Linear Instability

It is instructive to first examine the dynamical properties of weak perturbations in this system. They are governed by the linearized version of (6.2), viz.

The absolute vorticity and stream function fields of a pertinent basic state with uoo = 0.2 are shown in Fig. 1. The maximum nondimensional velocity is 1.05. The model Pacific jet has a maximum speed of ~63m/s and the model Atlantic jet has a maximum speed of ~37m/s. The two localized jets are made up of a zonal mean shear flow and a spectrum of planetary scale waves. They are ultimately sustained by radiative forcing in the presence of zonal inho-mogeneous surface conditions on earth. The proxy forcing is meant to mimic the net effect of all such factors.

The first task is to solve for the normal modes in the form of ^ = £(x, y)exp(ta). The eigenvalue a and the eigenfunction £ can only be solved numerically. The domain is depicted with I = 140 grid points in the x direction and J = 49 grid points in the y direction. The corresponding dimensional resolution is 5x = 220 km and 5y = 160 km. Such resolution has been verified to be adequate in the sense that the solution changes little when the domain is only depicted with I = 100 and J = 29. The finite difference version of (4) written in a matrix form is

Figure 1. Distribution of (a) the nondimensional absolute vorticity and (b) the nondimensional stream function of the reference state. The absolute vorticity value does not include the value of the Coriolis parameter in the middle of the domain.

where A and B are M x M matrices with M = IJ. p is an M vector containing the unknown £ at all grid points. In passing, it is noteworthy that for M = 6860, solving (5) by the matrix inversion method is a computationally demanding task. Such computation would not be feasible if we were to do so with a counterpart two-layer model.

The purpose of presenting a linear modal instability analysis is to delineate some fundamental and relevant properties of a basic two-localized-jet flow system. These properties help us infer the plausible locations of the storm tracks in a corresponding nonlinear system. They also help us develop a "feel" for what the individual disturbances might intrinsically look like. But they do not directly enable us to deduce any quantitative properties of the storm tracks in an equilibrated state, partly because the process of nonmodal instability rather than modal instability seems to be more relevant to storm track dynamics and, more important, storm tracks are a product of nonlinear dynamics.

3.1. Inviscid modal instability properties, a = 0

All eigenvalues for each parameter setting can be plotted as an ensemble of points on a (ar, ai) plane. Associated with a complex eigenvalue, a = ar + iai, we would have an eigenvector with complex value, p = pr + ipi. The normal mode can be evaluated as p = (pr cos ait — pi sin ait) exp(art), (6)

where pr (x,y) and pi (x,y) are the counterpart scalar fields of pr and pi. It embodies the information about the structure of the unstable mode under consideration. We will normalize pr and

Figure 2. Distribution of eigenvalues of the inviscid normal modes plotted as points on a (<ar = Re{<a}, ai = Im{a}) plane, for (a) Uoo = 0.2 corresponding to 14m/s and (b) Uoo = 0.0. The unit of time is LU-1 = 0.17 x 105 s.

of each normal mode to have unit domain-integrated kinetic energy.

The eigenvalues of the normal modes for the case of uoo = 0.2 without the influence of friction are shown in Fig. 2(a). These eigenvalues consist of two sets symmetrically distributed with respect to the ar axis. This symmetry stems from the fact that we may use either positive zonal wave numbers or negative zonal wave numbers in a spectral representation of the flow. It has been verified that each pair of corresponding modes in these two sets are structurally indistinguishable. It would therefore suffice to focus on the modes with ai > 0.

One subset of modes has distinctly higher frequency. The most unstable mode belongs to this subset and has a frequency ai ~ 2.4, which corresponds to a period of ^0.6 day. The structure of these high frequency modes reveals that they are almost exclusively associated with just one of the two jets. One example of such modes is shown in Fig. 3(a), which has (ar ,ai) = (0.0235, 2.42) and arises solely from the model Pacific jet. This mode has quite a short length scale and straddles the jet. It grows by extracting energy mostly from the meridional shear of the jet. We therefore refer to the high frequency modes as single-jet modes.

The other subset of normal modes in Fig. 2(a) has nondimensional frequencies ranging from 0.0 to ^ 0.4. The corresponding range of periods is quite large, from virtually being infinitely long to ^^ « 2 days. These modes consist of two groups. One group has lower frequency and larger growth rates. For example, one of these modes has a growth rate of @ r ~ 0.011 and a period of ai = 0.08. The corresponding e-folding time is ~11 days and the corresponding period is 12 days. Their structures suggest that these lower frequency modes tap energy from both jets. An example of them is shown in Fig. 3(b). The most intense parts of this mode are located at the exit region of the two jets where the stretching deformation is strong. Its length scale is longer than the synoptic scale. Such a mode is a counterpart of the normal modes found by Simmons et al. (1983) for an observed 300 mb January mean flow in their study of low frequency variability. We will therefore refer to those modes in this system as low-frequency modes.

Figure 3. Structure of (a) a single-jet mode with (ar,ai) = (0.0235, 2.42), (b) a low-frequency mode (ar,ai) = (0.011, 0.08) and (c) a synoptic-frequency mode with (ar, ai) = (0.009, 0.22) for the control case of Uoo = 0.2 without friction.

One example of the remaining group of modes in Fig. 2(a) is characterized by (or, oi) = (0.009,0.22) with a period of -5 days. The structure of this mode is shown in Fig. 3(c). It has synoptic temporal and spatial scales extending from one model oceanic region to the other. As such, we may think of them as the counterpart of storm track disturbances. We refer to them as synoptic-frequency modes. This particular mode has stronger intensity downstream of the model Pacific jet than downstream of the model Atlantic jet. But there is no basis to infer from this result that there will necessarily be a stronger model Pacific storm track since linear modal instability may not be pertinent.

A domain average zonal wind component, uoo, has a trivial effect on the instability of a zonally uniform shear flow because it merely advects each normal mode by that speed. However, the effect of uoo on the instability of a zonally varying shear flow is less obvious, and can be significant. We can ascertain the impact of this factor by comparing the result for uoo = 0.2 with that for uoo = 0 [Fig. 2(a) vs Fig. 2(b)]. Since the single-jet modes are largely under the local influence of one particular jet, the impact of uoo on their growth rates is relatively small. The impact of uoo on the low-frequency modes and synoptic-frequency modes is, however, much greater, as they are subject to the influence of both jets. For example, the most unstable mode for the case of uoo = 0.0 is a low-frequency mode with a growth rate of 0.036, whereas the corresponding low-frequency mode for the case of uoo = 0.2 has a growth rate of only 0.016. The presence of uoo reduces the growth rate of this mode by more than 50%. The stabilizing effect of uoo > 0 may be interpreted as a reduction of the residence time of disturbances in a two-jet flow (Pierrehumbert, 1984; Mak and Cai, 1989).

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