A Hierarchy of Models, from Planetary to Mesoscale, Within a Single

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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

• 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

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

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

The underlying equations...

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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

Red = Shallow-Atmosphere Approx (eliminate boxed terms) Blue = Hydrostatic Approx (eliminate boxed term) Green = Spherical Geopotential Approx (∂Φ/∂r = g (r) only)

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• 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

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• 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

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• 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

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• 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

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• 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

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Spheroidal Coordinates...

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• “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

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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

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SIMILAR ELLIPSES

Implementation...

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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

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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?

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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)

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• 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

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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

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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

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• 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

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• Basic derivation is the same

• “Trick”: use secondary meridional coordinate, ψ, geographic latitude (not orthogonal to vertical coordinate)

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

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Shallow-atmosphere...

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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

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• 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

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D iD

β φ φ A D _ 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)

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Departure points...

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• Departure points now take on greater signiﬁcance

• And they are also governed by a vector equation Dx = v Dt • So can we apply consistent approach (and avoid polar singularity issues)?

• Discrete vector form is ∆t x = x − (v + v ) D A 2 A D

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As before component form can be written as ∆t Mx = x − (v + Mv ) D A 2 A D T T Or, since xA = 0, 0, rA and xD = 0, 0, rD :

∆t M r = − [u + (M u + M v + M w )] 13 D 2 A 11 D 12 D 13 D ∆t M r = − [v + (M u + M v + M w )] 23 D 2 A 21 D 22 D 23 D ∆t M r = r − [w + (M u + M v + M w )] 33 D A 2 A 31 D 32 D 33 D

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© Crown copyright Met Office A Local Cartesian Transform Approach The result can be cast into a local Cartesian form: ∆t X = − (U + U ) DA 2 A DA ∆t Y = − (V + V ) DA 2 A DA ∆t Z = r − (W + W ) DA A 2 A DA where XDA ≡ MxD, and VDA ≡ MvD

are Departure point coordinates and velocities as seen in the Arrival-point Cartesian system

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From XDA,YDA,ZDA obtain departure point as:

−XDA tan (λA − λD) = ZDA cos φA − YDA sin φA

2 2 2 2 rD = XDA + YDA + ZDA

YDA cos φA + ZDA sin φA sin φD = rD

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Spheroidal case follows spherical case closely except:

T T xA = 0, 0, rA − dA 0, cos φA, sin φA and T T xD = 0, 0, rD − dD 0, cos φD, sin φD

jA kA A Polar axis

Y ψ Equator A C dA O R A

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• Applying M as before, but to modiﬁed position vectors, results in

XDA = M13rD YDA = M23rD + (dA − dD) cos φA XDA = M33rD + (dA − dD) sin φA

• Inversion is a little more complicated

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Form obtained by constraining departure point to remain on arrival sphere: ∆t x = x − (v + v ) + B (x + x ) D A 2 A D A D 1. Cartesian transformation as before XDA ≡ MxD, 2. but velocities transform using momentum shallow-atmosphere rotation matrix VDA ≡ QvD, 3. and 1 + M ∆t XH = − 33 VH + VH D 2 2 A D

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Single set of equations:

n+1 n (ui − α∆tΨi) = Mij uj + β∆tΨj D with similar ones for departure point equations Need only to specify: 1. ∇-operator in terms of the coordinate metric factors 2. Components of gravity vector (only vertical is non-zero)

3. Matrix elements Mij 4. Components of Earth’s rotation vector in chosen coordinate system

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Similar oblate spheroids (SOS) Spherical polars (ε ≡ common eccentricity) (ε = 0) (λ, ϕ, r) (λ, φ, r) λ = longitude λ = longitude Coords ϕ = meridional coordinate φ = latitude r = semi − major axis of spheroid r = radius of sphere r hλ = cos φ hr hλ = r cos φ Metric r sin φ cos φ hϕ = hφ = r factors hr sin ϕ cos ϕ 2 2 1/2 hr = 1 hr = 1 − ε sin φ ga a2 a2 Gravity g = 1/2 r g = ga r (1−2 sin2 φ) Geographic latitude φ = geographic latitude Notes = spherical polar ϕ → φ & r → r as ε → 0 latitude when ε = 0 Result is...

Identical numerical system encompassing hierarchy of model approximations:

1. Similar Oblate Spheroidal geopotential approximation 2. Deep-atmosphere Spherical geopotential approximation (no latitudinal variation of g) 3. Shallow-atmosphere approximation (non-Euclidean geometry) 4. Cartesian

Hydrostatic versions apply to (2)-(4)

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• Elements of approach form basis of deep-atmosphere, non- hydrostatic, spherical New Dynamics in MetUM

• Next version of dynamical core (ENDGame) will incorporate full functionality

• Allow clean assessment of impact of various approximations

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• Can relax Spherical Geopotential Approximation – Retain 2/3rds latitudinal variation of gravity

• Semi-Lagrangian method can be straightforwardly applied to non-spherical coordinate systems – Especially when use geographic latitude

• Eulerian component equations can be derived from the rotation matrix forms

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• Matrix view gives insight into the non-Euclidean nature of the shallow-atmosphere approximation – neglect of great circle curvature

• Therefore desirable to avoid shallow-atmosphere approximation: – avoid loss of some (signiﬁcant) Coriolis terms – and avoid non-Euclidean, space-distorting character

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Further details can be found in:

• Spheroidal coordinate systems for modelling global atmospheres White, Staniforth and Wood 2008 Q.J.R.Meteorol.Soc. 134 pp 261-270

• Rotation matrix treatment of vector equations in semi-Lagrangian models of the atmosphere I: Momentum equation Staniforth, White and Wood. 2010 Q.J.R.Meteorol.Soc. 136 pp 497-506

• Rotation matrix treatment of vector equations in semi-Lagrangian models of the atmosphere II: Kinematic equation Staniforth, White and Wood. 2010 Q.J.R.Meteorol.Soc. 136 pp 507-516