values in the ocean are
ρ
′
ρ c
= Oð10
− 3
Þ; hence, usually, this approach is valid in the
major part of the World Ocean. The quasi-static equation is written in square
brackets. Sometimes confusion arises because usually primes in density ρ′ and
pressure p′ are omitted, and it looks like the vertical acceleration should be compared with the gravity acceleration, but not with the reduced gravity one.
One of the possible sources of modeling errors in this approach is inaccurate
approximation of the processes associated with relatively high frequency vertical
motions, say with the internal waves. This is important in winter, when the
Brunt-Väisälä frequency is small and much less then the internal wave frequency.
Note, that on the Siberian shelf in the Kara Sea, near the river estuary, even the
annual-mean Brunt-Väisälä frequency exceeds 0.05 s
−1 , and should be even greater
in summer, hence actually quasi-static approximation is valid even for very fine
horizontal model resolution.
Traditionally the limit of the quasi-static approximation is also associated with
the ratio of vertical scale H to horizontal scale L; it is commonly assumed that this
limit is about 0.01. It means that a horizontal resolution of 1 km is the limit to
describe the upper layer processes with a vertical scale of 10 m. Our modeling
experience shows that this criterion is too rough and the H/L limit may be increased
up to 0.1 in the case of strong steady stratification mentioned above. It means that
the AO is a very complicated region for high resolution modeling, because various
regimes exist there, which should be taken into account in the same frame.
The other problem is associated with the unstable vertical motions like convection, which is extremely important in the high-latitude regions where cooling
occurs from above. Usually vertical convection caused by static instability is
parameterized with the assumption that convective plumes have horizontal diameters of the order of 100 m, and there are sufficiently many plumes on the model
characteristic horizontal scale to be taken into account statistically. In the case when
the model grid is fine enough (the size of an individual plume) one may expect that
the hydrostatic approximation will be incorrect.
And, the last but not the least source of uncertainty is the choice of turbulent
viscosity/diffusion terms F ⃗ , Q θ , Q S , which are process dependent. Here, we mention
some of such processes.
Mesoscale Eddies
A small diameter of quasi-geostrophic eddies is a characteristic feature of the Arctic
Ocean, which may be measured by the Rossby deformation radius. In the deep
Canada Basin the baroclinic Rossby deformation radius (R i, n =
NH
fn , N is the
Brunt-Väisälä frequency, H is the ocean depth, n is the vertical mode [9] is about
10 km for the first mode and 5 km for the second one [21], it is even smaller on the
Siberian shelf. This contrasts with the mid-latitudes of the World Ocean, where
these scales are about 50–100 km, and a model resolution of 15 km is quite enough
Arctic Ocean Modeling: The Consistent Physics on the Path …
561
ρ
′
ρ c
= Oð10
− 3
Þ; hence, usually, this approach is valid in the
major part of the World Ocean. The quasi-static equation is written in square
brackets. Sometimes confusion arises because usually primes in density ρ′ and
pressure p′ are omitted, and it looks like the vertical acceleration should be compared with the gravity acceleration, but not with the reduced gravity one.
One of the possible sources of modeling errors in this approach is inaccurate
approximation of the processes associated with relatively high frequency vertical
motions, say with the internal waves. This is important in winter, when the
Brunt-Väisälä frequency is small and much less then the internal wave frequency.
Note, that on the Siberian shelf in the Kara Sea, near the river estuary, even the
annual-mean Brunt-Väisälä frequency exceeds 0.05 s
−1 , and should be even greater
in summer, hence actually quasi-static approximation is valid even for very fine
horizontal model resolution.
Traditionally the limit of the quasi-static approximation is also associated with
the ratio of vertical scale H to horizontal scale L; it is commonly assumed that this
limit is about 0.01. It means that a horizontal resolution of 1 km is the limit to
describe the upper layer processes with a vertical scale of 10 m. Our modeling
experience shows that this criterion is too rough and the H/L limit may be increased
up to 0.1 in the case of strong steady stratification mentioned above. It means that
the AO is a very complicated region for high resolution modeling, because various
regimes exist there, which should be taken into account in the same frame.
The other problem is associated with the unstable vertical motions like convection, which is extremely important in the high-latitude regions where cooling
occurs from above. Usually vertical convection caused by static instability is
parameterized with the assumption that convective plumes have horizontal diameters of the order of 100 m, and there are sufficiently many plumes on the model
characteristic horizontal scale to be taken into account statistically. In the case when
the model grid is fine enough (the size of an individual plume) one may expect that
the hydrostatic approximation will be incorrect.
And, the last but not the least source of uncertainty is the choice of turbulent
viscosity/diffusion terms F ⃗ , Q θ , Q S , which are process dependent. Here, we mention
some of such processes.
Mesoscale Eddies
A small diameter of quasi-geostrophic eddies is a characteristic feature of the Arctic
Ocean, which may be measured by the Rossby deformation radius. In the deep
Canada Basin the baroclinic Rossby deformation radius (R i, n =
NH
fn , N is the
Brunt-Väisälä frequency, H is the ocean depth, n is the vertical mode [9] is about
10 km for the first mode and 5 km for the second one [21], it is even smaller on the
Siberian shelf. This contrasts with the mid-latitudes of the World Ocean, where
these scales are about 50–100 km, and a model resolution of 15 km is quite enough
Arctic Ocean Modeling: The Consistent Physics on the Path …
561