Brian F. Farrell and Petros J. Ioannou (1996), Journal of the Atmospheric Sciences 53(14), 2025–2040 — doi:10.1175/1520-0469(1996)053<2025:gstpia>2.0.co;2
Key takeaways
From the reading, NOTE-831.
- Eigenvalues govern only the long run. If A is normal, ‖e^{At}‖ = e^{λ^R_max t} and the spectrum decides stability for all time. If not, growth over time t is ‖e^{At}‖, the top singular value of the propagator. The eigenvalues bound it from below, not above (eq. 6).
- Two limits and the interval between. As t → 0 the growth rate is the numerical abscissa, the top eigenvalue of (A + A†)/2, which can be positive when every mode decays. As t → ∞ the leading mode wins, but its best excitation is its adjoint vector, not the mode. The physically relevant times lie between, where the SVD of the propagator gives the optimal perturbations.
- Large growth with every mode damped. A damped Eady model, stable at the chosen wavenumber, grows about a hundredfold in energy over four days. A channel with only decaying modes grows its two-day optimal 112-fold.
- Forced and stochastic responses are larger. Resonance is bounded by κ(E)/dist(iω, Λ(A)), not 1/dist. Under white noise the variance exceeds the sum over the poles, with equality only for normal A (eq. 22, cited to Ioannou 1995). In the 2 × 2 example, ⟨‖u‖²⟩ = 3/4 + cot²θ/12, of which 3/4 is the normal response.
- Norm-relative. Non-normality depends on the inner product, and some (usually unphysical) norm always makes the operator normal. The paper uses the energy norm.
Standing in the record
Read on 2026-10-14 in NOTE-831. It is the compact statement of non-modal stability for a fixed operator, and THEORY-323 is filed from it with Trefethen et al. (LIT-932): the eigenvalues of a non-normal operator govern only the long-run limit, and finite-time growth and forced variance can exceed what they predict by large factors.
It bears on CLAIM-186, whose note “settling does not rule out bursts”
states its central result at the right strength. Two things the reading
adds to that note. Under continual small disturbance a non-normal system
holds more standing variance than its spectrum predicts, so the effect
is not only bursts. And non-normality is relative to a norm, so a
reading across to an argument has to say what its measure of drift is.
The note’s further step, that directed coupling such as rests_on is a
standard source of non-normality, is not in this paper. Its source is
background shear; its 2 × 2 example is a one-way coupling and fits the
reading, but states no rule.
Part II (LIT-929) extends the theory to operators that change in time, where the conclusion changes: transient growth can become asymptotic instability (THEORY-325). Not held in the Anthology of the SOTA; it carries no instruction for machine-learning practice.