U.S. Dept. of Commerce / NOAA / OAR / PMEL / Publications

The upper ocean heat balance in the western equatorial Pacific warm pool during September-December 1992

Meghan F. Cronin and Michael J. McPhaden

Pacific Marine Environmental Laboratory, NOAA, Seattle, Washington

Journal of Geophysical Research, 102(C4), 8533-8553 (1997)
This paper is not subject to U.S. copyright. Published in 1997 by the American Geophysical Union.

5. The Niiler-Kraus Entrainment Mixing Parameterization

As shown in Figure 9, the residual of the heat budget is barely larger than its error (estimated through propagation of errors) and is therefore likely to involve more than just mixing. Thus we use a second, indirect method of estimating entrainment mixing using the turbulent energy balance. Assuming horizontally homogeneous and steady state turbulence in which the dissipation is proportional to the generation of turbulence, we have [Niiler and Kraus, 1977]

eq04.gif (2461 bytes) (4)

where w = dh/dt + w-h is the entrainment velocity (w 0); b = -g( - -h)/0 is the buoyancy of the vertically averaged layer relative to the base of the layer (g is gravity); h is the layer depth; Rib = bh |v| is the bulk Richardson number (v = v - v-h is the layer velocity relative to the base of the layer); m, n, and s are empirically determined efficiency factors; u* = (/) is the frictional velocity ( is wind stress); B is the surface buoyancy flux due to heat and freshwater fluxes adjusted for the surface penetrative radiation buoyancy flux J (38% of the incoming shortwave radiation); and is the extinction depth (taken to be 22.1 m). Following Niiler and Kraus, we assume a single exponential for the penetrative radiation in deriving (4) since for h >>> 1 m, the first term in (3) is essentially zero. Note, however, that our (4) differs from Niiler and Kraus [1977] by the inclusion of the term (h + 2) J e-h, which can be large when h   -1 ~ 22 m.

The efficiency factors m, n, and s express how much of the turbulence is dissipated within the layer rather than entrained across the layer depth. A value of m = 0, n = 0, or s = 0 would mean that all turbulence generated by wind stirring, free convection, or shear instability, respectively, is dissipated and unavailable for entrainment mixing. Consequently, the layer depth would be a material surface. As discussed by McPhaden [1982], the s parameter can be interpreted as a critical bulk Richardson number, above which shear production is effectively dissipated.

The Niiler-Kraus entrainment velocity is calculated from (4) using hourly data. As with the surface layer heat balance, the hourly estimate of entrainment cooling is then filtered with a 5-day triangular filter and subsampled once per day. Figure 11a shows the tendency rates for the three cases of purely wind-generated turbulence (m = 1, n = 0, s = 0), purely buoyancy-driven turbulence (m = 0, n = 1, s = 0), and purely shear-generated turbulence (m = 0, n = 0, s = 1). As one might expect from the strong correlation between surface fluxes and wind speed, the mixing rates due to wind stirring are quite similar to the mixing rate due to free convection, although free convection is a more efficient source of turbulence than the wind stirring. Shear-generated turbulence, on the other hand, is an extremely efficient source of turbulence only when the shear is sufficiently strong and the relative buoyancy is sufficiently low. At other times, the shear-generated turbulence is negligible.

 

fig11sm.gif (7265 bytes)

Figure 11. (a) Niiler and Kraus's [1977] parameterization of the entrainment mixing rate for the three cases of pure wind-generated turbulence (m = 1, n = 0, s = 0), pure free convection (m = 0, n = 1, s = 0), and pure shear-generated turbulence (m = 0, n = 0, s = 1). (b) Time series of the heat balance residual (solid curve) and the optimized Niiler-Kraus entrainment mixing rate (m = 0.4, n = 0.6, s = 0.6) (dashed curve). The heat balance residual includes entrainment mixing, diffusion, heat convergence due to the stratified and shear flow, and errors. The rms error of the observed residual is indicated by shading.

 

To determine the optimal empirical values, mixing rates were evaluated for the parameter range m = 0 - 2, n = 0 - 1, and s = 0 - 1. The 5-day filtered and subsampled mixing rates were then compared to the observed residual tendency rate for periods in which the 5-day filtered residual tendency rate was negative (primarily during the first month of the record). During this period, the mixing rate with parameters m = 0.4, n = 0.6, and s = 0.6 was found to have the minimal mean and rms difference and maximum correlation with the observed residual tendency rate. This optimal Niiler-Kraus mixing rate and the observed residual tendency rate are shown in Figure 11b. Both the large errors and the simplicity of the Niiler-Kraus parameterization preclude a detailed comparison. Nevertheless, as can be seen in Figure 11b, both time series imply higher mixing rates in September and October when the pycnocline was shallow, and lower mixing rates in November and December when the pycnocline was deep. Relatively low values of parameterized turbulent entrainment in late October indicate that the deepening of the pycnocline at that time (Figure 4c) was not due to turbulent mixing. This is consistent with our interpretation that the deepening is due to westerly wind-driven meridional Ekman convergence in the surface layer.


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