### Zonal Jets

Large-scale zonal jets are observed through the motion of the cloud bands in the atmospheres of the gas giant planets: Jupiter, Saturn, Uranus and Neptune. Jupiter's upper atmosphere is characterised by large-scale zonal jets, numerous vortices, and turbulent cloud patterns. It has strong westerly winds at its equator, and its zonal winds and jets are exceptionally stable, persisting over decades. We are formulating and analysing models for Jupiter's upper atmosphere using shallow water (SW) theory.

Numerical simulations of the rotating SW equations on a sphere capture many of the key qualitative features of Jupiter's upper atmosphere. They robustly produce the expected pattern of zonal jets, but the equatorial jet invariably flows in the wrong direction. Scott & Polvani (2008) incorporate large-scale energy dissipation by radiative relaxation into the rotating SW equations. Their model correctly gives a westerly equatorial jet, but the radiative relaxation destroys the mass and momentum conserving properties of the model. We propose the thermal SW equations as a new SW model for Jupiter. These equations extend the standard rotating SW equations by allowing horizontal temperature variations, so radiative relaxation may be included directly in the temperature equation without affecting mass or momentum conservation.

We have carried out and compared numerical simulations of the standard rotating SW equations, the Scott & Polvani (2008) model, and our new thermal SW equations in Cartesian geometry using Jovian parameter values. All models produce a mixture of robust zonal jets, vortices and turbulence. The thermal SW model gives the correct direction for the jet at the equator, like the Scott & Polvani (2008) model, but unlike the standard rotating SW equations. Our simulation programs are written to take advantage of the faster computation afforded by using a GPU (graphical processing unit) cluster. In the future we will extend the simulations of the thermal SW equations to spherical geometry. We will also extend the single-layer model described here by developing a multi-layered model. Approximating the continuous vertical structure using a small stack of layers with varying thicknesses will provide a more realistic model of Jupiter's atmosphere.

For more information contact Paul Dellar or Emma Warneford.

#### Key references in this area

**Cho, J. Y.-K. and Polvani, L. M.**(1996). The morphogenesis of bands and zonal winds in the atmospheres on the giant outer planets.*Science***273**, 335-337.**Scott, R. K. and Polvani, L. M.**(2008). Equatorial superrotation in shallow atmospheres.*Geophys. Res. Lett.***35**, L24202-5.

### Soil erosion

Erosion of topsoil is a serious problem in Australia, China and parts of the US. Although more than 99% of the world's food comes from the soil, experts estimate that each year more than 10m hectares (25m acres) of crop land are degraded or lost as rain and wind sweep away topsoil. Erosion is the removal of solids (sediment, soil, rock and other particles) in the natural environment. It usually occurs due to transport by wind, water, or ice and by down-slope creep of soil and other material under the force of gravity. This project is concerned paricularly with overland water flow soil erosion problems.

The Hairsine-Rose model describes the suspended sediment concentration of multiple particle sizes. Rainfall detachment, overland flow entrainment and gravity deposition are considered for modelling soil erosion mechanisms. The HR model is modified by combining the basic HR model with the St Venant equations (hydraulic) and the Exner equation (bed elevation) to simulate sediment transportation. Due to the instability of traditional finite difference methods for solving the hydraulic problem, a composite Liska-Wendroff scheme (LwLf4) was proposed for solving the modified HR model.

The modified HR model can be applied to many problems in sediment transportation. Of particluar interest are dune and anti-dune formation, rill formation, and the hysteresis loop. The hysteresis loop is defined as the curve representing sediment concentration against discharge in a rainfall event. Simulations show that the HR model successfully produces three basic types of hysteresis loops under different flow and soil surface conditions. Also of interest are vegetative buffer strips (filter strips), which are vegetated surfaces designed to trap sediment and to reduce overland flow from adjacent surfaces. Predictions of hydraulic jump formations in the modified HR model agree well with past experiment data for small slopes. By including the flow motion of different slopes, the predictions of deposited mass in hydraulic adjustment zone agree with Rose's experimental data.

At present these simulations are one-dimensional. The next step is to extend the HR model to two-dimensional applications, such as two-dimensional rill formation.

For more information contact Yiming Zhong.

### Strongly variable viscosity flows in mantle convection

The rocks exposed at the surface of the Earth are part of the crust. Beneath this crustal layer (which is on average 30km thick beneath continents and 6km thick beneath oceans) lies the mantle, which extends down some 2900km to the Earth's central core. The Earth releases internal heat by convection. Hot mantle rises to the surface and spreads laterally, transporting oceanic and continental plates. Speed of this motion is a few centimetres per year. The new lithosphere, created at the ocean spreading centres, cools and eventually sinks back into the mantle. This is mantle convection.

One of the most important dimensionless numbers in the study of convection is the Rayleigh number and one of the key factors within the Rayleigh number and in the overall study of mantle convection is the viscosity. The viscosity of the mantle is so large (average viscosity is about 10^{21} Pas) that it is called a slowly moving or creeping fluid and the more important fact is that mantle viscosity is not a constant but variable. Thus plate tectonics in the Earth is driven by convection of the Earth's mantle, which can be described by the high Rayleigh number thermal convection of a material whose viscosity is strongly dependent on temperature, pressure and stress. The strength of this dependence is such that numerical computations are unable to attain realistic parameter values, and, in particular, the pressure dependence of the viscosity has been very little studied. This project will extend asymptotic methods of solution, which have been successfully developed for purely temperature-dependent viscous flows, to the case where the viscosity depends strongly on both temperature and pressure. Particular issues of interest will be to investigate whether the core of the flow is isoviscous, whether thermal runaway occurs below the stagnant lid, and whether the surface yields plastically, causing subduction.

The figure shows a simulation of constant-viscosity mantle convection for Rayleigh number Ra = 10^{6}. The deep blue layers at the top represent the cold thermal boundary layers (i.e. tectonic plates) and the brown region at the bottom represents the hot thermal boundary layers at the core-mantle boundary. In the middle, we can see up-welling plumes and at both sides, we observe cold downwelling currents (subducting slabs).

For more information please contact Tania Khaleque or Andrew Fowler.

### Key references in this area

**Gerald Schubert, Donald L. Turcotte & Peter Olson**(2001). Mantle Convection in the Earth and Planets. Cambridge University Press.**David Bercovici**(Volume Editor) &**Gerald Schubert**(Editor-in-Chief) (2009). Treatise on Geophysics: Mantle Dynamics, Volume**7**, Elsevier.

### Estimating the mean global sea levels

Accurate knowledge about global sea levels (GSL) is available only two decades back in time, when the first oceanographic Satellite TOPEX/Poseidon was launched. Tide gauge records stretch back to the 1700s, but these measurements are scarce are scattered. Oceans exhibit important temporal and spatial co-variance structures on a multitude of scales, making it hard to estimate the GSL from tide gauge measurements.

Current methods focus on constructing empirical orthogonal eigenfunctions (EOFs), but we want to approach the problem as a state estimation problem, applying methods from atmospheric forecasting. Time has been spent mostly on studying Bayesian methods and inverse problems. A toy sea level model has been simulated and methods such as the Kalman filter, Ensemble Kalman filter, and the Particle filter have been implemented. Work has been done on formulating techniques with high efficiency that in certain cases can tackle very high-dimensional problems.

So far, implementation of the standard methods has been relatively straightforward. We hope to make significant and rapid progress on the GSL problem, contributing to current estimates with our results, and also giving some quantification of uncertainty. The techniques used for this problem are of course applicable to a wide range of problems, and another highly relevant area of application that we are looking at is modelling oil reservoirs.

### Key references in this area

**J.A. Church and N.J. White**(2006). A 20th century acceleration in global sea-level rise.*Geophysical Research Letters***33**(1):L01602.**C. Farmer**(2007). Bayesian field theory applied to scattered data interpolation and inverse problems.*Algorithms for Approximation*.*Part III.*147-166.**C.K. Wikle and L.M. Berliner**(2007). A bayesian tutorial for data assimilation.*Physica D***230**(1-2):1-16.

### Stochastic Parameterisation

The central aim of any atmospheric parametrisation scheme is to improve the forecasting skill of the atmospheric model in which it is embedded and to represent better our beliefs about the future state of the atmosphere, be this the weather in 5 days time or the climate in 50 years time. One aspect of this

goal is the accurate representation of uncertainty: a forecast should skilfully indicate the confidence the forecaster can have in his or her prediction. There are two main sources of error in atmospheric modelling: errors in the initial conditions and errors in the model's representation of the atmosphere. The ensemble forecast generated should explore these uncertainties, and a probabilistic forecast issued to the user. A probabilistic forecast is of great economic value as it allows reliable assessment of the risks associated with different decisions, which cannot be achieved using a deterministic forecast.

Simple chaotic systems such as the Lorenz 1996 model, are useful tools for testing methods for use in numerical weather simulations due to their transparency and computational cheapness. The full system is defined as the truth, while a truncated version is used as a testbed for parametrisation

schemes. Several stochastic parametrisation schemes are investigated, including additive and multiplicative Gaussian and AR(1) noise. Forecasts are started from perfect initial conditions, eliminating initial condition uncertainty. The stochastically generated ensembles are compared to perturbed parameter ensembles and deterministic schemes, using skill scores. Forecasting skill

is found to be linked to their ability to reproduce the climatology of the full model. This is important in a seamless prediction.

Of the several types of stochastic parametrisation schemes investigated using the Lorenz 1996 system, all show an improvement in weather and climate forecasting skill over deterministic parametrisations. This result is robust to error in measurement of the parameters; scanning over parameter space indicates a wide range of parameter settings give good skill scores.

Stochastic parametrisations have been shown to represent the uncertainty in a forecast due to model deficiencies accurately. This increase in forecast reliability is reflected by the increase in the weather prediction skill and improved climatology of the forecast model.

The Lorenz 1996 system is an excellent tool for testing developments in stochastic parametrisations. These ideas are now be applied to ensemble numerical weather prediction models at ECMWF.

For further information contact Irene Moroz.

### Key references in this area

**Lorenz, E.N.**(1996) Predictability - a problem partly solved. In Proceedings, Seminar on Predictability ECMWF, volume 1.**Arnold, H.M., Moroz, I.M. and Palmer, T.N.**(2012) Stochastic Parametrisations and Model Uncertainty in the Lorenz 1996 System. Phil. Trans. R. Soc. London. To appear.

### Synchronisation of the Quasi-biennial oscillation and the semi-annual mode

Time series of zonal mean zonal winds near the Equator from the ERA-40 and ERA-interim reanalysis datasets over a 50-year period reveal that in the stratosphere, the Quasi-biennial oscillation (QBO) is found to synchronize with the semi-annual oscillatory mode (SAO) almost all the time, but with a frequency ratio that changes erratically between 4:1, 5:1 and 6:1. A similar variable synchronization is also evident in the tropical troposphere between semi-annual and quasi-biennial cycles (known as TBOs). Mean zonal winds from ERA-40 and ERA-interim, and also time series of indices for the Indian and West Pacific monsoons, are commonly found to exhibit synchronization, with

SAO/TBO ratios that vary between 4:1 and 7:1.

While Coherent synchronization between the QBO and tropical TBO does not appear to persist for long intervals it suggests that both the QBO and tropical TBOs may be separately synchronized to SAOs that are themselves enslaved to the seasonal cycle, or to the annual cycle itself.

A combination of singular systems analysis and analytic phase techniques have been used by Read & Castrejón-Pita to investigate the possible occurrence in observations of coherent synchronization between quasi-biennial and semi-annual oscillations (QBOs; SAOs) in the stratosphere and troposphere.

With the aim of improving our understanding of these fluctuations – what drives them dynamically, and when are they likely to occur – we propose to approach this problem by analysing various candidate low-dimensional models of weakly coupled dynamical systems for similar behaviour. We will begin with the models of Holton and Lindzen (1972) and of Plumb (1977). These models will then form a basis of comparison to existing atmospheric models, in order to discern the fundamental physics driving both the synchronization and the fluctuations. It is hoped that such analysis will lead to a deeper understanding and appreciation of the subtle nonlinear dynamics of these coupled atmospheric windsystems.

For further information please contact Kylash Rajendran or Irene Moroz.

### Key references in this area

**Read, P. L. and Castrejón-Pita, A. A.**(2012), Phase synchronization between stratospheric and tropospheric quasi-biennial and semi-annual oscillations. Q.J.R. Meteorol. Soc. doi: 10.1002/qj.1872**Plumb, R.A.**(1977) The interaction of 2 internal waves with the mean flow Implications for the theory of the Quasi-Bienniel Oscillation. J.A.S. vol. 34, 1847-58.