On the instability of a buoyancy-driven downflow

• Flow transition in downslope gravity currents is theoretically modelled.
• Two branches of instability could occur for slope angle between 0° and 90°.
• A transitional slope angle, at which flow transition occurs, is identified.
• Critical Reynolds number for ensuing instability increases as the slope angle decreases.

Gravity currents flowing downslope, namely downflows, were observed to have a larger scale instability on high slope angles and such violent instability was absent for downflows on low slope angles. By linear theory, it is found that two branches of instability occur for slope angle in the range of 0° < θ < 90°. The ensuing instability is on the upper branch for low slope angles and on the lower branch for high slope angles. There also exists a transitional slope angle, θE ≈ 0.04°, at which the onset instability switches from one branch to the other. The scale of instability is found to increase and tend to skew towards the upper edge of the downflow as the ensuing instability switches from the upper branch to the lower one. Our findings surprisingly resonate with previously reported observations. Critical Reynolds number, below which the flow is stable to infinitesimal disturbances, is found to increase as the slope angle decreases. The role played by the bottom slope is essentially twofold. On one hand, the downslope component of gravity acts as the driving force for downflows. On the other hand, the wall-normal component of gravity acts for the stratification effect. Therefore, as the slope angle decreases, the driving force diminishes and the stratification intensifies, which can explain that the critical Reynolds number increases as the slope angle decreases. When a downflow propagates onto a sufficiently low slope angle, the low driving force and intensified stratification effect would make the downflow less prone to sustain a turbulent state of flow, which ultimately leads to the final stage of a gravity current event.

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