Modelling present-day basal melt rates for Antarctic ice shelves using a parametrization of buoyant meltwater plumes

Basal melting below ice shelves is a major factor in mass loss from the Antarctic Ice Sheet, which can contribute significantly to possible future sea-level rise. Therefore, it is important to have an adequate description of the basal melt rates for use in ice-dynamical models. Most current ice models use rather simple parametrizations based on the local balance of heat between ice and ocean. In this work, however, we use a recently derived parametrization of the melt rates based on a buoyant meltwater plume travelling upward beneath an ice shelf. This plume parametrization combines a non-linear ocean temperature sensitivity with an inherent geometry dependence, which is mainly described by the grounding-line depth and the local slope of the iceshelf base. For the first time, this type of parametrization is evaluated on a two-dimensional grid covering the entire Antarctic continent. In order to apply the essentially onedimensional parametrization to realistic ice-shelf geometries, we present an algorithm that determines effective values for the grounding-line depth and basal slope in any point beneath an ice shelf. Furthermore, since detailed knowledge of temperatures and circulation patterns in the ice-shelf cavities is sparse or absent, we construct an effective ocean temperature field from observational data with the purpose of matching (area-averaged) melt rates from the model with observed present-day melt rates. Our results qualitatively replicate large-scale observed features in basal melt rates around Antarctica, not only in terms of average values, but also in terms of the spatial pattern, with high melt rates typically occurring near the grounding line. The plume parametrization and the effective temperature field presented here are therefore promising tools for future simulations of the Antarctic Ice Sheet requiring a more realistic oceanic forcing

Turbulence modelling applied to the atmospheric boundary layer

Turbulent flows affected by buoyancy lie at the basis of many applications, both within engineering and the atmospheric sciences. A prominent example of such an application is the atmospheric boundary layer, the lowest layer of the atmosphere, in which many physical processes are heavily influenced by both stably stratified and convective turbulent transport. Modelling these turbulent flows correctly, especially in the presence of stable stratification, has proven to be a great challenge and forms an important problem in the context of climate models. In this thesis, we address this issue considering an advanced class of turbulence models, the so-called explicit algebraic models.In the presence of buoyancy forces, a mutual coupling between the Reynolds stresses and the turbulent heat flux exists, which makes it difficult to derive a fully explicit turbulence model. A method to overcome this problem is presented based on earlier studies for cases without buoyancy. Fully explicit and robust models are derived for turbulence in two-dimensional mean flows with buoyancy and shown to give good predictions compared with various data from direct numerical simulations (DNS), most notably in the case of stably stratified turbulent channel flow. Special attention is given to the problem of determining the production-to-dissipation ratio of turbulent kinetic energy, for which the exact equation cannot be solved analytically. A robust approximative method is presented to calculate this quantity, which is important for obtaining a consistent formulation of the model.The turbulence model derived in this way is applied to the atmospheric boundary layer in the form of two idealized test cases. First, we consider a purely stably stratified boundary layer in the context of the well-known GABLS1 study. The model is shown to give good predictions in this case compared to data from large-eddy simulation (LES). The second test case represents a full diurnal cycle containing both stable stratification and convective motions. In this case, the current model yields interesting dynamical features that cannot be captured by simpler models. These results are meant as a first step towards a more thorough investigation of the pros and cons of explicit algebraic models in the context of the atmospheric boundary layer, for which additional LES data are required. QC 20150522</p

Explicit algebraic turbulence modelling in buoyancy-affected shear flows

Turbulent flows affected by buoyancy forces occur in a large amount of applica-tions, from heat transfer in industrial settings to the effects of stratification inEarth’s atmosphere. The two-way coupling between the Reynolds stresses andthe turbulent heat flux present in these flows poses a challenge in the searchfor an appropriate turbulence model. The present thesis addresses this issueusing the class of explicit algebraic models.     Starting from the transport equations for the Reynolds stresses and the tur-bulent heat flux, an explicit algebraic framework is derived for two-dimensionalmean flows under the influence of buoyancy forces. This framework consistsof a system of 18 linear equations, the solution of which leads to explicit ex-pressions for the Reynolds-stress anisotropy and a scaled heat flux. The modelis complemented by a sixth-order polynomial equation for a quantity relatedto the total production-to-dissipation ratio of turbulent kinetic energy. Sinceno exact solution to such an equation can be found, various approximationmethods are presented in order to obtain a fully explicit algebraic model.     Several test cases are considered in this work. Special attention is given tothe case of stably stratified parallel shear flows, which is also used to calibratethe model parameters. As a result of this calibration, we find a critical Richard-son number of 0.25 in the case of stably stratified homogeneous shear flow,which agrees with theoretical results. Furthermore, a comparison with directnumerical simulations (DNS) for stably stratified channel flow shows an excel-lent agreement between the DNS data and the model. Other test cases includeunstably stratified channel flow and vertical channel flow with either mixed con-vection or natural convection, and a reasonably good agreement between themodel and the scarcely available, low-Reynolds-number DNS is found. Com-pared to standard eddy-viscosity/eddy-diffusivity models, an improvement inthe predictions is observed in all cases.     For each of the aforementioned test cases, model coefficients and additionalcorrections are derived separately, and a general formulation has yet to be given.Nevertheless, the results presented in this thesis have the potential of improvingthe prediction of buoyancy-affected turbulence in various application areas.QC 20130530</p

Turbulence modelling applied to the atmospheric boundary layer

Turbulent flows affected by buoyancy lie at the basis of many applications, both within engineering and the atmospheric sciences. A prominent example of such an application is the atmospheric boundary layer, the lowest layer of the atmosphere, in which many physical processes are heavily influenced by both stably stratified and convective turbulent transport. Modelling these turbulent flows correctly, especially in the presence of stable stratification, has proven to be a great challenge and forms an important problem in the context of climate models. In this thesis, we address this issue considering an advanced class of turbulence models, the so-called explicit algebraic models.In the presence of buoyancy forces, a mutual coupling between the Reynolds stresses and the turbulent heat flux exists, which makes it difficult to derive a fully explicit turbulence model. A method to overcome this problem is presented based on earlier studies for cases without buoyancy. Fully explicit and robust models are derived for turbulence in two-dimensional mean flows with buoyancy and shown to give good predictions compared with various data from direct numerical simulations (DNS), most notably in the case of stably stratified turbulent channel flow. Special attention is given to the problem of determining the production-to-dissipation ratio of turbulent kinetic energy, for which the exact equation cannot be solved analytically. A robust approximative method is presented to calculate this quantity, which is important for obtaining a consistent formulation of the model.The turbulence model derived in this way is applied to the atmospheric boundary layer in the form of two idealized test cases. First, we consider a purely stably stratified boundary layer in the context of the well-known GABLS1 study. The model is shown to give good predictions in this case compared to data from large-eddy simulation (LES). The second test case represents a full diurnal cycle containing both stable stratification and convective motions. In this case, the current model yields interesting dynamical features that cannot be captured by simpler models. These results are meant as a first step towards a more thorough investigation of the pros and cons of explicit algebraic models in the context of the atmospheric boundary layer, for which additional LES data are required. QC 20150522</p

Direct solution for the anisotropy tensor in explicit algebraic Reynolds stress models

A direct solution to a tensorial equation which constitutes a basis for explicit algebraic Reynolds stress models is derived. We consider equations linear and quasilinear in the strain tensor and show how the independent tensor groups emerge. Solution of an extended model with a linearly coupled active scalar, governed by a linear in anisotropy tensor equation, is also outlined.QC 20160314</p

Direct solution for the anisotropy tensor in explicit algebraic Reynolds stress models

A direct solution to a tensorial equation which constitutes a basis for explicit algebraic Reynolds stress models is derived. We consider equations linear and quasilinear in the strain tensor and show how the independent tensor groups emerge. Solution of an extended model with a linearly coupled active scalar, governed by a linear in anisotropy tensor equation, is also outlined.QC 20160314</p

Modelling present-day basal melt rates for Antarctic ice shelves using a parametrization of buoyant meltwater plumes

Basal melting below ice shelves is a major factor in mass loss from the Antarctic Ice Sheet, which can contribute significantly to possible future sea-level rise. Therefore, it is important to have an adequate description of the basal melt rates for use in ice-dynamical models. Most current ice models use rather simple parametrizations based on the local balance of heat between ice and ocean. In this work, however, we use a recently derived parametrization of the melt rates based on a buoyant meltwater plume travelling upward beneath an ice shelf. This plume parametrization combines a non-linear ocean temperature sensitivity with an inherent geometry dependence, which is mainly described by the grounding-line depth and the local slope of the ice-shelf base. For the first time, this type of parametrization is evaluated on a two-dimensional grid covering the entire Antarctic continent. In order to apply the essentially one-dimensional parametrization to realistic ice-shelf geometries, we present an algorithm that determines effective values for the grounding-line depth and basal slope in any point beneath an ice shelf. Furthermore, since detailed knowledge of temperatures and circulation patterns in the ice-shelf cavities is sparse or absent, we construct an effective ocean temperature field from observational data with the purpose of matching (area-averaged) melt rates from the model with observed present-day melt rates. Our results qualitatively replicate large-scale observed features in basal melt rates around Antarctica, not only in terms of average values, but also in terms of the spatial pattern, with high melt rates typically occurring near the grounding line. The plume parametrization and the effective temperature field presented here are therefore promising tools for future simulations of the Antarctic Ice Sheet requiring a more realistic oceanic forcing

An analytical derivation of ice-shelf basal melt based on the dynamics of meltwater plumes

The interaction between ice shelves and the ocean is an important process for the development of marine ice sheets. However, it is difficult to model in full detail due to the high computational cost of coupled ice- ocean simulations, so that simplified basal-melt parameterizations are required. In this work, a new analytical expression for basal melt is derived from the theory of buoyant meltwater plumes moving upward under the ice shelf and driving the overturning circulation within the ice-shelf cavity. The governing equations are nondimensionalized in the case of an ice shelf with constant basal slope and uniform ambient ocean conditions. An asymptotic analysis of these equations in terms of small slopes and small thermal driving, assumed typical for Antarctic ice shelves, leads to an equation that can be solved analytically for the dimensionless melt rate. This analytical expression describes a universal melt-rate curve onto which the scaled results of the original plume model collapse. Its key features are a positive melt peak close to the grounding line and a transition to refreezing further away. Comparing the analytical expression with numerical solutions of the plume model generally shows a close agreement between the two, even for more general cases than the idealized geometry considered in the derivation. The results show how the melt rates adapt naturally to changes in the geometry and ambient ocean temperature. The new expression can readily be used for improving ice-sheet models that currently still lack a sufficiently realistic description of basal melt

An analytical derivation of ice-shelf basal melt based on the dynamics of meltwater plumes

The interaction between ice shelves and the ocean is an important process for the development of marine ice sheets. However, it is difficult to model in full detail due to the high computational cost of coupled ice- ocean simulations, so that simplified basal-melt parameterizations are required. In this work, a new analytical expression for basal melt is derived from the theory of buoyant meltwater plumes moving upward under the ice shelf and driving the overturning circulation within the ice-shelf cavity. The governing equations are nondimensionalized in the case of an ice shelf with constant basal slope and uniform ambient ocean conditions. An asymptotic analysis of these equations in terms of small slopes and small thermal driving, assumed typical for Antarctic ice shelves, leads to an equation that can be solved analytically for the dimensionless melt rate. This analytical expression describes a universal melt-rate curve onto which the scaled results of the original plume model collapse. Its key features are a positive melt peak close to the grounding line and a transition to refreezing further away. Comparing the analytical expression with numerical solutions of the plume model generally shows a close agreement between the two, even for more general cases than the idealized geometry considered in the derivation. The results show how the melt rates adapt naturally to changes in the geometry and ambient ocean temperature. The new expression can readily be used for improving ice-sheet models that currently still lack a sufficiently realistic description of basal melt