Vertical Variable Research Articles

Nonlinear Convection in a Horizontal Layer with an Internal Heat Source

The problem of thermal convection is studied in a horizontal layer of fluid with an internal heat source which is restricted to vary linearly with the vertical variable. Square and hexagonal cells are found to be the only possible stable convection cells. Finite amplitude instability can occur for a range of the amplitude of convection. The presence of internal heating is found to be able to affect strongly the cell's size, the critical Rayleigh number, the heat transported by convection, the stability of the convective motion and the internal motion of the hexagonal cells.

Atmospheric oscillations—I

A general theory of atmospheric oscillations is developed in which, by expanding in a complete set of vector harmonics, the system of differential equations for the motion can be reformulated, giving a system of matrix differential equations in the vertical variable. General formulae are derived for the matrix elements using group-theoretical methods. The separable case is considered in detail and a matrix representation of the Hough functions and associated eigenvalues is obtained, from which the orthogonality property follows directly; detailed calculations of these eigenfunctions and eigenvalues are made for the lunar diurnal and semi-diurnal oscillations. In Paper II the method will include the determination of the forced tidal oscillations of the combined atmosphere-ionosphere system, with the full Coriolis force.

Application of the function sin ( X)/ X to solutions of twodimensional geophysical problems

A computing technique is described for routine processing of geophysical field readings A( k) observed in equidistant points along traverses across two-dimensional geological formations. For every point of observation x( k) a number of adjacent readings, N at each side, are included in the calculation. The arithmetical mean of these readings is taken as the temporary regional anomaly A ̄ (k) . The temporary local anomaly A 1 is defined as A(k + i) − A ̄ (k) for ¦ i ¦ ≤ N , and as zero for ¦ i ¦ > N. The local anomaly is approximated by an interpolation polynomial of infinite degree F, being a sum of unit functions E, each multiplied by its corresponding value A 1: F(x)= ∑ i=n N A 1(k+i)·E(λ,ξ,0) ξ= {x−x(k+i)} D E(λ, ξ, 0) = sin ( πξ λ ) πξ in which λ is the degree of smoothing and D the point spacing. If it is assumed that the observed potential field component has constant direction along a straight line any desired component or derivative of the field anywhere in a homogeneous semi-space may be computed with the aid of the potential of the harmonic unit function E( λ, ξ, gz), in which ζ is the normalized vertical variable. Routine calculations are performed by the formula : Q{x(k),z)= ∑ i=−N N C i·A(k+i) where Q is the desired geophysical quantity, and c i are coefficients which are generated by a special subprogram included in a general program for two-dimensional problems. Some applications on model data are demonstrated.