Matters of state and motion - Physical Processes in Earth and Environmental Sciences - page 34

VERTICAL MIXING

River water OUT

River water IN

Q rwout

Q rwin

Sea water IN

Q swout

Q swin

Sea water OUT

Fig. 2.23 Estuarine circulation: example of mass conservation in action.

2.5.3 Examples of volume and mass continuity

Plan Section

1 Delta or estuary channels are informative environments
within which to consider the workings of continuity
(Fig. 2.23). For any control volume the upstream dis-
charge of seawater decreases while the downstream input
of fresh river water decreases. A mass balance is brought
about by vertical mixing of seawater upward and freshwa-
ter downward.
2 It is instructive to apply the 3D volume continuity
expression for an incompressible fluid such as that found in
an idealized portion of fast-moving ocean, river, or tidal
shelf. It is usually fairly straightforward to measure the two
mean surface components of the local velocity but more
difficult to measure the time mean vertical velocity. We
compute this useful parameter from the basic conservation
expression in Cookie 3.
3 We finally touch upon divergence and convergence with
respect to sources and sinks. We stated that the continuity
expression depends upon the lack of sources or sinks
linked to the system in question. Two important cases arise
in hydrological, oceanographical, and meteorological
flows (Fig. 2.24; see also Cookie 3). Surface divergence of
streamlines, most obviously seen when flow is diverging
from a point implies that a source is present below the sur-
face, leading to a mass influx. Surface convergence of
streamlines to a point implies a sink is present and that
downwelling is occurring. An added complication for
meteorological flows is that vertical motions of fluid in
downwelling or upwelling situations also cause changes of
temperature and density, which cause feedback relevant to
the stability of a moving air mass.

Surface DIVERGENCE from a point is a source,
causes upwelling

Plan Section

Surface CONVERGENCE to a point is a sink,
causes downwelling

Fig. 2.24 Sources and sinks.

in the case of compressible gas flow or a thermally varying
flow, means there is one more degree of freedom than in
the case considered previously; we have: u 1 A 1 1
u 2 A 2 2 , so that any change in net mass outflow per unit
time (check the expression gives units MT 1 ) is now
caused by a change in density and/or velocity.
The full algebraic expression for 3D continuity is given
in Cookie 2 (the algebra looks hideous but is quite
logical).