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RAPID COMMUNICATIONS PHYSICAL REVIEW E VOLUME 62, NUMBER 4 OCTOBER 2000 Effect of a vertical magnetic field on turbulent Rayleigh-Be´nard convection S. Cioni, S. Chaumat,* and J. Sommeria Ecole Normale Supe´rieure de Lyon, Laboratoire de Physique, CNRS URA1325, 46, Alle´e d’Italie, 69364 Lyon Cedex 07, France ͑Received 3 July 2000͒ The effect of a vertical uniform magnetic field on Rayleigh-Be´nard convection is investigated experimen- tally. We confirm that the threshold of convection is in agreement with linear stability theory up to a Chan- drasekhar number QӍ4ϫ106, higher than in previous experiments. We characterize two convective regimes influenced by MHD effects. In the first one, the Nusselt number Nu proportional to the Rayleigh number Ra, which can be interpreted as a condition of marginal stability for the thermal boundary layer. For higher Ra, a second regime NuϳRa0.43 is obtained. PACS number͑s͒: 47.27.Te, 47.65.ϩa, 47.27.Gs We consider the Rayleigh-Be´nard problem of a fluid layer ary conditions for velocity ͑free slip or no slip͒. Note also heated from below and cooled from above. Convection oc- that the threshold does not depend on the electric boundary curs for a Rayleigh number ͓1͔ higher than a critical value conditions ͑i.e., wall conductivity͒, whatever the value of Q. ͑ ͒ Rac . We use an electrically conducting fluid mercury and Our experimental cell ͓9,10͔ is a vertical cylinder with impose a uniform vertical magnetic field B, resulting in eddy aspect ratio ⌫ϭD/Hϭ1, suitable to reach high Rayleigh currents and flow damping by the Lorentz force. Energy is numbers for a given available heating power. Due to the then dissipated by the Joule effect in addition to viscosity. As confinement by the lateral walls, the geometry differs from a consequence, the convection threshold increases with the the standard Rayleigh-Be´nard problem and from Nakaga- magnetic field, as predicted by the linear stability analysis of wa’s experiments ͑with thickness 3 to 6 cm and large aspect Thompson ͓2͔ and Chandrasekhar ͓3,4͔. We find a good ratio͒. In the absence of magnetic field, this difference results agreement with this linear theory, confirming the experi- in a higher convection threshold. However, the influence of ments of Nakagawa ͓5,6͔ and Jirlow ͓7͔. However, our main the lateral confinement is very weak in the turbulent regime. result is a characterization of turbulent magnetohydrodynam- The system is then mainly controlled by the thermal bound- ics ͑MHD͒ regimes occurring beyond the threshold. This ary layers, much thinner than the diameter. In the presence of problem is of interest for technical applications, as in the a vertical magnetic field the most unstable modes have a control of crystal growth, and for the convection in planetary small horizontal wavelength ͑in QϪ1/3), so we expect that cores or stars. Moreover this study can shed light on ordinary the lateral confinement is not significant even near the insta- convection, providing a means of external action on the sys- bility threshold. The cell is introduced in an electromagnetic tem. As pointed out by Bhattacharjee et al. ͓8͔, the different coil, producing a vertical uniform magnetic field, tunable regimes of ordinary turbulent convection have an MHD from 0 to 0.4 T, with uniformity better than 10Ϫ3. counterpart for which the scaling laws of the different re- In summary the dynamics is in principle determined by gimes can be more sharply distinguished. five nondimensional numbers, the Rayleigh number Ra, the The effect of the magnetic field is classically estimated by Chandrasekhar number Q, the Prandtl number Prϭ␯/␬, the the Chandrasekhar number Q, ratio of the damping times by magnetic Prandtl number ␩/␬, and the aspect ratio ⌫.We Joule and viscous effects respectively, vary Ra up to 3ϫ109 and Q up to 4ϫ106, while the last three parameters are fixed, with Prϭ0.025, ␩/␬ϭ2ϫ105 B2␴L2 Ӎϱ ⌫ϭ ͑ ⌫ ͒ Qϭ , ͑1͒ , and 1 but we expect that has little influence . ␳␯ The bottom plate is in copper ͑coated with nickel to pro- tect it from mercury͒ and is heated by an electrical resistor, where ␴ is the electrical conductivity and ␳ the mean fluid shaped in a double spiral to avoid generation of a magnetic Ϫ density. In our experiment, ␴ϭ1.04ϫ106 m⍀ 1 and ␳ field. The top plate, also in copper, is cooled by a water Ϫ ϭ1.36ϫ104 kg m 3,soQϭ3.21ϫ107 B2, with B in T. circulation, and is regulated in temperature ͑measured on the Note that Q is the square of the Hartmann number, initially axis of the cell, at 3 mm above the mercury layer͒. The introduced for duct flows. The magnetic diffusivity ␩ lateral wall is a 2-mm-thick stainless steel cylinder. The cell Ϫ ϭ0.8 m 2 is much larger than the thermal diffusivity, by a is thermally insulated from the outside by neoprene layers, factor 2ϫ105. As a consequence, the magnetic field pro- with a total loss coefficient 0.2W/K. duced by eddy currents can be neglected with respect to the The temperature is measured in each plate by eight ther- imposed field B. mistors, equally spaced in azimuth, at 5 cm from the cell The convection threshold Rac increases with the magnetic axis, and 3 mm from the plate surface in contact with mer- !␲2 ӷ field, and Rac Q when Q 1. This limit corresponds to cury. The temperature difference ⌬ between top and bottom negligible viscosity effects, and is independent of the bound- is obtained as an average over these probes. In each experi- ment, we set the heating power and the temperature of the top plate ͑in such a way that the mean temperature of the cell *Electronic address: [email protected] is close to the coil temperature, to minimize lateral heat 1063-651X/2000/62͑4͒/4520͑4͒/$15.00PRE 62 R4520 ©2000 The American Physical Society RAPID COMMUNICATIONS PRE 62 EFFECT OF A VERTICAL MAGNETIC FIELD ON... R4521 FIG. 1. Nu vs Ra with and without magnetic field. Without ͑ ͒ ͑ ͓͒ ͔ ͑ϩ͒ magnetic field: • data of Rossby 1968 12 , numerical simu- lations of Verzicco and Camussi ͑1996͓͒13͔, ͑᭺͒ data of Cioni et al. ͓9͔. With magnetic field, present experiment: ͑᭠͒ at Q FIG. 2. A comparison of the experimental and theoretical results ϭ7.22ϫ105 (Bϭ1500 G͒, ͑*͒ at Qϭ2.0ϫ106 ͑Bϭ2500 G͒, and on the critical Rayleigh numbers for the onset of instability. The ͑᭟͒ ϭ ϫ 6 ͑ ϭ ͒ ͑ϩ͒ at Q 3.93 10 B 3500 G . The two dashed lines represent theoretical (Rac ,Q) relation is shown by the solid line curve. the Ra0.43 scaling and the solid ones represent the Ra1 scaling. are the experimentally determined points of Nakagawa ͑1957͓͒5,6͔ and ͑*͒ are the experimental findings of the present work. loss͒. Then we measure the mean temperature difference ⌬ in the permanent regime, and deduce the Nusselt number ͓11͔ covers one decade at the highest magnetic field, 0.35 T. We Nu, ratio of the heat flux to the purely diffusive one. Our can understand this result by an argument of marginal stabil- main experimental result in this Rapid Communication is the ity of the thermal boundary layer, in the spirit of Malkus ͓16͔ determination of Nu as a function of the two parameters Ra and Howard ͓17͔. Turbulence is assumed to mix the tem- and Q. We also measure local temperature using a probe perature in the interior, confining the gradient to a thermal movable along the cell axis, providing vertical temperature boundary layer near the walls. The thickness ␭ of this bound- profiles and statistics of local fluctuations. ary layer is assumed to be maintained at the limit of stability. We first recall in Fig. 1 the results Nu versus Ra in the ͑ ͒ In other words, the Rayleigh number Ra␭ based upon this absence of a magnetic field in the same cell . The results of ϳ ͓ ͔ ͓ ͔ thickness remains equal to the critical one, Ra␭ Rac Rossby 12 and Cioni et al. 9 both fit very well with a law ϭ ϭ 0.26 4р р ϫ 8 const. In the absence of a magnetic field this yields the Nu 0.14Ra , in the range 10 Ra 4.5 10 . Numerical ϳ͑ 1/3 ͓ ͔ classical law Nu Ra/Rac) . Here Rac depends on the results 13 , performed in the same cylindrical geometry as ϳ␲2 4 magnetic field. From the Chandrasekhar theory Rac Q␭ in our experiment, confirm this law for Raу2ϫ10 ͑the nu- 2 2 for large Q␭ . Furthermore, using the fact that Q␭ϳQ␭ /L merical results at lower Ra differ from the Rossby experi- and NuϳL/␭ we deduce ͓8͔ ments due to the different aspect ratio͒. Takeshita et al. ͓14͔ found the same scaling in Ra, although their prefactor is larger by about 20%. Transitions to new regimes ͓9͔ are 1 Ra у ϫ 8 Nuϳ regime I. ͑2͒ obtained for Ra 4.5 10 . ␲2 Q The effect of a magnetic field reduces heat transfer as expected. With a magnetic field of 0.35 tesla, we are able to totally suppress convection, reaching Nuϭ1. The threshold is Therefore, this law is the MHD counterpart of the classical consistently obtained both by the change in the vertical tem- Ra1/3 law. We check that the dependence in Q is consistent perature profile ͑which is linear in the diffusive regime͒, and with Eq. ͑2͒, when comparing the results at Bϭ0.25 T and by the onset of temperature fluctuations. No hysteresis is Bϭ0.35 T ͑Fig.
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