^{1}

^{1}

^{*}

^{2}

The relationship of lateral eddy viscosity depending on length scale is estimated with the decay rate of mesoscale eddies identified from sea level anomaly of satellite observations. The eddy viscosity is expressed in terms of the mesoscale eddy parameters according to vortex dynamics. The census of mesoscale eddies shows, in general, that the eddy numbers obey the e-folding decay laws in terms of their amplitude, area and lifetime. The intrinsic values in the e-folding laws are used to estimate the lateral eddy viscosity. Dislike the previous theory that diffusivities are proportional to the length square, the eddy mixing rates (diffusivity and viscosity) from satellite mesoscale eddy datasets are proportional to rs to power of 1.8 (slightly less than 2), where
*r*
_{s} is the radius of eddy with radius larger than the Batchelor scale. Additionally, the extrapolation of the eddy mixing to the molecule scale implies that the above power laws may hold until the value of
*r*
_{s} is less than O (1 m). These mixing rates with the new parameterizations are suggested to use in numerical schemes. Finally, the climatological distributions of eddy viscosity are calculated.

In numerical ocean models, effects of the mesoscale eddies should be accurately represented or parameterized to study the general ocean circulations, since satellite observations [

For passive tracers from the oceanic observations, the isopycnal eddy diffusivity and viscosity may is on the order of 10^{3} m^{2}/s [^{2} m^{2}/s [^{4} m^{2}/s [

The diffusivities used in different approaches, e.g., the along-isopycnal diffusivity for tracers [_{e} is defined as k e = k m L e f f 2 / L min 2 , where L_{eff} is the equivalent length of a tracer contour that has been stretched by eddy stirring, L_{min} is the minimum possible length (a.k.a. the Batchelor scale) of such a contour, and k_{m} is the small-scale background diffusivity [_{OC} is similar to k_{e}, but L_{eff} and L_{min} are calculated from the spatial gradient of the tracer itself and the tracer’s disturbance, respectively [_{eff} and L_{min} can be calculated from the Sea Level Anomaly (SLA) data, and the small-scale background diffusivity k_{m} is set according to some field observations.

It is expected that the resulting diffusivities are independent of any unknown parameters [_{e} is proportional to k m 1 / 4 (exponent of 0.24 other than 0.25 used here), given a k_{m} in the range of 10 to 10^{2} m^{2}/s [_{OC} depends weakly on k m : k O C ∝ k m − 1 / 5 , given k_{m} in the range of 0.5 - 5 m^{2}/s [_{m}, but k_{e} is directly proportional to k_{m} and k_{OC} is inversely proportional to k_{m}.

Thus the simple relationship that diffusivities are proportional to the length square should be modified according to these previous studies. The above inconsistent results reveal the possibility that k_{m} is not proportional to L min 2 ; instead, it is proportional to L min n (n < 2), i.e., k m = k L min n , where k is the molecule-scale viscosity, because the time scale dependency on the length scale for the current range of spatial resolution range, as do k_{e} and k_{OC} to L_{eff} and other length scales. According to the above relations of k_{e} and k_{OC}, it is found that n = 3/2 for k_{m} in the range of 10 to 10^{2} m^{2}/s and n = 5/3 for k_{m} in the range of 0.5 to 5 m^{2}/s. When n = 5/3, it is the classic −5/3 power law of energy dissipation for the wave number in 2-dimensional or quasi-geostrophic flow [_{OC} become truly independent of k_{m} [

It is hypothesis that eddy viscosity, similar to eddy diffusivity, is proportional to L min n (n < 2). To test this hypothesis, the accumulated long-term satellite altimetry SLA data are used. Benefiting from studies of eddy statistics [

The paper is organized as follows. In Section 2, we first describe the eddy identification and tracking datasets used in our study, the relation between eddy viscosity and eddy properties based on vortex theory is established; in addition, the intrinsic eddy parameters are calculated from eddy statistics. In Section 3, we use two examples to present the method for estimating eddy viscosities; eddy viscosities in the South China Sea (SCS), the ACC and the global regions from different datasets are calculated. In Section 4, we discuss the relationships between the average parameters and the intrinsic parameters. In Section 5, we draw the conclusions.

The SLA data used here are the merged and gridded satellite product of MSLA (Maps of Sea Level Anomaly) produced and distributed by AVISO (http://www.aviso.oceanobs.com/) based on TOPEX/Poseidon, Jason 1, ERS?1, and ERS-2 data [

The first eddy dataset is based on the weekly SLA fields in Version 3 of the AVISO data taken from http://cioss.coas.oregonstate.edu/eddies/, which is based on the methods by Chelton et al. (2011) [

The second dataset is based on the 20-year (1993-2012) daily SLA fields of the AVISO data. The ocean eddies were identified by the SLA extremes and a sufficient number of neighboring regions, these criteria are similar to those of the previous method [

The third eddy dataset is also based on the weekly SLA fields of the AVISO data from 1993 to 2007, where the Okubo-Weiss method [

In this study, we directly calculate the eddy viscosity using formula from vortex dynamics. For an incompressible flow with density ρ and velocity u, the balance of eddy kinetic energy (EKE) e k = 1 / 2 ρ u ⋅ u is

d e k d t = F ⋅ u − Φ + B . (1)

where F is body force per unit mass, B is EKE flux at the boundary, and Φ is the dissipation rate

Φ = ρ v e ξ 2 . (2)

where v_{e} is the eddy viscosity and ξ = ∇ × u is vorticity. Most eddies in the open ocean are neutrally buoyant; thus, there is no body force and no net EKE flux at the boundary. Hence, the balance of EKE is,

1 ρ d e k d t = v e ξ 2 . (3)

To the lowest order approximation, oceanic mesoscale eddies have a universal profile [

h = A exp ( − r 2 2 r s 2 ) . (4)

where A and r_{s} are the amplitude and radius of maximum speed, respectively [

V θ = g A r f 2 r s 2 exp ( − r 2 2 r s 2 ) . (5)

where g and f are the gravity acceleration and Coriolis parameter. Thus, the total EKE of the eddy is

E = ∫ 0 2π ∫ 0 ∞ 1 2 V θ 2 r d r d θ = π 2 ( g A f ) 2 . (6)

The vorticity of the eddy in the geostrophic approximation is

ξ = 1 r d d r ( V θ r ) = 2 g A f r s 2 ( 1 − r 2 2 r s 2 ) exp ( − r 2 2 r s 2 ) . (7)

The enstrophy of the eddy is

Q = ∫ 0 2π ∫ 0 ∞ 1 2 ξ 2 r d r d θ = π ( g A f r s ) 2 . (8)

According to Equation (3), the EKE dissipation rate directly linked to the enstrophy [

d E d t = − 2 v e Q . (9)

For non-rotating fluid, the available gravitational potential energy (AGPE) of a vortex is small; it plays no important role in eddy dynamics, and thus often being ignored. However, AGPE in rotating stratified fluid plays quite important role. In fact, AGPE of the oceanic mesoscale eddy is larger than the EKE in general; on average, the AGPE is approximately 1.7 times the EKE [_{e} introduced in Equation (2) should be multiplied by a factor of C (ratio of total mechanical energy to EKE), and C~1 + 1.7 = 2.7, as discussed above. For the Gaussian-shaped eddy, this leads to

v e = − C 2 Q d E d t = − C r s 2 4 A d A d t = − C L a 4 π . (10)

where

L = π r s 2 / A = S / A . (11a)

a = d A / d t . (11b)

L, a and S are the length parameter, the amplitude decaying rate and the horizontal area of a circle with radius r_{s}. Thus, using observations and Equation (11), we can estimate the horizontal eddy viscosity.

There are several length scales in our analysis: the radius of maximum speed r_{s}, the e-folding decay radius r_{e}, and the effective radius r_{eff}. The effective radius r_{eff} is defined to be the radius of the circle that has the same area as the region within the eddy perimeter [_{s} = 0.44r_{eff} and r_{s} = 0.707r_{e}, are useful in comparing the results from different datasets [_{eff} in original datasets [

Note that lateral eddy viscosity discussed above is defined for the evolution of an eddy; thus, it may be interpreted as what an observer moving with the eddy can see. In this sense, this analysis can apply to the data collected by tracing the time evolution of individual eddies from satellite altimetry data. Thus, it is a parameter defined in the Lagrangian coordinates, so that it can be used as the viscosities for the whole domain in the ocean; however, whether such parameter can be directly used to the commonly used oceanic numerical models remains unclear because such models are defined in Eulerian coordinates.

As mentioned above, we need to calculate the decaying rate a of eddy amplitude. The simplest method is to estimate a from the time evolution of eddy amplitude. A simple example is shown in

A = a T + b . (12)

where T is the lifetime of the eddy. However, this regression is quite noisy, as shown in

The amplitude decay rate can be calculated from the eddy number distributions. In general, there are two different types of statistics in the literature (e.g., [

N a ( A ) = N a ( a T + b ) = N t ( T ) . (13)

On the other hand, if we have statistics of both N t ( T ) and N a ( A ) , we can use Equation (13) to derive the amplitude-lifetime relation. Specifically, the number of eddies in the global oceans may obey the following e-folding decay laws in terms of its amplitude, area and lifetime

N a ( A ) = N a 0 e − A / A i . (14a)

N s ( S ) = N s 0 e − S / S i . (14b)

N t ( T ) = N t 0 e − T / T i . (14c)

where A_{i}, S_{i} and T_{i} are the intrinsic eddy amplitude, area and lifetime, respectively, and N_{a0}, N_{s0} and N_{t0} are the eddy numbers when A = S = T = 0 . Then, substituting Equation (12) into Equation (14a), and using Equation (13), it yields to N a ( A ) = N a 0 e − A / A i = N a 0 e − ( a T + b ) / A i = N t 0 e − T / T i . Therefore, we can use Equation (13) to evaluate v_{e} with the intrinsic parameters.

a = A i / T i . (15a)

L = S i / A i . (15b)

v e = C 4 π S i T i . (15c)

We will use this method to calculate the eddy viscosity in this study. The intrinsic parameters are obtained from linear regression of e-folding decays (Figures 2-5). For Chelton dataset, the present linear regression of 16 weeks (

There are two long-lived eddies detected by automated eddy identification and tracking algorithm in the Li dataset (

rate a is approximately 0.276 cm/day (3.2 × 10^{−8} m/s) for the red lines and 0.234 cm/day (2.7 × 10^{−8} m/s) for the blue line. Meanwhile, the length parameter (L = S/A) is also plotted as the green curve in ^{2}/s).

Cyclonic eddy 2 moved from 85˚E, 33.5˚S to 67˚E, 36˚S from November 1995 to December 1997. During this time period, both the amplitude and the area of the eddy changed substantially. There are 6 time intervals when the amplitude monotonically decreased (^{−8} m/s) for the red lines and 0.232 cm/day (2.7 × 10^{−8} m/s) for the blue line. The eddy viscosities in these time periods are listed in ^{2}/s) are much more diverse in this case. Nevertheless, in these

Period | a (m/s) | A (m) | S/A (m) | v_{e} (m^{2}/s) |
---|---|---|---|---|

A1 | 3.2 × 10^{−8} | 0.32 | 5.9 × 10^{10} | 405 |

A2 | 3.2 × 10^{−8} | 0.34 | 5.0 × 10^{10} | 356 |

A3 | 2.7 × 10^{−8} | 0.27 | 7.3 × 10^{10} | 429 |

B1 | 4.5 × 10^{−8} | 0.30 | 5.2 × 10^{10} | 521 |

B2 | 2.7 × 10^{−8} | 0.14 | 3.2 × 10^{10} | 189 |

B3 | 4.5 × 10^{−8} | 0.135 | 2.7 × 10^{10} | 275 |

B4 | 2.7 × 10^{−8} | 0.14 | 3.0 × 10^{10} | 178 |

B5 | 4.5 × 10^{−8} | 0.21 | 3.6 × 10^{10} | 356 |

B6 | 2.7 × 10^{−8} | 0.20 | 3.6 × 10^{10} | 213 |

examples the eddy viscosities inferred from observations are relatively small, on the order of (~300 m^{2}/s).

Next, we use the data from Xiu et al. (2010) [^{−8} m/s, on average, for eddies observed in the SCS (^{2}, respectively [^{2} to the special area of 24,750 km^{2} (_{s} = 0.44r_{eff} [^{2}/s in the SCS (

As shown above, there is no significant difference for viscosity between the cyclonic and anticyclonic eddies. This is also similar to the results inferred from the data by Xiu et al. (2010) [

Dataset | A_{a} (cm) | T_{a} (d) | a (m/s) | S_{a} (km^{2}) | r_{a} (km) | v_{a} (m^{2}/s) |
---|---|---|---|---|---|---|

ACC-C | 7.62 | 78.9 | 1.1 × 10^{−8} | 3.2 × 10^{3} | 32 | 100 |

ACC-L | 16.7 | 73 | 2.6 × 10^{−8} | 5.06 × 10^{3} | 40 | 173 |

GO-C | 6.76 | 83.6 | 0.9 × 10^{−8} | 5.4 × 10^{3} | 41.5 | 162 |

GO-L | 16.7 | 76.6 | 2.5 × 10^{−8} | 9.0 × 10^{3} | 53.5 | 292 |

SCS-X | 15 | 174 | 1.0 × 10^{−8} | 24.8 × 10^{3} | 87.4 | 343 |

Next, we use the automatic identification and tracking data of the mesoscale eddies from the Li dataset to study the eddy viscosity. First, we use the data within the ACC region (45˚S - 65˚S). The eddy number vs amplitude distribution in the ACC region is plotted in _{i} = 8.7 cm. Similarly, the eddy number and their area obey the e-folding decay law, Equation (14b), with the intrinsic eddy area S_{i} = 2.8 × 10^{3} km^{2}, _{i} = 40 days, ^{2}/s in the ACC region.

Second, we apply this method to the global data. The census statistics are shown in _{02} = 38 cm, S_{02} = 36 × 10^{3} km^{2} and T_{02} = 78 days) separating the e-folding decay laws into two segments. The intrinsic eddy amplitude is A_{i1} = 9 cm and A_{i2} = 16cm for small amplitude and large amplitude eddies, these two segments join at amplitude of 38 cm (_{i1} = 5.7 × 10^{3} km^{2} and S_{i2} = 23 × 10^{3} km^{2} for small and large area eddies, respectively (_{i1} = 27 days and T_{i2} = 52 days for small and large eddies, respectively. Thus, the eddy viscosity v_{e} is 524 m^{2}/s and 1099 m^{2}/s for small and large eddies, respectively, in the global oceans.

The results discussed seem vary due to the difference in datasets created by different investigators. We also use the dataset calculated by Chelton et al. (2011) [^{ }be transformed to the special area with the regression relationship r_{s} = 0.44r_{eff} [_{i} = 6.5 cm, S_{i} = 2.2 × 10^{3} km^{2} and T_{i} = 8 weeks, respectively. Thus, the eddy viscosity is approximately 97 m^{2}/s, which is relatively smaller than that obtained from Li’s data in the ACC region (

Similarly, the census statistics of the global mesoscale eddies are also calculated (_{i1} = 7 weeks and T_{i2} = 16 weeks. The latter one is the same as the optimal value obtained by applying a stochastic model to the same dataset [^{2}/s and 270 m^{2}/s (approximately 1/3 and 1/4 of the above corresponding values) for the small and large eddies for the global oceans from the Chelton’s data.

It is concluded that viscosity calculated from different datasets is of the same order, although the difference is a bit large. Thus, using the intrinsic parameters rather than the census numbers is a better approach.

Lateral eddy viscosities derived in the discussion above are diverse in value, from 64 m^{2}/s to 1099 m^{2}/s. Additionally, these lateral eddy viscosities are approximately one or two orders of magnitude smaller than the canonical value of approximately 10^{3} - 10^{4} m^{2}/s [^{−}^{2} - 10^{2} km).

Dataset | A_{i} (cm) | S_{i} (km^{2}) | T_{i} (d) | a (m/s) | S_{i}/A_{i} (m) | v_{e} (m^{2}/s) |
---|---|---|---|---|---|---|

ACC-C | 6.5 | 2.2 × 10^{3} | 56 | 1.34 × 10^{-8} | 7.7 × 10^{10} | 97 |

ACC-L | 8.7 | 2.8 × 10^{3} | 40 | 2.52 × 10^{-8} | 3.2 × 10^{10} | 173 |

GO-C1 | 5 | 3.6 × 10^{3} | 49 | 1.19 × 10^{-8} | 1.9 × 10^{11} | 184 |

GO-C2 | 11.8 | 12 × 10^{3} | 119 | 1.15 × 10^{-8} | 2.7 × 10^{11} | 254 |

GO-L1 | 9 | 5.7 × 10^{3} | 27 | 3.86 × 10^{-8} | 6.3 × 10^{10} | 524 |

GO-L2 | 16 | 23 × 10^{3} | 52 | 3.57 × 10^{-8} | 1.4 × 10^{11} | 1099 |

This power law is also valid for the eddy diffusivities in the ocean. According to the observations [^{2}/s, 2 m^{2}/s, and 10^{3} m^{2}/s at scales of 0.1 to 1 km, 1 to 10 km, and 30 to 300 km, respectively. A recent study based on observations [^{2}/s and 5 m^{2}/s at scales of 1 km and 4 km, respectively. If we take these scales as radius r_{s}, the diffusivities also obey a power-law of k e = 1.4 × 10 − 6 r s 1.8 , as shown in _{eff}, the diffusivities obey a power-law k e = 7.2 × 10 − 6 r s 1.8 (figure not shown).

Both the eddy viscosity and diffusivity obey similar power laws, with slightly different constants in the front; in fact, viscosity is relatively smaller than the diffusivity. This can be understood in terms of the dissipation ratio Γ and the ratio of buoyancy flux to turbulence production [

As noted in the introduction, the diffusivities k_{e} and k_{OC} may be proportional to L e f f n (n < 2). Our results show that n = 3/2 for a k_{m} of 10 to 10^{2} m^{2}/s, n = 5/3 for a k_{m} of 0.5 - 5 m^{2}/s, and n = 9/5 for a k_{m} of 10^{−}^{2} to 10^{−}^{1} m^{2}/s. It seems that when k_{m} and the spatial resolution scale are sufficiently small, n tends to be 2, as we expected.

Additionally, we can extrapolate the above eddy mixing rates to small scales down to the molecule mixing rates, because it is well-known that the power laws are valid for a wide regime of O(10^{−}^{2} - 10^{2} km). Considering that the molecule diffusivity is 10^{−7} m^{2}/s and the molecule viscosity is 10^{−6} m^{2}/s, it seems that power laws for eddy mixing could be used when the scale is larger than 0.3 - 1 m. The extrapolation of the eddy mixing to molecule mixing implies that the above eddy mixing rates always hold until r_{s} is smaller than O (1 m). This extrapolation is physically sound, since that the turbulence is generally observed on scale larger than O (1 m) in fluid dynamics. Thus, the above eddy mixing is expected to hold in the scales of 10^{−}^{2} - 10^{2} km.

Finally, the intrinsic time has a relatively weaker relationship with the eddy scale. Because only time is directly proportional to length, the exponent n < 2 holds according to Equation (15c). Both present regression lines imply a 1/5- power law T i ∝ r s 1 / 5 under the condition of n = 9/5 for k_{m}. The larger scale eddies also have larger time scales.

The eddy viscosity power law discussed above can be used to calculate the climatological distribution of eddy viscosity. First, we calculate the viscosity of each eddy on each time snapshot using the eddy parameters from Chelton and Li datasets. Then, each point of the eddy within the eddy perimeter (as indicated by effective radius r_{eff}) is recorded by the same viscosity. Finally, we calculate the climatological eddy viscosity by averaging the total viscosities within the whole time.

^{2}m^{2}/s, consistent with the previously low values [

The present study shows a result of very low rate of viscosity than previous diffusivities [

In Section 3, we used the intrinsic eddy parameters to estimate the viscosity. We also used the averages of the eddy parameters to estimate the viscosity. To quantify the differences between these two approaches, we further compare the values obtained from these two different methods. According to the eddy census, the number of eddies obey the e-folding relations in Equation (14), the average parameters of eddies are

A a = 1 N ∫ A 0 ∞ A N a ( A ) d A = A 0 + A i . (16a)

S a = 1 N ∫ S 0 ∞ S N s ( S ) d S = S 0 + S i . (16b)

T a = 1 N ∫ T 0 ∞ T N t ( T ) d T = T 0 + T i . (16c)

where N is the total eddy number. It is noted that the results depend on the initial and intrinsic parameters, but they are independent of the total eddy number.

The initial values are given by the identification criteria, as mentioned in the data subsection. From the census data (Tables 4-6) the averages agree quite well with the sums of the initial and intrinsic parameters, Equation (16), although initial values in the different datasets are quite different. For example, for lifetimes > 16 weeks, the average lifetime is 32 weeks [

a a = A a / T a = A 0 + A i T 0 + T i = 1 + A 0 / A i 1 + T 0 / T i a . (17a)

L a = S a / A a = S 0 + S i A 0 + A i = 1 + S 0 / S i 1 + A 0 / A i L . (17b)

Dataset | A_{a} (cm) | A_{0} (cm) | A_{i} (cm) | (A_{0} + A_{i})/A_{a} |
---|---|---|---|---|

ACC-C | 7.62 | 1 | 6.5 | 1.0 |

ACC-L | 16.7 | 6 | 8.7 | 0.9 |

Dataset | S_{a} (km^{2}) | S_{0} (km^{2}) | S_{i} (km^{2}) | (S_{0}+ S_{i})/S_{a} |
---|---|---|---|---|

ACC-C | 3.2 × 10^{3} | 1.1 × 10^{3} | 2.2 × 10^{3} | 1.0 |

ACC-L | 5.06 × 10^{3} | 2.2 × 10^{3} | 2.8 × 10^{3} | 1.0 |

Dataset | T_{a} (days) | T_{0} (days) | T_{i} (days) | (T_{0} + T_{i})/T_{a} |
---|---|---|---|---|

ACC-C | 79 | 28 | 56 | 1.1 |

ACC-L | 73 | 30 | 40 | 1.0 |

Dataset | A_{02}/A_{i}_{2} | S_{02}/S_{i}_{2} | T_{02}/T_{i}_{2} |
---|---|---|---|

GO-C2 | 1.86 | 1.53 | 1.44 |

GO-L2 | 2.38 | 1.56 | 1.50 |

v a = − C 4 π L a a a = 1 + S 0 / S i 1 + T 0 / T i v e . (17c)

All of averages will be dependent on the intrinsic parameters, but independent of the artificial parameters A_{0}, T_{0} and S_{0}, under the condition of

A 0 / A i ≈ S 0 / S i ≈ T 0 / T i . (18a)

or,

A 0 / A i ≪ 1 , S 0 / S i ≪ 1 , T 0 / T i ≪ 1 . (18b)

For example (_{02}/S_{i} = 1.57 (_{02}/T_{i} = 1.5 (_{02}/S_{i} = 1.53 (_{02}/T_{i} = 1.44 (

In this study, different eddy datasets are used, and these datasets are derived from different identification and tracking algorithms. Moreover, the critical values used to identify the coherent structures as eddies are quite different in the various datasets. For this technical reason, the coherent structures associated with smaller amplitudes and sizes are not identified as eddies in the Li dataset, but they are identified as eddies in the Chelton data. Consequently, eddies in the Li dataset have larger amplitudes and sizes and relatively shorter lifetimes compared to those in the Chelton dataset (Tables 4-6).

As we can see from Equation (15), the larger the eddy size, the larger the viscosity is. This is the reason why the viscosities are larger in the Li dataset. Additionally, the larger eddies (e.g., the eddies in regime 2) also experience larger viscosities in the same datasets, as shown in

Both datasets show good consistency between the average and intrinsic parameters in Equation (16). The intrinsic parameters are independent of the critical values of identification, which are artificially chosen in the different datasets. It was expected that only the intrinsic parameters could be universal and independent of the mesoscale eddy datasets. The mixing rates, contrary to our expectations, are also universal and independent of the datasets.

Our method of estimating eddy viscosity from data is based on an implicit assumption that the dissipation of an eddy’s total energy is due to the viscosity only, and there is no net energy supplied from other mechanisms, such as wind forcing, energy genesis from the baroclinic instability of the flow and merging of ambient eddies, or the energy lost due to bottom friction, and the eddy splitting over time.

However, it is apparent from the two eddy examples shown above that eddy evolutions are not always characterized by monotonic decay with time. The amplitude of the eddy might increase from time to time (^{2}/s as inferred from the Li dataset, which is exactly the lowest value obtained from the example eddies in

The eddy viscosity in Equation (10) depends also on ratio C of total mechanical energy to EKE. To precisely estimate eddy viscosity, we need to use individual C for each eddy. In this study, we also use a constant C = 2.7, which is from a global average with 2-layer ocean model [

We test the hypothesis that eddy viscosity is proportional to r s n (n < 2) using eddy datasets. The dimensional eddy viscosities in different oceans obey the power law of v e = 10 − 6 r s 1.8 , which agrees well with the power law of observed diffusivities of k e = 1.4 × 10 − 6 r s 1.8 or k e = 7.2 × 10 − 6 r s 1.8 . Additionally, the extrapolation of the eddy mixing to molecule mixing implies that the above eddy mixing rates always hold until the value of r_{s} is less than O (1 m). Since such parameterization is valid from very small scale to very large scale, the mixing rates with the new parameterizations are suggested to use in numerical schemes. It is expected that the new parameterization may improve the numerical simulations accordingly. Compared with the larger value (10^{3} - 10^{4} m^{2}/s) of eddy viscosity commonly used in coarse and eddy permitting resolution models, lateral eddy viscosity inferred from satellite observations in the open ocean is on the order of 10^{2} - 10^{3} m^{2}/s. It implies that oceanic eddy mixing is more like strong diffusion than ordinary turbulence. The census of the mesoscale eddies shows, in general, that the eddy numbers obey e-folding decay laws in terms of their amplitude, area and lifetime, regardless of the regions and the choice of datasets. The present results are useful for the parameterizations in the numerical ocean models with horizontally variable resolutions.

We thank AVISO for providing the SLA data (http://www.aviso.oceanobs.com/). We thank Prof. R. X. Huang at WHOI for useful comments. This work was supported by the National Foundation of Natural Science (No. 41376017).

Li, Q.Y., Sun, L. and Xu, C. (2018) The Lateral Eddy Viscosity Derived from the Decay of Oceanic Mesoscale Eddies. Open Journal of Marine Science, 8, 152-172. https://doi.org/10.4236/ojms.2018.81008