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anticyclonic gyre asymmetries: laboratory and intermediate-model experiments
ofa~s andoceans ELSEVIER
Dynamics of Atmospheres and Oceans 27 (1997) 219-232
Cyclonic/anticyclonic gyre asymmetries: laboratory and intermediate-model experiments John E. Hart a,,, Bryan Adler b, Robert Leben c a Program in Atmospheric and Oceanic Sciences, Department of Astrophysical, Planetary. and Atmospheric Sciences, University of Colorado, Campus Box 391, Boulder, CO 80309-0391, USA b Department of Aerospace Engineering, University of Colorado, Boulder, CO 80309, USA c Colorado Center for Astrodynamics Research, University of Colorado, Boulder, CO 80309, USA Received 3 January 1996; revised 27 September 1996; accepted 7 October 1996
Abstract
Laboratory models of rapidly rotating geophysical flows often show significant asymmetries with respect to the sign of the gyre forcing. In this paper we focus on the instability of separated boundary currents and the resulting transition to time-dependent motion in a slightly sliced cylinder driven by a differentially rotating lid. This transition occurs more readily for cyclonic (co-rotating) gyre forcing, when compared with that observed for anticyclonic forcing, even though the system Rossby number is very small. Quasi-geostrophic models are invariant to changes in the sign of the forcing, so a more accurate theoretical framework must be used to capture the observed asymmetries. An intermediate model, which includes a second-order nonlinear Ekman suction relation, is proposed and integrated numerically. The results are in significantly better agreement with the laboratory observations, and simple diagnostics illustrate which of the higher-order physical effects are responsible for the enhanced instability of cyclonically forced gyres. © 1997 Elsevier Science B.V.
1. Introduction
There are many examples of rapidly rotating laboratory flows that show substantial differences in behavior if the sign of the forcing is changed. In two-layer f-plane models of baroclinic instability and the transition to baroclinic chaos, where the base rotation is defined to be cyclonic, it is observed that systems driven by a co-rotating 'wind stress' (e.g. a cyclonically revolving upper disk) are much more unstable, and aperiodic states
* Corresponding author. 0377-0265/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. Pll S 0 3 7 7 - 0 2 6 5 ( 9 7 ) 0 0 0 1 1 - 0
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are obtained at substantially weaker forcing than when a counter-rotating or anticyclonic wind stress is applied (Hart, 1985). This occurs in spite of the fact that the Rossby number e-= o~/2/2, based on the differential driving frequency oJ and the base rotation rate ~Q, is less than one-tenth. Quasi-geostrophic (QG) numerical simulations of the transition to baroclinic chaos more or less split the difference between these two situations, being more stable than the cyclonic laboratory gyres, and more unstable than the anticyclonic gyres (Mundt et al.. 1995). Similar comparisons have been obtained for the onset of separation and the transition to chaos in a homogeneous single-layer experiment on the wind-driven ocean circulation in a sliced annulus (Albaiz and Hart, 1991; Albaiz et al., 1993). In this one-layer fluid, the cyclonically forced system is more prone to instability and chaos than experiments with anticyclonic driving, and the quasi-geostrophic model predictions again lie somewhere between results of these two different laboratory realizations. Other examples of such 'gyre asymmetries' include differences in the nature of the barotropic instability of free shear layers resulting from changes in sign of the rotation of the driving disk (Hide and Titman, 1967), and the qualitatively different motions observed in the rotating annulus when the sign of the temperature difference is reversed. In this paper we seek a quantitative interpretation of this type of asymmetry, which manifests itself by having qualitatively different regimes or bifurcation points upon reversal of the direction of the forcing, in the specific problem of the transition to time dependence in mechanically driven one-layer flow over a shallow slope inside a rapidly rotating full cylinder. Results from a second-order-accurate 'intermediate' model, with errors of order E2 (as opposed to the e error in the quasi-geostrophic formulation), are compared with experiments on separation and instability in a slightly sliced full cylinder. It is likely that the symmetry-breaking physical effects found to be important here play a role in the other situations mentioned above. The laboratory experiments are summarized in Section 2, and the computational model is tormulated and tested against analytical solutions in Section 3. The numerical and experimental observations are compared in Section 4, and Section 5 contains a brief discussion of the underlying processes shaping these flows.
2. Experimental method and visualizations The experiment is shown schematically in Fig. I. It is a simple variant on the classic wind-driven ocean model of Beardsley (1969). A constant density fluid (water) is held in a slightly sliced cylinder of radius 22.5 cm and mean depth 14.5 cm. Motion is driven by a differentially rotating contact lid. The sign of the forcing is denoted by S (_+ 1), and the Rossby number E is held fixed at 0.05. The primary flow visualization was by video photography of a platelet suspension looking down through the top lid. In addition, a few runs included monitoring the voltage output of a hot-thermistor anemometer inserted about 1 cm into the sidewall boundary layer at the shallow side of the cylinder. Experiments were carried out by fixing w and ~Q such that the ratio gave a constant Rossby number of 0.05, then letting the flow evolve until the presence or absence of time dependence in the boundary layer region could be ascertained. After this, the sign
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of the forcing was impulsively flipped, and after another waiting period (about 1 h), the state of the system was again recorded on videotape. Typical values of the base period ranged from 1.5 to 3.0s, with the corresponding lid range being 15-30s. The base stability was better than one part in 1 0 4 , whereas that for the lid was about one part in 1 0 3 . Two important dimensionless numbers for this flow are the slope parameter 2 f2Ltan a ~,=
(1)
toH
and the non-dimensional bottom-friction spin-down time h-
toH 4 ~
(2)
Other parameters, such as the lateral Ekman number E -= v / 2 O f f , can be obtained from these, using the dimensional data given in Fig. 1. The main difference between the present experiments and those of Beardsley is that the slope a = t a n - l ( S h / 2 L) = 1.02 ° is very small. Therefore the topography only generates a perturbation to the basic axisymmetric interior rotation driven by the lid forcing, though it does induce enough spatial asymmetry to permit separation of the vertical boundary layer at the cylinder wall. For this small slope, the sidewall structure takes the form of a nonlinear Stewartson (1957) layer, as opposed to the Munk layers found in previous setups of this type which have a stronger topographic /3-effect. For example, if T = 0.56, e = 0.05, and h = 7.41, then E = 2.4 × l0 -6. If the boundary layers were linear, the non-dimensional (with respect to L) Munk layer e-folding thickness is ~m = 2 ( E / e T ) ~/3 = 0.089, and the Stewartson layer e-folding thickness is (5s = ( 2 A E / ~ ) 1 / 2 = 0.026. Thus the vorticity gradients generated at the sidewall are trapped in a narrow zone in which the topographic /3-effect is relatively weak. It is also useful to restate that in this particular
iiiiiiiiiiiiiiiiiiiiiiiiiT ,Y
,I
Fig. 1. Schematic representation of the laboratory experiment. The tank radius L = 22.5 cm, the mean depth H = 14.5cm, and the topography amplitude 8h = 0.8cm. The fluid is water, with viscosity, at the laboratory mean temperature of 22°C, v = 0.0095cm2 s -I . The driving disk is made of machined Plexiglas and is about 2.5cm thick. The gap between the disk and the sidewall varies, but the maximum spacing is 0.1 mm.
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223
experiment, which has no closed depth contours, the quasi-geostrophic model including topography has complete cyclonic/anticyclonic invariance, so the B-effect itself is not responsible for the observed asymmetries. If, however, the experiments had included a parabolic free surface, there would be an inherent difference between situations with cyclonic and anticylonic forcing even in the QG model. In the experiments we held y fixed at 0.56 and increased A in steps of about 2% over a range that typically takes the system from nearly axisymmetric steady motion to 0-dependent separated motion, followed by generation of time-dependent waves by barotropic instability of the separation bubble, and ultimately to chaotic behavior of the gyre as the eddies shed off the separation region teleconnect zonally around the gyre to the generation site itself. The details of this process, which occurs in a similar way in the annulus, were discussed by Albaiz et al. (1993). However, in that study detailed measurements of gyre asymmetries were not made, as only a couple of experiments with anticyclonic forcing were carried out. In addition, the annulus is not the simplest system for looking at this problem, as flow separation and instability arise at both sides of the apparatus. Because the boundary layer vorticity will necessarily be of opposite sign at the inner and outer walls of the annulus, the question of gyre asymmetry (which we hypothesize to be related to ageostrophic motions in the boundary layers) becomes clouded. In the full cylinder the sidewall layer vorticity is simply opposite to that of the lid driving, and it changes from anticyclonic to cyclonic as the forcing is flipped from a cyclonic (co-rotating) to an anticyclonic (counter-rotating) state. Of most interest to us is the change in the bifurcation point for the transition to time dependence under reversal of the lid differential rotation. Fig. 2 shows typical differences in the imagery for various values of A. It is clear from Fig. 2(a) and Fig. 2(b) that at A = 6.4 the cyclonic case has an obvious wavetrain emanating from a point at about nine o'clock (denoted by the arrow), upstream of the shallows, which are at the bottom of the photographs. On the other hand, the anticyclonic case shows no evidence of waves. Spatial symmetry arguments suggest that if such waves were to occur with anticyclonic forcing they should emanate from about the same place, but propagate downstream (i.e. clockwise) from nine o'clock to the deep end at the top of the picture. At substantially higher values of A, waves are indeed observed in both cases, but those associated with the anticyclonic forcing are weaker and more regular. In Fig. 2(c) the motion at a fixed point in space (e.g. at a velocity sensor) is more or less periodic, whereas in Fig. 2(d) it is fairly chaotic. The chaotic regime with cyclonic forcing is characterized by anticyclonic eddies that make it all the way around the container, finishing back at the generation site as weak disturbances, along with the emanation of anticyclonic vortices of different amplitudes out of a localized instability site that is perturbed by said waves. When the eddies extend further into the interior fluid, the enhanced propagation rates cause them to overtake the previous vortex, whence mergers
Fig. 2. Photographs through the upper lid. y = 0.56. (a) A = 6.4, anticyclonic forcing; (b) A = 6.4, cyclonic forcing; (c) h. = 7.59, anticyclonic forcing; (d) A = 7.59, cyclonic forcing. The shallow point of the gyre is at the bottom of each picture. The basic rotation is defined to be cyclonic (counter-clockwise).
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Fig. 3. Photographs of the evolution of anticyclonic eddies, y = 0.56, A= 7.80. The time interval between shots is about 10s. The shallow point of the tank is at the bottom of each picture.
can occur. Fig. 3 illustrates some of these processes. Detailed observations o f the flow transition show that the c y c l o n i c gyre b e c o m e s unsteady at A = 5.90 _+ 0.05, whereas the anticyclonic gyre bifurcates to oscillatory motion at A = 6.60 _+ 0.05. The meaning of the _+ numbers in this paper is that, for example, instability and unsteadiness is observed at A = 5.95 but not at A = 5.85, with an experimental parameter step size of 0.1. In addition, there is a m e a s u r e m e n t uncertainty in obtaining A from the laboratory setup o f about 2% o w i n g to possible errors in determining H and u, the latter of which depends on small r o o m temperature variations.
.I.E. Hart et al. / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
225
3. Model formulation and testing The model is an adaptation of that developed by Hart (1995), hereafter H95, to cylindrical geometry. The details are contained in this reference, so only a brief summary is given here. Basically, the flow is considered to be depth independent in the interior, outside the Ekman layers at the top and bottom, and it is assumed that inner z-dependent E 1/3 layers adjacent to the sidewall are passive and occupy only a small radial portion of the vertical E 1/4 Stewartson layers found there. In the experiments, the lateral Ekman number E = 3 × 10 - 6 SO that there should be a reasonably sizeable separation in the two layer widths. The interior motion is represented in terms of rotational and horizontally divergent parts associated with a streamfunction ~ ( r , O , t ) and a velocity potential ~b(r,O,t), respectively. The latter is presumed of order e with respect to the former, and the vorticity equation is developed by keeping all terms up to order e. In the quasi-geostrophic formulation, planetary vorticity is stretched by differential vertical velocities at the top and bottom of the columns, and these in turn are given by slope-induced motions and by an Ekman pumping that is determined by linear Ekman layer dynamics. To build an internally consistent e-accurate model (as opposed to QG, which has order E errors), it is necessary to include both the vertical velocities induced by interactions between the divergent flow and the bottom slope, as well as the contributions to the Ekman layer suction by nonlinear advections inside these boundary layers themselves. The requisite formulae were presented in H95. The governing prognostic equation for ~ ( r , 0 , t ) , based on a non-dimensionalization using L as the length scale, toL as the velocity scale, and to-1 as the time scale, is [
1
_
E(I+EV2~)]
4-A-(T - - E - ~ (a)
~Vz~ __ + J(a/t,V2a/t)
Ot
EV4~+(V~b'V)V2~+ • (b)
(l+eV2a/t) [ 1 [ i-'e~y -2--~-(S V2qt) + j ( y y , q t ) (c)
+(V~b" V ) y y + Aw2 (d)
(3)
The velocity potential is found by equating the depth-integrated horizontal divergence to the lowest-order vertical velocity difference at the top and bottom of the interior domain. In this way, ¢b(r,O,t) = 4)h + ~bv is determined from V2~h = e J ( y y , g r ) , and ~ =
4
~
(4)
These equations are solved with ~ = i ~ / ~ r = 04)h/Or = 0 at r = 1 and with regularity at r = 0. J represents the Jacobian advection operator (in cylindrical coordinates) and y = rcos 0 is the coordinate going from the axis to the shallow point. The nonlinear contribution to the Ekman suction used in this problem is the boundary layer version of the general equation in H95. Because the sidewall region is where the
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226
interior vorticity is highest, such a representation is appropriate (and simpler to implement), so we use + Aw~- = ~-~ ~ 14 ~-~r] + 18 rOr - 3 S - -~r
14l, Or 2 - 7 S r Or 2 + 11 r - 2 4
(5) where ~, is the rotational azimuthal velocity component OtF/Or. Eq. (5) includes reference to the particular forcing used in the experiments: solid rotation from above at rate S = +1. The quasi-geostrophic model is recovered by setting e = 0 everywhere except in the lateral diffusion term (the first on the right-hand side of Eq. (3)). Referring to Eq. (3), left to right, the higher-order effects are (a) a non-linear scaling of time because of isallobaric wind-induced Ekman suction (Young, 1973), (b) advection of vorticity by the divergent part of the flow, (c) stretching of total vorticity by the lowest-order vertical velocity difference from top to bottom of the vortex columns (numerator), and non-uniform height of vortex columns because of the slope (denominator; this latter effect is uniformly small), (d) stretching of planetary vorticity by the vertical velocity generated by the divergent part of the current moving up the slope, and (e) the stretching of planetary vorticity by the Ekman pumping owing to nonlinear advection in the Ekman layers. Eq. (4) shows that the divergent currents arise in response to the squeezing of fluid columns as they move over the slope, and by a similar vertical compression or expansion owing to the primary Ekman suction velocities. It should be noted that with Otlt/Or = 0 at r = 1, fluid appears to flow out of the sidewall (Ock~/Or 4= 0 at r = 1) in response to the mass flux coming out of the corner where the upper disk meets the sidewall. It is assumed that this flux comes out of the inner z-dependent sidewall structure and is distributed uniformly with depth. This 'intermediate' model is integrated numerically using a second-order conservative finite difference scheme where the elliptic equations are handled with a multigrid covariant fast Poisson solver. The technique is similar to that used by Albaiz et al. (1993). To verify the computational method and coding, axisymmetric simulations, obtained by setting y = 0 and integrating to a steady state, were compared with analytic boundary layer solutions of the model, as well as with results from a one-dimensional fourth-order Runge-Kutta shooting technique. We first note that when higher-order effects are included, the interior axisymmetric motion is still in solid rotation, but no longer at half the rate of the lid. The interior vorticity ~ is given, from Eqs. (3)-(5), by ~:i(S) =VzaP'i = - ~
l
]-~
l+Se-
100
]
Near the sidewall the vorticity becomes large and variable, but an analytic solution can be obtained if the boundary layer approximation t : / r << Oc/Or is made. Writing c = [ r - - v ' ( x ----(1 -- r ) / F ) ] ~ i / 2 , where /-2 = 2EA/E is the square of the linear Stewartson layer thickness, the flow near r = 1 is governed by the nonlinear equation •2 U' --
Ox 2
_
Ut
v
=-rlc
OU'
Ox'
13Se rl=- 40/-
J.E. Hart et al. / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
227
which is solved with v'(x ~ oo)= 0 and v'(x = 0 ) = 1. The radial shear at the wall is easily obtained by noting, following H95, that Or' q- 1 -= , where qe -q Ox ]r-I rt
=e
-'/2/2-
I
The two solutions of the transcendental equation for q give the shears for the two signs of S at any particular value of r/. Alternatively, the wall shear (or vorticity) can be obtained by numerically integrating Eqs. (3)-(5) from a small value of r where the vorticity is uniform and slightly different from ~i, out to the wall. The interior vorticity is adjusted (by a few parts per billion) until v ( r = 1)= 0. The results from the 1-D shooting method (using up to 1024 points), the boundary layer analysis, and integrations of the full numerical code are compared in Table 1. Apart from illustrating that the 2-D numerical model at an (r,O) resolution of 65 × 128 is acceptably accurate, Table 1 shows that the sidewall layer widens in the cyclonic (S = 1) case and is narrower (with higher cyclonic wall vorticity) in the situation with anticyclonic forcing by the top disk (S = - 1 ) . The primary physical effect that causes this is the advection of vorticity by the divergent wind. In the S = 1 case the Ekman flux flows out of the comer, down the wall, and then back into the interior. This radial inflow lifts the Stewartson layer off the wall. Adding to this is the fact that for the anticyclonic sidewall layer (cyclonic lid forcing) the total vorticity is reduced in the boundary layer so that the stretching (and narrowing) produced by the lowest-order linear part of the Ekman suction is smaller. The nonlinear Ekman suction counters these effects, but only at a 30% level. The higher-order terms are, of course, more significant in the boundary layer than in the interior, because the local Rossby number at r = 1 is larger than the bulk Rossby number • by a factor that is equal to the magnitude of the wall values shown in Table 1. That is, the local Rossby number in the Stewartson layers is of order one. The boundary layer approximation works pretty well for these layers, which are about 1 cm wide. It is poorest in the largest •, S = 1 case, as this has the widest sidewall layer. In summary, the higher-order effects shaping this
Table 1 V o r t i c i t y f o r a x i s y m m e t r i c f l o w ( b a s e p e r i o d 2.1 s) •
Full c o d e
Boundary layer
Shooting method
( r = 1)
A n a l y s i s ( r = 1)
( r = 1)
QG
S=I
S=-I
S=1
S=-1
S=I
S=-I
0.0676
- 12.26
21.745
- 12.79
22.01
- 12.29
21.76
0.05
- 13.432
20.254
- 13.71
20.49
- 13.43
20.25
16.67
0.042
- 13.922
19.65
- 14.18
- 13.92
19.64
16.67
r i (code, r = 0)
sri ( a n a l y s i s , r = 0 )
S=l
S=-I
S=l
S=-I
QG
0.0676
0.9797
- 1.0203
0.9803
- 1.0210
1.0
0.050
0.9851
- 1.0154
0.9853
- 1.0154
1.0
0.042
0.9874
- 1.0124
0.9876
- 1.0128
1.0
16.67
228
J.E. Hart et al. / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
n o n l i n e a r Stewartson layer are dominated by terms (c) and (d) of Eq. (3), which act together and in sum are about three times as big as the n o n l i n e a r E k m a n suction effect
(e).
4. Results
Fig. 4(a) shows the evolution of velocity in a typical computation. Most runs were started from rest and allowed to evolve to a statistically steady state. Each case took about 40 h on a Dec Alphastation. The model gives a very accurate description of the onset and nature of the time dependence for S = + 1. Fig. 4(b) and Fig. 4(c) show one comparison of the frequency spectra. As is typical for other parameters as well, the model produces the 0 . 0 5 5 H z wave observed in the experiments very nicely. The
0.6
r t --
'o~
[ lam:6.75
0.2
(8 01
,
200
100
300
600
400 500 time (seconds)
700
800
?
-10
(b) -15--
i
0.1
0}2
013
t 0.4
0:5
i 0.6
0.7
0:8
:
]
t 0.9
1
freq(Hz)
,
r -. o.oo
J
O.lO
' 1 0.20 Frequency in Hz
o.3o
~.'.1, r,.,[1 0.40
Fig. 4. Comparison of computational and experimental data at y = 0.56, A = 6.75. (a) Model time series of
velocity at the shallow end with r = 0.95: (b) model frequency spectrum after transients have decayed; (c) frequency spectrum of the voltage from a hot-thermistor speed sensor embedded in the boundary layer (i.e. located about l cm in from the wall) at mid-depth over the shallowest point in the tank. The amplitude is uncalibrated.
J.E. Hart et a l . / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
229
Fig. 5. Streamfunction plots for 3' = 0.56, A= 6.25 (top), A= 7.4 (bottom). The contouring is nonlinear, to bring out details of the waves and eddies near the outer wall. harmonic contents are similar, but it is necessary to keep in mind that the hot-thermistor probe is a nonlinear device with a limited high-frequency response. Fig. 5 illustrates that the location of the wave-generation site, and the scales and amplitudes of eddy motion, are close to what is seen in the laboratory photographs. The bifurcation points in the model were found by carrying out a large number of computer runs with T fixed at 0.56, and with A moving upwards in steps of 0.05. Inspecting the time-asymptotic state from these numerical experiments gives the bifurcation values A ~ Ac for transition from steady to periodic motion as follows: 5.925 + 0.025 ( S = 1, intermediate model), 6.35 ___0.025 (quasi-geostrophic model), 7.225 + 0.025 (S = - 1, intermediate model).
5. Summary and discussion Comparing flow bifurcations is a useful way to test a model with respect to experimental results. Flow states, in particular the transition between steady and
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J.E. Hart et al. / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
time-dependent motion, are easier to observe than are the details of a particular velocity field, say, which usually requires sophisticated measurement equipment. We looked at the transition to unsteadiness seen in a rapidly rotating slightly sliced cylinder, subject to both anticyclonic and cyclonic forcing by a differentially rotating lid. The observed transition points are significantly different in these two situations, leading to a 'gyre asymmetry' in which cyclonically forced gyres are more unstable and chaotic than their anticyclonic counterparts. Quasi-geostrophic models of these flows are invariant to changes in the sign of the forcing, and so cannot reproduce this asymmetry. A relatively simple second-order-accurate model is proposed, although the computation time with all the added term~ is more than double that required for the QG formulation. This 'intermediate model' fairly accurately reproduces the experimental bifurcation for cyclonic forcing (model Ac = 5.925 + 0.025 vs. 5.90 _+ 0.05 observed). The QG prediction A~. = 6.35 is significantly higher. The intermediate model does not, however, agree as well with the bifurcation seen for anticyclonic forcing (model A~ = 7.21 _+ 0.025 vs. 6.60 _+ 0.05 observed). In both situations the intermediate model is a little more stable than the experiments, suggesting that higher-order e z corrections might lower the theoretical bifurcation points for both S = +1 and S = - 1 . That the anticylonic case comparison is less favorable may be due to the fact that the laboratory system with retrograde forcing requires a higher differential rotation to excite the time-dependent modes, and closeup views of the wavy disturbances from the top lid show that these are accompanied by small-scale instabilities in the Ekman boundary layer. Such instabilities are not observed for the S = + 1 waves, partly because the transition occurs at a lower value of A, but more importantly because Ekman layer instabilities are much preferred in the retrograde (i.e. spin-down-like) experiments, a fact that is easily seen by conducting gyre asymmetry experiments with y = 0 (or see Savas, 1983; Savas, 1987; Lopez and Weidman, 1997). Thus the generally stabilizing effect of Ekman bottom-drag may be reduced in the anticyclonic experiments, leading to an earlier transition to time dependence as A is raised, compared with the theory, which assumes smooth boundary layers. In any case, the intermediate model certainly captures the flavor of the observed asymmetry, and the agreement in critical point and wave frequency for S = + 1 is remarkable, given that this is a depth invariant model of a flow that does have a z-dependent inner sidewall boundary layer structure (the nonlinear E ~/3 layers). It is possible to ask which of the new terms in Eq. (3) are responsible for the theoretical destabilization of the flow for S = + 1, relative to QG dynamics. Fig. 6 shows results of a few runs at the fixed value of A = 6.0 to see which effects lead to larger amplitudes (i.e. more instability) or vice versa. Term (c), the addition of the relative vorticity to planetary vorticity when considering vortex stretching by the linear Ekrnan layer suction and by the topography, has the most dramatic influence. Turning this term off stabilizes the motion (it becomes steady). The presence of this term leads to more instability because it reduces the bottom frictional damping in the sidewall region (for S = + 1 the vorticity in the boundary layer is negative). In addition, it increases the effect of the slope in generating a spatially non-axisymmetric interior gyre in the first place because for S = + 1 the interior relative vorticity is positive. The wave amplitudes both appear to increase when the nonlinear Ekman suction (e) and the advection of
J.E. Hart et al. / Dynamics of Atraospheres and Oceans 27 (1997) 219-232 2.66
231
t
Ix ~
iI \\
2.64
]
2.62
,
l
I1\
I\
I
] ,
\
I "+.
', X
l
2.6 "l
\X
I xx
'+
1
" "
llll1
ii ....
•
"+2.58 25+
"/ ..... (b) off
2.~/
.
.
.
.
.+ f
.. :
:
co>off
/I -- (d,e)°ff 14~65 1470 14~75 1480 time (seconds)
":" '::
'
."::1
1
14~85 1490 1495 1500
Fig. 6. Effects o f various terms in the m o d e l equation on the oscillations at a theoretical velocity sensor at r = 0.95 for 3' = 0.56, A = 6.0. The terms referred to are indicated in Eq. (3).
vorticity by the divergent current (b) are turned off. Thus it appears that these effects are stabilizing. Further calculations show that the presence or absence of (a) or (d) has essentially no consequence in this system. We have also recomputed the bifurcation point for S = + 1 with various terms in Eq. (3) switched off. With a A step size of 0.05 we find the following critical points for )t: 5.925 + 0.025 (all terms present), 5.825 (with full vorticity stretching and vorticity advection by the divergent flow; terms (d) and (e) off), 5.75 (full stretching only; terms (b), (d) and (e) off), and 6.55 (vorticity advection by divergent flow only; terms (c), (d) and (e) off). This reconfirms that full vortex stretching is strongly destabilizing (taking )k from a QG value of 6.35 down to 5.75), that vorticity advection by the divergent wind is moderately stabilizing, and that the nonlinear Ekman suction is weakly stabilizing (i.e. taking the critical point from 5.825 to 5.925). These results are consistent with the analytical arguments presented in H95, which suggested that term (c) would be most important in helping the flow to satisfy the 'necessary condition for separation' that there be an adverse pressure gradient along the wall. As separation precedes instability, the effect of term (c) on the interior dynamics is similar in importance to its effect in the sidewall layer in setting up conditions ripe for time-dependent wave formation. From these calculations it appears that a simple model that just adds full vorticity stretching to the QG case is capable of capturing the qualitative essence of the gyre asymmetry in our flow situation. Needless to say, such a model is dramatically simpler than that given by the full Eqs. (3)-(5). In the future it will be interesting to see if models of this type can explain the asymmetries observed in other more complicated laboratory flows, in particular the strong preference for baroclinic instability and chaos in a two-layer f-plane rotating cylinder subject to cyclonic (vs. anticyclonic) forcing. Another question brought up by
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J.E. Hart et al. / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
this work is that of how the presence of unstable or turbulent Ekman layers may affect large-scale separation and barotropic-baroclinic instability problems in rotating fluids.
Acknowledgements This research was funded by the National Science Foundation Grant ATM-9025087. The authors would especially like to thank Scott Kittelman for help with the laboratory experiments, and Michael Zulauf for coding an early version of the model.
References Albaiz, A., Hart, J.E., 1991. Ordered and chaotic states following boundary layer separation in a rotating fluid. Geophys. Astrophys. Fluid Dyn. 58, 3-23. Albaiz, A., Hart, J.E., Leben, R., 1993. Flow separation and chaos in a rotating annulus. Comput. Fluid Dyn. 2~ 59-72. Beardsley, R.C., 1969. A laboratory model of the wind driven ocean circulations. J. Fluid Mech. 38, 255-271. Hart, J.E., 1985. A laboratory study of baroclinic chaos on the f-plane. Tellus 37A, 286-296. Hart, J.E., 1995. Nonlinear Ekman suction and ageostrophic effects in rapidly rotating flows. Geophys. Astrophys. Fluid Dyn. 79, 201-222. Hide, R., Titman, C.W., 1967. Detached shear layers in a rotating fluid. J. Fluid Mech. 29, 289-306. Lopez, J.M., Weidman, P.D., 1997. Stability of stationary endwall boundary layers during spindown. J. Fluid Mech. in press. Mundt, M.D., Hart, J.E., Ohlsen, D.R., 1995. Symmetries, sidewalls, and the transition to chaos in baroclinic systems. J. Fluid Mech. 300, 311-338. Savas, O., 1983. Circular waves on a stationary disk in rotating flow. Phys. Fluids. 26, 3445-3448. Savas, O., 1987. Stability of Bodewadt flow. J. Fluid Mech. 183, 77-94. Stewartson, K., 1957. On almost rigid rotations. J. Fluid Mech. 3, 17-26. Young, J.A., 1973. A theory for isallobaric air flow in the planetary boundary layer. J. Atmos. Sci. 30, 1584-1592.
Dynamics of Atmospheres and Oceans 27 (1997) 219-232
Cyclonic/anticyclonic gyre asymmetries: laboratory and intermediate-model experiments John E. Hart a,,, Bryan Adler b, Robert Leben c a Program in Atmospheric and Oceanic Sciences, Department of Astrophysical, Planetary. and Atmospheric Sciences, University of Colorado, Campus Box 391, Boulder, CO 80309-0391, USA b Department of Aerospace Engineering, University of Colorado, Boulder, CO 80309, USA c Colorado Center for Astrodynamics Research, University of Colorado, Boulder, CO 80309, USA Received 3 January 1996; revised 27 September 1996; accepted 7 October 1996
Abstract
Laboratory models of rapidly rotating geophysical flows often show significant asymmetries with respect to the sign of the gyre forcing. In this paper we focus on the instability of separated boundary currents and the resulting transition to time-dependent motion in a slightly sliced cylinder driven by a differentially rotating lid. This transition occurs more readily for cyclonic (co-rotating) gyre forcing, when compared with that observed for anticyclonic forcing, even though the system Rossby number is very small. Quasi-geostrophic models are invariant to changes in the sign of the forcing, so a more accurate theoretical framework must be used to capture the observed asymmetries. An intermediate model, which includes a second-order nonlinear Ekman suction relation, is proposed and integrated numerically. The results are in significantly better agreement with the laboratory observations, and simple diagnostics illustrate which of the higher-order physical effects are responsible for the enhanced instability of cyclonically forced gyres. © 1997 Elsevier Science B.V.
1. Introduction
There are many examples of rapidly rotating laboratory flows that show substantial differences in behavior if the sign of the forcing is changed. In two-layer f-plane models of baroclinic instability and the transition to baroclinic chaos, where the base rotation is defined to be cyclonic, it is observed that systems driven by a co-rotating 'wind stress' (e.g. a cyclonically revolving upper disk) are much more unstable, and aperiodic states
* Corresponding author. 0377-0265/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. Pll S 0 3 7 7 - 0 2 6 5 ( 9 7 ) 0 0 0 1 1 - 0
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are obtained at substantially weaker forcing than when a counter-rotating or anticyclonic wind stress is applied (Hart, 1985). This occurs in spite of the fact that the Rossby number e-= o~/2/2, based on the differential driving frequency oJ and the base rotation rate ~Q, is less than one-tenth. Quasi-geostrophic (QG) numerical simulations of the transition to baroclinic chaos more or less split the difference between these two situations, being more stable than the cyclonic laboratory gyres, and more unstable than the anticyclonic gyres (Mundt et al.. 1995). Similar comparisons have been obtained for the onset of separation and the transition to chaos in a homogeneous single-layer experiment on the wind-driven ocean circulation in a sliced annulus (Albaiz and Hart, 1991; Albaiz et al., 1993). In this one-layer fluid, the cyclonically forced system is more prone to instability and chaos than experiments with anticyclonic driving, and the quasi-geostrophic model predictions again lie somewhere between results of these two different laboratory realizations. Other examples of such 'gyre asymmetries' include differences in the nature of the barotropic instability of free shear layers resulting from changes in sign of the rotation of the driving disk (Hide and Titman, 1967), and the qualitatively different motions observed in the rotating annulus when the sign of the temperature difference is reversed. In this paper we seek a quantitative interpretation of this type of asymmetry, which manifests itself by having qualitatively different regimes or bifurcation points upon reversal of the direction of the forcing, in the specific problem of the transition to time dependence in mechanically driven one-layer flow over a shallow slope inside a rapidly rotating full cylinder. Results from a second-order-accurate 'intermediate' model, with errors of order E2 (as opposed to the e error in the quasi-geostrophic formulation), are compared with experiments on separation and instability in a slightly sliced full cylinder. It is likely that the symmetry-breaking physical effects found to be important here play a role in the other situations mentioned above. The laboratory experiments are summarized in Section 2, and the computational model is tormulated and tested against analytical solutions in Section 3. The numerical and experimental observations are compared in Section 4, and Section 5 contains a brief discussion of the underlying processes shaping these flows.
2. Experimental method and visualizations The experiment is shown schematically in Fig. I. It is a simple variant on the classic wind-driven ocean model of Beardsley (1969). A constant density fluid (water) is held in a slightly sliced cylinder of radius 22.5 cm and mean depth 14.5 cm. Motion is driven by a differentially rotating contact lid. The sign of the forcing is denoted by S (_+ 1), and the Rossby number E is held fixed at 0.05. The primary flow visualization was by video photography of a platelet suspension looking down through the top lid. In addition, a few runs included monitoring the voltage output of a hot-thermistor anemometer inserted about 1 cm into the sidewall boundary layer at the shallow side of the cylinder. Experiments were carried out by fixing w and ~Q such that the ratio gave a constant Rossby number of 0.05, then letting the flow evolve until the presence or absence of time dependence in the boundary layer region could be ascertained. After this, the sign
J.E. Hart et al. / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
221
of the forcing was impulsively flipped, and after another waiting period (about 1 h), the state of the system was again recorded on videotape. Typical values of the base period ranged from 1.5 to 3.0s, with the corresponding lid range being 15-30s. The base stability was better than one part in 1 0 4 , whereas that for the lid was about one part in 1 0 3 . Two important dimensionless numbers for this flow are the slope parameter 2 f2Ltan a ~,=
(1)
toH
and the non-dimensional bottom-friction spin-down time h-
toH 4 ~
(2)
Other parameters, such as the lateral Ekman number E -= v / 2 O f f , can be obtained from these, using the dimensional data given in Fig. 1. The main difference between the present experiments and those of Beardsley is that the slope a = t a n - l ( S h / 2 L) = 1.02 ° is very small. Therefore the topography only generates a perturbation to the basic axisymmetric interior rotation driven by the lid forcing, though it does induce enough spatial asymmetry to permit separation of the vertical boundary layer at the cylinder wall. For this small slope, the sidewall structure takes the form of a nonlinear Stewartson (1957) layer, as opposed to the Munk layers found in previous setups of this type which have a stronger topographic /3-effect. For example, if T = 0.56, e = 0.05, and h = 7.41, then E = 2.4 × l0 -6. If the boundary layers were linear, the non-dimensional (with respect to L) Munk layer e-folding thickness is ~m = 2 ( E / e T ) ~/3 = 0.089, and the Stewartson layer e-folding thickness is (5s = ( 2 A E / ~ ) 1 / 2 = 0.026. Thus the vorticity gradients generated at the sidewall are trapped in a narrow zone in which the topographic /3-effect is relatively weak. It is also useful to restate that in this particular
iiiiiiiiiiiiiiiiiiiiiiiiiT ,Y
,I
Fig. 1. Schematic representation of the laboratory experiment. The tank radius L = 22.5 cm, the mean depth H = 14.5cm, and the topography amplitude 8h = 0.8cm. The fluid is water, with viscosity, at the laboratory mean temperature of 22°C, v = 0.0095cm2 s -I . The driving disk is made of machined Plexiglas and is about 2.5cm thick. The gap between the disk and the sidewall varies, but the maximum spacing is 0.1 mm.
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J.E. Hart et al. / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
223
experiment, which has no closed depth contours, the quasi-geostrophic model including topography has complete cyclonic/anticyclonic invariance, so the B-effect itself is not responsible for the observed asymmetries. If, however, the experiments had included a parabolic free surface, there would be an inherent difference between situations with cyclonic and anticylonic forcing even in the QG model. In the experiments we held y fixed at 0.56 and increased A in steps of about 2% over a range that typically takes the system from nearly axisymmetric steady motion to 0-dependent separated motion, followed by generation of time-dependent waves by barotropic instability of the separation bubble, and ultimately to chaotic behavior of the gyre as the eddies shed off the separation region teleconnect zonally around the gyre to the generation site itself. The details of this process, which occurs in a similar way in the annulus, were discussed by Albaiz et al. (1993). However, in that study detailed measurements of gyre asymmetries were not made, as only a couple of experiments with anticyclonic forcing were carried out. In addition, the annulus is not the simplest system for looking at this problem, as flow separation and instability arise at both sides of the apparatus. Because the boundary layer vorticity will necessarily be of opposite sign at the inner and outer walls of the annulus, the question of gyre asymmetry (which we hypothesize to be related to ageostrophic motions in the boundary layers) becomes clouded. In the full cylinder the sidewall layer vorticity is simply opposite to that of the lid driving, and it changes from anticyclonic to cyclonic as the forcing is flipped from a cyclonic (co-rotating) to an anticyclonic (counter-rotating) state. Of most interest to us is the change in the bifurcation point for the transition to time dependence under reversal of the lid differential rotation. Fig. 2 shows typical differences in the imagery for various values of A. It is clear from Fig. 2(a) and Fig. 2(b) that at A = 6.4 the cyclonic case has an obvious wavetrain emanating from a point at about nine o'clock (denoted by the arrow), upstream of the shallows, which are at the bottom of the photographs. On the other hand, the anticyclonic case shows no evidence of waves. Spatial symmetry arguments suggest that if such waves were to occur with anticyclonic forcing they should emanate from about the same place, but propagate downstream (i.e. clockwise) from nine o'clock to the deep end at the top of the picture. At substantially higher values of A, waves are indeed observed in both cases, but those associated with the anticyclonic forcing are weaker and more regular. In Fig. 2(c) the motion at a fixed point in space (e.g. at a velocity sensor) is more or less periodic, whereas in Fig. 2(d) it is fairly chaotic. The chaotic regime with cyclonic forcing is characterized by anticyclonic eddies that make it all the way around the container, finishing back at the generation site as weak disturbances, along with the emanation of anticyclonic vortices of different amplitudes out of a localized instability site that is perturbed by said waves. When the eddies extend further into the interior fluid, the enhanced propagation rates cause them to overtake the previous vortex, whence mergers
Fig. 2. Photographs through the upper lid. y = 0.56. (a) A = 6.4, anticyclonic forcing; (b) A = 6.4, cyclonic forcing; (c) h. = 7.59, anticyclonic forcing; (d) A = 7.59, cyclonic forcing. The shallow point of the gyre is at the bottom of each picture. The basic rotation is defined to be cyclonic (counter-clockwise).
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,LE. Hart et al. / Dynamics c~IA tmospheres and Oceans 27 (1997) 219-232
Fig. 3. Photographs of the evolution of anticyclonic eddies, y = 0.56, A= 7.80. The time interval between shots is about 10s. The shallow point of the tank is at the bottom of each picture.
can occur. Fig. 3 illustrates some of these processes. Detailed observations o f the flow transition show that the c y c l o n i c gyre b e c o m e s unsteady at A = 5.90 _+ 0.05, whereas the anticyclonic gyre bifurcates to oscillatory motion at A = 6.60 _+ 0.05. The meaning of the _+ numbers in this paper is that, for example, instability and unsteadiness is observed at A = 5.95 but not at A = 5.85, with an experimental parameter step size of 0.1. In addition, there is a m e a s u r e m e n t uncertainty in obtaining A from the laboratory setup o f about 2% o w i n g to possible errors in determining H and u, the latter of which depends on small r o o m temperature variations.
.I.E. Hart et al. / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
225
3. Model formulation and testing The model is an adaptation of that developed by Hart (1995), hereafter H95, to cylindrical geometry. The details are contained in this reference, so only a brief summary is given here. Basically, the flow is considered to be depth independent in the interior, outside the Ekman layers at the top and bottom, and it is assumed that inner z-dependent E 1/3 layers adjacent to the sidewall are passive and occupy only a small radial portion of the vertical E 1/4 Stewartson layers found there. In the experiments, the lateral Ekman number E = 3 × 10 - 6 SO that there should be a reasonably sizeable separation in the two layer widths. The interior motion is represented in terms of rotational and horizontally divergent parts associated with a streamfunction ~ ( r , O , t ) and a velocity potential ~b(r,O,t), respectively. The latter is presumed of order e with respect to the former, and the vorticity equation is developed by keeping all terms up to order e. In the quasi-geostrophic formulation, planetary vorticity is stretched by differential vertical velocities at the top and bottom of the columns, and these in turn are given by slope-induced motions and by an Ekman pumping that is determined by linear Ekman layer dynamics. To build an internally consistent e-accurate model (as opposed to QG, which has order E errors), it is necessary to include both the vertical velocities induced by interactions between the divergent flow and the bottom slope, as well as the contributions to the Ekman layer suction by nonlinear advections inside these boundary layers themselves. The requisite formulae were presented in H95. The governing prognostic equation for ~ ( r , 0 , t ) , based on a non-dimensionalization using L as the length scale, toL as the velocity scale, and to-1 as the time scale, is [
1
_
E(I+EV2~)]
4-A-(T - - E - ~ (a)
~Vz~ __ + J(a/t,V2a/t)
Ot
EV4~+(V~b'V)V2~+ • (b)
(l+eV2a/t) [ 1 [ i-'e~y -2--~-(S V2qt) + j ( y y , q t ) (c)
+(V~b" V ) y y + Aw2 (d)
(3)
The velocity potential is found by equating the depth-integrated horizontal divergence to the lowest-order vertical velocity difference at the top and bottom of the interior domain. In this way, ¢b(r,O,t) = 4)h + ~bv is determined from V2~h = e J ( y y , g r ) , and ~ =
4
~
(4)
These equations are solved with ~ = i ~ / ~ r = 04)h/Or = 0 at r = 1 and with regularity at r = 0. J represents the Jacobian advection operator (in cylindrical coordinates) and y = rcos 0 is the coordinate going from the axis to the shallow point. The nonlinear contribution to the Ekman suction used in this problem is the boundary layer version of the general equation in H95. Because the sidewall region is where the
J.E. Hart et aL / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
226
interior vorticity is highest, such a representation is appropriate (and simpler to implement), so we use + Aw~- = ~-~ ~ 14 ~-~r] + 18 rOr - 3 S - -~r
14l, Or 2 - 7 S r Or 2 + 11 r - 2 4
(5) where ~, is the rotational azimuthal velocity component OtF/Or. Eq. (5) includes reference to the particular forcing used in the experiments: solid rotation from above at rate S = +1. The quasi-geostrophic model is recovered by setting e = 0 everywhere except in the lateral diffusion term (the first on the right-hand side of Eq. (3)). Referring to Eq. (3), left to right, the higher-order effects are (a) a non-linear scaling of time because of isallobaric wind-induced Ekman suction (Young, 1973), (b) advection of vorticity by the divergent part of the flow, (c) stretching of total vorticity by the lowest-order vertical velocity difference from top to bottom of the vortex columns (numerator), and non-uniform height of vortex columns because of the slope (denominator; this latter effect is uniformly small), (d) stretching of planetary vorticity by the vertical velocity generated by the divergent part of the current moving up the slope, and (e) the stretching of planetary vorticity by the Ekman pumping owing to nonlinear advection in the Ekman layers. Eq. (4) shows that the divergent currents arise in response to the squeezing of fluid columns as they move over the slope, and by a similar vertical compression or expansion owing to the primary Ekman suction velocities. It should be noted that with Otlt/Or = 0 at r = 1, fluid appears to flow out of the sidewall (Ock~/Or 4= 0 at r = 1) in response to the mass flux coming out of the corner where the upper disk meets the sidewall. It is assumed that this flux comes out of the inner z-dependent sidewall structure and is distributed uniformly with depth. This 'intermediate' model is integrated numerically using a second-order conservative finite difference scheme where the elliptic equations are handled with a multigrid covariant fast Poisson solver. The technique is similar to that used by Albaiz et al. (1993). To verify the computational method and coding, axisymmetric simulations, obtained by setting y = 0 and integrating to a steady state, were compared with analytic boundary layer solutions of the model, as well as with results from a one-dimensional fourth-order Runge-Kutta shooting technique. We first note that when higher-order effects are included, the interior axisymmetric motion is still in solid rotation, but no longer at half the rate of the lid. The interior vorticity ~ is given, from Eqs. (3)-(5), by ~:i(S) =VzaP'i = - ~
l
]-~
l+Se-
100
]
Near the sidewall the vorticity becomes large and variable, but an analytic solution can be obtained if the boundary layer approximation t : / r << Oc/Or is made. Writing c = [ r - - v ' ( x ----(1 -- r ) / F ) ] ~ i / 2 , where /-2 = 2EA/E is the square of the linear Stewartson layer thickness, the flow near r = 1 is governed by the nonlinear equation •2 U' --
Ox 2
_
Ut
v
=-rlc
OU'
Ox'
13Se rl=- 40/-
J.E. Hart et al. / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
227
which is solved with v'(x ~ oo)= 0 and v'(x = 0 ) = 1. The radial shear at the wall is easily obtained by noting, following H95, that Or' q- 1 -= , where qe -q Ox ]r-I rt
=e
-'/2/2-
I
The two solutions of the transcendental equation for q give the shears for the two signs of S at any particular value of r/. Alternatively, the wall shear (or vorticity) can be obtained by numerically integrating Eqs. (3)-(5) from a small value of r where the vorticity is uniform and slightly different from ~i, out to the wall. The interior vorticity is adjusted (by a few parts per billion) until v ( r = 1)= 0. The results from the 1-D shooting method (using up to 1024 points), the boundary layer analysis, and integrations of the full numerical code are compared in Table 1. Apart from illustrating that the 2-D numerical model at an (r,O) resolution of 65 × 128 is acceptably accurate, Table 1 shows that the sidewall layer widens in the cyclonic (S = 1) case and is narrower (with higher cyclonic wall vorticity) in the situation with anticyclonic forcing by the top disk (S = - 1 ) . The primary physical effect that causes this is the advection of vorticity by the divergent wind. In the S = 1 case the Ekman flux flows out of the comer, down the wall, and then back into the interior. This radial inflow lifts the Stewartson layer off the wall. Adding to this is the fact that for the anticyclonic sidewall layer (cyclonic lid forcing) the total vorticity is reduced in the boundary layer so that the stretching (and narrowing) produced by the lowest-order linear part of the Ekman suction is smaller. The nonlinear Ekman suction counters these effects, but only at a 30% level. The higher-order terms are, of course, more significant in the boundary layer than in the interior, because the local Rossby number at r = 1 is larger than the bulk Rossby number • by a factor that is equal to the magnitude of the wall values shown in Table 1. That is, the local Rossby number in the Stewartson layers is of order one. The boundary layer approximation works pretty well for these layers, which are about 1 cm wide. It is poorest in the largest •, S = 1 case, as this has the widest sidewall layer. In summary, the higher-order effects shaping this
Table 1 V o r t i c i t y f o r a x i s y m m e t r i c f l o w ( b a s e p e r i o d 2.1 s) •
Full c o d e
Boundary layer
Shooting method
( r = 1)
A n a l y s i s ( r = 1)
( r = 1)
QG
S=I
S=-I
S=1
S=-1
S=I
S=-I
0.0676
- 12.26
21.745
- 12.79
22.01
- 12.29
21.76
0.05
- 13.432
20.254
- 13.71
20.49
- 13.43
20.25
16.67
0.042
- 13.922
19.65
- 14.18
- 13.92
19.64
16.67
r i (code, r = 0)
sri ( a n a l y s i s , r = 0 )
S=l
S=-I
S=l
S=-I
QG
0.0676
0.9797
- 1.0203
0.9803
- 1.0210
1.0
0.050
0.9851
- 1.0154
0.9853
- 1.0154
1.0
0.042
0.9874
- 1.0124
0.9876
- 1.0128
1.0
16.67
228
J.E. Hart et al. / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
n o n l i n e a r Stewartson layer are dominated by terms (c) and (d) of Eq. (3), which act together and in sum are about three times as big as the n o n l i n e a r E k m a n suction effect
(e).
4. Results
Fig. 4(a) shows the evolution of velocity in a typical computation. Most runs were started from rest and allowed to evolve to a statistically steady state. Each case took about 40 h on a Dec Alphastation. The model gives a very accurate description of the onset and nature of the time dependence for S = + 1. Fig. 4(b) and Fig. 4(c) show one comparison of the frequency spectra. As is typical for other parameters as well, the model produces the 0 . 0 5 5 H z wave observed in the experiments very nicely. The
0.6
r t --
'o~
[ lam:6.75
0.2
(8 01
,
200
100
300
600
400 500 time (seconds)
700
800
?
-10
(b) -15--
i
0.1
0}2
013
t 0.4
0:5
i 0.6
0.7
0:8
:
]
t 0.9
1
freq(Hz)
,
r -. o.oo
J
O.lO
' 1 0.20 Frequency in Hz
o.3o
~.'.1, r,.,[1 0.40
Fig. 4. Comparison of computational and experimental data at y = 0.56, A = 6.75. (a) Model time series of
velocity at the shallow end with r = 0.95: (b) model frequency spectrum after transients have decayed; (c) frequency spectrum of the voltage from a hot-thermistor speed sensor embedded in the boundary layer (i.e. located about l cm in from the wall) at mid-depth over the shallowest point in the tank. The amplitude is uncalibrated.
J.E. Hart et a l . / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
229
Fig. 5. Streamfunction plots for 3' = 0.56, A= 6.25 (top), A= 7.4 (bottom). The contouring is nonlinear, to bring out details of the waves and eddies near the outer wall. harmonic contents are similar, but it is necessary to keep in mind that the hot-thermistor probe is a nonlinear device with a limited high-frequency response. Fig. 5 illustrates that the location of the wave-generation site, and the scales and amplitudes of eddy motion, are close to what is seen in the laboratory photographs. The bifurcation points in the model were found by carrying out a large number of computer runs with T fixed at 0.56, and with A moving upwards in steps of 0.05. Inspecting the time-asymptotic state from these numerical experiments gives the bifurcation values A ~ Ac for transition from steady to periodic motion as follows: 5.925 + 0.025 ( S = 1, intermediate model), 6.35 ___0.025 (quasi-geostrophic model), 7.225 + 0.025 (S = - 1, intermediate model).
5. Summary and discussion Comparing flow bifurcations is a useful way to test a model with respect to experimental results. Flow states, in particular the transition between steady and
230
J.E. Hart et al. / Dynamics of Atmospheres and Oceans 27 (1997) 219-232
time-dependent motion, are easier to observe than are the details of a particular velocity field, say, which usually requires sophisticated measurement equipment. We looked at the transition to unsteadiness seen in a rapidly rotating slightly sliced cylinder, subject to both anticyclonic and cyclonic forcing by a differentially rotating lid. The observed transition points are significantly different in these two situations, leading to a 'gyre asymmetry' in which cyclonically forced gyres are more unstable and chaotic than their anticyclonic counterparts. Quasi-geostrophic models of these flows are invariant to changes in the sign of the forcing, and so cannot reproduce this asymmetry. A relatively simple second-order-accurate model is proposed, although the computation time with all the added term~ is more than double that required for the QG formulation. This 'intermediate model' fairly accurately reproduces the experimental bifurcation for cyclonic forcing (model Ac = 5.925 + 0.025 vs. 5.90 _+ 0.05 observed). The QG prediction A~. = 6.35 is significantly higher. The intermediate model does not, however, agree as well with the bifurcation seen for anticyclonic forcing (model A~ = 7.21 _+ 0.025 vs. 6.60 _+ 0.05 observed). In both situations the intermediate model is a little more stable than the experiments, suggesting that higher-order e z corrections might lower the theoretical bifurcation points for both S = +1 and S = - 1 . That the anticylonic case comparison is less favorable may be due to the fact that the laboratory system with retrograde forcing requires a higher differential rotation to excite the time-dependent modes, and closeup views of the wavy disturbances from the top lid show that these are accompanied by small-scale instabilities in the Ekman boundary layer. Such instabilities are not observed for the S = + 1 waves, partly because the transition occurs at a lower value of A, but more importantly because Ekman layer instabilities are much preferred in the retrograde (i.e. spin-down-like) experiments, a fact that is easily seen by conducting gyre asymmetry experiments with y = 0 (or see Savas, 1983; Savas, 1987; Lopez and Weidman, 1997). Thus the generally stabilizing effect of Ekman bottom-drag may be reduced in the anticyclonic experiments, leading to an earlier transition to time dependence as A is raised, compared with the theory, which assumes smooth boundary layers. In any case, the intermediate model certainly captures the flavor of the observed asymmetry, and the agreement in critical point and wave frequency for S = + 1 is remarkable, given that this is a depth invariant model of a flow that does have a z-dependent inner sidewall boundary layer structure (the nonlinear E ~/3 layers). It is possible to ask which of the new terms in Eq. (3) are responsible for the theoretical destabilization of the flow for S = + 1, relative to QG dynamics. Fig. 6 shows results of a few runs at the fixed value of A = 6.0 to see which effects lead to larger amplitudes (i.e. more instability) or vice versa. Term (c), the addition of the relative vorticity to planetary vorticity when considering vortex stretching by the linear Ekrnan layer suction and by the topography, has the most dramatic influence. Turning this term off stabilizes the motion (it becomes steady). The presence of this term leads to more instability because it reduces the bottom frictional damping in the sidewall region (for S = + 1 the vorticity in the boundary layer is negative). In addition, it increases the effect of the slope in generating a spatially non-axisymmetric interior gyre in the first place because for S = + 1 the interior relative vorticity is positive. The wave amplitudes both appear to increase when the nonlinear Ekman suction (e) and the advection of
J.E. Hart et al. / Dynamics of Atraospheres and Oceans 27 (1997) 219-232 2.66
231
t
Ix ~
iI \\
2.64
]
2.62
,
l
I1\
I\
I
] ,
\
I "+.
', X
l
2.6 "l
\X
I xx
'+
1
" "
llll1
ii ....
•
"+2.58 25+
"/ ..... (b) off
2.~/
.
.
.
.
.+ f
.. :
:
co>off
/I -- (d,e)°ff 14~65 1470 14~75 1480 time (seconds)
":" '::
'
."::1
1
14~85 1490 1495 1500
Fig. 6. Effects o f various terms in the m o d e l equation on the oscillations at a theoretical velocity sensor at r = 0.95 for 3' = 0.56, A = 6.0. The terms referred to are indicated in Eq. (3).
vorticity by the divergent current (b) are turned off. Thus it appears that these effects are stabilizing. Further calculations show that the presence or absence of (a) or (d) has essentially no consequence in this system. We have also recomputed the bifurcation point for S = + 1 with various terms in Eq. (3) switched off. With a A step size of 0.05 we find the following critical points for )t: 5.925 + 0.025 (all terms present), 5.825 (with full vorticity stretching and vorticity advection by the divergent flow; terms (d) and (e) off), 5.75 (full stretching only; terms (b), (d) and (e) off), and 6.55 (vorticity advection by divergent flow only; terms (c), (d) and (e) off). This reconfirms that full vortex stretching is strongly destabilizing (taking )k from a QG value of 6.35 down to 5.75), that vorticity advection by the divergent wind is moderately stabilizing, and that the nonlinear Ekman suction is weakly stabilizing (i.e. taking the critical point from 5.825 to 5.925). These results are consistent with the analytical arguments presented in H95, which suggested that term (c) would be most important in helping the flow to satisfy the 'necessary condition for separation' that there be an adverse pressure gradient along the wall. As separation precedes instability, the effect of term (c) on the interior dynamics is similar in importance to its effect in the sidewall layer in setting up conditions ripe for time-dependent wave formation. From these calculations it appears that a simple model that just adds full vorticity stretching to the QG case is capable of capturing the qualitative essence of the gyre asymmetry in our flow situation. Needless to say, such a model is dramatically simpler than that given by the full Eqs. (3)-(5). In the future it will be interesting to see if models of this type can explain the asymmetries observed in other more complicated laboratory flows, in particular the strong preference for baroclinic instability and chaos in a two-layer f-plane rotating cylinder subject to cyclonic (vs. anticyclonic) forcing. Another question brought up by
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this work is that of how the presence of unstable or turbulent Ekman layers may affect large-scale separation and barotropic-baroclinic instability problems in rotating fluids.
Acknowledgements This research was funded by the National Science Foundation Grant ATM-9025087. The authors would especially like to thank Scott Kittelman for help with the laboratory experiments, and Michael Zulauf for coding an early version of the model.
References Albaiz, A., Hart, J.E., 1991. Ordered and chaotic states following boundary layer separation in a rotating fluid. Geophys. Astrophys. Fluid Dyn. 58, 3-23. Albaiz, A., Hart, J.E., Leben, R., 1993. Flow separation and chaos in a rotating annulus. Comput. Fluid Dyn. 2~ 59-72. Beardsley, R.C., 1969. A laboratory model of the wind driven ocean circulations. J. Fluid Mech. 38, 255-271. Hart, J.E., 1985. A laboratory study of baroclinic chaos on the f-plane. Tellus 37A, 286-296. Hart, J.E., 1995. Nonlinear Ekman suction and ageostrophic effects in rapidly rotating flows. Geophys. Astrophys. Fluid Dyn. 79, 201-222. Hide, R., Titman, C.W., 1967. Detached shear layers in a rotating fluid. J. Fluid Mech. 29, 289-306. Lopez, J.M., Weidman, P.D., 1997. Stability of stationary endwall boundary layers during spindown. J. Fluid Mech. in press. Mundt, M.D., Hart, J.E., Ohlsen, D.R., 1995. Symmetries, sidewalls, and the transition to chaos in baroclinic systems. J. Fluid Mech. 300, 311-338. Savas, O., 1983. Circular waves on a stationary disk in rotating flow. Phys. Fluids. 26, 3445-3448. Savas, O., 1987. Stability of Bodewadt flow. J. Fluid Mech. 183, 77-94. Stewartson, K., 1957. On almost rigid rotations. J. Fluid Mech. 3, 17-26. Young, J.A., 1973. A theory for isallobaric air flow in the planetary boundary layer. J. Atmos. Sci. 30, 1584-1592.