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# FUND OF ATMOS DYNAMICS ATMO 601

Texas A&M

GPA 3.65

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This 8 page Class Notes was uploaded by Demarcus Schaden V on Wednesday October 21, 2015. The Class Notes belongs to ATMO 601 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/225957/atmo-601-texas-a-m-university in Atmospheric Sciences (ATM S) at Texas A&M University.

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Date Created: 10/21/15

Thermal Wind and temperature perturbations A perturbation of potential temperature 6 that lives along a boundary behaves in the same way as a delta function perturbation of potential vorticity would just inside of the boundary You can show this mathematically by integrating the de nition of qp over an in ntesimal region adjacent to one of the horizontal boundaries but it is much more useful to get a physical picture of why this is so On a lower boundary 6 anomalies behave like delta function qp anomalies of the same sign along an upper boundary they behave like qp anomalies of the opposite sign This set of notes illustrates how boundary 6 anomalies behave The presence of such an anomaly in a quasi balanced uid implies the presence of an anomalous circulation by the thermal wind relation Recall that the thermal wind equation is valid for geostrophic motions in a hydrostatic uid both of which conditions are met in the QC system By differentiating the hydrostatic equation p with respect to z and differentiating the de nition of geostrophic wind 1 8d 7 m f az with respect to p the thermal wind equation relating vertical wind shear to horizontal gradients of potential temperature may be obtained 71 86 81 774 m fH 8z 8 Here we swallowed all of extra letters in the hydrostatic equation involving the pressure and gas constant into a single one H R RCp H E i 1 Po Its just easier to write Doing the same thing for y 1 86 Bu 774 w fH 8y 8p So wherever there are gradients of 6 on a horizontal boundary eg heading into or com ing out of a potential temperature anomaly there will be an anomalous geostrophic wind which changes magnitude with height away from the boundary And as we noted earlier this relation also explains why the mean tropospheric winds are zonal increase with height and strongest in the mid latitudes there is a mean north south gradient of potential tem perature from the warm equatorial latitudes to the cold polar ones From the relations abov 7 oc 7 or In gure 1 a positive 6 anomaly is plotted in red on a lower boundary This anomaly decays in m y and 2 away from its center As suggested by the shrinking dashed circles directly above the surface anomaly the anomaly decays with height until at some altitude it vanishes In the plane of the lower boundary there are four points labeled A B C and D Let s consider the perturbation winds at each of these four points and how they change with height Heading east through point A the gradient of 6 is increasing it starts at zero some distance well west of the anomaly and increases heading east through point A to the center of the anomaly From the relation expressed by equation 3 3 gt 0 implies that an anomalous meridional wind will decrease in value with increasing pressure or increase in value with increasing height Obviously the anomalous wind owing to this temperature perturbation must decay by the same altitude as the temperature perturbation itself decays At this level there is no more horizontal gradient in 6 and thus there can be no perturbation motion at this height With this constraint let s return to how the wind changes with height above point A It is probably easiest to think of this in pressure coordinates Given that the perturbation wind must be zero at the altitude at which all perturbations vanish as one heads down toward the surface increasing pressure the value of v9 must decrease If it starts at zero it must become negative as one heads down to the surface Thus at point A there is a strong northerly wind blowing to the south Doing the same analysis for point C we see that to the east of the perturbation maximum the value of 6 decreases as one heads east through point C until at some point far to the east the perturbation vanishes Thus 3 lt 0 Again constrained by the fact that all perturbation motions must vanish at the same altitude as 6 does equation 3 tells us that an anomalous meridional wind will increase in value down from this pressure level to the surface Thus the wind is southerly at point C on the lower boundary and decreases in intensity as one heads up in height These winds are plotted in cyan in gure 1 Along the line connecting point D to point B the gradient of potential temperature per turbation is in the y direction From equation 4 the perturbation wind shears can be estimated at points B and D Using an analysis similar to that described above the pertur bation winds are plotted in gure 2 at and above point B in magenta and at and above point D in green Thus7 it should be clear from gure 2 that the existance of a positive potential temperature anomaly along a lower boundary implies by thermal wind the existance of a perturbation circulation which rotates cyclonically and decreases in intensity as one heads above the surface From equation 1 we can also learn that this temperature anomaly creates a neg ative geopotential perturbation in height coordinates7 a negative pressure perturbation Thus a positive potential temperature anomaly on a lower boundary behaves exactly as a positive potential vorticity anomaly would there is anomalous positive vorticity and anomalous negative geopotential negative pressure Examine gures 37 47 and 5 They feature a positive 6 anomaly7 but at the top boundary rather than the lower one By examining the gradient of potential temperature moving in and out of the anomaly7 and with the boundary condition that all perturbation winds vanish at some altitude below the upper surface7 see if you can explain why the winds rotate anticyclonically Obviously temperature perturbations that are negative will produce circulations opposite in direction to the ones examined here You might nd it helpful to sketch some of these plots altitude by which all perturbations have decayed X Figure l A positive potential temperature anomaly exists on the lower boundary here drawn in the plane containing the m and y axes The anomaly is drawn in red7 and decays in intensity away from its center in 7 y7 and 2 As suggested by the shrinking dashed circles above the lower boundary7 the temperature anomaly decays with height until at some altitude it vanishes Points A7 B7 C7 and D all lie in the same lower boundary plane as the 6 anomaly The meridional wind at and above points A and C owing to the gradient of 6 in the m direction is plotted in cyan The perturbation winds must vanish at the same altitude as the 6 anomaly altitude by which all perturbations have decayed X Figure 2 As in gure 17 but With perturbation Winds plotted at and above points B in magenta and D in green also At B and D7 the perturbation Winds are zonal as the temperature gradient is in the y direction see equation surface of top boundary y F altitude by which all perturbations have decayed Figure 3 Points E7 F7 G7 and H lie7 along With a positive potential temperature anomaly7 on the upper boundary of a uid The temperature anomaly is drawn in red7 and dotted Circles are drawn beneath the upper boundary to indicate that it decreases in intensity in 7 y7 and height until at some altitude beneath the top of the uid7 it vanishes entirely The zonal Winds owing to the gradient of 6 in the y direction are plotted at and below points F in magenta and H in green The perturbation Winds must vanish at the same altitude as the 6 anomaly surface of top boundary F altitude by which all perturbations have decayed Figure 4 As in gure 37 but With perturbation Winds plotted at points E in gold and G in cyan As the gradient of 6 is in the direction at and below points E and G7 the perturbation Winds are meridional lturface of top boundaIy altitude by which all perturbations have decayed Figure 5 Figures 3 and 4 combined

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