Geology 5640/6640 Introduction to Seismology 23 Mar 2015 © A.R. Lowry 2015 Last time: Normal Modes Free oscillations are stationary waves that result from interference of propagating waves For a string (length L, velocity v ) fixed at the endpoints, all propagating waves have eigenfrequencies  n = n  v/L: The Amplitudes A n in this equation relate to the source that excited the string: Propagating waves in the string can be represented by these normal modes. In the Earth, the equation is a leetle more complicated… Read for Wed 25 Mar: S&W (§3.1–3.3)

Geology 5640/6640 Seismology Last time: Normal Modes (Continued) On a sphere, free oscillations are described in terms of spherical harmonics as: Here n is radial order ( 0 for fundamental; > 0 for overtones); l (colatitude) and m (longitude) are surface orders; A lm n describe source displacement;  lm n are eigenfrequencies; & y l n (r) (at depth) and x lm (surface) are eigenfunctions. Spherical harmonics are basis functions on a sphere: orthonormal and can completely describe any function. 23 Mar 2015

Why study normal modes? A lm n are the excitation amplitudes, analogous to A n in the 1D (string) example… So from measurements of u one can get information about the source (provided the eigenfrequencies  lm n are known!) Conversely, given a source function A lm n and known  lm n, one can predict u … The modes form the basis vectors to describe displacements if one wants to model synthetic seismograms. The frequencies  lm n depend on density, shear modulus, and compressibility modulus of the Earth… so are used to get Earth structure.

Recall PREM is derived from normal modes!

Toroidal and spheroidal Using spherical harmonics (base on a spherical surface), we can separate the displacements into Toroidal (torsional) and spheroidal modes (as done with SH and P/SV waves): T : S : Radial eigenfunction Surface eigenfunction

Characteristics of the modes No radial component: tangential only, normal to the radius: motion confined to the surface of n concentric spheres inside the Earth. Changes in the shape, not of volume  Not observable using a gravimeter (but…) Do not exist in a fluid: so only in the mantle (and the inner core?) Horizontal components (tangential) et vertical (radial) No simple relationship between n and nodal spheres 0 S 2 is the longest (“fundamental”) Affect the whole Earth (even into the fluid outer core !) Toroidal modes n T m l : Spheroidal modes n S m l :

n, l, m … S : n : no direct relationship with nodes with depth l : # nodal planes in latitude m : # nodal planes in longitude ! Max nodal planes = l 0S020S02 T : n : nodal planes with depth l : # nodal planes in latitude m : # nodal planes in longitude ! Max nodal planes = l - 1 0T030T03

0 S 0 : « balloon » or « breathing » : radial only (20.5 minutes) 0 S 2 : « football » mode (Fundamental, 53.9 minutes) 0 S 3 : (25.7 minutes) Spheroidal normal modes: examples: Animation 0 S 2 from Hein Haak Animation 0 S 0/3 from Lucien Saviot 0 S 29 from: 0 S 29 : (4.5 minutes)... Rem: 0 S 1 = translation...

Toroidal normal modes: examples: 1 T 2 (12.6 minutes) 0 T 2 : «twisting» mode (44.2 minutes, observed in 1989 with an extensometer) 0 T 3 (28.4 minutes) Animation from Hein Haak Animation from Lucien Saviot Rem: 0 T 1 = rotation 0 T 0 = not existing

Solid inner core (1936) Fluid outer core (1906) Solid mantle Shadow zone Geophysics and normal modes Solidity demonstrated by normal modes (1971) Differential rotation of the inner core ? Anisotropy (e.g. crystal of iron aligned with rotation)?

Eigenfunctions Ruedi Widmer’s home page: shear energy density compressional energy density One of the modes used in 1971 to infer the solidity of the inner core: Part of the shear and compressional energy in the inner core Today, also confirmed by more modes and by measuring the elusive PKJKP phases

Eigenfunctions : 0 S l shear energy density compressional energy density l > 20: outer mantle l < 20: whole mantle Ruedi Widmer’s home page: Equivalent to surface Rayleigh waves

Eigenfunctions : S vs. T n = 10 nodal lines shear energy density compressional energy density T in the mantle only ! S can affect the whole Earth (esp. overtones) Ruedi Widmer’s home page: Deep earthquakes excite modes whose eigen functions are large at that depth

Eigenfunctions : 0 S l and 0 T l 0 S equivalent to interfering surface Rayleigh waves 0 T equivalent to interfering surface Love waves pics/SAWRAiGH.gif

The great Sumatra-Andaman Earthquake

300 km The great Sumatra-Andaman Earthquake 1300 km

Sumatra Earthquake: spectrum 0S30S3 0S20S2 2S12S1 0T40T4 0T30T3 0T20T2 0S40S4 1S21S2 0S00S0 Membach, SG C021, h h00

Sumatra Earthquake: time domain Membach, SG C021, Q factor 5327 Q factor M. Van Camp

Splitting No more degeneracy if no more spherical symmetry :  Coriolis  Ellipticity  3D Different frequencies and eigenfunctions for each l, m If SNREI (Solid Non-Rotating Earth Isotropic) Earth : Degeneracy: for n and l, same frequency for –l < m < l For each m = one singlet. The 2m+1 group of singlets = multiplet

Splitting  Rotation (Coriolis)  Ellipticity  3D Waves in the direction of rotation travel faster Waves from pole to pole run a shorter path (67 km) than along the equator Waves slowed down (or accelerated) by heterogeneities

Splitting: Sumatra M. Van Camp Membach SG-C021 0 S 2 Multiplets m=-2, -1, 0, 1, 2 “Zeeman effect”

Modes and Magnitude Time after beginning of the rupture: 00: (M W ) P-waves 7 stations 00: (M W )P-waves 25 stations 01: (M W )Surface waves 157 stations 04: (M W ) Surface waves (automatic) 19: (M W ) Surface waves (revised) Jan (M W )Free oscillations April (M W ) GPS displacements s surface waves