Objective Determination of Global Ocean Mixed Layer and Isothermal Layer Depths
Upper ocean is characterized by the existence of vertically quasi-uniform layer of temperature (T, isothermal layer) and density (ρ, mixed layer). Underneath it, there exists a layer with strong vertical gradient such as the thermocline (in temperature) and pycnocline (in density). The intense vertical turbulent mixing near the surface causes the vertically quasi-uniform layer. The mixed layer is a key component in studies of climate and the link between the atmosphere and deep-ocean. It directly affects the air-sea exchange of heat, momentum and gases. The mixed layer (or isothemal layer) depth, Hmix, is an important parameter which largely affects the evolution of the sea surface temperature (SST).
Three types of criteria are available to determine Hmix on the base of difference, gradient, and curvature. The difference criterion requires the deviation of T (or ρ) from its surface value to be smaller than a certain fixed value. The gradient criterion requires ∂T/ ∂z (or ∂ρ/ ∂z) to be smaller than a certain fixed value. The curvature criterion requires ∂2T/ ∂z2 (or ∂2ρ/∂z2) to be maximum at the base of mixed layer (z = -Hmix). Obviously, the difference and gradient criteria are subjective. For example, the criterion for determining Hmix for temperature varies from 0.5oC (Wyrtki 1964) to 0.8oC (Kara et al. 2000). Defant (1961) was among the first to use the gradient method. He uses a gradient of 0.015oC/m to determine Hmix for temperature of the Atlantic Ocean. Bathen (1972) chose 0.02oC/m, and Lukas and Lindstrom  used 0.025oC/m. The curvature criterion is an objective method, but is relatively hard to use for noisy profiling data since the curvature involves the calculation of the second derivative versus depth. The quality index (QImix) takes 1 for perfect identification and 0 for no identification. For these existing methods, QImix is 0.70 or less.
In this paper, a progressive linear fitting with maximum error ratio method is presented for objective determination of ocean mixed layer depth. Let density profile be represented by [ρ(zi)] and taken for illustration. A linear polynomial is used to fit the profile data from the first point neat the surface (z1) to a depth, zK (marked by a circle in Fig. 1). The original and fitted data are represented by (ρ1, ρ2, …, ρK) and (), respectively. The root-mean square (rms) error between the two is represented by E1. From the depth zK downward, pick up four data points: ρK+1, ρK+2, ρK+3, and ρK+4. The linear polynomial for data points (z1, z2, …, zK) is extrapolated into the depths (zK+1, zK+2, zK+3, zK+4): . The bias of the linear fitting for the four points is denoted by E2. If the depth zK is inside the mixed layer, the linear polynomial fitting is well representative for the data points (z1, z2, …, zK+4). The absolute value of bias (E2 = |Bias|) for the lowest four points are usually smaller than E1 since differences between observed and fitted data for the lowest four points may cancel each other. If the depth zK is located at the base of the mixed layer, E2 is large and E1 is small. If the depth zK is located below at the base of the mixed layer, both E1 and E2 are large. Thus, the criterion for determining the mixed layer depth can be described as the maximum ratio of E2(zk)/E1(zk), i.e., Hmix = -zk.
The global mixed (isothermal) layer depth dataset has been established from the Global Temperature and Salinity Profile Program (GTSPP) (1990 – present) using the recently developed objective method. The quality index (QI) is computed for each profile. Impact of temporal and spatial variability of the mixed (isothermal) layer depth on climate variability is also discussed.
Chu, P.C., and C.W. Fan, 2010: Progressive linear fitting with maximum error ratio for objective determination of ocean mixed layer depth. Journal of Atmospheric and Oceanic Technology, in press.
Supplementary URL: http://faculty.nps.edu/pcchu/