The relation of temperature and velocity to the slope of the tropopause in a steady zonal wind field is derived in spherical polar coordinates. Then, the tropopause “dip” just north of the jet streams in the westerlies is discussed.
The relation of temperature and velocity to the slope of the tropopause in a geostrophic wind field was first derived by V. Bjerknes.
(1) Further the relation of temperature and velocity to the slope of the tropopause in a gradient wind field was given by H. Ertel.
(2) The case treatd by H. Ertel is admittedly a simple one, so it may not be adequate for the problem of the tropopause funnel. However, such a consideration will be of value as a preliminary to the discussion of a steady zonal wind field, though it must be admitted that the analysis will not be essentially new from that given by V. Bjerknes.
The upper sign gives when
r=∞, so yields a solution which is continuous near straight isobars, while the lower sign gives an arrangement which is not available in the earth's atmosphere. Thus the only solution of Eq. (3 b) which can be extended out to a region of straight isobars is given by Thus the slope of the tropopause is generally steeper than the slopes of near-by isobaric surfaces in the temperate latitudes of the northern hemisphere.
Several recent papers from the University of Chicago have attempted to infer the vertical circulation pattern associated with well developed jet streams in the westerlies. _??_f there is to be a finite wind maximum, it is necessary that near the jet centre the slope of the isobaric surfaces steepens and above the jet centre the slope of the isobaric surfaces decreases upward. Since the atmosphere preserves hydrostatic equilibrium to a high degree of approximation, the meridional temperature gradient above the jet stream core must be opposite in direction from that of the lower levels. Thus, well developed jet streams are likely to happen in the region of the tropopause. Isobaric surfaces and the tropopause with steep slopes will be associated with the jet core in the westerlies. Thus, the tropopause dip will be found just north of the jet stream.
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