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1.
Bernard R. Durney 《Solar physics》1989,123(2):197-216
The following points are discussed:
Operated by the Association of Universities for Research in Astronomy, Inc. under contract with the National Science Foundation. 相似文献
| (i) | The dependence of the angular velocity, , on the spatial coordinates near the lower boundary, R c, of the solar convection zone (SCZ) can be obtained from an integration with respect to r of a sound approximation to the azimuthal equation of motion. Here P 2 (cos ) is the second-order Legendre polynomial and is the polar angle. Estimates of 0, 2 (the primes denote derivatives with respect to r), based on the best available values for the Reynolds stresses and anisotropic viscosity coefficients, suggest that 0 < 0,=">2 0 for r = R c. Since a reliable theory of anisotropic turbulent coefficients does not exist at present, positive values of 0 are conceivable. |
| (ii) | In the lower SCZ the latitudinal variations of the superadiabatic gradient vanish if is constant along cylinders. The uniformity of the superadiabatic gradient is, however, inconsequential: the physically meaningful rotation law is the one that insures the uniformity of the convective flux. |
| (iii) | With the exception of the polar regions, the angular momentum transport in thin azimuthal convective rolls is towards the equator. |
| (iv) | It is suggested that buoyancy uncorrelates horizontally separated regions in the lower SCZ preventing the generation of magnetic fields with small wave numbers: in consequence, the cycle magnetic field must be generated in a region of weak buoyancy whereas the lower SCZ generates a weak rather stochastic magnetic field. The dependence on rotation of these two types of magnetic field could differ. |
| (v) | In the context of helioseismology it is customary to expand the perturbations (induced by rotation) of the eigenfrequencies in the following form: , where the notation is standard. The observations reveal that to a good approximation a 1 is independent of l. It is shown that this is the case if is constant with r. For a simple viscous, rotating fluid in the steady state (r) is constant with r if the angular momentum loss vanishes. Let J(ri dr) be the angular momentum of a thin shell of radius r and thickness dr. Since , the constancy of (r) implies that each shell of radius r has the same angular momentum as if the Sun were rotating uniformly with an angular velocity given by . It is discussed whether, alternatively, the observations simply indicate that 0(r) is a slowly varying function of r. |
2.
A model is presented which describes the 3-dimensional non-radial solar wind expansion between the Sun and the Earth in a specified magnetic field configuration subject to synoptically observed plasma properties at the coronal base. In this paper, the field is taken to be potential in the inner corona based upon the Mt. Wilson magnetograph observations and radial beyond a certain chosen surface. For plasma boundary conditions at the Sun, we use deconvoluted density profiles obtained from synopticK-coronameter brightness observations. The temperature is taken to be 2 × 106 K at the base of closed field lines and 1.6 x 106K at the base of open field lines. For a sample calculation, we employ data taken during the period of the 12 November 1966 eclipse. Although qualitative agreement with observations at 1 AU is obtained, important discrepancies emerge which are not apparent from spherically symmetric models or those models which do not incorporate actual observations in the lower corona. These discrepancies appear to be due to two primary difficulties - the rapid geometric divergence of the open field lines in the inner corona as well as the breakdown in the validity of the Spitzer heat conduction formula even closer to the Sun than predicted by radial flow models. These two effects combine to produce conductively dominated solutions and lower velocities, densities, and field strengths at the Earth than those observed. The traditional difficulty in solar wind theory in that unrealistically small densities must be assumed at the coronal base in order to obtain observed densities at 1 AU is more than compensated for here by the rapid divergence of field lines in the inner corona. For these base conditions, the value ofβ(ratio of gas pressure to magnetic pressure) is shown to be significantly greater than one over most of the lower corona - suggesting that, for the coronal boundary conditions used here, the use of a potential or force-free magnetic field configuration may not be justified. The calculations of this paper point to the directions where future research on solar-interplanetary modelling should receive priority:
- better models for the coronal magnetic field structure
- improved understanding of the thermal conductivity relevant for the solar wind plasma.
3.
With the help of a model atmosphere of the Sun we evaluate the pole-equator difference in flux (as measured by Dicke and Goldenberg) assuming the following type of pole-equator temperature difference (T=T
e
–T
p
): (a) T 2K for > 0 (0 0.05); (b) T 10K for < 0.The small T at all optical depths given by (a) could, for example, be due to a pole-equator difference in effective temperatures. At small optical depths a difference in mechanical heating could give rise to the larger temperature difference given by (b). We compare the results of our calculations with Dicke and Goldenberg's observations.The National Center for Atmospheric Research is sponsored by the National Science Foundation. 相似文献
4.
As a consequence of the Taylor–Proudman balance, a balance between the pressure, Coriolis and buoyancy forces in the radial and latitudinal momentum equations (that is expected to be amply satisfied in the lower solar convection zone), the superadiabatic gradient is determined by the rotation law and by an unspecified function of r, say, S(r), where r is the radial coordinate. If the rotation law and S(r) are known, then the solution of the energy equation, performed in this paper in the framework of the ML formalism, leads to a knowledge of the Reynolds stresses, convective fluxes, and meridional motions. The ML-formalism is an extension of the mixing length theory to rotating convection zones, and the calculations also involve the azimuthal momentum equation, from which an expression for the meridional motions in terms of the Reynolds stresses can be derived. The meridional motions are expanded as U
r(r,)=P
2(cos)2(r)/r
2+P
4(cos)4(r)/r
2 +..., and a corresponding equation for U
(r,). Here is the polar angle, is the density, and P
2(cos), P
4(cos) are Legendre polynomials. A good approximation to the meridional motion is obtained by setting 4(r)=–H2(r) with H–1.6, a constant. The value of 2(r) is negative, i.e., the P
2 flow rises at the equator and sinks at the poles. For the value of H obtained in the numerical calculations, the meridional motions have a narrow countercell at the poles, and the convective flux has a relative maximum at the poles, a minimum at mid latitudes and a larger maximum at the equator. Both results are in agreement with the observations. 相似文献
5.
The power in the different modes of an expansion of the solar radial magnetic field at the surface in terms of Legendre polynomials,P , is calculated with the help of a solar dynamo model studied earlier. The model is of the Babcock–Leighton type, i.e., the surface eruptions of the toroidal magnetic field – through the tilt angle, , formed by the magnetic axis of a bipolar magnetic region with the east-west line – are the sources for the poloidal field. In this paper it is assumed that the tilt angle is subject to fluctuations of the form, = ()+ <> where <> is the average value and () is a random normal fluctuation with standard deviation which is taken from Howard's observations of the distribution of tilt angles. For numerical considerations, negative values of were not allowed. If this occurred, was recalculated. The numerical integrations were started with a toroidal magnetic field antisymmetric across the equator, large enough to generate eruptions, and a negligible poloidal field. The fluctuations in the tilt angle destroy the antisymmetry as time increases. The power of the antisymmetric modes across the equator (i.e., odd values of ) is concentrated in frequencies, p, corresponding to the cycle period. The maximum power lies in the =3 mode with considerable power in the =5 mode, in broad agreement with Stenflo's results who finds a maximum power at =5. For the symmetric modes, there is considerable power in frequencies larger than p, again in broad agreement with Stenflo's power spectrum. 相似文献
6.
The response of a layer to a horizontal shear flow at its top the surface was studied numerically as an initial value problem.
The geometry was Cartesian and the conservation equations were solved with the help of the Zeus-3D code. In the initial state,
the pressure, p, and density, ρ, of the layer were assumed to be related by a polytropic equation of index 1.14, which best approximates
the solar values in the region of interest. The values of p and ρ at the lower boundary of the layer, namely r=R
l=0.4 R
⊙, were taken to be the solar values. The upper boundary was chosen to be the base of the solar convection zone, r=R
c=0.7 R
⊙. The shear flow at the surface, v
φ(R
c), was proportional to the solar differential rotation, and acoustical oscillations were present in the layer.
It is shown that if the initial state is stable, a dynamical coupling between sound waves and the shear flow transmits the surface flow to the inner regions of the layer, even in the
absence of dissipation. The shear flow in the sublayer below the one at the surface is proportional to v
φ(R
c), to the time, and to the strength of the oscillations. The constant of proportionality is calculated from the numerical
integrations, performed for times of the order of 100 hr. Extrapolation of these results to longer times shows that the surface
shear flow is transmitted to the inner regions in a time of the order of of 30 000 years. If the initial state is unstable
to the vertical shear, the region of maximum instability depends also on the horizontal shear, and is located away from the
equator (where the vertical shear is maximum). As a consequence, the longitudinal flow below the surface shows two equidistant
maxima across the equator, located at intermediate latitudes. 相似文献
7.
B. Durney 《Solar physics》1972,26(1):3-7
The Sun's differential rotation can be understood in terms of a preferential stabilization of convection (by rotation) in the polar regions of the lower part of the convection zone (where the Taylor number is large). A significant pole-equator difference in flux () can develop deep inside the convection zone which would be unobservable at the surface, because can be very efficiently reduced by large scale meridional motions rising at the poles and sinking at the equator. This is the sense of circulation needed to produce the observed equatorial acceleration of the Sun. Differential rotation is generated, therefore, in the upper part of the convection zone (where the interaction of rotation with convection is small) and results as the convection zone adjusts to a state of negligible Taylor number.The National Center for Atmospheric Research is sponsored by the National Science Foundation. 相似文献
8.
If fluctuations in the density are neglected, the large-scale, axisymmetric azimuthal momentum equation for the solar convection zone (SCZ) contains only the velocity correlations
and
where u are the turbulent convective velocities and the brackets denote a large-scale average. The angular velocity, , and meridional motions are expanded in Legendre polynomials and in these expansions only the two leading terms are retained (for example,
where is the polar angle). Per hemisphere, the meridional circulation is, in consequence, the superposition of two flows, characterized by one, and two cells in latitude respectively. Two equations can be derived from the azimuthal momentum equation. The first one expresses the conservation of angular momentum and essentially determines the stream function of the one-cell flow in terms of
: the convective motions feed angular momentum to the inner regions of the SCZ and in the steady state a meridional flow must be present to remove this angular momentum. The second equation contains also the integral
indicative of a transport of angular momentum towards the equator.With the help of a formalism developed earlier we evaluate, for solid body rotation, the velocity correlations
and
for several values of an arbitrary parameter, D, left unspecified by the theory. The most striking result of these calculations is the increase of
with D. Next we calculate the turbulent viscosity coefficients defined by
whereC
ro
0
and C
o
0
are the velocity correlations for solid body rotation. In these calculations it was assumed that 2 was a linear function of r. The arbitrary parameter D was chosen so that the meridional flow vanishes at the surface for the rotation laws specified below. The coefficients v
ro
i
and v
0o
i
that allow for the calculation of C
ro
and C
0o
for any specified rotation law (with the proviso that 2 be linear) are the turbulent viscosity coefficients. These coefficients comply well with intuitive expectations: v
ro
1
and –v
0o
3
are the largest in each group, and v
0o
3
is negative.The equations for the meridional flow were first solved with
0 and
2 two linear functions of r (
0
1
= – 2 × 10 –12 cm –1) and (
2
1
= – 6 × 10 12 cm –1). The corresponding angular velocity increases slightly inwards at the poles and decreases at the equator in broad agreement with heliosismic observations. The computed meridional motions are far too large ( 150m s–1). Reasonable values for the meridional motions can only be obtained if
o (and in consequence ), increase sharply with depth below the surface. The calculated meridional motion at the surface consists of a weak equatorward flow for gq < 29° and of a stronger poleward flow for > 29°.In the Sun, the Taylor-Proudman balance (the Coriolis force is balanced by the pressure gradient), must be altered to include the buoyancy force. The consequences of this modification are far reaching: is not required, now, to be constant along cylinders. Instead, the latitudinal dependence of the superadiabatic gradient is determined by the rotation law. For the above rotation laws, the corresponding latitudinal variations of the convective flux are of the order of 7% in the lower SCZ. 相似文献
9.
Metamorphic differentiation associated with cleavage development in crenulated anisotropic rock fabrics is commonly due to a redistribution of mineral phases within a certain volume of the fabric. Such a volume redistribution can be explained by solution transfer of soluble minerals from sites of high chemical potential, fold limbs, to sites of low chemical potential, fold hinges. The processes involved are dissolution, diffusional transfer via grain-boundaries and redeposition. The driving force for the diffusion, differences in chemical potential, is relatable to stress and fabric variations around the microfolds. The rate of transfer is influenced by the initial solubility of the mineral grains, the kinetics of grain-boundary diffusion, the nature of grain contacts and the microfold wavelength. The sense of migration of dissolved species is dependent on spatial variations in the magnitudes of normal stress and mean stress combined with grain shape and grain orientation changes around the microfolds. 相似文献
10.
Measurements of total, incremental and progressive strains associated with the development of small scale crenulation cleavage in some low-grade metamorphic rocks from Australia and Switzerland are applied to a discussion of the mechanical significance of the cleavage.Limits are placed on the amount of incremental and total slip or simple shear possible along the cleavage by the observation that the XY principal plane trace of bulk total crenulation strain coincides within 4° of the crenulation cleavage trace in all cases where this strain has been measured or estimated. The measurements are made on eight specimens using deformed porphyroblasts, crystal fibres in pressure-shadows around pyrite and flattened folds and include deformations with coaxial and non-coaxial histories.Further measurements derived from pressure-shadow fibres (eight specimens) show that the style and orientation of incremental deformation are essentially independent of the crenulation cleavage, except for a limit (43°) to the obliquity of the principal incremental extension axis during a given cleavage episode. The only special deformation related to the cleavage is the coaxial one. An indication of passive cleavage behaviour at high strain is shown by the progressive strain history of one specimen. Evidence for passive rotation of a transected axial plane is shown by another. A model is proposed to account for these observations, especially the conditions necessary for initiation and continued development of a new cleavage fabric.Some further applications of existing strain measurement techniques are described: of the Rf/Øf method to heterogeneously superposed tectonic strains and of an improved procedure of t′α/α flattening analysis. 相似文献