Vertical Slice Model Research Articles

Geometric mechanics of the vertical slice model

The goals of this work are to: (i) investigate the dynamics of oceanic frontogenesis by taking advantage of the geometric mechanics underlying the class of Vertical Slice Models (VSMs) of ocean dynamics and (ii) illustrate the versatility and utility of deterministic and stochastic variational approaches by deriving several variants of wave–current interaction models which describe the effects of internal waves propagating within a vertical planar slice embedded in a 3D region of constant horizontal gradient of buoyancy in the direction transverse to the vertical plane.

Modeling the thermal processes within the ice shelf–ocean boundary current underlain by strong pycnocline underneath a cold-water ice shelf using a 2.5-dimensional vertical slice model

Oceanic melting beneath ice shelves is the main driver of the current mass loss of the Antarctic ice sheet. The resultant meltwater plumes influence the basal melting of ice shelves, and contribute to the development of the unique two-layer stratified ice shelf–ocean boundary currents underlain by warmer, saltier, and stationary source waters. However, knowledge of the dynamics and thermodynamics within these meltwater plumes, controlling the heat available for melting ice, remains outstanding. Here we investigate that important issue by developing a 2.5-dimensional nonhydrostatic vertical slice model with 1.5 m vertical resolution, and conduct the reference run based on a representative Ice Shelf Water (ISW)-High Salinity Shelf Water (HSSW) boundary current beneath the Amery Ice Shelf, East Antarctica. Based on that we identify two dominating vertical thermal processes regulating the local temperature: the turbulent diffusion and the shear instabilities-induced convection (i.e., vertical advection), and carry out a quantitative thermal budget analysis in the framework of plume model, including deriving an analytical expression for the entrainment-induced heat flux. Moreover, after evaluating the entrainment parametrizations, it is suggested that the common assumption of neglecting the velocity at the lower boundary of meltwater plume potentially leads to a considerable deviation from the real entrainment. The sensitivity of the simulated results to model configuration and model resolution are also investigated. The findings in this study imply that we need to improve the model resolution of current ocean cavity models to capture the interfacial processes between lighter–denser waters.

Numerical entropy conservation without sacrificing Charney–Phillips grid optimal wave propagation

AbstractConservation of entropy or potential temperature and accurate representation of wave propagation without computational modes are both desirable properties for a numerical model of the atmosphere. However, they appear to require different model formulations, forcing model developers to choose between them. Here it is shown that, by a straightforward modification of the horizontal entropy fluxes, numerical entropy conservation can be achieved without sacrificing accurate wave propagation. The result is confirmed by a numerical linear normal mode analysis for a simple but suitably modified finite‐volume scheme, and by buoyant bubble and gravity‐wave test cases in a vertical slice model using a suitably modified conservative semi‐Lagrangian transport scheme.

Two-dimensional isotropic inertia–gravity wave turbulence

We present an idealized study of rotating stratified wave turbulence in a two-dimensional vertical slice model of the Boussinesq equations, focusing on the peculiar case of equal Coriolis and buoyancy frequencies. In this case the fully nonlinear fluid dynamics can be shown to be isotropic in the vertical plane, which allows the classical methods of isotropic turbulence to be applied. Contrary to ordinary two-dimensional turbulence, here a robust downscale flux of total energy is observed in numerical simulations that span the full parameter regime between Ozmidov and forcing scales. Notably, this robust downscale flux of the total energy does not hold separately for its various kinetic and potential components, which can exhibit both upscale and downscale fluxes, depending on the parameter regime. Using a suitable extension of the classical Kármán–Howarth–Monin equation, exact expressions that link third-order structure functions and the spectral energy flux are derived and tested against numerical results. These expressions make obvious that even though the total energy is robustly transferred downscale, the third-order structure functions are sign indefinite, which illustrates that the sign and the form of measured third-order structure functions are both crucially important in determining the direction of the spectral energy transfer.

Inverse modeling of the natural state of geothermal reservoirs using adjoint and direct methods

The application of the adjoint and direct methods to inverse modeling of the natural state of a convective geothermal system is discussed. The methods have been applied to other subsurface modeling problems, but they have seldom been used in the geothermal context and not at all for high-enthalpy, two-phase systems.There are two important features of the natural state problem that make it an interesting challenge for inverse modeling. The first is that good downhole temperature measurements are typically available from 1 to 2 km deep wells, scattered over a large area of the geothermal system. Secondly, the geological structure of the system controls how the convective plume develops over geological time and thus downhole temperatures can be used to infer the large-scale permeability distribution. This interaction of large-scale, heat induced convection and geological structure determines the size and shape of hot geothermal systems and makes inverse modeling of their natural state particularly useful, no matter what method is used.Another important issue with natural state geothermal modeling is the challenge of the computational task involved. Most previous inverse modeling studies of the natural state of high-enthalpy geothermal reservoirs have used the derivative-based Levenberg-Marquardt method with derivatives of model outputs evaluated by finite differences. This is a very time-consuming process since many lengthy forward simulations of a transient approach to the steady natural state are required. In the present study adjoint and direct methods are used for the efficient evaluation of derivatives and prove to be very effective.A synthetic two-dimensional vertical slice model is used to test the applicability of the adjoint and direct methods. The results show that the methods offer much faster inversions compared to those based on finite differencing.

Vertical slice modelling of nonlinear Eady waves using a compatible finite element method

A vertical slice model is developed for the Euler–Boussinesq equations with a constant temperature gradient in the direction normal to the slice (the Eady–Boussinesq model). The model is a solution of the full three-dimensional equations with no variation normal to the slice, which is an idealised problem used to study the formation and subsequent evolution of weather fronts. A compatible finite element method is used to discretise the governing equations. To extend the Charney–Phillips grid staggering in the compatible finite element framework, we use the same node locations for buoyancy as the vertical part of velocity and apply a transport scheme for a partially continuous finite element space. For the time discretisation, we solve the semi-implicit equations together with an explicit strong-stability-preserving Runge–Kutta scheme to all of the advection terms. The model reproduces several quasi-periodic lifecycles of fronts despite the presence of strong discontinuities. An asymptotic limit analysis based on the semi-geostrophic theory shows that the model solutions are converging to a solution in cross-front geostrophic balance. The results are consistent with the previous results using finite difference methods, indicating that the compatible finite element method is performing as well as finite difference methods for this test problem. We observe dissipation of kinetic energy of the cross-front velocity in the model due to the lack of resolution at the fronts, even though the energy loss is not likely to account for the large gap on the strength of the fronts between the model result and the semi-geostrophic limit solution.

Use of the Gibbs thermodynamic potential to express the equation of state in atmospheric models

The thermodynamics of moist processes is complicated, and in typical atmospheric models numerous approximations are made. However, they are not always made in a self‐consistent way, which could lead to spurious sources or sinks of energy and entropy. One way to ensure self‐consistency is to derive all thermodynamic quantities from a thermodynamic potential such as the Gibbs function. Approximations may be made to the Gibbs function; these approximations are inherited by all derived quantities in a way that guarantees self‐consistency. Here, the feasibility of using the Gibbs function in an atmospheric model is demonstrated through the development of a semi‐implicit, semi‐Lagrangian vertical slice model, and its application to a standard buoyant bubble test case. The flexibility of the approach is also demonstrated by running the test case with four different equations of state corresponding to dry air, moist air that is saturated, a pseudo‐incompressible fluid, and an incompressible fluid. A recently presented ‘blended’ equation set that unifies the dry fully compressible case and the pseudo‐incompressible case is also easily accommodated.

Hamiltonian particle‐mesh simulations for a non‐hydrostatic vertical slice model

AbstractA Lagrangian particle method is developed for the simulation of atmospheric flows in a non‐hydrostatic vertical slice model. The proposed particle method is an extension of the Hamiltonian particle mesh (HPM) [Frank J, Gottwald G, Reich S. 2002. The Hamiltonian particle‐mesh method. In Meshfree Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering, Vol. 26, Griebel M, Schweitzer M (eds). Springer‐Verlag: Berlin Heidelberg; 131–142] and provides preservation of mass, momentum, and energy. We tested the method for the gravity wave test in Skamarock W, Klemp J. 1994. Efficiency and accuracy of the Klemp‐Wilhelmson time‐splitting technique. Monthly Weather Review 122: 2623–2630 and the bubble experiments in Robert A. 1993. Bubble convection experiments with a semi‐implicit formulation of the Euler equations. Journal of the Atmospheric Sciences 50: 1865–1873. The accuracy of the solutions from the HPM simulation is comparable to those reported in these references. A particularly appealing aspect of the method is in its non‐diffusive transport of potential temperature. The solutions are maintained smooth largely due to a ‘regularization’ of pressure, which is controlled carefully to preserve the total energy and the time‐reversibility of the model. In case of the bubble experiments, one also needs to regularize the buoyancy contributions. The simulations demonstrate that particle methods are potentially applicable to non‐hydrostatic atmospheric flow regimes and that they lead to a highly accurate transport of materially conserved quantities such as potential temperature under adiabatic flow regimes. Copyright © 2009 Royal Meteorological Society

On the importance of non-hydrostatic processes in determining tidally induced mixing in sill regions

The importance of using a non-hydrostatic model to compute tidally induced mixing and flow in the region of a sill is examined using idealized topography representing the sill at the entrance to Loch Etive. This site is chosen since detailed measurements were recently made there. Calculations are performed with and without the inclusion of non-hydrostatic dynamics using a vertical slice model for a range of sill widths corresponding to typical sill regions. Initial non-hydrostatic calculations showed that the model could reproduce the observed flow characteristics in the region. However, when calculations were performed using the model in hydrostatic form, the significant artificial convective mixing that occurred in order to remove density inversions led to excessively high vertical mixing. This influenced the computed temperature field and the intensity of the current jet that separated from the sill on its lee side. In addition it affected the magnitude and spatial characteristics of the lee waves generated on the lee side of the sill. Calculations with a range of sill widths, showed that as the sill width decreased the difference between the solution computed with the non-hydrostatic and hydrostatic model increased.

Analysis of a regularized, time‐staggered discretization applied to a vertical slice model

AbstractA regularized and time‐staggered discretization of the two‐dimensional, vertical slice Euler equation set is described and analysed. A linear normal mode analysis of the time‐discrete system indicates that unconditional stability is obtained, for appropriate values of the regularization parameters, for both the hydrostatic and non‐hydrostatic cases. Furthermore, when these parameters take their optimal values, the stability behaviour of the normal modes is identical to that obtained from a semi‐implicit discretization of the unregularized equations. © Crown Copyright 2006. Reproduced with the permission of the Controller of HMSO. Published by John Wiley & Sons, Ltd.

Symplectic Time‐Stepping for Particle Methods

AbstractThis paper surveys some of the fundamental properties of symplectic integration schemes for classical mechanics and particle methods in particular. The widely used Störmer‐Verlet method is discussed in detail and implications of conservation of symplecticity on long term simulations are outlined. The second part of the paper describes the application of a Lagrangian particle method and the Störmer‐Verlet time integrator to numerical weather prediction (NWP). A simple vertical slice model and non‐hydrostatic flow over orography are discussed in detail. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An ephemeral stream‐aquifer interaction model

The boundary integral equation method is applied to the interaction between an ephemeral stream and an unconfined aquifer through a semipervious streambed. The resulting two‐dimensional vertical slice model describes groundwater flow when the stream and aquifer are hydraulically disconnected or connected as well as the transition between these two states. It incorporates the Green‐Ampt equation with the hydraulically disconnected boundary conditions. The model was successfully calibrated using stream discharge and bore hydrograph data from the Little Para River recharge study, South Australia. It predicts with sufficient accuracy the inflow and outflow from the stream to the aquifer during fluctuating water levels in the stream.

Seismic Response of High Temperature Gas-Cooled Reactor Core with Block-Type Fuel, (III)

The resistance against earthquakes of high-temperature gas cooled reactors (HTGR) with block-type fuel, is not fully ascertained yet. Seismic studies must be made if such a reactor plant is to be installed in area with frequent earthquakes. In this paper experimental results of seismic behaviour of a half scale two-dimentional vertical slice model are presented. The following results were obtained: (1) with soft spring support of the side reflector structure the relative column displacement is larger than that for hard support but the impact force is smaller, (2) in the case of hard spring support the dowel force is smaller than for soft support, (3) the relative column displacement and impact force are smaller with the gas pressure simulation spring than without.

Seismic response of high temperature gas-cooled reactor core with block-type fuel, (III). Vibration experiment of two-dimensional vertical slice core model.

The resistance against earthquakes of high-temperature gas cooled reactors (HTGR) with block-type fuel, is not fully ascertained yet. Seismic studies must be made if such a reactor plant is to be installed in area with frequent earthquakes. In this paper experimental results of seismic behaviour of a half scale two-dimentional vertical slice model are presented. The following results were obtained: (1) with soft spring support of the side reflector structure the relative column displacement is larger than that for hard support but the impact force is smaller, (2) in the case of hard spring support the dowel force is smaller than for soft support, (3) the relative column displacement and impact force are smaller with the gas pressure simulation spring than without.