Density fronts are ubiquitous features of the ocean and atmosphere boundary layers. Boundary layers are characterised by strong surface fluxes of heat, water and momentum, and exhibit intense eddy fields that are associated with strong horizontal strains. Such boundary layer phenomena can drive the generation and sharpening of frontal density gradients in a process known as frontogenesis. Analytic models of frontogenesis have typically employed the two-dimensional front' configuration where the density front is assumed to be infinitely long and straight, such that gradients along the front may be neglected, and the mathematical problem reduced to two spatial dimensions. Hoskins and Bretherton (1972) used this configuration to demonstrate how a weak background strain flow, associated with a large scale weather system, can drive the collapse of a boundary front to a discontinuity in the inviscid equations in finite time. More recently, Blumen (2000) has used the same configuration to demonstrate how an unbalanced initial state --- associated with a rapidly applied boundary flux --- can trigger an adjustment process which drives frontogenesis on the boundary. These two types of frontogenesis are known as forced' and `spontaneous', respectively. Forced and spontaneous frontogenesis have typically been studied in isolation, despite it being well established that they can and do occur simultaneously. Furthermore, neither the Hoskins and Bretherton (1972) nor Blumen (2000) models include propagating inertia-gravity waves, despite recent observations and numerical simulations showing that these waves are often generated during active frontogenesis.

Here we formulate a generalised mathematical model for the classical two-dimensional density front subject to a simple background strain flow, as studied by Hoskins and Bretherton (1972) . This model firstly unifies the disparate frontogenesis theories of Hoskins and Bretherton (1972) and Blumen (2000). Secondly, the model extends these theories by permitting arbitrary initial conditions, stratification and strong strains. Thirdly, the model incorporates non-hydrostatic effects and unbounded domains. An important novel feature of the model is the accurate description of inertia-gravity wave generation during frontogenesis. We show that these waves can be generated both by the geostrophic adjustment of initial imbalances in a stratified ambient, and spontaneously due to the acceleration of the strain flow around the front. The generalised model thus provides a unified theory capable of describing frontogenesis and wave generation in the atmosphere and ocean boundary layers on a vast range of scales. In particular, the inclusion of strong strains permits the description of frontogenesis on the ocean submesoscale. The predictions of the generalised model are confirmed by comparison with a suite of fully non-linear numerical simulations.