Sensitivity kernels for inferring Lorentz stresses from normal-mode frequency splittings in the Sun

We begin by considering the full set of coupled ideal MHD equations in CGS units,

Here $\rho$ denotes the mass density, $\mathbf{v}$ the material velocity, $p$ the pressure, $\phi$ the gravitational potential, $c$ the speed of light, and $\mathbf{j}=(c/4\pi)\bm{\nabla}\times\mathbf{B}$ the current density; $\gamma$ is a ratio of specific heats determined by an adiabatic equation of state. The magnetic field, $\mathbf{B}$, is divergence free: $\bm{\nabla}\cdot\mathbf{B}=0$. The magnetic diffusivity term in the induction equation is dropped, assuming a large magnetic Reynolds number in the solar interior, of order $R_{m}\sim 10^{6}$ (Hood and Hughes, 2011). Hereafter, we resort to a background model which assumes the Sun to be spherically symmetric, non-rotating, non-magnetic, temporally stationary, capable of only sustaining adiabatic acoustic oscillations (model S in Christensen-Dalsgaard et al., 1996). We do not enlist the Poisson equation as we neglect perturbations to the gravitational potential $\phi$ — that is, we use the Cowling approximation.

In steady state with no background flow ($\mathbf{v}_{0}=\mathbf{0}$), the equilibrium force balance condition in model S is:

A subscript ‘0’ stands for zeroth order, static unperturbed fields. We do not subscript $\phi$ (or the gravitational acceleration $\mathbf{g}=-\bm{\nabla}\phi$) as the zeroth order gravitational potential (or field) is implied in the Cowling approximation. When we perturb model S by introducing magnetic field $\mathbf{B}\,$, the steady-state equilibrium force-balance also includes the Lorentz-force due to the zeroth order magnetic field:

A departure from this steady state equilibrium, via the introduction of perturbative forces, causes the system to respond by exhibiting free oscillations $\boldsymbol{\xi}(\boldsymbol{r},t)$. In Fourier space, these free oscillations can be reconstructed from a superposition of “normal-modes”, labelled by an index $k$, each of which have a characteristic frequency $\omega_{k}$ and spatial pattern $\boldsymbol{\xi}_{k}(\boldsymbol{r})$ with excitation amplitudes $a_{k}$:

Introducing first-order, time-dependent perturbations as outlined in Appendix A, we arrive at the linearized equation of motion for the above magnetohydrodynamic system:

where $\mathcal{L}\,_{0}$ is the linear acoustic wave operator for model-S while the operator $\delta\mathcal{L}\,$ captures the forcing due to Lorentz-stresses. We solve Eqn. (8) based on a perturbative analysis that involves the following two steps.

1. 1.

   We start with the unmagnetized case by setting $\mathbf{B}_{0}=\mathbf{0}$ and solve the eigenvalue problem $\rho_{0}\,\omega_{k,0}^{2}\,\boldsymbol{\xi}_{k,0}=\mathcal{L}\,_{0}\boldsymbol{\xi}_{k,0}$. This gives us the eigenfrequencies $\omega_{k,0}$ and eigenmodes $\boldsymbol{\xi}_{k,0}$ associated with the unperturbed model-S.

2. 2.

   We introduce a non-zero background magnetic field $\mathbf{B}_{0}$ and therefore “switch-on” the $\delta\mathcal{L}\,$ part of the wave operator in Eqn. (8). Consequently, to accommodate for the changes due to $\delta\mathcal{L}\,$, we introduce perturbed eigenfrequencies $\omega_{k}=\omega_{k,0}+\delta\omega_{k}$ and eigenfunctions $\boldsymbol{\xi}_{k}=\sum_{k^{\prime}}c_{k^{\prime}}\,\boldsymbol{\xi}_{k^{\prime},0}$. Note that we neglect the distortions in the solar structure induced due to the presence of magnetic fields. Accounting for this would involve mapping each point $\mathbf{r}$ on the model S spherical Sun onto a point $\mathbf{x}$ on the distorted Sun (Gough and Thompson, 1990).

The linearized equation of motion (see derivation of Eqn. A12) for the unmagnetized case is given by (also refer to Christensen–Dalsgaard, 2003):

where $c_{s}(r),\rho_{0}(r)$, and $g(r)$ are the sound speed, density, and gravity (directed radially inward) respectively, and $\bm{\nabla}$ denotes the covariant spatial derivative operator. For all ensuing calculations and derivations, we write Eqn. (9) in the form $\mathcal{L}_{0}\boldsymbol{\xi}_{k}=\rho_{0}\omega_{k}^{2}\boldsymbol{\xi}_{k}$, where the magnetically unperturbed wave operator $\mathcal{L}_{0}$ is self-adjoint (Goedbloed and Poedts, 2004). To solve for the eigenmodes of the unperturbed model S, the boundary conditions employed are: (a) $\boldsymbol{\xi}$ and the Eulerian pressure perturbation at $r=0$ is finite, and (b) the Lagrangian pressure perturbation at $r=R_{\odot}$ vanishes (see Section 17.6 in Cox, 1980). As already mentioned earlier, the Cowling approximation is used and hence the gravitational Poisson equation is not needed while finding the eigenmodes. We suppress the subscript ‘0’ in the unperturbed eigenfunctions $\boldsymbol{\xi}_{k,0}$ and eigenfrequencies $\omega_{k,0}$ for the rest of this paper. Unless specified otherwise, any instance of $\omega_{k}$ or $\boldsymbol{\xi}_{k}$ should be assumed to imply eigenfrequencies and eigenfunctions of Eqn. (9), respectively. The Sun is treated as a fluid body with vanishing shear modulus and hence is unable to sustain shear waves (although the presence of magnetic fields complicates this assumption). Thus, the eigenfunctions of the background model contain no toroidal components (see Chapter 8 of Dahlen and Tromp, 1998), rendering them purely spheroidal. We write the displacement field $\boldsymbol{\xi}(\boldsymbol{r})$ in the basis of vector spherical harmonics (and thereafter GSH) as follows

Here, $\boldsymbol{r}=(r,\theta,\phi)$ denote spherical polar coordinates, with basis vectors $(\hat{e}_{r},\hat{e}_{\theta},\hat{e}_{\phi})$ and $k=(n,\ell,m)$ where $n$ is the radial order, $\ell$ the angular degree, and $m$ the azimuthal order. The dimensionless lateral covariant derivative operator is denoted by $\boldsymbol{\bm{\nabla}}_{1}=\hat{\bm{e}}_{\theta}\,\partial_{\theta}+\hat{\bm{e}}_{\phi}\,(\sin\theta)^{-1}\partial_{\phi}$. The basis vectors in spherical polar coordinates are related to those in the GSH basis via

The vanishing toroidal component in (10) imposes the constraint that $\xi_{k}^{-}=\xi_{k}^{+}$ (Appendix C of Dahlen and Tromp, 1998). Owing to $\mathcal{L}\,_{0}$ being self-adjoint, the eigenvalues $\omega_{k}$ of the unperturbed state are real and the eigenfunctions $\boldsymbol{\xi}_{k}$ corresponding to distinct eigenvalues are orthogonal. For convenience, $\boldsymbol{\xi}_{k}$ is normalized to satisfy the orthonormality condition

Next, as derived in Appendix A (see Eqn [A13]), we introduce a magnetically induced perturbation through the operator $\delta\mathcal{L}\,$ where,

$\mathbf{B}_{0}$ is the zeroth-order background magnetic field and $\mathbf{B}_{1}=\bm{\nabla}\times(\boldsymbol{\xi}\times\mathbf{B}_{0})$ is the first order perturbation. Henceforth, we shall drop the subscript ‘0’ for the zeroth-order magnetic field. Therefore, any instance of $\mathbf{B}\,$ hereafter shall denote the zeroth-order magnetic field, unless specified otherwise. Like $\boldsymbol{\xi}$ (in Eqn [11]), we also expand the magnetic field and the corresponding Lorentz stress using the basis of generalized spherical harmonics $\boldsymbol{Y}_{st}^{N}$

where in Eqns. (15) and (16) $\mu$ and $\nu$ take values $-1$, 0, $+1$. In practice, the angular degree $s$ is usually truncated at some desired upper limit $s_{\rm{max}}$. Invoking the divergence-free condition on the magnetic field, $\bm{\nabla}\cdot\mathbf{B}\,=0$, we have $B_{st}^{+}+B_{st}^{-}=\partial_{r}(r^{2}B^{0}_{st})/(r\Omega_{s}^{0})$, where $\Omega_{\ell}^{\pm N}=\sqrt{\frac{1}{2}(\ell\pm N)(\ell\mp N+1)}$. The realness of $\mathbf{B}$ further imposes the constraint $B_{st}^{\mu\,*}=(-1)^{t}B_{s\bar{t}}^{\bar{\mu}}$, where overbars represent negatives (i.e., ${\bar{t}}=-t$), a property that follows from its realness condition. An equivalent constraint on the Lorentz stress $\boldsymbol{\mathcal{H}}$, which accounts for a solenoidal $\mathbf{B}\,$ field, seems unlikely. Therefore, as elucidated in Appendix D.2, we begin constructing our $\boldsymbol{\mathcal{H}}$ by choosing a divergence-free $\mathbf{B}\,$ field. Eqn. (D18) relates components of $\boldsymbol{\mathcal{H}}$ to $\mathbf{B}\,$. $\boldsymbol{\mathcal{H}}$, by construction, satisfies the symmetry property $h^{\mu\nu}_{st}=h^{\nu\mu}_{st}$ ($\because\boldsymbol{\mathcal{H}}=\mathbf{B}\,\mathbf{B}\,$), and $h^{\mu\nu\,*}_{st}=(-1)^{t}h_{s\bar{t}}^{\bar{\mu}\bar{\nu}}$. The introduction of magnetic perturbations introduces a set of surface terms which can be found in Eqn. (C4). Further details about the importance of these terms and why their contribution is negligible for the Model S (Christensen-Dalsgaard et al., 1996) can be found in Appendix C.

The eigenfrequencies of the unmagnetized state (eqn [9]) are denoted by ${}_{n}\omega{}_{\ell}$. They are independent of the azimuthal order $m$ because of the spherical symmetry of the background model (we use model S of Christensen–Dalsgaard, 2008). Therefore, each multiplet ${}_{n}\mathrm{S}_{\ell}$ is $2\ell+1$ degenerate. The introduction of (non-spherically symmetric) perturbations breaks this degeneracy and gives rise to frequency splitting. The aim of our forward problem is to calculate the perturbed eigenfrequencies under the action of Lorentz stresses. In calculating frequency shifts of an erstwhile degenerate multiplet ${}_{n_{0}}\mathrm{S}_{\ell_{0}}$, we apply the formalism of quasi-degenerate perturbation theory. Therefore, perturbations in the eigenfunctions and eigenfrequencies of a target multiplet ${}_{n_{0}}\mathrm{S}_{\ell_{0}}$ are assumed to arise from eigenmodes whose unperturbed eigenfrequencies $\omega_{k}$ lie within the range $[\omega_{\rm{ref}}-\Delta\omega,\omega_{\rm{ref}}+\Delta\omega]$, where the reference angular frequency $\omega_{\rm{ref}}$ is chosen to be ${}_{n_{0}}\omega_{\ell_{0}}$, and $\Delta\omega$ defines a window in frequency around $\omega_{\rm{ref}}$. Any mode with frequency $\omega_{k}$ outside the range $[\omega_{\rm{ref}}-\Delta\omega,\omega_{\rm{ref}}+\Delta\omega]$ is assumed to not “talk” to multiplet ${}_{n_{0}}\mathrm{S}_{\ell_{0}}$. Carrying out the standard quasi-degenerate perturbation analysis discussed in Appendix B, frequency splitting within each multiplet is given by the following expression:

where the matrix $\Lambda_{k^{\prime}k}=\int_{\odot}\mathrm{d}^{3}\mathbf{r}\,\,\boldsymbol{\xi}_{k^{\prime}}^{*}\cdot\delta\mathcal{L}\boldsymbol{\xi}_{k}=\braket{{\boldsymbol{\xi}_{k^{\prime}}}}{\delta\mathcal{L}\boldsymbol{\xi}_{k}}$ is called the coupling matrix and $\delta\omega^{2}=2\,\omega{}_{\mathrm{ref}}\,\delta\omega$. Here, $\delta\omega$ represents the shift in eigenfrequencies about the reference frequency $\omega_{\mathrm{ref}}$. The shifted eigenfrequencies, in general, do not carry an unperturbed mode index $k^{\prime}$ because $n^{\prime},\ell^{\prime},m^{\prime}$ cease to behave as good quantum numbers. For conciseness, it is common to define $\mathcal{Z}_{k^{\prime}k}=\Lambda_{k^{\prime}k}-(\omega_{\rm{ref}}^{2}-\omega_{k}^{2})\,\delta_{k^{\prime}k}$ as the supermatrix. The degenerate eigenmode with unperturbed frequency ${}_{n}\omega{}_{\ell}$ is denoted by the triplet $k=(n,\ell,m)$. The derivation of these matrix elements is outlined in Appendix C. The simplified form of the coupling matrix is written as a radial integral of $h_{st}^{\mu\nu}(r)$ components weighted by sensitivity kernels ${}_{k^{\prime}k}\mathcal{B}_{st}^{\mu\nu}(r)$ according to

Explicit expressions and symmetry relations for corresponding sensitivity kernels ${}_{k^{\prime}k}\mathcal{B}^{\mu\nu}_{st}$ are stated in Appendix C.1. Hereafter, we shall drop the mode subscripts $k^{\prime}k$ in ${}_{k^{\prime}k}\mathcal{B}^{\mu\nu}_{st}$ for notational simplicity. Expressions for Lorentz-stress components $h^{\mu\nu}_{st}$ may be found in Appendix D.2. In writing Eqn. (19) we use the symmetry in the Lorentz-stress tensor-expansion components, $h^{\mu\nu}_{st}=h^{\nu\mu}_{st}$, and the fact that there are only four independent components of $\mathcal{B}_{st}^{\mu\nu}$. The eigenvalues $\delta\omega^{2}$ of the supermatrix $\mathcal{Z}_{k^{\prime}k}$ are given by Eqn. (17). These relate to $\Omega_{p}$, the perturbed frequencies, via $\Omega_{p}=(\omega_{\rm{ref}}^{2}+\delta\omega^{2}_{p})^{1/2}$ where $p$ denotes a new set of labels different from the unperturbed label $k$. In the absence of spherical or azimuthal symmetry and cross-mode coupling, $n,\ell,m$ are no longer ‘good’ quantum numbers and therefore inhibit the direct mapping of the perturbed modes with the erstwhile unperturbed modes. Note that for the purpose of inverse problems, the proper method of parameterizing the inversion is to use $\boldsymbol{\mathcal{H}}/\rho$ and corresponding effective kernels $\rho\mathcal{B}$. This is discussed in further detail in Appendix C. The forward problem of Eqn. (19) remains unchanged under this consideration.

A general spheroidal p-mode wavefield in the spectral domain can be expanded in the basis of the unperturbed spheroidal p-modes as $\xi(\boldsymbol{r,\omega})=\sum_{k}\varphi_{k}(\omega)\boldsymbol{\xi}_{k}(\boldsymbol{r})$. Following the concise notation used in Hanasoge et al. (2017), we henceforth write $\varphi_{k}(\omega)$ as $\varphi_{k}^{\omega}$, which contains the phase and amplitude of mode $k$. The time series of the spherical harmonic oscillations at the solar surface (and hence $\varphi_{k}^{\omega}$) are supplied by the HMI and MDI missions. The change in this wavefield $\boldsymbol{\xi}$ in the presence of a perturbation can be expressed as

As shown in Woodard (2016) and Hanasoge et al. (2017), the expectation value of the cross-spectral correlation signal can be modelled as:

where $\langle A\rangle$ represents the statistical expectation value of parameter $A$, $N_{k}$ is the mode-amplitude normalization and $R_{k}^{\omega}=(\omega^{2}-\omega_{k}^{2})^{-1}$. The singularity at $\omega=\omega_{k}$ is avoided by introducing a tiny imaginary component to $\omega_{k}$ (which is representative of negligible attenuation). This results in damping of the mode. The damping rate $\gamma_{k}\sim 1/\tau$ where $\tau$ is the e-folding lifetime of the mode. The lifetime or damping rate can be accurately computed by modeling the power spectrum of modes in frequency and fitting a Lorenzian profile (Schou and Brown, 1994). If $\Delta$ is the FWHM then $\tau=(\pi\Delta)^{-1}$. The observed lifetime of modes in the Sun vary from days to a few months (Korzennik et al., 2013). Therefore, an estimation of $\langle\varphi_{k^{\prime}}^{\omega^{\prime}}\delta\varphi_{k}^{\omega*}+\delta\varphi_{k^{\prime}}^{\omega^{\prime}}\varphi_{k}^{\omega*}\rangle$ from observations allow us to recover elements in $\Lambda_{k^{\prime}k}$.

The elements $\Lambda_{k^{\prime}k}$ thus obtained define a non-sparse matrix for a general non-axisymmetric field. This is unlike the $a$-coefficient formalism, which is restricted to an inverse problem for axisymmetric fields and therefore has a diagonal coupling matrix. Because $\Lambda_{k^{\prime}k}$ has eigenvalues $\delta\omega^{2}=2\omega_{\rm{ref}}\,\delta\omega$, where $\omega_{\rm{ref}}$ is a fiducial reference frequency (refer to Appendix B). One can then choose to frame the following statement for an inverse problem (Dahlen and Tromp, 1998, Section 14.2.9 and 14.3.5):

where the coefficients $c_{st}^{n^{\prime}\ell^{\prime}n\ell}$ remain to be determined. The solid angle integration is carried out over the complete surface of a sphere. The integral involving three spherical harmonics can be conveniently expressed in terms of Wigner 3-$j$ symbols as follows  (Dahlen and Tromp, 1998, C.225):

where $\gamma_{\ell}=\sqrt{(2\ell+1)/(4\pi)}$. Upon comparing Eqn. (22) with Eqn. (18), we find that

where $\mathcal{B}_{st}^{\mu\nu}=4\pi(-1)^{m^{\prime}}\gamma_{\ell^{\prime}}\gamma_{s}\gamma_{\ell}\bigg{(}\begin{smallmatrix}\ell^{\prime}&s&\ell\\ -m^{\prime}&t&m\end{smallmatrix}\bigg{)}\,\mathcal{G}_{s}^{\mu\nu}$ (Eqn. [C6]). As $\mathcal{G}_{s}^{\mu\nu}$ does not have an $m$ or $m^{\prime}$ dependence, the coefficient $c_{st}^{n^{\prime}\ell^{\prime}n\ell}$ only depends on the multiplet labels — $(n^{\prime},\ell^{\prime})$ and $(n,\ell)$. Doing away with the $m$-dependence reduces the inverse problem in Eqn. (22) for each submatrix of dimension $(2\ell^{\prime}+1)\times(2\ell+1)$ to that for a single element — $c_{st}^{n^{\prime}\ell^{\prime}n\ell}$ — as shown in Eqn. (25). Again, while inverting for the components of the Lorentz stress tensor from $c_{st}^{n^{\prime}\ell^{\prime}n\ell}$, the effective kernels should be $\rho\mathcal{G}_{s}^{\mu\nu}$ for obtaining $h_{st}^{\mu\nu}/\rho$. The $c_{st}^{n^{\prime}\ell^{\prime}n\ell}$ are the so-called structure coefficients. It is commonplace in terrestrial seismology to use the structure coefficients to construct the ‘generalized splitting functions’

These splitting functions allow for a convenient form of visualization of the splitting due to self-coupling or cross-coupling of multiplets under a generic non-axisymmetric structure perturbation (the Lorentz-stress field in our case). As mentioned in Dahlen and Tromp (1998), the function $\eta(\theta,\phi)$ at any location $(\theta,\phi)$ on the surface illustrates the local radially averaged internal 3D structure, as “seen” through the kernels. A demonstrative plot of splitting functions due to the self-coupling of mode ${}_{2}\mathrm{S}_{8}\,$ and its cross-coupling with mode ${}_{3}\mathrm{S}_{7}\,$ is shown in Figure 5.

It may be noted that $\eta^{n^{\prime}\ell^{\prime}n\ell}(\theta,\phi)$ is purely real. This can be verified by using (a) the kernel property $\mathcal{G}_{s}^{\mu\nu}=(-1)^{\ell^{\prime}+\ell+s}\mathcal{G}_{s}^{\bar{\mu}\bar{\nu}}$ with $\ell^{\prime}+\ell+s=$ even because of the selection rule in Eqn. (25), (b) realness of $\boldsymbol{\mathcal{H}}$, i.e., $h_{st}^{\mu\nu}=(-1)^{t}h_{s\bar{t}}^{\bar{\mu}\bar{\nu}}$, and (c) the GSH property $Y_{st}^{N*}=(-1)^{N+t}Y_{s\bar{t}}^{\bar{N}}$.

The extent of coupling between two multiplets depends primarily on the proximity of their unperturbed frequencies, i.e., $|{}_{n_{1}}\omega_{\ell_{1}}-{}_{n_{2}}\omega_{\ell_{2}}|$. The more separated they are in frequency, the weaker is the coupling. As shown in equation (143) of Lavely and Ritzwoller (1992) this separation and hence the importance of considering cross-coupling can be quantified through $\nu_{\mathrm{CC}}$ the “coupling strength coefficient”. When a multiplet ${}_{n}\mathrm{S}_{\ell}$ is well separated ($\nu_{CC}<1nHz$) from any other multiplet in the frequency domain, contribution due to modes in other multiplets become significantly weak and therefore ${}_{n}\mathrm{S}_{\ell}$ can potentially be treated as isolated in frequency. Under such circumstances one can use what is known as the “isolated-multiplet” approximation where only the modes with different azimuthal orders $m$ belonging to the same multiplet ${}_{n}\mathrm{S}_{\ell}$ are considered for perturbation analysis. In such cases, $\omega_{\rm{ref}}={}_{n}\omega{}_{\ell}$ and Eqn. (17) reduces to

Note the change in the index from $k=(n,\ell,m)$ to $m$. This is because we restrict the coupling to a specific multiplet and therefore $n$ and $\ell$ remain good quantum numbers whereas $m$ does not.

We use a $50\,\mu$ Hz frequency window for the quasi-degenerate perturbation analysis. It can be shown that differential rotation induces a coupling strength coefficient $\nu_{CC}\approx$ $1nHz$ between the interacting modes within a frequency window of $30\,\mu$ Hz (Lavely and Ritzwoller, 1992). Differential rotation is the strongest perturbation over the SNRNMAIS Sun. Therefore, all multiplets capable of causing a $\nu_{CC}\geq 1nHz$ due to magnetic perturbations, lie within a window of $\Delta\omega=50\,\mu$ Hz. Therefore, any mode ${}_{n^{\prime}}\rm{S}_{\ell^{\prime}}$ that has an unperturbed frequency ${}_{n^{\prime}}\omega{}_{\ell^{\prime}}$ in the window ${}_{n}\omega{}_{\ell}-\Delta\omega\leq{}_{n^{\prime}}\omega{}_{\ell^{\prime}}\leq{}_{n}\omega{}_{\ell}+\Delta\omega$ is considered to couple non-trivially with ${}_{n}\rm{S}_{\ell}$ and hence is included in the eigenspace of modes considered for cross-coupling. We refer to this eigenspace as $\mathcal{K}$. We consider a Lorentz stress-field consisting of angular degrees $s=\{0,1,2,3,4,5,6\}$ containing all the azimuthal orders $t$ for a given $s$. Therefore, modes with $|\ell^{\prime}-\ell|\leq 6$ are coupled. A uniform-strength Lorentz-stress field (varying as a function of $r$) is allocated to all the spatial wavenumber channels

Here, $h_{0}(r)=b_{0}^{2}(r)$, with $b_{0}(r)$ being the profile of the ‘total’ field shown in Figure 4. In addition, we impose the realness condition of $\boldsymbol{\mathcal{H}}$ , $h^{\mu\nu\,*}_{st}=(-1)^{t}h_{s\bar{t}}^{\bar{\mu}\bar{\nu}}$ , and the symmetry $h_{st}^{\mu\nu}=h_{st}^{\nu\mu}$. In Figure 4, the field strength $b_{0}(r)$ as a function of radius is inspired from the references cited in Section 3.3. Allocating this to all the components $h_{st}^{\mu\nu}(r)$ certainly guarantees a general magnetic field with a total strength significantly exceeding current estimates of the solar internal field strength. Therefore, in the ensuing calculation, the extent of mode-coupling should be an over-estimation. Thus, modes labelled ‘coupled’ could, in reality, be ‘isolated’ but modes labelled ‘isolated’ are expected to be good for carrying out degenerate perturbation analysis.

In Figure 1 we color-code multiplets to indicate the departure of frequency shifts obtained from quasi-degenerate perturbation $\delta{}_{n}\omega{}_{\ell m}^{Q}$ as compared to shifts obtained from a degenerate perturbation $\delta{}_{n}\omega{}_{\ell m}^{D}$. As the perturbations are not axisymmetric, the azimuthal order does not stay well defined after coupling and hence prevents a one-one mapping of unperturbed to perturbed modes. This hinders an explicit comparison of frequency shifts on a singlet-by-singlet basis. Hence, to quantify the departure of $\delta{}_{n}\omega{}_{\ell m}^{Q}$ from $\delta{}_{n}\omega{}_{\ell m}^{D}$ we calculate the Frobenius norm of these frequency shifts:

The gray-scale intensity in Figure 1 indicates the relative offset of $L_{2}^{\text{QDPT}}$ as compared to $L_{2}^{\text{DPT}}$ for a mode ${}_{n}\rm{S}_{\ell}$ marked as an ‘o’. Larger offset indicates stronger relative offset and are marked as a bigger and darker ‘o’. For visual clarity, we have set a lower threshold on the size of the ‘o’ — any mode with less than $5\%$ offset has the same size. Certain ${}_{n}\rm{S}_{\ell}$ have unperturbed frequencies proximal to adjacent multiplets ${}_{n^{\prime}}\rm{S}_{\ell^{\prime}}$ prior to perturbation. Depending on the magnitude of perturbation and its sensitivity, modes of such ${}_{n}\rm{S}_{\ell}$ become strongly entangled with adjacent modes. Because the labels $n,\ell,m$ lose meaning after perturbation, it is impossible to disentangle modes belonging to our target multiplet from those belonging to its adjacent multiplets. For such multiplets we cannot obtain $\delta{}_{n}\omega{}_{\ell m}^{\text{Q}}$ and therefore avoid plotting them in Figure 1. We refer to these as ‘invisible modes’ in the context of this figure.

We plot four different cases, considering the effect of independent components of $\boldsymbol{\mathcal{H}}$ in each. This helps in explicitly demonstrating which multiplets have a non-trivial difference between $\delta{}_{n}\omega{}_{\ell m}^{\text{Q}}$ and $\delta{}_{n}\omega{}_{\ell m}^{\text{D}}$ under the influence of a particular Lorentz-stress component. For example, we plot the effect due to the presence of only $h_{st}^{00}$ or only $h_{st}^{+-}$ in Figure 1(c) and Figure 1(d) respectively. In Figure 1(a), we plot the effect of the presence of $h_{st}^{--}$ and $h_{st}^{++}$ as these are coupled through the realness of $\boldsymbol{\mathcal{H}}$. Similarly for Figure 1(b) we consider the presence of $h_{st}^{0-}$ and $h_{st}^{0+}$. Because we choose the same radial profile for all $h_{st}^{\mu\nu}$ (refer Eqn [37]), differences in the four panels primarily reflect the effect of sensitivity kernels $\mathcal{B}_{st}^{\mu\nu}$ for changing $\mu,\nu$. We find the maximum number of invisible modes in Figure 1(d) corresponding to $h_{st}^{+-}$. This is because $\mathcal{B}_{st}^{+-}$ is at least an order of magnitude larger than other components (Figures 2 and 3), thereby causing multiplets to overlap in frequency and hindering post perturbation mode identification. $\mathcal{B}_{st}^{00}$ and $\mathcal{B}_{st}^{--}$ have comparable magnitudes, are smaller than $\mathcal{B}_{st}^{+-}$ but larger than $\mathcal{B}_{st}^{0-}$. Therefore, the presence of $h_{st}^{00}$ (Figure 1(c)) and $h_{st}^{--},h_{st}^{++}$ (Figure 1(a)) induce an intermediate number of invisible modes and $h_{st}^{0-},h_{st}^{0+}$ (Figure 1(b)) induce the least number of invisible modes. It should be noted that $\mathcal{B}_{st}^{0-}$ is the weakest component of the sensitivity kernel and therefore we expect it to couple modes the least. However, the number of darker ‘o’s due to the presence of $h_{st}^{0-},h_{st}^{0+}$ is more than the other cases because: (i) it contains the least number of invisible modes as many of its strongly coupled multiplets are invisible in the other components. Hence, many of the dark ‘o’s in Figure 1(b) are actually invisible in Figure 1(a), (b), and (d), and (ii) the gray-scale intensity depends on the relative offset $(L_{2}^{\text{QDPT}}-L_{2}^{\text{DPT}})/L_{2}^{\text{DPT}}$. The lowest sensitivity of $h_{st}^{0-},h_{st}^{0+}$ causes the least frequency shifts and therefore $L_{2}^{\text{DPT}}$ is much smaller in magnitude, resulting in seemingly larger relative offsets.

Returning to the question we had sought to address earlier, because of the large number of small, white ’o’ symbols across all the components, we can assert that there are multiplets which can be treated as isolated. However, because Lorentz-stress in the Sun comes as a combination of all the four cases, a multiplet can be regarded as free of cross-coupling only when it is isolated in all the four cases. Thus, although there are many isolated-multiplets in Figures 1(a), (b), and (c) the number is restricted to those which are deemed isolated in (d). So, in the patch of $n,\ell$ we have chosen, we do find modes that can be treated as isolated. Therefore, in the following sections we explore the simplifications to the forward and inverse problem while restricting our analysis to these isolated-multiplets. However, as mentioned earlier, it should be noted that our choice of assigning the same strength to all $h_{st}^{\mu\nu}$ is supposed to cause an over-estimation of the coupling and therefore an under-estimation of the number of isolated-multiplets. A similar plot showing the validity of isolated-multiplet assumption when considering differential rotation as the sole perturbation can be found in Figure 8 under Appendix G.

Figure 2 shows sensitivity kernels ($\rho\,\mathcal{A}_{s}^{\mu\nu}$) for the $a_{s}^{n\ell}$ coefficients due to density-scaled axisymmetric Lorentz-stress $h_{s0}^{\mu\nu}/\rho$ (and therefore effectively $h_{s0}^{\mu\nu}$), where $s=2$. We have chosen three distinct modes — (5,110); (4,60); (2,10). The kernels have been plotted down to a depth of $r=0.68R_{\odot}$. This is because the equatorial location of the tachocline is $\sim 0.7R_{\odot}$ with a thickness of $\sim 0.04R_{\odot}$ (Howe, 2009). The sensitivity for the isotropic or on-diagonal component $h_{20}^{+-}$ is larger, by at least an order of magnitude, than all the other components. The other isotropic component $h_{20}^{00}$ and the anisotropic or off-diagonal components $h_{20}^{--}$ have comparable sensitivities. The anisotropic component $h_{s0}^{0-}$ has the least sensitivity. This is in agreement with Hanasoge (2017) with an additional finding that $h_{20}^{--}$ has sensitivities comparable to $h_{20}^{00}$. This difference is because of the correction terms included in this study to the Lorentz-stress sensitivity kernels (Eqn. C3). Within the outer envelop $r\gtrsim 0.9R_{\odot}$, the mode ${}_{5}\rm{S}_{110}$ is dominantly sensitive followed by ${}_{4}\rm{S}_{60}$, while ${}_{2}\rm{S}_{10}$ is the weakest of all components. This trend of sensitivity across $n,\ell$ is more evident in Figure 3. Also, for the same kernel component, the mode possessing the maximum sensitivity changes with depth. For $h_{20}^{--},{}_{2}\rm{S}_{10}$ becomes dominantly sensitive for $r<0.9R_{\odot}$. This holds true for the other components as well (see Figure 6) where kernels have been plotted from 0.95 $R_{\odot}$ – 0.68 $R_{\odot}$. This shows that there are broadly two category of modes — surface and depth sensitive.

Figure 3 shows the trend in variation of the sensitivity kernels $\rho\,\mathcal{A}_{2}^{\mu\nu}$ for self-coupling of spheroidal modes ${}_{n}\rm{S}_{\ell}$ for radial order $n=1$–5 and every $10^{\rm{th}}$ angular degree $\ell=10$–110. For a fixed radial order $n$, the near-surface sensitivity increases steadily with angular degree $\ell$. This increase in sensitivity is greatest for the kernel component $\mathcal{A}_{2}^{+-}$ which corresponds to the sensitivity for $\langle B_{\theta}B_{\theta}\rangle$ and $\langle B_{\phi}B_{\phi}\rangle$. Although weaker than $\mathcal{A}_{2}^{+-}$, the kernel components $\mathcal{A}_{2}^{00}$ and $\mathcal{A}_{2}^{--}$ demonstrates a similar trend in sensitivity for increasing $\ell$. $\mathcal{A}_{2}^{00}$ corresponds to the sensitivity of the field component $\langle B_{r}B_{r}\rangle$ and $\mathcal{A}_{2}^{--}$ to $\langle B_{\theta}B_{\theta}\rangle$, $\langle B_{\phi}B_{\phi}\rangle$ and $\langle B_{\theta}B_{\phi}\rangle$. Therefore, the kernels have the largest sensitivity to the magnetic pressure and cross-terms of the angular components $\langle B_{\theta}B_{\phi}\rangle$ of Lorentz-stress. A faster rise in sensitivity is seen with increasing radial order $n$ for a fixed angular degree $\ell$. The cross-terms between radial and angular components, viz., $\langle B_{r}B_{\theta}\rangle$ and $\langle B_{r}B_{\phi}\rangle$ have poor sensitivities ($\mathcal{A}_{2}^{0-}$) and therefore more challenging to invert for.