Resonant interactions between a marginally unstable wave and one or two pairs of slightly damped waves in a quasi-geostrophic two-layer flow on a beta plane are investigated. It is found that a system of waves that is stable under strictly inviscid conditions is destabilized by slight viscosity provided the initial energy of the participating waves exceeds a certain threshold value. In the linear phase viscosity can trigger parametric instability. When weakly nonlinear interactions are considered viscosity can lead to explosive instability, whereby the amplitudes of the waves grow without bound at a finite time. The unbounded growth is not always halted by increasing the number of participating waves or by considering higher-order nonlinear effects. The phenomenon of explosive instability whose significance lies in the rapid growth of the amplitudes of the waves on the approach to the singularity may explain the rapid growth of certain events of cyclogenesis recently observed.