Fixed Rossby Waves: Quasigeostrophic Explanations and Conservation of Potential Vorticity

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April Cooper
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1 Fixed Rossby Waves: Quasigeostrophic Explanations and Conservation of Potential Vorticity 1. Observed Planetary Wave Patterns After upper air observations became routine, it became easy to produce contour plots of upper air patterns. Long term mean contour plots showed that in the Northern Hemisphere winter, three major long wave troughs/ridges appeared on average in the middle troposphere (Fig. 1), whereas the same patterns were not observed in the Southern Hemisphere (Fig. 2). Evidently, there are factors at work in the Northern Hemisphere not active in the Southern Hemisphere. Figure 1: Average January 500 mb Heights, , Northern Hemisphere

2 Figure 2: Average January 500 mb Heights, , Southern Hemisphere In addition, it was noted that the three mean long waves observed in the Northern Hemisphere weaken with height (Fig. 3). In other words, their amplitudes dampen with height until the flow is nearly zonal above 200 mb or so. 2

3 Figure 3a: Average January 200 mb Heights, , Northern Hemisphere Our understanding of these phenomena came out of the work of Carl Rossby. We have already looked at some of the background on this in previous handouts. 2. Implications of Conservation of Absolute Circulation: Conservation of Potential Vorticity Last semester, we derived the simplified vorticity equation from the Principle of Conservation of Absolute Circulation. Recall that vorticity is circulation per unit area, or ζ a = C a A (1) 3

Figure 3b: Average January 700 mb Heights, 1975-2006, Northern Hemisphere If there are no solenoids, and no friction,

4 Figure 3b: Average January 700 mb Heights, , Northern Hemisphere If there are no solenoids, and no friction, assume that absolute circulation is conserved. Thus, (1) becomes d(ζ a A) dt = dc a dt = 0 (2) or ζ a A = Constant (3) 4

5 Equation (3) can be used to derive the Simplified Vorticity Equation. Then by assumption of non-divergence, the equation reduces to Conservation of Absolute Vorticity, which can then be used to understand Rossby Waves. We now use Equation (3) (which assumes barotropy no variation in density) to develop another important conservation principle. The principle of conservation of mass states that the volume of an air column (in which density does not vary) is constant, or Or Volume = Area of Air Column X Depth = Constant V = A z = k (4a) A= k/ z (4b) Put Equation (4b) into (3) to obtain a relationship between the absolute vorticity and the depth of the air column by embedding both constants on the right side of the equation. ζ a z =ζ Pot = k (5) Equation (5) states that the ratio of the absolute vorticity to the depth of the air column (in a barotropic system) is constant. This ratio is known as Potential Vorticity and the fact that, given the constraints used in obtaining the equation, the ratio does not change for a given air parcel is known as Conservation of Potential Vorticity. Since the derivation of Equation (5) was predicated on Conservation of Absolute Circulation, it is important to note that Equation (5) will help you to understand characteristics of only the features in the large scale flow that are barotropic or equivalent barotropic (such as Rossby Waves), and not baroclinic waves. Since absolute circulation tends to be conserved, so too does Potential Vorticity tend to be conserved. 5

6 With that in mind, consider a zonal jet stream in which there is no northward variation in u (no horizontal speed shear) approaching a mountain range (see Fig. 4). Figure 4: Plan and Cross-section Views showing differential development of leeside trough due to Conservation of Potential Vorticity As the depth of the air column decreases, Equation (5) states that its absolute vorticity must also decrease. Since the case considered here is at the core of the jet stream in which there is no horizontal shear, this decrease in absolute vorticity must show up as either anticyclonic curvature relative vorticity and/or a decrease in latitude (f). In either case, a southward turn will develop (a ridge). Downwind of the mountain range the opposite occurs, leading to troughs down wind of major mountain ranges. In order for the topography to have this effect, the mountain range must have a width and depth of synoptic-scale dimensions and be oriented at right 6

7 angles to the flow. That is to say, 1000 km or so in diameter and a good fraction of the troposphere in depth. The complex of the highlands of western North America and eastern Asia fulfill these criteria. Studies have shown that the level above which the underlying topography has no effect on flow patterns (the so-called nodal surface) is around the 200 mb level, or the top of the troposphere. Note that, in this model, the air stream approaches its original latitude downwind of the mountains at a 45 degree angle. This means that it will overshoot its original latitude and produce a Rossby Wave trough. Downstream from the trough axis, the wind will again be approaching its original latitude, again at a 45 degree angle, and will again overshoot. Thus this process should, in theory, produce a train of Rossby Waves. In reality, such an infinite train of waves will not occur because there are torques, including friction, that will act to dampen the waves fairly quickly. But it is possible that the Rossby Wave trough that appears, on average, over eastern Europe is really part of a train of waves stimulated by the western highlands of North America. 3. Conservation of Isentropic Potential Vorticity (IPV) Conversion of (5) into isentropic coordinates yields gζ ( aθ θ ) = g ζ + f p θ ( ) θ p ( ) =ζ ErtlsPotVort = k (6) The quantity to the left of the equals sign is known as Ertl s Potential Vorticity or Isentropic Potential Vorticity (IPV). The factor theta / p is related to the depth of the air column as shown in Fig. 5 below. Recall that since potential temperature increases with height (except in the unusual circumstance of absolute instability), the term, theta/ p is negative. Ertl s Potential vorticity is conserved for frictionless, adiabatic (isentropic) flow and is generally positive (since absolute vorticity is almost always positive). 7

In Fig. 5, consider the bounds of the air column at 200 mb and at the surface. Since the motion is adiabatic, the two streamlines shown are also isentropes.

8 In Fig. 5, consider the bounds of the air column at 200 mb and at the surface. Since the motion is adiabatic, the two streamlines shown are also isentropes. Note that on the windward side of the mountain the spacing between the isentropes gets smaller_hence theta / p gets more largely negative. In order for the product in (6) to remain constant, the absolute vorticity must decrease. In the core of the jet (in which there is no shear relative vorticity), this can only be accomplished if the Coriolis parameter experienced by the air column is decreased; i.e, the air column turns southward. This produces a ridge on the upstream side of the mountain. Figure 5: Cross-section s showing effect of topography on depth of air columns and vertical gradient of potential temperature for westerly crossmountain flow. On the downstream side of the mountain, the vertical gradient of potential temperature is decreased, and, therefore, the absolute vorticity of the air column must increase, to keep the product in (6) constant. Thus, a trough that weakens with height is found on the eastern side of major mountain ranges, in these circumstances. The factor - theta / p is also directly related to the static stability parameter. In the diagram above vertical shrinking and horizontal divergence will lead to an increase in - theta / p. This is consistent with what we learned last semester: that the more stable the atmosphere, the more closely spaced the isentropes in vertical cross-section. From the definition of PV above, one can see that the units are Kkg -1 m 2 s -1. For the purposes of contouring on maps, it is convenient to define one PV unit as 1 PVU= 10-6 Kkg -1 m 2 s -1. Since IPV is conserved in adiabatic frictionless flow, we will see that it can be used to trace large scale motions in the atmosphere. 8

9 4. Implications of the QG-omega Equation The simplified Equation of Continuity (synoptic-scaling) is ω p = V = DIV h (1) which says that layer horizontal divergence is related to the vertical motion field. Under weak synoptic forcing (no or weak differential vorticity advection and no or weak temperature advection, and with no frictional and/or diurnal heating effects), there are no forcing terms on the right hand side of the equation. 2 ω + f 2 0 σ or 2 ω = f 2 0 σ 2 ω 2 p 2 = 0 2 ω 2 p 2 (2) The QG omega equation can be rewritten as given in (3) [with the substitution of equation (1) into the right hand side of (2)]. 2 ω = f 2 0 σ 2 ω 2 p 2 ( DIV h) p (3) Inverting the Laplacian on the left gives the approximate equation 9

10 ω ( DIVh ) p (4) which says that in the layer in which omega is positive (subsidence), then the derivative on the right returns a positive value. In the case of leeside sinking, there is no omega at the nodal surface, and there is great sinking at the surface. If the nodal surface is at 200 mb, a finite difference version of the far right hand term says that divergence must become more positive with height (more negative with decreasing height) if downward motion is occurring. Since subsidence associated with topography is zero at the nodal surface and maximum at the ground, this implies that there is no divergence at the nodal surface. For divergence to increase with height from the ground, this implies that convergence occurs at the ground on the lee side of mountain ranges. 5. Leeside Low Development and Anchored Rossby Waves The effects of the phenomena described in sections 2, 3 and 4 above manifest themselves east of major mountain ranges as the development of so-called "leeside lows" or troughs at the surface that weaken with height (Fig. 6). Figure 6. GFS forecast for 12 UTC 16 March 2009, (left to right) 300 mb, 500 mb, and Mean Sealevel. At the latitude and longitude of the Rocky Mountains near zonal flow is occurring at 300 mb, a deformation to the streamlines evident at 500 mb, and a profoundly deep leeside low is found at the surface. 10

11 These have warm core and have, essentially, the structure of a thermal low. These lows are not baroclinic and are not associated with pre-existing temperature or vorticity advection. However, once such a low is in place (often in the spring and early summer east of the Rockies, for example) significant moisture advection can occur east of it, and, actually, a synoptic scale "warm front" like feature can develop as well. At the larger scale, this implies that long wave troughs should occur east of major mountain ranges. Again, "major" is defined as a mountain range having a depth through at least the middle troposphere, and a width at least as large as the width of the jet. Thus, stationary waves as shown in Figs. 1 and 3 can be explained on the basis of (4). 11

Using simplified vorticity equation,* by assumption 1 above: *Metr 430 handout on Circulation and Vorticity. Equations (4) and (5) on that handout

Using simplified vorticity equation,* by assumption 1 above: *Metr 430 handout on Circulation and Vorticity. Equations (4) and (5) on that handout Rossby Wave Equation A. Assumptions 1. Non-divergence 2. Initially, zonal flow, or nearly zonal flow in which u>>v>>w. 3. Initial westerly wind is geostrophic and does not vary along the x-axis and equations