Abstract
Climate science involves various functional relations and needs to investigate the dominance or relative importance of the variables in their relation. In our previous studies, we examined several problems in which causal relations are established, showing how the dependent quantity is affected by two or more independent variables. With linear fitting, the relative contributions of the variables to the variation of the quantity are compared. In this study, we examine constraint relation, which is a simultaneous multivariate relation, with all variables in the relation being equal in position. The relation can generally be nonlinear. To be convenient for examining the dominance, plane equation fitting can be used to linearize the relation. The equation of state for ideal dry air is investigated as a simple case of the relation. For this special case, a linearized relation can be obtained from both the fitting and the derivation. The scale analysis tool used in dynamic meteorology is applied here for the dominance analysis. Through comparing the scales of the terms, we can simplify the equation. The simplified relations correspond, respectively, to Charle’s law, Boyle’s law, and Gay-Lussac’s law. The geographical preferences of the different dominance patterns are exhibited. In addition, when considering the change of the variable that is smallest in scale, we can identify which factor is the dominator. The ideal gas law is intentionally chosen as the example, since the relation is simple in form, and the results of dominance can be deduced analytically. A comparison demonstrates that the methods used here for the dominance analysis are reliable.
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