Geoscience Reference
In-Depth Information
8
The equations of atmospheric turbulence
8.1 Introduction
We shall begin by deriving the governing equations for a dry atmosphere. In the
shallow-fluid limit they become those traditionally used for laboratory turbulence
with heat transfer and buoyancy. We then generalize to moist air (dry air and water
vapor) and to cloud air (dry air, water vapor, and water droplets).
Density variations impact turbulence in the lower atmosphere in two ways. First,
the decrease of density with height (about 10% in the first kilometer) makes us
rethink some of our intuitive notions about turbulent mixing. Second, we'll see that
the density fluctuations produced by the surface heat transfer can make daytime
and nighttime turbulence as different as night and day.
8.2 The governing equations for a dry atmosphere
8.2.1 An isentropic, hydrostatic base state
Most of the vertical decrease of temperature, pressure, and density in the lower
atmosphere can be described through the hydrostatics of an ideal gas. We begin
†
with a motionless, adiabatic
base state
, denoted with a subscript zero, and write the
ideal gas law
for a dry atmosphere as
p
0
=
ρ
0
R
d
T
0
,
(8.1)
with
R
d
the gas constant for dry air. In
Section 8.3
we generalize
(8.1)
to include
water vapor and liquid water. The base-state variables depend only on height
(
x
3
or
z
).
In
Part I
we used a modified pressure whose vertical gradient includes the gravity
term in the equation of motion. Because we are now allowing density to vary we
shall return to the traditional pressure. The vertical component of the momentum
†
This development is an adaptation and extension of that of
Lumley and Panofsky
(
1964
).
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