18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
Spin up/down in linearly stratified fluid
ROMANI M.1 2, SOMMERIA J.2 & LONGHETTO A.1
1 Department of General Physics, University of Turin - Turin, Italy
2 Laboratoire LEGI-Coriolis, CNRS - Grenoble, France
Electronic mail: romani@to.infn.it
Abstract :
Vertical transport of horizontal momentum due to baroclinic instabilities is investigated in a rotating stratified fluid.
The experiments were carried out in the large rotating tank of the Coriolis-LEGI laboratory in Grenoble with flat
bottom and linear density stratification. The mean flow was generated by increasing (spin up) or decreasing (spin
down) the rotation rate of the platform. The velocity fields were measured by PIV (Particle Image Velocimetry)
technique. The dimensionless parameters that are important in determining the observed response are the Burger
number and the Rossby number.
Our experimental study reveals two very different behaviors. For O(1) Burger numbers, the bulk of the flow
remains axisymmetric with a very slow decay. We have checked that the decay is well described by the usual
vertical diffusion law, except in a region near the boundaries. The mismatch pertains to the presence of Ekman
pumping/suction, but this mechanism is confined by stratification to a relatively shallow layer.
For Burger numbers ? O(1), baroclinic instabilities are observed. The increased vertical momentum transport
leads to a much faster spin than the previous case. Neither the diffusion nor the Ekman pumping/suction can be
considered responsible for this enhanced transport of momentum and we interpret it as the effect of baroclinic
instabilities. Eady?s theory is used to analyse the quasi-geostrophic baroclinic stability problem and a model is
proposed to estimate the azimuthal velocity decay. The results are presented and discussed.
Key-words : spin up/down; rotating flow; stratified flow
1 Introduction
The spin of a fluid in a container represents an important process as it describes an adjustment
mechanism which arises frequently in both the atmosphere and oceans. The modern era of spin
up studies was initiated by Greenspan & Howard (1963), who considered the effect on a flow of
small changes in the rotation rate of its container. They showed that the spin up takes place in
three time scales corresponding to three distinct physical mechanisms: (1) the Ekman boundary
layers development as a result of the stresses on the rigid horizontal boundaries; (2) the spin up
of the fluid caused by a secondary, meridional circulation due to an imbalance between viscous,
centrifugal and pressure forces; (3) the viscous decay of any residual motion.
The literature has shown that the stratification adds a wealth of flow behaviors and instabili-
ties. The continuous model restricts the role of viscosity to boundary layer regions adjacent only
to solid surfaces and the interior core region can be regarded as essentially inviscid in character
unless the effect of viscosity permeates the entire fluid, which is not plausible if the cinematic
viscosity is O(10?6) m2 s?1, as the fluid usually considered.
A theoretical analysis of the linear spin-up process for a stably stratified fluid was presented
by Holton (1965), which proved to be qualitatively correct but had some inaccuracies in the
treatment of the sidewall layer, and by Pedlosky (1967), but he incorrectly predicted that spin
up would then be achieved by a diffusive mechanism. The correct description of the linear spin
up process was later presented by Walin (1969), who predicted that the effect of a stable strati-
fication was to restrict the recirculation of the fluid from the Ekman layers to a localized region
near the horizontal boundaries.
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
Buzyna & Veronis (1971), Saunders & Beardsley (1975), and Lee (1975) performed a se-
quence of experiments in a cylinder filled with a linearly stratified fluid. These three studies
clearly showed that the decay rate was faster than predicted by the linearized theory of Walin.
Later experiments by Greenspan (1980) confirmed the intricate nature of stratified spin up.
Spin down received less attention in the past, but it has been long understood that the resul-
ting flow may be subjected to sidewall instabilities due to an imbalance between centrifugal and
pressure gradient forces. The development of this type of instabilities was discussed by Max-
worthy (1971). Later, computations and experiments by Neitzel & Davis (1981) and Mathis &
Neitzel (1985) verified that such centrifugal instabilities arise, leading to the formation of an
array of Taylor-G?rtler vortices along the sidewall. The most recent treatments of this problem
are by Lopez (1996) and Lopez & Weidman (1996). It appears from their studies that spin down
is a problem with various instability mechanisms and a complex dynamics.
The spin problem is clearly central to an understanding of the dissipative mechanisms for
large scale motions, where rotational effects are dominant. Moreover, the stratification causes
vertical shear in the azimuthal flow: the baroclinic instability may taps the available potential
energy of the sloping isopycnal surfaces, thereby indirectly the kinetic energy of the mean flow.
The formation of large scale eddies through the development of the instabilities thus provides
an additional mechanism for the transport of angular momentum from solid boundaries to the
bulk of the fluid.
The onset of the baroclinic instabilities during spin up has been studied by Smirnov et al.
(2005), but they did not address its effect on the azimuthal velocity decay. The main goal of our
work is thus to deepen the role of the baroclinic instability as source of momentum transport
and to model the resulting dissipation on the relative flow.
2 Experimental Procedure
A series of spin-up and spin-down experiments, described in fig 1, were performed in the large
cylindrical rotating tank, 13 m diameter, of the Coriolis-LEGI laboratory.
EXP Tf ?f ?? Ro Bu
UP 1 50 0.126 +?f/12 0.08 0.06
DW 1 50 0.126 ??f/12 0.08 0.06
UP 2 50 0.126 +?f/6 0.17 0.06
DW 2 50 0.126 ??f/6 0.17 0.06
UP 3 100 0.063 +?f/6 0.17 0.26
DW 3 100 0.063 ??f/6 0.17 0.26
UP 4 200 0.031 +?f/3 0.32 1.16
DW 4 200 0.031 ??f/3 0.32 1.16
Figure 1: List of performed experimental runs. Tf is the
final (reference) rotation period of the tank [s], ?f is the final
rotation rate of the tank [rad s?1], ?? is the change in rota-
tion rate of the tank [rad s?1], Ro and Bu are the Rossby and
Burger number [?].
2.50
vertical
laser
sheet
horizontal
laser
sheet
camera
1.0
1.0 profilers
0.52.50
camera
6.50
0.60
windows
1.63
1.00
Figure 2: Experimental set-up, top view. The change
in the rotation rate of the tank was performed during a
time interval of approximatively five second and may be
considered as impulsive.
The tank was filled while rotating with linearly stratified fluid. The stratification was cha-
racterized by the buoyancy frequency N, defined by N2 ? (g/D)(??/?0) ? 0.49 s?2, with
g ? |vectorg| the gravity acceleration, D = 60 cm the total depth of the fluid layer, ? and ?0 the
background and reference densities. The buoyancy force on a small fluid element displaced
from equilibrium is proportional to N2. When N is large, the buoyancy force is strong and
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
tends to inhibit motion parallel to vectorg. The Coriolis force, by contrast, tends to make motions
transverse to vector? vertically uniform. When vector? and vectorg are parallel, as in our case, the two forces
compete and their relative importance is measured by the Burger number.
In order to avoid spurious perturbations caused by shear stress exerted by the air on the
fluid surface during the filling, the tank was covered with a floating foam canopy, removed
just before the experiments. The fluid was seeded with light-scattering particles as to use PIV
(Particle Image Velocimetry) system and the flow evolution was monitored for about 100 rota-
tion periods. There were two laser sheets, a horizontal one, 2.5 m?2.5 m, and a vertical one,
0.6 m?0.6 m, as sketched in fig. 2. The horizontal laser sheet could scan seven different ver-
tical positions in the bulk of the fluid. Images were taken by a digital CCD camera with spatial
resolution 1024 pixels ? 1024 pixels. Each velocity field was obtained from a burst of four
images, allowing a choice of an optimum time interval for the image cross-correlation. Such
bursts were repeated every 5 s at each horizontal level providing a measurement periodicity at
a given height of 35 s. A similar procedure was used for the fixed vertical laser sheet.
3 Experimental Results
Following Smirnov et al., the nature of the flow considered depends on six dimensional param-
eters, i.e., the rotation rate, ?; the change in the rotation rate, ??; the buoyancy frequency, N;
the horizontal length scale, L; the depth of the fluid layer, D; the kinematic viscosity, ?. Using
dimensional analysis, the problem depends on four dimensionless parameters which we take as
the Rossby number, Ro ? ??/?; the Burger number, Bu ? (ND/fL)2, with f ? 2? the
Coriolis parameter; the (vertical) Ekman number, Ev ? ?/(?D2); the aspect ratio ? ? D/L.
Recognizing that the fluid depth was kept constant and the Ekman number was very small,
Ev = O(10?4), the conditions for the flow development were primarily determined by the pa-
rameters Ro and Bu.
We chose to investigate the experiments UP-DW 1 and UP-DW 4, characterized respectively
by Bu = 0.06 and Bu = 1.16. The temporal evolution of the azimuthal velocity 1 in the bulk
of the fluid is shown in Fig.s 3-4. In the exp. UP 4 the velocity vector field is characterized by
circular streamlines quite perfectly oriented along the azimuthal direction for all the duration
of the acquisition. The effect of this behavior is clear in Fig. 3, black line: the stability of
0 2000 4000 6000 8000 10000 12000?7
?6
?5
?4
?3
?2
?1
0
1
2
t
v
Figure 3: (left) Spin up - azimuthal
velocity [cm s?1] temporal evolution [s]:
red = UP 1 black = UP 4 - averaged in a
small box at r = 535 cm h = 40 cm.
Figure 4: (right) Spin down - azimuthal
velocity [cm s?1] temporal evolution [s]:
red = UP 1 black = UP 4 - averaged in a
small box at r = 535 cm h = 40 cm. 0 2000 4000 6000 8000 10000?2
?1
0
1
2
3
4
5
6
7
t
v
the flow implies a slow velocity decay with fluctuations of the order of 10?1 cm s?1. In the
exp. UP 1 the velocity vector field shows two different regimes: at the outset of the exp. just
a relative small discrepancy is evident by respect to the previous case, and the streamlines pre-
serve a circular shape. In the temporal evolution - Fig. 3, red line - it corresponds to a series of
growing oscillations, quite regular in period, superimposed to a faster decay trend than the exp.
UP 4. During the second stage, after about 55 reference rotation periods (t ? 2750 s), a drastic
change occurs in the flow development, characterized by nonaxisymmetric disturbances which
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
can even lead to local reversal flow. The result on the azimuthal velocity temporal evolution
consists in wider and lesser regular oscillations, superimposed to a slower decay trend.
It?s therefore clear that the system has experienced a more rapid transport of momentum
from the boundaries to the bulk of the fluid in the exp. UP 1 than in UP 4.
During spin down the fluid motion was perturbed by sidewall instabilities, while it was al-
ways centrifugally stable in the case of spin up. Fig. 4, black line, shows that, even if the fluid
was baroclinically stable, the system has experienced a faster decay. In fact, the effect of the
centrifugal instabilities was to decelerate the fluid more rapidly by extracting energy from the
mean flow. We decided to not linger over it too long since this topic is beyond our scope.
The exp.s UP 1 and UP 4 are characterized by a different change in the rotation rate of the
tank, which means different Ro numbers, and a different reference rotation period, which im-
plies different Bu numbers. In both the exp.s Ro is relatively small and, since the effects of a
weak nonlinearity were considered modest 2, we supposed to be responsible for this discrepancy
in the flow development the Burger number.
First, we decided to investigate the mechanism of momentum transport in absence of ba-
roclinic instabilities. Since the baroclinic instability requires Bu < O(1), we considered the
exp.s UP 4 (Bu = 1.16). We used the finite differences method of solving the vertical diffusion
equation for the azimuthal velocity of the flow, with no-slip condition on the bottom and no-flux
condition on the free surface. Fig. 5 shows a comparison between experimental and numerical
0 20 40 60 80 100 120 140?6
?5
?4
?3
?2
?1
0
T
v
ts
(a) z = 6 cm
0 20 40 60 80 100 120 140?6
?5
?4
?3
?2
?1
0
T
v
(b) z = 13 cm
0 20 40 60 80 100 120 140?6
?5
?4
?3
?2
?1
0
T
v
(c) z = 20 cm
0 20 40 60 80 100 120 140?6
?5
?4
?3
?2
?1
0
T
v
(d) z = 26 cm
0 20 40 60 80 100 120 140?6
?5
?4
?3
?2
?1
0
T
v
(e) z = 33 cm
0 20 40 60 80 100 120 140?6
?5
?4
?3
?2
?1
0
T
v
(f) z = 40 cm
Figure 5: Spin up: Bu = 1.16 Ro = 0.32 - azimuthal velocity [cm s?1]: black = exp. blue = num. - temporal evolution (T is the
reference rotation period) at different vertical locations.
results for the azimuthal velocity at different vertical locations. At lower levels - fig.s 5(a), 5(b)
- the exp. velocity decays initially at a faster rate than the model and as the distance from the
bottom increases the decay rate weakens progressively - fig.s 5(c), 5(d). This result was inter-
preted as showing the effect of the confinement of the meridional circulation by stratification,
in accordance with the spatial structure of the streamfunction corresponding to the secondary
circulation velocity field in Buzyna & Veronis. Later in time, no further Ekman pumping occurs
because there is no difference in the local angular velocity between Ekman layer and adjacent
2A review on this topic is available in Duck & Foster (2001)
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
fluid: the decay rate of the residual motion is then well expressed by viscous diffusion since the
two curves show the same slope - fig.s 5(a), 5(b), 5(c), 5(d). Our analysis is in agreement with
the work of Walin, which recognized that the effect of the meridional circulation, in stratified
fluid, is to bring only a partial spin up of the interior, while the total adjustment to the final
state of solid body rotation is accomplished by viscous diffusion. The time ts required for the
partial spin up is ts ? r/(?f?S2)1/2, where r is the radius of the tank and S ? N/f: using
the parameters of UP 4 we obtain ts ? 16,5 Tf - fig. 5(a). In the bulk of the fluid - fig.s
5(e), 5(f) -, where the secondary circulation doesn?t penetrate, the exp. velocity decays faster
than the model, according with the work of Hyun et al. (1981): the non-uniform spin up in
stratified fluid produces azimuthal velocity gradients in the interior, which introduce viscous
diffusion sooner than anticipated by Walin. A qualitative comparison between our exp. case
(SD/L ? 1.08) and the num. results of Hyun et al. (SD/L ? 1.03) shows a good agreement.
Neither the diffusion nor the secondary circulation can be considered responsible for the
enhanced transport of momentum experienced in the case of low Burger number, therefore the
effect of the instabilities was to decelerate the relative flow more rapidly.
Next, we considered the exp. UP 1 (Bu = 0.06). We focussed on the growing oscilla-
tions, due to the onset of the instabilities, developed just after the change in the rotation rate
of the tank. In accordance with the normal mode analysis, the perturbations were interpre-
ted as wave-like. In order to estimate the wavelength of the instability, the azimuthal velocity
0 10 20 30 40 50?6.5
?6
?5.5
?5
?4.5
?4
?3.5
?3
?2.5
T
v
Figure 6:
(left) Spin up - temporal
evolution of azimuthal velocity [cm s?1]
averaged in two boxes along the same
streamline just after the change in the ro-
tation rate of the tank.
Figure 7: (right) The growth rate,
kci, for the most unstable Eady mode.
The solid vertical line is in correspon-
dence of the experimental value kexp ?
0.55 m?1.
0 0.5 1 1.50
0.01
0.02
0.03
0.04
0.05
0.06
k [m]
kc i
temporal profile was averaged in small boxes along the same circular streamline - fig. 6. In
agreement with the relative velocity of the flow and its radial profile, we interpreted the minima
(less negative points) as the velocity of the mean flow, while the maxima as the effect of the
superimposed velocity of the baroclinic waves. Knowing the frequency of the signal as well as
the time spent by the perturbation to move between the two boxes, we estimated the wavelength
of the instability. A comparison with Eady?s model shows a fine agreement - fig. 7.
We supposed that the faster velocity decay experienced in the case of low Burger number
could be connected to the adjustment of the isopycnal surfaces due to a conversion of available
potential energy in kinetic energy of the perturbations: an important role should be played by
the radial velocity associated to this process.
Using Boussinesq approximation in Navier-Stokes eq. for an inviscid, incompressible, stra-
tified fluid, we proposed a simple model for the rate of change of the azimuthal velocity of the
flow, averaged along the dimensions of the system:
?
?t(?u?) ? 2?f ???vr (1)
where u? and ur are the azimuthal and radial velocities of the flow. The transport term related
to the vertical Reynolds stress was found negligible according to the quasi-geostrophic theory.
Fig. 8 shows the radial velocity and its cumulative sum for the exp.s UP 1 and UP 4. In the
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
0 500 1000 1500 2000 2500 3000
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
1
t
u
Figure 8: (left) Temporal evolution
of radial velocity [cms?1] - black = UP
4, red = UP 1 - averaged in a rectangular
patch and its cumulative sum - blu = UP
4, green = UP 1.
Figure 9: (right) UP 1: tem-
poral evolution of azimuthal velocity
[cm s?1] - black = exp. averaged in
a rectangular patch, green = num. Num.
filtered data using a windows size: blu
= 10, red = 30 elements.
0 10 20 30 40 50 60?12
?10
?8
?6
?4
?2
0
2
T
u
case of Bu = O(1), baroclinically stable flow, the radial velocity - black line - remains almost
zero for all the duration of the acquisition. The cumulative sum - blue line -, which provides
information on its net effect, confirms this result. In the case of Bu ? O(1), the radial velocity
was affected by baroclinic instabilities: its profile - red line - is characterized by an oscillation
trend in the range 0.1 ? 0.4 cm s?1 and the cumulative sum - green line - shows that the net
effect is a positive radial velocity. This result is in accordance with the expectation since the
isopycnal surfaces were sloped towards the lateral wall of the tank and the measurements were
taken in the upper part of the fluid, where the lighter fluid spreaded over the heavier one.
Fig. 9 shows a comparison between experimental and numerical results for the azimuthal
velocity during the regime corresponding to the growing oscillations: the model - green line -
shows oscillations with the same period of the measurements - black line - but too wide. Better
results were achieved filtering the data using different windows size to obtain a running average
which could smooth the oscillations.
This work confirms that in the case of spin up with Bu ? O(1), the transport of momentum
is governed by baroclinic instabilities. Although the stratification produces a confinement of
the meridional circulation, the adjustment of the isopycnal surfaces allows a radial velocity field
which enhances the transport of momentum with respect to the diffusion process.
References
Buzyna G. Veronis G 1971 Spin up of a stratified fluid: theory and experiment J. Fluid Mech. 50
Duck P. W. Foster M. R. 2001 Spin up of homogeneous and stratified fluid Annu. Rev. Fluid Mech. 33
Greenspan H. P. 1980 A note on the spin up from rest of a stratified fluid Geophys. Astrophys. Fluid Dyn. 15
Greenspan H. P. Howard L. N. 1963 On a time-dependent motion of a rotating fluid J. Fluid Mech. 17
Holton J. R. 1965 The influence of viscous boundary layers on transient motions in a stratified rotating fluid J. Atmos. Sci. 22
Hyun J. N. Fowlis W. W. Warn-Varnas A. 1981 Numerical solutions for the spin up of a stratified fluid J. Fluid Mech. 117
Lee S. M. 1975 An investigation of stratified spin-up using a rotating laser doppler velocimeter MS thesis. Fla. State Univ.
Lopez J. M. 1996 Flow between a stationary and a rotating disk shrouded by a co-rotating cylinder Phys. Fluids 8
Lopez J. M. Weidman P. D. 1996 Stability of stationary endwall boundary layers during spin down J. Fluid Mech. 326
Mathis D. M. Davis S. H. 1985 Experiments on impulsive spin down to rest Phys. Fluids 28
Maxworthy T. 1971 Boundary layer stability and turbulence observation by flow visualization using dense Al flake suspension
Turbulence measurements in liquids; Proceedings of the Symposium, Univ. of Missouri
Neitzel P. D. Davis S. H. 1981 Centrifugal instabilities during spin down to rest in finite cylinders. Numerical experiments. J.
Fluid Mech. 102
Pedlosky J. 1967 The spin up of a stratified fluid J. Fluid Mech. 28
Saunders K. D. Beardsley R. C. 1975 An experimental study of the spin up of a thermally stratified rotating flow J. Geophys.
Fluid Dyn. 7
Smirnov S. A., Baines P. G., Boyer D. L., S. Voropayev S. I., Srdic-Mitrovic A. N. 2005 Long-time evolution of linearly
stratified spin-up flows in axisymmetric geometries Phys. Fluids 17
Walin G. 1969 Some aspects of time-dependent motion of a stratified rotating fluid J. Fluid Mech. 36
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