ENDGame Dynamical Core
Thomas 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
DCMIP
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Design Philosophy
• Met Office philosophy: use unapproximated equations; use
numerics to do "filtering"
• Fully compressible, nonhydrostatic models do not filter the
acoustic 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
DCMIP
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Operational requirements (NWP)
Global 25km model (2010):
• Forecast to: 7 days 3 hours
• Timestep: = 10mins → 1026 time steps
• Resolution N 512L70 → 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
DCMIP
c Crown Copyright 2012 – p.4/13
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 for
fast processes
DCMIP
c Crown Copyright 2012 – p.5/13
Equations
uw
cpd θvd ∂Π
Dr u uv tan φ
− 2Ω sin φv +
−
=−
+ 2Ω cos φw + S u
Dt
r
r cos φ ∂λ
r
vw Dr v
cpd θvd ∂Π
u2 tan φ
+ 2Ω sin φu +
+ Sv
−
=−
Dt
r
r cos φ ∂φ
r
2
u +v
∂Π ∂Φ
Dr w
+ cpd θvd
+
=
Dt
∂r
∂r
r
Dr
2
ρd r cos φ + ρd r 2 cos φ
Dt
DCMIP
2
+ 2Ω cos φu + S w
∂
u
∂ h v i ∂w
+
+
=0
∂λ r cos φ
∂φ r
∂r
Dr θvd
= Sθ
Dt
c Crown Copyright 2012 – p.6/13
Spatial Discretisation
• C-grid horizontal staggering + Charney Philips vertical grid
i=1
i=3/2
i=2
w, θ
w, θ
u
v, π, ρ
w, θ
i=5/2
u
v, π, ρ
w, θ
k=2
k=3/2
k=1
k=1/2
k=0
• Uses height as vertical coordinate - terrain following at lower
boundary
• Discretise equations using second order centred finite
differences
∂F
Fi − Fi−1
=
∂x i−1/2
∆x
DCMIP
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Temporal Discretisation
• Two-time-level Semi-Implicit Semi-Lagrangian discretisation
t
Dq
n+1
n
= F (q) → qA
− qD
= ∆tF (q)
Dt
• Option to time offcentre terms
t
G = αGn+1 + (1 − α)GnD
DCMIP
c Crown Copyright 2012 – p.8/13
Temporal Discretisation
• Use iterative approach to handle nonlinear terms
h
i
Dq
(k+1)
(k)
n
− qD
= F (q) → qA
= α∆t L(q)(k+1) + (F (q) − L(q))
Dt
+(1 − α)∆tF (q)nD
• for centred scheme (α = 1/2) at convergence this reduces to a
second order accurate Crank-Nicolson scheme
• in practice α > 1/2 and scheme is not run to convergence (fixed
number of iterations used)
DCMIP
c Crown Copyright 2012 – p.9/13
Departure points
For semi-Lagrangian scheme need to solve trajectory equation
Dx
Dt
=u
• ENDGame uses local cartesian departure point scheme
n
• xD = xA − ∆t un+1
+ u (xD )
A
2
• This gives a doubly implicit equation (depends on xD , un+1 )
• This is solved in an iterative manner
DCMIP
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Iterative algorithm
Do time-step loop:
- given (η̇, θ, w, u, v, ρ, π)n at level n
- compute slow physics terms, (radiation, gwd etc.)
Do outer-loop iteration:
n
n
- compute SL departure points (xnD , yD
, zD
) 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 π n+1
- update estimate for prognostic variables at timelevel n + 1
Enddo
Enddo
Enddo
DCMIP
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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
DCMIP
c Crown Copyright 2012 – p.12/13
Any Questions?
DCMIP
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