ENDGame Dynamical Core - Earth System CoG€¦ · Iterative algorithm Do time-step loop: -...
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ENDGame Dynamical Core Thomas Melvin and Markus Gross Met Office, Exeter, Devon c Crown Copyright 2012 Met Office – p.1/13
Transcript of ENDGame Dynamical Core - Earth System CoG€¦ · Iterative algorithm Do time-step loop: -...
ENDGame Dynamical CoreThomas Melvin and Markus Gross
Met Office, Exeter, Devon
c© Crown Copyright 2012 Met Office – p.1/13
ENDGame...
• Even• Newer• Dynamics for• General• atmospheric• modelling of the• environment
...is the next dynamical core for the Unified Model (UM)• single model for:
• Weather forecasts (25Km → 1Km, hours-days)• Climate simulations (100Km 10-100 years)• Research tool > 10m
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Design Philosophy
• Met Office philosophy: use unapproximated equations; usenumerics to do "filtering"
• Fully compressible, nonhydrostatic models do not filter theacoustic modes• Have to be handled implicitly if wish to avoid severe
restriction on time step• Deep atmosphere models have twice as many Coriolis terms to
handle• Larger stencil if terms handled implicitly which stability
requires for two-time-level scheme• But, more accurate; more general (eg planetary atmospheres)• Do not want to introduce any computational modes• Does not need diffusion/filtering for stability
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Operational requirements (NWP)
Global 25km model (2010):• Forecast to: 7 days 3 hours• Timestep: = 10mins → 1026 time steps• Resolution N512L70 → 1024× 768× 70 = 55M grid points• To run in 60 minute slot, including data assimilation and output
36 (≈ 25) times bigger than running 5 years ago
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Features
• Evolution of the current New Dynamics dynamical core• Uses a latitude-longitude grid• Optional ellipsoidal geopotential approximation• Switches to allow hydrostatic/shallow atmosphere
approximations• Improved handling of Coriolis terms on staggered grid• (Almost) same grid as New Dynamics - C-grid + Charney Philips
but with v-at-poles• Fully implicit (iterative) scheme for all non-advective terms• Consistent SL scheme for all variables (+SLICE option)• Physics coupling:
• Parallel split for slow processes, sequential and iterative forfast processes
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Equations
Dru
Dt−
uv tanφ
r− 2Ω sinφv +
cpdθvdr cosφ
∂Π
∂λ= −
(uw
r+ 2Ωcosφw
)
+ Su
Drv
Dt−
u2 tanφ
r+ 2Ω sinφu+
cpdθvdr cosφ
∂Π
∂φ= −
(vw
r
)
+ Sv
Drw
Dt+ cpdθvd
∂Π
∂r+
∂Φ
∂r=
(
u2 + v2)
r+ 2Ωcosφu+ Sw
Dr
Dt
(
ρdr2 cosφ
)
+ ρdr2 cosφ
(
∂
∂λ
[
u
r cosφ
]
+∂
∂φ
[v
r
]
+∂w
∂r
)
= 0
DrθvdDt
= Sθ
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Spatial Discretisation
• C-grid horizontal staggering + Charney Philips vertical grid
k=0
k=1/2
k=1
k=3/2
k=2
i=1 i=3/2 i=2 i=5/2
w, θ
w, θ
w, θ
w, θ
v, π, ρv, π, ρ uu
• Uses height as vertical coordinate - terrain following at lowerboundary
• Discretise equations using second order centred finitedifferences
(
∂F
∂x
)
i−1/2
=Fi − Fi−1
∆x
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Temporal Discretisation
• Two-time-level Semi-Implicit Semi-Lagrangian discretisation
Dq
Dt= F (q) → qn+1
A − qnD = ∆tF (q)t
• Option to time offcentre terms
Gt= αGn+1 + (1− α)Gn
D
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Temporal Discretisation
• Use iterative approach to handle nonlinear terms
Dq
Dt= F (q) → q
(k+1)A − qnD = α∆t
[
L(q)(k+1) + (F (q)− L(q))(k)
]
+(1− α)∆tF (q)nD
• for centred scheme (α = 1/2) at convergence this reduces to asecond order accurate Crank-Nicolson scheme
• in practice α > 1/2 and scheme is not run to convergence (fixednumber of iterations used)
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Departure points
For semi-Lagrangian scheme need to solve trajectory equation DxDt = u
• ENDGame uses local cartesian departure point scheme
• xD = xA −∆t2
[
un+1A + un (xD)
]
• This gives a doubly implicit equation (depends on xD, un+1)• This is solved in an iterative manner
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Iterative algorithm
Do time-step loop:- given (η, θ, w, u, v, ρ, π)n at level n- compute slow physics terms, (radiation, gwd etc.)Do outer-loop iteration:
- compute SL departure points (xnD, ynD, znD) using (u, v, w)n
and latest estimate for (u, v, w)n+1
- interpolate time level n terms to departure points for required fields- compute predictors for timelevel n+ 1 fields for use by fast physics terms- compute fast physics increments, (convection, boundary layer etc.)Do inner-loop iteration:
- evaluate Coriolis and nonlinear terms- solve Helmholtz problem for πn+1
- update estimate for prognostic variables at timelevel n+ 1
EnddoEnddo
EnddoDCMIP c© Crown Copyright 2012 – p.11/13
Conservation
• ENDGame is not inherently conservative:• Solve advective form of equations• SL interpolation is inherently dissapitive• Charney-Phillips grid staggers tracers wrt. mass
• However, option to use SLICE -• Semi-Lagrangian Inherently Conserving and Efficient for
local conservation of mass and tracers• Alternativly can use a posteriori correction schemes• Need to use correctors to conserve other quantities - e.g energy
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Any Questions?
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