SLICE-3D: A three-dimensional conservative semi-lagrangian scheme for transport problems. Mohamed Zerroukat. Met Office, Exeter, UK

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1 SLICE-D: A three-dimensional conservative semi-lagrangian scheme for transport problems Mohamed Zerroukat (Nigel wood & Andrew Staniforth) Met Office, Exeter, UK SRNWP-NT, Zagreb, c Crown Copyright

2 Content SLICE-D specifications Brief review of SLICE-D & SLICE-D SLICE-D Test results Conclusions c Crown Copyright

3 SLICE-D specifications. SLICE-D is a conservative and computationally efficent semi-lagrangian scheme for the transport of scalars.. SLICE-D uses multiple sweeps of a one-dimensional remapping algorithm along Eulerian and Lagrangian directions. This cascade-type splitting, and flow-dependent, strategy reduces the splitting error compared to traditional fixed direction-based splitting.. SLICE-D does not compute at any stage, or needs to know, about the complex geometry of irregular Lagrangian volumes. This constitute an andvantage by comparison to volume-based multi-dimensional remapping.. SLICE-D uses a C-grid wind-density staggering arrangements.. SLICE-D would work by taking the vertical as the first cascade direction. This is for following reasons: (a) since the grid spacing in the vertical is known and is the same at each timestep, a higher-order remapping scheme can be applied efficiently as various factors can be precomputed and stored c Crown Copyright

4 SLICE-D specifications (b) it is possible to impose exactly a global max/min that is non-zero since the cross-sectional area is known exactly.. Having cascaded in the vertical, we are left with N (the number of vertical levels) SLICE-D problems in the (deformed) horizontal. SLICE-D in turn is a series of horizontal sweeps of the same d-remapping algorithm.. All distances used SLICE-D are D physical distances.. SLICE-D computes some D and D interesctions. This overhead is wind (or deformed geometry) dependent and independent of scalars. Therefore, it is anticipated that this overhead will be relatively small when SLICE-D is used for a large number of physical/chemical species. c Crown Copyright

5 SLICE-D approach d x Eulerian grid point Eulerian CV boundaries Departure point Lagrangian CV boundaries Departure of x n+ t n+ M i n+ M i n+ M i+ ρ (x,t n+ ) x i / x i / x i+/ x i+/ t d M i d Mi d Mi+ ρ (x,t n ) n t x d i / x d i / x d i+/ d x i+/ x c Crown Copyright

6 Reconstruction (PCM) The Piecewise Cubic Method ( Zerroukat, Wood and Staniforth, Q. J. R. Meteo. Soc., ) that satisfies: ρ i (ξ) = a i + b i ξ + c i ξ + d i ξ, ξ [, ] () ρ i = ρ i (ξ) dξ, ρ i () = ρ i /, ρ i () = ρ i+/, dρ i dξ = δρ i () ξ=/ c Crown Copyright

7 Reconstruction (PCM) where a i = ρ i /, b i = ρ i / + ρ i δρ i, c i = 9ρ i / ρ i+/ ρ i + δρ i, d i = ρ i / + ρ i+/ δρ i. Evaluation of the unknown cubic parameters { ρ i /, ρ i+/, δρ i } () ρ (x) L ρ i ρ ^ (x) i / ρ (x) i ^ ρ (x) i+/ ρ R i ECV i xi / xi / xi / xi xi+/ x i+/ xi+/ x c Crown Copyright

8 Reconstruction (PSM) PCM + (δρ i = ρ i+/ ρ i / ) > Parabola (Piecewise Parabolic Method (PPM) of Colella and Woodward, J. Comput. Phys. 9): that satisfies ρ i (ξ) = a i + b i ξ + c i ξ, ξ [, ] () ρ i = ρ i (ξ) dξ, ρ i () = ρ i /, ρ i () = ρ i+/ () Parabolic Spline Method (PSM) ( Zerroukat, Wood and Staniforth, Int. J. Num. Meth. Fluids, ) dρ i dx = dρ i+ dx ( ρ xi+/ h i + + i h i h i+ ) ρ i+ + ( ρ h i+ = ρi + ρ ) i+ i+ h i h i+ () PSM is more accurate than PPM but with a computational cost of only % of that of PPM c Crown Copyright

9 PSM (Monotonicity). Non-monotonic behaviour at the grid scale (a) The estimates { ρ i /, i =,,... } at the cell boundaries are tested for undershoots and overshoots with respect to { ρ i, i =,,...} (b) The non-monotonic values are corrected (the closest ρ i ) c Crown Copyright 9

10 PSM (Monotonicity). Non-monotonic behaviour at the sub-grid scales (a) The parabola is tested for non-monotonic behaviour within the interval ξ [, ] (b) The parabola is modified to the limiting case of monotonicity (discontinuities) (c) If the limiting case is not possible = piecewise constant profile c Crown Copyright

11 SLICE-D Figure : Superposition of Lagrangian and Eulerian grids away from poles. c Crown Copyright

12 SLICE-D (polor cap region) Figure : Superposition of Lagrangian and Eulerian grids in the vicinity of the north pole. c Crown Copyright

13 SLICE-D To take into account for the inaccuracy of the pivoting assumption for a region near the poles (polar cap), the density field is corrected (while maintaining mass conservation), using standard Semi-Lagrangian solution as a guide, i.e., ρ n+ i,j = ρ c i,j = N φ Nλ l=k N φ l=k ρ SSL i,j, for i =,..., N λ ; j = k,..., N φ, () ρ i,j, otherwise m= ρ l,m EA l,m Nλ m= ρssl l,m EA l,m c Crown Copyright

14 SLICE-D SLICE-D remapping. Given the Eulerian masses m i,j,k for the spherical Eulerian Control-Volumes (ECV i,j,k ). ECV i,j,k is delimited by λ i / and λ i+/ in the λ-direction; by φ j / and φ j+/ in the φ-direction; and by r k / and r k+/ in the r-direction (V olume = / δr δλ δ(sin φ)). The union of all the grid-points (λ i /, φ j /, r k / ) for a constant k /, forms a discrete regular spherical Eulerian Shell (ES) k / (k = / (surface) and k = K +/ (top surface)); The union of all the grid points (λ d i /, j /,k /, φd i /, j /,k /, rd i /, j /,k / ) form a discrete deformed closed (irregular) Lagrangian Shell (LS) k /. (ES + wind) > LS.. Each LS k / (including bottom and top surfaces) intersects each column (i, j) at a hight r k /, k =,..., K + (vertical intersection computation).. Given the set of masses {m i,j,k, k =,..., K}, for each column (i, j), perform a SLICE-D to compute the intermediate masses { m i,j,k, k =,..., K} of the Intermediate Vertical Control Volume (IV CV ij,k ). IV CV ij,k is delimited by λ i / and λ i+/ in the λ-direction; by φ j / and φ j+/ in the φ-direction; and by r k / and r k+/in the r-direction.. Given the intermediate masses { m i,j,k, i =,..., N; j =,..., M} of a spherical annuli k =,..., K (volume between two LS s k / and k + /), use SLICE-D to remap mass from the IVCVs to its final Lagrangian CVs (LCVs). c Crown Copyright

15 SLICE-D (cont.) Lagrangian(blue) Eulerian (red) grids & vertical intersections (+megenta)(original POLAR) Lagrangian(blue) Eulerian (red) grids & vertical intersections (+megenta)(original POLAR)... NP NP NP NP NP SP SP SP SP SP c Crown Copyright

16 SLICE-D (cont.) Lagrangian(blue) Eulerian (red) grids & vertical intersections (+megenta)(modified CARTESIAN) Lagrangian(blue) Eulerian (red) grids & vertical intersections (+megenta)(modified CARTESIA c Crown Copyright

17 SLICE-D (cont.) Lagrangian(blue) Eulerian (red) grids & vertical intersections (+megenta)(modified CARTE Lagrangian(blue) Eulerian (red) grids & vertical intersections (+megenta)(modified CARTESIAN) c Crown Copyright

18 SLICE-D (cont.) Lagrangian(blue) Eulerian (red) grids & vertical intersections (+megenta)(modified CARTESIAN) c Crown Copyright

19 SLICE-D (cont.) grid formed by the d departure passed to slice d for levels to Lagrangian(blue) Eulerian (red) grids & vertical intersections (+megenta)(modified CARTESI c Crown Copyright 9

20 SLICE-D Intesections. Computation of intersections of the spherical shells with the vertical columns. This invloves computing the height (r) at which each deformed spherical shell intersects each vertical column using a bilinear interpolation. These are needed for vertical remapping.. Intersections in the SLICE-D are intersections of great circles with latitudes circles. These are needed for the second and third cascade remappings.. These extra computations are geometry dependent. These are done only once per time step irrespective of the number of scalars. Therefore, this extra cost can be offset by increasing the number of scalars. c Crown Copyright

21 Test Problems (A Hadley cell circulation test) Wind for centre points.9... zl yl c Crown Copyright

22 Test Problems (Hadley cells circulation test) c Crown Copyright

23 Test Problems (Hadley cells circulation test) c Crown Copyright

24 Test Problems (A Rossby wave test). Wind for centre points. yl.. xl c Crown Copyright

25 Test Problems (A Rossby wave test) c Crown Copyright

26 Test Problems (A Rossby wave test) c Crown Copyright

27 Test Problems (superposed Rossby and Hadley tests) Wind for centre points.. zl.. xl.. yl.. c Crown Copyright

28 Test Problems (superposed Rossby and Hadley tests) c Crown Copyright

29 Test Problems (superposed Rossby and Hadley tests) c Crown Copyright 9

30 Test Problems (superposed Rossby and Hadley tests) c Crown Copyright

31 Test Problems (rotated superposed waves) Wind for centre points.. zl..... yl. xl c Crown Copyright

32 Test Problems (rotated superposed waves) c Crown Copyright

33 Conclusions. SLICE-D has been tested on a number of idealised tests and the results show that the scheme is more accurate and efficent compared to tri-cubic interpolating non-conservative SL.. The scheme (code) is now beeing optimised, implemented, and tested within the UM system.. The scheme will be an integral part of the next UM dynamical core, and likely to be the default transport algorithm for climate, chemistry and other models. c Crown Copyright