In (3.19), we used the equality Re z ∇ $_{x}$z = 1 2 ∇ x ( | z | 2 ) . In (3.20), we used the equality Re z ∇ $_{y}$z = 1 2 ( ∇ y | z | 2 ) and Re z ∇ $_{x}$z = 1 2 ( ∇ x | z | 2 ) . Hence, we can rewrite (3.13) where I$_{7}$ = − 4 s $^{3}$Re ∫ Ω ( |∇ $_{y}$ϕ | 2 − |∇ $_{x}$ϕ | 2 ) z ( ∇ $_{y}$ϕ · ∇ $_{y}$z − ∇ $_{x}$ϕ · ∇ $_{x}$z ) dxdydt = 2 s 3 ∫ Ω | z | 2 |∇ $_{y}$ϕ | 2 − |∇ $_{x}$ϕ | 2 (Δ$_{y}$ ϕ − Δ$_{x}$ ϕ ) dxdydt − 2 s 3 ∫ Γ$_{y}$ ( ∂$_{ν}$ϕ ) | z | 2 ( |∇ $_{y}$ϕ | 2 − |∇ $_{x}$ϕ | 2 ) dS$_{y}$dxdt +2 s 3 Ω | z | 2 ∇ $_{y}$ϕ · ∇ y ( |∇ $_{y}$ϕ | 2 − |∇ $_{x}$ϕ | 2 ) dxdydt +2 s 3 ∫ Γ$_{x}$ ∂$_{ν}$ϕ | z | 2 ( |∇ $_{y}$ϕ | 2 − |∇ $_{x}$ϕ | 2 ) dS$_{x}$dydt − 2 s 3 Ω | z | 2 ∇ $_{x}$ϕ · ∇ x ( |∇ $_{y}$ϕ | 2 − |∇ $_{x}$ϕ | 2 ) dxdydt. (3.20) I$_{8}$ = − 2 s $^{3}$Re ∫ Ω ( |∇ $_{y}$ϕ | 2 − |∇ $_{x}$ϕ | 2 ) (Δ$_{y}$ ϕ − Δ$_{x}$ ϕ ) zzdxdydt = − 2 s 3 ∫ Ω ( |∇ $_{y}$ϕ | 2 − |∇ $_{x}$ϕ | 2 ) (Δ$_{y}$ ϕ − Δ$_{x}$ ϕ ) | z | 2 dxdydt. (3.21) 2 Re ( P + s z, P − s z ) L $^{2}$(Ω) = J$_{1}$ + J$_{2}$ + J$_{3}$ + J$_{4}$ + J$_{5}$ + J$_{6}$ + B$_{0}$, J$_{1}$ = − 2 s Im ∫ Ω z∂$_{t}$ ( ∇ $_{y}$ϕ ) · ∇ $_{y}$zdxdydt + 2 s Im ∫ Ω z∂$_{t}$ ( ∇ $_{x}$ϕ ) · ∇ $_{x}$zdxdydt, J$_{2}$ = 4 s Re n ∑ i,j =1 ∫ Ω ϕ$_{y}$$_{i}$$_{y}$$_{j}$ z$_{y}$$_{j}$z$_{y}$$_{i}$dxdydt − 4 s Re m ∑ j =1 n ∑ i =1 ∫ Ω ϕ$_{y}$$_{i}$$_{x}$$_{j}$ z$_{y}$$_{i}$z$_{xj}$dxdydt, J$_{3}$ = − s ∫ Ω | z | $^{2}$Δ$_{y}$ (Δ$_{y}$ ϕ − Δ$_{x}$ ϕ ) dxdydt, J$_{4}$ = − 4 s Re m ∑ j =1 n ∑ i =1 ∫ Ω ϕ$_{y}$$_{i}$$_{x}$$_{j}$ z$_{y}$$_{i}$z$_{x}$$_{j}$ dxdydt + 4 s Re n ∑ i,j =1 ∫ Ω ϕ$_{x}$$_{i}$$_{x}$$_{j}$z$_{x}$$_{i}$z$_{x}$$_{j}$dxdydt, 11