Instability of Ekman Flow at large Taylor Number

A liquid of mean depth H is rotating at constant angular velocity (ω) inside a cylindrical annulus whose radial width is large compared to H and small compared to the mean radius. An equilibrium Ekman flow is then produced inside the boundaries by (for example) symmetrically forcing liquid into the annulus at the outer rim and removing the same quantity at the inner circumference. In response to the radial pressure gradient a geostrophic azimuthal velocity ( u ) is established above the inflowing viscous boundary layer; the latter extending to a characteristic height h E = ( n? /ω) 1/2 above the bottom surface of the annulus. This is a study of the shearing instability of an Ekman flow whose components are uniform in both horizontal directions. We first discuss the anticipated similarities and differences of the stability of this “rotational” boundary layer and their counterparts in hydrodynamical flows which have no Coriolis force. Thus we note that the available kinetic energy in the boundary layer of the mean Ekman flow can be released by “pure-boundary” modes whose characteristic dimensions are of order h E and whose onset condition only depends on a boundary layer Reynolds number R = Uh Eμ −1 . This paper establishes the theoretical possibility of a qualitatively different energy releasing process, apparently confined to gyroscopic boundary layers, by “body-boundary” modes and suggests that for certain values of the Taylor number ( E = ω H 2 n? −1 ) these may occur at lower Reynolds numbers than the pure boundary species. The energetic picture of this instability is as follows: Pressure fluctuations in the mean boundary layer, whose radial wavelength is much larger than h E but much smaller than H and whose azimuthal wavelength is much larger than H , are partially (quasi-hydrostatically) transmitted vertically where they are accompanied by quasi-geostrophic azimuthal components of perturbation velocity and ageostrophic radial components. The coupling or this interior component with the boundary component of the perturbation influences the energy which is transferred to the latter from the radial (i.e. cross-isobaric) component of the mean field, and more than compensates for the burden of driving the perturbation against the stabilizing viscous forces in the interior of the fluid. Perhaps the most noteworthy aspect of these modes is that they can transport mean energy and momentum from the deep interior into the boundary, thereby making it available for further dissipation or to replenish the mean radial energy. The pure boundary modes, on the other hand, can only alter the geostrophic momentum in the interior by producing horizontal convergences of the mean flow in the friction layer. We have used two slowly “converging” asymptotic expansions to obtain solutions of the characteristic value problem and have supplemented these results with error estimates to indicate the quantitative limitations in the range of physical interest. For large values of the Taylor number a disturbance of fixed horizontal wavelength will be unstable for values of the Reynolds number occurring in discrete ranges which are separated by other ranges in which the disturbance transfers energy to the mean flow. These unstable ranges or “scales” correspond to the set of eigenfunctions having different vertical structure. On the other hand for fixed supercritical Reynolds number there is a preferred radial wavelength in the n th scale which is given by 2π m n −1 E 1/4 h E , where mn approaches a numerical constant as E → ∞. The corresponding minimum Reynolds number for the marginal stability of the nth eigenfunction is R n = x n E1/4, where x n is a numerical constant which only depends on the integer n, in the asymptotic limit E → ∞. For the “largest scale of motion” n = 1, this asymptotic limit is easily reached under laboratory conditions (e.g. E = 2,500). Because of the limitations of the second asymptotic expansion we can only estimate the numerical constants ( x 1 , m 1 ) as: m 1 = 10°, x 1 < 11. Thus, for example, when E = 2,500 the Ekman flow should not be stable for Reynolds numbers as large as 8 × 10‘. For larger values of E the Reynolds number which is necessary to destabilize the n = 1 scale increases. However smaller scales of motion (larger n ) are preferred as E increases since it is found that x n+1 < x n in the asymptotic limit E →. The present theory is unable to determine that finite value of E when the second scale of motion becomes preferred over n = 1. It is believed that a manifestation of this instability may be produced in the laboratory by causing a small decrease in the rate of rotation of a cylindrical vessel containing a shallow layer of water. Under certain conditions the decaying relative motion has the form of a time-modulated Ekman flow. By means of a permanganate solution placed in the boundary layer, the breakdown of this symmetric flow and the formation of thin rolls has been observed. We have also correlated experimental measures of the average rate of decay of relative angular momentum with a detailed laminar calculation (not included herein). For the converging Ekman flow, starting at a value of E of about 2,500, the measurements indicate a systematically greater mean decay rate than one would expect if the motion were laminar and symmetric. For the range of Reynolds numbers which were used in the experiment this critical point varied substantially less with changes in U (or R ) than with changes in ω (or E ). From these qualitative and quantitative observations we have concluded that the energy source for the instability is in the mean Ekman flow but that it cannot be entirely explained as a boundary layer phenomenon. It is suggested that the body-boundary modes discussed in the instability theory provide the mechanism for the increased transfer of angular momentum from the interior of the fluid to the lower boundary. DOI: 10.1111/j.2153-3490.1960.tb01327.x