Onset of convection in a rotating semi‐infinite fluid: Theory and experiment

Abstract

We look at the development of the first plumes that emerge from a convectively unstable boundary layer by modelling the process as the instability of a fluid with a time‐dependent mean density field. The fluid is semi‐infinite, rotating, dissipative ‐ characterized by the ratio of its viscosity to thermal diffusivity (Prandtl number Pr = ν/κ) ‐ and initially homogeneous. A constant destabilizing heat flux is applied at the boundary and the stability of the evolving density field is investigated both mathematically and in laboratory experiments.

Using a “natural convective” scaling, we show that the behaviour of the non‐dimensional governing equations depends on Pr and the parameter γ = f(ν/B)^1/2, where f is the Coriolis parameter, and B is the applied buoyancy flux. For the ocean, γ ≈ 0.1, whilst for the atmosphere γ ≈ 0.01. In the absence of rotation, the behaviour of the differential equations is independent of B, depending only on Pr. The boundary‐layer Rayleigh number (Ra_bl) is also independent of B. We show that Ra_bl, evaluated at the onset of rapid vertical motion, depends on the form of the perturbation.

Due to the time‐dependence of the mean density field, analytic instability analysis is difficult, so we use a numerical technique. The governing equations are transformed to a stretched vertical coordinate and their stability investigated for a particular form of perturbation function. The model predictions are, for the ocean: instability time ～2–4 h, density difference ～0.002–0.013 kg m^‐3, boundary‐layer thickness ～50–75 m and horizontal scale ～200–300 m; and for the atmosphere: instability time ～10 min, temperature difference ～2.0–3.0°C, boundary‐layer thickness ～400–500 m and horizontal scale ～1.5–2.0 km.

Laboratory experiments are performed to compare with the numerical predictions. The time development of the mean field closely matches the assumed analytic form. Furthermore, the model predictions of the instability timescale agree well with the laboratory measurements. This supports the other predictions of the model, such as the lengthscales and buoyancy anomaly.