michael baldauf deutscher wetterdienst, offenbach, germany

Deutscher Wetterdienst

1FE 13 – 20.04.23

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

report ‘Oct. 2007 – Sept. 2008’ / final report

COSMO General Meeting, Cracow15.-19.09.2008

Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany


2FE 13 – 20.04.23

Tasks of the Priority Project ‚Runge-Kutta‘:

Repair detected model deficiencies:1. Looking at pressure bias4. Advection of moisture quantities in conservation form6. Deep valleys7. Different filter options for orography14. DFI for RK

New developments:8. Higher order discretization in the vertical for Runge Kutta scheme9. Physics coupling scheme10. Testing of alternative fast wave scheme13. Divergence damping in a truely 3D-version

Tool development:2. Continue RK case studies3. Conservation tool5. Investigation of convergence

11. Development of a more conservative dynamics (planned)12. Development of an efficient semi-implicit solver in combination with RK time integration scheme (planned)


3FE 13 – 20.04.23

Task 1: Looking at pressure bias(Torrisi, Zängl)

Goals: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.Reasons and solutions?

Leapfrog

RKstarting point of the task:

Talk by Lucio Torrisi


5FE 13 – 20.04.23

Task 2: Continue RK case studies(deMorsier,Torrisi)Identify problems of the RK scheme

Several unstable cases found in previous winter periods (e.g. ‚13. Jan. 2004‘) most of them could be simulated with Semi-Lagrange Adv. for moisture variables

Winter storms Kyrill ('18.01.2007') and Lothar ('26.12.1999') simulated with MeteoCH new pre-operational model chain (2.2 km and 6.6 km): new configuration: 1.) WRF-like RK3 used (instead of TVD-RK3) (as found at DWD for the Kyrill case)2.) Semi-Lagrange-Adv. for moisture (instead of Bott-scheme)3.) new level distribution especially in boundary layer (cures problems with TKE scheme)


6

Vertical level distribution

Test-chains in July 2007 using operational (L60.2) and a new (L60.1) vertical level distributions

Three test cases 12.7.2006 (convection) 23.12.2006 (fog) 18.1.2007 (Kyrill)

BL

TP


7

TKE InstabilityL60 v1L60 v2

23.12.2006 (fog)gridpoint (60,180)

• Strong checkerboard instability in TKE-field• Same effect also 12.7.2006 (convection) and 18.1.2007 (Kyrill)


8

Stability of TKE-diffusion If equation is solved explicitly, stability constraints apply

In COSMO, the diffusion constant is limited

Default value for securi = 0.85 is wrong!!!

Alternative: vertically implicit (Crank-Nicholson) scheme was implementedthis cures the most problems; some artefacts remain (stability functions?)

O. Fuhrer


9FE 13 – 20.04.23

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)


10FE 13 – 20.04.23

Task 4: Advection of moisture quantities in conservation form(Förstner, Baldauf)

Status: two schemes available

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

Task was finished in Sept. 2006 because implemented schemes (Bott-2, Bott-4) behaved well

But in the meanwhile: stability problems occured in some cases (steep orography!)revival of the task necessary!

Semi-Lagrange-scheme (backtraj. 2nd order, tri-cubic interpolation)multiplicative filling algorithm for global conservation


11FE 13 – 20.04.23

Task 5: investigation of convergence(Ceci, Vitagliano)

Goals: determination of the spatial and temporal order of convergence of the RK-scheme in combination with advection schemes of higher order.

Test cases:• linear, 2D, hydrostatic mountain flow (h=10 m, a=10 km)• linear, 2D, non-hydrostatic mountain flow (h=10 m, a=500 m)• nonlinear, 2D mountain flows (dry case) (h=500 m, a=10 km)• linear, 3D mountain flow• nonlinear mountain flows with precipitation


12FE 13 – 20.04.23

Starting point: compressible Euler equations

Preconditions:• no friction• only adiabatic processes (in particular no phase changes)• ideal gas law• cp=const., cV=const., R=const.• no Coriolis force• all movements take place on a plane (no earth curvature)these preconditions can easily be fulfilled by a dynamical core (‚switches‘)

Only 2 approximations will be made:1. linearisation (1/Fr<<<1 very flat mountains; not too small U)2. the assumption that kz=const (see below; not absolutely necessary)

confidence into the accuracy of the linear solution for comparison with numerical models

Deliver a program to calculate linear analytic solutions(e.g. for convergence tests)

(M. Baldauf)


13FE 13 – 20.04.23

Stationary case (=0)From perturbation equations: express u', v', ' and p' by w' equation 2nd order for w'(kx, ky, z):

with coefficient functions:

The only approximation so far is linearisation!


14FE 13 – 20.04.23

Example 2: 2D-test case from Schaer et al (2002)

w [m/s]


15FE 13 – 20.04.23

Example 3: 3D Gaussian Hill

w [m/s]


16FE 13 – 20.04.23

Initialization of the perturbation pressure field

The present initialization of the perturbation pressure field (executed in src_artifdata for idealized simulations; otherwise in int2LM) is not exactly consistent with the discretized buoyancy term in the vertical momentum equation

The error is too small to be noticeable in real-case applications; however, it becomes evident in idealized simulations with constant flow and a very low mountain (or no mountain at all)

To fix the problem, a new initialization procedure has been developed by solving the discretized vertical wind equation (for dw/dt = 0) for p‘; ideally, this would ensure strict absence of buoyancy at the lateral model boundaries

G. Zängl


17FE 13 – 20.04.23

Simulation with flat surface, u = 10m/s, and fixed relaxation b.c.‘s, t = 12 hFields: θ (contour interval 2 K), w (colours)

Old p‘ initializationError amplitude: 1 mm/s

New p‘ initializationError amplitude: 10-4 mm/s


18Krakow - September, 15th 2008

Gaussian mountain height=750 m size=10 km

Horizontal resolution 4 km

3th order upwind 5th order upwind

W

0.700.500.300.10

-0.10-0.30-0.50-0.70

Gaussian Mountainh=750[m] a=10[km] N=0.01[1/s] U=10[m/s]

x=y=4km z=100mSolution at Y=0 symmetry plane - RK3 UP3

W

0.700.500.300.10

-0.10-0.30-0.50-0.70



3D TEST CASES: HYDROSTATIC FLOW

P. L. Vitagliano, G. Ceci


19Krakow - September, 15th 2008




W

0.700.500.300.10

-0.10-0.30-0.50-0.70



W

0.700.500.300.10

-0.10-0.30-0.50-0.70





20

W

0.700.500.300.10

-0.10-0.30-0.50-0.70



W

0.700.500.300.10

-0.10-0.30-0.50-0.70








21FE 13 – 20.04.23

CONVERGENCE OF KINETIC ENERGY

DX [km]

Err

or

No

rm

10-3 10-2 10-1 100 101 10210-6

10-5

10-4

10-3

10-2

10-1

L2L1L02nd order

NON-HYDROSTATIC TESTRK3 TVD

KINETIC ENERGY

DX [km]

Err

or

No

rm

10-3 10-2 10-1 100 101 10210-6

10-5

10-4

10-3

10-2

10-1

L2L1L02nd order

NON-HYDROSTATIC TESTRK3

KINETIC ENERGY



22FE 13 – 20.04.23

CONVERGENCE OF KINETIC ENERGY

DX [km]

Err

or

No

rm

10-2 10-1 100 101 10210-2

10-1

100

101

102

L2L1L02nd order

NON-LINEAR HYDROSTATIC TESTRK3 TVD

KINETIC ENERGY

DX [km]

Err

or

No

rm

10-2 10-1 100 101 10210-2

10-1

100

101

102

L2L1L02nd order

NON-LINEAR HYDROSTATIC TESTRK3

KINETIC ENERGY


23FE 13 – 20.04.23

• slightly less than 2nd order spatial convergence (fast waves scheme dominates)• TVD and non-TVD 3 stages Runge Kutta show similar behaviour• time step has minor effect (if any) on spatial convergence• important issues with upper and lateral boundary condition• difficult to compare with analytical solutions, due to b.c.

Conclusions from Convergence Tests:

3D mountain flow

2D mountain flow


24FE 13 – 20.04.23

Task 6: deep valleysGoal:detection of the reason for the unrealistic ‚cold pools‘ in Alpine valleys+ Task 7: Different filter options for orography(Förstner, Torrisi, Reinhardt, deMorsier)

Status: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.

example: For the COSMO-DE first the orography is filtered globally to remove scales approximately smaller than 4-. In a second step a stronger filter (5-) is used for all points with a step of the orography still bigger than 625 m.


25FE 13 – 20.04.23

Spurious noise over mountains in a resting atmosphere

Tests reveal a 2Δz structure in the horizontal and vertical wind field

Depending on the difference between base state and actual temperature profile, it can take more than 12 h until the noise reaches a significant amplitude

Afterwards, it rapidly grows within a time scale of a few hours until some sort of saturation is reached

Tests indicate that a modified discretization of the dw/dz term in the pressure tendency equation may damp the noise

(G. Zängl)

Setup of test experiments: mountain with h = 1500 m, a = 5 km; Δx = 1 km, no ambient winds; results are shown for t = 24 h


26FE 13 – 20.04.23

Results with implicit 2nd-order vertical advection θ (contour interval 1 K), u (colours)

standard discretization with damping discretization


27FE 13 – 20.04.23

Results with implicit 2nd-order vertical advection θ (contour interval 1 K), w (colours)



28FE 13 – 20.04.23

Results for quasi-linear flow over a mountain, h = 300 m, u = 10 m/s θ (contour interval 1 K), u (colours)



29FE 13 – 20.04.23

Spurious noise over mountains in a resting atmosphere

In the modified version, the term is not only evaluated between half-levels but also between full-levels (which damps 2Δz waves), followed by a weighting of both terms

A weight of 0.05 of the damping discretization turned out to suffice for eliminating the noise

Normally very small impact on flow dynamics, but stability problems over steep topography in the presence of strong winds


30FE 13 – 20.04.23

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)

Work to do: best combination with time integration scheme?


31FE 13 – 20.04.23

Comparison of the two implicit vertical advection schemesTest with constant vertical velocity; initial cone distribution

implicit cent. diff. 2nd order implicit cent. diff. 3rd order


32FE 13 – 20.04.23

Task 9: Physics coupling scheme(deMorsier, Förstner)

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

Work to do:Is the problem cured now also in the moist turbulence case with the

improvement of the TKE-Diffusion (solution of O. Fuhrer)?

Problems in new physics-dynamics coupling (NPDC) (=WRF-like coupling):

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


33FE 13 – 20.04.23

Task 10: Testing of alternative fast wave scheme(Torrisi, Gassmann)

Goals:• p‘T‘-RK-scheme• ‚shortened-RK2‘-scheme (Gassmann)• this allows the use of the ‚radiative upper boundary condition‘ (RUBC)

Properties of A. Gassmann dyn. core:• Splitting up of vertical advection of p*/T into fast/slow mode equations and consistent boundary conditions • Vertical average to half levels: mass weighted mean (in RK simple mean) and base-state consistent formulation of the discrete w-equation • Different horizontal pressure gradient discretization• Divergence in conservative flux-form• Slightly different buoyancy term • No artificial divergence damping


34FE 13 – 20.04.23

Status:• The fast waves part (Gassmann) is combined with the Leapfrog scheme in LM 3.21• Original Gassmann dynamical core poses stability problems in several cases!• Gassmann fast waves part in RK3 worked in only 1 case• ‚shortened RK2‘-scheme (Gassmann (2002), Gassmann and Herzog (2007)) is

implemented into LM 3.21 using the fast waves solver of RK3 and the RK3 advection/physics subroutines

• Preliminary investigation of this dynamical core (L. Torrisi)tested in real cases for a five days period: similar results to the RK3 splitting method

• Separate inspection of divergence in conservation form and vertical staggering• Implementation questions pointed out:

•Splitting of contravariant vertical velocity

poses problems in formulation of lower boundary conditions

Overall assessment: needs too much work to bring to operational use


35FE 13 – 20.04.23

Task 13: Divergence damping in a true 3D-version(Baldauf)

Description:Cases occured, where the up to now used 'quasi-2D' divergence filtering lead to unstable results. But a complete abandoning of the divergence filtering (as proposed by A. Gassmann for her dynamical core) also leads to several instabilities. This was also shown by stability analyses of the RK-core by M. Baldauf. P. Prohl (DWD) could demonstrate, that the Bryan-Fritsch- test case of a rising warm bubble is unstable with 'quasi-2D' divergence damping but becomes stable only with a full 3D (=isotropic) version (realised with a preliminary explicit formulation). For operational use an implicit version of 3D divergence damping is necessary.


36FE 13 – 20.04.23

*

Twice Digital Filter Initialization

An initialization scheme

Adiabatic backward integration

Diabatic forward integration

Task 14: DFI for RK (L. Torrisi)


37FE 13 – 20.04.23

• Some modifications (mostly in the adv functions ) are needed to run DFI with RK core. They are in: - dfi_initialization.f90: add initialization of rho_snow - src_runge_kutta.f90: correction in wind Rayleigh damping - src_advection_rk.f90: changes in cfl control and changes in adv function interfaces - fast_waves_rk.f90: changes in adv function interfaces - numeric_utilities_rk.f90: changes in adv function interfaces and correction to run with DFI

• All the odd order advection operators are changed to run in the backward integration of the DFI. The odd order advection operators implicitly contain a dissipative term that needs a special treatment in the backward integration of the DFI. The dissipative terms are treated as the horizontal diffusion operator in the backward integration of DFI (when dt<0 , -1 is multiplied to the dissipative term). For example: 5th order velo*ds/dx operator = 6th order velo*ds/dx operator + dissipative term * SIGN(1.,dt)

Sign(1.,dt)*

Digital Filter Initialization in RK core (L. Torrisi)


38FE 13 – 20.04.232h 2h

DFI seems to work well using a 7 km grid spacing

Digital Filter Initialization in RK core


39FE 13 – 20.04.23 1h 1h

Using a 2.8km grid spacing

DFI works only with explicit

vertical advection

COSMO-IT (2.8km)

Digital Filter Initialization in RK core


40FE 13 – 20.04.23

Optimization of horizontal advection:

up to COSMO 4.3: 'advection operators' = a subroutine acting on every single grid point compiler has problems to optimize loops

since COSMO 4.4: advection routines using 'field operations'(and additionally the DFI modifications of Lucio Torrisi)

Efficiency gain for routine COSMO-DE at DWD (IBM):• speedup of the horizontal advection alone: ~ 3 times faster • overall reduction of model run time: ~ 1 Min. / 20 Min. ~ 5%

Furthermore, some inconsistencies using metrical factors could be repaired: acrlat(j,1) acrlat(j,2) lent to an error of ~ -0.05% in the term v dw/dy

(M. Baldauf)


41FE 13 – 20.04.23

Introduction of RK-scheme into operational models

DWD:COSMO-DE (2.8km): since 16.04.2007COSMO-EU (7km): planned for ~Q4/2008 (if pressure bias problems removed) (weak artificial horizontal diffusion, SL-scheme, new aver. reference pressure)

MeteoCH:COSMO-S2: operational since April 2008COSMO-S7:

CNMCA:COSMO-IT (2.8km): since Oct. 2007COSMO-ME (7km): in next future


42FE 13 – 20.04.23

Publications of the PP 'Runge-Kutta'

Reviewed articles

• M. Baldauf (2008): Stability analysis for linear discretisations of the advection equation with Runge-Kutta time integration, J. Comput. Phys. 227, 6638-6659

Other articles• M. Baldauf (2008): A Tool for Testing Conservation Properties in the COSMO-Model (LM),

COSMO-Newsletter 7, 7-17• J. Förstner, M. Baldauf, A. Seifert (2006), Courant Number Independent Advection of the

Moisture Quantities for the LMK, COSMO-Newsletter 6, 51-64• L. Torrisi (2006): Sensitivity experiments with the Runge-Kutta time integration scheme, COSMO-

Newsletter No. 6

Final report: Draft version (12.09.2008) available


43FE 13 – 20.04.23

Thanks to all contributing scientists (in alphabetical order): Michael Baldauf1, Gabriella Ceci2, Jochen Förstner1, Oliver Fuhrer4,Almut Gassmann5,Hans-Joachim Herzog1,Guy deMorsier4,Thorsten Reinhardt 7,Gdaly Rivin6,Lucio Torrisi3,Pier Luigi Vitagliano2, Günther Zängl1 1 Deutscher Wetterdienst (DWD), Germany2 Centro Italiano Ricerche Aerospaziali (CIRA), Italy3 Centro Nazionale di Meteorologia e Climatologia Aeronautica (CNMCA), Italy4 MeteoSchweiz, Switzerland5 Max-Planck-Insitut, Hamburg, Germany6 Federal Service for Hydrometeorology and Environmental Monitoring, Russia7 Universität Köln, Germany


44FE 13 – 20.04.23


45FE 13 – 20.04.23

List of people contributing to the project during Oct. 2007 - Sept. 2008:(alphabetical order)

• Michael Baldauf (DWD, D)• Gabriella Ceci (CIRA, I)• Oliver Fuhrer (MeteoCH, CH)• Lucio Torrisi (CNMCA, I)• Pier Luigi Vitagliano (CIRA, I)• Gdaly Rivin (Roshydromet, RU)• Günther Zängl (DWD,D)

Additional meeting of PP-RK-Group during the LM-User-Workshop, Langen, 05.03.2008


4604.09.2007

(Mn-Mn-1) / t

total surface flux

total moisture mass M = x dV

Weisman-Klemp (1982)-test case

without physical parameterisation(only advection & Condensation/Evap.)

Semi-Lagrange-Adv. for qx

with multiplicative filling

x := (qv + qc )

Res.

timestep

violation in moisture conservation (?)

Task 3:


48FE 13 – 20.04.23

HYDROSTATIC LINEAR / NON LINEARa = 10 kmH = 10 m / 500 mTime = 60 h / 100 hdt = 2.5”Domain size 500x19.5 km2

Horizontal resolution = 4km, 2km, 1km, 500m, 250m, 125m

NON HYDROSTATICa = 500 mH = 10 mTime = 100 hdt = 2.5”Domain size 250x19.5 km2

Horizontal resolution = 1km, 500m, 250m, 125m, 62.5m

Convergence tests

• All test cases runned again with constant time step = 2.5”• Test cases repeated with non-TVD 3-stage RK

G. Ceci, P. L. Vitagliano


49FE 13 – 20.04.23

CONVERGENCE OF VERTICAL VELOCITY w

DX [km]

Err

or

No

rm

10-2 10-1 100 101 10210-6

10-5

10-4

10-3

10-2

L2L1L02nd order

HYDROSTATIC TESTRK3 TVD

VERTICAL VELOCITY

DX [km]

Err

or

No

rm

10-2 10-1 100 101 10210-6

10-5

10-4

10-3

10-2

L2L1L02nd order

HYDROSTATIC TESTRK3

VERTICAL VELOCITY



50FE 13 – 20.04.23

CONVERGENCE OF VERTICAL VELOCITY w

DX [km]

Err

or

No

rm

10-3 10-2 10-1 100 10110-5

10-4

10-3

10-2

10-1

L2L1L02nd order

NON-HYDROSTATIC TESTRK3 TVD

VERTICAL VELOCITY

DX [km]

Err

or

No

rm

10-3 10-2 10-1 100 10110-5

10-4

10-3

10-2

10-1

L2L1L02nd order

NON-HYDROSTATIC TESTRK3

VERTICAL VELOCITY

michael baldauf deutscher wetterdienst, offenbach, germany

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