Model of Convective Taylor Columns in Rotating Rayleigh-Be ´nard Convection Ian Grooms, 1 Keith Julien, 1 Jeffrey B. Weiss, 2 and Edgar Knobloch 3 1 Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309, USA 2 Department of Atmospheric and Oceanic Sciences, University of Colorado, Boulder, Colorado 80309, USA 3 Department of Physics, University of California, Berkeley, California 94720, USA (Received 15 February 2010; published 1 June 2010) Observations, and laboratory and numerical studies, of fluid flows with strong rotation and thermal forcing often show long-lived convective Taylor columns (CTCs) which carry a large portion of the vertical heat and mass fluxes. However, owing to experimental and numerical challenges, these structures remain poorly understood. Here we present a nonlinear, analytical multiscale model of CTCs in the context of rotating Rayleigh-Be ´nard convection that successfully matches numerical simulations and provides a new multiscale interpretation of the Taylor-Proudman constraint. DOI: 10.1103/PhysRevLett.104.224501 PACS numbers: 47.55.pb, 47.32.Ef, 47.27.De, 47.27.te Many geophysical and astrophysical phenomena involve fluids under the combined influence of rotation and thermal forcing. Examples include the Sun and stars [1], giant planets [2], and Earth’s oceans [3]. As is common in rotating fluids, these phenomena display self-organization into coherent features which dominate the flow. Simple models of coherent features in other complex fluids have led to major advances in understanding and modeling [4– 6]. In this Letter, we present a new, analytical, multiscale model of coherent convective columns which successfully describes the structures seen in laboratory experiments [7– 9] and quantified in numerical simulations [10–12]. Furthermore, the multiscale nature of the approach pro- vides new insight into the role of the Taylor-Proudman (TP) constraint, improving on previous single-scale interpretations. The essence of rotating, thermally forced flow is cap- tured by rotating Rayleigh-Be ´nard convection, i.e., con- vection in a layer of Boussinesq fluid confined between flat, horizontal, rigidly rotating upper and lower boundaries held at fixed temperatures, with the temperature of the lower boundary higher than that of the upper boundary by an amount T> 0. In this framework, the controlling nondimensional parameters are the (convective) Rossby number (Ro), which measures the strength of rotation relative to thermal forcing, and the Rayleigh number Ra / T, which measures the strength of the thermal forcing. Rotating Rayleigh-Be ´nard convection has been exten- sively studied using linear [13], weakly nonlinear [14], and fully nonlinear approaches [15], in addition to numerical [10,11,16] and laboratory [7–9] experiments. These studies attest to the simple fact that the flow tends to self-organize into regions of upwelling hot and downwelling cold fluid, whether the flow is turbulent, as in the case of highly supercritical plume-dominated convection [17], or laminar, as in the case of planform convection [13,18]. In each case, the flow must somehow accommodate the TP constraint. This constraint, formulated theoretically by Taylor [19] following earlier experiments by Proudman [20], explains why rapid rotation tends to inhibit flow variation along the axis of rotation. For high enough Rayleigh numbers the TP effect is overwhelmed by thermal forcing, while for strong enough rotation the constraint completely inhibits vertical variation of the flow. Between these extremes the compet- ing influences of thermal forcing and rotational (Ekman) friction have a profound effect on the morphology of the flow structures, and for certain parameter regimes long- lived columnar convective structures are observed to form [7–9]. In a recent paper, Portegies et al. [21] presented a linear model of these structures. In this Letter we construct a self-consistent nonlinear model that agrees well with recent numerical simulations [11,12]. Hereafter we call thesestructures ‘‘convective Taylor columns’’ (CTCs). The TP constraint requires that, away from the confining boundaries, the CTCs are nearly uniform vertically. The columns must, however, overcome the TP constraint near the boundaries where the vertically moving fluid slows down, resulting in a strong vertical variation in velocity. It is commonly believed that the required deceleration at the boundaries is accomplished by Ekman boundary layers, i.e., viscous boundary layers influenced by rotation (e.g., [22]). While it is certainly true that Ekman layers can in theory cause strong vertical variations, recent simulations [12] support a different explanation. The essential morphology of CTCs is that they are tall and thin. This suggests an approach with horizontal scales that are much smaller than the dominant vertical scale. Recent work [23] shows that this scale separation arises naturally in the low Rossby number (rapid rotation) re- gime. In this regime the Ekman layers are passive [18], but the flow nonetheless organizes into CTCs. Recent simula- tions [12,21] show that the CTCs in this regime are shielded by a sheath of opposite vorticity and hence inter- act only weakly. The presence of this shield appears to be a characteristic property of the CTCs in the rapid rotation limit; Fig. 1 shows that this shield is visible in the tem- perature field as well. Owing to the multiscale nature of the flow, the TP constraint requires vertical variations to be PRL 104, 224501 (2010) PHYSICAL REVIEW LETTERS week ending 4 JUNE 2010 0031-9007= 10=104(22)=224501(4) 224501-1 Ó 2010 The American Physical Society