Vorticity in natural coordinates

Transcription

1 Vorticity in natural coordinates (see Holton pg 95, section 4.2.) Let s consider the vertical vorticity component only, i.e. ζ kˆ ω, we have ω u dl kˆ ω lim --- lim curve is in xy plane Let s use our natural coordinate system again, HOWEVE this time let s apply it to a snapshot (i.e. streamlines of the flow) rather than trajectories, here the unit vector, is not the same in Stokes Thm! u u streamlines D tˆ B d tˆ displace D Let s look at the circulation around the BD segment, the segment consists of streamlines lines and pieces to the streamlines. By convention, we evaluate in a counter-clockwise orientation, So what direction is dl in w.r.t the various segments? B: dl is in the same direction as u B: dl is in direction to u D: dl is in opposite direction of u D: dl to u. So knowing this we have U dl + U dl + U dl + U dl B B D D

2 Thus, V ( + d) V upper ( ) curve upper curve We once again can use Taylor Series expansion to relate the velocity on the upper curve to that on the curve, V upper V HOT Plugging in for V upper above, we get V ( + d) V upper ( ) V ( + d) V HOT Using the geometry from our figure d, or, V ( ) d ( HOT) V HOT V HOT Note that the area of our enclosed curve is: + δ using and noting that δ, r dr ( dθ) -- [ 2 + δ 2 + 2δ 2 ] 2 +δ For an infinitesimal loop ~ so we have ~, and the circulation becomes

3 V HOT ecall from Stokes theorem that we divide the circulation by the area, and in the limit as the area goes to zero, we obtain the normal component of vorticity (in this case it is the vertical vorticity since we are in the xy plane), V HOT ζ lim --- lim V What is /? From our figure above we have or / /, where is the radius of curvature of the streamline, thus as ->, we have β s -- and the vertical vorticity in natural coordinates can be written: V , (Holton Eq. 4.9) curvature shear We can do the same for the x and y vorticity components and get a similar decomposition.

4 PTIL PPLITIONS Jet streak vorticity During the lecture I mentioned that the 4-quadrant jet streak model applied to straight flow only (i.e. s approaches infinity - pure shear vorticity only) - technically this is not correct. The air parcel must experience acceleration as it enters and leaves the jet streak. In order to accomodate this, we assume a straight line geopotential through the center of the jet (with curved flow off axis as drawn in the figure below). Note that there is a change in the geopotential gradient normal to the direction of motion however - hence is not constant for a parcel traversing the jet. y x Φ is to left of flow 3 hpa hgts ζ > MX WINDS ζ < Φ Φ cyclonic side of jet 3 mb < > anticyclonic side of jet Φ+ Φ ecall that we can decompose the hz velocity into a geostrophic component and ageostrophic component, u u ag + u g, and v v ag + v g, where it is assumed that v ag << v g, and u ag << u g. The inviscid momentum equations can be rewritten as du p + fv fv dt ρ x g + fv fv ( v g ) fv ag dv p fu fu dt ρ y g fu fu u g fu ag We can apply these equations to our jet streak (known as 4-quadrant model) (frictionless, assume air moves thru jet). cceleration in the entrance region, and decelleration in the exit region produce the following ageostrophic flow: Φ Φ L convergence (sinking) y x Φ MX WINDS divergence (rising) H 3 mb (rising) divergence < > convergence(sinking) Φ+ Φ

5 You should know which equation was applied above to get the ageostrophic wind vectors (red arrows). This is consistent with our 2-D equations of motion in natural coordinates, recall (Holton 3.9): dv dt Φ s parcel entering the jet will develop an ageostrophic wind component from high heights to low heights hence Φ s < and dv/dt > (parcel accelerates). The opposite occurs for a parcel leaving the jet. Keep in mind that this is a highly idealized model - and applies for a non-evolving height field (not reality!), where we ve assumed that the parcel travels through the jet streak, i.e. the parcel is moving faster than the jet streak itself.