To generalize (8), we consider the shear-free convection 
case once for the case without vertical temperature differences 
(heat number = 0) and once for the case without vertical salin- 
ity differences (salinity number = 0). From this consideration, 
minimum values a, and a; result for the heat number or the 
salinity number. Due to reasons of symmetry, a; = a; = dj. 
so that (9) becomes 
dh > ds >aps 
in the case with double diffusion. 
4 Model results 
The main motivation for the investigation of this problem was 
the occurrence of unrealistic surface currents during a storm 
event in the CMEMS-Baltic MFC NRT-forecast product ver- 
sion 1 (Fig. 1, right side), which did not occur anymore after 
including stability and realizability criteria described in 
Chapter 3 and the following change to product version 3 
(Fig. 1, left side). Of course, these criteria were not the only 
difference between version 1 and version 3, so that the com- 
parison ofthese versions is not suitable for demonstrating their 
influence. Therefore, this is demonstrated in the following by 
using the operational BSH setup during a storm event in 
December 2013, whereby the model runs only differ in their 
different turbulence closure schemes. 
4.1 (Physical) model validation 
By chance, temporary unrealistic surface currents during a 
storm event in December 2013 were also discovered in the 
operational setup of the BSH (Fig. 2, left column). The fact 
that the water levels at the tide gauges were simulated 
Fig. 1 (taken from CMEMS 66°N 
(n.d.)) Surface currents for V3 
and V1 products: direction are 
shown in arrows; magnitude is 
coloured; snapshot during strong 
wind event (13 December 2014 4 63°N 
UTC) 
HBM V3 prodı" 
50°N 
Ocean Dynamics 
correctly led to the conclusion that the water transports as a 
whole were correct, but the turbulence scheme in this situation 
was locally not correctly solved or unstable. The assumption 
that this was a local instability was also confirmed by the €- 
test (see Chapter 4.2). 
Moreover, the assumption was confirmed by looking at the 
current and eddy diffusivity profiles at 6.55° E/55.58° N 
(Fig. 3, left column). Tests with Canuto et al. (2010)-based 
closure schemes (Figs. 2 and 3, middle and right column) 
show that all used closure schemes provide comparable cur- 
rent patterns before the peak of the storm and therefore in a 
more or less common situation (Figs. 2 and 3, first row). The 
Jdiffusivity profiles of the closure schemes without explicit 
realizability and stability checks, on the other hand, already 
look partly nonphysical (Fig. 3, first row, left and middle 
column). In the described storm case, the diffusivity profiles 
{hen appear to become completely unstable, so that a clear 
stratification of the currents occurs at a depth of approximately 
5-20 m. While enormous surface currents are directed to- 
wards the east, the water flows into the opposite direction at 
depth, so that the entire water transport fits again. Realistic 
oarofiles and current patterns in all situations are only achieved 
with a closure scheme that has been extended by stability and 
realizability criteria (10), (11) and (12) described in 
Chapter 3.2 (Fig. 2 and 3, right column). 
4.2 g-Tests/technical model validation 
The e-test is an originally technical test in which the results of 
short model runs (e.g. 24-h simulation) are compared, the only 
difference between these runs being the compiling of the 
source code. It is done with different compilers and/or with a 
different set of compiler flags. The results of these runs are 
compared point by point, and the maximum differences are 
the £’s, where small £’s indicate both technically and 
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