Thermal wind and temperature perturbations

Transcription

1 Thermal wind and temerature erturbations Robert Lindsay Korty Massachusetts Institute of Technology October 15, 2002 Following the work of Bretherton (1966), we showed in class that a boundary otential temerature erturbation behaves in the same way as a delta function erturbation of otential vorticity would just inside of the boundary. By integrating the definition of seudo-otential vorticity over an infintesimal region adjacent to one of the horizontal boundaries, we showed the mathematical equivalence of a boundary otential temerature anomaly and a delta function q anomaly just inside the boundary. In the case of a lower boundary, the temerature anomalies behave like delta function q anomalies of the same sign; at the uer boundary, they behave like q anomalies of the oosite sign. It can be useful to have an illustration of how boundary θ anomalies behave. The resence of such an anomaly in a quasi-balanced fluid imlies the resence of an anomalous circulation by the thermal wind relation. Recall that the thermal wind equation is valid for geostrohic motions in a hydrostatic fluid. By differentiating the hydrostatic equation, φ = θ π, (1) with resect to and differentiating the definition of geostrohic wind, v g = 1 f φ, (2) with resect to, the thermal wind equation relating vertical wind shear to horizontal gradients of otential temerature may be obtained: 1 θ fπ = v g. (3) Similarly, 1 θ fπ y = u g. (4) So wherever there are gradients of θ on a horizontal boundary (e.g., heading into or coming out of a otential temerature anomaly), there will be an anomalous geostrohic wind, which changes magnitude with height away from the boundary. (Note that this relation also

2 elains why the mean troosheric winds are zonal, increase with height, and strongest in the mid-latitudes: there is a mean north-south gradient of otential temerature from the warm, equatorial latitudes to the cold, olar ones. From the relations above, dθ dy dug d or + dug.) dz In figure 1, a ositive otential temerature anomaly is lotted in red on a lower boundary. This anomaly decays in, y, and z 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. I have labeled four oints in the lower boundary lane containing the θ anomaly: A, B, C, and D. Let us consider the erturbation winds at each of these four oints, and how they change with height. As one heads east through oint A, the gradient of otential temerature anomaly is increasing (it starts at zero some distance well west of the anomaly and increases as one heads east, assing through oint A, to the center of the anomaly). From the relation eressed θ by equation (3), > 0 imlies that an anomalous, meridional wind will decrease in value with increasing ressure (or increase in value with increasing height). Obviously, the anomalous wind owing to this temerature erturbation must decay by the same altitude as the temerature erturbation itself decays. (At this level, there is no more horizontal gradient in θ, and thus there can be no erturbation motion at this height.) With this constraint, let us now return to how the wind changes with height above oint A. It is robably easiest to think of this in ressure coordinates. Given that the erturbation wind must be zero at the altitude at which all erturbations vanish, as one heads down toward the surface (increasing ressure), the value of v g must decrease. If it starts at zero, it must become negative as one heads down to the surface. Thus, at oint A, there is a strong northerly wind (blowing to the south). A similar analysis may be conducted at oint C, to the east of the erturbation maimum. To the east of the maimum, the value of θ decreases as one heads east through oint C until at some oint, far to the east, the erturbation vanishes. Thus, θ < 0. Again constrained by the fact that all erturbation motions must vanish at thesamealtitudeasθ does, equation (3) tells us that an anomalous meridional wind will increase in value down from this ressure level to the surface. Thus, the wind is southerly at oint C on the lower boundary, and decreases in intensity as one heads u in height. These winds are lotted in cyan in figure 1. Along the line connecting oint D to oint B, the gradient of otential temerature erturbation is in the y direction. From equation (4) the erturbation wind shears can be estimated at oints B and D. Using an analysis similar to that described above, I have lotted the erturbation winds in figure 2 at and above oint B (in magenta) and at and above oint D (in green). Thus, it should be clear from figure 2 that the eistance of a ositive otential temerature anomaly along a lower boundary imlies by thermal wind the eistance of a erturbation circulation which rotates cyclonically and decreases in in-

3 tensity as one heads above the surface. From equation (1) we can also learn that this temerature anomaly creates a negative geootential erturbation (in height coordinates, a negative ressure erturbation). Thus a ositive otential temerature anomaly on a lower boundary behaves eactly as a ositive otential vorticity anomaly would: there is anomalous ositive vorticity and anomalous negative geootential (negative ressure). I would encourage you to eamine figures 3, 4, and 5. They feature a ositive θ anomaly, but at the to boundary rather than the lower one. By eamining the gradient of otential temerature moving in and out of the anomaly, and with the boundary condition that all erturbation winds vanish at some altitude below the uer surface, see if you can elain why the winds rotate anticyclonically. Obviously temerature erturbations that are negative will roduce circulations oosite in direction to the ones eamined here. Try sketching some of these lots, if it would be useful.

4 z y A B D C Figure 1: A ositive otential temerature anomaly eists on the lower boundary (here drawn in the lane containing the and y aes). The anomaly is drawn in red, and decays in intensity away from its center in, y, and z. As suggested by the shrinking dashed circles above the lower boundary, the temerature anomaly decays with height until at some altitude it vanishes. Points A, B, C, and D all lie in the same lower boundary lane as the θ anomaly. The meridional wind at and above oints A and C owing to the gradient of θ in the direction is lotted in cyan. The erturbation winds must vanish at the same altitude as the θ anomaly.

5 z y A B D C Figure 2: As in figure 1, but with erturbation winds lotted at and above oints B (in magenta) and D (in green) also. At B and D, the erturbation winds are zonal as the temerature gradient is in the y direction (see equation (4)).

6 y E surface of to boundary F H G z Figure 3: Points E, F, G, and H lie, along with a ositive otential temerature anomaly, on the uer boundary of a fluid. The temerature anomaly is drawn in red, and dotted circles are drawn beneath the uer boundary to indicate that it decreases in intensity in, y, and height until at some altitude beneath the to of the fluid, it vanishes entirely. The zonal winds owing to the gradient of θ in the y direction are lotted at and below oints F (in magenta) and H (in green). The erturbation winds must vanish at the same altitude as the θ anomaly.

7 y E surface of to boundary F H G z Figure 4: As in figure 3, but with erturbation winds lotted at oints E (in gold) and G (in cyan). As the gradient of θ is in the direction at and below oints E and G, the erturbation winds are meridional.

8 y E surface of to boundary F z H G Figure 5: Figures 3 and 4 combined.