^{1,a)}, D. L. Boyer

^{1}and P. G. Baines

^{2}

### Abstract

Laboratory experiments were conducted on spin-up of a linearly stratified fluid in a rotating axisymmetric annular channel formed by two cylindrical coaxial walls and a flat bottom. Secular as well as instantaneous variation in rotation speeds was investigated for a range of Rossby numbers , where is the change in the rotation rate and is the final rotation rate of the annulus. The experimental studies reported by Smirnov *et al.* [S. A. Smirnov, P. G. Baines, D. L. Boyer, S. I. Voropayev, and A. N. Srdic-Mitrovic, “Long-time evolution of linearly stratified spin-up flows in axisymmetric geometries,” Phys. Fluids17, 016601 (2005)] were extended to (i) explore the density structure of the corner regions formed adjacent to the inner and outer sidewalls of the annular channel during spin-up, and to (ii) investigate the role of the boundary conditions at the vertical sidewalls in the development of nonaxisymmetric disturbances and formation of large columnar eddies at late spin-up times. The latter was achieved by introducing roughness elements in the form of vertical prisms at the inner sidewall. Observations demonstrated that isopycnals (surfaces of constant density) experience large vertical displacements near the lateral boundaries during spin-up. The density gradient reduces to near zero in the corner regions, where the fluid is stirred, and increases above/below them near the outer/inner sidewalls, respectively. The relative height of the corner regions ( is the depth of the fluid layer) was found to be determined only by the relative values of the Rossby and Burger numbers and follows the experimental dependence . A flow stability regime diagram is presented as a function of the Rossby and Burger numbers. Introduction of roughness elements at the inner sidewall did not alter significantly the process and time scales of stratified spin-up, large eddy formation, and subsequent relaxation to the initial state obtained with smooth sidewalls. This finding confirms that the growth of instability in the sidewall shear layers studied by Smirnov *et al.* does not depend on viscosity.

The authors would like to thank the National Science Foundation (NSF) under Grant No. OCE-0137197 for support of this research. We are also grateful to Professor H. J. S. Fernando for support of this research under the ONR Grant No. N00014-0-1-0626.

I. INTRODUCTION

II. LABORATORY SETUP

III. EXPERIMENTAL RESULTS

IV. DISCUSSION AND CONCLUSIONS

### Key Topics

- Eddies
- 36.0
- Flow instabilities
- 19.0
- Rotating flows
- 18.0
- Stratified flows
- 13.0
- Boundary value problems
- 11.0

## Figures

Schematics of the experimental facility. Inner (1) and outer (2) cylindrical walls form an axisymmetric annular channel of the width . The motion of a linearly stratified rotating fluid layer of the depth was video recorded from above using a camera (3) fixed relative to the rotating reference frame. and axes (in cm) form a local Cartesian coordinate system in the horizontal plane (when axis is orthogonal to both inner and outer sidewalls, i.e., it points in the radial direction).

Schematics of the experimental facility. Inner (1) and outer (2) cylindrical walls form an axisymmetric annular channel of the width . The motion of a linearly stratified rotating fluid layer of the depth was video recorded from above using a camera (3) fixed relative to the rotating reference frame. and axes (in cm) form a local Cartesian coordinate system in the horizontal plane (when axis is orthogonal to both inner and outer sidewalls, i.e., it points in the radial direction).

Dye visualization of the stratified spin-up flow in the annulus. Top view. Parameters: , , , and . Dimensionless time . The direction of the axis is from the center of the tank outwards, so that the inner and outer sidewalls of the annulus are located at the bottom and top parts of the image respectively. The direction of the spin-up current is from left to right.

Dye visualization of the stratified spin-up flow in the annulus. Top view. Parameters: , , , and . Dimensionless time . The direction of the axis is from the center of the tank outwards, so that the inner and outer sidewalls of the annulus are located at the bottom and top parts of the image respectively. The direction of the spin-up current is from left to right.

The same as in Fig. 2. Parameters: , , , and . Dimensionless time (a) 16 and (b) 62.

The same as in Fig. 2. Parameters: , , , and . Dimensionless time (a) 16 and (b) 62.

Velocity vector field corresponding to stratified spin-up. Top view. Parameters: , , , and . Coordinates and are in cm. Coordinates and correspond to the inner sidewall boundary. Dimensionless time (a) 16 and (b) 64. The streamlines are shown by the thick solid lines with the arrows representing the direction of the spin-up flow.

Velocity vector field corresponding to stratified spin-up. Top view. Parameters: , , , and . Coordinates and are in cm. Coordinates and correspond to the inner sidewall boundary. Dimensionless time (a) 16 and (b) 64. The streamlines are shown by the thick solid lines with the arrows representing the direction of the spin-up flow.

The same as in Fig. 2. Parameters: , , , and . Dimensionless time (a) 5, (b) 13, (c) 35, and (d) 50.

The same as in Fig. 2. Parameters: , , , and . Dimensionless time (a) 5, (b) 13, (c) 35, and (d) 50.

The same as in Fig. 4. Parameters: , , , and . Dimensionless time (a) 5, (b) 21, and (c) 48.

The same as in Fig. 4. Parameters: , , , and . Dimensionless time (a) 5, (b) 21, and (c) 48.

The same as in Fig. 2. Parameters: , , , and . Dimensionless time (a) 7, (b) 10, and (c) 45.

The same as in Fig. 2. Parameters: , , , and . Dimensionless time (a) 7, (b) 10, and (c) 45.

The same as in Fig. 4. Parameters: , , , and . Dimensionless time (a) 5, (b) 12, and (c) 64.

The same as in Fig. 4. Parameters: , , , and . Dimensionless time (a) 5, (b) 12, and (c) 64.

The same as in Fig. 2. Parameters: , , , and . Dimensionless time (a) 24, (b) 64, and (c) 77.

The same as in Fig. 2. Parameters: , , , and . Dimensionless time (a) 24, (b) 64, and (c) 77.

The same as in Fig. 2. Parameters: , , , and . Dimensionless time (a) 3 and (b) 11.

The same as in Fig. 2. Parameters: , , , and . Dimensionless time (a) 3 and (b) 11.

Flow regime diagram. The squares correspond to the experiments in the annulus reported in Ref. 1, the triangles represent the current data. The open symbols—unstable regime, and solid symbols—stable regime.

Flow regime diagram. The squares correspond to the experiments in the annulus reported in Ref. 1, the triangles represent the current data. The open symbols—unstable regime, and solid symbols—stable regime.

Vertical salinity profiles 1.5 cm from the outer sidewall for the same experiment as shown in Fig. 10. Dimensionless time is shown in the legend.

Vertical salinity profiles 1.5 cm from the outer sidewall for the same experiment as shown in Fig. 10. Dimensionless time is shown in the legend.

Vertical salinity profiles 2.5 cm from the rough inner sidewall. Parameters: , , and . Dimensionless time is shown in the legend.

Vertical salinity profiles 2.5 cm from the rough inner sidewall. Parameters: , , and . Dimensionless time is shown in the legend.

Vertical profiles of the salinity anomaly at 1.5 (solid symbols) and 8 cm (open symbols) from the outer sidewall. Parameters: (solid symbols) , , and ; (open symbols) , , and . Dimensionless time is shown in the legend.

Vertical profiles of the salinity anomaly at 1.5 (solid symbols) and 8 cm (open symbols) from the outer sidewall. Parameters: (solid symbols) , , and ; (open symbols) , , and . Dimensionless time is shown in the legend.

Vertical profiles of the salinity anomaly at 1.5 cm from the outer sidewall. Parameters: (solid symbols) , , and ; (open symbols) , , and . Dimensionless time is shown in the legend.

Vertical profiles of the salinity anomaly at 1.5 cm from the outer sidewall. Parameters: (solid symbols) , , and ; (open symbols) , , and . Dimensionless time is shown in the legend.

The same as in Fig. 15. Parameters: (solid symbols) , , and ; (open symbols) , , and . Dimensionless time is shown in the legend.

The same as in Fig. 15. Parameters: (solid symbols) , , and ; (open symbols) , , and . Dimensionless time is shown in the legend.

Dependence of the corner region height (measured at 1.5 cm from the outer sidewall) on the Rossby and Burger numbers. The corresponding boundary conditions are shown in the legend. Characteristic error bars are shown.

Dependence of the corner region height (measured at 1.5 cm from the outer sidewall) on the Rossby and Burger numbers. The corresponding boundary conditions are shown in the legend. Characteristic error bars are shown.

Relation between dimensionless eddy formation time and density relaxation time . The vertical level of observations is confined within the range of 3.5–5 cm from the bottom of the tank. The boundary conditions are shown in the legend. Characteristic error bars are shown.

Relation between dimensionless eddy formation time and density relaxation time . The vertical level of observations is confined within the range of 3.5–5 cm from the bottom of the tank. The boundary conditions are shown in the legend. Characteristic error bars are shown.

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