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EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-13687, https://doi.org/10.5194/egusphere-egu21-13687, 2021Semi-Lagrangian advectionmodels for quasi-uniform nodeson the sphereTakeshi Enomoto and Koji OgasawaraKyoto UniversityThis work was supported by JSPS KAKENHI Grant Number 21K03662 and“Program for Promoting Researches on the Supercomputer Fugaku” (Elucidation of solar and planetary dynamics and evolution)

Introductionradial basis functions (RBF) and spherical helix RBF enables the use of unstructured quasi-uniform nodes to solve hyperbolicdifferential equations on the sphere (Flyer and Wright 2007, 2009). Spherical helix nodes (Bauer 2000) are fast to generate without arbitrariness. Application of RBF interpolation in semi-Lagrangian advection is proposed. This study compares uniformity of the different quasi-uniform nodes. describes RBF-based Eulerian and semi-Lagrangian advection models. compares the RBF-based advection models in a challenging test case suite.

Uniformity of nodes on the sphere

Spherical helixGeneration of quasi-uniform nodes on the sphere (Bauer 2000)n 512 The spherical helix equationλ mθ mod 2πλ: longitude, θ: colatitude,m dλ/dθ: gradient of spiral With Δs Δθ, m nπ .2k 1z 1 cosθ,k 1,2, nk kn # of nodes is flexible (can add by one).ΔsΔθ

UniformityDeparture from equal weight 4π/n in percentMinimum energy(Flyer and Wright 2007)a)Geodesic optimized with springdynamics (Tomita et al. 2002)Less uniform at verticesSpherical Helix (Bauer 2000)b)c)e)f)Large departurewith ptsLess uniform near thepoles due to largecurvature of helixn 2562

Model description

Advection models on a sphereusing radial basis functions (RBF) Multiquadric (MQ)221 (εr) is chosen over Gaussian (GA) exp( (εr) ) RBF for stability The shape parameter fixed to the inverse ofthe geometric mean distance Eulerian advection the RBF derivative operator (Flyer and Wright 2007) Fourth-order Runge–Kutta Semi-Lagrangian advection Upstream trajectory (Ritchie 1987) Interpolation: radial basis functions

Eulerian modelFlyer and Wright 2007, 2009 The derivative operators for advection Dx, Dy, Dz are prepared in Cartesian 1coordinates, e.g. Dx BxA where an element of Bx isdϕk(rjk)1T.bjk (xjxj xk xk)rjk dr The model integrates h··· x Dx h y Dy h z Dz h. t

Semi-Lagrangian modelusing RBF interpolation The model solves dh/dt 0 or h(t Δt, x) h(t, x uΔt), the value of theupstream point, is interpolated using RBF. Tracer h is approximated by RBF expansion:h(x) n j 1cjϕ(rj). The coefficients are determined by solving the collocation condition Ac h,where A is an n n interpolation matrix whose element is an RBF aij ϕ( xi xj ). NB. A is determined by the node distribution and time independent.

Challenging test case suite

Non-divergent deformational flowA standard test case (Lauritzen et al. 2012)reversal at T/2 Solidbodyπt2π cos(θ)u(λ, θ, t) κ sin (λ′)sin(2θ) cos (T)T22ψ(χ) 0.8χ 0.9πtv(λ, θ, t) κ sin(2λ′)sin(θ) cos(T)

Model configurations MQ RBF is used both in Eulerian and semi-Lagrangian models. n’s are chosen to match the degree of freedom (DOF) for spectral models usingquadratic grids (Flyer and Wright 2007). n’s to match DOF for spectral models using linear grids are also tested withsemi-Lagrangian models. λITTL‘Quadratic’(T 1)23 1.5 ’(TL 1)236001440057600CN 5 only

Numerical order of convergenceEulerian vs Semi-Lagrangian models using MQ RBF‘Quadratic’ (n 1600,6400,25600)Faster convergenceand larger ‘minimal’resolution with semiLagrangian model3rd4thscheme Gaussian CosineNeEU1.792.221.00SL5.512.981.52

Numerical order of convergenceSensitivity to time step length of semi-Lagrangian model using MQ RBF‘Quadratic’ (n 1600,6400,25600)Slower convergenceimplies trajectorycalculation error3rd4thCFLGaussian CosineNe52.872.891.6515.512.981.52

‘Filament’ preservationEulerian vs Semi-Lagrangian models using MQ RBFEulerian1.5 n 6400Semi-LagrangianA(τ, t)ℓf(τ, t) 100 A(τ, t 0)Lauritzen et al. 20140.75 n 25600Filament preservationcomparable betweenEulerian and semiLagrangian models

‘Filament’ preservationSensitivity to the number of nodes in semi-Lagrangian model using MQ RBFn 6400n 14400A(τ, t)ℓf(τ, t) 100 A(τ, t 0)1.5 n 256000.75 n 57600Lauritzen et al. 2014Approach to 100% as# of nodes increases

Transport of ‘rough’ distributionShapes are comparablyresolved, but less noisywith semi-LagrangianEulerian vs semi-Lagrangian model using MQ RBFCFL 1n 64001.5 n 256000.75 L0.180.82-0.200.27

Transport of ‘rough’ distributionSlow but steadyconvergenceSensitivity to the number of nodes in semi-Lagrangian model using MQ RBFCFL 51.5 0.75 .190.23576000.140.70-0.190.17

Preservation of pre-existing functional relationEulerian vs semi-Lagrangian models using MQ RBFEuleriann 6400overshootinglo‘re80.al’mngSmaller mixing errorswith semi-Lagrangiancosine bellsLauritzen and Thuburn 201190.ixi rangepreservingunmixing2lulr n 25600Semi-Lagrangian‘correlated’ cosine bellsCFL 1

P. H. Lauritzen et al.: Results from standarPreservation of pre-existing functional relationSensitivity to the number of nodes in semi-Lagrangian model using MQ RBFCFL 5n 64001.5 n 14400CSLAM (Lauritzen et al. 2010) spectral bicubic (Enomoto 2008)P. H. Lauritzen et al.: Results from standard test case suiten 256000.75 n 57600RBF-based semi-Lagrangianmodels are competitive or betterNB. sizes of red dots are differentLauritzen et al. 2014

SummaryAdvantages of RBF-based semi-Lagrangian models A semi-Lagrangian model was constructed on quasi-uniform nodes along aspherical helix on the sphere using radial basis function (RBF) for interpolation. The semi-Lagrangian model outperforms the Eulerian model and is stable withMultiquadric RBF, at least for n 57600. The semi-Lagrangian model using MQ RBF has cubic convergence, maintainsfilaments, produces small errors and preserves the functional relation, iscompetitive with the state of art advection models, at least up to n 57600.

Discussions Alternatively, sparsity of interpolation matrix (e.g.Gaussian, Gaspari–Cohn) could be exploitedwith a parallel sparse solver.1e-03loses the exponential convergence.singleparallel1e-05 RBF-FD allows for large n (Flyer et al. 2012), buttime s The size of the interpolation matrix increases2with n , e.g. 25GB for n 57600.LAPACKScaLAPACKLis1e-011e 01How to deal with a large interpolation matrix?10002000500010000number of nodes With this approach, one of the author(K. Ogawawara) is working on the parallelshallow water model.Wall-clock time (average of 10measurements) for solving linear systemsusing single (solid) and four (broken) coresmeasured on MacBook Pro (15-inch 2016,2.9GHz quad-core Intel Core i7)Enomoto 2020, DPRI annuals

Introduction radial basis functions (RBF) and spherical helix RBF enables the use of unstructured quasi-uniform nodes to solve hyperbolic differential equations on the sphere (Flyer and Wright 2007, 2009). Spherical helix nodes (Bauer 2000) are fast to generate without arbitrariness. Application of RBF interpolation in semi-Lagrangian advection is proposed.