Document Type : Research
University of Hormozgan
Department of Marine Science, Imam Khomeini University of Marine Sciences, Noshahr, Iran
In this study, the effect of jet width on baroclinic instability is discussed, while baroclinic instability 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, 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 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 new width on baroclinic instability is identified in a two-dimensional model, in which the vertical 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 increase 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 as 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 by 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 decreases 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.