Impact of uncertainties in the horizontal density gradient upon low resolution global ocean modelling

Abstract

In this study, it is shown (i) that, as a result of the nonlinearity of the seawater equation of state, unresolved scales represent a major source of uncertainties in the computation of the large-scale horizontal density gradient from the large-scale temperature and salinity fields, and (ii) that the effect of these uncertainties can be simulated using random processes to represent unresolved temperature and salinity fluctuations. The results of experiments performed with a low resolution global ocean model show that this parameterization has a considerable effect on the average large-scale circulation of the ocean, especially in the regions of intense mesoscale activity. The large-scale flow is less geostrophic, with more intense associated vertical velocities, and the average geographical position of the main temperature and salinity fronts is more consistent with observations. In particular, the simulations suggest that the stochastic effect of the unresolved temperature and salinity fluctuations on the large-scale density field may be sufficient to explain why the Gulf Stream pathway systematically overshoots in non-stochastic low resolution ocean models.

Introduction

One of the most salient feature of today’s state-of-the-art ocean models is that they are essentially deterministic models, in the sense that they do not involve random numbers to represent uncertainties in the model equations, parameters and forcing, or to simulate the effect of unresolved processes. Yet, this deterministic model dynamics is known to become chaotic as soon as mesoscale eddies are resolved by the model, so that the simulated mesoscale flow can only be viewed as one random realization sampled from a large set of possibilities. It is thus only in a statistical sense that the mesoscale can be compared to the real world, and it is only as a stochastic process that the effect of the mesoscale in the model can be analysed. Mesoscale fluctuations indeed produce a considerable effect on the general circulation of the ocean (Zhai et al., 2004, Penduff et al., 2010), with prominent contributions to momentum, heat and salt fluxes, which cannot be easily parameterized in low resolution models.

As a general rule, the effect of uncertainties or unresolved processes (even if unbiased) does not average to zero in a nonlinear model. For instance, if the wind is fluctuating or if it is uncertain, then neglecting the fluctuations or the uncertainties systematically underestimates the air–sea momentum flux (proportional to the square of the wind speed). In the same way, the average effect of the mesoscale fluctuations does not vanish in the two nonlinear terms of the primitive equations: the advection term and the equation of state. Concerning the advection term, this effect was originally parameterized in ocean models using empirically specified horizontal diffusion (Bryan et al., 1979), and afterwards using more and more sophisticated advection/diffusion operators (see Griffies et al., 2000 for a review). Concerning the equation of state, the effect of the mesoscale temperature and salinity fluctuations on the large-scale density field is generally ignored, maybe because it cannot be easily parameterized using a deterministic formulation. However, it can easily be argued (see Section 3.1), that, in low resolution ocean models, the resulting approximation in the large-scale density is a major source of uncertainties in the horizontal pressure gradient, and thus in the horizontal momentum balance equation.

A different point of view can also be adopted to deal with model uncertainties. Rather than parameterizing their mean effect in the model, they can be explicitly simulated by including a random forcing in the model equations. This can be done to produce ensemble forecasts (Buizza et al., 1999, Palmer et al., 2005) or to simulate model error in ensemble data assimilation methods (Evensen, 1994). In such applications, the random forcing is not only responsible for the dispersion of the ensemble; it can also produce a significant mean effect in the simulations (Berner et al., in press, Williams, 2012, Palmer, 2012). In this study, the same kind of approach is used to simulate the uncertainties that unresolved mesoscale temperature and salinity fluctuations produce on the large-scale horizontal density field. The objective is to propose a simple (empirically specified) stochastic parameterization of these uncertainties (in Section 3), and to evaluate the impact that this parameterization may have on the ocean circulation (in Section 4), as simulated by a low resolution global model configuration (described in Section 2).