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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