Inertial Instability The following is not meant to be a self-contained tutorial. It is meant to accompany active discussion and demonstration in the classroom. 1. Perspective from the Horizontal Equations of Motion (Momentum Equations) du dv z = g + fv (1) x x component z = g fu (2) y y component Suppose that there is no horizontal variation of the height field along the x- axis. du = fv = f dy Integrating (3) with respect to time gives (3) u fy = 0 (4) where u is the west or east wind that would develop if the air parcel is displaced a distance y, north or south. u fy = M (5) where M is defined as the absolute momentum which is conserved following the motion of the air parcel (from conservation of absolute angular momentum).
Since we started by assuming base state geostrophic flow, the air parcel in initial position will be in geostrophic balance. But it will not be so if it is displaced north or south. (Classroom demonstation). Meanwhile the undisturbed base state is still in geostrophic balance at every latitude. u g fy = M g (6) The u-component of the geostrophic wind is obtained from (2) u g = g f z y (7) divide both sides of equation (2) by f and substitute (3) into the result. Substituting (5) and (6) into (8) gives dv = f (u g u) (8) dv = f (M M g) (9) which resembles a lifted index for upright convection. The concept that will help the student understand this is embedded in parcel theory for upright convection. In parcel theory, the air parcel is taken vertically out of its base state environment that remains undisturbed. Vertical accelerations develop if the parcel s temperature at a given pressure is different than the environmental temperature at the same pressure elevation. The measure of that instability is CAPE, which is a function of the parcel s temperature excess relative to the temperature of the environmental air around it. But, also, instability can be viewed on the basis of how much
greater than air parcel s potential temperature is relative to the potential temperature of the environment. 2. Inertial Stability Inertial stability is the classic term for what is now known as symmetric stability. The student can understand this by using parcel theory, but for purely lateral motions, or at least the component of the motion that is lateral. Equation (9) provides the student with an expression that at first looks like it contains a lifted index, except that the index measures the difference between the parcel s momentum after it is deflected out of the base state current initially in geostrophic balance. We re making the argument with respect to accelerations along the y axis because mostly the jet stream and mid and upper tropospheric flow during time of peak magnitude is westerly. Examining equation (9) we note that for situations in which the right hand side returns a positive value (meaning the parcel s M is less than the geostrophic M) an air parcel deflected out of the base state initial position will continue to accelerate northward. This is the case of intertial instability. For situations in which the right hand side returns a negative value (meaning the parcel s M is greater than the geostrophic M) an air parcel deflected out of the base state initial position will be restored in an oscillation to its initial latitude. This is the case of interital stability.
3. Symmetric Instability Since mostly the cross-sections of geostrophic momentum resemble those shown in Figure 3.8, it is almost always true that lateral motions result in the case of interital stability. But suppose that a combination of inertial stability and static instability exists so that a consideration of the vector accelerations due to both ends up producing a laterally unstable wave. This is the general concept that is foundation for symmetric instability. Symmetric instability most often occurs when the slopes for the geostrophic momentum contours are nearly exactly the same as shown in Fig. 3.8, but when the slopes of the potential temeprature contours are such that a combination of lateral motion and vertical motion produces symmetric instability. In the case shown in Fig. 3.9, note that as the parcel is taken lateraly out of the base state current northward, it will experience upright instabiliy (shown as an acceleration vector pointing upward) because its potential temperature is greater than that of the environment. At the same time, although the parcel is symmetrically stable because its momentum is greater than the background geostrophic momentum, this new vertical acceleration creates a net motion such that the parcel is unstable both with respect to upright motion and symmetric motion. In equation (9), the acceleration returned would be positive. Hence, the air parcel would continue to be accelerated northward, and for each time step, the added acceleration would produce near exponential growth of the deviation.
The issue one must confront is that such a combination of potential temperature lapse rates and horizontal variation of geostrophic momentum as shown in Fig. 3.9 is rare. Yet, observations show the development of convective motions in bands do occur. This leads us to a discussion of conditional symmetric instability (CSI).