where Then, we obtain and Before estimating J$_{3}$ , we can directly verify J$_{1}$ = − 2 s Im ∫ Ω z∂$_{t}$ ( ∇ $_{y}$ϕ ) · ∇ $_{y}$zdxdydt + 2 s Im ∫ Ω z∂$_{t}$ ( ∇ $_{x}$ϕ ) · ∇ $_{x}$zdxdydt = − 2 s Im Ω ( − 2 γ $^{2}$βt ) zϕ ∇ $_{y}$ψ · ∇ $_{y}$zdxdydt +2 s Im ∫ Ω − 2 γ $^{2}$βt zϕ ∇ $_{x}$ψ · ∇ $_{x}$zdxdydt (3.22) J$_{2}$ = m ∑ i,j =1 Re ∫ Ω 4 sϕ$_{y}$$_{i}$$_{y}$$_{j}$ z$_{y}$$_{j}$ z$_{y}$$_{i}$dxdydt − Re m ∑ j =1 n ∑ i =1 ∫ Ω 4 sz$_{y}$$_{i}$z$_{x}$$_{j}$ ϕ$_{y}$$_{i}$$_{x}$$_{j}$ dxdydt = m i,j =1 Re Ω 4 sγϕ ( ψ$_{y}$$_{i}$$_{y}$$_{j}$ + γψ$_{y}$$_{i}$ψ$_{y}$$_{j}$ ) z$_{y}$$_{j}$ z$_{y}$$_{i}$dxdydt + ∫ Ω 4 sγ $^{2}$ϕ |∇ $_{y}$ψ · ∇ $_{y}$z | 2 dxdydt − m ∑ j =1 n ∑ i =1 ∫ Ω 4 sγ $^{2}$ϕ ( ∇ $_{y}$ψ · ∇ $_{y}$z ) ( ∇ $_{x}$ψ · ∇ $_{x}$z ) dxdydt. (3.23) Δ$_{y}$ ( ϕd$_{2}$ ( ψ )) = (Δ$_{y}$ ϕ ) d$_{2}$ ( ψ ) + 2 ∇ $_{y}$ϕ · ∇ y ( d$_{2}$ ( ψ )) + ϕ Δ$_{y}$ ( d$_{2}$ ( ψ )) = γϕ (Δ$_{y}$ ψ ) d$_{2}$ ( ψ ) + γ $^{2}$ϕ |∇ $_{y}$ψ | 2 d$_{2}$ψ + 2 γϕ ∇ $_{y}$ψ · ∇ $_{y}$( d$_{2}$ ( ψ )) + ϕ Δ$_{y}$ ( d$_{2}$ ( ψ )) , Δ$_{y}$ ( ϕd$_{1}$ ( ψ )) = (Δ$_{y}$ ϕ ) d$_{1}$ ( ψ ) + 2 ∇ $_{y}$ϕ · ∇ y ( d$_{1}$ ( ψ )) + ϕ Δ$_{y}$ ( d$_{1}$ ( ψ )) = γϕ (Δ$_{y}$ ψ ) d$_{1}$ ( ψ ) + γ $^{2}$ϕ |∇ $_{y}$ψ | 2 d$_{1}$ψ + 2 γϕ ∇ $_{y}$ψ · ∇ $_{y}$( d$_{1}$ ( ψ )) + ϕ Δ$_{y}$ ( d$_{1}$ ( ψ )) . d$_{1}$ ( ψ ) = Δ$_{y}$ ψ − Δ$_{x}$ ψ, d$_{2}$ ( ψ ) = |∇ $_{y}$ψ | 2 − |∇ $_{x}$ψ | 2 . 13