It is thought that changes in the concentration of cloud-active aerosol can alter the precipitation efficiency of clouds, thereby changing cloud amount and, hence, the radiative forcing of the climate system. Despite decades of research, it has proved frustratingly difficult to establish climatically meaningful relationships among the aerosol, clouds and precipitation. As a result, the climatic effect of the aerosol remains controversial. We propose that the difficulty in untangling relationships among the aerosol, clouds and precipitation reflects the inadequacy of existing tools and methodologies and a failure to account for processes that buffer cloud and precipitation responses to aerosol perturbations. P recipitation-mediated aerosol-cloud relationships are often called 'lifetime effects', and stem from the hypothesis that changes in the aerosol lead to changes in the precipitation efficiency (or colloidal stability) of clouds 1 , which in turn changes cloud amount. The potential importance of such effects for cloud radiative forcing is especially evident in shallow marine cloud systems The complexity of the climate system, and the limitations of the tools we have to study it, have led to the development of two schools of scientific enquiry into the lifetime hypothesis described above. The first takes the lifetime effect as given and attempts to measure its trace statistically, either in observations or by enforcing relationships among the aerosol, clouds and precipitation, in large-scale models. The second takes the effect as hypothetical and endeavours to test it or the various assumptions upon which it is based, for instance through dedicated field or modelling studies that explore how individual clouds or fields of clouds respond to changes in the ambient aerosol. The two approaches naturally differ in scale, but also conceptually. Large-scale modelling studies are often premised on clouds and precipitation being strongly sensitive to the aerosol. However, such premises ignore mechanisms that, according to small-or regime-scale studies, absorb (or offset) some of the effects of aerosol perturbations and, hence, buffer the system. This disjunction, combined with a growing understanding of other limitations of the tools employed in large-scale surveys (that is, satellite observations and large-or global-scale models), explains why the statistical effect of the aerosol on clouds and precipitation remains so controversial. Here we propose that the sensitivity of clouds and precipitation to changes in the aerosol is regime dependent, and that although we expect lifetime effects to be on average weaker than implied by simple arguments (that is, buffered), substantial effects may still emerge in specific circumstances or regimes. Hence, research aimed at untangling the effects of the aerosol on clouds and precipitation should intensify efforts to understand those cloud and precipitation regimes in which the signature of lifetime effects is likely to be clearest. Cloud lifetime hypotheses What has come to be known as the cloud lifetime effect was put forward twenty years ago 3 . Initially it was formulated in specific terms for one particular cloud regime, as illustrated in Lifetime hypotheses encapsulate three basic ideas: that the number concentration of cloud droplets increases with anthropogenic aerosol burden; that the precipitation efficiency of shallow clouds decreases monotonically as a function of the concentration of cloud droplets; and that cloud albedo is a strictly decreasing function of precipitation efficiency. The first idea is also the basis for the albedo effect, which rests on the radiative signature of such changes alone. The second idea has a long history in the literature 1 and has served as the basis for much research, and even more speculation, on the efficacy of weather modification by glaciogenic or hygroscopic cloud seeding 11 . The

The accurate parameterization of moist convection presents a major challenge for the accurate prediction of weather and climate through numerical models. Superparameterization is a promising recent alternative strategy for including the effects of moist convection through explicit turbulent fluxes calculated from a cloud-resolving model. Basic scales for cloud-resolving modeling are the microscales on the order of 10 km in space on time scales on the order of 15 min, where vertical and horizontal motions are comparable and moist processes are strongly nonlinear (meso-gamma scale). In this paper, systematic multiscale asymptotic analysis is utilized to develop simplified microscale mesoscale dynamic (MMD) models for interaction between the microscales and spatiotemporal mesoscales on the order of 100 km and 2.5 h (meso-beta scale). The new MMD models lead to a systematic framework for superparameterization for numerical weather prediction (NWP) generalizing the traditional column modeling framework. The MMD formulation also provides a flexible systematic framework for devising new parameterization strategies for NWP interme-diate between the two extremes of column modeling and detailed cloud-resolving modeling. It is also established here that these MMD models fit crudely into the recent systematic multiscale framework developed to explain the observed larger-scale statistical self-similarity of tropical convection, and therefore provide a systematic framework for superparameterization. Finally, it is shown that the new MMD models have the structure of a heterogeneous multiscale method so that many numerical techniques recently developed in the applied mathematics literature can be applied to this formulation. 1.

One of the unexplained striking features of tropical convection is the observed statistical self-similarity in clusters, superclusters, and intraseasonal oscillations through complex multiscale processes ranging from the mesoscales to the equatorial synoptic scales to the intraseasonal/planetary scales. Here new multispatial-scale, multitime-scale, simplified asymptotic models are derived systematically from the equatorial primitive equations on the range of scales from mesoscale to equatorial synoptic to planetary/intraseasonal, which provide a useful analytic framework for addressing these issues. New mesoscale equatorial synoptic dy-namical (MESD) models and balanced MESD (BMESD) models are developed for the multitime, multi-space interaction from mesoscales to equatorial synoptic scales; new multitime versions of the intraseasonal planetary equatorial synoptic dynamics (IPESD) models are developed for multiple spatiotemporal inter-actions on equatorial synoptic scales and planetary scales. The mathematical character derived below for all these simplified models explicitly demonstrates that the main nonlinear interactions across scales are quasi-linear where eddy flux divergences of momentum and temperature from nonlinear advection from the smaller-scale spatiotemporal flows as well as mean source effects accumulate in time and drive the waves on the successively larger spatiotemporal scales. Furthermore, these processes that transfer energy to the next larger, longer, spatiotemporal scales are self-similar in a suitable sense established here. On the other hand, the larger scales set the environment for this transport through processes such as mean advection of the smaller scales. 1.

Superparameterization (SP) is a large-scalemodeling systemwith explicit representation of small-scale and mesoscale processes provided by a cloud-resolving model (CRM) embedded in each column of a large-scale model. New efficient sparse space–time algorithms based on the original idea of SP are presented. The large-scale dynamics are unchanged, but the small-scale model is solved in a reduced spatially periodic domain to save the computation cost following a similar idea applied by one of the authors for aquaplanet simulations. In addition, the time interval of integration of the small-scale model is reduced systematically for the same purpose, which results in a different couplingmechanism between the small- and large-scale models. The new algorithms have been applied to a stringent two-dimensional test suite involving moist convection interacting with shear with regimes ranging from strong free and forced squall lines to dying scattered convection as the shear strength varies. The numerical results are compared with the CRMand original SP. It is shown here that for all of the regimes of propagation and dying scattered convection, the large-scale variables such as hori-zontal velocity and specific humidity are captured in a statistically accurate way (pattern correlations above 0.75) based on space–time reduction of the small-scale models by a factor of 1/3; thus, the new efficient al-gorithms for SP result in a gain of roughly a factor of 10 in efficiency while retaining a statistical accuracy on the large-scale variables. Even the models with 1/6 reduction in space–time with a gain of 36 in efficiency are able to distinguish between propagating squall lines and dying scattered convection with a pattern correlation above 0.6 for horizontal velocity and specific humidity. These encouraging results suggest the possibility of using these efficient new algorithms for limited-area mesoscale ensemble forecasting. 1.

The authors evaluate whether the structure and intensity of simulated squall lines can be explained by “RKW theory, ” which most specifically addresses how density currents evolve in sheared environments. In contrast to earlier studies, this study compares output from four numerical models, rather than from just one. All of the authors ’ simulations support the qualitative application of RKW theory, whereby squall-line structure is primarily governed by two effects: the intensity of the squall line’s surface-based cold pool, and the low- to midlevel environmental vertical wind shear. The simulations using newly developed models generally support the theory’s quantitative application, whereby an optimal state for system structure also optimizes system intensity. However, there are significant systematic differences between the newer nu-merical models and the older model that was originally used to develop RKW theory. Two systematic differences are analyzed in detail, and causes for these differences are proposed. 1.

Clouds in the tropics can organize the circulation on planetary scales and profoundly impact long range seasonal forecasting and climate on the entire globe, yet contemporary operational computer models are often deficient in representing these phenomena. On the other hand, contemporary observations reveal remarkably complex coherent waves and vortices in the tropics interacting across a bewildering range of scales from kilometers to ten thousand kilometers. This paper reviews the interdisciplinary contributions over the last decade through the modus operandi of applied mathematics to these important scientific problems. Novel physical phenomena, new multi-scale equations, novel PDEs, and numerical algorithms are presented here with the goal of attracting mathematicians and physicists to this exciting research area.

ABSTRACT It is known that gravity waves in the troposphere, which are often excited by preexisting convection, can favor or suppress the formation of new convection. Here it is shown that in the presence of wind shear or barotropic wind, the gravity waves can create a more favorable environment on one side of preexisting convection than the other side. Both the nonlinear and linear analytic models developed here show that the greatest difference in favorability between the two sides is created by jet shears, and little or no difference in favorability is created by wind profiles with shear at low levels and no shear in the upper troposphere. A nonzero barotropic wind (or, equivalently, a propagating heat source) is shown to also affect the favorability on each side of the preexisting convection. It is shown that these main features are captured by linear theory, and advection by the background wind is the main physical mechanism at work. These processes should play an important role in the organization of wave trains of convective systems (i.e., convectively coupled waves); if one side of preexisting convection is repeatedly more favorable in a particular background wind shear, then this should determine the preferred propagation direction of convectively coupled waves in this wind shear. In addition, these processes are also relevant to individual convective systems: it is shown that a barotropic wind can lead to near-resonant forcing that amplifies the strength of upstream gravity waves, which are known to trigger new convective cells within a single convective system. The barotropic wind is also important in confining the upstream waves to the vicinity of the source, which can help ensure that any new convective cells triggered by the upstream waves are able to merge with the convective system. All of these effects are captured in a two-dimensional model that is further simplified by including only the first two vertical baroclinic modes. Numerical results are shown with a nonlinear model, and linear theory results are in good agreement with the nonlinear model for most cases.

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Abstract Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east- and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden–Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity