Context. Protoplanetary disks are known to host spiral features that are observed in scattered light, ALMA continuum and more recently in CO gas emission and gas dynamics. It is however unknown if spirals in gas and dust trace the same morphology. Aims. We study the morphology and amplitude of dusty spirals as function of Stokes number and the underlying mechanisms causing a difference from gas spirals. We then construct a model to relate the deviation from Keplerian rotation in the gas to a perturbation in surface density of gas and dust. Methods. We use FARGO-3D with dust implementation to numerically study the spirals, after which the results are interpreted using a semi-analytical model. This model is tested on observational data to predict the perturbation of the spiral in gas dynamics based on the continuum data. Results. We find that the pitch angle of a spiral does not differ significantly between gas and dust. The amplitude of the dust spiral decreases with Stokes number (St) and starts to fade out at a typical St > 0.1 as the dust becomes decoupled from the gas. The semianalytical model provides an accurate and fast representation of the spiral in surface density of the dust from the gas. We find a spiral in the TW Hya velocity residual map, never seen before, which is a feature in the vertical velocity and has a kink at the continuum gap, yielding strong evidence for a planet at 99 au. Conclusions. We built a model that gives an estimate of the underlying dynamics of dust in a spiral, which can serve as a proof of planetary origin of spirals and can be a probe for the Stokes number in the disk.
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