Thorpe Analysis

The moorings provide a time series of coarsely resolved vertical temperature profiles. We use the collocated temperature and salinity measurement (Figure 2.4), to infer salinity at all the sensors depths and to calculate potential density profiles. A noticeable feature of these profiles is the frequent occurrence of statically unstable patches, or overturns. The detection and analysis of these overturns provides an estimate of mixing. Levine and Boyd (2005) used this method to estimate dissipation and mixing at a mooring near the Kaena Ridge crest, and Finnigan et al. (2002) performed this analysis on CTD data at two locations near the Hawaiian Ridge.

We detect an overturn using the following algorithm. First, inflection points in the density profile are detected where the density profile changes from stable to unstable (point A in Figure 5.1). The lower extent of the overturn (point B) is taken as the next point below point A where the density of A and B are equal. The point with the minimum density between point A and B is taken as point C, and the upper limit of the overturn (point D) is taken as the point above A that has the same density as C. Any small overturn contained within a bigger overturn is discarded, and overlapping overturns are combined into one. Similar results are obtained using potential temperature instead of potential density.

Within each detected overturn of vertical size $H$ , the unstable profiles are reordered into stable ones. Each sample $\rho_{n}$ initially at a depth $z_{n}$ is assigned a new depth $z_{m}$ in the reordered profile. The difference $d'_{n}=z_{m}-z_{n}$ is called the Thorpe displacement (Thorpe, 1977), and the Thorpe scale is defined as the root mean square of this quantity for each re-ordered overturn :

$$L_{T}=\overline{({d'^{2}})}^{1/2}$$ (11)

The Thorpe scale can be related to another measure of turbulent dissipation, the Ozmidov length scale (Ozmidov 1965)

$$L_{0}=\epsilon^{1/2}N^{-3/2}$$ (12)

Several studies have shown a linear relationship between $L_{T}$ and $L_{0}$ (Ferron et al., 1998; Dillon, 1982). Dissipation can be estimated by

$$\epsilon=a^{2}N^{3}L_{T}^{2}$$ (13)

where $a=L_{0}/L_{T}$ . The value of $a$ used in previous studies varies between 0.65 and 0.95 (see Finnigan et al. (2002) for a review). For the sake of consistency with the mooring study of Levine and Boyd (2005), we use $a=0.8$ in this analysis. Levine and Boyd (2005) also discuss the validity of the Thorpe scale method for estimating dissipation.

For each detected overturn of size $H_{i}$ , we calculate the dissipation $\epsilon_{i}$ using equation (5.1.3). For each profile, the average dissipation for the bottom layer is $\epsilon=\frac{1}{H}\sum\epsilon_{i}H_{i}$ where $H= 200m$ .

In (5.1.2) and (5.1.3), $N$ is obtained from the reordered profile, and therefore is always real. Following Osborn (1980), we also compute the vertical eddy diffusivity coefficient $K_\rho$ by assuming that the turbulent kinetic energy balance is between shear production, buoyancy loss and turbulent dissipation,

$$K_{\rho}=\Gamma\epsilon \bar{N}^{-2}$$ (14)

The mixing efficiency $\Gamma$ is taken equal to 0.2, and $\bar{N}$ is now the background stratification.