Modeling of ionospheric perturbation by 2004 Sumatra tsunami

Authors


Abstract

[1] A complete model is proposed to analyze the electron density perturbation caused by tsunami-induced gravity waves. Loss mechanisms of thermal conduction, viscosity, and ion drag are considered in deriving the dispersion relation of the atmospheric gravity waves (AGWs). This model is then used to analyze the electron density perturbation in the ionosphere caused by the Sumatra tsunami on 26 December 2004. It is found that the AGWs move horizontally at the same speed with that of the tsunami and are trapped at about 400 km high. The simulation results well explain that about 13 min after the tsunami triggers AGWs, electron density perturbation in the ionosphere can be detected by satellites that pass over.

1. Introduction

[2] In 1952, Daniels [1952] claimed that the acoustic energy generated by ocean waves was large enough to fluctuate the ion density in the ionosphere. In the 1960s, Hines [1960] proposed a theory of atmospheric gravity waves (AGWs) in the ionosphere, where the atmosphere is assumed isothermal, irrotational, lossless, windless, and horizontally stratified. By using an incoherent scattering technique to study large-scale traveling ionospheric disturbances (TIDs) [Thome, 1964], it is found that the TIDs start to propagate horizontally with reduced amplitude above certain height, implying there are some loss mechanisms in the upper atmosphere.

[3] Pitteway and Hines [1963] proposed a propagation model of gravity waves in the atmosphere, including viscosity and thermal conduction. Friedman [1966] proposed a ducting model of AGWs, including leakage of energy in a thermally stratified atmosphere. Other models of AGW including viscosity, heat flow, ion drags, Coriolis forces have also been proposed [Midgley and Liemohn, 1966; Hines, 1968; Liu and Yeh, 1969; Volland, 1969]. Dispersion relation of anelastic gravity waves under the influence of molecular viscosity and thermal conduction have been derived [Vadas and Fritts, 2005, 2006; Vadas, 2007].

[4] In 1976, Peltier and Hines [1976] pointed out the possibility of detecting tsunamis by monitoring their ionospheric signature. High-frequency (HF) radio sounder has been used to study acoustic waves in the ionosphere induced by earthquakes (Alaska, USA, 1964 [Row, 1967]; Hachinohe, Japan, 1968 [Yuen et al., 1969]), volcano eruptions, nuclear detonation (Novaya Zemlya, 1961), and mine blasts. Najita and Weaver [1974] studied ionospheric signatures left by a Rayleigh wave on the solid earth surface, and concluded that an earthquake might generate a tsunami. Rayleigh waves on ocean surface have also been studied using HF Doppler sounding [Najita and Yuen, 1979].

[5] Speed of AGWs can be measured using MST (mesosphere, stratosphere, thermosphere) radars [Kuo et al., 1993]. In 2004, Artru et al. [2004] proposed a method to measure the vertical oscillation of ionospheric layers caused by seismic surface waves using ground-based HF Doppler sounder. The seismic acoustic wave is assumed to propagate in an isothermal, hydrostatic atmosphere under an adiabatic process, considering viscosity but neglecting heat conduction. The measured speed and amplification factor agree with the theory reasonably well up to an altitude of 150 km, but the speed above certain height is much slower than predicted.

[6] Although AGWs can be detected by using ionospheric sounding techniques, the source of AGWs is difficult to locate due to multiple hops of HF signals. By using the technique of transionospheric sounding, very high frequency (VHF) radio signals are sent from satellites to measure the Faraday rotation to detect the AGWs. However, the time of event must be known a priori in order to determine the speed of AGWs.

[7] Global positioning system (GPS) has been utilized to detect AGW-induced TIDs. In 1998, Calais and Minster [1996] analyzed the ionospheric disturbances during the 1994 Northridge earthquake. Afraimovich et al. [2001] proposed a method to detect and locate the source of ionospheric disturbance based on a global network of GPS receivers. In 2003, Ducic et al. [2003] reconstructed a Rayleigh wave profile induced by the 2002 Denali earthquake in California using GPS networks.

[8] On 23 June 2001, a tsunami-induced ionospheric perturbation above Peru was recorded for the first time by the GPS dense Japanese network (GEONET) [Artru et al., 2005a]. The ionospheric perturbation induced by the 2004 Sumatra tsunami (Mw = 9.3 at 0:58:50 UT, 3.3°N, 95.8°E) has been widely discussed [Artru et al., 2005b; Liu et al., 2006a, 2006b; Ablain et al., 2006; Occhipinti et al., 2006, 2008]. Artru et al. [2005b] used a simple isothermal and lossless model to estimate the travel time of AGW, then verified it with the perturbation of total electron content (TEC) measured with GPS. In the work of Liu et al. [2006a, 2006b], TEC was recorded with ground-based GPS receivers. On the other hand, they also simulated the earthquake-induced tsunami and its aftershocks, and their results are consistent with those of Najita and Weaver [1974] and Artru et al. [2005a].

[9] In the work of Ablain et al. [2006], a data mapping technique is used to extract tsunami-induced signals. Occhipinti et al. [2006, 2008] proposed a model of tsunami-induced AGWs propagating from the ocean surface to the top of ionosphere, in which the coupling mechanisms between tsunami, atmosphere, and ionosphere are included. Their results agree reasonably well with the satellite altimetry data of Jason 1 and Topex/Poseidon. Computer tomography technique has also been applied to display three-dimensional images of ionospheric perturbation [Garcia et al., 2005; Lognonné et al., 2006; Lee et al., 2008].

[10] In the work of Shinagawa et al. [2007], another mechanism of ionospheric perturbation in the 2004 Sumatra earthquake was proposed. A single period of sinusoidal displacement is superposed upon the sea surface above the epicenter and the aftershock region, with the period of 10 s and vertical velocity amplitude of 1 m/s. However, such wave component was not observed in the open sea during the 2004 Sumatra tsunami, and the acoustic energy launched by this mechanism is too small compared with that of AGWs induced by a traveling tsunami.

[11] Model of nondispersive shallow-water wave can be applied to tsunamis traveling in an open sea, while nonlinear Boussinesq and full Navier-Stokes equations must be resorted to if more complex seabeds are put into consideration [Mader, 2004; Eckart, 1960]. Models on the 2004 Sumatra tsuanmi have been proposed based on the method of splitting tsunamis (MOST) and finite difference bathymetry [Titov and Synolakis, 1998; Titov et al., 2005, Hebert et al., 2007], in which the epicenter and seabed profile are considered. Bathymetry on a large geographical scale can be used to measure the tsunami profile for the first few cycles. Satellite-borne altimetry can also be used to measure the sea surface profile with fair resolution, but a full picture is not yet available due to incomplete coverage of satellites.

[12] Between 2 and 9 h after the 2004 Sumatra earthquake, the satellite-borne altimetry in Jason 1, Topex/Poseidon, Envisat, and Geosat-Follow-On (GFO) recorded profiles of sea surface along their nadir tracks across the Indian ocean [Smith et al., 2005; Gower, 2005, 2007]. The data of total electron content (TEC) were also recorded by other systems (http://nasascience.nasa.gov/missions/jason-1?searchterm=Jason+1; ftp://podaac.jpl.nasa.gov/pub/sea_surface_height/jason/igdr/data/; http://modelweb.gsfc.nasa.gov/models/msis.html; http://omniweb.gsfc.nasa.gov/vitmo/msis_vitmo.html; http://podaac.jpl.nasa.gov/; http://nasascience.nasa.gov/missions/topex-poseidon?searchterm=topex).

[13] In this work, a complete model is proposed to analyze tsunami-induced perturbation in both the atmosphere and the ionosphere. Several well-established models are also used to construct a more realistic propagation environment. The tsunami profile of the 2004 Sumatra event will be reconstructed based on the altimetry data of Jason 1.

2. Tsunami-Induced Atmospheric Gravity Waves

[14] Tsunami waves in the open sea behave like shallow-water waves with little dispersion. Their speed is about 210 m/s in the Pacific Ocean (average depth of 4,500 m) or 195 m/s in the Indian Ocean (average depth of 3,900 m). Their peak-to-trough amplitude and period are related to the strength and duration of the earthquake. Typically, the amplitude is about 1 m, and the period is on the order of 10–30 min.

[15] A tsunami can be decomposed into many time harmonics with different wave numbers. Hence, we will first analyze the AGWs induced by one of these harmonics. Assume the wind is frictionless, hence the wind normal to the tsunami-atmosphere interface vanishes, namely,

equation image

where equation image is the wind velocity, and equation image is the unit normal vector to the interface. Set the mean sea level at z = 0, and the tsunami profile is h(x), then (1) is reduced to

equation image

where C is the traveling speed of tsunami, equation imagez (x, z) is the z component of velocity perturbation of the neutral particles.

[16] Assume the atmosphere is irrotational, compressible, and spherically stratified. Within the region of interest, the spherical layers can be approximated as planar layers to simplify the mathematics. If thermal conduction, viscosity, and ion drag are considered, the equations of momentum conservation, mass conservation, heat flow, and the state become [Nappo, 2002; Yeh and Liu, 1972]

equation image
equation image
equation image
equation image

where ρ is the mass density of neutral particles, p is the pressure, g is the gravitational constant, c is the speed of sound, R = 8.314472 JK−1 mol−1 is the gas constant, m is the mean mole mass of neutral particles, σ is the thermal conductivity, T is the absolute temperature, γ = 1.4 is the ratio of specific heat, cv is the specific heat at constant volume, νin is the ion-neutral collision frequency, ρp is the plasma mass density, equation image is the neutral particle velocity, equation imagei is the ion velocity, and equation image is the viscous stress tensor.

[17] If a harmonic component of tsunami propagates in the direction of equation imagekx + equation imageky, all the induced quantities will take the form of

equation image

where the z dependence is included in f(z).

[18] The atmosphere is approximated as isothermal layers piled up vertically, within each layer the temperature, mass density and chemical properties are constant and only slightly different from those in adjacent layers. Hence, WKB technique can be applied by extracting a factor eequation image out of f (z). Thus, a matrix equation can be derived from (3)(6), and a dispersion relation between (kx, ky, Kz) and ω can be obtained.

[19] The wave number Kz can be further decomposed into Kz = kz + j/2H with kz = kzr + j kzi, where H is the scale height, and kzi = 0 in a lossless atmosphere. If each layer is much thinner than H, f (z) in (7) reduces to

equation image

where the last exponential term is called the amplification factor.

3. Chemical Reactions in F Region

[20] In the F region, the change rate of electron density, ∂Ne/∂t, is related to the photoionization rate q, the chemical loss rate of electrons, L (Ne), and the mean velocity of electrons, equation imagee, as [Hooke, 1968]

equation image

In the presence of AGWs, each of these quantities are the superposition of an equilibrium value (with subscript 0) and a perturbation (with prime), namely, Ne = Ne0 + Ne, q = q0 + q′, and L = L0 + L′. Equation (9) can thus be linearized as

equation image

Similar equations hold for the ions.

[21] The most important reactions from the altitude of 140 km to several H above the F2 peak are

equation image

where (X, Y) = (N, O) and (O, O), and n (XY) is the number density of XY. In the presence of AGWs, the number densities become n (O+) = n0 (O+) + n′ (O+) and n (XY+) = n0 (XY+) + n′ (XY+), and the rate β becomes β = β0 + β′.

[22] The perturbation of ion velocity consists of a drift part and a diffuse part, the former is determined by the electric field and the geomagnetic field, while the latter is determined by the ambient temperature. The diffuse part can be ignored, and the drift part caused by the AGW-induced electric field is also negligible [Hines, 1960, 1955]. In the F region, the ion-neutral collision frequency is much lower than the gyrofrequency of ions, hence the velocity perturbation of ions is about the same as that of neutral particles along the geomagnetic field lines.

[23] In the F region well above 200 km at height, Ne0n0 (O+) ≫ n0 (XY+) [Hooke, 1968], q′ is small, hence q0β0Ne0 and ∇ · [n0 (O+) equation image0 (O+)] ≃ β0Ne0 [Rishbeth and Barron, 1960]. The perturbation of electron density can be expressed as [Hooke, 1968]

equation image

where I is the geomagnetic dip angle, and

equation image

equation imageB is the direction of local geomagnetic field.

4. Effect of Gravity Waves on Photoionization Rates

[24] The photoionization rate is contributed by the constituent particles in the atmosphere as [Hooke, 1968]

equation image

where ni is the number density of the ith constituent particle, σi (λ) is the absorption cross section of the ith constituent particle at wavelength λ, and s′ (λ) is the radiation flux over the wavelength band , ℓ is the coordinate defined along the radiation path. The AGWs perturb the photoionization rate by perturbing the number density ni and the radiation flux. The latter effect is important, especially at altitudes lower than the height of maximum photoionization.

[25] The equilibrium number density n0 basically follows the distribution

equation image

where ng is the equilibrium number density of an ionizable particle at a reference height, n0 (ζ) is that of the same particle at a height ζ above the reference, and H is the associated scale height. An AGW perturbs the number density to become

equation image

where A is the amplitude, equation image′ = equation imagex′ + equation imagey′ + equation imagez′, and equation image = equation imagekx + equation imageky + equation imageKz.

[26] If σ is independent of λ, (11) can be reduced to

equation image

where S = equation images′ (λ) is the total radiation flux. Substituting (12) and (13) into (14), we have

equation image

where S is the radiation flux at a reference point equation image far above the atmosphere. The solution to (15) is

equation image

where χ and equation image are the zenith angle and azimuthal angle, respectively, of the sun, zm is the height of maximum photoionization rate with eequation image = ngσH sec χ. Note that ln S (equation image, t) is a time harmonic, but S (equation image, t) is not.

[27] By substituting (13) and (16) into (14), the photoionization rate can be derived as

equation image

The perturbation of radiation flux by the AGWs is much smaller than the equilibrium value, which implies that

equation image

[28] Comparing (13) and (17), we have

equation image

[29] Assume that α′ = 0 and β′ is mainly caused by the number density perturbation n′ [Hooke, 1968], thus

equation image

5. Simulation With Time Harmonic Model

[30] NO+, O2+, and O+ are the majority of ions in the ionosphere, and O2, N2, and O are the majority of neutral particles in the atmosphere. In this work, MSIS model is used to simulate the background distributions of O, N2, and O2 (MSIS), MSIS and IRI models are used to simulate the background distributions of electrons, O+, O2+, and NO+ in the ionosphere [Bilitza, 1990].

[31] The process of charge exchange between ions and neutral particles can be resonant or nonresonant [Dalgarno et al., 1958; Schunk and Nagy, 1980]. The plasma mass density is related to its constituents as

equation image

where NA = 6.02214 × 1023 mole−1 is the Avogadro number. The effective ion-neutral collision frequency ν (z) = νin (ρp0/ρ0) has a profile similar to that of Ne0 (z). Note that the ion-neutral collision frequency is about 1 Hz in the F region.

[32] The Sutherland's formula for an ideal gas is used to estimate the dynamic viscosity as [Eckart, 1960; Dalgarno and Smith, 1962]

equation image

where T0 = 291.15 K is the reference temperature at the ground level, C0 = 120 K is the Sutherland's constant, and η0 = 18.27 × 10−6 kg m−1s−1 is the reference viscosity at T0. The thermal conductivity is related to the dynamic viscosity as

equation image

where ζ ≃ 2.

[33] Viscosity in gases arises principally from the molecular diffusion that transports momentum between layers of flow. The atmospheric viscosity in (18) is Sutherland's law for dynamic viscosity of an ideal gas [White, 1974; Starr, 1966]. In the atmosphere under about 100 km of altitude, the eddy viscosity is important, but the dynamic viscosity dominates above that height. In this work, the eddy viscosity is neglected and only the dynamic viscosity is considered at all altitudes.

[34] Figure 1 shows the profile of Kz (z), the results of lossless model are also shown for comparison. The profiles calculated using both models are different above 150 km, where the loss mechanisms become more significant. Note that in the lossless model, kzi = 0 and Im {Kz} = 1/2H at all altitudes.

Figure 1.

Propagation constant at different heights. Solid curve, our model; dashed curve, lossless model. (a) kzr, (b) kzi, and (c) kzi + 1/2H.

[35] The temperature profile above 150 km barely changes with altitude, which implies a large thermal conductivity therein, hence the loss due to heat flow must be considered. As shown in Figure 1b, the contribution of loss mechanisms to kzi becomes comparable to 1/2H above 134 km, turning kzi + 1/2H to negative above 230 km, as shown in Figure 1c.

[36] Figure 2 shows the amplification factor of AGWs, where we choose kx = 3.655 × 10−5 m−1, close to that of the dominant harmonic in the 2004 Sumatra tsunami. The amplification factor is significant in 95 < z < 440 km, with the peak value of 3.258 × 104 at z = 222 km. This result is compatible to the observations made by Thome [1964]. The amplification factor calculated using the lossless model is close to that of our model below 200 km [Artru et al., 2004], but keeps increasing with altitude above 200 km [Hines, 1960], which is nonphysical and also causes numerical problem [Artru et al., 2004; Occhipinti et al., 2006].

Figure 2.

Amplification factor of AGW. Solid curve, our model; dashed curve, lossless model.

[37] The real part of kz in both models is the same below 150 km and approaches zero at about 550 km, above which the wave becomes evanescent. Note that negative kzr implies that AGWs are backward waves [Yeh and Liu, 1972]. The velocity profile over the wavelengths of 10 to 60 km has been studied, and the maximum amplitude appears between 70 to 140 km above sea level [Midgley and Liemohn, 1966].

[38] Figure 3 shows the profile of vertical velocity uz of neutral particles. The magnitude of uz is less than 100 m/s at all altitudes, with a maximum value of about 70 m/s, which agrees with the observations of Hooke [1968]. The uz calculated using the lossless model is larger than 100 m/s, and exceeds 1000 m/s above 400 km. The perturbation of electron number density is strongly related to the ion velocity, and the latter is affected by the neutral particle velocity equation image. Thus, the accuracy in the estimation of equation image determines the accuracy in the calculation of Ne. The lossless model becomes inaccurate above 200 km, and usually requires normalization on the perturbed quantities to avoid numerical difficulties [Liu et al., 2006a, 2006b; Occhipinti et al., 2006, 2008].

Figure 3.

Vertical velocity uz of neutral particles. (a) Lossless model and (b) our model. Solid curve, real part; dashed curve, imaginary part; dotted curve, amplitude.

[39] Figure 4 shows the vertical kinetic energy density, defined as ρ0uzu*z. The kinetic energy calculated using our model decays significantly above 150 km, but that using the lossless model is almost a constant from 150 to 600 km, and grows monotonically above 600 km due to the inaccurate amplification factor.

Figure 4.

Vertical kinetic energy density of AGWs. Solid curve, our model; dashed curve, lossless model.

[40] Figure 5 shows the magnitude of electron density perturbation, ∣Ne∣ The AGWs significantly perturb the electron density at altitudes from 100 to 450 km, with the peak perturbation of 1.962 × 1011e/m3 at 238 km, which is 37.23% of Ne0. At 303 km, the height of maximum Ne0, the electron density perturbation is 14.28% of Ne0. The magnitude of Ne predicted using the lossless model grows with altitude and exceeds Ne0 above 255 km.

Figure 5.

Magnitude of electron density perturbation ∣Ne∣. Solid curve, our model; dashed curve, lossless model; dotted curve, equilibrium electron density Ne0.

[41] Up to now, the tsunami is assumed to take a time harmonic waveform which extends to infinity along its propagation direction. The induced AGW also has the time harmonic waveform without space divergence, hence its amplitude is larger than that induced by a real tsunami of finite extent.

[42] Figures 6 and 7 show the electron density perturbation Ne and the total electron density Ne = Ne0 + Ne, respectively. The total electron content (TEC) is decomposed as TEC = TEC0 + TEC′ with

equation image

where the integration path ℓ is a straight line vertical to the sea surface, extending from the sea level to z = 1330 km, slightly above Jason 1 at 1322–1327 km (Jason 1). The perturbation of TEC is within ±7.5% that of TEC0. The TEC profile has the same wavelength as that of the tsunami.

Figure 6.

Distribution of electron density perturbation Ne (e/m3). Ne = 1.96 × 1011e/m3 at black ellipsis, and Ne = −1.96 × 1011e/m3 at gray ellipse; difference between two adjacent contours is 2.5 × 1010e/m3.

Figure 7.

(top) Percentage variation of total electronic content, ɛ = TEC′/TEC0. (bottom) Distribution of total electron density Ne = Ne0 + Ne (e/m3). Maximum of Ne = 7.54 × 1011e/m3 is marked with black ellipsis, and difference between two adjacent contours is 4.5 × 1010e/m3.

[43] Peak fluctuation of 20–40% in ionospheric electron density has been reported when applying the perturbation analysis [Hooke, 1968; Huang and Sofko, 1998]. The electron perturbation in the 2004 Sumatra event is about 10%, making the perturbation analysis reasonable in this work.

6. Reconstruction of Sumatra Event

[44] The distribution equation imagez (x, z) is related to its Fourier transform uz (kx, z) as

equation image

Substitute the harmonic waveform of tsunami

equation image

into (2), then take the Fourier transform to obtain

equation image

where δ(x) is the Dirac delta function which can be expressed as

equation image

[45] Applying the WKB technique, uz (kx, z) can be derived as

equation image

Taking the inverse Fourier transform of (19) gives

equation image

The electron density perturbation induced by the AGWs is derived by taking the inverse Fourier transform of (10) as

equation image

Its time domain expression is

equation image

Since the dispersion in the open sea tsunami is neglected, the wave number kx is approximated as kx = ω/C [Mader, 2004], hence (20) can be reduced to

equation image

Note that Kz (−kx) = −Re{Kz (kx)} + jIm {Kz (kx)}.

[46] If the spectral components of the tsunami fall within kxminkxkxmax, then the integral in (21) can be calculated with numerical method like Gauss quadrature technique [Press et al., 1992].

[47] In the work of Shinagawa et al. [2007], acoustic waves generated by the 2004 Sumatra earthquake are simulated using a nonhydrostatic model of Shinagawa and Oyama [2006], which was originally used to analyze the thermospheric dynamics near an auroral arc. The horizontal wind in the thermosphere blows at nearly supersonic speed during intense storms, making this model incompatible with the assumption of hydrostatic equilibrium in the 2004 Sumatra event. However, it is pointed out that a horizontal wind blow is equivalent to a Doppler shift of the AGW frequency [Yeh and Liu, 1972]. In the equatorial and midlatitude to low-latitude regions, the background wind is not as strong as that in the polar region, making the hydrostatic approximation more realistic.

[48] Numerical approaches like finite difference time domain (FDTD) have been applied to study ionospheric perturbation by acoustic waves triggered by the 2004 Sumatra earthquake, rather than by AGWs induced by the tsunami [Sun et al., 1995; Shinagawa and Oyama, 2006; Walterscheid et al., 1985; Walterscheid and Lyons, 1992]. Our analytical approach, based on reasonable approximations, delivers a complete picture of the whole event and reconstructs the key features of tsunami, its induced AGWs, and ionospheric perturbation with reasonable accuracy.

[49] Figure 8 shows the tsunami wavefronts at different hours after the earthquake. The 2004 Sumatra tsunami was triggered by an earthquake beginning at 0:58:50 UT, 26 December, with the epicenter at 3.3°N, 95.8°E. The tsunami traveled at the speed of 191 m/s (700 km/h), with peak-to-trough amplitude of 1 m. About 2 h after the earthquake, the first wavefront was detected by Jason 1 along track 129 at about (4.56° S, 84.12° E), marked by A. The propagation direction of tsunami at point A intercepts track 129 with an angle of about 30°.

Figure 8.

Tsunami wavefronts at different hours after the earthquake; p1 marks the epicenter (http://nctr.pmel.noaa.gov/indo_1204.html).

[50] Figure 9a shows the sea level profile detected by Jason 1. The sea level reaches a maximum of about 60 cm, and the peak-to-trough amplitude is about 1 m. The tsunami profile along equation image will be reconstructed from the data shown in Figure 9a.

Figure 9.

(a) Sea level profile detected along track 129 around 0300 UT, 26 December 2004 (http://nctr.pmel.noaa.gov/indo_1204.html). (b) Mapped sea level profile along equation image on the ϕ-equation image plane as shown in Figure 11; the portion between D′ and D″ cannot be reconstructed. (c) Solid curve, mapped profile; dashed curve, smoothed profile excluding tsunami wave components which excite attenuated AGWs.

[51] Figure 10a shows the wavefronts on a spherical surface, triggered by a point source which is arbitrarily located at (ϕ′ = 0°, θ′ = 0°). Assume the wave speed is independent of the propagation direction, hence the wavefronts appear as concentric circles. A point on the spherical surface with Cartesian coordinates (x′, y′, z′) can be transformed to the spherical coordinates as

equation image

where R is the spherical radius, θ′ = π/2 − θ1 with θ1 the angle between equation image and the z′ axis, equation image′ is the angle between the projection of equation image on the x′ − y′ plane and the x′ axis.

Figure 10.

(a) Wavefronts on a spherical surface, triggered by a point source. (b) Mapping onto ϕ′-θ′ plane.

[52] Figure 10b shows the mapping of wavefronts onto the ϕ′-θ′ plane with

equation image

Within the range equation image ≤ 30°, the wavefronts appear almost like concentric circles on the ϕ′-θ′ plane. The region of interest shown in Figure 9a falls between 5°S and 7°N, hence the wavefronts can be approximated as concentric circles. To simplify the analysis without changing major physical features, the curvature of earth surface is neglected in this region.

[53] Figure 11 shows the ϕ-θ plane mapped from Figure 8, with the epicenter p1 at the origin. The shaded area is mapped from the open sea area bounded by two dashed lines as shown in Figure 8. During the first 2 h after the earthquake, reflections of tsunami from coastlines can be ignored. Track 129 of Jason 1 appears almost as a straight line segment in the equation image-θ plane. Point A marks the intercept point at (4.56° S, 84.12° E), and the sea level profile shown in Figure 9a is recorded along line segment equation image. A point C (equation image, equation image) along equation image is mapped to a point C′ (−equation image, 0 ) along equation image. The tsunami profile h(x) along equation image is thus reconstructed as shown in Figure 9b, which contains 682 data points at space intervals of 1 km, with a resolution of 3 × 10−7 m−1 in the kx spectrum.

Figure 11.

Coordinate system with the epicenter at the origin; the data along equation image are mapped from those along equation image.

[54] Figure 12 shows the spectrum of the tsunami profile with ∣kx∣ less than the cutoff wave number of AGWs, namely, ∣kx∣ ≤ 9.3 × 10−5 m−1. The spectral intensity out of this range is at least 20 dB below the maximum intensity of 116.9 m2 at kx = ±1.2 × 10−5 m−1. Figure 9c shows the tsunami profile after filtering out the spectral components above the cutoff wave number.

Figure 12.

Spectrum of the tsunami profile shown in Figure 9c; cutoff wave number of AGWs is about 9.3 × 10−5 m−1. (a) Amplitude. (b) Solid curve, real part; dashed curve, imaginary part.

7. Simulation on Sumatra Tsunami

[55] Figure 13 shows the amplification factor of AGW spectral components at different altitudes. The AGW components with longer wavelength (smaller kx) can propagate to a higher altitude than those with shorter wavelength (larger kx). The AGWs with kx in the range of 2–4 × 10−5 m−1 have larger amplification factor, and will perturb the ionosphere more significantly. The vertical velocity fluctuation in the ionosphere induced by AGWs reaches a maximum magnitude of about 80 m/s, which agrees with the measurement using Doppler sounder [Liu et al., 2006a].

Figure 13.

Amplification factor of AGWs at different altitudes and different kxs. The square marks the peak value of 3.66 × 104, the triangle marks the value of 4.63 × 103, the arrow indicates the sense of decreasing value, and the difference between two adjacent contours is 4 × 103.

[56] The vertical group velocity vgz (kx, z) of AGW component at kx can be estimated as

equation image

where ω = C kx. The transit time for the AGWs to propagate from the sea level to height z is

equation image

Numerical method is used to calculate the derivative with step size Δkx = 3 × 10−7 m−1, and the integral with step size Δz = 1 km.

[57] Figures 14a and 14b show the travel time for AGW components to reach specific heights. On the basis of Figure 14a, it takes about 2–11 min for AGWs with kx = 1–7 × 10−5 to reach 100 km high, the average speed is 152 m/s. It takes about 30 min for AGWs with kx = 1.5–4.5 × 10−5 to reach 300 km high, the average speed is 167 m/s. Each AGW component slows down above certain altitude, and these altitudes are linked by a dashed line. AGW components with larger kx take longer time to reach a given altitude. The travel time calculated with both models is close below 100 km. Above 100 km, loss mechanisms become prominent, and the difference between these two models becomes obvious.

Figure 14.

Travel time of AGW components from sea level to different altitudes. (a) Our model and (b) lossless model.

[58] Figure 15 shows the equi-phase contours of the AGWs induced by the Sumatra tsunami. Different AGW components have different vertical velocity, hence the AGW waveform is distorted while it propagates upward. The real part of Kz reduces zero at certain height, causing the phase front to travel horizontally above 400 km.

Figure 15.

(top) Equi-phase contours of AGWs; the arrow points in the sense of phase progression. (bottom) Tsunami propagation to the right at speed C.

[59] Figure 16 shows the electron density perturbation equation imagee. The peak-to-peak deviation of TEC is about 3.3 TECU (1 TECU = 1016e/m2), and the changing rate is 0.11 TECU/30 s, which agrees with the GPS observation [Liu et al., 2006a, 2006b]. The maximum deviation is about 10% of the equilibrium TEC. As pointed out by Hooke [1968] and Otsuka et al. [2006], the TEC perturbation is mainly caused by the AGWs traveling north–south bound. The AGWs traveling east–west bound are perpendicular to the geomagnetic field, hence having less effect on the electron density perturbation.

Figure 16.

(top) Percentage deviation of total electron content. (Bottom) Electron density perturbation equation imagee. Local maxima are marked with black ellipsis, and local minima are marked with gray ellipsis. The global maximum is 2.0279 × 1011e/m3, and the global minimum is −1.7202 × 1011e/m3; difference between two adjacent contours is 3 × 1010e/m3.

[60] Figure 15 shows that the tsunami wavefront is 1600 km away from the epicenter, the ionospheric disturbance falls behind the tsunami wavefront by about 150 km horizontally. The time lag is about 13 min, which is about the AGW travel time from the sea level to the ionosphere.

[61] The location with peak TEC perturbation falls behind the tsunami wavefront by about 750 km horizontally. The TEC perturbation spans over 1000 km, hence is observable at one site for about 1.5 h.

[62] The tsunami wavefronts are approximated as concentric circles in our model, hence the simulated TEC profile is rotationally symmetric with respect to the epicenter. Figure 17 shows the simulated TEC profile along track 129 of Jason 1, the TEC profile recorded along the same track is also shown for comparison. These two profiles have the same order of magnitude and look similar. Their difference can be accounted for by the geomagnetic field, local ionosphere conditions and neutral wind profiles [Yeh and Liu, 1972].

Figure 17.

Comparison between simulated TEC profile and recorded TEC profile. Solid curve, our model; dashed curve, recorded along track 129 of Jason 1. Top label is universal time, and bottom label is latitude.

8. Conclusions

[63] A complete model is presented to simulate the ionospheric perturbation caused by tsunami-induced AGWs. Loss mechanisms like thermal conduction, viscosity, and ion drag are considered so that the proposed model can be extended to the ionosphere. This model is also used to analyze the ionospheric perturbation caused by the 2004 Sumatra tsunami. An open sea tsunami profile in a radial direction from the epicenter has been reconstructed based on altimetry data from satellite. The simulation results well explain the TEC perturbation reported in the literature.

Acknowledgments

[64] This work was sponsored by the National Science Council, Taiwan, ROC, under contract NSC 96-2221-E-002-163 and the National Taiwan University, Taiwan, ROC, under contract 97R0329.