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Closed-form linear stability conditions for magneto-convection

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J. Fluid Mech. (2003), vol. 490, pp c 2003 Cambridge University Press DOI: /S Printed in the United Kingdom 333 Closed-form linear stability conditions for magneto-convection
J. Fluid Mech. (2003), vol. 490, pp c 2003 Cambridge University Press DOI: /S Printed in the United Kingdom 333 Closed-form linear stability conditions for magneto-convection By R. C. K L OOSTERZIEL 1 AND G. F. CARNEVALE 2 1 School of Ocean & Earth Science & Technology, University of Hawaii, Honolulu, HI 96822, USA 2 Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093, USA (Received 17 January 2003 and in revised form 9 April 2003) Chandrasekhar (1961) extensively investigated the linear dynamics of Rayleigh Bénard convection in an electrically conducting fluid exposed to a uniform vertical magnetic field and enclosed by rigid, stress-free, upper and lower boundaries. He determined the marginal stability boundary and critical horizontal wavenumbers for the onset of convection as a function of the Chandrasekhar number or Hartmann number squared. No closed-form formulae appeared to exist and the results were tabulated numerically. We have discovered simple expressions that concisely describe the stability properties of the system. When the Prandtl number Pr is greater than or equal to the magnetic Prandtl number Pm the marginal stability boundary is R 2/3 ]where R is the Rayleigh number and R c =(27/4)π 4 is Rayleigh s famous critical value for the onset of stationary convection in the absence of a magnetic field ( =0). When Pm Pr the marginal stability boundary is determined by this curve until intersected by the curve described by the curve = π 2 [R R 1/3 c = 1 π 2 [ Pm 2 (1 + Pr) Pr 2 (1 + Pm) R ( ) 1/3 ( (1 + Pr)(Pr + Pm) Pm 2 (1 + Pr) Pr 2 Pr 2 (1 + Pm) ) 2/3 R 1/3 c R 2/3 ]. An expression for the intersection point is derived and also for the critical horizontal wavenumbers for which instability sets in along the marginal stability boundary either as stationary convection or in an oscillatory fashion. A simple formula is derived for the frequency of the oscillations. Also we show that in the limit of vanishing magnetic diffusivity, or infinite electrical conductivity, the system is unstable for sufficiently large R. Instability in this limit always sets in via overstability. 1. Introduction Thompson (1951) and Chandrasekhar (1952) were the first to consider Rayleigh Bénard (RB) convection of electrically conducting fluids (i.e. liquid metals) in the presence of an external uniform magnetic field. Here we revisit this classical linear RBproblem with stress-free, rigid upper and lower boundaries. The top and bottom are considered perfect heat conductors, maintained at constant (different) temperatures, with the higher temperature at the bottom. The horizontal domain is unbounded, the Boussinesq approximation is made, incompressibility is assumed and the kinematic viscosity, diffusivity, electrical conductivity and magnetic diffusivity are constant. This work was inspired by our recent success in establishing closed-form linear stability conditions for Rayleigh-Bénard convection subjected to Coriolis forces (see Kloosterziel & Carnevale 2003). Although there are some analogies between the two 334 R. C. Kloosterziel and G. F. Carnevale cases, the stability properties are quite different. However, no comparison will be made here. In 2.1 we briefly discuss the general properties of the cubic polynomial which determines the eigenvalues p for normal-mode perturbations with an assumed time dependence exp(pt). In we show that the linear stability results tabulated in chapter IV of Chandrasekhar s (1961) monograph for this easy case of so-called free free boundaries in the vertical can be summarized with simple closed-form formulae. In 2.5 we discuss the limit of zero magnetic diffusivity or infinite electrical conductivity. In 3wesummarize the results and discuss some additional matters of possible interest. 2. Linear stability In the standard RB-problem density variations are caused by temperature variations. In the unperturbed system, the temperature distribution between the bottom at z =0 and top at z = d, with z the vertical coordinate, is linear with a constant adverse temperature gradient β 0, so that the temperature at the bottom is higher than at the top. Density is assumed to vary with temperature via a constant coefficient of volume expansion α. Thefluid is electrically conducting and subjected to a uniform external vertical magnetic field H which coincides with the direction of the gravitational acceleration g, taken along the z-axis. The equations governing the RB-problem for an incompressible fluid under the Boussinesq approximation are discussed by Chandrasekhar (1961). Linearizing the equations about the basic motionless state, a set of coupled linear equations is derived for the evolution of smallamplitude velocity and temperature perturbations and the accompanying magnetic field perturbations and electric current density. Equations (101), (102) and (105) in chapter IV, 42, can be used to derive a single equation for the vertical component w of the velocity perturbations. This is a third-order differential equation with respect to time t. The stability of the system is investigated by introducing perturbations w exp[pt +i(k x x + k y y)] sin(nπz/d) whichisthe correct form for the boundary conditions at the top and bottom mentioned above (i.e. rigid and stress free). The vertical wavenumber takes the values n =1, 2,...Substitution of w of the form given above in the third-order equation just mentioned yields a cubic polynomial with the exponential time factor p as variable: p 3 + B p 2 + C p + D =0, where p =(d 2 /ν)p (2.1) and Pr + Pm + PrPm B = (a 2 + n 2 π 2 ), Here C = 1 PrPm PrPm [ (1 + Pr + Pm)(a 2 + n 2 π 2 ) 2 + n 2 π 2 Pr a2 PmR a 2 + n 2 π 2 D = 1 PrPm [(a2 + n 2 π 2 ) 3 +(a 2 + n 2 π 2 )n 2 π 2 a 2 R]. ], (2.2) R = gαβd4, = σµ2 H 2 d 2, Pr = ν and Pm = ν (2.3) κν ρ 0 ν κ η are the Rayleigh number, the Chandrasekhar number (or Hartmann number squared), the Prandtl number and the magnetic Prandtl number, respectively, and a = ( 1/2d kx 2 + ky) 2 Closed-form linear stability conditions for magneto-convection 335 is the non-dimensional horizontal wavenumber. Note that p has been nondimensionalized with the time scale d 2 /ν where ν is the kinematic viscosity. ρ 0 is the constant reference density of the fluid which arises in the context of the Boussinesq approximation. κ is the coefficient of thermal diffusivity, σ the electrical conductivity, η the magnetic diffusivity and µ the magnetic permeability. σ, η and µ obey the relation σ =(µη) 1. R,, Pr and Pm characterize the system and a and n the perturbations. If for given {R,, Pr, Pm} for all perturbations the three roots of (2.1) have Re p 0there is stability. When for certain perturbations there is at least one root with Re p 0, there is instability. With each root of the cubic there is an associated combination of a flow field, temperature and current distribution and a magnetic field, which we refer to as modes, although not explicitly considered here. The hyper-surface in the space spanned by {R,, Pr, Pm} separating stable systems from unstable systems defines the marginally stable states. As explained by Chandrasekhar (1961), when crossing from the stable to the unstable side, instability can set in either as stationary convection in which case one root of (2.1) is p =0,orinanoscillatory fashion when there are two purely imaginary, complex conjugate roots. This is referred to as overstability and the associated modes are called overstable modes. Modes associated with p = 0arecalled convective modes The eigenvalues It is important to note that B in (2.2) is always positive (assuming non-zero Pr and Pm). As discussed in detail by Kloosterziel & Carnevale (2003) positive B implies that the marginal stability boundary can be determined by mere examination of the coefficients B,C and D without actually solving the cubic: stability/instability is entirely determined by the signs of D and BC D. Sincethe coefficients B,C and D in (2.1) are real, either the three roots are real or one root is real and the other two are complex conjugates. When D 0 there will be a positive real root, which implies instability. The other two roots always have Re p 0, that is, they are either both negative real or, when complex conjugates, their real part is negative. When D =0, p =0 is a rootsothatthere is at least one convective mode while when D 0there will be a negative real root and therefore at least one damped mode. For D 0the properties of the other two roots are determined by the sign of BC D. Thedetails are given in table 1, where we list the signs of the real part of each root of the cubic for all combinations of D and BC D. Inthistable unstable simply means that at least one root has Re p 0. Stable means that all three roots have Re p 0, i.e. all modes are damped. This only occurs when both D 0andBC D 0. Convection indicates that at least one root is p =0while none of the other two roots has Re p 0 nor are they purely imaginary. A complex conjugate pair of purely imaginary roots only occurs when D 0 and BC D = 0. That has been indicated with overstable. In the overstable case the non-dimensional frequency ω is determined by ω 2 = C = D/B (see Kloosterziel & Carnevale 2003) The convection curve Equation (2.2) shows that for given a,n and, wehave D 0whenR is large enough, which according to table 1 implies instability. For small enough R on the other hand D 0, while D =0 when (x + n 2 π 2 ) 3 +(x + n 2 π 2 )n 2 π 2 xr =0 where x a 2 (2.4) 336 R. C. Kloosterziel and G. F. Carnevale D 0 D =0 D 0 p 0 p 0 p 0 BC D 0 Re p 0 p =0 Re p 0 Re p 0 p 0 Re p 0 unstable unstable unstable p 0 p =0 p 0 BC D =0 Re p 0 p =0 p =+iω Re p 0 p 0 p = iω unstable convection overstable p 0 p =0 p 0 BC D 0 Re p 0 Re p 0 Re p 0 Re p 0 Re p 0 Re p 0 unstable convection stable Table 1. Diagram summarizing the properties of the three roots of the cubic (2.1) when B 0. Whenever the sign of Re p is given, p can either be real or is one of a complex conjugate pair. In the overstable case the frequency ω is determined by ω 2 = C or ω 2 = D/B. or, alternatively, when R = x + n2 π 2 [(x + n 2 π 2 ) 2 + n 2 π 2 ]. (2.5) x As a function of x this has a minimum where x R =0, which yields 2x 3 +3n 2 π 2 x 2 =(n 2 π 2 ) 3 +(n 2 π 2 ) 2. (2.6) Chandrasekhar (1961) numerically determined the positive real root x c of this cubic as a function of, setting n =1 since this yields the smallest critical Rayleigh number. Substituting x = x c back in (2.5) he obtained the critical Rayleigh number R (c) c (). This is how table XIV with R (c) c () anda c (c) ()=xc 1/2 values in chapter IV of Chandrasekhar (1961) was compiled. An alternative approach that makes analytical progress possible is the following (see figure 1a). The condition D =0 or (2.4) can also be written as f (z)= n 2 π 2 R where f (z)=z 3 +(n 2 π 2 R)z and z = x + n 2 π 2. (2.7) Since d z f =3z 2 +(n 2 π 2 R), it follows that d z f 0forallz when n 2 π 2 R, which implies that (2.7) can only be satisfied for one negative z-value. Since x 0, only z 0 are relevant and in this case D 0forallx 0. However, when n 2 π 2 R there is a local minimum and a maximum, i.e. points where d z f =0, at ( ) R n2 π 2 1/2 z ± = ± and f (z ± )= (4/27) 1/2 (R n 2 π 2 ) 3/2. (2.8) 3 If f (z + )= n 2 π 2 R,thenthe curve f (z) istangenttothehorizontal line n 2 π 2 R at z = z + (see figure 1a) and the square of the critical horizontal wavenumber is simply x c = z + n 2 π 2.Solving f (z + )= n 2 π 2 R,wefindthat this occurs when = (c) c (R,n)= 1 [ R R 1/3 n 2 π 2 c (n)r 2/3], (2.9) Closed-form linear stability conditions for magneto-convection 337 (a) (b) 0 z 3 (R n 2 π 2 )z D 0 c (c) (R,n =1) D = 0 D 0 instability n 2 π 2 R c (c) (R,n = 2) 0 z z + R c (1) R R c (2) Figure 1. (a) Thecritical Chandrasekhar number c and corresponding critical wavenumber a c = xc 1/2 are determined by the condition that the curve f (z)=z 3 (R n 2 π 2 )z is tangent to the horizontal line n 2 π 2 R at z = z + with z = x + n 2 π 2 and z + the positive z-value for which d z f (z)=0 (see text). (b) Thecritical Chandrasekhar number (c) c as a function of R for n =1 and n =2. The leftmost curve (c) c (R,n=1), given by (2.11), starts on the R-axis at R = R c =(27/4)π 4 and for n 1atr c (n)=n 4 R c.for{r, } left of the curve (c) c (R,n=1) D 0forall{a,n}. Onthecurve, D =0 only when {a,n} = {a c (c) (R), 1}, witha c (c) (R) givenby (2.12), and D 0forallother{a,n}. Forpoints {R, } to the right of the leftmost curve there are {a,n} for which D 0, implying instability, e.g. there will be a positive real root of (2.1). where R c (n)=(27/4)n 4 π 4.Thatis,forfixedR, thisisthe critical -value for which D =0 at z = z + = x c + n 2 π 2,butD 0forallotherz 0. For smaller there is a range of positive z-values for which D 0. Substituting given by (2.9) back in the expression for z + (2.8), we find that x c =(n 2 π 2 R/2) 1/3 n 2 π 2. (2.10) In the (R, )-plane the curves (c) c (R,n) start on the R-axis (where =0) at R = R c (n). The leftmost curve has n =1, asshowninfigure 1(b). It follows that D 0 for all {a,n} when {R, } is to the left of the curve (c) c (R)= 1 π 2 [ R R 1/3 c R 2/3], (2.11) where R c = R c (n =1)=(27/4)π 4.Thisistheinverse of the (implicit) relation R (c) c () numerically determined by Chandrasekhar (1961). When {R, } lies exactly on the curve (2.11), D =0 for {a,n} = {a c (c) (R), 1} with a c (c) (R)= [ (π 2 R/2) 1/3 π 2] 1/2, (2.12) which follows from (2.10) with n =1 (remember that x = a 2 ), but D 0forallother {a,n}. Thus,when {R, } lies on (2.11) there is one convective mode ( p =0) when {a,n} = {a c (c) (R), 1} while for all other {a,n} at least one mode is damped ( p 0; see table 1). For {R, } to the right of the curve (2.11) there are a and n for which D 0 and there will be unstable modes ( p 0; see table 1). It may be noted that according to (2.11) (c) c =0 when R = R c.putting R = R c in (2.12), one finds that a c (c) (R c )=(π 2 /2) 1/2. R = R c and a c (c) =(π 2 /2) 1/2 are the famous critical Rayleigh number and horizontal wavenumber found by Rayleigh (1916) in the classical problem without a magnetic field ( = 0) or any other external forces. 338 R. C. Kloosterziel and G. F. Carnevale 2.3. The overstability curve When {R, } lies to the left of the convection curve (2.11) stability is not guaranteed. According to table 1 we have to investigate the sign of BC D. Itfollows with (2.2) that BC D =0 when (x + n 2 π 2 ) 3 + (x + n2 π 2 )n 2 π 2 Pr 2 (1 + Pr)(Pr + Pm) xpm 2 R =0. (2.13) (1 + Pm)(Pr + Pm) Note that (2.13) follows from (2.4) by substituting Pm 2 R (1 + Pm)(Pr + Pm) for R, and Pr 2 (1 + Pr)(Pr + Pm) for (2.14) in (2.4). Thus, the critical wavenumber and Chandrasekhar number for which BC D =0 can be found in an analogous fashion as for D =0 by redefining R and in (2.7) and (2.8), and then solving for f (z +)= n 2 π 2 R,wheref(z)=z 3 +(n 2 π 2 R )z and ( R ) z + n 2 π 2 1/2 =, R = 3 We then find that BC D =0 along the curve (o) c (R, Pr, Pm)= 1 [ Pm 2 (1 + Pr) π 2 Pr 2 (1 + Pm) R ( (1 + Pr)(Pr + Pm) Pr 2 when {a,n} = {a (o) c (R, Pr, Pm), 1}, where a c (o) (R, Pr, Pm)= Pm 2 R (1 + Pm)(Pr + Pm), = ) 1/3 ( Pm 2 (1 + Pr) Pr 2 (1 + Pm) Pr 2 (1 + Pr)(Pr + Pm). ) 2/3 R 1/3 c R 2/3 ] (2.15) [ ( π 2 Pm 2 ) 1/3 ] 1/2 R π 2. (2.16) 2(1 + Pm)(Pr + Pm) For all other {a,n}, BC D 0when{R, } lies on (2.15). When {R, } is to the left of (2.15), BC D 0forall{a,n}, andwhen{r, } is to the right of (2.15), BC D 0forsome{a,n}. Thecurve (o) c (R, Pr, Pm) starts on the R-axis at ( ) Pr + Pm + PrPm R = R s = 1+ R Pm 2 c. (2.17) This follows from setting (o) c =0 in (2.15). Clearly R s R c for all Pr, Pm 0. Thus, (R, Pr, Pm) startsonther-axis to the right of the convection curve (2.11) since that one starts at R = R c.figure2shows this for two cases that are representative for all combinations of Pr 0andPm 0. The superscript (o) in(2.15) and (2.16) indicates that these are critical numbers for which there are overstable modes, as will be seen shortly. Consider the factor for all finite, non-zero Pr and Pm the curve (o) c δ = Pm2 (1 + Pr) (2.18) Pr 2 (1 + Pm) which multiplies R in (2.15), and ( ) 1/3 ( (1 + Pr)(Pr + Pm) Pm 2 ) 2/3 (1 + Pr) γ =, (2.19) Pr 2 Pr 2 (1 + Pm) Closed-form linear stability conditions for magneto-convection 339 (a) Pm Pr D 0 BC D 0 stable D 0 unstable i (b) Pm Pr D 0 BC D 0 stable c (o) (II) (III) c (c) c (c) c (o) c (c) 0 R c R s R Figure 2. (a)graphshowing that when Pm Pr the overstability curve (o) c (R, Pr, Pm)(2.15) does not cut the convection curve (c) c (R) (2.11). The overstability curve starts at R = R s given by (2.17), the convection curve at R = R c.totheleft of (c) c (R) (thick line) for all {a,n} both D 0andBC D 0, implying stability. To the right D 0forsome{a,n} which implies instability (see table 1). This example was created with Pm =0.98 and Pr =1. (b) ForPm Pr the curves cut at {R, } = {R i, i } given by (2.20). R i and i decrease with increasing Pm with Pr held fixed (see figure 3). In region (I) for some {a,n} D 0, BC D 0, in (II) D 0, BC D 0and in (III) D 0, BC D 0, each combination implying instability (see table 1). This example was created using Pm =2 and Pr =1. 0 R c (I) R s R i R which multiplies R 1/3 c R 2/3 in (2.15). With a little algebra it follows that γ δfor all finite Pr, Pm 0. Further it is seen that δ 1whenpm Pr. Since for large positive R the convection curve is (c) c R/π 2 while the overstability curve (o) c δr/π 2, it follows that when δ 1theymustintersect at some point in the (R,)-plane. Equating (2.11) to (2.15), we find that the curves intersect at {R, } = {R i, i } with ( ) 3 γ 1 R i (Pr, Pm)= R c, δ 1 i (Pr, Pm)= 1 ( ) 2 [( ) ] γ 1 γ 1 1 R π 2 c. δ 1 δ 1 (2.20) When Pm Pr, the terms multiplying R c in (2.20) are positive because then γ δ 1. Thus, the intersection occurs at R i R c and i 0. An example is shown in figure 2(b). For Pr = Pm (δ =1), the intersection point {R i, i } is formally at infinity. When Pm Pr we have δ 1and the curves never intersect at any point in the positive quadrant of the (R,)-plane. A representative example is shown in figure 2(a). The expressions (2.18) and (2.19) for δ and γ show that when we take the limit Pm while keeping Pr fixed, the term (γ 1)/(δ 1) 1. Thus, with (2.20) it is seen that in this limit, R i R c and i 0. This also follows with (2.17), i.e. in the limit Pm, R s R c.thislimit is further discussed below in 2.5. Figure 3 shows arepresentative example of how R i and i vary with increasing Pm while keeping Pr fixed The marginal stability boundary When Pm Pr consider the boundary consisting of the convection curve (2.11) drawn as a thick line in figure 2(a). For all points {R, } to the left ofthisboundary D 0andBC D 0forperturbations with any {a,n}. According to table 1 the system is stable for such R and values, i.e. the three roots of the cubic have Re p 0 for all {a,n}. Fromthediscussion in 2.2 it should be clear that for all {R,} to 340 R. C. Kloosterziel and G. F. Carnevale 5R c R i R c i 0 Pr Pm 25Pr Figure 3. Example of R i (Pr, Pm) and i (Pr, Pm) givenby(2.20) for fixed Pr and variable Pm Pr. Thisgraph was created using Pr =1. the right of the boundary there are perturbations with wavenumbers {a,n} for which D 0, which implies instability. The boundary composed of the convection curve (2.11) for R c R R i and the overstability curve (2.15) for R R i separates the stable region from the unstable region in the (R,)-plane when Pm Pr. Thisboundary is drawn as a thick line in figure 2(b). Again for all points {R, } to the left of this boundary D 0and BC D 0forperturbations with any {a,n} and for such R and the system is stable. From the discussion in 2.2 and 2.3 it follows that in each of the regions marked as (I), (II) or (III) in figure 2(b) either D 0and/orBC D 0forcertain {a,n}. According to table 1 therefore for {R,} to the right of the boundary the system is unstable. When {R, } lies on either boundar
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