Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation

We study the linearized dynamics of the Kolmogorov flow $U^{(0)}=(a\sin x_2,0)$ for the two-dimensional incompressible Navier--Stokes equations with horizontal dissipation $ν\partial_1^2$. Unlike the fully dissipative case, this anisotropic system admits $U^{(0)}$ as an exact unforced steady state.