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# Numerical simulations of evolving atmospheric vortices using 'tangent plane' approximations

Numerical simulations of evolving atmospheric phenomena are considered. The height of the vortices is small with respect to their width and depending on the atmospheric phenomenon being considered can have a diameter of hundreds if not thousands of kilometres. They can therefore be thought of as large flat structures in a shallow atmosphere. A weakly compressible atmosphere is assumed for both Earth-bound and Saturn simulations. The atmospheric fluid motion is subject to the Coriolis pseudo-force, due to atmospheres being in a non-inertial rotating reference frame. The simulations involve reducing the fully spherical nature of the atmosphere to a localised region, so that the commonly used 'tangent plane' approximations apply. The advantage of using 'tangent plane' approximations is that necessary spheroidal effects can be studied using Cartesian based equations. Three types of 'tangent plane' approximations are used, (i) the f-plane, where the Coriolis parameter is assumed constant over the entire region; (ii) the ˜í‚â§-plane, where the Coriolis parameter varies linearly with latitude and (iii) the ˜í¬•-plane, which is a high latitude approximation where quadratic effects of the Coriolis term are accounted for. Large-scale low-pressure systems in the atmosphere are occasionally observed to possess Kelvin-Helmholtz fingers spiralling outwards, and an example is shown in this thesis. However, these structures are hundreds of kilometres long, so that they are necessarily affected strongly by non-linearity. They are evidently unstable and are commonly observed to dissipate after a few hours, and in rare cases may last for days. A model for this phenomenon is presented in this thesis, based on the usual f-plane equations of meteorology, assuming an atmosphere governed by the ideal gas law. Large-amplitude perturbations are accounted for, by retaining the equations in their non-linear forms, and these are then solved numerically using a spectral method. Finger formation is modelled as an initial perturbation to the nth Fourier mode, and the numerical results show that the fingers grow in time, developing structures that depend on the particular mode. Results are compared with predictions of the ˜í‚â§-plane theory and there is close alignment with f-plane results at mid-latitudes. An idealized vortex in the northern hemisphere is considered, but the results are at least in qualitative agreement with an observation of a system in the southern hemisphere. Vortices in the atmosphere are rarely observed to be singular entities. Thus the non-linear behaviour of interacting mid-latitude vortices is also investigated. The vortices studied are coupled binary systems and the high- or low-pressure in each vortex is modelled initially using an exponential function. Non-linear results in the f-plane approximation are discussed at mid-latitudes. It is found that the vortices do or do not interact, depending on their initial radii and the location of their centres. A scaling law is found numerically for the ratio of these two quantities, which determines whether interaction does occur at the approximate mid-latitude 43¬¨‚àûN. An approximate rule has been developed, to generalize the scaling law to other latitudes. Atmospheric vortices are rarely circular structures and have been observed to have a definite polygonal form. Saturn's North Polar Hexagon is an example of such a vortex, and was discovered by Godfrey [31] who pieced together map projections of images captured by the Voyager mission to unveil a hexagonal structure over the north pole of Saturn. This thesis attempts to answer whether or not a hexagonal structure can be formed through anti-cyclones impinging on the dominant eastward circumpolar flow and is in part based upon the proposed theory by Allison et al. [1] that the Hexagon may be the result of at least one impinging anti-cyclone perturbing a circumpolar jet centrally located around the 76¬¨‚àûN latitude. A high-latitude ˜í¬•-plane approximation is used to simulate the interaction between an initially circular circumpolar jet and at least one perturbing anti-cyclone. The simulations with one perturbing anti-cyclone failed to form a hexagonal structure; yet by including an additional anti-cyclone it was found that depending on the strength, location and radius of the perturbing anti-cyclones a hexagonal feature could develop. However, the longevity and drift rate of the actual Hexagon must be attributed to other factors not considered in this thesis.

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Copyright 2017 the author Chapters 1 and 4 appear to be, in part, the equivalent of a post-print version of an article published as: Cosgrove, J. M., Forbes, L. K., 2016. The formation of large-amplitude fingers in atmospheric vortices, ANZIAM journal, 57(4), 395-416 Chapters 1 and 5 appear to be, in part, the equivalent of a post-peer-review, pre-copyedit version of an article published in Journal of engineering mathematics. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10665-016-9871-9 Chapter 6 appears to be the equivalent of a pre-print version of an article that article has been accepted for publication in Monthly notices of the Royal Astronomical Society, published by Oxford University Press on behalf of the Royal Astronomical Society. The version of record, Cosgrove, J. M., Forbes, L. K. 2017. A ˜í¬•-plane simulation of anticyclones perturbing circumpolar flows to form a transient north polar hexagon, Monthly notices of the Royal Astronomical Society, 469(4), 4133‚Äö-4147, is available online at: https://doi.org/10.1093/mnras/stx1170## Repository Status

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