| 85 | | Due to the large stencil of WS5, additional ghost layers are necessary on each lateral boundary of each processor subdomain to avoid local data dependencies. Therefore, the exchange of ghost layers is adapted to a dynamic number of ghost layers. For the bottom and top boundaries, lateral non-cyclic boundaries as well as near topography walls the order of the discrertization is successively degraded from WS5 to WS3 to a 1^st^ order scheme. The degradation is controlled by three-dimensional bit flags, where several bits mark the order of the advection scheme in the ''x''-, ''y''-, and ''z''-direction. Technically, in case of topography, advective fluxes are calculated for the first-, third-, and fifth-order discretization at each grid point, while the respective bit controls which discretization is finally used. |
| | 85 | Due to the seven-point stencil of WS5, three ghost layers are required on each lateral boundary of the processor subdomains, in order to avoid local data dependencies. Therefore, the exchange of ghost layers is adapted to a dynamic number of ghost layers, depending on the applied advection scheme. Near the bottom and top boundaries, as well as near lateral non-cyclic boundaries and near topography walls, the order of the discretization is successively degraded from fifth order to a third-order scheme and further to a first-order scheme. The degradation is controlled by three-dimensional bit flags, where several bits mark the order of the advection scheme in the ''x''-, ''y''-, and ''z''-direction. Technically, in case of WS-scheme advective fluxes are calculated for the first-, third-, and fifth-order descretization, while the respective bit controls which discretization is finally used. |
| 452 | | The statistical evaluation of turbulent fluxes should be consistent with the discretization in the prognostic equations because otherwise some unphysical effects occur. For example the computation of the turbulent fluxes as variances and covariances induces some conspicuous kinks in the vertical heat and momentum fluxes near the surface, while the temperature and velocity profiles show no conspicuity. In order to compute the turbulent fluxes as they appear in the prognostic equations, the fluxes are computed in the advection routines, buffered and then reused for the statistics. To receive the turbulent and not the mean signal and to remove the influence of Galilei transformation, the centered fraction of the flux ''F,,i+1/2,,'' has to be multiplied with a factor |
| 453 | | {{{ |
| 454 | | #!Latex |
| 455 | | \begin{align*} |
| 456 | | \frac{u_{i+\frac{1}{2}} - \overline u}{u_{i+\frac{1}{2}} - u_{i, Galilei}} |
| 457 | | \end{align*} |
| 458 | | }}} |
| 459 | | and the dissipative fraction with a factor |
| 460 | | {{{ |
| 461 | | #!Latex |
| 462 | | \begin{align*} |
| 463 | | \frac{|u_{i+\frac{1}{2}} - \overline u|}{|u_{i+\frac{1}{2}} - u_{i, Galilei}|}, |
| 464 | | \end{align*} |
| 465 | | }}} |
| 466 | | |
| 467 | | where ''u'' denotes the corresponding velocity component. Furthermore, the turbulent fluxes are evaluated on each Runge-Kutta substep and weighted with the respective Runge-Kutta coefficients to remove dependencies of the Runge-Kutta substeps. The interpretation of the turbulent fluxes as variances and covariances is no longer valid when using WS5. For other advection schemes, like the PW-scheme, the interpretation of turbulent fluxes as co/variances is still valid, because the discretization is alike the computation of the co/variances. |
| | 452 | In case of lower order advection scheme, e.g. the Piascek-Williams scheme, the turbulent flux of a quantity '''s''' is calculated by the so-called spatial eddy covariance method, |
| | 453 | {{{ |
| | 454 | #!Latex |
| | 455 | \begin{align*} |
| | 456 | <\overline{w's'}>\,=\,<\,\overline{(w - \overline{w}) (s -\overline{s})}\, > \,, |
| | 457 | \end{align*} |
| | 458 | }}} |
| | 459 | where w is the vertical velocity. The overbar denotes the spatial mean, while the angle brackets denote a temporal mean. |
| | 460 | However, in case of higher order advection scheme, the statistical evaluation of a turbulent flux is not consistent with the way how the flux is treated in the model by the advection scheme, due to the different spatial stencil. As a consequence, in case of higher-order advection scheme, some unphysical kinks occur in the turbulent flux profiles if spatial eddy covariance is used, especially near the ground where the turbulent transport is poorly resolved. This is also known as statistical evaluation problem. \\ |
| | 461 | In order to circumvent this problem in case of higher-order advection scheme, turbulent fluxes are calculated the same way as they are treated by the advection scheme, i.e. they are calculated directly in the advection routines and are reused later for statistical output. |
| | 462 | This case, the statistical output reads as: |
| | 463 | {{{ |
| | 464 | #!Latex |
| | 465 | \begin{align*} |
| | 466 | <\overline{w's'}>\,=\,<\, \overline{F_{i-\frac{1}{2}}^{5}(s)\,\cdot\,\frac{u_{i+\frac{1}{2}} - \overline{u}}{u_{i+\frac{1}{2}}} } \, > \,, |
| | 467 | \end{align*} |
| | 468 | }}} |
| | 469 | where the second term removes the mean part of the transport. \\ |
| | 470 | In case of higher-order advection, turbulent fluxes are evaluated on each Runge-Kutta substep and weighted with the respective Runge-Kutta coefficients to remove dependencies of the Runge-Kutta substeps. |
