Lecture 2: Maintenance of the the main reservoirs
Four lectures on the mechanical energy
budget of the World Ocean
• Lecture 1: What are the mechanical energy reservoirs in the
World Ocean?
•
Lecture 2: How are the energy reservoirs
maintained? What’s the rate of forcing of
the geostrophically balance flow?
• Lecture 3: Pathways to dissipation.
• Lecture 4: Relation of mechanical energy dissipation and
abyssal mixing.
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Lecture 2: Maintenance of the the main reservoirs
Recurring themes:
• Geostrophically balanced flow is of central interest.
• Critical Assessment of two influential review articles (Wunsch
and Ferrari , 2004; Ferrari and Wunsch, 2009).
• Ultimately Wunsch and Ferrari (2004); Ferrari and Wunsch
(2009) are interested in the energy available to drive abyssal
mixing.
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Lecture 2: Maintenance of the the main reservoirs
Outline of lesson 2, Tuesday Feb 22, 2011
• Brief statement on the three fundamental energy equations.
• How are the principle energy reservoirs of the World Ocean
maintained?
• How is the geostrophically balanced flow maintained?
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Lecture 2: Maintenance of the the main reservoirs
Five main reservoirs of mechanical energy
In order of decreasing timescale:
1. General Circulation (20 × 1024 J): Long time mean.
2. Mesoscale Eddies (13 × 1018 J): Days and longer.
3. Internal Waves (1.4 × 1018 J): |f | < ω < N
4. Internal Tides (0.1 × 1018 J): Discrete frequencies mostly
semidurnal and diurnal.
5. Surface Waves and Turbulence (11 × 1018 J): Seconds to
minutes.
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Lecture 2: Maintenance of the the main reservoirs
The 3 fundamental energy equations
• Wunsch and Ferrari (2004) list three energy equations: kinetic
energy KE, the gravitational potential energy PE, and the
internal energy I.
• The internal energy is not usually considered in the mechanical
energy budget, but they need to include it because they allow
for fluid expansion, which converts internal energy back to
mechanical energy.
• We’ll just summarize the important points. See Vallis (2006,
section 1.10.2) for discussion of inviscid energetics.
• Working with simplified equations, e.g. Boussineq, primitive,
quasi-geostrophic etc., one finds very different discussion of
energetics. But we should be able to understand the energetics
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Lecture 2: Maintenance of the the main reservoirs
of the real/complete equations. If the dominant terms of the
real system are not represented in the simplified equations,
then that’s cause for concern about the usefulness of the
simplified equations.
• Boussinesq equations do not obey energy conservation! (Vallis,
2006, section 2.4.3).
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Lecture 2: Maintenance of the the main reservoirs
7
Derivation of the KE equation
• Start with the most general system, compressible
Navier-Stokes. (Not making the incompressibility assumption,
the Boussinesq approximation, nor the hydrostatic
approximation!)
• NS in tensor notation (i = 1,2, or 3):
2
ρ
∂ui
∂ui
∂p
∂Φ
∂ ui
1
∂
+ uj
=−
−ρ
+ µ 2 + ( µ + µ2 )
∂t
∂xj
∂xi
∂xi
∂xj
3
∂xi
∂uj
∂xj
(1)
where ρ is the in situ density, p is pressure, Φ is the
gravitational potential, µ is the dynamic viscosity and µ2 is the
“2nd viscosity” or bulk viscosity coefficient. We’re not used to
Lecture 2: Maintenance of the the main reservoirs
8
seeing the last term because it drops out for incompressible flow
∂uj
=0
∂xj
but we’re not making this incompressible assumption.
• Multiplying the NS momentum equations by velocity, get on
the LHS:
2
∂ ui
∂ui
=ρ
(2)
ρui
∂t
∂t 2
• But then must add KE times the density equation:
u2i ∂ρ
2 ∂t
to obtain the rate of change of KE per unit volume.
2
∂
ui
ρ
∂t
2
(3)
Lecture 2: Maintenance of the the main reservoirs
KE equation
• After some manipulation one finds the full equation (Wunsch
and Ferrari , 2004, (10)):
∂
∂E
∂ρE
+
uj (p + ρE) − µ
=
(4)
∂t
∂xj
∂xj
2
∂uj
∂Φ
∂ui
+p
− ρui
−µ
(5)
∂xj
∂xi
∂xj
where E ≡ u2i /2 and we have dropped terms involving µ2
because they are much smaller.
• The viscosity term leads to two terms – a diffusion of KE and
and dissipation of KE.
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Lecture 2: Maintenance of the the main reservoirs
Internal and Potential Energy equations
• There are two other fundamental energy equations. But all we
really need to know is that there are equal but opposite sign
terms to those on the RHS of the KE equation.
• Therefore we interpret the RHS of the KE equation as
providing the conversion terms – they don’t create nor destroy
energy but simply convert energy between KE and PE and
Internal.
• Specific internal energy, I (Wunsch and Ferrari , 2004, (11)) :
2
∂ρI
∂
∂uj
∂ui
+
[flux] = −p
+µ
(6)
∂t
∂xj
∂xj
∂xj
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Lecture 2: Maintenance of the the main reservoirs
11
• Gravitational PE (Wunsch and Ferrari , 2004, (12)):
∂
∂Φ
∂
∂ρΦ
+
[ρΦ] = ρui
+ρ
Φ
∂t
∂xj
∂xi
∂t tide
(7)
Lecture 2: Maintenance of the the main reservoirs
12
Volume integrated KE equation
• Recall:
∂
∂E
∂ρE
+
uj (p + ρE) − µ
∂t
∂xj
∂xj
= +p
∂uj
∂Φ
− ρui
−µ
∂xj
∂xi
∂ui
∂xj
2
• Integrating this equation over volume would give the rate of
change of total KE.
• The divergence term on the LHS integrates easily using Gauss’s
theorem and gives the only sources of energy. The flux of KE
across the free surface is apparently minuscule, leaving only
pressure work on the moving free surface and viscose stress
working on surface currents.
Lecture 2: Maintenance of the the main reservoirs
13
Volume integrated KE equation
• Integrate over entire ocean:
Z
∂ρE
dV = time rate of change of total KE of ocean
V ∂t
where dV = dx dy dz is a volume element.
• Integral of divergence term, use Gauss’s theorem:
Z
Z
∇ · [~u(p + ρE) − µ∇E] dV = (p + ρE)(~u − ~us ) · n̂ − µ∇E · n̂ dA
V
S
where S is the surface bounding the ocean (including the free
surface at the air-sea boundary and the fixed surface at the
seafloor), where the free surface has velocity ~us , and dA is an
element of area on the surface S.
• Flux of KE across the free surface, associated with
Lecture 2: Maintenance of the the main reservoirs
precipitation and evaporation, is minuscule:
Z
(ρE)(~u − ~us ) · n̂ dA
S
• Pressure work on the free surface:
Z
p(~u − ~us ) · n̂ dA
S
High-frequency drives surface waves, large but limited to
near-surface energy reservoir. Low-frequency they claim is
balanced exactly by conversion terms (cite unpublished article.)
Comprehensive discussion of surface pressure forcing of
low-frequency motions (Wunsch and Stammer , 1997). Not the
dominate driver of Rossby waves and eddies.
• “Diffusion of KE” is the viscose stress working on the
boundaries:
Z
−µ∇E · n̂ dA
S
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Lecture 2: Maintenance of the the main reservoirs
On the free-surface, this is the wind work on the surface flow
and at the solid boundaries is the viscose dissipation in the
bottom boundary layers. We’ll consider in detail later, and
again usual to consider different timescales separately.
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Lecture 2: Maintenance of the the main reservoirs
Conversion into/out of KE
• Viscosity causes a sign-definite loss of KE to I.
• Expansion against pressure converts I to KE.
– They claim this is small (not clear to me).
– Absent in Boussinesq equations.
– It’s “reversible” (i.e. it is associated with a reversible
thermodynamic process) but does that mean that there is
no net, time-averaged, conversion in the ocean? (I’ve never
heard anyone claim one way or the other.)
• Only non-negligible forcing of potential energy is tidal.
Implication – only net conversion from PE to KE is for tidal
motions. (This should be contrasted with the Boussinesq
equations, which have surface forcing of PE (Vallis, 2006) and
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Lecture 2: Maintenance of the the main reservoirs
the QG system for which there is surface forcing of available
PE (APE)).
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Lecture 2: Maintenance of the the main reservoirs
Rate of forcing by winds
In order of decreasing timescale:
1. Winds force General Circulation: 0.8 TW, 0.9 ± 0.05 TW
(Scott and Xu, 2009).
2. Mesoscale Eddies: 0.2 TW (Wunsch and Ferrari , 2004),
negative (Hughes and Wilson, 2008), indistinguishable from
zero (Scott and Xu, 2009).
3. Internal Waves: 0.5 to 0.7 TW (Alford , 2003; Watanabe and
Hibiya, 2002), too large (Plueddemann and Farrar , 2006).
4. Surface waves and turbulence: 20 TW (Wunsch and Ferrari ,
2004), 62.4 TW (Wang and Huang, 2004a,b).
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Lecture 2: Maintenance of the the main reservoirs
19
Rate of wind forcing of the surface
circulation
• The time-averaged,
stress is:
Z
Ẇ =
Z
=
area-integrated, rate of work by surface
~τs · ~us dx dy
~τ s · ~us dx dy +
(8)
Z
τ~0 s · u~0 s dx dy
(9)
where ~τs is the surface stress, ~us is the surface velocity, overbar
is a time average and prime denotes the anomaly relative to the
time average.
• And the anomaly component can be further decomposed into
contributions from different timescales.
Lecture 2: Maintenance of the the main reservoirs
20
• High-frequency component drives surface waves and turbulence.
• On subinertial timescales, we expect the surface velocity to
have two main components, the geostrophic component and the
Ekman component, and this is supported by observations (Rio
and Hernandez , 2004). We argue now that the rate of working
on the geostrophic component is the more important one, since
this builds potential energy of the general circulation while the
remainder is dissipated in the Ekman layer.
• Decompose the surface velocity into geostrophic (ug , vg ) and
Ekman (uE , vE ) components:
us = ug + uE ,
(10)
vs = vg + v E .
(11)
Lecture 2: Maintenance of the the main reservoirs
21
• So the rate of work by surface stress has two contributions
Z
Ẇ = ~τs · ~us dx dy
(12)
Z
Z
= ~τs · ~ug dx dy + ~τs · ~uE dx dy
(13)
= Ẇg + ẆE
• Let’s look more closely at Ẇg , we’ll find it is related to the
building of the gravitational PE.
(14)
Lecture 2: Maintenance of the the main reservoirs
Rate of wind forcing of the Geostrophic
flow
• Simply substitute the definition of geostrophic velocity into the
above expression, and integrate by parts to obtain:
Z
Ẇg = ~τs · ~ug dx dy
(15)
Z
0
0
−1
∂p
1
∂p
= τx
+ τy
dx dy
(16)
ρ0 f ∂y
ρ0 f ∂x
y
Z
0
x
−p
∂τ
∂τ
=
−
dx dy
(17)
ρ0 f ∂x
∂y
Z
~τs
0
= −p ∇ ×
dx dy
(18)
ρ0 f
• We have ignored the line integral around the boundary of
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Lecture 2: Maintenance of the the main reservoirs
terms of the form:
Z
X
east
X
west
−1 x 0
τ p
ρ0 f
23
Y
north
dx
Y
south
on the basis that if the area of integration is large enough, the
interior term will dominate, for the interior scales as the square
of the linear dimension while the boundary terms scale linearly
with the linear dimension.
• Note the p0 is the “dynamic pressure”, that is the hydrostatic
pressure anomalies from the average on a level surface.
• The definition of the Ekman velocities
1 ∂τ y
uE =
,
ρ0 f ∂z
1 ∂τ x
vE = −
,
ρ0 f ∂z
(19)
(20)
• Integrating from a depth −H assumed to be below the Ekman
Lecture 2: Maintenance of the the main reservoirs
layer (where stress is small) to the
Z η
Z η
1
uE dz =
−H
−H ρ0 f
Z η
Z η
1
vE dz =
−H
−H ρ0 f
24
surface η:
∂τ y
τsy
dz =
∂z
ρ0 f
∂τ x
τsx
dz = −
∂z
ρ0 f
• We can relate the curl of the wind stress to the Ekman
pumping velocity by
Z
~τs
0
dx dy
Ẇg = −p ∇ ×
ρ0 f
Z η
Z
= −p0
∇H · ~uE dz dx dy
−H
Z η
Z
∂wE
= +p0
dz dx dy
−H ∂z
Z
= −p0 wE (−H) dx dy
(21)
(22)
(23)
(24)
(25)
(26)
Lecture 2: Maintenance of the the main reservoirs
25
where we have assumed that the horizontal divergence of the
geostrophic flow is zero, so we only have Ekman pumping
vertical velocities wE .
• This is justified because
∂ug
∂vg
∇H · ~ug =
+
∂x
∂y
0
0 2
2
−p
∂
p
∂
+
=
∂x∂y ρ0 f
∂x∂y ρ0 f
0
1
1 ∂p ∂
=
ρ0 ∂x ∂y f
1 ∂p0 1 ∂f
=−
ρ0 f ∂x f ∂y
vg β
vg cot (lat)
=−
=−
f
a
= O(0.06m s−1 /6371 × 103 m) = O(10−8 )s−1
(27)
(28)
(29)
(30)
(31)
(32)
Lecture 2: Maintenance of the the main reservoirs
26
• This should be compared to a typical Ekman pumping velocity
divided by the Ekman layer depth:
R0
∇H · uE dz
wE
−H
=
(33)
H
H
1
~τs
= ∇×
(34)
H
ρ0 f
0.1Pa
(35)
=O
−3 −4 −1
6
3
10 m10 kg m 10 s 10m
= O(10−7 ) s−1
(36)
Lecture 2: Maintenance of the the main reservoirs
The rate of increase in potential energy
due to Ekman pumping
• Consider a density stratified fluid such as the ocean, where
ρ0 (x, y, z) is the density anomaly relative to the area averaged
density at geopotential level z, ρ(z),
ρ(x, y, z) = ρ(z) + ρ0 (x, y, z)
• On long timescales we can assume these density anomalies are
in hydrostatic balance:
∂p0
ρg=−
∂z
0
where the p0 is again the dynamic pressure, which is the
departure from the area averaged pressure at that z level.
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Lecture 2: Maintenance of the the main reservoirs
• The gravitational potential energy only changes when we raise
the centre of mass of the fluid, so the rate of increase of PE of
a fluid element is
wρ0 g dz dx dy
• Let’s integrate over the geostrophic interior, from a depth
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Lecture 2: Maintenance of the the main reservoirs
D H up to the base of the Ekman layer at −H:
Z Z Z −H
˙ E=
PE
wρ0 g dz dx dy
Z Z
=
Z Z
=
Z Z
=
−D
Z −H
29
(37)
∂p0
−w
dz dx dy
(38)
∂z
−D
!
Z −H
∂w
−H
[−wp0 ]−D +
p0
dz dx dy
(39)
∂z
−D
!
Z −H
β
−p0 (−H)wE (−H) +
p0 vg dz dx dy
f
−D
(40)
Lecture 2: Maintenance of the the main reservoirs
Z
−H
−D
β
0
p vg dz dx dy =
f
30
Z
−H
−D
Z −H
=
−D
Z −H
=
−D
0
1
∂p
β
0
p
dz dx dy
ρo f ∂x f
(41)
1 ∂p02 β
dz dx dy
2ρo f ∂x f
(42)
1
β
[p02 ]xxE
dz dy
W
2ρo f
f
(43)
Lecture 2: Maintenance of the the main reservoirs
Summary
• Wind stress working on the free surface of the ocean provides
one of the important forcing mechanism of the ocean.
• Wind stress work can be decomposed into work on geostrophic
flow, Ekman flow and others, and based on timescale.
• Wind stress working on surface geostrophic flow forces the
interior PE.
• We didn’t show this, but the friction associated with the
Ekman layer absorbs the wind work on the surface Ekman
currents.
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Lecture 2: Maintenance of the the main reservoirs
Estimates of wind power input to
geostrophic flow in literature
• Previous estimates Wunsch (1998); Huang et al. (2006); von
Storch et al. (2007) ALL used NCEP wind stress!
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Lecture 2: Maintenance of the the main reservoirs
Recent improvements
• Improvements in data allow us to make better estimate than
ten years ago, and allow us to assess the error.
• ~τ : Most significant improvement is availability of multi-year,
near global wind stress fields from satellite based
scatterometers(Kelly, 2004).
• ~u0g : Combining altimeter data from multiple satellites greatly
improved resolution of mesoscale eddies(Pascual et al., 2007).
• ~ug : Geoid is greatly improved by GRACE mission, and the
mean sea surface (MSS) has much higher resolution.
• ~ug : Combine hydrographic and surface drifter data with the
MSS from altimetry relative to the geoid – improved estimates
of mean circulation(Rio and Hernandez , 2004; Niiler et al.,
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Lecture 2: Maintenance of the the main reservoirs
2003).
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Lecture 2: Maintenance of the the main reservoirs
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Lecture 2: Maintenance of the the main reservoirs
36
Wind power input to surface geostrophic flow
mW/m
2
Lecture 2: Maintenance of the the main reservoirs
37
τ
Currents
eddy-WPI [TW]
WPI [TW]
area [%]
QS TL
Aviso upd
0.024 (0.233)
0.91 (1.19)
100
QS TL
GT mean
0.024
0.90
93.5
QS TL
NMM
0.024
0.90
97.7
QS TL
Aviso ref
0.025
0.91
100
QS LP
Aviso upd
0.022
0.86
100
GSSFT2 a
Aviso upd
0.088
0.95
86.6
NCEP2
Aviso upd
0.076 (0.116)
1.02 (1.09)
82.0 (100)
ERA-40 a
Aviso upd
0.087 (0.173)
0.99 (1.28)
81.9 (100)
NCEP2 a
Aviso ref
0.040
0.98
81.9
NCEP2
Aviso ref
0.079
1.02
82.0
Lecture 2: Maintenance of the the main reservoirs
details ...
Table 1. Time: a = 1/2/1994 - 12/31/1999, b = 1/2/2000 12/31/2005; Wind stress: QS = QuikSCAT, TL = Tang and Liu,
LP = Large and Pond; Currents: upd = updated, ref = reference,
GT mean = replace AVISO mean dynamic topography with
GRACE-Tellus mean dynamic topography poleward of 3◦ , NMM
mean = replace AVISO mean dynamic topography with
Maximenko mean dynamic topography poleward of 3◦ ; Numbers in
parentheses do not take into account the bias arising from the
surface current effect on stress described above. The area of good
data was 3.14 × 1014 m2 .
... please see (Scott and Xu, 2009).
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Lecture 2: Maintenance of the the main reservoirs
For Thursday, Feb. 24
• Energy transport to the abyss.
• Routes to dissipation.
• Estimates of bottom dissipation.
• “Inverse cascade” of geostrophic turbulence inhibits the
forward cascade of geostrophically balanced flow to small-scale
dissipation.
• Relation to mixing (if time permits).
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Lecture 2: Maintenance of the the main reservoirs
References
Alford, M. H. (2003), Improved global maps and 54-year history of
wind-work on ocean inertial motions, Geophys. Res. Lett., 30,
art. no. 1424.
Ferrari, R., and C. Wunsch (2009), Ocean Circulation Kinetic
Energy: Reservoirs, Sources, and Sinks, Ann. Rev. Fluid Mech.,
41, 253–282.
Huang, R. X., W. Wang, and L. L. Liu (2006), Decadal variability
of wind-energy input to the world ocean, Deep-Sea Res. II, 19,
31–41.
Hughes, C. W., and C. Wilson (2008), Wind work on the
geostrophic circulation: an observational study of the effect of
small scales in the wind stress, J. Geophys. Res., 113, C02,016.
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Lecture 2: Maintenance of the the main reservoirs
Kelly, K. A. (2004), Wind data: A promise in peril, Science, 303,
962–963.
Niiler, P. P., N. A. Maximenko, and J. C. McWilliams (2003),
Dynamically balanced absolute sea level of the global ocean
derived from near-surface velocity observations, Geophys. Res.
Lett., 30 (22), 2164.
Pascual, A., M.-I. Pujol, G. Larnicol, P.-Y. LeTraon, M. H. Rio,
and F. Hernandez (2007), Mesoscale mapping capabilities of
multisatellite altimeter missions: First results with real data in
the Mediterranean Sea, J. Mar. Sys., 65, 190–211.
Plueddemann, A., and J. T. Farrar (2006), Observations and
models of the energy flux from the wind to mixed-layer inertial
currents, Deep-Sea Res. II, 53, 5–30.
Rio, M.-H., and F. Hernandez (2004), A mean dynamic topography
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Lecture 2: Maintenance of the the main reservoirs
computed over the world ocean from altimetry, in-situ
measurements and a geoid model, J. Geophys. Res., 109 (C12),
Art. No. C12,032.
Scott, R. B., and Y. Xu (2009), An update on the wind power
input to the surface geostrophic flow of the world ocean,
Deep-Sea Res. I, 56 (3), 295–304.
Vallis, G. K. (2006), Atmospheric and Oceanic Fluid Dynamics:
Fundamentals and Large-scale circulation, 744 pp., Cambridge
University Press.
von Storch, J.-S., H. Sasaki, and J. Marotzke (2007),
Wind-generated power input to the deep ocean: an estimate
using a 1/10◦ general circulation model, J. Phys. Oceanogr.,
37 (3), 657–672.
Wang, W., and R. X. Huang (2004a), Wind energy input to the
surface waves, J. Phys. Oceanogr., 34, 1276–1280.
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Lecture 2: Maintenance of the the main reservoirs
Wang, W., and R. X. Huang (2004b), Wind energy input to the
Ekman layer, J. Phys. Oceanogr., 34, 1267–1275.
Watanabe, M., and T. Hibiya (2002), Global estimates of the
wind-induced energy flux to inertial motions in the surface mixed
layer, Geophys. Res. Lett., 29, 10.1029/2001GL014,422.
Wunsch, C. (1998), The work done by the wind on the oceanic
general circulation, J. Phys. Oceanogr., 28, 2332–2340.
Wunsch, C., and R. Ferrari (2004), Vertical mixing, energy, and the
general circulation of the oceans, Ann. Rev. Fluid Mech., 36,
281–314.
Wunsch, C., and D. Stammer (1997), Atmospheric loading and the
oceanic “inverted barometer” effect, Rev. Geophys., 35 (1),
79–107.
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