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

Abstract: 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 $ν\partial_1<sup>2$. Unlike the fully dissipative case, this anisotropic system admits $U<sup>{(0)}$ as an exact unforced steady state. On each horizontal Fourier mode $k$, the dissipation reduces to the scalar $-νk<sup>2$ and commutes with the advection, so that the linearized semigroup factors exactly into $e<sup>{-νk<sup>2t}$ times the inviscid Euler group. For $|k|&gt;1$, we prove two-sided bounds showing that the decay rate is exactly $νk<sup>2$, uniformly in the shear amplitude $a$; hence no enhanced dissipation occurs. This reflects a fundamental mismatch: the shear transfers enstrophy to high vertical frequencies, which horizontal dissipation does not detect. At the critical modes $|k|=1$, the inviscid group grows like $\sqrt{2at}$, producing a transient amplification of size $(a/eν)<sup>{1/2}$ before decay at the rate $ν$. The horizontally independent modes form an infinite-dimensional undamped kernel. For $0<|k|<1$, the viscous spectrum is an exact translate of the inviscid one, and the mode is linearly unstable if and only if $aΛ(k)&gt;νk<sup>2$. In particular, when $L&gt;2π$, the flow is linearly unstable for all sufficiently large $a$.