The LES model, developed by Deardorff (1980) and Moeng (1984) for the atmospheric boundary layer, dynamically solves for unresolved (subgrid) turbulent kinetic energy (TKE). Several adaptations of this model are currently used to model oceanic boundary layers, including high-latitude deep convection in response to storm forcing (Harcourt et al 2001), and forced convection in more shallow free-slip ocean boundary layers (e.g. Skyllingstad and Denbo, 1995; Wang et al, 1996).
Following Moeng (1984), the advection terms of the momentum equations are computed prognostically from vorticity components
Terms involving both subgrid and resolved TKE are absorbed into a modified nonhydrostatic pressure . b is the buoyancy force, and the mean acceleration is removed from the right-hand side of the vertical momentum equation to maintain a prescribed mean vertical velocity.
The LES uses a finite series expansion of the equation of state about a potential reference state with density chosen to minimize truncation errors. After removing the horizontally averaged vertical acceleration , buoyancy fluctuations account for mean scalar dependence in the expansion coefficients, thermobaricity, cabbeling and a halobaric effect through , , and , as modifications to the standard approximation of constant thermal and haline expansion coefficients and . The salinity-related components of cabbeling and the dependence of the haline expansion coefficient on mean salinity are small and generally omitted.
The evolution of potential temperature is computed prognostically from
with corresponding equations for salinity and Lagrangian tracers. The velocity components are subject to the condition of incompressibility, which is applied to the momentum equations to solve diagnostically for . Unresolved stresses are modeled in terms of the symmetric resolved strain rate and local nonlinear eddy viscosity ; scalar fluxes in terms of resolved gradients and eddy diffusivity :
Parameterizations eddy viscosity diffusivity, and , depend upon subgrid TKE , its associated dissipation length scale , turbulent eddy Prandtl number and constant . The length scale of unresolved motion is equal to the resolution scale , unless reduced below that by stability or by proximity to a wall boundary. Parameterizations for the dissipation length scale for stably stratified turbulence are discussed below in the description of simulations. Resolution scale is governed either explicitly by a spatial filter, implicitly by the numerical scheme, or by a combination of the two.
The subgrid TKE equation
includes unresolved buoyant production as well as unresolved shear production and transport, and parameterizes dissipation as . When a portion of the turbulent inertial subrange is resolved, constants and of the subgrid parameterization are determined by integrals over the Kolmogorov energy spectrum for unresolved TKE smaller than
The thermodynamic ice-ocean interaction parameterizations used by McPhee (2000) in 1-D Local Turbulence Closure (LTC) simulations of the Weddell Sea Mixed Layer have been implemented in the LES to simulate the freezing and melting of ice. A prescribed mean surface stress derived from observations is applied at the upper model domain, and the velocity boundary condition is nonslip in fluctuations with respect to the mean horizontal velocity at the surface. Heat flux through the ice is also derived from ANZFLUX measurements, and the model ocean heat flux at the upper boundary is
where the freezing point is the boundary temperature is the LES temperature in the first grid level adjacent to the boundary, and is a nondimensional Stanton number, adjusted for consistency with vertical grid size . Freezing and melting of ice is then determined by the imbalance between heat fluxes in the ice and in the ocean.
Several LES runs have been carried out to test the stability of the upper ocean under conditions observed during ANZFLUX at certain times when ocean observations were not made. In general, they agree with the LTC results of McPhee (2000), showing the upper ocean becoming unstable to a process of thermobaric parcel detrainment, but at an earlier point in time. These simulations do not yet account for cabbeling effects, as this term was omitted in the preliminary simulations for comparison with LTC results.
Figures 1-3 below summarize some results from the most recent simulation, for which a short animation is available online at http://www.oc.nps.edu/thermobaricity under Numerical Experiments, Simulation 3. This simulation was initialized from averaged typical profiles measured in 1994 on year days 216 and 217, and forced with time-dependent surface stress deduced from observations. High wind in the first two days results in shear production of mixed layer TKE with classical forced mixed layer scaling, (Figure 1, top). Thermobarically unstable parcels are subsequently detrained from the mixed layer in plumes with with small vertical TKE, large negative skewness (Figure 1, second from top) and positive values of below . Large negative values of buoyant TKE production (Figure 1, third plot) resulting from the conversion of TKE to entrainment work, are followed by significant release of potential energy below the mixed layer as the detrained thermobaric plumes become more negatively buoyant with depth. The fourth plot in Figure 1 tracks the net buoyant production of TKE above and below separately, and demonstrates that without thermobaricity the detrained plumes would be a net sink, rather than a source of TKE. This sensitivity (lower right) to the second order terms in the expanded equation of state arises from the high level of compensation between the heat and salinity fluxes (lower left).
Including the cabbeling term in the equation of state will probably delay thermobaric instability, owing to the negative sign of for the plumes. Mean TKE below the mixed layer is relatively small, but the skewness of the thermobaric plumes is large and negative, and individual plumes (Figure 2) have large downward velocities concentrated in structures about 100m in horizontal scale.
We propose that this preliminary work on LES model development be extended through quantitative verification comparisons with existing data sets from polar boundary layers below ice, and subsequently be used to make predictions about upper ocean instabilities for use in designing future experiments in the Weddell Sea.
Figure 1: The top figure shows vertical TKE integrated over depth, separately within and below the mixed layer, along with a representative one-third of the dimensional surface stress forcing scale . The profile timeseries of mean skewness is illustrated in the second plot, and buoyancy flux -the work rate for buoyant TKE production- is shown in the third plot. The fourth plot shows the net work rate due to buoyancy production of TKE separately within and below the mixed layer, both with and without thermobaricity. The lower right profile also shows the effect of thermobaricity on buoyant TKE production, averaged over the entire simulation, and the lower left plot decomposes into heat and salinity flux components.
Figure 2: Vertical cross sections of potential temperature (left plot) and vertical velocity (right plot), on about day 220 of the simulation, illustrating a detraining thermobaric plume.
Preliminary LES model development has been guided by earlier development of the 1-D LTC model. While this has allowed some preliminary simulations, comparisons between the modified LES and observations of turbulence below the ice are necessary to verify the thermodynamic parameterizations of ice-ocean interaction. Data for verification will come from past observations of both ANZFLUX and SHEBA experiments. Comparisons will include vertical fluxes and dissipation rates as well as the evolution hydrographic profiles. Figure 3 demonstrates the equivalence of LES subgrid TKE dissipation and dissipation observations from a Turbulence Instrument Cluster (TIC) during the ANZFLUX experiment (n.b. two different time frames) by Tim Stanton and Miles McPhee.
Figure 3: Time series of measured dissipation profiles (left plot) from the two final days of observation East of Maud Rise, along with LES subgrid dissipation (right plot) from the simulation of the subsequent two days, initialized from observed hydrographic profiles. Model resolution is 6.25m.
The two quantities are equivalent above the instrument noise level because the LES resolution scale is the same order as the length scale characterizing the TIC response, and because most of the dissipation is at much smaller length scales. We anticipate three verification simulations, two based on ANZFLUX observations and one based on SHEBA results, each about one week in duration. Ice-ocean parameterizations will be tuned to give the best agreement in these comparisons.
Modeling for predictions of upper ocean instabilities will focus on what may have happened in the area East of Maud Rise after the end of ANZFLUX ocean observations. This region is of interest because hydrographic analysis indicates a weak barrier to thermobaric instability if the upper ocean layer, and because satellite observations indicate significant areas of open water (Drinkwater, 1997) in 1994, and large polynyas in other years. The goal of this modeling is to use the dynamics of the LES predict more accurately the point at which instabilities occur and the form they will take. Furthermore, timeseries from ensembles of virtual measurements embedded in the LES will predict the shape of observed thermobaric plumes and other features of the upper ocean, and assess the random and systematic uncertainties associated with different observing systems. Embedded measurements will include Lagrangian, isobaric, and tethered Lagrangian drifter ensembles as well as Eulerian point and volume-averaged timeseries representing ice-fixed measurements.
In short, these embedded measurements will be used to assess the effectiveness of an array of experimental strategies, thus aiding in the design of future field expeditions. These virtual measurement ensembles will simulate observations in about six LES cases of one to two weeks in duration each, initialized from different 1994 ANZFLUX CTD profiles from East of Maud Rise, and forced with observed time-dependent surface forcing also derived from observations.
The LES code is currently optimized for parallelvector machines, where up to twothirds of the load is scalable. Each LES case, with 105 - 106 grid points model up to several weeks of simulation time (about 105 timesteps), and will require approximately 800 cpuhours on a Cray J90 or SV1, or 200 GAU in NCAR's general accounting units. In core memory requirements are moderate at 30MWords per simulation. These computer resources will be obtained through applications for the use of both NCAR and DOD high performance computing resources that are available to carry out NSF funded research. Large Eddy Simulations will be carried out by Harcourt and Lherminier, LTC simulations by McPhee. Analysis of model output for verification and prediction will be done by Harcourt, Lherminier, Stanton and McPhee.
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