Geography Reference
In-Depth Information
CHAPTER 2
Basic Conservation Laws
Atmospheric motions are governed by three fundamental physical principles:
conservation of mass, conservation of momentum, and conservation of energy.
The mathematical relations that express these laws may be derived by considering
the budgets of mass, momentum, and energy for an infinitesimal
control volume
in the fluid. Two types of control volume are commonly used in fluid dynamics.
In the
Eulerian
frame of reference the control volume consists of a parallelepiped
of sides δx, δy, δz, whose position is fixed relative to the coordinate axes. Mass,
momentum, and energy budgets will depend on fluxes caused by the flow of fluid
through the boundaries of the control volume. (This type of control volume was
used in Section 1.4.1.) In the
Lagrangian
frame, however, the control volume con-
sists of an infinitesimal mass of “tagged” fluid particles; thus, the control volume
moves about following the motion of the fluid, always containing the same fluid
particles.
The Lagrangian frame is particularly useful for deriving conservation laws, as
such laws may be stated most simply in terms of a particular mass element of the
fluid. The Eulerian system is, however, more convenient for solving most problems
because in that system the field variables are related by a set of partial differential
equations in which the independent variables are the coordinates x, y, z, and
t
.
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