Nonhydrostatic Modeling with NICAM

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Nonhydrostatic Modeling with NICAM Nonhydrostatic Modeling with NICAM HIROFUMI TOMITA Research Institute for Global Change, JAMSTEC Advanced Institute for Computational Science, RIKEN Contents Numerical method on NICAM Dynamical core Horizontal discretization • Icosahedral grid – Modified by spring dynamics( Tomita et al. 2001, 2002 JCP) Nonhydrostatic framework • Mass and total energy conservation – ( Satoh 2002, 2003 MWR ) Computational strategy Domain decomposition in parallel computer Example of computational performance Problems Current problems Future problems Summary NICAM project NICAM project ( ~2000 ) Aim : • Construction of GCRM for climate simulation – Suitable to the future computer environment » Massively parallel supercomputer » First target : Earth Simulator 1 40TFLOPS in peak 640 computer nodes Strategy: • Horizontal discretization : – Use grid of icosahedarl grid (quasi-homogeneous) – Anyway, 2nd accuracy over the globe! • Dynamics: – Non-hydrostatic » Mass & total energy conservation • Physics : – Import from MIROC ( one of Japan IPCC models ) except for microphysics. Horizontal discretization Grid arrangement Arakawa A-grid type Velocity, mass • triangular vertices Control volume • Connection of center of triangles – Hexagon – Pentagon at the icosahedral vertices Advantage Easy to implement no computational mode • Same number of grid points for vel. and mass Disadvantage Non-physical Glevel-3 grid & control volume 2-grid scale structure • E.g. bad geostrophic adjustment Horizontal differential operator e.g. Divergence 1. ector : given at VPi u()Pi 2. Interpolation of u at Qi u()()()αPPP+ β u+ γ u u()Q ≈ 0 i 1+ mod(i , 6 ) i α+ β + γ 3. Gauss theorem 6 1 u()()QQi +1 u + mod(i , 6 ) ∇ •u()P0 ≈ ∑bi •ni AP()0 i=1 2 2nd order accuracy? NO Æ Allocation points is not gravitational center ( default grid ) Modified Icosahedral Grid (1) Reconstruction of grid by spring dynamics To reduce the grid-noise 1. STD-grid : Generated by the recursive grid division. 2. SPRING DYNAMICS : Connection of gridpoints by springs 6 dw0 ∑k()− die d i −α w0 M = i=1 dt dr w = 0 0 dt SPR-grid: Solve the spring dynamics Æ The system calms down to the static balance Modified Icosahedral Grid (2) Gravitational-Centered Relocation To make the accuracy of numerical operators higher 1. SPR-grid: Generated by the spring dynamics. Æ ● 2. CV: Defined by connecting the GC of triangle elements. Æ ▲ 3. SPR-GC-grid: The grid points are moved to the GC of CV. Æ ● Æ The 2nd order accuracy of numerical operator is perfectly guaranteed at all of grid points. Improvement of accuracy of operator est functionT Heikes & Randall(1995) Error norm Global error 1 / 2 I(,)(,)λ() x θ− x λ2 θ l() x = { [ T ]} 2 2 1 / 2 I{}[] x(,)Tλ θ Local error maxallλ,λx θ ( θ ,− )T x λ ( θ , ) l∞() x = maxallλ, θx(,)λ T θ STD-grid SPR-GC-grid L_2 norm Almost 2nd-order(●) Perfect 2nd-order(○) l_inf norm 2nd order(Not ▲) Perfect 2nd-order(△) Nonhydrostatic framework Design of our non-hydrostatic modeling Governing equation Full compressible system • Acoustic wave Æ Planetary wave Flux form • Finite Volume Method • Conservation of mass and energy Deep atmosphere • Including all metrics terms and Coriolis terms Solver Split explicit method • Slow mode : Large time step • Fast mode : small time step HEVI ( Horizontal Explicit & Vertical Implicit ) • 1D-Helmholtz equation Governing Equations Å MODE STAFL.H.S. : Æ Å MODE R.H.S. : SLOWÆ ∂ Vh ∂ ⎛ W 3 Vh ⎞ +R ∇h ⋅ +⎜ +G ⋅ ⎟ = 0 ⎜ 1 / 2 ⎟ (1) ∂t γ∂ ξ⎝ G γ ⎠ ∂ P ∂ ⎛ 3 P ⎞ +Vh ∇ h + ⎜G ⎟ =ADVh + Coriolis F (2) ∂t γ∂ ξ⎝ ⎠ γ ∂ ∂ ⎛ P ⎞ W + γ 2 ⎜ +⎟ Rg =ADV + F (3) ⎜ 1 / 2⎟ 2 z Coriollis ∂t ∂ξ⎝ G γ ⎠ ∂ ⎛ Vh ⎞ ∂ ⎡ ⎛ W 3 Vh ⎞⎤ Eh ⎜ h ⎟ h⎜ G ⎟ + ∇⎜ ⋅ ⎟ + ⎢ ⎜ 1 /+ 2 ⋅ ⎟⎥ ∂t ⎝ γ⎠ ∂ ξ⎣ ⎝ G γ ⎠⎦ Vh ⎡ P ∂ ⎛ PW3 ⎞⎤ 2 ∂ ⎛ P ⎞ − ⋅ ∇h +⎜G ⎟ − γ ⎜ +⎟Wg = Q ⎢ ⎜ ⎟⎥ ⎜ 1 / 2⎟ 2 heat (4) R ⎣ γ∂ ξ⎝ ⎠ γ⎦ R ∂ξ⎝ G γ ⎠ Prognostic variables Metrics • density RG=γ2 1 / 2ρ 1 / 2 ⎛ ∂z ⎞ G = ⎜ ⎟ 2 1 / 2 ⎝ ∂ξ ⎠ • horizontal momentum Vh =γG ρvh x, y 3 • vertical momentum W=γ2 G 1 / 2ρ w G=() ∇hξ z 2 1 / 2 H() z− z • internal energy E=γ G ρ in e ξ = s H− s z Temporal Scheme ( in the case of RK2) Assumption : the variable at t=A is known. ¾ Obtain the slow mode tendency S(A). HEVI solver 1. 1st step : Integration of the prog. var. by using S(A) from A to B. ¾ Obtain the tentative values at t=B. ¾ Obtain the slow mode tendency S(B) at t=B. 2. 2nd step : Returning to A, Integration of the prg.var. from A to C by using S(B). Æ Obtain the variables at t=C Small Step Integration step integration, there In small are 3 steps: 1. Horizontal Explicit Step ¾ of horizontal momentumUpdate HEVI 2. Implicit StepVertical ¾ of vertical Updates momentum and density. 3. Energy Correction Step ¾ Update of energy Horizontal Explicit Step • explicitly momentum is updated byHorizontal ⎡ t+ n Δτ [t , or+ t Δ / 2t ⎤ ] +t( n + 1 )τ Δ t+ n Δτ ⎛ P ∂ ⎛ 3 P ⎞⎞ ⎛ ∂Vh ⎞ Vh =Vh + Δτ ⎢⎜ −h ∇⎜G − ⎟⎟ + ⎜ ⎟ ⎥ ⎜ γ∂ ξ⎜ ⎟ γ⎟ ∂t ⎣⎢⎝ ⎝ ⎠⎠ ⎝ slow⎠ mode⎦⎥ Fast mode Slow mode : given Small Step Integration (2) Step Implicit Vertical • The equations of R,W, and E can be written as: RR+t( n + 1 )τ Δ− t+ n Δτ ∂ ⎛W+t( n + 1 )τ Δ⎞ + ⎜ ⎟ = G ⎜ 1 / 2 ⎟ R (6) Δτ ∂ξ ⎝ G ⎠ W+t( n + 1 )τ Δ−Wt+ n Δτ ∂ ⎛ P+t( n + 1 )τ Δ⎞ + γ 2 ⎜ ⎟R+ +t( n + 1 )τ Δg= G (7) ⎜ 1 / 2 2⎟ z Δτ ∂ξ⎝ G γ ⎠ PP+t( n + 1 )τ Δ− t+ n Δτ ∂ ⎡⎛W+t( n + 1 )τ Δ⎞ ⎤ R R + ⎜ ⎟c2t+ n Δτ + dW+t( n + 1 )τ Δ~ g = d G (8) ⎢⎜ 1 / 2 ⎟ s ⎥ E Δτ ∂ξ ⎣⎝ G ⎠ ⎦ CV CV – Eqs.(6), (7), Coupling and (8), we can obtain the 1D-Helmholtz equation for W: W+t( n + 1 )τ Δ ∂ ⎡ 1 ∂ ⎛ W+t( n + 1 )τ Δ⎞⎤ ⎡ ∂ ⎛ R W+t( n + 1 )τ Δ⎞⎤ g∂ ⎛ W+t( n + 1 )τ Δ⎞ − ⎜Δτ2c t+ 2 n Δτ ⎟ − ⎜Δτ 2 d g~ ⎟+ Δτ 2 ⎜ ⎟ 2 ⎢1 / 2 2 ⎜ s 1 / 2 ⎟⎥ ⎢ ⎜ 1 / 2 2 ⎟⎥ 2 ⎜ 1 / 2 ⎟ γ ξ∂ ⎣G γ∂ ⎝ ξ G ⎠⎦ ⎣∂ξ ⎝ CV G γ ⎠⎦ γ∂ ξ⎝ G ⎠ = R.H.S.(source term) (9) • Eq.(9) Æ W • Eq.(6) Æ R • Eq.(8) Æ E Small Step Integration (3) Energy Correction Step (Total eng.)=(Internal eng.)+(Kinetic eng.)+(Potential eng.) • consider the equation of total energyWe ∂ ⎡ Vh ⎤ ∂ ⎡ ⎛ W 3 Vh ⎞⎤ +E ∇ ⋅h h + k + Φ+ +h + k Φ⎜ G +⎟ ⋅ = 0 total ⎢()⎥ ⎢()⎜ 1 / 2 ⎟⎥ (10) ∂t ⎣ γ⎦ ∂ ξ⎣ ⎝ G γ ⎠⎦ where 2 1 / 2 Etotal = ργG(+ ein + k ) Φ • Additionally, Eq.(10) is solved as +t( n + 1 )τ Δ t+ n Δτ Etotal = Etotal +t( n + 1 )τ Δ ⎡ ⎡ Vh ⎤ ∂ ⎡ ⎛ W 3 Vh ⎞⎤⎤ − Δτ ∇h ⋅h + k ++ Φ +h + k Φ⎜ G +⎟ ⋅ ⎢ ⎢()⎥ ⎢()⎜ 1 / 2 ⎟⎥⎥ ⎣⎢ ⎣ γ⎦ ∂ ξ⎣ ⎝ G γ ⎠⎦⎦⎥ – Written by a flux form. • The kinetic energy and potential energy: Æ known by previous step. • Recalculate the internal energy: +t( n + 1 )τ Δ +t( + n 1 Δ )τ(t 1 )+ n +τ 2 Δ 1 / 2+t n + (τ Δ 1 ) E =Etotal ρ − G γ ( k + Φ) Large Step Integration Large step tendecy has 2 main parts: 1. Coliolis term ¾ Formulated straightforward. 2. Advection term • We should take some care to this term because of curvature of the earth Advection of momentum the Use of a catesian coordinate in which the origin is center of the earth. The advection of term Vh and W is calculated as follows. 1. Construct the 3-dimensional momentum V using Vh and W. 2. this vector as 3 components as (Express V1,V2,V3) in a fixed coordinate. ¾ These components are scalars. 3. Obtain a vector which contains 3 divergences as its components. Æ 1 / 2 2 ∇ ⋅()vVVV1 ,, ∇ v ⋅2 3 ∇ v ⋅ wherevi= V i ( γ G / ρ) 4. Split again to a horizontal vector and a vertial components. Æ ADVh , ADVz Computational strategy and performance Computational strategy(1) (0) region division level 0 (1) region division level 1 Domain decomposition 1. By connecting two neighboring icosahedral triangles, 10 rectangles are constructed. (rlevel-0) 2. For each of rectangles, 4 sub-rectangles are generated by connecting (2) region division level 2 (3) region division level 3 the diagonal mid-points. (rlevel-1) 3. The process is repeated. (rlevel-n) Computational strategy(2) Load balancing Example ( rlevel-1 ) # of region : 40 # of process : 10 Situation: Polar region: Less computation Equatorial region: much computation Each process manage same color regions Cover from the polar region and equatorial region. Avoid the load imbalance Computational strategy(3) Vectorization Structure in one region Icosahedral grid Æ Unstructured grid? Treatment as structured grid Æ Fortran 2D array Æ vectorized efficiently! 2D array Æ 1D array Higher vector operation length Computational Performance (1) Computational performance Depend on… • Computer architecture, degree of code tuning….. Performance on the old Earth Simulator Earth Simulator • Massively parallel super-computer based on NEC SX-6 architecture. – 640 computational nodes. – 8 vector-processors in each of nodes. – Peak performance of 1CPU : 8GFLOPS – Total peak performance: 8X8X640 = 40TFLOPS – Crossbar network Target simulations for the measurement • 1 day simulation of Held & Suarez dynamical core experiment Computational Performance (2) Scalability of our model (NICAM) --- strong scaling Configuration • Horizontal resolution : glevel-8 •30km resolution • Vertical layers : 100 Fixed • Computer nodes : increases from 10 to 80. Results Green : ideal speed-up line Red : actual speed-up line Computational Performance (3) Performance against the horizontal resolution --- weak scaling The elapse time should increase by a factor of 2. glevel Number of PNs Elapse Time Average Time GFLOPS (grid intv.) (peak performance) [sec] [msec] (ratio to peak[%]) 6 5 48.6 169 140 (120km) (320GFLOPS) (43.8) 7 20 97.4 169 558 (60km) (1280GFLOPS) (43.6) 8 80 195 169 2229 (30km) (5120GFLOPS) (43.5) 9 320 390 169 8916 (15km) (20480GFLOPS) Configuration As the grid level increases, # of gridpoints : X 4 Results # of CPUs : X 4 The elapse time increases by a factor of 2.
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