A hierarchy of models, from planetary to mesoscale, within a single switchable numerical framework Nigel Wood, Andy White and Andrew Staniforth Dynamics Research, Met Office © Crown copyright Met Office Outline of the talk • Overview of Met Office’s Unified Model • Spherical Geopotential Approximation • Relaxing that approximation • A method of implementation • Alternative view of Shallow-Atmosphere Approximation • Application to departure points © Crown copyright Met Office Met Office’s Unified Model Unified Model (UM) in that single model for: Operational forecasts at ¾Mesoscale (resolution approx. 12km → 4km → 1km) ¾Global scale (resolution approx. 25km) Global and regional climate predictions (resolution approx. 100km, run for 10-100 years) + Research mode (1km - 10m) and single column model 20 years old this year © Crown copyright Met Office Timescales & applications © Crown copyright Met Office © Crown copyright Met Office UKV Domain 4x4 1.5x4 4x4 4x1.5 1.5x1.5 4x1.5 4x4 1.5x4 4x4 Variable 744(622) x 928( 810) points zone © Crown copyright Met Office The ‘Morpeth Flood’, 06/09/2008 1.5 km L70 Prototype UKV From 15 UTC 05/09 12 km 0600 UTC © Crown copyright Met Office 2009: Clark et al, Morpeth Flood UKV 4-20hr forecast 1.5km gridlength UK radar network convection permitting model © Crown copyright Met Office © Crown copyright Met Office © Crown copyright Met Office The underlying equations... c Crown Copyright Met Office © Crown copyright Met Office Traditional Spherically Based Equations Dru uv tan φ cpdθv ∂Π uw u − − (2Ω sin φ) v + = − − (2Ω cos φ) w + S Dt r r cos φ ∂λ r 2 D v u tan φ cpdθv ∂Π vw r + + (2Ω sin φ) u + = − + Sv Dt r r ∂φ r D w ∂Π ∂Φ u2 + v2 r + c θ + = + (2Ω cos φ) u + Sw Dt pd v ∂r ∂r r Red = Shallow-Atmosphere Approx (eliminate boxed terms) Blue = Hydrostatic Approx (eliminate boxed term) Green = Spherical Geopotential Approx (∂Φ/∂r = g (r) only) c Crown Copyright Met Office © Crown copyright Met Office The Spherical Geopotential Approximation • All geopotentials - including the Earth’s surface - are represented by concentric spheres • Apparent gravity acts towards the centre of the Earth • To prevent spurious vorticity sources/sinks g cannot vary with latitude (White et al., 2005) • Nearly all numerical models of the global atmosphere are based on the SGA c Crown Copyright Met Office © Crown copyright Met Office Why relax the SGA? • Good to get Earth’s shape right(ish) in models! • Good to include the small ( ≈ 0.5%) increase of g from Equator to Poles without spurious vorticity sources/sinks • Most likely to be evident in long term climate simulations • Good to quantitatively test the accuracy of the spherical geopotential approximation • Important though that signal is not influenced by different numerics c Crown Copyright Met Office © Crown copyright Met Office The Shape of the Earth • The Earth is approximately an oblate spheroid: – Equatorial radius a = 6378 km – Polar radius c = 6357 km • Ellipticity small a − c 1 ε ≡ ≈ a 298 • But a − c = 21 km, more than twice the height of Everest c Crown Copyright Met Office © Crown copyright Met Office Geopotential near rotating fluid spheroid • Classical problem (Clairaut in 18th century) to account for the observed increase of g with latitude • Re-examined by White et al. (2008) to find geopotentials • Two small parameters: a − c 1 ε ≡ ≈ a 298 2 3 Ω a centrifugal force 1 m ≡ ∼ ≈ , γME Newtonian gravity 289 • To O (ε, m) geopotentials are spheroids " # a2 R3 x2 + z2 1 + (2 − m) + m = R2 R2 a3 c Crown Copyright Met Office © Crown copyright Met Office Variation of Gravity • Near to the surface R = a g 5 P = 1 + m − gEq 2 • Earth’s actual mass distribution m = (to within 3%): g 3 P = 1 + ⇒ 0.5% increase gEq 2 • For uniform mass distribution m = 4/5 (Newton) g P = 1 + gEq captures 2/3rds of the variation c Crown Copyright Met Office © Crown copyright Met Office Spheroidal Coordinates... c Crown Copyright Met Office © Crown copyright Met Office Confocal Oblate Spheroids • “Oblate spheroidal coordinates” of e.g. Gill (1982) and Gates (2004) are in fact confocal oblate spheroidal coordinates: x2 z2 + = d2 cosh2 η sinh2 η with d fixed, η variable • Match to Earth’s surface ⇒ gP 3 = tanh η0 ≈ 1 − =6 1 + ! gEq 2 • Separation along minor axis greater than along major axis c Crown Copyright Met Office © Crown copyright Met Office z d x CONFOCAL ELLIPSES and HYPERBOLAS Similar ellipses Equation is: x2 + (1 + µ) z2 = r2 Now µ fixed, r variable Match to Earth’s surface ⇒ µ = 2 and g P = (1 + µ)1/2 ≈ 1 + gEq • Sign right! • Captures 2/3rds actual variation for actual Earth • Exact for an Earth with uniform mass distribution c Crown Copyright Met Office © Crown copyright Met Office z x SIMILAR ELLIPSES Implementation... c Crown Copyright Met Office © Crown copyright Met Office Semi-Lagrangian scalar advection Consider DF = R Dt Integrate along trajectory: Z t+∆t DF F t+∆t (x) − F t (x − U∆t) Z t+∆t Rt+∆t (x) + Rt (x − U∆t) dt = = Rdt ≈ t Dt ∆t t 2 i.e. ∆t t+∆t ∆t t F − R = F + R 2 A 2 D c Crown Copyright Met Office © Crown copyright Met Office Semi-Lagrangian vector advection For the vector equation Dv = S Dt Integrate along trajectory to obtain, as before: ∆t t+∆t ∆t t v − S = v + S 2 A 2 D All well and good... But what of components? c Crown Copyright Met Office © Crown copyright Met Office The Problem of Curvilinear Coordinates... The familiar problem that Dv D D (i.v) Du i. ≡ i. (ui + vj + wk) =6 = Dt Dt Dt Dt translates into the SL equivalent that iA. (vD) =6 (iA.vA)D [ = uD] In fact vD = uDiD + vDjD + wDkD so that, e.g., iA. (vD) = uD (iA.iD) + vD (iA.jD) + wD (iA.kD) c Crown Copyright Met Office © Crown copyright Met Office The Matrix Formulation • Extending this to all three directions: u uD v = M vD w D wD where iA.iD iA.jD iA.kD M ≡ jA.iD jA.jD jA.kD kA.iD kA.jD kA.kD • M transforms – from: vector components in the departure-point frame – to: vectors in the arrival-point frame c Crown Copyright Met Office © Crown copyright Met Office The Consequence... Component form of momentum equations can be written as: ∆t t+∆t ∆t t v − S = M v + S 2 A 2 D where T XA ≡ XA,YA,ZA T XD ≡ XD,YD,ZD • No explicit metric terms • No singularity at the pole c Crown Copyright Met Office © Crown copyright Met Office M for Spherical Polar Coordinates M11 = cos∆λ, M12 = sinφD sin∆λ, M13 = − cosφD sin∆λ M21 = − sin φA sin ∆λ M22 = cos φA cos φD + sin φA sin φD cos ∆λ M23 = cos φA sin φD − sin φA cos φD cos ∆λ M31 = cos φA sin ∆λ M32 = sin φA cos φD − cos φA sin φD cos ∆λ M33 = sin φA sin φD + cos φA cos φD cos ∆λ c Crown Copyright Met Office © Crown copyright Met Office Where have the Metric Terms Gone? • SL form holds for finite displacements • Reduces to Eulerian form as displacement (∆λ ≡ λA − λD) and ∆t → 0: • In this limit: ∆λ uA sin ∆λ → ∆λ, → , uD → uA, vD → vA ∆t rA cos φA and so: uAvA tan φA M12vD = vD sin φD sin ∆λ → ∆t rA • Each off-diagonal element of M generates one metric term c Crown Copyright Met Office © Crown copyright Met Office Similar Oblate Spheroidal Form • Basic derivation is the same • “Trick”: use secondary meridional coordinate, ψ, geographic latitude (not orthogonal to vertical coordinate) Z jA kA A Polar axis Y ψ Equator A C dA O R A • Significantly M has exactly the same form but with φ → ψ • Contrast Eulerian formulation where (for COS at least) two extra metric terms appear c Crown Copyright Met Office © Crown copyright Met Office Shallow-atmosphere... c Crown Copyright Met Office © Crown copyright Met Office Shallow-Atmosphere Approximation D u uv tan φ cpdθv ∂Π uw r − − (2Ω sin φ) v + = − − (2Ω cos φ) w + Su Dt r r cos φ ∂λ r 2 D v u tan φ cpdθv ∂Π vw r − + (2Ω sin φ) u + = − + Sv Dt r r ∂φ r D w ∂Π ∂Φ u2 + v2 r + c θ + = + (2Ω cos φ) u + Sw Dt pd v ∂r ∂r r • Traditional approach: – Eliminate all boxed red terms – Replace r with a in all algebraic expressions c Crown Copyright Met Office © Crown copyright Met Office Shallow-atmosphere - a different perspective • M is a rotation matrix • Can be factored in infinite number of ways • One is of particular interest: 1. Rotate departure-point UVT about radial direction to line up i with Great Circle connecting rD with rA 2. Rotate UVT along Great Circle arc about new j direction 3. Rotate UVT about new radial direction to line up i direction c Crown Copyright Met Office © Crown copyright Met Office γ D N kA j A γ A j D i k A D Great circle arc A D iD β φ φ D A _ λ A λD A PARTICULAR DECOMPOSITION OF M Shallow-atmosphere - a different perspective • Result is M = ACD • Shallow-atmosphere approximation is obtained by C → I • M → AD ≡ Q: p q 0 Q = −q p 0 0 0 1 with M + M p = 11 22 1 + M33 and M − M q = 12 21 1 + M33 giving the same result as in Temperton et al. (2001) c Crown Copyright Met Office © Crown copyright Met Office Departure points..