Abstract: This study gives a closed-form expression for the aerodynamic damping matrix, incorporating the multi-mode aeroelastic coupling of blades and tower, to enable rapid and accurate quantification of aerodynamic damping ratios for bottom-fixed wind turbines. A 14-degree-of-freedom nonlinear aero-elastic model of the wind turbine system integrating tower multi-mode dynamics is developed using multibody dynamics and the modal-based formulation. The aerodynamic damping matrix of the system is analytically derived through the linearization of aerodynamic loads, facilitating the evaluation of multi-mode aeroelastic coupling and aerodynamic damping. The findings demonstrate that the aerodynamic damping matrix is accurate when tower top rotation, multi-mode coupling and blade flexibility are accounted for. Analysis of a 5 MW baseline wind turbine reveals that the aerodynamic damping ratio of the blade flap-wise modes can reach 86%, markedly exceeding the maximum value of 2.3% of the edgewise modes. The modal aerodynamic damping ratios of the tower in the fore-aft direction are significantly higher than those in the side-side direction, where the latter exhibits a peak value of only 0.4%. The third side-side modal aerodynamic damping ratios of the tower remain negative across all operational points, suggesting a risk of aeroelastic instability.