P. Perona

Hierarchy of models for meandering rivers and related morphodynamic processes

We review the importance of the physical mechanisms involved in river meandering by comparing some existing linear models and extensions thereof. Such models are hierarchically derived from a common and general mathematical framework and then analyzed with a detailed discussion of the physical processes and relevant hypotheses that are involved. Experiments and field data are also used to discuss the related morphodynamic processes. The analysis of the models shows the importance of the closure of secondary currents especially in the modeling of eddy viscosity. This aspect confirms the usefulness of using simplified models for some practical applications, provided the secondary currents are modeled in detail. On the other hand, the free response of the sediments, the phase lag of secondary currents, and the momentum redistribution due to the coupling between the main and the transverse flow are shown to be less relevant. Hence the second-order models, which neglect the effect of superelevation induced by the topography-driven lateral flow on the longitudinal flow, can reasonably be considered a good approximation for both predictive analysis and the computation of the resonant conditions. Finally, the analysis of higher harmonics suggests that the multilobed pattern can intrinsically be present in both second- and fourth-order models.

On the long‐term behavior of meandering rivers

In spite of notable advances in the description of river morphodynamics, the long-term dynamics of meandering rivers is still an open question, in particular, regarding the existence of a possible statistical steady state and its scaling properties induced by the competing action of cutoffs and reach elongation. By means of extensive numerical simulations, using three fluid dynamic models of different complexity and analysis of real data from the Amazon, North America, and Russia, we show that the reach cutoffs, besides providing stability and self-confinement to the meander belt, also act as a dynamical filter on several hydrodynamic mechanisms, selecting only those that really dominate the long-term dynamics. The results show that the long-term equilibrium conditions are essentially governed by only one spatial scale (proportional to the ratio of the river depth and the friction coefficient) and one temporal scale (proportional to the square of the spatial scale divided by the river width, the mean longitudinal velocity, and the erodibility coefficient) that contain the most important fluid dynamic quantities. The ensuing statistical long-term behavior of meandering rivers proves to be universal and largely unaffected by the details of the fluid dynamic processes that govern the short-term river behavior.

A stochastic process for the interannual snow storage and melting dynamics

A continuous time process of alternating stochastic growths and deterministic decays is proposed as a simple model for the interannual dynamics of snow water equivalent (SWE) storage and melting. The related stationary properties are studied, and an integral difference equation for the probability density function of the state variable is derived. The corresponding solution is obtained by both solving the master equation numerically and simulating the process with a Monte Carlo scheme. An analytical solution is also obtained for the inverse problem, i.e., when the pdf of growths that lead to accumulation is reconstructed from that of the process. The models statistical properties help explain the probabilistic character of the end of summer snowlines and their intrinsic variability with the geographic location and climatic factors. Possible applications to model the interannual SWE cover evolution or to reconstruct the past precipitation characteristics are also discussed.

HYDROLOGICAL AND GEOMORPHOLOGICAL SIGNIFICANCE OF RIPARIAN VEGETATION IN DRYLANDS

Drylands are regions encompassing hyperarid, arid, semiarid, or subhumid climatic conditions (see also Chap. 1). They include cold and warm subtropical deserts, savannas, and the Mediterranean environments. Our focus here is on warm drylands, which are generally characterized by the existence of a well-defined dry season dominated by subtropical high pressure (Malanson 1993) and a rainy season with an average precipitation of less than 700 mm/year. Such regions cover approximately 50% of the continents, with about 20% of the world’s population living in these areas (Le Houerou 1982; Nanson et al. 2002). This explains the growing scientific interest in the study of drylands. Here, we focus on the interactions between fluvial geomorphology and riparian vegetation. These interactions act at different spatial and temporal scales, suggesting the existence of an intrinsic and remarkable sensitivity of riparian ecosystems to hydrological and geomorphological modifications. In this respect, geomorphological resilience to disturbances of either climatic or anthropic origin has recently been questioned (Tooth 2018). Dryland riparian ecosystems are (spatially) linear oases playing the role of humid spots in dryland regions (see Tooth and McCarthy (2007) for a review) used by people and wildlife (Fig. 10.1). However, such ecosystems have been affected by heavy anthropogenic disturbances and risks associated with the encroachment of invasive riparian species, with great reductions in spatial extent (up to 80%, as in certain USA sites) with respect to presettlement times (Smith et al. 1991; Tooth 2000a, b; Salinas et al. 2000; O’Connor 2001; Pettit et al. 2001; Williams et al. 2013). This also sets the risk of reducing common property resources in drylands, e.g., water bodies and related ecological functions being benefited by a community or a group of communities (Gaur et al. 2018) and ecosystem species (McGinnes et al. 2010).

Stochastic dynamics of snow avalanche occurrence by superposition of Poisson processes

We study the dynamics of systems with deterministic trajectories randomly forced by instantaneous discontinuous jumps occurring according to two different compound Poisson processes. One process, with constant frequency, causes instantaneous positive random increments, whereas the second process has a state-dependent frequency and describes negative jumps that force the system to restart from zero (renewal jumps). We obtain the probability distributions of the state variable and the magnitude and intertimes of the jumps to zero. This modelling framework is used to describe snow-depth dynamics on mountain hillsides, where the positive jumps represent snowfall events, whereas the jumps to zero describe avalanches. The probability distributions of snow depth, together with the statistics of avalanche magnitude and occurrence, are used to explain the correlation between avalanche occurrence and snowfall as a function of hydrologic, terrain slope and aspect parameters. This information is synthesized into a ‘prediction entropy’ function that gives the level of confidence of avalanche occurrence prediction in relation to terrain properties.