The Korteweg-de Vries (KdV) equation is widely recognized as a simple model for unidirectional weakly nonlinear dispersive waves on the surface of a shallow body of fluid. While solutions of the KdV equation describe the shape of the free surface, information about the underlying fluid flow is encoded into the derivation of the equation, and the present article focuses on the formulation of mass, momentum and energy balance laws in the context of the KdV approximation. The densities and the associated fluxes appearing in these balance laws are given in terms of the principal unknown variable η representing the deflection of the free surface from rest position. The formulae are validated by comparison with previous work on the steady KdV equation. In particular, the mass flux, total head and momentum flux in the current context are compared to the quantities Q, R and S used in the work of Benjamin and Lighthill (Proc. R. Soc. Lond. A 224:448-460, 1954) on cnoidal waves and undular bores.

Depth-integrated long-wave models, such as the shallow-water and Boussinesq equations, are standard fare in the study of small amplitude surface waves in shallow water. While the shallow-water theory features conservation of mass, momentum and energy for smooth solutions, mechanical balance equations are not widely used in Boussinesq scaling, and it appears that the expressions for many of these quantities are not known. This work presents a systematic derivation of mass, momentum and energy densities and fluxes associated with a general family of Boussinesq systems. The derivation is based on a reconstruction of the velocity field and the pressure in the fluid column below the free surface, and the derivation of differential balance equations which are of the same asymptotic validity as the evolution equations. It is shown that all these mechanical quantities can be expressed in terms of the principal dependent variables of the Boussinesq system: the surface excursion η and the horizontal velocity w at a given level in the fluid.

The energy loss in the shallow-water theory for an undular bore is thought to be due to oscillations that carry away the energy lost at the front of the bore. Using a higherorder dispersive model equation, this expectation is confirmed through a quantitative study which shows that there is no energy loss if dispersion is accounted for.

a b s t r a c tMonitoring of the marine environment for indications of a leak, or precursors of a leak, will be an intrinsic part of any subsea CO 2 storage projects. A real challenge will be quantification of the probability of a given monitoring program to detect a leak and to design the program accordingly. The task complicates by the number of pathways to the surface, difficulties to estimate probabilities of leaks and fluxes, and predicting the fluctuating footprint of a leak. The objective is to present a procedure for optimizing the layout of a fixed array of chemical sensors on the seafloor, using the probability of detecting a leak as metric. A synthetic map from the North Sea is used as a basis for probable leakage points, while the spatial footprint is based on results from a General Circulation Model. Compared to an equally spaced array the probability of detecting a leak can be nearly doubled by an optimal placement of the available sensors. It is not necessarily best to place the first in the location of the highest probable leakage point, one sensor can monitor several potential leakage points. The need for a thorough baseline in order to reduce the detection threshold is shown.

Risk‐based monitoring requires quantification of the probability of the design to detect the potentially adverse events. A component in designing the monitoring program will be to predict the varying signal caused by an event, here detection of a gas seep through the seafloor from an unknown location. The Bergen Ocean Model (BOM) is used to simulate dispersion of CO2 leaking from different locations in the North Sea, focusing on temporal and spatial variability of the CO2 concentration. It is shown that the statistical footprint depends on seep location and that this will have to be accounted for in designing a network of sensors with highest probability of detecting a seep. As a consequence, heterogeneous probabilistic predictions of CO2 footprints should be available to subsea geological CO2 storage projects in order to meet regulations.