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Numeric developments in COSMO

SRNWP / EWGLAM-MeetingDubrovnik, 08.-12.10.2007

Michael Baldauf1, Jochen Förstner1, Uli Schättler1

Pier Luigi Vitagliano2, Gabriella Ceci2 , Lucio Torrisi3, Ronny Petrik4

1Deutscher Wetterdienst, 2CIRA-Institute, 3USMA (Rome), 4Max-Plank-Institut Hamburg

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Outlook

• Terrain following• Leapfrog time integration• Runge-Kutta time integration (COSMO Priority Project)

• ‚operational version‘• Stability considerations (Winter storm ‚Kyrill‘, ...)• p'T'-dynamics• Moisture advection• Deep / shallow atmosphere• Physics/Dynamics coupling

• alternatives (A. Gassmann)• Semi-implicit (S. Thomas, ...)

• LM-Z (COSMO Priority Project)

Dynamical cores in the COSMO model:

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COSMO-EU(LME)GME

COSMO-DE(LMK)

The operational Model Chain of DWD: GME, COSMO-EU and -DE

(since 16. April 2007)

hydrostaticparameterised convectionx 40 km368642 * 40 GPt = 133 sec., T = 7 days

non-hydrostaticparameterised convectionx = 7 km665 * 657 * 40 GPt = 40 sec., T= 78 h

non-hydrostaticresolved convectionx = 2.8 km421 * 461 * 50 GPt = 25 sec., T = 21 h

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COSMO - Working Group 2 (Numerics)

COSMO Priority Project 'LM-Z'

several improvements on the code:• prevent decoupling of z-grid (dynamics) and tf-grid (physics) by

'nudging'• implicit vertical advection increase in time step• tendencies of data assimilation are now also transformed to the

z-grid

Comparison of LM-Z and an older version of LM (COSMO-model) (e.g. without prognostic precipitation)

--> report: end 2007

Collaboration with Univ. of Leeds started

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COSMO-Priority Project ‚Runge-Kutta‘:

1. New Developments1. NEW: Divergence damping in a 3D-(isotropic) version2. NEW: DFI for RK3. Advection of moisture quantities in conservation form4. Higher order discretization in the vertical5. Physics coupling scheme 6. Testing of alternative fast wave scheme7. Development of a more conservative dynamics (planned)8. Development of an efficient semi-implicit solver in combination with RK

time integration scheme (planned)2. Developing diagnostic tools

1. Conservation inspection tool (finished)2. Investigation of convergence

3. Known problems1. Looking at pressure bias2. Deep valleys 3. (Different filter options for orography) (finished)

COSMO - Working Group 2 (Numerics)

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Numerics and Dynamics - COSMO-DE developments

Grid structure horizontal: Arakawa C, vertical: Lorenz

Prognostic Var. cartesian components u, v, w, p’,T’ (LME: T)Time integration time-splitting between fast and slow modes

- 3-timelevels: Leapfrog (+centered diff.) (Klemp, Wilhelmson, 1978)- 2-timelevels: Runge-Kutta: 2. order, 3. order (Wicker, Skamarock, 1998, 2002)

Fast modes (=sound waves, buoyancy, divergence filtering)centered diff. 2. order, vertical implicit, (p’T’-Dyn.)

Advection for u,v,w,p’,T’: horizontal. adv.: upwind 3., 5. order / centered diff. 4. 6. ordervertical adv.: implicit 2. order

for qv, q

c, q

i, qr, qs, qg, TKE:

LME: qv, qc: centered diff. 2nd orderqi: 2nd ord. flux-form advection scheme qr, qs: semi-lagrange (tri-linear interpol.)

Courant-number-independent (CNI)-advection:- Bott (1989) (2., 4. order), in conservation form- Semi-Lagrange (tricubic interpol.)

Other slow modes (optional: complete Coriolis terms)Smoothing 3D divergence damping

horizontal diffusion 4. order applied only in the boundary relaxation zoneslope dependent orographic filtering

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Stability considerationsWinter storm ‚Kyrill‘, 18.01.2007

crash of all COSMO-DE (2.8 km)-runs from 03, 06, 09, ... UTC

two measures necessary:• timestep:

• old: t = 30 sec. (winter storm ‚Lothar' could be simulated) • new: t = 25 sec

• time integration scheme: • old: TVD-RK3

(Shu, Osher, 1988)

• new: 3-stage 2nd order RK3 (Wicker, Skamarock 2002)

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COSMO-DE (2.8 km), 18.01.2007

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COSMO-DE (2.8 km), 18.01.2007

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Von-Neumann stability analysis of a 2-dim., linearised Advection-Sound-Buoyancy-system

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Crank-Nicholson-parameter for buoyancy terms in the p‘T‘-dynamics=0.5 (‚pure‘ Crank-Nic.) =0.6 =0.7

=0.8 =0.9 =1.0 (pure implicit)

choose =0.7 as the best value Csnd = cs t / x

Ca

dv

= u

T

/

x

amplification

factor

• RK3-scheme(WS2002)

• upwind 5th order• Sound: =0.6 x/ z=10 T/ t=6

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What is the influence of divergence filtering ?

• fast processes (operatorsplitting):• sound (Crank-Nic., =0.6), • divergence damping (vertical implicit) • no buoyancy

• slow process: upwind 5. order• time splitting RK 3. order (WS2002-Version)• aspect ratio: x / z=10T / t=6

--> Divergence damping is needed in this dynamical core!

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Cdiv=0.025

Cdiv=0.05 Cdiv=0.1 Cdiv=0.15

Influence of Cdiv

Cdiv = xkd * (cs * t/ x)2

~0.35

stability limit by long waves (k0)

Cdiv=0

Csnd = cs t / x

Ca

dv

= u

T

/

x

amplification

factor

Cdiv = div t/x2

in COSMO-model:

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Advantages of p'T'-dynamics over p'T-dynamics

1. Improved representation of T-advection in terrain-following coordinates

2. Better representation of buoyancy term in fast waves solver

Terms (a) and (b) cancel analytically, but not numerically

using T:

Buoyancy term alone generates an oscillation equation:

= g/cs

= a = acoustic cut-off frequencyusing T':

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Idealised test case:Steady atmosphere with mountain

base state: T0, p0

deviations from base state: T', p' 0 introduces spurious circulations!

point 1.): 'improved T-advection' ...

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LeapfrogRunge-Kutta

old p*-T-dynamics

contours: vertical velocity w isolines: potential temperature

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contours: vertical velocity w isolines: potential temperature

Runge-Kuttaold p*-T-Dynamik

Runge-Kuttanew p*-T*-Dynamik

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Climate simulations

start: 1. july 1979 + 324 h (~2 weeks)

results: accumulated precipitation (TOT_PREC) and PMSL

(simulations: U. Schättler, in cooperation with the CLM-community)

Problems:

unrealistic prediction of pressure and precipitation distribution

strong dependency from the time step

These problems occur in the Leapfrog and the (old) Runge-Kutta-Version

(both p'T-dynamics) but not in the semi-implicit solver or the RK-p'T'-dynamics.

assumption: point 2.) 'treatment of the buoyancy term' improves this case

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Leapfrog – t = 75s Leapfrog – t = 90s RR(mm/h)

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RK (p*-T) – t = 150s RK (p*-T) – t = 180s RR(mm/h)

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LF (semi-implizit) – t = 75s LF (semi-implizit) – t = 90s RR(mm/h)

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RK (p*-T*) – t = 150s RK (p*-T*) – t = 180s RR(mm/h)

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Advection of moisture quantities qx

• implementation of the Bott (1989)-scheme into the Courant-number independent advection algorithm for moisture densities (Easter, 1993, Skamarock, 2004, 2006)

• ‚classical‘ semi-Lagrange advection with 2nd order backtrajectory and tri-cubic interpolation (using 64 points) (Staniforth, Coté, 1991)

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Problems found with Bott (1989)-scheme in the meanwhile:

2.) Strang-splitting ( 'x-y-z' and 'z-y-x' in 2 time steps) produces 2*dt oscillationsSolution: proper Strang-Splitting ('x-y-2z-y-x') in every time step solves the problem, but nearly doubles the computation time

1.) Directional splitting of the scheme:Parallel Marchuk-splitting of conservation equation for density can lead to a complete evacuation of cellsSolution: Easter (1993), Skamarock (2004, 2006), mass-consistent splitting

3.) metric terms prevent the scheme to be properly mass conserving <-- Schär–test case of an unconfined jet and ‚tracer=1‘ initialisation(remark: exact mass conservation is already violated by the 'flux-shifting' to make the Bott-scheme Courant-number independent)

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COSMO-ITA 2.8 km: comparisonRK+Bott / RK+Semi-Lagrange

RK+SL for light precipitation:TS is larger, whereas FBI is smaller than that for RK+Bott.

Moreover, RK+SL has slightly less domain-averaged precipitation and larger maximum prec. values than RK.

L. Torrisi

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Moisture transport in COSMO model:DWD: COSMO-DE: Bott-scheme used

COSMO-EU: SL scheme planned operationallyMeteoCH: COSMO-S2 and COSMO-S7: SL scheme used pre-operationallyCNMCA: COSMO-ITA 2.8: SL-scheme used pre-operationally

Semi-Lagrangian advection in COSMO-model

‚classical‘ semi-Lagrange advection (Staniforth, Coté, 1991) with 2nd order backtrajectory and tri-cubic interpolation (using 64 points)

SL is not positive definite clipping necessary 'multiplicative filling' (Rood, 1987) combines clipping with global conservation

problem: global summation is not ‚reproducible‘ (dependent from domain decomposition) -> solution: REAL -> INTEGER mapping

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Momentum equations for deep atmosphere (spherical coordinates):

Deep / shallow atmosphere

shallow atmosphere approximation: • r ~ a• neglect terms in advection and Coriolis force

deep atmosphere terms are implemented in COSMO 3.21

additionally:• introduce a hydrostatic, steady base state• transformation to terrain following coordinates

• diploma thesis R. Petrik, Univ. Leipzig• White, Bromley (1995), QJRMS• Davies et al. (2005), QJRMS

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Test case Weisman, Klemp (1982):

warm bubble in a base flow with vertical velocity shear + Coriolis force

wmax

RRdx= 2 km

precipitation distribution‚deep‘ (shaded), ‚shallow‘ (isolines) RRdeep- RRshallow (shaded)

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Case study ‚12.08.2004‘

summary for precipitation forecast in ‚deep‘, convection resolving models:• additional advection terms: not relevant• additional Coriolis terms:

• have a certain influence, but don't seem to be important for COSMO-DE application • could be important for simulations near the equator

(Diploma thesis R. Petrik)

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Physics coupling scheme

original idea: problems with reduced precipitation could be due to a nonadequate coupling between physics scheme and dynamics

Work to do:• what are the reasons for the failure of the WRF-PD-scheme in LM?

(turbulence scheme?)• Test different sequences of dynamics and physics (especially physics after

dynamics)

test tool (Bryan-Fritsch-case) is developed in PP ‚QPF‘, task 4.1

Problems in new physics-dynamics coupling (NPDC):

Negative feedback between NPDC and operational moist turbulence parameterization (not present in dry turbulence parameterization)

2-z - structures in the specific cloud water field (qc)

2-z - structures in the TKE field, unrealistic high values, where qc > 0

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Physics (I)• Radiation• Shallow Convection• Coriolis force• Turbulence

DynamicsRunge-Kutta [ (phys) + (adv) fast waves ]

‚Physics (I)‘-Tendencies: n(phys I)

Physics (II)• Cloud Microphysics

Physics-Dynamics-CouplingPhysics-Dynamics-Couplingn = (u, v, w, pp, T, ...)n

n+1 = (u, v, w, pp, T, ...)n+1

* = (u, v, w, pp, T, ...)*

‚Physics (II)‘-Tendencies: n(phys II)

+ n-1(phys II)

- n-1(phys II)

Descr. of Advanced Research WRF Ver. 2 (2005)

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Plans (RK-core, short, medium range)

• 3D- (isotropic) divergence filtering in fast waves solver

• implicit advection of 3. order in the verticalbut: implicit adv. 3. order in every RK-substep needs ~ 30% of total computational time! planned: use outside of RK-scheme (splitting-error?, stability with fast waves?)

• Efficiency gains by using RK4?

• Development of a more conservative dynamics (rho’-Theta’-dynamics?)

• diabatic terms in the pressure equation (up to now neglected, e.g. Dhudia, 1991)

• radiation upper boundary condition (non-local in time )

• continue diagnostics:

• convergence (mountain flows)

• conservation: mass, moisture variables, energy

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up1 cd2 up3 cd4 up5 cd6

Euler 1 0 0 0 0 0LC-RK2 0.5 0 0.437 0 0 0LC-RK3 0.419 0.577 0.542 0.421 0.478 0.364LC-RK4 0.348 0.707 0.436 0.515 0.433 0.446LC-RK5 0.322 0 0.391 0 0.329 0LC-RK6 0.296 0 0.385 0 0.311 0LC-RK7 0.282 0.252 0.369 0.184 0.323 0.159

Stability limit of the ‚effective Courant-number‘ for advection schemes

Ceff := C / s, s= stage of RK-scheme

Baldauf (2007), submitted to J. Comput. Phys.

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Higher order discretization on unstructured grids using Discontinuous Galerkin methodsUniv. Freiburg: Kröner, Dedner, NN.,   DWD: Baldauf

In the DFG priority program 'METSTROEM' a new dynamical core for the COSMO-model will be developed. It will use Discontinuous Galerkin methods to achieve higher order, conservative discretizations. Currently the building of an adequate library is under development. The work with the COSMO-model will start probably at the end of 2009. This is therefore base research especially to clarify, if these methods can lead to efficient solvers for NWP.

start: 2007, start of implementation into COSMO: 2009  

Plans (long range)

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Analytical solution (Klemp-Lilly (1978) JAS)

Investigation of convergence

solution with a damping layer of 85 levels and nRΔt=200.

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CONVERGENCE OF VERTICAL VELOCITY w

DX

DW

10-2 10-1 100 101 10210-6

10-5

10-4

10-3

10-2

L1L02nd order

HYDROSTATIC FLOW

L1 = average of errors

L = maximum error

Convergence slightly less than 2. order.(2. order at smaller scales?)

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NON LINEAR HYDROSTATIC FLOW

Stable and stationary solution of this non-linear case!

DX

DW

10-2 10-1 100 101 10210-4

10-3

10-2

10-1

100

L1L02nd order

NON LINEAR HYDROSTATIC FLOW

Convergence of vertical velocity wL1 = average of absolute errors

L = maximum error

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Operational timetableof the

DWD model suiteGME, COSMO-EU, COSMO-DE

and WAVE

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Equation system of LM/LMK in spherical coordinates

additionally:• introduce a hydrostatic, steady base state• Transformation to terrain-following coordinates• shallow/deep atmosphere

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(from spatial discretization of advection operator)

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How to handle the fast processes with buoyancy?

with buoyancy (Cbuoy

= adt = 0.15, standard atmosphere)

• different fast processes:1. operatorsplitting (Marchuk-Splitting): ‘Sound -> Div. -> Buoyancy‘2. partial adding of tendencies: ‘(Sound+Buoyancy) -> Div.')3. adding of all fast tendencies: ‘Sound+Div.+Buoyancy‘

• different Crank-Nicholson-weights for buoyancy:=0.6 / 0.7

• RK3-scheme• slow process: upwind 5. order• aspect ratio: dx/dz=10• dT/dt=6

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‘Sound -> Div. -> Buoyancy‘ ‘(Sound+Buoyancy) -> Div.') ‘Sound+Div.+Buoyancy'

=0.6

=0.7

curious result: operator splitting of the fast processes is not the best choice, better: simple addition of tendencies.

Csnd = cs t / xC

ad

v =

u

T /

x

amplification

factor

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balance equation for scalar :

Task 3: Conservation(Baldauf)Tool for inspection of conservation properties will be developed.

temporal change

flux divergence

sources / sinks

integration area = arbitrarily chosen cuboid (in the transformed grid, i.e. terrain-following)Status: available in LM 3.23:

• Subr. init_integral_3D: define cuboid (in the transformed grid!), prepare domain decomp.

• Function integral_3D_total: calc. volume integral V ijk Vijk

• Subr. surface_integral_total: calc. surface integrals V jijk * Aijk

• prelimineary idealised tests were carried out

• report finished; will be published in the next COSMO-Newsletter Nr. 7 (2007)

Task is finished

(Study of conservation properties will be continued in collaboration with MPI-Hamburg, see WG2 Task 2.10.1)

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(Mn-Mn-1) / t

total surface flux

total moisture mass M = x dV

Weisman-Klemp (1982)-test case

without physical parameterisation(only advection & condensation/evaporation)

Semi-Lagrange-Adv. for qx

with multiplicative filling

x := (qv + qc )

Res.

timestep

violation in moisture conservation (?)

Task 3:

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total moisture mass M = x dV

(Mn-Mn-1) / t

total surface flux

Res.

Weisman-Klemp (1982)-test case with warmer bubble (10 K)

without physical parameterisation,without Condensation/Evap.

Semi-Lagrange-Adv. for qx

with multiplicative filling

x := (qv + qc )

Residuum 0 advection seems to be ‚conservative enough‘

possible reasons for conservation violation:

saturation adjustment conserves specific mass (and specific energy)but not mass (and energy) itself !

timestep

Task 3:

Baldauf (2007), COSMO-Newsletter Nr. 7

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COSMO-ITA: RK+SL / RK+new Bott

RK+new Bott has a larger FBI for all precipitation thresholds than RK+SL (= COSMO-ITA operational run).

Moreover, RK+new Bott has a deterioration in MSLP bias and RMSE after T+12h.

SL

Bott

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Idealized 1D advection test

analytic sol.implicit 2. orderimplicit 3. orderimplicit 4. order

C=1.580 timesteps

C=2.548 timesteps

Verbesserte Vertikaladvektion für

dynamische Var. u, v, w, T, p‘