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Reformulation of the LM fast-waves equation part
including a radiative upper boundary condition
Almut Gassmann and Hans-Joachim Herzog
(Meteorological Institute of Bonn University, DWD Potsdam)
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Time integration is divided into 2 parts
1. Fast waves (gravity and sound waves)
2. Slow tendencies (including advection)
Short review…
n+1n*
F(n*)
nF(n)
Fast waves and slow tendencies
improper mode separation improper combination in
case of the Runge-Kutta-variants
Further numerical shortcomings
divergence damping vertical implicit weights:
symmetry: buoyancy term <-> other implicit terms
lower boundary condition
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2-time-level scheme (KW-RK2-short)
Comments on the Murthy-Nanundiah-test (Baldauf 2004)
The test relies on the equation
whose stationary solution is known as g . Baldauf claimed that the KW-RK2-short-scheme was not suitable since the splitting into fast forcing and slow relaxation does not yield the correct stationary solution. BUT: The splitting into slow forcing and fast relaxation does.What is our approch alike? -> slow forcing and fast relaxation Forcing: physical and advective processes, nonlinear onesRelaxation: wave processes, linear ones
gdtd
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2-dimensional linear analysis of the fast wave part
Vertical advection of background pressure and temperature
These terms are essential for wave propagation and energy
consistency
•Which is the state to linearize around?
•LM basic state (current)
or•State at timestep „n“, slow mode backgroundBrunt-Vaisala-Frequency for
the isothermal atmosphere
scale height
variables are scaled to get rid of the density
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Time scheme for fast waveshorizontal explicit – vertical implicit
divergence damping
symmetric implicitness(treatment as in other
implicit terms)vertical temperature advection
Remark: Acoustic and gravity waves are not neatly separable!
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Divergence dampingRelative phase change
Phase speeds of gravity waves are distorted.
With divergence damping Without divergence damping
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Symmetric implicitnessAmplification factor
unsymmetric
symmetric
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Vertical advection of temperatureRelative phase change
Phase speeds are incorrect. The impact in forecasts can hardly be estimated.
Without T-advection (nonisothermal atmosphere) With T-advection
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Conclusions from linear analysis No divergence damping! Symmetric implicit formulation! Vertical temperature advection
belongs to fast waves as well as vertical pressure advection! Further conclusion: state to linearize
around is state at time step „n“ and not the LM base state!
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Appropriate splitting
with
slow tendencies fast wavesin vertical advection for
perturbation pressure or temperature
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•prescribe Neumann boundary conditions
with access to
which is also used to derive surface pressure
„fast“ LBC•prescribe w(ke1) via
•prescribe metrical term
in momentum equation via
(Almut Gassmann, COSMONewsletter 4, 2004, 155-158)
Lower boundary conditionfast waves
„slow and fast“ LBC
In that way we avoid any computational boundary condition.
slow tendencies
„slow“ LBC
•prescribe
via
out of fast waves
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• Gassmann, Meteorol Atmos Phys (2004):“An improved two-time-level split-explicit integration scheme for non-hydrostatic compressible models“ • Crank-Nicolson-method is used for vertical advection.• Runge-Kutta-method RK3/2 is used for horizontal advection only and should not be mixed up with the fast waves part. • Splitting errors of mixed methods (Wicker-Skamarock-type) are larger.
Splitting slow modes and fast waves
Gain of efficiency: • No mixing of slow-tendency computation with fast waves• No mixing of vertical advection with Runge-Kutta-steps
K42
K5.33
K0
Background profile
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Mountain wave with RUBC
w-field
isothermalbackgroundand base state
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Strong sensitvity of surface pressure at the lee side of the Alps, if different formulations of metric terms in the wind divergence are used
Conservation form (not used in the default LM version), Direct control over in- and outflow across the edges
Alternative representation (used in the default LM version)
Divergence and metric terms
GvGa
uGaG
D )cos()cos(
1)cos(
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w
Gv
Ga
JuGa
Jva
ua
D 1)cos(
)cos()cos(
1)cos(
1
p
u
u
G
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Southerly flow over the Alps12UTC, 3. April 2005, Analysis
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Pressure problem at the lee side of the Alps – Reference LM
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Pressure problem at the lee side of the Alps – 2tls ALM
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Cross section: pressure problem
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Significance?
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Potential temperature
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Potential temperature
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Northerly wind over the Alps
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Northerly wind over the Alps
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Moisture profiles in Lindenberg with different LM-Versions (Thanks to Gerd Vogel)
7-day mean with oper. LM version and new version, dx=2.8 km
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Conclusions and plans Conclusions
The presented split-explicit algorithm is fully consistent and proven by linear analysis.
It needs no artificial assumptions and thus overcomes intuitive ad hoc methods.
Divergence formulation in terrain following is a very crucial point.
Plans Higher order advection and completetion for more
prognostic variables Further realistic testing Comparison with Lindenberg data
