## Abstract

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 -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, run-up can be well predicted by various wave models, e.g. nonlinear shallow water equations [1–4]. When breaking and inundation are involved, Navier–Stokes 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 [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 (*M*_{2} 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 -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 sech^{2}*γx*. The magnetic field induced by single waves and -waves can be obtained directly by selecting a corresponding wavenumber *γ*. Characteristic of the magnetic field induced by single and -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.

## 2. Explicit series solutions of the kinematic dynamo problem

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.

### (a) Review of the magnetic field due to harmonic water waves

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
2.1where the superscripts (*x*) and (*z*) represent the components of the magnetic field in each direction, *K*=(*μσ*)^{−1}, *μ*=4*π*×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 , 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. 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*)=*C*_{1} e^{−kz}+*C*_{2} e^{kz}. From the limited value of *b*(*z*) in the far field, we get *C*_{2}=0 in the air region over the sea surface and *C*_{1}=0 in the Earth below the seafloor. We have the following condition for the magnetic field at the air and Earth boundaries:
2.3

These two conditions can be viewed as the boundary conditions for *B*^{(x)} in the ocean layer when considering the continuity of *B*^{(x)} and on both sides of the two interfaces between air, the Earth and the ocean. In the ocean layer, the governing equation is
2.4with *A*≡|**F**⋅∇*u*|. Substituting the general solution *B*^{(x)}=*b*(*z*) e^{−i(ωt−kx)} into (2.4)), we have
2.5Defining , 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 §2*b*. The value at the sea surface is
2.6

Using , the following can be obtained: 2.7

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 .

For ocean swells of periods around 10 s, Weaver [21] simplified as 1+i*β*/2 when *β* was a small parameter defined as *β*=4*πσg*^{2}/*ω*^{3}. However, this approximation is not suitable for tsunami waves. The relationship between wavenumber *k* and angular frequency *ω* is the linear dispersion equation . We will introduce another kind of perturbation analysis in §2*c*. The solution for *B*^{(z)} is similar and is given in §2*d*.

### (b) Magnetic field of the general tsunami wave model

Without loss of generality, a typical single tsunami wave model is chosen as
2.8in 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 . We define the relative wave height *α*=*H*/*h* and phase *θ*=*γ*(*x*−*c*_{0}*t*) in the following analysis. The horizontal velocity component of the general tsunami wave field can be approximated by *u*=*c*_{0}*ζ*/*h* for tiny-amplitude 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 *u*_{z}=0. Substituting the horizontal velocity into (2.1), we get
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
2.10using a Fourier transform of , 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,
2.11with Fourier integration by substituting *A* by spectrum *A*_{0}(*k*) expressed in (2.10).

Integral (2.11) can be numerically integrated easily since *A*_{0}(*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 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*/ 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 expansion. Power series expansions with the small parameters *κh* and *kh* for terms in (2.7) are
2.12and
2.13recall the definition .

For leading-order approximation, the values of both and are equal to 1. Therefore,
2.14where is the phase velocity of the long wave. Variable *c*_{d}=2*K*/*h* is defined as the *lateral diffusion speed* by Tyler [30]. For typical sea water, *K*=10^{7}/16*π*, and values of *c*_{d} and *c*_{0} are shown in figure 1. For most parameters in open ocean, *c*_{d} and *c*_{0} have comparable values. In fact, the magnetic Reynolds number can be defined as *R*_{m}=2*c*_{0}/*c*_{d} when choosing long-wave celerity *c*_{0} 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 *R*_{m}=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 *R*_{m}∝*h*^{3/2}. In fact, the coefficients *β*_{00} and *β*_{01} can be expressed in magnetic Reynolds number with minor manipulations.

The second- and fourth-order approximations of *b*(0) with respect to *kh* are
2.15with
and
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 *c*_{d} and *c*_{0} 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 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 , the integration of the leading-order and the second-order approximation of *b*(0) can be simplified as
2.17and
2.18Here, is the *n*th-order polygamma function, see [35].

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 −2*αγc*_{0}*F*, reads
2.19in which *β*_{nj} can be calculated from the series expansion of and 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 *γh* 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 *γh*=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 *γh*=0.087, which is in agreement with the error analysis shown in figure 2. For the magnetic field induced by a tsunami wave, the second-order 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 *w*_{z}=−*u*_{x}. For a small-amplitude wave, the horizontal derivative of the vertical wave velocity *w*_{x}∼*αu*_{x} 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
2.20which 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 -waves

### (a) Single waves

In this section, we study the magnetic field induced by a single wave proposed by Madsen & Schäffer [36],
3.1Here, *Ω*=2*π*/*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*/*c*_{0}, applying the analytical solution to a single wave is straightforward. For a single wave of period *T*=13 min, the dispersive parameters are *γh*=0.081 for water depth *h*=1000 m and *γh*=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 *γh*, 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.

### (b) -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 -wave when it is induced by a high-magnitude earthquake in deep ocean as shown by the numerical results of Zhao *et al.* [37]. -waves may occur in both deep and shallow water regions, and are representative models of realistic tsunamis. The profile of an -wave changes little over a long distance during oceanic transmission. -waves caused by two counter-acting single waves are chosen as follows:
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 -wave is represented by *μ*=1. A generalized -wave is represented by 0<*μ*<1. The horizontal velocity can be approximated as *u*=*c*_{0}*ζ*_{s}/*h*. The velocity gradient *u*_{x} is
3.3

The Fourier transform of *u*_{x} is
3.4

Three typical -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 *u*_{x} 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
3.5where . The magnetic fields induced by various -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.

## 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
4.1Given the values of the local peak's position *x*_{i} and length scale *γ*_{i}, the amplitudes *A*_{i} 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 -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 , we have when *x*=0 and *z*=0 for a position just over the sea surface at height *ζ*. The value of exactly. With these conditions, and setting *kh*=0 for very long water waves, the magnetic field expressed in (2.14) can be simplified to
4.2which 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 high-order 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 *γh*=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 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 . 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,
4.3The 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 -wave and Sumatra tsunami. The magnetic fields predicted by various models and field data are shown in figure 9.

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.

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.

## 5. Remarks and conclusions

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 IGW-induced 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 by the front of the tsunami waves. Therefore, the possibility of detecting tsunami waves using magnetic anomalies is verified.

Based on the spatial and temporal behaviour of magnetic signals over the ocean, a small-amplitude transoceanic tsunami is expected to be detected before it propagates into shallow water regions, which can help to improve the accuracy of early warning system and avoid unnecessary evacuation events.

## Acknowledgements

We thank Keller Stroke from NOAA for providing 2010 Chilean tsunami wave elevation 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.

- Received January 17, 2013.
- Accepted May 30, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.