  Solar differential rotation exhibits a prominent feature: its cyclic
variations over the solar cycle, referred to as zonal flows or torsional
oscillations, are observed throughout the convection zone. Given the challenge
of measuring magnetic fields in subsurface layers, understanding deep torsional
oscillations becomes pivotal in deciphering the underlying solar dynamo
mechanism. In this study, we address the critical question of identifying
specific signatures within helioseismic frequency-splitting data associated
with the torsional oscillations. To achieve this, a comprehensive forward
modeling approach is employed to simulate the helioseismic data for a dynamo
model that, to some extent, reproduces solar-cycle variations of magnetic
fields and flows. We provide a comprehensive derivation of the forward modeling
process utilizing generalized spherical harmonics, as it involves intricate
algebraic computations. All estimated frequency-splitting coefficients from the
model display an 11-year periodicity. Using the simulated splitting
coefficients and realistic noise, we show that it is possible to identify the
dynamo wave signal present in the solar zonal flow from the tachocline to the
solar surface. By analyzing observed data, we find similar dynamo wave patterns
in the observational data from MDI, HMI, and GONG. This validates the earlier
detection of dynamo waves and holds potential implications for the solar dynamo
theory models.
