The purpose of this Appendix is to prove claim (16.120) from section 16.3.

We need to postulate the following assumption:

Assumptions: d > 0 and P ₂ ? ₀ is uniformly bounded in time.

Remark: The first condition is equivalent to saying that there is real dissipation in the system. The second condition is satisfied provided that ₀ is uniformly bounded in time.

In order to prove the smallness of we have to deal with the interaction term between different surface spherical harmonics. This is not a trivial task. We recall from Jones (1985, page 180), formula (46), that the Euler s equation on the unit sphere can be written in the form in terms of the coefficients of the surface spherical harmonics

where (see Jones, 1985, page 175, equation (30))

? = ? ? is the vorticity ^{[ ]}, and there are restrictions on the summation indices which must satisfy (see Jones, 1985 page 193)

Since the non-linear term in Euler s equation is J( ?, ? ?), we must have

We may then identify the contribution from different sources

1. Contribution from the interaction between the first shell and the second shell. This must correspond to

   
   

2. Contribution from the interaction between the second shell and higher modes

   
   

3. Contribution from the interaction between higher modes ( ? 3),

   
   

4. Contribution from the interaction of the ground energy shell and higher modes (