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Statistical properties of $d$-dimensional incompressible flows with andwithout cylindrical reduction are studied, leading to several explanations andconjectures about turbulent flows and passive scalars, such as thede-correlation between the flow and scalar, reduction of passive scalarintermittency in the bottleneck regime, et al. The absolute-equilibriumanalyses assure the correctness of a recent numerical result. It is impliedthat passive scalar(s) in two-dimensional (2D) space can be fundamentallydifferent to those in $d>2$, concerning the correlations with the flow, whichis not considered in the celebrated Kraichnan model. The possibility of genuineinverse transfer to large scales of 2D passive scalar energy, together with theadvection energy, is indicated. The compressible situation is also brieflyremarked in the end, in particular the absence of density in a nontrivialCasimir which, without boundary contribution, also vanishes for $d=4$.

Statistical mechanics of $d$-dimensional flows and cylindrically reduced passive scalars