Dynamical Subgrid-scale Parameterizations for Quasigeostrophic Flows using Direct Numerical Simulations

In this thesis, parameterizations of non-linear interactions in quasigeostrophic (QG) flows for severely truncated models (STM) and Large Eddy Simulations (LES) are studied. Firstly, using Direct Numerical Simulations (DNS), atmospheric barotropic flows over topography are examined, and it is established that such flows exhibit multiple equilibrium states for a wide range of parameters. A STM is then constructed, consisting of the large scale zonal flow and a topographic mode. It is shown that, qualitatively, this system behaves similarly to the DNS as far as the interaction between the zonal flow and topography is concerned, and, in particular, exhibits multiple equilibrium states. By fitting the analytical form of the topographic stationary wave amplitude, obtained from the STM, to the results obtained from DNS, renormalized dissipation and rotation parameters are obtained. The usage of renormalized parameters in the STM results in better quantitative agreement with the DNS.¶ In the second type of problem, subgrid-scale parameterizations in LES are investigated with both atmospheric and oceanic parameters. ..

Local Martingale and Pathwise Solutions for an Abstract Fluids Model

We establish the existence and uniqueness of both local martingale and local pathwise solutions of an abstract nonlinear stochastic evolution system. The primary application of this abstract framework is to infer the local existence of strong, pathwise solutions to the 3D primitive equations of the oceans and atmosphere forced by a nonlinear multiplicative white noise. Instead of developing our results specifically for the 3D primitive equations we choose to develop them in a slightly abstract framework which covers many related forms of these equations (atmosphere, oceans, coupled atmosphere-ocean, on the sphere, on the {\beta}-plane approximation etc and the incompressible Navier-Stokes equations). In applications, all of the details are given for the {\beta}-plane approximation of the oceans equations

Using Satellite Data to Determine Empirical Relationships between Volcanic Ash Source Parameters

Poor knowledge of dispersion model source parameters related to quantities such as the total fine ash mass emission rate, its effective spatial distribution, and particle size distribution makes the provision of quantitative forecasts of volcanic ash a difficult problem. To ameliorate this problem, we make use of satellite-retrieved mass load data from 14 eruption case studies to estimate fine ash mass emission rates and other source parameters by an inverse modelling procedure, which requires multidimensional sampling of several thousand trial simulations with different values of source parameters. We then estimate the dependence of these optimal source parameters on eruption height. We show that using these empirical relationships in a data assimilation procedure leads to substantial improvements to the forecasts of ash mass loads, with the use of empirical relationships between parameters and eruption height having the added advantage of computational efficiency because of dimensional reduction. In addition, the use of empirical relationships, which encode information in satellite retrievals from past case studies, implies that quantitative forecasts can still be issued even when satellite retrievals of mass load are not available in real time due to cloud cover or other reasons, making it especially useful for operations in the tropics where ice and water clouds are ubiquitous

Improving Ensemble Volcanic Ash Forecasts by Direct Insertion of Satellite Data and Ensemble Filtering

Improved quantitative forecasts of volcanic ash are in great demand by the aviation industry to enable better risk management during disruptive volcanic eruption events. However, poor knowledge of volcanic source parameters and other dispersion and transport modelling uncertainties, such as those due to errors in numerical weather prediction fields, make this problem very challenging. Nonetheless, satellite-based algorithms that retrieve ash properties, such as mass load, effective radius, and cloud top height, combined with inverse modelling techniques, such as ensemble filtering, can significantly ameliorate these problems. The satellite-retrieved data can be used to better constrain the volcanic source parameters, but they can also be used to avoid the description of the volcanic source altogether by direct insertion into the forecasting model. In this study we investigate the utility of the direct insertion approach when employed within an ensemble filtering framework. Ensemble members are formed by initializing dispersion models with data from different timesteps, different values of cloud top height, thickness, and NWP ensemble members. This large ensemble is then filtered with respect to observations to produce a refined forecast. We apply this approach to 14 different eruption case studies in the tropical atmosphere. We demonstrate that the direct insertion of data improves model forecast skill, particularly when it is used in a hybrid ensemble in which some ensemble members are initialized from the volcanic source. Moreover, good forecast skill can be obtained even when detailed satellite retrievals are not available, which is frequently the case for volcanic eruptions in the tropics

Parameterization of subgrid-scale energy injection in oceanic flows

Dissipative terms are usually employed to parameterize subgrid fluxes of energy and enstrophy in turbulent flows. However, in certain flows, such as oceanic flows, when the dissipation is calculated self-consistently it turns out to be negative and hence numerically unstable. A solution to this problem is offered in this study in the form of a stochastic subgrid scale parametrization scheme. It is a spectral scheme employing matrices that generalize the classical eddy dissipation and stochastic forcing variance to include vertical transfers. Using a baroclinic model with typical oceanic parameters the scheme is able to maintain the resolved-scale spectra exceptionally well when the energy injection scale is in the subgrid scales. This work has implications for ocean climate modelling where the resolution is typically too coarse to resolve the energy injection due to baroclinic instability. References Frederiksen, J. S. and S. M. Kepert, Dynamical subgrid-scale parameterizations from direct numerical simulations, J. Atmos. Sci., 63, 2006, 3006--3019. doi:10.1175/JAS3795.1 Zidikheri, M. J. Dynamical Subgrid-scale parameterizations for Quasigeostrophic Flows using Direct Numerical Simulations, PhD Thesis, The Australian National University, 2008. Bourke, W., An efficient, one-Level, primitive-equation spectral model, Mon. Weath. Rev., 100, 1972, 683--689. doi:10.1175/1520-0493(1972)100<0683:AEOPSM>2.3.CO;2 Leith, C. E., Atmospheric predictability and two-dimensional turbulence, J. Atmos. Sci., 28, 1971, 145--161. doi:10.1175/1520-0469(1971)028<0145:APATDT>2.0.CO;2 Kraichnan, R. H., Eddy viscosity in two and three dimensions, J. Atmos. Sci., 33, 1976, 1521--1536. doi:10.1175/1520-0469(1976)033<1521:EVITAT>2.0.CO;2 Frederiksen, J. S., and A. G. Davies, Eddy viscosity and stochastic backscatter parameterizations on the sphere for atmospheric circulation models, J. Atmos. Sci., 54, 1997, 2475--2492. doi:10.1175/1520-0469(1997)054<2475:EVASBP>2.0.CO;2 Salmon, R., Two-layer quasi-geostrophic turbulence in a simple special case, Geophys. Astrophys. Fluid Dynamics, 10, 1978, 25--52. doi:10.1080/0309192780824262

Computationally efficient methods for climate model inversion

The dynamical equations of a model are used to obtain the `forcing function', which is a representation of climate change drivers, from an observed climatic anomaly. This inversion problem is mathematically difficult because of the two-way interaction between the mean field and transient eddies; this is known as the turbulence closure problem. The first method that we explore for overcoming the closure problem involves iteratively nudging a climate simulation towards the observed climate. We demonstrate how this method is used to successfully calculate the climatic forcing function. The second method that we explore involves finding approximations to the turbulence closure problem. In this method, the transient eddy feedback term in the mean field equation is represented as a linear combination of the mean fields and a constant term. We demonstrate that the closure method yields a good approximation to the climatic forcing function. This forcing function is then used as an improved first estimate in the iterative method, thereby yielding a scheme that converges very quickly to the correct solution in only a few iteration steps. References G. C. Hegerl, T. R. Karl, M. Allen, N. L. Bindoff, N. Gillett, D. Karoly, X. Zhang and F. Zwiers. Climate change detection and attribution: beyond mean temperature signals. J. Clim., 19:5058--5077, 2006. doi:10.1175/JCLI3900.1 S. Corti, A. Giannini, S. Tibaldi and F. Molteni. Patterns of low-frequency variability in a three-level quasi-geostrophic model. Clim. Dyn., 13:883--904, 1997. doi:10.1007/s003820050203 C. E. Leith. Climate response and fluctuation dissipation. J. Atmos. Sci., 32:2022--2026, 1975. doi:10.1175/1520-0469(1975)032<2022:CRAFD>2.0.CO;2 T. L. Bell. Climate sensitivity from fluctuation dissipation: some simple model results. J. Atmos. Sci., 37:1700--1707, 1980. doi:10.1175/1520-0469(1980)037<1700:CSFFDS>2.0.CO;2 A. S. Gritsun. Fluctuation-dissipation theorem on attractors of atmospheric models. Russ. J. Numer. Anal. Math. Modelling, 16:115--133, 2001. A. Gritsun and G. Branstator. Climate response using a three-dimensional operator based on the fluctuation-dissipation theorem. J. Atmos. Sci., 64:2558--2575, 2007. doi:10.1175/JAS3943.1 J. S. Frederiksen. Subgrid-scale parametrisations of eddy-topographic force, eddy viscosity, and stochastic backscatter for flow over topography. J. Atmos. Sci., 56:1481--1494, 1999. doi:10.1175/1520-0469(1999)056<1481:SSPOET>2.0.CO;2 J. S. Frederiksen. Statistical dynamical closures and subgrid modeling for inhomogeneous QG and 3D turbulence. Entropy, 14:32--57, 2012. doi:10.3390/e14010032 J. S. Frederiksen. Self-energy closure for inhomogeneous turbulent flows and subgrid modeling. Entropy, 14:769--799, 2012. doi:10.3390/e14040769 R. Salmon. Lectures on Geophysical Fluid Dynamics, 1998. Oxford University Press. M. J. Zidikheri and J. S. Frederiksen. Inverse method for attribution of climate change. ANZIAM J., 52:C823--C836, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3930. I. M. Held and M. J. Suarez. A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models. Bull. Amer. Meteor, Soc., 75:1825--1830, 1994. doi:10.1175/1520-0477(1994)075<1825:APFTIO>2.0.CO;2 J. Namias, X. Yuan and D. R. Cayan. Persistence of North Pacific sea surface temperature and atmospheric flow patterns. J. Clim., 1:682--703, 1988. doi:10.1175/1520-0442(1988)001<0682:PONPSS>2.0.CO;

Stochastic subgrid modelling for atmospheric large eddy simulations

Dynamical subgrid scale parameterizations of stochastic backscatter and eddy dissipation have been calculated for typical atmospheric turbulent flows on the sphere. A methodology based on a stochastic model representation of the subgrid scale eddies in direct numerical simulations, and with wide applicability to fluid flows, has been employed. Large eddy simulations incorporating these subgrid scale parameterizations are found to have energy spectra that compare closely with the results of higher resolution direct numerical simulations for both barotropic and baroclinic turbulent flows. References O'Kane, T.J. and J.S. Frederiksen, Statistical dynamical subgrid-scale parameterizations for geophysical flows, Physica Scripta, 78, 2008, in press. Frederiksen, J.S., Subgrid-scale parameterizations of eddy-topographic force, eddy viscosity, and stochastic backscatter for flow over topography, J. Atmos. Sci. 56, 1999, 1481--1494. doi:10.1175/1520-0469(1999)056<1481:SSPOET>2.0.CO;2 Frederiksen, J.S., and A.G. Davies, Eddy viscosity and stochastic backscatter parameterizations on the sphere for atmospheric circulation models, J. Atmos. Sci., 54, 1997, 2475--2492. doi:10.1175/1520-0469(1997)054<2475:EVASBP>2.0.CO;2 Frederiksen, J.S. and S.M. Kepert, Dynamical subgrid-scale parameterizations from direct numerical simulations, J. Atmos. Sci., 63, 2006, 3006--3019. doi:10.1175/JAS3795.1 Frederiksen, J.S. and M.R. Dix and A.G. Davies, The effects of closure-based eddy diffusion on the climate and spectra of a GCM, Tellus A, 55, 2003, 31--44. doi:10.1034/j.1600-0870.2003.201329.

Subgrid parameterisations for high resolution atmospheric flows

Numerical and computational methods are developed for the Large Eddy Simulation of atmospheric flows on a sphere with a spectral quasigeostrophic model. The subgrid scales of motion are parameterised using a net eddy viscosity that is derived from a high resolution reference Direct Numerical Simulation with $504$~zonal and total wavenumbers or $1536 \times 768$ grid points (longitude by latitude). Simulations are undertaken for a wide range of truncation wavenumbers to determine the influence of resolution on the net eddy viscosity. A universal scaling law for these coefficients is established for application to large eddy simulations of more general geophysical flows. References J. S. Frederiksen. Precursors to blocking anomalies: the tangent linear and inverse problems. J. Atmos. Sci., 55:2419--2436, 1998. doi:10.1175/1520-0469(1998)055<2419:PTBATT>2.0.CO;2 J. S. Frederiksen. Subgrid-scale parameterizations of eddy-topographic force, eddy viscosity and stochastic backscatter for flow over topography. J. Atmos. Sci, 56:1481--1493, 1999. doi:10.1175/1520-0469(1999)056<1481:SSPOET>2.0.CO;2 J. S. Frederiksen and A. G. Davies. Eddy viscosity and stochastic backscatter parameterizations on the sphere for atmospheric circulation models. J. Atmos. Sci, 54:2475--2492, 1997. doi:10.1175/1520-0469(1997)054<2475:EVASBP>2.0.CO;2 J. S. Frederiksen and S. M. Kepert. Dynamical subgrid-scale parameterizations from {Direct Numerical Simulations}. J. Atmos. Sci., 63:3006--3019, 2006. doi:10.1175/JAS3795.1 C. E. Leith. Stochastic backscatter in a subgrid-scale model: Plane shear mixing layer. Phys. Fluids, 2(3):297--299, 1990. doi:10.1063/1.857779 J. Smagorinsky. General circulation experiments with the primitive equations: I. the basic experiment. Mon. Wea. Rev., 91(3):99--164, 1963. doi:10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2 M. J. Zidikheri and J. S. Frederiksen. Stochastic subgrid parameterizations for simulations of atmospheric baroclinic flows. J. Atmos. Sci., 66:2844--2858, 2009. doi:10.1175/2009JAS303

Inverse method for attribution of climate change

A new inverse method for determining the anomalous mean forcing functions responsible for climate change is presented. The method closes the mean field equations by representing the second order statistical moments as linear functions of the mean field. The coefficients of the linear parameterisation are determined by least-squares regression. The technique successfully reproduces the anomalous forcing functions responsible for the observed change in climate simulations between the periods July 1949--1968 and 1975--1994. References J. S. Frederiksen and C. S. Frederiksen. Interdecadal changes in southern hemisphere winter storm track modes. Tellus, 59A:599--617, 2007. doi:10.1111/j.1600-0870.2007.00264.x J. S. Frederiksen. Growth and vacillation cycles of disturbances in southern hemisphere flows. J. Atmos. Sci., 38:1360--1375, 1981. doi:10.1175/1520-0469(1981)038<1360:GAVCOD>2.0.CO;2 J. S. Frederiksen. Subgrid-scale parameterizations of eddy-topographic force, eddy viscosity, and stochastic backscatter for flow over topography. J. Atmos. Sci., 56:1481--1494, 1999. doi:10.1175/1520-0469(1999)056<1481:SSPOET>2.0.CO;2 T. J. O'Kane and J. S. Frederiksen. The QDIA and regularized QDIA closures for inhomogeneous turbulence over topography. J. Fluid. Mech., 504:133--165, 2004. doi:10.1017/S0022112004007980 J. S. Frederiksen and T. J. O'Kane. Inhomogeneous closure and statistical mechanics for Rossby wave turbulence over topography. J. Fluid. Mech., 539:137--165, 2005. doi:10.1017/S0022112005005562 J. S. Frederiksen. Precursors to blocking anomalies: The tangent linear and inverse problems. J. Atmos. Sci., 55:2419--2436, 1998. doi:10.1175/1520-0469(1998)055<2419:PTBATT>2.0.CO;2 M. J. Zidikheri and J. S. Frederiksen. Stochastic subgrid parameterizations for simulations of atmospheric baroclinic flows. J. Atmos. Sci., 66:2844--2858, 2009. doi:10.1175/2009JAS3036.