scholarly journals The Budyko functions under non-steady state conditions: new approach and comparison with previous formulations

2016 ◽  
Author(s):  
Roger Moussa ◽  
Jean-Paul Lhomme
Keyword(s):  
Steady State ◽  
Aridity Index ◽  
Soil Water Storage ◽  
Additional Parameter ◽  
Simple Relationship ◽  
Evaporation Ratio ◽  
Feasible Domain ◽  
Physically Based ◽  
New Formulation ◽  
Long Timescales

Abstract. The Budyko functions relate the evaporation ratio E / P (E is evaporation and P precipitation) to the aridity index Φ = Ep / P (Ep is potential evaporation) and are valid on long timescales under steady state conditions. A new formulation physically based (noted ML) is proposed to extend the Budyko framework under non-steady state conditions taking into account the change in soil water storage S. The ML formulation introduces an additional parameter S* = S / Ep and can be applied with all classical Budyko functions. In the standard Budyko space (Ep / P, E / P), and for the particular case where the Fu-Zhang equation is used as a Budyko function, the ML formulation yields similar results to the analytical solution of Greve et al. (2016), and a simple relationship can be established between their respective parameters. Then, the ML formulation is extended to the space [(Ep / (P + S), E / (P + S)] and compared to the formulations of Chen et al. (2013) and Du et al. (2016). We show that the ML and Greve et al. formulations have similar upper feasible domain but their lower feasible domain is different from those of Chen et al. (2103) and Du et al. (2016). Moreover, the domain of variation of Ep / (P + S) differs: it is bounded by an upper limit 1 / S* in the ML formulation, while it is bounded with a lower limit in Chen et al.'s and Du et al.'s formulations.

scholarly journals The Budyko functions under non-steady-state conditions

2016 ◽  
Vol 20 (12) ◽  
pp. 4867-4879 ◽  
Author(s):  
Roger Moussa ◽  
Jean-Paul Lhomme
Keyword(s):  
Steady State ◽  
Dimensionless Parameter ◽  
Lower Boundary ◽  
Aridity Index ◽  
Simple Relationship ◽  
Upper Limit ◽  
Evaporation Ratio ◽  
Physically Based ◽  
Terrestrial Water ◽  
Long Timescales

Abstract. The Budyko functions relate the evaporation ratio E ∕ P (E is evaporation and P precipitation) to the aridity index Φ  =  Ep ∕ P (Ep is potential evaporation) and are valid on long timescales under steady-state conditions. A new physically based formulation (noted as Moussa–Lhomme, ML) is proposed to extend the Budyko framework under non-steady-state conditions taking into account the change in terrestrial water storage ΔS. The variation in storage amount ΔS is taken as negative when withdrawn from the area at stake and used for evaporation and positive otherwise, when removed from the precipitation and stored in the area. The ML formulation introduces a dimensionless parameter HE  =  −ΔS ∕ Ep and can be applied with any Budyko function. It represents a generic framework, easy to use at various time steps (year, season or month), with the only data required being Ep, P and ΔS. For the particular case where the Fu–Zhang equation is used, the ML formulation with ΔS  ≤  0 is similar to the analytical solution of Greve et al. (2016) in the standard Budyko space (Ep ∕ P, E ∕ P), a simple relationship existing between their respective parameters. The ML formulation is extended to the space [Ep ∕ (P − ΔS), E ∕ (P − ΔS)] and compared to the formulations of Chen et al. (2013) and Du et al. (2016). The ML (or Greve et al., 2016) feasible domain has a similar upper limit to that of Chen et al. (2013) and Du et al. (2016), but its lower boundary is different. Moreover, the domain of variation of Ep ∕ (P − ΔS) differs: for ΔS  ≤  0, it is bounded by an upper limit 1 ∕ HE in the ML formulation, while it is only bounded by a lower limit in Chen et al.'s (2013) and Du et al.'s (2016) formulations. The ML formulation can also be conducted using the dimensionless parameter HP = −ΔS ∕ P instead of HE, which yields another form of the equations.

scholarly journals The Budyko framework beyond stationarity

2015 ◽  
Vol 12 (7) ◽  
pp. 6799-6830 ◽  
Author(s):  
P. Greve ◽  
L. Gudmundsson ◽  
B. Orlowsky ◽  
S. I. Seneviratne
Keyword(s):  
Water Balance ◽  
Aridity Index ◽  
Additional Parameter ◽  
Economic Sectors ◽  
First Order ◽  
Wide Range ◽  
Additional Water ◽  
New Formulation ◽  
Storage Term ◽  
Land Water

Abstract. Water availability is of major importance for a wide range of socio-economic sectors. Over land, the partitioning of precipitation (P) into evapotranspiration (E) and runoff (Q) is the key process to assess hydrological conditions. For climatological averages, the Budyko framework provides a simple first order relationship to estimate the evaporative index E / P as a function of the aridity index (Ep / P, with Ep denoting potential evaporation). However, a major downside of the Budyko framework is its limitation to steady state conditions, being a result of the assumption of a closed land water balance. Nonstationary processes coming into play at other than mean annual catchment scales are thus not represented. Here we propose an analytically derived new formulation of the Budyko curve including an additional parameter being implicitly related to the nonlinear storage term of the land water balance. The new framework is comprehensively analysed, showing that the additional parameter leads to an upward rotation of the original supply limit and therefore implicitly represents the amount of additional water available for evaporation. The obtained model is further validated using standard datasets of P, E and Ep. It is shown that the model is capable to represent first-order seasonal dynamics within the hydroclimatological system.

scholarly journals A two-parameter Budyko function to represent conditions under which evapotranspiration exceeds precipitation

2016 ◽  
Vol 20 (6) ◽  
pp. 2195-2205 ◽  
Author(s):  
Peter Greve ◽  
Lukas Gudmundsson ◽  
Boris Orlowsky ◽  
Sonia I. Seneviratne
Keyword(s):  
Aridity Index ◽  
Additional Parameter ◽  
Economic Sectors ◽  
Wide Range ◽  
Terrestrial Water ◽  
Additional Water ◽  
New Formulation ◽  
Storage Change ◽  
Two Parameter ◽  
Original Water

Abstract. A comprehensive assessment of the partitioning of precipitation (P) into evapotranspiration (E) and runoff (Q) is of major importance for a wide range of socio-economic sectors. For climatological averages, the Budyko framework provides a simple first-order relationship to estimate water availability represented by the ratio E / P as a function of the aridity index (Ep∕P, with Ep denoting potential evaporation). However, the Budyko framework is limited to steady-state conditions, being a result of assuming negligible storage change in the land–water balance. Processes leading to changes in the terrestrial water storage at any spatial and/or temporal scale are hence not represented. Here we propose an analytically derived modification of the Budyko framework including a new parameter explicitly representing additional water available to evapotranspiration besides instantaneous precipitation. The modified framework is comprehensively analyzed, showing that the additional parameter leads to a rotation of the original water supply limit. We further evaluate the new formulation in an example application at mean seasonal timescales, showing that the extended framework is able to represent conditions in which monthly to annual evapotranspiration exceeds monthly to annual precipitation.

The Effect of Lubricant Inertia in Journal-Bearing Lubrication

1957 ◽  
Vol 24 (4) ◽  
pp. 494-496
Author(s):  
J. F. Osterle ◽  
Y. T. Chou ◽  
E. A. Saibel
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Journal Bearings ◽  
Hydrodynamic Theory ◽  
Simple Relationship ◽  
Inertia Effect ◽  
Laminar Regime ◽  
Steady State Operation

Abstract The Reynolds equation of hydrodynamic theory, modified to take lubricant inertia into approximate account, is applied to the steady-state operation of journal bearings to determine the effect of lubricant inertia on the pressure developed in the lubricant. A simple relationship results, relating this “inertial” pressure to the Reynolds number of the flow. It is found that the inertia effect can be significant in the laminar regime.

scholarly journals Matching the Turc–Budyko functions with the complementary evaporation relationship: consequences for the drying power of the air and the Priestley–Taylor coefficient

2016 ◽  
Author(s):  
Jean-Paul Lhomme ◽  
Roger Moussa
Keyword(s):  
Shape Parameter ◽  
Functional Relationship ◽  
Functional Dependence ◽  
Aridity Index ◽  
Taylor Coefficient ◽  
Potential Evaporation ◽  
Catchment Scale ◽  
Varying Coefficient ◽  
Actual Evaporation ◽  
Long Timescales

Abstract. The Turc–Budyko functions B1(Φp) are dimensionless relationships relating the ratio E/P (actual evaporation over precipitation) to the aridity index Φp = Ep/P (potential evaporation over precipitation). They are valid on long timescales at catchment scale with Ep generally defined by Penman's equation. The complementary evaporation (CE) relationship stipulates that a decreasing actual evaporation enhances potential evaporation through the drying power of the air which becomes higher. The Turc–Mezentsev function with its shape parameter λ is chosen as example among various Turc–Budyko curves and the CE relationship is implemented in the form of the Advection-Aridity model. First, we show that there is a functional dependence between the Turc–Budyko curve and the drying power of the air. Then, we examine the case where potential evaporation E0 is calculated by means of the Priestley–Taylor equation with a varying coefficient α0. Introducing the CE relationship into the Turc–Budyko function leads to a new transcendental form of the Turc–Budyko function B1'(Φ0) linking E/P to Φ0 = E0/P. The two functions B1(Φp) and B1'(Φ0) are equivalent only if α0 has a specified value which is determined. The functional relationship between the Priestley–Taylor coefficient, the Turc–Mezentsev shape parameter and the aridity index is specified and analysed.

scholarly journals Earth System Economics: a bio-physical approach to the human component of the Earth System

2020 ◽  
Author(s):  
Eric Galbraith
Keyword(s):  
Time Allocation ◽  
Subjective Experience ◽  
Population Level ◽  
Well Being ◽  
Simple Relationship ◽  
Earth System ◽  
State Variables ◽  
The Earth ◽  
Physically Based ◽  
Suitable Framework

Abstract. The study of humans has largely been carried out in isolation from the study of the non-human Earth system. This isolation has encouraged the development of incompatible philosphical, aspirational, and methodological approaches that have proven very difficult to integrate with those used for the non-human remainder of the Earth system. Here, an approach is laid out for the scientific study of humans that is intended to facilitate integration with non-human processes by striving for a consistent physical basis, for which the name Earth System Economics is proposed. The approach is typified by a foundation on bio-physical state variables, quantification of time allocation amongst available activities at the population level, and an orientation towards measuring human experience. A suitable framework is elaborated, which parses the Earth system into four classes of state variables, including a neural class that would underpin many societal features. A working example of the framework is then illustrated with a simple numerical model, considering a global population that is engaged in one of two waking activities: provisioning food, or doing something else. The two activities are differentiated by their motivational factors, outcomes on state variables, and associated subjective experience. Although the illustrative model is a gross simplification of reality, the results suggest a simple relationship to predict first order changes in the human population size, and how neural characteristics and subjective experience can robustly emerge from model dynamics, including transient golden ages. The approach is intended to provide a flexible and widely-applicable strategy for understanding the human-Earth system, appropriate for physically-based assessments of the past and present, as well as long-term model projections that are oriented towards improving human well-being.

Assessment of variability of soil Infiltration Characteristics in Forage Cover

2020 ◽  
Vol 7 (03) ◽  
Author(s):  
AKRAM AHMED ◽  
A. K. PAL ◽  
V. K. PANDEY ◽  
MAHENDRA PRASAD ◽  
ASHUTOSH UPADHYAYA
Keyword(s):  
Steady State ◽  
Sandy Loam ◽  
Infiltration Rate ◽  
Panicum Maximum ◽  
Soil Infiltration ◽  
Infiltration Rates ◽  
Bare Land ◽  
Physically Based ◽  
Loam Soil ◽  
Horton Model

In India, very limited knowledge of soil infiltration characteristics in forages are available. In this study, infiltration characteristics of land covered by six forages have been studied with respect to bare land in sandy loam soil. Two empirical (Kostiakov and Horton) and two physically-based (Phillip and Green‒Ampt) models have been employed to estimate infiltration characteristics and compared with observed field infiltration data. The steady-state infiltration rates measured in forages and bare land were significantly (p less than 0.05) different. The highest average steady-state infiltration rate was measured in Panicum maximum (9.00 cm h-1) followed by TSH (7.40 cm h-1) and least was recorded in Cenchrus ciliaris (2.65 cm h-1) whereas the average steady-state infiltration rate recorded for bare land was 1.90 cm h-1. Results showed that the Kostiakov and Phillip model simulated the field infiltration characteristics with higher accuracy than the two other models except for Chrysopogonfulvus and bare land in which the Horton model outperformed other models. Higher steady-state infiltration rates in forages were attributed to more porosity measured in the soils under forages as compared to bare land.

Vegetation carrying capacity of arid regions: on the fraction of rainfall sheltered from surface evaporation

2020 ◽  
Author(s):  
Dani Or ◽  
Peter Lehmann ◽  
Samuel Bickel ◽  
Simone Fatichi
Keyword(s):  
Carrying Capacity ◽  
Arid Regions ◽  
Potential Evapotranspiration ◽  
Rainfall Variability ◽  
Aridity Index ◽  
Arid Lands ◽  
Soil Water Storage ◽  
Annual Rainfall ◽  
Rainfall Events ◽  
Surface Evaporation

<p>Arid lands represent one third of terrestrial surfaces with ecosystems uniquely adapted to water limitations. Arid regions are characterized by low rainfall and sparse vegetation with potential evapotranspiration (ET<sub>0</sub>) exceeding annual rainfall (P) and surface evaporation dominating water losses. The objective was to quantify the fraction of rainwater sheltered from surface evaporation to estimate arid region vegetation carrying capacity. The surface evaporation capacitor (SEC) model was used to quantify surface evaporation from the climatic record of rainfall and potential evaporation. The SEC uses soil-specific active evaporation depth where only rainfall events that exceed its critical capacitance result in leakage into deeper layers. This “leakage” becomes protected from surface evaporation and may support vegetation or inter-annual storage. Focusing on arid regions (aridity index P/ET<sub>0</sub>< 0.2) we illustrate the strong correlation between evaporation-protected rainwater and net primary productivity (NPP) using typical values of water use efficiency. SEC-estimated NPP values were in good agreement with observations and predictions by a state-of-the art ecohydrological model (T&C). Evaporation-protected soil water storage is generated during a few large rainfall events that exceed surface capacitance. This leakage increases with increasing rainfall variability, potentially enhancing vegetation carrying capacity by diverting larger fractions of rainfall from surface evaporation to vegetation-supporting “leakage”. The potential increase in carrying capacity and resulting vegetation cover are greatly influenced by (i) the change in rainfall variability, (ii) soil type, and (iii) surface features that concentrate or divert runoff. We discuss implications of this mechanism for global greening of arid lands and woody plant encroachment.</p>

Relationship Between the External Force and the Specific Work of Fracture R for Steady-State Tearing and Peeling of Elastoplastic Materials

2001 ◽  
Vol 68 (5) ◽  
pp. 758-765 ◽  
Author(s):  
J. H. Liu ◽  
A. G. Atkins ◽  
G. Jeronimidis
Keyword(s):  
Steady State ◽  
External Force ◽  
Elastoplastic Material ◽  
Specific Work ◽  
Simple Relationship ◽  
Elastoplastic Materials ◽  
Self Similar ◽  
The Relationship ◽  
Work Of Fracture ◽  
Specific Work Of Fracture

A simple relationship is obtained between the external force F and the fracture toughness R for thin sheets in steady state elastoplastic combined tearing and peeling along self-similar paths. The relationship depends only on the material properties (E, σy, and α for an elastoplastic material with linear hardening) and strip cross section (B and H). An earlier analysis (which incorporates transient tearing and peeling) requires lengthy computations over the whole length of the strip. The present analysis avoids that complication. Experiments in steady-state agree with the theory.

A Novel Mixing Plane Method Using Nonreflecting Boundary Conditions for Multirow Analysis in Turbomachines

2016 ◽  
Vol 138 (7) ◽  
Author(s):  
Fernando Gisbert ◽  
Roque Corral
Keyword(s):  
Boundary Condition ◽  
Steady State ◽  
Boundary Conditions ◽  
Reverse Flow ◽  
Plane Boundary ◽  
Normal Velocity ◽  
Nonreflecting Boundary Conditions ◽  
Plane Normal ◽  
Convergence Problems ◽  
New Formulation

A new formulation of the mixing plane boundary condition to analyze the steady-state interaction between adjacent rows of a turbomachine, used in conjunction with steady two-dimensional nonreflecting boundary conditions, is presented. Existing mixing plane formulations rely on the differences between some variables at the interface of adjacent rows to determine the boundary condition. These differences are driven to zero as the case is converged to the steady state. By contrast, the proposed approach determines the differences that result in the conservation of mass, momentum, and energy after the boundary condition is enforced, ensuring conservation at any instant during the iterative process. The reverse flow within the mixing plane boundary is naturally treated, but both inlet and outlet boundary conditions fail when the mixing plane normal velocity tends to zero, giving rise to sharp variations of the fluid variables that must be properly limited to prevent convergence problems. Some examples will be given to demonstrate the ability of the new method to resolve these cases while preserving the boundary condition robustness.

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