About these Movies

Sidewall boundary conditions are all free slip to ignore boundary layer effects. The flow is released from rest and a solitary wave of depression propagates towards the slope, as shown in this two dimensional simulation with a free surface: 2d soliton In this simulation, the free surface is shown magnified by 100 times. While the free surface disturbances propagate back and forth quickly, a depression at the free surface propagates above and at the speed of the solitary wave as it moves towards the slope. This simulation is clearly nonhydrostatic, as the breaking dynamics would not ensue with a hydrostatic model, as shown in the following comparison: hydro vs. nonhydrostatic. In this simulation the three methods described in the numerical method above are used. As shown in the hydrostatic case, the interfacial wave does not break, while the quasi and fully nonhydrostatic cases exhibit wave breaking. The fully nonhydrostatic case differs from the quasihydrostatic case in that the phase speed of the solitary wave is slightly slower for the fully nonhydrostatic case. This results from the exact computation and hence the exact nonhydrostatic effect on the interfacial wave, which inherently slows it down. Hydrostatic compuations tend to speed up phase propagation in regions where the flow is inherently nonhydrostatic. A more detailed view of the three dimensional breaking dynamics for the fully nonhydrostatic case can be seen in this movie: 3d break detail. The three dimensionality of the flow is evident in this image. It accounts for most of the dissipative mechanisms associated with interfacial wave breaking. Hence, a two dimensional simulation does not dissipate nearly as much energy as a three dimensional simulation. A more detailed description of the breaking dynamics can be found in detailed dynamics.

Shear and Convective Instabilities

• Pure Convective Instability (Ra > 0, Ri = -Infinity)

  A statically unstable situation occurs in a stratified fluid when a heavy fluid is lifted above a light fluid. This unstable situation develops a convective instability as the heavier fluid moves into the lighter fluid beneath it and generates Rayleigh-Taylor billows. A purely convective instability can be generated by releasing a statically unstable fluid from rest and allowing the heavier fluid to move into the lighter fluid beneath it and forcing the lighter fluid above it, as shown in pure convective (Ra = 6 million, Re = 300, Ri = 0). Characteristic of this flow as with all convectively dominated flows is the notion that all scales of motion grow without bound. That is, there is no critical wavenumber that maximizes the growth rate of the instability. Instead, the growth rate increases with decreasing wavelength. Hence the small disturbances quickly grow in amplitude and the flow progresses into large scale turbulence that does not effectively mix out the fluid.

  Pure Shear Instability (Ra = 0, 0 < Ri < 1/4)

  A statically stable stratified fluid can be unstable with the onset of shear across the interface. If the shearing stress is large enough, then the Richardson number of the flow can decrease below the stable value of 1/4 and generate Kelvin-Helmholtz billows, as shown in pure shear (Ra = 0, Re = 5000, Ri = 0.038). In this animation, the stratification is stable, but the shearing force is large enough to cause the most unstable wavelengths to grow and develop into billows. If the shear is not large enough, then the stratification prevents disturbances from growing without bound. The tradeoffs between these two forces is quantified as the Richardson number. High Richardson numbers (above 1/4) imply strong stratification and weak succeptibility to shear instabilities, while low Richardson numbers (below 1/4) imply weak stratification and strong shear. As can be seen in the animation, there is a dominant wavelength at which disturbance grow the fastest. Smaller waves do not grow, nor do larger waves. This is characteristic of all pure shear instabilities. A disturbance with a small wavelength decays as a result of viscosity, while one that is too large requires too much shear to cause it to overturn. Hence only one wavelength dominates the flow after the instability sets in, and rather than being mixed effectively, only large scale stirring dominates the flow.

  Mixed Instability (Ra > 0, -Infinity < Ri < 0)

  In the above two examples, either a convective instability or a shear instability stirs and mixes the two fluids of differing densities. In either case, with enough time, the fluids in each container would have been completely mixed. But in each case, large scale turbulence dominated the flow field just after the onset of the instabilites. Small scale turbulence decayed quickly, and mixing occured through molecular diffusion across relatively weak gradients within the flow field stirred by the large scale turbulence. Small scale turbulence is what more effectively mixes out two fluids of differing densities, since it increases the gradients by which molecular diffusion acts. But if only small scale turbulence exists, it is quickly overtaken by viscosity. Hence an ideal situation that maximizes the efficiency is one in which large scales feed energy into small scales which efficiently accomplish the mixing. This occurs in a mixed instability, which arises when an unstable density gradient exists at a sheared interface between two fluids, creating a situation with a negative Richardson number (Ri < 0).

  • Shear dominated

    If the shear is too great, the shear instability dominates the flowfield and large scale stirring develops, weakening the mixing efficiency, as shown in shear dominated mixed (Ra = 3.4 million, Re = 3600, Ri = -0.038). In this animation it can be seen that even though the flow is statically unstable, convective instabilities do not develop because the large scale stirring resulting from the shear instability dominates the flow field.

    Buoyancy dominated

    The efficiency with which a mixed instability mixes two fluids of differing densities can be increased by decreasing the Richardson number below Ri = -0.25, as shown in convectively dominated mixed (Ra = 31 million, Re = 1800, Ri = -1.34). In this situation mixing occurs more rapidly because small scales persist throughout the flow field as the large scale stirring occurs as a result of the shear instability. The large scale motion feeds energy into the small scales by lifting heavier fluid above lighter fluid and creating statically unstable regions which are succeptible to convective instabilities. If the shear instability is too strong, then the statically unstable regions do not last long enough for the convective instability to cause them to overturn, as was shown in shear dominated mixed (Ra = 3.4 million, Re = 3600, Ri = -0.038). Hence if the time scale of the convective instability is on the order of or shorter than the shear instability, then the flow will be mixed more efficiently. If the time scale of the shear instability is too long, however, then the flow returns to the pure convective instability, which does not effectively mix the fluid.

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