Geoscience Reference
In-Depth Information
where D
Dt indicates a derivative following air parcel motion; v is the three-
dimensional wind velocity; v h is the horizontal component of the wind velocity;
=
is the density of moist air, which includes water vapor and other water substances
such as liquid water and ice; p is pressure; and t is time. An air parcel is a tiny
blob of air, a collection of air molecules. This air parcel is carried along with the
three-dimensional wind and always contains the same collection of air molecules
(i.e., the same collection of material). Unlike a mass in rigid body mechanics,
however, the parcel may change shape.
The acceleration induced by the Coriolis force fU, where f ¼ 2
O
sin
, U is
the horizontal wind speed,
is the latitude. It follows that accelerations induced by the Coriolis force are sig-
nificant for time scales 1
O
is the rotation rate of the Earth about its axis, and
f . The Coriolis force is therefore not included unless
the time scale of the phenomenon considered is at least 6 h. For a typical horizon-
tal wind speed 10m s 1 at midlatitudes, where f 10 4 s 1 , the acceleration due
to the Coriolis force is 10 3 ms 2 , while the acceleration following air parcel
motion for convection is 10m s 1 /10min 10m s 1 /10 3 s ¼ 10 2 ms 2 , which is
an order of magnitude greater. Thus, for ordinary convective storms, the Earth's
rotation plays no direct role, while for long-lived, mesoscale convective systems
Earth's rotation must be taken into account, but is a modifying—not necessarily a
fundamental—factor in nearly all cases.
Molecular and turbulent friction are @
=
2 U
z 2 , where
=@
is the molecular/
eddy coecient of viscosity;
for molecular viscosity is small and the turbulent
term is significant typically only in the lowest kilometer or so, where
z 2 is
relatively large. For the sake of simplicity and in an effort to focus on the smallest
number of important physical processes possible, turbulent and molecular friction
are not included here or in most subsequent equations except much later when
tornadoes are discussed; it is thus assumed that all the variables are time and
space averaged for the scales of motion we are considering and that subgrid-scale
turbulence is ignored.
2 U
@
=@
2.1.2 Buoyancy and the vertical equation of motion: defying gravity
The inviscid vertical equation of motion (without including the effects of Earth's
rotation) is as follows:
Dw
=
Dt ¼ 1
= @
p
=@
z g
ð 2
:
2 Þ
where w is the vertical component of the wind (the vertical velocity); and g is the
acceleration of gravity (which is taken to be a constant).
According to (2.2), vertical accelerations in convective clouds are simply a
result of the imbalance between the vertical pressure gradient force and the force
of gravity. The pressure can be decomposed into a part that is hydrostatic, which
is associated with a balance between the vertical pressure gradient force and
gravity, and a part that is non-hydrostatic, which is not. It is the vertical gradient
of the latter that is responsible for vertical accelerations. It can be assumed that
the base state of the atmosphere on the large scale is hydrostatic and that,
Search WWH ::




Custom Search