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The Ageostrophic Wind Equation

The Ageostrophic Wind Equation. Remember from before: The “forcing” terms in the QG omega equation are geostrophic “Weather” results from ageostrophic motions that act as a need to restore thermal wind balance

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The Ageostrophic Wind Equation

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  1. The Ageostrophic Wind Equation • Remember from before: • The “forcing” terms in the QG omega equation are geostrophic • “Weather” results from ageostrophic motions that act as a need to restore thermal wind balance • These motions are a secondary response due to the imbalance brought about by advection of the primary flow Geostrophic forcing Ageostrophic response

  2. The Ageostrophic Wind Equation • Knowing and understanding the ageostrophic wind is a powerful concept in weather analysis and forecasting • The ageostrophic wind will allow us to better identify areas of CONV and DIV aloft • Typically apply the ageostrophic wind at the jet level, however, it can also apply at the surface • Remember, with the jet stream: • Strong curvature regions are important • Identifying areas of acceleration and deceleration • In fact, the above is MORE important than our knowledge of 4-quadrant jet theory for identifying areas of DIV/CONV aloft

  3. Ageostrophic Wind Definition The horizontal wind can be partitioned into a geostrophic and ageostrophic component: **The ageostrophic wind represents the difference between what the wind is actually doing and what it would be doing if it were in a perfect geostrophic balance Basically, any wind blowing at an angle to isoheights/isobars will have an ageostrophic component Wind blowing at a 90° angle to isobars/heights is strictly ageostrophic

  4. Ageostrophic Wind Equation (Natural Coordinates) • Term 1: Isallohypsic or isallobaric Acceleration Term • Height tendency / pressure tendency term • Term 2: Centripetal or Centrifugal Acceleration Term • Curvature Term • Important along troughs/ridges • Term 3: Speed Divergence Term • Speed Acceleration Term • Look for significant changes in velocity along the flow • Important in entrance/exit regions of jet streaks • Term 4: Vertical Advection Term • Relates to the thermal field • Note the omega  vertical motion in pressure coordinates • Term 5: Friction Term • Most important at the surface • Friction always opposes the wind Term 1 Term 2 Term 3 Term 4 Term 5

  5. Natural Coordinate System Review • Natural coordinates are another way of representing direction. It is based on the relative motion of the object of interest, rather than a fixed coordinate plane • Three unit vectors: • t: oriented parallel to direction of velocity vector • n: oriented perpendicular to the velocity vector and points to the left of the flow • k: directed vertically upward • We also define “s” as the horizontal distance along a curve followed by an air parcel • “R” is the radius of curvature following the parcel motion • If the center of curvature is in the positive n direction, then R>0

  6. Ageostrophic Wind Equation (Natural Coordinates) • Term 1: Isallohypsic or isallobaric Acceleration Term • Height tendency / pressure tendency term • Term 2: Centripetal or Centrifugal Acceleration Term • Curvature Term • Important along troughs/ridges • Term 3: Speed Divergence Term • Speed Acceleration Term • Look for significant changes in velocity along the flow • Important in entrance/exit regions of jet streaks • Term 4: Vertical Advection Term • Relates to the thermal field • Note the omega  vertical motion in pressure coordinates • Term 5: Friction Term • Most important at the surface • Friction always opposes the wind Term 1 Term 2 Term 3 Term 4 Term 5

  7. Term 1Isallohypsic or Isallobaric Acceleration Term • An isallohypse is a contour of constant height change • An isallobar is a contour of constant pressure change • **Height or pressure changes produces an ageostrophic wind

  8. Term 1 • For a constant pressure surface: **Gradient of the height change with time**

  9. Term 1 • For a constant height surface(i.e. surface charts) **Gradient of the pressure change with time**

  10. Term 1Isallohypsic or Isallobaric Acceleration Term • Board Example… • The contour defines a gradient of height change • The gradient vector points toward lower values • Based on our equation, we also find the Vag is always oriented along the gradient vector • i.e., Vag due to term 1 points towards the lowest pressure/height falls

  11. Term 1Isallohypsic or Isallobaric Acceleration Term • Since f is in the denominator of Term 1, Vag has a latitudinal dependence. • Since f is small at low latitudes, a 30 m height change at 20N over a 100km distance will produce a larger Vag than the same height change at 60N • i.e., a smaller height change at low latitudes will produce a larger Vag • A developing hurricane (rapid height/pressure falls) produces a massive Vag (also occurs at low latitudes!)

  12. Term 1Isallohypsic or Isallobaric Acceleration Term • Let’s look at term 1 as it pertains to divergence. • Remember, divergence is defined as the ageostrophic component of the wind • Geostrophic wind is non-divergent • Board notes • **Laplacian of the height change field will define divergence

  13. Term 1Isallohypsic or Isallobaric Acceleration Term • Recall that the Laplacian operator is positive in local minima regions and negative in local maxima (“opposite operator”) • Thus, a height fall center would have a positive Laplacianand a height rise center would have a negative Laplacian • Example with falling heights • Example with rising heights

  14. Term 1Pressure/Height Tendency Term • Vag due to term 1 points toward the lowest pressure/height falls • Vag due to term 1 points away from the highest pressure/height rises

  15. Ageostrophic Wind Equation (Natural Coordinates) • Term 1: Isallohypsic or isallobaric Acceleration Term • Height tendency / pressure tendency term • Term 2: Centripetal or Centrifugal Acceleration Term • Curvature Term • Important along troughs/ridges • Term 3: Speed Divergence Term • Speed Acceleration Term • Look for significant changes in velocity along the flow • Important in entrance/exit regions of jet streaks • Term 4: Vertical Advection Term • Relates to the thermal field • Note the omega  vertical motion in pressure coordinates • Term 5: Friction Term • Most important at the surface • Friction always opposes the wind Term 1 Term 2 Term 3 Term 4 Term 5

  16. Term 2: Centripetal or Centrifugal Acceleration (curvature) Term Change in a parcel direction along parcel’s path Rs = radius of curvature along parcel’s path Rs > 0 for cyclonic curvature (trough) Rs < 0 for anticyclonic curvature (ridge) t is positive along direction of flow So, in reality, only Rs will change the sign of our term

  17. Term 2: Centripetal or Centrifugal Acceleration (curvature) Term Positive for cyclonic curvature Negative for anticyclonic curvature Curvature of the flow defines the sign of this term!

  18. Term 2: Centripetal or Centrifugal Acceleration (curvature) Term • Example for cyclonic flow • So, for cyclonic flow, Vag is oriented in the the –t direction • i.e., AGAINST the direction of motion • This means that for cyclonic curvature, the actual wind blows at less than geostrophic (since Vag opposes motion). • Thus cyclonic flow is sub-geostrophic flow • Hey! This matches what we talked about before

  19. Term 2: Centripetal or Centrifugal Acceleration (curvature) Term • Example for anticyclonic flow • So, for anticyclonic flow, Vag is oriented in the the +t direction • i.e., WITH the direction of motion • This means that for anticyclonic curvature, the actual wind blows at more than geostrophic (since Vag is with the actual motion). • Thus anticyclonic flow is super-geostrophic flow • Hey! This also matches what we talked about before

  20. Term 2: Centripetal or Centrifugal Acceleration (curvature) Term • Term 2 is a major reason why we get CONV behind troughs and DIV ahead of troughs at upper levels

  21. Term 2: Curvature Term (Supergeostrophic Winds) (Supergeostrophic Winds) Ridge Ridge CONV DIV 970 dam 990 dam 970 dam 990 dam Trough (Subgeostrophic Winds) 300-mb Isobaric Surface

  22. Term 2: Centripetal or Centrifugal Acceleration (curvature) Term • Term 2 is a major reason why we get CONV behind troughs and DIV ahead of troughs at upper levels • Vag(2) does not work well at mid-levels • Need to know DIV/CONV at rigid lids (at ground or near tropopause) • Use term 2 near the jet stream level • It’s very useful for estimating jet stream DIV/CONV • Important to state the fact that winds accelerate out of a trough, thus causing speed DIV, has nothing to do with term 2 • It is purely due to curvature

  23. Term 2: Centripetal or Centrifugal Acceleration (curvature) Term • The stronger the curvature, the stronger the Vag • A negative tilt trough is often oriented so curvature is more severe • Thus term 2 is stronger, and often more concentrated • Significant DIV • Explosive cyclogenesis • Opposite can occur with positive tilt trough • Often concentrated CONV on backside • Significant ridge and surface high pressure development

  24. Term 2 - Example

  25. Term 2 - Example

  26. Ageostrophic Wind Equation (Natural Coordinates) • Term 1: Isallohypsic or isallobaric Acceleration Term • Height tendency / pressure tendency term • Term 2: Centripetal or Centrifugal Acceleration Term • Curvature Term • Important along troughs/ridges • Term 3: Speed Divergence Term • Speed Acceleration Term • Look for significant changes in velocity along the flow • Important in entrance/exit regions of jet streaks • Term 4: Vertical Advection Term • Relates to the thermal field • Note the omega  vertical motion in pressure coordinates • Term 5: Friction Term • Most important at the surface • Friction always opposes the wind Term 1 Term 2 Term 3 Term 4 Term 5

  27. Term 3: Speed Divergence (Acceleration) Term • In this term dV/ds = acceleration of the air parcel • This term tells us: • For a parcel changing speed along its path (in the tangential direction along s), an ageostrophic flow will be induced normal to the path of the parcel

  28. Term 3: Speed Divergence (Acceleration) Term • For a parcel accelerating along its path: • Board notes • Vag will be induced in the +n direction • Thus, the actual wind (V) is turned toward the left • **Cross contour flow usually means • The contours are wrong, or • You have an ageostrophic wind component

  29. Term 3: Speed Divergence (Acceleration) Term • For a parcel decelerating along its path: • Board notes • Vag will be induced in the -n direction • Thus, the actual wind (V) is turned toward the right • Term 3 is the only term working in the 4-quadrant jet theory • All 4 other terms are not accounted for

  30. Term 3: Speed Divergence (Acceleration) Term(Jet Theory) A B 50 kts CONV DIV 100 kts Entrance Region Exit Region JET DIV CONV B’ A’

  31. Term 3: Speed Divergence (Acceleration) Term(Jet Theory) A B 50 kts CONV DIV 100 kts Entrance Region Exit Region JET DIV CONV B’ A’ CONV DIV DIV CONV Warm Air Rising & Cooling Cold Air Sinking & Warming Cold Air Rising & Cooling Warm Air Sinking & Warming JET JET A A’ B B’

  32. Term 3: Speed Divergence (Acceleration) Term(Jet Theory) • Thermally Direct Circulation (Review) • Warm air rising / cold air sinking • Generates Kinetic Energy • Jet accelerates in entrance region • Vertical motion pattern weakens thermal gradient • Restores thermal wind balance • Rising warm air cools warm side of jet • Sinking cold air warms cool side of jet • Height gradient relaxes • Jet streak progrades out of area

  33. Term 3: Speed Divergence (Acceleration) Term(Jet Theory) • Thermally Indirect Circulation (Review) • Cold air rising / warm air sinking • Destroys Kinetic Energy • Jet decelerates in exit region • Vertical motion pattern strengthens thermal gradient • Restores thermal wind balance • Rising cold air cools cold side of jet • Sinking warm air warms warm side of jet • Height gradient strengthens • Jet streak progrades into area

  34. Term 3: Speed Divergence (Acceleration) Term(Jet Theory) 50 kts CONV DIV 100 kts Entrance Region Exit Region JET DIV CONV Remember that PVA/NVA is an indicator of DIV/CONV But it’s actually the Vag that makes the DIV/CONV possible

  35. Term 2 - Example

  36. Term 2 & Term 3 • Term 2 – Curvature term • Cyclonic curvature, Vag due to term 2 points against the flow (-t direction) • Anticyclonic curvature, Vag due to term 2 points with the flow (+t direction) • Term 3 – Speed Acceleration Term • Winds accelerating along parcel path, Vag due to term 3 points in +n direction • Winds deccelerating along parcel path, Vag due to term 3 points in –n direction • Term 2 + Term 3 are very important at the jet level!

  37. Term 2 + 3 - Example

  38. Term 2 + 3 - Example

  39. Term 2+3 - Example

  40. Ageostrophic Wind Equation (Natural Coordinates) • Term 1: Isallohypsic or isallobaric Acceleration Term • Height tendency / pressure tendency term • Term 2: Centripetal or Centrifugal Acceleration Term • Curvature Term • Important along troughs/ridges • Term 3: Speed Divergence Term • Speed Acceleration Term • Look for significant changes in velocity along the flow • Important in entrance/exit regions of jet streaks • Term 4: Vertical Advection Term • Relates to the thermal field • Note the omega  vertical motion in pressure coordinates • Term 5: Friction Term • Most important at the surface • Friction always opposes the wind Term 1 Term 2 Term 3 Term 4 Term 5

  41. Term 4: Vertical Advection Term • Term 4 involves vertical motion (omega) and the vertical wind shear (dV/dp)

  42. Term 4: Vertical Advection Term • If we assume that the vertical shear of the geostrophic wind is much larger than the vertical shear of the ageostrophic wind, we can relate the vertical shear to the horizontal temperature gradient through the thermal wind principle • So, Term 4 becomes: • Board notes

  43. Term 4: Vertical Advection Term • Solving the cross-product, we come up with some simple conceptual relationships • These relationships hinge on the sign of omega

  44. Term 4: Vertical Advection Term • When: • The sign of omega > 0 • We have subsidence and Vag via term 4 points in the direction of warmer air • i.e., Vag is directed against the thermal gradient • The sign of omega < 0 • We have lifting and Vag via term 4 points in the direction of colder air • i.e., Vag is directed along the thermal gradient

  45. Term 4: Vertical Advection Term • Term 4 example: • Board notes

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