In this thesis I study a variety of two-dimensional turbulent systems using a mixed analytical, phenomenological and numerical approach. The systems under consideration are governed by the two-dimensional Navier-Stokes (2DNS), surface quasigeostrophic (SQG), alpha-turbulence and magnetohydrodynamic (MHD) equations. The main analytical focus is on the number of degrees of freedom of a given system, defined as the least value $N$ such that all $n$-dimensional ($n$ ≥ $N$) volume elements along a given trajectory contract during the course of evolution. By equating $N$ with the number of active Fourier-space modes, that is the number of modes in the inertial range, and assuming power-law spectra in the inertial range, the scaling of $N$ with the Reynolds number $Re$ allows bounds to be put on the exponent of the spectrum. This allows the recovery of analytic results that have until now only been derived phenomenologically, such as the $k$[superscript(-5/3)] energy spectrum in the energy inertial range in SQG turbulence. Phenomenologically I study the modal interactions that control the transfer of various conserved quantities. Among other results I show that in MHD dynamo triads (those converting kinetic into magnetic energy) are associated with a direct magnetic energy flux while anti-dynamo triads (those converting magnetic into kinetic energy) are associated with an inverse magnetic energy flux. As both dynamo and anti-dynamo interacting triads are integral parts of the direct energy transfer, the anti-dynamo inverse flux partially neutralises the dynamo direct flux, arguably resulting in relatively weak direct energy transfer and giving rise to dynamo saturation. These theoretical results are backed up by high resolution numerical simulations, out of which have emerged some new results such as the suggestion that for alpha turbulence the generalised enstrophy spectra are not closely approximated by those that have been derived phenomenologically, and new theories may be needed in order to explain them.

In this thesis, parameterizations of non-linear interactions in quasigeostrophic (QG) flows for severely truncated models (STM) and Large Eddy Simulations (LES) are studied. Firstly, using Direct Numerical Simulations (DNS), atmospheric barotropic flows over topography are examined, and it is established that such flows exhibit multiple equilibrium states for a wide range of parameters. A STM is then constructed, consisting of the large scale zonal flow and a topographic mode. It is shown that, qualitatively, this system behaves similarly to the DNS as far as the interaction between the zonal flow and topography is concerned, and, in particular, exhibits multiple equilibrium states. By fitting the analytical form of the topographic stationary wave amplitude, obtained from the STM, to the results obtained from DNS, renormalized dissipation and rotation parameters are obtained. The usage of renormalized parameters in the STM results in better quantitative agreement with the DNS.¶ In the second type of problem, subgrid-scale parameterizations in LES are investigated with both atmospheric and oceanic parameters. This is in the context of two-level QG flows on the sphere, mostly, but not exclusively, employing a spherical harmonic triangular truncation at wavenumber 63 (T63) or higher. The methodology that is used is spectral, and is motivated by the stochastic representation of statistical closure theory, with the ‘damping’ and forcing covariance, representing backscatter, determined from the statistics of DNS. The damping and forcing covariance are formulated as 2 × 2 matrices for each wavenumber. As well as the transient subgrid tendency, the mean subgrid tendency is needed in the LES when the energy injection region is unresolved; this is also calculated from the statistics of the DNS. For comparison, a deterministic parameterization scheme consisting of 2×2 ‘damping’ parameters, which are calculated from the statistics of DNS, has been constructed. The main difference between atmospheric and oceanic flows, in this thesis, is that the atmospheric LES completely resolves the deformation scale, the energy and enstrophy injection region, and the truncation scale is spectrally distant from it, being well in the enstrophy cascade inertial range. In oceanic flows, however, the truncation scale is in the vicinity of the injection scale, at least for the parameters chosen, and is therefore not in an inertial range. A lower resolution oceanic LES at T15 is also examined, in which case the injection region is not resolved at all.¶ For atmospheric flows, it is found that, at T63, the matrix parameters are practically diagonal so that stratified atmospheric flows at these resolutions may be treated as uncoupled layers as far as subgrid-scale parameterizations are concerned. It is also found that the damping parameters are relatively independent of the (vertical) level, but the backscatter parameters are proportional to the subgrid flux in a given level. The stochastic and deterministic parameterization schemes give comparably good results relative to the DNS. For oceanic flows, it is found that the full matrix structure of the parameters must be used. Furthermore, it is found that there is a strong injection of barotropic energy from the subgrid scales, due to the unresolved, or partially resolved, baroclinic instability injection scales. It is found that the deterministic parameterization is too numerically unstable to be of use in the LES, and instead the stochastic parameterization must be used to obtain good agreement with the DNS. The subgrid tendency of the ensemble mean flow is also needed in some problems, and is found to reduce the available potential energy of the flow.

Two codes have been developed and implemented for use on massively parallelsuper computers to simulate two-dimensional and quasigeostrophic turbulence.The codes have been found to scale well with increasing resolution and width ofthe simulations. This has allowed for the highest resolution simulations of two-dimensional and quasigeostrophic turbulence so far reported in the literature.The direct numerical simulations have focused on the statistical characteristicsof turbulent cascades of energy and enstrophy, the role of coherent vorticesand departures from universal scaling laws, theoretized more than 40 yearsago. In particular, the investigations have concerned the enstrophy and energycascade in forced and decaying two-dimensional turbulence. Furthermore, theapplicability of Charney’s hypotheses on quasigeostrophic turbulence has beentested. The results have shed light on the flow evolution at very large Reynoldsnumbers. The most important results are the robustness of the enstrophycascade in forced and decaying two-dimensional turbulence, the unexpecteddependency on an infrared Reynolds number in the spectral scaling of theenergy spectrum in the inverse energy cascade, and the validation of Charney’spredictions on the dynamics of quasigeostrophic turbulence. It has also beenshown that the scaling of the energy spectrum in the enstrophy cascade isinsensitive to intermittency in higher order statistics, but that corrections mightapply to the ”universal” Batchelor-Kraichnan constant.

<p>Two codes have been developed and implemented for use on massively parallelsuper computers to simulate two-dimensional and quasigeostrophic turbulence.The codes have been found to scale well with increasing resolution and width ofthe simulations. This has allowed for the highest resolution simulations of two-dimensional and quasigeostrophic turbulence so far reported in the literature.The direct numerical simulations have focused on the statistical characteristicsof turbulent cascades of energy and enstrophy, the role of coherent vorticesand departures from universal scaling laws, theoretized more than 40 yearsago. In particular, the investigations have concerned the enstrophy and energycascade in forced and decaying two-dimensional turbulence. Furthermore, theapplicability of Charney’s hypotheses on quasigeostrophic turbulence has beentested. The results have shed light on the flow evolution at very large Reynoldsnumbers. The most important results are the robustness of the enstrophycascade in forced and decaying two-dimensional turbulence, the unexpecteddependency on an infrared Reynolds number in the spectral scaling of theenergy spectrum in the inverse energy cascade, and the validation of Charney’spredictions on the dynamics of quasigeostrophic turbulence. It has also beenshown that the scaling of the energy spectrum in the enstrophy cascade isinsensitive to intermittency in higher order statistics, but that corrections mightapply to the ”universal” Batchelor-Kraichnan constant.</p>

This thesis presents a numerical exploration of the dynamics governing rotating flow driven by a surface stress in the " sliced cylinder " model of Pedlosky & Greenspan (1967) and Beardsley (1969), and its close relative, the " sliced cone " model introduced by Griffiths & Veronis (1997). The sliced cylinder model simulates the barotropic wind-driven circulation in a circular basin with vertical sidewalls, using a depth gradient to mimic the effects of a gradient in Coriolis parameter. In the sliced cone the vertical sidewalls are replaced by an azimuthally uniform slope around the perimeter of the basin to simulate a continental slope. Since these models can be implemented in the laboratory, their dynamics can be explored by a complementary interplay of analysis and numerical and laboratory experiments. ¶ In this thesis a derivation is presented of a generalised quasigeostrophic formulation which is valid for linear and moderately nonlinear barotropic flows over large-amplitude topography on an f-plane, yet retains the simplicity and conservation properties of the standard quasigeostrophic vorticity equation (which is valid only for small depth variations). This formulation is implemented in a numerical model based on a code developed by Page (1982) and Becker & Page (1990). ¶ The accuracy of the formulation and its implementation are confirmed by detailed comparisons with the laboratory sliced cylinder and sliced cone results of Griffiths (Griffiths & Kiss, 1999) and Griffiths & Veronis (1997), respectively. The numerical model is then used to provide insight into the dynamics responsible for the observed laboratory flows. In the linear limit the numerical model reveals shortcomings in the sliced cone analysis by Griffiths & Veronis (1998) in the region where the slope and interior join, and shows that the potential vorticity is dissipated in an extended region at the bottom of the slope rather than a localised region at the east as suggested by Griffiths & Veronis (1997, 1998). Welander's thermal analogy (Welander, 1968) is used to explain the linear circulation pattern, and demonstrates that the broadly distributed potential vorticity dissipation is due to the closure of geostrophic contours in this geometry. ¶ The numerical results also provide insight into features of the flow at finite Rossby number. It is demonstrated that separation of the western boundary current in the sliced cylinder is closely associated with a " crisis " due to excessive potential vorticity dissipation in the viscous sublayer, rather than insufficient dissipation in the outer western boundary current as suggested by Holland & Lin (1975) and Pedlosky (1987). The stability boundaries in both models are refined using the numerical results, clarifying in particular the way in which the western boundary current instability in the sliced cone disappears at large Rossby and/or Ekman number. A flow regime is also revealed in the sliced cylinder in which the boundary current separates without reversed flow, consistent with the potential vorticity " crisis " mechanism. In addition the location of the stability boundary is determined as a function of the aspect ratio of the sliced cylinder, which demonstrates that the flow is stabilised in narrow basins such as those used by Beardsley (1969, 1972, 1973) and Becker & Page (1990) relative to the much wider basin used by Griffiths & Kiss (1999). ¶ Laboratory studies of the sliced cone by Griffiths & Veronis (1997) showed that the flow became unstable only under anticyclonic forcing. It is shown in this thesis that the contrast between flow under cyclonic and anticyclonic forcing is due to the combined effects of the relative vorticity and topography in determining the shape of the potential vorticity contours. The vorticity at the bottom of the sidewall smooths out the potential vorticity contours under cyclonic forcing, but distorts them into highly contorted shapes under anticyclonic forcing. In addition, the flow is dominated by inertial boundary layers under cyclonic forcing and by standing Rossby waves under anticyclonic forcing due to the differing flow direction relative to the direction of Rossby wave phase propagation. The changes to the potential vorticity structure under strong cyclonic forcing reduce the potential vorticity changes experienced by fluid columns, and the flow approaches a steady free inertial circulation. In contrast, the complexity of the flow structure under anticyclonic forcing results in strong potential vorticity changes and also leads to barotropic instability under strong forcing. ¶ The numerical results indicate that the instabilities in both models arise through supercritical Hopf bifurcations. The two types of instability observed by Griffiths & Veronis (1997) in the sliced cone are shown to be related to the western boundary current instability and " interior instability " identified by Meacham & Berloff (1997). The western boundary current instability is trapped at the western side of the interior because its northward phase speed exceeds that of the fastest interior Rossby wave with the same meridional wavenumber, as discussed by Ierley & Young (1991). ¶ Numerical experiments with different lateral boundary conditions are also undertaken. These show that the flow in the sliced cylinder is dramatically altered when the free-slip boundary condition is used instead of the no-slip condition, as expected from the work of Blandford (1971). There is no separated jet, because the flow cannot experience a potential vorticity " crisis " with this boundary condition, so the western boundary current overshoots and enters the interior from the east. In contrast, the flow in the sliced cone is identical whether no-slip, free-slip or super-slip boundary conditions are applied to the horizontal flow at the top of the sloping sidewall, except in the immediate vicinity of this region. This insensitivity results from the extremely strong topographic steering near the edge of the basin due to the vanishing depth, which demands a balance between wind forcing and Ekman pumping on the upper slope, regardless of the lateral boundary condition. The sensitivity to the lateral boundary condition is related to the importance of lateral friction in the global vorticity balance. The integrated vorticity must vanish under the no-slip condition, so in the sliced cylinder the overall vorticity budget is dominated by lateral viscosity and Ekman friction is negligible. Under the free-slip condition the Ekman friction assumes a dominant role in the dissipation, leading to a dramatic change in the flow structure. In contrast, the much larger depth variation in the sliced cone leads to a global vorticity balance in which Ekman friction is always dominant, regardless of the boundary condition.