Theoretical and Applied Mechanics Letters 5 (2015) 187–190

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Theoretical and Applied Mechanics Letters

journal homepage: www.elsevier.com/locate/taml

Letter On the magnetic anomaly at during the 2010

Benlong Wang a,b,∗, Xiaoyu Guo a, Hua Liu a,b, Cheng Gong c a Department of Engineering Mechanics, 800 Dongchuan Road, Shanghai 200240, China b Ministry of Education Key Laboratory of Hydrodynamics, 800 Dongchuan Road, Shanghai 200240, China c Chinese Aviation Radio and Electronic Research Institute (CARERI), Shanghai 200233, China article info a b s t r a c t

Article history: A magnetic anomaly was recorded at Easter Island on 27 February 2010 during the Chile tsunami event. Received 3 August 2015 The physics of the magnetic anomaly is analyzed using kinematic dynamo theory. Using a single wave Accepted 10 August 2015 model, the space and time behavior of the magnetic field is given. By joint analysis of the magnetic Available online 8 September 2015 observations, tide gauge data and numerical results of the global tsunami propagation, we show the *This article belongs to the Fluid Mechanics close resemblance between the predicted spatial and temporal magnetic distributions and the field data, indicating the magnetic anomaly at Easter Island was actually induced by the motion of seawater under Keywords: tsunami waves. Similarity between the field magnetic data at Easter Island during 2010 Chile tsunami and Magnetic anomaly Tsunami sea surface level is verified with realistic tsunami propagating model. Nonlinear shallow water equations © 2015 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).

2010 Chile earthquake occurred off the coast of central Chile of 20–30 nT. For some internal wave in tropical area, measured on February 27, 2010, at 06:34 UTC, having a magnitude of 8.8 magnetic fields value may reach 100 nT [6]. Tyler [7] compared and on the . Large tsunami was generated found similarities between the results of a numerical prediction of and propagated across the south Pacific ocean. A tiny magnetic lunar M2 tide (i.e. twice tide cycles in one day) and scalar magnetic signal oscillation was captured by the magnetic station named IPM records measured by satellite CHAMP. It has been shown for the after UTC 11:30 am when the tsunami wave passed through Easter first time that the ocean flow makes a substantial contribution Island as shown in Fig. 1. The occurrence of oscillation of magnetic to the geomagnetic field at satellite altitude. For the magnetic field is coincident with the arriving time of tsunami wave, which anomaly induced by 2010 Chile tsunami, it is the first time that suggests the tight connection between the seawater’s motion and field magnetic data induced by seawater’s motion under tsunami variation of magnetic field variation. was captured. However, the shifted sea surface record at Deep- Magnetic anomaly induced by seawater’s motion is recognized ocean Assessment and Reporting of (DART) 51406 with and known in very early days [1–3]. Seawater is an electrically a time lag were imperfectly used to verify the existence of the conducting media which generates small electromagnetic fields similarity due to the lack of accurate seismic data [8,9]. as it flows through the Earth’s main magnetic field. Gravity Employing global tsunami transport modeling, at now we can surface water wave is a common flow form in ocean. Around simulate the sea surface elevation during the 2010 tsunami event O(0.1)–O(1) nT magnetic field can be generated by the Kelvin accurately. We start with modeling the sea surface elevation dur- waves after submarine and surface ship at cruise speed [4,5]. ing the selected tsunami events, employing the phase resolved Electromagnetic fields tend to enhance when wave periods and global tsunami propagation equations. A benchmarked numeri- lengths increase. Waves of period 16–17 s and amplitude of 1 m cal package GeoClaw [10], an adaptive numerical solver of non- can induce magnetic anomaly of 1 nT. Swells of period 10–20 s linear shallow water equations in spherical coordinates, was used can induced a magnetic field of 103–104 times of the one’s induced to simulate the 2010 Chile tsunami propagation across the south by wind waves of 5 s. Typical tide flow can induce magnetic field Pacific ocean, which has been successfully used on 2004 Suma- tra tsunami and 2012 Tohoku tsunami [11]. ETOPO 2-min global dataset at the National Geophysical Data Center (NGDC) is used for ∗ Corresponding author at: Department of Engineering Mechanics, 800 the bathymetry in offshore region at a resolution of 10 min in lat- Dongchuan Road, Shanghai 200240, China. itude and longitude, and 1-min dataset is used near the epicenter. E-mail address: [email protected] (B. Wang). Initial sea surface deformation was computed by applying Okada’s http://dx.doi.org/10.1016/j.taml.2015.08.001 2095-0349/© 2015 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 188 B. Wang et al. / Theoretical and Applied Mechanics Letters 5 (2015) 187–190

40 2 32412 20

0 0 –20 0 2 4 6 8

Magnetic field (nT) Magnetic field –2 40 6168101214 Simulations 51406 20 UTC time (h) DART data 0 Fig. 1. Vertical component of total main magnetic field at IPM during February 27, 2010 Chile tsunami event. Mean value of -19 000 nT has been subtracted. Magnetic –20

gauge station IPM is operated by the Bureau Central de Magnetisme Terrestre, (cm) Water elevation 4681012 France, locates at (−27.200° S, −109.420° W). formulas for 180 sub-fault using the fault parameter [12]. With 20 E the refined adaptive mesh, good agreements at DART 32412 and W N 51406 are obtained as shown in Fig. 2. Although there is no DART 0 S data at Easter Island, we have confidence that the numerical re- –20 sults of GeoClaw could predict reliable sea surface elevation since 475 6 the position IPM locates between DART 32412 and 51406 along the Time (h) tsunami’s path. On a mild slope bathymetry, leading waves from shoaling of a transient wave usually appear like a capital letter N Fig. 2. Sea surface elevation during February 27, 2010 Chile tsunami event. DART or M [13,14]. In the same situation, numerical wave gauges from 51406 and 32412 are two buoys for real time tsunami monitoring systems operated GeoClaw record similar tsunami shapes as shown in Fig. 2 when the by National Oceanic and Atmospheric Administration (NOAA) center for tsunami tsunami approaching Easter Island. However, a single wave profile research. Sea surface elevations at four numerical wave gauges around Easter Island, wherein E, W, N and S are abbreviation of each directions. These four positions are was captured by DART buoy, where the water is deep and there is two degrees away from Easter Island. no wave shoaling effects. Snapshots of sea surface elevation during the tsunami event are given in Fig. 3. theory, the induced magnetic field could be predicted when the As shown in Fig. 2, the numerical model nearly perfectly cap- seawater velocity is known. tures the period and amplitude of the first wave at site DART 51406. Harmonic solutions of linear kinematic dynamo problem for The simulated tsunami arrived approximately 6 min before the ob- linear progressive waves were obtained by many works [2,3]. served tsunami. Such a phase shift has been previously discussed Focusing in the tsunami induced magnetic field, two kinds of one and corresponds to the dispersive effects of natural tsunami waves. dimensional simplified models are derived: the first one is based However, the nonlinear shallow-water equations used in present on the projection of Maxwell equation at vertical direction [15]; simulations neglect dispersive effects and therefore over-predict taken the dispersive effects into account an elaborated model was the phase speed. From the linear frequency dispersion relation, the proposed recently [9]. The attenuation of the induced magnetic phase speed of tsunami waves can be estimated by field could be well predicted by these two models for long waves.   At sea surface, both of these two models reduce to a simple  1 2 2 4 4 formulae: the induced magnetic field linearly depends on the sea C0 = gh 1 − k h + O(k h ) . (1) 2 surface elevation. The other relative parameters include the local main magnetic field, the local water depth and conductivity of the By estimating the average ocean depth as 4000 m and the leading seawater. wavelength as 200 km, we obtain a correction due to dispersive When considering the induced magnetic field at the sea surface effects of 1 k2h2 ≈ 0.01. This correction would result in approx- 2 only, the dispersive model and long-wave approximation reduce imately a 1% change in the arrival time of the leading wave; for to the same expression for long waves: present study, this corresponds to a 6-min difference, which nearly equals the time lag in Fig. 2 for DART 51406. C0 ζ B(z) = F (z) . (2) At DART 32412, the main feature of the sea surface elevation is C0 + iCd h captured. The slight discrepancy is attributed to the interaction of Here, h is the local water depth and ζ is the sea surface el- large-scale resonance from the coast to the edge of the continental evation obtained from the numerical simulations with nonlinear shelf. With validation using data from the two DART gauges, shallow water equations. At a distance of two degrees from Easter the numerical tsunami propagation model provides a reliable sea Island, the average water depth is taken as h = 3000 m. F (z) is surface elevation for the leading wave near Easter Island. The the vertical component of the magnetic field of Earth. The value trailing tsunami waves near Easter Island, however, cannot be well of F (z) = 19600 nT at Earth’s surface at station IPM on Easter Is- predicted by nonlinear shallow-water equations due to the weak land is used in this analysis, although it may vary slowly over a dispersive properties. Therefore, the magnetic field induced by the timescale√ of several hours or even over periods of months or years. trailing waves is excluded from our analysis. C0 = gh is the phase speed of the tsunami, and Cd = 2K/h is the Generation of magnetic field by a conductive fluid’s motion is lateral diffusion speed of the magnetic field, where K = (µσ )−1 known as the dynamo problem. The complete dynamo problem can be interpreted as a diffusion coefficient determined by the per- consists of the equations of motion and Maxwell equations, which meability of free space, µ = 4π × 10−7 N/A2, and the electri- are coupled by the action of the Lorentz forces. Due to the cal conductivity of seawater, σ = 4 S/m. Consequently, we set tiny magnetic strength caused by seawater’s motion, the Lorentz K = 107/(16π) A2m/(NS) in the following analysis. With these forces exerted on seawater can be neglected, which lead to an parameters, we can easily calculate the vertical magnetic field B(z) equilibrium magnetic field. Therefore the hydrodynamic problem (e.g. the results shown in Fig. 4) from the sea surface elevation ζ and dynamo problem are decoupled. Using kinematic dynamo by multiplying a coefficient of 4.1 at Easter Island. B. Wang et al. / Theoretical and Applied Mechanics Letters 5 (2015) 187–190 189

Sea surface 1 elevation (m) Surface at 5.00 hours 0° 0.30 0.5

0.25 0 ° –10 0.20 –0.5

0.15 (nT) Magnetic field –20° –1 0.10 1112 13 14 UTC time (h) –30° 0.05 Fig. 4. Predicted and observed amplitude are almost congruent. The north and 0.00 west numerical wave gauges information are used to estimate the magnetic field, –40° due to the bathymetry effects are fully developed when the tsunami passed by the –0.05 island. The slow variation of main Earth magnetic field is filtered. In the trailing part after 12 o’clock, the short dispersive tsunami waves are not be accurately predicted –50° –0.10 with nonlinear shallow water equations. Consequently, the periodic variation of magnetic signal could not be predicted. –0.15 –60° The space–time behavior of magnetic anomalies induced by –130° –120° –110° –100° –90° –80° –70° –0.20 tsunamis in open oceans has been investigated. It was found Surface at 5.00 hours that the magnitude of the magnetic anomaly induced by a 0.2 m –22° tsunami is on the order of 1.0 nT at the sea surface. Even using a simple 1D kinematic dynamo model, the induced magnetic field is well predicted, see Fig. 4, which indicating the essence of the –24° phenomenon is well captured. Focusing on tsunami waves, it has been verified in a very real –26° sense that the seawater’s motion induced an observable magnetic field, although it is a tiny signal comparing with the background main Earth magnetic field. This has important implications: a –28° small amplitude transoceanic tsunami is expected to be on time detected before it propagates into shallow water region, which may serve as a makeup especially at regions such as Indian Ocean –30° and South China Sea, where tsunami monitoring systems have not been setup yet but high risk is identified for the next devastating –32° tsunami [16]. In broader terms, it encourages future studies on the multidiscipline interactions associated with tsunami as well as other kinds of long waves in ocean. Based on the spatial and –34° temporal behaviors of the magnetic signal over the ocean surface, –114° –112° –110° –108° –106° –104° –102° seawater’s motions of certain scales are expected to be detected remotely by unmanned near space airship or satellite [7,17], which Surface at 5.50 hours –22° may provide novel approaches for scientific research.

Acknowledgments –24° We thank Keller Stroke from NOAA for providing 2010 Chile –26° tsunami wave elevation at DART 32412 and 51406. The geomag- netic data are obtained from the Bureau Central de magnetisme Terrestre (http://www.bcmt.fr). Supports by the Shanghai Leading –28° Academic Discipline Project (No. B206) and National Natural Sci- ence Foundation of China (No. 11272210) are greatly appreciated.

–30° References

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