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Appendix A: Background and first-order contributions to the momentum influx

Here we evaluate Ṗ₂ and Ṗ₁, the first two coefficients on the right-hand side of Eq. (55). We will show that both coefficients are zero. The first is (A.1)Substituting f₂ = β   sin   θ/2 , this becomes The background flow, while it certainly carries z-momentum, makes no net contribution to the influx.

The integral expression for the second coefficient is lengthier, and we write it as (A.2)where and It is convenient to split up the integral in Eq. (A.2) by region. For the downstream part, which we denote as , we use Eqs. (15) and (16) for f₁ and g_-1. We find, after algebraic simplification, that (A.3)We now use the identity

to conclude that (A.4)Similarly, the upstream part is (A.5)so that (A.6)For the intermediate piece, to be denoted , we use Eqs. (21) and (22) for f₁ and g_-1. Although we have established that C =  +1 , we will see that is independent of this parameter, so we retain the original notation. After algebraic manipulation, we find (A.7)Using the identity we infer that (A.8)In summary, we have (A.9)

© ESO, 2014