Local instability in a periodically forced sliced cylinder

ofa~s andoceans ELSEVIER

Dynamicsof Atmospheresand Oceans 25 (1996) 87-107

Local instability in a periodically forced sliced cylinder Michael Zulauf a,l John E. Hart b,, Robert Leben Michael Mundt b,2

a

a Department of Aerospace Engineering and Colorado Center for Astrodynamics Research, University of Colorado, Boulder, CO 80309, USA b Program in Atmospheric and Oceanic Sciences, University of Colorado, BouMer, CO 80309, USA

Received 16 February 1995; revised9 October 1995; accepted 6 November 1995

Abstract

This paper investigates the dynamics of mesoscale eddy generation by instability of time-varying flows. Laboratory experiments on oscillatory motion over topography in a rapidly rotating cylinder have shown that isolated mesoscale eddies, which form in the sidewall boundary layer during certain phases of the forcing cycle, are associated with the onset of chaotic behavior in this system. This paper explores the origin of these eddies by performing computational simulations of the flow, and then interpreting the results of the calculations using spatially localized and quasi-static linear stability theory. For most of the experimental parameter space the quasi-geostrophic simulations are in excellent agreement with the laboratory observations. The eddies arise as a barotropic shear flow instability in regions of sPace and at times where the inflection points of the instantaneous large-scale flow are farthest from the sidewall, and where Fjortoft's theorem is strongly satisfied. At finite amplitude, advection of the local wavetrains up the bottom slope strengthens the anticyclonic eddies. These then merge, leading in most circumstances to a single strong anticyclonic vortex that can leave the sidewall and penetrate the interior. When parameters are such that the eddy persists all the way around the basin and back to the local instability region, the flow is observed to become chaotic.

* Corresponding author. Present address: Departmentof Meteorology, Universityof Utah, Salt Lake City, UT 84112, USA. 2 Present address: Departmentof Marine Sciences, Universityof California, Santa Cruz, CA 95060, USA. 0377-0265/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0377-0265(96)00476-9

88

M. Zulauf et al. / lkynamics of Atmospheres and Oceans 25 (1996) 87-107

1. Introduction This paper describes computational simulations and theoretical analysis of periodically forced flow over topography in a rotating fluid. The central goal of this study is to understand the mechanism for the formation of isolated mesoscale eddy structures that are observed to occur in laboratory experiments on oscillatory motions in a 'sliced-cylinder', a configuration which combines simple axisymmetric lateral confinement with asymmetric topography. Beardsley (1969) studied motion that is driven by a differentially rotating lid in a rapidly rotating cylinder with a uniform sloping bottom. The uniform slope generates a topographic /3-effect, and the laboratory model provided observations of the formation and separation of western-intensified boundary currents. Beardsley (1975) later investigated Rossby-wave resonance with periodic lid forcing, and Krishnamurti (1981) described additional experiments with both steady and oscillatory driving. These works focused on systems that have no closed depth contours. At low Rossby number the interior motions are depth-invariant and constrained to flow along the isobaths, so that fluid columns eventually run into the sidewall. This situation therefore leads to slow interior currents and strong western boundary jets. Our study of the dynamics of periodically forced rotating fluids is motivated by an interest in oceanographic coastal currents. Such flows can have small Rossby number, and are often subject to fluctuating wind stresses. One main difference between this situation and those described above is that the coastal currents often move over closed depth contours, typically running parallel to the coast, but which may be perturbed by canyons or by capes that break the alongshore symmetry. The basic response is no longer in Sverdrup balance, and strong oscillatory alongshore currents can be excited. These feel the irregularities in the bottom topography in a manner that generates Rossby waves which can nonlinearly self-interact to affect the basic currents. In contrast to the 'sliced-cylinder' experiments, our flows include a free upper surface, which naturally deforms into a dynamically significant parabola under the influence of the mean rotation. In combination with a uniform bottom slope, this geometry leads to a domain with many closed geostrophic contours, but with an important along-coast wavenumberone asymmetry associated with the sloping bottom. Previous studies of oscillatory flow in this latter type of rotating basin have focused on motions with scales similar to the basin itself. Periodic sloshing motions over topography in a rotating fluid in a /3-plane channel can lead to rectified currents, and Samelson and Allen (1987) proposed a low-order model to explain observations like those of Denbo and Allen (1983) of along-coast-averaged currents off the coast of California. The low-order theory predicts a chaotic response to sinusoidal wind forcing under certain circumstances that were delineated by Allen et al. (1991). This result, in part, stimulated our laboratory experiments on periodically forced flows over topography in a rotating cylinder. The experiments by Pratte and blart (1991) (hereafter PH) showed that although low-order models capture several aspects of the oscillatory flow, the observed chaotic motions, which occur abruptly as parameters are changed, are not a reflection of spatially simple large-scale dynamics. Rather, small mesoscale eddies, born at special isolated locations in the experiments, are precursors to the observed aperiodicity. These eddies appear central to the onset of spatio-temporal chaos, and it is of

M. Zulauf et a l . / Dynamics of Atmospheres and Oceans 25 (1996) 87-107

89

interest to determine their origin. The two obvious mechanisms are boundary current separation and shear instability of the time-varying basic flows. Satellite observations have documented the formation of anticyclonic eddies along the continental slope in the eastern Bering Sea (Paluszkiewicz and Niebauer, 1984). The eddies are also evident in buoy trajectories (Stabeno and Reed, 1994) and satellite altimetry (Leben and Fox, 1992). Proposed mechanisms for the formation of these eddies include instabilities, wind forcing, and topographic interaction (Kinder et al., 1980). A large-scale three-layer numerical study of the region with periodic forcing by monthly climatological winds predicts annual reversals of the Bering Slope current, along with the generation and westward propagation of the eddies in the vicinity of the shelf break (Overland et ai., 1994). In the simulation, the eddies form in embayments along the continental slope in phase with the seasonal (summer) minimum of the cyclonic wind forcing. Verification of this seasonal signal has been difficult with the sparse observational database and the strong interannual variations in the wind forcing and basin boundary inflows. It is instructive to view the laboratory and mechanistic modeling experiments described here in light of these observational and computational results. From a more fundamental perspective, our particular model is of interest because it is asymmetric in space and time. That is, instability and eddy formation may be expected to be localized in space by the bottom topography, and to occur only during certain phases of the forcing cycle. This might be compared, for example, with classical instability problems in steady parallel flows which are invariant with respect to changes in zonal coordinate and time origin. The laboratory experiments of PH show that eddies arise for a wide range of parameters, but are indeed born at particular locations along the bathymetry and only amplify during particular phases of the forcing cycle. A useful tool for analyzing such flows is numerical solution of the one-layer quasi-geostrophic vorticity equation. Based on a successful comparison with the observable experimental states and structures, we are motivated to use the computational model to investigate important facets of the local dynamics that are made accessible because of the more detailed diagnostics available in the numerical simulations. We use the computed flow fields to generate time-localized azimuthal velocity profiles that are averaged over a small range in polar angle 0. These quasi-static, locally parallel profiles are then tested for linear instability. The results are in good agreement with experiment, and this suggests that an elemental description of the dynamics in such systems can be obtained by a combination of simulation or observation of the general motion, along with a stability analysis of isolated segments of the flow.

2. Formulation and numerical method

Fig. 1 illustrates the geometry of the problem. A homogeneous fluid (water) with mean depth H is contained in a cylinder of radius L. The time-averaged rotation 120 leads to a parabolic free surface that gives a depth (or potential vorticity) gradient perpendicular to the coastal sidewall. Alongshore bottom topographic variations desymmetrize the basin in the azimuthal direction. PH used a wavenumber-two Bessei function

90

M. Zulauf et al./ Dynamws of Atmospheres and Oceans 25 (1996) 87-107

H

O=

Ah "/T

L g2o(1 - 6 cos(cJ t)) Fig. I. Cross-section of the experimental system. Water is contained in a sliced cylinder that is rotated about its axis at a rate that has a mean part .62o and a periodic modulation with amplitude S and frequency ~o. The mean fluid depth is H and the radius is L. The topography has amplitude Ah, with the shallow side defined to be at 0 = 0, and the deep end is at 0 = 7r. The angle 0 is measured in the direction of the mean rotation (counter-clockwise). shape, as well as a uniform slope, to perform this function. W e choose the latter here because there is not much qualitative difference in the response b e t w e e n the two cases, but the situation with the slope is a little easier to interpret. In the oceanographic setting the motion is assumed to be driven by an oscillatory alongshore wind stress. This is difficult to i m p l e m e n t in laboratory experiments such as these, which require a free surface, but, as discussed below, essentially the same effect is attained by m o d u l a t i n g the basic rotation of the apparatus by a small a m o u n t S. PH visualized the flows by seeding the free surface with floats. Fig. 2 shows some typical examples. W h e n S is small, or the modulation period is short so that the frequency based Rossby n u m b e r e = w / 2 ~2o is large, the motion is relatively simple. The gyre center migrates around periodically and a m e a n prograde zonal flow is generated. Fig. 2(a) illustrates one phase of this motion. W h e n S is increased at fixed e, or when • is decreased at fixed 8, then a short wavetrain (two or three peaks with a zonal w a v e n u m b e r of about 16) appears in the sidewall b o u n d a r y layer. It is important to note that the phasing of this apparent instability is such that these waves appear at the deep end of the tank and become o b v i o u s at a phase angle about ~ - / 2 past the time (t = 0 + 2nTr) when the basic rotation is at a m i n i m u m . This local wavetrain grows, but usually one pair of vortices emerge from the amplification stage and are advected alongshore. W h e n , under the m o d u l a t i o n action, the rotation rate increases, a basic flow arises which is opposite to the direction of rotation, and the vortex couple is swept away from the deep end. Because vertical vortex tubes are compressed as they move toward shallower regions, the anticyclonic vortex is strengthened and the cyclonic one is wiped out. What emerges is a single anticyclonic eddy, or in some circumstances a pair of anticyclones, that can enter the interior (e.g. Fig. 2(b) at 0 = 0). At supercritical

Fig. 2. Streak photographs of float particles on the free surface. (a) S = 0.05, E = 0.055; (b) S = 0.05, E = 0.036; (c) S = 0.05, E = 0.022. In this and all other planform views the deep side of the tank (0 = ,n') is on the right, and the shallow side is on the left (0 = 0).

M. Zulauf et a l . / Dynamics of Atmospheres and Oceans 25 (1996) 87-107

91

M. Zulauf et al./ Dynamics of Atmospheres and Oceans 25 (1996) 87-107

92

parameter settings with respect to this sidewall eddy generation, eddies can last more than one forcing cycle (especially if the modulation period is long), leading to a complex turbulent flow filled with eddies (Fig. 2(c)). The basic theoretical model for these phenomena is the depth-independent quasi-geostrophic vorticity equation written in coordinates attached to the cylinder. Hart (1990) (see also PH) derived this equation and discussed some results for the large-scale weakly nonlinear flows that result when the non-axisymmetric topography amplitude is relatively small. The derivation of the governing equation centers on small flow Rossby number S, and small ~, conditions that are reasonably well satisfied in the experiments. The non-dimensional quasi-geostrophic vorticity equation is (V2-F)

dO-~t= J ( V 2 ~ ' ~ ) + J[-H~-rcOs( O ) - -Ht r2'~

-

E, S QV2qj + --& V4~O- - - s i n ( t ) E

(1)

E

where q,(r,O,t) is the streamfunction for the geostrophic motion based on the mean rotation so that ~'O-= v = aqJ/l Or, v~- u = -Otp/raO. The parameters are defined by

F = 4O2L2/gH E = ~o/200

Q = u/f~-~o/eH

H b = Ah/H H~ = f2oL2/2 gH E~. = u/2f2o Le F is the rotational Froude number and determines the relative dynamic free-surface fluctuation amplitude, e is the Rossby number based on the forcing frequency w, Q is the forcing period divided by the spin-down time, H b is the non-dimensional bottom topography amplitude, H t is the non-dimensional amplitude of the free surface parabola, and E L is the lateral Ekman number. Here u is the kinematic fluid viscosity, and the other parameters are defined in Fig. 1. The non-dimensionalization uses w- ~ as the time scale, wL as the velocity scale, and L as a horizontal length scale. The right-hand side of Eq. (1) expresses the time rate of change of potential vorticity. This is due to advection of relative vorticity (the first term on the right), stretching of mean planetary vorticity by motion across isobaths (the second Jacobian term), spin-down owing to suction out of the bottom Ekman layer (the third), and lateral friction (the fourth). The last term reflects the modulated rotation-rate forcing, which appears as a periodic source of vorticity in the fluid. The easiest physical interpretation of this effect is simply to realize that as the cylinder rotation speeds up and slows down a purely inviscid fluid would appear to an observer attached to the cylinder to slosh back and forth as a solid body. Now it is well known that a surface wind stress induces an Ekman pumping just below a sub-surface Ekman boundary layer, and that this suction velocity is proportional to the curl of the wind stress. This effiux stretches the uniform mean planetary vorticity in a manner that generates relative motion. The forcing term in Eq. (1) that arises from modulating the basic rotation is then equivalent to an oscillatory surface wind stress that has a curl which is constant in space.

M. Zulauf et al. / Dynamics of Atmospheres and Oceans 25 (1996) 87-107

93

In the analysis that follows many of the dimensional constants are fixed to the laboratory values: ~0 = 2.85 r a d s - I , v = 0.01 cm 2 s - l , H = 10cm, L = 22.5 cm. These parameters give a Froude number F - - 1.64. If the streamfunction field is described in terms of Bessel functions, the lowest radial-mode wavenumber-one fluctuations (which are the naturally excited ones with this topography) have V 2 -- 3.832 = 14.6 :~ F. Based on this, the calculations reported here neglect the small dynamic deviations of the free surface (or, F--* 0), although keeping in mind, however, that the free surface equilibrium parabola is crucial because of its associated topographic polar/3-effect. When free surface deviations are included, additional mass conservation constraints are needed and the computation becomes more complicated. With one exception, all the data in this paper were obtained with H b = 0.1. This leaves 3 and E as the two main dimensionless variables. We note that with the above laboratory parameter values, H, = 0.21. The interesting dynamics occur when 8 / E is of order one, with ~--0.03. Thus, because H b / E is fairly large, the motions excited by the periodic forcing will be nonlinear. For later comparison it is, however, useful to note that if Hb//~= 0 there is an exact axisymmetric solution of Eq. (1) in which the flow is purely azimuthal and the advections are identically zero. This is the Stokes-Stewartson layer flow l , ( r , t ) = 2E(1 + 0 2) {rcos(t) - r Q s i n ( t )

- Re[ ( 1 + iQ) 1,( rA)exp( it)//l,( A)]} with A - I/E( Q + i ) / / E

L , I 1

(2)

denoting the Bessel function, and R e the real part.

In a separate paper (Hart and Mundt, 1996) the stability of such flow has been discussed. It is much more stable than the motions that accrue in the sliced cylinder because, as shown below, alongshore non-axisymmetric topography variations generate changes in the basic flow that are highly destabilizing. We solve Eq. (1) numerically on a polar grid with regularity imposed at r = 0 and with no-slip at r - - 1. The calculation technique is similar to that used by Albaiz et al. (1993) so a detailed exposition is not given here, except to say that a second-order conservative difference scheme is used and the elliptic equations are solved using a multigrid covariant Laplacian fast Poisson solver. The solver treats the grid singularity at the origin as an interior point in the multigrid procedure, allowing solutions to be found in the interior of the cylinder. The grid typically has 65 points in radius and 129 in azimuth. Exponential grid stretching in radius was used to resolve the boundary layers near the wall. If a uniform grid with N points is defined by j-1 rJ=N_l,

j= 1,2,...,N

then the stretched grid radial variable is given by 1 - ears

r~.= 1 - - e a

94

where

M. Zulauf et al./ Dynamics of Atmo,wheres and Oceans 25 (1996) 87-107 in this w o r k

a = 4. C a l c u l a t i o n s

external parameter settings employed

with higher

resolution

suggest

that for the

h e r e this r e s o l u t i o n is a d e q u a t e . R u n s w e r e started

f r o m initial c o n d i t i o n s o b t a i n e d f r o m the final states at n e a r b y p o i n t s in p a r a m e t e r s p a c e .

045

-(a)

040

0.35

0 30

~ 0.25 E -5 0.20

015

010

OO5

1.2

i

J

40

45

i

i

50 55 T (dimensionless)

60

1

i

65

70

(b)

1.0

0.8

~o.6 -o

\

0.4

/

0.2

0.0

70

75

80 T

i

i

i

L

85

90

95

100

(dimensionless)

Fig. 3. Time histories of flow speed at a fixed point three-quarters of the way out from the axis at 0 = ~. (a) 6 = 0.02, e = 0.027, H b = 0.05; (b) 6 = 0.05, E = 0.037, H b = 0.1. Time is shown in units of the dimensionless forcing modulation period (27r).

M. Zulauf et al. / Dynamics of Atmospheres and Oceans 25 (1996) 87-107

95

All computations were carried out on a DEC Alphastation, on which a simulation over an elapsed time of order 100 forcing periods takes about 6h.

3. Comparison of computational simulations with laboratory experiments The numerical simulations reproduce most of the laboratory observations, and offer an opportunity to diagnose the underlying dynamics of eddy generation in this system. Because the flow is periodically forced, the possible states of motion are either periodic (possibly with multiple periods), or aperiodic. Fig. 3 shows two typical time traces of flow speed at a 'computer' probe located at r = 0.5 and 0 = 1.6 radians. In Fig. 3(a) the flow is periodic, and in Fig. 3(b) it is chaotic. Laboratory data from PH (their Figs. 2(a) and 15) are essentially the same. The small differences in harmonic content are thought to be due to nonlinearity in the hot thermistor probe used in the experiments. Fig. 4 compares the transition points between periodic and chaotic states. PH observed abrupt transitions between these two regimes. Although we tried very hard to observe secondary periodic regimes (i.e. period doubling or quasi-periodicity) between purely periodic states and the chaotic equilibria in the simulations, by taking very small parameter steps and increasing the resolution, it appears that the simulations too, for all practical purposes, undergo an abrupt transition. The nearly vertical transition line in Fig. 4 at E--0.04 is very accurately reproduced by the model, and although the simulation underestimates the tendency to have a chaotic response at small E and low 8, the basic structure of the regime diagram is nicely replicated. It should be noted that the quasi-geostrophic model is expected to be accurate to O ( 8 , H b , H t ) . Thus errors of

0.080

/

0.050

I

CHAOT//C

0.040

o.oso

/ ~ 1 . ~ ~J --

--

~

/

P E R I O D I C

0.020 . . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . . . . . . . . 0.010 0.020 0.030 0.040 0.050 E

Fig. 4. Regime diagram for the transition between periodic and chaotic flow. The dashed curve is from the experiments of Pratte and Hart (1991), and the continuous line indicates the transition points for the computational simulations.The computationswere performedon a parametergrid with about 20% changes in 8 and • between runs, except near the transition lines where closer steps were taken.

96

M. Zulauf et al. / Dynamics of Atmospheres and Oceans 25 (1996) 87-107

(a)

0 = 3~/2

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0=0

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M. Zulauf et al. / Dynamics of Atmospheres and Oceans 25 (1996) 87-107

(c)

97

0 = 3r/2 ;;77SS~S;772~Z~SS~S2ZZ~?~

;S;77;;777;;;:7:::::SzZZ2~ZZ~

,i i!iiiiiiiiiiiiii!!!!!iiiii °o=,iiiiiii?iii!i!i::::i::iii!;!i iiiiiii! . . . . . . . . . . . . . . . . . . . . . . .

.

,

,

¢

t

l

0=re/2

I Fig. 5. Comparisons of laboratory and computed flow fields. Each panel shows the experimental velocity vector field using PIV at the top. Below these are (left to right) the computed streamfield at the same forcing cycle phase, the exact analytic axisymmetric flow obtained from the basic quasi-geostrophic equation without topography, and the locally averaged (over 22°), numerically simulated swirl velocity T,(r,Oo,to) at the base angle 00 corresponding to the pin-wheel axes. The deep side is at 0 = ~ , and the shallow edge is at 0 = 0. The time tp is the forcing phase angle measured from the rotation rate minimum. (a) ~ = 0.05, • = 0.055, tp = 0; (b) ,5 = 0.05, ~ = 0.047, tp = 0; (c) 8 = 0.05, ~ = 0.047, tp = I r / 2 .

around 10 or 20% might be anticipated, so the fact that the regime diagrams match up as well as they do is very encouraging. To provide more detailed comparisons between the simulations and the experiments, we repeated some of the laboratory runs using a PIV (particle image velocimetry) visualization scheme. The free surface is seeded with a relatively dense distribution of aluminum flakes. A small quantity of alcohol is added to the water to help disperse the particles. The spatial cross-correlation matrix of two sequential 512 × 480 video frames is constructed, using videographic separation of a few tenths of a second, which is very much shorter than the forcing period of 20-50 s. Peaks in the local spatial correlation matrices are related to surface flow velocity, and once these are interpolated onto cylindrical coordinates a velocity vector map is obtained that can be compared with instantaneous streamfields from the computer model. Fig. 5 shows a set of typical comp~isons between the simulated and laboratory flows, at times when the system has settled into its final state. This figure contains laboratory PIV fields and computed flow streamlines, and illustrates the theoretical

98

M. Zula uf et al. / Dynamics o/Atmospheres and Oceans 25 (1996) 87-107

axisymmetric Stokes-Stewartson profile, as well as the local azimuthally averaged swirl velocity -~(r,Oo,t o) from the numerical simulations. This latter object, which will be featured in the stability calculations of Section 4, is obtained by picking a base angle 00, as shown on the pinwheel, at a specific forcing phase angle, and then averaging u symmetrically about this base angle over 22 ° . Fig. 5(a) shows one phase of the periodic oscillation just outside the chaotic region of parameter space at 6 = 0.05. The model reproduces the structure of the oscillation very well, and point-by-point comparisons show that the flow speeds are correct to better than + 10% in regions where the experimental data is good. The PIV method works best when there is a uniform distribution of particles on the surface. On occasion, the aluminum flakes stick to the cylinder wall, leaving a void near the boundary so that dropout regions occur in the PIV results (e.g. the lower left of Fig. 5(a)). This does not usually lead to significant difficulty in interpretation. By examining several snapshots it can be seen that there is a mean (time average over a forcing period) prograde motion in the tank. The mechanism for this mean rotation is associated with nonlinear rectification of the Rossby waves excited by the flow over the topography (Hart, 1990; PH). One can also note that during phases where the motion is nearly axisymmetric the exact Stokes-Stewartson solution gives a moderately accurate picture of the motion, although being a linear solution it cannot describe any wave-induced zonal rotation. Fig. 5(b) and (c) illustrate different phases of the forcing cycle in a region of parameter space with weak instability. The Eulerian motion is periodic. Fig. 5(b), taken when the basic rotation is at a minimum, demonstrates that the flow no longer is spatially smooth as in the previous case. An eddy that was generated during the previous cycle appears at about 0 = 7r/2. Fig. 5(c) ~hows the flow configuration at a time associated with significant linear instability at 0 = 7r (see Section 4 below). The remnant of the eddy in Fig. 5(b) is still visible although it will decay and be replaced by a new eddy emanating from the deep end. Of most significance for the instability calculation is the strong shear, reversed flow, and changes in sign of vorticity as one leaves the wall at the deep end, 0 = rr. It is useful to notice how different the topographically shaped flow is from the axisymmetric solution. Fig. 6(a), (b) and (c) show computer simulations of the evolution of the eddy generation and propagation process for slightly smaller (more supercritical) E, where the features are more visible. In Fig. 6(a), one can see small waves forming on the streamlines in the reversed flow region at about r = 0.9, 0--- 0.97r. The waves amplify, with one being favored, and the train is advected with the motion up the slope (Fig. 6(b)). The anticyclone intensifies by planetary vortex compression and a strong eddy appears at the shallow end (Fig. 6(c)). The PIV laboratory velocity field for this time (see Fig. 6(d)) illustrates again the excellent agreement between the experimental and simulated fields. Feature tracking of the computationally simulated eddies was carried out. Instability and eddy generation similar to that described above can occur outside the chaotic regime of Fig. 4. Although the band of parameters where this occurs is relatively narrow, so that the transition lines of Fig. 4 give a fairly good indication of where mesoscale instability in the boundary layer occurs, it is interesting that a periodic regime with small-scale eddies appears possible. The feature tracking indicates that when E is fixed so that the

M. Zulauf et al, / Dynamics of Atmospheres and Oceans 25 (1996) 87-107

(a)

99

(b) ..222-.2.--

'

, .................

:;~:":

.,,,~x~

t .......... iii ...............

:::::::::::::::::::::::::::::

:

,,

?~H

' ~!i:::~iiiiiiii!iiiiiiii ~iiiiiii!~ • , ......................

(c) Fig. 6. Computed streamline patterns for 6 = 0.05, ~ = 0.037, and PIV velocity vector field corresponding to pattern (c).

:;;~,~

(d) tp =

1r/2 (a),

tp =

"rr (b), tp = 21r (c). (~l)

system is to the right of the transition curve, for example, the eddies do not persist all the way around to the generation region. They either decay or are reabsorbed into the boundary layer. At longer forcing periods (smaller E) the eddies are stronger and the tracks can be followed all the way around to the deep end. It appears that this is an example of a zonal teleconnection where the local instability region, being perturbed by previous eddies, emits instabilities at slightly different times and spatial locations, thus leading to major chaotic deviations in subsequent eddy tracks. Albaiz et al. (1993) reported a similar mechanism associated with chaotic dynamics owing to boundary layer separation in a connected re-entrant geometry subject to steady forcing.

4. Local quasi-static stability theory In an effort to further interpret the numerical and experimental results we construct a local stability theory. Although this is a somewhat crude approximation to the actual

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M. Zulauf et al./ Dynamics qf Atmosphere,~ and Oceans 25 (1996) 87-107

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M. Zulauf et a l . / Dynamics o f Atmospheres and Oceans 25 (1996) 87-107

(c)

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0.0

~:

0.0

~, -o.~

L.I

-0.6

.............. i 0.0 0.2 0.4 0.6 0.8 1.0 time (forcing periods) 0=~

o Jc

" ........ '"

~. '.\

~-0.2

-o,<

2.0

.

.

.

.

.

.

.

.

.

.

.

.

101

.

.

"

,t'i t.5

L ~'"

-0.4 -0.6 0.0 0.2 0.4 0.6 0.8 1.0 time (forcing periods)

0=3n/2

0.2~ 0.4 " " '

"

'

"

'

"

0.0

o.o)

o.5 '/

i.

-0.5

0.0 0.2 0.4 0.6 0.8 .0 time (forcing periods)

-0.6 . . . . . . . . . . . . . . . . . . . 0.0 0.2 0.4 0.6 0.8 1.0 time (forcing periods)

Fig. 7. Growth rate curves for Steal from the quasi-static model. For each case four angles are shown (0 = 0 is the shallow end, 0 = er is the deep end; see Fig. 1). The local time t o is measured in tenths of the forcing period, with zero being the time of m i n i m u m rotation and 0.5 being the time of m a x i m u m basic rotation. Each profile -P(r,Oo,t o) is tested for growing modes at wavenumber k = 8 (continuous line), k = 12 (dotted line), k = 16 (dashed line), k = 20 ( d a s h - d o t t e d line) and k = 24 (long-dashed line). (a) 8 = 0.05, ~ = 0.055; (b) (5 = 0.05, E = 0.047; (c) 8 = 0.05, ~ = 0.037.

instability process, the success of the theory leads us to believe that it does at least capture the essential physical mechanism. The stability calculation proceeds as follows. Results from high-resolution fully nonlinear stretched-grid computations are used to extract local azimuthal swirl profiles ~(r, Oo,t o) as described above. Assuming that this swirl is the dominant part of the flow at any base angle 0 0, especially near the wall where the instability occurs, the governing Eq. (1) is linearized about "b. The gradient of the topography is taken to be a function of r but is assumed constant in 0 when referred to the base angle 0 0. After taking disturbances with form ~b = q)(r)e ikO+st we obtain the stability equation O~

EL

[ s + ( ik-P/r) - Q][12 _ (kE/r2)] ~0- i k ~ p ~r = --~-[ l a - ( k2/r2)]2~o

O~p Oh

~oOh

- + ik Or ~rO0 rEOr

(3)

M. Zulauf et al./ Dytulmics qf Atmo,~pheres and Oceans 25 (1996) 87-107

t02

where / 2 = (02/Or 2) + [l/r O/Or] and h is the total nondimensional depth with its gradients being evaluated at radius r and base latitude 0 o. The mean vorticity gradient is defined by

c3r

0r

+ -r

(4)

A solution is sought with boundary conditions of regularity of q~ at r = 0 and at r = 1. This is essentially a quasi-static, locally parallel-flow stability problem. It is solved using a high resolution finite-difference method on the same stretched grid as was used in the fully nonlinear computer simulations. Because the basic flow is concentrated near the wall and has very large vorticity gradients, the perturbation vorticity equation is dominated by advection of relative vorticity. The effects of the topography on the perturbations, although included in the numerical solution of Eq. (3), are negligible. With this in mind, it is useful to note that the resulting frictionless form of Eq. (3), with regularity imposed at the origin and impermeability imposed at the wall, can be used to derive the usual necessary conditions for inviscid instability that the basic state vorticity gradient must change sign in the domain (the inflection point theorem), and that (~ - -~s)O~/Or must be negative somewhere in the domain (Fjortoft's theorem), where P~ is the velocity at the inflection point of 7. Fig. 7 shows the results for several E moving across the periodic-to-chaotic transition at 8 = 0.05. In each instance a range of typical zonal wavenumbers k is investigated, with basic profiles taken from four angles and eight times. In Fig. 7(a) it is observed that only the deep end is slightly unstable. Though the growth rates are small enough that the quasi-static approximation (in which the waves are assumed to grow quickly before the basic state changes very much) may not be well satisfied, the laboratory experiments are in agreement with the spirit of this result. No significant eddy generation was observed in the laboratory for this value of E. When E is decreased to 0.047, for which weak waves are observed in the laboratory, it is seen (Fig. 7(b)) that the deep end is strongly destabilized at wavenumbers 20 and 24, at a time centered on a phase angle of ~-/2. It should be noted that this is precisely the time and spatial location at which eddy formation is observed in the laboratory. The position 0 = 7r/2 (at the bottom of the planviews in Fig. 5) is the second most unstable location of the four studied, with the greatest growth occurring at a somewhat later time. This may help explain some of the eddy amplification as the waves that originate in the deep end of the basin are carried downstream. The instability causes growth of high-wavenumber (e.g. mesoscale) disturbances with significant perturbation amplitudes concentrated in the shear zones near the walls. In Fig. 7(b), the external forcing is outside the chaotic region for the total flow, again illustrating that it is possible to have mesoscale eddies without having chaos over a narrow range of external conditions. Fig. 7(c) indicates that local instability is expected at more than one point in the more supercritical (small E) flows. The most unstable wavenumber is about 16, as observed, and again arises predominantly at the deep side of the tank during the forcing phase when the rotation rate is increasing from its minimum value. At this E another instability would seem to be possible a phase factor "rr later, but at the shallow end of the tank (0 = 0). Time lapse video films of the surface floats in the laboratory show that this is = Oq~/Or = 0

M. Zulauf et a l . / Dynamics of Atmospheres and Oceans 25 (1996) 87-107

103

precisely what happens. Namely, at first the eddy generation occurs only once during a forcing period, hut at highly supercritical conditions, the shallow end starts to see instability too, although the eddies so generated are not as vigorous as their deep-end counterparts. It may also be significant that the basic profiles used in this calculation, which were extracted by spatially averaging the full numerical simulation locally, are probably affected by the instability itself. Nonetheless, the growth rates are larger than v e l o c i t y (V)

vetoclty (V)

0.2

0,2

0.1

0.1

0.0

0.0

-0,1

-0.1

-0.2

-0.2 I

-0.3 0.0

-0.3 0.64

0.88 r

0.97

1.0

relative v o r t l c i t y grad. (Otn/cqr) 8000 t

0.0

0.64

0.88 r

0.97

1.0

(aw/ar)

r~lotive v o r t i c i t y grad. 1.2xtO ' ' ' '

'"

1.0x104 6000 : 8.0x103 4000

6,0xi03 4.0x103

2000

2.0x103 0 0 -2000 0.0

i

i

i

0.64

0.88 r

0.97

1.0

-2.0x103 0.0

(V-Vs)*aw/Or

i

0.64

0.88 r

J i

0.97

1.0

(V-Vs),a~/ar

2000

1500

A

500 1000

1000 500

500

0

0

-500

- 500 0.0

J

-

0.64

0.88 r

(a)

0.97

~.0

t

000 0.0

i

i

0,64

0.88

m

0.97

1.0

r

(b)

Fig. 8. Basic flow quantities of interest in the instability calculation. The panels show the locally a v e r a g e d swirl velocity, the relative vorticity gradient for this swirl (which is almost exactly t h e s a m e a s t h e potential vorticity gradien0 a n d t h e Fjortoft diagnostic, where Vs is the swirl velocity at the inflection point at r = 0.98. T h e grid is strongly stretched so that details in the sidewall boundary layer can be discerned. (a) 8 = 0.05, = 0.055, tp = , r / 2 , 0 = 3 7 r / 2 ; (b) 8 = 0.05, ¢ = 0.047, tp = ~ / 2 , 0 = ft.

104

M. Zulauf et al. / Dynamics of Atmospheres and Oceans 25 (1996) 87-107

those in Fig. 7(b), for example. A possible interpretation is that the wave-perturbed flows are more unstable, leading to the suggestion that the instability may be explosive. A term-by-term analysis of the perturbation model. Eq. (3), shows that vorticity advection is the most important process in the instability, followed by lateral vorticity diffusion, with weak contributions from bottom drag and topographic stretching. This suggests that a simple interpretation of the instability can be obtained by looking at the particular basic flows involved. Fig. 8(a) shows the key quantities for an essentially stable location and time. Comparison with the unstable basic state in Fig. 8(b) demonstrates that the unstable zone is associated with inflection points that are farther from the boundary, and with a basic flow that much more strongly satisfies the necessary condition for instability expressed by Fjortoft's theorem. By studying a large number of such diagrams it is determined that the physical conditions favoring instability are indeed precisely these two situations: inflection points substantially removed from the wall and significant negativity of the Fjortoft diagnostic (-b- -Ps)O~/Or.

5. Conclusions

Computer simulations of laboratory flows on periodically forced motion over topography in a rapidly rotating cylinder are described. The simulations accurately model most of the experimental observations, including simple periodic sloshing flows, periodic flows with mesoscale eddy generation, and chaotic motions involving one or more strong eddies. For example, Fig. 9 shows a couple of frames from a fully chaotic flow that can be compared with typical observations as in Fig. 2(c). The computations indicate that a single eddy generation site is preferred at low supercriticality, but as the parameters move further into the chaotic regime, a secondary site arises on the opposite side of the basin. The chaos is directly associated with eddies that circumnavigate the basin and come back to interact with the instability site. The details of this interaction, which probably causes erratic shifts in the spatial and temporal phases of the instability, need further investigation. Of the two processes, boundary current separation and local instability, which were reasonable candidates for the mesoscale eddy generation in these experiments, the latter seems the appropriate mechanism. The experimental (and computational) observations are that the eddy generation starts locally in space and time with a short wavetrain of typically one or two waves. These waves then grow, interact, and at finite amplitude the source region emits one or two strong anticyclones. Calculation of the growth rates for linear azimuthally wavy disturbances to locally steady and axisymmetric swirls (taken from the full numerical model) give substantial growth rates at just those positions and times when wave generation is observed in the laboratory experiments. These occur at azimuthal angles where the inflection point in the basic state azimuthal velocity has moved somewhat away from the wall so that frictional effects are lowered and normal perturbation velocities can advect vorticity across it, and where the Rayleigh and Fjortoft necessary conditions for shear instability are strongly satisfied. The shear instability itself is largely independent of the bottom topography as well as the surface-topographic fl-effect. However, these latter two physical ingredients are crucial in setting up 'basic

M. Zulauf et a l . / Dynamics of Atmospheres and Oceans 25 (1996) 87-107

0=0

105

0=~

Fig. 9. Computed flow streamlines at two phase angles in a strongly chaotic regime. 8 = 0.05, • = 0.022.

M. Zulauf et al./ Dynamics of Atmospheres and Oceans 25 (1996) 87-107

107

Beardsley, R.C., 1975. A 'sliced-cylinder' laboratory model of the wind-driven ocean circulation. Part 2. Oscillatory forcing and Rossby-wave resonance. J. Fluid Mech., 69:41-64. Denbo, D.W. and Allen, J.S., 1983. Mean flow generation on a continental margin by periodic wind forcing. J, Phys. Oceanogr., 13: 78-92. Hart, J.E., 1990. On oscillatory flow over topography in a rotating fluid. J. Fluid Mech., 214: 437-554. Hart, J.E. and Mundt, M., 1996. The stability of the oscillatory Stokes-STewartson layer. J. Fluid Mech., 311: 119-140. Kinder, T.H., Schumacher, LD. and Hansen, D.V., 1980. Observations of a baroclinic eddy: an example of mesoscale variability in the Bering Sea. J. Phys. Oceangogr., 10: 1228-1245. Krishnamurti, R., 1981. Laboratory modeling of the oceanic response to monsoonal winds. In: J. Lighthill and R.P. Pearce (Editors), Monsoon Dynamics. Cambridge University Press, Cambridge, pp. 557-576. Leben, R. and Fox, C.A., 1992. Altimetric studies of circulation variability in the Bering Sea. Abstract, EOS, 73: 125. Overland, J.E., Spillane, M.C, Huriburt, H.E. and Wallcrafi, A.J., 1994. A numerical study of the circulation of the Bering Sea basin and exchange with the North Pacific Ocean. J. Phys. Oceanogr., 24: 736-758. Paluszkiewiez, T. and Niebauer, H.J., 1984. Satellite observations of circulation in the eastern Bering Sea. J. Geophys. Res., 89: 3663-3678. Pratte, J.M. and Hart, J.E., 1991. Experiments on periodically forced flow over topography in a rotating fluid. J. Fluid Mech., 229: 87-114. Samelson, R.M. and Allen, J.S., 1987. Quasi-geostrophic topographically generated mean flow over a continental margin. J. Phys. Oceanogr., 17: 2043-2064. Stabeno, P.J. and Reed, R.K., 1994. Circulation in the Bering Sea Basin observed by satellite-tracked drifters: 1986-1993. J. Phys. oceanogr., 24: 848-854.