Solar Inertial Oscillations

• Solar inertial oscillations are global, low-frequency motions in the Sun driven by the Coriolis force, exhibiting quasi-toroidal flow patterns.
• They are classified into families such as equatorial Rossby modes, high-latitude inertial modes, and critical-latitude modes, each highlighting distinct dynamical behaviors.
• Advanced modeling and helioseismic techniques diagnose how differential rotation, turbulent viscosity, and baroclinicity influence these modes, probing the deep convection zone.

Solar inertial oscillations are global, low-frequency motions of the rotating solar interior whose dominant restoring force is the Coriolis force. In the Sun they are predominantly quasi-toroidal, with periods on the order of the solar rotation period, and they occupy a spectrum that includes equatorial Rossby modes, high-latitude inertial modes, critical-latitude modes, high-frequency retrograde modes, and prograde columnar convective modes. Because their frequencies and eigenfunctions depend sensitively on differential rotation, turbulent viscosity, superadiabaticity, and latitudinal entropy gradients, they provide a diagnostic window on the deep convection zone that complements acoustic helioseismology (Gizon et al., 2021, Gizon et al., 2024, Mukhopadhyay et al., 28 Jan 2025).

1. Definition, classification, and canonical theory

In rotating stellar fluid dynamics, inertial modes are oscillations whose restoring force is the Coriolis force. In the solar case, the leading-order flow is largely horizontal, with negligible radial displacement at leading order in the small-Rossby-number, weakly compressible limit. A convenient representation uses toroidal vector spherical harmonics,

$\mathbf{T}_{\ell m}=\nabla\times(rY_{\ell m}),$

so that the velocity field may be written

$$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$

with

$$Y_{\ell m}(\theta,\phi)=P_\ell^m(\cos\theta)e^{im\phi}.$$

For classical $r$-modes, the flow is purely toroidal to leading order (Joshi et al., 23 Jan 2025).

The canonical frequency bound for inertial waves in a uniformly rotating fluid is

$-2\Omega \le \omega \le 2\Omega.$

A distinguished spherical subclass is the Rossby or $r$-mode family, whose rotating-frame dispersion relation is

$\omega_r=-\frac{2m\Omega}{\ell(\ell+1)},$

with $m\ge 1$ and $\ell\ge m$. For sectoral modes $(\ell=m)$, this reduces to $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$0. The negative sign denotes retrograde propagation in the rotating frame (Joshi et al., 23 Jan 2025, Bhattacharya et al., 2023).

Solar observations and modeling separate the inertial spectrum into several families. Equatorial Rossby modes are quasi-toroidal and concentrated near the equator. High-latitude inertial modes have strongest power at latitudes $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$1 and, in several models, kinetic-energy density peaking near the base of the convection zone. Critical-latitude modes are organized around latitudes where the local corotation condition is met. Additional branches include high-frequency retrograde modes, which are non-toroidal and propagate substantially faster than equatorial Rossby modes, and prograde columnar convective modes, also termed thermal Rossby or Busse modes (Gizon et al., 2021, Gizon et al., 2024, Bekki et al., 2022).

These oscillations are distinct from solar $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$2-modes and $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$3-modes. $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$4-modes are pressure-restored acoustic oscillations with five-minute periods, while $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$5-modes are buoyancy-restored and are primarily associated with the radiative interior. Inertial modes are Coriolis-restored, have frequencies of order the rotation rate, and couple directly to large-scale vorticity, shear, and baroclinicity (Nguyen et al., 2024, Gizon et al., 2024).

2. Governing equations, frames, and approximations

A central complication is frame dependence. If $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$6 is a mode frequency in a local co-rotating frame with angular speed $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$7, then

$$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$8

and in a frame rotating at $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$9,

$$Y_{\ell m}(\theta,\phi)=P_\ell^m(\cos\theta)e^{im\phi}.$$0

For $$Y_{\ell m}(\theta,\phi)=P_\ell^m(\cos\theta)e^{im\phi}.$$1-modes, $$Y_{\ell m}(\theta,\phi)=P_\ell^m(\cos\theta)e^{im\phi}.$$2, so the observed sign depends on both retrograde propagation and the chosen tracking rate (Joshi et al., 23 Jan 2025).

Linear solar models typically use either the fully compressible equations, the anelastic approximation, or—more problematically—the Boussinesq approximation. In anelastic form, the continuity equation is

$$Y_{\ell m}(\theta,\phi)=P_\ell^m(\cos\theta)e^{im\phi}.$$3

and a poloidal–toroidal decomposition enforces that constraint while retaining density stratification. The momentum and entropy equations are solved in spherical shells with prescribed $$Y_{\ell m}(\theta,\phi)=P_\ell^m(\cos\theta)e^{im\phi}.$$4, turbulent viscosity $$Y_{\ell m}(\theta,\phi)=P_\ell^m(\cos\theta)e^{im\phi}.$$5, and thermal diffusivity $$Y_{\ell m}(\theta,\phi)=P_\ell^m(\cos\theta)e^{im\phi}.$$6, often with stress-free, impenetrable radial boundaries and zero entropy flux (Bhattacharya et al., 2023, Bhattacharya et al., 2022).

Differential rotation enters in two distinct ways. First, it Doppler-shifts frequencies by terms of order $$Y_{\ell m}(\theta,\phi)=P_\ell^m(\cos\theta)e^{im\phi}.$$7. Second, it couples spherical-harmonic components and creates critical layers. In several formulations, one introduces a Doppler-shifted local frequency

$$Y_{\ell m}(\theta,\phi)=P_\ell^m(\cos\theta)e^{im\phi}.$$8

or equivalently $$Y_{\ell m}(\theta,\phi)=P_\ell^m(\cos\theta)e^{im\phi}.$$9, depending on sign convention. Critical latitudes occur where the intrinsic frequency vanishes locally, and these regions strongly distort eigenfunctions and can trap power (Bekki et al., 2022, Bhattacharya et al., 2023).

Thermal-wind balance links differential rotation to baroclinicity: $r$0 This relation is central for the high-latitude modes, whose growth and geometry depend on the latitudinal entropy gradient implied by the solar rotation profile (Gizon et al., 2021, Bekki et al., 2022).

A major methodological conclusion is that the anelastic and fully compressible models produce almost identical solar inertial eigenmodes for both uniform rotation and helioseismic differential rotation, whereas the Boussinesq approximation produces significantly different non-toroidal modes because it removes the density stratification and hence the compressional $r$1-effect. The Boussinesq or incompressible approximations therefore cannot be used to model solar inertial modes accurately, except for the nearly purely toroidal $r$2-modes under uniform rotation (Mukhopadhyay et al., 28 Jan 2025).

3. Detection on the Sun and extraction of eigenfunctions

The modern observational program rests primarily on long HMI/SDO time series. Ring-diagram analyses of near-surface horizontal flows over 2010–2020 revealed narrow, latitude-coherent peaks that established the existence of equatorial Rossby modes, high-latitude inertial modes, and critical-latitude modes. The observed inertial spectrum extends across $r$3–10 in the early global survey, with equatorial Rossby frequencies close to the classical $r$4 scaling after differential-rotation corrections, and with representative high-latitude frequencies such as the $r$5 symmetric mode near $r$6 nHz in the Carrington frame (Gizon et al., 2021).

A complementary route uses local correlation tracking of small magnetic features in HMI line-of-sight magnetograms from May 2010 to September 2020. In that approach, magnetic network elements are treated as passive tracers of the near-surface horizontal flow, with tracking performed in the Carrington frame and with Postel projections on a grid spaced by $r$7 in latitude and longitude. The local flow is inferred from the displacement that maximizes the short-lag cross-covariance,

$r$8

$r$9

with $-2\Omega \le \omega \le 2\Omega.$0 h, $-2\Omega \le \omega \le 2\Omega.$1, and $-2\Omega \le \omega \le 2\Omega.$2 h. Fourier selection in $-2\Omega \le \omega \le 2\Omega.$3, narrow band-pass filtering, and singular-value decomposition then yield

$-2\Omega \le \omega \le 2\Omega.$4

For the $-2\Omega \le \omega \le 2\Omega.$5 high-latitude mode, this procedure reveals a pronounced power excess at $-2\Omega \le \omega \le 2\Omega.$6 nHz in the Carrington frame, strongest for $-2\Omega \le \omega \le 2\Omega.$7, together with a smooth latitudinal eigenfunction consistent with quasi-toroidal inertial flow and with improved signal-to-noise toward the poles relative to ring-diagram measurements (Joshi et al., 23 Jan 2025).

Normal-mode coupling provides radial information that local flow tracking does not. Using eight years of HMI Doppler data from 2010–2017, same- and different-degree acoustic-mode couplings reveal a high-latitude $-2\Omega \le \omega \le 2\Omega.$8 inertial mode at $-2\Omega \le \omega \le 2\Omega.$9 nHz that is present throughout the convection zone. Sectoral Rossby modes are visible down to $r$0, and their amplitudes increase with depth down to around $r$1 before decreasing. The same analysis also showed that apparent departures of Rossby latitudinal eigenfunctions from sectoral spherical harmonics can be explained by spatial leakage and even pure noise in non-sectoral components, so inferences about latitudinal structure require caution (Mandal et al., 2024).

The $r$2 high-latitude mode also leaves a direct magnetic signature. In HMI and GONG line-of-sight magnetograms, high-latitude oscillations at $r$3–$r$4 are detected at $r$5 nHz in the Earth frame, the synodic counterpart of the Carrington-frame $r$6 nHz mode. The oscillations are predominantly symmetric across the equator, have peak magnetic amplitude up to $r$7 gauss, and show a spatial pattern consistent with simplified calculations in which the background radial magnetic field is advected by the mode’s horizontal flow (Heinemann et al., 1 Oct 2025).

4. Differential rotation, stratification, and the organization of the spectrum

The simplest Rossby–Haurwitz law is not sufficient once solar differential rotation is included. Helioseismic measurements extended to $r$8 show that sectoral equatorial modes become progressively less retrograde than the thin-shell prediction at high $r$9. A linear anelastic spectral model with realistic solar differential rotation reproduces this behavior by producing several nearly linear ridges in frequency versus $\omega_r=-\frac{2m\Omega}{\ell(\ell+1)},$0. In that model, the fundamental $\omega_r=-\frac{2m\Omega}{\ell(\ell+1)},$1 sectoral ridge is deflected strongly in the retrograde direction, while another ridge lies strikingly close to the observed frequencies, suggesting that the measured high-$\omega_r=-\frac{2m\Omega}{\ell(\ell+1)},$2 solar sectoral modes may correspond not to the fundamental Rossby–Haurwitz solutions but to higher-$\omega_r=-\frac{2m\Omega}{\ell(\ell+1)},$3 modes (Bhattacharya et al., 2023).

The physical reason is that the Doppler-like $\omega_r=-\frac{2m\Omega}{\ell(\ell+1)},$4 term grows linearly with $\omega_r=-\frac{2m\Omega}{\ell(\ell+1)},$5, whereas the classical sectoral Rossby contribution scales as $\omega_r=-\frac{2m\Omega}{\ell(\ell+1)},$6. At sufficiently large $\omega_r=-\frac{2m\Omega}{\ell(\ell+1)},$7, shear dominates the frequency budget, pushing modes toward less retrograde or even slightly prograde behavior in the tracking frame. Differential rotation also generates critical latitudes for all $\omega_r=-\frac{2m\Omega}{\ell(\ell+1)},$8 in solar-like profiles and couples multiple $\omega_r=-\frac{2m\Omega}{\ell(\ell+1)},$9-channels within each $m\ge 1$0, altering both eigenfunctions and depth localization (Bhattacharya et al., 2023).

High-latitude modes respond even more strongly to baroclinicity. In the first global inertial-mode survey, high-latitude and critical-latitude modes were matched with 2D shell eigenfunctions only when the deep convection zone was nearly adiabatic and the turbulent viscosity was small. That study obtained the constraints

$m\ge 1$1

for the lower half of the convection zone, and found that the high-latitude modes have maximum kinetic-energy density near the base of the convection zone (Gizon et al., 2021).

A more realistic fully compressible shell calculation later showed that three effects are especially important. First, for $m\ge 1$2, diffusivities $m\ge 1$3 radically alter the radial dependence of $m\ge 1$4 equatorial Rossby modes away from the classical $m\ge 1$5 expectation. Second, mixed modes exist that connect $m\ge 1$6 equatorial Rossby modes with prograde columnar convective modes. Third, the observed $m\ge 1$7 high-latitude mode is reproduced when the latitudinal entropy gradient required by thermal-wind balance is included, making it a baroclinically unstable mode with modeled Carrington-frame frequency $m\ge 1$8 nHz and growth time of about $m\ge 1$9 months (Bekki et al., 2022).

A recurrent misconception is that solar inertial modes are adequately modeled without density stratification. The detailed comparison of compressible, anelastic, and Boussinesq models shows that this is false for non-toroidal families: the absence of the compressional $\ell\ge m$0-effect in Boussinesq calculations strongly distorts high-frequency retrograde modes, mixed modes, high-latitude modes, and prograde columnar modes. An acceptable baseline setup is instead anelastic dynamics together with solar differential rotation (Mukhopadhyay et al., 28 Jan 2025).

5. Inertial-mode helioseismology and inverse problems

One motivation for studying solar inertial oscillations is that they enable inverse problems that are complementary to those based on $\ell\ge m$1-mode splittings. In a surface-toroidal model, the horizontal velocity is written in terms of a stream function $\ell\ge m$2 through $\ell\ge m$3, and after Fourier transform in time and longitude one obtains the scalar equation

$\ell\ge m$4

with

$\ell\ge m$5

$\ell\ge m$6

Under random convective forcing, the covariance of $\ell\ge m$7 is related to the Green’s function of this operator, which turns inertial-wave observations into a passive imaging problem for differential rotation and effective viscosity (Nguyen et al., 2024).

Synthetic inversions with adjoint-based gradients show that covariance imaging contains enough information to recover both $\ell\ge m$8 and $\ell\ge m$9. In the surface model, excellent agreement between truth and reconstruction was achieved in about $(\ell=m)$0 iterations, even when initialized far from the true rotation profile and viscosity. The same framework makes clear why $(\ell=m)$1 and $(\ell=m)$2 can be disentangled: $(\ell=m)$3 primarily shifts phase and frequency through the advection and shear terms, while $(\ell=m)$4 controls damping through the bi-Laplacian operator (Nguyen et al., 2024).

A mathematically more formal development treated the same toroidal problem as a fourth-order scalar equation on the sphere, proved Fredholm well-posedness under explicit conditions on differential rotation, verified the tangential cone condition, and showed convergence of iterative regularization methods. Numerical experiments with Nesterov–Landweber iteration demonstrated robust joint reconstruction of viscosity and differential rotation under full, partial, and noisy surface observations (Nguyen et al., 28 Jul 2025).

The most direct inertial-mode helioseismic inference so far uses the $(\ell=m)$5 high-latitude mode itself as a localized probe of solar rotation. Starting from the HMI/SDO reference rotation profile averaged over 2010–2024, the linear sensitivity kernel of that mode was found to peak at latitude $(\ell=m)$6 and radius $(\ell=m)$7, with full widths of $(\ell=m)$8 and $(\ell=m)$9. Using the observed Carrington-frame frequency $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$00 nHz and a validated eigenvalue solver, the inferred local rotation rate is

$$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$01

which exceeds the reference $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$02-mode estimate at that location by $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$03 nHz. This is the first example of spatially resolved inertial-mode helioseismology (Dey et al., 19 May 2026).

6. Nonlinear dynamics, radiative coupling, and open problems

Linear theory explains the existence and morphology of the modes, but nonlinear simulations address excitation, saturation, and feedback on the mean flow. Global EULAG simulations of a subadiabatic, differentially rotating shell show that inertial modes can arise spontaneously through baroclinic instability. In a shear sequence DR05–DR40, the instability threshold lies between $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$04 and $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$05 relative shear, the growth rate increases with shear and saturates beyond about $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$06, and the first modes to appear are high-latitude retrograde polar vortices that originate near the tachocline and grow outward. In the Sun-like DR20 case, the dominant intrinsic frequencies are $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$07, $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$08, and $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$09 nHz for $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$10, $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$11, and $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$12, respectively, close to helioseismic values for polar modes (Souza-Gomes et al., 7 Nov 2025).

Those nonlinear calculations also show that equatorial Rossby branches are weak unless seeded, whereas arbitrary perturbations can excite Rossby modes across all available wave numbers via direct and inverse cascades. At stronger shear, Reynolds stresses become dynamically important, transport angular momentum poleward, accelerate the poles, and reduce the imposed radial shear; this feedback coincides with the saturation of the instability growth rate (Souza-Gomes et al., 7 Nov 2025).

Fully nonlinear rotating-convection simulations offer a different perspective. In a Yin–Yang spherical-shell calculation covering about $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$13 solar years, equatorial Rossby modes with no radial node were stochastically excited with amplitudes of a few $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$14 and linewidths of about $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$15–$$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$16 nHz, comparable to solar measurements. The same simulation also contained prograde columnar convective modes, mixed Rossby–columnar modes, and a high-latitude $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$17 mode, although the latter was much weaker than on the Sun. This suggests that additional ingredients—stronger baroclinicity, magnetic effects, or more realistic near-surface structure—may be required to reproduce the observed polar amplitudes (Bekki et al., 2022).

The role of the radiative interior appears to be subtle. Linear Dedalus computations extending the domain down to $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$18 show that including the radiative zone changes the real frequencies of most convection-zone inertial modes by less than $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$19 nHz and leaves surface eigenfunctions almost unchanged, while significantly increasing damping because modes penetrate into the overshoot layer, where viscous dissipation is large. The radiative interior itself supports a complete $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$20-mode spectrum for all allowed $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$21 and radial nodes. When a radiative-zone Rossby mode lies within about $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$22 nHz of a convection-zone inertial mode of the same north–south symmetry, mixed modes arise, but their large mode mass in the radiative interior makes stochastic excitation difficult (Mukhopadhyay et al., 14 Dec 2025).

Several issues remain unresolved. A systematic mode taxonomy across the full $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$23 plane is still incomplete; high-$$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$24 sectoral branches may correspond to higher radial order, but their depth localization remains in tension with surface-sensitive observations (Bhattacharya et al., 2023). Spatial leakage and pure noise can masquerade as non-sectoral latitudinal structure in normal-mode coupling analyses (Mandal et al., 2024). The passive-tracer assumption in magnetic-feature tracking can be biased by anchoring, emergence, cancellation, and morphology changes (Joshi et al., 23 Jan 2025). Magnetic fields are omitted from most current forward models, even though the magnetic modulation of the $$\mathbf{u}(r,\theta,\phi,t)\approx \Re\left\{a_{\ell m}(r)\,\mathbf{T}_{\ell m}(\theta,\phi)e^{-i\omega t}\right\},$$25 high-latitude mode and the likely importance of Lorentz forces in the tachocline indicate that magnetically modified inertial dynamics will be necessary for a complete theory (Heinemann et al., 1 Oct 2025, Gizon et al., 2024).

Taken together, these results place solar inertial oscillations at the center of a developing program in deep-interior diagnostics. The observational evidence now extends from near-surface flow maps to normal-mode coupling, magnetic-feature tracking, and direct magnetic-field oscillations; the modeling now spans linear eigenvalue theory, passive-imaging inversions, and nonlinear global simulations. The cumulative picture is that solar inertial modes are both probes of the convection zone and active participants in its dynamics (Gizon et al., 2024).