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- Pressure Coordinates: Advantage and Disadvantage
- Momentum Equation è Balanced Flows
- Thermodynamic & Momentum Eq.s èThermal Wind Balance
- Continuity Equation è Surface Pressure Tendency
- Trajectories and Streamlines
- Ageostrophic Motion
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- From the hydrostatic equation, it is clear that a single valued
monotonic relationship exists between pressure and height in each
vertical column of the atmosphere.
- Thus we may use pressure as the
independent vertical coordinate.
- Horizontal partial derivatives must be evaluated holding p constant.
- è How to treat
the horizontal pressure gradient force?
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- Thus in the isobaric coordinate system the horizontal pressure gradient
force is measured by the gradient of geopotential at constant pressure.
- Density no longer appears explicitly in the pressure gradient force;
this is a distinct advantage of the isobaric system.
- Thus, a given geopotential gradient implies the same geostrophic wind at
any height, whereas a given horizontal pressure gradient implies
different values of the geostrophic wind depending on the density.
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- Vertical Velocity in the Z-coordinate is w, which is defined as dz/dt:
- w > 0 for ascending motion
- w < 0 for descending
motion
- Vertical velocity in the P coordinate is ω (pronounced as “omega”),
which is defined as dp/dt:
- ω < 0 for ascending
motion
- ω > 0 for descending
motion
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- Following a control volume (δV= δxδyδz = -δxδyδp/ρg
using hydrostatic balance), the mass of the volume does not change:
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- This form is similar to that on the Z-coordinate, except that there is a
strong height dependence of the stability measure (Sp), which is a minor
disadvantage of isobaric coordinates.
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- Rossby number is a non-dimensional measure of the magnitude of the
acceleration compared to the Coriolis force:
- The smaller the Rossby number, the better the geostrophic balance can be
used.
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- “Geo” è Earth
- “Strophe” è
Turing
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- At any point on a horizontal surface, we can define a pair of a system
of natural coordinates (t, n), where t is the length directed downstream
along the local streamline, and n is distance directed normal to the
streamline and toward the left.
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- Because the Coriolis force always acts normal to the direction of
motion, its natural coordinate form is simply in the following form:
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- If the horizontal scale of a disturbance is small enough, the Coriolis
force may be neglected compared to the pressure gradient force and the
centrifugal force. The force balance normal to the direction of flow
becomes in cyclostrophic balance.
- An example of cyclostrophic scale motion is tornado.
- A cyclostrophic motion can be either clockwise or counter-clockwise.
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- Horizontal frictionless flow that is parallel to the height contours so
that the tangential acceleration vanishes (DV/Dt = 0) is called gradient
flow.
- Gradient flow is a three-way balance among the Coriolis force, the
centrifugal force, and the horizontal pressure gradient force.
- The gradient wind is often a better approximation to the actual wind
than the geostrophic wind.
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- It is important to distinguish clearly between streamlines, which give a
“snapshot” of the velocity field at any instant, and trajectories, which
trace the motion of individual fluid parcels over a finite time
interval.
- The geopotential height contour on synoptic weather maps are streamlines
not trajectories.
- In the gradient balance, the curvature (R) is supposed to be the
estimated from the trajectory, but we estimate from the streamlines from
the weather maps.
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- The thermal wind is a vertical shear in the geostrophic wind caused by a
horizontal temperature gradient. Its name is a misnomer, because the
thermal wind is not actually a wind, but rather a wind gradient.
- The vertical shear (including direction and speed) of geostrophic wind
is related to the horizontal variation of temperature.
- è The thermal
wind equation is an extremely useful diagnostic tool, which is often
used to check analyses of the observed wind and temperature fields for
consistency.
- It can also be used to estimate the mean horizontal temperature
advection in a layer.
- Thermal wind blows parallel to the isotherms with the warm air to the
right facing downstream in the Northern Hemisphere.
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- For synoptic-scale motions, the vertical velocity component is typically
of the order of a few centimeters per second. Routine meteorological
soundings, however, only give the wind speed to an accuracy of about a
meter per second.
- Thus, in general the vertical velocity is not measured directly but must
be inferred from the fields that are measured directly.
- Two commonly used methods for inferring the vertical motion field are
(1) the kinematic method, based on the equation of continuity, and (2)
the adiabatic method, based on the thermodynamic energy equation.
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- We can integrate the continuity equation in the vertical to get the
vertical velocity.
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- The adiabatic method is not so sensitive to errors in the measured
horizontal velocities, is based on the thermodynamic energy equation.
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- Barotropic Atmosphere
- èno temperature
gradient on pressure surfaces
- isobaric surfaces are also the isothermal surfaces
- density is only function of pressure ρ=ρ(p)
- no thermal wind
- no vertical shear for geostrophic
winds
- geostrophic winds are independent
of height
- you can use a one-layer model to
represent the barotropic atmosphere
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- Baroclinic Atmosphere
- ètemperature
gradient exists on pressure surfaces
- density is function of both
pressure and temperature ρ=ρ(p, T)
- thermal wind exists
- geostrophic winds change with height
- you need a multiple-layer model
to represent the baroclinic atmosphere
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