A 2D Finite Volume Non-hydrostatic Atmospheric Model ...

®¬¬1, juS iju v1, jv1, jt}}}3.3 Computing the gradientsThe evaluation of the complete state at the cell interfaces in (3.10) and (3.11) requires an approximationto the gradients of the deviation from the local hydrostatic background state along the computational dimensionsi and j. We calculate the deviations in the center of all neighbor cells of cell c ij. For the easterncell this deviation isŠ Q iQ ihQ ij s‹ iwhere ‹ i 1, j is the value of the local vertical coordinate ž evaluated at the center of cell c i 1, j. For theother three cells the deviation is calculated analogously. From this four deviations we can compute fourgradients, two along each computational dimension. How this is done in detail for curvilinear grids takingmetric terms into account is shown in [6]. To avoid spurious oscillations the two gradients along thesame computational direction are smoothed using a gradient limiter function. To smoothen the gradientsg 1and g 2we have used either the minmod limiter3.13~“ g 1,g 2” : •12 “ sign “ g 1”Sš sign “ g 2”.” — min “.­ g 1­ , ­ g 2­ ” ,3.14~the monotonized central limiter1s g 1,©g©2: t u 2 sign g sign s 1g tSx s 2min 2 min © © © g 1g x 2t.t/ s ¯ g 1, g 2, t © 2 °ss 3.15tor the Van Leer (sometimes also called WENO (weighted essentially non oscillatory)) limiterw “ g 2” — g 1š w “ g 1” — g 2“ g 1,g 2: ” • w g “ 1w ”Sš g “ 2”3.16~where w is the regulated absolute value w Ÿ g i S¡2¢ Ÿ g i2 £ ¤, and ¥’¦ 10§6 .3.4 Calculation of the source termAs seen in section 2.4 the source term inside cell c ijcan be approximated with second order by1P h c ijn dS.ij©c ij©Sª(«s 3.17tTo perform a discrete integration of the source term in each cell the hydrostatic background pressure hasto be defined at the interfaces. In section 3.5 the hydrostatic background state is defined for each cell asa function of the local vertical coordinate ¨ . The discrete integration of the vertical momentum componentof the source term then reads±wS ij t s t² ² 1c ij4 ³k 1hP ij sR‹ k t/zh k n . ´k iju vAll three other components of the source term of the inhomogeneous Euler equations (2.1) equal zero.s 3.18t9