Recent Advances Concerning Certain Class of Geophysical Flows

This paper is devoted to reviewing several recent developments concerning certain class of geophysical models, including the primitive equations (PEs) of atmospheric and oceanic dynamics and a tropical atmosphere model. The PEs for large-scale oceanic and atmospheric dynamics are derived from the Navier-Stokes equations coupled to the heat convection by adopting the Boussinesq and hydrostatic approximations, while the tropical atmosphere model considered here is a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture. We are mainly concerned with the global well-posedness of strong solutions to these systems, with full or partial viscosity, as well as certain singular perturbation small parameter limits related to these systems, including the small aspect ratio limit from the Navier-Stokes equations to the PEs, and a small relaxation-parameter in the tropical atmosphere model. These limits provide a rigorous justification to the hydrostatic balance in the PEs, and to the relaxation limit of the tropical atmosphere model, respectively. Some conditional uniqueness of weak solutions, and the global well-posedness of weak solutions with certain class of discontinuous initial data, to the PEs are also presented.Comment: arXiv admin note: text overlap with arXiv:1507.0523

Initial-boundary value problem for 2D temperature-dependent tropical climate model

It is well known that the tropical climate model is an important model to describe the interaction of large scale flow fields and precipitation in the tropical atmosphere. In this paper, we address the issue of global well-posedness for 2D temperature-dependent tropical climate model in a smooth bounded domain. Through classical energy estimates and De Giorgi-Nash-Moser iteration method, we obtain the global existence and uniqueness of strong solution in classical energy spaces. Compared with Cauchy problem, we establish more delicate a priori estimates with exponential decay rates. To the best of our knowledge, this is the first result concerning the global well-posedness for the initial-boundary value problem in 2D tropical climate model.Comment: 20 page

Upwards Tropospheric Influence on Tropical Stratospheric Upwelling

The response of the stratosphere to a steady geopotential forcing is considered in two separate theoretical models. Solutions to the linearized quasi-geostrophic potential vorticity equations are first used to show that the vertical length scale of a tropopause geopotential anomaly is initially shallow, but significantly increased by diabatic heating from radiative relaxation. This process is deemed as geostrophic adjustment of the stratosphere to tropospheric forcing. Idealized, time-dependent calculations show that tropopause geopotential anomalies can appreciably rise in the stratosphere on time scales of a couple months. A previously developed, coupled troposphere-stratosphere model is introduced and modified to further understand how tropospheric geopotential forcing can induce upwelling in the stratosphere. Solutions to steady, zonally-symmetric sea-surface-temperature forcings in the linear $\beta$-plane model show that the upwards stratospheric penetration of the thermally induced tropopause geopotential anomaly is controlled by a non-dimensional parameter that depends on the ratio between the time scale of wave-drag to that of radiation. It is also shown that the horizontal scale of the tropopause geopotential anomaly modulates the vertical scale of the anomaly. When Earth-like non-dimensional parameters are used, the theoretical model predicts stratospheric temperature anomalies around two times larger in magnitude than those in the boundary layer, approximately in line with observational data. The results are argued to show that wave-drag alone may not suffice to explain certain observed features of the lower stratosphere, foremost of which is the anti-correlation between sea-surface temperature and lower stratospheric temperature

Stratospheric Modulation of the MJO through Cirrus Cloud Feedbacks

Recent observations have indicated significant modulation of the Madden Julian Oscillation (MJO) by the phase of the stratospheric Quasi-Biennial Oscillation (QBO) during boreal winter. Composites of the MJO show that upper tropospheric ice cloud fraction and water vapor anomalies are generally collocated, and that an eastward tilt with height in cloud fraction exists. Through radiative transfer calculations, it is shown that ice clouds have a stronger tropospheric radiative forcing than do water vapor anomalies, highlighting the importance of incorporating upper tropospheric/lower stratospheric processes into simple models of the MJO. The coupled troposphere-stratosphere linear model previously developed by the authors is extended by including a mean wind in the stratosphere and a prognostic equation for cirrus clouds, which are forced dynamically and allowed to modulate tropospheric radiative cooling, similar to the effect of tropospheric water vapor in previous formulations. Under these modifications, the model still produces a slow, eastward propagating mode that resembles the MJO. The sign of zonal mean wind in the stratosphere is shown to control both the upward wave propagation and tropospheric vertical structure of the mode. Under varying stratospheric wind and interactive cirrus cloud radiation, the MJO-like mode has weaker growth rates under stratospheric westerlies than easterlies, consistent with the observed MJO-QBO relationship. These results are directly attributable to an enhanced barotropic mode under QBO easterlies. It is also shown that differential zonal advection of cirrus clouds leads to weaker growth rates under stratospheric westerlies than easterlies. Implications and limitations of the linear theory are discussed

The relationship between sea surface temperature anomalies and atmospheric circulation in GCM experiments

Several 19-year integrations of the Hamburg version of the ECMWF/T21 general circulation model driven by the monthly mean sea surface temperature (SST) observed in 1970-1988 were examined to study extratropical response of the atmospheric circulation to SST anomalies in the Northern Hemisphere in winter. In the first 19-years run SST anomalies were prescribed globally (GAGO run), and in two others SST monthly variability was limited to extratropical regions (MOGA run) and to tropics (TOGA run), respectively. A canonical correlation analysis (CCA), which select from two time-dependent fields optimally correlated pairs of patterns, was applied to monthly anomalies of SST in the North Alantic and Pacific Oceans and monthly anomalies of sea level pressure and 500 hPa geopotential height in the Northern Hemisphere. In the GAGO run the best correlated atmospheric pattern is global and is characterized by north-south dipole structures of the same polarity in the North Atlantic and the North Pacific sectors. In the MOGA and TOGA experiments the atmospheric response is more local than in the GAGO run with main centers in the North Atlantic and North Pacific, respectively. The extratropical response in the GAGO run is not equal to the sum of the responses in the MOGA and TOGA runs. The artificial meridional SST gradients at 25 degrees 30 degrees N probably influence the results of the MOGA and TOGA runs. The atmopsheric modes found by the CCA were compared with the normal modes of the barotropic vorticity equation linearized about the 500 hPa. winter climate. The normal modes with smallest eigenvalues are similar to the model leading variability modes and canonical patterns of 500 hPa geopotential height. The corresponding eigenvectors of the adjoint operator, which represent an external forcing optimal for exciting normal modes, have a longitudinal structure with maxima in regions characterized by enhanced high frequency baroclinic activity over both oceans

Atmospheric planetary wave response to external forcing

The tools of observational analysis, complex general circulation modeling, and simpler modeling approaches were combined in order to attack problems on the largest spatial scales of the earth's atmosphere. Two different models were developed and applied. The first is a two level, global spectral model which was designed primarily to test the effects of north-south sea surface temperature anomaly (SSTA) gradients between the equatorial and midlatitude north Pacific. The model is nonlinear, contains both radiation and a moisture budget with associated precipitation and surface evaporation, and utilizes a linear balance dynamical framework. Supporting observational analysis of atmospheric planetary waves is briefly summarized. More extensive general circulation models have also been used to consider the problem of the atmosphere's response, especially in the horizontal propagation of planetary scale waves, to SSTA

Mathematical Theory and Modelling in Atmosphere-Ocean Science

Mathematical theory and modelling in atmosphere-ocean science combines a broad range of advanced mathematical and numerical techniques and research directions. This includes the asymptotic analysis of multiscale systems, the deterministic and stochastic modelling of sub-grid-scale processes, and the numerical analysis of nonlinear PDEs over a broad range of spatial and temporal scales. This workshop brought together applied mathematicians and experts in the disciplinary fields of meteorology and oceanography for a wide-ranging exchange of ideas and results in this area with the aim of fostering fundamental interdisciplinary work in this important science area