Energetics of lateral eddy diffusion/advection：Part Ⅲ. Energetics of horizontal and isopycnal diffusion/advection

Gravitational Potential Energy （GPE） change due to horizontal/isopycnal eddy diffusion and advection is examined. Horizontal/isopycnal eddy diffusion is conceptually separated into two steps： stirring and sub scale diffusion. GPE changes associated with these two steps are analyzed. In addition, GPE changes due to stirring and subscale diffusion associated with horizontal/isopycnal advection in the Eulerian coordinates are analyzed. These formulae are applied to the SODA data for the world oceans. Our analysis indicates that horizontal/isopycnal advection in Eulerian coordinates can introduce large artificial diffusion in the model. It is shown that GPE source/sink in isopycnal coordinates is closely linked to physical property distribution, such as temperature, salinity and velocity. In comparison with z-coordinates, GPE source/sink due to stir ring/cabbeling associated with isopycnal diffusion/advection is much smaller. Although isopycnal coordi nates may be a better choice in terms of handling lateral diffusion, advection terms in the traditional Eule rian coordinates can produce artificial source of GPE due to cabbeling associated with advection. Reducing such numerical errors remains a grand challenge.     

Energetics of lateral eddy diffusion/advection：Part Ⅳ. Energetics of diffusion/advection in sigma coordinates and other coordinates

Gravitational potential energy （GPE） source and sink due to stirring and cabbeling associated with sigma dif fusion/ advection is analyzed. It is shown that GPE source and sink is too big, and they are not closely linked to physical property distribution, such as temperature, salinity and velocity. Although the most frequently quoted advantage of sigma coordinate models are their capability of dealing with topography; the exces sive amount of GPE source and sink due to stirring and cabbeling associated with sigma diffusion/advec tion diagnosed from our analysis raises a very serious question whether the way lateral diffusion/advection simulated in the sigma coordinates model is physically acceptable. GPE source and sink in three coordinates is dramatically different in their magnitude and patterns. Overall, in terms of simulating lateral eddy diffu sion and advection isopycnal coordinates is the best choice and sigma coordinates is the worst. The physical reason of the excessive GPE source and sink in sigma coordinates is further explored in details. However, even in the isopycnal coordinates, simulation based on the Eulerian coordinates can be contaminated by the numerical errors associated with the advection terms.     

Energetics of lateral eddy diffusion/advection：Part Ⅰ. Thermodynamics and energetics of vertical eddy diffusion

Two important nonlinear properties of seawater thermodynamics linked to changes of water density, cabbeling and elasticity(compressibility), are discussed. Eddy diffusion and advection lead to changes in density; as a result, gravitational potential energy of the system is changed. Therefore, cabbeling and elasticity play key roles in the energetics of lateral eddy diffusion and advection. Vertical eddy diffusion is one of the key elements in the mechanical energy balance of the global oceans. Vertical eddy diffusion can be conceptually separated into two steps: stirring and subscale diffusion. Vertical eddy stirring pushes cold/dense water upward and warm/light water downward; thus, gravitational potential energy is increased. During the second steps, water masses from different places mix through subscale diffusion, and water density is increased due to cabbeling. Using WOA01 climatology and assuming the vertical eddy diffusivity is equal to a constant value of 2×103 Pa2/s, the total amount of gravitational potential energy increase due to vertical stirring in the world oceans is estimated at 263 GW. Cabbeling associated with vertical subscale diffusion is a sink of gravitational potential energy, and the total value of energy lost is estimated at 73 GW. Therefore, the net source of gravitational potential energy due to vertical eddy diffusion for the world oceans is estimated at 189 GW.     

Appendix.Why GPE source due to sigma diffusion is so large

The GPE source and sink due to stirring and cabbeling along sigma surfaces is much larger than that associated with hori zontal diffusion or isopycnal diffusion. This must be linked to the special way of handling lateral diffusion in the sigma mod els. We argue that the largeness of source and sink associated with sigma diffusion is due to the fact that in sigma models, warm water parcels from shallow levels is assumed to be mixed with cold water parcel from deeper levels. Mixing water par cels with such a large difference in properties can leads to big change in density and thus strong source and sink of GPE. As an example, we demonstrate the large GPE source asso ciated with sigma-surface stirring related to steep topography. For simplicity, we will use a simple one-dimensional periodic topography with valleys and seamounts lining up one next to another, as shown in Fig. A1. The pressure grid over the valley stations is taken as the standard pressure grid for the commonly used climatological datasets, such as WOA01 and WOA09, i.e. the center of the bottom grid in the valley is at 5500 （104 Pa）, with the corresponding sea floor at 5 750 （104 PaL Thus, the cen- ters of grid box in the vertical direction at the valley station are at O, 10, 20, 30, 50, 75, 100, 125, 150, 200, 250, 300, 400, 500, 600, 700, 800, 900, 1000, 1 100, 1200, 1300, 1400, 1500, 1750, 2000, 2500, 3000, 3500, 4000, 4500, 5000, 5500 （104 Pa）.     

Formation and transport of intermediate water masses in a model of the Pacific Ocean

A basin-wide ocean general circulation model of the Pacific Ocean was used to investigate how the interior restoration in the Okhotsk Sea and the isopycnal diffusion affect the circulation and intermediate water masses. Four numerical experiments were conducted, including a run with the same isopycnal and thickness diffusivity of 1.0×103 m2/s, a run employing the interior restoration of temperature and salinity in the Okhotsk Sea with a time scale of 3 months, a run that is the same as the first run except for the enhanced isopycnal mixing, and a final run with the combination of the restoration in the Okhotsk Sea and large isopycnal diffusivity. Simulated results show that the intermediate water masses reproduced in the first run are relatively weak. An increase in isopycnal diffusivity can improve the simulation of both Antarctic and North Pacific intermediate waters, mainly increasing the transport in the interior ocean, but inhibiting the outflow from the Okhotsk Sea. The interior restoration generates the reverse current from the observation in the Okhotsk Sea, whereas the simulation of the temperature and salinity is improved in the high latitude region of the Northern Hemisphere because of the reasonable source of the North Pacific Intermediate Water. A comparison of vertical profiles of temperature and salinity along 50°N between the simulation and observations demonstrates that the vertical mixing in the source region of intermediate water masses is very important.     

Numerical computation of tidal currents in the eastern and southern parts of the East China Sea

Numerical simulation on diurnal and semidiurnal tidal currents is made for the eastern and southern parts of the East China Sea bounded by the Tsushima, islands of Kyushu and Ryukyu in the east and extending to the central Taiwan Strait in the south. The numerical computation shows that the rapid variation in the seabed topography near the Okinawa Trough may cause computational instability and this instability can be removed by setting a suitable lateral eddy viscosity. The computed results are in good agreement with the observations and indicate that the tidal currents in the greater part of the area studied rotate clockwise. An M2 current-amphidromic point exists in the southern part of the Korea Strait and another one appears in the west of Okinawa with less certainty. K1 and M2 cotidal charts based on the observed and computed tidal currents in the sea and the observed tides on the coast are also given.     

Radium-traced nutrient outwelling from the Subei Shoal to the Yellow Sea: Fluxes and environmental implication

The Subei Shoal is the largest sandy ridge in the southern Yellow Sea and is important source for nutrient loading to the sea. Here, the nutrient fluxes in the Subei Shoal associated with eddy diffusion and submarine groundwater discharge（SGD） were assessed to understand their impacts on the nutrient budget in the Yellow Sea. Based on the analysis of 223 Ra and 224 Ra in the field observation, the offshore eddy diffusivity mixing coefficient and SGD were estimated to be 2.3×10⁸ cm     

模态分离的环流模式POM的垂向速度算法的误差估计

A time splitting technique is common to many free surface ocean models. The different truncation errors in the equations of the internal and external modes require a numerical adjustment to make sure that algorithms correctly satisfy continuity equations and conserve tracers quantities. The princeton ocean model （POM） has applied a simple method of adjusting the vertical mean of internal velocities to external velocities at each internal time step. However, due to the Asselin time filter method adopted to prevent the numerical instability, the method of velocity adjustment used in POM can no longer guarantee the satisfaction of the continuity equation in the internal mode, though a special treatment is used to relate the surface elevation of the internal mode with that of the external mode. The error is proved to be a second-order term of the coefficient in the Asselin filter. One influence of this error in the numerical model is the failure of the kinetic boundary condition at the sea floor. By a regional experiment and a quasi-global experiment, the magni- tudes of this error are evaluated, and several sensitivity tests of this error are performed. The characteristic of this error is analyzed and two alternative algorithms are suggested to reduce the error.     

2012年夏季海南岛东岸上升流区的混合观测

The turbulent mixing in the upwelling region east of Hainan Island in the South China Sea is analyzed based on in situ microstructure observations made in July 2012. During the observation, strong upwelling appears in the coastal waters, which are 3℃ cooler than the offshore waters and have a salinity 1.0 greater than that of the offshore waters. The magnitude of the dissipation rate of turbulent kinetic energy ε in the upwelling region is O(10–9 W/kg), which is comparable to the general oceanic dissipation. The inferred eddy diffusivity K_ρ is O(10–6 m~2/s), which is one order of magnitude lower than that in the open ocean. The values are elevated to K_ρ≈O(10–4 m~2/s) near the boundaries. Weak mixing in the upwelling region is consistent with weak instability as a result of moderate shears versus strong stratifications by the joint influence of surface heating and upwelling of cold water.The validity of two fine-scale structure mixing parameterization models are tested by comparison with the observed dissipation rates. The results indicate that the model developed by Mac Kinnon and Gregg in 2003 provides relatively better estimates with magnitudes close to the observations. Mixing parameterization models need to be further improved in the coastal upwelling region.     

Model study of the upper circulation in the Sulu Sea and its relation to the South China Sea circulation

According to the satellite remote sensing monthly mean sea surface temperature data from 1998 to 2002, it is shown that, the Sulu Sea is dominated by a cold eddy both in summer and winter. A coupled single-layer/two-layer model is employed here to study the summery and wintry characteristics and dynamic mechanism of the upper circulation in the Sulu Sea. According to the numerical experiments, it is found that, due to the local monsoon stress cud, the upper circulation in the Sulu Sea is dominated by a weak anticyclonic eddy in summer and a strong cyclonic eddy in winter. Once there is a large outflow via the Sibutu Passage flowing out of the Sulu Sea in summer, the upper circulation in the Sulu Sea may be dominated by a cyclonic eddy instead of the normal anti- cyclonic one. Moreover, in summer, the water exchange between the Sulu Sea and South China Sea via the Mindoro and Balabac Straits might have some effect on the separation position and strength of the northward western boundary current next to the Indo - China Peninsula in the southern South China Sea.     

Numerical issues for coupling biological models with isopycnal mixing schemes

In regions of sloping isopycnals, isopycnal mixing acting in conjunction with biological cycling can produce patterns in the nutrient field which have negative values of tracer in light water and unrealistically large values of tracer in dense water. Under certain circumstances, these patterns can start to grow unstably. This paper discusses why such behavior occurs. Using a simple four-box model, it demonstrates that the instability appears when the isopycnal slopes exceed the grid aspect ratio (Δz/Δx). In contrast to other well known instabilities of the CFL type, this instability does not depend on the time step or time-stepping scheme. Instead it arises from a fundamental incompatibility between two requirements for isopycnal mixing schemes, namely that they should produce no net flux of passive tracer across an isopycnal and everywhere reduce tracer extrema. In order to guarantee no net flux of tracer across an isopycnal, some upgradient fluxes across certain parts of an isopycnal are required to balance downgradient fluxes across other parts of the isopycnal. However, these upgradient fluxes can cause local maxima in the nutrient field to become self-reinforcing. Although this is less of a problem in larger domains, there is still a strong tendency for isopycnal mixing to overconcentrate tracer in the dense water. The introduction of eddy-induced advection is shown to be capable of counteracting the upgradient fluxes of nutrient which cause problems, stabilizing the solution. The issue is not simply a numerical curiosity. When used in a GCM, different parameterizations of eddy mixing result in noticeably different distributions of nutrient and large differences in biological production. While much of this is attributable to differences in convection and circulation, the numerical errors described here may also play an important role in runs with isopycnal mixing alone.     

Southern Ocean transformation in a coupled model with and without eddy mass fluxes

A coupled air–sea general circulation model is used to simulate the global circulation. Different parameterizations of lateral mixing in the ocean by eddies, horizontal, isopycnal, and isopycnal plus eddy advective flux, are compared from the perspective of water mass transformation in the Southern Ocean. The different mixing physics imply different buoyancy equilibria in the surface mixed layer, different transformations, and therefore a variety of meridional overturning streamfunctions. The coupled‐model approach avoids strong artificial water mass transformation associated with relaxation to prescribed mixed layer conditions. Instead, transformation results from the more physical non‐local, nonlinear interdependence of sea‐surface temperature, air–sea fluxes, and circulation in the model's atmosphere and ocean. The development of a stronger mid‐depth circulation cell and associated upwelling when eddy fluxes are present, is examined. The strength of overturning is diagnosed in density coordinates using the transformation framework.     

Vertical diffusion and oxygen consumption during stagnation periods in the deep North Aegean

Ventilation of the deep basins of the North Aegean Sea takes place during relatively scarce events of massive dense water formation in that region. In the time intervals between such events, the bottom waters of each sub-basin are excluded from interaction with other water masses through advection or isopycnal mixing and the only process that changes their properties is diapycnal mixing with overlying waters. In this work we utilize a simple one-dimensional model in order to estimate the vertical eddy diffusion coefficient K_ρ based on the observed rate of change of density and stratification. Vertical diffusivity is estimated for each of three sub-basins of the North Aegean, one of convex shape of the seabed and the other two of concave topography. It is noteworthy that the convex sub-basin exhibited much higher vertical diffusivity than the two concave sub-basins, a fact consistent with theoretical predictions that internal-wave-induced mixing is higher over the former shape of seabed. Furthermore, the estimates of K_ρ are exploited in computing the vertical transport of dissolved oxygen through diffusion and the rate of oxygen consumption by decaying organic matter. The different levels of the estimated diffusion and oxygen consumption rates testify to the dynamical and biogeochemical characteristics of each basin.     

Effects in a climate model of slope tapering in neutral physics schemes

In many global ocean climate models, mesoscale eddies are parameterized as along isopycnal diffusion and eddy-induced advection (or equivalently skew-diffusion). The eddy-induced advection flattens isopycnals and acts as a sink of available potential energy, whereas the isopycnal diffusion mixes tracers along neutral directions. While much effort has gone into estimating diffusivities associated with this closure, less attention has been paid to the details of how this closure (which tries to flatten isopycnals) interacts with the mixed layer (in which vertical mixing tries to drive the isopycnals vertical). In order to maintain numerical stability, models often stipulate a maximum slope S_max which in combination with the thickness diffusivity A_gm defines a maximum eddy-induced advective transport A_gm1S_max. In this paper, we examine the impact of changing S_max within the GFDL global coupled climate model. We show that this parameter produces significant changes in wintertime mixed layer depth, with implications for wintertime temperatures in key regions, the distribution of precipitation, and the vertical structure of heat uptake. Smaller changes are seen in details of ventilation and currents, and even smaller changes as regards the overall hydrography. The results suggest that not only the value of the coefficient, but the details of the tapering scheme, need to be considered when comparing isopycnal mixing schemes in models.     

A Generation Mechanism for Mesoscale Eddies in the Kuril Basin of the Okhotsk Sea: Baroclinic Instability Caused by Enhanced Tidal Mixing

Mesoscale eddies, particularly anticyclonic ones, are dominant features in the Kuril Basin of the Okhotsk Sea. In 1999, both surface drifter and hydrographic observations caught the same anticyclonic eddy northwest of Bussol’ Strait, which has a diameter of ∼100 km, typical surface velocity of 0.2–0.3 m s⁻¹, and less dense core extending to a depth of ∼1200 m. Based on an idea that the generation of mesoscale eddies is caused by strong tidal mixing in and around Kuril Straits, we have conducted a series of three-dimensional numerical model experiments, in which strong tidal mixing is simply parameterized by increasing coefficients of vertical eddy viscosity and diffusivity along the eastern boundary. Initially, a regular series of disturbances with a wavelength of ∼70 km starts to develop. The disturbances can be clearly explained by a linear instability theory and regarded as the baroclinic instability associated with the near-surface front formed in the region between the enhanced mixing and offshore regions. In the mature phase, the disturbances grow large enough that some eddies pinch off and advect offshore (westward), with the scale of disturbances increasing gradually. Typical eddy scale and its westward propagation speed are ∼100 km and ∼0.6 km day⁻¹, respectively, which are consistent with the observations by satellites. The westward propagation can be explained partly due to nonlinear effect of self-offshore advection and partly due to the β-effect. With the inclusion of the upper ocean restoring, the dominance of anticyclonic eddy, extending from surface to a depth of ∼1200 m, can be reproduced.     

Large eddy simulation of stratified mixing in two-dimensional dam-break problem in a rectangular enclosed domain

Mixing in both coastal and deep ocean emerges as one of the important processes that determines the transport of pollutants, sediments and biological species, as well as the details of the global thermohaline circulation. Both the observations, due to their lack in space and time resolution, and most coastal and general circulation models due to inadequate physics, can only provide partial information about oceanic mixing processes. A new class of nonhydrostatic models supplemented with physically based subgrid-scale (SGS) closures, or so-called large eddy simulation (LES), is put forth as another tool of investigation to complement observational and large-scale modeling efforts.However, SGS models have been developed primarily for homogeneous, isotropic flows. Here, four SGS models based on Smagorinsky eddy viscosity and diffusivity are tested for stratified flows in the context of 2D dam-break problem in a rectangular enclosed domain. This idealized testbed leads to a number of simplifications about the initial conditions, boundary conditions and geometry, while exhibiting the dynamically complex characteristics of stratified flows involving the interaction of shear-induced mixing and internal waves. Direct numerical simulations (DNS) at high resolutions are taken as benchmark solutions. Under-resolved simulations without SGS terms (so-called DNS¹) are used to quantify the impact of SGS stresses. The performance of LES is assessed by using the time evolution of the volume fraction of intermediate density water masses generated by mixing. The simulations are conducted using a nonhydrostatic high-order spectral element model Nek5000 developed to exhibit minimal numerical dissipation and dispersion errors, which is advantageous to quantify accurately the impact of SGS stresses.It is found that all tested SGS models lead to improved results with respect to those from DNS¹. Also, SGS models allow for simulations with coarse resolutions that blow up in DNS¹ due to lack of adequate dissipation where needed. The SGS model in which the vertical eddy diffusion is modulated via a function that depends on the Richardson number Ri shows the most faithful reproduction of mixed water masses at all resolutions tested.The sensitivity of the results to the tunable parameter of the SGS model, to changes in the Ri-dependent function and resolution of the turbulent overturning scales is shown.