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Balanced Flow

Balanced Flow. The momentum equation in natural coordinates:. Let’s break this up into component equations:. If that the flow is parallel to the height contours, then.

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Balanced Flow

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  1. Balanced Flow

  2. The momentum equation in natural coordinates: Let’s break this up into component equations: If that the flow is parallel to the height contours, then 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.

  3. Rossby Number where U and L are, respectively, characteristic velocity and length scales of the phenomenon and f = 2 Ω sin φ is the Coriolis frequency. 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.

  4. 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. Pure geostrophic flow is uncommon in the atmosphere, but the geostrophic flow is a good approximation when R0 is small:

  5. Inertial flow Inertial flow occurs in the absence of a PGF or This type of flow follows circular, anticyclonic paths sincefRis negative Time to complete a circle: • is one half rotation  is one full rotation/day called a half-pendulum day

  6. Power Spectrum of kinetic energy at 30 m in the ocean near Barbados (13N) Pure inertial oscillation is rare in the atmosphere but common in the oceans where transient wind stress drives currents

  7. 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 Coriolisforce or R0 is much larger than 1. A synoptic scale wave: NO A tornado: YES

  8. In cyclostrophicflow, circulation can rotate counterclockwise or clockwise (anticyclonic andcyclonic 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.

  9. Gradient flow: a three-way balance among CO, PGF and CEN This expression has a number of mathematically possible roots, not all of which conform to reality Is V a nonnegative real number?

  10. the unit vector is everywhere normal to the flow and positive to the left of the flow, and  is the geopotential height is the height gradient in the direction of R is the radius of curvature following parcel motion 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

  11. V is always positive in the natural coordinate system Solutions for For radical to be positive Therefore: is always negative. Cyclonic high V = negative = UNPHYSICAL

  12. Solutions for Positive root physical Radical > Negative root unphysical Anticyclonic low Called an “anomalous low” it is rarely observed (technically since f is never 0 in mid-latitudes, anticyclonic tornadoes are actually anomalous lows outward Increasing in direction (low)

  13. Solutions for Positive root physical Radical > Negative root unphysical Cyclonic Called an “regular low” it is commonly observed (synoptic scale lows to cycloni-cally rotating dust devils all fit this category inward decreasing in direction (low)

  14. Positive Root Solutions for or radical is imaginary Antiyclonic Then or outward decreasing in direction (high) CEN>CO/2, so CEN>PGF Called an “anomalous high” (CEN>PGF)

  15. Negative Root Solutions for or radical is imaginary Antiyclonic therefore outward Called a “regular high” : PGF exceeds the centrifugal force decreasing in direction (high) PGF CEN

  16. Condition for both regular and anomalous highs For a regular high, we have 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

  17. Force Balance in a Regular Low and a Regular High What if V is non-zero at small radii?

  18. Note pressure gradients in vicinity of highs and lows

  19. 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?

  20. The ageostrophic wind in natural coordinates Note that We have For cyclonic flow (fR> 0) gradient wind is less than geostrophicwind (V<Vg) For anticyclonic flow (fR< 0) gradient wind is greater than geostrophicwind V>Vg.

  21. Summary Centrifugal Force PGF Coriolis Force Geostrophic Balance: PGF = CO Inertial Balance: CEN = CO CyclostrophicBalance: CEN = PGF Gradient Balance: CEN + PGF + CO = 0

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