Transient growth

Transient growth is finite-time amplification in a stable linear system. For a non-normal generator it can occur even when every eigenvalue has negative real part, because nonorthogonal eigenmodes can interfere constructively.

For $M=\left(\begin{array}{cc}-1/R& 0\\ 1& -2/R\end{array}\right)$ with $R>0$, write $s=t/R$. Its matrix exponential is $B=\left(\begin{array}{cc}a& 0\\ R(a-a^2)& a^2\end{array}\right)$, $a=e^{-s}$. The maximizing unit initial state is the largest-eigenvalue eigenvector of $B^{\ast }B$. Its component ratio is the displayed expression, with $d=(e^s+1)/(2R)+(R/2)(e^s-1)$. This follows by solving the quadratic eigenvector equation for the symmetric two-by-two Gram matrix. The state tends to $(1,0{)}^T$ for $t\gg R$.

In the optimal initial state for triangular stable shear, expansion of $e^{t/R}$ gives the displayed leading parameter with relative error $O(t/R)$. Keeping both terms distinguishes a large-$R$ shear regime from the limit $t\to 0$ at fixed $R$. The latter direction is the largest-eigenvalue direction of the Hermitian part of $M$; the former can instead have large absolute time and substantial shear amplification even though $t/R$ is small.

For $x^{\prime }=Lx$ and $E=\Vert x{\Vert }_2^2$, the Hermitian matrix $H=(L+L^{\ast })/2$ gives $\dot{E}=2x^{\ast }Hx$. Thus some initial condition grows immediately iff the largest eigenvalue of $H$ is positive. All trajectories have nonincreasing energy iff $H$ is a negative semidefinite matrix. This differs from requiring negative real parts for the eigenvalues of $L$: a non-normal matrix can satisfy modal decay while allowing transient growth.

For the matrix exponential $A(t)=e^{Lt}$, maximizing the energy ratio over nonzero initial conditions givesThe Rayleigh quotient achieves its maximum at an eigenvector of $A^{\ast }A$ with largest eigenvalue, equivalently a right singular vector of $A$. Such an optimal disturbance is generally not an eigenvector of $L$.

For $L=\left(\begin{array}{cc}{\lambda }_1& 0\\ 1& {\lambda }_2\end{array}\right)$ with distinct real negative eigenvalues, the matrix exponential is $A=\left(\begin{array}{cc}a& 0\\ b& d\end{array}\right)$, where $a=e^{{\lambda }_{1}t}$, $d=e^{{\lambda }_{2}t}$ and $b=(a-d)/({\lambda }_1-{\lambda }_2)$. The optimal energy amplification of a linear system isIts eigenvectors are nonorthogonal, and immediate energy growth occurs exactly when ${\lambda }_1{\lambda }_2<1/4$. If ${\lambda }_2>{\lambda }_1$, putting $\alpha =({\lambda }_2-{\lambda }_1{)}^{-1}$ gives $G(t)\sim (1+{\alpha }^2)e^{2{\lambda }_{2}t}$. The optimal asymptotic initial direction $(\alpha ,1)$ is an adjoint eigenvector, while the eventual state aligns with the right eigenvector $(0,1)$.

For $L=\left(\begin{array}{cc}-1/R& 1\\ 0& -1/(4R)\end{array}\right)$ with $R>1$, put $s=\sqrt{1-R^{-2}}$ and $q=(5-3s)/(5+3s)$. The optimal energy amplification of a linear system reaches its maximum atThe maximizing initial orientation is ${\rho }^{+}$ and the final orientation is ${\rho }^{-}$ from the orientation interval for transient energy growth. Taking $R\to \infty$ gives the displayed viscous-time and Reynolds-squared asymptotics. These constants belong to this specified triangular model, not to every non-normal matrix.

For $L=\left(\begin{array}{cc}-1/R& 1\\ 0& -1/(4R)\end{array}\right)$ and $R>1$, write $\rho =x_2/x_1$. The energy derivative is $x_1^2[-1/R+\rho -{\rho }^2/(4R)]$, positive exactly between the displayed orientations. Linear trajectories cross the sector from ${\rho }^{+}$ to ${\rho }^{-}$ because $\dot{\rho }=\rho [3/(4R)-\rho ]$. An energy-neutral rotational nonlinearity can reverse the crossing direction; the growth sector itself remains the same.

A nonlinear term $f(|x|)Nx$ with real skew-symmetric matrix $N$ contributes zero to the derivative of the quadratic energy $|x{|}^2/2$. It changes orientation rather than magnitude directly. If a separate linear generator is a non-normal matrix, that reorientation can still alter later transient growth, so energy cancellation alone does not make the nonlinear evolution equivalent to the linear one.