# Simulation of the Arctic—North Atlantic Ocean Circulation with a Two-Equation K-Omega Turbulence Parameterization

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## Abstract

**:**

## 1. Introduction

- to present a new splitting algorithm for solving $k-\omega $ turbulence equations that allows to reduce the complete system to the stages of transport-diffusion and generation-dissipation;
- to find an analytical solution of $k-\omega $ equations at the generation-dissipation splitting stage that is impossible for $k-kl$ and $k-\u03f5$ closures;
- to demonstrate possibilities of controlling the obtained analytical solution of $k-\omega $ equations through its coefficients by means of different physical factors. We take into account such physical factors as climatic annual mean buoyancy frequency (AMBF) and Prandtl number as function of the Richardson number to goal this aim;
- to compare results of numerical experiments with the INMOM + $(k-\omega )$ to data on the hydrographic structure of the North Atlantic and Arctic Ocean to demonstrate physical effects of the accounting AMBF and variations of the Prandtl number.

## 2. Model and Methods

#### 2.1. Equations of the Ocean General Circulation Model

#### 2.2. The Two-Equation K-Omega Turbulence Model

#### 2.3. Boundary and Initial Conditions

#### 2.4. Numerical Algorithm

- The symmetrized form gives the form of the adjoint operator, which is close to the original one.
- This form leads to the finite difference approximation retaining the main properties typical of original differential operators (symmetry, skew-symmetry, nonnegativeness).
- From the form naturally follows the splitting of the problem operator into the sum of simple nonnegative operators.

## 3. Scenarios of Numerical Experiments

**Basic experiment EX1.**In the EX1, in coefficients A and B we put $Pr=1$, and for turbulent exchange coefficients, Pr is calculated according to [39] in the form:

**Experiments EX2.**The climatic annual mean buoyancy frequency ${N}_{Cl}$ (AMBF) is used in coefficients A and B for the EX2. AMBF is reliably defined by the World Ocean Atlas data using the climatic annual mean potential density for the whole period of observations: ${\rho}_{Cl}=\rho ({T}_{Cl},{S}_{Cl})$ [40], where ${T}_{Cl}$ is the potential temperature [41] and ${S}_{Cl}$ is the salinity [42]. The simple argument is used in the favour of taking AMBF into account. Annual mean vertical density gradients are lower than instantaneous ones in the warm season and are higher in the cold season. This favors to the mixing balance in the ocean within the annual cycle. In coefficients A and B, instead of ${N}^{2}$, we used:

**Experiment EX3.**The aim of the EX3 is to study the sensitivity of the model solution to the Prandtl number variation. For neutral or unstable density stratification, we use either ${Pr}_{0}=0.7143$ [43] or ${Pr}_{0}=1.0$ [39]. For stable stratification, the Prandtl number is usually given as a function of the Richardson number $Ri$ [39,43]. It is assumed that the Prandtl number depends on the Richardson number:

- ${a}_{P}=0.6790$, ${b}_{P}=3.5062$, ${c}_{P}=0.2716$ and ${Pr}_{0}=1$;
- ${a}_{P}=1.3227$, ${b}_{P}=2.2487$, ${c}_{P}=0.2116$ and ${Pr}_{0}=0.7143$.

## 4. Discussion

#### 4.1. Comparison to the Generalized Observational Data

**Deviation of simulated temperature and salinity from the climatic data in the ocean upper layer.**Denote the differences “model minus climate” for T and S as $dT$ and $dS$ respectively for the period 1963–2009. $dT$ are in the range $\pm {(0.2-0.7)}^{\circ}$C in the predominant part of the domain for the EX1, EX2 and EX3 in 0–30 m layer (Figure 1). Temperature in the EX1 is more than a one degree lower than the climatic data in the Equatorial currents, Sargasso and Norwegian seas. Using the AMBF in the EX2 significantly reduces $dT$ in comparison to the EX1 (see Figure 1a,b). Negative $dT$ of more than 2 ${}^{\circ}$C disappear in the EX2 on the Equator and in the Tropics. Square of negative deviations more than 1 ${}^{\circ}$C are halved. As well, negative $dT$ decrease significantly in the Caribbean sea and African upwelling area. There is a significant decrease in positive $dT$ in the Gulf Stream, the North Atlantic and Labrador currents. Negative $dT$ in the EX3 at the Equator, in the Caribbean and Norwegian Seas, in the African upwelling region decrease even more (see Figure 1c). However, the positive $dT$ in the EX1 and EX2 change to the negative ones in the EX3 for the North Atlantic Current. It is due to the additional entrainment of cold water from the seasonal thermocline by decreasing the Prandtl number relative to the EX1 and EX2.

**Vertical structure of the upper ocean layer.**We choose the “climatic” period of 30 years 1980–2009 to compare simulated T and S profiles to the ones from the Ocean Atlas. The most observations were performed for this period. We compare the profiles of simulated T and S mean for thirty February and August for 1980–2009 with the climatic monthly mean ${T}_{Cl}$ and ${S}_{C1}$ for February and August [41,42]. February is period of the greatest loss of buoyancy by the ocean. August is month of the greatest inflow of buoyancy from the atmosphere to the ocean. We choose the regions with the most different mixing conditions.

#### 4.2. Sensitivity of Ocean Model Characteristics to the Changes in Mixing Parametrization

**Thickness of the ocean upper mixed layer.**Distribution of the maximum UML depth in the North Atlantic is shown in the Figure 6. This distribution is typical for all winters of the whole period 1948–2009. Maximum UML depth is defined as the level where water density differs from the sea surface density less than 0.15 kg/m${}^{3}$. UML depth is highly sensitive to the used parameterizations (Figure 6). UML depth reaches 2–3 km in the Labrador, Norwegian and Greenland seas for the EX1. Area of these regions are sharply reduced for the EX2. UML depth is nowhere greater than 3 km for the EX3. Distribution and values of the UML depth are most similar to their estimates from observations [44] for the EX2 and EX3. EX3 reproduces a deeper UML than the EX1 and EX2 in the frontal zone of the Gulf Stream (Figure 6). The reason for this is in the greater diffusivity of heat and salt due to the smaller Prandtl numbers in the zone of decaying turbulence at relatively small UML depths.

**Ocean currents.**Changes in the fields of temperature, salinity and density associated with the change in the mixing parameterization cover almost the entire upper half of the baroclinic layer (Figure 3, Figure 4 and Figure 5). This causes a high sensitivity of the circulation and SSH to the change of the mixing parameterization. The greatest changes in ocean currents happen by the transition from parameterization EX4 using a simple dependence of the turbulent coefficients on the Richardson number to EX1–EX3 where STA is used.

**Sea surface height.**SSH is also sensitive to the change of parameterizations. SSH is associated with the annual cycle of sea ice extent. Figure 8 shows the SSH simulated in the EX4 (a), EX1 (b), EX2 (c) and EX3 (d), for the 62nd year of integration in April 2009. Sea ice extent reached its maximum in this month for the Arctic Ocean and Nordic Seas (NSIDC data). Beaufort Gyre is here the most characteristic feature of the circulation and SSH field. SSH distribution is more similar to the reconstruction according to the climatic data [38] for the EX2 relative to the other experiments. Beaufort Gyre is better expressed in the SSH field for EX2. Unrealistic overestimation of SSH was obtained for the EX4 in the region of the Siberian shelf. The greatest difference in the maximum SSH between the EX2 and EX1 is 15 cm in the Beaufort Gyre. EX3 reproduces high SSH values on the periphery of the Beaufort Gyre due to the steric effect as the result of the greater entrainment of dense water from the pycnocline into the upper ocean layer (Figure 8d). So, SSH is reproduced most realistically in the EX2 with taking into account the AMBF in the equations of turbulence (Figure 8c).

**Heat flux at the ocean surface.**Heat fluxes at the upper boundary of the ocean are calculated using the model SST. Model SST depends on turbulent mixing in its upper layer. Thus, the forcing for the OGCM also depends on the used mixing parameterization. Typical example for the sum of the latent $LE$ and sensible ${Q}_{T}$ heat fluxes shows that the space distributions of the heat fluxes are close to each other in the EX1, EX2 and EX3 for the period of maximum heat loss by the ocean in February (Figure 9). However, quantitative differences of fluxes can reach 50–200 W/m${}^{2}$ in some regions. For example, sensitivity of the heat fluxes to the change of the mixing parameterization revealed in the subtropics. Low heat losses $LE+{Q}_{T}<50$ W/m${}^{2}$ for EX1 changes to greater ones for the EX2 (Figure 9a,b). This is due to the following process. Turbulence zone penetrates to deeper layers in the preceding warm period for the EX2. Greater accumulation of heat in the upper ocean layer occurs. More significant water-air temperature differences for the EX2 relative to the EX1 arises in the winter.

#### 4.3. Turbulent Energy and Omega

#### 4.4. Numerical Aspects, Data Assimilation

## 5. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AMBF | Annual Mean Buoyancy Frequency |

INMOM | Institute of Numerical Mathematics Ocean Model |

OGCM | Ocean General Circulation Model |

STA | Splitting Turbulence Algorithm |

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**Figure 1.**Average for 1963–2009 temperature deviations (in ${}^{\circ}$C) in the 0–30 m layer from climatic data [41] in the experiments EX1 (

**a**), EX2 $({\alpha}_{Cl}=0.9)$ (

**b**) and EX3 (

**c**). Coordinates of the model: the geographical grid is replaced by a two-pole orthogonal (poles on the geographical equator at points with coordinates 120${}^{\circ}$ W and 60${}^{\circ}$ E). The outlines of continents and islands are shown.

**Figure 3.**T (in ${}^{\circ}$C) and S (in PSU) profiles for February and August averaged for 1980–2009 in the Sargasso Sea (center of the one-degree model cell near 36${}^{\circ}$ N, 51${}^{\circ}$ W): the columns are February (

**left**) and August (

**right**), rows are temperature (

**top**) and salinity (

**bottom**). Black solid thick line with crosses represents the climate; Green solid line with dark circles—EX1; Black dash/double dot line with dark circles—EX2; Black solid thin line with hollow circles—EX3; Red solid line with hollow squares—EX4.

**Figure 4.**T (in ${}^{\circ}$C) and S (in PSU) profiles for February and August averaged for 1980–2009 in the North Atlantic Current (center of the one-degree model cell near 52${}^{\circ}$ N, 38${}^{\circ}$ W): the columns are February (

**left**) and August (

**right**), rows are temperature (

**top**) and salinity (

**bottom**). Black solid thick line with crosses represents the climate; Green solid line with dark circles—EX1; Black dash/double dot line with dark circles—EX2; Black solid thin line with hollow circles—EX3; Red solid line with hollow squares—EX4.

**Figure 5.**T (in ${}^{\circ}$C) and S (in PSU) profiles for February and August averaged for 1980–2009 in the recirculation region of the West Spitsbergen Current (center of the one-degree model cell near 75${}^{\circ}$ N, 2${}^{\circ}$ W): the columns are February (

**left**) and August (

**right**), rows are temperature (

**top**) and salinity (

**bottom**). Black solid thick line with crosses represents the climate; Green solid line with dark circles—EX1; Black dash/double dot line with dark circles—EX2; Black solid thin line with hollow circles—EX3; Red solid line with hollow squares—EX4.

**Figure 6.**Maximum thickness of the upper mixed layer (in meters) for the 50th year of integration (February 1997) in the EX1 (

**a**), EX2 (

**b**), EX3 (

**c**) in model coordinates.

**Figure 7.**The average in the 0–50 m layer for thirty February of 1980–2009: the difference in the velocities between the EX1 and EX4 (EX1 minus EX4) (

**a**); the velocity in the EX1 (

**b**). The model coordinates are used (see the caption under Figure 1). The direction is shown by vectors, and the magnitude in cm/s is shown by shading.

**Figure 8.**Sea surface height (in cm plus 30) in the EX4 (

**a**), EX1 (

**b**), EX2 (

**c**) and EX3 (

**d**) in April 2009 (The 62nd year of integration). April 2009 is the month of the maximum sea ice extent for the Arctic Ocean and Nordic Seas according NSIDC data. The model coordinates are used (see the caption under Figure 1, the point with coordinates (0,0) corresponds to the North Pole).

**Figure 9.**The sum of latent and sensible heat fluxes (W/m${}^{2}$) at the ocean surface in February for the 50th year of the runs (1997) in the experiments EX1 (

**a**), EX2 (

**b**) and EX3 (

**c**).

**Figure 10.**Turbulent kinetic energy (cm${}^{2}$/s${}^{2}$) in the 0–30 m layer in August 2009 (the 62nd year of integration). The model coordinates are used (see the caption under Figure 1).

Experiments | Prandtl Number | Computation of k and $\mathit{\omega}$ | Accounting AMBF | Viscosity and Diffusivity |
---|---|---|---|---|

EX1 | $\begin{array}{c}Pr=\left(\right)open="\{"\; close>\begin{array}{c}1,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Ri\le 0.2\hfill \\ 5Ri,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}0.2Ri2\hfill \\ 10,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Ri\ge 2\hfill \end{array}& ,\end{array}{\displaystyle Ri=\frac{{N}^{2}}{{G}^{2}}}$ | $\begin{array}{c}A={G}^{2}-{N}^{2},\hfill \\ B={c}_{1}^{\omega}{G}^{2}-{c}_{3}^{\omega}{N}^{2}\hfill \end{array}$ | $\begin{array}{c}{\alpha}_{M}=1,\hfill \\ {\alpha}_{Cl}=0\hfill \end{array}$ | $\begin{array}{c}{\displaystyle {K}_{U}=\frac{k}{\omega},}\hfill \\ {\displaystyle {K}_{T}={K}_{S}=\frac{{K}_{U}}{Pr}}\hfill \end{array}$ |

EX2 | $Pr=\left(\right)open="\{"\; close>\begin{array}{c}1,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Ri\le 0.2\hfill \\ 5Ri,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}0.2Ri2\hfill \\ 10,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Ri\ge 2\hfill \end{array}$ | $\begin{array}{c}A={G}^{2}-{N}_{s}^{2},\hfill \\ B={c}_{1}^{\omega}{G}^{2}-{c}_{3}^{\omega}{N}_{s}^{2},\hfill \\ {N}_{s}^{2}={\alpha}_{M}{N}_{M}^{2}+{\alpha}_{Cl}{N}_{Cl}^{2}\hfill \end{array}$ | $\begin{array}{c}{\alpha}_{M}=0.1,\hfill \\ {\alpha}_{Cl}=0.9\hfill \end{array}$ | $\begin{array}{c}{\displaystyle {K}_{U}=\frac{k}{\omega},}\hfill \\ {\displaystyle {K}_{T}={K}_{S}=\frac{{K}_{U}}{Pr}}\hfill \end{array}$ |

EX3 | $\begin{array}{c}Pr=\left(\right)open="\{"\; close>\begin{array}{c}{Pr}_{0},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Ri\le 0.2\hfill \\ {a}_{P}R{i}^{2}+{b}_{P}Ri+\hfill \\ +{c}_{P},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}0.2Ri2\hfill \\ 10,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Ri\ge 2\hfill \end{array}\end{array}\begin{array}{c}{a}_{P}=1.3227,\phantom{\rule{0.166667em}{0ex}}{b}_{P}=2.2487,\\ {c}_{P}=0.2116,\phantom{\rule{4pt}{0ex}}{Pr}_{0}=0.7143\end{array}$ | $\begin{array}{c}{\displaystyle A={G}^{2}-\frac{1}{Pr}{N}^{2},}\hfill \\ {\displaystyle B={c}_{1}^{\omega}{G}^{2}-\frac{{c}_{3}^{\omega}}{Pr}{N}^{2}}\hfill \end{array}$ | $\begin{array}{c}{\alpha}_{M}=1,\hfill \\ {\alpha}_{Cl}=0\hfill \end{array}$ | $\begin{array}{c}{\displaystyle {K}_{U}=\frac{k}{\omega},}\hfill \\ {\displaystyle {K}_{T}={K}_{S}=\frac{{K}_{U}}{Pr}}\hfill \end{array}$ |

EX4 | No | No | No | $\begin{array}{c}{\displaystyle {K}_{U}=\frac{{K}_{0}}{{(1+5Ri)}^{2}}+{K}_{b},}\hfill \\ {\displaystyle {K}_{T}={K}_{S}=}\hfill \\ {\displaystyle =\frac{{K}_{U}}{1+5Ri}+{K}_{Tb},}\hfill \\ {\displaystyle {K}_{0}=100\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}c{m}^{2}/s,}\hfill \\ {\displaystyle {K}_{b}=1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}c{m}^{2}/s,}\hfill \\ {\displaystyle {K}_{Tb}=0.05\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}c{m}^{2}/s}\hfill \end{array}$ |

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**MDPI and ACS Style**

Moshonkin, S.; Zalesny, V.; Gusev, A.
Simulation of the Arctic—North Atlantic Ocean Circulation with a Two-Equation K-Omega Turbulence Parameterization. *J. Mar. Sci. Eng.* **2018**, *6*, 95.
https://doi.org/10.3390/jmse6030095

**AMA Style**

Moshonkin S, Zalesny V, Gusev A.
Simulation of the Arctic—North Atlantic Ocean Circulation with a Two-Equation K-Omega Turbulence Parameterization. *Journal of Marine Science and Engineering*. 2018; 6(3):95.
https://doi.org/10.3390/jmse6030095

**Chicago/Turabian Style**

Moshonkin, Sergey, Vladimir Zalesny, and Anatoly Gusev.
2018. "Simulation of the Arctic—North Atlantic Ocean Circulation with a Two-Equation K-Omega Turbulence Parameterization" *Journal of Marine Science and Engineering* 6, no. 3: 95.
https://doi.org/10.3390/jmse6030095