Subvisible cirrus clouds – a dynamical system approach
Abstract. Ice clouds, so-called cirrus clouds, occur very frequently in the tropopause region. A special class are subvisible cirrus clouds with an optical depth lower than 0.03, associated with very low ice crystal number concentrations. The dominant pathway for the formation of these clouds is not known well. It is often assumed that heterogeneous nucleation on solid aerosol particles is the preferred mechanism although homogeneous freezing of aqueous solution droplets might be possible, since these clouds occur in the low-temperature regime T < 235 K. For investigating subvisible cirrus clouds as formed by homogeneous freezing we develop a reduced cloud model from first principles, which is close enough to complex models but is also simple enough for mathematical analysis. The model consists of a three-dimensional set of ordinary differential equations, and includes the relevant processes as ice nucleation, diffusional growth and sedimentation. We study the formation and evolution of subvisible cirrus clouds in the low-temperature regime as driven by slow vertical updraughts (0 < w ≤ 0. 05 m s−1). The model is integrated numerically and also investigated by means of theory of dynamical systems. We found two qualitatively different states for the long-term behaviour of subvisible cirrus clouds. The first state is a stable focus; i.e. the solution of the differential equations performs damped oscillations and asymptotically reaches a constant value as an equilibrium state. The second state is a limit cycle in phase space; i.e. the solution asymptotically approaches a one-dimensional attractor with purely oscillatory behaviour. The transition between the states is characterised by a Hopf bifurcation and is determined by two parameters – vertical updraught velocity and temperature. In both cases, the properties of the simulated clouds agree reasonably well with simulations from a more detailed model, with former analytical studies, and with observations of subvisible cirrus, respectively. The reduced model can also provide qualitative interpretations of simulations with a complex and more detailed model at states close to bifurcation qualitatively. The results indicate that homogeneous nucleation is a possible formation pathway for subvisible cirrus clouds. The results motivate a minimal model for subvisible cirrus clouds (SVCs), which might be used in future work for the development of parameterisations for coarse large-scale models, representing structures of clouds.