Hi all, As part of some recent work I've done on the 2010 Nashville Flood, I was able to construct a solver for the Sawyer-Eliassen Equation which can be easily applied to the storm currently moving up the East Coast. Attached you'll find two horizontal plots from 12Z this morning (Nov. 27). The first shows geopotential height and potential temperature at 850mb. From this plot, it is evident that strong baroclinicity is associated with the cold front. Consequently, I decided to take a cross section and solve for the Sawyer-Eliassen Equation along a line from eastern Wisconsin, through southern Virginia, and into the Atlantic Ocean. As further background, the second plot below shows that at 250 mb, a geostrophic jet maximum was located along the Florida-Georgia border with the chosen cross section located in a region of geostrophic deceleration. Furthermore, the cross section is taken through an area of geostrophic cold air advection at 500mb, which will be important for determining the magnitude of the shearing term of the Sawyer-Eliassen circulation. With that said, I went ahead and solved for the secondary circulations associated with the various forcing terms in the Sawyer-Eliassen Equation. To give a brief intro on how to interpret the plots (and please excuse their messy nature), the x-axis of the plot is directed towards colder air (e.g. towards -85 degrees longitude), the black contours (both solid and dashed) are the Sawyer-Eliassen streamfunction, red contours are potential temperature, green are the geostrophic isotachs beginning at 30 m/s (the cross section is oriented such that the wind is blowing out of the page). The flow associated with the Sawyer-Eliassen streamfunction is defined such that larger values of the streamfunction are to the left of the flow. The third plot shows the full forcing (both geostrophic and diabatic) solution to the Sawyer-Eliassen Equation. For the most part, the solution depicts a thermally indirect circulation that is shifted towards the anticyclonic shear side of the jet, such that ascent (dashed blue contours) is positioned directly beneath the jet core. Furthermore, this ascent is located directly along the warm side of the cold front at low-levels. At 12Z this location of ascent from the solution appears to agree well with a linear band of precipitation that stretched through central and eastern Virginia. The utility of the Sawyer-Eliassen Equation, though, is best realized in its ability to separate the contributions by the separate forcing terms. The fourth plot shows the circulation associated with just the geostrophic forcing function (recall that this function is a sum of both a shearing and stretching deformation term). In this plot, it is evident that the circulation is still thermally indirect in nature, with the strongest ascent observed in the middle to upper troposphere in the vicinity of the jet core. The ascent that was present at lower tropospheric levels is absent in this solution, indicating that the diabatic component may be a significant contributor to that portion of the vertical motion field. The fifth plot illustrates this, as the presence of latent heating (blue contours), drive the formation of dipole circulations centered around -75 degrees longitude. The combination of the circulations would act to drive ascent right at the leading edge of the frontal boundary. One can then imagine these vertical motions coupling to those in the middle and upper troposphere to promote a column of tropospheric-deep ascent. The geostrophic circulation can be broken down even further into the portions driven specifically by the individual shearing and stretching deformation terms. The sixth plot shows that the shearing deformation forcing acts to drive a thermally indirect circulation that straddles the jet core, with ascent on the poleward side of the jet. Here we can also see that the presence of geostrophic cold air advection (blue contours) in cyclonic shear is the predominant forcing for this circulation at mid-levels (the shearing term is the product of the along-front temperature gradient and the across-front shear). The seventh (and final) plot shows that the stretching term acts to drive a thermally direct circulation that is situated about the strong region of baroclinicity associated with the cold front. The primary driver of this circulation is the combination of confluence (blue contours indicate across-front geostrophic winds; solid = wind is going in the positive x-direction; dashed = wind is going in the negative x-direction) in the presence of a strong temperature gradient. Interestingly, considering that these two circulations sum up to the total geostrophic forcing, it appears that the stretching circulation counteracts the poleward-most portion of the shearing circulation. Given that the shearing circulation appears to be the most dominant (in terms of the absolute magnitude of the streamfunction and its spatial distribution), the sum of the two circulations results in a "shift" of the geostrophic circulation towards the anticyclonic shear side of the jet. These are just a few observations I picked up from examining these plots and excuse me for being a little longwinded. I'd definitely be interested to hear what other comments or observations anyone else may have on these plots! Sincerely, Andrew Winters Graduate Research Assistant - UW-Madison ------------------------------------ ------------------------------------ ------------------------------------ Hi Andrew, Many thanks for posting your S-E diagnostics from 1200 UTC today. I have attached the corresponding NCAR/RAL 250 hPa analysis so that everyone can see the difference between the 250 hPa derived geostrophic and analyzed observed 250 hPa winds. Your fourth plot (sum of the geostrophic shearing and deformation terms) nicely shows that midlevel ascent is shifted to the anticyclonic shear side of the jet, consistent with the observed along-flow midlevel cold-air advection as you indicated. Your cross-jet circulation calculations are remarkably similar to the idealized and real-data two- and three-dimensional circulation diagnostics presented by Shapiro (1981), Keyser and Pecnick (1985), and Keyser et al. (1992). Seeing these derived cross-jet S-E circulations in real time is a nice treat. I would also add, based on a comparison of your derived 250 hPa geostrophic wind field and the attached 250 hPa wind analysis, that the shift of the midlevel ascent maximum to the anticyclonic shear side of the jet is consistent with the observed anafront structure of today's storm. Mel, Gary, Brian and Fred, Overall, today's storm appears to have underperformed (relative to the hype). As Rich Grumm always likes to say, the best coastal storms always have large negative standardized low-level u-wind anomalies (strong easterly wind components). Large negative u-wind anomalies have been missing in action in this storm as will be the case for a jet-driven ana front event. This storm is also a good example of why the phasing of northern and southern stream disturbances alone is insufficient to guarantee the occurrence of a major wrapped-up coastal storm. The maintenance of an anafront structure, at least while the elongated surface cyclone was situated south of 40 N, is consistent with Andrew Winter's S-E diagnostics, and is also consistent with the absence of strong negative u-wind anomalies. Phasing without cyclonic wave breaking (CWB) with the northern disturbance makes it very difficult to generate strong negative u-wind anomalies. CWB is forecast to develop with the current storm, but not until near 1200 UTC 28 Nov when the surface storm is moving poleward of 50 N. So, was the delayed onset of CWB one factor that contributed to the relatively large ensemble uncertainty associated with this storm? To illustrate this large uncertainty I have attached Kyle MacRitchie's map of GEFS 500 hPa geopotential height ensemble mean and spread for 72 h forecasts verifying 1200 UTC 27 Nov. Note the large GEFS spread associated with the northern and southern disturbances. With regard to the point about coastal fronts raised by Fred, this case is a good example of how we can sometimes be prisoners of our own terminology. In this case, the inland baroclinic zone that was approximately parallel to the coast is quite deep and appears to be primarily connected to dynamical processes associated with a well-defined 150 kt equatorward jet-entrance region (or eastward jet-entrance region for a more north-south oriented jet) and secondarily to low-level forcing associated with differential diabatic heating (as shown by Andrew Winter's circulation diagnostics discussed above). Coastal fronts are typically shallow mesoscale features. That said, there clearly were a number of "coastal front like" signatures in the surface temperature and wind fields that appear to be related (in part) to topographic forcing from the Carolinas northward. To what extent differential diabatic heating and differential roughness, important players during shallow mesoscale coastal frontogenesis, played a role in the evolution of this inland baroclinic zone is unknown to me. Shapiro, M. A., 1981: Frontogenesis and Geostrophically Forced Secondary Circulations in the Vicinity of Jet Stream-Frontal Zone Systems. J. Atmos. Sci., 38, 954–973. Keyser, Daniel, Michael J. Pecnick, 1985: Diagnosis of Ageostrophic Circulations in a Two-Dimensional Primitive Equation Model of Frontogenesis. J. Atmos. Sci., 42, 1283–1305. Keyser, Daniel, Brian D. Schmidt, Dean G. Duffy, 1992: Quasigeostrophic Diagnosis of Three-Dimensional Ageostrophic Circulations in an Idealized Baroclinic Disturbance. Mon. Wea. Rev., 120, 698–730. Lance ------------------------------------ ------------------------------------ ------------------------------------ Hello Andrew and MAP list. Interesting analysis! I've always wanted a real solver for these elliptic diagnostics. Ideally in 3D even. I made a few IDV cross sections of GFS 0.5deg analysis omega and ageostrophic winds and model rainrate, trying to match your section: There do seem to be two bands of upward motion (blue in both sections, viewed from the NE) ahead (SE) of the green jet core: The third plot shows the full forcing (both geostrophic and diabatic) solution to the Sawyer-Eliassen Equation. For the most part, the solution depicts a thermally indirect circulation that is shifted towards the anticyclonic shear side of the jet, such that ascent (dashed blue contours) is positioned directly beneath the jet core. Furthermore, this ascent is located directly along the warm side of the cold front at low-levels. At 12Z this location of ascent from the solution appears to agree well with a linear band of precipitation that stretched through central and eastern Virginia. The ageostrophic winds in this 3D section viewed from the northeast are strongest through the jet core: That's because this is a section through the jet entrance region, where the 40 m/s isosurface strongly resembles a turkey leg: (or is it just me?) But wait you ask, isn't this section downstream of the FL-GA border jet? No, that's the GEOSTROPHIC WIND jet (in Andrew's plot, or filled contours here): The true wind jet (75 m/s isosurface so it is more like a banana than a turkey leg) is far downstream from the geostrophic jet. The Coriolis force on the ageostrophic flow governs the total wind acceleration, not the geostrophic acceleration, right? DV/Dt = f v_a (x k-hat). So we should prefer the total windspeed for this kind of thinking, right? And ageostrophic wind contains all the divergence (except for a small beta term) so that is what connects to or "explains" the momentum-forced part of omega, right? QG might be tying our hands too much. But then there's some strong ageostrophic flow along-jet too. (little arrows poking out of the xsection) so it isn't as 2-dimensional as ya might wish. It'd be so great to have full 4D diagnostics, and 3D steady-state (trajectory=streamline for gradient wind) approximations to that, and QG approximations to that, and so on down to these 2D exercises. Some young code wiz could probably learn to compute all those from a Best Time Series analyses sequence, and put the results into a legal datafile. I'd happily host it on my RAMADDA web server, and we could all overplot these right atop the analysis they were computed from. Scale analysis of all sorts of interpretive terms, everywhere all the time, in real flows. Nifty dream! Key is finding the diagnostics and coding wiz, for real 4D Biggish Numerics... Andrew? Or one of you Albanians? Brian Mapes