A stable treatment of conservative thermodynamic variables for semi-implicit semi-lagrangian dynamical cores Kevin Viner Naval Research Laboratory, Monterey, CA September 26, 2012 Kevin Viner (NRL) PDE s on the sphere 2012 1 / 21
Introduction Navy Global Environmental Model (NAVGEM) 3TL SL-SI hydrostatic spectral transform horizontal, finite-difference vertical with Lorenz staggering hybrid pressure coordinates {u, v, θ v, p s } prognostic variables Currently T359L50 with t = 360s Requires heavy implicit biasing (α = 0.8) for stability...why? Kevin Viner (NRL) PDE s on the sphere 2012 2 / 21
SI scheme in NAVGEM Introduction Robert [3] semi-implicit scheme with {p s, T } ref = {800hPa, 300K } du dt dv = fu cos2 (Φ) dt a 2 dθ v = tt {τd} dt dp 1 s = [D p dt η + η = fv 1 φ a 2 λ c pθ v π a 2 λ tt a 2 {γ θ v φ µ c pθ v cos 2 (Φ) 0 tt () = [α() + () 0 + (1 α)() ] a 2 λ + δ p s λ } ( η p η )]dη tt{νd} π µ tt a 2 {γ θ v µ + δ p s µ } where {γ, δ, τ, ν} are prebuilt reference matrices. Performing a spectral transform allows reduction to a single Helmholtz equation in terms of D which can be solved through an eigenvalue decomposition of the mass matrix Γ = (γτ + δν). Kevin Viner (NRL) PDE s on the sphere 2012 3 / 21
Introduction Stability Conditions for SL advection SL removes the limiting CFL condition t < α x u associated with Eulerian advection The new stability condition is t < α x u, known as the Lipschitz or deformation CFL condition Shown with traditional linear backward trajectories from a regular grid... x D x D Crossing trajectories represent a non-unique solution...not allowed! i-1 i i+1 i+2 x xa A Kevin Viner (NRL) PDE s on the sphere 2012 4 / 21
semi-implicit scheme Introduction Fast mode terms are treated implicitly in Eulerian models to relax the CFL condition In SL models this has the effect relaxing the deformation CFL condition Generally, for a 3TL scheme (extension to 2TL is straightforward) for some prognostic variable ψ: EUL ψ t SL dψ dt = S 0 + F 0 + tt F L = F 0 M + ttf L What happens if F and S are not separable? Kevin Viner (NRL) PDE s on the sphere 2012 5 / 21
Problem SI-SL using potential temperature (or other conservative thermodynamic variable) Consider the "energy conversion term" in the thermodynamic equation for temperature (e.g., ECMWF s IFS) and potential temperature (e.g., NAVGEM, UKMO s UM) or similarly for enthalpy (e.g., NCEP s GFS-SL, Juang 2011 [1]) in a model utilizing a hybrid pressure coordinate η: F N = κωt p F N = η θ η = S vert The time discretization for the potential temperature case is then: dθ dt = tt F L. However, it has been noted by "The Joy of U.M. 6.3" (Staniforth et. al. 2006 [4]) that the lack of a semi-implicitly treated energy conversion term on the RHS results in an unstable scheme. Kevin Viner (NRL) PDE s on the sphere 2012 6 / 21
Solutions Potential Remedies Circumvent the problem with reduced accuracy smoothing: Decentering (α > 0.5) increased diffusion increased time filter Return the missing nonlinear term to RHS: Vertically non-interpolating scheme for θ (see UKMO UM New Dynamics documentation) SL advection of perturbation θ (see UKMO UM ENDGame documentation, Wood et. al 2010 [2]) Kevin Viner (NRL) PDE s on the sphere 2012 7 / 21
Solutions Vertically non-interpolating scheme for θ Project the departure point onto the nearest vertical level, treat the remainder in an Eulerian semi-implicit fashion on RHS. x A k ' x D k+1 x Stable in linear stability analysis. However, unstable in practice. UKMO forced to result to heavy decentering to maintain stability (α 1) D Kevin Viner (NRL) PDE s on the sphere 2012 8 / 21
Solutions SL advection of perturbation θ as done in ENDGame The predictor-corrector and iterative Helmholtz solver applied in the UM are quite useful for this particular problem: Use solution from previous time step as basic state to keep θ vd small Eulerian advection of basic state is treated semi-implicitly by the predictor-corrector (No CFL worries) Iterative Helmholtz solver can be used to reduce the dependence of the discretization on the choice of basic state d(θ vd θvd 0) + v θ 0 dt vd = 0 Kevin Viner (NRL) PDE s on the sphere 2012 9 / 21
Solutions SL advection of perturbation θ as done is NAVGEM Since our SI scheme forces us to use θ v throughout our coupled equation set, in addition to gaining stability we have the opportunity to: reduce error in vertical integration of geopotential gradient terms treat variation in θ v due to terrain in an Eulerian fashion to eliminate terrain resonance Borrow idea from Wu et al 2008 [5] and most limited area models to define and extract hydrostatically balanced basic state a single vertical profile is used which represents a nonlinear regression to the U.S. standard atmosphere θ 0 = θ 0 (p) Kevin Viner (NRL) PDE s on the sphere 2012 10 / 21
Solutions NAVGEM implementation, continued du dt dv dt = fv 1 a 2 φ λ c pθv a 2 π λ = fu cos2 (Φ) φ a 2 µ c pθ v cos 2 (Φ) π µ a 2 dθ v dt = ω dθ 0 dp 1 dp s = [D p dt 0 η + p ( η η η )]dη dq = 0 dt φ π = c pθ v η p η = p t 1 [D p η + η ( U p cos 2 (Φ) λ + V p µ )]dη 0 Kevin Viner (NRL) PDE s on the sphere 2012 11 / 21
Experimental Setup Experimental setup Perform a few experiments for with and without modifications described: Single one month cold start Varied time step Cycled DA tests for 7 weeks (snapshots) SL details and physics may vary across experiments, but only difference in settings between compared results are modification discussed herein. Kevin Viner (NRL) PDE s on the sphere 2012 12 / 21
Results Stratospheric Warming: monthly mean zonal winds Control Perturbation Analysis c.f., GMAO MERRA Analysis for January 2011 Thanks to Y.-J. Kim Kevin Viner (NRL) PDE s on the sphere 2012 13 / 21
Results Stability: T239L60 Control Case with α = 0.80 Kevin Viner (NRL) PDE s on the sphere 2012 14 / 21
Results Stability: T239L60 Perturbation θ with α = 0.60 Kevin Viner (NRL) PDE s on the sphere 2012 15 / 21
Results 2012070400+120hrs pressure vertical velocity at 0.2mb: Control case (with limiters) Kevin Viner (NRL) PDE s on the sphere 2012 16 / 21
Results 2012070400+120hrs pressure vertical velocity at 0.2mb: Perturbation case (α = 0.55, no limiters) Kevin Viner (NRL) PDE s on the sphere 2012 17 / 21
Results Advantages/Limitations Advantages: Semi-implicit scheme is stable and effective Allows larger time steps without issue Reduced decentering retains high frequency motions that drive stratospheric/mesospheric circulations Better agreement with satellite observations at upper levels Limitations: choice of basic state is subjective and allows large variations in θ v Use without predictor-corrector leaves behind explicit nonlinear vertical advection term requiring weak decentering to avoid CFL violations over steep terrain Kevin Viner (NRL) PDE s on the sphere 2012 18 / 21
Results Influence of Basic State?...structure of θ Kevin Viner (NRL) PDE s on the sphere 2012 19 / 21
Results Influence of Basic State?...(θ pert θ ctrl ) 2012070400 Kevin Viner (NRL) PDE s on the sphere 2012 20 / 21
Summary Conclusions Application of a standard SISL scheme to a conservative thermodynamic variable leads to an unstable centered scheme The only way to realize stability using a centered scheme is to extract the vertical advection of said variable from the SL trajectory and treat it in an Eulerian manner Results show significant improvements above 10mb where decentering was most detrimental Future work will look to create more uniform θ v and attempt a more implicit treatment once we move to a two-time-level scheme Kevin Viner (NRL) PDE s on the sphere 2012 21 / 21
Summary A. White J. Thuburn T. Allen T. Davies M. Diamantakis M. Dubal M. Gross T. Melvin C. Smith M. Zerroukat N. Wood, A. Staniforth. Endgame formulation v3.01. Technical report, U.K.M.O. A. Robert. The integration of a spectral model of the atmosphere by the implicit method, journal =. A. Staniforth, A. White, N. Wood, J. Thuburn, M. Zerroukat, E. Cordero, and T. Davies. The joy of u.m. 6.3- model formulation, unified model documentation paper no. 15. Technical report, U.K.M.O. Tongwen Wu, Rucong Yu, and Fang Zhang. A modified framework for the atmospheric spectral model and its application. Kevin Viner (NRL) PDE s on the sphere 2012 21 / 21
Summary J. Atmos. Sci., 65:2235 2253, 2008. Kevin Viner (NRL) PDE s on the sphere 2012 21 / 21