Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany

19.09.2006

Aktionsprogramm 2003

M. Baldauf, DWD 1

COSMO Priority Project:Further developments of the Runge-Kutta Time Integration Scheme

COSMO General Meeting

Bukarest, 18.-21.09.2006

Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany

19.09.2006


M. Baldauf, DWD 2

List of people contributing to the project: (alphabetical order)

• Michael Baldauf (DWD, D)• Gabriella Ceci (CIRA, I)• Guy deMorsier (MeteoCH, CH)• Jochen Förstner (DWD, D)• Almut Gassmann (Univ. Bonn, D) (FTE not counted)

• Paola Mercogliano (CIRA, I)• Thorsten Reinhardt (DWD, D)• Lucio Torrisi (CNMCA, I)• Pier Luigi Vitagliano (CIRA, I)• Klaus Stephan (DWD, D) (FTE not counted)

• Matthias Raschendorfer (DWD, D) (FTE not counted)

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Task 1: Looking at pressure bias(Torrisi, Förstner)verifications of LM 7 km runs showed a higher positive pressure bias for the RK core than for the Leapfrog core, whereas other variables show comparable behaviour.

Work done:5-day verifications were done for several model

configurations

only little impact on PMSL by:• physics coupling• advection of qx (Bott, Semi-Lagrange)

most significant impact on PMSL:• new dynamical bottom boundary condition (DBBC)

( Task 6)• p‘T‘-dynamics in the RK-coreboth measurements reduce the pressure bias

verification area

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M. Baldauf, DWD 4

Task 1: Looking at pressure bias

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positive impact of p‘T‘-dynamics


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positive impact of DBBC


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Status: verifications carried out

Work to do: longer verifications periods should be inspectedUpper air verificationsLateral boundary conditions pressure bias improved by another fast waves solver?? ( --> task 10)

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Task 2: Continue RK case studies(Torrisi, deMorsier)extensive verification of the other tasks

Status: test case carried out; verifications were made

Work to do: test cases should be continued

Work done:case study 4.-8.Dez. 2004 (inversion in 800...1600m)was carried out: penetration of the stratus in Alpine valleys in 2km- sim. better performed by RK-core compared to LF-core

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M. Baldauf, DWD 9

Status: • integral over a volume (arbitrary square-stone): ready

• Subr. init_integral_3D: define square-stone (in the transformed grid!), domain decomp. • Function integral_3D_total: calc. volume integral• Function integral_3D_cond: calc. vol. integral over individual processor

Work to do• flux integral over the surface

balance equation for scalar :

Task 3: Conservation(Baldauf)Tool for inspection of conservation properties will be developed. Integration area = arbitrarily chosen cubus (in the transformed grid, i.e. terrain-following)

temporal change

flux divergence

sources / sinks

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M. Baldauf, DWD 10

Task 4: Advection of moisture quantities in conservation form(Förstner)implementation of a Courant-number independent advection algorithm for the moisture densities

Status: implemented schemes (Bott-2, Bott-4) behave well(Semi-Lagrange-scheme as a testing tool is also available)

task finished

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Transport of TracerTransport of Tracerin a Real Case Flow Fieldin a Real Case Flow Field

initinit

Bott (2Bott (2ndnd))“Flux Form“Flux Form

- DIV”- DIV”+ Clipping+ Clipping

Bott (2Bott (2ndnd))“Conserv. “Conserv.

Form”Form”

semi-semi-LagrangeLagrange((tri-cubic)tri-cubic)+ Clipping+ Clipping

PP RK 2.1.4 + 2.1.3

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Task 5: investigation of convergence(Ceci, Vitagliano, Baldauf)determination of the spatial and temporal order of convergence of the RK-scheme

in combination with advection schemes of higher order.

Planned test cases:• linear mountain flows (2D, 3D)• nonlinear mountain flows (dry case)• nonlinear mountain flows with precipitation

Status: implementation of LM and test environment. First tests with linear mountain flow.

Kinetic energy (v.l. 120,160)

8,700E-04

8,900E-04

9,100E-04

9,300E-04

9,500E-04

9,700E-04

0 2 4 6 8 10

Horizontal Resolution [km]

Kin

eti

c e

ne

rgy

[m

^2

/s^

2]

Work to do: determine L2, L – errors of KE, w, ..., dependent from x, t, ... for the tests cases

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Task 6: deep valleys(Förstner, Torrisi, Reinhardt, deMorsier)detection of the reason for the unrealistic ‚cold pools‘ in Alpine valleys

Task 7: Different filter options for orography(Förstner)

Status: the orography filtering is now sufficiently weak for DWD-LMK applications (max. slopes 30% allowed)

The reason for the cold pools was identified: metric terms of the pressure gradient

Dynamical Bottom boundary condition (DBBC) (A. Gassmann (2004), COSMO-Newsl.) and a slope-dependent orography-filtering cures the problem to a certain extent.

Proposal for future work:inspect the limitations of the terrain following coordinate for steeper slopes, e.g. • for application of aLMo 2 (MeteoCH) in Alpine region• for future LMK ~1 km horizontal resolution

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M. Baldauf, DWD 14

starting point after 1 h after 1 h

modified version:pressure gradient on z-levels, if

|metric term| > |terrain follow. term|

cold pool – problem in narrow valleys

is essentially induced by pressure gradient term

T (°C)

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Dynamic BottomBoundary...

... Condition...... for metric pressure gradient term in equation for u- and v-component.Gaßmann (COSMO Newsletter No. 4)

“(Positive) Pressure Bias Problem”blue: Old Bottom Boundary Cond.red: Dynamic Bottom Boundary Cond.(Figures by Torrisi, CNMCA Rom)

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Improved vertical advection for the dynamic var. u, v, w, T (or T‘), p‘

motivation: resolved convection

• vertical advection has increased importance => use scheme of higher order (compare: horizontal adv. from 2. order to 5. order)

• => bigger w (~20 m/s) => Courant-crit. is violated =>implicit scheme or CNI-explicit scheme

up to now: implicit (Crank-Nicholson) advection 2. order (centered differences)

new: implicit (Crank-N.) advektion 3. order LES with 5-banddiagonal-matrix

but: 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?)

Task 8: Higher order discretization in the vertical for RK-scheme(Baldauf)

Status: implicit scheme of 3. order implemented (5-banddiagonal solver, ...)

Work to do: best combination with time integration scheme?test suite; verification

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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

Task 8: Improved vertical advektion for dynamic var. u, v, w, T, p‘

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case study ‚25.06.2005, 00 UTC‘

total precipitation sum after 18 hwith vertical advection 2. order

difference total precpitation sum after 18 h‚vertical advection 3. order – 2. order‘

Task 8: Improved vertical advektion for dynamic var. u, v, w, T, p‘

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Task 9: Physics coupling scheme(Förstner, Stephan, Raschendorfer)original task: problems with reduced precipitation could be due to a nonadequate coupling between physics scheme and dynamics

Status: NPDC-scheme analogous to WRF was implemented.Problems occuring reduced variant is used now

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

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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Task 10: Testing of alternative fast wave scheme(Gassmann, Förstner, Baldauf)• p‘T‘-RK-scheme• ‚shortened-RK2‘-scheme (Gassmann)• this allows the use of the ‚radiative upper boundary condition‘ (RUBC)

Status:• p‘T‘-RK-scheme is already tested and is used now in LMK• ‚shortened RK2‘-scheme works• RUBC is tested in idealized and one real test case

Work to do:• implement ‚shortened RK2‘ version in official LM version• experiments especially with RUBC• tests both versions in CLM-application (dx=18 km)

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contours: vertical velocity isolines: potential temperature

Runge-Kuttaold p*-T-dynamics

Runge-Kuttanew p*-T*-dynamics

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Choose CN-parameters for buoyancy in p‘T‘-dynamics from stability analysis=0.5 (‚pure‘ Crank-Nic.) =0.6 =0.7

=0.8 =0.9 =1.0 (purely implicit)

choose =0.7 as the best value

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• grid length: x = 2.8 km

• direct simulation of the coarser parts of deep convection

• interactions with fine scale topography

• timestep t=30 sec.

• 421 x 461 x 50 grid points~ 1200 * 1300 * 22 km³lowest layer in 10 m above ground

• forecast duration: 18 hstarted at 0, 3, 6, 9, 12, 15, 18, 21 UTC

• center of the domain 10° E, 50° N

• boundary values from LME(x = 7 km)

LMK (Lokal-Modell Kürzestfrist)

in pre-operational mode at DWD since 14.08.2006

Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany

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