First passage time statistics of Brownian motion with purely time dependent drift and diffusion

Systems where resource availability approaches a critical threshold are common to many engineering and scientific applications and often necessitate the estimation of first passage time statistics of a Brownian motion (Bm) driven by time-dependent drift and diffusion coefficients. Modeling such systems requires solving the associated Fokker-Planck equation subject to an absorbing barrier. Transitional probabilities are derived via the method of images, whose applicability to time dependent problems is shown to be limited to state-independent drift and diffusion coefficients that only depend on time and are proportional to each other. First passage time statistics, such as the survival probabilities and first passage time densities are obtained analytically. The analysis includes the study of different functional forms of the time dependent drift and diffusion, including power-law time dependence and different periodic drivers. As a case study of these theoretical results, a stochastic model for water availability from surface runoff in snowmelt dominated regions is presented, where both temperature effects and snow-precipitation input are incorporated

Delay-induced rebounds in CO_{2} emissions and critical time-scales to meet global warming targets

While climate science debates are focused on the attainment of peak anthropogenic CO2 emissions and policy tools to reduce peak temperatures, the human‐energy‐climate system can hold “rebound” surprises beyond this peak. Following the second industrial revolution, global per capita CO_{2} emissions (c_{c}) experienced a punctuated growth of about 100% every 60 years, mainly attributable to technological development and its global spread. A model of the human‐energy‐climate system capable of reproducing past punctuated dynamics shows that rebounds in global CO_{2} emissions emerge due to delays intrinsic to the diffusion of innovations. Such intrinsic delays in the adoption and spread of low‐carbon emitting technologies, together with projected population growth, upset the warming target set by the Paris Agreement. To avoid rebounds and their negative climate effects, model calculations show that the diffusion of climate‐friendly technologies must occur with lags one‐order of magnitude shorter (i.e., ∼6 years) than the characteristic timescale of past punctuated growth in c_{c}. Radically new strategies to globally implement the technological advances at unprecedented rates are needed if the current emission goals are to be achieved

A note on aerosol sized particle deposition onto dense and tall canopies situated on gentle cosine hills

Micrometeorological measurements of aerosol sized dry particle deposition velocity ( V d ) onto forested canopies have significantly advanced over the past two decades and now include both—airborne and stationary platforms. However, the interpretation of these  V d measurements still relies on stationary and planar homogeneous flow assumptions only appropriate to flat-terrain conditions. Simplified model calculations were used to examine how variations in hill height ( H ) introduce biases in  V d when assumptions appropriate to flat terrain are applied to periodic and gentle 2-D cosine topography covered with tall and dense forested canopies. It was shown that increasing  H reduced the variability in  V d for all aerosol sized particle diameters ( d p ) inside the canopy when the hill slope ( H / L ) remained constant (=0.1), where  L is the cosine hill half-length. At the landscape scale, as may be monitored from airborne platforms, assumptions appropriate to flat-terrain appear accurate with increasing  H for a constant and gentle H/ L (= 0.1). Inside the canopy, variability in  V d tends to be larger than above the canopy for all  H values and  d p classes. DOI: 10.1111/j.1600-0889.2011.00528.

Effects of Gentle Topography on Forest‐Atmosphere Gas Exchanges and Implications for Eddy‐Covariance Measurements

Abstract: The interpretation of tower‐based eddy‐covariance (EC) turbulent flux measurements above forests hinges on three key assumptions: (1) steadiness in the flow statistics, (2) planar homogeneity of scalar sources or sinks, and (3) planar homogeneity in the flow statistics. Large eddy simulations (LESs) were used to control the first two so as to explore the break‐down of the third for idealized and real gentle topography such as those encountered in Amazonia. The LES runs were conducted using uniformly distributed sources inside homogeneous forests covering complex terrain to link the spatial patterns of scalar turbulent fluxes to topographic features. Results showed strong modulation of the fluxes by flow features induced by topography, including large area with negative fluxes compensating “chimney” regions with fluxes almost an order of magnitude larger than the landscape flux. Significant spatial heterogeneity persisted up to at least two canopy heights, where most eddy‐covariance measurements are performed above tall forests. A heterogeneity index was introduced to characterize and contrast different scenarios, and a topography categorization was shown to have predictive capabilities in identifying regions of negative and enhanced fluxes

Relation between the spectral properties of wall turbulence and the scaling of the Darcy-Weisbach friction factor

Empirical formulas describing the Darcy-Weisbach friction factor remain indispensable for applications in sciences and engineering dealing with turbulent flows. Despite their practical significance, these formulas have remained without theoretical interpretation for many decades. To close this knowledge gap, much research has been devoted to the development of the so-called "spectral link"introduced in the early 2000s. Such a theory is entirely based on elegant phenomenological arguments that make no contact with equations describing turbulent wall flows. The spectral link spawned alternative approaches, now labeled "cospectral budget"(or CSB) models, that describe how turbulent eddies contribute to wall stresses. The CSB overcomes some of the shortcomings of the phenomenological approach and is here employed to provide a thorough clarification of the link between spectral properties of velocity fluctuations and the scaling of friction factors in turbulent pipe flows in the hydraulically smooth and fully rough regimes

Closure Schemes for Stably Stratified Atmospheric Flows without Turbulence Cutoff

Two recently proposed turbulence closure schemes are compared against the conventional Mellor-Yamada (MY) model for stably stratified atmospheric flows. The Energy and Flux-Budget (EFB) approach solves the budgets of turbulent momentum and heat fluxes and turbulent kinetic and potential energies. The Cospectral Budget (CSB) approach is formulated in wavenumber space and integrated across all turbulent scales to obtain flow variables in physical space. Unlike the MY model, which is subject to a "critical gradient Richardson number," both EFB and CSB models allow turbulence to exist at any gradient Richardson number R-t and predict a saturation of flux Richardson number (R-f -> R-fm) at sufficiently large R-i. The CSB approach further predicts the value of Rim and reveals a unique expression linking the Rotta and von Karman constants. Hence, all constants in the CSB model are nontunable and stability independent. All models agree that the dimensionless sensible heat flux decays with increasing R-i. However, the decay rate and subsequent cutoff in the MY model appear abrupt. The MY model further exhibits an abrupt cutoff in the turbulent stress normalized by vertical velocity variance, while the CSB and EFB models display increasing trends. The EFB model produces a rapid increase in the ratio of turbulent potential energy and vertical velocity variance as Rim is approached, suggesting a strong self-preservation mechanism. Vertical anisotropy in the turbulent kinetic energy is parameterized in different ways in MY and EFB, but this consideration is not required in CSB. Differences between EFB and CSB model predictions originate from how the vertical anisotropy is specified in the EFB model.Peer reviewe

Inverse Cascade Evidenced by Information Entropy of Passive Scalars in Submerged Canopy Flows

Turbulent mixing of scalars within canopies is investigated using a flume experiment with canopy-like rods of height h mounted to the channel bed. The data comprised a time sequence of high-resolution images of a dye recorded in a plane parallel to the bed at z/h= 0.2. Image processing shows that von Kármán wakes shed by canopy drag and downward turbulent transport from upper canopy layers impose distinct scaling regimes on the scalar spectrum. Measures from information theory are then used to explore the dominant directionality of the interaction between small and large scales underlying these two spectral regimes, showing that the arrival of sweeps from aloft establishes an inertial-range spectrum with forward “information” cascade. In contrast, wake growth with downstream distance leads to persistent upscale transfer (inverse cascade) of scalar variance, which hints at their nondiffusive character and the significance of the stem diameter as an active length scale in canopy turbulence

Seasonal hysteresis of surface urban heat islands

Temporal dynamics of urban warming have been extensively studied at the diurnal scale, but the impact of background climate on the observed seasonality of surface urban heat islands (SUHIs) remains largely unexplored. On seasonal time scales, the intensity of urban–rural surface temperature differences (ΔTs) exhibits distinctive hysteretic cycles whose shape and looping direction vary across climatic zones. These observations highlight possible delays underlying the dynamics of the coupled urban–biosphere system. However, a general argument explaining the observed hysteretic patterns remains elusive. A coarse-grained model of SUHI coupled with a stochastic soil water balance is developed to demonstrate that the time lags between radiation forcing, air temperature, and rainfall generate a rate-dependent hysteresis, explaining the observed seasonal variations of ΔTs. If solar radiation is in phase with water availability, summer conditions cause strong SUHI intensities due to high rural evaporative cooling. Conversely, cities in seasonally dry regions where evapotranspiration is out of phase with radiation show a summertime oasis effect controlled by background climate and vegetation properties. These seasonal patterns of warming and cooling have significant implications for heat mitigation strategies as urban green spaces can reduce ΔTs during summertime, while potentially negative effects of albedo management during winter are mitigated by the seasonality of solar radiation