# 10.23.10: Sensitivity of the decomposition to the definition of the surface mixed layerΒΆ

Using the test (one surface mode, a barotropic and second baroclinic mode for one specific meridional wavelength), I show how sensitive the surface value of the surface mode is to the definition of the bottom of the surface mixed layer (SML). The bottom is defined as the depth for which T reaches a threshold value. T is the ratio of N/f over the max of N/f where N is the buoyancy frequency and f is the local Coriolis parameter.

In Figs. 1 and 2, we compare the case without SML with the cases with SML with T = 0.5 and 0.95. With T = 0.5, there is a depth range for which N, the buoyancy frequency is decreasing. In this depth range, dE/dz is also decreasing upward (red curve in Fig. 1b) explaining its local maximum. With T = 0.95, we avoid the transition from large to weak N and dE/dz looks everywhere below the SML exponentially decaying; its surface value has also changed. The surface value is thus tightly dependent on the depth used to define the SML (dashed lines in Fig. 1). This can be seen in the different values of the surface mode (at the surface but also at depth) for the potential density (Fig. 2). I have also found (not shown) that performing the calculation over a depth range that completely exclude the SML gives the most accurate decomposition (where the true decomposition is the one given by the case without the SML).

Figure 1: (a) Squared buoyancy frequency and (b) dE/dz (the surface mode for the potential density) for one particular wavelength: no SML (blue), SML with T = 0.5 (red) SML with T = 0.95. T is the threshold to define the bottom of the SML; it is the ratio of N/f over the max of N/f where N is the buoyancy frequency and f is the local Coriolis parameter.

Figure 2: Total, surface and interior component of the potential density: (a) no SML, (b) SML with T = 0.5 and (c) SML with T = 0.95. Notice how sensitive is the surface value of the surface component to T.

This might be the cause of why the surface value of the surface mode in potential density in the decomposition of the OFES fields tend to be larger than the actual surface potential density. Indeed, in Fig. 3, we perform the decomposition for the Gulf Stream in OFES for three cases: T=0.5 (left), T=0.95 (middle) and the depth range used to compute the surface mode is 150-400 m instead of from below the SML to 400 m as in the first two cases (right).

Figure 3: Total, surface and interior component of the potential density at one location in the Gulf Stream region for the OFES fields: (a) T = 0.5, (b) T = 0.95 and (c) the depth range used to compute the surface mode is 150-400 m instead of from below the SML to 400 m as in the first two cases. Notice how the reconstruction is more accurate either with a more constrained definition of the SML (as in b) or by avoiding completely the SML in the calculation (as in c).