The strange energetics in gravity currents

n fluid flows driven down a slope by their excess density, the ability to transport sediment is dependent on the turbulent energy. Our work showed that the energetics within these currents does not behave at all how one might expect. This is the story of how we came to this realisation.
Published in Earth & Environment
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What are gravity currents?

Gravity currents are fluid flows driven by a (small) excess density with respect to the surrounding fluid. This density difference can be created by the current being colder, saltier, or full of particles, which either drives the fluid to move downhill or else spread out across a flat terrain. Examples the first two classes include cold fronts in the atmosphere, dense oceanic currents, and ventilation flows in buildings. However, it is the particle driven currents which most interest us here. Turbidity currents are mixtures of sediment and sea water that can travel at speeds over 1m/s for up to 1000km from the cost to the deep ocean, a single event can transport over 100km3 of sediment, making them one of the most significant and hazardous fluid flows on the planet. Pyroclastic density currents are produced by volcanos when the concentration of dense particulates overwhelms the buoyancy of the hot gas, and the ash cloud collapses down the side of the volcano and races out across the terrain. Research in this area aims to understand the dynamics of these currents better, so that their flow may be predicted accurately and efficiently.

The research plan

Historically, models of gravity currents have used a set of simplifying assumptions inherited from models of ‘open channel’ flows in rivers and canals, and these models have been surprisingly successful considering how different the flows may appear. However, differences between open channel flows and gravity currents have been repeatedly found, and insufficiencies in these simple models highlighted. This led Rob Dorrell to suspect that there are differences between the mechanisms that suspend particles in gravity currents and in rivers, and that these should have an influence on the flow.

When Rob took on Sojiro Fukuda as a PhD student, understanding the suspension mechanisms was one of the key research goals. Due to the ongoing pandemic there wasn’t the opportunity to undertake new experiments at this stage. Instead, Sojiro performed a thorough search of the literature and compiled all the available data into the most complete dataset on gravity currents to date, which has been an asset for several research projects performed by our team.

The discovery

Once Sojiro’s dataset was nearing completion, Rob asked Sojiro to test out a few of the results from conventional models of gravity currents based on wisdom from open channel flows, in particular the balances in the self-acceleration model. One of the figures they plotted in included here as Figure 1. What this figure reveals is very surprising. The trend is highly nonlinear (not the log scales of the axes), showing that the established models for gravity currents do not capture the internal energetics. Instead, the dynamics of real gravity currents are non-linearly related to the quantities in the self-acceleration model. How can we understand this result?

Improving the dataset

Examining the data in figure 1, it is clear that multiple clusters exist with gaps between, and closing these data gaps would provide greater confidence in both this result and other results based on the dataset. With Covid restrictions lifting, Sojiro undertook to close the data gaps by performing a targeted set of experiments, in collaboration with Marijke de Vet, Elena Bastianon, Roberto Fernández, and Xuxu Wu. These experiments were performed at the Total Environment Simulator based at The Deep, an aquarium in Hull, a photo of which is shown in Figure 2. An example experiment is shown in Figure 3, with the current flowing down a flume surrounded by measurement probes. These were combined with experiments performed by Marijke de Vet in the same facility, and newly published field results, further enhancing the dataset.

Gravity current energetics

A long-standing idea in the study of turbidity currents is that of autosuspension. This aims to answer the following question: what stops the sediment from falling out of the current? After all, the sediment is denser than the water that its flowing with, and if the mixture were to be stationary then we would rapidly find that all the particles had dropped to the ground. The basic answer is that the turbulent vortices in the flow stir the fluid, so that the higher concentration regions lower down in the flow get mixed up with the lower concentration regions high up, preventing the particles from falling out. This process takes energy out of the turbulence, which needs to be replenished if the current is to persist, and the autosuspension criterion is essentially that the amount of turbulent energy generated by the flow needs to be larger than the amount of energy lost to holding up the particles. Consequently, figure that Sojiro had plotted revealed that there was something wrong with our understanding of the turbulent energy in the current.

This is what Rob, Sojiro, William D. MacCaffrey, Hajime Naruse, Daniel R. Parsons and Ed Skevington, who had recently joined institute as a postdoc, tried to understand. We needed to ensure that all of the effects that were already understood were included in the analysis. This meant adjusting the plot to include the energy associated with the mixing of the current with the surrounding water, which only served to make the non-linear trend even clearer, see figure 4. We could also show that the amount of extra energy needed was substantial, especially when accounting for the turbulent energy lost to viscosity. At this stage the range of data sources that were confirming the result was very broad, including real turbidity currents, along with experimental results in different facilities with different types of sediment, and even experiments with real pyroclastic material. Because the same trend fitted all of the types of data, the explanation needed to be some effect present in all of the flows we had examples of.

Based on work led by Rob, we know that including the internal vertical variations in velocity and concentration in the modelled flow could have a substantial influence on the dynamics of the current. This work had been perused further by both Sojiro and Ed, investigating what the vertical structure is and what effect it has on the flow. After some careful thought and discussion, we concluded that this vertical variation was the only prominent effect in all of the flows that was omitted from standard models. This was our conclusion, that the internal energetics of gravity currents are strongly influenced by the internal structure of the current, and that models will need to include this structure if they are to accurately predict how much sediment the currents carry, and thereby their speed, travel distance, and duration.

Figure 1: An early version of the plot comparing available flow energy (x-axis) with the energy required to keep sediment in suspension (y-axis) according to models based on fluvial systems. Data for a river is included in grey with a black best fit line, while data for gravity currents is in colour with a red best fit line. It is clear from this figure that the two systems are very different.

The Deep, Hull, the location of the experiments

Figure 2: The Deep, Hull, the location of the experiments (image from https://www.visithull.org/venue/the-deep/)

 

Figure 3: The experiments, showing a gravity current passing, and measurement probes which collect data from the current.

Figure 4: The final version of Figure 1. The x-axis is the amount of sediment that would be carried if the standard modelling approach was correct, while the y-axis is the actual amount of sediment carried by the current. There is a strongly non-linear relationship between the two (red line) which is very different to the best fit of the fluvial data (black dotted line).

 

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