The Diurnal Temperature Wave with a Coefficient of Diffusivity Which Varies Periodically with Time and Exponentially with Height

The article of record as published may be found at http://dx.doi.org/10.1175/1520-0469(1958)0152.0.CO;2The partial differential equation for heat diffusion is numerically integrated by the Runge-Kutta method. Solutions are·obtained for the diurnal temperature variation with a bounded coefficient of eddy diffusivity which varies periodically with time and exponentially with height. The surface wave is represented by the sum of a diurnal and a semidiurnal harmonic wave. The results may be interpreted to apply over a fairly broad range of diffusivity values and height. With appropriate choices of the various parameters, reasonably good agreement is obtained between theoretical and observational values of amplitude reduction and phase lag as functions of height and time

Additional comments

The article of record as published may be found at http://dx.doi.org/10.1175/1520-0469(1951)0082.0.CO;2My initial comments applied to Dr. Hsieh's original paper. His reply contains an additional assumption not made previously; namely, that the vortex always consists of the same fluid particles. Also, he now considers a fixed region in the vicinity of the center, rather than following a "fictitious point at the surface moving so as to remain directly under the center of the upper vortex." In view of these changes, it appears desirable to add a few remarks

Finite difference approximations for the determination of dynamic instability

Approximate forms of the vorticity and thermal equations are linearized and combined to yield a second-order partial differential equation for the amplitude of harmonic perturbations. Finite-difference approximations for the derivatives yield a homogeneous system of algebraic equations; and the condition that its determinants vanish for a non-trivial solution yields the “frequency” equation, which may be solved to give the phase velocities of the harmonic waves. Solutions are obtained for zonal currents in which the wind varies vertically and horizonally and for a variety of conditions with respect to grid distances, latitude and current width. Generally speaking, the computations showed that decreasing the latitude and shear and increasing the static stability were all destabilizing influences, not without some exceptions, however. In addition, very short waves were found to be stable; however, instability was found for very long waves, including a retrogressive unstable mode. Moreover, multiple unstable modes were found for many wavelengths. Calculations based on actual observations of the jet stream in December show it to be dynamically unstable, both baroclinically and barotropically, with one mode of maximum instability at a wavelength of about 3000 to 4000 km and a secondary maximum at about 10,000 km

The effects of sensible heat exchange on the dynamics of baroclinic waves

The article of record as published may be found at http://dx.doi.org/10.1111/j.2153-3490.1967.tb01472.xA diabatic two‐level model with variable static stability is investigated with respect to the dynamic stability and thermal structure of harmonic perturbations. The exchange of sensible heat is assumed to be proportional to the difference in temperature between the air and the underlying surface. This type of diabatic heating reduces the instability of short and medium waves and shifts the maximum instability to a shorter wave length than the corresponding adiabatic model; however, the instability of long waves is increased. Solution of the initial value problem for various initial phase differences between the stream function wave, thermal wave and static stability waves show the importance of these parameters with respect to the growth characteristics which are complex. Limiting angles for a 4000–km wave length show an 85° lag of the thermal wave behind the stream wave and the latter lagging the vertical velocity and static stability waves by about 90° and 110° degrees respectively; but no significant differences are found between the adiabatic and diabatic cases. For an 8000–km wave length, the thermal wave lags the stream wave by about 30° and the stream wave, in turn, lags the static stability and vertical velocity waves by 140° with adiabatic flow, while with heat exchange the corresponding figures are 25° and 105°

Some Recent Advances in Numerical Weather Prediction

The article of record as published may be found at http://dx.doi.org/10.1175/1520-0493(1975)1032.0.CO;2Presented at the Third Symposium on Scientific Reviews, 54th Annual Meeting of the American Meteorological Society, January 8-11, 1974, Honolulu, Hawaii.Recent developments in numerical weather prediction during the past several years are briefly summarized for the nonspecialist

Some further results on convective currents

The article of record as published may be found at http://dx.doi.org/10.3402/tellusa.v12i4.9417An earlier theory of steady state, saturated convective currents is modified to include the drag of the condensed liquid water on the rising air. The system of equations is numerically integrated for several cases in which the entrainment is allowed to assume negative values. Results appear to indicate improvement over the earlier model

Dynamic instability in barotropic flow

The dynamic stability of single-jet and double-jet zonal currents is investigated for several quasi-barotropic models by a finite difference method and by means of finite Fourier series. The two methods give generally similar results, but some moderate differences occur when the mesh length is decreased or the number of Fourier components is increased. Also the stability characteristics of single-jet and double-jet profiles differ considerably

Numerical Prognosis Including Non-Adiabatic Warming

The article of record as published may be found at http://dx.doi.org/10.1175/1520-0469(1960)0172.0.CO;2A model for numerical prediction of the 1000-mb surface is developed which includes a term expressing the interchange of sensible heat between the air and the underlying surface as well as the effect of terraininduced vertical motion. In spite of the crudeness of the non-adiabatic representation, the model shows a definite improvement over a similar adiabatic model when the two are compared in a series of prognoses. Moreover, when monthly mean isotherms may be used to represent the temperature of the underlying surface, the non-adiabatic term may be combined with the orographic term and the earth's vorticity so that there is no work added to the prognostic routine

A Note on the Diurnal Wind Variation

The article of record as published may be found at http://dx.doi.org/10.3402/tellusa.v13i3.9500In recent correspondence HARRIS (1960) suggested that the discrepancy between observed winds and those computed from a theoretical model for the diurnal wind variation (HALTINER 1959) may be due, at least in part, to the diurnal pressure variation which had been omitted in the theoretical model

Higher-order geostrophic wind approximations

Two iterative methods are described for obtaining horizontal winds from the pressure-height field by means of higher-order geostrophic approximations for the purpose of improving upon the geostrophic wind. The convergence properties of the iterative methods are discussed; and in a simple theoretical ease, one of the methods is found to diverge with strong cyclonic motion. Both iterative methods were applied to analyzed 500-mb, height charts and obtained from the balance equation. However in a few locations continued iteration led to increasing differences between successively computed winds: i.e., the methods appeared to diverge. In fact, wind values in adjacent areas gradually tended to be corrupted. This lack of convergence, occuring mainly in areas of negative vorticity and additionally in the ease of method II in areas of strong cyclonic vorticity, was associated with the development of excessive horizontal wind divergence, which after three or four iterations sometimes exceeded the relative vorticity. Stream functions were computed by relaxing the relative vorticity of the winds obtained by methods I and II, generally after one iteration. These were compared to the stream function obtained by solving the balance equation and no significant difference were noted. Barotropic forecasts prepared from the streatm functions derived from the two methods are essentially the same as forecasts with the stream function obtained from the balance equation