the semi-lagrangian technique: current status and future ... · a very stable formulation...
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![Page 1: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/1.jpg)
The semi-Lagrangian technique: current status andfuture developments
Michail Diamantakis
ECMWF
4 September 2013
(thanks to: Adrian Simmons, Niels Borman and Richard Forbes)
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 1 / 42
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Outline
1 Semi-Lagrangian technique and history
2 Semi-Lagrangian numerics in the upper atmosphereTrajectory equation and numerical noiseExtratropical tropopause cold bias
3 The mass conservation .... headache
4 Concluding
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 2 / 42
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Outline
1 Semi-Lagrangian technique and history
2 Semi-Lagrangian numerics in the upper atmosphereTrajectory equation and numerical noiseExtratropical tropopause cold bias
3 The mass conservation .... headache
4 Concluding
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 3 / 42
![Page 4: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/4.jpg)
Semi-Lagrangian modeling in NWP
Semi-Lagrangian: an established numerical technique for solvingthe atmospheric transport equations.
Many global and regional weather & climate prediction models use asemi-Lagrangian semi-implicit numerical formulation (SLSI):
ARPEGE(MeteoFrance)/IFS(ECMWF)/ALADIN, UM(UKMO),HIRLAM, SL-AV(Russia)
GEM(Environment Canada), GFS(NCEP)
GSM(JMA)
HADGEM(UK), C-CAM(Australia) ...
What is the reason for this?
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 4 / 42
![Page 5: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/5.jpg)
Semi-Lagrangian modeling in NWP
Semi-Lagrangian: an established numerical technique for solvingthe atmospheric transport equations.
Many global and regional weather & climate prediction models use asemi-Lagrangian semi-implicit numerical formulation (SLSI):
ARPEGE(MeteoFrance)/IFS(ECMWF)/ALADIN, UM(UKMO),HIRLAM, SL-AV(Russia)
GEM(Environment Canada), GFS(NCEP)
GSM(JMA)
HADGEM(UK), C-CAM(Australia) ...
What is the reason for this?
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 4 / 42
![Page 6: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/6.jpg)
A very stable formulation
Semi-Lagrangian advection:
Unconditionally stable
Good phase speeds with little numerical dispersion
Simplicity: the nonlinear advective terms are “absorbed” by theLagrangian derivative operator and essentially the advection problemis turned to an interpolation one!
Combines virtues of Lagrangian and Eulerian approach
Semi-implicit timestepping:
Wide (unconditional) stability for fast forcing terms: gravity +acoustic waves (in non-hydrostatic models)
SL+SI: ∆t determined by desired accuracy and not limited by stability
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 5 / 42
![Page 7: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/7.jpg)
A very stable formulation
Semi-Lagrangian advection:
Unconditionally stable
Good phase speeds with little numerical dispersion
Simplicity: the nonlinear advective terms are “absorbed” by theLagrangian derivative operator and essentially the advection problemis turned to an interpolation one!
Combines virtues of Lagrangian and Eulerian approach
Semi-implicit timestepping:
Wide (unconditional) stability for fast forcing terms: gravity +acoustic waves (in non-hydrostatic models)
SL+SI: ∆t determined by desired accuracy and not limited by stability
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 5 / 42
![Page 8: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/8.jpg)
A very stable formulation
Semi-Lagrangian advection:
Unconditionally stable
Good phase speeds with little numerical dispersion
Simplicity: the nonlinear advective terms are “absorbed” by theLagrangian derivative operator and essentially the advection problemis turned to an interpolation one!
Combines virtues of Lagrangian and Eulerian approach
Semi-implicit timestepping:
Wide (unconditional) stability for fast forcing terms: gravity +acoustic waves (in non-hydrostatic models)
SL+SI: ∆t determined by desired accuracy and not limited by stability
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 5 / 42
![Page 9: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/9.jpg)
A very stable formulation
Semi-Lagrangian advection:
Unconditionally stable
Good phase speeds with little numerical dispersion
Simplicity: the nonlinear advective terms are “absorbed” by theLagrangian derivative operator and essentially the advection problemis turned to an interpolation one!
Combines virtues of Lagrangian and Eulerian approach
Semi-implicit timestepping:
Wide (unconditional) stability for fast forcing terms: gravity +acoustic waves (in non-hydrostatic models)
SL+SI: ∆t determined by desired accuracy and not limited by stability
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 5 / 42
![Page 10: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/10.jpg)
A very stable formulation
Semi-Lagrangian advection:
Unconditionally stable
Good phase speeds with little numerical dispersion
Simplicity: the nonlinear advective terms are “absorbed” by theLagrangian derivative operator and essentially the advection problemis turned to an interpolation one!
Combines virtues of Lagrangian and Eulerian approach
Semi-implicit timestepping:
Wide (unconditional) stability for fast forcing terms: gravity +acoustic waves (in non-hydrostatic models)
SL+SI: ∆t determined by desired accuracy and not limited by stability
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 5 / 42
![Page 11: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/11.jpg)
A very stable formulation
Semi-Lagrangian advection:
Unconditionally stable
Good phase speeds with little numerical dispersion
Simplicity: the nonlinear advective terms are “absorbed” by theLagrangian derivative operator and essentially the advection problemis turned to an interpolation one!
Combines virtues of Lagrangian and Eulerian approach
Semi-implicit timestepping:
Wide (unconditional) stability for fast forcing terms: gravity +acoustic waves (in non-hydrostatic models)
SL+SI: ∆t determined by desired accuracy and not limited by stability
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 5 / 42
![Page 12: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/12.jpg)
A very stable formulation
Semi-Lagrangian advection:
Unconditionally stable
Good phase speeds with little numerical dispersion
Simplicity: the nonlinear advective terms are “absorbed” by theLagrangian derivative operator and essentially the advection problemis turned to an interpolation one!
Combines virtues of Lagrangian and Eulerian approach
Semi-implicit timestepping:
Wide (unconditional) stability for fast forcing terms: gravity +acoustic waves (in non-hydrostatic models)
SL+SI: ∆t determined by desired accuracy and not limited by stability
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 5 / 42
![Page 13: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/13.jpg)
Some History
1959 Wiin-Nielsen combines Lagrangian and Eulerian approach ina barotropic and a simple baroclinic model:
there is a fixed (Eulerian) grid to keep parcels evenlydistributed and to accurately calculate spatial derivativesair parcels arrive always at fixed grid points at the endof each timestep
1960s Robert develops semi-implicit time integration for HPE NWPmodels
1984 Robert & Ritchie combine semi-implicit and semi-Lagrangianmethod on a multilevel HPE model using 90 min timestep!
1991 IFS becomes a SLSI spectral model which allows a bigincrease in horizontal resolution
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 6 / 42
![Page 14: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/14.jpg)
Some History
1959 Wiin-Nielsen combines Lagrangian and Eulerian approach ina barotropic and a simple baroclinic model:
there is a fixed (Eulerian) grid to keep parcels evenlydistributed and to accurately calculate spatial derivativesair parcels arrive always at fixed grid points at the endof each timestep
1960s Robert develops semi-implicit time integration for HPE NWPmodels
1984 Robert & Ritchie combine semi-implicit and semi-Lagrangianmethod on a multilevel HPE model using 90 min timestep!
1991 IFS becomes a SLSI spectral model which allows a bigincrease in horizontal resolution
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 6 / 42
![Page 15: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/15.jpg)
Some History
1959 Wiin-Nielsen combines Lagrangian and Eulerian approach ina barotropic and a simple baroclinic model:
there is a fixed (Eulerian) grid to keep parcels evenlydistributed and to accurately calculate spatial derivativesair parcels arrive always at fixed grid points at the endof each timestep
1960s Robert develops semi-implicit time integration for HPE NWPmodels
1984 Robert & Ritchie combine semi-implicit and semi-Lagrangianmethod on a multilevel HPE model using 90 min timestep!
1991 IFS becomes a SLSI spectral model which allows a bigincrease in horizontal resolution
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 6 / 42
![Page 16: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/16.jpg)
Some History
1959 Wiin-Nielsen combines Lagrangian and Eulerian approach ina barotropic and a simple baroclinic model:
there is a fixed (Eulerian) grid to keep parcels evenlydistributed and to accurately calculate spatial derivativesair parcels arrive always at fixed grid points at the endof each timestep
1960s Robert develops semi-implicit time integration for HPE NWPmodels
1984 Robert & Ritchie combine semi-implicit and semi-Lagrangianmethod on a multilevel HPE model using 90 min timestep!
1991 IFS becomes a SLSI spectral model which allows a bigincrease in horizontal resolution
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 6 / 42
![Page 17: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/17.jpg)
Some History
1959 Wiin-Nielsen combines Lagrangian and Eulerian approach ina barotropic and a simple baroclinic model:
there is a fixed (Eulerian) grid to keep parcels evenlydistributed and to accurately calculate spatial derivativesair parcels arrive always at fixed grid points at the endof each timestep
1960s Robert develops semi-implicit time integration for HPE NWPmodels
1984 Robert & Ritchie combine semi-implicit and semi-Lagrangianmethod on a multilevel HPE model using 90 min timestep!
1991 IFS becomes a SLSI spectral model which allows a bigincrease in horizontal resolution
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 6 / 42
![Page 18: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/18.jpg)
Some History
1959 Wiin-Nielsen combines Lagrangian and Eulerian approach ina barotropic and a simple baroclinic model:
there is a fixed (Eulerian) grid to keep parcels evenlydistributed and to accurately calculate spatial derivativesair parcels arrive always at fixed grid points at the endof each timestep
1960s Robert develops semi-implicit time integration for HPE NWPmodels
1984 Robert & Ritchie combine semi-implicit and semi-Lagrangianmethod on a multilevel HPE model using 90 min timestep!
1991 IFS becomes a SLSI spectral model which allows a bigincrease in horizontal resolution
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 6 / 42
![Page 19: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/19.jpg)
The “economics” behind the introduction of SLSI ...
At the beginning of 1991 IFS is a spectral Eulerian model on a fullGaussian grid running at T106/L19 resolution
Resolution increase and super-computer upgrade
A 4×CPU computer upgrade is planned
It was estimated then that increasing resolution to T213/L31 wouldrequire 12×CPU. That was a serious underestimate!
A numerical analysis miracle
Change to SLSI numerics and to a reduced Gaussian grid came to rescue
the planned resolution/model upgrade was made possible at the givenhardware
with the new model elapsed time for a 10d forecast at new increasedresolution was reduced from > 24hrs to 4 hours!
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 7 / 42
![Page 20: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/20.jpg)
The “economics” behind the introduction of SLSI ...
At the beginning of 1991 IFS is a spectral Eulerian model on a fullGaussian grid running at T106/L19 resolution
Resolution increase and super-computer upgrade
A 4×CPU computer upgrade is planned
It was estimated then that increasing resolution to T213/L31 wouldrequire 12×CPU. That was a serious underestimate!
A numerical analysis miracle
Change to SLSI numerics and to a reduced Gaussian grid came to rescue
the planned resolution/model upgrade was made possible at the givenhardware
with the new model elapsed time for a 10d forecast at new increasedresolution was reduced from > 24hrs to 4 hours!
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 7 / 42
![Page 21: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/21.jpg)
The “economics” behind the introduction of SLSI ...
At the beginning of 1991 IFS is a spectral Eulerian model on a fullGaussian grid running at T106/L19 resolution
Resolution increase and super-computer upgrade
A 4×CPU computer upgrade is planned
It was estimated then that increasing resolution to T213/L31 wouldrequire 12×CPU. That was a serious underestimate!
A numerical analysis miracle
Change to SLSI numerics and to a reduced Gaussian grid came to rescue
the planned resolution/model upgrade was made possible at the givenhardware
with the new model elapsed time for a 10d forecast at new increasedresolution was reduced from > 24hrs to 4 hours!
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 7 / 42
![Page 22: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/22.jpg)
Semi-Lagrangian advection
Let ρ be the air density and ρχ the density of a tracer transported by awind field V
Continuity equation in non-conservative form
DρχDt
= −ρχ∇ · V,Dρ
Dt= −ρ∇ · V, D
Dt=
∂
∂t+ V · ∇
Using specific ratios φχ = ρχ/ρ:
DφχDt
= 0⇒φt+∆tχ − φtχ,d
∆t= 0⇒ φt+∆t
χ = φtχ,d
d : departure point (d.p.), i.e. spatial location of field at time t
A simplified linear equation: non-linear advection terms absent
To compute φt+∆t : (i) find d (ii) interpolate φt to d
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 8 / 42
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Semi-Lagrangian advection
Let ρ be the air density and ρχ the density of a tracer transported by awind field V
Continuity equation in non-conservative form
DρχDt
= −ρχ∇ · V,Dρ
Dt= −ρ∇ · V, D
Dt=
∂
∂t+ V · ∇
Using specific ratios φχ = ρχ/ρ:
DφχDt
= 0⇒φt+∆tχ − φtχ,d
∆t= 0⇒ φt+∆t
χ = φtχ,d
d : departure point (d.p.), i.e. spatial location of field at time t
A simplified linear equation: non-linear advection terms absent
To compute φt+∆t : (i) find d (ii) interpolate φt to d
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 8 / 42
![Page 24: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/24.jpg)
Basic steps in Semi-Lagrangian algorithm
SolveDφ
Dt= 0 given a grid r and a velocity field V
1 Fluid parcels are always assumed to “arrive” on r at t + ∆t
2 For each grid-point solve backward trajectory equation for the d.p.:
Dr
Dt= V(r, t)⇒ rt+∆t︸ ︷︷ ︸
arrival g .p.
− rtd︸︷︷︸unknown d .p.
=
∫ t+∆t
tV(r, t)dt
3 Remap (interpolate) φ to rd (Lagrangian grid) to obtain
φt+∆t = φtd
Usual techniques employed
Midpoint rule and fixed point iteration for solving trajectory equation
Cubic-Lagrange interpolation normally used to obtain φtd
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 9 / 42
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Basic steps in Semi-Lagrangian algorithm
SolveDφ
Dt= 0 given a grid r and a velocity field V
1 Fluid parcels are always assumed to “arrive” on r at t + ∆t
2 For each grid-point solve backward trajectory equation for the d.p.:
Dr
Dt= V(r, t)⇒ rt+∆t︸ ︷︷ ︸
arrival g .p.
− rtd︸︷︷︸unknown d .p.
=
∫ t+∆t
tV(r, t)dt
3 Remap (interpolate) φ to rd (Lagrangian grid) to obtain
φt+∆t = φtd
Usual techniques employed
Midpoint rule and fixed point iteration for solving trajectory equation
Cubic-Lagrange interpolation normally used to obtain φtd
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 9 / 42
![Page 26: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/26.jpg)
Computing trajectories
“Classic” scheme: midpoint using time extrapolation
Midpoint discretization:
r − rd = ∆t V
(r + rd
2, t +
∆t
2
), V = (u, v , η)
1 Extrapolate velocity field: Vt+∆t/2 = 1.5Vt − 0.5Vt−∆t
2
rd(0) = r
Interpolate Vt+∆t/2 to rM ≡ 0.5[r + r(`−1)d ]
Update: rd(`) = r −∆t Vt+∆t/2
∣∣rM
, ` = 1, 2, . . .
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 10 / 42
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SLSI time-stepping in IFS
General adiabatic prognostic equation
DX
Dt= F . Let F = N + L, N ≡ F − L where,
L: fast terms linearised on a reference state, N: nonlinear slow terms
Fast linear terms L are treated implicitly
Non-linear terms can be extrapolated at t + ∆t/2 using:
Nt+∆t/2 ≈ 1.5Nt − 0.5Nt−∆t
2nd order two time-level discretization
X t+∆t − X td
∆t=
1
2
(Ltd + Lt+∆t
)+ N
t+∆t/2M , M : trajectory midpoint
Prognostic variables are eliminated to derive a constant coefficientHelmholtz equation for D: cheap and easy to solve in spectral space
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 11 / 42
![Page 28: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/28.jpg)
SLSI time-stepping in IFS
General adiabatic prognostic equation
DX
Dt= F . Let F = N + L, N ≡ F − L where,
L: fast terms linearised on a reference state, N: nonlinear slow terms
Fast linear terms L are treated implicitly
Non-linear terms can be extrapolated at t + ∆t/2 using:
Nt+∆t/2 ≈ 1.5Nt − 0.5Nt−∆t
2nd order two time-level discretization
X t+∆t − X td
∆t=
1
2
(Ltd + Lt+∆t
)+ N
t+∆t/2M , M : trajectory midpoint
Prognostic variables are eliminated to derive a constant coefficientHelmholtz equation for D: cheap and easy to solve in spectral space
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 11 / 42
![Page 29: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/29.jpg)
SLSI time-stepping in IFS
General adiabatic prognostic equation
DX
Dt= F . Let F = N + L, N ≡ F − L where,
L: fast terms linearised on a reference state, N: nonlinear slow terms
Fast linear terms L are treated implicitly
Non-linear terms can be extrapolated at t + ∆t/2 using:
Nt+∆t/2 ≈ 1.5Nt − 0.5Nt−∆t
2nd order two time-level discretization
X t+∆t − X td
∆t=
1
2
(Ltd + Lt+∆t
)+ N
t+∆t/2M , M : trajectory midpoint
Prognostic variables are eliminated to derive a constant coefficientHelmholtz equation for D: cheap and easy to solve in spectral space
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 11 / 42
![Page 30: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/30.jpg)
Iterative Centred Implicit SLSI solver
Extrapolation results to a weak instability Durran & Reinecke (MWR2003), Cordero et al (QJRMS, 2005)
This is mostly noticed in strong jet areas in the stratosphere
An iterative approach (P. Benard, MWR 2003) can be used to obtaina stable semi-implicit formulation:
I Once a predictor for prognostic variable X becomes available
X (P) ≈ X t+∆t , X (P) : solution at the end of an iteration
then time-interpolation can be used instead of extrapolation:
Use Vt+∆t/2 = 0.5[V(P) + Vt
]in trajectory iterations
Use N t+∆t/2 = 0.5[N(P) + N t
]in:
X t+∆t − X td
∆t=
1
2
(Ltd + Lt+∆t
)+ N
t+∆t/2M
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 12 / 42
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Iterative Centred Implicit SLSI solver
Extrapolation results to a weak instability Durran & Reinecke (MWR2003), Cordero et al (QJRMS, 2005)
This is mostly noticed in strong jet areas in the stratosphere
An iterative approach (P. Benard, MWR 2003) can be used to obtaina stable semi-implicit formulation:
I Once a predictor for prognostic variable X becomes available
X (P) ≈ X t+∆t , X (P) : solution at the end of an iteration
then time-interpolation can be used instead of extrapolation:
Use Vt+∆t/2 = 0.5[V(P) + Vt
]in trajectory iterations
Use N t+∆t/2 = 0.5[N(P) + N t
]in:
X t+∆t − X td
∆t=
1
2
(Ltd + Lt+∆t
)+ N
t+∆t/2M
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 12 / 42
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SETTLS: stable extrapolation (Hortal, QJRMS 2002)
AssumedX
dt= R and expand:
X (t + ∆t) ≈ Xd(t) + ∆t
Rd (t)︷ ︸︸ ︷(dX
dt
)d
+∆t2
2
R(t)−Rd (t−∆t)
∆t︷ ︸︸ ︷(d2X
dt2
)AV
⇓
X (t + ∆t) = Xd(t) +∆t
2{R(t) + [2R(t)− R(t −∆t)]d}
Use same formula for:
Trajectories: r(`)d = r − 0.5∆t
[V t +
(2V t − V t−∆t
)∣∣∣d (`−1)
]Non-linear RHS terms: N
t+∆t/2M = 0.5
[Nt +
(2Nt − Nt−∆t
)∣∣∣d
]Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 13 / 42
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SETTLS: stable extrapolation (Hortal, QJRMS 2002)
AssumedX
dt= R and expand:
X (t + ∆t) ≈ Xd(t) + ∆t
Rd (t)︷ ︸︸ ︷(dX
dt
)d
+∆t2
2
R(t)−Rd (t−∆t)
∆t︷ ︸︸ ︷(d2X
dt2
)AV
⇓
X (t + ∆t) = Xd(t) +∆t
2{R(t) + [2R(t)− R(t −∆t)]d}
Use same formula for:
Trajectories: r(`)d = r − 0.5∆t
[V t +
(2V t − V t−∆t
)∣∣∣d (`−1)
]Non-linear RHS terms: N
t+∆t/2M = 0.5
[Nt +
(2Nt − Nt−∆t
)∣∣∣d
]Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 13 / 42
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Outline
1 Semi-Lagrangian technique and history
2 Semi-Lagrangian numerics in the upper atmosphereTrajectory equation and numerical noiseExtratropical tropopause cold bias
3 The mass conservation .... headache
4 Concluding
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 14 / 42
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Outline
1 Semi-Lagrangian technique and history
2 Semi-Lagrangian numerics in the upper atmosphereTrajectory equation and numerical noiseExtratropical tropopause cold bias
3 The mass conservation .... headache
4 Concluding
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 15 / 42
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Stratospheric noiseSETTLS greatly improves stability, however, occasionally, solutionbecomes noisy in the upper atmosphere (near strong jets)Testing shows that noise originates from the calculation of thevertical component of the departure pointUntil recently solution used “smoothing of vertical velocities” (leastsquare interpolation)
(a) 5hPa D at T+24 (16/01/12) (b) 1hPa D at T+24 (29/12/12)
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 16 / 42
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Techniques to solve stratospheric noise problem
Iterative scheme (ICI) - too expensive
Off-centring - damps gravity waves/impact on accuracy
Smoothing vertical velocities - impact on accuracy (temperature)
Using “at selected grid-points” non-extrapolated trajectory scheme(SETTLS limiter) for computing vertical component of the d.p.
(c) 5hPa D at T+24 (16/01/12) (d) 1hPa D at T+24 (29/12/12)
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 17 / 42
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Sudden stratospheric warming episode 15/01/2012
instabilities → noisy solution upper atmosphere → rejection of satellite obs
(e) Control FC (f) Smoothing η
(g) SETTLS traj limiter
SETTLS Limiter:
ηd =
{η − ∆t
2
(ηt + ηtd
),
∣∣∇t ηt∣∣ > β|η| t,t−∆t
SETTLS, otherwise
0 < β < 2, ∇t ηt = ηt − ηt−∆t
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 18 / 42
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Outline
1 Semi-Lagrangian technique and history
2 Semi-Lagrangian numerics in the upper atmosphereTrajectory equation and numerical noiseExtratropical tropopause cold bias
3 The mass conservation .... headache
4 Concluding
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 19 / 42
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Extra-tropical cold bias
{Results from cubic QMRMS norm FC-ERAI
}→
90°S60°S30°S0°N30°N60°N90°N
200
400
600
800
1000
Pre
ssur
e (h
Pa)
237.5
237.
5
212.5
212.5212.5
262.5
4 Dates: 20000801, ... Averaging Period Start: 200009 Length: 12 MonthsClimate Forecast (fwp9) - ERAIDifference: Zonal Mean Average T (n=4)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1-0.5
1
2
3
4
5
6
7
8
9
10
Stenke et all (Clim Dyn 2008)
Water vapor (q) overestimation by SL models in lower extratropicalstratosphere leads to radiative cooling and noticeable cold bias
“Considerable numerical meridional diffusion in presence of sharpgradients”
I Interpolation in the horizontal is acting on model levels intersectingdownward sloping tropopause (from tropics to extra-tropics)
No bias in pure Lagrangian models (non-diffusive)
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 20 / 42
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Cold bias investigations with IFS
A dynamically sensitive problem
Dynamical model components affecting cold bias
Large sensitivity to (horizontal part) of q-interpolationI the more diffusive the interpolation the worstI switching on/off quasi-monotone limiter
Large sensitivity wrt to d.p. calculation
Moderately sensitive wrt to timestep
Moderately sensitive wrt to semi-implicit options (ICI, off-centring)
Problem persists at low and high resolution
Current results suggest that bias can be reduced by improvinginterpolation scheme and accuracy of d.p. calculation
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 21 / 42
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Quasi-monotone limiting in cubic interpolation
Bermejo & Staniforth limiter (MWR, 1992)
φd = max(min(φd , φ+), φ−), φ+ = max
i∈{Ni}{φi}, φ−= min
i∈{Ni}{φi}
Ni = {set of neighbouring points surounding d}
φ+
φ−
IFS qm-interpolation in (λ, θ, η) coordinates:Interp in λ and limit → Interp in θ and limit → Interp in η and limit
The limiter can also be applied at the very end of interpolation:Interp in λ → Interp in θ → Interp in η → limit in (λ, θ, η)
Limiting at the end seems to be beneficial for the cold bias
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 22 / 42
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Differences from control FC: testing QM limiter
90°S60°S30°S0°N30°N60°N90°N
200
400
600
800
1000
Pre
ssur
e (h
Pa)
237.5
212.5212.5
262.5
4 Dates: 20000801, ... Averaging Period Start: 200009 Length: 12 MonthsClimate Forecast (fws6) - (fulg)Difference: Zonal Mean Average T (n=4)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1-0.5
1
2
3
4
5
6
7
8
9
10
(h) QM off / neg fixer on
90°S60°S30°S0°N30°N60°N90°N
200
400
600
800
1000
Pre
ssur
e (h
Pa)
237.5
212.5212.5
262.5
4 Dates: 20000801, ... Averaging Period Start: 200009 Length: 12 MonthsClimate Forecast (fxho) - (fulg)Difference: Zonal Mean Average T (n=4)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1-0.5
1
2
3
4
5
6
7
8
9
10
(i) Bermejo & Staniforth QM
90°S60°S30°S0°N30°N60°N90°N
200
400
600
800
1000
Pre
ssur
e (h
Pa)
237.5
212.5212.5
262.5
4 Dates: 20000801, ... Averaging Period Start: 200009 Length: 12 MonthsClimate Forecast (fx4f) - (fulg)Difference: Zonal Mean Average T (n=4)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1-0.5
1
2
3
4
5
6
7
8
9
10
(j) QM + vert quasi-conserv filter
←
�
�
�
�1. Point i at lev k is limited.
2. Mass removed/added by limiter
added/removed at point i , k + 1
3. Point i ,k + 1 is limited ...
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 23 / 42
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Trajectory sensitivities
90°S60°S30°S0°N30°N60°N90°N
200
400
600
800
1000
Pre
ssur
e (h
Pa)
237.5
237.
5
212.5
212.5212.5
262.5
4 Dates: 20000801, ... Averaging Period Start: 200009 Length: 12 MonthsClimate Forecast (fxaf) - ERAIDifference: Zonal Mean Average T (n=4)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1-0.5
1
2
3
4
5
6
7
8
9
10
(k) smoothing η - ERAI
90°S60°S30°S0°N30°N60°N90°N
200
400
600
800
1000
Pre
ssur
e (h
Pa)
237.5
237.
5
212.5
212.5212.5
262.5
4 Dates: 20000801, ... Averaging Period Start: 200009 Length: 12 MonthsClimate Forecast (fxao) - ERAIDifference: Zonal Mean Average T (n=4)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1-0.5
1
2
3
4
5
6
7
8
9
10
(l) SETTLS lim - ERAI
90°S60°S30°S0°N30°N60°N90°N
200
400
600
800
1000
Pre
ssur
e (h
Pa)
237.5
212.5212.5
262.5
4 Dates: 20000801, ... Averaging Period Start: 200009 Length: 12 MonthsClimate Forecast (fxaf) - (fulg)Difference: Zonal Mean Average T (n=4)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1-0.5
1
2
3
4
5
6
7
8
9
10
(m) smoothing η - CNTL fc
90°S60°S30°S0°N30°N60°N90°N
200
400
600
800
1000
Pre
ssur
e (h
Pa)
237.5
212.5212.5
262.5
4 Dates: 20000801, ... Averaging Period Start: 200009 Length: 12 MonthsClimate Forecast (fxao) - (fulg)Difference: Zonal Mean Average T (n=4)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1-0.5
1
2
3
4
5
6
7
8
9
10
(n) SETTLS limiter - CNTL fc
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 24 / 42
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Outline
1 Semi-Lagrangian technique and history
2 Semi-Lagrangian numerics in the upper atmosphereTrajectory equation and numerical noiseExtratropical tropopause cold bias
3 The mass conservation .... headache
4 Concluding
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 25 / 42
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Mass Conservation
Mass conservation becomes increasingly important
Has always been for long (climate) integrations
Moist tracer conservation important for microphysics/convection
Increasing importance of “environmental/chemical” forecasts
Eulerian flux-form models
Conservation straightforward: change of mass in volume ∆A = total fluxthrough its faces ∆S from neighbouring grid-boxes
∂
∂t
∫∆A
ρdV = −∫
∆A∇ · (ρU)dV ⇒ ρt+∆t − ρt =
∆t
∆A
∫∆S
ρU · ηdS
Semi-Lagrangian schemes are not formally conserving: equations appliedon grid-points rather than volumes - there is no flux-counting
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 26 / 42
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Mass Conservation
Mass conservation becomes increasingly important
Has always been for long (climate) integrations
Moist tracer conservation important for microphysics/convection
Increasing importance of “environmental/chemical” forecasts
Eulerian flux-form models
Conservation straightforward: change of mass in volume ∆A = total fluxthrough its faces ∆S from neighbouring grid-boxes
∂
∂t
∫∆A
ρdV = −∫
∆A∇ · (ρU)dV ⇒ ρt+∆t − ρt =
∆t
∆A
∫∆S
ρU · ηdS
Semi-Lagrangian schemes are not formally conserving: equations appliedon grid-points rather than volumes - there is no flux-counting
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 26 / 42
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Mass Conservation
Mass conservation becomes increasingly important
Has always been for long (climate) integrations
Moist tracer conservation important for microphysics/convection
Increasing importance of “environmental/chemical” forecasts
Eulerian flux-form models
Conservation straightforward: change of mass in volume ∆A = total fluxthrough its faces ∆S from neighbouring grid-boxes
∂
∂t
∫∆A
ρdV = −∫
∆A∇ · (ρU)dV ⇒ ρt+∆t − ρt =
∆t
∆A
∫∆S
ρU · ηdS
Semi-Lagrangian schemes are not formally conserving: equations appliedon grid-points rather than volumes - there is no flux-counting
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 26 / 42
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Accurate continuity discretization in IFS
D
Dt(ln ps) = RHS ,
Splitln ps = l
′+ l∗
where l∗ = φs/(RdT
)the orography time-independent part and solve
Dl ′
Dt= [RHS ] +
1
RdTVh · ∇φs
Benefits of this formulation:
Advected field is smoother and therefore better air mass conservationdue to reduced interpolation error
Aleviates orographic resonance problem
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 27 / 42
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Global mass conservation error in IFS
In NWP forecasts we have been not paying too much attention in massconservation. Why?
An accurate numerical scheme may not be mass conserving but itsmass conservation error should be small
In a 10 day IFS forecast at T1279 horizontal resolution total air massapproximately increasing by 0.015% of its initial value.
Global conservation errors in tracer advection are larger and dependon the smoothness of the field, i.e. smoother fields such as ozone andspecific humidity have much smaller conservation errors than fieldswith sharp features such as cloud fields
Global tracer-mass conservation error reduces with increasingresolution
Not reduced when timestep is reduced
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 28 / 42
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Global mass conservation error in IFS
In NWP forecasts we have been not paying too much attention in massconservation. Why?
An accurate numerical scheme may not be mass conserving but itsmass conservation error should be small
In a 10 day IFS forecast at T1279 horizontal resolution total air massapproximately increasing by 0.015% of its initial value.
Global conservation errors in tracer advection are larger and dependon the smoothness of the field, i.e. smoother fields such as ozone andspecific humidity have much smaller conservation errors than fieldswith sharp features such as cloud fields
Global tracer-mass conservation error reduces with increasingresolution
Not reduced when timestep is reduced
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 28 / 42
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Global mass conservation error in IFS
In NWP forecasts we have been not paying too much attention in massconservation. Why?
An accurate numerical scheme may not be mass conserving but itsmass conservation error should be small
In a 10 day IFS forecast at T1279 horizontal resolution total air massapproximately increasing by 0.015% of its initial value.
Global conservation errors in tracer advection are larger and dependon the smoothness of the field, i.e. smoother fields such as ozone andspecific humidity have much smaller conservation errors than fieldswith sharp features such as cloud fields
Global tracer-mass conservation error reduces with increasingresolution
Not reduced when timestep is reduced
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 28 / 42
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Global mass conservation error in IFS
In NWP forecasts we have been not paying too much attention in massconservation. Why?
An accurate numerical scheme may not be mass conserving but itsmass conservation error should be small
In a 10 day IFS forecast at T1279 horizontal resolution total air massapproximately increasing by 0.015% of its initial value.
Global conservation errors in tracer advection are larger and dependon the smoothness of the field, i.e. smoother fields such as ozone andspecific humidity have much smaller conservation errors than fieldswith sharp features such as cloud fields
Global tracer-mass conservation error reduces with increasingresolution
Not reduced when timestep is reduced
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 28 / 42
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Timeseries of global tracer-mass conservation error
(o) T159 L91 (p) T1279 L91
(q) T1279 L137
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 29 / 42
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Approaches in improving mass conservation
How to improve mass conservation error, particularly for tracers?
Interpolation
Improving interpolation schemes. In IFS quasi-cubic Lagrange with aquasi-monotone filter is used. Alternative ones have been recentlytested:
I cubic splines in the vertical: very good for smooth fields such as ozonebut worst for rough fields
I cubic Hermite in the vertical with derivative limiting: improvedconservation in specific humidity
I improving quasi-monotone limiters can also have positive impact
Mass conservative advection schemes
mass fixers
inherently conserving schemes
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 30 / 42
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Approaches in improving mass conservation
How to improve mass conservation error, particularly for tracers?
Interpolation
Improving interpolation schemes. In IFS quasi-cubic Lagrange with aquasi-monotone filter is used. Alternative ones have been recentlytested:
I cubic splines in the vertical: very good for smooth fields such as ozonebut worst for rough fields
I cubic Hermite in the vertical with derivative limiting: improvedconservation in specific humidity
I improving quasi-monotone limiters can also have positive impact
Mass conservative advection schemes
mass fixers
inherently conserving schemes
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 30 / 42
![Page 57: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/57.jpg)
Approaches in improving mass conservation
How to improve mass conservation error, particularly for tracers?
Interpolation
Improving interpolation schemes. In IFS quasi-cubic Lagrange with aquasi-monotone filter is used. Alternative ones have been recentlytested:
I cubic splines in the vertical: very good for smooth fields such as ozonebut worst for rough fields
I cubic Hermite in the vertical with derivative limiting: improvedconservation in specific humidity
I improving quasi-monotone limiters can also have positive impact
Mass conservative advection schemes
mass fixers
inherently conserving schemes
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 30 / 42
![Page 58: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/58.jpg)
SL mass conservative advection schemes
Global Mass Fixers
Low cost algorithms
Easy to implement in existing models: no need to change modelformulation
a-posterior fixes: diagnose global mass conservation error and correctthe interpolated field to ensure total mass remains unchanged
Criticism: they are somewhat ad-hoc and global in nature
Inherently conserving
Theoretically desirable schemes: both local and global conservation isachieved for tracers and air mass without inserting artificial fixes
They can be very expensive and implementation in an operationalmodel is a complex task
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 31 / 42
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SL mass conservative advection schemes
Global Mass Fixers
Low cost algorithms
Easy to implement in existing models: no need to change modelformulation
a-posterior fixes: diagnose global mass conservation error and correctthe interpolated field to ensure total mass remains unchanged
Criticism: they are somewhat ad-hoc and global in nature
Inherently conserving
Theoretically desirable schemes: both local and global conservation isachieved for tracers and air mass without inserting artificial fixes
They can be very expensive and implementation in an operationalmodel is a complex task
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 31 / 42
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Mass Fixers
Proportional fixers: adjust each g.p. value by the same proportion
There are more sophisticated approaches which correct locally (verysmall corrections when solution is smooth, larger when not):
I quasi-monotone Bermejo & Conde fixer (MWR 2002)I quasi-monotone Priestley fixer (MWR 1993)I Mac Gregor’s fixer (C-CAM, CSIRO Tech Rep 70)I Zerroukat fixer (JCP, 2010)
The above fixers have been recently implemented in IFS
Mass Fixer CPU overhead(5 tracers)
BC 1%PRqm 2%PR 3.5%JMG 0.75%Ze 0.85%
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Local behaviour of Bermejo & Conde fixer
(r) T+24 q field at ≈ 700 hPa (s) BC fixer correction
Corrections are concentrated in areas of large gradients
The quality of forecast is maintained (verification scores neutral)
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Long integrations with fixers
90°S60°S30°S0°N30°N60°N90°N
200
400
600
800
1000
Pre
ssur
e (h
Pa)
237.5
237.
5
212.5
212.5212.5
262.5
4 Dates: 20000801, ... Averaging Period Start: 200009 Length: 12 MonthsClimate Forecast (fx9p) - ERAIDifference: Zonal Mean Average T (n=4)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1-0.5
1
2
3
4
5
6
7
8
9
10
(t) Control - no fixer
90°S60°S30°S0°N30°N60°N90°N
200
400
600
800
1000
Pre
ssur
e (h
Pa)
237.5
237.
5
212.5
212.5212.5
262.5
4 Dates: 20000801, ... Averaging Period Start: 200009 Length: 12 MonthsClimate Forecast (fxkh) - ERAIDifference: Zonal Mean Average T (n=4)
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1-0.5
1
2
3
4
5
6
7
8
9
10
(u) BC fixer on all
RMS norm of FC-ERAI
EXP ID TR 200 TR 700 TR 925
CNTL 1.4085 0.7214 0.7881BC fixer 1.2825 0.6812 0.7795MacGregor fixer 1.3006 0.6778 0.7596
EXP ID NH 200 NH 700 SH 200 SH 700
CNTL 1.9332 0.9664 1.7280 0.8341BC fixer 1.9730 0.9414 1.6726 0.8258MacGregor fixer 2.0665 0.9645 1.6120 0.7906
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Preserving functional relationships between tracers
3-d version of DCMIP case 11: q2 = 0.9− 0.8q21
(v) qm cubic (w) BC fixer
(x) PR fixer (y) JMG fixer
Adding the fixers didn’t distort functional relationship
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 35 / 42
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Summary on mass fixers
Global mass conservation at low cost and no deterioration of forecastquality
They correct locally where interpolation error is expected to be largerI difference between high order and low order interpolation scheme is
used to construct a weight which determines how much to correct
They can preserve monotonicity or positive-definitiness
When applied on point sources they can be diffusive (they removemass due to large gains)
Work underway to determine impact on other chemical tracers
No matter how good a fixer algorithm can be will never be “perfect”:I cannot be truly locally conservingI inconsistent transport of different species
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 36 / 42
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Summary on mass fixers
Global mass conservation at low cost and no deterioration of forecastquality
They correct locally where interpolation error is expected to be largerI difference between high order and low order interpolation scheme is
used to construct a weight which determines how much to correct
They can preserve monotonicity or positive-definitiness
When applied on point sources they can be diffusive (they removemass due to large gains)
Work underway to determine impact on other chemical tracers
No matter how good a fixer algorithm can be will never be “perfect”:I cannot be truly locally conservingI inconsistent transport of different species
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 36 / 42
![Page 66: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/66.jpg)
Summary on mass fixers
Global mass conservation at low cost and no deterioration of forecastquality
They correct locally where interpolation error is expected to be largerI difference between high order and low order interpolation scheme is
used to construct a weight which determines how much to correct
They can preserve monotonicity or positive-definitiness
When applied on point sources they can be diffusive (they removemass due to large gains)
Work underway to determine impact on other chemical tracers
No matter how good a fixer algorithm can be will never be “perfect”:I cannot be truly locally conservingI inconsistent transport of different species
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 36 / 42
![Page 67: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/67.jpg)
Summary on mass fixers
Global mass conservation at low cost and no deterioration of forecastquality
They correct locally where interpolation error is expected to be largerI difference between high order and low order interpolation scheme is
used to construct a weight which determines how much to correct
They can preserve monotonicity or positive-definitiness
When applied on point sources they can be diffusive (they removemass due to large gains)
Work underway to determine impact on other chemical tracers
No matter how good a fixer algorithm can be will never be “perfect”:I cannot be truly locally conservingI inconsistent transport of different species
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 36 / 42
![Page 68: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/68.jpg)
Summary on mass fixers
Global mass conservation at low cost and no deterioration of forecastquality
They correct locally where interpolation error is expected to be largerI difference between high order and low order interpolation scheme is
used to construct a weight which determines how much to correct
They can preserve monotonicity or positive-definitiness
When applied on point sources they can be diffusive (they removemass due to large gains)
Work underway to determine impact on other chemical tracers
No matter how good a fixer algorithm can be will never be “perfect”:I cannot be truly locally conservingI inconsistent transport of different species
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 36 / 42
![Page 69: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/69.jpg)
Summary on mass fixers
Global mass conservation at low cost and no deterioration of forecastquality
They correct locally where interpolation error is expected to be largerI difference between high order and low order interpolation scheme is
used to construct a weight which determines how much to correct
They can preserve monotonicity or positive-definitiness
When applied on point sources they can be diffusive (they removemass due to large gains)
Work underway to determine impact on other chemical tracers
No matter how good a fixer algorithm can be will never be “perfect”:I cannot be truly locally conservingI inconsistent transport of different species
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 36 / 42
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Inherently conserving schemes: SLICE
Finite volume semi-Lagrangian continuity:
D
Dt
∫A(t)
ψdA = 0⇒∫Aψn+1
dA =
∫Ad
ψndA, ψ = air/tracer density
SLICE-3D (Zerroukat & Allen QJRMS, 2012)
ψn+1
=Mt
d
A, Mt
d =
∫Ad
ψndA
No need to compute Lagrangian (departure) volume Ad . Instead:
1 Compute departure points as usual to define Lagrangian controlvolumes (CV) corresponding to Eulerian CVs
2 Remap: compute mass Mnd of LCVs using 1-dimensional cascade
interpolation at each direction
SLICE is available as an option in new UKMO ENDGame model
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CSLAM (Lauritzen, JCP 2010)
αk` = αk ∩ A`, ` = 1, 2, . . . , Lk
ψn+1k =
1
Akψk,d
nαk , ψ
nk,dαk =
Lk∑`=1
∫ ∫αk`
f`dA =∑`=1
∑i+j≤2
c(i ,j)` w
(i ,j)k`
For each Eulerian grid cell find corresponding departure (Lagrangian)grid cell by computing departure points of Eulerian vertices
Continuous sub-grid-scale representation f` of ψ with massconservation as an integral constraint:∫ ∫
A`
f`dA = ψ`A`, ` = 1, 2, . . . , Lk
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The difficulty with inherently conserving schemes
Computational expense is a limiting factor for their application inoperational meteorology
However, they may be competitive for the multiple-tracer advectionproblem
I e.g. CSLAM is currently developed as an offline transport package formulti-tracer advection in climate model CAM-SE
I also see LMCSL 3-D (Sorensen, GeoSci. Model Dev. 2013) forHIRLAM
Performance of such schemes in presence of complex terrainI Departure volumes are computed using simple algorithms not coupled
with continuity: along the trajectory the gas volume maydeform/compress/expand
I Alternative (expensive) trajectory calculations: Thuburn et al (QJRMS,2010), Cossette & Smolarkiewicz (Computer & Fluids 2011)
I Accurate (no staright line) representations of departure cell boundariesare even more expensive to compute
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 39 / 42
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The difficulty with inherently conserving schemes
Computational expense is a limiting factor for their application inoperational meteorology
However, they may be competitive for the multiple-tracer advectionproblem
I e.g. CSLAM is currently developed as an offline transport package formulti-tracer advection in climate model CAM-SE
I also see LMCSL 3-D (Sorensen, GeoSci. Model Dev. 2013) forHIRLAM
Performance of such schemes in presence of complex terrainI Departure volumes are computed using simple algorithms not coupled
with continuity: along the trajectory the gas volume maydeform/compress/expand
I Alternative (expensive) trajectory calculations: Thuburn et al (QJRMS,2010), Cossette & Smolarkiewicz (Computer & Fluids 2011)
I Accurate (no staright line) representations of departure cell boundariesare even more expensive to compute
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 39 / 42
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The difficulty with inherently conserving schemes
Computational expense is a limiting factor for their application inoperational meteorology
However, they may be competitive for the multiple-tracer advectionproblem
I e.g. CSLAM is currently developed as an offline transport package formulti-tracer advection in climate model CAM-SE
I also see LMCSL 3-D (Sorensen, GeoSci. Model Dev. 2013) forHIRLAM
Performance of such schemes in presence of complex terrainI Departure volumes are computed using simple algorithms not coupled
with continuity: along the trajectory the gas volume maydeform/compress/expand
I Alternative (expensive) trajectory calculations: Thuburn et al (QJRMS,2010), Cossette & Smolarkiewicz (Computer & Fluids 2011)
I Accurate (no staright line) representations of departure cell boundariesare even more expensive to compute
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 39 / 42
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Outline
1 Semi-Lagrangian technique and history
2 Semi-Lagrangian numerics in the upper atmosphereTrajectory equation and numerical noiseExtratropical tropopause cold bias
3 The mass conservation .... headache
4 Concluding
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 40 / 42
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A view of the future of SLSI techniques
High resolution global NWP models on massively parallel machines
Efficiency of (any) advection scheme on a regular lat/lon grid on thesphere deteriorates: convergence of meridionals, anisotropy near poles
SLSI and Eulerian SI advection schemes on quasi-uniform grids(cubed sphere, icosahedral, reduced Gaussian etc) are in betterposition provided that:
I For semi-implicit types efficient elliptic solver is available (multigrid?)I They are mass conservative
For multi-tracer applications even expensive inherently conserving SLmay be more efficient comparing to Eulerian flux-form
Spectral transform SLSI: critical issue cost of transpositions + massconservation
Climate models
Likely that existing climate models built on SLSI techniques willcontinue being used in climate modeling for long time
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 41 / 42
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A view of the future of SLSI techniques
High resolution global NWP models on massively parallel machines
Efficiency of (any) advection scheme on a regular lat/lon grid on thesphere deteriorates: convergence of meridionals, anisotropy near poles
SLSI and Eulerian SI advection schemes on quasi-uniform grids(cubed sphere, icosahedral, reduced Gaussian etc) are in betterposition provided that:
I For semi-implicit types efficient elliptic solver is available (multigrid?)I They are mass conservative
For multi-tracer applications even expensive inherently conserving SLmay be more efficient comparing to Eulerian flux-form
Spectral transform SLSI: critical issue cost of transpositions + massconservation
Climate models
Likely that existing climate models built on SLSI techniques willcontinue being used in climate modeling for long time
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 41 / 42
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A view of the future of SLSI techniques
High resolution global NWP models on massively parallel machines
Efficiency of (any) advection scheme on a regular lat/lon grid on thesphere deteriorates: convergence of meridionals, anisotropy near poles
SLSI and Eulerian SI advection schemes on quasi-uniform grids(cubed sphere, icosahedral, reduced Gaussian etc) are in betterposition provided that:
I For semi-implicit types efficient elliptic solver is available (multigrid?)I They are mass conservative
For multi-tracer applications even expensive inherently conserving SLmay be more efficient comparing to Eulerian flux-form
Spectral transform SLSI: critical issue cost of transpositions + massconservation
Climate models
Likely that existing climate models built on SLSI techniques willcontinue being used in climate modeling for long time
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 41 / 42
![Page 79: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/79.jpg)
A view of the future of SLSI techniques
High resolution global NWP models on massively parallel machines
Efficiency of (any) advection scheme on a regular lat/lon grid on thesphere deteriorates: convergence of meridionals, anisotropy near poles
SLSI and Eulerian SI advection schemes on quasi-uniform grids(cubed sphere, icosahedral, reduced Gaussian etc) are in betterposition provided that:
I For semi-implicit types efficient elliptic solver is available (multigrid?)I They are mass conservative
For multi-tracer applications even expensive inherently conserving SLmay be more efficient comparing to Eulerian flux-form
Spectral transform SLSI: critical issue cost of transpositions + massconservation
Climate models
Likely that existing climate models built on SLSI techniques willcontinue being used in climate modeling for long time
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 41 / 42
![Page 80: The semi-Lagrangian technique: current status and future ... · A very stable formulation Semi-Lagrangian advection: Unconditionally stable Good phase speeds with little numerical](https://reader034.vdocuments.mx/reader034/viewer/2022042416/5f3222986ac263706b76688c/html5/thumbnails/80.jpg)
A view of the future of SLSI techniques
High resolution global NWP models on massively parallel machines
Efficiency of (any) advection scheme on a regular lat/lon grid on thesphere deteriorates: convergence of meridionals, anisotropy near poles
SLSI and Eulerian SI advection schemes on quasi-uniform grids(cubed sphere, icosahedral, reduced Gaussian etc) are in betterposition provided that:
I For semi-implicit types efficient elliptic solver is available (multigrid?)I They are mass conservative
For multi-tracer applications even expensive inherently conserving SLmay be more efficient comparing to Eulerian flux-form
Spectral transform SLSI: critical issue cost of transpositions + massconservation
Climate models
Likely that existing climate models built on SLSI techniques willcontinue being used in climate modeling for long time
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 41 / 42
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Future work plans for the ECMWF SL scheme
Efforts to improve the IFS SLSI scheme continue
Focusing on areas such as:I Mass conservationI Investigating interpolation techniques and limitersI Numerical schemes for solving SL trajectory equationsI Model biases due to dynamics such as the cold bias
Thank you for your attention!
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 42 / 42
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Future work plans for the ECMWF SL scheme
Efforts to improve the IFS SLSI scheme continue
Focusing on areas such as:I Mass conservationI Investigating interpolation techniques and limitersI Numerical schemes for solving SL trajectory equationsI Model biases due to dynamics such as the cold bias
Thank you for your attention!
Michail Diamantakis (ECMWF) The semi-Lagrangian technique: current status and future developments4 September 2013 42 / 42