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We describe a framework for the simultaneous estimation of model parameters in a partial differential equation using sparse observations. Markov Chain Monte Carlo sampling is used in a Bayesian framework to estimate posterior probability distributions for each parameter. We describe the necessary components of this approach and its broad potential for application in models of unsteady processes. The framework is applied to three case studies, of increasing complexity, from the field of cohesive sediment transport. We demonstrate that the framework can be used to recover posterior distributions for all parameters of interest and the results agree well with independent estimates (where available). We also demonstrate how the framework can be used to compare different model parameterizations and provide information on the covariance between model parameters. Plain Language Summary: We describe a framework for the simultaneous estimation of multiple unobserved parameters by combining observations of a tracer with a numerical model. This framework uses Bayesian inference techniques established in statistical literature to estimate the unobserved parameters of interest used in the model with uncertainty quantification. We explain the key components of this framework in simple terms to encourage its use for analyzing other unsteady processes and performing quantitative inference on parameters that are difficult or impossible to measure directly. We then demonstrate the framework's efficacy by applying it to three case studies from the field of cohesive sediment transport that all use the transport equation (advection‐diffusion). Inferred parameter values show good agreement with independent estimates, where available. Key Points: Probabilistic framework to estimate unobserved erosion model parameters using sparse measurements collected above the seabedGeneral approach can be updated with any model parameterization and quantitatively comparedThe framework is applicable to many similar data sets with both unsteady or quasi‐steady forcing and response [ABSTRACT FROM AUTHOR]

We present a statistical method for reconstructing continuous background density profiles that embeds incomplete measurements and a physically intuitive density stratification model within a Bayesian hierarchal framework. A double hyperbolic tangent function is used as a parametric density stratification model that captures various pycnocline structures in the upper ocean and offers insight into several density profile characteristics (e.g., pycnocline depth). The posterior distribution is used to quantify uncertainty and is estimated using recent advances in Markov chain Monte Carlo sampling. Temporally evolving posterior distributions of density profile characteristics, isopycnal heights, and nonlinear ocean process models for internal gravity waves are presented as examples of how uncertainty propagates through models dependent on the density stratification. The results show 0.95 posterior interval widths that ranged from 2.5% to 4% of the expected values for the linear internal wave phase speed and 15%–40% for the nonlinear internal wave steepening parameter. The data, collected over a year from a through-the-column mooring, and code, implemented in the software package Stan, accompany the article. [ABSTRACT FROM AUTHOR]

Non‐linear internal waves (NLIW) are important to processes such as heat transfer, nutrient replenishment and sediment transport on continental shelves. Our unique field observations of shoaling NLIW of elevation revealed a variety of different wave shapes, varying from relatively symmetric waves, to waves with either steepened leading‐ or trailing‐faces; many had evidence of trapped cores. The wave shape was related to the position of maximum density overturns and diapycnal mixing. We observed both shear (where sheared currents overcome the stabilizing effects of stratification) and convective (where the local velocity exceeds the wave propagation speed) instabilities. The elevated diapycnal mixing (>10−3 m2s−1) and heat flux (>500 Wm−2) were predominantly local to the NLIW of elevation packets, and were transported onshore 10s kilometers with the wave packets. We demonstrate that wave steepness may be a useful bulk property for the parameterization of wave‐averaged diapycnal heat flux. Plain Language Summary: Predicting the distribution of constituents such as nutrients, heat, sediment and pollutants on the continental shelf is key to processes such as: the safe operation of offshore infrastructure; understanding the variation in primary productivity; the prediction of marine heat waves; and environmental impact assessments of new activities. Non‐linear internal waves, waves that travel within the stratified water column, are likely to have a significant impact on constituent transport; however, they are poorly understood. Here we present unique observations that quantify the properties of a class of these nonlinear internal waves, known as waves of elevation. These waves can break in a similar way to surface waves at the beach, leading to dramatic increases in turbulent mixing over horizontal length scales of 10s kilometers. We have used our observations to define the amount of vertical heat transport induced by the breaking waves as a function of wave steepness. The vertical heat transport can be used as a proxy to estimate the vertical nutrient transport. This study has made a step change in quantifying the impact of NLIW of elevation on vertical transport. Key Points: Nonlinear internal waves of elevation can form both convective and shear instabilities, resulting in enhanced diapycnal mixing and heat fluxBreaking continues as the nonlinear waves propagate onshore, influencing diapycnal mixing and vertical heat flux over 10s of kilometersThe elevated heat flux associated with the nonlinear internal waves of elevation increases with the wave steepness [ABSTRACT FROM AUTHOR]