Green's theorem and Green's functions for the steady-state cosmic-ray equation of transport

Green's Theorem is developed for the spherically-symmetric steady-state cosmic-ray equation of transport in interplanetary space. By means of it the momentum distribution functionF _o(r,p), (r=heliocentric distance,p=momentum) can be determined in a regionr _arr_bwhen a source is specified throughout the region and the momentum spectrum is specified on the boundaries atr _a andr _b . Evaluation requires a knowledge of the Green's function which corresponds to the solution for monoenergetic particles released at heliocentric radiusr _o , Examples of Green's functions are given for the caser _a =0,r _b = and derived for the cases of finiter _a andr _b . The diffusion coefficient is assumed of the form = _o(p)r ^b . The treatment systematizes the development of all analytic solutions for steady-state solar and galactic cosmic-ray propagation and previous solutions form a subset of the present solutions.     

Some features of the solar microwave emission and their connection with geomagnetic activity

A statistical investigation of the slowly varying component of the 9.1-cm solar radio emission is based upon the Stanford radioheliograms covering the years 1962–66. On the average the peak value of the brightness temperature T _b is proportional to the area covered by the corresponding spot group. However, in individual cases the observed T _b is definitely lower or higher than is to be expected from the size of the spot group. We introduce the concept microwave importance I _m of the spot group, which is the T _b to be expected from the Zürich class and spot number; and the concept relative brightness B _r, which is the ratio of the observed T _b to I _m. This leads to the distinction of faint, normal and bright sources with B _r 0.8, 0.8 < B _r < 1.2 and B _r 1.2 respectively. B _r is correlated with the maximum magnetic-field strength H observed in the spot group and with the flux-density spectrum of the source. The yearly average of B _r and the average flux-density spectrum vary with the phase of the solar cycle.An analysis of the results is based upon the electron-density distribution in the condensation, which was visible at the solar limb during the eclipse of February 5, 1962, and on an adopted temperature distribution with a central value of 4 × 10⁶ K. The computed T _b, including gyro-resonance absorption, agrees with the value derived from the microwave importance of the spot group and B _r = 0.5, which shows that in the current gyro-resonance models the electron density is underestimated. The variation of T _b with the size of the spot group can be explained by varying the dimensions of the condensation in area and in height, if the central density and temperature remain constant. The statistical relationships between B _r and H and between B _r and the flux-density spectrum yield a model for the differences between faint and bright sources: B _r increases with the contribution from the gyro-resonance absorption and with the central electron density.For paper I see Solar Phys. 7, 448. For paper III see Solar Phys. 8, 450.     

On the reconstruction of the nonlinear force-free coronal magnetic field from boundary data

Using a simple model in which the corona is represented by the half-space domain = {z > 0} and the photosphere by the boundary plane = {z = 0}, we discuss some important aspects of the general problem of the reconstruction of the magnetic field B in a small isolated coronal region from the values of the vector B¦ measured by a magnetograph over its whole basis. Assuming B to be force-free in : (i) we derive a series of relations which must be necessarily satisfied by the boundary field B¦ , and then by the magnetograph data if the force-free assumption is actually correct; (ii) we show how to extract directly from the measured B¦ some useful informations about the energy of B in and the topological structure of its field lines; (iii) we present a critical discussion of the two methods which have been proposed so far for computing effectively B in from B¦ .     

Transport equations for low-energy solar particles in evolving interplanetary magnetic fields

Two new forms of a simplified Fokker-Planck equation are derived for the transport of low-energy solar energetic particles in an evolving interplanetary magnetic field, carried by a variable radial solar wind. An idealised solution suggests that the invariant anisotropy direction reported by Allum et al. (1974) may be explained within the conventional theoretical framework. The equations may be used to relate studies of solar particle propagation to solar wind transients, and vice versa.     

The effect of interplanetary acceleration on the propagation of energetic solar particles in prompt events

Crank-Nicholson solutions are obtained to the time-dependent Fokker-Planck equation for propagation in the interplanetary medium following a point in time injection of energetic solar particles and including the acceleration terms $$\frac{\partial }{{\partial T}}\left( {D_{TT} \frac{{\partial U}}{{\partial T}}} \right) - \frac{\partial }{{\partial T}}\left( {\frac{{D_{TT} U}}{{2T}}} \right)$$ . The diffusion coefficient in kinetic energyD _TT is allowed to be either independent of radial distance,R(AU), or follow the lawD _TT=D₀T²R ₀ ² /(A²+R²) in either case with the 1 AU value ofD _TT at 10 MeV ranging between 10^?4 (MeV)² s^?1 and zero. The spatial diffusion mean free path at the Earth's orbit is fixed at λ_‖ AU at 10 MeV according to numerical estimates made by Moussas and Quenby. However, a variety ofR dependences are allowed. Reasonable agreement with experimental data out to 4 AU is obtained with the above values ofD _TT and the spatial diffusion coefficientK _r=K₀R^?2 forR«1 andK _r=K₀R^0.4 forR»1 AU. It is only in the decay phases of prompt events as seen at 2–4 AU that significant differences in the temporal behaviour of the events can be distinguished, depending on the value ofD _TT chosen within the above range. Experimental determination of the decay constant is difficult.     

Numerical studies of the transport of solar protons in interplanetary space

Numerical solutions of the Fokker-Planck equation governing the transport of solar protons are obtained using the Crank-Nicholson technique with the diffusion coefficient represented by K_r=K₀r^b where r is radial distance from the Sun and b can take on positive or negative values. As b ranges from +1 to ?3, the time to the observation of peak flux decreases by a factor of 5 for 1 MeV protons when $\text{V}\text{K}{}_0\text{=}\text{3 AU}{}^{\mathrm{b?1}}$ where V is the solar wind speed. The time to peak flux is found to be very insensitive to assumptions concerning the solar and outer scattering boundary conditions and the presence of exponential time decay in the flux does not depend on the existence of an outer boundary. At $\text{V}\text{K}{}_0\text{? 15 AU}{}^{\mathrm{b?1}}$, 1 MeV particles come from the Sun by an almost entirely convective process and suffer large adiabatic deceleration at b?0 but for b=+1, large Fermi acceleration is possible at all reasonable $\text{V}\text{K}{}_0$ values. Implications of this result for the calculation and measurement of particle diffusion coefficients is discussed. At $\text{b?0}$, the pure diffusion approximation to transport overestimates by a factor 2 or more the time to peak flux but as b becomes more negative, the additional effects of convection and energy loss become less important.     

Wave and stability properties of black holes

In this paper we transform the wave equation governing gravitational perturbations of a Schwarzschild black hole from its standard Schrödinger or Regge-Wheeler form to a Klein-Gordon type wave equation. This latter form reveals immediately that incoming waves with frequencies () _cml , a critical frequency, are completely reflected (transmitted). This process is entirely due to the radial variation of the cut-off frequency inherent in the dispersive nature of the wave propagation properties of gravitational perturbations of the Schwarzschild metric. Moreover, those high-frequency waves ( _cml) which penetrate through the region near the Schwarzschild radiusr _sare, on crossing this event horizon, attenuated by a factor exp (–r _s/c), thereby dumping most of their energy and momentum into the black hole. It is shown that in the vicinity ofr _sthe metric is locally unstable. This feature and the wave absorption process indicate that the neighbourhood aroundr _sis dynamically active, and, as well as acting like a Hawking-type particle creator, will behave as a wave emitter in order to relax the stresses on the metric.     

Restricted quatum-mechanical three-body problems

On the basis of a perturbative procedure in which the eigenfunctions of a helium-like ion are expanded in the Hilbert space built up from the eigenfunctions of an electron in two fixed Coulomb charges,all asymptotic eigenfunctions are constructed for the Schrödinger equation of helium-like ions. If the nuclear chargeZ ₁ is not less than 2, then our asymptotic considerations clarify the singularities of the Schrödinger equation of a helium-like ion (atr ₁=0, ,r ₂=0, ,r ₁₂=0, ), while in the case ofZ ₁=1 (negative hydrogen ion)r ₂=0 will not be treated in this paper. The established order (inr ₂) of asymptotics at 0 or in an exceptional case the zeroth-order term of the functions (actually the coefficients of an expansion of the desired eigenfunctions) as one of the electron coordinates (r ₂, the distance of the two fixed Coulomb charges, a parameter of the set of the basic functions) enables us to classify the eigenfunctions of a helium-like ion. This classification resembles the classification scheme for one-electron configurations. The asymptotics forr ₂ indicate bounded, pseudobounded (auto-ionizing) and free states. (Doubly ionized continuum states are not discussed here.) The use of the asymptotic solutions is indicated for the complete solution of the problem which may be either numerical integration or a variational procedure.Neutral muonic helium is included in the discussion.     

On the relative roles of unipolar and mixed-polarity fields

Away from plages, solar magnetic fields may be classified as unipolar or as of mixed polarity, though the distinction is strictly arbitrary. The dividing line used here is 0.4 ¦B _minor/B _major¦ 1, where average fields of major and minor polarities are measured over large areas. Some of their statistical properties and cyclical variations are detailed. In unipolar regions, 3 B _major 50 G, B _minor 0.1 B _major, and ¦B¦ 1.1 B _major. In regions of mixed polarity, 3.5 ¦B¦ 10 G.Below latitudes of ± 60°, mixed polarities predominate for about 5 yr around sunspot minimum. For several years around sunspot maximum, unipolar fields fill the 20°–40° zone completely, and occupy about 75% of the 0°–20° and 40°–60° zones.The polar unipolar fields are weak on the whole (Bmajor 4 G for 6 typical days in 1976–79), with small regions having stronger fields at times, probably not exceeding B _major = 10 G. Again B _minor 0.1 B _major. There is no direct way at present of measuring properties of polar mixed fields, such as may occur around sunspot maximum, but by inference ¦B¦ 2 to 5 G.Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

A discussion of Hill's method of secular perturbations and its application to the determination of the zero-rank effects in non-singular vectorial elements of a planetary motion

Knowledge of the perturbations of zero-rank is essential for the understanding of the behavior of a planetary or cometary orbit over a long interval of time. Recent investigations show that these zero-rank perturbations can cause large oscillations in both the shape and position of the orbit. At present we lack a complete analytical theory of these perturbations that can be applied to cases where either the eccentricity or inclination is large or has large oscillations. For this reason we here develop formulas for the numerical integration of the zero-rank effects, using a modified Hill's theory and suitable vectorial elements. The scalar elements of our theory are the two components of Hamilton's vector in a moving ideal reference frame and the three components of Gibb's rotation vector in an inertial system. The integration step can be taken to be several hundred years in the planetary or cometary case, and a few days in the case of a near-Earth space probe. We re-discuss Hill's method in modern symbolism and by applying the vectorial analysis in a pseudo-euclidean spaceM ₃, we obtain a symmetrical computational scheme in terms of traces of dyadics inM ₃. The method is inapplicable for two orbits too close together. In Hill's method the numerical difficulty caused by such proximity appears in the form of a small divisor, whereas in Halphen's method it appears as a slow convergence of a hypergeometric series. Thus, in Hill's method the difficulty can be watched more directly than in Halphen's method. The methods of numerical averaging have, at the present time, certain advantages over purely analytical methods. They can treat a large range of eccentricities and orbital inclinations. They can also treat the free secular oscillations as well as the forced ones, and together with their mutual cross-effects. At the present time, no analytical theory can do this to the full extent.Basic Notations m the mass of the disturbed body - M the mass of the Sun - f the gravitational constant - f(M+m) - r the heliocentric position vector of the disturbed body - r |r| - r ⁰ the unit vector alongr - n ⁰ the unit vector normal tor and lying in the orbital plane of the disturbed body - a the semi-major axis of the orbit of the disturbed body - e the eccentricity of the orbit of the disturbed body - g the mean anomaly of the disturbed body - the eccentric anomaly of the disturbed body - p a(1–e ²) - P ₁ the unit vector directed from the Sun toward the perihelion of the disturbed body - P ₂ the unit vector normal toP ₁ and lying in the orbital plane of the disturbed body - s - the true orbital longitude of the disturbed body, reckoned from the departure point of the ideal system of coordinates - X the true orbital longitude of the perihelion of the disturbed body in the ideal system of coordinates reckoned from the departure point - the angular distance of the ascending node from the departure point - R ₁,R ₂,R ₃ the unit vectors along the axes of the ideal system of coordinates,R ₁ andR ₂ are in the osculating orbital plane of the disturbed body,R ₃ is normal to this plane. The intersection ofR ₁ with the celestial sphere is the departure point - R ₃ P ₁×P ₂ - S ₁,S ₂,S ₃ the initial values ofR ₁,R ₂,R ₃, respectively - q the Gibb's vector. This vector defines the rotation of the orbital plane of the disturbed body from its initial position to the position at the given timet - m the mass of the disturbing body - r the heliocentric position vector of the disturbing body - a the semi-major axis of the orbit of the disturbing body - e the eccentricity of the orbit of the disturbing body - g the mean anomaly of the disturbing body - the eccentric anomaly of the disturbing body - P₁ the unit vector directed from the Sun toward the perihelion of the disturbing body - P₂ the unit vector normal toP₁ and lying in the orbital plane of the disturbing body - A₁ a P₁ - A₂ - |r–r|     

Observed oddities in the lines H,K, b and Hβ

We compare microphotometer intensity traces perpendicular to dispersion in simultaneous spectrograms of good spatial resolution traced at various 's in each of the lines. Cross correlations between the different traces show the following: (a) For each _K there is a corresponding _{b 1}at which the coefficient of correlation, r, is a maximum, usually > 0.8. (b) No such high correlations are found between H and H. (c) Comparison of traces in the continuum and at all observed 's in K, H, b₁, b₂ show a range of 's in each line over which r is very significantly negative, while H shows no such peculiarity.     

Possible effects of anisotropy ofG on celestial orbits

Will (1971) has discussed a possible anisotropy in the gravitational constantG. Suppose that the attractive gravitational force between two particles of massesm ₁ andm ₂ is given by the usual expressionF=–Gm ₁ m ₂ r/r ³, wherer is the separation vector. Ifc is the velocity of light in vacuo and if 1 _r r/r, he expresses the anisotropy byG=G [1+(v·1 r/c)²], whereG is a constant,v is identified practically as the velocity of the Sun around the galaxy, and 1. Will's suggestion is to look for such an effect in the laboratory.The purpose of the present paper is to look for such an effect in the solar system, wherem ₁ andm ₂ become the masses of the Sun and a planet or of the Earth and the Moon. For simplicity I consider only those planets whose orbits are close to the ecliptic, so that the angle betweenv and the plane of the ecliptic is about 59°.With the above force, the resulting two-body problem is completely solvable. The results are these. If =1, there is an increase in mean motion of 7 parts in 10⁸, a periodic fluctuation in true longitude with period half that of the orbit and amplitude ranging possibly from 0.01 to 0.02, and periodic fluctuations in the radius vector, with period also one half that for the orbit. The amplitudes are: 2.7 km for Mercury, 5.1 km for Venus, 7.0 km for Mars, 18 m for the Moon about the Earth, and 28 cm for a close artificial satellite with inclination 23°. The more conservative estimate <0.0115 would reduce these values by the factor 70.     

Satellite attitude acquisition by momentum transfer-the controlled wheel speed method

An implementation of the momentum transfer method for spacecraft attitude acquisition of momentum wheel stabilized geostationary satellites is presented, in which the wheel speed is varied in a predeterminable manner to reduce the nutation usually associated with the method. The implementation is found to be capable of achieving the transfer to the desired zero nutation end point with 5° to 20° of residual nutation in practical situations without additional nutation damping. The transfer time is typically 10–30 min. The implementation is described in terms of the momentum sphere and energy ellipsoid. The detailed functional dynamics and parametric relationships are given in terms of phase planes and elliptic integral solutions. Feasibility of the implementation is shown to be dependent on the moment of inertia configuration and the degree to which tolerances on moments of inertia, satellite spin rate, and wheel speed are predeterminable.Nomenclature A, B, C Principal moments of inertia of the total spacecraft - b ₁,b ₂,b ₃ Unit vectors parallel to principal axes - E, E(t) Energy of the spacecraft body (excludes the spin energy of momentum wheel) - E ₀ Initial energy of the spacecraft (wheel not spinning) - E _c A constant arbitrary value ofE(t) - E _i Energy after an impulse which causes to equal zero, Equation (13) - E ₂ The constant value ofE during Stage 2 - E ₃ Energy at the end of Stage 3 - E _s The separatrix value ofE, i.e.E when =0 - H ₀,H ₀ Angular momentum vector of the total spacecraft - H _x ,H _y ,H _z Variable components ofH ₀ resolved with respect tob ₁,b ₂, andb ₃ - J Moment of inertia of the momentum wheel - L The slope of the wheel speed profile, Equation (31) - L _y Momentum wheel torque - m Direction cosine of the angle, , betweenH ₀ andb ₂ - m ₁,m ₂ Constant values ofm - O The centre of the momentum sphere - Oxyz Spacecraft coordinate axes - P Represents a state on the momentum sphere and energy ellipsoid - s Momentum wheel speed - s ₂ Constant value ofs during Stage 2 - s _i Wheel speed after an impulse which causes to be zero - t Time - t ₁, t ₂, t ₃ Elapsed time for Stages 1, 2, and 3 respectively - ₁ to ₄ Roots of Equation (19) (poles on the phase plane) - Parameter defined in Equation (9) - cos^–1 m - ₂ The value of at the termination of Stage 2 - _min The minimum of the periodic function (t) (during Stage 2) - _x , _y , _z Angular rates of spacecraft body - ₀ Initial value of _z - K _m Curvature of the momentum sphere - K _xy ,K _yz Curvatures of the principal lines of the energy ellipsoid at (O,H ₀,O), in theOH _x H _y ) and (OH _y H _z ) planes respectively     

Simulation for electron acceleration by DC electric field in the presence of ion sound waves and associated hard X-ray emission

Time-dependent Fokker-Planck equation was numerically solved to demonstrate the dynamics of electrons in a uniform coronal loop with an applied axial DC electric field in the presence of ion-sound waves. This electric field is attributed to an anomalous resistivity due to the ion-sound turbulence caused by an initially given critical current density.The electron momentum distribution becomes a steady state in the whole turbulent region in a short time for which some electrons can be accelerated to the maximum electric potential K _c. The steady energy distribution of electrons flowing out the end of the turbulent region has a very hard power-law-like spectrum with an index of about 0.75. The associated hard X-rays from a thick target also show a hard spectrum with a photon spectral index of 1.3. In order for to be much greater as observed in impulsive X-ray bursts, it is required that the source is a sum of many elementary loops with a power-law-like distribution in K _c with an index = – + 2.5.     

One-armed oscillations of a non-selfgravitating polytropic disk

One-armedglobal oscillations in a non-selfgravitating polytropic disk rotating around a star are investigated. The unperturbed disk is axisymmetric, geometrically thin, and extends infinitely in the radial direction keeping its thickness constant. Perturbations considered are inviscid and adiabatic. It is found that there are one-armed retrograde wave modes which are trapped in an inner region of the disk. The eignefrequency of the lowest order mode is given by _K(r _s)(z ₀/r _s)₂, wherer _s is the radius of the central star,z ₀ is the half-thickness of the disk, and _K(r _s) is the Keplerian angular frequency at the surface of the star.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

Orientation of the galactic magnetic field in the vicinity of the solar system

It is found from analysis of the position angles of the plane of polarization of about 3000 stars (¦b¦ 5° andP 0.5%) that the angle between the magnetic field and the equatorial plane of the galaxy is approximately 0–5°. The distance within which the local magnetic fields of the galaxy have a greater effect on the position angles of the plane of polarization than the galactic magnetic field is estimated to be about 500 pc. The effect of the galactic magnetic field becomes dominant for distancesr 1000 pc.Translated fromAstrofizika, Vol. 39, No. 4, pp. 553–559, November, 1996.     

Coronal magnetic-field model with non-spherical source surface

Previous global models of coronal magnetic fields have used a geometrical construction based on a spherical source surface because of requirements for computational speed. As a result they have had difficulty accounting for (a) the tendency of full magnetohydrodynamic (MHD) models to predict non-radial plasma flow out to r 10r and (b) the appreciable magnitude, 3, of B _r , (the radial component of B) consistently observed at r 1 AU. We present a new modelling technique based on a non-spherical source surface, which is taken to be an isogauss of the underlying potential field generated by currents in or below the photosphere. This modification of the source surface significantly improves the agreement between the geometrical construction and the MHD solution while retaining most of the computational ease provided by a spherical source surface. A detailed comparison between the present source-surface model and the MHD solution is made for the internal dipole case. The resulting B field agrees well in magnitude and direction with the coronal B field derived from the full MHD equations. It shows evidence of the slightly equatorward meridional plasma flow that is characteristic of the MHD solution. Moreover, the B field obtained by using our non-spherical source surface agrees well with that observed by spacecraft in the vicinity of the Earth's orbit. Applied to a solar dipole field with a moment of 1 G-r ³ , the present model predicts that B _r at r 1 AU lies in the range of 1–2 and is remarkably insensitive to heliomagnetic latitude. Our method should be applicable also to more general (i.e., more realistic) configurations of the solar magnetic field. Isogauss surfaces for two representative solar rotations, as calculated from expansions of observed photospheric magnetic-field data, are found to show large and significant deviations from sphericity.     

Time-dependent Green's functions of the cosmic ray equation of transport

Two spherically symmetric time-dependent Green's functions of the equation of transport for cosmic rays in the interplanetary region are derived by transform techniques. The solar wind velocity is assumed radial and of constant speedV. In the first model the radial diffusion coefficient =₀ r (₀ constant), and in the second solution =₀= constant. The solutions are for monoenergetic, impulsive release of particles from a fixed heliocentric radius. Integration of the solutions over timet, fromt=0 tot=, gives the steady-state Green's functions obtained previously.     

Coronal and interplanetary transport of solar energetic protons and electrons

We present a new method to separate interplanetary and coronal propagation, starting from intensity variations observed by spaceprobes at different heliolongitudes. In general, a decrease in absolute intensities is observed simultaneously with an increase in temporal delays. The coupling of these two effects can be described by Reid's model of coronal diffusion and can in principle be used to determine the two coronal time constants, diffusion time t _c and escape time A. In addition, a least-squares fit method is used to determine the parameters of interplanetary transport, assuming a radial dependence as (r) = ₀(r/1 AU)^b. The method is applied to the two solar events of 27 December, 1977 and 1 January, 1978 which were observed by the spaceprobes Helios 1, Helios 2, and Prognoz 6. Energetic particle data are analysed for 13–27 MeV protons and -0.5 MeV electrons. For the regions in space encountered during these events the mean free path of electrons is smaller than that of protons. Straight interpolation between the two rigidities leads to a rather flat rigidity dependence (P) P ⁿ with n = 0.17–0.25. This contradicts the prediction of a constant mean free path or of the transition to scatter-free propagation below about 100 MV rigidity. In three of the four cases the mean free path of 13–27 MeV protons is of the order 0.17 AU, the mean free path of electrons of the order 0.06 AU. For protons we find b - 0.7 for the exponent of the radial variation.The concept of two different coronal propagation regimes is confirmed. It is remarkable that in both regimes electrons are transported more efficiently than protons. This holds for the temporal delay as well as for the amplitude decrease. This is in contrast with the long existing concept of rigidity independent transport and puts severe limits to any model of coronal transport. For the December event all three spaceprobes are in the fast propagation regime up to an angular distance of 62°. For protons we find a finite delay even in the fast propagation region, corresponding to a coronal delay rate of about 0.8 hr rad^-1 up to 60° angular distance. In contrast, relativistic electrons may reach this distance within a few minutes.The fast transport of electrons and the different behaviour of electrons and protons is in contradiction to the expanding bottle concept. An explanation of coronal transport by shock acceleration directly on open field lines could in principle work in case of protons in the fast propagation region, but would fail in case of the electrons. The fast and efficient transport of electrons is most likely due to a region of field lines extending over a wide range of longitudes directly from the active region into interplanetary space. The much slower transport of both particle types at large azimuthal distances can neither be explained by direct access to open field lines not by the direct shock acceleration concept. A possible explanation is the loop reconnection model in a modified version, allowing for a faster lateral transport of electrons.Now at AEG, 2000 Wedel, F.R.G.     

The Energy Equation in the Lower Solar Convection Zone

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.