Unbalanced instabilities of rapidly rotating stratified shear flows. (English) Zbl 1165.76334

Summary: The linear stability of a rotating stratified inviscid horizontal plane Couette flow in a channel is studied in the limit of strong rotation and stratification. Two dimensionless parameters characterize the flow: the Rossby number \(\epsilon\), defined as the ratio of the shear to the Coriolis frequency and assumed small, and the ratio \(s\) of the Coriolis frequency to the buoyancy frequency, assumed to satisfy \(s \leq 1\). An energy argument is used to show that unstable perturbations must have large, \(O(\epsilon^{-1})\), wavenumbers. This motivates the use of a WKB-approach which, in the first instance, provides an approximation for the dispersion relation of the various waves that can propagate in the flow. These are Kelvin waves, trapped near the channel walls, and inertia-gravity waves with or without turning points.
Although the waves have real phase speeds to all algebraic orders in \(\epsilon\), we establish that the flow is unconditionally unstable. This is the result of linear resonances between waves with oppositely signed wave momenta. Three modes of instabilities are identified, corresponding to the resonance between (i) a pair of Kelvin waves, (ii) a Kelvin wave and an inertia-gravity wave, and (iii) a pair of inertia-gravity waves. Whilst all three modes of instability are active when the Couette flow is anticyclonic, mode (iii) is the only possible instability mechanism when the flow is cyclonic.
We derive asymptotic estimates for the instability growth rates. These are exponentially small in \(\epsilon\), i.e. of the form Im \(\omega = a\exp(-\Psi /\epsilon)\) for some positive constants \(a\) and \(\Psi\). For the Kelvin-wave instabilities (i), we obtain analytic expressions for \(a\) and \(\Psi\); the maximum growth rate, in particular, corresponds to \(\Psi = 2\). For the other types of instabilities, we make the simplifying assumption \(s \ll 1\) and find that the maximum growth rates correspond to \(\Psi=2.80\) for (ii) and \(\Psi= \sqcap\) for (iii). The asymptotic results are confirmed by numerical computations. These reveal, in particular, that the instabilities (iii) have much smaller growth rates in cyclonic flows than in anticyclonic flows, even though \(\Psi = \sqcap\) in both cases.
Our results highlight the limitations of the so-called balanced models, widely used in geophysical fluid dynamics, which filter out Kelvin and inertia-gravity waves and hence predict the stability of Couette flow. They are also relevant to the stability of Taylor-Couette flows and of astrophysical accretion disks.


76E07 Rotation in hydrodynamic stability
76U05 General theory of rotating fluids
76B70 Stratification effects in inviscid fluids
Full Text: DOI arXiv


[1] DOI: 10.1017/S0022112083000270 · Zbl 0521.76041
[2] DOI: 10.1007/BF00876229
[3] DOI: 10.1175/JAS3426.1
[4] Papaloizou, Mon. Not. R. Astr. Soc. 225 pp 267– (1987) · Zbl 0622.76122
[5] Narayan, Mon. Not. R. Astr. Soc. 228 pp 1– (1987) · Zbl 0646.76121
[6] DOI: 10.1063/1.527544 · Zbl 0628.58012
[7] DOI: 10.1175/JPO2770.1
[8] DOI: 10.1017/S0022112094002946 · Zbl 0822.76037
[9] DOI: 10.1103/PhysRevLett.86.5270
[10] DOI: 10.1051/0004-6361:200400065
[11] DOI: 10.1063/1.1785132 · Zbl 1187.76340
[12] DOI: 10.1017/S0022112006000644 · Zbl 1098.76031
[13] DOI: 10.1175/1520-0485(1998)0282.0.CO;2
[14] DOI: 10.1017/S0022112092003215 · Zbl 0760.76029
[15] DOI: 10.1017/S0022112079000495 · Zbl 0398.76033
[16] DOI: 10.1017/S0022112099004279 · Zbl 0938.76026
[17] DOI: 10.1175/1520-0485(1990)0202.0.CO;2
[18] Yavneh, J. Fluid Mech. 448 pp 1– (2001)
[19] DOI: 10.1002/qj.49712152313
[20] Warn, Atmos. Ocean 35 pp 135– (1997)
[21] DOI: 10.1175/1520-0469(2004)0612.0.CO;2
[22] Satomura, J. Met. Soc. Japan 60 pp 227– (1982)
[23] Satomura, J. Met. Soc. Japan 59 pp 168– (1981)
[24] Satomura, J. Met. Soc. Japan 59 pp 148– (1981)
[25] DOI: 10.1017/S0022112089001138 · Zbl 0669.76066
[26] DOI: 10.1051/0004-6361:20020853 · Zbl 1062.85517
[27] DOI: 10.1017/S0022112097006216 · Zbl 0911.76023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.