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rspa.royalsocietypublishing.org
Space–time behaviour of
magnetic anomalies induced
by tsunami waves in open
ocean
Benlong Wang1,2 and Hua Liu1,2
Research
Cite this article: Wang B, Liu H. 2013
Space–time behaviour of magnetic anomalies
induced by tsunami waves in open ocean. Proc
R Soc A 469: 20130038.
http://dx.doi.org/10.1098/rspa.2013.0038
Received: 17 January 2013
Accepted: 30 May 2013
Subject Areas:
geophysics, fluid mechanics,
ocean engineering
Keywords:
kinematic dynamo problem, tsunami,
single wave, N -wave
Author for correspondence:
Benlong Wang
e-mail: [email protected]
1 School of Naval Architecture, Ocean and Civil Engineering,
Shanghai Jiao Tong University, Shanghai 200240,
People’s Republic of China
2 MOE Key Laboratory of Hydrodynamics, Shanghai 200240,
People’s Republic of China
The magnetic anomaly induced by an inhomogeneous
velocity field under tsunami waves in open ocean is
investigated. With asymptotical analysis, an explicit
series solution of the kinematic dynamo problem
is established for weak dispersive water waves.
The magnetic field induced by typical tsunami
models, including single wave and N -wave, can be
directly obtained using the proposed series solution.
The characteristics of the magnetic field induced
by two realistic tsunami events are investigated.
By analysis, the magnetic magnitude induced by a
1 m high tsunami is estimated as of the order of
10 nT at the sea surface, which depends on the wave
parameters as well as the Earth’s magnetic field. The
space and time behaviour of the magnetic field shows
fair similarity with the field data at Easter Island
during the 2010 Chile tsunami.
1. Introduction
Tsunamis induced by submarine mass failures or
earthquakes have become a great concern, especially
after the 2004 Sumatra earthquake and the 2011 Tohoku
earthquake. To improve the anti-disaster capabilities of
human beings, more in-depth knowledge of the onset
and characteristics of earthquake-induced tsunamis is
needed.
Most tsunami-induced damage is caused by run-up
and inundation along coastal lines. Given the initial
conditions and/or offshore boundary conditions, runup can be well predicted by various wave models,
e.g. nonlinear shallow water equations [1–4]. When
breaking and inundation are involved, Navier–Stokes
2013 The Author(s) Published by the Royal Society. All rights reserved.
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rspa.royalsocietypublishing.org Proc R Soc A 469: 20130038
equations are appropriate [5]. An obvious fact is that the well-developed governing equations
depend heavily on proper initial conditions arising from coupled seafloor–water wave motion.
Therefore, a key for better early warning is the real-time detection of offshore sea-level
variations. Approaches in two categories can be taken to obtain this information, as summarized
by Synolakis et al. [6].
The first approach is determining the quantitative features of early seafloor deformation from
teleseismic data. In fact, the seismic tsunami warning method has a long history, and is still one
of the most basic elements of the tsunami warning system in operation. W-phase inversion can
provide a reliable base for effective and rapid tsunami warning. The W-phase data can be collected
within 15–30 min after the origin time of the earthquake. The time response strongly depends
on the proximity of the seismic network to the rupture zone. Nowadays, the accurate source
parameters of the earthquake, such as moment tensor and dip angle, can be obtained within
6 min using the W-phase source inversion algorithm with pre-computed Green’s functions [7,8].
For the purpose of regional tsunami warning, more rapid methods are desired. For instance, the
first tsunami arrived at coastal areas only about 15 min after the origin time of the 2011 Tohoku
earthquake. A quick response is the first challenge for a regional warning system. Complicated
coupling among seismic motion, seafloor ruptures and transient water waves is another main
obstacle for accurate assessment of tsunami from seismic data.
Another approach is obtaining data from the monitoring network of real-time sea-level data,
which is the most direct method to obtain hydrodynamical data. However, sea-level monitoring
networks require vast investments and have not yet been built up in many regions, e.g. the Indian
Ocean and South China Sea, among others. Sometimes, tide gauge records can be used to analyse
giant tsunamis [9,10], provided that the sampling rate is adequate for measuring the amplitude
and resolving most of the tsunami features. One exception is the Indian Ocean tsunami, for which
the wave amplitude obtained by satellites that passed over the ocean a couple of hours after the
earthquake was about 60 cm [11]. The wavelength of the first two leading waves were estimated
to be 160 and 240 km from satellite signals.
The importance of instrumental tsunameter measurements was emphasized by Synolakis &
Bernard [12] for future research after the Boxing Day tsunami. Some novel methods have to be
considered as supplements.
Electronic or magnetic detection is one of the most attractive and promising candidates.
Ionospheric remote sensing of the tsunami signature in total electron content (TEC) could
provide new tools for offshore tsunami detection [13–15]. Much evidence on ionospheric
disturbances has been accumulated, including for the 1964 Alaskan earthquake, the 2004 Sumatra
earthquake [13] and the 2011 Japan earthquake [16,17]. Tsunami induces, by dynamic coupling,
the propagation of internal gravity waves (IGWs) in the ocean, atmosphere and ionosphere
system. The plasma perturbation induces a magnetic variation of the order of a few nano Tesla
[18,19] at ionospheric altitudes.
In contrast to plasma perturbations in the ionosphere layer, a small electromagnetic field
can also be generated when sea water flows through the Earth’s main magnetic field. Magnetic
anomalies induced by the sea-water motion has been recognized and known from very early
days [20]. Compared with the strong quasi-steady background of the Earth’s magnetic field,
the magnetic field induced by ocean flow is rather weak. It is hard to detect, and has not
been used until new sensitive techniques have become available for scientific research and field
measurements. Magnetic variations associated with ocean waves and swells were studied by
Weaver [21] using a relatively concise approach following other pioneering work. Larsen [22]
formulated the solution for electrical and magnetic fields induced by long and intermediate ocean
waves. Magnetic detection of ocean flow is still in its infancy, and so efforts should concentrate on
the spatial and temporal behaviour of the flow-induced magnetic field. Thus, tiny signals could
possibly be extracted from the main Earth’s magnetic field.
In fact, there has been some progress on magnetic detection for various ocean waves in recent
years. For gravity waves with periods of several seconds, a magnetic field of O(0.1)–O(1) nT can
be induced by Kelvin ship waves from submarines and surface ships travelling at cruise speeds
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Generation of a magnetic field by a conductive fluid’s motion is known as the dynamo problem.
The complete dynamo problem consists of the equations of motion and Maxwell equations, which
are coupled by the action of Lorentz forces. Owing to the tiny magnetic strength caused by water
waves in the ocean, the Lorentz forces exerted on water are neglected in most cases, which leads
to an equilibrium magnetic field. Therefore, the hydrodynamic problem and the dynamo problem
are decoupled.
..................................................
2. Explicit series solutions of the kinematic dynamo problem
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[23–25]. Given that the accuracy of a modern airborne superconducting quantum interference
device magnetic transducer is of the order of 10−6 nT, such weak magnetic fields can, in principle,
be detected.
Electromagnetic fields tend to be enhanced when wave periods and lengths increase. The wave
heights and periods of ocean swells are usually several metres and around 13 s. These ocean swells
can generate magnetic signals up to 5 nT [26]. Typical tide flow can induce a magnetic field of 20–
30 nT. For some internal waves in tropical areas, the induced magnetic field may reach 100 nT [27].
Tyler et al. [28] compared and found similarities between the results of a numerical prediction of a
lunar semi-diurnal tide (M2 tide) and scalar magnetic records by satellite CHAMP. It was shown,
for the first time, that the ocean flow makes a substantial contribution to the geomagnetic field at
satellite altitude.
The wave period of harmful tsunamis is around O(10) min. The wavelength may extend from
tens to hundreds of kilometres. The wave period of a tsunami is longer than a typical wind
wave and swell, but shorter than that of internal waves and tidal waves. However, the velocity
magnitude of a tsunami is much larger than that of an internal wave, especially at the upper
ocean layer. The tsunami-induced magnetic field above the sea surface should easily be detected
by magnetic transducers on board ships or airborne platforms. Little is known on the spatial
distribution of the magnetic field caused by real tsunami waves, especially the spatial decay
character. Accordingly, it is hard to assess the possibility of whether such tiny signals could be
captured by the sensors in satellites.
Opportunities and difficulties exist simultaneously for detecting tsunamis by magnetic
anomalies [29]. The most important point is the space–time behaviour of the magnetic field
induced by tsunamis. Preliminary results on the 2004 Indian Ocean tsunami were obtained
numerically using a global magnetic solver [30]. Using sea-surface data from the satellite Jason-1
for the 26 December 2004 tsunami, a maximum magnitude of about 20 nT near the magnetic pole
was estimated for unit sea-surface displacement; this provides a possibility for detecting tsunamis
in open ocean. An encouraging point is that measurable magnetic signals were captured in the
recent Chile tsunami (27 February 2010) at Easter Island [31], which represents a milestone in
understanding tsunami-induced electromagnetic effects.
The purpose of this work is to investigate the spatial and temporal behaviour of a magnetic
field induced by various tsunami waves in realistic wave forms. The magnetic fields induced
by single waves and N -waves are studied. The wave form is important because it describes
the scales of geometric attenuation. Indeed, it is important to get this scale right because, in
assessing the potential for magnetic remote sensing of the tsunami, understanding the geometric
attenuation is vital. Furthermore, understanding the details of the tsunami magnetic field creates
possibilities for better filtering to extract the (somewhat tiny) tsunami-induced signal from the
magnetic record.
The mathematical modelling for both the hydrodynamic and electromagnetic problems is
discussed in §2. A series solution of the kinematic dynamo problem is established for a general
wave in the form of sech2 γ x. The magnetic field induced by single waves and N -waves can be
obtained directly by selecting a corresponding wavenumber γ . Characteristic of the magnetic field
induced by single and N -waves are investigated in §3. Two examples of realistic tsunami waves
are studied in §4 to investigate the spatial and temporal behaviour of the induced magnetic field.
Conclusions and remarks are summarized in the final section.
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(a) Review of the magnetic field due to harmonic water waves
4
∇ 2 B(x) = 0
and
∇ 2 B(z) = 0.
(2.2)
Supposing the magnetic component can be expressed as a harmonic function B(x) =
b(z) e−i(ωt−kx) , then the solution can be obtained as b(z) = C1 e−kz + C2 ekz . From the limited value
of b(z) in the far field, we get C2 = 0 in the air region over the sea surface and C1 = 0 in the Earth
below the seafloor. We have the following condition for the magnetic field at the air and Earth
boundaries:
(x)
(x)
(2.3)
Bz (x, 0) = −kB(x) (x, 0) and Bz (x, −h) = kB(x) (x, −h).
These two conditions can be viewed as the boundary conditions for B(x) in the ocean layer
(x)
when considering the continuity of B(x) and Bz on both sides of the two interfaces between air,
the Earth and the ocean. In the ocean layer, the governing equation is
∂
− K∇ 2 B(x) = A e−i(ωt−kx) ,
(2.4)
∂t
with A ≡ |F · ∇u|. Substituting the general solution B(x) = b(z) e−i(ωt−kx) into (2.4), we have
A
ω
b(z) = − .
(2.5)
b (z) − k2 − i
K
K
Defining κ = k2 − iω/K, the solution of this ODE for b(z) can be obtained with boundary
condition (2.3). A is assumed to be a constant at this moment, and will be justified in §2b. The
value at the sea surface is
−1
A
κK
Kκ 2
A sinh(κh/2)
=
+
.
(2.6)
b(0) =
κK(kcosh(κh/2) + κ sinh(κh/2))
k tanh κh/2
k
Using Kκ 2 /k = Kk(1 − iω/Kk2 ) = Kk − i gk tanh kh/k, the following can be obtained:
tanh kh −1
A
2K κh/2
− i gh
.
(2.7)
b(0) =
kK +
k
h tanh κh/2
kh
..................................................
where the superscripts (x) and (z) represent the components of the magnetic field in each
direction, K = (μσ )−1 , μ = 4π × 10−7 N A−2 is the permeability of free space and σ = 2–6 S m−1
is the electrical conductivity of the sea water depending on the salinity. The mean conductivity
of the oceans is chosen as 4.0 S m−1 in this paper. The main magnetic field of the Earth is
F = F(cos I cos θi − cos I sin θ j + sin Ik), in which F is the magnitude of the Earth’s magnetic field,
I is the dip angle and θ is the angle between the wave propagation direction and the magnetic
meridian. At the Earth’s surface, the total intensity F varies from 24 000 to 66 000 nT. The dip
angle I is the angle at which the magnetic field lines intersect the surface of the Earth. This angle
ranges from 0◦ at the equator to 90◦ at the poles. A mean value of 40 000 nT is chosen in §3 in
both the horizontal and vertical directions for convenience. Variable v = (u, w) is the water wave
velocity in the vertical plane. For harmonic progressive waves, the horizontal velocity has a form
u = |u| e−i(ωt−kx) .
Outside the ocean water layer, the electromagnetic properties of the media are assumed to be
insulating. Hence, the Maxwell equation reduces to the Laplace equation, i.e.
rspa.royalsocietypublishing.org Proc R Soc A 469: 20130038
Harmonic solutions of the linear kinematic dynamo problem for linear progressive waves have
been obtained by many authors, e.g. Weaver [21], Larsen [22] and Tyler [30], among others.
Herein, the solution procedure is briefly described.
A two-dimensional problem in a vertical plane is considered with a horizontal x-axis and
a vertical z-axis pointing upwards. The linear kinematic dynamo problem for a magnetic field
vector B = (B(x) , B(z) ) can be solved using
∂
(2.1)
− K∇ 2 B = F · ∇v,
∂t
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Without loss of generality, a typical single tsunami wave model is chosen as
ζ = Hsech2 γ (x − c0 t),
(2.8)
in which γ is a parameter measuring the wavenumber. For extremely small-amplitude tsunami
waves in a geophysics sense, the phase speed can be approximated by c0 = gh. We define the
relative wave height α = H/h and phase θ = γ (x − c0 t) in the following analysis. The horizontal
velocity component of the general tsunami wave field can be approximated by u = c0 ζ /h for tinyamplitude long waves in open ocean. For such small-amplitude linear long waves, the horizontal
velocity u is uniform from the ocean floor to the free surface (Lamb [32] and Mei [33]). Therefore,
the derivative uz = 0. Substituting the horizontal velocity into (2.1), we get
∂
− K∇ 2 B(x) = −2αγ c0 F(x) sech2 θ tanh θ.
∂t
(2.9)
Herein the governing equation in the horizontal direction is given with a source term
expressed as the wave velocity. The horizontal velocity gradient spectrum versus wavenumber
of the wave in the form of (2.8) is
A0 (k) =
ik2
kπ
csch ,
3
2γ
2γ
(2.10)
using a Fourier transform of sech2 θ tanh θ , i.e. the source term on the right-hand side of (2.9).
Using the solution of one harmonic component (2.6), the horizontal magnetic field B(x) induced
by general waves in the form of (2.8) can be obtained,
B(x) =
B(x) (x, z)
= Re
−2αγ c0 F(x)
∞
0
e−kz
A0 (k) sinh(κh/2)
eikx dk,
κK(kcosh(κh/2) + κ sinh(κh/2))
(2.11)
with Fourier integration by substituting A by spectrum A0 (k) expressed in (2.10).
Integral (2.11) can be numerically integrated easily since A0 (k) is a smooth and narrow support
function and there is no singularity in the integration. However, the numerical integration of
(2.11) does not provide any intuitive insights on how wave parameters affect the magnetic field,
but various simplifications are possible. Next, we will investigate the integral in explicit form for
weak dispersive waves.
(c) Approximation of b(0) for weak dispersive waves
The expression tanh kh/kh in (2.7) is a function related to the dispersion properties for water
waves. The parameter kh, the ratio of water depth to wavelength, measures the dispersive effects.
For long waves in water, both κh and kh are small numbers. The approximation of kh/tanh kh
was studied by Madsen et al. [34] for modelling the dispersive effects of shallow water waves. In
a similar approach, the expression of b(0) can be further simplified using a regular Taylor series
..................................................
(b) Magnetic field of the general tsunami wave model
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After B(x) is solved in the ocean region, the solution in air above the sea surface can be obtained
using the boundary value b(0) at the air–sea interface, i.e. B(x) (x, z) = b(0) e−kz e−i(ωt−kx) . Note k ≥ 0
in this analysis due to the condition of the limited value at z = ∞.
√
For ocean swells of periods around 10 s, Weaver [21] simplified 1 + iβ as 1 + iβ/2 when β
was a small parameter defined as β = 4π σ g2 /ω3 . However, this approximation is not suitable for
tsunami waves. The relationship between wavenumber k and angular frequency ω is the linear
dispersion equation ω2 = gk tanh kh. We will introduce another kind of perturbation analysis in
§2c. The solution for B(z) is similar and is given in §2d.
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600
6
300
200
100
0
2000
4000
6000
h (m)
8000
10 000
Figure 1. Comparison of diffusion speed cd (solid line) and phase speed c0 (dashed line) of a long wave versus water depth h.
expansion. Power series expansions with the small parameters κh and kh for terms in (2.7) are
1 κh 2
1 κh 4
κh/2
=1+
−
+ O(κ 6 h6 )
tanh κh/2
3 2
45 2
=1+
and
1 (k2 − iω/K)2 h4
1 (k2 − iω/K)h2
−
+ O(κ 6 h6 )
3
4
45
24
1
tanh kh
19
= 1 − (kh)2 +
(kh)4 + O(k6 h6 );
kh
6
360
(2.12)
(2.13)
recall the definition κ = k2 − iω/K.
For leading-order approximation, the values of both (κh/2)/(tanh κh/2) and tanh kh/kh are
equal to 1. Therefore,
⎫
−1
⎪
kh
A
. A
(0)
⎪
cd 1 +
− ic0
=
b (0) =
(β00 + β01 kh) ,⎪
⎬
k
2
kc0
(2.14)
⎪
c0
c0 cd
⎪
⎪
β00 =
and β01 = −
,
⎭
cd − ic0
2(cd − ic0 )2
where c0 = gh is the phase velocity of the long wave. Variable cd = 2K/h is defined as the lateral
diffusion speed by Tyler [30]. For typical sea water, K = 107 /16π , and values of cd and c0 are shown
in figure 1. For most parameters in open ocean, cd and c0 have comparable values. In fact, the
magnetic Reynolds number can be defined as Rm = 2c0 /cd when choosing long-wave celerity c0
as the velocity scale and the water depth to be the length scale of the tsunami flow, together with
magnetic diffusivity K. Magnetic Reynolds numbers take the values of Rm = 0.50, 1.41 and 3.98 for
water depths of 1000, 2000 and 4000 m, respectively. The magnetic Reynolds number varies from
the shallow water region to the deep water region following Rm ∝ h3/2 . In fact, the coefficients β00
and β01 can be expressed in magnetic Reynolds number with minor manipulations.
The second- and fourth-order approximations of b(0) with respect to kh are
−1
A
kh k2 h2
kh k2 h2
(2)
+
−
b (0) =
cd 1 +
− ic0 1 +
k
2
12
6
6
. A
=
(β20 + β21 kh + β22 k2 h2 ),
kc0
(2.15)
with
β20 =
c0
,
cd − ic0
β21 = −
c0 (3cd − ic0 )
,
6(cd − ic0 )2
β22 =
c0 (6c2d − 7c20 − 9 icm c0 )
36(cd − ic0 )3
..................................................
400
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c (m s–1)
500
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(a)
(b)
0.04
7
0.0008
0.0002
0.05
0.01
–0.0002
0
0.05
0.10
0.15
kh
0.20
0.25
0.30
0.10
0.15
0.20
0.25
0.30
kh
–0.0004
Figure 2. (a,b) Relative errors of coefficients in leading- and second-order series expansions of b(0) and b(2) . Solid, dotted and
dashed-dotted lines are coefficients of cd in shallow, intermediate and deep water regions, respectively, the dashed line is the
coefficient of c0 .
and
k2 h2 c20
A
k4 h4
kh k2 h2
+
+
−
b (0) =
cd 1 +
k
2
12
720 c2d
720
−1
kh k2 h2
k3 h3
19k4 h4
.
−
−
+
− ic0 1 +
6
6
30
360
(4)
(2.16)
The relative errors of the leading-order and the second-order approximations can be estimated
by comparing with the fourth-order approximation or the exact integral formulation (2.11). For
weak dispersive long waves in water, almost all components of the harmonics have a dispersion
parameter kh < π/10. Comparing b(4) , the maximum relative errors are less than 1 and 4 per
cent for the coefficients of cd and c0 if the leading-order approximation is used, as shown in
figure 2, which implies the leading-order approximation can predict the magnetic field with a
reasonable accuracy. In the comparison, the ratio of c20 /c2d takes the values of 0, 1 and 10 for
shallow, intermediate and deep water depth, respectively, corresponding to the values shown
in figure 1. If the second-order approximation is used, the relative errors are all less than 0.1
per cent, as shown in figure 2. Apparently, b(2) is a more accurate approximation than b(0) . From
another point of view, the approximation has a fast convergence rate as there is little difference
between the second and the fourth approximations with respect to the small dispersive parameter
kh considered in this context.
By introducing a new variable ξ = 12 + (γ /π )(−ix + z), the integration of the leading-order and
the second-order approximation of b(0) can be simplified as
B(x) = Re
∞
0
e−kz b(0) (0)eikx dk =
1 1
iγ h
Re
iβ
ψ
(ξ
)
−
ψ
(ξ
)
β
00 1
01 2
π
π 2 γ c0
(2.17)
and
B(x) = Re
∞
e−kz b(2) (0) eikx dk
0
iγ h
iγ 2 h2
1 1
β21 ψ2 (ξ ) −
Re iβ20 ψ1 (ξ ) −
β22 ψ3 (ξ ) .
= 2
π
π γ c0
π2
Here, ψn (z) = (−1)n+1 n!
∞
k=0 (k + z)
−(n+1)
is the nth-order polygamma function, see [35].
(2.18)
..................................................
0.0004
R 0.02
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0.0006
0.03
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B(x)(nT)
8
60
x (km)
–40
–20
20
40
–20
–40
Figure 3. Comparisons of magnetic magnitude B(x) induced by a wave of length 60 km in a water layer of h = 1000 m,
F = 40 000 nT. Dashed line, leading-order approximation with b(0) ; solid line, second-order approximation with b(2) (0); open
circle, full numerical integration of (2.11).
Using this approach, the series solution can be obtained by the summation of high-order terms
of kh. Consequently, the final expression of B(x) , after multiplication by a factor of −2αγ c0 F, reads
B
(x)
⎡
⎤
j
n
γ
h
2
(x)
= − 2 αF × Re ⎣iβn0 ψ1 (ξ ) − i
βnj
ψj+1 (ξ )⎦ ,
π
π
(2.19)
j=1
in which βnj can be calculated from the series expansion of (κh/2)/ tanh κh/2 and tanh kh/kh
in (2.7).
The series expansion solution can be thought of as the approximation of the integration for
a specified velocity field of wave form (2.8). In practice, the small parameter γ h results in a
rapid convergence of the series expression of the magnetic field. A direct comparison between the
approximate solution and the numerical integration is conducted to further justify the accuracy
of the truncated series expansion of (2.11), i.e. formulae (2.17) and (2.18). To include more
dispersive effects, we select a large value of γ h = 0.087, which corresponds to a large relative
amplitude solitary wave α = H/h = 0.01 for h = 1000 m, although this is rare in a realistic case. The
wavelength is around 60 km, which is shorter than most components of realistic tsunami waves,
see the 2004 Sumatra tsunami and the 2010 Chile tsunami in §4. The magnetic field distributions
are almost indistinguishable in figure 3, even for γ h = 0.087, which is in agreement with the
error analysis shown in figure 2. For the magnetic field induced by a tsunami wave, the secondorder approximation (2.18) is accurate enough, i.e. n = 2 in (2.19), as proved by the order analysis
mentioned previously.
(d) Magnetic field induced by long waves in the vertical direction
From the continuity equation for incompressible fluids, we have wz = −ux . For a small-amplitude
wave, the horizontal derivative of the vertical wave velocity wx ∼ αux is a higher order small
term, and can be neglected on the right-hand side of the governing equation (2.1), provided that
F(x) and F(z) are values of the same magnitude, except in the low geographical latitude regions
near the Equator. With these considerations, the governing equation describing the kinematic
dynamo problem in the vertical direction can be approximated as
∂
.
− K∇ 2 B(z) = (F(x) wx + F(z) wz ) = F(z) wz = −F(z) ux ,
∂t
(2.20)
..................................................
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40
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(a) 500
(b)
0.1
0.1
(c)
0.2
0.5
9
z (km)
1
0.5
0.2
200
100
0
1000
6 4
2000
3000
0.5
1
2
6
8
4000 1000
2
1
2
4
2000
h (m)
6
8
3000
4000
1000
h (m)
2000
3000
4
8
4000
h (m)
Figure 4. Maximum absolute magnetic magnitude caused by 1 m single waves with different periods: (a) T = 5, (b) 13 and (c)
30 min. Unit of contour label: nano Tesla. The Earth’s magnetic field is F = 40 000 nT.
which is similar to the equation in the horizontal direction (2.9). Therefore, the magnetic field of
the vertical direction takes the minus value of the horizontal one multiplied by a factor F(z) /F(x) .
For the vertical component of the magnetic field B(z) , the governing equation (2.20) takes the
same form as that derived by Tyler [30], who projected the Maxwell equations only in the vertical
direction perpendicular to the sea surface.
3. Magnetic field induced by single waves and N -waves
(a) Single waves
In this section, we study the magnetic field induced by a single wave proposed by Madsen &
Schäffer [36],
ζs = Hsech2
Ω
(x − c0 t).
c0
(3.1)
Here, Ω = 2π/T, where T is the wave period. For a giant tsunami, the wave period is
usually around 10 min. We compare three different waves with periods T = 5, 13 and 30 min.
Following the previous definition, γ = Ω/c0 , applying the analytical solution to a single wave is
straightforward. For a single wave of period T = 13 min, the dispersive parameters are γ h = 0.081
for water depth h = 1000 m and γ h = 0.163 for h = 4000 m. Obviously, kh is not a small number,
and the dispersive effect is notable. This is also the reason why a linear dispersive model is
preferred for the prediction of tsunamis propagation in some circumstances. The second-order
approximation still has good accuracy for large values of γ h, since the relative error is quite small,
as shown in figure 2.
The attenuation of magnetic magnitude in the vertical direction appears to have a strong
connection with the tsunami period. At sea-surface level, the maximum value of the magnetic
field is around 8nT for the three scenarios. However, the magnetic field induced by a short wave
decays much faster than the others, as shown by the different subplots in figure 4. At 100 km
above the sea surface, the magnetic field is around 0.5 nT for T = 5 min, in contrast to 2.0 nT for
T = 30 min.
Sea-water depth also plays an important role in the attenuation of magnetic field. At low
altitude over oceans of different depths, there is little difference in the magnetic magnitude.
However, at high altitudes, the variation is apparent. The vertical decay rate over deep ocean
is much slower than that over shallow ocean for the same tsunami wave, providing relatively
good opportunities for early warnings.
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1.0 hS (m)
(a)
(b)
ux (s–1)
10
1×10–6
–100
–50
50
100
50
–50
x (km)
150
100
x (km)
150
–1×10–6
–2×10–6
–0.5
–3×10–6
–1.0
Figure 5. (a,b) Surface and velocity gradients for various N -waves. Solid line, μ = 0; dotted line, μ = 0.3; dashed line
μ = 1.0.
(b) N -waves
When triggered by low magnitude earthquakes, e.g. magnitude 6.5, a tsunami wave appears
similar to a wave train of Airy wave form, as predicted by Madsen & Schäffer [36] using both
linear Korteweg–de Vries equations and highly dispersive Boussinesq equations. A tsunami may
behave as an N -wave when it is induced by a high-magnitude earthquake in deep ocean as shown
by the numerical results of Zhao et al. [37]. N -waves may occur in both deep and shallow water
regions, and are representative models of realistic tsunamis. The profile of an N -wave changes
little over a long distance during oceanic transmission. N -waves caused by two counter-acting
single waves are chosen as follows:
ζs = Hsech2
Ω
Ω
(x − c0 t) − μHsech2 (x − c0 t − x0 ).
c0
c0
(3.2)
Similar to the single wave, we use T = 13 min as a characteristic wave period. Parameter
μ controls the amplitude ratio of the positive and negative waves. An isosceles N -wave is
represented by μ = 1. A generalized N -wave is represented by 0 < μ < 1. The horizontal velocity
can be approximated as u = c0 ζs /h. The velocity gradient ux is
ux = −2αc0 γ (sech2 γ x tanh γ x − μsech2 γ (x − x0 ) tanh γ (x − x0 )).
The Fourier transform of ux is
ûx = −2αc0 γ
ik2
kπ
−ikx0
(1
−
μ
e
csch
)
.
2γ
2γ 3
(3.3)
(3.4)
Three typical N -waves are selected for comparison. The amplitude is normalized and the
phase is shifted to match with the crest height at x = 0. The amplitudes H are 1.048 and 1.165
for μ = 0.3 and 1.0, respectively. For μ = 1, the minimum value of ux is about twice that for μ = 0,
as shown in figure 5.
Using the series solution (2.19), the magnetic field can be calculated using
⎡
⎤
j
2
γ
h
2
(x)
(x)
β2j
ψj+1 (ξ )⎦
B = − 2 αF × Re ⎣iβ20 ψ1 (ξ ) − i
π
π
⎡
j=1
2
+ 2 μαF(x) × Re ⎣iβ20 ψ1 (ξ ) − i
β2j
π
2
j=1
γh
π
j
⎤
ψj+1 (ξ )⎦ ,
(3.5)
where ξ = 12 − (iγ (x − x0 )/π ) + γ z/π . The magnetic fields induced by various N -waves are
shown in figure 6 at the sea-surface level z = 0.
There is a significant difference on the run-up between leading depression and leading
elevation tsunamis [36,38]. By inverse analysis of the magnetic signals, either leading depression
or leading elevation tsunamis can be determined, which can help when predicting the maximum
run-up.
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10
B(x)(nT)
11
–50
50
100
–5
Figure 6. Comparisons of magnetic field B(x) at sea-surface level z = 0 for various N -waves. Solid line, μ = 0; dotted line,
μ = 0.3; dashed line μ = 1.0.
Table 1. Best-fit wave parameters for modelling the 2004 Sumatra tsunami.
..........................................................................................................................................................................................................
Ai (cm)
60
60
−45
40
−50
50
−40
γi (km )
0.021
0.046
0.015
0.05
0.012
0.005
0.002
xi (km)
0
−130
−350
−460
−550
−780
−1280
..........................................................................................................................................................................................................
−1
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
4. One-dimensional magnetic field modelling coupled with tsunami wave
observation
(a) 2004 Sumatra tsunami
Seven single waves similar to (3.2) are superimposed to simulate the Sumatra tsunami observed
by the Jason satellite [11]. After projecting the relative sea level measured by Jason-1 onto the
wave propagating direction, the real 2004 Sumatra tsunami profile can be modelled by
ζs (x) =
7
Ai sech2 γi (x − xi ).
(4.1)
i=1
Given the values of the local peak’s position xi and length scale γi , the amplitudes Ai can be
obtained by least-squares regression. The wave parameters of the seven components are listed
in table 1. Using these parameters, a comparison between the fitting surface profile and the
measured data is shown in figure 7.
The average water depth is 5000 m in this region. The horizontal component of the main
magnetic field is taken to be 40 000 nT during the calculation. The corresponding magnetic field
can be calculated in a similar approach as for N -waves (3.5), and is shown in figure 7. The
magnetic signal due to the Sumatra tsunami might reach 3 nT for the vertical magnetic component
at sea level, as estimated using a simple formula by Tyler [30]. Using the wave field calculated by
shallow water equations, Manoj et al. [39] predicted the magnetic strength to be a similar level as
that of Tyler [30]. From the World Magnetic Model 2005–2010, the vertical magnetic intensity is
around 20 000 nT at this site. The peak values of the horizontal magnetic field shown in figure 7
agree well with the values predicted by Tyler [30] and Manoj et al. [39] at sea-surface level when
the ratio F(z) /F(x) is taken into account.
Recalling the definition of ξ = 12 + (γ /π )(−ix + z), we have ξ = 12 when x = 0 and z = 0 for
a position just over the sea surface at height ζ . The value of ψ1 ( 12 ) = π 2 /2 exactly. With these
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(a)
100
–2 –1 0
–1
0 –1
–1 0
1
12
0
z (km)
40
2
–1
–2
20
1
–3
0
2
–2
0
–3
–3
1
32
43
(b)
z (km)
0.5
0
0.5
–2000
–1500
–1000
–500
0
500
x (km)
Figure 7. (a) Distribution of magnetic field B(x) caused by realistic tsunami wave. Unit of contour label: nano Tesla. (b) Seasurface level: thick solid line, composite wave model; thin dot line, field data measured by satellite Jason-1.
conditions, and setting kh = 0 for very long water waves, the magnetic field expressed in (2.14)
can be simplified to
c0
ζ
,
(4.2)
B(z) = F(z)
c0 + icd h
which is the same as the formula (eqn (14)) in [30] at the sea surface. The simple formula
proposed by Tyler [30] can estimate the magnetic field induced by a single long wave accurately.
The magnetic field B(z) decays at rate e−kz in the formula proposed by Tyler [30]; herein, the
wavenumber k takes the value of the harmonic components after Fourier transform of the surface
elevation.
Formula (4.2) relates to the sea surface in a linear manner, and there are no dispersive effects
taken into account. To further investigate the influence of the dispersive effects, we consider both
long-wave and short-wave components in a realistic Sumatra tsunami signal (4.1). The magnetic
distributions at altitude of z = 100 km and z = 0 km are compared in figure 8.
The long-wave model (4.2) predicts a reasonably accurate result compared with the highorder approximations proposed in the present work. There are slight differences between the
magnetic field distributions predicted by these two formulae. First of all, there is a phase lag
between the long-wave model and the dispersive model. Next, the difference at sea-surface level
is apparent for the shortest wave components in (4.1), for which γ h = 0.23. The intensity of the
second magnetic peak (located at x = −130 km) is overestimated by (4.2). Using the second-order
approximation, the second magnetic peak is located at the trough of the first wave’s magnetic
field. Therefore, a drop of the magnetic intensity takes place for the second magnetic peak.
(b) 2010 Chilean tsunami
As mentioned by Manoj et al. [31], a tiny magnetic signal was captured by magnetic station IPM
during the Chile 2010 tsunami passing through Easter Island, which shows a striking similarity
to the sea-level record at tidal gauge DART 51406 real-time tsunami monitoring systems. There
is also another tidal gauge DART 32412 at a similar distance as DART 51406 away from Easter
Island. As the location of DART 32412 is closer to the epicentre and coastal line of Chile, the
transient effects of the tsunami wave are significant, and the sea-water-level record is not suitable
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13
B(x)(nT)
(b)
2
1
0
–1
–2
–2000
–1500
–1000
x (km)
–500
0
Figure 8. Magnetic field B(z) induced by the Sumatra 2004 tsunami, with parameters h = 5000 m and F (z) = 20 000 nT. (a)
At sea level z = 0 km; (b) at high altitude of z = 100 km. Solid line is present work result and dashed-dotted line is calculated
using Tyler’s formulation (equation (4.2)).
Table 2. Best-fit wave parameters on modelling 2010 Chile tsunami.
..........................................................................................................................................................................................................
Ai (cm)
−1.5
29.0
−11.7
7.4
−10.7
1.0
−5.2
7.3
Ωi (h )
15
12
17
28
25
50
36
33
ti (h)
0.340
0.530
0.674
0.765
0.842
0.900
0.948
1.010
..........................................................................................................................................................................................................
−1
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
Ai (cm)
−2.5
6.6
−5.3
6.8
−3.6
5.0
−4.0
−2.8
Ωi (h )
40
33
40
38
40
40
40
45
ti (h)
1.070
1.121
1.173
1.228
1.275
1.315
1.366
1.415
..........................................................................................................................................................................................................
−1
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
Ai (cm)
2.8
−4.0
2.5
−7.0
2.2
−8.2
1.4
−7.2
Ωi (h )
40
40
45
50
60
50
60
45
ti (h)
1.462
1.500
1.528
1.570
1.605
1.643
1.680
1.712
..........................................................................................................................................................................................................
−1
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
Ai (cm)
−5.3
2.7
−5.3
3.7
−2.5
6.5
−1.2
2.8
Ωi (h )
50
60
60
58
60
52
60
66
ti (h)
1.772
1.807
1.840
1.875
1.900
1.937
1.971
1.992
..........................................................................................................................................................................................................
−1
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
for a quasi-steady-state study. The distance between IPM and DART 51406 is 2650 km. The
water depths are about 3000 and 4480 m at these two sites. The average tsunami velocity can be
estimated using gh. Hence, we use the shifted field record at DART 51406 with a time lag of 3 h
and 50 min as the local wave history at Easter Island. Using the measured field data, the surface
profile of the tsunami wavefront during the first 2 h can be decomposed into 32 components using
(4.3) with least-squares regression,
ζs (t) =
32
Ai sech2 Ωi (t − ti ).
(4.3)
i=1
The wave parameters of each component are shown in table 2.
Once the sea-surface elevation is known in the form of single waves, the magnetic field can be
calculated in a routine procedure as N -wave and Sumatra tsunami. The magnetic fields predicted
by various models and field data are shown in figure 9.
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1
0
–1
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B(x)(nT)
(a)
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(a) 0.3
14
z (m)
0
–0.1
(b)
1.0
B(x)(nT)
0.5
0.0
–0.5
–1.0
–1.5
11.0
11.5
12.0
12.5
13.0
time (h)
Figure 9. (a) Sea-surface elevation at Easter Island on 27 February 2010, shifted 3 h and 50 min from DART 51406, shown
by the dashed line. The solid line is the curve following equation (4.3). Local water depth h = 3000 m. (b) Comparison on
tsunami-induced magnetic field B(z) in the vertical direction. The thick solid line is estimated by the present work, and the
dashed line is calculated using Tyler’s formulation (equation (4.2)). The thin line is field data at gauge station IPM operated by
the Bureau Central de Magnetisme Terrestre, France. The main magnetic field of the Earth F (z) = −19 600 nT. (Online version
in colour.)
Qualitatively, the signal pattern is reproduced by both the present model and Tyler’s model
(equation (4.2)) at the time interval between 11.30 and 12.00. After 12.00 o’clock, the magnitude
and frequency of the magnetic signal induced by relatively short-wave components are not well
predicted for both the present model and Tyler’s model. Since the weak dispersive effects are
taken into account in the present model, the long-wave assumption is verified to be reasonable
for predicting a tsunami-generated magnetic field.
However, it is still insufficient for both the present model and Tyler’s model to accurately
predict the amplitude and waveform of the 2010 Chile tsunami-induced magnetic field. There
are several aspects that could influence the comparison between the predicted and measured
magnetic histories at Easter Island.
Firstly, we used the sea-surface-level history at DART 51406, which is located at 2650 km
northwest of Easter Island. The wavelength of each component is around 50–200 km. Generally,
the wave form will not change as the long wave travels for a finite distance. Shoaling effects are
neglected in the present analysis.
Secondly, local wave deformation is not taken into account. From the geographical data,
the ocean depth changes from 1000 m at the near-shore region to 3000 m offshore around
Easter Island, which may cause local scattering of tsunami waves. Furthermore, diffraction and
refraction effects may cause some other uncertainties in estimation of local wave height when
the tsunami waves pass through Easter Island. As indicated by equation (2.19), the magnetic
magnitude directly depends on the relative surface amplitude α. Therefore, the amplitude due to
local wave deformation is another source of discrepancy.
Thirdly, the proposed formulations are derived from plane waves in the open ocean. For the
realistic tsunami waves considered in this section, the wave pattern is more like cylindrical waves
rather than plane waves. The corresponding three-dimensional effects should not be expected to
be resolved by the present analysis.
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The magnetic field induced by the electrically conducting sea-water motion beneath tsunami
waves decays along the vertical direction in the same way as (2.11). The ideal detection position
should be as low as possible. The existing magnetic stations may provide such a service, e.g. the
data obtained by IPM at Easter Island during the 2010 Chile tsunami. In addition to traditional
shipboard and airborne platforms, unmanned near-space airships, e.g. DARPA’s ISIS [40], are
another economical and long endurance platform for environmental monitoring. Near space is
the region between 20 and 100 km above sea level. Inside near space, the magnetic magnitude
induced by typical harmful tsunamis can reach several nano Tesla, which can be observed easily
by magnetic detection instruments.
Satellites are another candidate for remote detecting of magnetic systems. The low Earth orbit
(LEO) satellite is around 300–1200 km, which is expected to be a suitable altitude for a satellite
detection system. A variety of different types of satellites use LEO, including communication
satellites, Earth monitoring satellites and the International Space Station. As can be seen in
figure 4, the magnetic magnitude can reach 0.2–0.5 nT at altitudes between 300 and 500 km
for a single wave of period 13 min and wave height 1 m. For satellites Ørsted and CHAMP,
scalar and vector resolutions are already better than 0.3 and 0.5 nT [41,42]. Swarm will further
improve spatial and temporal resolutions of multi-satellite missions and make simultaneous
measurements [43]. With the development of these new instruments and methods, we are
optimistic that the magnetic field induced by a giant tsunami can be detected by the LEO satellite
in the near future.
In addition to the magnetic field induced by the motion of sea water, there are other
mechanisms that can induce electromagnetic anomalies during a tsunami event. A relevant
example is the TEC variation in the ionospheric layer due to the atmospheric IGWs generated
by a tsunami. The influence of TEC variation in the ionospheric layer may influence the tiny
magnetic anomalies at satellite altitudes. In this work, the space above the sea surface is assumed
to be a vacuum. The interaction between water-motion-induced magnetic anomalies and IGWinduced TEC anomalies needs to be further investigated. In addition, it should be pointed out
that geomagnetic storms and solar activity of small time periods could restrict the practical
utility of tsunami electromagnetic monitoring. In addition to magnetic magnitude, the spatial and
temporal characteristics are important to identify the magnetic signals induced by the tsunami
water motion from the complex geomagnetic field.
In summary, this work has established an asymptotic approach to predicting the magnetic
field induced by tsunami waves. By introducing a general wave in the form of a hyperbolic
secant square with wavenumber γ , an explicit series solution of the magnetic field has been
established for linear kinematic dynamo problems. Fast convergent rates of the series solution
were verified from the analysis. The second-order approximation formula is sufficiently accurate
for predicting the tsunami-induced magnetic field. The magnetic field induced by a single wave
has been studied as a fundamental solution. For more practical applications, realistic tsunami
waves have complex wave forms and can be treated as the summation of several single-wave
components. Using realistic field sea-surface-level data, the magnetic field has been predicted
for the 2004 Indian Ocean tsunami. Agreement of the magnetic field at sea level has been
obtained by comparison with other full numerical models. Finally, the proposed formulation
has been used to investigate field magnetic signals at Easter Island during the 2010 Chile
tsunami. The present asymptotical model can qualitatively predict the magnetic field induced
..................................................
5. Remarks and conclusions
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In spite of these uncertainties, the main features of the magnetic history can be reasonably
predicted using the proposed formulation, at least partially for the wavefront. The frequency and
distribution qualitatively agree with the measured magnetic data. To make a more reasonable and
precise comparison, a real-time history of the local sea-water surface at Easter Island from oceanic
tsunami simulation models is necessary.
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at DART 51406. The geomagnetic data are obtained from the Bureau Central de Magnetisme Terrestre
(http://www.bcmt.fr). We would also like to acknowledge discussions with Prof. Y. S. He on the potential
applications of magnetic characters. Support by the Shanghai Leading Academic Discipline Project (no. B206)
and the National Natural Science Foundation of China (no. 11272210) is greatly appreciated.
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16
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