Towards understanding the mean annual water-energy balance equation based on an Ohms-type approach

Xu Shan, Xingdong Li, Hanbo Yang
2019 Hydrology and Earth System Sciences Discussions  
<p><strong>Abstract.</strong> The Budyko hypothesis has been widely used to describe precipitation partitioning at the catchment scale. Many empirical and analytical formulas have been proposed to describe the Budyko hypothesis. Based on dimensional analysis and mathematic reasoning, previous studies gave an analytical derivation, i.e., the Mezentsev-Choudhury-Yang (MCY) equation. However, few hydrological processes were involved in the derivation. Therefore, this study firstly defines a
more » ... ly defines a catchment network to describe water vapor transformation and transportation using the Lagrangian particle tracking method; and then proposes the generalized flux of water vapor, which can be expressed as the ratio of potential difference with resistance. Furthermore, this study obtains a new constraint for the mean annual water-energy balance, <span style="border-bottom: 1px solid #000; vertical-align: 90%; font-size: .7em; color: #000;"> 1 </span><span style="margin-left: -1.3em; margin-right: .5em; vertical-align: -15%; font-size: .7em; color: #000;"><i>f</i>(<i>E</i>)</span> &amp;thinsp;=&amp;thinsp; <span style="border-bottom: 1px solid #000; vertical-align: 90%; font-size: .7em; color: #000;"> 1 </span><span style="margin-left: -1.3em; margin-right: .5em; vertical-align: -15%; font-size: .7em; color: #000;"><i>f</i>(<i>E</i><sub>0</sub>)</span> &amp;thinsp;+&amp;thinsp; <span style="border-bottom: 1px solid #000; vertical-align: 90%; font-size: .7em; color: #000;"> 1 </span><span style="margin-left: -1.3em; margin-right: .5em; vertical-align: -15%; font-size: .7em; color: #000;"><i>f</i>(<i>P</i>)</span> with <i>E</i>, <i>E</i><sub>0</sub> and <i>P</i> being evaporation, potential evaporation and precipitation, respectively, and <i>f</i>( ) being a function of generalized flux, based on an analogy of the Ohms-type approach and the homogeneity assumption, i.e., the generalized flux has the same form for both water vapor transportation and chase transformation, and in other words, precipitation and potential evaporation have an equalized effect on evaporation. According to this constraint, the MCY equation can be obtained when the generalized flux <i>f</i>( ) is a power function. In addition, this study suggests a more general expression <i>E</i> &amp;thinsp;=&amp;thinsp; <span style="border-bottom: 1px solid #000; vertical-align: 90%; font-size: .7em; color: #000;"> <i>P</i>(<i>b</i>+<i>kE</i><sub>0</sub>) </span><span style="margin-left: -5em; margin-right: 0.5em; vertical-align: -15%; font-size: .7em; color: #000;">[<i>p</i><sup><i>n</i></sup>+(<i>b</i>+<i>kE</i><sub>0</sub><sup><i>n</i></sup>]<sup>1/<i>n</i></sup></span> under conditions without the homogeneity constraint, where <i>E</i>, <i>E</i><sub>0</sub> and <i>P</i> are evaporation, potential evaporation and precipitation, respectively, and <i>n</i>, <i>k</i> and <i>b</i> are constants (MCY equation when <i>b</i>&amp;thinsp;=&amp;thinsp;0 and <i>k</i>&amp;thinsp;=&amp;thinsp;1).</p>
doi:10.5194/hess-2019-283 fatcat:7v3ox6seafhyzfynezzu4ji5yi