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UC Santa Cruz 2010 International Summer Institute for Modeling in Astrophysics UC Santa Cruz 2010 International Summer Institute for Modeling in Astrophysics Title Taming jets in magnetised fluids Permalink https://escholarship.org/uc/item/5ff523wv Authors Kosuga, Yusuke Brummell, Nicholas Publication Date 2010-09-01 eScholarship.org Powered by the California Digital Library University of California Taming Jets in magnetized fluids Y. Kosuga1 and N. H. Brummell2 1)Center for Astrophysics and Space Sciences and Department of physics, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093 2)Department of Applied Mathematic and Statistics, University of California at Santa Cruz, 1156 High Street, Santa Cruz, CA 95064 Effects of a uniform horizontal magnetic field on jets dynamics in 2D Boussinesq turbulence, i.e. Howard-Krishnamurti problem are studied with a numerical simula- tion. For a fixed fluid and magnetic diffusivity, it is shown that as the imposed field strength becomes larger jets start behaving in a more organized way, i.e. achieve sta- tionary state and are finally quenched. The time evolution of total stress, Reynolds stress, Maxwell stress is examined and all the stresses are shown to vanish when jets are quenched. The quenching of jets is confirmed for different values of magnetic diffusivity, albeit the required field strength increases. It is also shown that the in- clusion of overstable modes reinforces jets where Maxwell stress overcomes Reynolds stress. For a larger imposed field jets are shown to quench. A possible mechanism for the transition to the reinforcement of jets by Maxwell stress is discussed based on the transition in the most unstable mode in the underlying turbulence. 1 I. INTRODUCTION Turbulence and subsequent turbulence driven large scale flows or Jets are ubiquitous phenomena in nature. In the convective region in the sun, turbulence is driven by the convective instability and the differential rotation result1. In planets, the Jupiter or the earth for example, geostrophic turbulence with Rossby waves is excited due to the energy input from the Sun, etc, and produces the famous zonal bands of Jets, i.e. zonal flows2,3. The combination of turbulence and large scale flows are observed in laboratory plasmas as well. In tokamaks, a device to confine plasmas magnetically, inhomogeneity in temperature or density excites drift wave turbulence and the drift wave turbulence in turn produces large scale flows called intrinsic rotation4,5 and zonal flows6,7, which are analogous to the differential rotation in the sun and zonal jets on planets, respectively. Understanding basic physics of these systems with turbulence and jets is an important issue, since of course these systems are physically interesting and challenging to model, but also are quite beneficial from a practical point of view: in fusion devices these large scale flows are believed to play an important role to achieve better confinement by reducing turbulent transport of heat and particle and improving stability of tokamak plasmas. To be more specific, we focus on the issue of the generation and quenching of jets in 2D turbulence with magnetic field. A theory for beta plane MHD turbulence was formulated8 and numerical work9 was done to confirm the quenching of jets by the inclusion of magnetic field in beta plane turbulence; however, they used arbitrary forcing to excite turbulence. Here, instead, we consider a problem of jets dynamics in 2D convective turbulence, which is a natural forcing mechanism, with horizontal magnetic field, where horizontal here means the direction of jets which is normal to the direction with a temperature gradient. The prob- lem with a vertical magnetic field, i.e. the problem of the magneto-convection was treated and understood fairly well10{12. The research of jets in 2D convective turbulence without magnetic field was pioneered by Krishnamurti et al13 who observed the tilting of convective cell, formation of jets, chaotic motion which destroys well-defined structures of cells, by in- creasing the Rayleigh number for different values of the Prandtl number. Here the Rayleigh number measures the strength of temperature difference in a system and the Prandtl num- ber is defined to be the ratio of a fluid viscosity and a thermal diffusivity. Howard et al14 modeled the phenomena as a dynamical system with truncated Fourier components which 2 includes the famous Lorenz system for convection as a subsystem. The transition between different flow patterns was described as the transition between fixed points and especially the transition from steady convective cell to jets with tilted steady cells was understood as an instability of the fixed point which represents a steady convective solution. Numer- ical experiments also confirmed the similar behavior and further discovered different type of behavior such as a pulsating traveling wave15. The problem was extended to include a horizontal magnetic field and understand the transition of flow patterns discovered by the simulation for 2D convective turbulence with compressibility (anelastic approximation) and a horizontal magnetic field16,17. They observed the transition of flow patterns from a chaotic pattern to conductive state where the instability is switched off, by increasing the 18 ≡ 2 2 Chandrasekhar number Q vAd =(νη) at a fixed Rayleigh number. However, the result is somewhat trivial, since the increasing of the Chandrasekhar number or the increasing of the field strength for a fixed magnetic diffusivity, leads to the stabilization of the linear convective instability, the very source which drives rich behaviors in the system, i.e. the formation of jets, oscillation of jets and so on. And also the dynamics of jets, which is the main focus in this report, was not the focus of the work by Lantz et al. In this report, we examine the effect of horizontal magnetic fields on the transition be- tween flow patterns in 2D convective turbulence, i.e. Howard & Krishnamurti (HK) problem. We especially focus on the behavior of jets, i.e. whether horizontal magnetic fields weaken, quench, or possibly enhance jets. In doing so, we start from HK problem with a given parameter set, and change the values of the horizontal magnetic field imposed on a sys- tem and magnetic diffusivity, with a fixed criticality. First we focus on the cases without overstable modes, which exist only when η > κ, where η and κ is magnetic diffusivity and thermal conductivity respectively, and show that the jets are quenched by increasing the Chandrasekhar number. We calculate the balance between Reynolds stress and Maxwell stress for different values of Q and show that Reynolds stress always exceeds Maxwell stress. Hence the jets are quenched due to the cancellation of Reynolds stress by Maxwell stress. We also show the results with different magnetic diffusivities and show that qualitative pic- ture of jets quenching agree with those obtained by Tobias et al9. We also calculate the case when overstable modes exist in a system and show that Maxwell stress can exceed Reynolds stress for high enough Q. We give an `guesstimation' for the value of the transition between Reynolds stress driven regime and Maxwell stress regime. 3 The reminder of the report is organized as follows. In the section 2 we review the set of equations, conservation relations, and linear stability problems to identify important parameters in the system and different modes possible one could have. In the section 3, we show the simulation results on the dynamics of jets with the horizontal magnetic fields. The section 4 is conclusions and discussions. II. FORMULATION In this section we give the set of equations for 2D Boussinesq system with a horizontal magnetic field and review basic properties of them. Specifically we give a brief discussion on the conserved quantities in the system and linear stability problem. For the conservation relations, we discuss the conservation of energy, magnetic potential and cross helicity. Of these we pay a special attention to magnetic potential and cross helicity, which leads to Zeld- vich theorem and a local conservation of momentum between waves and flows, respectively. For the stability problem, we derive dispersion relations for critical modes and show that there are two possibilities for the onset of the instability; one is a stationary cell which is analogous to the usual hydrodynamic convection and the other is an overstabile mode which is unique to the Rayleigh-Benard convection with magnetic fields. Some of the discussions given here are beyond the scope of the report, but I just do it any way since it would be helpful for me to summarize them! A. 2D Boussinesq system with a uniform horizontal magnetic field We consider a simple 2D Boussinesq system with a uniform horizontal magnetic field. As a coordinate system, we take x being the horizontal direction, y being the neglected direction, z being the vertical direction with temperature difference. We consider a box with a height d in z, a length L in x. The set of equations are momentum, temperature and induction equations: ( ) 1 δρ 1 @ v + v · rv = − rp − 1 + gz^ + B · rB + νr2v (1) t ρ tot ρ 4πρ 2 @tT + v · rT = κr T (2) 2 @tB = r × (v × B) + ηr B (3) 4 where δρ/ρ = −α(T − T0) denotes the effect of density variation due to the thermal ex- pansion, which leads to the buoyant force. Introducing the stream function and magnetic potential to write v =y ^ × r , B =y ^ × rA (since r · v = r · B = 0), taking curl of the equation of motion and integrating induction equation we can rewrite the equations as 1 @ r2 + J( ; r2 ) = gα@ T + B · rr2A + νr2r2 (4) t x 4πρ 2 @tT + J( ; θ) = κr T (5) 2 @tA + J( ; A) = ηr A (6) 0 0 where J(A; B) ≡ @xA@zB − @zA@xB. Setting T = T (z) + θ and A = −B0z + A , where the prime denotes a fluctuation part of physical quantities, we get 2 0 0 2 0 0 2 0 0 2 0 4 0 @tr + J( ; r ) = σR@xθ + σζQ@xr A + σζQJ(A ; r A ) + σr (7) 0 0 0 0 2 0 @tθ + J( ; θ ) = @x + r θ (8) 0 0 0 0 2 0 @tA + J( ;A ) = @x + ζr A (9) Here the physical quantities are normalized as t ! t=(κ−1d2), l ! l=d, θ ! θ=∆T (∆T ≡ 0 0 dj@zT (z)j), A ! A =dB0 and the dimensionless numbers are the Prandtl number σ ≡ ν/κ, the Rayleigh number R ≡ gα∆T d3=(νκ), the magnetic Prandtl number ζ ≡ η/κ and the ≡ 2 2 Chandrasekhar number Q B0 d =(νη4πρ).
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