Scale Analysis of the Equations of Motion

Transcription

1 hen are curvature terms important? Important for GCMs and large-scale (global) weather. ere, we consider sub-global scale referred to as synoptic scale. Scale Analysis of the Equations of Motion e use typically observed values of synoptic-scale features to perform scale analysis. variable description value L z length scale ~ radius low/high pressure 000 km Vertical length scale ~ scale height 0 km z velocity scale ~ u or v component 0 ms - Vertical velocity scale ~ w component cms - δp h hz pressure diff. ~ change in p over L 0 mb δp h /ρ ρ is typically kg m m 2 s -2 δp v /ρ vertical pressure diff. ~ change in p over 0 5 m 2 s -2 T h (=L/) time for parcel travelling at to cover L 0 6 m/0ms - =0 5 s T v (=/w) time for a parcel moving at w to cover 0 4 m/0-2 ms - =0 6 s (0 days!) T v and T h are referred to as advective time scales. Estimating order of magnitudes for our terms in the equations and, x --- y x y L ---, , and - x y L The local time derivative terms

2 and, - or T h or ---- whichever is larger T h T v ---- whichever is larger T v Order of magnitude theory, i.e. normal math order o mag math + = 2 + ~ ++=3 ++ ~ = ~ =5 ++++=5 (too close to call) -=0 -=0 (maybe not! ARNING) sing our scaling parameters in the previous table, we note that the magnitude of Dt is given by the largest term (i.e. either the advective terms of local tendency term), thus looking at the component equation we have /T h 2 /L / - Dt = + u + v + w 0 4 x y (e can show that Dv/Dt~0-4 as well). Note that above we showed, 2 - L > --- or L ---- Store this into your memory as we will revisit this a little later when looking at mass conservation. A typical value for the mid latitude magnitude of the Coriolis is f 0 = 2Ωsinφ = 2Ωcosφ = sin45 0 4

3 Also -> tan45=.0 (for curvature terms) Apx earth s radius -> a~6000 km ~ 0 7 m orizontal eqns of motion: x-eq /Dt -2Ωvsinφ 2Ωwcosφ uw/a -uvtanφ/a y-eq Dv/Dt 2Ωusinφ vw/a u 2 tanφ/a scale 2 /L f 0 f 0 /a 2 /a ( ρ) p x ( ρ) p y δp ( ρl) OM Thus we see that the largest terms in the hz eqns of motion are the Coriolis and the pressure gradient! Thus, to a first order of magnitude, our x and y component eqns reduce to fv -- p, and fu -- p ρ x ρ y where f=2ωsinφ. These are the geostrophic relationship, i.e. the Coriolis and pressure gradient terms balance one another. In chapter 3 (section 3.2.2), we ll see what this means from a dynamical perspective. In vector notation (i.e. combine the two eqns above), Or, fkˆ v = -- p ρ v v g = ---- kˆ p = u ρf g î + v g ĵ LO p LO IG IG u kˆ p kˆ out of paper

4 Thus, from the eqn and figure we have -Geostrophic wind vector is parallel to isobars. - Buys Ballots's law: ith geostrophic wind at your back, low pressure is to your left (in N. hemisphere) - Geostrophic wind speed is proportional to magnitude of pressure gradient (packing of isobars). Note however that for the same pressure gradient, the geostrophic wind speed is greater equatorward (/f). Geos wind is hypothetical wind, i.e. the wind you'd have in an atmos in a pure geos balance. The true wind is nearly equal to geostrophic wind for synoptic flows above the boundary layer (i.e. for z>km) ere, u u g, v v g with 0-5% error. The actual wind differs from geostrophic wind near fronts, and near the earth's surface. Y IS TIS IS? Note that in general, if the weather is interesting it is not generally in geostrophic balance! And more on geosgtrophic balance... The geos balance is a diagnostic relation btw pressure gradient and velocity. Thus, knowledge of the horizontal pressure distribution exactly determines the velocity field. Not a bad eyeball estimate of wind speed and direction for large-scale flow (synoptic or larger). For a prognostic set of horizontal equations that are more representative of the true atmosphere, we keep the next most important term (i.e. the acceleration), - fv -- p Dv = f( v v Dt ρ x g ) and -, fu -- p = fu ( u Dt ρ y g ) where we have substituted the geostrophic relationship for the pressure gradient. The above eqn can be used for prognostic meteorology (predictions, Y?) One issue is that we know from our scaling arguments earlier that the acceleration terms (/Dt and Dv/Dt) are, for large-scale flows, an order of magnitude smaller than the pressure gradient and Coriolis terms. Thus, the acceleration terms are the resultant of a slight difference between two (much) larger terms. STDENTS: hat does this mean in terms of observations of u, v, and p? They best be accurate!

5 e can get an estimate of the relative order of magnitude of these terms by taking the ratio of the acceleration term and Coriolis term (see table for scaling values), - -- Dt Rossby # (famous Met C. G. Rossby) fv L f fl The ratio of /Dt to fu is the same! NON-DIMNENSIONAL (as are all of the scaling numbers in meteorology, e.g. Froude, Reynolds etc.) IF R >, then accelerations are important! IF R <, then Coriolis dominates For small Rossby # s the flow is apx geostrophic. hat about our atmosphere? /fl = 0/(0-4 x0 6 )= 0 -. Therefore, << fl and to a first Order, the atmosphere is geostrophic! hat about a tornado? = 00 ms-, L = 00 m, thus /fl = 0 2 /(0-4 x0 2 ) = 0 4 Thus we can neglect the Coriolis force for a tornado! Coriolis force is important to the tornado indirectly in that it affects the way the winds turn with height in the atmosphere (e.g. hodographs). Certain wind profiles are more conducive than others to tornado formation, thus the Coriolis force affects "initial conditions, i.e the environment in which the tornadic thunderstorm develops. Recall curvature terms (e.g. uw/a ~ 0-5 and 0-8 ) are smaller than acceleration, Coriolis force, and pgf terms (~0-3 ). e can safely neglect them for synoptic scale flows. This is equivalent to treating spherical x,y,z coords as if they were Cartesian x,y,z coords. Flat earth society!