Here we concern ourselves with the derivation of a system of evolution equations for slowly varying amplitude of a BAROCLINIC wave packet. The self-induced transparency, sine-Gordon, and nonlinear Schrodinger equations, all of which possess soliton solutions, each arise for different in viscid limits. The presence of viscosity, however, alters the form of the evolution equations and changes the character of the solutions from highly predictable soliton solutions to unpredictable chaotic solutions. When viscosity is weak, equations related to the Lorenz attractor equations obtain, while for strong viscosity the Ginzburg-Landau equation obtain.

Atmospheric flows are often associated with long waves which, in the presencet of fronts (regions with large spatial gradients of temperature or density) can produce vortices as a result of instability. Vortex motion also can lead to front formation. These processes are often associated with potential vorticity advection, which is nonlinear and, hence, not easily computable. Laboratory experiments (physical simulation) can help us to understand such complex processes in atmosphere and ocean. We present some laboratory experiments in a' rotating fluid in which temperature (or density) gradients are imposed producing wavy circulations that can, under certain conditions, become unstable and result in vortices. One of the experiments involves rotating annulus in which a circular layer of uniform depth is subjected to a radial temperature gradient. Thermal Rossby and Taylor number values similar to those in large scale atmospheric flows are chosen for these experiments. The results show that BAROCLINIC instability can produce wavy motion, which grows and produces flow pattern as well as vortices which are comparable to the atmospheric flow pattern. In another experiment, a BAROCLINIC vortex is produced by injecting a buoyant fluid on the free surface of a denser layer of fluid (usually satlier water) in solid-body rotation. Flow visualization and measurement show that for a relatively small source Froude number when the size of the vortex is about four times the internal Rossby radius of deformation of the environment, the vortex goes unstable and usually turns in two vortices (with circumferential wave number 2) and some of the potential energy of the flow is released into the kinetic energy of the flow. Such behavior is also observed in the polar stratospheric vortex, which has implications for the formation of ozone hole in polar atmosphere.      

In this study, the effect of jet width on BAROCLINIC instability is discussed, while a BAROCLINIC instability problem is solved using a quasi-geostrophic (QG) model on a β-plane. To solve the QG equations on the β-plane, the finite difference method is applied in the vertical and meridional directions. Boundary conditions in this problem are considered for both vertical and meridional directions. Indeed, two hard boundaries at the surface of the Earth and tropopause are chosen for the vertical, with non-flux conditions at the upper and lower boundaries along the meridian. After discretization along both meridian and vertical directions, the equation takes the form of Sturm–Liouville, particularly the eigenvalue of the resulting Sturm – Liouville equation is the imaginary part of the phase velocity. Using the Matlab software, the eigenvalue instability equation can be solved. In this study, the effect of jet stream width on BAROCLINIC instability is investigated. In addition, jet streams with different widths are defined and the growth rate of atmospheric waves is calculated. The jet stream equation has a sinusoidal shape in the meridional direction, but an exponential form in the vertical, in which the jet width is adjusted using the sine-wave parameter. Once built according to the desired width, the problem is solved and the rate of the growth of atmospheric waves is obtained. The jet has a limited effect on the growth of atmospheric waves. The effect of the jet on the BAROCLINIC instability is such that a disturbance with meridional wavenumber is imposed on the problem. The meridional wavenumber causes a decrease of the growth rate at the desired zonal wavenumber. For this reason, we conclude that the jet has a limited effect on the growth rate of BAROCLINIC instability. The effect of the width on BAROCLINIC instability is identified in a two-dimensional model, in which the vertical extent is an independent variable in the problem, such that the solution is very similar to the combination of Eady (1949) and Charney (1947) models. Using a quasi-terrestrial linear model, they concluded that jet streams width, increases the growth rate of waves. Their results are inconsistent with ours due to application of one-dimensional model in their study. They noted that jet stream introduces increasing or decreasing wind shear, and with increasing wind shear, an increasing growth rate of BAROCLINIC instability is observed. However, this result cannot be generalized for a two-dimensional problem, in which for a range of latitudes, which is called a channel, the jet velocity at the bottom of the channel starts from a minimum, but increases to the maximum value in the middle of the channel and again decreases to the same value at the top of the channel. However, in a one-dimensional problem, only the jet stream core is considered, such that BAROCLINIC instability is solved only on the vertical direction in the jet core. Thus, the effect of jet stream on BAROCLINIC instability in a two-dimensional framework is conducted here. The instability problem is solved using the jet stream shown in Figure 1. According to Lindzen (1993), in the presence of a jet stream, the meridional wavenumber is equivalent to the inverse of the width of the jet, which increases as the jet width decreases, such that an increase in the meridional wavenumber is associated with a slowdown of the jet stream, following Eady (1949). Initially, by reducing the jet width to 2400 kilometers, the growth rate also decreases. However, reduction of the jet width to a certain extent (i. e., 3240 km) results in a decrease of the growth rate, while further decrease of the jet width is associated with an increase of the growth rate (e. g., for jet stream with widths of 2400 and 1710 km). Thus, the widest jet stream is associated with the maximum growth rate for wavenumbers between 6 and 13, while the narrowest jet stream is associated with the maximum growth rate for wavenumbers between 13 and 20. The relationship between the jet bandwidth and velocity of the jet center based on observational data over the Pacific is discussed below. A linear relationship (34) is obtained between velocity of the jet core and the observed jet width. Velocity of the jet core increases with the decline of the jet width (Table 4). Width and velocity of the jet in Table 4 are plotted in the numerical scheme, in which real and imaginary parts of the phase velocity are calculated when the jet core velocity is increased following a decrease of the jet width, which results in an enhancing of the growth of atmospheric waves. Therefore, under real conditions, in which width and velocity of the jet core are represented in Table 3, a meridional constraint can no longer be introduced.

The behavior of the oceans and the atmosphere in mid-latitudes may be considered as a small departure from the background rotation of the Earth as a solid body. This provides a ground for the quasigeostrophic (QG) approximation, which is obtained by a formal expansion of the primitive equations in Rossby number that measures the intensity of such departures from the background rotation. The resulting equations, though much simpler than the full set, are still complex enough that it is not always clear what they imply about the nature of their solutions. Therefore, further simplifications have been sought in particular contexts, looking for more tractable models. A model of this kind constructed based on the assumption of a uniform interior QG potential vorticity is discussed in this paper. A further simplification may be obtained by assuming uniform stratification in the atmosphere/ocean. This model has been proposed for explaining some aspects of instability in the atmosphere by Charney and Eady and is used in this paper for studying some effects of wind shear on BAROCLINIC instability.In the Eady model, the wind shear on the lower (ground surface) and upper (tropopause surface) boundaries plays a determining role in the occurrence of instability. However, in the classic form of the BAROCLINIC instability theory of Charney and Eady, wind shear is considered constant with height, and therefore the effect of variations in wind shear on the ground and tropopause surfaces are not covered.According to the Charney–Stern–Pedlosky theorem, with uniform interior potential vorticity, for instability to occur, the wind shear at the upper and lower boundaries must be of the same sign. This theorem provides the necessary condition for instability but gives no further information on the effect of the wind shear at the two boundaries.Then here, the objective is to assess the effects of the wind shear on Eady-like models, that is, models with uniform interior QG potential vorticity. After examining a quadratic vertical zonal wind profile for the basic state as a special case, the arbitrary variation of the wind shear at the two boundaries is studied in an Eady-like model. It is shown that for each wavenumber, there are upper and lower bounds, respectively denoted by g1 and g2, for the ratio of the tropopause wind shear `uzH to the earth's surface wind shearuz `uz0, beyond which instability cannot occur. That is, for instability the ratio must be in the interval g1>`uzH/`uz0>g2 which serves as an additional necessary condition for instability. Considering all the wavenumbers, the lowest value for g2 is found to be 0.3. With nondimensional wavenumbers k*=(2p/L) LR in which L and LR are respectively the dimensional wavelength and Rossby deformation radius, for k*>2.4, instability occurs provided that the wind shear at the lower boundary is greater than that at the upper boundary. For k* between 1 and 1.4, g1 tends to infinity which means that for instability there is no restriction on the magnitude of the wind shear at the upper boundary.

A review of past work in the subject areas of latent heat release in extratropical cyclones, within the concept of the potential vorticity framework or "PV thinking" is presented. It is also aimed to assess to what extent the conventional BAROCLINIC instability theory can be applied to extratropical cyclones involving intense latent heat release. The main results of the previous studies concerning the effect of latent heat release on extratropical cyclones DYNAMICS can be divided into two categories, depending on whether the impact of the diabatically generated PV anomaly on the BAROCLINIC DYNAMICS was very weak or strong. In the weak cases, cyclogenesis is primarily driven by BAROCLINIC DYNAMICS, with latent heat release playing a secondary role. Latent heating influences the BAROCLINIC DYNAMICS as simply by superposing a positive PV anomaly near the cyclone center without significantly altering the PV structure elsewhere. On the other hand, a few studies reveal that latent heat release can enhance largely the cyclone intensity, particularly when the surface thermal gradients are weak and alter significantly the structure of upper-level PV and surface thermal anomalies. The low-level diabatically produced PV anomaly is able to substitute for the absent surface warm anomaly.

Using the global data of NOAA provided by the Iran Meteorological Organization, the energetics of the wave packets detected in part I of this paper are studied in detail. To this end, within the wave packets detected, six troughs with downstream development were selected and for each the eddy kinetic energy as well as the other important terms in the energy budget relation were computed for the life time of the troughs. The ageostrophic flux convergence, total convergence flux, BAROCLINIC conversion, barotropic conversion, and the residual are the terms computed for the study. The energetics results show that in nearly all the cases involving downstream development, BAROCLINIC conversion plays a significant role as either a source in the mature state or a sink in the decaying state of the troughs. The initial perturbation is generated by BAROCLINIC conversion and subsequently initiates the downstream development. The main cause of downstream development is the ageostrophic flux convergence by which energy is radiated from the upstream of the existing perturbation to its downstream. This process leads to the decay of the existing wave and generation of a new wave in its downstream.

In mid-latitudes, the surface cyclones and anticyclones are mainly generated by the action of mid to upper tropospheric BAROCLINIC waves. To understand the mechanisms of formation and intensification of these weather systems, the study of BAROCLINIC wave packets is essential. In the first part of this two-part paper, using the global data from NOAA for Feb. 2003 provided by Iran Meteorological Organization, a synoptic-dynamic study to detect BAROCLINIC wave packets and determine their evolution is undertaken. Hovmoller diagrams and complex demodulation as applied to various dynamical quantities have been used to detect wave packets and determine their characteristics. For the upper tropospheric wave packets detected in Hovmoller diagrams, group velocity is greater than the phase speed. This characteristic of wave packets is consistent with downstream development of the waves. The results indicate that the blocking action in the east of the Pacific and Atlantic Oceans in the first 10 days of Feb. inhibits the presence of BAROCLINIC wave packets and their related low-level activities. The wave packets detected by Hovmoller and complex demodulation methods have been compared. For the period 10th to 15th of Feb. the two methods consistently give the same packets. Tracking the wave packets by using complex demodulation point to the intensification and weakening of the waves over, respectively, the west and east of the Pacific and Atlantic Oceans, which is studied in more detail in the second part by dynamical analysis considering the energetics of the waves.

Due to the stochastic nature of wind energy, allocating an appropriate investment incentive for wind generation technology (WGT) is a complicated issue. We propose an improvement on the traditional incentive, known as capacity payment mechanism (CPM), to reward the wind generators based on their performance exogenously affected by the wind energy potential of the location where the turbines are installed, and therefore, lead the investments towards locations with more generation potential. In CPM, a part of investment cost of each generator is recovered through fixed payments. However, in our proposal, wind generators are rewarded according to dynamic forecasts of the wind energy potential of the wind farm where they are located. We use an auto-regressive moving average (ARMA) model to forecast the wind speed fluctuations in long-term while capturing the auto-correlation of wind velocity variation in consecutive time intervals. Using the system DYNAMICS (SD) modelling approach a competitive electricity market is designed to examine the efficiency of the proposed incentive. Performing a simulation analysis, we conclude that while a fixed CPM for wind generation can decrease the loss of load durations and average prices in long-term, the proposed improvement can provide quite similar results more efficiently.