Numerical simulations of thermal convection in a rapidly rotating spherical shell cooled inhomogeneously from above

Abstract

Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell with and without inhomogeneous temperature anomalies on the top boundary have been carried out using a three-dimensional, time-dependent, spectral-transform code. The spherical shell of Boussinesq fluid has inner and outer radii the same as those of the Earth's liquid outer core. The Taylor number is 10⁷, the Prandtl number is 1, and the Rayleigh number R is 5R_c (R_c is the critical value of R for the onset of convection when the top boundary is isothermal and R is based on the spherically averaged temperature difference across the shell). The shell is heated from below and cooled from above; there is no internal heating. The lower boundary of the shell is isothermal and both boundaries are rigid and impermeable. Three cases are considered. In one, the upper boundary is isothermal while in the others, temperature anomalies with (l,m) = (3,2) and (6,4) are imposed on the top boundary. The spherically averaged temperature difference across the shell is the same in all three cases. The amplitudes of the imposed temperature anomalies are equal to one-half of the spherically averaged temperature difference across the shell. Convective structures are strongly controlled by both rotation and the imposed temperature anomalies suggesting that thermal inhomogeneities imposed by the mantle on the core have a significant influence on the motions inside the core. The imposed temperature anomaly locks the thermal perturbation structure in the outer part of the spherical shell onto the upper boundary and significantly modifies the velocity structure in the same region. However, the radial velocity structure in the outer part of the shell is different from the temperature perturbation structure. The influence of the imposed temperature anomaly decreases with depth in the shell. Thermal structure and velocity structure are similar and convective rolls are more columnar in the inner part of the shell where the effects of rotation are most dominant.     

Non-axisymmetric α2Ω-dynamo waves in thin stellar shells

Linear α²Ω-dynamo waves are investigated in a thin turbulent, differentially rotating convective stellar shell. A simplified one-dimensional model is considered and an asymptotic solution constructed based on the small aspect ratio of the shell. In a previous paper Griffiths et al. (Griffiths, G.L., Bassom, A.P., Soward, A.M. and Kuzanyan, K.M., Nonlinear α²Ω-dynamo waves in stellar shells, Geophys. Astrophys. Fluid Dynam., 2001, 94, 85–133) considered the modulation of dynamo waves, linked to a latitudinal-dependent local α-effect and radial gradient of the zonal shear flow. These effects are measured at latitude θ by the magnetic Reynolds numbers R _α f(θ) and R _Ω g(θ). The modulated Parker wave, which propagates towards the equator, is localised at some mid-latitude θ_p under a Gaussian envelope. In this article, we include the influence of a latitudinal-dependent zonal flow possessing angular velocity Ω_*(θ) and consider the possibility of non-axisymmetric dynamo waves with azimuthal wave number m. We find that the critical dynamo number D _c?=?R _α R _Ω is minimised by axisymmetric modes in the αΩ-limit (Rα→0). On the other hand, when Rα?≠?0 there may exist a band of wave numbers 0?m?m ^? for which the non-axisymmetric modes have a smaller D _c than in the axisymmetric case. Here m ^? is regarded as a continuous function of R _α with the property m?→0 as R _α→0 and the band is only non-empty when m??>1, which happens for sufficiently large R _α. The preference for non-axisymmetric modes is possible because the wind-up of the non-axisymmetric structures can be compensated by phase mixing inherent to the α²Ω-dynamo. For parameter values resembling solar conditions, the Parker wave of maximum dynamo activity at latitude θ_p not only propagates equatorwards but also westwards relative to the local angular velocity Ω_*(θ _p ). Since the critical dynamo number D _c?=?R _α R _Ω is O (1) for small R _α, the condition m ^??>?1 for non-axisymmetric mode preference imposes an upper limit on the size of |dΩ_*/dθ|.     

Necessary conditions for the magnetohydrodynamic dynamo

Abstract

A magnetohydrodynamic, dynamo driven by convection in a rotating spherical shell is supposed to have averages that are independent of time. Two cases are considered, one driven by a fixed temperature difference R and the other by a given internal heating rate Q. It is found that when q, the ratio of thermal conductivity to magnetic diffusivity, is small, R must be of order q ^?4/3 and Q of order q ^?2 for dynamo action to be possible; q is small in the Earth's core, so it is hoped that the criteria will prove useful in practical as well as theoretical studies of dynamic dynamos. The criteria can be further strengthened when the ohmic dissipation of the field is significant in the energy balance. The development includes the derivation of two necessary conditions for dynamo action, both based on the viscous dissipation rate of the velocity field that drives the dynamo.     

Transitions to chaotic thermal convection in a rapidly rotating spherical fluid shell

Abstract

Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell heated from below and within have been carried out with a nonlinear, three-dimensional, time-dependent pseudospectral code. The investigated phenomena include the sequence of transitions to chaos and the differential mean zonal rotation. At the fixed Taylor number T _a =10⁶ and Prandtl number Pr=1 and with increasing Rayleigh number R, convection undergoes a series of bifurcations from onset of steadily propagating motions SP at R=R _c = 13050, to a periodic state P, and thence to a quasi-periodic state QP and a non-periodic or chaotic state NP. Examples of SP, P, QP, and NP solutions are obtained at R = 1.3R _c , R = 1.7 R _c , R = 2R _c , and R = 5 R _c , respectively. In the SP state, convection rolls propagate at a constant longitudinal phase velocity that is slower than that obtained from the linear calculation at the onset of instability. The P state, characterized by a single frequency and its harmonics, has a two-layer cellular structure in radius. Convection rolls near the upper and lower surfaces of the spherical shell both propagate in a prograde sense with respect to the rotation of the reference frame. The outer convection rolls propagate faster than those near the inner shell. The physical mechanism responsible for the time-periodic oscillations is the differential shear of the convection cells due to the mean zonal flow. Meridional transport of zonal momentum by the convection cells in turn supports the mean zonal differential rotation. In the QP state, the longitudinal wave number m of the convection pattern oscillates among m = 3,4,5, and 6; the convection pattern near the outer shell has larger m than that near the inner shell. Radial motions are very weak in the polar regions. The convection pattern also shifts in m for the NP state at R = 5R _c , whose power spectrum is characterized by broadened peaks and broadband background noise. The convection pattern near the outer shell propagates prograde, while the pattern near the inner shell propagates retrograde with respect to the basic rotation. Convection cells exist in polar regions. There is a large variation in the vigor of individual convection cells. An example of a more vigorously convecting chaotic state is obtained at R = 50R _c . At this Rayleigh number some of the convection rolls have axes perpendicular to the axis of the basic rotation, indicating a partial relaxation of the rotational constraint. There are strong convective motions in the polar regions. The longitudinally averaged mean zonal flow has an equatorial superrotation and a high latitude subrotation for all cases except R = 50R _c , at this highest Rayleigh number, the mean zonal flow pattern is completely reversed, opposite to the solar differential rotation pattern.     

Nonlinear dynamics of boussinesq convection in a deep rotating spherical shell-i

Abstract

Convection in a rotating spherical shell has wide application for understanding the dynamics of the atmospheres and interiors of many celestial bodies. In this paper we review linear results for convection in a shell of finite depth at substantial but not asymptotically large Taylor numbers, present nonlinear multimode calculations for similar conditions, and discuss the model and results in the context of the problem of solar convection and differential rotation. Detailed nonlinear calculations are presented for Taylor number T = 10⁵, Prandtl number P = 1, and Rayleigh number R between 1 |MX 10⁴ and 4 |MX 10⁴ (which is between about 4 and 16 times critical) for a shell of depth 20% of the outer radius. Sixteen longitudinal wave numbers are usually included (all even wave numbers m between 0 and 30) the amplitudes of which are computed on a staggered grid in the meridian plane.

The kinetic energy spectrum shows a peak in the wave number range m = 12–18 at R = 10⁴, which straddles the critical wave number m = 14 predicted by linear theory. These are modes which peak near the equator. The spectrum shows a second strong peak at m = 0, which represents the differential rotation driven by the peak convective modes. As R is increased, the amplitude of low wave numbers increases relative to high wave numbers as convection fills in in high and middle latitudes, and as the longitudinal scale of equatorial convection grows. By R = 3 |MX 10⁴, m = 8 is the peak convective mode. There is a clear minimum in the total kinetic energy at middle latitudes relative to low and high, well into the nonlinear regime, representing the continued dominance of equatorial and polar modes found in the linear case. The kinetic energy spectrum for m > 0 is maintained primarily by buoyancy work in each mode, but with substantial nonlinear transfer of kinetic energy from the peak modes to both lower and higher wave numbers.

For R = 1 to 2 |MX 10⁴, the differential rotation takes the form of an equatorial acceleration, with angular velocity generally decreasing with latitude away from the equator (as on the sun) and decreasing inwards. By R = 4 |MX 10⁴, this equatorial profile has completely reversed, with angular velocity increasing with depth and latitude. Also, a polar vortex which has positive rotation relative to the reference frame (no evidence of which has been seen on the sun) builds up as soon as polar modes become important. Meridional circulation is quite weak relative to differential rotation at R = 10⁴, but grows relative to it as R is increased. This circulation takes the farm of a single cell of large latitudinal extent in equatorial regions, with upward flow near the equator, together with a series of narrower cells in high latitudes. It is maintained primarily by axisymmetric buoyancy forces. The differential rotation is maintained at all R primarily by Reynolds stresses, rather than meridional circulation. Angular momentum transport toward the equator for R = 1–2 |MX 10⁴ maintains the equatorial acceleration while radially inward transport maintains the opposite profile at R = 4 |MX 10⁴.

The total heat flux out the top of the convective shell always shows two peaks for the range of R studied, one at the equator and the other near the poles (no significant variation with latitude is seen on the sun), while heat flux in at the bottom shows only a polar peak at large R. The meridional circulation and convective cells transport heat toward the equator to maintain this difference.

The helicity of the convection plus the differential rotation produced by it suggest the system may be capable of driving a field reversing dynamo, but the toroidal field may migrate with lime in each cycle toward the poles and equator, rather than just toward the equator as apparently occurs on the sun.

We finally outline additions to the physics of the model to make it more realistic for solar application.     

On the frontal condition for finite amplitude α2Ω-dynamo wave trains in stellar shells

Abstract

In a previous paper, Bassom et al. (Proc. R. Soc. Lond. A, 455, 1443–1481, 1999) (BKS) investigated finite amplitude αΩ-dynamo wave trains in a thin turbulent, differentially rotating convective stellar shell; nonlinearity arose from α-quenching. There asymptotic solutions were developed based upon the small aspect ratio ε of the shell. Specifically, as a consequence of a prescribed latitudinally dependent α-effect and zonal shear flow, the wave trains have smooth amplitude modulation but are terminated abruptly across a front at some high latitude θ_F. Generally, the linear WKB-solution ahead of the front is characterised by the vanishing of the complex group velocity at a nearby point θ_f; this is essentially the Dee–Langer criterion, which determines both the wave frequency and front location.

Recently, Griffiths et al. (Geophys. Astrophys. Fluid Dynam. 94, 85–133, 2001) (GBSK) obtained solutions to the α²Ω-extension of the model by application of the Dee—Langer criterion. Its justification depends on the linear solution in a narrow layer ahead of the front on the short O(θ_f—θ_F) length scale; here conventional WKB-theory, used to describe the solution elsewhere, is inadequate because of mode coalescence. This becomes a highly sensitive issue, when considering the transition from the linear solution, which occurs when the dynamo number D takes its critical value D _c corresponding to the onset of kinematic dynamo action, to the fully nonlinear solutions, for which the Dee—Langer criterion pertains.

In this paper we investigate the nature of the narrow layer for α²Ω-dynamos in the limit of relatively small but finite α-effect Reynolds numbers R _α, explicitly ε^½ ? R ² _α ? 1. Though there is a multiplicity of solutions, our results show that the space occupied by the corresponding wave train is generally maximised by a solution with θ_f—θ_F small; such solutions are preferred as evinced by numerical simulations. This feature justifies the application by GBSK of the Dee—Langer criterion for all D down to the minimum D _min that the condition admits. Significantly, the frontal solutions are subcritical in the sense that |D _min| ≤ |D _c|; equality occurs as the α-effect Reynolds number tends to zero. We demonstrate that the critical linear solution is not connected by any parameter track to the preferred nonlinear solution associated with D _min. By implication, a complicated bifurcation sequence is required to make the connection between the linear and nonlinear states. This feature is in stark contrast to the corresponding results for αΩ-dynamos obtained by BKS valid in the limit R _{2 α} ? ε^½, which, though exhibiting a weak subcriticality, showed that the connection follows a clearly identifiable nonbifurcating track.     

An antidynamo theorem for partly symmetric flows

Abstract

It is shown that magnetic fields generated by flows v _r,(r,t)e_r+v_T where v_T is an arbitrary toroidal component (e_r˙v_T≡V≡v_T≡0), cannot be maintained indefinitely against ohmic dissipation. The poloidal field variable max |r ² B _r| is shown to decay strictly monotonically with an undetermined decay rate. A bound on the growth of the toroidal field norm ∥T∥₁ is established solely dependent on the rate of conversion of poloidal to toroidal field, so that when the poloidal field is negligible then ∥T∥₁ decays strictly monotonically. The main application of these results is to models of stellar evolution based on axisymmetric differential rotation and spherically symmetric contraction. This symmetric velocity theorem overlaps with two already known theorems, namely the toroidal velocity theorem where v _r≡0 and the radial velocity theorem where v_T≡0. The new theorem does not entirely include the already established ones, principal differences being in the rates of decay and the field variables for which the decay is proven (see Table 1).     

Properties of a nonlinear solar dynamo model

Abstract

A simple nonlinear model is developed for the solar dynamo, in which the real convective spherical shell is approximated by a thin flat slab, and only the back-reaction of the field B on the helicity is taken into account by choosing the simple law α = α(1-ζB ²), where α and ζ are constants, to represent the decrease in generation coefficient ζ with increasing field strength. Analytic expressions are obtained for the amplitude of the field oscillation and its period, T, as functions of the deviation d - d_CT of a dynamo number d from its critical value d_cr for regeneration. A symmetry is found for the case of oscillations of small constant amplitude: B(t+½T)= -B(t). A Landau equation is obtained that describes the transition to such oscillations.     

Hydromagnetic dynamo model with spiral convection

Summary The present paper deals with a hydromagnetic dynamo model of the generation mechanism of the Earth's magnetic field. An attempt has been made at selecting a flow-velocity field in the Earth's core which would satisfy the condition 0 for regenerating the field according to [2], and which would yield a velocity field pattern on the core surface as given in the papers by Kahle et al. [9]. These conditions are satisfied by the velocityv=V ₁+U ₂ ^c –V ₂ ^c and, geometrically, this velocity field is represented in space by a spiral convective motion. On the core surface two downflows and two upflows with the corresponding rotating cells may then be found. Only the axisymmetric harmonic component regeneration of the magnetic field has been considered. Adequate regeneration equations have been obtained by means of Braginski's method of quantity estimates in order of magnitude.     

The flux ejection dynamo effect

Abstract

The mean-field effects of cyclonic convection become increasingly complex when the cyclonic rotation exceeds ½-π. Net helicity is not required, with negative turbulent diffusion, for instance, appearing in mirror symmetric turbulence. This paper points out a new dynamo effect arising in convective cells with strong asymmetry in the rotation of updrafts as against downdrafts. The creation of new magnetic flux arises from the ejection of reserve flux through the open boundary of the dynamo region. It is unlike the familiar α-effect in that individual components of the field may be amplified independently. Several formal examples are provided to illustrate the effect. Occurrence in nature depends upon the existence of fluid rotations of the order of π in the convective updrafts. The flux ejection dynamo may possibly contribute to the generation of field in the convective core of Earth and in the convective zone of the sun and other stars.     

Generation of magnetic fields by convection in a rotating sphere,I

The magnetohydrodynamic dynamo problem is solved for an electrically conducting spherical fluid shell with spherically symmetric distributions of gravity and heat sources. The dynamics of motions generated by thermal buoyancy are dominated by the effects of rotation of the fluid shell. Dynamos are found for low and intermediate values of the Taylor number, T ? 10⁵, if the scale of the nonaxisymmetric component of the velocity field is sufficiently small. The generation of magnetic fields of quadrupolar symmetry is preferred at Rayleigh numbers close to the critical value R_c for onset of convection. As the Rayleigh number increases, the generation of dipolar magnetic fields becomes preferred.     

Mercury's magnetic field: a thermoelectric dynamo?

Permanent magnetism and conventional dynamo theory are possible but problematic explanations for the magnitude of the Mercurian magnetic field. A new model is proposed in which thermoelectric currents driven by temperature differences at a bumpy core-mantle boundary are responsible for the (unobserved) toroidal field, and the helicity of convective motions in a thin outer core (thickness 10² km) induces the observed poloidal field from the toroidal field. The observed field of 3 × 10⁻⁷ T can be reproduced provided the electrical conductivity of Mercury's semiconducting mantle approaches 10³ Ω⁻¹ m⁻¹. This model may be testable by future missions to Mercury because it predicts a more complicated field geometry than conventional dynamo theories. However, it is argued that polar wander may cause the core-mantle topography to migrate so that some aspects of the rotational symmetry may be reflected in the observed field.     

The rotational quenching of the rotation-induced kinetic alpha-effect

Abstract

A theory of the non-diffusive anisotropic kinetic alpha-effect (“Γ-effect”) for densitystratified rotating turbulent fluids is developed. No limitations on the rotation rate are imposed and the fully nonlinear dependence of the Γ-effect on the angular velocity is studied. When the Coriolis number, ω? = 2τ ω, is small the dimensionless “dynamo number”, Cτ, characterising the power of the Γ-effect, grows with ω?. The dependence, however, reaches a maximum for ω? ～ 2. For still higher rotation rates C_Λ decreases as 1/ω?. In opposition, the corresponding number, C_x, of the hydromagnetic α² -dynamo problems remains finite for very large ω?. Hence, for fast rotation the hydrodynamic Γ-effect is small while the hydromagnetic α-effect remains large. In consequence, the large-scale magnetic and velocity structures are expected to be generated with roughly equal power in slowly rotating objects. In the rapid rotators, however, generation of the large-scale flows is problematic.     

Theoretical estimates of the westward drift

Virtually all dynamo models may be expected to give rise to a permanent differential rotation between mantle and core. Weak conductivity in the mantle permits small leakage currents which couple to the radial component of the magnetic field, producing a Lorentz torque. Mechanical equilibrium is achieved when a zero net torque is established at a critical rotation rate. An estimate of the drift is determined easily given the magnetic field structure predicted by any dynamo model. The result for the drift rate at the core-mantle interface along the equator is given by the product of three factors

$\text{U}{}_{\mathrm{\phi 1}}\text{=U}{}_{\mathrm{\phi }}\text{R}\text{λ}{}_{\mathrm{*}}\text{L}{}_{\mathrm{*}}$

The first of these is a geometrical factor which depends only on the structural character of the field. For a variety of model fields, this factor ranges from 16 to 35. The second factor is the ratio of r.m.s. toroidal to poloidal field. This ratio is an (implicitly) adjustable parameter of both α² and α-ω dynamos, and is a measure of the relative efficiency of the generation process for each component. The third (dimensional) term is the ratio of core magnetic diffusivity to core radius, 10^?4 cm s^?1.The result is essentially independent of the value of mantle diffusivity and its effective depth. The sign of the result may be positive or negative. For α² dynamos a westward drift is produced by choosing α > 0 in the Northern Hemisphere, which constitutes a dynamical assertion about the dynamo process. For an r.m.s. toroidal field of the order of 15 Gs, based on fairly general considerations, a drift rate comparable to observation is expected.     

Tuning of the mean-field geodynamo model

Parker’s two-dimensional (2D) dynamo model with an algebraic form of nonlinearity for the α-effect is considered. The model uses geostrophic distributions for the α-effect and differential rotation, which are derived from the three-dimensional (3D) convection models. The resulting configurations of the magnetic field in the liquid core are close to the solutions in Braginsky’s Z-model. The implications of the degree of geostrophy observed in the 3D dynamo models for the behavior of the mean magnetic field are explored. It is shown that the reduction in geostrophy leads to magnetic field reversals accompanied by the relative growth of the nondipole component of the field on the surface of the liquid core. The simulations with a random α-effect which causes turbulent pulsations are carried out. The approach is capable of producing realistic sequences of magnetic reversals.     

Kinematic Dynamos in the Kerr Metric

It is known that a sharp decrease in the angular velocity of the accretion disk around a black hole could in principle produce a kinematic axisymmetric dynamo, in contrast to the classical situation described by Cowling's antidynamo theorem. Here the effect of a nontrivial poloidal velocity of the disk is studied, showing that a strong gradient of this velocity enhances the possibilities of a working dynamo.     

A unified approach to a class of slow dynamos

Abstract

Dynamo action in a highly conducting fluid with small magnetic diffusivity η is particularly sensitive to the topology of the flow. The sites of rapid magnetic field regeneration, when they occur, appear to be located at the stagnation points or in regions where the particle paths are chaotic. Elsewhere only slow dynamo action is to be expected. Two such examples are the nearly axially symmetric dynamo of Braginsky and the generalisation to smooth velocity fields of the Ponomarenko dynamo. Here a method of solution is developed, which applies to both these examples and is applicable to other situations, where magnetic field lines are close to either closed or spatially periodic contours. Particular attention is given to field generation in the neighbourhood of resonant surfaces where growth rates may be intermediate between the slow diffusive and fast convective time scales. The method is applied to the case of the two-dimensional ABC-flows, where it is shown that such intermediate dynamo action can occur on resonant surfaces.     

Hydromagnetic waves in rapidly rotating spherical shells generated by poloidal decay modes

Abstract

This paper presents the first attempt to examine the stability of a poloidal magnetic field in a rapidly rotating spherical shell of electrically conducting fluid. We find that a steady axisymmetric poloidal magnetic field loses its stability to a non-axisymmetric perturbation when the Elsasser number A based on the maximum strength of the field exceeds a value about 20. Comparing this with observed fields, we find that, for any reasonable estimates of the appropriate parameters in planetary interiors, our theory predicts that all planetary poloidal fields are stable, with the possible exception of Jupiter. The present study therefore provides strong support for the physical relevance of magnetic stability analysis to planetary dynamos. We find that the fluid motions driven by magnetic instabilities are characterized by a nearly two-dimensional columnar structure attempting to satisfy the Proudman-Taylor theorm. This suggests that the most rapidly growing perturbation arranges itself in such a way that the geostrophic condition is satisfied to leading order. A particularly interesting feature is that, for the most unstable mode, contours of the non-axisymmetric azimuthal flow are closely aligned with the basic axisymmetric poloidal magnetic field lines. As a result, the amplitude of the azimuthal component of the instability is smaller than or comparable with that of the poloidal component, in contrast with the instabilities generated by toroidal decay modes (Zhang and Fearn, 1994). It is shown, by examining the same system with and without fluid inertia, that fluid inertia plays a secondary role when the magnetic Taylor number T_m ? 10⁵. We find that the direction of propagation of hydromagnetic waves driven by the instability is influenced strongly by the size of the inner core.     

Forces on spheres moving horizontally in a rotating stratified fluid

Abstract

Measurements have been made of the net horizontal force F acting on a sphere moving with horizontal velocity U (Reynolds numbers in the range 10²-10⁴) through a stratified fluid rotating about a vertical axis with uniform angular velocity Ω. In both homogeneous and stratified rotating fluids with small Rossby number R(R = U/Ωa ? 1 where a is the radius of the sphere) the force F is of magnitude 2ΩρUV (where ρ is the density of the fluid and V is the volume of the sphere). In a homogeneous fluid the relative directions of F and U were found to depend on the quantity F = 8Ωa ²/UD (where D is the depth of the fluid in which the object is placed (Mason, 1975)). In a rotating stratified fluid the relative directions of F and U are found to depend on the inverse Froude number k(k = Na/U where N ² = (g/δ)?ρ/?z) provided D > 4aΩ/N. In a homogeneous fluid with F ? 1 the force F is mainly in the U direction (a drag force due to inertial wave radiation) and is ～ ?0.4 |MX 2ΩρUV For F ? 1 a “Taylor column” occurs and the force, in correspondence with theoretical expectations, is ～ - 2Ω |MX UρV In a rotating stratified fluid with N ～2Ω and k ? 1 the force F is mainly in the U direction but is roughly one half of that occurring in the homogeneous situation with F ? 1 (tentatively explained as due to the evanescence of inertia-gravity disturbances). In a rotating stratified fluid with k ? 1 the flow should have no vertical motion (as with F ? 1) and again in correspondence with theoretical expectations the drag is ～ ?2 Ω |MX UρV. In a non-rotating stratified fluid the drag coefficient C _D(C _D = F U/½?ρU ²) was measured in the range k = 0.1 to 10 and had a maximum value ～ 1.2 for k ～ 3.