Numerical continuation is used to investigate stationary spatially localized states in two-dimensional thermosolutal convection in a plane horizontal layer with no-slip boundary conditions at top and bottom. Convectons in the form of 1-pulse and 2-pulse states of both odd and even parity exhibit homoclinic snaking in a common Rayleigh number regime. In contrast to similar states in binary fluid convection, odd parity convectons do not pump concentration horizontally. Stable but time-dependent localized structures are present for Rayleigh numbers below the snaking region for stationary convectons. The computations are carried out for (inverse) Lewis number τ = 1/15 and Prandtl numbers Pr = 1 and Pr1.

1.
J. S.
Turner
, “
The coupled turbulent transports of salt and heat across a sharp density interface
,”
Int. J. Heat Mass Transfer
8
,
759
(
1965
).
2.
R. W.
Schmitt
, “
Double diffusion in oceanography
,”
Annu. Rev. Fluid Mech.
26
,
255
(
1994
).
3.
W. R.
Wilcox
, “
Transport phenomena in crystal growth from solution
,”
Prog. Cryst. Growth Charact. Mater.
26
,
153
(
1993
).
4.
U.
Hansen
and
D. A.
Yuen
, “
Nonlinear physics of double-diffusive convection in geological systems
,”
Earth-Sci. Rev.
29
,
385
(
1990
).
5.
E.
Knobloch
, “
Nonlinear diffusive instabilities in differentially rotating stars
,”
Geophys. Astrophys. Fluid Dyn.
22
,
133
(
1982
).
6.
H. E.
Huppert
and
D. R.
Moore
, “
Nonlinear double-diffusive convection
,”
J. Fluid Mech.
78
,
821
(
1976
).
7.
D. R.
Moore
,
J.
Toomre
,
E.
Knobloch
, and
N. O.
Weiss
, “
Period doubling and chaos in partial differential equations for thermosolutal convection
,”
Nature
303
,
663
(
1983
).
8.
O.
Batiste
,
E.
Knobloch
,
A.
Alonso
, and
I.
Mercader
, “
Spatially localized binary-fluid convection
,”
J. Fluid Mech.
560
,
149
(
2006
).
9.
A.
Bergeon
and
E.
Knobloch
, “
Periodic and localized states in natural doubly diffusive convection
,”
Physica D
237
,
1139
(
2008
).
10.
G. W.
Hunt
,
M. A.
Peletier
,
A. R.
Champneys
,
P. D.
Woods
,
M. A.
Wadee
,
C. J.
Budd
, and
G. J.
Lord
, “
Cellular buckling in long structures
,”
Nonlinear Dyn.
21
,
3
(
2000
).
11.
J.
Burke
and
E.
Knobloch
, “
Homoclinic snaking: structure and stability
,”
Chaos
17
,
037102
(
2007
).
12.
D.
Lo Jacono
,
A.
Bergeon
, and
E.
Knobloch
, “
Spatially localized binary fluid convection in a porous medium
,”
Phys. Fluids
22
,
073601
(
2010
).
13.
A. E.
Deane
,
E.
Knobloch
, and
J.
Toomre
, “
Traveling waves in large-aspect-ratio thermosolutal convection
,”
Phys. Rev. A
37
,
1817
(
1988
).
14.
A. A.
Predtechensky
,
W. D.
McCormick
,
J. B.
Swift
,
A. G.
Rossberg
, and
H. L.
Swinney
, “
Traveling wave instability in sustained double-diffusive convection
,”
Phys. Fluids
6
,
3923
(
1994
).
15.
J. D.
Crawford
and
E.
Knobloch
, “
Symmetry and symmetry-breaking bifurcations in fluid mechanics
,”
Annu. Rev. Fluid Mech.
23
,
341
(
1991
).
16.
E.
Knobloch
,
D. R.
Moore
,
J.
Toomre
, and
N. O.
Weiss
, “
Transitions to chaos in two-dimensional double-diffusive convection
,”
J. Fluid Mech.
166
,
409
(
1986
).
17.
E.
Knobloch
,
A. E.
Deane
,
J.
Toomre
, and
D.
Moore
, “
Doubly diffusive waves
,”
Contemp. Math.
56
,
203
(
1986
).
18.
A. E.
Deane
,
E.
Knobloch
, and
J.
Toomre
, “
Traveling waves and chaos in thermosolutal convection
,”
Phys. Rev. A
36
,
2862
(
1987
).
19.
C. S.
Bretherton
and
E. A.
Spiegel
, “
Intermittency through modulational instability
,”
Phys. Lett. A
96
,
152
(
1983
).
20.
L. S.
Tuckerman
, “
Steady-state solving via Stokes preconditioning: Recursion relations for elliptic operators
,” in
11th International Conference on Numerical Methods in Fluid Dynamics
, edited by
D.
Dwoyer
,
M.
Hussaini
, and
R.
Voigt
, Lecture Notes in Physics Vol.
323
(
Springer
,
Berlin
,
1989
), p.
573
.
21.
C. K.
Mamun
and
L. S.
Tuckerman
, “
Asymmetry and Hopf bifurcation in spherical Couette flow
,”
Phys. Fluids
7
,
80
(
1995
).
22.
G. E.
Karniadakis
,
M.
Israeli
, and
S. A.
Orszag
, “
High-order splitting methods for the incompressible Navier–Stokes equations
,”
J. Comp. Phys.
97
,
414
(
1991
).
23.
A.
Bergeon
and
E.
Knobloch
, “
Natural doubly diffusive convection in three-dimensional enclosures
,”
Phys. Fluids
14
,
92
(
2002
).
24.
D.
Funaro
,
Polynomial Approximation of Differential Equations
(
Springer
,
New York
,
1991
).
25.
M.
Deville
,
P. F.
Fischer
, and
E.
Mund
,
High-order Methods for Incompressible Fluid Flow
(
Cambridge University Press
,
New York
,
2002
).
26.
R.
Peyret
,
Spectral Methods for Incompressible Viscous Flow
, Applied Mathematical Sciences Vol.
148
(
Springer
,
New York
,
2002
).
27.
A.
Quarteroni
and
A.
Valli
,
Domain Decomposition Methods for Partial Differential Equations
(
Oxford Science Publications
,
1999
).
28.
C.
Canuto
,
M. Y.
Hussaini
,
A.
Quarteroni
, and
T. A.
Zang
,
Spectral Methods in Fluid Dynamics
(
Springer
,
New York
,
1988
).
29.
A.
Bergeon
and
E.
Knobloch
, “
Spatially localized states in natural doubly diffusive convection
,”
Phys. Fluids
20
,
034102
(
2008
).
30.
A.
Bergeon
,
J.
Burke
,
E.
Knobloch
, and
I.
Mercader
, “
Eckhaus instability and homoclinic snaking
,”
Phys. Rev. E
78
,
046201
(
2008
).
31.
P. D.
Woods
and
A. R.
Champneys
, “
Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate Hamiltonian–Hopf bifurcation
,”
Physica D
129
,
147
(
1999
).
32.
P.
Coullet
,
C.
Riera
, and
C.
Tresser
, “
Stable static localized structures in one dimension
,”
Phys. Rev. Lett.
84
,
3069
(
2000
).
33.
M.
Beck
,
J.
Knobloch
,
D. J. B.
Lloyd
,
B.
Sandstede
, and
T.
Wagenknecht
, “
Snakes, ladders and isolas of localized patterns
,”
SIAM J. Math. Anal.
41
,
936
(
2009
).
34.
E.
Knobloch
, “
Convection in binary fluids
,”
Phys. Fluids
23
,
1918
(
1980
).
35.
P.
Kolodner
,
C. M.
Surko
, and
H.
Williams
, “
Dynamics of traveling waves near the onset of convection in binary fluid mixtures
,”
Physica D
37
,
319
(
1989
).
36.
V.
Steinberg
,
J.
Fineberg
,
E.
Moses
, and
I.
Rehberg
, “
Pattern selection and transition to turbulence in propagating waves
,”
Physica D
37
,
359
(
1989
).
37.
J.
Burke
and
E.
Knobloch
, “
Snakes and ladders: localized states in the Swift–Hohenberg equation
,”
Phys. Lett. A
360
,
681
(
2007
).
38.
J.
Burke
and
E.
Knobloch
, “
Multipulse states in the Swift–Hohenberg equation
,” Discrete Contin. Dyn. Syst. (Suppl.), 109 (
2009
).
39.
A.
Alonso
,
O.
Batiste
,
E.
Knobloch
, and
I.
Mercader
, “
Convectons
,” in
Localized States in Physics: Solitons and Patterns
, edited by
O.
Descalzi
,
M.
Clerc
,
S.
Residori
, and
G.
Assanto
(
Springer
,
Berlin
,
2010
), p.
109
.
40.
C.
Martel
,
E.
Knobloch
, and
J. M.
Vega
, “
Dynamics of counterpropagating waves in parametrically forced systems
,”
Physica D
137
,
94
(
2000
).
41.
T. M.
Schneider
,
J. F.
Gibson
, and
J.
Burke
, “
Snakes and ladders: Localized solutions of plane Couette flow
,”
Phys. Rev. Lett.
104
,
104501
(
2010
).
42.
O.
Batiste
and
E.
Knobloch
, “
Simulations of oscillatory convection in 3He-4He mixtures in moderate aspect ratio containers
,”
Phys. Fluids
17
,
064102
(
2005
).
43.
P.
Kolodner
, “
Stable and unstable pulses of traveling-wave convection
,”
Phys. Rev. A
43
,
2827
(
1991
).
44.
W.
Barten
,
M.
Lücke
,
M.
Kamps
, and
R.
Schmitz
, “
Convection in binary fluid mixtures. II. Localized traveling waves
,”
Phys. Rev. E
51
,
5662
(
1995
).
45.
A.
Spina
,
J.
Toomre
, and
E.
Knobloch
, “
Confined states in large-aspect-ratio thermosolutal convection
,”
Phys. Rev. E
57
,
524
(
1998
).
46.
P.
Kolodner
, “
Extended states of nonlinear traveling-wave convection. II. Fronts and spatiotemporal defects
,”
Phys. Rev. A
46
,
6452
(
1992
).
47.
M.
van Hecke
,
C.
Storm
, and
W.
van Saarloos
, “
Sources, sinks and wavenumber selection in coupled CGL equations and experimental implications for counter-propagating wave systems
,”
Physica D
134
,
1
(
1999
).
48.
P.
Coullet
,
T.
Frisch
, and
F.
Plaza
, “
Sources and sinks of wave patterns
,”
Physica D
62
,
75
(
1993
).
49.
A. E.
Deane
,
E.
Knobloch
, and
J.
Toomre
, “
Erratum: Traveling waves in large-aspect-ratio thermosolutal convection
[Phys. Rev. A 37, 1817 (
1988
)],” Phys. Rev. A 38, 1661 (1988).
50.
L.
Pastur
,
M.-T.
Westra
, and
W.
van der Water
, “
Sources and sinks in 1d traveling waves
,”
Physica D
174
,
71
(
2003
).
You do not currently have access to this content.