### Abstract:

The effect of non-zero, but small, viscosity and diffusivity on the marginal stability
of a stably stratified shear flow is examined by making perturbations around the
neutral solution for an inviscid and non-diffusive flow. The results apply to turbulent
flows in which horizontal and vertical turbulent transports of momentum and buoyancy
are represented by eddy coefficients of viscosity and diffusivity that vary in the
vertical (z) direction. General expressions are derived for the modified phase speed
and the growth rate of small disturbances as a function of wavenumber. To first
order in their coefficients, the effect on the phase speed of adding viscosity and
diffusivity is zero. Growth rates are found for two mean flows when the horizontal
or vertical coefficients of viscosity and diffusivity vary in z in such a way that the
rates can be found analytically. The first flow, denoted as a ‘Holmboe flow’, has a
velocity and density interface: the mean horizontal velocity and the density are both
proportional to tanh az, where a is proportional to the inverse of the interface thickness.
The second, ‘Drazin flow’, has a similar velocity variation in z but uniform density
gradient. The analytical results compare favourably with numerical calculations. Small
horizontal coefficients of viscosity and diffusivity may affect disturbances to the flow
in opposite ways. Although the effect of uniform vertical coefficients of viscosity is to
decrease the growth rates, and uniform vertical coefficients of diffusivity increase them,
cases are found in which, with suitably chosen z dependence, vertical coefficients of
viscosity (or diffusivity) may cause a previously neutral disturbance to grow (or to
diminish); viscosity may destabilize a stably stratified shear flow. The introduction of
viscosity and diffusivity may consequently increase the critical Richardson number to
a value exceeding 1=4. While some patterns of behaviour are apparent, no simple rule
appears to hold about whether flows that are neutral in the absence of these effects
(viscosity or diffusivity) will be stabilized or destabilized when they are added. One
such rule, namely our conjecture that viscosity is always stabilizing and that diffusivity
is destabilizing, is explicitly refuted.