Flows in a rotating annular tank [J. Sommeria, S. D. Meyers, and H. L. Swinney, Nonlinear Topics in Ocean Physics, edited by A. Osborne (North Holland, Amsterdam, 1991); Nature (London) 337, 58 (1989); T. H. Solomon, W. J. Holloway, and H. L. Swinney, Phys. Fluids A 5, 1971 (1993); J. Sommeria, S. D. Meyers, and H. L. Swinney, Nature (London) 331, 689 (1989)] with a sloping bottom (that simulates a barotropic atmosphere’s Coriolis force with a topographic β-effect [J. Pedlosky, Geophysical Fluid Dynamics, 2nd ed. (Springer, Berlin, 1986)]) produce eastward and westward jets, i.e., azimuthal flows moving in the same or opposite direction as the annulus’ rotation. Flows are forced by pumping fluid in and out of two concentric slits in the bottom boundary, and the direction of the jets depends on the direction of the pumping. The eastward and westward jets differ, with the former narrow, strong, and wavy. The jets of Jupiter and Saturn have the same east–west asymmetry [P. S. Marcus, Ann. Rev. Astron. Astro. 431, 523 (1993)]. Numerical simulations show that the azimuthally-averaged flow differs substantially from the non-averaged flow which has sharp gradients in the potential vorticity q. They also show that the maxima of the eastward jets and Rossby waves are located where the gradients of q are large, and the maxima of the westward jets and vortex chains are located where they are weak. As the forcing is increased the drift velocities of the two chains of vortices of the eastward jet lock together; whereas the two chains of the westward jet do not. Inspired by a previously published, [P. S. Marcus, Ann. Rev. Astron. Astro. 431, 523 (1993)] piece-wise constant-q model of the Jovian jets and based on numerical simulations, a new model of the experimental flow that is characterized by regions of undisturbed flow and bands of nearly uniform q separated by sharp gradients is presented. It explains the asymmetry of the laboratory jets and quantitatively describes all of the wave and vortex behavior in the experiments including the locking of the vortex chains of the eastward jet. The simulations and new model contradict the predictions of a competing, older model of the laboratory flow that is based on a Bickley jet; this raises concerns about previous calculations of Lagrangian mixing in the laboratory experiments that used the Bickley model for the fluid velocity. The new model, simulations and laboratory experiments all show that jets can be formed by the mixing and homogenization of q. The relevance of this to the jets of Jupiter is discussed.

1.
J. Sommeria, S. D. Meyers, and H. L. Swinney, “Experiments on vortices and Rossby waves in eastward and westward jets,” in Nonlinear Topics in Ocean Physics, edited by A. Osborne (North-Holland, Amsterdam, 1991), p. 227.
2.
J.
Sommeria
,
S. D.
Meyers
, and
H. L.
Swinney
, “
Laboratory model of a planetary eastward jet
,”
Nature (London)
337
,
58
(
1989
).
3.
T. H.
Solomon
,
W. J.
Holloway
, and
H. L.
Swinney
, “
Shear flow instabilities and Rossby waves in barotropic flow in a rotating annulus
,”
Phys. Fluids A
5
,
1971
(
1993
).
4.
J.
Sommeria
,
S. D.
Meyers
, and
H. L.
Swinney
, “
Laboratory simulation of Jupiter’s Great Red Spot
,”
Nature (London)
331
,
689
(
1989
).
5.
J. Pedlosky, Geophysical Fluid Dynamics, 2nd ed. (Springer, Berlin, 1986).
6.
P. S.
Marcus
, “
Jupiter’s Great Red Spot and other vortices
,”
Annu. Rev. Astron. Astrophys.
431
,
523
(
1993
).
7.
In the atmospheric literature eastward jets, which flow from west to east, are “westerlies.”
8.
P. S.
Marcus
and
C.
Lee
, “
Jupiter’s Great Red Spot and zonal winds as a self-consistent, one-layer, quasigeostrophic flow
,”
Chaos
4
,
269
(
1994
).
9.
S. S.
Limaye
, “
Jupiter: New estimates of the mean zonal flow at the cloud level
,”
Icarus
65
,
335
(
1986
).
10.
B. A.
Smith
,
L.
Soderblom
,
R.
Batson
,
P.
Bridges
,
J.
Inge
,
H.
Masursky
,
E.
Shoemaker
,
R.
Beebe
,
J.
Boyce
,
G.
Briggs
,
A.
Bunker
,
S. A.
Collins
,
C. J.
Hansen
,
T. V.
Johnson
,
J. L.
Mitchell
,
R. J.
Terrile
,
A. F.
Cook
, II
,
J.
Cuzzi
,
J. B.
Pollack
,
G. E.
Danielson
,
A. P.
Ingersoll
,
M. E.
Davies
,
G. E.
Hunt
,
D.
Morrison
,
T.
Owen
,
C.
Sagan
,
J.
Veverka
,
R.
Storm
, and
V. E.
Suomi
, “
A new look at the Saturnian system: The Voyager 2 Images
,”
Science
215
,
504
(
1982
).
11.
D. A.
Godfrey
, “
A hexagonal feature around Saturn’s North Pole
,”
Icarus
76
,
335
(
1988
).
12.
D.
del-Castillo-Negrete
and
P. J.
Morrison
, “
Chaotic transport by Rossby waves in shear flow
,”
Phys. Fluids A
5
,
948
(
1993
).
13.
P. B.
Rhines
and
W. R.
Young
, “
Homogenization of potential vorticity in planetary gyres
,”
J. Fluid Mech.
122
,
347
(
1982
).
14.
Numerical calculations are spectral with Fourier modes in the azimuthal and Chebyshev modes in the radial direction. The method has no splitting errors due to the fractional steps. Details are in Refs. 18 and 20.
15.
For all laboratory experiments and numerical calculations s=−0.1, H=18.7, Rin=10.8, Rout=43.2, R2slit=35.1, and 16<P⩽100 in CGS units. The values of ν, f, and R1slit vary. In the annular geometry β<0 while in the Cartesian (which replaces r with (−y))β>0. For pumping rates near Plock, the Reynolds number of the vortex chains (defined as the circulation of a vortex divided by ν) is ∼500, the Rossby number (based on their vorticities divided by f) is ∼0.1 and the Ekman spin-down time τ≃T where T is the characteristic turn-around time of a vortex.
16.
P. S.
Marcus
, “
Numerical simulations of Jupiter’s Great Red Spot
,”
Nature (London)
331
,
693
(
1988
).
17.
Another, but less general, way to show how pumping creates F is to note that pumping creates a weak, ageostrophic, radial velocity that starts at one slit and ends at the other: vag≃±P/2πrH for R1slit<r<R2slit and is zero elsewhere. Note that ∇⋅vagr̂≠0 and the vag is not part of the QG U in equation (2) (∇⋅U≡0, so there is a stream function: U=ẑ×∇ψ). The rotation of the annulus acts upon the ageostrophic velocity to create a Coriolis acceleration vagr̂×fẑ which is equivalent to F in equation (3). One could include the Coriolis acceleration due to U, U×fẑ, in equation (2), but it is equal to ∇(fψ), so it can be incorporated into the definition of Π/ρ (when ρ is constant) and is not included.
18.
C. Lee, “Basic instability and transition to chaos in a rapidly rotating annulus on a β-plane ,” Ph. D. dissertation, University of California at Berkeley, 1993.
19.
P. S.
Marcus
, “
Spatial self-organization of vortices in chaotic shearing flows
,”
Nucl. Phys. B, Proc. Suppl.
2
,
127
(
1988
).
20.
P. S.
Marcus
, “
Vortex dynamics in a shearing zonal flow
,”
J. Fluid Mech.
215
,
393
(
1990
).
21.
L. M.
Polvani
,
J.
Wisdom
,
E.
DeLong
, and
A. P.
Ingersoll
, “
Simple dynamical models of Neptune’s Great Dark Spot
,”
Nature (London)
249
,
1393
(
1990
).
22.
L. M. Polvani and J. Touma, “A note on recent experiments with Rossby waves on eastward jets” in Nonlinear Phenomena in Atmospheric and Ocean Sciences, edited by G. Carnevale and R. Pierrehumbert (1991).
23.
P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, 1981).
24.
For the westward jet model (in an unbounded domain) χ is given by: χ(1+k0)−2=[χ(1−k0)+2]ηe−2k(a+b) where η≡[k−k1±(k1+k)e−2(Y2slit−a)k1]/[k+k1±(k−k1)e−2(Y2slit−a)k1] where k1≡k1+2/{k(a+b)[k(a+b)−χ]} and k0=1−2/[kχ(a+b)]. The upper (lower) sign in the definition of η is for eigenmodes that are symmetric (anti-symmetric) about y=0. For eastward jets, the eigenmodes are always symmetric; otherwise, there is no disturbance of the contours of constant q at y=0 which is contrary to the laboratory and the numerically calculated flows, but for westward jets the eigenmodes could be antisymmetric, although in the initial-value calculations and experiments they are always symmetric. For Y2slit=a, η is +1 for symmetric and −1 for anti-symmetric modes. Tables III and IV are computed with symmetric modes. For eastward jets with (with one or two discontinuities) χ approximately satisfies χ(k0+1)−2=[χ(k0−1)+2]e−kk0(|Y1slit−Y2slit|−2a) with fractional errors of O(e−2k(a+b)/k2(a+b)2). If there are boundaries within a distance Lb of the discontinuities, then there are additional fractional errors in both expressions for χ of order e−2kLb. Under most circumstances the wavenumbers of interest are large and the discontinuities are far from the boundaries, so k(a+b) and kLb are large, the fractional errors are small, and the boundary locations do not usually affect c. For eastward jets, the fractional errors in [ū(yd)−c̄] in equation (27) contain the errors inherent in χ from above as well as errors of O(e−2kk0(Y2slit−a)). For the westward jets, equation (27) has fractional errors of O(e−2k(a+b)/k2(a+b)2,e−2kLb) for 2k(Y2slit−a)≫1 and O(e−2k(a+b),e−2kLb) otherwise.
25.
The dispersion relation for a flow as in fig. 8b has been previously published.
L. D.
Talley
, “
Radiating barotropic instability
,”
J. Phys. Oceanogr.
13
,
972
(
1983
).
26.
Our model flows also have Rossby waves that are not localized to the locations where there are jumps in . These eigenmodes are sines and cosines in y rather than exponentials. They exist in regions where ū is constant and are similar to the usual Rossby waves in a channel with c̄≃ū−β/[k2+(nπD)2] where n is an integer and D is the width of the region of constant ū. Since there is no forcing in the regions of constant ū, there would be no reason for these modes to be excited. Indeed, our numerical calculations show no evidence of these channel Rossby waves, so we do not consider them.
27.
Δq2 and c2 are further reduced by the proximity of Rout to R2slit. Note that Δq2 is equal to β(a2+b2). The value of b2 in the numerical calculation is smaller than that predicted by the model because the model’s value of (R2slit+b2) is greater than Rout which is impossible.
28.
E. Weeks and T. Solomon (private communication, 1995).
29.
The location of the outer discontinuity in q is much closer to Rout than a Rossby wavelength, typically 5% of a wave length or less than 2 cm.
30.
C. Lee and P. S. Marcus, “Eastward and westward jets in a rotating annulus model of a barotropic atmosphere,” in the Proceedings of the CERCA Workshop in Semi-analytic Methods in Turbulence, edited by K. Coughlin (Montreal, Canada, 1997).
31.
The large uncertainties in the averaged jovian velocities make it necessary to infer q. We carried out run-down experiments where the initial flow consists of an axisymmetric, east–west velocity superposed with noise. We varied the initial velocity and examined the speeds, shapes, and locations of the vortices that formed.6 The best results were with the initial q of the east–west velocity approximately equal to a step-function. Use of the traditional sinusoidal east–west velocity was particularly bad. When the initial q was approximately a step function, the chains of cyclonic and anti-cyclonic vortices were intertwined and centered at the westward jets, similar to the observed jovian vortex streets. An initially sinusoidal east–west velocity produced vortex chains widely spaced in latitude with each chain located near a local zero of the east–west velocity. Inclusion of a finite Rossby deformation radius in the QG equations or use of the shallow-water equations made these results even more pronounced.
32.
The authors have also used a model with finite Rossby deformation radius to examine these jets.8 In this case ū is a series of connected cosh functions rather than parabolas.
33.
S. Kim, “Dynamics of the Great Red Spot and Jovian Zonal Flows,” Ph. D. thesis, University of California at Berkeley, 1996).
34.
P. S. Marcus and C. Lee, “Two-dimensional turbulence and the formation of β-jets with forcing and dissipation” (in preparation).
35.
R. L.
Panetta
, “
Zonal jets in wide baroclincally unstable regions–Persistence and scale selection
,”
J. Atmos. Sci.
50
,
2073
(
1993
).
36.
S. A.
Condie
and
P. B.
Rhines
, “
A convective model for the zonal jets in the atmospheres of Jupiter and Saturn
,”
Nature (London)
367
,
711
(
1994
).
37.
G.
Williams
, “
Jovian and comparative atmospheric modeling
,”
Adv. Geophys.
28A
,
381
(
1985
).
38.
J. Y.-K.
Cho
and
L. M.
Polvani
, “
The morphogenesis of bands and zonal winds in the atmospheres on the giant outer planets
,”
Science
273
,
335
(
1996
).
39.
J. Y.-K.
Cho
and
L. M.
Polvani
, “
The emergence of jets and vortices in freely evolving, shallow-water turbulence on a sphere
,”
Phys. Fluids
8
,
1531
(
1996
).
40.
Computations with parabolic, Gaussian, and top-hat w(r) with the same Lslit give the same Pcrit to within 1%, where Lslit is the full width at half maximum of w(r). Figures 13 and 14 used a parabolic w(r). In principle, one could use the experimental values of Pcrit with fig. 14 to determine the best value of Lslit to use in numerical computations. However, this method cannot be used at P>Pcrit because the “best” value of Lslit depends on P: Lslit controls the vortex layer’s thickness. As Lslit increases, the ω(r) at each slit goes from a delta function to a function with width approximately equal to Lslit. With ν→0 equation (5) shows that u of the primary flow is approximately independent of Lslit, so at each layer
q≃∓fτP2πHRislitLslit+βRislit.
As the layers roll-up into discrete vortices, the q of each vortex is approximately conserved at its initial value. Thus, Lslit determines the strength of the vortices in the chains. With ν→0 equation (5) also shows that the potential circulation of the layers and chains are nearly independent of Lslit; therefore, the areas of the vortices must depend on Lslit.
This content is only available via PDF.
You do not currently have access to this content.