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,

Daniel Reinert 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

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

Daniel Reinert 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

Daniel Reinert  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.

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

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

Daniel Reinert 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

Daniel Reinert  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 ?

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 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:

Daniel Reinert II.FFSL-implementation

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 Approximations according to Miura (2007) unknown mass point of control volume i

Daniel Reinert 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

Daniel Reinert 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

Daniel Reinert IV.Results: Linear vs. quadratic reconstruction

Daniel Reinert  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° )

Daniel Reinert 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

Daniel Reinert 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

Daniel Reinert 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

Daniel Reinert 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

Daniel Reinert 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.

Daniel Reinert 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

Daniel Reinert Thank you for your attention !!

Daniel Reinert 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.