1. Deutscher Wetterdienst Flux form semi-Lagrangian transport in ICON: construction and results of idealised test cases Daniel Reinert Deutscher Wetterdienst Workshop on the Solution of Partial Differential Equations on the Sphere Potsdam, 26.08.2010

2. Daniel Reinert 26.08.2010 Outline I.Flux form semi-Lagrangian (FFSL) scheme for horizontal transport  A short recap of the basic ideas II.FFSL-Implementation  approximations according to Miura (2007)  Aim: higher order extension III.Results: Linear vs. quadratic reconstruction  2D solid body rotation  2D deformational flow IV.Summary and outlook

3. Daniel Reinert 26.08.2010 I.Flux form semi-Lagrangian scheme (FFSL) for horizontal transport

4. Daniel Reinert 26.08.2010 FFSL: A short recap of the basic ideas  Scheme is based on Finite-Volume (cell integrated) version of the 2D continuity equation.  Assumption for derivation: 2D cartesian coordinate system  Starting point: 2D continuity equation in flux form  Problem: Given at time t 0 we seek for a new set of at time t 1 = t 0 +Δt as an approximate solution after a short time of transport.  Control volume (CV): triangular cells  Discrete value at mass point is defined to be the average over the control volume

5. Daniel Reinert 26.08.2010  In general, the solution can be derived by integrating the continuity equation over the Eulerian control volume A i and the time interval [t 0,t 1 ]. FFSL: A short recap of the basic ideas  No approximation as long as we know the subgrid distribution ρq and the velocity field (or a ie ) analytically.  applying Gauss-theorem on the rhs, assuming a triangular CV, … … we can derive the following FV version of the continuity equation.

6. Daniel Reinert 26.08.2010 control volume 1.Let this be our Eulerian control volume (area A i ), with area averages stored at the mass point Physical/graphical interpretation ?

7. Daniel Reinert 26.08.2010 trajectories control volume Physical/graphical interpretation ? 2.Assume that we know all the trajectories terminating at the CV edges at n+1

8. Daniel Reinert 26.08.2010 control volume 3.Now we can construct the Lagrangian CV (known as „departure cell“) Physical/graphical interpretation ? In a ‚real‘ semi-Lagrangian scheme we would integrate over the departure cell

9. Daniel Reinert 26.08.2010  We apply the Eulerian viewpoint and do the integration just the other way around:  Compute tracer mass that crosses each CV edge during Δt.  Material present in the area a ie which is swept across corresponding CV edge Physical/graphical interpretation ?

10. Daniel Reinert 1.Determine the departure region a ie for the e th edge 2.Determine approximation to unknown tracer subgrid distribution for each Eulerian control volume 3.Integrate the subgrid distribution over the (yellow) area a ie. 26.08.2010 Basic algorithm example for edge 1 Note: For tracer-mass consistency reasons we do not integrate/reconstruct ρ(x,y)q(x,y) but only q(x,y). Mass flux is provided by the dynamical core. The numerical algorithm to solve for consists of three major steps:

11. Daniel Reinert 26.08.2010 II.FFSL-implementation

12. Daniel Reinert 1.Departure region a ie :  aproximated by rhomboidally shaped area.  Assumption: v=const on a given edge 2.Reconstruction of q n (x,y):  SGS tracer distribution approximated by 2D first order (linear) polynomial.  conservative weighted least squares reconstruction 3.Integration:  Gauss-Legendre quadrature  No additional splitting of the departure region. Polynomial of upwind cell is applied for the entire departure region. 26.08.2010 Approximations according to Miura (2007) unknown mass point of control volume i

13. Daniel Reinert 26.08.2010 Default: first order (linear) polynomial (3 unknowns) Stencil for least squares reconstruction Possible improvement - Higher order reconstruction Test: second order (quadratic) polynomial (6 unknowns) 3-point stencil 9-point stencil  improvements to the departure regions appear to be too costly for operational NWP  we investigated possible advantages of a higher order reconstruction. CV

14. Daniel Reinert 26.08.2010 3-point stencil Gnomonic projection Plane of projection Equator How to deal with spherical geometry ?  All computations are performed in local 2D cartesian coordinate systems  We define tangent planes at each edge midpoint and cell center  Neighboring points are projected onto these planes using a gnomonic projection.  Great-circle arcs project as straight lines

15. Daniel Reinert 26.08.2010 IV.Results: Linear vs. quadratic reconstruction

16. Daniel Reinert 26.08.2010  Uniform, non-deformational and constant in time flow on the sphere.  Initial scalar field is a cosine bell centered at the equator  After 12 days of model integration, cosine bell reaches its initial position  Analytic solution at every time step = initial condition Solid body rotation test case Error norms (l 1, l 2, l ∞ ) are calculated after one complete revolution for different resolutions Example of flow in northeastern direction (  =45° )

17. Daniel Reinert 26.08.2010 Setup  R2B4 (≈ 140km)  CFL≈0.5   =45°  flux limiter  conservative reconstruction L 1 = 0.392E-01 L 2 = 0.329E-01 L 1 = 0.887E-01 L 2 = 0.715E-01 Shape preservation Errors are more symmetrically distributed for the quadratic reconstruction. quadratic linear

18. Daniel Reinert 26.08.2010 Setup  R2B4 (≈ 140km)  CFL≈0.5   =45°  flux limiter  non-conservative reconstruction quadratic linear L 1 = 1.293E-01 L 2 = 1.133E-01 L 1 = 0.854E-01 L 2 = 0.699E-01 Non-conservative reconstruction Conservative reconstruction: important when using a quadratic polynomial

19. Daniel Reinert 26.08.2010 Convergence rates (solid body)  Quadratic reconstruction shows improved convergence rates and reduced absolute errors. quadratic, conservative linear, non-conservative Setup:  CFL≈0.25   =45° (i.e. advection in northeastern direction)  flux limiter

20. Daniel Reinert 26.08.2010 Deformational flow test case  based on Nair, D. and P. H. Lauritzen (2010): A class of Deformational Flow Test- Cases for the Advection Problems on the Sphere, JCP  Time-varying, analytical flow field  Tracer undergoes severe deformation during the simulation  Flow reverses its course at half time and the tracer field returns to the initial position and shape  Test suite consists of 4 cases of initial conditions, three for non-divergent and one for divergent flows. t=0 T t=0.5 T t=T Example: Tracer field for case 1

21. Daniel Reinert 26.08.2010 Convergence rates (deformational flow)  C≈0.50  flux limiter quadratic, conservative linear, non-conservative  Superiority of quadratic reconstruction (absolute error, convergence rates) less pronounced as compared to solid body advection. But still apparent for l ∞.  Possible reason: departure region approximation (rhomboidal) does not account for flow deformation.

22. Daniel Reinert 26.08.2010 V.Summary and outlook  Implemented a 2D FFSL transport scheme in ICON (on triangular grid)  Based on approximations originally proposed by Miura (2007) for hexagonal grids  Pursued higher order extension of the 2 nd order ‚Miura‘ scheme by using a higher order (i.e. quadratic) polynomial reconstruction.  Quadratic reconstruction led to  improved shape preservation and reduced maximum errors  improved convergence rates (in particular L ∞ )  reduced dependency of the error on CFL-number  For more challenging deformational flows, the superiority was less marked  conservative reconstruction is essential when using higher order polynomials Outlook:  Which reconstruction (polynomial order) is the most efficient one?  Possible gains from cheap improvements of the departure regions?  Implementation on hexagonal grid comparison

23. Daniel Reinert 26.08.2010 Thank you for your attention !!

24. Daniel Reinert 26.08.2010 Model error as a function of CFL (solid body)  Fixed horizontal resolution: R2B5 (≈ 70 km)  variable timestep  variable CFL number  flux limiter Flow orientation angle: α=0°Flow orientation angle: α=45°  Linear rec.: increasing error with increasing timestep and strong dependency on flow orientation angle.  Quadratic rec.: errors less dependent on timestep and flow angle.