Modal and Nonmodal Growths of Symmetric Perturbations in Unbounded Domain

Abstract

Abstract Modal and nonmodal growths of nonhydrostatic symmetric perturbations in an unbounded domain are examined in comparison with their hydrostatic counterparts. It is shown that the modal growth rate is a function of a single internal parameter s, the slope of the cross-band wave pattern. The maximum nonmodal growth of total perturbation energy norm is produced, also as a function of s, by an optimal combination of one geostrophic neutral mode and two paired nongeostrophic growing and decaying (or propagating) modes in the unstable (or stable) region. The hydrostatic approximation inflates the maximum modal growth rate significantly (or boundlessly) as the basic-state Richardson number Ri is small (or → 0) and inflates the maximum nonmodal growth rate significantly (or boundlessly) as |s| is large (or → ∞). Inside the unstable region, the maximum nonmodal growth scaled by the modal growth is a bounded increasing function of growth time τ but reduces to 1 at (Ri, s) = (¼, −2) where the three modes become orthogonal to each other. Outside the unstable region, the maximum nonmodal growth is a periodic function of τ and the maximum growth time τm is bounded between ¼ and ½ of the period of the paired propagating modes. The scaled maximum nonmodal growth reaches the global maximum at s = −Ri−1 ± Ri−1(1 − Ri)1/2 (the marginal-stability boundary) for any τ if Ri ≤ 1, or at s = −1 ± (1 − Ri−1)1/2 for τ = τm if Ri > 1. When the neutral mode is filtered, the nonmodal growth becomes nongeostrophic and smaller than its counterpart growth constructed by the three modes but still significantly larger than the modal growth in general. The scaled maximum nongeostrophic nonmodal growth reaches the global maximum at s = −Ri−1 ± Ri−1(1 − Ri)1/2 for any τ if Ri ≤ 1, or at s = −Ri−1/2 for τ = τm if Ri > 1. Normalized inner products between the modes are introduced to measure their nonorthogonality and interpret their constructed nonmodal growths physically.