Studies of stratified spin-up experiments in enclosed cylinders have reported the presence of small pockets of well-mixed fluids but quantitative measurements of the mixedness of the fluid has been lacking. Previous numerical simulations have not addressed these measurements. Here we present numerical simulations that explain how the combined effect of spin-up and thermal boundary conditions enhances or hinders mixing of a fluid in a cylinder. The energy of the system is characterized by splitting the potential energy into diabatic and adiabatic components, and measurements of efficiency of mixing are based on both, the ratio of dissipation of available potential energy to forcing and variance of temperature. The numerical simulations of the Navier–Stokes equations for the problem with different sets of thermal boundary conditions at the horizontal walls helped shed some light on the physical mechanisms of mixing, for which a clear explanation was absent.

1.
J. S.
Allen
, “
Upwelling and coastal jets in a continuously stratified ocean
,”
J. Phys. Ocean.
3
,
245
257
(
1973
).
2.
P. F.
Linden
and
G. J. F.
van Heijst
, “
Two-layer spin-up and frontogenesis
,”
J. Fluid Mech.
143
,
69
94
(
1984
).
3.
T. O.
Manley
and
H.
Hunkins
, “
Mesoscale eddies of the Arctic Ocean
,”
J. Geophys. Res.
90
,
4911
4930
, doi: (
1985
).
4.
J. C.
McWilliams
, “
Submesoscale, coherent vortices in the ocean
,”
Rev. Geophys.
23
,
165
182
, doi: (
1985
).
5.
S.
Monismith
, “
An experimental study of the upwelling response of stratified reservoirs to surface shear stress
,”
J. Fluid Mech.
171
,
407
439
(
1986
).
6.
D. B.
Olson
, “
Rings in the ocean
,”
Annu. Rev. Earth Planet. Sci.
19
,
283
311
(
1991
).
7.
C.
Garrett
,
P.
MacCready
, and
P.
Rhines
, “
Boundary mixing and arrested Ekman layers: Rotating stratified flow near a sloping boundary
,”
Annu. Rev. Fluid Mech.
25
,
291
323
(
1993
).
8.
F. Y.
Moulin
and
J.-B.
Flór
, “
On the spin-up by a rotating disk in a rotating stratified fluid
,”
J. Fluid Mech.
516
,
155
180
(
2004
).
9.
W. R.
Peltier
and
C. P.
Caufield
, “
Mixing efficiency in stratified shear flows
,”
Annu. Rev. Fluid Mech.
35
,
135
167
(
2003
).
10.
J. S.
Turner
,
Buoyancy Effects in Fluids
(
Cambridge University Press
,
Cambridge
,
1973
).
11.
J. S.
Turner
, “
Turbulent entrainment: The development of the entrainment assumption, and its application to geophysical flows
,”
J. Fluid Mech.
173
,
431
471
(
1986
).
12.
J. J.
Sturman
,
G. N.
Ivey
, and
J. R.
Taylor
, “
Convection in a long box driven by heating and cooling on the horizontal boundaries
,”
J. Fluid Mech.
310
,
61
87
(
1996
).
13.
T.
Maxworthy
, “
Convection into domains with open boundaries
,”
Annu. Rev. Fluid Mech.
29
,
327
371
(
1997
).
14.
S. B.
Dalziel
,
M. D.
Patterson
,
C. P.
Caufield
, and
I. A.
Coomaraswamy
, “
Mixing efficiency in high-aspect-ratio Rayleigh-Taylor experiment
,”
Phys. Fluids
20
,
065106
(
2008
).
15.
K. B.
Winters
and
W. R.
Young
, “
Available potential energy and buoyancy variance in horizontal convection
,”
J. Fluid Mech.
629
,
221
230
(
2009
).
16.
H. J. S.
Fernando
, “
Turbulent mixing in stratified fluids
,”
Annu. Rev. Fluid Mech.
23
,
455
493
(
1991
).
17.
E. J.
Strang
, and
H. J. S.
Fernando
, “
Entrainment and mixing in stratified shear flows
,”
J. Fluid Mech.
428
,
349
386
(
2001
).
18.
J. R.
Wells
and
K. R.
Helfrich
, “
A laboratory study of localized boundary mixing in a rotating stratified fluid
,”
J. Fluid Mech.
516
,
83
113
( 10
2004
).
19.
J.
Pedlosky
,
Geophysical Fluid Dynamics
(
Springer–Verlag
,
Berlin
,
1987
).
20.
H. P.
Greenspan
, “
A note on the spin-up from rest of a stratified fluid
,”
Geophys. Fluid Dyn.
15
,
1
5
(
1980
).
21.
J.-B.
Flór
,
M.
Ungarish
, and
J. W. M.
Bush
, “
Spin-up from rest in a stratified fluid: Boundary flows
,”
J. Fluid Mech.
472
,
51
82
(
2002
).
22.
J.-B.
Flór
,
J. W. M.
Bush
, and
M.
Ungarish
, “
An experimental investigation of spin-up from rest of a stratified fluid
,”
Geophys. Fluid Dyn.
98
,
277
296
(
2004
).
23.
S. A.
Smirnov
,
J. R.
Pacheco
, and
R.
Verzicco
, “
Numerical simulations of nonlinear thermally stratified spin-up in a circular cylinder
,”
Phys. Fluids
22
,
116602
(
2010
).
24.
J. R.
Pacheco
and
R.
Verzicco
, “
Formation of columnar baroclinic vortices in thermally-stratified non-linear spin-up
,”
J. Fluid Mech.
702
,
265
285
(
2012
).
25.
K. B.
Winters
,
P. N.
Lombard
,
J. J.
Filey
, and
E. A.
D'Asaro
, “
Available potential energy and mixing in density–stratified fluids
,”
J. Fluid Mech.
289
,
115
128
(
1995
).
26.
J. L.
Thiffeault
, “
Using multiscale norms to quantify mixing and transport
,”
Nonlinearity
25
,
R1
R44
(
2012
).
27.
J. R.
Pacheco
, “
Mixing enhancement in electro-osmotic flows via modulation of electric fields
,”
Phys. Fluids
20
,
093603
(
2008
).
28.
J. R.
Pacheco
,
A.
Pacheco-Vega
, and
K. P.
Chen
, “
Mixing-dynamics of a passive scalar in a three-dimensional microchannel
,”
Int. J. Heat Mass Transfer
54
,
959
966
(
2011
).
29.
Y.-H.
Tseng
and
Joel H.
Ferziger
, “
Mixing and available potential energy in stratified flows
,”
Phys. Fluids
13
,
1281
1293
(
2001
).
30.
R.
Verzicco
and
P.
Orlandi
, “
A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates
,”
J. Comput. Phys.
123
,
402
414
(
1996
).
31.
R.
Verzicco
and
R.
Camussi
, “
Transitional regimes of low-Prandtl thermal convection in a cylindrical cell
,”
Phys. Fluids
9
,
1287
1295
(
1997
).
32.
R.
Verzicco
,
F.
Lalli
, and
E.
Campana
, “
Dynamics of baroclinic vortices in a rotating stratified fluid: A numerical study
,”
Phys. Fluids
9
,
419
432
(
1997
).
33.
S. A.
Smirnov
,
J. R.
Pacheco
, and
R.
Verzicco
, “
Three-dimensional vortex visualization in stratified spin-up
,”
J. Visualization
13
,
81
84
(
2010
).
34.
J. R.
Pacheco
,
J. M.
Lopez
, and
F.
Marques
, “
Pinning of rotating waves to defects in finite Taylor–Couette flow
,”
J. Fluid Mech.
666
,
254
272
(
2011
).
35.
J. R.
Pacheco
,
A.
Ruiz-Angulo
,
R.
Zenit
, and
R.
Verzicco
, “
Fluid velocity fluctuations in a collision of a sphere with a wall
,”
Phys. Fluids
23
,
063301
(
2011
).
36.
D. K.
Lilly
, “
On the instability of Ekman boundary flow
,”
J. Atmos. Sci.
23
,
481
494
(
1966
).
37.
D. R.
Caldwell
and
C. W.
Van Atta
, “
Characteristics of Ekman boundary layer instabilities
,”
J. Fluid Mech.
44
,
79
95
(
1970
).
38.
I.
Kanda
, “
A laboratory study of columnar baroclinic vortices in a continuously stratified fluid
,”
Dyn. Atmos. Oceans
38
,
69
92
(
2004
).
39.
C. R.
Doering
and
J.-L.
Thiffeault
, “
Multiscale mixing efficiencies for steady sources
,”
Phys. Rev. E
74
,
025301
(
2006
).
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