Modeling the Effect of Elasticity on Internal
and External Waves
Celal S Konor
Department of Atmospheric Science
Colorado State University, Fort Collins, CO, USA
and
Akio Arakawa
Department of Atmospheric and Oceanic Sciences
University of California, Los Angeles, CA, USA
Workshop on Global High-Resolution Modeling, Fort Collins, CO, 15-17 June 2010
Wednesday, June 16, 2010
1
Why “elasticity” instead of “compressibility”?
Compressibility sounds like density change due to pressure change.
Elastic is opposite of anelastic.
Wednesday, June 16, 2010
2
Elasticity
Do we need it?
Yes
No
Don’t know / Don’t care
Why do we need it?
What are the consequences if we do not have it?
How much of it do we need?
How can we get it?
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3
Outline
A short introduction of the method we use to address the questions
A short introduction of the system of equations we use in this study
Behavior of (free and forced) internal and external modes in linearized
systems
Errors and distortions due to (vertical) discretization
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4
Inclusion of Elasticity into Different Systems
Elastic systems
Anelastic systems
FULLY-COMPRESSIBLE SYSTEM:
ANELASTIC SYSTEM:
Dw
= cp
Dt
NONHYDROSTATIC
COMPRESSIBLE
Elastic
t
g
z
(
( v)
=
w)
z
HYDROSTATIC
0 = cp
COMPRESSIBLE
Elastic
qs
t
0=
(
+b
z
(
0v)
Dw
= cp
NONHYDROSTATIC
Dt
(
v
( )
=
INCOMPRESSIBLE
0
0
w)
z
PSEUDO-INCOMPRESSIBLE SYSTEM:
g
z
Dw
= cp
Dt
Anelastic
QUASI-HYDROSTATIC SYSTEM:
qs
NONHYDROSTATIC
qs
qs
w
)
z
INCOMPRESSIBLE
Anelastic
0=
(
z
)
v +
+b
(
w
)
z
QUASI-HYDROSTATIC ANELASTIC SYSTEM:
UNIFIED SYSTEM:
NONHYDROSTATIC
HYDROSTATIC EQ.
COMPRESSIBLE
Elastic
Wednesday, June 16, 2010
HYDROSTATIC
Dw
= cp
Dt
qs
0 = cp
(
qs
z
w
)=
z
(
Anelastic
g
z
qs v
INCOMPRESSIBLE
)
qs
t
0 = cp
0=
qs
z
g
(
( v)
qs
qs
w
)
z
Quasi-hydrostatic = Quasi-static = Primitive
5
Elastic systems
FULLY-COMPRESSIBLE SYSTEM:
NONHYDROSTATIC
COMPRESSIBLE
Elastic
Dw
= cp
Dt
t
=
z
( v)
g
(
w)
z
The most basic system of equations we use in atmospheric models.
All variables are predicted.
Includes acoustic waves.
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6
Elastic systems
QUASI-HYDROSTATIC SYSTEM:
HYDROSTATIC
COMPRESSIBLE
Elastic
0 = cp
qs
t
=
qs
g
z
(
( v)
qs
qs
w
)
z
Excludes acoustic waves.
Subscript qs denotes quasi-hydrostatic portion of the variables.
Continuity equation can be diagnostic.
There two ways to obtain w (in the z vertical coordinate).
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Elastic systems
UNIFIED SYSTEM:
Dw
= cp
Dt
NONHYDROSTATIC
HYDROSTATIC EQ.
COMPRESSIBLE
qs
0 = cp
(
Elastic
qs
z
w
)=
z
g
z
(
qs
v
)
qs
t
Unifies the anelastic and quasi-hydrostatic systems.
Continuity equation is diagnostic.
Excludes acoustic waves.
No need for a basic or mean state.
Subscript qs denotes quasi-hydrostatic portion of the variables.
Prefix  denotes nonhydrostatic portion of the variables.
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Anelastic systems
ANELASTIC SYSTEM:
Dw
= cp
NONHYDROSTATIC
Dt
INCOMPRESSIBLE
Anelastic
0=
(
0
0
+b
z
v)
(
0
w)
z
Continuity equation is diagnostic.
Excludes acoustic waves.
Prefix  denotes nonhydrostatic portion of the variables.
Needs a basic state.
Surface pressure is a diagnostic quantity.
Wednesday, June 16, 2010
9
Anelastic systems
PSEUDO-INCOMPRESSIBLE SYSTEM:
Dw
= cp
NONHYDROSTATIC
Dt
INCOMPRESSIBLE
Anelastic
0=
(
z
)
v +
+b
(
w
)
z
Continuity equation is diagnostic.
Excludes acoustic waves.
Prefix  denotes nonhydrostatic portion of the variables.
Needs a mean state.
Surface pressure is a diagnostic quantity.
Wednesday, June 16, 2010
10
Elastic Systems
The acoustic waves are filtered with the unified system without any modification to
the dispersion of the inertia-gravity and Lamb waves.
The acoustic waves are filtered with the quasi-hydrostatic system while the dispersion
of waves with smallest horizontal scales is greatly modified.
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Anelastic Systems
Lack of elasticity does not negatively affect the dispersion of free internal modes on
a mid-latitude f-plane with the anelastic system.
Dispersion of the deepest modes is modified with the pseudo-incompressible system.
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Vertical Structure of Free Modes
The vertical structure of free modes on a mid-latitude f-plane is distorted with the
anelastic system due to lack of recognition of vertical stability.
Most likely, lack of elasticity is not the cause of the phase error.
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13
Elastic Systems
Anelastic Systems
Lack of elasticity modifies the dispersion of ultra-long waves with the anelastic and
pseudo-incompressible systems.
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14
Hypothesis:
Fast retrogression speed of ultra-long waves appearing in the early
NWP runs (obtained by the numerical solutions of the hemispheric
barotropic vorticity equation) is related to lack of elasticity.
Wednesday, June 16, 2010
15
The Error in Numerical Forecasts Due to Retrogression of
Ultra-Long Waves
(With Hemispheric Barotropic Forecast Model)
(Wolff, P. M., 1958, MWR)
Speed of incompressible Rossby wave: c = − β
Initial Wavenumber 1
k
k → 0, c → − ∞
2
48-Hour Forecast Wavenumber 1
---
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+
+
16
Confirmation:
Adding an ad hoc procedure representing the effect of elasticity to the
barotropic vorticity equation slows down the ultra-long waves.
Projection:
We may need an “ad hoc elasticity” included in a global anelastic
model.
Wednesday, June 16, 2010
17
Elasticity of Forced Modes
Continuity equation:
∂ρ
= HMC + VMC
∂t
Divergent horizontal motion must necessarily be elastic.
Definition of Elasticity:
1 ∂ρ HMC
ε≡
=
+1
VMC ∂t VMC
For purely horizontal motion,  is infinity.
For anelastic motion,  is zero.
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singularity
Elasticity of Forced External Modes
n is a real number
for forced modes
A singularity forms in the field of  where the motion is purely horizontal.
As n approaches to zero, elasticity approaches to approx. -1.35, which means that
HMC becomes 2.35 times larger than WMC for small values of n. A large portion
of HMC is compensated by elasticity.
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Errors Due to Vertical Discretization
 needs to be averaged to
determine buoyancy b
 needs to be vertically averaged
to determine density 
Vertical  advection needs to be
averaged
w needs to be vertically averaged
to calculate vertical advection of 
 needs to be averaged in the
hydrostatic equation
L and CP stand for Lorenz and Charney-Phillips, respectively
Wednesday, June 16, 2010
20
Vertically Discrete Fully-Compressible and Unified
Systems
For the fully-compressible system, vertically propagating acoustic modes are not shown
in this figure.
The two-z-wave is not recognized and the dispersion of modes with high vertical
wavenumbers is greatly modified with the L-grid.
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21
Vertically Discrete Fully-Compressible System
Acoustic waves
Acoustic waves
Acoustic waves
nmax=160
nmax=160
The performance of the L- and CP-grids is highly comparable in simulating vertically
propagating acoustic waves.
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22
Vertically Discrete Quasi-Hydrostatic System
nmax=160
Wednesday, June 16, 2010
nmax=160
23
Vertically Discrete Anelastic System
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24
Vertically Discrete Fully-Compressible, Unified
and Quasi-Hydrostatic Systems
The two-z-wave is not recognized and the dispersion of modes with high vertical
wavenumbers is greatly modified with the L-grid.
Wednesday, June 16, 2010
25
Conclusions
Lack of elasticity does not seem to affect the dispersion of inertia-gravity
modes.
There is a problem with the vertical structure of these modes with the
anelastic system but this is not due to the lack of elasticity.
Lack of elasticity greatly affects the dispersion of the (ultra-long) Rossby
waves.
Elasticity appears to be important for the very deep forced modes.
For the vertical discretization, the use of the CP-grid is advantageous
over the L-grid.
Bottom line: Effects of elasticity may play a more important role in
global models than that in regional models.
Wednesday, June 16, 2010
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