Abstract
A correction for the vertical gradient of air density has been incorporated into a skewed probability density function formulation for turbulence in the convective boundary layer. The related drift term for Lagrangian stochastic dispersion modelling has been derived based on the well-mixed condition. Furthermore, the formulation has been extended to include unsteady turbulence statistics and the related additional component of the drift term obtained. These formulations are an extension of the drift formulation reported by Luhar et al. (Atmos Environ 30:1407–1418, 1996) following the well-mixed condition proposed by Thomson (J Fluid Mech 180:529–556, 1987). Comprehensive tests were carried out to validate the formulations including consistency between forward and backward simulations and preservation of a well-mixed state with unsteady conditions. The stationary state CBL drift term with density correction was incorporated into the FLEXPART and FLEXPART-WRF Lagrangian models, and included the use of an ad hoc transition function that modulates the third moment of the vertical velocity based on stability parameters. Due to the current implementation of the FLEXPART models, only a steady-state horizontally homogeneous drift term could be included. To avoid numerical instability, in the presence of non-stationary and horizontally inhomogeneous conditions, a re-initialization procedure for particle velocity was used. The criteria for re-initialization and resulting errors were assessed for the case of non-stationary conditions by comparing a reference numerical solution in simplified unsteady conditions, obtained using the non-stationary drift term, and a solution based on the steady drift with re-initialization. Two examples of “real-world” numerical simulations were performed under different convective conditions to demonstrate the effect of the vertical gradient in density on the particle dispersion in the CBL.
Highlights
The need to use a density correction for Lagrangian stochastic (LS) particle modelling of dispersion in the atmospheric boundary layer was discussed by Thomson (1995) for the random displacement model and by Stohl and Thomson (1999) for stochastic models of the particle velocity
Following Stohl and Thomson (1999) a density correction has been incorporated into the skewed probability density function (PDF) formulation for convective boundary layer (CBL) turbulence proposed by Luhar et al (1996), and the related drift term for a Lagrangian stochastic dispersion model has been obtained based on Thomson (1987) well-mixed condition
The stationary state CBL drift term with density correction has been incorporated into the FLEXPART and FLEXPART-WRF models
Summary
The need to use a density correction for Lagrangian stochastic (LS) particle modelling of dispersion in the atmospheric boundary layer was discussed by Thomson (1995) for the random displacement model and by Stohl and Thomson (1999) for stochastic models of the particle velocity. Here we extend the work of Stohl and Thomson (1999) including a density correction in the model of Luhar et al (1996), formulated using the well-mixed criterion and assuming a skewed PDF based on the sum of two Gaussians PDFs. Since backward-in-time modelling is extensively used in many applications The re-initialization allows the use of the drift term formulated for steady and horizontally homogeneous conditions in non-stationary and horizontally inhomogeneous conditions This was needed here since the numerical formulation of the steady and horizontally homogenous drift for the skewed PDF was found to be more sensitive than the standard Gaussian formulation to the time varying turbulence statistics and this, occasionally, introduced the possibility of a serious numerical degradation for the simulated particle trajectories. Here the main cause of instability appears to be the inconsistency between the formulation of the model (i.e. steady drift) and the time-varying turbulence statistics, and not the numerical implementation of the model itself, or the intrinsic dynamical instability of the stochastic differential equation since, with a sufficiently small timestep, no instability was observed in the case of the time-varying drift formulation
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