Geostrophic Flow

1
Geostrophic Flow

• For synoptic scale motion in the atmosphere above
  the friction layer, two terms dominate the
  equations and are approximately in balance
• Coriolis force
• Pressure gradient force
• This balance is known as geostrophic equilibrium
• This holds for both the x and y component
  equations

2
Geostrophic Flow

• We can write the geostrophic equations as
• The subscripts g on the velocity components are
  to emphasize that the winds associated with the
  solution to these equations is the geostrophic
  wind
• Note that the meridional geostrophic velocity
  depends on the zonal pressure gradient, and that
  the zonal geostrophic velocity depends on the
  meridional pressure gradient

3
Geostrophic Flow

• We can also write the geostrophic equations with
  respect to the height of a constant pressure
  surface
• f is the Coriolis parameter f 2?sin?
• g is the Earths gravity

4
Geostrophic Flow

• We can combine the two scalar equations into a
  vector one
• (6.2)
• The geostrophic equation is diagnostic
• This means that if we know the height field, we
  can determine the present wind field
• But we can NOT use the equation to forecast
  future values of the wind
• The wind speed is proportional to the magnitude
  of the pressure gradient, and it direction is
  parallel to the isobars
• Low pressure is to the left in the Northern
  Hemisphere
• This result is also known as Buys Ballot Law

5
Geostrophic Flow Implications

• 1) Thermodynamics controls the dynamics
• The geostrophic wind is determined by the
  pressure field in the atmosphere
• But the pressure field in the atmosphere is
  determined by the mass distribution
• And the mass distribution depends on the density
  field
• And the density field is set by the temperature
  field through the ideal gas law

6
Geopotential

• Geopotential - defined such that the change in
  the geopotential, d?, over the vertical distance,
  dz, is given by the amount of work done (or
  potential energy gained) in raising a mass M
  through a vertical distance dz against the force
  of gravity
• Note that units are energy per unit mass (J kg-1
  or m2 s-1)
• The approximation comes because of small
  variations of g with height, which we will ignore

7
Geopotential

• We can also write the geostrophic wind equations
  in terms of the geopotential

8
Geostrophic Thermal Wind

• The geostrophic thermal wind is not actually a
  wind
• For example, it units are not m/s
• Actually it is the vertical shear of the
  geostrophic wind
• NOT the vertical shear of the actual wind
• We can thus write a geostrophic thermal wind
  equation

9
Geostrophic Thermal Wind

• The thermal wind vector lies in the plane of the
  constant pressure surface
• It is also perpendicular to the temperature
  gradient
• Therefore, the thermal wind vector runs parallel
  to the isotherms
• Thus, in the free atmosphere the tendency for the
  wind to run parallel to isotherms (or thickness
  contours) is consistent with the geostrophic idea
• The thermal wind tends to rotate the wind vector
  into alignment with the isotherms

10
Geostrophic Thermal Wind
11
Geostrophic Thermal Wind

• Notes
• Only in regions where the wind vector cuts across
  the isotherms do we have temperature advection
• The thermal wind vector points with colder air on
  its left
• Above the friction layer, merely by observing the
  turning of the wind with height on a sounding, we
  can infer whether warm or cold advection is
  occurring locally
• On soundings with little change in wind direction
  with height, no temperature advection is occurring