a a
Low frequency acoustic waves from explosive
sources in the atmosphere
C. Millet1, C. Roblin1,2, J.-C. Robinet2 and X. Gloerfelt21 : CEA, Paris, France
2 : SINUMEF, Paris, France
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 1
a a
E. Blanc, CEA, Private Communication
Misty Picture Event (White Sands Missile Range, NM), May 14 1987
4.7 kt of ANFO at ground level (= 3.8 kt TNT)
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 2
Data from CEA / Sandia Nat. Lab. / Los Alamos Nat. Lab.a a
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 3
Data from CEA / Sandia Nat. Lab. / Los Alamos Nat. Lab.a a
18 19 20 21 22 23−1000
0
1000
2000
Kinney law
Admin Park 7.26 km
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 4
1 Dispersion-Relation-Preserving methodsa a
◦ High order finite difference DRP (Dispersion-Relation-Preserving) schemes
→ optimized spatial and time discretization
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
k∆x
k∆x
' ikf̂
ikf̂ '
1
∆x
8
<
:
NX
j=−M
ajeijk∆x
9
=
;
f̂
E =
Z π2
ε
˛
˛
˛
˛
˛
˛
iα −
NX
j=−M
ajeijα
˛
˛
˛
˛
˛
˛
d(ln α)
fx(x) '
1
∆x
NX
j=−M
ajf(x + j∆x)
(5,5)
(0,6)
(2,4)
For a given (M, N), the coefficients aj are chosen to minimize the error E over
the range of waves with wavelength longer than four ∆x (or α = k∆x < π/2)
→ radiation and outflow boundary conditions
→ perfect reflexion at the ground ⇒ py(y = 0) = 0
→ fully non linear Euler equations
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 5
1 Dispersion-Relation-Preserving methodsa a
◦ Normal mode solution
→ numerical integration of the Rayleigh (or wave) equation
−0.01 0 0.01 0.02 0.03 0.04−1
−0.5
0
0.5
1
1.5
2
2.5x 10−3
0
50
100
150
200
250
300
-1 -0.5 0 0.5 10
50
100
150
200
250
300
-0.4 -0.2 0 0.2 0.4
1
2
1
2
20 Hz
40 Hz φ1 φ2
Residue theorem
k-plane
p(r, z) ∼
X
n
φn(z)H(1)0 (knr)kn
Dk(kn)
D(kn; ω) = 0 : dispersion relation
Im(kn) < 0 : amplified wave
Im(kn) > 0 : damped wave
→ complex boundary conditions→ linearized Euler equations
◦ Ray tracing
→ the dispersion relation is given by the eikonal equation
→ high frequency approximation of the wave equation
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 6
Temperature and wind profiles at ground zeroa a
−60 −40 −20 0 200
0.5
1
1.5
2
2.5 x 105
−20 0 20 400
0.5
1
1.5
2
2.5 x 105
−20 0 200
0.5
1
1.5
2
2.5 x 104
0 500 10000
0.5
1
1.5
2
2.5 x 105
radiosonde
rocketsonde
ECMWF
MSISE 90HWM 93
damping zonev
u
: empirical global model of horizontal winds (HWM 93)
: radiosonde upper air observations at Stallion
: European Centre for Medium-range Weather Forecasts
u (m/s) v (m/s) T (K)
Z (m)
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 7
2 The atmospheric waveguidea a
0 200 400 600 800 10000
100
200
300
400
500
600
700
800
River Side
Silver City
Alpine
White River
Roosevelt
Lake Havasu
Las Vegas
Time−Distance/340 (s)
Dist
ance
(km
) 0 200 400 600 800 10000
100
200
300
400
500
600
700
800
River Side
Silver City
Alpine
White River
Roosevelt
Lake Havasu
Las Vegas
Time−Distance/340 (s)Di
stan
ce (k
m)
dis
tance
(km
)
Roosevelt (416 km West)
: travel-time by the ray tracing
time - distance/340 (sec)
distance (km)
alt
itude
(km
)
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 8
2 The atmospheric waveguidea a
0 200 400 600 800 10000
100
200
300
400
500
600
700
800
River Side
Silver City
Alpine
White River
Roosevelt
Lake Havasu
Las Vegas
Time−Distance/340 (s)
Dist
ance
(km
) 0 200 400 600 800 10000
100
200
300
400
500
600
700
800
River Side
Silver City
Alpine
White River
Roosevelt
Lake Havasu
Las Vegas
Time−Distance/340 (s)Di
stan
ce (k
m)
Stratospheric phase IS
Thermospheric phase IT
IS
IT
dis
tance
(km
)
Roosevelt (416 km West)
: travel-time by the ray tracing
time - distance/340 (sec)
distance (km)
alt
itude
(km
)
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 9
2 The atmospheric waveguidea a
0 200 400 600 800 10000
100
200
300
400
500
600
700
800
River Side
Silver City
Alpine
White River
Roosevelt
Lake Havasu
Las Vegas
Time−Distance/340 (s)
Dist
ance
(km
) 0 200 400 600 800 10000
100
200
300
400
500
600
700
800
River Side
Silver City
Alpine
White River
Roosevelt
Lake Havasu
Las Vegas
Time−Distance/340 (s)Di
stan
ce (k
m)
Stratospheric phase IS
Thermospheric phase IT
cusp singularity
IS
IT
dis
tance
(km
)
Roosevelt (416 km West)
: travel-time by the ray tracing
time - distance/340 (sec)
distance (km)
alt
itude
(km
)
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 10
3 Three-dimensional waveguidesa a
0.1 Hz
1.0 Hz1.0 Hz
Normal mode solution
◦ tropospheric phase
limitations of ray tracing
f = 1.0 Hz
◦ finite amplitude
f → ∞
◦ 3D waveguide
◦ caustics
◦ shadow zones
O. Gainville, CEA, Private Communication
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 11
3 Three-dimensional waveguidesa a
0.1 Hz
1.0 Hz1.0 Hz
limitations of ray tracing
f → ∞
◦ 3D waveguide
◦ caustics
◦ shadow zones
O. Gainville, CEA, Private Communication
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 12
3 Three-dimensional waveguidesa a
0.1 Hz
0.1 Hz1.0 Hz
limitations of ray tracing
f → ∞
◦ 3D waveguide
◦ caustics
◦ shadow zones
O. Gainville, CEA, Private Communication
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 13
4 Acoustic and instability wavesa a
0 1 2 3 4x 10−4
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.8 0.805 0.81 0.815 0.82 0.825 0.83 0.835 0.84 0.845 0.85−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
X: 0.8281Y: −0.07798
Re
Im
0.1 Hz
1.0 Hz
x 1E−04
4800 sec
5600 sec
400 sec
300 sec
acoustics
Kelvin Helmholtz
∼ u
∼ c + u
k2 - plane
multiscale spectral collocation technique
Normal modes by the
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 14
Conclusionsa a
◦ The Misty Picture experiment was studied using a DRP finite
difference scheme to solve the low frequency component of non linear
Euler equations over large distances. Most of phases were identified
using 2D calculations.
◦ Realistic temperature and winds profiles are required to compute
waveforms that are in agreement with measurements.
◦ The method is capable to capture features of Kelvin-Helmholtz insta-
bility waves and acoustics phases. Very good agreements were obtained
with the normal mode solution in the linear regime. Three-dimensional
tropospheric phases were identified.
59th Annual Meeting of the APS Division of Fluid Dynamics Nov. 19-21, 2006 15
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