OCEANIC BUBBLE PLUME MODELLING
(Click here for a formal write-up of the associated experiment)
Gases are transferred between the atmosphere and the ocean via mechanisms such as aerosol release, direct exchange across the ocean surface and bubble mediated exchange within the water column. Not only do entrained bubbles increase the effective surface area of the atmosphere-ocean interface, the gases within the bubbles are subjected to increased pressure due to Laplace and hydrostatic pressures which means that small bubbles facilitate a faster rate of gas exchange than would already exist at the surface of the ocean.
Once a bubble cloud has been generated by a breaking wave there are four physical processes that shape the bubble population distribution function. These are fragmentation, gas flux, buoyancy and coalescence. Fragmentation and buoyancy tend to remove larger bubbles from the population whilst bubble coalescence increases the number of larger bubbles and decreases the number of small bubbles. Gas flux across the bubble wall can occur in both directions depending on the relative concentrations of the gas in both the liquid and gas phases together with the bubble size and bubble depth. These four processes combine to shape the final bubble population distribution function and so are considered to be instrumental in defining the position of the global maximum evident in historical bubble population measurements (Figure 1).
Figure 1: A typical bubble population distribution function. The coloured arrows indicate the effect different physical processes have on moving the global peak of the distribution towards a particular limit. The effect of dissolution can impose a positive and a negative rate of radius change depending on liquid saturation levels, bubble radius and bubble depth.
The formulation of these physical mechanisms and the defining of further physical attributes common to the oceanic surf zone enables the creation of a mathematical model that simulates the evolution of an artificial bubble cloud. In order to simulate the effect of turbulence within the surf zone a random motion is applied to each individual bubble within three-dimensional space. In the vertical plane this random motion was further modified by the characteristic rise velocity of the bubble. Once the spatial position of each bubble within the cloud has been defined, the model is then able to calculate the amount of gas that would flux across the bubble wall. By combining this information with the hydrostatic pressure, defined by the bubble’s depth, the change in the bubble’s radius is calculated. Before adjusting the random walk position of the bubbles again, those bubbles that have either reduced in size to a radius smaller than 1 mm or that risen to the ocean surface are removed from the simulation.
Figure 2: A flow diagram of the major operations preformed by the Matlab code towards simulating the evolution of an artificial bubble population in the surf zone.
Once a bubble has been entrained into the ocean it is either removed by rising to the surface or by completely dissolving its gas into the surrounding liquid. Oceanic bubbles rarely disappear altogether as their surface layer is contaminated by surface active agents. As the bubble reduces in size via either gas flux or hydrostatic pressure, these contaminants pack more densely on the reduced surface area of the smaller bubble so preventing these microbubbles from totally dissolving into solution.
Bubble radius change within the model is mathematically based on the ideal gas equation (1).
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The pressure of the gas within the bubble is denoted P, V is the volume of the bubble, n is the number of moles of gas within the bubble, Rmg is the molar gas constant and T is the temperature of the gas in degrees Kelvin. Derivations of this equation allow for the calculation of the bubble radius change depending on the molar change and the pressure change of the gas within the bubble given that the temperature remains constant.
The pressure of the gas within the bubble is composed of the hydrostatic, atmospheric and Laplace pressures. For the purposes of this derivation the atmospheric and hydrostatic pressures, Pah, are couple in equation (2).
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The second term of equation (2) represents the over pressure subjected to gas within the bubble by the Laplace pressure. The Laplace pressure is caused by the bubble surface tension g and is proportional to the equilibrium bubble radius, R0.
In order to derive the change in bubble size from the ideal gas equation it is necessary to express the volume, V, of the bubble in terms of its radius.
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(3) |
The volume-pressure term of the ideal gas equation (1) can now be represented as two terms;
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the first expresses the PV relation in terms of atmospheric and hydrostatic pressure, the second in terms of the bubble surface tension.
By taking the time derivative of equation (4),
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it can be seen that there are three terms that are functions of time.
The product rule must be applied to the left hand side of equation (5), to give
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(6) |
Terms that have been differentiated with respect to time are represented with dots. The ideal gas equation can now be expressed as equation (7).
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Rearrangement of equation (7) together with the substitution of
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(8) |
where,
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gives,
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The square brackets of equation (9)
represent the balance of gaseous pressures on either side of the bubble
surface. Equation (10) shows that the calculation of the rate of change of
bubble size with respect to time consists of
a contribution due to gas flux (
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and a contribution due to changes in
hydrostatic pressure (
).
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(11) |
The main assumption at the base of all these calculations is that the gas within the bubbles are ideal in nature and are initially composed of nitrogen and oxygen in diatomic form with the same molar ratio, x, as that found in the atmosphere. This ratio will change as the bubble evolves in the water column but initially x = x0 = 0.215. The equation that defines the evolution of the molar ratio is given by (12).
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In equation (12), Rmg is the universal gas constant, p is the hydrostatic pressure and T is the temperature of the liquid in degrees Kelvin. It is assumed that any temperature differential between the gas and the liquid phases of the system are negligible because the diffusion coefficient of heat is an order of magnitude greater than that of the gases and because of this the temperature of the gases soon reach that of the liquid1. The term Di represents the diffusion coefficients, Ki represents the absorption coefficients and Ni the Nusselt numbers of the each gas within the bubble. The subscript i are given the values 1 or 2 depending on whether the gas considered is oxygen (i=1) or nitrogen (i=2). The partial pressure of the individual gas in the liquid far from the bubble is termed pi0.
The vertical position of the bubble in the water column is determined by the resolution of the downward component of the water turbulence velocity and the bubbles’ rise velocity. To complete the description of the bubble population model, the bubble rise velocity must be defined.
By the nature of the density differential between the atmosphere and the ocean, bubbles are buoyant. Therefore in a static fluid, a bubble of a certain radius will have a characteristic rise speed (wb). Apart from being a function of the bubble’s radius, the rise speed also depends on the cleanliness of the bubble’s surface. A dirty bubble will contain a surface active film which will make the bubble behave very much like a rigid body1. A fuller study into the nature of a bubble’s surface is given in the thesis by Patro2, where the three different types of surfactant are discussed together with an overview of the hydrodynamic effects of a dirty bubble surface.
The formulation for the bubbles’ rise velocity follows that presented by Thorpe1.
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where,
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(14) |
where,
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(15) |
Figure 3: A plot of a dirty bubble’s rise speed taken from equation (13).
The governing equations were coded into Matlab and calculations were performed based on the operations presented in Figure 2. The evolution of the bubble plume is presented below in terms of an animation. Each frame of the animation represents a second in time. The initial bubble size distribution was determined from passive measurements made in the surf zone3. These entrained bubbles were introduced into the simulation every 10 s. The spatial position of each bubble can be seen in the left hand plot whilst the cumulative bubble size distributions of the top 1 m and bottom 1 m are presented in the top and bottom plots respectively on the right hand side of the figure.
Figure 4: An animation of the simulated bubble plume evolution in the surf zone. The air-water interface is assumed to be flat and at depth=0 m. The seabed is assumed to be flat and at depth=2 m. The Every frame of the animation represents a second of the bubble plume's evolution. The left hand figure shows the spatial position of each individual bubble whilst the right hand side plots show the cumulative bubble population distributions for the top and bottom 1 m of the water column.
CLICK HERE for a similar animation in water infinitely deep (i.e. no seabed).
1. Thorpe, S. A., A model of turbulent diffusion of bubbles below the sea surface. J. Phys. Ocean. 14 (1984).
2. Patro, R. K., The role of large bubbles in air-sea gas exchange, (National University of Ireland, 2000).
3. Leighton, T. G. et al., The Hurst Spit experiment: The characterisation of bubbles in the surfzone using multiple acoustic techniques. 227-34, (Institute of Acoustics, Southampton Oceanography Centre, 2001).
This page was developed by MD Simpson under supervision of TG Leighton
