The goal of the international ITER burning plasma experiment is to produce 500MW of fusion power for 50MW of heating power. A baseline scenario with plasma current 15 MA is being considered as the highest priority for achieving this goal of fusion gain Q=10. The burning plasma experiments to be attempted in ITER will differ from previous experiments in that non-thermal fusion alpha-particles must be highly confined, both to ensure machine safety, and to attain conditions required for new burning plasma physics studies. These 3.5MeV non- Maxwellian fusion alpha-particles have characteristic transit speeds that are super-Alfvénic, thereby allowing resonant destabilisation of Alfven normal modes. The toroidal Alfvén eigenmodes (TAEs) have been shown to be particularly dangerous, with up to 70% of neutral beam power being lost to machine components in some experiments due to modes of this type. For assessing the TAE-induced re-distribution of alpha-particles in the ITER Baseline scenario, we have conducted the most comprehensive nonlinear study so far of all relevant TAEs in this scenario, computing their coupled growth and final amplitudes with realistic calculations of the thermal Landau and radiative damping.