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Review Articles

Instability in Stratified Shear Flow: Review of a Physical Interpretation Based on Interacting Waves

[+] Author and Article Information
Jeffrey R. Carpenter

Eawag, Swiss Federal Institute of
Aquatic Science and Technology,
Surface Waters – Research and Management,
Kastanienbaum 6047, Switzerland;
Department of Geology and Geophysics,
Yale University,
New Haven, CT 06520-8109
e-mail: jeffcarp@gmail.com

Edmund W. Tedford

Marine Science Institute,
University of California, Santa Barbara,
Santa Barbara, CA 93106-9620
e-mail: ttedford@eos.ubc.ca

Eyal Heifetz

Department of Geophysical,
Atmospheric and Planetary Sciences,
Tel Aviv University,
Tel Aviv 69978, Israel;
Department of Meteorology (MISU),
Stockholm University,
Stockholm SE-106 91, Sweden
e-mail: eyalh@cyclone.tau.ac.il

Gregory A. Lawrence

Department of Civil Engineering,
University of British Columbia,
Vancouver, BC, V6T 1Z4, Canada
e-mail: lawrence@civil.ubc.ca

Here we are neglecting a part of the solution referred to as the continuous spectrum, which is not essential for the purposes of this paper. For more information the reader is referred to Schmid and Henningson [4].

The Rayleigh and Fjørtoft theorems can also be interpreted in terms of the conservation of pseudomomentum and pseudoenergy, respectively, and the interested reader is referred to Heifetz et al. [11] for further details.

Although the work of Garcia [23] was published 6 years previous to Holmboe's [14] paper, Garcia directly acknowledges Holmboe with the wave interaction concept in an acknowledgments section that appears (unusually) in the main body of the paper.

The stability of this particular flow may also be interpreted in terms of an energetics perspective. The stable flow is due to the lack of a shear growth source—the waves must either be embedded within the shear or at its sides in order to be able to extract energy from it. See Harnik and Heifetz [37] and Rabinovich et al. [29] for further details.

It is more appropriate to refer to this vorticity-vorticity wave interaction as the Rayleigh [22] mechanism, however, it has become conventional in the stratified shear flow literature to use KH after Kelvin [40] and Helmholtz [41], and we shall stick with this notation in the remainder of the paper.

1Corresponding author.

Manuscript received May 20, 2011; final manuscript received October 17, 2012; published online January 23, 2013. Editor: Harry Dankowicz.

Appl. Mech. Rev 64(6), 060801 (Jan 23, 2013) (17 pages) doi:10.1115/1.4007909 History: Received May 20, 2011; Revised October 17, 2012

Instability in homogeneous and density stratified shear flows may be interpreted in terms of the interaction of two (or more) otherwise free waves in the velocity and density profiles. These waves exist on gradients of vorticity and density, and instability results when two fundamental conditions are satisfied: (I) the phase speeds of the waves are stationary with respect to each other (“phase-locking“), and (II) the relative phase of the waves is such that a mutual growth occurs. The advantage of the wave interaction approach is that it provides a physical interpretation to shear flow instability. This paper is largely intended to purvey the basics of this physical interpretation to the reader, while both reviewing and consolidating previous work on the topic. The interpretation is shown to provide a framework for understanding many classical and nonintuitive results from the stability of stratified shear flows, such as the Rayleigh and Fjørtoft theorems, and the destabilizing effect of an otherwise stable density stratification. Finally, we describe an application of the theory to a geophysical-scale flow in the Fraser River estuary.

Copyright © 2011 by ASME
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References

Figures

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Fig. 1

(a) Piecewise profiles used to demonstrate vorticity and internal gravity waves. The interface of density and vorticity is located at zℓ. (b) The vertical velocity eigenfunction w∧(z) for vorticity and internal gravity waves. Since w∧(z) is generally complex it has both an amplitude |w∧(z)|=|A|e-k|z-zℓ| and a phase given by tan(θw)=Im(w∧)/Re(w∧)=Im(A)/Re(A).

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Fig. 2

Dispersion relation for the profiles in Fig. 1 showing (a) the phase speed cr(k), and (b) the frequency ωr(k) for vorticity and internal gravity waves. The vorticity wave mode shown here has a leftward intrinsic propagation and is denoted by V- (thin line), whereas the gravity wave modes are indicated by G± (thick lines) and have intrinsic propagation in both directions.

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Fig. 3

Structure of the vorticity wave broken down into three important fields: interface displacement η˜, vorticity perturbation q˜, and the vertical velocity w˜. These fields are all sinusoidal with wavenumber k and illustrated on the right. Throughout the remainder of the paper we will use the more compact notation shown on the left, with circular arrows denoting crests (clockwise) and troughs (counterclockwise) in q˜ and similarly with the vertical arrows denoting crests and troughs in the w˜-field. These diagrams will be referred to as wave field diagrams.

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Fig. 4

Structure of the rightward-propagating internal gravity wave broken down into its important fields. The notation is the same as in Fig. 3 except we have also plotted the generation of baroclinic vorticity, described by Eq. (16), as the dashed line. In addition to the rotational arrows, the q˜-field is indicated by the gray shading. This is because once wave interactions are accounted for the q˜-field is not generally directly related to the η˜-field, as is the case in the vorticity wave (see Sec. 5).

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Fig. 5

Illustration of the relationship between the η˜ and w˜ fields in a growing and rightward-propagating interfacial wave

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Fig. 6

The piecewise shear layer profile on left along with the eigenfunction amplitude of the two interface contributions to the total eigenfunctions |w∧(z)| and |ψ∧(z)|, which are related by w∧=-ikψ∧. The wave field diagram illustrating the unstable interaction of vorticity waves is shown on the right for the particular value of k = 0.4, corresponding to the maximum growth rate. As in Fig. 3, the vertical arrows indicate the peaks and troughs in the w˜-fields at each interface, due only to the displacement of that interface. In addition, we also show the vorticity q˜ and displacement η˜ of each interface.

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Fig. 7

Dispersion relation for the piecewise shear layer. Upper panel shows the real frequency ωr(k) of the dispersion relation with the dashed line indicating unstable (ℜ) modes, solid lines denote the two stable (S) modes, and the gray lines correspond to the isolated vorticity waves of the upper (V1-) and lower (V2+) interfaces. The bottom panel shows the growth rate ωi(k) of the unstable mode.

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Fig. 8

(a) The smooth `tanh’ shear layer velocity profile U*(z*) (gray) and the associated vorticity gradient distribution U*''(z*) (black). (b) Comparison between the growth rates of the smooth tanh-profile and the piecewise linear profile. Growth rates for both the traditional scaling of U*(z*) (dashed line) and the vorticity interface scaling (thick solid line) are shown, along with the piecewise result (thin line) for comparison.

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Fig. 9

(a) Dispersion relation ωr(k) for two stable interacting vorticity waves used to illustrate the Rayleigh inflection point condition. As in Fig. 2 the gray lines indicate the dispersion of waves in isolation from each other. (b) Wave field diagram showing that it is not possible for the waves to cause mutual growth in one another—regardless of the relative phase difference. Here an arbitrary phase difference has been chosen for illustration purposes.

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Fig. 10

(a) Dispersion relation for two stable vorticity waves that demonstrate Fjørtoft's extension (notation as in Fig. 2). (b) Wave field diagram showing the reinforcement of the intrinsic wave propagation by the advection of the mean flow profile that is stable by Fjørtoft's extension.

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Fig. 11

Profiles of U(z) and Q(z) for the triangular jet flow, together with the wave field diagram at k = 0.5

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Fig. 12

Dispersion relation for the triangular jet. The frequencies of the isolated vorticity waves (gray curves, V1,2,3), the stable waves (solid curves, S1,2), and the unstable jet mode (dashed curve, J) are shown in (a), with the growth rate in (b).

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Fig. 13

(a) Profiles of U(z) and ρ¯(z) that demonstrate the unstable Holmboe interaction between a vorticity wave (at z = 1) and an internal gravity wave (located at z = 0). (b) The wave field diagram showing the structure of the unstable mode with J = 2, k = 1.87. The gray shading is used to indicate the q˜2 field, which may have a phase difference from the η˜2 field. (c) Stability diagram showing the Holmboe unstable region. Contours are of growth rate (ωi) with an interval of 0.03, and the thick solid lines indicate the stability boundary with stable regions in gray. The resonance condition is denoted by the dot-dashed line and follows very closely the unstable region. An example of the dispersion relation ωr(k) shown in (d) is taken at J = 2.

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Fig. 14

(a) Profiles that demonstrate a stable interaction between a gravity wave and a vorticity wave. An example of the dispersion relation ωr(k) is shown in (b) for J = 0.25. See Fig. 2 for further notation.

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Fig. 15

(a) Dimensionless profile of background velocity U(z) that demonstrates the unstable Taylor–Caulfield interaction between two internal gravity waves (at z=±1). (b) The wave field diagram showing the structure of the unstable mode with J = 2, k = 1. (c) Stability diagram showing the Taylor–Caulfield unstable region (see Fig. 13(c) for notation). An example of the dispersion relation ωr(k) shown in (d) is taken at J = 2.

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Fig. 16

(a) Profiles for Holmboe's [14] symmetric (ε = 0) stratified shear layer. (b) The associated symmetric stability diagram where the gray area represents stable regions, with the thick dark line denoting the stability boundary, as well as the transition from propagating to stationary modes. The thin dark lines are contours of growth rate, ωi, with spacing of 0.03, and the thin gray contours represent the phase speed cr, with spacing 0.2. (c), (d) Profiles and stability diagram for the asymmetric stratified shear layer with ε = 0.25. In both (b), (d) the dashed-dotted line represents the resonance condition between vorticity and gravity waves.

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Fig. 17

Profiles and stability diagram for the smooth profiles of Holmboe's [14] symmetric (ε = 0) stratified shear layer with R = 5. All notation the same as in Fig. 16. Oscillations appear on the high-k stability boundary due to issues in numerical resolution (see Ref. [43] for more details).

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Fig. 18

Profiles measured in the Fraser River estuary. (a) Density and N*2, (b) velocity and vorticity gradients, along with the displacement eigenfunction from the most amplified wavenumber of the linear stability predictions in (c). An echosounding image is shown in (d) where instability waves can be seen at the depth where the eigenfunction predicts a maximum displacement (indicated by the horizontal line in all panels). The wavelength of maximum growth rate is shown by the horizontal bar in (d). (Figure is modified from Tedford et al. [45].)

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