1 Supporting Online Materials 2 Minami, T. and H. Toh (2013), Two-dimensional simulations of the tsunami 3 dynamo effect using the finite element method 4 5 1. Observed Magnetic Tsunami Signals at NWP 6 [1] 7 induced by the 2011 Tohoku earthquake tsunami. Figure S1 shows the 3-hour plots of 8 the time series of the vector magnetic field and horizontal tilts at NWP and the vector 9 magnetic fields observed at three land geomagnetic observatories operated by Japan 10 Meteorological Agency (JMA). In the top panel, we can find that the variations reach as 11 large as 3 nT for both by and bz about 100 minutes after the earthquake, which coincides 12 well with the estimated time of tsunami arrival (ETA). These variations can, therefore, 13 be magnetic signals of the tsunami passage. The middle panel indicates that motions of 14 the seafloor instrument are due mainly to seismic waves, which started a few minutes 15 after the earthquake, and it is hard to explain the large magnetic signature at ETA by 16 instrumental motions. In the bottom panel, there are time series of three vector magnetic 17 field variations observed at Memambetsu (MMB), Kakioka (KAK), and Kanoya (KNY). 18 The vector geomagnetic fields show similar profiles among the three observatories, Here we demonstrate how the magnetic field variations observed at NWP are 19 which implies that external sources, such as ionospheric electric currents, caused 20 large-scale magnetic variations near the Japanese Islands after the earthquake. 21 Comparing the magnetic variations at NWP with those at the three land observatories 22 before ETA, it is evident that the land geomagnetic variations as large as 5 nT with a 23 period of about 40 minutes attenuate to less than 1 nT at NWP with a depth of 5616 m. 24 Consequently, we can claim that there were no external geomagnetic variations more 25 than 1 nT at NWP for 3 hours after the earthquake, because the amplitudes of the land 26 geomagnetic variations decreased towards ETA, and the magnetic variations at NWP 27 was larger than 3 nT. 28 29 Fig. S1 The 3-hour plots of the observed time series at the time of the 2011 Tohoku 30 earthquake. (top to bottom) The vector geomagnetic field observed at NWP, the 31 horizontal tilts at NWP, and the three vector geomagnetic fields observed at land 32 geomagnetic observatories: Memambetsu (MMB; 43.910oN, 144.189oE ), Kakioka 33 (KAK; 36.232oN, 140.186oE ), and Kanoya (KNY; 31.424oN, 130.88oE ). In each panel, 34 the x-, y-, and z-components indicate the northward, eastward, and downward 35 components, respectively. All the time series in the three panels are high-pass filtered 36 with a 1-hour cut off period. 37 38 2. Tsunami dynamo effects in TM mode 39 [3] 40 tsunami propagating in the y-direction and oceanic flows in the y-/z-direction coupling 41 with the x-component of the geomagnetic main field. It is expected that emfs driven in 42 the y,z-plane induce the y-/z-component of the electric field and the x-component of the 43 magnetic field. However, we cannot actually detect tsunami-induced EM fields in TM 44 mode, if the oceanic flows are incompressible (Larsen, 1971). 45 shown as follows.. 46 [4] 47 ocean, Here we consider tsunami dynamo effects in TM mode. Figure S2 shows a The reason can be Let us start with the induction equation in terms of the magnetic field in the ๐๐ = ∇ × (๐ฏ × ๐) + ๐พ∇2 ๐. ๐๐ก (S1) 48 Here ๐, ๐ฏ, and K are the magnetic field, the oceanic flow, and the magnetic diffusivity, 49 respectively. If we assume oceanic flows are incompressible, ∇ โ ๐ฏ = 0, and recall ∇ โ 50 ๐ = 0, the source term of Eq. (S1) reduces to ∇ × (๐ฏ × ๐) = (๐ โ ๐)๐ฏ − (๐ฏ โ ๐)๐. (S2) 51 Here we used a vector identity, ∇ × (๐ × ๐) = ๐(๐ โ ๐) − ๐(๐ โ ๐) + (๐ โ ๐)๐ − (๐ โ 52 ๐)๐ and decomposed the magnetic field, ๐, into the geomagnetic main field and the 53 tsunami-induced magnetic field, ๐ + ๐, assuming ∇2 ๐ = 0, ∇ × ๐ = 0, ๐๐ ⁄๐๐ก = 0, 54 and |๐ | โซ |๐|. Using Eq. (S2), the x-component of Eq. (S1) is reduced to ๐๐๐ฅ = −(๐ฏ โ ๐)๐๐ฅ + ๐พ∇2 ๐๐ฅ , ๐๐ก (S3) 55 since vx = 0. If we assume tsunamis to be linear long waves, orders of each term in Eq. 56 (S3) are estimated as follows: ∂ 1 ~ ~10−3 , ∂t T |๐ฏ โ ๐|~c K∇2 = A ~10−5 , h2 K ~10−2 , h2 (S4) (S5) (S6) 57 where T, c, A, and h are the period, the phase velocity, and the height of the tsunami, and 58 the ocean depth, respectively. Here we set T, c, A, h, and K to 103 s, 200 m/s, 1 m, 103 m, 59 and 10-6 m2s-1, respectively. Eqs. (S4) through (S6) indicate that the first term in r.h.s of 60 Eq. (S1) is negligible compared with the second term. This means that tsunami-induced 61 EM fields are very small in TM mode. 62 [5] 63 the contribution of ๐ฏ × ๐ , cannot remain in Eq. (S3), while, in TE mode, the term is 64 very significant because The source term in TM mode becomes insignificant because (๐ โ ๐)๐ฏ, which is |(๐ โ ๐)๐ฏ|/ |(๐ฏ โ ๐)๐| โซ 1. (S7) 65 Larsen (1971) simply demonstrated why emfs driven by ๐ฏ × ๐ vanish in TM mode, by 66 evaluating a line integral of ๐ฏ × ๐ about an arbitrary loop in y,z-plane using Stokes’ 67 theorem: ฬ๐๐ฆ๐๐ง = โฌ(๐ โ ๐)๐ฃ๐ฅ ๐๐ฆ๐๐ง = 0. โฎ(๐ฏ × ๐ ) โ ๐๐ = โฌ[∇ × (๐ฏ × ๐ )] โ ๐ (S8) 68 Equation (S8) indicates the net emf driven by ๐ฏ × ๐ about any y,z-loop vanishes. In 69 other words, short circuits of emfs don’t allow the electric potential difference to exist in 70 incompressible oceanic flows. Consequently, the term, ๐ฏ × ๐ , cannot contribute to 71 motional inductions in TM mode. 72 73 Fig. S2 74 propagate in the positive y-direction, which contains oceanic flows in the y- and 75 z-directions independent of x-coordinate. In the presence of the x-component of 76 geomagnetic main field, emfs in the y,z-plane are driven, inducing electric fields in 77 y,z-plane and the x-component of the magnetic field. Configuration of motional induction in TM mode. Plane tsunami waves 78 79 3. Numerical mesh for tsunami dynamo simulation 80 [2] 81 and tsunami-induced EM signals in the northwest Pacific. In both hydrodynamic and 82 EM calculations, the same mesh shown in Fig. S3 was used. It is noticeable that the 83 actual bathymetry is expressed well by the triangular mesh. Furthermore, the fine mesh 84 around the seafloor and in the ocean enabled us to calculate accurate particle motions of 85 the ocean flows and tsunami-induced electromagnetic (EM) fields at the same time. Figure S3 shows the numerical mesh adopted to reproduce sea level changes 86 87 Fig. S3 The numerical mesh for 2-D FEM calculations using triangular elements. The 88 blue and yellow regions indicate the ocean and the solid Earth beneath the seafloor, 89 respectively. The red line denotes the sub-faults based on Maeda et al. (2011) for 90 calculations of the initial sea surface elevations. 91 92 93 4. Electromagnetic signals of a solitary wave 94 [8] 95 appreciate effects of the first arrival of tsunamis. For mitigation of tsunami disasters, it We here consider a simple case of a solitary tsunami wave, in order to 96 is meaningful to investigate effects of the first wave because it is probably most 97 devastating. For simplicity, we adopt a Gaussian waveform and the long-wave 98 approximation for the solitary wave. Assuming that the solitary wave propagates in the 99 positive y-direction, the wave form and the corresponding sea water flow distribution 100 can be expressed as follows: ๐ = ๐ดexp (− ๐ข๐ฆ = ๐ (๐ฆ − ๐๐ก)2 ), 2๐ 2 ๐ด (๐ฆ − ๐๐ก)2 exp (− ). โ 2๐ 2 (S9) (S10) 101 Here, ๐ = √๐โ is the phase velocity, where h, A, and χ are the ocean depth, the height 102 of the solitary wave, and the horizontal scale length of the tsunami dynamic field. In this 103 circumstance, induced EM fields are considered to be functions of ๐ฆ − ๐๐ก. Then we can 104 derive a relation between derivatives of ๐๐ง : ๐๐๐ง ⁄๐๐ก = −๐ ๐๐๐ง ⁄๐๐ฆ. Using this relation, Eq. 105 (6) in the main text is reduced to ( ๐2 ๐2 ๐ ๐ + + ๐๐๐ ) ๐๐ง = ๐๐ (๐น๐ง ๐ข๐ฆ ). 2 2 ๐๐ฆ ๐๐ง ๐๐ฆ ๐๐ฆ (S11) 106 Equation (S11) can easily be solved numerically by the finite element method (FEM) 107 because of its independence of time. The boundary conditions for all the components 108 mentioned in the main text (i.e., nil conditions on the boundaries far from the center 109 grid) are adopted here again. Hence, all the EM components in TE mode (bz, by, and ex) 110 induced by the solitary wave in Eqs. (S9) and (S10) can be calculated using Eqs. (8) and 111 (9) in the main text in addition to Eq. (S11). In the calculation, we assumed a flat ocean 112 of 5000m deep, χ = 20 km for the solitary wave, namely, a wave with a horizontal 113 extent of approximately 100 km, and half-space insulators above and beneath the 114 seafloor. The vertical component of the geomagnetic main field at NWP, -37000 nT, was 115 assigned to Fz. 116 [9] 117 and a sum of the inducing v×B field and the induced horizontal electric field (ex) when 118 the solitary wave passes. In Fig. S4, the profile of bz on the seafloor is very similar to an 119 upside-down wave-form of the solitary wave, although bz is not symmetric due to the 120 effect of self-induction. This similarity also appears in Tyler’s (2005) frequency-domain 121 solution. Namely, the relation between the sea level change, ๐ , and the vertical 122 magnetic field, bz, can be expressed as; Figure S4 shows the vertical and horizontal magnetic components (bz and by) ๐๐ง ≈ ๐ ๐น ๐ ๐(๐ง+โ) , โ ๐ง (๐ง < −โ) (S12) 123 in deep oceans. Here, z is upward positive and ๐ง = −โ corresponds to the flat seafloor. 124 This equation indicates that bz and ๐ vary almost in-phase as if the vertical magnetic 125 field were frozen into the tsunami flow. 126 induced magnetic field (by) and the total electric field (ex + v×B ), it looks difficult to 127 interpret them by the analytical solution in frequency domain, because Tyler’s (2005) As for the horizontal component of the 128 solution requires the same amplitude for bz and by. Furthermore, by initially starts to rise 129 prior to the arrival of the tsunami wave and the bz variation. As shown in Fig. 4 in the 130 main text, the initial rise in seafloor by is caused by the induced electric field, ex, which 131 slightly precedes the inducing v×B field and points the direction opposite to v×B. In the 132 bottom panel of Fig. S4, the preceding phase in ex is clearly seen above and beneath the 133 ocean layer thanks to the absence of the inducing v×B field. Although the induced ex is 134 weaker than the inducing v×B, difference in phase between them makes a positive 135 x-component current in front of v×B, which causes an initial rise in seafloor by prior to 136 the bz variations. 137 138 Fig. S4 Magnetic and electric fields induced by a rightward propagating solitary 139 tsunami which has a Gaussian wave-form. In the lower three panels, the sea surface and 140 seafloor are depicted by horizontal black lines at depths of 0 km and -5 km, respectively. 141 The red and blue colors denote positive and negative values of each component, 142 respectively. Profiles of each EM component on the seafloor are also drawn by black 143 solid lines. In the lowest panel, the sum of the inducing v×B field and the induced 144 electric field, ex, is shown. The vertical blue dashed line corresponds to the center of the 145 symmetric wave-form, where the center of the inducing v×B field resides as drawn in 146 the lowest panel. 147 148 5. Estimation of the error due to the Coriolis force 149 Here, we estimate the error due to the Coriolis force in the tsunami dynamics. The 150 governing momentum equation in terms of the particle motion including the Coriolis 151 force is expressed as, ๐๐ก ๐ฏ + (๐ฏ โ ๐๐ป )๐ฏ = −๐๐๐ป ๐ − 2๐ × ๐ฏ, (S13) 152 where ๐ฏ, ๐, and ๐ are the depth averaged horizontal vector velocity, the sea surface 153 elevation, and the angular velocity of the Earth’s rotation, respectively. ๐๐ก and ๐๐ป are the 154 derivative in terms of time and the horizontal coordinate, respectively. Since we 155 neglected the 2nd term on the r.h.s. of Eq. (S13) in the main text, here we compare the 156 magnitudes of those terms quantitatively. Amplitudes of the first and second term on 157 the r.h.s of Eq. (S13) are estimated as follows: |๐๐๐ป ๐| = ๐๐ด⁄๐ = 5 × 10−5, (S14) |2๐ × ๐ฏ| = 2๐บsin๐ โ ๐ฃ = 4 × 10−6 . (S15) 158 Here g, A, ๐, ๐, and v are the gravity acceleration, the wave height, the wavelength, the 159 latitude, the particle velocity, respectively. We adopted ๐ = 9.8 m/s2, ๐ด = 1m, ๐ = 160 200 km, θ = 45 °, and ๐ฃ = 4 cm/s, assuming the 2011 off the Tohoku earthquake 161 tsunami passing through the northwest Pacific. Equation (S14) and (S15) show that the 162 effects of the Coriolis force is less than 10 % of that of the pressure gradient. This order 163 estimation is consistent with other preceding numerical studies (e.g., Dao and Tkalich, 164 2007). Furthermore, it is reported that the Coriolis force significantly influences the 165 height of the tsunami rather than the arrival time (Shuto, 1991). We, therefore, can say 166 that there are at most 10 % errors in the tsunami height due to the Colioris force in our 167 simulations. 168 169 References 170 Dao, M. H. and P. Tkalich (2007), Tsunami propagation modeling – a sensitivity study, 171 172 173 Nat. Hazards Earth Syst. Sci., 7, 741-754. Shuto, N. (1991), Numerical Simulation of Tsunamis – Its Present and Near Future, Natural Hazards, 4, 171-191.