(a) Marginal stability (σ = 0) contour in dimensionless wavenumber–time space in an unconfined domain; (b) zoomed-in view around the critical time and critical wavenumber . The base state is unstable in the shaded region above the contour and is stable under it. The dashed line represents the large wavenumber cutoff of . Perturbation modes with wavenumbers larger than this cutoff do not become unstable.
(a) and (b) vs. or ; (c) and (d) vs. or . The aspect ratio A is unity in (a) and (c) so that both and range from 0 to . The dashed line represents or . The behavior of the system is very different depending on whether it lies to the left or the right of the dashed line, as we describe in the text.
vs. for various values of the aspect ratio A: (a) A = 10; (b) A = 5; (c) A = 2; (d) A = 0.5; (e) A = 0.2; (f) A = 0.1. The dashed line represents . Large aspect ratios can significantly reduce boundary effects for both small and large values of .
vs. in Cartesian geometries at two different aspect ratios: (a) A = 1; (b) A = 0.1. The dashed line represents . Finer gridding in numerical simulations may be necessary to capture the earliest-growing mode in domains where is small, especially when A < 1.
The indices i and j of the wavenumber of the earliest-growing mode as a function of the width and the aspect ratio A: (a) the index i in the direction; (b) the index j in the direction. The size in this direction ranges from 0 to ; (c) the index j plotted using the same colorbar axis as in (a). The indices generally increase with .
The indices l and m of the wavenumber of the earliest-growing mode as a function of the diameter : (a) l; (b) m. The earliest-growing mode is usually non-axisymmetric (m ≠ 0).
Patterns in the plane of the earliest-growing mode for various with A = 1: (a) ; (b) ; (c) ; (d) ; (e) ; (f) . Perturbations tend to be highly localized in very positive (red) or very negative (blue) regions. In most cases, there are channels that connect positive regions to each other and other channels that connect negative regions to each other. Pau et al. 50 have observed similar channels in their numerical simulations.
Patterns in the plane of the earliest-growing mode for at two different aspect ratios: (a) A = 2; (b) A = 0.5. Patterns are asymmetric with respect to an interchange of and . Two-dimensional modes tend to be preferred when one of the dimensions is small, as it is in (b).
Top-down and bird's-eye views of patterns in the plane of the earliest-growing mode for smaller values of with : (a), (b) ; (c), (d) ; (e), (f) .
Top-down and bird's-eye views of patterns in the plane (for clarity, only one quarter of the plane is shown) of the earliest-growing mode for larger values of with : (a), (b) ; (c), (d) ; (e), (f) .
The indices i and j of for various widths with A = 1.
The indices l, m of and the zeros b lm for various diameters .
Article metrics loading...
Full text loading...