1. Rossby and Ertel 2. Conservation and invertibility - a useful combination 2.1 Action at a distance

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1 Met Office College - Course Notes Potential vorticity Contents 1. Rossby and Ertel 2. Conservation and invertibility - a useful combination 2.1 Action at a distance 3. Conceptual models of development using PV 3.1 Cyclonic development 4. Ways of presenting and using PV 4.1 Water vapour imagery and PV 5. Summary 5.1 References Crown Copyright. Permission to quote from this document must be obtained from The Principal, Met Office College, Shinfield Park, Reading, RG2 9AU Page 1 of 15 Last saved date: 17 December 2007 FILE: MS-TRAIN-COLLEGE-WORK-J:\POTENTIAL_VORTICITY.DOC

2 Met Office College 1. Rossby and Ertel C.-G. Rossby was an early pioneer in the use of vorticity as a way of looking at atmospheric flow. In the 1930s he took a key step in realising that the vertical component of absolute vorticity, ζ a, is the most important for large-scale flow in the atmosphere. He further showed that ( f + ζ θ )/ Δ = constant, where Δ is an expression for the thickness of a layer equal to δp/g, ζ θ is the relative 'isentropic vorticity' given by ( v/ x) θ ( u/ y) θ, i.e. vorticity computed from winds along an isentropic surface. Ertel (1942), working independently, derived the more general expression 1 P = ζ θ ρ a where ζ a is the three dimensional absolute vorticity vector. P is known as 'Ertel's potential vorticity', or 'potential vorticity' for short, and is often abbreviated to PV. For adiabatic, frictionless motion P is conserved following the motion. There are no further qualifications to this statement, such as those attached to quasi-geostrophic theory; it applies to fully three-dimensional, non-hydrostatic motion. 2. Conservation and invertibility - a useful combination PV can be looked upon as the product of the static stability ( θ/ z) and the 'quasi-vertical' (i.e. normal to the isentropes) component of absolute vorticity, ζ a, and can be closely approximated by 1 P a ρ ζ θ z It is easily verified using a tephigram that the closer together the isentropic surfaces are in the vertical (i.e. the larger θ/ z is) the greater the stability is, so as the stability decreases, the isentropes are pulled apart. In addition, it can be seen that the warmer an air mass is, the lower a given isentropic surface is. Since under the adiabatic assumption we can think of parcels of air as having their tops and bottoms defined by isentropic surfaces, we can see that vorticity increases through the stretching effect, whereas increasing static stability causes squashing and a compensating decrease in vorticity. Page 2 of 15

3 Potential Vorticity θ + Δθ cold θ Figure 1 A parcel of air flowing from a cold, statically stable region to a warm, statically unstable region acquiring positive relative vorticity through PV conservation. Taking values of PV based on a motionless atmosphere with a temperature structure given by the Standard Atmosphere, PV increases with poleward progression due to latitudinal variation in f, and increases somewhat with height due to variation in ρ. Much the most marked change, though, occurs across the tropopause due to the rapid increase in static stability encountered on penetrating the stratosphere. Tropospheric values are generally less than around 1.5 PV units (10 m s K kg ) and jump to values typically in excess of 4 PV units. The high static stability of the stratosphere makes it a reservoir of high PV, and the fact that gradients of PV are greatest around the tropopause make anomalies at this level very important. Given a wind field, it is possible to retrieve the geopotential height distribution. The inversion process requires certain assumptions, and it needs information on boundary conditions. The simplest assumption would be that of geostrophic balance, though something more realistic such as gradient wind balance could be used; boundary conditions are necessary since the wind only gives information on pressure gradients, so geopotential height values at the edges of the field are needed together with the gradients to give actual geopotential values in the Figure 2 Cross section of isentropes (thin lines, increasing in value upwards) and tropopause. or PV=2 surface, (thick line) associated with idealised upper level PV anomalies, cyclonic (left) and anticyclonic (right). Shown also are anomaly positions (shaded) and isotachs (solid into page, dashed out of page). Based on Thorpe (1985). Page 3 of 15

4 Met Office College interior. In a similar, but more powerful, way, PV can be said to be invertible. By this is meant that, using specified balance conditions (e.g. geostrophic balance) and boundary conditions, all the other dynamic fields may be obtained (e.g. temperature, geopotential, wind, vertical velocity). The two virtues of invertibility and conservation make the PV a powerful tool in understanding atmospheric flow and development. 2.1 Action at a distance In the inversion of a field of vorticity to gain the geopotential distribution, point values of geopotential do not depend simply on the local value of vorticity. Since vorticity can be looked upon as the curvature of the height field, it affects the geopotential height of the region around it. An extension to this principle to three dimensions is seen in the omega equation; omega depends not just on the local quasigeostrophic forcing but on that above and below, as well as laterally at a distance. In a similar way, PV can be shown to affect dynamic fields remotely; the mathematical relationship between PV and the derived dynamical fields is similar to that of an electric charge and the field of electrical potential, there being a 1/r fall-off in influence with distance where the vertical component of r is scaled so that N r = Δx + Δy + Δ z 2 f In the above expression N 2 is the buoyancy frequency, being a measure of static stability equal to g θ. So PV anomalies exert maximum θ z influence at their location, but also induce effects in the atmosphere remote from their position, the vertical distance being greater with more unstable lapse rates. An upper level positive PV anomaly can induce a cyclonic circulation down through the atmosphere, perhaps to the surface. The nature of the invertibility principle is such that small PV features have a weak effect on their surroundings compared with large scale features of the same strength. 2 2 Figure 2 shows schematically isentropes and tropopause associated with positive and negative upper PV anomalies. From the potential temperature distribution, it can be seen that the positive anomaly is accompanied by relatively cold air in the upper troposphere (since isentropes bow upwards), but relatively warm air in the lower stratosphere, these being the reverse in the case of the negative anomaly. It can also be seen that thermal wind balance requires that the flow associated with the positive anomaly become more cyclonic with height until the tropopause is reached (since the air is cold), whilst that associated with the negative anomaly becomes more anticyclonic with height. Page 4 of 15

5 Potential Vorticity In the PV anomalies, vorticity and stability are contributing to the anomalous PV, whereas away from the anomaly they tend to counteract each other. Thus with the positive PV anomaly, centred in the lower stratosphere, the stability is high and the vorticity is high. Below this in the mid to upper troposphere, vorticity is still appreciably high, so stability has to be low, evidenced by the large vertical spacing between isentropes. 3. Conceptual models of development using PV Consider a broad airstream, free of significant PV or surface θ anomalies, flowing under a stationary upper PV anomaly, such as the cyclonic anomaly shown in Figure 2. The invertibility principle tells us that the induced cyclonic vortex extending down towards the lower troposphere will stay in place, despite the tendency of the pre-existing flow to advect it downstream. Light can be cast on how it is that the advection is cancelled by imagining a column of air whose upper and lower surfaces are defined by the lowest two isentropes in the figure. It can be seen that stretching must occur as the air passes underneath the anomaly and the isentropic surfaces configure themselves in response. Vortex stretching creates cyclonic vorticity, cancelling out the negative vorticity advection, and also results in adiabatic cooling, offsetting the warm advection. Downstream cancellation of advection occurs once again with vortex squashing and descent of air flowing out from under the PV anomaly. It is more realistic to change reference frame and think of the upper anomaly as being advected by a strong upper level flow and overtaking lower level air. This has the effect of drawing the isentropes upwards and apart, leading to ascent, cooling and cyclonic spin-up of lower level air. A helpful analogy of a positive PV anomaly as a magnet, and the isentropic surfaces flexible magnetic sheets, which are attracted towards a positive anomaly and repelled by a negative anomaly. The closer they are to the anomalies, the stronger the attractive or repulsive force is. In addition to vertical motion caused by bodily ascent as the isentropic surfaces bulge upwards underneath the moving PV anomaly, air flowing relative to the sloping isentropes causes vertical motion, referred to as isentropic upgliding. With the usual latitudinal temperature gradient and induced cyclonic flow, the movement of both air relative to the theta surfaces and the theta surfaces themselves leads to ascent ahead of an eastward moving disturbance, and descent behind. This is shown in Figure 3, along with the case of an eastwards moving anticyclonic anomaly. Notice that is an anomaly, or locally different value, of PV that is required, since if the whole of the upper troposphere were equally 'magnetic', there would be no bulging upwards ordownwards by the isentropic surfaces. Page 5 of 15

6 Met Office College 10 km θ surface 0 km Figure 3 Topography of isentropic surfaces associated with eastward moving upper PV anomalies; the dark lines mark the intersection of the tropopause with the 10 km level, separating air of PV > 2 units to the north from air of PV < 2 units to the south. Shown also are system relative isentropic up- and downgliding (arrows directed along θ surfaces) and vertical motion due to induced bulging of the isentropic surface (vertical arrows) when viewed from earth relative perspective. Dashed line shows intersections of cross section shown in Figure Cyclonic development Figure 4, taken from Hoskins et al. (1985), shows a useful schematic view of cyclogenesis. An upper PV anomaly arrives over a zone of baroclinicity at the surface. + + cold warm Figure 4 Schematic picture of cyclogenesis associated with the arrival of an upper PV anomoly over a low level baroclinic zone. The dark line tropopause, thin lines surface isotherms. (After Hoskins et al. 1985) The induced cyclonic circulation at low levels causes a warm anomaly in the surface temperature field just ahead of the upper feature. This warm anomaly brings about its own cyclonic circulation, which reinforces that being induced from above. In addition, a circulation generated at upper levels by the warm anomaly surface feature may reinforce the upper PV Page 6 of 15

7 Potential Vorticity anomaly by advecting high values of PV equatorwards. While the upper and lower systems remain in favourably relative position, with the lower feature just ahead of the upper, they feed positively off each other's development. The tendency for the upper feature to be advected ahead of the surface development may be counteracted by the induced upper, equatorward advection of high PV occurring slightly to the rear of the upper feature. vertical gradient of θ decreasing -ve PV anomaly θ + Δθ θ z = 10 km z = 5 km vertical gradient of θ increasing +ve PV anomaly z = 0 Figure 5 Latent heat release warming the mid-troposphere and redistributing PV. Figure 5 shows a column of air in which ascent of moist air is leading to condensation and latent heat release in mid troposphere. The warming at this level lowers the θ surface locally, increasing θ in the lower troposphere and decreasing it in the upper troposphere. This creates a positive PV anomaly low down and a negative one aloft. Total PV in the column is unchanged, but the PV has been concentrated at low levels, intensifying cyclonic vorticity here whilst the flow at upper levels becomes more anticyclonic. This process can be important in intensifying the low-level circulation associated with a developing depression, particularly where there is warm, low-level air of high wetbulb potential temperature associated with the system. PVA - cold warm Figure 6 A pre-existing low level cyclonic disturbance leads through induced ascent and latent heat release to an upper negative PV anomaly with ascent on its upstream side. Page 7 of 15

8 Met Office College Mansfield (1994) noted that, on occasion, development appears to be initiated from below, creating an upper PV anomaly in an originally straight gradient (Figure. 6). He proposed a possible sequence of events in which ascent associated with a surface depression leads to latent heat release and thence to a negative PV anomaly aloft. Ascent on the upwind side of the negative anomaly (as shown in Figure 3) could then reinforce the surface depression through vortex stretching. 4. Ways of presenting and using PV Rather than looking at different levels of the atmosphere according to their height, geopotential or pressure, it is more natural to look at fields on an isentropic surface, since air is constrained to move along such surfaces while the flow is adiabatic (though it must be remembered that the surface itself can also move up and down and become distorted or even folded). A chart of PV distribution on an isentropic surface would show a PV anomaly by a localised region of PV of significantly different values from those around it, provided the θ surface chosen intersected it. Equally, since PV is conserved, it is natural to look along a PV surface. Reference to Figure 2 shows that a chart constructed on the PV=2 surface, essentially the dynamic tropopause surface, would show a high PV anomaly by low values of θ (though it should not be called a cold anomaly since it is probable that the tropopause temperature is locally higher here) and a low PV anomaly by high values of θ. If one is interested in upper tropospheric forcing, the PV=2 level is arguably more useful, since it focuses attention on the area of interest, whereas an isentropic surface may not intersect the tropopause in the right region to show fully the structure of a PV anomaly. Whether it is θ on a PV surface or PV on a θ surface that is being used, such representations of atmospheric structure have the advantage over, say, geopotential or temperature on a pressure surface, because of the conservation properties of θ and PV. Features are better resolved, and can be followed in time with more certainty, clarifying our perception of important processes such as cut-off low development, blocking and cyclogenesis. A chart that has been found to be of particular use is one that shows variation of θ over a PV=2 surface together with 850 hpa wet-bulb potential temperature. This way of looking at development can give a lucid picture of the type of cyclogenesis process depicted in Figure 4. In allowing information for more than one level to be depicted, it emphasises the importance of the relative positions of surface and upper features and permits more accurate assessment of the likelihood of constructive interaction. By way of illustration, Figures. 7 and 8 show fields of theta on PV=2, 900 hpa θ w, and mean sea level pressure for two situations, 24 hours apart. In Figure 7, a low θ anomaly (in other words a high PV anomaly) has engaged a low level baroclinic zone to the west of Ireland, in the manner Page 8 of 15

9 Potential Vorticity of Figure 4, and the associated depression is deepening. Further along the frontal zone to the west there are two more low pressure areas of Page 9 of 15

10 Met Office College Figure 7 Theta on PV=2 surface, background shades, coldest (i.e. highest PV) to the north; mean sea level pressure, black contours, every 4 hpa; 900 hpa θ w, white contours, every 2 C. 0000UTC 2 November Figure 8 As above, but for 0000UTC 3 November Page 10 of 15

11 Potential Vorticity similar depth to each other, one in mid Atlantic and the other on the extreme west of the chart. It can be seen immediately that the shallow depression in mid Atlantic is not likely to develop, being far from any portion of the high PV reservoir. However, high PV to the south and west of Newfoundland is coming over the baroclinic zone associated with the far western centre, showing that development is likely. A cutoff feature of high PV which has evidently previously separated itself from the high PV polar air is over the western Mediterranean. Figure 8 confirms that, 24 hours later, the western depression has deepened considerably, by about 20 hpa. In the mean time, the shallow feature in the central Atlantic has filled slightly, by about 4 hpa. The depression that was to the west of Ireland has tracked into Norway and deepened by 16 hpa or so. However, most of this deepening took place in the first 12 hours, and it can be seen that the occlusion process has caused the upper PV anomaly to lose contact with strong baroclinicity at the surface, limiting development of the feature. This is the typical situation, the growth of the average warm sector depression being limited by its removal from the warm air and the main baroclinic zone. The feature in the far west is less typical, baroclinicity just to the north of the depression being maintained by shearing frontogenesis and a continuing supply of polar air drawn down the western flank of the low. At the same time, the upper PV anomaly has not yet lost contact with the polar reservoir. There is a tendency for such features to cut themselves off by advection of low PV air round their eastern then northern peripheries in the cyclonic flow they induce, as has probably happened some time previously to the feature over the Mediterranean (now in the extreme southeastern corner of the chart). However, the PV anomaly causing strong surface development in the west has kept a link with the source of high PV, though this is increasingly tenuous. The trough disruption/cut-off process is notoriously finely balanced, being one of the more frequent causes of poor model forecasts. The PV view of the process is more distinct than that afforded by the usual fields, the critical factor being the amount of PV drawn into the feature before it becomes separated from the source. 4.1 Water vapour imagery and PV Of course, it is the case that if the model is realistic in its simulation of atmospheric development, it should produce realistic PV fields (though the interpolation onto θ or PV levels may introduce inaccuracies, especially at high levels, where model layers are thickest). Looking at model output of PV may increase our understanding of what is forecast to happen, but this in itself will not allow us to correct or adjust the model forecast. What is needed is observational evidence of the PV distribution to compare with the model predictions or initial fields. It turns out that water vapour imagery can provide just this type of evidence. Page 11 of 15

12 Met Office College Showing data measured by radiometers sensitive to radiation in the μm band (ESA, 1987), Meteosat water vapour images provide information, albeit indirectly, of water vapour content at levels normally associated with the upper troposphere. Coding of images elements follows the convention of standard infra-red imagery, with low brightness temperatures appearing as light shades, and high brightness temperatures looking dark. Since the spectral response is in one of the bands in which water vapour absorbs and emits well, high humidity in the upper troposphere will result in low measured brightness temperatures. The response in humid air has a contribution from a broad spread of heights from around 700 to 300 hpa, with a peak at about 400 hpa, so, to an extent, the grey-scale of an image feature is correlated with humidity at around this height, dark indicating dry and light indicating moist. However, this oversimplifies the picture; light tones can either be associated with high moisture content aloft, or with dry air overlying a very cold surface. Dark tones can be associated with dryness aloft or with a very warm atmosphere that is not necessarily dry. The link with PV distribution comes about through the twin properties of stratospheric air high PV and low humidity. Low θ on the PV=2 surface suggests a low tropopause, and therefore that a significant portion of the depth in which water vapour contributes most to measured radiances is essentially stratospheric air, and thus appears dark. Major cloud masses produced by dynamic ascent appear white in water vapour imagery (liquid and ice cloud in the upper troposphere provide at least as strong a white response as water vapour). As shown earlier, the latent heat release accompanying the formation of such cloud leads to low PV aloft. Page 12 of 15

13 Potential Vorticity Figure 9 Water vapour image for 0300 UTC 3 March 1994 overlain with isopleths of forecast height of PV=2 surface at 50 dam intervals. Figure 10 As above, but 0600 UTC 9 January (There are missing PV data over S. France.) Page 13 of 15

14 Met Office College Figures. 9 and 10 show examples of water vapour imagery overlain with PV data. This time, the PV information presented is the height of the PV=2 level in decametres. Reference to Figure 2 shows that positive upper PV anomalies are associated with depressions in the PV=2 surface, whilst negative upper PV anomalies are accompanied by domes in this surface. Although the height of the PV=2 surface is not strictly conserved, Mansfield found it a useful field for presentation in operational use, and points out (1994) that the at-a-distance induction effect means that the height of the PV surface is related to the extent of direct influence of the upper anomaly on surface developments. Figures. 9 and 10 show the main PV=2 height gradient to lie along the cloud edge/ moist boundary, being associated with the front at upper levels. Frontal cloud forms mostly within the air mass of sub-tropical origin, and would therefore be expected to be associated with air of low PV. In addition, latent heat release concentrates low PV aloft, shown by the marked ridges in the topography of the PV=2 surface where the characteristic cyclonic comma clouds are found. Mansfield discovered that gradient of PV=2 height tends to be concentrated on the warm side of dark strips associated with the bases of upper troughs, such as is seen in Figure 10 at 35 W, whereas on the forward sides of troughs, the gradient extends along the cloud edge. The dry strips may be due to descent in the cold air limb of the thermally direct circulation associated with frontogenesis, or with more active descent sometimes found when the flow cuts across the jet as marked by the isotachs from its cold side, giving strong upper tropospheric negative vorticity advection to the cold side of the jet, and thus descent. Intrusions of dry, stratospheric air of high PV have been associated for some time with rapid surface cyclogenesis and these show well in both the water vapour imagery and PV fields as slots of dry/high PV air tucking in behind and cutting through the cloud head/pv ridge associated with the depression. PV fields overlain with the imagery can give evidence to verify or cast doubt upon the forecast/analysed positions and strength of the intrusions of high PV, and perhaps modify the forecast fields in the light of apparent errors. Page 14 of 15

15 Potential Vorticity 5. Summary PV offers a useful alternative to conventional fields showing temperature, geopotential etc. Whereas features such as upper troughs in the geopotential tend to develop and decay, making them difficult to follow with time, PV is changed only slowly by diabatic effects; to a close approximation development can be ignored as advective changes dominate. Ascent can be identified as arising from potential vorticity advection, which can be abbreviated to PVA (strictly speaking you need an increase in cyclonic PV advection with height) plus flow up sloping isentropic surfaces, or isentropic upgliding. A PV anomaly with strong flow across it will induce mid-tropospheric ascent and falling pressure at the surface through PVA. If it engages a surface baroclinic zone isentropic upgliding comes into play and positive feedback is possible, with the developing surface and upper anomalies reinforcing each other. Development is halted by either removal of the upper anomaly from the low-level baroclinicity (e.g. by the occlusion process) or by removal of the upper PV anomaly from the polar air mass reservoir. Positive PV can be shown by either low potential temperature on PV=2, high PV on an isentropic surface or a low PV=2 surface. 5.1 References Hoskins, B.J., MacIntyre, M.E. and Robertson, A.W., 1985: On the Use and Significance of Isentropic Potential Vorticity Maps. Quart. J.R. Met. Soc. 111: Mansfield, D., 1994: The Use of Potential Vorticity in Forecasting Cyclones. Paper presented at Bergen Conference, July Thorpe, A.J., 1985: Diagnosis of balanced vortex structure using potential vorticity. J. Atmos. Sci., 42: Page 15 of 15