  We study instability of unidirectional flows for the linearized 2D
Navier-Stokes equations on the torus. Unidirectional flows are steady states
whose vorticity is given by Fourier modes corresponding to a single vector
$\mathbf p \in \mathbb Z^{2}$. Using Fourier series and a geometric
decomposition allows us to decompose the linearized operator $L_{B}$ acting on
the space $\ell^{2}(\mathbb Z^{2})$ about this steady state as a direct sum of
linear operators $L_{B,\mathbf q}$ acting on $\ell^{2}(\mathbb Z)$ parametrized
by some vectors $\mathbf q\in\mathbb Z^2$. Using the method of continued
fractions we prove that the linearized operator $L_{B,\mathbf q}$ about this
steady state has an eigenvalue with positive real part thereby implying
exponential instability of the linearized equations about this steady state. We
further obtain a characterization of unstable eigenvalues of $L_{B,\mathbf q}$
in terms of the zeros of a perturbation determinant (Fredholm determinant)
associated with a trace class operator $K_{\lambda}$. We also extend our main
instability result to cover regularized variants (involving a parameter
$\alpha>0$) of the Navier-Stokes equations, namely the second grade fluid
model, the Navier-Stokes-$\alpha$ and the Navier-Stokes-Voigt models.
