Session Q30: Geophysical Fluid Dynamics: Stratified Flows II

Numerical simulations of a Minimal Ocean Mixing Systems (MOMS)

Diapycnal mixing is a critical component of the ocean energy budget and one of the processes which regulate the Meridional Overturning Circulation.

In analogy with the Minimal Channel Unit concept in boundary layer turbulence, we consider a Minimal Ocean Mixing System (MOMS), that is the smallest system which exibits a GM-like spectrum, i.e., a Zakharov-Kolmogorov solution of the associated wave kinetic equation at large to intermediate scales, and a hard isotropic turbulent regime at scales smaller than the Ozmidov scale. The idea is to move away from studying individual mixing events in favor of considering a system where mixing is "naturally" driven by the interaction of random internal waves.   

To simulate a MOMS, we use the Stratified Ocean Model with Adaptive Refinement to simulate  the flow in a uniformly stratified rectangular tank with aspect ratio 3, forced by a wavemaker that injects energy at low frequency. Over several hundreds forcing periods, nonlinearities create an internal wave field characterized by a GM spectrum. Episodically, instabilities develop which drive irreversible mixing. By monitoring the energy in the system, we can calculate the mixing efficiency of the system, as opposed to the mixing efficiency of individual mixing events.       

Gravity currents from moving sources

Emerging technologies such as deep-sea mining and geoengineering pose fundamentally new questions regarding the dynamics of gravity currents. Such activities can continuously release dense sediment plumes from moving locations, thereafter propagating as gravity currents. Here, we present the results of idealized numerical simulations of this novel configuration, and investigate the propagation of a gravity current that results from a moving source of buoyancy, as a function of the ratio of source speed to buoyancy velocity. We show that above a certain value of this ratio, the flow enters a supercritical regime in which the source moves more rapidly than the generated current, resulting in a statistically steady state in the reference frame of the moving source. Once in the supercritical regime, the current goes through a second transition beyond which fluid in the head of the current moves approximately in the direction normal to the direction of motion of the source, and the time evolution of the front in the lateral direction is well described by an equivalent constant volume lock-release gravity current. We use our findings to gain insight into the propagation of sediment plumes released by deep-sea mining collector vehicles, and present proof-of-concept tow-tank laboratory experiments of a model deep-sea mining collector discharging dense dyed fluid in its wake. The experiments reveal the formation a wedge-shaped gravity current front which narrows as the ratio of collector to buoyancy velocity increases. The time-averaged front position shows good agreement with the results of the numerical model in the supercritical regime.   

Turbulent Structure of Rotating Oil Plumes under Stratification

The overall fate and transport of oil after a deep-water blowout depends heavily on the vertical distribution of effluent, set predominantly by turbulent plume dynamics.  Of primary modeling importance is understanding how long effluent parcels remain in the water column, what fraction reach the surface and what is the surface expression of individual effluent constituents. Previous laboratory experiments and numerical simulations of thermal and gas bubble plumes have shown, that system rotation significantly alters plume behavior even at oceanographic Rossby numbers. In the current study, we turn our attention to modelling liquid-liquid mixtures of water and oil allowing for weak slip of the oil phase relative to water. High-fidelity turbulence simulations employing high-order spectral-element methods (SEM) and a computational domain that includes fully turbulent effluent input from a resolved pipe, are used to study the evolving  structure of the oil plumes in the presence of background stratification and rotation. The multiphase oil plume is modeled using a standard mixture model for the liquid phases in a Boussinesq, Eulerian-Eulerian framework while incorporating a non-linear model for the effective phase slip.  We concentrate on the effects of varying both the Rossby number and slip speed on the overall level of sub-surface oil trapping of and on changing both the number and thickness of the multiple intrusion layers formed.   

Simulations of settling marine aggregates in a stratified fluid.

Settling marine aggregates play an important role in transporting dissolved carbon dioxide (CO2) from the surface ocean to the deep ocean. While sinking, they accumulate in thin layers where sharp density stratifications are present which become nutrient hotspots for bacteria and animal activity.  We present a method to simulate settling aggregates in a stratified fluid. We assemble fractal aggregates as collections of cubic particles to model the marine aggregates.  The flow around the aggregates is computed in the limit of zero Reynolds number using a boundary integral method. Due to variable density, a term involving a volume integral is added to the boundary integral formulation. For rapid computation in three dimensions, we use the fast multipole method with a modified Laplace's kernel. We couple the solution with the advection-diffusion equation to track the heat density or salt concentration in time. We use this method to quantify how the presence of stratification affects the settling speed of the aggregates.   

Marangoni forces on oil droplets rising in a stratified fluid

Subsurface trapping for phenomenon rising and settling in a stratified fluid is present in various aspects of nature, including marine snow aggregates, smoke plumes in thermal inversions, and rising oil spills. Oceanographic studies on the 2010 Deepwater Horizon oil spill (McNutt, 2012) estimated that approximately 40 percent of spilled oil was trapped beneath the ocean surface, primarily in regions with strong oceanic density gradients. We are conducting experiments on single oil droplets rising  in sharply stratified fluids. Previous studies have shown that Marangoni forces from surface tension gradients may play a role in the trapping phenomenon. This work aims to verify the role and significance of interfacial tension in a systematic study of oil droplets rising through a sharply stratified fluid at intermediate Reynolds numbers.   

Numerical simulations of oil droplets rising in a sharply stratified fluid

Rising droplets and bubbles in stratified fluids are a physical feature of many atmospheric and oceanic systems. For example, the Deepwater Horizon oil spill in 2010 resulted in large plumes being trapped as they rose through stratified layers in the Gulf of Mexico. To begin understanding how and why these plumes became trapped, we produced high fidelity numerical simulations of a single oil droplet rising in a stratified ambient flow. The numerical methods we use include a modified pressure correction projection method on adaptive non-graded octree grids and a coupled level set-reference map method to capture the moving interface. These simulations are compared against recent experimental results, which characterized the velocity and dynamics of the retention of a droplet rising in stratification. Through simulations, we provide a detailed analysis of the forces acting on the droplet.   

Flows induced by diffusion in the presence of permeable solids

Results for flows induced by diffusion in the presence of solids immersed in a background density stratification are extended to include solids which allow for significant internal diffusion and flux of the solute across the fluid-solid interface. The strength of the induced flow as a function of the ratio of the diffusivities of the solute in the solid and fluid is predicted theoretically and shown to agree with simulations. The dynamic problem is investigated for constrained and freely settling solids whose internal concentration differs from the background, and a quasi-steady flow is investigated for solids in a linear background stratification which settle due to inward flux of the solute.   

Not Participating

We study resonant triad interactions between discrete internal wave modes m and n at a constant frequency ω and mode q at a frequency 2ω in a uniformly stratified shear flow. Using a weakly nonlinear framework, we obtain the inhomogeneous version of the Taylor-Goldstein-Haurwitz equation for the superharmonic wave vertical structure, the divergence of which indicates resonance. In the absence of shear, not all the locations (in the parameter space) where the horizontal wavenumber condition (k_m + k_n = k_q) is satisfied are the resonant locations[1,2]. It is shown using a weak shear asymptotic theory that even the presence of an arbitrarily small shear will result in all those locations, that are earlier non-resonant but satisfy k_m + k_n = k_q, will now become resonant. These include the resonances due to self-interaction and resonances close to ω ≈ 0 as well. For an ocean-like background shear flow, we identify all the possible resonant locations in the frequency(ω) - Richardson number(Ri) plane. We find that the total number of resonant triads are of the same order at small Ri as in the case of the Ri → ∞, although attaining a maximum at moderate Ri.

References:

1. Varma, D. and Mathur, M. (2017). JFM, 824, 286-311.

2. Vanneste, J. and Vial, F. (1994). GAFD, 78(1-4), 115-141.   

Self-resonance and stability in uniformly stably stratified shear flow

Stratified flows are of great importance from a geophysical point of view because of their ability to support generating and propagating internal waves (IWs). For instance, oceanic IWs act as a primary source for transferring heat, energy, and momentum throughout the ocean. Understanding interactions among IWs are necessary to know the energy transfer mechanism among IWs or IWs and surrounding flow. We aim to study self-resonances among IWs in an ideal situation consisting of uniform shear flow with uniform stratification in a vertically confined two-dimensional channel. In the absence of shear, self-resonance is not possible among IWs in a uniformly stratified medium. In contrast, shear instigates self-resonance among IWs, and thus, plays a crucial role in energy transfer. First, we show exact self-resonances and obtain associated parameters by solving the dispersion relation. Next, we study the stability analysis of self-resonating IWs using the amplitude evolution equations, which is the first step towards understanding the energy transfer mechanism through IWs in the ocean. From the analytical solution of amplitude equations, we analyze the role of stratification on the stability of self-resonating IWs and hence, the energy transfer among IWs and their higher-order harmonics.   

A data-driven model for the prediction of local dissipation rates in stratified turbulent flows

We construct a probabilistic artificial neural network (ANN) model to calculate local values of the turbulent energy dissipation rate $\varepsilon$ in strongly stratified decaying turbulence using a limited number of inputs. In general, simplified models for turbulent dissipation are important from a practical perspective as calculating the true value requires that 9 velocity derivatives be resolved simultaneously. It is common to apply theoretical `surrogate models' for $\varepsilon$ that use one or two velocity derivatives as inputs based on assumptions of homogeneity and isotropy, though these models can often be inaccurate in a geophysical setting where the stabilising presence of a vertical stratification generates large-scale anisotropy in the flow. We argue that our ANN model has two main advantages over isotropic surrogates. Firstly, we find it to be robust over multiple stratified turbulent regimes, in particular outperforming surrogate models when the turbulence has decayed and the flow is in a characteristic `layered' state. Secondly, the probabilistic nature of the model - which comes from sampling output values from a learnt distribution - captures underlying turbulence intermittency and as a result can produce estimates of uncertainty along with predictions. We discuss the implications of our work for the determination of dissipation rates from oceanographic data.