Two-level Quasi-geostrophic Model Research Articles

Indian summer monsoon cyclogenesis and its variability

MONSOON disturbances forming over the north Bay of Bengal and moving towards the WNW across northern India are the main rain producers of the Indian summer monsoon. There are some (monsoon) seasons when these disturbances fail to develop in adequate number and some that do form move towards the northwest or north—these are periods of drought. Understanding how monsoon disturbances develop and why they sometimes do not is an interesting scientific and an important economic problem. Studies so far on monsoon cyclogenesis1,2 have shown that the monsoon atmosphere is not baroclinically unstable. Shukla1 obtained barotropically unstable upper tropospheric modes and emphasised the role of the conditional instability of the second kind (CISK) but failed to get a preferred scale of maximum growth for lower tropospheric disturbances. R.N.K. et al.2 found slowly growing barotropically unstable modes in the lower troposphere corresponding to monsoon disturbances. Mak3 studied the baroclinic instability of monsoon zonal flow with a superposed meridional component, but the meridional winds required for instability were large compared to observed winds. Lorenz4 studied the barotropic instability of Rossby waves and found that shorter waves can be barotropically unstable. V.S. et al.5 showed that the superimposition of a long stationary wave, of the scale of wave number two, on monsoon zonal flow yielded fast growing upper tropospheric modes and lower tropospheric modes of scale 2,500 km, a doubling time of 5 days and slow westwards movement. These correspond well to the observed monsoon disturbances. We report here a combined barotropic–baroclinic stability analysis of monsoon zonal flow with a superimposed long stationary wave (global monsoon) using a two-level quasi-geostrophic model. We show that the phase of the global monsoon wave is a factor in deciding whether a particular year is good for monsoons or for drought.

Seasonal variation and latitudinal distribution of the instability of two-level quasi-geostrophic waves in horizontal shear

The instability of a simplified, combined baroclinic-barotropic, two-level quasi-geostrophic model is investigated. The results are applied to the observed mean zonal wind, which varies with season and latitude, by approximating the sphericity of the earth by infinite tangent ?-planes. Maximum instability is found to exist in winter at about 37° N, for zonal wave number 7, changing to zonal wave number 6 in higher middle latitudes and to lower zonal wave numbers in polar latitudes. The latitudinal and spectral bands of instability shrink gradually from winter through spring, reach a minimum in summer, and gradually widen in fall. DOI: 10.1111/j.2153-3490.1973.tb00608.x

Numerical simulation of quasi-geostrophic turbulence

The development of turbulence is studied numerically through the use of a simplified two-level quasi-geostrophic model. The domain occupied by the fluid has 64 times 64 grid points in the horizontal and obeys cyclic boundary conditions in the x - and y -directions. The turbulence is generated by a heating function composed of the 2-dimensional fourier components having the larger wave number, m = 8. Three cases are investigated for two values of H , the amplitude of the heating function and two values of v , the coefficient of eddy viscosity. Quasistationary spectra are obtained in each case. The slopes of the spectra for 8 ? m ? 16 on a log-log graph vary from ?1.7 to ?3.4 depending on v and H . The nonlinear cascade of available potential energy is found to be toward larger wavenumbers. The nonlinear cascade of kinetic energy is from components for which 8 ? m ? 14, mainly toward 1 ? m ? 7, the larger scales. The gains for m > 14 resulting from the cascade is relatively small. These transfers are in qualitative agreement with atmospheric observations. Gains of enstrophy are observed to be fairly constant for 14 ? m ? 28. In an inertial subrange they should be zero. Relatively small gains are recorded for m < 7 while the spectral region 8 ? m ? 14 acts as source. Except for the wavenumber of transition from negative to positive values the distribution of losses and gains of enstrophy due to the inertial cascade is similar to that resulting from calculations based on atmospheric observations. It is concluded that the two dimensional turbulence theories enunciated by Fjortoft, Kraichnan and Leith pertain to the model in this study and the atmosphere (so far as our model resembles the atmosphere) in the prediction of the dominant direction of the nonlinear cascades of kinetic energy and enstrophy in wavenumber space. Inertial subranges in which the cascades result in zero changes are neither observed in the numerical experiments of this study nor in calculations based on atmospheric observations in the spectral region 8 ? m ? 16. The experiments described in this paper were performed prior to the theory proposed by Charney and so were not designed to be a test of it. However, some assumptions and conclusions are discussed in the light of that theory. DOI: 10.1111/j.2153-3490.1973.tb00607.x

Numerical simulation of quasi-geostrophic turbulence

The development of turbulence is studied numerically through the use of a simplified two-level quasi-geostrophic model. The domain occupied by the fluid has 64 times 64 grid points in the horizontal and obeys cyclic boundary conditions in the x - and y -directions. The turbulence is generated by a heating function composed of the 2-dimensional fourier components having the larger wave number, m = 8. Three cases are investigated for two values of H , the amplitude of the heating function and two values of v , the coefficient of eddy viscosity. Quasistationary spectra are obtained in each case. The slopes of the spectra for 8 ? m ? 16 on a log-log graph vary from ?1.7 to ?3.4 depending on v and H . The nonlinear cascade of available potential energy is found to be toward larger wavenumbers. The nonlinear cascade of kinetic energy is from components for which 8 ? m ? 14, mainly toward 1 ? m ? 7, the larger scales. The gains for m > 14 resulting from the cascade is relatively small. These transfers are in qualitative agreement with atmospheric observations. Gains of enstrophy are observed to be fairly constant for 14 ? m ? 28. In an inertial subrange they should be zero. Relatively small gains are recorded for m < 7 while the spectral region 8 ? m ? 14 acts as source. Except for the wavenumber of transition from negative to positive values the distribution of losses and gains of enstrophy due to the inertial cascade is similar to that resulting from calculations based on atmospheric observations. It is concluded that the two dimensional turbulence theories enunciated by Fjortoft, Kraichnan and Leith pertain to the model in this study and the atmosphere (so far as our model resembles the atmosphere) in the prediction of the dominant direction of the nonlinear cascades of kinetic energy and enstrophy in wavenumber space. Inertial subranges in which the cascades result in zero changes are neither observed in the numerical experiments of this study nor in calculations based on atmospheric observations in the spectral region 8 ? m ? 16. The experiments described in this paper were performed prior to the theory proposed by Charney and so were not designed to be a test of it. However, some assumptions and conclusions are discussed in the light of that theory. DOI: 10.1111/j.2153-3490.1973.tb00607.x

EFFECT OF SURFACE FRICTION ON THE STRUCTURE OF BAROTROPICALLY UNSTABLE TROPICAL DISTURBANCES

The structure of barotropically unstable disturbances in the Tropics is studied with a two-level quasi-geostrophic model. An Ekman layer is attached to the lower boundary. The equations are linearized and the most unstable mode is found numerically by using the initial value technique. Computations are made for a shear-zone wind profile and a jet profile; these fields are independent of height. The disturbance structure is found to be critically dependent on the absolute vorticity gradient in the mean flow. The predicted disturbance structures contain a number of features that are observed in tropical wave disturbances.

A numerical integration of a two-level quasi-geostrophic model with variable Rossby parameter

Circulation patterns were obtained from a two-level quasi-geostrophic model, in which the Rossby parameter varied with latitude. The initial basic state was characterized by a constant zonal flow at both reference level. The results we re compared with the pattern obtained in a series of experiments with a similar model, in which the Rossby parameter was assumed to be constant. the initial zonal flow in these experiments was assumed in one case to be constant with latitude, in the second casse to have a symmetrical jet centered on the central latitude, and in the thrid case to have its maximum in the southern half of the region. the results of the experiment with a variable Rossby parameter have the closest resemblance to the patterns observed in the real atmospehre. In particular, during the mature stages of the run, the distribution of the zonallyaveraged eddytransport of momentum corresponded quite well to patterns actually observed in the atmosphere.

The developing circulation patterns on a β-plane obtained in the case of an initial basic flow varying sinusoidally with latitude

Circulation patterns were obtained from a two-level quasigeostrophic model and an initial state characterized by a sinusoidal latitudinal variation of the zonal flow. The results were compared to a similar numerical experiment with an initial constant zonal flow.

Numerical integration of a quasi-geostrophic atmospheric model with an asymmetric zonal current

A two-level quasi-geostrophic model on a β-plane, with an initial state characterized by an asymmetric zonal current was integrated numerically for a period of 14 days. Various field variables were computed and compared with the corresponding distributions obtained in two experiments with the same model and with initial states characterized by a constant zonal flow and by a flow varying sinusoidally with latitude.

The Effect of Variable Coriolis Parameter in a Planetary Circulation Generated by a Two-Parameter Quasigeostrophic Model

Abstract The governing equations of a two-level quasigeostrophic model were integrated twice for periods of about 10 days, once including variation of the Coriolis parameter (the β term) and once excluding it. As expected, the exclusion of the β-term resulted in a quicker rate of development. The evolving perturbation had almost the same pattern in the two cases, but the final distribution of the mean zonal winds was different. In the β=0 case the zonal velocities in the central region tended towards a barotropic pattern, while a baroclinic jet was formed in the case where β≠0. The numerical integration is also compared with linear theory.