Earth s flattening : which impact for meteorology and climatology?

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1 Earth s flattening : which impact for meteorology and climatology? Pierre Bénard Météo-France CNRM/GMAP CNRS Aug. 2014, Montreal P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 1 / 27

2 INTRODUCTION Need of a model of the Earth To write governing equations, one needs (namely): - the shape of the Earth - the shape of Centrifugal forces - the shape of gravity field -... = Hence one needs to first define/choose the assumed shape of the gravity (or geopotential) field P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 2 / 27

3 INTRODUCTION First of all: terminology What is gravity and geopotential, etc., in Meteorology - Gravity : g = the apparent gravity felt by a resting object in the rotating frame (e.g. lying on the floor) - Vertical line = line along apparent gravity (plumb) - Geopotential : g = φ - Horizontal surface = to vertical = iso-geopotential (spirit level) = CONSEQUENCES: - Gravity contains Newtonian and Centrifugal components: g = g N +g C - All surfaces/lines have complicated shapes - Since g = φ, the intensity of gravity is inversely to the vertical separation of geopotentials, locally. P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 3 / 27

4 INTRODUCTION Ω Vertical line ( // to g ) g bigger g smaller Horizontal line ( iso geopotential ) P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 4 / 27

5 INTRODUCTION Currently the model is: Spherical Geopotential Approximation (SGA) - Earth s shape = spherical - All geopotential surfaces = spherical = - Vertical lines are straight radial lines - Vertical separation of geopotentials : uniform - Surface gravity : uniform (g 0 = , some average) Underlying hypothesis This is equivalent to neglect almost all centrifugal effects of earth s rotation P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 5 / 27

6 INTRODUCTION REALITY - Non spherical Earth: Departure best sphere : ±10 km, best ellipsoid : ±100 m Ellipticity ǫ = (a c)/a 0.3% geometrical errors z - Non uniform Gravity : Meridional variations Pole : g P 9.83, Eq. : g E 9.78 Departure mean gravity : ±0.2% gravity errors a 2. a c x P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 6 / 27

7 INTRODUCTION Why relaxing SGA? - Authors claim : systematic and cumulative errors impact on long and climate forecasts? when do they stop to be neglectible? - Also, modelling giant planets atmospheres :... for time being, still considered spherical!! - Thus, it is not useless to prepare and try to evaluate. P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 7 / 27

8 OUTLINE Introduction Consistent Formulations Geometric and gravity errors Estimation of errors in a shallow-water model Estimation of gravity errors in a 3D NWP model Conclusions P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 8 / 27

9 CONSISTENT FORMULATIONS Dynamical Consistency - Definition Definition : Formal respect of conservation principles for axial angular momentum, energy and potential vorticity Dynamically-consistent formulation of gov. equations lead to constraints (cannot write anything which looks good ). Example: bluntly setting g = g(ϕ) in SGA (to better mimic reality) is inconsistent. in SGA, g must be meridionally constant. P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 9 / 27

10 CONSISTENT FORMULATIONS Dynamical Consistency - Theory Big work ( ) mostly UKMO, and also French (LMD, M-F) teams, to better understand non-spherical dynamically-consistent systems Find Governing Equations systems which ensure dynamical consistency (with proofs...) Find appropriate curvilinear coordinate systems for a concretely tractable mathematical description and use. Derivation from Hamiltonian Mechanics formalism helps a lot. Shallow-water, 3D shallow-atmosphere, and 3D deep-atmosphere variants... if interested, look for Staniforth, White, Bénard (Q.J.R. Meteor. Soc), Dubos (JAS), mostly 2014, and refs therein P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 10 / 27

11 CONSISTENT FORMULATIONS Dynamical Consistency - EGA In practice: Ellipsoidal Geopotential Approximation (EGA) instead of SGA all geopotentials are assumed to be ellipsoids (no obvious need to go to actual irregular geoid geometry) EGA allows reasonably tractable equations P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 11 / 27

12 CONSISTENT FORMULATIONS - EGA why reasonably? For families of concentric geopotential ellipsoids respecting the correct ratio g Pole /g Eq = 9.83/9.78, vertical lines have no direct analytical expressions (recall: these lines are simple straight lines in SGA) Spheroidal coordinates (λ,ϕ,r) where r = r(φ) (preferred in order to avoid projection of g on horizontal components). orthogonal horizontal/vertical curvilinear coordinates system. Computation of metrics OK. BUT transformation from Cartesian coordinates non-analytic Requires either polynomial approximation or numerical solution. (not considered as a big problem, though) P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 12 / 27

13 ILLUSTRATION Shallow-Water system in geodetic latitude ϕ, no orog. ( ) du R dt a 2 cosϕ u +2Ω R v sinϕ+ a 2 cosϕ g(ϕ) H λ = 0 ( ) dv R dt + a 2 cosϕ u +2Ω R 3 usinϕ+ a 2 c 2 cosϕ ϕ [g(ϕ)h] = 0 ( dh R dt +H u a 2 cosϕ λ + R 3 v a 2 c 2 cosϕ ϕ Rtanϕ ) a 2 v = 0 where : d dt = t + Ru a 2 cosϕ λ + R3 v a 2 c 2 ϕ R = a 2 cos 2 ϕ+c 2 sin 2 ϕ c a P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 13 / 27

14 OUTLINE Introduction Consistent Formulations Geometric and gravity errors Estimation of errors in a shallow-water model Estimation of gravity errors in a 3D NWP model Conclusions P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 14 / 27

15 Geometric and gravity errors Geometric error - Length of Montreal parallel (45 N) on sphere: km - Length of ACTUAL Montreal parallel (45 N) : km - A pressure low travelling eastward at given V is too fast on sphere... Gravity error - g = 9.806m/s 2 on sphere - g = 9.78 to g = 9.83m/s 2 on actual earth - Propagation of a wave sensitive to gravity will be distorted on sphere... separate evaluation - gravity and geometry errors may be assessed separately by setting only correct geometry or correct gravity (Bénard, 2014, Q. J. Roy. Meteor. Soc.) P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 15 / 27

16 OUTLINE Introduction Consistent Formulations Geometric and gravity errors Estimation of errors in a shallow-water model Estimation of gravity errors in a 3D NWP model Conclusions P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 16 / 27

17 Estimation of errors in a shallow-water model Hauriwtz wave (e.g. Thuburn and Li, 2000, Q. J. R. Meteor. Soc.) - Quasi stationary analytical wave for SW - rotation (eastward), vacillations, becomes instable after 30 days P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 17 / 27

18 Estimation of errors in a shallow-water model Longitude Phase (compared to EGA reference) SGA (total error) gravity error geometry error EGA (ref) P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 18 / 27

19 Estimation of errors in a shallow-water model Standard Real case (from Williamson s test set) 10 days forecast, height field at 250 hpa P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 19 / 27

20 Estimation of errors in a shallow-water model Standard Real case (from Williamson s test set) 10 days forecast, height field at 250 hpa: P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 20 / 27

21 Estimation of errors in a shallow-water model Standard Real case (from Williamson s test set) - Longitude phase shift for some features - modified intensity - modified shape - global RMS after 10 days : about 30 m for Z gravity error still dominates P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 21 / 27

22 OUTLINE Introduction Consistent Formulations Geometric and gravity errors Estimation of errors in a shallow-water model Estimation of gravity errors in a 3D NWP model Conclusions P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 22 / 27

23 Estimation of errors in a 3D NWP model estimation of gravity error only: - spherical planet - shallow-atmosphere approximation - but meridionally-variable gravity ( m/s 2 ) Physically consistent (Bénard, 2014, Q. J. R. Meteor. Soc). Allows spherical geometry (at any level), and variable gravity. P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 23 / 27

24 Estimation of errors in a 3D NWP model Forecast scores against TEMP (Radio-soundings) sample : 2 months (April+May 2014) Contour=0.5m Puzzling results?!? weak significance after 72 h P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 24 / 27

25 Estimation of errors in a 3D NWP model Evaluation against verifying analysis shows: - scores/improvements significant only below 72 h differences/score improvements quite weak - at large forecast times, (larger) potential improvements are buried in (much larger) forecasts errors and become unsignificant P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 25 / 27

26 OUTLINE Introduction Consistent Formulations Geometric and gravity errors Estimation of errors in a shallow-water model Estimation of gravity errors in a 3D NWP model Conclusions P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 26 / 27

27 CONCLUSION Governing Equations are now obtained in various frameworks Allows an estimation of errors in SGA and benefits from EGA Gravity errors seems to slightly dominate 3D shows immediate (but weak) score improvement at short ranges At medium/long range (larger) improvements seem to be buried in (much larger) forecast errors and become not significant. Seemingly no panics with SGA, but reasonably need to prepare the set for future NWP in EGA... P. Bénard (Météo-France) Earth s flattening : which impact? 16/21 Aug 2014, Montreal 27 / 27

( u,v). For simplicity, the density is considered to be a constant, denoted by ρ 0

( u,v). For simplicity, the density is considered to be a constant, denoted by ρ 0 ! Revised Friday, April 19, 2013! 1 Inertial Stability and Instability David Randall Introduction Inertial stability and instability are relevant to the atmosphere and ocean, and also in other contexts