Salt-finger convection provides a key mixing process in geophysical and astrophysical fluid flows. Because of its small characteristic spatial scale and slow diffusive time scale, this process must be parameterized in geophysical and astrophysical models, where relations linking background gradients to fluxes are required. To obtain such relations, most authors study the dependence of temperature and salinity fluxes on fixed background gradients. Using the reduced model derived by Xie et al. [“A reduced model for salt-finger convection in the small diffusivity ratio limit,” Fluids 2(1), 6 (2017)] for salt-finger convection in the limit of small diffusivity and large density ratios, this paper considers the conjugate problem where the fluxes are fixed, but the mean gradients are permitted to adjust in response. In small domains, the fixed-flux condition leads to stable single-mode solutions, which are not achievable with fixed-gradient conditions. In large domains, with statistically steady saturated states, the relations between mean fluxes and mean gradients are identical for both sets of conditions. The fixed-flux condition provides a new perspective for understanding the resulting statistically steady states by identifying two distinct regimes with the same dissipation rate. We find that the statistically steady dynamics select the state with the smaller Rayleigh ratio Ra subject to the constraint Ra > 1, ensuring that the background state is linearly unstable. The fixed-flux formulation results in a more potent restoring mechanism toward the statistically steady state, with a smaller variance, skewness, and characteristic time scale than in the fixed-gradient setup. This distinctive feature can be used as a diagnostic to determine whether in situ salt-finger convection is flux-driven or gradient-driven.

1.
J. M.
Brown
,
P.
Garaud
, and
S.
Stellmach
, “
Chemical transport and spontaneous layer formation in fingering convection in astrophysics
,”
Astrophys. J.
768
,
34
(
2013
).
2.
F. H.
Busse
, “
The optimum theory of turbulence
,”
Adv. Appl. Mech.
18
,
77
121
(
1978
).
3.
E.
Calzavarini
,
D.
Lohse
,
F.
Toschi
, and
R.
Tripiccione
, “
Rayleigh and Prandtl number scaling in the bulk of Rayleigh-Bénard turbulence
,”
Phys. Fluids
17
,
055107
(
2005
).
4.
E.
Calzavarini
,
C. R.
Doering
,
J. D.
Gibbon
,
D.
Lohse
,
A.
Tanabe
, and
F.
Toschi
, “
Exponentially growing solutions in homogeneous Rayleigh-Bénard convection
,”
Phys. Rev. E
73
,
035301
(
2006
).
5.
C. F.
Chen
and
J. S.
Turner
, “
Crystallization in a double-diffusive system
,”
J. Geophys. Res.
85
,
2573
2593
, (
1980
).
6.
J.
Toomre
,
E.
Knobloch
,
D. R.
Moore
, and
N. O.
Weiss
, “
Transitions to chaos in two-dimensional double-diffusive convection
,”
J. Fluid Mech.
166
,
409
448
(
1986
).
7.
M. R. E.
Proctor
,
E.
Knobloch
, and
N. O.
Weiss
, “
Heteroclinic bifurcations in a simple model of double-diffusive convection
,”
J. Fluid Mech.
239
,
273
292
(
1992
).
8.
P.
Garaud
, “
What happened to the other Mohicans? The case for a primordial origin to the planet-metallicity connection
,”
Astrophys. J.
728
,
L30
(
2011
).
9.
P.
Garaud
and
N.
Brummell
, “
2D or not 2D: The effect of dimensionality on the dynamics of fingering convection at low Prandtl number
,”
Astrophys. J.
815
,
42
(
2015
).
10.
I.
Grooms
and
K.
Julien
, “
Multiscale models in geophysical fluid dynamics
,”
Earth Space Sci.
5
,
668
675
(
2018
).
11.
J. Y.
Holyer
, “
The stability of long, steady, two-dimensional salt fingers
,”
J. Fluid Mech.
147
,
169
185
(
1984
).
12.
L. N.
Howard
and
G.
Veronis
, “
The salt-finger zone
,”
J. Fluid Mech.
183
,
1
23
(
1987
).
13.
T. M.
Joyce
, “
Marginally unstable salt fingers: Limits to growth
,”
J. Mar. Res.
40
(
suppl
.),
291
306
(
1982
).
14.
E.
Knobloch
and
M. R. E.
Proctor
, “
Nonlinear periodic convection in double-diffusive systems
,”
J. Fluid Mech.
108
,
291
316
(
1981
).
15.
E.
Knobloch
and
N. O.
Weiss
, “
Bifurcations in a model of double-diffusive convection
,”
Phys. Lett. A
85
,
127
130
(
1985
).
16.
L. St.
Laurent
and
R. W.
Schmitt
, “
The contribution of salt fingers to vertical mixing in the North Atlantic Tracer release experiment*
,”
J. Phys. Oceanogr.
29
,
1404
1424
(
1999
).
17.
E.
Knobloch
,
L. N.
Da Costa
, and
N. O.
Weiss
, “
Oscillations in double-diffusive convection
,”
J. Fluid Mech.
109
,
25
43
(
1981
).
18.
F.
Paparella
and
J.
von Hardenberg
, “
Clustering of salt fingers in double-diffusive convection leads to staircaselike stratification
,”
Phys. Rev. Lett.
109
,
014502
(
2012
).
19.
M. R. E.
Proctor
and
J. Y.
Holyer
, “
Planform selection in salt fingers
,”
J. Fluid Mech.
168
,
241
253
(
1986
).
20.
T.
Radko
, “
Equilibration of weakly nonlinear salt fingers
,”
J. Fluid Mech.
645
,
121
143
(
2010
).
21.
T.
Radko
,
Double-Diffusive Convection
(
Cambridge University Press
,
2013
).
22.
T.
Radko
and
D. P.
Smith
, “
Equilibrium transport in double-diffusive convection
,”
J. Fluid Mech.
692
,
5
27
(
2012
).
23.
T.
Radko
and
M. E.
Stern
, “
Finite-amplitude salt fingers in a vertically bounded layer
,”
J. Fluid Mech.
425
,
133
160
(
2000
).
24.
T.
Radko
and
M. E.
Stern
, “
Salt fingers in three dimensions
,”
J. Mar. Res.
57
,
471
502
(
1999
).
25.
A. M.
Rucklidge
, “
Chaos in models of double convection
,”
J. Fluid Mech.
237
,
209
229
(
1992
).
26.
R. W.
Schmitt
, “
Double diffusion in oceanography
,”
Annu. Rev. Fluid Mech.
26
,
255
285
(
1994
).
27.
C. Y.
Shen
, “
Equilibrium salt-fingering convection
,”
Phys. Fluids
7
,
704
717
(
1995
).
28.
S.
Stellmach
,
A.
Traxler
,
P.
Garaud
,
N.
Brummell
, and
T.
Radko
, “
Dynamics of fingering convection. Part 2 The formation of thermohaline staircases
,”
J. Fluid Mech.
677
,
554
571
(
2011
).
29.
M. E.
Stern
, “
The ‘salt-fountain’ and thermohaline convection
,”
Tellus
12
,
172
175
(
1960
).
30.
M. E.
Stern
, “
Collective instability of salt fingers
,”
J. Fluid Mech.
35
,
209
218
(
1969
).
31.
M. E.
Stern
, “
Maximum buoyancy flux across a salt finger interface
,”
J. Mar. Res.
34
,
95
110
(
1976
).
32.
M. E.
Stern
and
J.
Simeonov
, “
The secondary instability of salt fingers
,”
J. Fluid Mech.
533
,
361
380
(
2005
).
33.
A.
Traxler
,
P.
Garaud
, and
S.
Stellmach
, “
Numerically determined transport laws for fingering (‘thermohaline’) convection in astrophysics
,”
Astrophys. J.
728
,
L29
(
2011
).
34.
A.
Traxler
,
S.
Stellmach
,
P.
Garaud
,
T.
Radko
, and
N.
Brummell
, “
Dynamics of fingering convection. Part 1 Small-scale fluxes and large-scale instabilities
,”
J. Fluid Mech.
677
,
530
553
(
2011
).
35.
J. S.
Turner
, “
Double-diffusive phenomena
,”
Annu. Rev. Fluid Mech.
6
,
37
54
(
1974
).
36.
J. S.
Turner
, “
Multicomponent convection
,”
Annu. Rev. Fluid Mech.
17
,
11
44
(
1985
).
37.
S.
Vauclair
, “
Metallic fingers and metallicity excess in exoplanets’ host stars: The accretion hypothesis revisited
,”
Astrophys. J.
605
,
874
879
(
2004
).
38.
G.
Veronis
, “
On finite amplitude instability in thermohaline convection
,”
J. Mar. Res.
23
,
1
17
(
1965
).
39.
A. P.
Willis
,
Y.
Duguet
,
O.
Omel’chenko
, and
M.
Wolfrum
, “
Surfing the edge: Using feedback control to find nonlinear solutions
,”
J. Fluid Mech.
831
,
579
591
(
2017
).
40.
J.-H.
Xie
,
B.
Miquel
,
K.
Julien
, and
E.
Knobloch
, “
A reduced model for salt-finger convection in the small diffusivity ratio limit
,”
Fluids
2
(
1
),
6
(
2017
).
41.
J.-H.
Xie
,
K.
Julien
, and
E.
Knobloch
, “
Subcritical saturation of the magnetorotational instability through mean magnetic field generation
,”
Mon. Not. R. Astron. Soc.
474
,
3451
3465
(
2018
).
42.
J.-H.
Xie
,
K.
Julien
, and
E.
Knobloch
, “
Jet formation in salt-finger convection: A modified Rayleigh-Bénard problem
,”
J. Fluid Mech.
858
,
228
263
(
2019
).
43.
Y.
Yang
,
E. P.
van der Poel
,
R.
Ostilla-Mónico
,
C.
Sun
,
R.
Verzicco
,
S.
Grossmann
, and
D.
Lohse
, “
Salinity transfer in bounded double diffusive convection
,”
J. Fluid Mech.
768
,
476
491
(
2015
).
44.
Y.
Yang
,
R.
Verzicco
, and
D.
Lohse
, “
Vertically bounded double diffusive convection in the finger regime: Comparing no-slip versus free-slip boundary conditions
,”
Phys. Rev. Lett.
117
,
184501
(
2016
).
45.
Y.
Yang
,
R.
Verzicco
, and
D.
Lohse
, “
Scaling laws and flow structures of double diffusive convection in the finger regime
,”
J. Fluid Mech.
802
,
667
689
(
2016
).
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