Within the context of the well-known interpretation in terms of the wave interaction [P. G. Baines and H. Mitsudera, J. Fluid Mech. 276, 327 (1994); J. R. Carpenter et al., Phys. Fluids 22, 054104 (2010)], instability of sharply stratified (so that the vertical scale of density variation is much smaller than the scale Λ of velocity shear) flows with inflection-free velocity profiles should be treated as Holmboe’s instability. In such flows with a relatively weak stratification (when the bulk Richardson number J < (/Λ)3/2), eigenoscillations (i.e., Holmboe waves) have much the same phase velocities in a broad spectral range. This creates favorable conditions for a wide variety of three-wave interactions, in contrast to the homogeneous boundary layers where subharmonic resonance is the only effective three-wave process. In the paper, evolution equations are derived which describe three-wave interactions of Holmboe waves and have the form of nonlinear integral equations. Analytical and numerical methods are both used to find their solutions in different cases, and it is shown that at the nonlinear stage disturbances increase, as a rule, explosively. Some possible relations of the results obtained with those of numerical simulations and laboratory experiments are briefly discussed.

1.
R.
Rosner
,
A.
Alexakis
,
Y.
Young
,
J.
Truran
, and
W.
Hillebrand
, “
On the C/O enrichment of novae ejecta
,”
Astrophys. J. Lett.
562
,
L177
(
2002
).
2.
A.
Alexakis
,
A. C.
Calder
,
A.
Heger
,
E. F.
Brown
,
L. J.
Dursi
,
J. W.
Truran
,
R.
Rosner
,
D. Q.
Lamb
,
F. X.
Timmes
,
B.
Fryxell
,
M.
Zingale
,
P. M.
Ricker
, and
K.
Olson
, “
On heavy element enrichment in classical novae
,”
Astrophys. J.
603
,
931
(
2004
).
3.
P.
Garaud
, “
Latitudinal shear instability in the solar tachocline
,”
Mon. Not. R. Astron. Soc.
324
,
68
(
2001
).
4.
H.
Luce
,
S.
Fukao
,
F.
Dalaudier
, and
M.
Crochet
, “
Strong mixing events observed near the tropopause with the MU radar and high-resolution balloon techniques
,”
J. Atmos. Sci.
59
,
2885
(
2002
).
5.
N. M.
Gavrilov
,
S.
Fukao
,
H.
Hashiguchi
,
K.
Kita
,
K.
Sato
,
Y.
Tomikawa
, and
M.
Fujiwara
, “
Combined MU radar and ozonesonde measurements of turbulence and ozone fluxes in the tropo-stratosphere over Shigaraki, Japan
,”
Geophys. Res. Lett.
33
,
L09803
, doi:10.1029/2005GL024002 (
2006
).
6.
D.
Farmer
and
L.
Armi
, “
The flow of Mediterranean water through the Strait of Gibraltar
,”
Prog. Oceanogr.
21
,
1
(
1998
).
7.
S.
Yoshida
,
M.
Ohtani
,
S.
Nishida
, and
P. F.
Linden
, “
Mixing processes in a highly stratified river
,” in
Physical Processes in Lakes and Oceans, Coastal and Estuarine Studies
(
American Geophysical Union
,
Washington, DC
,
1998
), Vol.
54
, pp.
389
400
.
8.
E. W.
Tedford
,
J. R.
Carpenter
,
R.
Pawlowicz
,
R.
Pieters
, and
G. A.
Lawrence
, “
Observation and analysis of shear instability in the Fraser River estuary
,”
J. Geophys. Res.
114
,
C11006
, doi:10.1029/2009JC005313 (
2009
).
9.
J.
Holmboe
, “
On the behaviour of symmetric waves in stratified shear layers
,”
Geofys. Pub.
24
,
67
(
1962
).
10.
P.
Hazel
, “
Numerical studies of the stability of inviscid stratified shear flows
,”
J. Fluid Mech.
51
,
39
(
1972
).
11.
W. D.
Smyth
and
W. R.
Peltier
, “
The transition between Kelvin-Helmholtz and Holmboe instability: An investigation of the over-reflection hypothesis
,”
J. Atmos. Sci.
46
,
3698
(
1989
).
12.
W. D.
Smyth
and
W. R.
Peltier
, “
Three-dimensional primary instabilities of a stratified, dissipative, parallel flow
,”
Geophys. Astrophys. Fluid Dyn.
52
,
249
(
1990
).
13.
S.
Ortiz
,
J.-M.
Chomaz
, and
T.
Loiseleux
, “
Spatial Holmboe instability
,”
Phys. Fluids
14
,
2585
(
2002
).
14.
A.
Alexakis
, “
On Holmboe’s instability for smooth shear and density profiles
,”
Phys. Fluids
17
,
084103
(
2005
).
15.
A.
Alexakis
, “
Marginally unstable Holmboe modes
,”
Phys. Fluids
19
,
054105
(
2007
).
16.
J. R.
Carpenter
,
E. W.
Tedford
,
M.
Rahmani
, and
G. A.
Lawrence
, “
Holmboe wave fields in simulation and experiment
,”
J. Fluid Mech.
648
,
205
(
2010
).
17.
J. R.
Carpenter
,
N. J.
Balmforth
, and
G. A.
Lawrence
, “
Identifying unstable modes in stratified shear layers
,”
Phys. Fluids
22
,
054104
(
2010
).
18.
P. G.
Baines
and
H.
Mitsudera
, “
On the mechanism of shear flow instabilities
,”
J. Fluid Mech.
276
,
327
(
1994
).
19.
C. P.
Caulfield
, “
Multiple linear instability of layered stratified shear flow
,”
J. Fluid Mech.
258
,
255
(
1994
).
20.
A. D. D.
Craik
,
Wave Interactions and Fluid Flows
(
Cambridge University Press
,
Cambridge
,
1985
).
21.
A. V.
Timofeev
, “
Oscillations of inhomogeneous plasma and fluid flows
,”
Usp. Fiz. Nauk
102
,
185
(
1970
), in Russian.
22.
A. A.
Andronov
and
A. L.
Fabrikant
, “
Landau damping, wind waves, and whistle
,” in
Nonlinear Waves
, edited by
A. V.
Gaponov-Grekhov
(
Nauka
,
Moscow
,
1979
), pp.
68
104
, in Russian.
23.
A.
Fabrikant
, “
Plasma-hydrodynamic analogy for waves and vortices in shear flows
,”
Sound-Flow Interactions. Lecture Notes in Physics
, edited by
Y.
Aurégan
 et al. (
Springer
,
New York
,
2002
), Vol.
586
, pp.
192
209
.
24.
S. A.
Maslowe
, “
Critical layers in shear flows
,”
Annu. Rev. Fluid Mech.
18
,
405
(
1986
).
25.
M. E.
Goldstein
, “
The role of nonlinear critical layers in boundary layer transition
,”
Philos. Trans. R. Soc. London, Ser. A
352
,
425
(
1995
).
26.
X.
Wu
and
P. A.
Stewart
Interaction of phase-locked modes: a new mechanism for the rapid growth of three-dimensional disturbances
,”
J. Fluid Mech.
316
,
335
(
1996
).
27.
S. M.
Churilov
and
I. G.
Shukhman
, “
Critical layer and nonlinear evolution of disturbances in weakly supercritical shear layer
,”
Izv., Atmos. Oceanic Phys.
31
,
534
(
1996
).
28.
M. E.
Goldstein
and
S.-W.
Choi
, “
Nonlinear evolution of interacting oblique waves on two-dimensional shear layers
,”
J. Fluid Mech.
207
,
97
(
1989
);
M. E.
Goldstein
and
S.-W.
Choi
, “
Corrigendum
,”
ibid.
216
,
659
(
1990
).
29.
X.
Wu
,
S. S.
Lee
, and
S. J.
Cowley
On the weakly nonlinear three-dimensional instability of shear layers to pairs of oblique waves: the Stokes layer as a paradigm
,”
J. Fluid Mech.
253
,
681
(
1993
).
30.
S. M.
Churilov
, “
Nonlinear stage of instability development in a stratified shear flow with an inflection–free velocity profile
.”
Phys. Fluids
21
,
074101
(
2009
).
31.
S. M.
Churilov
, “
On the stability of stratified shear flows with a monotonic velocity profile without inflection points
,”
Izv., Atmos. Oceanic Phys.
40
,
725
(
2004
).
32.
S. M.
Churilov
, “
Stability analysis of stratified shear flows with a monotonic velocity profile without inflection points
,”
J. Fluid Mech.
539
,
25
(
2005
).
33.
S. M.
Churilov
, “
Stability analysis of stratified shear flows with a monotonic velocity profile without inflection points. Part 2. Continuous density variation
,”
J. Fluid Mech.
617
,
301
(
2008
).
34.
S. M.
Churilov
, “
Three–dimensional instability of shear flows with inflection–free velocity profiles in stratified media with a high Prandtl number
,”
Izv., Atmos. Oceanic Phys.
46
,
159
(
2010
).
35.
P. G.
Drazin
and
W. H.
Reid
,
Hydrodynamic Stability
(
Cambridge University Press
,
Cambridge
,
2004
).
36.
H. B.
Squire
, “
On the stability of three-dimensional disturbances of viscous flow between parallel walls
,”
Proc. R. Soc. London, Ser. A
142
,
621
(
1933
).
37.
Smyth and Peltier (Ref. 12) were the first who have demonstrated such an opportunity for Holmboe waves in a stratified mixing layer with Λ/ℓ = 3.
38.
A.
Alexakis
, “
Stratified shear flow instabilities at large Richardson numbers
,”
Phys. Fluids
21
,
054108
(
2009
).
39.
With the exception of a continuous spectrum of the so-called Van Kampen—Case modes, see Refs. 21 and 23.
40.
M. E.
Goldstein
and
S. S.
Lee
, “
Fully coupled resonant-triad interaction in an adverse-pressure-gradient boundary-layer
,”
J. Fluid Mech.
245
,
523
(
1992
).
41.
D. W.
Wundrow
,
L. S.
Hultgren
, and
M. E.
Goldstein
, “
Interaction of oblique instability waves with a nonlinear plane wave
,”
J. Fluid Mech.
264
,
343
(
1994
).
42.
It is interesting to note that in two-layer flow (with N(Z) = δ(Z) and zero thickness of the pycnocline), Eq. 26a can be written in the integro-differential form as well. Indeed, differentiating them twice in τ we obtain a*(Kn)d2An/dτ2+Kn2R̃An=d2n/dτ2, where n is the right-hand side of the n-th equation.
43.
X.
Wu
,
P. A.
Stewart
, and
S. J.
Cowley
, “
On the catalytic role of the phase-locked interaction of Tollmien–Schlichting waves in boundary-layer transition
,”
J. Fluid Mech.
590
,
265
(
2007
).
44.
A.
Erdelyi
,
Higher Transcendental Functions
(
McGraw-Hill
,
New York
,
1953
), Vol.
1
.
45.
If M = 2 we deal with isolated harmonic-subharmonic triad(s) that grows super-exponentially (in accordance with Eq. 30), rather than explosively.
You do not currently have access to this content.