Co-latitude

The standard coordinates on the surface of a sphere of radius r, i.e., the spherical coordinates, are the longitude and the co-latitude 6 (co-latitude is n/2 minus the latitude). Orthogonality over the surface of a sphere of two distinct functions f(,0) and g(4 0) takes the usual form but in two dimensions... 

If in spherical polar coordinates the polarization vector if/ makes an angle t/z with the x axis, and the field vector R makes a spherical co-latitude angle 6 with the z axis and an azimuthal angle (f> with the x axis, then... 

The total number of amino-acid pairs at a given grid point whose atoms are within 7 A of each other and in which each molecule contributes one member of the pair, appears in the numerator and that of hydrophobic amino-acid pairs appears in the denominator. The numbers in parentheses correspond to values at the outcome of the coarse grid search before the optimization of the contacts. All other numbers correspond to optimized contacts. (-) correspond to contacts which do not contain the Pq residue. The star indicates the longitude and co-latitude of the best up down contact. (Complete parameters for this contact are given in the legend of Table II.)... 

The symbol Q denotes a unit vector in actual space (i.e., a direction) and can be specified by giving two angles, a co-latitude 0 and an azimuth therefore, O =, ... 

The direction (unit vector) is specified by means of the co-latitude... 

The locally stable equilibrium configurations of the surface Coulomb problem satisfy the additional constraint that the associated energies are local minima. Specifically, if the charge positions are described by spherical coordinates—the co-latitudes 0 s d>. < tt, and longitudes -7T < 0, TT—then the Coulomb energy (2.1) is E(i, 0,), 1 < j < N and the equilibrium condition (2.2) is equivalent to... 

Similar heterogeneous reactions also can occur, but somewhat less efticientiy, in the lower stratosphere on global sulfate clouds (ie, aerosols of sulfuric acid), which are formed by oxidation of SO2 and COS from volcanic and biological activity, respectively (80). The effect is most pronounced in the colder regions of the stratosphere at high latitudes. Indeed, the sulfate aerosols resulting from emptions of El Chicon in 1982 and Mt. Pinatubo in 1991 have been impHcated in subsequent reduced ozone concentrations (85). 

Temperatures are average existing at 40 latitude and are rounded to even numbers. jNegative exponent shows number of spaces the decimal point must be moved to the left. Courtesy Ingersoil-Rand Co. 

Temperalures are average existing al 40° latitude and are rounded lo even numbers. jNegalive exponent shows number of spaces Ihe decimal point must be moved lo Ihe left. Courtesy Ingersoll-Rand Co. 

Figure 8. Steady-state isopleths calculated from clean/polluted air chemical mechanism using NMHC and NO source rates on axes. Mechanism was integrated to steady-state for each calculation using 1/4 the equinoxial light intensity at 45° latitude. A. [O3] ppb. B. [HO ] 10 molecules cm . C. [HO2] 10 molecules cm D. [NO ] ppt. E. [CO] ppb. D. CH4 ppm.

Now we demonstrate one interesting feature of the magnitude of the surface force as a function of the latitude. First, assuming that the fluid Earth is almost a sphere with a radius a, we can represent the distance r as r — a cos 1. Then, Equation (2.18) has the form... 

A and B are constants and they are independent of time. Bearing in mind that = X + iy, it is simple to find functions x t) and y t), which describe a motion of the pendulum on the earth s surface. In accordance with Equation (3.88) a solution is represented as a product of two functions. The first one characterizes a swinging of the pendulum with the angular velocity p, which depends only on the gravitational field and the length /, while the second is also a sinusoidal function and its period is defined by the frequency of the earth s rotation and the latitude of the point, (Foucault s pendulum). In order to understand the behavior of the pendulum at the beginning consider the simplest case when a rotation is absent, co — 0. Then, we have... 

Vrf + r — 2r1r2 cos (0 — 02) in polar coordinates. Other coordinate systems are sometimes used. For example, on the surface of a sphere latitude and longitude prove useful. 

As a result, the point of interest D on the earth at latitude 0 2 rotates eastward with velocity Ve cos 0 2, which will contribute to the Doppler... 

TIjc Coriolis parameter/is defined by/ = 2a> cos (j>, where o> is the earth s rotation rate and is the latitude. 

Carbon monoxide, which is predominantly produced during combustion processes, may exhibit an 0 excess of up to 7.5%c in summer at high northern latitudes (Rockmann et al. 1998). The major source of this fractionation is its atmospheric removal reaction CO + OH = CO2 + H2, in which the remaining CO gains excess 0. 

Brenninkmeijer CAM, Lowe DC, Manning MR, Sparks RJ, van Velthoven PFJ (1995) The C, C and 0 isotopic composition of CO, CH4 and CO2 in the higher southern latitudes and lower stratosphere. J Geophys Res 100 26163-26172 Broecker WS (1974) Chemical oceanography. Harcourt Brace Jovanovich, New York Brooker R, Blundy J, James R (2004) Trace element and Li isotope systematics in zabargad peri-dotites evidence of ancient subduction processes in the Red Sea mantle. Chem Geol 212 179-204... 

Water vapor concentrations have also been used to show that stratospheric air in the midlatitudes cannot all have originated via the tropical pump, i.e., path I in Fig. 12.3. For example, Dessler et al. (1995b) have shown that water vapor concentrations in the lowermost stratosphere at 37.4°N, 122.1°W are higher than expected for an air mass that has passed through the cold tropical tropopause. Their data are consistent with path II, although as they point out, these measurements do not exclude path III, which represents convective transport from the troposphere to the stratosphere at mid and high latitudes. Lelieveld et al. (1997) report aircraft measurements of CO, 03, and HNO-, over western Europe that suggest that tropospheric air can be mixed into the lower stratosphere. 

An example of the use of this technique is the measurement of NO and N02 made near 50°N latitude by Ridley and co-workers. Summertime measurements of NO, N02, 03, temperature, and the photolysis rate of N02 showed that NO and NO were in photochemical steady state (65). However, the abundances of NOx (NO + N02) were observed to be ten times smaller in winter than in summer at altitudes between 20 and 28 km (63). At altitudes above 28 km, the abundances of NOt were similar in both winter and summer. Considering the trajectories of air at different altitudes, they were able to determine that N205 must be the wintertime reservoir species, as was predicted in a number of previous studies. 

These studies stimulated the investigation of the use of TOMS data for the determination of the tropospheric ozone column amount by related techniques. Hudson, Thompson and co-workers have developed and refined a technique called the tropical tropospheric ozone (TTO) method (Hudson et al., 1995 Kim et al., 1996 Hudson and Thompson, 1998). This technique utilises a Fourier analysis to identify the range of latitudes for which the method is applicable by using the recognition of a planetary wave pattern to estimate stratospheric and background tropospheric ozone. 

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