G470_AtmBioEx_7_Turb_Motion

# G470_AtmBioEx_7_Turb_Motion - Mathematical Description of...

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Mathematical Description of Turbulent Motion and Transport Preliminaries. .. follow largely notation of Stull (1988) establish frame of reference definitions and rules of notation Taylor’s Hypothesis (“frozen turbulence” in Lagrangian and Eulerian framework) Conservation of (Air) Mass: Continuity equation (in turbulent flow) Conservation of a Scalar Admixture (in turbulent flow)

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System of Reference and Notation (i) streamwise x x 1 u u 1 i i 1 spanwise (+ = right) y x 2 v u 2 j i 2 vertical (+ = up) z x 3 w u 3 k i 3 Space Coordinates x y z Motion Coordinates u v w Local Coordinate System (following the mean flow): Note: This local coordinate system is not Cartesian (i.e., fixed, rectangular), since it follows the flow. These are natural coordinates, in quasi-Cartesian form. direction distance velocity unit vector right hand systems
System of Reference and Notation (ii) Definitions: field variable: a measurable quantity that is continuous and differentiable in every point air parcel: an infinitesimal volume of fluid that is • large enough to contain a statistically significant number of fluid molecules (a continuous medium) • small enough for gradients ( ρ , v , T , etc.) to be approximately linear or negligible derivatives: as an abbreviation , the following notation is used x x ϕ &

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Einstein Summation Rule The Einstein summation rule is a convention to simplify indexing in vector or tensor notation, as follows: • If a given index ([i,j,k] or [1,2,3]) occurs only once in a term of an equation, the equation is valid separately for each possible value of the index. This usually means that the equation represents three analog equations. • If a given index occurs twice in one term of an equation, the term is implicitly summed over all possible values of the index. For example, the following equation is written in indexed form: r ξ & = - & i i u t ± 1 = - - - 1 2 3 1 2 3 u u u t x x x which expands to If a given index occurs double in one term of an equation, the same index cannot occur single in any term of the same equation. However, the equation can have terms that do not contain that given index at all (e.g., the left side of the above equation is not indexed). These rules serve as a check for the correctness of the equation. (see, e.g., Stull 1988, p. 57ff)
Temperature exactly inversely proportional to its temperature. In boundary-layer and micrometeorology vertical mixing. In such vertical motions the pressure, and thus the temperature changes (due to adiabatic expansion or compression). To emphasize the temperature differences common reference pressure, assuming that the reference pressure is reached by adiabatic the pressure were brought to a given reference level. Commonly, the reference pressure is taken as P 0 = 1000 hPa, and the resulting process temperature is called the potential temperature . The relationship between the potential temperature,

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## This note was uploaded on 01/18/2011 for the course INTENSIVE 6 taught by Professor Null during the Spring '10 term at Indiana.

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G470_AtmBioEx_7_Turb_Motion - Mathematical Description of...

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