1 Introduction

Turbulence closure models are needed in a dynamic framework to calculate grid turbulence for the purpose conserving momentum and kinetic energy in global ocean climate simulations. A variety of momentum closure schemes have been developed so far, such as the Laplacian viscosity closure, the biharmonic viscosity closure,the Leith closure (Leith, 1996), the Smagorinsky closure(Smagorinsky, 1963; Lilly, 1967), the anticipated vorticity method closure, the Lagrange mean α model closure(Holm et al., 1998; Chen et al., 1998), etc. Graham and Ringler (2013) evaluated the six types of closures used in 2D f-plane N-S equations by computing the sub-grid enstrophy spectrum and flux transport in all six closures.Ilicak et al. (2012) introduced explicit Friction operators,i.e., the local adaptive momentum closure scheme, to close the momentum equations. In this scheme, when the local grid Reynolds number is sufficiently small, the spurious dianeutral mixing of ocean models is reduced.

The model for prediction across scales (MPAS) is a coupling model to develop atmosphere, ocean, and other earth-system simulation components for global and regional climate and weather studies (http://mpas-dev.github.io). MPAS-Ocean (hereafter MPAS-O) was jointly developed by the National Center for Atmospheric Research(NCAR) and Los Alamos National Laboratory (LANL) in the United States (Ringler et al., 2013). Petersen et al.(2014) have evaluated the arbitrary Lagrangian-Eulerian vertical coordinate method in the MPAS-O model by using the relationship of spurious dianeutral mixing and grid Reynolds number. Reckinger et al. (2015) have examined the sensitivity of the overflow simulations to ver-tical and horizontal numerical parameters. Zhao and Liu(2016) have evaluated the horizontal resolutions, vertical resolution, horizontal advection schemes and grid characteristics of the models/simulations. However, the performance of the three local adaptive momentum closures used in the MPAS-O, i.e., the biharmonic viscosity (▽4),the Laplacian viscosity (▽2), and the Leith in the MPASO, is unknown. In this paper, the performances of the three closure schemes in the MPAS-O in a 3D baroclinic periodic channel case study are evaluated using the method of reference potential energy (RPE) (Winters et al., 1995;Ilicak et al., 2012). The horizontal kinetic energy and enstrophy spectrum obtained in the three respective closure schemes are analyzed.

This paper is organized as follows. Section 2 introduces the MPAS-O model. The 3D baroclinic eddy channel test and the RPE method are also described. Section 3 introduces the Laplacian viscosity, biharmonic viscosity, and Leith closure schemes. Section 4 compares the performance of the three closure schemes on the basis of the relative change of RPE (RCRPE), the RPE time change rate(RPETCR), the RMS of the horizontal kinetic energy, the spectra of the horizontal kinetic energy and enstrophy,and the optimum parameters. Section 5 focuses on discussion of the Leith scheme, and conclusions are given in Section 6.

2 Model, Test Case, and Method

2.1 Ocean Model

The governing equations of MPAS-O are hydrostatic,incompressible Boussinesq approximations. The MPAS-O uses non-structural, spherical, centroidal Voronoi tessellations (Skamarock et al., 2012) and C-grid staggering of the state variables for the horizontal discretization and an arbitrary Lagrangian-Eulerian vertical coordinate. In this study, the z* coordinate is applied because it can effectively reduce the spurious vertical mixing that arises from surface gravity waves. A split-explicit time integration scheme is employed. The flux-corrected transport scheme proposed by Zalesak (1979) is applied to ensure monotonicity by blending higher- and lower-order flux. The Leith (1996) scheme is used for the horizontal turbulence closure to ensure the conservation of mass, tracer, potential vorticity, and energy.

汽轮机装置在运行中出现异响及振动现象，为常见的一类故障现象。异响及振动故障现象的出现，对于汽轮机运行中的稳定性，以及组件之间的配合度影响较大。分析当前在具体作业中汽轮机异响及振动现象的出现，主要造成的不良现象体现为：气流激荡及转子异响振动；汽轮机运行温度异常，热变形，设备组件运行摩擦加剧。最终分析各类不良现象的出现，造成汽轮机在运行中出现了严重的温度升高现象，同时由于转子内部受温不均匀，产生的转子热变形，引起的设备停机现象也为常见的一类不良现象。

2.2 Test Case: The 3D Baroclinic Eddy Channel

Here we consider the rotating effect and take the fplane approximation, i.e., f = 1.2 × 10−4 s−1 (about 55˚ latitude). This test case is designed as an idealized case of the Antarctic Current near the polar region, which is similar to that in Ilicak et al. (2012). The model domain is a planar channel with zonal periodic boundary conditions.The latitudinal length is 433km, and the longitudinal width is 160 km. The depth of the channel is 1000 m, and the bottom is flat. The north and south boundaries are fixed. No-slip boundary conditions along the north and south are adopted for the velocity field. The initial temperature profile in the vertical direction declines along a linear gradient, while a cosine-shaped temperature perturbation with a wavelength of 120 km in the zonal direction is used to generate mesoscale eddies under the condition of rotation at the center of the channel. The disturbance in the third wave trough (110 km < x < 130 km) is to trigger baroclinic instability. The above settings constitute a baroclinic instability channel that favors the development of mesoscale eddies. Quadratic bottom drag is applied, and the drag coefficient is set to Cd = 0.01. The first baroclinic Rossby radius of deformation is 20 km. The horizontal resolution is 4 km, and the vertical direction is evenly divided into 20 layers (Fig.1). The initial state is at rest. The simulation covers a period of 200 days after the development of baroclinic instability. The time step is 300 s.

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2.3 Assessment Method

Fig.1 Snapshots of temperature profiles in the y-z plane (a) and sea surface temperature at the initial time (b) from the baroclinic eddy test. Here the horizontal resolution is 4 km, and the vertical resolution is 50 m with x = 28 km.

The Reynolds number is a dimensionless number used in fluid mechanics to represent the situation of fluid flow.It is the ratio between the forces of inertia and viscosity in physics. The grid Reynolds number is employed in numerical fluid experiments. In fluid simulation using the grid model, a high Reynolds number can intensify the grid scale energy and associated noise. Bryan et al. (1975)and Griffies (2004) analyzed the characteristics of high-Reynolds-number flow in the stable state based on the discretion of advection diffusion equation with secondorder accuracy. They found that in order to adequately dissipate the spurious grid scale energy, the grid Reynolds number must be less than 2 (Re∆<2). The grid Reynolds number can be obtained by dimensional analysis, which is expressed as:

where U indicates the RMS of the horizontal kinetic energy, ∆ is the horizontal resolution, and νh is the numerical viscosity in units of m2 s-1.

Winters et al. (1995) suggested that the RPE can describe well the spurious dianeutral mixing of the ocean model. Ilicak et al. (2012) found that the decrease in spurious dianeutral mixing is proportional to the grid Reynolds number prior to saturation. In the present study,relative changes in reference potential energy (RCRPE)and RPETCR (Ilicak et al., 2012) are calculated by Eqs.(2) and (3), respectively,

and the relationship between the RPETCR and grid Reynolds number from the three closure schemes are compared to evaluate their performance.

3 Three Momentum Closure Schemes

3.1 Laplacian Viscosity Closure (▽2)

The Laplacian viscosity turbulent closure in MPAS-O is expressed as (Petersen et al., 2013):

Eq. (8) can be obtained on the basis of dimensional analysis of Eqs. (5) and (7) as follows:

here ρm is the maximum grid density in the study area,and its value is set to 1 because the homogenization grid in this article is used. νh is the Laplacian viscosity coefficient with units of m2 s−1. The grid Reynolds number is calculated by Eq. (1).

Fig.11 is the enstrophy diagram based on filter analysis of the simulations using the three types of closure schemes. The same average RPETCR of 0.00011 W m−2,which corresponds to ref 3 in Fig.7, was used in the three closure schemes. The horizontal kinetic energy was integrated first in the vertical direction, and the Fourier decomposition and filter were then applied to the vertically integrated horizontal kinetic energy. Here waves 1–3, 4–6,and 7–9 represent large-, intermediate-, and small-scale patterns, respectively. As seen in Fig.11, waves 1–3 weaken over time in all three closure schemes, i.e., all three closure schemes are dissipative. Among the three closure schemes, the enstrophy simulated by the Leith scheme is relatively weak. The intensity (duration) of the enstrophy of waves 4–6 simulated by the Laplacian viscosity closure scheme is between that simulated by the biharmonic viscosity and by the Leith closure scheme(the shortest). Eddies simulated by the Leith closure scheme are the most stable and continuous and last the longest. The strength and stability of the viscous enstrophy for waves 7–9 simulated by the biharmonic viscosity scheme are the strongest, followed by those obtained by simulation with the Laplacian viscosity scheme. Eddies simulated by the Leith closure are also moderately stable.

Fig.2 Snapshots of sea surface temperature from the baroclinic eddy test with the Laplacian viscosity closure on the 60th day and the horizontal viscosity of 5 m2 s−1(a), 50 m2 s−1 (b), and 200 m2 s−1 (c). The corresponding averaged grid Reynolds numbers are 99, 8.4 and 1.88,respectively, which are averaged over the 200 days.

3.2 Biharmonic Viscosity Closure (▽4)

The biharmonic viscosity turbulent closure in MPAS-O is expressed as

here ρm ranges from 0 to 1. Because a uniform grid is adopted in this study, the value of ρm is 1. is the biharmonic operator coefficient for the biharmonic viscosity closure, which can be expressed as

here ∆x is the characterizedmotion scale, and ∆x0 is the grid interval. The units of are m4 s−1. The biharmonic operator is similar to the Laplacian operator, and the right-hand side of Eq. (5) can be written as Eq. (7) by assuming an equivalent parameter ν* as follows:

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Generally, the order of magnitude of the characteristic scale is the same as that of the grid interval, thus we have

Substituting Eq. (9) into Eq. (1), the corresponding grid Reynolds number can be written as

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Fig.3 Same as Fig.2, but for results obtained with the biharmonic viscosity closure. Here the biharmonic viscosity coefficients are 1.6×107 m4 s−1 (a), 9.0×107 m4 s−1(b), and 9.9×108 m4 s−1 (c). The corresponding grid Reynolds numbers are 7.9, 93 and 515, respectively.

Fig.3 shows the simulated SST from the baroclinic eddy test with the biharmonic viscosity closure at the 60th day. The biharmonic viscosity coefficients are 1.6 × 107,9.0 × 107, and 9.9 × 108 m4 s−1 in Figs.3a, b and c, respectively, and the corresponding averaged grid Reynolds numbers are 505, 86 and 7, respectively. As shown in Fig.3, multi-scale eddies can be simulated well with all three sets of viscosity coefficients and Reynolds numbers in the biharmonic viscosity closure. In general, the smaller the grid Reynolds number, the more numerous and stronger the simulated eddies. In addition, the clearer the interface between cold and warm water, the greater the temperature gradient. However, unlike the Laplacian viscosity closure,a smaller grid Reynolds number always corresponds to stronger dissipation in the biharmonic viscosity closure,but the area of cold water is not the largest. This result suggests that large-scale dissipation is weaker in the biharmonic viscosity closure than that in the Laplacian viscosity closure. Small-scale dissipation is dominant in the biharmonic viscosity closure. The dissipation in the biharmonic viscosity closure may be more scale dependent.The grid Reynolds numbers in Fig.3c are similar to those in Fig.2b, but more eddies are well simulated by the biharmonic viscosity closure, and the temperature interface between cold and warm water is clearer, indicating that the dissipation in the biharmonic viscosity closure is stronger than that in the Laplacian viscosity closure.

3.3 The Leith Closure

here L is the characteristic length scale. Substituting Eq.(13) into Eq. (1), we obtain the corresponding grid Reynolds number, which is expressed as

In Eq. (11), ω is the relative vorticity, u is the horizontal velocity, ∆x0 is the horizontal grid interval, and Γ is a dimensionless parameter. The right-hand side of (11) can be written as

The Leith viscosity scale can be obtained by dimensional analysis of Eqs. (11) and (12) and expressed as:

The Leith (1996) scheme is employed in MPAS-O. The Leith closure can be expressed as

To test the Leith closure, in this section baroclinic eddies are simulated by first setting the characteristic length scale to the grid interval, which is 4000 m, with different Г values. The grid Reynolds numbers corresponding to different Г values are 2.5, 0.5 and 0.2, respectively. The simulated results at the 60th day are presented in Fig.4,which shows that the interface between the warm and cool water becomes clearer and smoother while its southward motion is inhibited as the value of the parameter Г increases. In addition, the temperature over the warm water increases, and the temperature gradient increases. For example, the maximum temperature in Fig.4c reaches 12.9℃, while it is only 12.8℃ in Fig.4a. Fig.4b clearly shows more mesoscale vortexes with some submesoscale structures. However, when Г is too large, the number of eddies decreases and the sub-mesoscale eddies disappear (Fig.4c). This result suggests that when the characteristic length scale is held constant, the larger the parameter Г, the greater the momentum dissipation. Meanwhile, the smaller the advection velocity, the weaker the spurious mixing.

It has been shown in Section 4.3.1 that L and Г are a pair of coupled parameters. Then what is the function of the parameter L? In this section the horizontal kinetic energy with Г=0.5 and L=2, 4, and 10 km are first integrated vertically. The Fourier decomposition is then applied to the results, and the first three waves (Fig.16) are analyzed. As shown in Fig.16, waves 1, 2, and 3 are significant when L=2 km. The kinetic energy decreases as L increases. When Г=0.5, there exists certain energy dissipation, but the scales on which the dissipation occurs are different. Figs.16g–i shows clearly that the kinetic energies of waves 1 and 2 greatly reduced, whereas the wave energy peak for wave 3 occurs between the 60th and 70th day. Note that such peaks are absent in Figs.16c and f. It is speculated that L plays a key role on a large scale in transferring the kinetic energy from waves 1 and 2 to wave 3, i.e., in the Leith closure, part of the energy is absorbed from large-scale motions and transferred to small-scale ones.

To test the impact of the characteristic length L on the Leith scheme, a series of 200-day simulations are conducted with Г = 0.5, and L is set to 2 km, 4 km, and 10 km.Fig.5 shows snapshots of the simulated SST on the 60th day in the above three experiments. As can be seen in Fig.5,with a fixed value of Г, the temperature interface between cold and warm water becomes clearer and smoother with the increase in the value of the L, and the eddies become stronger. However, unlike the impact of Г on the moving speed of the temperature interface, the greater the L value,the faster the interface moves southward. This is probably because part of the kinetic energy on scales smaller than L are transferred to kinetic energy on scales larger than L,which moves the interface southward.

Fig.4 Same as Fig.2, but for results obtained with the Leith closure, which uses the same characteristic length scale L=4 km, and the parameter Г is 0.4 (a), 2 (b) and 5(c). The corresponding grid Reynolds numbers are 2.5,0.5 and 0.2, respectively.

Fig.5 Same as Fig.4, but the same Г is used, and the characteristic length scale L is 2 km (a), 4 km (b), and 10 km (c). The corresponding grid Reynolds numbers are 0.5, 2 and 12.5, respectively.

4 Performance Comparison Among Momentum Closure Schemes

In order to compare the three momentum closure schemes, i.e., the Laplacian viscosity, biharmonic viscosity and Leith scheme, three closure schemes with different parameters are used to simulate baroclinic eddies in this section. The horizontal resolution is 4 km, and the RPE method is employed to evaluate the simulation results. Tables 1, 2 and 3 show the specified parameters and corresponding grid Reynolds numbers at the 50th, 100th,and 200th days and the averaged grid Reynolds numbers during 200 days for the Laplacian viscosity, biharmonic viscosity, and Leith schemes, respectively. Note that the grid Reynolds number of the Leith closure scheme in Table 3 has nothing to do with the velocity, i.e., the grid Reynolds number at each individual day is the same.

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4.1 Temporal Changes in the RCRPE

Fig.6 shows temporal changes in the RCRPE simulated in the baroclinic eddy tests with the Laplacian viscosity,biharmonic viscosity, and Leith closure scheme. As shown in Fig.6, the RCRPE increases with time, which suggests that spurious dianeutral mixing is increasing. However,the RCRPE in general will decrease when any of the above three turbulence closures are used. Spurious dianeutral mixing can be restrained effectively by increasing the Laplacian viscosity coefficient or biharmonic viscositycoefficient or by increasing the dimensionless parameter Г in the Leith closure scheme.

Table 1 The parameter settings and corresponding grid Reynolds numbers for the Laplacian viscosity closure

Notes: Exp. indicates the experiment numbers, νh is the lateral viscosities, and Re∆_50, Re∆_100, Re∆_200 and Aver_Re∆ indicate the grid Reynolds numbers on the 50th, 100th and 200th days and the averaged grid Reynolds numbers during the 200 days, respectively.

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Table 2 The parameter settings and corresponding grid Reynolds numbers for the biharmonic viscosity closure

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Table 3 The parameter settings and corresponding grid Reynolds numbers for the Leith closure

Exp. L (km) Г Aver_Re∆ LГ (km)1 4 0.003 333 0.012 2 4 0.015 67 0.06 3 4 0.07 14 0.28 4 4 0.4 2.5 1.6 5 4 0.5 2 2 6 4 0.6 1.67 2.4 7 4 0.625 1.6 2.5 8 4 1 1 4 9 4 2 0.5 8 10 4 5 0.2 20 11 2 0.01 25 0.02 12 2 0.1 2.5 0.2 13 2 0.5 0.5 1 14 2 1 0.25 2 15 2 1.25 0.2 2.5 16 10 0.0007 8929 0.007 17 10 0.002 3125 0.02 18 10 0.018 347 0.18 19 10 0.09 69 0.9 20 10 0.2 31 2 21 10 0.25 25 2.5 22 10 0.5 12.5 5

Fig.6 Part of the RCRPE changes over time in the baroclinic eddy test with the Laplacian viscosity closure, biharmonic viscosity closure, and Leith closure.

4.2 Temporal Changes in the RPE

Fig.7 shows the average RPETCR over 200 days of simulation in the baroclinic eddy tests using different horizontal momentum closures. The average grid Reynolds numbers during the 200-day period of simulation and the specific parameter settings used in the tests can be found in Tables 1, 2 and 3. Fig.7 shows clearly that with the same grid Reynolds number, the dissipation in the biharmonic viscosity scheme is the largest among the three schemes, while that in the Laplacian viscosity (Leith)scheme is moderate (the weakest). The RPETCR in the Leith closure scheme is the maximum among the three schemes and larger than that in the other two schemes by one order of magnitude. All three turbulence closure schemes show a phenomenon of RPETCR saturation with a saturation value of approximately 2.0 × 10−3 W m−2 for RPETCR, which corresponds to the reference line ‘ref 1’in Fig.7. The grid Reynolds number thresholds differ between the three turbulence closure schemes when the RPETCR reaches saturation. The critical grid Reynolds number is the largest in the biharmonic viscosity scheme,followed by that in the Laplacian viscosity closure scheme,and that in the Leith closure scheme is the smallest. This result differs from the conclusions of Ilicak et al. (2012)and Petersen et al. (2014), who used various horizontal resolutions and viscosity coefficients in four tests to simulate lock-exchange flow, overflow, internal waves,and baroclinic eddies. Ilicak et al. (2012) analyzed the RPETCR and grid Reynolds numbers at the end of the simulation, while Petersen et al. (2014) investigated their average values. They found the saturation phenomenon in the first three tests but not in the baroclinic eddy test. This is probably because rotation in the baroclinic eddy simulation test could promote enstrophy cascading and energy transfer and conversion among different scales. The dissipation of energy of large-scale movements is roughly the same as that of energy contained in small-scale movements in the Laplacian viscosity closure scheme. Therefore, it is difficult to reach the RPETCR saturation. The energy dissipation in the Leith and biharmonic viscosity closure schemes focuses on small-scale motions, and their dissipation of energy of motion on large scales is weak,which makes it easy to reach the RPETCR saturation in the Leith and biharmonic viscosity scheme.

Fig.7 Changes in the 200-day averaged RPETCR (W m−2)with different grid Reynolds numbers from the baroclinic eddy test using various momentum closure configurations, in which the RPETCR of the reference line ref 1, ref 2 and ref 3 are 0.002 W m−2, 0.00025 W m−2,and 0.00011 W m−2, respectively.

The grid Reynolds number threshold for the RPETCR to reach saturation may differ, because the characteristic length scale in the Leith closure scheme may also differ.If the characteristic length scale is set to the grid interval of 4 km, the RPETCR could reach saturation when Re∆>14; if the characteristic length scale is set to 2 km (i.e., the characteristic length scale is smaller than the grid distance), the RPETCR could reach saturation when Re∆>1;and if the characteristic length scale is set to 10 km (i.e.,the characteristic length scale is larger than the grid distance), the RPETCR could reach saturation when Re∆>347.Differences in the grid Reynolds number threshold for the above three characteristic length scales are up to one or two orders of magnitude, which is equivalent to the difference in magnitudes of energy of movements on different scales. Based on the above results, it can be deduced that for the Leith closure scheme, the dissipation of energy of movements on scales larger than the grid interval is strong when the characteristic length scale is greater than the horizontal grid interval (L>∆x0). As a result, the energy dissipation strengthens overall. On the contrary,the dissipation of energy focuses on movements on scales smaller than the grid interval when the characteristic length scale is smaller than the horizontal grid interval (L<∆x0);thereby, the dissipation weakens in general. As a result,the grid Reynolds number threshold for the RPETCR to reach saturation becomes smaller.

4.3 Temporal Changes in the RMS of the Horizontal Kinetic Energy

Fig.8 shows temporal changes in the RMS of the horizontal kinetic energy simulated in the baroclinic eddy test with different closure schemes but with the same RPETCR. The units are m s−1. The values of RPETCR are 0.002, 0.00025 and 0.00011 W m−2, respectively, corresponding to the three reference lines shown in Fig.7.Temporal changes in the RMS of the horizontal kinetic energy with the viscosity coefficient 0.001 m2 s−1 are taken as the reference. As depicted in Fig.8, the RMS of the horizontal kinetic energy first increases and then decreases with time. Such a changing pattern clearly reflects the baroclinic adjustment process for the flow field and potential height field to reach geostrophic equilibrium through energy dispersion by inertia-gravity waves. The baroclinic adjustment process is more complex than the typical baroclinic adjustment after the start of baroclinic instability. This is because there is more than one initial disturbance in the baroclinic eddy test case. In addition to the cosine wave disturbance, whose wavelength is 120 km in the horizontal direction, a temperature disturbance is added to the third wave trough (110 km < x < 130 km).Vertical stratification also exists in the test. During the 40–50 days of the simulation, the RMS of horizontal kinetic energy has two peaks. The life span of the first wave is shorter than that of the second. The two wave peaks correspond to the two different initial disturbances, i.e.,the cosine temperature disturbance, whose wavelength is 120 km, and the disturbance in the third wave trough,whose scale is 20km. Different lengths of time are needed for the two disturbances to develop. The first peak corresponds to the baroclinic adjustment, whereas the second peak corresponds to the impact of the development of the mesoscale eddy and the interaction between eddies. The second peak has a higher magnitude and longer duration than the first, because the kinetic energy of the eddy is larger than that of the environmental flow.

Fig.8 RMS of the kinetic energy in the baroclinic eddy test for different closures with the same RPETCR.RPETCR corresponds to three reference lines shown in Fig.6, in which the same line style indicates the reference line with the same RPETCR, while the same color indicates the same closure. The black line is the reference line of RMS of the horizontal kinetic energy with aviscosity coefficientand the RPETCR corresponds to ref 2 shown in Fig.7. Because the three lines for the Leith scheme corresponding to ref 2 with L=2are the same, their RMS curves of the horizontal kinetic energy overlap and are shown as a single line in cyan.

As seen in Fig.8, the first wave peak corresponding to the baroclinic adjustment appears at almost the same time in all test cases, but its duration differs. The three closure curves corresponding to ref 1 are basically consistent when the RPETCR decreases. Accompanied by the increase in the kinetic energy dissipation, the first peak of the RMS of the horizontal kinetic energy decreases, and its life cycle shortens. Corresponding to ref 2 and ref 3,the magnitude of the peak in the biharmonic viscosity closure scheme is the highest, followed by that in the Laplacian viscosity scheme, and the magnitude of the peak in the Leith scheme is the lowest. For different closure schemes, the value and duration of the second peak in the RMS of the kinetic energy differ. The second peak simulated by the Laplacian viscosity closure scheme appears at roughly the same time (i.e., the 30th day), and the magnitude of the peak decreases as the viscosity coefficient increases. The second peak corresponding to ref 1 and ref 2 appears later in the simulation with the biharmonic viscosity closure scheme than that in the simulation with the Laplacian viscosity closure scheme around the 35–40th days. In addition, the magnitude of the peak is slightly higher than that corresponding to ref 1 in the simulation by the Laplacian viscosity closure scheme, and the peak corresponding to ref 3 appears earlier than that corresponding to ref 1 in the simulation by the Laplacian viscosity scheme on the 25th day. The magnitude of the peak is basically equal to that corresponding to ref 1 in the simulation by the Laplacian viscosity scheme. The simulated second peak of the RMS of the horizontal kinetic energy that corresponds to ref 1 using the Leith closure agrees with that simulated by the Laplacian viscosity closure scheme. The peak corresponding to ref 2 and ref 3 appears on the 40–50th days, later than the appearance of the peak corresponding to ref 1. Note that the magnitude of the peak corresponding to ref 2 is lower than that corresponding to ref 1, and the magnitude of the peak corresponding to ref 3 is higher than that corresponding to ref 1 in the simulation with the Laplacian viscosity closure scheme.

Several peaks occur after the occurrence of the second one in all simulations, except those simulations in which the peak corresponds toref 2 by the Laplacian viscosity and Leith closure schemes. The magnitudes of those peaks that occur later are smaller than that of the second peak. This result demonstrates the periodicity of the tests.At the end of the simulation (on the 200th day), the RMS of the kinetic energy simulated by the Laplacian viscosity closure scheme still shows that the magnitude of the peak corresponding to ref 1 is the highest, followed by that corresponding to ref 2, and the magnitude of the peak corresponding to ref 3 is the smallest. The RMS of the kinetic energy corresponding to ref 2 in the simulation with the biharmonic viscosity closure and Leith closure scheme are the highest, followed by that corresponding to ref 1, and the RMS of the kinetic energy corresponding to ref 3 is the smallest. Overall, for all three closure schemes,the greater the dissipation of kinetic energy in the early stage of the baroclinic geostrophic adjustment, the greater the kinetic energy obtained at the end of the adjustment.This is reflected in the fact that either the magnitude of the peak is large or the decay lasts longer. The magnitude of the third peak corresponding to ref 2 simulated by the Laplacian viscosity closure and Leith closure scheme is higher than that of the second peak, which suggests that the flow field may regain some energy from the potential height field because of the interaction between eddies in the mixing process. We speculate that overall, the energy is dissipative in the Laplacian viscosity closure scheme,which explains why the peak value of kinetic energy decreases as the viscosity coefficient increases. The dissipation in the biharmonic viscosity closure scheme strongly depends on scales, with most dissipation occurring for energy of movements on small scales. Thereby, with the same RPETCR, the kinetic energy obtained in the simulation with the biharmonic viscosity closure scheme is the highest. With the same RPETCR, the magnitude of the second peak corresponding to ref 3 in the simulation with the Leith closure scheme is higher than that simulated with the other two closure schemes. This is probably because in the Leith closure scheme, part of the kinetic energy is transferred to large-scale kinetic energy when the sub-grid scale kinetic energy is dissipated.

4.4 Spectra of Kinetic Energy and Enstrophy

In order to study the dependence of different closure schemes on the spatial scale, the vertically integrated horizontal kinetic energy and enstrophy simulated by the three closure schemes are analyzed based on results of the baroclinic eddy test. In this test, the same RPETCR is applied. The vertically integrated horizontal kinetic energy and enstrophy are averaged over the simulation time,and the spatial information in the horizontal direction is kept, while the average flow is removed. The Fourier decomposition is then conducted (Figs.9 and 10). RPETCR are set to 0.002 and 0.00025 W m−2, corresponding to the lines of ref 1 and ref 2 in Fig.7, respectively. Due to the temporal average and vertical integration of the kinetic energy and enstrophy, the time information and vertical scale information are removed from the two fields. The abscissa represents ‘frequency’ in Figs.9 and 10. Here‘frequency’ is related to the horizontal scale, and a higher frequency corresponds to a smaller horizontal scale.

As can be seen in Fig.9, when the average RPETCR reaches saturation, i.e., the effect of the closure in momentum equations is very small, the kinetic energy and enstrophy spectra simulated by the three closure schemes are similar for each individual frequency. All of the spectra present a decreasing trend when the frequency increases except for those with a frequency of 0. This result indicates that the kinetic energy and enstrophy decrease when the horizontal scale of the motion decreases, and the frequency increases. Overall, the kinetic energy and enstrophy simulated by the biharmonic viscosity closure scheme are the largest on each individual scale. In particular, the kinetic energy on the largest scale is much larger than those simulated by the Laplacian viscosity closure and Leith closure schemes. The kinetic energy and enstrophy simulated by the Laplacian viscosity closure scheme are in between the results from the biharmonic viscosity and Leith closure scheme on all scales except the largest one.

Table 4 lists the energy changes on the 200th day compared to that at the initial time in the baroclinic eddy simulations by the three closures. The values in the table are obtained on the basis of results using the optimal parameters and RPETCR of levels of ref 1 and ref 2, respectively. As can be seen from Table 4, the relative change in the total energy (RCE) during the whole simulation with the RPETCR of the ref 1 level is positive, suggesting thatspurious numerical energy is introduced. When the RPETCR declines to the ref 2 level, the relative change in total energy during the whole simulation becomes negative, implying that energy is dissipated gradually. Theoretically, the RCE should be zero when energy is conserved. The RPE increases due to mixing; therefore,RCRPE is positive. α is the ratio of the total energy change to the reference potential energy change, and it reflects the change in total energy when the increase in RPE is the same. As seen in Table 4, the RCE is one order of magnitude lower than RCRPE. Therefore, it is appropriate to dissipate a small amount of physical energy in order to suppress spurious numerical mixing. When α is negative, the smaller the absolute value of α, the more energy is conserved and the better the corresponding setting of the closure. Therefore, we calculated the corresponding α for all simulations (for simplicity, only some of the values are listed in Table 4). We concluded that the optimal value of the viscosity coefficient lies within the range of [0.1, 1] m2 s−1, and the optimal value of the biharmonic viscosity coefficient is within the range of [2.5×105, 5.0×105] m4 s−1. For the Leith closure with the characteristic scale of 4 km, the optimal value of Г is within[0.2, 0.4]. The optimal parameters proposed by Graham and Ringler (2013) are not within the above-mentioned range. The optimal parameter values may be different for different resolutions or physical processes. It is uncertain to what degree the energy dissipation is allowed. In order to determine the optimal parameters for the three closures in a 3D baroclinic model, more detailed studies are necessary in the future.

Fig.9 Fourier spectra of the horizontal kinetic energy and enstrophy simulated in the baroclinic eddy test with the three closures, i.e., the Laplacian viscosity closure (top panels), biharmonic viscosity closure (middle panels), and Leith closure (bottom panels), respectively, under the same average RPETCR 0.002 W m−2, which corresponds to ref 1 shown in Fig.7. The vertically integrated results are then temporarily averaged, the average flow is thereby removed, and the marked grid Reynolds number is the 200-day averaged value.

Fig.10 Same as Fig.9, except that the RPETCR is 0.00025 W m−2, which corresponds to ref 2 shown in Fig.7.

When the average RPETCR is 0.00011 W m−2, there exists strong dissipation of the horizontal kinetic energy on all scales in the three closure schemes. However, the dissipation of the kinetic energy in the biharmonic viscosity closure scheme is relatively small, especially considering that more kinetic energy on the largest scale and sub-largest scale remain. All three closure schemes show a strong dissipation of enstrophy on all scales. The enstrophy simulated in the Laplacian viscosity closure scheme tends to be almost zero, whereas only a small amount of large-scale enstrophy remains in the biharmonic viscosity and Leith closure schemes (figure not shown).

Fig.2 shows the simulated sea surface temperature(SST) at the 60th day. The horizontal resolution is 4 km,and the viscosities are 5 m2 s−1, 50 m2 s−1, and 200 m2 s−1,respectively. The averaged grid Reynolds number during the 200-day integration period is 93, 7.5 and 1.35, respectively. As shown in Fig.2, the smaller the horizontal viscosity, the more disordered the simulated results and the smaller the SST gradient, indicating that spurious mixing is strong (Fig.2a). When the horizontal viscosity is sufficiently large, mesoscale and small-scale eddies can be well simulated, and the surface temperature gradient increases. Meanwhile, the vortices become strengthened and the number of eddies subsequently increases. Eventually,the interface between cold and warm water becomes clear,and the area with SST warmer than 12.9℃ becomes larger (Fig.2b). Note that sub-mesoscale eddies occur in the lower left corner of Fig.2b, which are similar to the vortex filament of the cascade. When the viscosity is sufficiently large, the simulated mesoscale eddies decrease,but the scale of these eddies increases and approaches the baroclinic Rossby deformation radius (Fig.2c). Meanwhile,the southward motion of the warm water interface is weak.This result indicates that spurious dianeutral mixing is stronger when the horizontal viscosity is smaller. When the viscosity increases, the dissipation of kinetic energy and the smoothing effect of the viscosity become stronger,and small-scale eddies are almost all dissipated.

Fig.11 Fourier decomposition filter of the enstrophy simulated in the baroclinic eddy tests with the same average RPETCR of 0.00011 W m−2, which corresponds to ref 3 shown in Fig.7 and the three closures: Laplacian viscosity closure (top panels), biharmonic viscosity closure (middle panels), and Leith closure (bottom panels). The left column is for 1–3 waves, the middle column is for 4–6 waves, and the right column is for 7–9 waves. The grid Reynolds number is the value averaged over 1–200 days.

4.5 Simulated Results with the Optimal Parameters

With three different levels of RPETCR, all three closures dissipate enstrophy. When the RPETCR declines from ref 1 to ref 3, the enstrophy simulated by the three closures decreases gradually, but there are differences in the changes in horizontal kinetic energy. The horizontal kinetic energies corresponding to ref 1 and ref 2 simulated by the Laplacian viscosity closure are similar to each other, and the horizontal kinetic energy corresponding to ref 2 increases slightly. However, the simulated horizontal kinetic energy corresponding to ref 3 is far lower than those corresponding to ref 1 and ref 2. The kinetic energy simulated by the biharmonic viscosity and Leith closure corresponding to ref 2 (ref 1) is the largest(moderate), and that corresponding to ref 3 is the smallest.The above result implies that there exists an optimal value for the parameters used in the three closures, which can make the RPETCR as small as possible, i.e., spurious mixing can be as little as possible, and the damage to the physical process can be reduced to a minimum. Graham and Ringler (2013) obtained the optimal parameters based on the 2D barotropic primitive equation. They calculated the sub-grid transport of the enstrophy spectrum and flux errors of different closures and found that the viscosity coefficient νh_optimal =11 m2 s−1, biharmonic viscosity coefficient ν4_optimal =1.1×109 m4 s−1, and Leith dimensionless parameters Γoptimal =1 are optimal for the closure schemes.We applied the above optimal parameters to simulate the test case over a period of 200 days and obtained average RPETCR of 1.9 × 10−4, 1.08 × 10−4, and 1.8 × 10−4 W m−2,respectively. These values correspond to the results denoted by the black symbols in Fig.7. The averaged RPETCR in the biharmonic viscosity closure corresponds to ref 3 shown in Fig.7. The above optimal parameters obtained in the 2D barotropic model were then applied to 3D baroclinic eddy simulation. Results indicate that both the decreasing RPE and the increasing available potential energy are the largest in the biharmonic viscosity closure,followed by that in the Leith closure, and that in the Laplacian viscosity closure is the smallest. Note that the optimal parameters obtained in the 2D barotropic model may not be applicable to the 3D baroclinic model.

4.5.1 Temporal changes in the RMS of the horizontal kinetic energy

Fig.12 depicts temporal changes in the RMS of the horizontal kinetic energy simulated in the baroclinic eddy test by the three closures with the above-mentioned optimal parameters. The result shown by the black dotted line,corresponding to a viscosity of 0.001 m2 s−1 (similar to Fig.8), is taken as the reference line. As seen in Fig.12,the overall temporal trend for change in the RMS of the horizontal kinetic energy corresponding to the optimal parameters is similar to that represented by the reference line. There are four peaks, among which the latter three peaks simulated by the biharmonic viscosity closure appear earlier than that indicated by the reference line, and the second and third peaks are higher than that of the reference line. The peaks simulated by the Leith and Laplacian viscosity closures occur later than that of the reference line, and the second and third peaks simulated by the Leith closure are comparable to that of the reference line.All three peaks simulated by the Laplacian viscosity closure are lower than that of the reference line. Possible reasons for the above results may be that the enstrophy dissipation is the smallest in the biharmonic viscosity closure, followed by that in the Leith closure, and the enstrophy dissipation is the largest in the Laplacian viscosity closure. As a result, mesoscale eddies first developed in the simulation of biharmonic viscosity, and their strength is the greatest, while mesoscale eddies developed last and with the weakest intensity in the simulation of viscous closure. The simulation by the Leith closure is intermediate between the two. A possible explanation for these results is that the RPETCR used in the biharmonic viscosity closure corresponds to the level of ref 3 in Fig.7,while those used in the Laplacian viscosity and Leith closure are higher than that in the biharmonic viscosity closure. The lower the RPETCR, the lower the numerical mixing and the stronger the kinetic energy dissipation at the same time. Therefore, the kinetic energy simulated by the biharmonic viscosity closure is the smallest at end of the simulation, whereas there is little difference between those simulated by the Laplacian viscosity and Leith closures.

Fig.12 Same as Fig.8, except that the optimal parameters obtained in the 2D barotropic model are applied.

4.5.2 Changes in sea surface temperature

Fig.13 shows the surface temperature in baroclinic eddy tests using the three momentum closures on the 50th,100th and 200th days with the optimal parameters proposed by Graham and Ringler (2013). As shown in Fig.13,because the RPETCR used in the biharmonic viscosity closure is the optimal value of 0.00011Wm−2 corresponding to ref 3 in Fig.7, whereas the value of RPETCR used in the Laplacian viscosity and Leith closures is between ref 2 and ref 3, the interface between warm and cool water on the 50th and 100th days is clearer and smoother in the simulation by the biharmonic viscosity closure than that in the simulation by the Laplacian viscosity and Leith closures. The large-scale dissipation in the three closures is similar, but large differences are found in the mesoscale dissipation among the three closure schemes. The interface between warm and cold water simulated by the three closures moves southward the same distance on the 50th day. However, as shown in Fig. 12, the RMS of the horizontal kinetic energy on the 50th day in the simulation by the Laplacian viscosity closure is the largest, followed by that from the biharmonic viscosity closure, and that in the simulation by the Leith closure is the smallest. As seen in Fig.13, two mesoscale eddies have formed, and a large eddy is developing in the results simulated by the Laplacian viscosity closure, while two mesoscale eddies and a sub-mesoscale eddy are found in the result by the biharmonic viscosity closure. In contrast, only a mesoscale eddy and a sub-mesoscale eddy are found in the result of the Leith closure simulation, and two ridges of disturbance occur on the interface between cold and warm water. It is well known that the kinetic energy in the vortex generally is greater than that in the surrounding environment; therefore, differences in the kinetic energy on the 50th day can be represented by the number, scale, and intensity of the eddies.

（2）依托山水资源发展旅游经济。漳河水库位于湖北省荆门、宜昌、襄阳三市交界处，地处古三国战场的中心地带，是张家界、古隆中、长江三峡等旅游热线的黄金腹地。漳河水库按照“科学规划，统一管理，严格保护，永续利用”的总体要求，旅游项目开发与投入不断加大，重点完成观音岛、长青岛、观音寺、宾馆半岛景区和漳河旅游港游客中心建设，实现景区统一管理。2016年被评定为国家级“AAAA”景区，经营收入和接待游客人数连续三年增长速度均超过30%。

The front simulated by the Leith closure on the 100th day moves southward the farthest and reaches the south boundary, followed by the simulation from the biharmonic viscosity closure, and the southward moving distance of the front simulated by the Laplacian viscosity closure is the shortest. Accordingly, the RMS of the horizontal kinetic energy simulated by the Leith closure is the largest as shown in Fig.12 on the 100th day, followed by that from the biharmonic viscosity closure, and the RMS of the horizontal kinetic energy is the smallest in the simulation by the Laplacian viscosity closure. The strength of the cold eddy is the strongest in the simulation by the Leith closure, followed by that in the biharmonic viscosity closure, and that in the simulation by the Laplacian viscosity closure is the weakest.

Fig.13 Snapshots of SST from the baroclinic eddy test using the three closures on the 50th, 100th and 200th days with the optimal parameters obtained in the 2D barotropic model, νh_optimal =11 m2 s−1, ν4_optimal =1.1×109 m4 s−1, and Γoptimal =1, for the Laplacian viscosity closure(left column), the biharmonic viscosity closure (middle column), and the Leith closure (right column). The marked grid Reynolds numbers are averaged values over 1–200 days.

The entire model area is filled with warm water on the 200th day in simulations by all three closures. Note that water warmer than 12.9℃ can still be found in the simulation by the biharmonic viscosity closure, indicating that spurious mixing in the biharmonic viscosity is the weakest among the three closures. This is consistent with the fact that RPETCR is the lowest (ref 3) in the biharmonic viscosity closure. Two mesoscale cold eddies are simulated by the biharmonic viscosity and Laplacian viscosity closures, while three mesoscale cold eddies are formed in the simulation by the Leith closure. The cold eddies simulated by the Laplacian viscosity closure are weak,while those simulated by the biharmonic viscosity and the Leith closure are quite strong.

4.5.3 Effects on total energy

Two variables, i.e., the relative change of energy (RCE)and the ratio of the total energy change to the reference potential energy (RPE) change, are defined in this study to investigate the effects of the three closures on the simulation of total energy. The RCE is defined similarly to the definition of RCRPE and expressed as:

here the En(t) is the sum of the potential, internal, and kinetic energies in the entire space at a given time t, and Cv=4096 J (kg ℃) −1 is the specific heat capacity of sea water. The ratio of the total energy change to the RPE change is written as:

As seen in Fig.10, when the RPETCR decreases to 0.00025 W m−2, spurious mixing is reduced by the same degree. Under this condition, certain degrees of dissipation can be found in the results of the three closure schemes. In general, dissipation is the greatest in the Laplacian viscosity closure scheme, followed by that in the Leith closure scheme, and finally the biharmonic viscosity closure scheme. Dissipation of the horizontal kinetic energy and enstrophy of motions on all scales is found in the Laplacian viscosity closure scheme, and this dissipation is the largest as shown in Figs.10a and b. The dissipation of horizontal kinetic energy mainly concentrates on small scales in the biharmonic viscosity closure scheme, and the horizontal kinetic energy on large scales basically remains unchanged. The peak value of kinetic energy on the largest scale is reduced, whereas the frequency band is slightly wider in the simulation with the biharmonic viscosity closure scheme, suggesting that the biharmonic viscosity closure scheme has a strong smoothing effect on the horizontal kinetic energy on the largest scale as shown in Fig.10c. Enstrophy is dissipated on smaller scales, whereas that on the largest scale increases.This result indicates that the enstrophy on the largest scale is somehow compensated in the biharmonic viscosity closure scheme (Fig.10d). The dissipation of horizontal kinetic energy and enstrophy is concentrated on smaller scales, whereas less dissipation occurs on larger scales in the Leith closure scheme. The dissipation of enstrophy in the Leith closure scheme is between that in the Laplacian viscosity and biharmonic viscosity closure schemes, but dissipation of the horizontal kinetic energy on the largest scale increases (Figs.10e and f). This is because the Leith closure scheme can transfer part of the kinetic energy and enstrophy from small scales to the largest one.

Table 4 Changes in energy parameters in the baroclinic eddy simulations for different closures

Notes: ▽2_optimal, ▽4_optimal, and Leith_optimal mean the Laplacian viscosity, biharmonic viscosity, and Leith closure with the optimal parameter, respectively. ▽2_ref 1 (▽2_ref 2), ▽4_ref 1(▽4_ref 2), and Leith_ref 1 (Leith_ref 2) indicate the Laplacian viscosity, biharmonic viscosity, and Leith closure with the level of ref 1 (ref 2), respectively.

?

5 Analysis of the Leith Scheme

5.1 Coupling Parameters

In Fig.8, we find that the temporal changes in RMS of the kinetic energy simulated by the Leith scheme with L=2 km and Г=1.25 is the same as that with L=4 km and Г=0.625, or L=10 km and Г=0.25. As shown in Fig.7, the averaged RPETCR corresponding to the three Leith sets overlap on the reference line ref 2, i.e., it is completely equal in the three closures. Results of Fourier spectrum analysis of the horizontal kinetic energy and enstrophy(Fig.14) are exactly the same as that shown by the third line in Fig.10 for the Leith closure simulation with L=4 km and Г=0.625. In the three settings, the same parameter Г is used; therefore, we speculate that the parameters L and Г are coupled in the Leith closure, and spurious dianeutral mixing is a function of the product of L and Г.The RPETCR as a function of the corresponding LГ shown in Table 3 is depicted in Fig.15, which indicates that the three curves with different L are basically similar,and the points corresponding to L=2.5 km and L=2 km in the three curves appear to overlap. Thus, when the RPETCR does not reach saturation, spurious dianeutral mixing is inversely proportional to the product of L and Г.The greater the product of L and Г, the less the spurious dianeutral mixing.

5.2 Parameter L

Fig.14 Same as Fig.10, except that different settings are used in the Leith closure.

色彩课程属于小学美术教学的基础性课程，在这一阶段对学生的色彩感知能力进行培养是一项艰巨而受负责的任务。美术具有一定的艺术性和创造性，色彩是美术学习中不可或缺的重要组成部分，绚烂的色彩不仅能够激发学生的想象力、创造力，也能够在长期的学习中对学生进行潜移默化的影响，使学生在学习中逐渐树立正确的学习态度，并愿意积极主动地去领悟和感知多彩的世界。因此，本文主要对小学美术教学中，学生色彩感知能力的培养方法进行了探讨。

Fig.15 RPETCR as a function of the product of L and Г in the baroclinic eddy test simulated with the Leith closure.

Fig.16 Same as Fig.11 except waves 1 (left panels), 2 (middle panels), and 3 (right panels) are simulated by the Leith closure with Г = 0.5 and L = 2 km (top panels), L = 4 km (middle panels), and L = 10 km (bottom panels), respectively.

5.3 Parameter Г

In this section, we set L equal 4 km, and Г is set to 0.003, 0.2, and 2 in turn. The horizontal kinetic energy is vertically integrated, and the Fourier decomposition is applied to the results. In this study we analyze the first three waves (Fig.17). As depicted in Fig.17, the kinetic energy of waves 1, 2 and 3 all decrease with the increase in Г, suggesting that dissipation in the Leith closure depends on the value of Г. However, it is worth noting that the kinetic energy of waves 1 and 2 decrease gradually with the increase in Г, and the simulated eddies last for a longer time. With the increase in Г from 0.003 to 0.2, the kinetic energy of wave 3 intensifies, and wave 3 lasts longer due to the dissipation of spurious numerical mixing. With the increase in Г from 0.2 to 2, the kinetic energy of wave 3 weakens, and wave 3 lasts for a shorter time, which is attributed to the excessive dissipation of the physical energy. This is consistent with the analysis presented in Section 4.4 and 4.5.

Fig.17 Same as Fig.16, except that L = 4 km, and Г = 0.003 (top panels), Г = 0.2 (middle panels), and Г = 2 (bottom panels), respectively.

6 Conclusions

The MPAS-O model was applied to a 3D baroclinic periodic channel case to compare and evaluate the performance of three closure schemes, i.e., the Laplacian viscosity closure, the biharmonic viscosity closure, and the Leith closure, by calculating RCRPE, RPETCR, horizontal grid Reynolds numbers, and the RMS of the kinetic energy. The spectra of the horizontal kinetic energy and enstrophy simulated with the optimal parameters were also analyzed. The major conclusions are as follows.

1) Overall, the RCRPE is reduced, and spurious dianeutral mixing is well controlled by using the three types of closures.

2) There is a saturation phenomenon for the RPETCR in the baroclinic eddy test. The critical grid Reynolds number corresponding to the saturation of RPETCR varies and depends on the local adaptation closures. The critical grid Reynolds number is the largest for the biharmonic viscosity closure, followed by that for the Laplacian viscosity closure, and that for the Leith closure is the smallest.

3) Spurious dianeutral mixing is a function of the grid Reynolds number. Under the condition when the RPETCR does not reach saturation, spurious dianeutral mixing can be effectively controlled in all three closures by using different settings that can reduce the local grid Reynolds number.

4) There is a certain degree of damage to the physical processes when the local adaptation turbulence closure is applied to control spurious dianeutral mixing, and the damage to spinning process (enstrophy) is greater than that to the advection process (kinetic energy).

在现场作业工作完成后，要对现场进行仔细清理，并且注意要恢复现场的有关安全防护措施，还原原有的设备定置，防止由于作业而给现场留下日后的安全隐患。

由表1可见，各参试品种(系)的全生育期为143～154d，其中以青海12号全生育期最长，为154d；青海13号全生育期最短为143d。

5) The degree of damage to the physical process is different in the three closures under the same average RPETCR. For example, a) the viscous dissipation of the kinetic energy and enstrophy is very strong on each scale;b) for the biharmonic viscosity closure, dissipation is scale dependent and focuses on small scales. Part of the enstrophy on small scales can be transferred to enstrophy on large scales via energy cascade. At the same time, part of the enstrophy can be converted into the horizontal kinetic energy of the flow; c) for the Leith closure, its dissipation of the kinetic energy is concentrated on small scales. In addition, part of the kinetic energy on small scales can be transferred to large-scale kinetic energy.The dissipation in the biharmonic viscosity closure is between those in the Laplacian viscosity and biharmonic viscosity closures.

6) The scale-dependence of dissipation in the Leith closure is controlled by the characteristic length scale L.The larger the characteristic length scale L, the larger the dissipated scale, and dissipation will intensify. The transfer of energy from large to small scales is subsequently enhanced. On the contrary, the smaller the L, the smaller the dissipated scale, and dissipation weakens. The dissipative intensity in the Leith closure is controlled by the dimensionless parameter Г: the greater the value of Г, the stronger the dissipation.

7) The 3D baroclinic case is simulated with the optimal parameters obtained from the 2D barotropic model simulation. All three closures dissipate total energy. The dissipation in the biharmonic viscosity is the strongest, followed by that in the Leith closure, and that in the Laplacian viscosity closure is the weakest. The average spurious dianeutral mixing in the biharmonic viscosity closure is lower than that in the Leith and Laplacian viscosity closures. The mesoscale eddies develop the fastest, and the peak of RMS of the horizontal kinetic energy is the highest after baroclinic adjustment in the biharmonic viscosity closure, which is due to the weakest dissipation of enstrophy in this scheme. The mesoscale eddies develop the slowest, and the peak of RMS of the horizontal kinetic energy is the lowest in the Laplacian viscosity closure.Results obtained from the Leith closure are in between those from the Laplacian viscosity and biharmonic viscosity closures.

2.4.3 固定不牢。导管安置后,固定胶布因患者的汗液和分泌物污染而失去粘性,护士未及时更换,导致管道滑脱。

8) Г and L are a pair of coupling parameters in the Leith closure. When the product of Г and L is the same,the RPETCR and the simulated results are the same. The inhibition of spurious dianeutral mixing is inversely proportional to the product Г and L in the Leith closure.

“臭四眼，你就在这里待着吧！”走到僻角，“霸王龙”弗兰克大力地拉开一扇门，然后冲死党们使了个眼色。他的死党们嘻嘻哈哈地笑了起来，摁着步凡的那个人松开手，然后在步凡反应过来之前，抬起大脚踹向他的屁股，将他踹进了屋里。

Results of the 3D baroclinic model shown in this paper may not be optimal because the optimal parameters obtained from the 2D barotropic model are used. Because the physical process in the 3D baroclinic model is relatively complicated, and we only run the model with a horizontal resolution of 4 km, optimal parameters cannot be determined in this article. More detailed studies will be conducted in our future work.

我这一走，这门亲事自是结不成了。街上的人不会怪越家，只会骂我母亲教子无方。母亲是要强的人，出了这事，她的脸真没地方搁了。奶奶年纪大了，都由母亲照料，要是母亲有个闪失，这个家就毁了。我这一走，兵是当不成了，抗击胡虏、报效国家，更是谈不上了。但我若不走，岂不是对不住乔瞧？

Acknowledgement

The work was sponsored by the National Natural Science Foundation of China Program (No. 41175089).

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