We describe an instability that appears at the front
of laminar gravity currents as they
intrude into a viscous, miscible ambient fluid. The instability causes
a current to
segment into fingers aligned with its direction of flow. In the case of
currents flowing
along a rigid floor into a less dense fluid, the case of primary interest
here, two
mechanisms can produce this instability. The first is gravitational and
arises because
the nose of the gravity current is elevated above the floor and overrides
a buoyantly
unstable layer of ambient liquid. The second is a form of viscous fingering
analogous
to a Saffman–Taylor instability in a Hele-Shaw cell. Whereas the
ambient fluid must
be more viscous than the current in order for the latter instability to
occur, the
gravitational instability can occur even if the ambient fluid is less viscous,
as long as
it is viscous enough to elevate the nose of the current and trap a layer
of ambient fluid.
For the gravitational mechanism, which is most important when the current
and
ambient fluids have comparable viscosities, the wavelength when the instability
first
appears is proportional to a length scale constructed with the viscosity,
the flux and the
buoyancy. The Saffman–Taylor-type mechanism is most important when
the ambient
liquid is much more viscous than the current. We have carried out experiments
with
miscible fluids in a Hele-Shaw cell that show that, at the onset of instability,
the ratio
of the finger wavelength to the cell width is a constant approximately
equal to 2. This
result is explained by using the principle that the flow tends to minimize
the dissipation
associated with the finger perturbation. For the gravity currents with
high viscosity
ratios, the ratio of the wavelength to the current thickness is also a
constant of about
2, apparently consistent with the same mechanism. But, further analysis
of this
instability mechanism is required in order to assess its role in wavelength
selection for
gravity currents.
Shape of spreading and leveling gravity currents in a Hele-Shaw cell with flow-wise width variation
