| Abstract: In regions of sloping isopycnals,
isopycnal mixing acting in conjunction with biological cycling can produce
patterns in the nutrient field which have negative values of tracer in
light water and unrealistically large values of tracer in dense water.
Under certain circumstances, these patterns can start to grow unstably.
This paper discusses why such behavior occurs. Using a simple four-box
model, it demonstrates that the instability appears when the isopycnal
slopes exceed the grid aspect ration (delta z / delta x).
In contrast to other well known instabilities of the CFL type, this instability
does not depend on the time step or time-stepping scheme. Instead
it arises from a fundamental incompatibility between two requirements for
isopycnal mixing schemes, namely that they should produce no net flux of
passive tracer across an isopycnal and everywhere reduce tracer extrema.
In order to guarantee no net flux of tracer across an isopycnal, some upgradient
fluxes across certain parts of an isopycnal are required to balance downgradient
fluxes across other parts of the isopycnal. However, these upgradient
fluxes can cause local maxima in the nutrient field to become self-reinforcing.
Although this is less of a problem in larger domains, there is still a
strong tendency for isopycnal mixing to overconcentrate tracer in the dense
water. The introduction of eddy-induced advection is shown to be
capable of counteracting the upgradient fluxes of nutrient which cause
problems, stabilizing the solution. The issue is not simply a numerical
curiosity. When used in a GCM, different parameterizations of eddy
mixing result in noticeably different distributions of nutrient and large
differences in biological production. While much of this is attributable
to differences in convection and circulation, the numerical errors described
here may also play an important role in runs with isopycnal mixing alone. |