etd@IISc Collection:http://hdl.handle.net/2005/282014-11-11T09:04:58Z2014-11-11T09:04:58ZA Variable Resolution Global Spectral Method With Finer Resolution Over The TropicsJanakiraman, Shttp://hdl.handle.net/2005/9402010-11-19T20:30:23Z2010-11-18T18:30:00ZTitle: A Variable Resolution Global Spectral Method With Finer Resolution Over The Tropics
Authors: Janakiraman, S
Abstract: Variable resolution method helps to study the local scale phenomenon of interest within the context of global scale atmosphere/ocean dynamics. Global spectral methods based on spherical harmonics as basis functions are known to resolve a given function defined on the sphere, in an uniform manner. Though known for its mathematical elegance and higher order accuracy, global spectral methods are considered to be restrictive for developing mesh-refinement strategies. The only mesh refinement strategy available until now is due to the pioneering work of F. Schmidt. Schmidt transformation can study only one region with higher resolution.
The study of tropical dynamics is an interesting theme due to the presence of teleconnections between various phenomena, especially Indian Monsoon and the El-Nino. The Inter-Tropical Convergence Zone (ITCZ)is a continental scale phenamenon. It is in the ITCZ, many monsoon systems and tropical cyclones do occur. To study such phenomena under variable resolution method, high resolution is required in the entire tropical belt. Hitherto such a kind of mesh refinement strategies were not available in global spectral models. In this work, a new variable resolution method is developed that can help to study the tropical sub-scale phenomena with high resolution, in global spectral models.
A new conformal coordinate transformation named ’High resolution Tropical Belt Transformation(HTBT)’ is developed to generate high resolution in the entire tropical belt. Mathematical demonstrations are given to show the existence of additional conformal transformations available on the sphere, indicating additional degrees of freedom available to create variable resolution global spectral method.
Variable resolution global spectral method with high resolution over tropics is created through HTBT. The restriction imposed by Schmidt’s framework that the map-ping factor of the transformation need to have a finite-decomposition in the spectral space of the transformed domain is relaxed, by introduction of a new framework. The new framework uses transformed spherical harmonics Bnm as basis for spectral computations. With the use of FFT algorithm and Gaussian quadrature, the efficiency of the traditional spectral method is retained with the variable resolution global spectral method.
The newly defined basis functions Bnm are the eigenvalues of the transformed Laplacian operator . This property of Bnm provide an elegant direct solver for the transformed Helmholtz operator on the sphere. The transformed Helmholtz equations are solved accurately with the variable resolution method.
Advection experiments conducted with variable resolution spectral transport scheme on the HTBT variable grid produces near-dispersion free advection on the tropical belt. Transport across homogeneous resolution regions produce very less dispersion errors. Transport of a feature over the poles result in severe grid representation errors. It is shown that an increase in resolution around the poles greatly reduces this error. Transport of a feature from a point close to poles but not over it, does not produce such representation errors. Fourth-order Runge-Kutta scheme improves the accuracy of the transport scheme. The second order Magazenkov time-scheme proves to be better accurate than the leap-frog scheme with Asselin filter.
The non-divergent barotropic vorticity equation is tested with two exact solutions namely Rochas solution and Rossby-Haurwitz wave solutions. Each of the solution tests certain unique and contrasting characteristic of the system. The numerical behaviour of the solutions show non-linear interactions in them.
The singularity at the poles, arising due to the unbounded nature of the latitudinal derivative of the map factor of HTBT, triggers Gibbs phenomena for certain functions. However the recent advances in spectral methods, especially spectral viscosity method and Boyd-Vandeven filtering strategy provide ways to control the Gibbs oscillation and recover higher accuracy; make the variable resolution global spectral method viable for accurate meteorological computations.2010-11-18T18:30:00ZA Two Dimensional Plume In A Rotating FluidRaju, Jampana V Shttp://hdl.handle.net/2005/17292012-05-30T11:14:29Z2012-05-29T18:30:00ZTitle: A Two Dimensional Plume In A Rotating Fluid
Authors: Raju, Jampana V S2012-05-29T18:30:00ZTime Splitting Methods Applied To A Nonlinear Advective EquationShrivathsa, Bhttp://hdl.handle.net/2005/4172010-04-28T22:29:10Z2009-03-09T11:41:23ZTitle: Time Splitting Methods Applied To A Nonlinear Advective Equation
Authors: Shrivathsa, B
Abstract: Time splitting is a numerical procedure used in solution of partial diﬀerential equations whose solutions allow multiple time scales. Numerical schemes are split for handling the stiﬀness in equations, i.e. when there are multiple time scales with a few time scales being smaller than the others. When there are
such terms with smaller time scales, due to the Courant number restriction, the computational cost becomes high if these terms are treated explicitly.
In the present work a nonlinear advective equation is solved numerically using diﬀerent techniques based on a generalised framework for splitting methods.
The nonlinear advective equation was chosen because it has an analytical solution making comparisons with numerical schemes amenable and also because its nonlinearity mimics the equations encountered in atmospheric
modelling. Using the nonlinear advective equation as a test bed, an analysis of the splitting methods and their inﬂuence on the split solutions has been made.
An understanding of inﬂuence of splitting schemes requires knowledge of behaviour of unsplit schemes beforehand. Hence a study on unsplit methods has also been made.
In the present work, using the nonlinear advective equation, it shown that the three time level schemes have high phase errors and underestimate energy (even though they have a higher order of accuracy in time). It is also found that the leap-frog method, which is used widely in atmospheric modelling, is the worst among examined unsplit methods. The semi implicit method, again a popular splitting method with atmospheric modellers is the worst among examined split methods.
Three time-level schemes also need explicit ﬁltering to remove the computational mode. This ﬁltering can have a signiﬁcant impact on the obtained numerical solutions, and hence three-time level schemes appear to be
unattractive in the context of the nonlinear convective equation. Based on this experience, splitting methods for the two-time level schemes is proposed. These schemes realistically capture the phase and energy of the nonlinear advective equation.2009-03-09T11:41:23ZSurface Energy Budget Over A Land Surface In The TropicsArunchandra, S Chttp://hdl.handle.net/2005/5122010-04-28T23:42:06Z2009-05-21T07:01:12ZTitle: Surface Energy Budget Over A Land Surface In The Tropics
Authors: Arunchandra, S C
Abstract: Atmospheric convection is sensitive to the nature of the surface and its temperature. Both dry (without cloud) and moist (with cloud) convections depend on the surface temperature. Surface temperature is of critical importance in several practical applications like human comfort and crop cultivation. In the climate change scenario too, variations in the surface temperature take the center stage. Therefore, prediction of surface temperature is important. The evolution of the temperature is governed by the energy equation and the surface temperature by the surface energy balance. Important components of the surface energy balance are radiation (incoming solar radiation, reflected solar radiation, incoming and outgoing longwave radiation), sensible and latent heat fluxes and heat flux into the ground (called ground heat flux). A large number of individual and collective observations have been carried out in the past to understand the atmospheric boundary layer and the surface energy budgets. However a major share of the observations is from mid-latitudes. There have been few experiments carried out in India, for example, MONTBLEX, LASPEX, etc. One common drawback among these experiments is that the data time series is discontinuous and continuous measurements covering an entire season are lacking. Moreover these measurements were not comprehensive and hence did not allowed to calculate complete surface energy balance – in some cases radiation data is not available while in some humidity data. Therefore, continuous time series of sufficient duration and covering all variables needed to look at the seasonal energy balance based on measurements alone is missing in the Indian context. New programmes with the main objective of predicting convection are being planned in India. For example, PROWNAM (Prediction of Regional Weather with Observational Meso-Network and Atmospheric Modeling) is aimed at predicting the short term weather at SHAR and STORM (Severe Thunderstorms – Observations and Regional Modeling) aims to predict the occurrence of severe thunderstorms in the northeastern India. In both these programmes, measurement of all components of surface energy balance is one of the main objectives. However, the minimum configuration and data accuracy requirements for the flux towers, sensitivity of computed fluxes on data accuracy have not been carefully evaluated. This thesis is aimed at filling this gap.
As a part of my work, a 10 m high micrometeorological tower was installed in an open area within the Indian Institute of Science (IISc) Air Field. Temperature, relative humidity and wind speed and direction instruments were mounted at two levels, 2 m and 8 m. All components of radiation were measured. Data, sampled every 5 s and averaged for 2 minutes were continuously stored, starting May 2006 onwards. Soil temperature was measured at 4 depths, 5 cm, 10 cm, 15 cm and 20 cm. In addition, a sonic anemometer capable of measuring 3 components of velocity and air temperature was installed at 2 m height, and data was collected for more than a month to enable the calculation of momentum and buoyancy fluxes using the Eddy correlation method (ECM).
The present work evaluated the sensitivity of the fluxes for small calibration errors and quantified the minimum data accuracies and configuration needed for flux measurement with the Profile method (PM). After applying corrections, the comparison of fluxes from PM and ECM are in good agreement. The complete long-term surface energy balances is calculated in terms of source and sink. One aspect that emerges from the observation is that the seasonal variation in the sink term is relatively small (150-170 Wm-2) whereas the source term shows much larger variation from 180-250 Wm-2. A method has been implemented by which the ground surface temperature can be estimated by using the subsurface temperature timeseries by the method of Fourier decomposition and using the Fourier heat conduction equation. In addition we can compute the thermal diffusivity of the soil by using the amplitude and phase information of the sub-surface soil time series. The estimated temperatures from this method and one that estimated from radiation method are in good agreement with the maximum difference being less than 0º C.2009-05-21T07:01:12Z