Skip to main content Accessibility help
×
Home

Optical altimetry: a new method for observing rotating fluids with applications to Rossby and inertial waves on a polar beta-plane

  • P. B. RHINES (a1), E. G. LINDAHL (a1) and A. J. MENDEZ (a1)

Abstract

The entire free-surface elevation field of a rotating fluid in the laboratory can be imaged and analysed, by using it as a parabolic Newtonian telescope mirror. This ‘optical altimetry’ readily achieves a precision of better than 1 μm of surface elevation. The surface topography corresponds to the pressure field just beneath the surface. It is the streamfunction for the geostrophic hydrostatic circulation, which can be resolved to better than 0.1 mm s−1. Still and animated images thus produced, of the entire surface elevation field, are of value in themselves, and using a projected image (a speckle pattern), have the promise of providing quantitative slope and height field data recovered by PIV (particle imaging velocimetry) techniques. With homogeneous fluid, geostrophic flow is the same at all depths. Yet of equal interest are sheared stratified rotating flows where the surface pressure is associated with inertial waves, convection, and other motions, geostrophic or ageostrophic.

Although the technique is designed for experiments in which Coriolis effects are strong, it is possible to use reflective imaging for flows at such high Rossby number that Coriolis effects are negligible, and hence this becomes a tool of more general interest in non-rotating fluid dynamics (for example, illuminating surface gravity waves).

Examples are given, involving (i) the Taylor–Proudman effect with very slow flows over topography; (ii) quasi-geostrophic and inertial-wave flows over a mountain (f-plane); (iii) inertial waves generated by oscillatory forcing; (iv) Kelvin waves (v) free oscillatory Rossby waves on a polar β-plane; and (vi) stationary waves, blocking, jets and wakes with β-plane zonal flow past a mountain. Movies are available with the online version of the paper.

Copyright

References

Hide All
Adrian, R. J. 2005 Twenty years of particle image velocimetry. Exps. Fluids 39, 59169.
Afanasyev, Y. D., Rhines, P. B. & Lindahl, E. G. 2007 Altimetric imaging velocimetry: investigation of emission of inertial and Rossby waves by baroclinally unstable flows. J. Atmos. Sci. submitted.
Barbour, D. A. 2000 Understanding Foucault; a primer for beginners. http://www.atmsite.org/contrib/Harbour/Foucault.html.
Dabiri, D. & Gharib, M. 2001 Simultaneous free-surface deformation and near-surface velocity measurements. Exps. Fluids 30, 381390.
Fu, L. L., Cazenave, A. 2001 Satellite Altimetry and Earth Sciences, A Handbook of Techniques and Applications. International Geophysics Series, vol. 69. Academic, San Diego.
Hart, J. E. & Kittelman, S. 1986 A method for measuring interfacial wave fields in the laboratory. Geophys. Astrophys. Fluid Dyn. 36, 179185.
Haidvogel, D. & Rhines, P. B. 1983 Waves and circulation driven by oscillatory winds in an idealized ocean basin. Geophys. Astrophys. Fluid Dyn. 25, 165.
Hide, R., Ibbetson, A. & Lighthill, M. J. 1968 On slow transverse flow past obstacles in a rapidly rotating fluid. J. Fluid Mech. 32, 251272.
Holford, J. M. & Dalziel, S. B. 1996 Measurements of layer-depth in a two-layer flow. Appl. Sci. Res. 56, 191207.
Kunze, E. & Boss, E. 1998 A model for vortex-trapped internal waves. J. Phys. Oceanogr. 28, 21042115.
Lighthill, M. J. 1967 On waves generated in dispersive systems by travelling forcing effects, with applications to the theory of rotating fluids. J. Fluid Mech. 27, 725752.
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.
McCartney, M. 1975 Inertial Taylor columns on a β-plane. J. Fluid Mech. 68, 7195.
Newton, I. & Huygens, C. 1672 An accompt of a new catadioptical telescope invented by Mr. Newton. Phil. Trans. R. Soc. Lond. 81.
Rhines, P. B. 1969 a Slow oscillations in an ocean of varying depth. Part 1 Abrupt topography. J. Fluid Mech. 37, 161189.
Rhines, P. B. 1969 b Slow oscillations in an ocean of varying depth. Part 2. Islands and seamounts. J. fluid Mech. 37, 190205.
Rhines, P. B. 1977 The dynamics of unsteady currents. In The Sea, vol. 6, (ed. Goldberg, E. D.), pp. 189318. John Wiley.
Rhines, P. B. 1979 Geostrophic turbulence. Annu. Rev. Fluid Mech. 11, 404441.
Rhines, P. B. 1989 Deep planetary circulation over topography: simple models of mid-ocean flows. J. Phys. Oceanogr 19, 14491470.
Rhines, P. B. 2006 a Geophysical Fluid Dynamics laboratory, University of Washington, http://www.ocean.washington.edu/research/gfd
Rhines, P. B. 2006 b Wiki: laboratory fluid experiments for research and teaching. http://gfd.ocean.washington.edu/wiki
Rhines, P. B. 2006 c Jets and Orography: Idealized Experiments with Tip-Jets and Lighthill blocking J. Atmos. Sci. submitted.
Williams, P. D., Read, P. L. & Haine, T. W. N. 2004 Measuring the internal interface height field at high resolution in the rotating, two-layer annulus. Geophys. Astrophys. Fluid Dyn. 98, 453471.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Related content

Powered by UNSILO
Type Description Title
VIDEO
Movies

Rhines et al. supplementary movie
Animation of Figure 6: by peridiocally varying the rotation rate of the system the fluid oscillates back and forth over two circular obstacles, generating inertial waves, Rossby waves, lee vortices and Taylor columns. Time is compressed by a factor of approximately 10 in this and succeeding videos.

 Video (6.6 MB)
6.6 MB
VIDEO
Movies

Rhines et al. supplementary movie
Animation of Figure 6: by peridiocally varying the rotation rate of the system the fluid oscillates back and forth over two circular obstacles, generating inertial waves, Rossby waves, lee vortices and Taylor columns. Time is compressed by a factor of approximately 10 in this and succeeding videos.

 Video (3.3 MB)
3.3 MB
VIDEO
Movies

Rhines et al. supplementary movie
Animation of Figure 11: Kelvin waves and inertial waves generated by high-frequemcy oscillation of the table rotation with a shallow fluid layer (H = 5 cm). rate, in the presence of a small (1 cm high) spherical-cap mountain near 6 o'clock. The Kelvin waves propagate with the speed of long gravity waves anticlockwise about the cylinder, trapped along the outer boundary. Note the inertial wave crests moving opposite to the group velocity, hence toward their source. The circular crests are the inertial wave reflected from the circular boundary, whose energy propagates inward toward the center. There are also visible small, cyclonic eddies left of the forcing region which are shed from the oscillating mountain. At larger amplitude these become a dominant part of the response.

 Video (2.5 MB)
2.5 MB
VIDEO
Movies

Rhines et al. supplementary movie
Animation of Figure 11: Kelvin waves and inertial waves generated by high-frequemcy oscillation of the table rotation with a shallow fluid layer (H = 5 cm). rate, in the presence of a small (1 cm high) spherical-cap mountain near 6 o'clock. The Kelvin waves propagate with the speed of long gravity waves anticlockwise about the cylinder, trapped along the outer boundary. Note the inertial wave crests moving opposite to the group velocity, hence toward their source. The circular crests are the inertial wave reflected from the circular boundary, whose energy propagates inward toward the center. There are also visible small, cyclonic eddies left of the forcing region which are shed from the oscillating mountain. At larger amplitude these become a dominant part of the response.

 Video (1.0 MB)
1.0 MB
VIDEO
Movies

Rhines et al. supplementary movie
Animation of Figure 13: Monochrome imaging of surface elevation of a rotating fluid, using optical altimetery (full colour altimetry with quantitative velocity, vorticity and potential vorticity has now been achieved, Afanasyev et al., 2007). Rossby waves on a polar beta-plane generated by periodically oscillating the table rotation rate with a single frequency, in the presence of a spherical cap mountain at 8 o'clock. Note the short Rossby waves to the east, and long Rossby waves to the west (clockwise), which spiral into the North Pole, and fleeting, very short inertial waves to the west, yet not to the east. A dark radial line (the image of a wire near the light source) distorts, showing the azimuthal slope of the free surface (and hence the radial component of geostrophic velocity). Time compression: 12x. Omega = 2.23 -1; oscillation frequency = 0.16 s-1. At smaller amplitude this experiment generates waves of this same frequency; however at the present amplitude we are making time-dependent lee waves to the east of the mountain (the short waves), with a spectrum of intrinsic frequencies, while the long waves west of the mountain are in fact nearly a sharp response at the forcing frequency.

 Video (5.2 MB)
5.2 MB
VIDEO
Movies

Rhines et al. supplementary movie
Animation of Figure 13: Monochrome imaging of surface elevation of a rotating fluid, using optical altimetery (full colour altimetry with quantitative velocity, vorticity and potential vorticity has now been achieved, Afanasyev et al., 2007). Rossby waves on a polar beta-plane generated by periodically oscillating the table rotation rate with a single frequency, in the presence of a spherical cap mountain at 8 o'clock. Note the short Rossby waves to the east, and long Rossby waves to the west (clockwise), which spiral into the North Pole, and fleeting, very short inertial waves to the west, yet not to the east. A dark radial line (the image of a wire near the light source) distorts, showing the azimuthal slope of the free surface (and hence the radial component of geostrophic velocity). Time compression: 12x. Omega = 2.23 -1; oscillation frequency = 0.16 s-1. At smaller amplitude this experiment generates waves of this same frequency; however at the present amplitude we are making time-dependent lee waves to the east of the mountain (the short waves), with a spectrum of intrinsic frequencies, while the long waves west of the mountain are in fact nearly a sharp response at the forcing frequency.

 Video (2.5 MB)
2.5 MB
VIDEO
Movies

Rhines et al. supplementary movie
Animation of Figure 14: eastward (anticlockwise) flow over a spherical cap mountain at 2 o'clock. Development of lee Rossby waves, jet-like concentration of flow, spiral flow above the mountain (an arrested topographic Rossby wave), and an upstream (eastward, clockwise) blocking of the circulation. A growing cyclonic wake involving eastward counter-flow forms behind the mountain. The table rotation is steadily ramped downward to maintain the zonal flow, which accounts for the gradually change of the lighting, as the parabolic free surface relaxes. We have left the glass top from the experiment to promote evaporative convection, which produces the mottled pattern of tiny convective cyclones upstream of the mountain. Note how these show the larger scale flow field. In regions of strong shear we see instead convective sheets (tall 'rolls') which thus outline regions of strong shear. The fluid surface deformation is of order 1 micron (10-6 m. in these features, and is of order 1 mm. in the most intense flows.

 Video (919 KB)
919 KB
VIDEO
Movies

Rhines et al. supplementary movie
Animation of Figure 14: eastward (anticlockwise) flow over a spherical cap mountain at 2 o'clock. Development of lee Rossby waves, jet-like concentration of flow, spiral flow above the mountain (an arrested topographic Rossby wave), and an upstream (eastward, clockwise) blocking of the circulation. A growing cyclonic wake involving eastward counter-flow forms behind the mountain. The table rotation is steadily ramped downward to maintain the zonal flow, which accounts for the gradually change of the lighting, as the parabolic free surface relaxes. We have left the glass top from the experiment to promote evaporative convection, which produces the mottled pattern of tiny convective cyclones upstream of the mountain. Note how these show the larger scale flow field. In regions of strong shear we see instead convective sheets (tall 'rolls') which thus outline regions of strong shear. The fluid surface deformation is of order 1 micron (10-6 m. in these features, and is of order 1 mm. in the most intense flows.

 Video (1.0 MB)
1.0 MB

Optical altimetry: a new method for observing rotating fluids with applications to Rossby and inertial waves on a polar beta-plane

  • P. B. RHINES (a1), E. G. LINDAHL (a1) and A. J. MENDEZ (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.