\right\rVert}_2^2 + \nu {\left\lVert  |\nabla| M_A \omega_S \right\rVert}_2^2 \notag\\
 &\qquad = \sqrt{\nu} {\left\lVert  |\nabla|^{1/2} M_A \omega_S \right\rVert}_2^2 + \sum_k \int_\eta  M_A(k,\eta)^2 \partial_\eta \hat{\omega}_S(k,\eta) \, k \, \overline{\hat{\omega}_S(k,\eta)}  d\eta \notag\\
 &\qquad \ \ + \sum_{k,l} \int_\eta \! \int_\xi M_A(k,\eta) \overline{\hat{\omega}_S(k,\eta)} M_A(l,\xi)  \hat{\omega}_S(l,\xi) M_A(k-l,\eta-\xi) \hat{\omega}_S(k-l,\eta-\xi)   \notag\\
 &\qquad \qquad \qquad \qquad \times  \frac{(l,\xi)^\perp \cdot (k,\eta)}{ {\left\vertl,\xi\right\vert}^2} \frac{M_A(k,\eta)}{M_A(l,\xi) M_A(k-l,\eta-\xi)} d\xi d\eta\notag\\
 &\qquad = \sqrt{\nu} {\left\lVert  |\nabla|^{1/2} M_A \omega_S \right\rVert}_2^2 + T_A + N_A
 \label{eq:step:a:1}\end{aligned}$$ where $T_A$ is the shear transport term and $N_A$ is the nonlinear term. Upon integrating by parts in $\eta$ and bounding $|\partial_\eta M_A(k,\eta)|$ we immediately obtain $$\begin{aligned}
|T_A| &\lesssim \sqrt{\nu} \; t {\left\lVert  |\nabla|^{1/2} M_A \omega^\nu_S \right\rVert}_2^2.
\label{eq:step:a:3}\end{aligned}$$ In order to bound the nonlinear term $N_A$ we use the triangle inequality $$|M_A(k,\eta)| \leq |M_A(l,\xi)| |M_A(k-l,\eta-\xi)|,$$ followed by , which implies $$\begin{aligned}
|N_A| 
& \leq \sum_{k,l} \int_\eta \! \int_\xi {\left\vertk,\eta\right\vert} M_A(k,\eta) |\hat{\omega}_S(k,\eta)| M_A(k-l,\eta-\xi) |\hat{\omega}_S(k-l,\eta-\xi)| \notag\\
&\qquad \qquad \times {\left\vertl,\xi\right\vert}^{\frac{\delta}{(2+\delta)}} M_A(l,\xi)  |\hat{\omega}_S(l,\xi)|   \frac{{\bf 1}_{|(l,\xi)|\geq 1}}{{\left\vertl,\xi\right\vert}^{1+\frac{\delta}{(2+\delta)}}}   d\xi d\eta \notag\\
&\quad + \left|  \sum_k \int_{\eta} \int_\xi  k\,  M_A(k,\eta)  \overline{\hat{\omega}_S(k,\eta)} M_A(k,\eta-\xi) \hat{\omega}_S(k,\eta-\xi)  M_A(0,\xi)  \hat{u}_S(0,\xi)  {\bf 1}_{|\xi|\leq 1}  d\xi d\eta \right| \notag\\
&\lesssim_\delta {\left\lVert |\nabla| M_A \omega_S \right\rVert}_2  {\left\lVert M_A \omega_S \right\rVert}_2 {\left\lVert |\nabla|^{\frac{\delta}{2+\delta}} M_A \omega_S \right\rVert}_{2} 
+ {\left\lVert |\nabla| M_A \omega^\nu_S \right\rVert}_2 {\left\lVert M_A \omega^\nu_S 