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...om Gaussian field that is homogeneous in space and φ1(t) a random Gaussian field that is inhomogeneous only in time. Amplitude mapping closure requires φ1(t) = 0, which fails in the asymptotic behaviour of conditional dissipation rate due to lack of an independent time-evolving reference field. Following the standard method [12, 17], we can formulate the mapping equation ∂Y ∂t + ∂Y ∂φ1 〈 dφ1 dt ∣∣∣∣Z 〉 = −〈u∇Y |Z〉 + κ〈∇2Y |Z〉. (14) The conditional moment 〈u∇Y |Z〉 vanishes due to the assumption of homogeneity. Its analytical model for inhomogeneous ‘white noise’ velocity fields can be found in [18]. Other conditional averages in (14) can be evaluated by homogeneity of the velocity and scalar fields and Gaussianity of the reference fields, 〈 dφ1 dt ∣∣∣∣Z 〉 = φ1 2〈φ21〉 d〈φ21〉 dt , 〈∇2Y |Z〉 = −φ0 〈φ 2 0x〉 〈φ20〉 ∂Y ∂φ0 + 〈φ20x〉 ∂2Y ∂φ20 . (15) An exact solution can be obtained from (14) with the evaluated conditional averages (15) and the initial condition (5) which requires φ1(0) = 0, Y = 1 2 [ 1 + erf ( φ0 − γeτ√ 2 Σ ) + ( 1 − exp [ φ1 − 〈φ1〉 + ∫ t 0 〈 dφ1 dt ∣∣∣∣Z 〉 dt ]) (2〈Z〉 − 1) ] . (16) We will calculate the CDR χ(Z, t) = κ〈(∇Y )2|Z〉, where ensemble average is taken over the level s...