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Supporting Online Materials
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Minami, T. and H. Toh (2013), Two-dimensional simulations of the tsunami
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dynamo effect using the finite element method
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1. Observed Magnetic Tsunami Signals at NWP
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[1]
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induced by the 2011 Tohoku earthquake tsunami. Figure S1 shows the 3-hour plots of
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the time series of the vector magnetic field and horizontal tilts at NWP and the vector
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magnetic fields observed at three land geomagnetic observatories operated by Japan
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Meteorological Agency (JMA). In the top panel, we can find that the variations reach as
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large as 3 nT for both by and bz about 100 minutes after the earthquake, which coincides
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well with the estimated time of tsunami arrival (ETA). These variations can, therefore,
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be magnetic signals of the tsunami passage. The middle panel indicates that motions of
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the seafloor instrument are due mainly to seismic waves, which started a few minutes
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after the earthquake, and it is hard to explain the large magnetic signature at ETA by
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instrumental motions. In the bottom panel, there are time series of three vector magnetic
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field variations observed at Memambetsu (MMB), Kakioka (KAK), and Kanoya (KNY).
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The vector geomagnetic fields show similar profiles among the three observatories,
Here we demonstrate how the magnetic field variations observed at NWP are
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which implies that external sources, such as ionospheric electric currents, caused
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large-scale magnetic variations near the Japanese Islands after the earthquake.
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Comparing the magnetic variations at NWP with those at the three land observatories
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before ETA, it is evident that the land geomagnetic variations as large as 5 nT with a
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period of about 40 minutes attenuate to less than 1 nT at NWP with a depth of 5616 m.
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Consequently, we can claim that there were no external geomagnetic variations more
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than 1 nT at NWP for 3 hours after the earthquake, because the amplitudes of the land
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geomagnetic variations decreased towards ETA, and the magnetic variations at NWP
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was larger than 3 nT.
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Fig. S1 The 3-hour plots of the observed time series at the time of the 2011 Tohoku
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earthquake. (top to bottom) The vector geomagnetic field observed at NWP, the
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horizontal tilts at NWP, and the three vector geomagnetic fields observed at land
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geomagnetic observatories: Memambetsu (MMB; 43.910oN, 144.189oE ), Kakioka
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(KAK; 36.232oN, 140.186oE ), and Kanoya (KNY; 31.424oN, 130.88oE ). In each panel,
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the x-, y-, and z-components indicate the northward, eastward, and downward
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components, respectively. All the time series in the three panels are high-pass filtered
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with a 1-hour cut off period.
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2. Tsunami dynamo effects in TM mode
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[3]
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tsunami propagating in the y-direction and oceanic flows in the y-/z-direction coupling
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with the x-component of the geomagnetic main field. It is expected that emfs driven in
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the y,z-plane induce the y-/z-component of the electric field and the x-component of the
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magnetic field. However, we cannot actually detect tsunami-induced EM fields in TM
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mode, if the oceanic flows are incompressible (Larsen, 1971).
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shown as follows..
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[4]
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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
𝜕𝐁
= ∇ × (𝐯 × 𝐁) + 𝐾∇2 𝐁.
𝜕𝑡
(S1)
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Here 𝐁, 𝐯, and K are the magnetic field, the oceanic flow, and the magnetic diffusivity,
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respectively. If we assume oceanic flows are incompressible, ∇ ∙ 𝐯 = 0, and recall ∇ ∙
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𝐁 = 0, the source term of Eq. (S1) reduces to
∇ × (𝐯 × 𝐁) = (𝐅 ∙ 𝛁)𝐯 − (𝐯 ∙ 𝛁)𝐛.
(S2)
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Here we used a vector identity, ∇ × (𝐀 × 𝐁) = 𝐀(𝛁 ∙ 𝐁) − 𝐁(𝛁 ∙ 𝐀) + (𝐁 ∙ 𝛁)𝐀 − (𝐀 ∙
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𝛁)𝐁 and decomposed the magnetic field, 𝐁, into the geomagnetic main field and the
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tsunami-induced magnetic field, 𝐅 + 𝐛, assuming ∇2 𝐅 = 0, ∇ × 𝐅 = 0, 𝜕𝐅⁄𝜕𝑡 = 0,
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and |𝐅| ≫ |𝐛|. Using Eq. (S2), the x-component of Eq. (S1) is reduced to
𝜕𝑏𝑥
= −(𝐯 ∙ 𝛁)𝑏𝑥 + 𝐾∇2 𝑏𝑥 ,
𝜕𝑡
(S3)
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since vx = 0. If we assume tsunamis to be linear long waves, orders of each term in Eq.
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(S3) are estimated as follows:
∂ 1
~ ~10−3 ,
∂t T
|𝐯 ∙ 𝛁|~c
K∇2 =
A
~10−5 ,
h2
K
~10−2 ,
h2
(S4)
(S5)
(S6)
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where T, c, A, and h are the period, the phase velocity, and the height of the tsunami, and
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the ocean depth, respectively. Here we set T, c, A, h, and K to 103 s, 200 m/s, 1 m, 103 m,
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and 10-6 m2s-1, respectively. Eqs. (S4) through (S6) indicate that the first term in r.h.s of
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Eq. (S1) is negligible compared with the second term. This means that tsunami-induced
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EM fields are very small in TM mode.
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[5]
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the contribution of 𝐯 × 𝐅, cannot remain in Eq. (S3), while, in TE mode, the term is
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very significant because
The source term in TM mode becomes insignificant because (𝐅 ∙ 𝛁)𝐯, which is
|(𝐅 ∙ 𝛁)𝐯|/ |(𝐯 ∙ 𝛁)𝐛| ≫ 1.
(S7)
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Larsen (1971) simply demonstrated why emfs driven by 𝐯 × 𝐅 vanish in TM mode, by
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evaluating a line integral of 𝐯 × 𝐅 about an arbitrary loop in y,z-plane using Stokes’
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theorem:
̂𝑑𝑦𝑑𝑧 = ∬(𝐅 ∙ 𝛁)𝑣𝑥 𝑑𝑦𝑑𝑧 = 0.
∮(𝐯 × 𝐅) ∙ 𝑑𝒔 = ∬[∇ × (𝐯 × 𝐅)] ∙ 𝒙
(S8)
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Equation (S8) indicates the net emf driven by 𝐯 × 𝐅 about any y,z-loop vanishes. In
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other words, short circuits of emfs don’t allow the electric potential difference to exist in
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incompressible oceanic flows. Consequently, the term, 𝐯 × 𝐅, cannot contribute to
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motional inductions in TM mode.
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Fig. S2
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propagate in the positive y-direction, which contains oceanic flows in the y- and
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z-directions independent of x-coordinate. In the presence of the x-component of
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geomagnetic main field, emfs in the y,z-plane are driven, inducing electric fields in
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y,z-plane and the x-component of the magnetic field.
Configuration of motional induction in TM mode. Plane tsunami waves
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3. Numerical mesh for tsunami dynamo simulation
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[2]
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and tsunami-induced EM signals in the northwest Pacific. In both hydrodynamic and
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EM calculations, the same mesh shown in Fig. S3 was used. It is noticeable that the
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actual bathymetry is expressed well by the triangular mesh. Furthermore, the fine mesh
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around the seafloor and in the ocean enabled us to calculate accurate particle motions of
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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
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Fig. S3 The numerical mesh for 2-D FEM calculations using triangular elements. The
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blue and yellow regions indicate the ocean and the solid Earth beneath the seafloor,
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respectively. The red line denotes the sub-faults based on Maeda et al. (2011) for
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calculations of the initial sea surface elevations.
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4. Electromagnetic signals of a solitary wave
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[8]
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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
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is meaningful to investigate effects of the first wave because it is probably most
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devastating. For simplicity, we adopt a Gaussian waveform and the long-wave
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approximation for the solitary wave. Assuming that the solitary wave propagates in the
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positive y-direction, the wave form and the corresponding sea water flow distribution
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can be expressed as follows:
𝜂 = 𝐴exp (−
𝑢𝑦 = 𝑐
(𝑦 − 𝑐𝑡)2
),
2𝜒 2
𝐴
(𝑦 − 𝑐𝑡)2
exp (−
).
ℎ
2𝜒 2
(S9)
(S10)
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Here, 𝑐 = √𝑔ℎ is the phase velocity, where h, A, and χ are the ocean depth, the height
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of the solitary wave, and the horizontal scale length of the tsunami dynamic field. In this
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circumstance, induced EM fields are considered to be functions of 𝑦 − 𝑐𝑡. Then we can
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derive a relation between derivatives of 𝑏𝑧 : 𝜕𝑏𝑧 ⁄𝜕𝑡 = −𝑐 𝜕𝑏𝑧 ⁄𝜕𝑦. Using this relation, Eq.
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(6) in the main text is reduced to
(
𝜕2
𝜕2
𝜕
𝜕
+
+ 𝜇𝜎𝑐
) 𝑏𝑧 = 𝜇𝜎 (𝐹𝑧 𝑢𝑦 ).
2
2
𝜕𝑦
𝜕𝑧
𝜕𝑦
𝜕𝑦
(S11)
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Equation (S11) can easily be solved numerically by the finite element method (FEM)
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because of its independence of time. The boundary conditions for all the components
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mentioned in the main text (i.e., nil conditions on the boundaries far from the center
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grid) are adopted here again. Hence, all the EM components in TE mode (bz, by, and ex)
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induced by the solitary wave in Eqs. (S9) and (S10) can be calculated using Eqs. (8) and
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(9) in the main text in addition to Eq. (S11). In the calculation, we assumed a flat ocean
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of 5000m deep, χ = 20 km for the solitary wave, namely, a wave with a horizontal
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extent of approximately 100 km, and half-space insulators above and beneath the
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seafloor. The vertical component of the geomagnetic main field at NWP, -37000 nT, was
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assigned to Fz.
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[9]
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and a sum of the inducing v×B field and the induced horizontal electric field (ex) when
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the solitary wave passes. In Fig. S4, the profile of bz on the seafloor is very similar to an
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upside-down wave-form of the solitary wave, although bz is not symmetric due to the
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effect of self-induction. This similarity also appears in Tyler’s (2005) frequency-domain
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solution. Namely, the relation between the sea level change, 𝜂 , and the vertical
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magnetic field, bz, can be expressed as;
Figure S4 shows the vertical and horizontal magnetic components (bz and by)
𝑏𝑧 ≈
𝜂
𝐹 𝑒 𝑘(𝑧+ℎ) ,
ℎ 𝑧
(𝑧 < −ℎ)
(S12)
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in deep oceans. Here, z is upward positive and 𝑧 = −ℎ corresponds to the flat seafloor.
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This equation indicates that bz and 𝜂 vary almost in-phase as if the vertical magnetic
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field were frozen into the tsunami flow.
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induced magnetic field (by) and the total electric field (ex + v×B ), it looks difficult to
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interpret them by the analytical solution in frequency domain, because Tyler’s (2005)
As for the horizontal component of the
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solution requires the same amplitude for bz and by. Furthermore, by initially starts to rise
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prior to the arrival of the tsunami wave and the bz variation. As shown in Fig. 4 in the
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main text, the initial rise in seafloor by is caused by the induced electric field, ex, which
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slightly precedes the inducing v×B field and points the direction opposite to v×B. In the
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bottom panel of Fig. S4, the preceding phase in ex is clearly seen above and beneath the
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ocean layer thanks to the absence of the inducing v×B field. Although the induced ex is
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weaker than the inducing v×B, difference in phase between them makes a positive
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x-component current in front of v×B, which causes an initial rise in seafloor by prior to
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the bz variations.
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Fig. S4 Magnetic and electric fields induced by a rightward propagating solitary
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tsunami which has a Gaussian wave-form. In the lower three panels, the sea surface and
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seafloor are depicted by horizontal black lines at depths of 0 km and -5 km, respectively.
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The red and blue colors denote positive and negative values of each component,
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respectively. Profiles of each EM component on the seafloor are also drawn by black
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solid lines. In the lowest panel, the sum of the inducing v×B field and the induced
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electric field, ex, is shown. The vertical blue dashed line corresponds to the center of the
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symmetric wave-form, where the center of the inducing v×B field resides as drawn in
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the lowest panel.
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5. Estimation of the error due to the Coriolis force
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Here, we estimate the error due to the Coriolis force in the tsunami dynamics. The
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governing momentum equation in terms of the particle motion including the Coriolis
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force is expressed as,
𝜕𝑡 𝐯 + (𝐯 ∙ 𝛁𝐻 )𝐯 = −𝑔𝛁𝐻 𝜂 − 2𝛀 × 𝐯,
(S13)
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where 𝐯, 𝜂, and 𝛀 are the depth averaged horizontal vector velocity, the sea surface
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elevation, and the angular velocity of the Earth’s rotation, respectively. 𝜕𝑡 and 𝛁𝐻 are the
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derivative in terms of time and the horizontal coordinate, respectively. Since we
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neglected the 2nd term on the r.h.s. of Eq. (S13) in the main text, here we compare the
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magnitudes of those terms quantitatively. Amplitudes of the first and second term on
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the r.h.s of Eq. (S13) are estimated as follows:
|𝑔𝛁𝐻 𝜂| = 𝑔𝐴⁄𝜆 = 5 × 10−5,
(S14)
|2𝛀 × 𝐯| = 2𝛺sin𝜃 ∙ 𝑣 = 4 × 10−6 .
(S15)
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Here g, A, 𝜆, 𝜃, and v are the gravity acceleration, the wave height, the wavelength, the
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latitude, the particle velocity, respectively. We adopted 𝑔 = 9.8 m/s2, 𝐴 = 1m, 𝜆 =
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200 km, θ = 45 °, and 𝑣 = 4 cm/s, assuming the 2011 off the Tohoku earthquake
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tsunami passing through the northwest Pacific. Equation (S14) and (S15) show that the
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effects of the Coriolis force is less than 10 % of that of the pressure gradient. This order
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estimation is consistent with other preceding numerical studies (e.g., Dao and Tkalich,
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2007). Furthermore, it is reported that the Coriolis force significantly influences the
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height of the tsunami rather than the arrival time (Shuto, 1991). We, therefore, can say
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that there are at most 10 % errors in the tsunami height due to the Colioris force in our
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simulations.
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References
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Dao, M. H. and P. Tkalich (2007), Tsunami propagation modeling – a sensitivity study,
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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.