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3.3 Response to an impulsive source

Of central importance to the theory of local helioseismology is the concept of Green’s functions. The Green’s functions are the impulse responses of the solar model and solve
ℒGi (r,t;r′,t′) = ˆei(r ′) δD(r − r′)δD(t − t′), (11 )
where δD is the Dirac delta function in one or two dimensions. For each i = 1,2,3, the ′ ˆei(r ) is the unit vector in the i− th direction at ′ r, and ℒ is the wave operator for the displacement (see Equation 7View Equation). For example, in the non-magnetic case ℒ as given by Equation (5View Equation). Because it describes a displacement vector, Gi is a vector. Taken together, the three Green’s vectors Gi form a Green’s tensor, {Gi} j. The function Gi(r, t;r ′,t′) j is the displacement in the j− th direction at (r, t) that results from a unit source acting in the i− th direction at ′ ′ (r ,t)

There are three approaches to solving Equation (11View Equation). In the case of very simple problems it is sometimes possible to solve analytically for the Green’s functions in the Fourier domain (see, e.g., Gizon and Birch, 2002Jump To The Next Citation Point). Another, more general approach, is direct numerical solution (Section 3.3.1). An efficient approximate solution is normal mode summation (Section 3.3.2).

3.3.1 Direct solution in plane-parallel models

For plane-parallel steady models with horizontal translation invariance, the Green’s functions will be of the form i G (x − x ′,t − t′,z,z ′) where now r = (x, z) and r′ = (x′,z′) are the decompositions of r and r ′ into horizontal, x and x′, and vertical, z and z′, components. In this case we can write the Green’s functions as the inverse Fourier transforms of Fourier domain Green’s functions

i ′ ∫ 2 i ′ i(k⋅x−ωt) G (x,t,z,z ) = d k dω G (k,ω, z,z )e , (12 )
where k is the horizontal wavevector and ω is the temporal angular frequency. We can then obtain the equation satisfied by i G (k,ω,z, z′) from Equation (11View Equation), for any particular choice of ℒ. The result will be an inhomogeneous set of ordinary differential equations (ODEs) in the variable z for the components of Gi. These ODEs can then be integrated numerically to obtain Gi (k,ω, z,z′) for given k, ω, and ′ z. Care must be taken with the treatment of the delta function on the right hand side of Equation (11View Equation); this can treated by folding the computational domain so that the jump condition across the delta function becomes a non-local boundary condition (Birch et al., 2004Jump To The Next Citation Point).

3.3.2 Normal-mode summation approximation

Dahlen and Tromp (1998Jump To The Next Citation Point) give an excellent discussion of the normal-mode summation approach to the computation of Green’s functions. For a detailed discussion in the context of helioseismology, see Birch et al. (2004Jump To The Next Citation Point). The basic notion of the normal-mode summation approximation is that the Green’s function can be represented as a sum over normal modes. Intuitively, the idea is to compute the amplitude to which a particular source excites each mode in the model, and then to let each mode evolve in time. For example, for the case of undamped modes, Dahlen and Tromp (1998Jump To The Next Citation Point) write

∑ Gij(r, t;r ′,t′) = sβj(r)sβi (r′)sin [ω β(t − t′)], t > t′. (13 ) β
Here the index i refers to the direction of the source, and the index j to the component of the response that we are looking at. For t < t′ the Green’s function is zero. The sum over β is taken over all normal modes of the background model. By normal mode we mean a solution of Equation (4View Equation) of the form s(r)e− iωt, with the normalization ∫ ⊙dr ρ0 ∥s∥2 = 1. In the case of spherical background models the modes are described by the radial order n the angular degree l and the azimuthal order m. In this case we have β = (n,l,m ). For the case of plane-parallel models, modes can be labeled by a radial order n and a horizontal wavevector k. In this case we have β = (n,k). The situation is somewhat more complicated for the case of non-adiabatic modes, although it can still be addressed in much the same way (see Chapter six of Dahlen and Tromp, 1998).
View Image

Figure 3: Real (red) and imaginary (blue) parts of the horizontal component of normal-mode and numerical Green’s functions (Gh) for −1 k = 1 Mm, ν = 3.92 mHz (just above the n = 2 resonance), zsrc = − 3.7 Mm, and a vertical momentum source. The Green’s function has been scaled by the square root of the background density. The horizontal axis is acoustic depth in minutes. The vertical scale is arbitrary. The source depth is shown by the solid blue vertical line. The photosphere is shown by the vertical blue dashed line. The solid curves show the numerical results. The real part of the numerical result has a discontinuity at the source depth while the imaginary part is continuous there. The dashed curves show the normal-mode summation approximation. Notice that the normal-mode approximation is continuous at the source depth. This is because we have used only a finite number of modes (radial orders not greater than 15). From Birch et al. (2004Jump To The Next Citation Point).

Figure 3View Image compares Green’s functions computed numerically and approximated via normal-mode summation. Notice that the exact numerical result shows a discontinuity at the source depth; this comes from the jump across the delta function on the right hand side of Equation (11View Equation). The normal mode approximation, being a finite sum of continuous functions, is everywhere continuous.

In the case of plane-parallel translation-invariant isotropic models where the only restoring forces are pressure and gravity, it can be shown that the Green’s functions, in the Fourier domain, can be decomposed as

i ′ i ′ ˆ i ′ G (k,ω,z, z) = ˆz G z(k, ω,z,z ) + ikG h(k,ω, z,z ). (14 )
This decomposition is useful as now for any source we need only to compute two components of the response rather than three. Also notice that Gi z and Gi h depend only on the wavenumber k, defined by k = kˆk. A similar decomposition can be done for the dependence on the source direction (Birch et al., 2004Jump To The Next Citation Point).

3.3.3 Green’s functions for the observable

For the remainder of this review, it will be convenient to have a Green’s function for the response of the observable to a wave source. We denote the observable wavefield by the scalar Φ. For most current helioseismology work, the observable is the line-of-sight Doppler velocity. As a result, we define

Φ (x,t) = ℱ {ˆℓ ⋅ v(x, z ,t)} , (15 ) o
where v(x,zo,t) is the Eulerian velocity at horizontal location x, at depth zo, and time t. The line-of-sight unit vector ˆℓ may depend on x. The operator ℱ describes the filter used in the data analysis, which includes the time window, instrumental effects, and other filtering.

As will become obvious in the following sections, it is convenient to introduce a Green’s function for the observable Φ (see Equation 15View Equation), given by:

{ [ ]} 𝒢i(x,t;r′,t′) = ℱ ˆℓ ⋅ ∂tGi (x,zo,t;r′,t′) + Gi (x, zo,t;r′,t′) ⋅ ∇v0 (x,zo) . (16 )
The function i 𝒢 is the response of the observable to a unit source acting in the i− th direction. Notice that for the special case when the steady background flow v0 is constrained to be horizontal at the surface and the line of sight is vertical (ˆℓ = zˆ), we simply have
i ′ ′ { i ′ ′} 𝒢 (x,t;r ,t) = ℱ ∂tG z(x,zo,t;r ,t) . (17 )

For plane-parallel steady models with horizontal translation invariance, the Green’s functions for the observable are of the form 𝒢i (x, t;r′,t′) = 𝒢i(x − x ′,t − t′,z′) where r′ = (x′,z′). In this case we can write the Fourier domain Green’s functions for the observable as 𝒢i(k,ω, z′), according to

∫ i ′ 2 i ′ i(k⋅x−ωt) 𝒢 (x, t,z) = d k dω 𝒢 (k,ω,z )e . (18 )
In the short notation i ′ 𝒢 (k,ω, z), the ′ z always refers to the vertical position of the source of excitation.
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