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
Ageostrophic
response
Geostrophic 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)
æ k ¶V ö æ V 2 ö æ V ¶V ö æ w k ¶V ö æ k
ö
Vag = ç ´ ÷ - ç
t ÷+ç
n÷ + ç
´ ÷-ç ´ F÷
è f ¶t ø è fRs ø è f ¶s ø è f ¶p ø è f
ø
Term 1
•
Term 2
Term 3
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
Term 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
Ageostrophic Wind Equation
(Natural Coordinates)
æ k ¶V ö æ V 2 ö æ V ¶V ö æ w k ¶V ö æ k
ö
Vag = ç ´ ÷ - ç
t ÷+ç
n÷ + ç
´ ÷-ç ´ F÷
è f ¶t ø è fRs ø è f ¶s ø è f ¶p ø è f
ø
Term 1
•
Term 2
Term 3
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
Term 5
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:
k ¶V -g ¶z g é ¶z ù
´
» 2 Ñ » 2 ê-Ñ ú
f ¶t
f
¶t f ë ¶t û
**Gradient of the
height change
with time**
Term 1
• For a constant height surface (i.e. surface charts)
k ¶V a é ¶p ù
´
» 2 ê-Ñ ú
f ¶t
f ë ¶t û
**Gradient of the
pressure change with
time**
Term 1
Isallohypsic 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
Term 1
Isallohypsic 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!)
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
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)
æ k ¶V ö æ V 2 ö æ V ¶V ö æ w k ¶V ö æ k
ö
Vag = ç ´ ÷ - ç
t ÷+ç
n÷ + ç
´ ÷-ç ´ F÷
è f ¶t ø è fRs ø è f ¶s ø è f ¶p ø è f
ø
Term 1
•
Term 2
Term 3
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
Term 5
Term 2: Centripetal or Centrifugal Acceleration
(curvature) Term
-V
-V
t=
t
æ ¶b ö
fRs
fç ÷
è ¶s ø
2
2
¶b
Rs =
¶s
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
Term 2: Centripetal or Centrifugal Acceleration
(curvature) Term
¶b
¶s
Positive for cyclonic curvature
Negative for anticyclonic curvature
Curvature of the flow defines the sign of this term!
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
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
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
Term 2: Curvature Term
(Supergeostrophic Winds)
(Supergeostrophic Winds)
Ridge
Ridge
CONV
970
dam
DIV
990
dam
970
dam
Trough
(Subgeostrophic Winds)
300-mb Isobaric Surface
990
dam
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
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
Term 2 - Example
Term 2 - Example
Ageostrophic Wind Equation
(Natural Coordinates)
æ k ¶V ö æ V 2 ö æ V ¶V ö æ w k ¶V ö æ k
ö
Vag = ç ´ ÷ - ç
t ÷+ç
n÷ + ç
´ ÷-ç ´ F÷
è f ¶t ø è fRs ø è f ¶s ø è f ¶p ø è f
ø
Term 1
•
Term 2
Term 3
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
Term 5
Term 3: Speed Divergence (Acceleration) Term
æ V ¶V ö
Vag = ç
n÷
è f ¶s ø
• 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
• 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
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
Term 3: Speed Divergence (Acceleration) Term
(Jet Theory)
A
B
CONV
DIV
Entrance
Exit
JET
Region
DIV
Region
CONV
A’
B’
Term 3: Speed Divergence (Acceleration) Term
(Jet Theory)
A
B
CONV
DIV
Entrance
Exit
JET
Region
Region
DIV
CONV
B’
A’
CONV
Cold Air
Sinking &
Warming
A
DIV
JET
Warm Air
Rising &
Cooling
A’
DIV
Cold Air
Rising &
Cooling
B
CONV
JET
Warm Air
Sinking &
Warming
B’
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
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)
CONV
Entrance
DIV
But it’s actually
the Vag that makes
the DIV/CONV
possible
Exit
JET
Region
Remember that
PVA/NVA is an
indicator of
DIV/CONV
DIV
Region
CONV
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 2 + 3 - Example
Term 2 + 3 - Example
Term 2+3 - Example
Ageostrophic Wind Equation
(Natural Coordinates)
æ k ¶V ö æ V 2 ö æ V ¶V ö æ w k ¶V ö æ k
ö
Vag = ç ´ ÷ - ç
t ÷+ç
n÷ + ç
´ ÷-ç ´ F÷
è f ¶t ø è fRs ø è f ¶s ø è f ¶p ø è f
ø
Term 1
•
Term 2
Term 3
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
Term 5
Term 4: Vertical Advection Term
æ w k ¶V ö
Vag = ç
´ ÷
¶p ø
è f
• 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 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
Term 4: Vertical Advection Term
• Term 4 example:
– Board notes
Ageostrophic Wind Equation
(Natural Coordinates)
æ k ¶V ö æ V 2 ö æ V ¶V ö æ w k ¶V ö æ k
ö
Vag = ç ´ ÷ - ç
t ÷+ç
n÷ + ç
´ ÷-ç ´ F÷
è f ¶t ø è fRs ø è f ¶s ø è f ¶p ø è f
ø
Term 1
•
Term 2
Term 3
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
Term 5
Term 5: Friction Term
æk
ö ak
Vag = - ç ´ F ÷ =
´Vg
èf
ø f
• Where:
– a = friction constant which depends on the roughness of
the terrain
– “a” increases with roughness
Term 5: Friction Term
æk
ö ak
Vag = - ç ´ F ÷ =
´Vg
èf
ø f
• 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
æk
ö ak
Vag = - ç ´ F ÷ =
´Vg
èf
ø f
•
•
•
•
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