110
H.E.M. Meier and A. Höglund
4.2.1.4 Insolation
The divergence of absorbed intensity I of the penetrated short-wave radiation is
heating the water column (Eq. (4.5)). This effect can contribute to summertime
warming of up to 2 ◦ C in 10 m depth. The solar intensity is parameterized according
to Paulson and Simpson (1977) with two extinction lengths
I = Q SW
R SW exp
z
ζ 1
+ (1 − R SW ) exp
z
ζ 2
(4.28)
with R SW = 0.64, ζ 1 = 1.78 m and ζ 2 = 3.26 m. The short-wave incoming radiation
Q SW is calculated according to Bodin (1979). Climatological data from Dera (1992,
see his Table 5.3.1) have been used to optimize the unknown constant R SW and
the extinction lengths ζ 1 and ζ 2 utilizing a least-squares fit. The available data are
average monthly means of solar energy over the entire spectrum reaching particular
depths in the southern Baltic. The optimization procedure has been carried out as
described in Paulson and Simpson (1977). For comparison, the most turbid optical
water type III for oceans according to Jerlov (1968) uses the values R SW = 0.78,
ζ 1 = 1.14 m and ζ 2 = 7.9 m.
4.2.1.5 Horizontal Mixing
Horizontal viscosity and diffusivity are parameterized using a harmonic approach
(4.9)–(4.12). More advanced but not necessarily better parameterizations are scaledependent approaches like biharmonic friction or the Smagorinsky (1963) scheme.
For long-term simulations it is important that diffusivity is chosen as small as
possible. Hence, the coefficients for harmonic viscosity and diffusivity amount to
A M = 2 × 10 2 m 2 /s and A H = 0, respectively. Minimum values for horizontal friction are necessary to ensure numerical stability of the discretization schemes.
4.2.1.6 Turbulence Model
Subgrid-scale vertical mixing is parameterized using a turbulence closure scheme
(Rodi 1980) with flux boundary conditions to include the effect of turbulence enhanced in the uppermost layer due to breaking surface gravity waves and a parameterization for breaking internal waves (Meier 2001). The approach follows the socalled k–ε model with two additional prognostic equations for turbulent kinetic energy, TKE (k), and dissipation of TKE (ε) that need to be solved at every grid point
of the 3D model:
∂k
∂t
−
∂
∂z
ν t
σ k
∂k
∂z
= P + G − ε,
(4.29)
∂ε
∂t
−
∂
∂z
ν t
σ ε
∂ε
∂z
= c ε1
ε
k
(P + c ε3 G) − c ε2
ε 2
k
(4.30)
H.E.M. Meier and A. Höglund
4.2.1.4 Insolation
The divergence of absorbed intensity I of the penetrated short-wave radiation is
heating the water column (Eq. (4.5)). This effect can contribute to summertime
warming of up to 2 ◦ C in 10 m depth. The solar intensity is parameterized according
to Paulson and Simpson (1977) with two extinction lengths
I = Q SW
R SW exp
z
ζ 1
+ (1 − R SW ) exp
z
ζ 2
(4.28)
with R SW = 0.64, ζ 1 = 1.78 m and ζ 2 = 3.26 m. The short-wave incoming radiation
Q SW is calculated according to Bodin (1979). Climatological data from Dera (1992,
see his Table 5.3.1) have been used to optimize the unknown constant R SW and
the extinction lengths ζ 1 and ζ 2 utilizing a least-squares fit. The available data are
average monthly means of solar energy over the entire spectrum reaching particular
depths in the southern Baltic. The optimization procedure has been carried out as
described in Paulson and Simpson (1977). For comparison, the most turbid optical
water type III for oceans according to Jerlov (1968) uses the values R SW = 0.78,
ζ 1 = 1.14 m and ζ 2 = 7.9 m.
4.2.1.5 Horizontal Mixing
Horizontal viscosity and diffusivity are parameterized using a harmonic approach
(4.9)–(4.12). More advanced but not necessarily better parameterizations are scaledependent approaches like biharmonic friction or the Smagorinsky (1963) scheme.
For long-term simulations it is important that diffusivity is chosen as small as
possible. Hence, the coefficients for harmonic viscosity and diffusivity amount to
A M = 2 × 10 2 m 2 /s and A H = 0, respectively. Minimum values for horizontal friction are necessary to ensure numerical stability of the discretization schemes.
4.2.1.6 Turbulence Model
Subgrid-scale vertical mixing is parameterized using a turbulence closure scheme
(Rodi 1980) with flux boundary conditions to include the effect of turbulence enhanced in the uppermost layer due to breaking surface gravity waves and a parameterization for breaking internal waves (Meier 2001). The approach follows the socalled k–ε model with two additional prognostic equations for turbulent kinetic energy, TKE (k), and dissipation of TKE (ε) that need to be solved at every grid point
of the 3D model:
∂k
∂t
−
∂
∂z
ν t
σ k
∂k
∂z
= P + G − ε,
(4.29)
∂ε
∂t
−
∂
∂z
ν t
σ ε
∂ε
∂z
= c ε1
ε
k
(P + c ε3 G) − c ε2
ε 2
k
(4.30)