### Abstract:

The Taylor–Goldstein (T–G) equation is extended to include the effects of small-scale
turbulence represented by non-uniform vertical and horizontal eddy viscosity and
diffusion coefficients. The vertical coefficients of viscosity and diffusion, A[subscript V] and K[subscript V],
respectively, are assumed to be equal and are expressed in terms of the buoyancy
frequency of the flow, N, and the dissipation rate of turbulent kinetic energy per unit
mass, ε, quantities that can be measured in the sea. The horizontal eddy coefficients,
A[subscript H] and K[subscript H], are taken to be proportional to the dimensionally correct form, ε[superscript 1/3]∫[superscript 4/3],
found appropriate in the description of horizontal dispersion of a field of passive
markers of scale ∫. The extended T–G equation is applied to examine the stability
and greatest growth rates in a turbulent shear flow in stratified waters near a sill,
that at the entrance to the Clyde Sea in the west of Scotland. Here the main effect
of turbulence is a tendency towards stabilizing the flow; the greatest growth rates of
small unstable disturbances decrease, and in some cases flows that are unstable in the
absence of turbulence are stabilized when its effects are included. It is conjectured that
stabilization of a flow by turbulence may lead to a repeating cycle in which a flow
with low levels of turbulence becomes unstable, increasing the turbulent dissipation
rate and so stabilizing the flow. The collapse of turbulence then leads to a condition
in which the flow may again become unstable, the cycle repeating. Two parameters are
used to describe the ‘marginality’ of the observed flows. One is based on the proximity
of the minimum flow Richardson number to the critical Richardson number, the other
on the change in dissipation rate required to stabilize or destabilize an observed flow.
The latter is related to the change needed in the flow Reynolds number to achieve zero
growth rate. The unstable flows, typical of the Clyde Sea site, are relatively further
from neutral stability in Reynolds number than in Richardson number. The effects of
turbulence on the hydraulic state of the flow are assessed by examining the speed and
propagation direction of long waves in the Clyde Sea. Results are compared to those
obtained using the T–G equation without turbulent viscosity or diffusivity. Turbulence
may change the state of a flow from subcritical to supercritical.