so-called ‘pseudomomentum rule’, i.e. the force is approximately equal to minus the local dissipation rate of the wave's pseudomomentum. However, it is also shown that this holds only if the underlying dissipation mechanism is momentum-conserving. This requirement has important implications for numerical simulations, and these are discussed.The novelty of the results presented here is that they have been derived within a uniform theoretical framework, that they are not restricted to small wave amplitude, ray-tracing or JWKB-type approximations, and that they also include wave dissipation by breaking, or shock formation. The theory is tested carefully against shock-capturing nonlinear numerical simulations, which includes the detailed study of a wavetrain subject to slowly varying bottom topography. The theory is also cross-checked in the appropriate asymptotic limit against recently formulated weakly nonlinear theories. In addition to the general finite-amplitude theory, detailed small-amplitude expressions for the main results are provided in which the explicit appearance of Lagrangian fields can be avoided. The motivation for this work stems partly from an on-going study of high-altitude breaking of internal gravity waves in the atmosphere, and some preliminary remarks on atmospheric applications and on three-dimensional stratified versions of these results are given.

scholarly journals Spectral analysis of the transition to turbulence from a dipole in stratified fluid

2012 ◽  
Vol 713 ◽  
pp. 86-108 ◽  
Author(s):  
Pierre Augier ◽  
Jean-Marc Chomaz ◽  
Paul Billant
Keyword(s):  
Reynolds Number ◽  
Spectral Analysis ◽  
Kinetic Energy ◽  
Numerical Simulations ◽  
Scaling Laws ◽  
Stratified Fluid ◽  
Shear Instability ◽  
Transition To Turbulence ◽  
Small Scale ◽  
Wide Range

AbstractWe investigate the spectral properties of the turbulence generated during the nonlinear evolution of a Lamb–Chaplygin dipole in a stratified fluid for a high Reynolds number $Re= 28\hspace{0.167em} 000$ and a wide range of horizontal Froude number ${F}_{h} \in [0. 0225~0. 135] $ and buoyancy Reynolds number $\mathscr{R}= Re{{F}_{h} }^{2} \in [14~510] $. The numerical simulations use a weak hyperviscosity and are therefore almost direct numerical simulations (DNS). After the nonlinear development of the zigzag instability, both shear and gravitational instabilities develop and lead to a transition to small scales. A spectral analysis shows that this transition is dominated by two kinds of transfer: first, the shear instability induces a direct non-local transfer toward horizontal wavelengths of the order of the buoyancy scale ${L}_{b} = U/ N$, where $U$ is the characteristic horizontal velocity of the dipole and $N$ the Brunt–Väisälä frequency; second, the destabilization of the Kelvin–Helmholtz billows and the gravitational instability lead to small-scale weakly stratified turbulence. The horizontal spectrum of kinetic energy exhibits a ${{\varepsilon }_{K} }^{2/ 3} { k}_{h}^{\ensuremath{-} 5/ 3} $ power law (where ${k}_{h} $ is the horizontal wavenumber and ${\varepsilon }_{K} $ is the dissipation rate of kinetic energy) from ${k}_{b} = 2\lrm{\pi} / {L}_{b} $ to the dissipative scales, with an energy deficit between the integral scale and ${k}_{b} $ and an excess around ${k}_{b} $. The vertical spectrum of kinetic energy can be expressed as $E({k}_{z} )= {C}_{N} {N}^{2} { k}_{z}^{\ensuremath{-} 3} + C{{\varepsilon }_{K} }^{2/ 3} { k}_{z}^{\ensuremath{-} 5/ 3} $ where ${C}_{N} $ and $C$ are two constants of order unity and ${k}_{z} $ is the vertical wavenumber. It is therefore very steep near the buoyancy scale with an ${N}^{2} { k}_{z}^{\ensuremath{-} 3} $ shape and approaches the ${{\varepsilon }_{K} }^{2/ 3} { k}_{z}^{\ensuremath{-} 5/ 3} $ spectrum for ${k}_{z} \gt {k}_{o} $, ${k}_{o} $ being the Ozmidov wavenumber, which is the cross-over between the two scaling laws. A decomposition of the vertical spectra depending on the horizontal wavenumber value shows that the ${N}^{2} { k}_{z}^{\ensuremath{-} 3} $ spectrum is associated with large horizontal scales $\vert {\mathbi{k}}_{h} \vert \lt {k}_{b} $ and the ${{\varepsilon }_{K} }^{2/ 3} { k}_{z}^{\ensuremath{-} 5/ 3} $ spectrum with the scales $\vert {\mathbi{k}}_{h} \vert \gt {k}_{b} $.

Simulation of Interactions Between Microbubbles and Turbulent Flows

1994 ◽  
Vol 47 (6S) ◽  
pp. S70-S74 ◽  
Author(s):  
M. R. Maxey ◽  
E. J. Chang ◽  
L. -P. Wang
Keyword(s):  
Turbulent Flows ◽  
Isotropic Turbulence ◽  
Small Scale ◽  
Air Bubbles ◽  
Homogeneous Isotropic Turbulence ◽  
Mean Flow ◽  
Drag Forces ◽  
Spherical Inclusions ◽  
Average Rise ◽  
Surrounding Fluid

Microbubbles formed by small air bubbles in water are characterized as spherical inclusions that are essentially rigid due to the effects of surfactants, and respond to the action of drag forces and added-mass effects from the motion relative to the surrounding fluid. Direct numerical simulations of homogeneous, isotropic turbulence are used to study the effects of the small-scale, dissipation range turbulence on microbubble transport and in particular the average rise velocity of microbubbles. It is found that microbubbles rise significantly more slowly than in still fluid even in the absence of a mean flow, due to a strong interaction with the small-scale vorticity. The way in which microbubbles might modify the underlying turbulence by the variations in their local distribution is discussed for dilute, dispersed systems and some estimates for the enhanced viscous dissipation given.

scholarly journals Numerical Simulations of Melt-Driven Double-Diffusive Fluxes in a Turbulent Boundary Layer beneath an Ice Shelf

2020 ◽  
Author(s):  
Leo Middleton ◽  
Catherine A. Vreugdenhil ◽  
Paul R. Holland ◽  
John R. Taylor
Keyword(s):  
Boundary Layer ◽  
Numerical Simulations ◽  
Isotropic Turbulence ◽  
Small Scale ◽  
Relevant Parameter ◽  
Double Diffusive Convection ◽  
Ice Shelf ◽  
Diffusive Convection ◽  
Forced Turbulence ◽  
Double Diffusive

AbstractThe transport of heat and salt through turbulent ice shelf-ocean boundary layers is a large source of uncertainty within ocean models of ice shelf cavities. This study uses small-scale, high resolution, 3D numerical simulations to model an idealised boundary layer beneath a melting ice shelf to investigate the influence of ambient turbulence on double-diffusive convection (i.e. convection driven by the difference in diffusivities between salinity and temperature). Isotropic turbulence is forced throughout the simulations and the temperature and salinity are initialised with homogeneous values similar to observations. The initial temperature and the strength of forced turbulence are varied as controlling parameters within an oceanographically relevant parameter space. Two contrasting regimes are identified. In one regime double-diffusive convection dominates, and in the other convection is inhibited by the forced turbulence. The convective regime occurs for high temperatures and low turbulence levels, where it is long-lived and affects the flow, melt rate and melt pattern. A criterion for identifying convection in terms of the temperature and salinity profiles, and the turbulent dissipation rate, is proposed. This criterion may be applied to observations and theoretical models to quantify the effect of double-diffusive convection on ice shelf melt rates.

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