4 Studying the Baltic Sea Circulation with Eulerian Tracers
107
with the advection operator
Γ (μ) =
1
R cos φ
∂
∂λ
(uμ) +
∂
∂φ
(vμ cos φ)
+
∂
∂z
(wμ),
(4.8)
with μ = (T , S, u, v, 1), the Coriolis parameter f = 2Ω sin φ, Earth’s radius
R = 6370 km, angular speed of rotation of the Earth Ω = 2π/86164 rad/s, acceleration of gravity g = 9.81 m/s 2 , reference density of water ρ 0 = 10 3 kg/m 3 , specific
heat capacity of water c pw = 4.186 × 10 3 J/(kg K) and solar insolation I .
This system of partial differential equations (4.1)–(4.7) can be solved for the
seven dependent variables: velocity components u, v, w, pressure p, potential temperature T , salinity S and density ρ as a function of time t, latitude φ and longitude
λ as well as water depth z (usually interpreted as <0) if an equation of state (4.7)
and boundary conditions are prescribed. The vertical coordinate z is positive upward and zero at the sea surface. F u , F v , F T and F S denote divergences of turbulent
fluxes which are parameterized according to
F u =
∂
∂z
ν t (z)
∂u
∂z
+ A M ∇
2 u,
(4.9)
F v =
∂
∂z
ν t (z)
∂v
∂z
+ A M ∇
2 v,
(4.10)
F T =
∂
∂z
ν t
σ t
(z)
∂T
∂z
+ A T ∇
2 T ,
(4.11)
F S =
∂
∂z
ν t
σ t
(z)
∂S
∂z
+ A T ∇
2 S,
(4.12)
with
∇
2 μ =
1
R 2 cos 2 φ
∂ 2 μ
∂λ 2 +
1
R 2 cos φ
∂
∂φ
∂μ
∂φ
cos φ
.
(4.13)
Additional metric terms in F u and F v are neglected, A M and A T denote horizontal
viscosity and diffusivity coefficients, respectively, ν t is the turbulent vertical friction
coefficient and σ t is the turbulent Prandtl number. As timescales of barotropic and
baroclinic processes are different, it is more efficient rather than integrating Eqs.
(4.1)–(4.7) to introduce an external and internal mode with different time steps. For
details the reader is referred to Killworth et al. (1991), Meier et al. (1999). The
prognostic equations are discretized on an Arakawa B grid (Mesinger and Arakawa
1976).
The lateral boundary conditions are ‘no slip’ for momentum and isolation for
tracer:
u = v =
∂T
∂n
=
∂S
∂n
= 0,
(4.14)
where n is a normal vector to the coast. At the ocean surface (z = 0) the boundary
conditions are
107
with the advection operator
Γ (μ) =
1
R cos φ
∂
∂λ
(uμ) +
∂
∂φ
(vμ cos φ)
+
∂
∂z
(wμ),
(4.8)
with μ = (T , S, u, v, 1), the Coriolis parameter f = 2Ω sin φ, Earth’s radius
R = 6370 km, angular speed of rotation of the Earth Ω = 2π/86164 rad/s, acceleration of gravity g = 9.81 m/s 2 , reference density of water ρ 0 = 10 3 kg/m 3 , specific
heat capacity of water c pw = 4.186 × 10 3 J/(kg K) and solar insolation I .
This system of partial differential equations (4.1)–(4.7) can be solved for the
seven dependent variables: velocity components u, v, w, pressure p, potential temperature T , salinity S and density ρ as a function of time t, latitude φ and longitude
λ as well as water depth z (usually interpreted as <0) if an equation of state (4.7)
and boundary conditions are prescribed. The vertical coordinate z is positive upward and zero at the sea surface. F u , F v , F T and F S denote divergences of turbulent
fluxes which are parameterized according to
F u =
∂
∂z
ν t (z)
∂u
∂z
+ A M ∇
2 u,
(4.9)
F v =
∂
∂z
ν t (z)
∂v
∂z
+ A M ∇
2 v,
(4.10)
F T =
∂
∂z
ν t
σ t
(z)
∂T
∂z
+ A T ∇
2 T ,
(4.11)
F S =
∂
∂z
ν t
σ t
(z)
∂S
∂z
+ A T ∇
2 S,
(4.12)
with
∇
2 μ =
1
R 2 cos 2 φ
∂ 2 μ
∂λ 2 +
1
R 2 cos φ
∂
∂φ
∂μ
∂φ
cos φ
.
(4.13)
Additional metric terms in F u and F v are neglected, A M and A T denote horizontal
viscosity and diffusivity coefficients, respectively, ν t is the turbulent vertical friction
coefficient and σ t is the turbulent Prandtl number. As timescales of barotropic and
baroclinic processes are different, it is more efficient rather than integrating Eqs.
(4.1)–(4.7) to introduce an external and internal mode with different time steps. For
details the reader is referred to Killworth et al. (1991), Meier et al. (1999). The
prognostic equations are discretized on an Arakawa B grid (Mesinger and Arakawa
1976).
The lateral boundary conditions are ‘no slip’ for momentum and isolation for
tracer:
u = v =
∂T
∂n
=
∂S
∂n
= 0,
(4.14)
where n is a normal vector to the coast. At the ocean surface (z = 0) the boundary
conditions are