Balanced Flow. Natural Coordinates

Transcription

1 Balanced Flow The pessue and velocity distibutions in atmospheic systems ae elated by elatively simple, appoximate foce balances. We can gain a qualitative undestanding by consideing steady-state conditions, in which the fluid flow does not vay with time, and by assuming thee ae no vetical motions. To exploe these balanced flow conditions, it is useful to define a new coodinate system, known as natual coodinates. Natual Coodinates Natual coodinates ae defined by a set of mutually othogonal unit vectos whose oientation depends on the diection of the flow. Unit vecto points along the diection of the flow. t Unit vecto is pependicula to the flow, with positive to the left. Unit vecto points upwad. 1

2 To detemine acceleation following the fluid motion, Hoizontal velocity: is the hoizontal speed, which is a nonnegative scala defined by ds, whee s ( x, y, t ) is the cuve followed by a fluid pacel moving in the hoizontal plane. d d ( ) d d + d δt δs δ δψ δ δψ t+δt δψ adius of cuvatue (positive in positive n diection) > 0 if ai pacels tun towad left < 0 d ds d d ds ds if ai pacels tun towad ight (taking limit as δs 0) <0 δs t n > 0

3 d d d d + d + vecto fom of acceleation following fluid motion in natual coodinates f f pφ + s d s + f (always acts nomal to flow) pessue gadient component equations of hoizontal momentum equation (isobaic) in natual coodinate system d s + f Balance of foces paallel to flow. Balance of foces nomal to flow. Fo motion paallel to geopotential height contous, 0, which means s that the speed is constant following the motion. If the geopotential gadient nomal to the diection of motion is constant along a tajectoy, the nomal component equation implies that the adius of cuvatue is also constant. t When these assumptions ae met we can define seveal simple categoies of balanced flow that depend on the elative contibutions of the thee tems in the nomal component equation. 3

4 Geostophic Flow Staight-line flow paallel to the height contous ( ± ). 0 + f Hoizontal components of foce and pessue gadient foce ae in exact balance. f g Φ 0 g Φ 0 + ΔΦ Inetial Flow If the geopotential field is unifom on a constant pessue suface, then + f 0 f Because unifom geopotential implies constant speed, then the adius of cuvatue is constant if we assume f is appoximately constant. Ai pacels will follow cicula paths in anticyclonic otation with peiod π π P f 4

5 Cyclostophic Flow If the hoizontal scale of an atmospheic distubance is sufficiently small, such as in a tonado, watespout, o dust devil, then the tem will be substantially smalle than the pessue gadient and centifugal tems: 0 + f 1 Thee ae fou possible oientations fo the diection of cuvatue and the pessue gadient: > 0, < 0 is eal < 0, < 0 > 0, > 0 is imaginay < 0, > 0 is imaginay is eal > 0, < 0, < 0 > 0 Only low pessue systems can have cyclostophic flow. Cyclostophic flow can eithe be clockwise o counteclockwise. 5

6 Cyclostophic Flow in the eal Wold Because the acceleation is neglected in cyclostophic flow, thee should be no pefeed diection of otation. In fact, watespouts and (especially) dust devils can show both clockwise and counteclockwise otation. Most (but not all) tonadoes in the Nothen Hemisphee otate counteclockwise, because they develop fom lage, otating supecell thundestoms. Because of thei elatively lage scale (~10 km), supecell thundestoms do feel the effects of acceleation. Gadient Flow Fo lage-scale weathe distubances in which the flow is not equied to be in a staight line, a thee-way balance among the, centifugal, and pessue gadient foces exists. + f f f 4 ± 1 Thee ae fou possible oientations fo the diection of cuvatue and the pessue gadient, each with two oots: > 0 < 0 > 0 < 0 Positive oot: unphysical Negative oot: unphysical Positive oot: egula low Negative oot: unphysical Positive oot: anomalous low Negative oot: unphysical Positive oot: anomalous high Negative oot: egula high 6

7 egula low egula high H > 0, < 0, anomalous low positive oot < 0, < 0, negative oot anomalous high H < 0, > 0, positive oot < 0, < 0, positive oot Geostophic Wind vs. Gadient Wind + f + f fg 0 g 1+ f Stat t with balance of foces nomal to flow. ewite using definition of geostophic wind. Solve fo atio of geostophic wind to gadient wind. ossby numbe Fo nomal cyclonic flow (f > 0) the geostophic wind oveestimates the gadient wind, while fo anticyclonic flow (f < 0) the geostophic wind undeestimates the gadient wind. The geostophic wind is a good estimate of the gadient wind when the ossby numbe is small. 7