Scalar mean location-conditioned

Note that (6.190) contains a number of conditional expected values that must be evaluated from the particle fields. The Lagrangian VCIEM model follows from (6.86), and has the same form as the LIEM model, but with the velocity, location-conditioned scalar mean (0 U, X )(U. X. t) in place of location-conditioned scalar mean ( X )(X. t) in the final term on the right-hand side of (6.190). 

In order to simulate (6.194) and (6.195) numerically, it will be necessary to estimate the location-conditioned mean scalar field < />. Y )(.v. t) from the notional particles X(ni(j), (p t) for n e 1,..., Nv. In order to distinguish between the estimate and the true value, we will denote the former by

notional particles used in the simulation. Likewise, the subscript M is a reminder that the estimate will depend on the number of grid cells (M) used to resolve the mean fields across the computational domain. 

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