Applications of ‘IPV’ thinking for time-dependent dynamical processes (p. 202, Bluestein, 1993) The purpose of this discussion is to utilize ‘IPV’ thinking to explain the motions and development of synoptic-scale weather systems The basic concepts to be discussed include: • The atmospheric structure consists of a superposition of upper-level positive and negative IPV anomalies, positive and negative surface potential-temperature anomalies, along with a basic flow. The more conventional interpretation is the atmospheric structure consists of upper-level troughs and ridges, along with surface cyclones and anticyclones. Basic ideas of time-dependent dynamical processes (Continued) • Gradient-wind balance holds to a first-order approximation. We assume that the magnitudes of the anomalies (perturbations) are weak enough, so that quasi-geostrophic theory is valid: The diagnostic equation relating PV to the wind field (eq. 1.9.29) has a linear operator. Additionally, the atmosphere is statically stable so that the equation (1.9.29) is elliptic. Therefore, the total wind field that is induced by all of the PV anomalies is the sum of the wind fields induced by each anomaly separately. For typical synoptic- scale anomalies and for typical static stabilities, the induced wind fields extend throughout the depth of the troposphere. Basic ideas of time-dependent dynamical processes (Continued) • Both diabatic heating and friction are ignored, so that potential vorticity is conserved. Therefore, potential vorticity anomalies are advected on isentropic surfaces and account for local changes in the potential vorticity. Basic ideas of time-dependent dynamical processes (Continued) • Each of the potential vorticity anomaly’s induced wind field will therefore change the distribution of PV. • The consequent new distribution of PV is associated with new induced wind fields, which will change the distribution of PV, etc. The motion of upper-level troughs and ridges in the baroclinic westerlies • Consider, as in the following figure, a series of alternating positive and negative upper-level PV anomalies in the east-west direction, and inserted in a uniform westerly flow: Potential vorticity inversion may be used to understand the motions of troughs and ridges (as for Fig. 1.149: • Potential vorticity N maxima and minima, correspond, max min max min respectively to troughs and ridges N • instantaneous winds Consider a PV reference state (as for Fig. 1.150): • Consider the PV larger PV PV+2dPV contours at right N with increasing PV PV+dPV northward (owing primarily to increase PV of the Coriolis PV-dPV parameter) Consider the introduction of alternating PV anomalies (as for Fig. 1.151): • The sense of the wind field that is induced by larger PV the PV anomalies N • There will be a propagation to the left or to the west (largest PV+2dPV effect for large L anomalies + - PV+dPV + • This effect is opposed PV by the eastward advective effect East The previous figure shows the following: 1. The locations of the maximum southerly component of the induced wind are L/4 to the west of the most poleward parcel displacements (whose locations are the sites of the negative PV anomalies, or ridges). 2. The locations of the maximum northerly component of the induced wind are L/4 to the west of the most equatorward parcel displacements (whose locations are the sites of the positive PV anomalies, or troughs). Therefore: • The induced wind field advects lower PV northward just to the east of the PV maxima, and high PV southward just to the west of PV maxima. • Consequently, the wave pattern in the PV field, as well as its induced velocity field, propagates to the west. Propagation effects as a function of scale: • Large-scale PV anomalies induce relatively strong wind fields. • Small-scale PV anomalies induce relatively weak wind fields. • Consequently, the westward propagation effect is greatest for long waves, and the smallest for short waves Consider the effect of adding a basic westerly advecting wind: • This basic current acts to advect the entire wave pattern to the east (eastward). • Consequently, the effect of eastward advection in dominant in short waves. • Whereas, the effect of westward propagation is dominant in long waves. • Long waves tend to retrogade to the west, while short waves travel to the east. Movement of surface cyclones and anticyclones on level terrain (as in Fig. 1.152): Consider a reference state of potential temperature: North - + Consider that air parcels are displaced alternately poleward and equatorward within the east-west channel. Potential temperature is conserved for isentropic processes (as in Fig. 1.153) Since =0 at the surface, potential temperature changes Occur due to advection only North - - + L/4 L/4 + The previous slide shows the maximum cold advection occurs one quarter of a wavelength east of cold potential temperature anomalies, with maximum warm advection occurring one-quarter of a wavelength east of the warm potential temperature anomalies. The entire wave travels (propagates), with the cyclones and anticyclones propagates eastward. Just as with traditional quasi-geostrophic theory, surface cyclones Travel from regions of cold advection to regions of warm advection. Surface anticyclones travel from regions of warm advection to regions of cold advection. Note that we did not need to consider explicitly the effects of vertical motion, as we did when we used isobaric, quasi-geostrophic reasoning. Orographic effects on the motions of surface cyclones and anticyclones Consider a statically stable reference state in the vicinity of mountains as shown below, with no relative vorticity on a potential Temperature surface (as in Fig. 1.154) z + - x Note that cyclones and anticyclones move with higher terrain to their right, in the absence of any other effects (as in Fig. 1.155). + - N - + Mountain Range The formation of upper-level systems; baroclinic instability (pp. 208-211) • Consider a two-layer atmosphere (Fig. 156.a), in which in each layer, we have an alternating train of positive and negative PV anomalies (From Bluestein, 1993) Top layer: • PV increases to the north mostly because of increase in the Coriolis parameter to the North. • Additionally, the static stability increases to the North • Also, the temperatures decrease to the north with the horizontal temperature gradient being concentrated in the center of the channel (with accompanying strong thermal wind). Therefore, there is cyclonic shear to the North, and anticyclonic shear to the South. This relative vorticity gradient is much stronger near the tropopause, than is found in the lower troposphere. Bottom layer: • The PV gradient is oriented towards the South in the lower troposphere • The justification for this opposite sense of the gradient is the existence of warm, low-level air to the south, with increasing cyclonic shear, and higher static stability (with isentropes becoming more packed together near the ground in a warm anomaly). At the interface, assume there is no basic current: • The basic current is easterly in the lower layer • The basic current is westerly in the upper layer Because of this two layer structure: • Upper-level disturbances will propagate to the west • Lower-level disturbances will propagate to the east • Upper-level disturbances will advect to the east • Lower-level disturbances will advect to the west If the disturbances are relatively small: • The effects of advection overwhelm those effects of propagation • Therefore, disturbances in the lower layer will travel to the west • And disturbances in the upper layer will travel to the east • The disturbances in each layer will travel in opposite directions. However: • The upper-level PV anomalies induce vortices in the lower layer, affecting the distribution of PV in the lower layer • The lower-level PV anomalies induce vortices in the upper layer, affecting the distribution of PV in the upper layer With the slight westward shift with elevation of the anomalies: • The wind fields in the top layer induced by PV anomalies in the top layer and in the bottom layer result in a greater northward component of motion just west of the PV minima - and a greater southward component of motion west of the PV maxima + than would occur in the absence of the wind field induced by the lower layer. Therefore, the rate of westward propagation of upper-level PV anomalies is increased, and the net rate of eastward motion is reduced • Similarly, the sum of the wind fields in the bottom layer induced by the PV anomalies in the bottom and top layers results in a greater northward component of induced wind east of the PV maxima + and a greater southward component of motion east of the PV minima - than would occur in the absence of the wind field induced by the upper layer alone Therefore, the rate of eastward propagation is increased below, and the net rate of westward motion of the lower wavetrain is reduced. Therefore, the wavetrains try to ‘lock’ onto one another: Each prevents the other from racing off in the opposite direction Let us assume that the wavetrains were shifted more downstream, so that there is less tilt in the vertical, so that the wavetrains were more in phase with each other: • The effects of wind fields induced by lower wavetrain on upper wavetrain, plus the effects of wind induced by upper on lower wavetrains would act to increase the individual propagation speeds. • Therefore, the propagation effects would increase in each layer, so that the wavetrains would move into a configuration in which they were again tilted more westward with height. Conversely, if the wavetrains were shifted upstream so that more westward tilt was shown, the propagation effects would decrease, and advection by the basic current would restore the wavetrains to their original phase. Therefore, there is an optimal phase difference for which the two wavetrains may lock onto one another • For very short wavelengths, however, propagation could never be significant, if the basic current were strong, and the wavetrains could not lock onto one another • For very long wavelengths, propagation would always overwhelm the effects of advection, and the wavetrains would still not lock onto one another • Therefore, for a given vertical shear, the two wavetrains can lock onto one another only for a certain range of wavelengths. • If L is within range for which the wavetrains can lock onto one another, then total induced velocity pattern is L/4 out of phase with the displacement pattern • Therefore, the locations at which the PV contours are displaced farthest to the north are subjected to more northward displacements, while locations at which PV contours are displaced farthest to the south are subjected to more southward displacement. • Therefore, the waves grow in ampitude • Therefore, for a certain range of wavelengths, depending on the vertical shear, troughs and ridges will grow in amplitude if they lean westward with height • Additionally, using PV thinking, if the wavetrains lean eastward with increasing height, then for a certain range of wavelengths, the two wavetrains can lock onto one another, and decay in amplitude Effect of static stability on baroclinic instability: • For a given wavelength, the depth of the layer affected by a PV anomaly increases as the static stability decreases • Therefore, the effect of propagation is enhanced at low static stabilities, because the wind field induced by a wavetrain at one level on the other level is enhanced. • Therefore, while the induced wind field is weak for typical static stabilities and short wavelengths, it is relatively strong if the static stability is low enough • Thus, it may be possible for short wave wavetrains (which could not lock onto one another at typical static stabilities) to lock onto one another. • Furthermore, for long waves, induced winds are also stronger for lower static stabilities. • The induced winds may become so strong, that long wave wavetrains that could lock onto each other at typical static stabilities cannot do so at lower static stabilities, because the propagation effects are too strong. • Therefore, the effect of lower static stability is to reduce the scales at which baroclinic instability occurs. • We would expect to find shorter wavelengths growing in an environment of weak static stability, such as is the case over relatively warm oceans during the winter, in which small, intense cyclogenesis occurs. References: • Bluestein, H. B., 1993: Synoptic-dynamic meteorology in midlatitudes. Volume II: Observations and theory of weather systems. Oxford University Press. 594 pp. • Hoskins, B. J., M. McIntyre, and A. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877-946.