Ocean Dynamics 
3 Realizability and stability 
3.1 General explanation 
Of course, models are always a simplification of reality. In 
particular, numerical models actually solve only mathematical 
equations, so unfortunately such models are not realizable in 
general, meaning that the quantities, which are positive by 
definition, may become negative due to the model implemen- 
tation. Explicitly implemented realizability criteria in the form 
of limiters are intended to ensure that the physically relevant 
model state resp. the physically relevant parameter range 1s 
not left, so that the model does not predict unphysical values, 
such as, for example, negative variances, diffusion coeffi- 
cients, length scales, tracer concentrations, masses and 
volumes. 
In practice stability and realizability are often difficult to 
distinguish, both are local characteristics, which might be trig- 
gered (in different directions) by small numeric discrepancies 
(e.g. due to more or less aggressive compiler tuning flags). 
They can spread, disappear or even lead to model crashes. 
Therefore, the goal is to define explicit criteria that guarantee 
robust numeric. 
3.2 Additional stability/realizability criteria 
As described in Berg (2012), the three vertical diffusivities, 
namely, K,, for momentum, K, for heat and K, for salinıty can 
be expressed as 
k* 
K; m 2—S8i; with ı = m,h,s 
€ 
where & is the turbulent kinetic energy and € is the dissipation 
rate of turbulent kinetic energy. The dimensionless functions 
S; are the structure or stability functions, which depend on 
three variables and can be written as 
Si = Siıl(am, An, as) with i= m,h,s and (2) 
Am = (72, dh = T”Ra, As — TR: (3) 
Equation (3) describes the dimensionless shear number, 
heat number and salinity number with the dynamic dissipation 
time scale 7 = 2£ ‚ the mean shear X, RR’ = ars and 
R, = as. In this case, z is the water depth, 7 the water 
temperature, S the salinity and ars = — Or denotes the 
thermal and the haline concentration coefficient, respectively, 
where p is the water density. It is important to mention that, 
according to Canuto et al. (2002) and Canuto et al. (2010), the 
buoyancy production N* can be expresses as follows: 
N? 
Ru -R 
(4) 
In contrast to the structure functions (2), which are used in 
his study and which include double diffusion, structure func- 
tions without double diffusion are of the form 
Si = Sıl(dam, An) with i = m, n (5) 
where x stands for buoyancy and a, = (7N)* for the dimension- 
less buoyancy number. For the functions (5), the following 
stability and realisability criteria are explicitly stated in 
Umlauf and Burchardt (2005): 
At first all vertical diffusivities and therefore all structure 
functions must be greater or equal a background value, which 
must be greater or equal zero, i.e. 
Ki>ci>0 with i = m, n and constants c;. (6) 
The condition guaranteeing increasing effective vertical 
shear anisotropy with increasing dimensionless vertical shear 
number can be formulated as 
= (Kımlam, an)an'/:) >0 
(7) 
The value of a, must be greater or equal the value a”, = a, 
which describes the value in shear-free convective conditions 
for the turbulence equilibrium, in which the buoyancy produc- 
tion G equals the dissipation rate €, Le. 
an >a, (8) 
To prevent oscillation between two mathematically possi- 
ble values of a„, monotonicity of“ Ya with respect to a, 
must be insured for negative a„: 
ö (* (-)) 0 
(9) 
In Umlauf and Burchardt (2005), it is shown that (8) al- 
ways implies (9), so that (9) is not an additional condition 
from a numerical point of view. Finally the velocity variances 
must be positive, with the only critical condition being 
(10) 
where <w,, w> is the vertical velocity variance. 
In the original implementation in HBM, condition (6) is 
always fulfilled by definition. Conditions (7), (8) and (10), 
however, are not explicitly queried and—as it turned out— 
not always fulfilled. It is therefore necessary to extend these 
conditions to the used structure functions including double dif- 
fusion. While (10) is very easy to implement due to the numer- 
ical descriptions of Canuto et al. (2010) (<w, w> is listed ex- 
plicitly in the equations), (7) is also relatively easy to expand to 
On (Kımlam, Ah, as)an'/:) >0 
(11) 
A Springer