In this paper, a nonlinear multi-scale interaction model is used to examine how the planetary waves associated with eddy-driven blocking wave packets propagate through the troposphere in vertically varying weak baroclinic basic westerly winds (BWWs). Using this model, a new one-dimensional finite-amplitude local wave activity flux (WAF) is formulated, which consists of linear WAF related to linear group velocity and local eddy-induced WAF related to the modulus amplitude of blocking envelope amplitude and its zonal non-uniform phase. It is found that the local eddy-induced WAF strengthens the convergence of linear WAF and reduces its divergence to favor blocking during the blocking growth phase. But during the blocking decay phase, enhanced WAF convergence occurs in the blocking downstream region and in the upper troposphere when BWW is stronger in the upper troposphere than in the lower troposphere, which leads to enhanced upward-propagating tropospheric wave activity, though the linear WAF plays a major role. This is not seen for a zonally uniform eddy forcing.
A perturbed inverse scattering transform method is used to solve the blocking envelope amplitude equation. It is found that the finite amplitude WAF represents a modified group velocity related to the variations of blocking soliton amplitude and zonal wavenumber caused by local eddy forcing. Using this amplitude equation solution, it is revealed that under local eddy forcing the blocking wave packet is nearly non-dispersive during its growth phase, but strongly dispersive during the decay phase for vertically increased BWWs, leading to strong eastward and upward propagation of planetary waves in the downstream upper troposphere.