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A streamwise-independent cross-stream disturbance in an inviscid parallel shear flow has time-independent wall-normal velocity. It advects the base streamwise momentum, producing the displayed linear growth of streamwise velocity and quadratic growth of perturbation kinetic energy. This is algebraic amplification, not an exponentially unstable normal mode; it exemplifies transient growth from non-normal modes.

Under the convention $e^{i(kx-\omega t)}$, a mode's amplitude grows as $e^{{\omega }_{i}t}$. The linear stability analysis of real-wavenumber disturbances is therefore summarized byAt ${\omega }_{i,\max }=0$ there is marginal or neutral hydrodynamic stability: no normal mode grows exponentially, but some may fail to decay. If “linearly stable” includes neutral modes, its condition is ${\omega }_{i,\max }\le 0$. These modal statements do not rule out transient growth from non-normal modes or algebraic growth.

Every generalized frequency in the inviscid Couette continuous spectrum is real: ${\omega }_{\xi }=\alpha \xi$ for real $\alpha ,\xi$. Thus every time factor has unit modulus and there is no exponentially growing normal mode:The vorticity equation also gives $q(z,t)=q_0(z)e^{-i\alpha zt}$, preserving its $L^2$ norm. For fixed nonzero $\alpha$, the bounded Dirichlet Green function inversion therefore gives a uniform bound on the reconstructed velocity in terms of the initial vorticity norm. Superpositions can still rearrange their velocity energy and display transient growth from non-normal modes; neutral hydrodynamic stability does not require every velocity component to decrease monotonically. Treating only smooth discrete modes would miss this continuous neutral family.

Both eigenvalues of the linear generator, $-1/R$ and $-1/(4R)$, are negative. Nevertheless the non-normal matrix transfers the slowly decaying second component into the first before viscous decay wins. Modal decay coexists with energy amplification of order $R^2$ over a time of order $R$.

This illustrates transient growth from non-normal modes: large Reynolds number permits much greater disturbance amplification even though the linear stability analysis finds no unstable eigenvalue. The energy-neutral rotational nonlinearity can redirect the amplified state and affect its subsequent evolution. The linear gain alone establishes neither sustained nonlinear instability nor a nonlinear transition threshold.

An Orr-Sommerfeld mode has $\hat{v}\ne 0$ satisfying the Orr-Sommerfeld equation, with accompanying vorticity satisfying the forced Squire equation. A Squire mode has $\hat{v}=0$ and $\hat{\eta }\ne 0$, so its Squire equation is homogeneous.

For a Squire mode multiply that homogeneous equation by $\overline{\eta }$ and integrate over the channel. Under the no-slip condition $\eta =0$ at the walls, integration by parts givesSince $U,\alpha$ are real, taking the real part provesThe numerator is strictly positive for a nontrivial finite-channel mode, and $Re>0$. Thus Squire modes are exponentially damped in the convention $e^{-i\omega t}$. This does not rule out transient growth from non-normal modes in the coupled system.

For $q(t)=B{q}_0$, the energy ratio is the Rayleigh quotient of the Hermitian positive-definite matrix $B^{†}B$:Expanding $q_0$ in the orthonormal right singular vectors makes this a weighted average of ${\sigma }_j^2$. HenceAny nonzero vector in the largest right-singular subspace is optimal if ${\sigma }_1$ is repeated. Its amplified state is proportional to the paired left singular vector. For a non-normal matrix $L$, this optimal initial direction need not be an eigenvector of $L$, explaining transient growth from non-normal modes.

The energy isThusThis quadratic form can be negative for positive $c_1,c_2$ precisely whenThe decaying vectors must therefore be sufficiently nonorthogonal and oppositely directed. This is transient growth from non-normal modes.

With $c_2=1$ and $\mu =c_1/c_2$, maximizing $E^{\prime }(0)$ givesso

A Squire mode has zero wall-normal velocity and nonzero wall-normal vorticity, satisfying the homogeneous Squire equation. For real base velocity, positive finite Reynolds number, and zero boundary terms,for a nontrivial mode in a finite channel with no-slip walls. This follows by multiplying the homogeneous equation by $\overline{\eta }$, integrating by parts and taking the real part. Such modes are damped even when the coupled velocity-vorticity system can exhibit transient growth from non-normal modes.