^{1}and William R. Young

^{1}

### Abstract

We develop a nonlinear stability method, the energy-enstrophy method, that is specialized to two-dimensional hydrodynamics and basic state flows consisting of a single Helmholtz eigenmode. The method is applied to a -plane flow driven by a sinusoidal body force and retarded by drag with damping time scale . The standard energy method [H. Fukuta and Y. Murakami, J. Phys. Soc. Jpn.64, 3725 (1995)] shows that the laminar solution is monotonically and globally stable in a certain portion of the -parameter space. The method proves nonlinear stability in a larger portion of the -parameter space than does the energy method. Moreover, by penalizing high wavenumbers, the method identifies a most strongly amplifying disturbance that is more physically realistic than that delivered by the energy method. Linear instability calculations are used to determine the region of the -parameter space where the flow is unstable to infinitesimal perturbations. There is only a small gap between the linearly unstable region and the nonlinearly stable region, and full numerical solutions show only small transient amplification in that gap.

This work was supported by the National Science Foundation by Grant Nos. OCE07-26320 and OCE02-20362.

I. INTRODUCTION

II. FORMULATION OF THE STABILITY PROBLEM

A. Dynamics of the disturbance

B. Linear instability, global stability, and monotonic global stability

III. LINEAR INSTABILITY

A. Tracing the neutral surface

B. Gill’s inequality

C. The inviscid case,

D. The effects of small viscosity

E. A comment on the viscously controlled limit

IV. THE ENERGY METHOD

A. The inviscid case,

B. The viscous case

V. THE ENERGY-ENSTROPHY METHOD

A. The method,

B. The variational problem

C. Limitations of the method

VI. CONCLUSION AND DISCUSSION

### Key Topics

- Flow instabilities
- 17.0
- Viscosity
- 13.0
- Inequalities
- 9.0
- Vortex stability
- 7.0
- Eigenvalues
- 6.0

## Figures

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 .

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 .

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).

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).

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.

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.

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).

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).

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).

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).

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

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

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.

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.

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

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

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, .

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, .

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

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

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

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

## Tables

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.

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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