Abstract
The linear convective instability of a mixture of dry air, water vapour and liquid water, with a stable unsaturated layer residing on an unstable saturated layer, is studied. It may serve as a prototype model for understanding the instability that causes mixing at the top of stratocumulus cloud or fog. Such a cloud-clear air interface is modelled as an infinitely thin saturation interface where radiative and evaporative cooling take place. The interface position is determined by the Clausius–Clapeyron equation, and can undulate with the evolution of moisture and temperature. In the small-amplitude regime two physical mechanisms are revealed. First, the interface undulation leads to the undulation of the cooling source, which destabilizes the system by superposing a vertical dipole heating anomaly on the convective cell. Second, the evolution of the moisture field induces non-uniform evaporation at the interface, which stabilizes the system by introducing a stronger evaporative cooling in the ascending region andvice versain the descending region. These two mechanisms are competing, and their relative contribution to the instability is quantified by theoretically estimating their relative contribution to buoyancy flux tendency. When there is only evaporative cooling, the two mechanisms break even, and the marginal stability curve remains the same as the classic two-layer Rayleigh–Bénard convection with a fixed cooling source.
Highlights
A thermally unstable layer lying over a solid surface and below a stable layer renders a two-layer convection set-up that is frequently seen in geophysical flow
For a mixture of dry air and water such as the atmosphere, a special case is when the lower layer is saturated to water vapour, with liquid water suspending in the air as droplets
The assumption of identical diffusivity for T∗, q∗v, q∗l and q∗t was proposed by Bretherton (1987), and has been used in many cloud-top mixing simulations (Siems et al 1990; Mellado et al 2009; Mellado González 2010; de Lozar & Mellado 2015; Schulz & Mellado 2018). It leads to a drastic simplification: evaporative cooling is determined by the liquid water diffusive flux at the interface
Summary
A thermally unstable layer lying over a solid surface and below a stable layer renders a two-layer convection set-up that is frequently seen in geophysical flow. The cooling at the interface drives turbulence, which entrains the upper-layer dry air into the cloud layer. This is an important factor that leads to the breakup of stratocumulus and fog (Lilly 1968). The lower rigid lid, which does not exist for the real cloud-top mixing layer, artificially set a length scale for the largest overturning eddy (Siems et al 1990) It may qualitatively represent the cloud bottom, or the surface of a fog layer, but not in a strict sense due to their multiscale and turbulent nature.
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