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Energy-enstrophy stability of -plane Kolmogorov flow with drag
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Image of FIG. 1.
FIG. 1.

Disturbance energy from the numerical solution of Eq. (1) with and . The results are normalized by the laminar energy . The solid and dashed curves illustrate the transient growth in when the laminar solution is globally stable, but not monotonically globally stable. The dot-dashed curve illustrates monotonic global stability. The dotted curve shows the development of a linearly unstable perturbation; in this case the ratio equilibrates at around 0.35 as .

Image of FIG. 2.
FIG. 2.

This shows as a function of and ; dark areas are regions of large values. The values of in the titles are obtained by substituting and the peak value, , into Eqs. (18a) and (18b).

Image of FIG. 3.
FIG. 3.

Results from linear stability analysis of Eq. (1). For the inviscid case , the solid line is the neutral curve. The point is where the most unstable mode changes character. For , the dashed line is the neutral curve.

Image of FIG. 4.
FIG. 4.

Variation of (a) , defined in Eq. (16), and (b) wavenumber of the most unstable mode along the linear stability neutral curve. The dashed lines in (a) show the piecewise linear approximation (21a) and (21b).

Image of FIG. 5.
FIG. 5.

Eigenvalue of the gravest mode of Eq. (29), . The minimum is at and . The inset shows a cut along (solid line); decreases monotonically from at to as . The dashed line is the approximation (B8).

Image of FIG. 6.
FIG. 6.

The functional (26) evaluated at the trial function (31) with different . The inset shows the trial function with .

Image of FIG. 7.
FIG. 7.

A summary showing three different stability boundaries for two-dimensional Kolmogorov flow. The dashed curve is the neutral curve of the linear stability documented in Sec. III; the dotted curve is the energy-stability boundary in Eq. (32); the solid curve is the stability boundary in Eq. (40). The point at is where the most unstable linear mode changes from a stationary disturbance to a traveling wave.

Image of FIG. 8.
FIG. 8.

This shows energy-stability boundaries in the parameter plane. The flow is monotonically globally stable if the parameters locate it above the appropriate curve.

Image of FIG. 9.
FIG. 9.

This shows the dependence of the wavenumber of the most dangerous disturbance on . (The most dangerous disturbance has for all .) If then . As decreases, increases and ultimately, as shown in the inset, .

Image of FIG. 10.
FIG. 10.

Trajectories of some illustrative solutions of Eq. (6) in the -plane; all solutions have and . The dashed line marks the set in Eq. (35).

Image of FIG. 11.
FIG. 11.

The most dangerous -disturbance (47) calculated using the final row of Table I.


Generic image for table
Table I.

The estimate of in the final column quickly converges to the numerical value indicated in Eq. (38) as more terms in the Fourier series (44) are retained.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Energy-enstrophy stability of β-plane Kolmogorov flow with drag