Gravity Wave Symposium
Superpressure Balloon Studies of Atmospheric Gravity Waves in the
Stratosphere
Robert Vincent Jennifer Haase, Albert Hertzog, Weixing Zhang
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Gravity Wave Symposium
Introduction• Super pressure balloons are a powerful way to derive
gravity wave momentum fluxes
• Problems occur at periods shorter than about 20 min
• High precision GPS measurements offer a way to derive momentum fluxes at short periods
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Gravity Wave Symposium
SPB Equation of Motion
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Following Nastrom (1980) the equation of motion in the vertical direction is:(MB + ηMa)
�2ζ �b
�t2 = �g(MB � Ma) � 12 ρaCdAB
��ζ �
b�t � w�
� ����ζ �b
�t � w���� +
(MB + ηMa)�w�
�ti.e. net force = buoyancy force + drag force + dynamic force
This can be simplified to�2ζ �
b�t2 = �ω2
Bζ�b + 2
3gR � A�
�ζ �b
�t � w�� ����ζ �
b�t � w�
��� + �w�
�t (1)where
ω2B = 2g
3T
��T�z + g
Ra
�
A = Cd4r
R = wave induced density perturbationωB = Buoyancy frequency
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For a gravity wave of intrinsic frequency ω and vertical velocity amplitude wothe instantaneous vertical velocity is w� = woe�iωt
R = ρ�
ρ = i N2
gωw�
N2 = gT
�gcp + �T
�z
�
(1) can be solved numerically
Example:Wave with 15 min intrinsic period, wo=1 ms�1
Red curve is numerical solution
Response contains only odd harmonics
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Wave packet τ = 15 min
Harmonic distortion increases as wave amplitude increases
Gravity Wave Symposium
Quasi-Analytic Solution
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If the vertical displacement of air parcel is ζ � then let ζ �bζ � = Z = |Z|eiφ
(1) becomes:�ω2ζ �
b = �ω2Bζ
�b + 2
3N2ζ � � A(�iωζ �
b + iωζ �)| � iωζ �b + iωζ �| � ω2ζ �
Hence:
Z =23N
2�ω2�iAω2ζoYω2B�ω2�iAω2ζoY
(2)
where Y = |1 � Z|
Iterative solution for (2) with quick convergence.
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Note the limiting values of |Z| and φ when the balloon is on itsequilibrium density surface (EDS) - isopycnic balloon.
From (2) |ZEDS| � 2N2
3ω2B
=
�gcp
+ dTdz
�
( gRa
+ dTdz )
� 0.3
For referenceAntarctic:
N = 0.023 rad s�1 τN = 4.6 minωB = 0.034 rad s�1 τB = 3.1 minZEDS = 0.31
Equator:N = 0.025 rad s�1 τN = 4.2 minωB = 0.035 rad s�1 τB = 3.0 minZEDS = 0.34
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��� ζbζ���
��� ζbζ���
φ
φ
τ = 60 min
τ = 15 minNote significant phase shift, φ(ω),when τ � 20 min
Gravity Wave Symposium
SPB Measurements
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Horizontal position: x(t), y(t) � u(t), v(t)
Vertical position: ζb
Pressure: p�T = p� + dp
dz ζ�b
p� = p�T + ρgζ �
b
Temperature: T�T = T� + dT
dz ζ�b
Process observations in wavelet space as a function of (ω, t)
Im�pTu||
�= �ρHN2
ω Re�u�
||w�
Systematic errors in flux retrieval for τ � 20 min
Gravity Wave Symposium
Characteristics of GW Generated by Convection
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Significant part of momentum flux spectrum at periods less than 20 min
Typical phase speeds c � 0 � 60 ms�1
Typical horizontal wavelengths λh � 10 � 100 km
Gravity Wave Symposium
High-Resolution GPS Observations
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Zhang et al., (2016) Improvement of stratospheric balloon GPS positioning andthe impact on gravity wave parameter estimation for the Concordiasi campaignin Antarctica, JGR, (under review).
�h � 0.1 m�u � 0.003 ms�1
��b � 0.2 m
Gravity Wave Symposium
I - Direct Computation of w′
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(Simple version)
1. In wavelet space find peak value of u2|| � tmax, �max
2. Estimate peak value of �b at this location
3. Find range of periods that are significant
4. Make an initial guess of �o = �b/ZEDS
5. Compute Z from (2), derive new guess of �o = �b/Re(Z)
6. Iterate until convergence � Zf � �(�, t) = �b(�, t)/Zf
7. Compute fluxes directly in wavelet space
Gravity Wave Symposium
II -Direct Computation of w′
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u�|| � �
b
Input
Output
� � before phase correction � � after phase correction
Notice distortion
τ = 12 minc = 25 ms�1
θ = 45�
u|| = 2 ms�1
Gravity Wave Symposium
III-Direct Computation of w′
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Flux Retrievals - Single Wave
τ = 10 minc = 26 ms�1
θ = 135�
Old Method New Method
With noise
Without noise
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Flux Retrievals - Multiple Waves
Up going
Down going
Gravity Wave Symposium
Conclusions• Objective method for converting vertical SPB
displacements to vertical parcel displacements
• Requires high resolution GPS measurements of SPB position in horizontal and vertical
• Estimate fractional wave fluxes to ~2%
• Results important for flux retrievals over convection
• Easy to extend to derive temperature perturbations
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