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A baroclinic instability test
case for dynamical cores
of GCMs
Christiane Jablonowski (University of Michigan / GFDL)
David L. Williamson (NCAR)
AMWG Meeting, 3/20/06
Overview
• Basic Idea & Design Goals
• Derivation of the test case & discussion of the initital conditions
• 4 dynamical cores: NCAR CAM3 & DWD’s GME
• Results of the test case
– Steady-state conditions
– Evolution of the baroclinic wave
– Uncertainty of the ensemble of reference solutions
• Conclusions
The Idea & Design Goals
• Goal: development of a dynamical core test case (without physics, dry & prescribed orography) that – is easy to apply
– is idealized but as realistic as possible
– gives quick results
– starts from an analytic initial state, suitable for all grids
– triggers the evolution of a baroclinic wave
• Designed for primitive equation models with pressure-based vertical coordinates (hybrid, sigma or pure pressure coordinate)
Derivation of the Initial Conditions
• Initial data required: u, v, T, ps, s
• Find a steady-state, balanced solution of the PE eqns: prescribe u, v and the surface pressure ps
• Plug prescribed variables into PE equations and derive the– Geopotential field : based on the momentum equation for v
(integrate), calculate surface geopotential s
– Temperature field: based on the hydrostatic equation
dv
dtu2 tana
1
a
RdT ln p
fu
RdT
p
p
'
RdT
'
Initial Conditions• v = 0 m/s
• ps = 1000 hPa
• u & s
Initial Temperature Field
Perturbation
• Overlaid perturbation (at each model level) triggers the evolution of a baroclinic wave over 10 days
• Suggested: pertubation of the zonal wind field ‘u’ orthe vorticity and divergence (for models in - form)
Characteristics of the Initial Conditions
• Instability mechanisms:– Baroclinic instability - vertical wind shear
– Barotropic instability - horizontal wind shear
• But:– Statically stable
– Inertially stable
– Symmetrically stable
Characteristics
• Static stability
Characteristics
• Symmetric stability & Inertial stability
Test Strategy
• Initialize the dynamical core with the analytic initial conditions (balanced & steady state)
• Let the model run over 30 days (if possible without explicit diffusion)
• Does the model maintain the steady state?
• Perturb the initial conditions with a small, but well-resolved Gaussian hill perturbation
• 10-day simulation: Evolution of a baroclinic wave
Step 1:
Step 2:
Model Intercomparison
• Eulerian dynamical core (EUL), spectral
• Semi-Lagrangian (SLD), spectral
• Finite Volume (FV) dynamical core (NCAR/NASA/GFDL)
• Icosahedral model GME (2nd order finite difference approach)
NCAR CAM3:
German Weather Service (DWD):
Resolutions
• Default: 26 vertical levels (hybrid) with model top ≈ 3hPa.In addition: 18 and 49 vertical levels were tested.
• EUL & GME: varying horizontal diffusion coefficients K4, no explicit diffusion in SLD and FV simulations.
EUL / SLD (truncation)
FV (lat x lon, degrees)
GME(min. grid distance)
T21 4 x 5 ≈ 440 km (ni = 16)
T42 2 x 2.5 ≈ 220 km (ni = 32)
T85 1 x 1.25 ≈ 110 km (ni = 64)
T170 0.5 x 0.625 ≈ 55 km (ni =128)
T340 0.25 x 0.3125 ≈ 26 km (ni = 256)
Steady-State Test Case
• Maintenance of the zonal-mean initial state (u wind)
Wave number 5 effect
Decentering
parameter effect,with =0 EUL is matched
Steady-State Test Case: GME• GME shows a truncation error with wave number 5
• Artifact of computational grid and low-order num. method
Steady-State Test Case as a Debugging Tool
• Discovery of a flaw in the SLD dynamical core (old CAM2 version):
• systematic decrease in the zonal wind speed over time
• Here: plotted for 30 days with an older version of the test case (umax = 45m/s)
Baroclinic Waves: 30-day Animation
Movie: Courtesy ofFrancis X. Giraldo, NRL, Monterey
Surface pressure
Surface temperature
North-polar stereographicprojection
Evolution of the Baroclinic Wave
• Perturbation gets organized over the first 4 days and starts growing rapidly from day 6 onwards
FV 0.5 x 0.625 L26 dycore
T (850 hPa)ps
Evolution of the Baroclinic Wave• Explosive cyclogenesis after day 7• Baroclinic wave breaks after day 9
FV 0.5 x 0.625 L26 dycore
T (850 hPa)ps
Convergence with Resolution• Surface pressure starts converging at 1 x 1.25 degrees
FV L26 dycore, Day 9
Model Intercomparison at Day 9
• Second highest resolutions, L26
• ps fields visually very similar
• Spectral noise in EUL and SLD
Model Intercomparison at Day 9
• ps fields visually almost identical
• Differences only at small scales
850 hPa Vorticity at Day 7
• Differences in the vorticity fields grow faster than ps diff.
850 hPa Vorticity at Day 9• Small-scale differences easily influenced by diffusion• Spectral noise in EUL and SLD (L26)
Impact of explicit diffusion• EUL T85L26 with K4 increased by a factor of 10 (1 x 1016 m4/s)• No spectral noise, but severe damping of the circulation
Model Intercomparisons: Uncertainty
• Estimate of the uncertainty in the reference solutions across all four models using l2(ps)
Single-Model Convergence• Single-model uncertainty stays well below the uncertainty
across models
• Models converge within the uncertainty for the resolutions T85 (EUL & SLD), 1x1.25 (FV), GME (55km / ni=128)
Uncertainty in the Relative Vorticity• Estimate of the uncertainty in the reference solutions
using l2[ (850 hPa)]
• Errors grow faster, but conclusions are the same
Vertical Resolutions• Model runs with 18 and 49 levels at mid-range horizontal
resolutions are compared to the default 26-level runs
• Uncertainty stays well below the uncertainty across the models
Phase Errors• Phase errors diminish with increasing resolutions
• Phase error at lower resolutions is substantial for GME, attributes to the relatively ‘late’ convergence of GME (55 km)
Energy Fixer: SLD Dynamical Core
• Baroclinic wave test revealed problem in the energy fixer of the SLD dynamical core (old CAM2 version)
SLD problem with the energy fixer
corrected
Conclusions
• Goal: Development of an easy to use baroclinic wave test case that serves as a debugging tool and and fosters model intercomparisons
• Test of the models as they are used operationally (no extra diffusion, no special tuning of parameters)
• Established an ensemble of reference solutions and their uncertainty
• Models converge within the uncertainty at the resolutions EUL & SLD T85, FV 1 x 1.25, GME (55km/ni=128)
• Convergence characteristics the same for T or variable• Accessibility: We make the ensemble of solutions (ps)
available to all interested modeling groups and offer to compute l2(ps) norms
Publications
• Jablonowski, C. and D. L. Williamson, 2006a: A Baroclinic
Wave Test Case for Dynamical Cores of General Circulation
Models: Model Intercomparisons, NCAR Technical Note TN-
469+STR, 89 pp. (available online at
http://www.library.ucar.edu/uhtbin/cgisirsi/TRm6NSmtE3/0/261320020/503/7631
(shortly: also available on NCAR’s CAM3 web page)
• Jablonowski, C. and D. L. Williamson, 2006b: A Baroclinic
Instability Test Case for Atmospheric Model Dynamical Cores,
Quart. J. Roy. Meteor. Soc., in review