Wave breaking in the ABC model

Wave breaking will appear as a convergence of eddy momentum flux in the zonal momentum equations of the model. Write the wind variables as the sum of a zonal mean and a perturbation:
(1) u = u + δu,  v = v + δv,  w = w + δw,  ρ̃’ = ρ̃’ + δρ̃’,  b’ = b’ + δb’, 
(do not confuse primes with deltas - here primes are deviations from the reference and deltas are deviations from the zonal mean). Note that δu = 0 (this is true for all quantities). In all of the following, suppose that the zonal mean satisfies the following balances:
(2) C(∂ρ̃’)/(∂x) = fv and C(∂ρ̃’)/(∂z) = b’
(geostrophic and hydrostatic balance respectively).

Eddy momentum flux and the zonal wind

Substituting (1↑) into the zonal momentum equation and using (2↑) gives:
(∂(u + δu))/(∂t) + B⎡⎣(u + δu)(∂(u + δu))/(∂x) + (w + δw)(∂(u + δu))/(∂z)⎤⎦ + C(∂(ρ̃’ + δρ̃’))/(∂x) − f(v + δv)  =  0,        (∂(u + δu))/(∂t) + B⎡⎣(u + δu)(∂(u + δu))/(∂x) + (w + δw)(∂(u + δu))/(∂z)⎤⎦ + C(∂δρ̃’)/(∂x) − δv  =  0.
Now take the zonal mean and develop the result:
(∂(u + δu))/(∂t) + B⎡⎣(u + δu)(∂(u + δu))/(∂x) + (w + δw)(∂(u + δu))/(∂z)⎤⎦ + C(∂δρ̃’)/(∂x) − δv  =  0,        (∂u)/(∂t) + B⎡⎣δu(∂δu)/(∂x) + δw(∂δu)/(∂z) + w(∂u)/(∂z)⎤⎦  =  0.

Eddy momenum flux and the vertical wind

Substituting (1↑) into the vertical momentum equation and using (2↑) gives:
(∂(w + δw))/(∂t) + B⎡⎣(u + δu)(∂(w + δw))/(∂x) + (w + δw)(∂(w + δw))/(∂z)⎤⎦ + C(∂(ρ̃’ + δρ̃’))/(∂z) − b’ − δb’  =  0,        (∂(w + δw))/(∂t) + B⎡⎣(u + δu)(∂(w + δw))/(∂x) + (w + δw)(∂(w + δw))/(∂z)⎤⎦ + C(∂δρ̃’)/(∂z) − δb’  =  0.
Now take the zonal mean and develop the result:
(∂(w + δw))/(∂t) + B⎡⎣(u + δu)(∂(w + δw))/(∂x) + (w + δw)(∂(w + δw))/(∂z)⎤⎦ + C(∂δρ̃’)/(∂z) − δb’  =  0,        (∂w)/(∂t) + B⎡⎣δu(∂δw)/(∂x) + δw(∂δw)/(∂z) + w(∂w)/(∂z)⎤⎦  =  0.

Eddy momentum flux and buoyancy

Substituting (1↑) into the buoyancy equation:
(∂(b’ + δb’))/(∂t) + B⎡⎣(u + δu)(∂(b’ + δb’))/(∂x) + (w + δw)(∂(b’ + δb’))/(∂z)⎤⎦ + A2(w + δw) = 0.
Now take the zonal mean and develop the result:
(∂(b’ + δb’))/(∂t) + B⎡⎣(u + δu)(∂(b’ + δb’))/(∂x) + (w + δw)(∂(b’ + δb’))/(∂z)⎤⎦ + A2(w + δw)  =  0,        (∂b’)/(∂t) + B⎡⎣δu(∂δb’)/(∂x) + δw(∂δb’)/(∂z) + w(∂b’)/(∂z)⎤⎦ + A2w  =  0.

The continuity equation

Substituting (1↑) into the continuity equation gives:
(∂(ρ̃’ + δρ̃’))/(∂t) + B⎡⎣(∂(ρ̃’ + δρ̃’)(u + δu))/(∂x) + (∂(ρ̃’ + δρ̃’)(w + δw))/(∂z)⎤⎦ = 0.
Now take the zonal mean and develop the result:
(∂(ρ̃’ + δρ̃’))/(∂t) + B⎡⎣(∂(ρ̃’ + δρ̃’)(u + δu))/(∂x) + (∂(ρ̃’ + δρ̃’)(w + δw))/(∂z)⎤⎦  =  0,        (∂ρ̃’)/(∂t) + B⎡⎣(∂(δρ̃’δu))/(∂x) + (∂(δρ̃’δw))/(∂z) + (∂(ρ̃’w))/(∂z)⎤⎦  =  0.