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We have developed the dynamical model of a clumpy torus in an active galacticnucleus (AGN) and compared to recent ALMA observations. We present $N$-bodysimulations of a torus in the field of a supermassive black hole (SMBH), madeof up to $N=10^5$ gravitationally interacting clouds. As initial conditions, wechoose random distributions of the orbital elements of the clouds with acut-off in the inclination to mimic the presence of wind cones produced at theearly AGN stage. When the torus reaches an equilibrium, it has a doughnutshape. We discuss the presence of box orbits. We have then constructed thevelocity and velocity dispersion maps using the resulting distributions of theclouds at equilibrium. The effects of torus inclination and cloud sizes areduly analyzed. We discuss the obscuration effects of the clouds using a raytracing simulation matching the model maps to ALMA resolution. By comparing themodel with the observational maps of NGC 1068 we find that the SMBH mass is$M_\text{smbh}=5\times 10^6 M_\odot$ for the range of the torus inclinationangles $45^\circ - 60^\circ$. We also construct the velocity dispersion mapsfor NGC 1326 and NGC 1672. They show that the peaks in the ALMA dispersion mapsare related to the emission of the torus throat. Finally, we obtain thetemperature distribution maps with parameters that correspond to our modelvelocity maps for NGC 1068. They show stratification in temperaturedistribution with the shape of the high temperature region as in the VLTI/MIDImap.

Dynamical model of an obscuring clumpy torus in AGNs: I. Velocity and velocity dispersion maps for interpretation of ALMA observations