a theoretical and experimental study of the self-similarity concept

4Tamsalu & Myrberg MERI No. 37:3-13, 1998profiles will be shown and the results of the analyses of experimental data will be compared with theory.In the last section, the main conclusions of the study will be given.The reader interested in self-similarity in more detail is referred to the papers of Barenblatt (1996) andTamsalu & al. (1997).1. MATERIAL AND METHODS1.1 Theoretical derivation of the flux self-similarityThe vertical structure of temperature in the seasonal thermocline will be written in the following form:where:∂T∂q=− (1)∂t∂zT(z,t) is the temperature, q = w' T' is the temperature flux, t is time, z is the vertical coordinate directeddownwards. Substituting the non-dimensional coordinate:z−h()tς=H−h()tequation (1) can be written as follows:(2)∂Th T q( H−h) −( − ς)∂ ∂ ∂1 = −(3)∂t∂t∂ς ∂ςwhere:h(t) is the thickness of the upper mixed layer and H is the bottom of the seasonal thermocline. The followingexpression for the non-dimensional temperature (θ) and for the non-dimensional temperature flux(Q) are proposed:T1() t −T(, t z)θ=T()t −1T H(4)Q q h() t − q (,=t z )q () t −qhHwhere:T 1 (t) is the temperature in the upper mixed layer, T H is the temperature at the bottom of the seasonalthermocline (z=H), q h (t) is the temperature flux at the level z=h, q H is the temperature flux at the levelz=H. Here we suppose that q h (t) >> q H . Then (5) can be written as follows:(5)Q q h() t − q (,=t z )q () thUsing (4) and (5a), equation (3) takes the form:(5a)where:( 1−θ) 1 −( 1− ς)∂θ ∂Qaa2=∂ς ∂ς(6)