Ole Peters

A complexity view of rainfall.

We show that rain events are analogous to a variety of nonequilibrium relaxation processes in Nature such as earthquakes and avalanches. Analysis of high-resolution rain data reveals that power laws describe the number of rain events versus size and number of droughts versus duration. In addition, the accumulated water column displays scale-less fluctuations. These statistical properties are the fingerprints of a self-organized critical process and may serve as a benchmark for models of precipitation and atmospheric processes.

The Transition to Strong Convection

Recent work has shown that observations of tropical precipitation conform to properties associated with critical phenomena in other systems. Here some of these universal properties are used to probe the physics of tropical convection empirically, providing potential tests for models and parameterizations. The power-law pickup of ensemble average precipitation as a function of column water vapor w occurs above a critical value wc whose temperature dependence is determined for layer-integrated tropospheric temperature or saturation value. This dependence differs from the simplest expectations based on column saturation. Rescaling w by wc permits a collapse of precipitation-related statistics to similar functional dependence for all temperatures. The sharp precipitation variance peak at wc, obtained without detailed vertical structure information, appears consistent with arguments that onset requires a deep moist layer. Sea surface temperature (SST) is found not to have a significant effect on the precipitation pickup. The effect of SST on the climatological precipitation occurs via the frequency of occurrence of w values as the system spends a larger fraction of time near criticality over regions of warm SST. Near and above criticality, where most precipitation occurs, the w distribution is highly constrained by the interaction with convection, with a characteristic sharp drop at criticality. For precipitating points, the distribution has a Gaussian core with an approximately exponential tail akin to forced advection‐diffusion problems. The long tail above wc, implying relatively frequent strong events, remains similar through the range of tropospheric temperature and SST spanning tropical large-scale conditions. A simple empirical closure illustrates time decay properties.

Rain: Relaxations in the sky

We demonstrate how, from the point of view of energy flow through an open system, rain is analogous to many other relaxational processes in nature such as earthquakes. By identifying rain events as the basic entities of the phenomenon, we show that the number density of rain events per year is inversely proportional to the released water column raised to the power of 1.4. This is the rain equivalent of the Gutenberg-Richter law for earthquakes. The event durations and the waiting times between events are also characterized by scaling regions, where no typical time scale exists. The Hurst exponent of the rain intensity signal H=0.76>0.5. It is valid in the temporal range from minutes up to the full duration of the signal of half a year. All of our findings are consistent with the concept of self-organized criticality, which refers to the tendency of slowly driven nonequilibrium systems towards a state of scale-free behavior.

Rethinking convective quasi-equilibrium: observational constraints for stochastic convective schemes in climate models

Convective quasi-equilibrium (QE) has for several decades stood as a key postulate for parametrization of the impacts of moist convection at small scales upon the large-scale flow. Departures from QE have motivated stochastic convective parametrization, which in its early stages may be viewed as a sensitivity study. Introducing plausible stochastic terms to modify the existing convective parametrizations can have substantial impact, but, as for so many aspects of convective parametrization, the results are sensitive to details of the assumed processes. We present observational results aimed at helping to constrain convection schemes, with implications for each of conventional, stochastic or ‘superparametrization’ schemes. The original vision of QE due to Arakawa fares well as a leading approximation, but with a number of updates. Some, like the imperfect connection between the boundary layer and the free troposphere, and the importance of free-tropospheric moisture to buoyancy, are quantitatively important but lie within the framework of ensemble-average convection slaved to the large scale. Observations of critical phenomena associated with a continuous phase transition for precipitation as a function of water vapour and temperature suggest a more substantial revision. While the systems attraction to the critical point is predicted by QE, several fundamental properties of the transition, including high precipitation variance in the critical region, need to be added to the theory. Long-range correlations imply that this variance does not reduce quickly under spatial averaging; scaling associated with this spatial averaging has potential implications for superparametrization. Long tails of the distribution of water vapour create relatively frequent excursions above criticality with associated strong precipitation events.

Mesoscale Convective Systems and Critical Clusters

Abstract Size distributions and other geometric properties of mesoscale convective systems (MCSs), identified as clusters of adjacent pixels exceeding a precipitation threshold in satellite radar images, are examined with respect to a recently identified critical range of water vapor. Satellite microwave estimates of column water vapor and precipitation show that the onset of convection and precipitation in the tropics can be described as a phase transition, where the rain rate and likelihood of rainfall suddenly increase as a function of water vapor. This is confirmed in Tropical Rainfall Measuring Mission radar data used here. Percolation theory suggests that cluster properties should be highly sensitive to changes in the density of occupied pixels, which here translates into a rainfall probability, which in turn sensitively depends on the water vapor. To confirm this, clusters are categorized by their prevalent water vapor. As expected, mean cluster size and radius of gyration strongly increase as the ...

Universality of rain event size distributions

We compare rain event size distributions derived from measurements in climatically different regions, which we find to be well approximated by power laws of similar exponents over broad ranges. Differences can be seen in the large-scale cutoffs of the distributions. Event duration distributions suggest that the scale-free aspects are related to the absence of characteristic scales in the meteorological mesoscale.

The time resolution of the St Petersburg paradox

A resolution of the St Petersburg paradox is presented. In contrast to the standard resolution, utility is not required. Instead, the time-average performance of the lottery is computed. The final result can be phrased mathematically identically to Daniel Bernoullis resolution, which uses logarithmic utility, but is derived using a conceptually different argument. The advantage of the time resolution is the elimination of arbitrary utility functions.

Ergodicity Breaking in Geometric Brownian Motion

Geometric Brownian motion (GBM) is a model for systems as varied as financial instruments and populations. The statistical properties of GBM are complicated by nonergodicity, which can lead to ensemble averages exhibiting exponential growth while any individual trajectory collapses according to its time average. A common tactic for bringing time averages closer to ensemble averages is diversification. In this Letter, we study the effects of diversification using the concept of ergodicity breaking.

ATMOSPHERIC CONVECTION AS A CONTINUOUS PHASE TRANSITION: FURTHER EVIDENCE

We present further methods to investigate in how far atmospheric precipitation can be described as a continuous phase transition. Previous work has shown a scale-free range in the rainfall event size distribution and a suggestive power-law pickup in the rain rate above a critical level of instability. Here we examine an additional technique for estimating critical parameters, we investigate the rain rate pickup for an example of an extreme event, namely satellite observations of Hurricane Katrina, and develop an analysis of fluctuations in the rain rate to estimate uncertainties in the tuning parameters relevant for the transition.

Avalanche behavior in an absorbing state Oslo model

Self-organized criticality can be translated into the language of absorbing state phase transitions. Most models for which this analogy is established have been investigated for their absorbing state characteristics. In this paper, we transform the self-organized critical Oslo model into an absorbing state Oslo model and analyze the avalanche behavior. We find that the resulting gap exponent D is consistent with its value in the self-organized critical model. For the avalanche size exponent tau an analysis of the effect of the external drive and the boundary conditions is required.