Very basic tsunami physics...

Transcription

1 Very basic tsunami physics... Energy Bottom uplift & Waterberg formation E R M E T = 1 2 ρglλ(δh)2 L ~ 1 6 m; λ ~ 1 4 m; δh ~ 5m E R 1 18 J 1 2 E T Center of mass falls... Wavelength λ H ~ 4; H a ~ 3 13 λ >> H >> a a Potential energy goes to tsunami energy Tsunami is a shallow-water gravity wave with great wavelength and tiny amplitude

2 Navier-Stokes equations Newton s law + Conservation of matter + Viscosity ρ v t + ρ(v grad)v = grad(p) ρgrad(φ) + ( ) +ηδv + (η + η )grad div(v)

3 Gravity waves: dispersion F(z) = 2Ae -kh cosh[ k(z + h) ] and the boundary at the top gives the dispersion relation for incompressible, irrotational, small amplitude gravity waves: ω 2 = kg[ tanh(kh) ] deep water (kh goes to infinity) ω 2 = kg shallow water (kh goes to zero) ω 2 = k 2 gh c = g k = gλ u = ω k = 1 2 c = 1 2 2π g k = 1 2 gλ 2π c = gh u = ω k = c = gh

4 Tsunami eigenvalues & eigenfunctions

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6 Modal approach - sketch Equations of elastic motion with gravity + boundary conditions FULL coupling between the fluid and solid layers Eigenvalues & Eigenfunctions x = x 1 x 2 x 3 x N-1 x N =X! x Seismic source excitation Tsunami mode propagation in LHM z Propagation factor Excitation factor Receiver factor exp( -iπ/4) ( ) = U X, ϕ, z, ω, t 8π exp[ iω( t - τ) ] J χ( h s, ϕ) R ( ω ) ωc v g I 1 s u( z, ω) v g I 1 X

7 Modal approach: formulation Direction of propagation x z -l z -l+1 Free surface l-th liquid layer EQUATIONS OF MOTION α 2 ( u) - ge z u = 2 u t 2 z -j z -j+1 j-th liquid layer ocean α 2 ( u) - β 2 ( u) = 2 u t 2 z -1 z z 1 1-st liquid layer 1-st solid layer BOUNDARY CONDITIONS α 2 u - gw = z m z m+1 m-th solid layer solid ( ) = w -j-1 ( z -j ) u -j ( z -j ) = u -j-1 ( z - j ) p -j ( z -j + w - j ) = p -j-1 ( z -j + w -j-1 ) w -j z -j z z N-1 z N (N-1)-th solid layer halfspace w -1 ( z ) = w 1 ( z ) p -1 ( z ) = σ 1 ( z ) = τ 1 ( z ) w m ( z m ) = w m+1 ( z m ) u m ( z m ) = u m+1 ( z m ) Reference 1-D model σ m ( z m ) = σ m+1 ( z m ) τ m ( z m ) = τ m+1 ( z m )

8 Modal approach: Eigenvalues velocity (km/s).2.1 p_4 p_6 p_8 g_4 g_6 g_ depth (km) frequency (Hz) Eigenfunctions of the radial and vertical (normalized to 1 at the freesurface) component of motion at frequency equal to.7 Hz, in the fluid. The curves for three crustal models 1, 2 and 3, are totally overlapped; on the bottom, the eigenfunctions in the solid layers are shown depth (km) _x 1_z 2_x 2_z 3_x 3_z

9 Modal approach: excitation spectra.4 amplitude spectra x_ss z_ss x_ds z_ds Amplitude spectra calculated for a double couple source (1 13 Nm seismic moment) at 5 km from the receiver: a) for pure strike-slip and pure dip-slip; b) for a liquid layer 4, 6 and 8 km thick; frequency (Hz) c) for different crustal model and source depths..3.4 x_4 1_9 amplitude spectra.2.1 x_6 x_8 z_4 z_6 z_8 amplitude spectra.3.2 1_14 3_9 3_ frequency (Hz) frequency (Hz)

10 Modal approach: 2D tsunami motion U( X,!,z,", t) = exp -i#/4 ( ) 8# exp[ i" ( t - X/c) ] X $ ( h s,!)r(") u z," "c v g I 1 v g I 1 ( ) U( X,!,z,",t) = exp -i#/4 ( ) 8# exp[ i" ( t - $ )] J %( h s,!)r(") "c v g I 1 s u z," ( ) v g I 1 X SHOALING FACTOR " W( X 2,,! ) w(,!) W( X 1,,! ) = 2 v g I 1 % $ 1 ' $ w(,!) 1 v g I # 1 ' 2 & J 1 J 2! H 4 1 H 2

11 Example: Synthetic signals for the tsunami mode (vertical component) excited by a dip-slip mechanism with M = Nm. h s = 14 km; h s = 34 km. 2.5 z (m) z (m) -2 2 X=5 km X=5 km z (m) z (m) z (m) z (m) -2 2 X=2 km X=2 km z (m) z (m) time (h) time (h) For each of the two source-receiver distances considered, the upper trace refers to the 1-D model and the lower trace to a laterally varying model. In the laterally varying model the liquid layer is getting thinner with increasing distance from the source, with a gradient of.175 and the uppermost solid layer is compensating this thinning.

12 Example:Sketch of a laterally heterogeneous model for a realistic scenario. Synthetic mareograms (vertical) calculated at various distances along the section. The extension of zone C is 5 km x X=5 km X=15 km -2 2 z A B C D X=65 km -2 1 ds_16_7.7 2 ds_16_8.1 ds_2_8.1 X=69 km 1 ss_16_ X=75 km distance (km) -2 3 time (h)

13 Measurement of tsunami waves Tide gauges can measure TW along the coast... Tsunami records and their f-t diagram: solid line (E) is the time of main shock, dashed line (TA) is Tsunami arrival The 26 December 24 Sumatra Tsunami: Analysis of Tide Gauge Data from the World Ocean Part 1. Indian Ocean and South Africa Alexander B. Rabinovich and Richard E. Thomson

14 Measurement of tsunami waves Tide gauges can measure TW along the coast, but their detection in open ocean is challenging, due to their wavelengths and amplitudes. ocean bottom sensors (pressure gauges & seismometers) Seismic Records of the 24 Sumatra and Other Tsunamis: A Quantitative Study Emile A. Okal

15 Measurement of tsunami waves Tide gauges can measure TW along the coast, but their detection in open ocean is challenging, due to their wavelengths and amplitudes. ocean bottom sensors hydrophones (towards high frequency bands...) a) Raw time series b) spectrogram c) close-up of the tsunami branch and comparison with w 2 =gktanh(kh) Quantification of Hydrophone Records of the 24 Sumatra Tsunami Emile A. Okal, Jacques Talandier and Dominique Reymond

16 Measurement of tsunami waves Tide gauges can measure TW along the coast, but their detection in open ocean is challenging, due to their wavelengths and amplitudes. ocean bottom sensors (pressure gauges or seismometers) sea level measurement (GPS receivers on buoys) satellite altimetry NOAA

17 Tsunami signature in the ionosphere By dynamic coupling with the atmosphere, acousticgravity waves are generated Traveling Ionospheric Disturbances (TID) can be detected and monitored by high-density GPS networks

18 Tsunami signature in the ionosphere Hines (196): atmospheric Internal Gravity Waves Peltier & Hines (1972): can generate ionospheric signatures in the plasma Lognonné et al. (1998): Analytical Coupled model Artru et al. (25): ionospheric imaging can detect tusnami signatures. GPS JAPAN net was used to map Chilean Tsunami of 21 Occhipinti et al. (26): Sumatra tsunami mapped Three-dimensional waveform modeling of ionospheric signature induced by the 24 Sumatra tsunami Giovanni Occhipinti, Philippe Lognonné, E. Alam Kherani and Helene Hebert GRL, 26, 33

19 Tsunami signature in the ionosphere Tsunami-generated IGWs and the response of the ionosphere to neutral motion at 2:4 UT. Normalized vertical velocity Perturbation in the ionospheric plasma

20 Tsunami signature in the ionosphere The TEC (Total Electron Content) perturbation induced by tsunami-coupled IGW is superimposed on a broad local-time (sunrise) TEC structure.