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 A geostrophic response G eostrophic forcing

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

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

Ageostrophic Wind Equation (Natural Coordinates) Term 1Term 2Term 3Term 4Term 5 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

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

Ageostrophic Wind Equation (Natural Coordinates) Term 1Term 2Term 3Term 4Term 5 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 Isallohypsic 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

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

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

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 Term 1 Isallohypsic or Isallobaric Acceleration Term

Since f is in the denominator of Term 1, V ag 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 V ag than the same height change at 60N – i.e., a smaller height change at low latitudes will produce a larger V ag A developing hurricane (rapid height/pressure falls) produces a massive V ag (also occurs at low latitudes!) Term 1 Isallohypsic 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 Term 1 Isallohypsic 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 Laplacian and a height rise center would have a negative Laplacian Example with falling heights Example with rising heights Term 1 Isallohypsic or Isallobaric Acceleration Term

Term 1 Pressure/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

Ageostrophic Wind Equation (Natural Coordinates) Term 1Term 2Term 3Term 4Term 5 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 2: Centripetal or Centrifugal Acceleration (curvature) Term Change in a parcel direction along parcel’s path R s = radius of curvature along parcel’s path R s > 0 for cyclonic curvature (trough) R s < 0 for anticyclonic curvature (ridge) t is positive along direction of flow So, in reality, only Rs will change the sign of our term

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

Term 2: Centripetal or Centrifugal Acceleration (curvature) Term Example for cyclonic flow So, for cyclonic flow, V ag 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 V ag opposes motion). Thus cyclonic flow is sub-geostrophic flow – Hey! This matches what we talked about before

Term 2: Centripetal or Centrifugal Acceleration (curvature) Term Example for anticyclonic flow So, for anticyclonic flow, V ag 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 V ag is with the actual motion). Thus anticyclonic flow is super-geostrophic flow – Hey! This also matches what we talked about before

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

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

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 V ag(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

Term 2: Centripetal or Centrifugal Acceleration (curvature) Term The stronger the curvature, the stronger the V ag 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

Term 2 - Example

Ageostrophic Wind Equation (Natural Coordinates) Term 1Term 2Term 3Term 4Term 5 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 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

Term 3: Speed Divergence (Acceleration) Term For a parcel accelerating along its path: – Board notes V ag 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

Term 3: Speed Divergence (Acceleration) Term For a parcel decelerating along its path: – Board notes V ag 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

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

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

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 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 Term 3: Speed Divergence (Acceleration) Term (Jet Theory)

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

Term 2 - Example

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!

Term Example

Ageostrophic Wind Equation (Natural Coordinates) Term 1Term 2Term 3Term 4Term 5 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 4: Vertical Advection Term Term 4 involves vertical motion (omega) and the vertical wind shear (dV/dp)

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

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

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

Term 4: Vertical Advection Term Term 4 example: – Board notes

Ageostrophic Wind Equation (Natural Coordinates) Term 1Term 2Term 3Term 4Term 5 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 5: Friction Term Where: – a = friction constant which depends on the roughness of the terrain – “a” increases with roughness

Term 5: Friction Term By the right-hand rule, Vag (due to friction) will always be directed in the direction of +n i.e., Vag is always directed down the pressure gradient along, or to the left, of the direction of geostrophic flow

Term 5: Friction Term Frictionless flow example a = small - example a = large - example a = infinity - example

Term 5: Friction Term As a first approximation in the boundary layer: – Board notes Thus, for cyclonic flow, where geostrophic relative vorticity is greater than zero: – Board notes

Term 5: Friction Term This effect is often called “boundary layer pumping” – Due to frictional effects near the surface This boundary layer pumping results in low-level convergence and lifting This shallow lifting near the surface is responsible for producing low-cloud decks (such as stratocumulus) associated with low-level cyclonic flow around surface lows and surface troughs (For more details see Eckman Spiral section in Holton, Chapter 5)

Term 5: Friction Term For anticyclonic flow, where geostrophic relative vorticity is less than zero: – Board Notes This effect (due to friction near the surface) results in low-level divergence and subsidence – Board notes

Term 5: Friction Term This is many times related to the clear skies commonly associated with low-level anticyclonic flow around a surface high or surface ridge