Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
The influence of centrifugal buoyancy in rotating convection.<br />
F. Marques ∗ ,J.M. Lopez † , O. Batiste ∗ and I. Mercader ∗<br />
Rotating thermal convection is a paradigm problem that incorporates fundamental<br />
processes of great importance to atmospheric and oceanic circulations, as well as of<br />
astrophysical importance. We analyze rotating convection in a simple geometrical<br />
setting, consisting of an enclosed circular cylinder of depth-to-radius ratio Γ=d/R =<br />
1.0, rotating at an angular rate ω, with the bottom endwall at temperature T0 +∆T<br />
and the top endwall at T0. The nondimensional parameters are the Coriolis parameter<br />
Ω=ωd 2 /ν, theRayleigh number Ra = αg∆Td 3 /νκ, the Prandtl number σ = ν/κ<br />
and the Froude number Fr = ω 2 R/g. Three-dimensional numerical simulations using<br />
a spectral code have been made in this geometry.<br />
Traditionally, density variation was only incorporated in the gravitational buoyancy<br />
term and not in the centrifugal buoyancy term. Inthislimit, corresponding to<br />
Fr → 0, the governing equation admit a trivial conduction solution, resulting in important<br />
simplifications of the problem. Linear stability analysis neglecting centrifugal<br />
buoyancy in an enclosed rotating cylinder finds that typically the onset of thermal<br />
convection from the conduction state is to a so-called wall mode which consists of alternating<br />
hot and cold thermals rising and decending in the cylinder boundary layer 1 .<br />
Experiments to test this linear theory have needed to be carefully designed in order<br />
to minimize the effects of the neglected centrifugal buoyancy, and have found good<br />
agreement with the theory for the onset of convection 2 .<br />
The presence of centrifugal buoyancy changes the problem in a fundamental manner.<br />
The buoyancy force has a radial component that destroys the horizontal translation<br />
invariance assumed in the unbounded theoretical treatments of the problem.<br />
Furthermore, the Z2 reflection symmetry about the cylinder mid-height is also destroyed.<br />
The centrifugal buoyancy drives a large scale circulation in which the cool<br />
denser fluid is centrifuged radially outward and the hot less dense fluid is centrifuged<br />
radially inward 3 . This large scale circulation exists for any ∆T = 0,andsothereis<br />
no trivial conduction state when the centrifugal buoyancy is incorporated.<br />
For small Froude numbers the transition from the base axisymetric state to 3D<br />
flow happens around Ra ≈ 7.500. We have found that for moderate Froude numbers<br />
(Fr ≥ 4) the centrifugal buoyance delays this transition to much larger Rayleigh<br />
numbers, Ra ≈ 50.000. At intermediate Fr the transition to 3D flow happens via<br />
four different Hopf bifurcations, with complex interactions between the bifurcated<br />
states,resulting in different coexisting branches of 3D solutions. How the centrifugal<br />
buoyancy and the gravitational buoyancy interact and compete, and the manner in<br />
which the flow becomes three-dimensional is different along each branch. The main<br />
conclusion is that centrifugal buoyancy changes quantitatively and qualitatively the<br />
flow dynamics, and should not be neglected in rotating convection problems.<br />
∗Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain.<br />
† Department of Mathematics and Statistics, Arizona State University, Tempe AZ 85287, USA.<br />
1Goldstein et. al., J. Fluid Mech. 248, 583–604 (1993).<br />
2Zhong et. al., Phys. Rev. Lett. 67, 2473–2476 (1991) and J. Fluid Mech. 249, 135–159 (1993).<br />
3Barcilon &Pedlosky, J. Fluid Mech. 29, 673–690 (1967); Homsy & Hudson, J. Fluid Mech. 35,<br />
33–52 (1969).<br />
17