Statistical mechanics and stochastic convective parameterisation

Using elementary considerations from statistical mechanics, it is suggested that convective variability in equilibrium can be characterised as a system of convective updraughts, randomly located in space, and with an exponential distribution of mass flux. The predictions for the magnitude and distribution of convective variability that result from this argument are confirmed to a high degree of quantitative accuracy in simulations in a cloud-resolving model. A stochastic cumulus parameterisation is described that imposes this variability. Initial tests in a single column model conform to expectations. 1. Statistics of convective equilibrium Cumulus parameterisation schemes are usually based on the assumption of an equilibrium, in which the large scale interacts deterministically with the statistical properties of the convection. For equilibrium to be realised, there must be a scale separation, where the large-scale flow is slowly varying over times and distances large enough to average out the effects of the individual convective clouds. If the resolution of the model using the parameterisation is finer than the intrinsic scales of the convection the behaviour of the convection in a given grid box will be stochastic, but with the known equilibrium statistics. It is clearly of interest to identify the relevant scales: the time scale of adjustment to equilibrium, the length scale required to average out convective variability and the magnitude of the fluctuations about equilibrium that occur in a small region of a given size. In this note, we develop such a statistical characterisation of the convective scale behaviour, and use it to develop a stochastic parameterisation scheme. To do this, we need to consider the underlying processes at the “microscopic” level, that is to say, at the level of the individual clouds. An essential part of this programme is to test the statistical theory and parameterisation against simulations from a cloud-resolving model (CRM) that has been run to equilibrium with a uniform forcing. The setup is summarised below, and details may be found in the references (Cohen and Craig 2004, 2005). • Resolution: 2 km x 2 km x 50 levels • Domain: 128 x 128 km (256 x 256 km for some experiments) • Boundary conditions: doubly periodic, fixed SST of 300 K • Forcings: fixed tropospheric cooling of 2, 4, 8, 12, 16 K/day, vertical wind shear of 0, 0.2 m/s/km, 2 m/s/km 2. Timescale of Adjustment to Equilibrium Following Cohen and Craig (2004), consider the response of a convecting atmosphere to a step function perturbation in cooling rate. As shown in Fig. 1, the convective mass flux adjusts in two stages: an initial rapid adjustment (in the first hour or so), followed by a slow evolution over many days. The rapid shift CRAIG ET AL.: STATISTICAL MECHANICS AND STOCHASTIC CONVECTIVE PARAMETERISATION 108 represents the adjustment to a new equilibrium state, while the slow evolution is associated with changes in tropospheric moisture and can be considered as a drift in the large-scale environment of the convection. Figure 1: Convective mass flux as a function of time for a simulation in equilibrium with a constant forcing of -8 K/day, changed to -16 K/day at day 11. An important result is that the time scale is proportional to the spacing of the clouds, being more rapid when the clouds are densely packed. This is shown in Fig. 2 which shows the evolution of the mass flux, when a forcing of -8 K/day is changed to forcings of -4, -12, -16 K/day. It can be seen in Fig. 2(b) that when the mass flux is rescaled by its quasi-equilibrium value (after rapid adjustment) and the time axis by cloud spacing distance, the curves become essentially identical.