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Balanced Flow The momentum equation in natural coordinates: r r2 ) ) dV ) V ) ) i n i n fV n s dt R n Let’s break this up into component equations: dV s dt V 2 fV R n 0 If that the flow is parallel to the height contours, then dV dt s 0 Under these conditions, the flow is uniquely described by the equation in the yellow box. If PGF normal to the flow direction is a constant, then the radius of curve is also a constant. Rossby Number R0 U fL where U and L are, respectively, characteristic velocity and length scales of the phenomenon and f = 2 Ω sin φ is the Coriolis frequency. V V fk V V t fk V U 2 L fU U (L U ) fU U fL R0 U fL R0 Ratio of advection to the CF Ratio of local acceleration to the CF A small Rossby number signifies a system which is strongly affected by the Coriolis force, and a large Rossby number signifies a system in which inertial and centrifugal forces dominate. V 2 R fV n 0 Geostrophic flow Geostrophic flow occurs when the PGF = CO, implying that R Strictly speaking, for geostrophic flow to occur the flow must be straight and parallel to the latitude circles. Vg 1 f n Pure geostrophic flow is uncommon in the atmosphere, but the geostrophic flow is a good approximation when R0 is small: V 2 R 0 ~ R / fV V fR V 2 fV R n 0 Inertial flow Inertial flow occurs in the absence of a PGF R V V fR or f This type of flow follows circular, anticyclonic paths since fR is negative Time to complete a circle: t 2 R V is one half rotation is one full rotation/day sin 2 R fR 0 . 5 day sin 2 f 2 2 sin sin called a half-pendulum day Power Spectrum of kinetic energy at 30 m in the ocean near Barbados (13N) 0 . 5 day sin 13 2 . 23 days Pure inertial oscillation is rare in the atmosphere but common in the oceans where transient wind stress drives currents V 2 fV R n 0 Cyclostrophic flow When the horizontal scale of the motion is small (e.g., tornados, dust devils, water spouts), the Coriolis force can be neglected: Flow is approximately cyclostrophic when the Centrifugal force is much larger than the Coriolis force or R0 is much larger than 1. V 2 R 0 ~ / fV R A synoptic scale wave: A tornado: V fR V fR V fR 10 m s 10 4 s 1 10 s 1 6 0 .1 NO 10 m 100 m s 4 1 1 3 10 m 1000 YES 1 V 2 R n 0 2 V R n In cyclostrophic flow, circulation can rotate counterclockwise or clockwise (anticyclonic and cyclonic tornadoes and smaller vortices are observed), but it is always associated with a low. The centrifugal force points away from the center of curvature so the PGF must point toward the center of curvature. V 2 R fV n 0 Gradient flow: a three-way balance among CO, PGF and CEN 2 2 fR f R V R 2 4 n 1/2 This expression has a number of mathematically possible roots, not all of which conform to reality Is V a nonnegative real number? the unit vector nis everywhere normal to the flow and positive to the left of the flow, and is the geopotential height n is the height gradient in the direction of n R is the radius of curvature following parcel motion R 0 R 0 n n directed toward center of curve (counterclockwise flow) directed toward outside of curve (clockwise flow) Let’s consider the Northern Hemisphere (f>0): R>0: cyclonic R<0: anticyclonic V is always positive in the natural coordinate system f R V R 2 n 4 fR Solutions for n 2 1/ 2 2 0, Therefore: Cyclonic high 2 R 0 For radical to be positive f 2R2 R 2 n 4 fR f R 4 2 R 1/ 2 is always negative. V = negative = UNPHYSICAL n V Solutions for n 0, f R R 2 4 n fR R0 Anticyclonic low R 0 n outward Increasing in n direction (low) 2 2 1/ 2 Radical > fR 2 Positive root physical Negative root unphysical Called an “anomalous low” it is rarely observed (technically since f is never 0 in mid-latitudes, anticyclonic tornadoes are actually anomalous lows ) n V Solutions for Cyclonic n inward n 0, f R R 2 4 n fR R 0 2 2 1/ 2 Radical > fR 2 Positive root physical Negative root unphysical R 0 decreasing in n direction (low) Called an “regular low” it is commonly observed (synoptic scale lows to cycloni-cally rotating dust devils all fit this category ) n V Solutions for n 0, f R R 2 4 n fR 2 2 R0 2 f R decreasing in 2 R 4 Antiyclonic R 0 n outward 1/ 2 Then n direction (high) V Positive Root n fR 2 or or radical is imaginary V 2 R fV 2 CEN>CO/2, so CEN>PGF Called an “anomalous high” (CEN>PGF) ) n V Solutions for n Antiyclonic R 0 n outward decreasing inn f R R 2 4 n fR 2 2 1/ 2 Negative Root 0, R0 2 f R 2 R 4 n therefore direction (high) or radical is imaginary V fR 2 Called a “regular high” : PGF exceeds the centrifugal force ) n PGF CEN Condition for both regular and anomalous highs f 2 R 2 V R 2 n 4 1/2 fR 2 f R 4 2 R n For a regular high, we have 2 f R 0 4 R n 0 2 R n n f 2 R 4 This is a strong constraint on the magnitude of the pressure gradient force in the vicinity of high pressure systems Close to the high, the pressure gradient must be weak, and must disappear at the high center Force Balance in a Regular Low and a Regular High What if V is non-zero at small radii? Note pressure gradients in vicinity of highs and lows Force Balance in a Regular Low and a Regular High If PGF has the same magnitude, which one, the high or the low, has stronger wind speed? The ageostrophic wind in natural coordinates V 2 fV R Note that Vg We have V 2 R 0 n 1 f n f V V g 0 V Vg V 2 fR For cyclonic flow (fR > 0) gradient wind is less than geostrophic wind (V<Vg) For anticyclonic flow (fR < 0) gradient wind is greater than geostrophic wind V>Vg. Summary V 2 fV R Centrifugal Force n 0 PGF Coriolis Force Geostrophic Balance: PGF = CO Inertial Balance: CEN = CO Cyclostrophic Balance: CEN = PGF Gradient Balance: CEN + PGF + CO = 0