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Quasigeostrophic model of the instabilities of the Stewartson layer in flat and depth-varying containers
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Image of FIG. 1.
FIG. 1.

Sketch of the nested Stewartson layers in the split-sphere geometry. The bold arrows show the Ekman pumping inside these layers.

Image of FIG. 2.
FIG. 2.

Basic state flow for in the split-sphere geometry. (a) Radial profiles; (b) radial profiles rescaled by the asymptotic size of the Stewartson layer.

Image of FIG. 3.
FIG. 3.

North-pole views of the radial velocity for the first unstable mode at for various geometries. The picture on the left is for , and the one on the right is for . The split radius is shown by a dotted circle: (a) shows the split-sphere case; (b) shows the constant case ; (c) shows the flat container case ; (d) shows the spherical case with flat disks.

Image of FIG. 4.
FIG. 4.

Stability threshold data (given in the first appendix table) as a function of the Ekman number. (a) Represents the threshold; (b) shows the critical wave number; (c) is the raw frequency at the threshold; (d) is the frequency corrected for advection effects. Dotted lines indicate the asymptotic behavior.

Image of FIG. 5.
FIG. 5.

Sketch of the experimental setup with flat disks. The spherical shell geometry can be realized with a central inner sphere replacing the two disks on the shaft (see Fig. 6).

Image of FIG. 6.
FIG. 6.

Instabilities of the Stewartson layer slightly above the threshold as seen in our experiments at . The shaft holding the disks or inner sphere is along the rotation axis. The features shown by flakes are aligned with the rotation axis. Top left panel shows a stable axisymmetric flow driven by two corotating disks, that becomes unstable on the top right panel as the disks rotate faster . At the bottom, we show the unstable flow driven by a rotating inner sphere for , illuminated with a meridional light sheet on the left (equator view), and with a light sheet parallel to the equator on the right (north pole view).

Image of FIG. 7.
FIG. 7.

Experimental determination of the stability threshold for two geometries, compared with the numerical calculations using a quasigeostrophic model and with 3D calculations of Hollerbach (Ref. 13).

Image of FIG. 8.
FIG. 8.

Critical Rossby numbers obtained by Hollerbach [2003, summary of his Figs. 4 and 8(a), here denoted by ] showing the impact of the slope on the stability threshold. We also plotted the corresponding results obtained with our QG model (basic state shear layer obtained with an inner core, instabilities computed without) including the critical mode. When removing the inner core (ic), the threshold drops, and positive and negative Rossby numbers get closer. Our QG model is close to the fully 3D results of Hollerbach (Ref. 13).

Image of FIG. 9.
FIG. 9.

Basic state axisymmetric flow profiles [solution of Eq. (14)], as a function of radius for and a split-sphere geometry with . (a) Angular velocity ; (b) vorticity ; (c) radial derivative of vorticity .


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Generic image for table
Generic image for table
Generic image for table
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Table I.

Comparison between the experimental results of Ref. 9 (denoted by ) and our numerical results.

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Table II.

Comparison between the experimental results of Ref. 10 (denoted by ) and our numerical results.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Quasigeostrophic model of the instabilities of the Stewartson layer in flat and depth-varying containers