Intercomparison of the Charnock and COARE bulk wind stress formulations for coastal ocean modelling.

The accurate parameterisation of momentum and heat transfer across the air-sea interface is vital for realistic simulation of the atmosphere-ocean system. In most modelling applications accurate representation of the wind stress is required to numerically reproduce surge, coastal ocean circulation, surface waves, turbulence and mixing. Different formulations can be implemented and impact the accuracy of: the instantaneous and long-term residual circulation; the surface mixed layer; and the generation of wave-surge conditions. This, in turn, affects predictions of storm impact, sediment pathways, and coastal resilience to climate change. The specific numerical formulation needs careful selection to ensure the accuracy of the simulation. Two wind stress parameterisation widely used in the ocean circulation and the storm surge communities respectively are studied with focus on an application to the NW region of the UK. Model-observation validation is performed at two nearshore and one estuarine ADCP stations in Liverpool Bay, a hypertidal region of freshwater influence (ROFI) with vast intertidal areas. The period of study covers both calm and extreme conditions to test the robustness of the 10 m wind stress component of the Coupled Ocean–Atmosphere Response Experiment (COARE) bulk formulae and the standard Charnock relation. In this coastal application a realistic barotropic-baroclinic simulation of the circulation and surge elevation is setup, demonstrating greater accuracy occurs when using the Charnock relation, with a constant Charnock coefficient of 0.0185, for surface wind stress during this one month period.


Introduction
For realistic simulation of the atmosphere-ocean system an accurate parameterisation of momentum and heat transfer across the air-sea interface is required. Coupled modelling systems are often used to include interactions across this boundary (e.g. Liu et al., 2011). In oceanographic applications the parameterisation of the surface roughness is very important. The presence of waves can influence the coastal circulation and also sea surface temperatures through modification of the surface turbulence and mixed layer depth (Moon, 2005). The presence of ocean surface waves also modifies the wind profile itself (Large et al., 1995), requiring oceanatmosphere model coupling to capture the feedback (Kukulka and Hara, 2008). Climate variability is also influenced by ocean-atmosphere interaction, model coupling is therefore essential to better understand the global climate and future change (Neelin et al., 1994). In barotropic storm surge models the wind stress is often represented by the Charnock (1955) relation between the wind speed and the surface roughness. In coupled wave-circulation models this parameterisation is often extended with a wave-related stress (Janssen, 1991) to properly account for the existence of the wave field (see Mastenbroek et al., 1992).
In non-coupled ocean models it is common to apply bulk formulae to atmospheric forcing fields to determine surface fluxes for baroclinic-barotropic circulation studies (e.g., Holt and Proctor, 2008). In the absence of ocean-atmosphere coupling the implementation of different formulations can impact the accuracy of the instantaneous and long-term residual circulation, and also the generation of coastal wave-surge conditions. This, in turn, affects predictions of storm impact, sediment pathways, and coastal resilience to climate change. To enable the best prediction of coastal circulation, we investigate two wind stress formulae widely used in the ocean circulation and the storm surge communities. We focus on an application to Liverpool Bay, which is in the NW region of the UK and a case study of specific interest for improving the understanding of sediment pathways around the UK. At this location both the Charnock (1955) relation and the COARE (Coupled Ocean-Atmosphere Response Experiment) bulk formulae (Fairall et al., 2003) have previously been used in separate studies of storm surge extremes (Brown and Wolf, 2009) and freshwater influence (O'Neill et al., 2012), but not compared.
Model comparisons against ADCP (Acoustic Doppler Current Profiler) data are employed at two nearshore and one estuarine location in Liverpool Bay, which is a hypertidal region of freshwater influence (ROFI) from three large estuary systems (Polton et al., 2011;Verspecht et al., 2009), and which is characterized by a maximum tidal range O(10) m, maximum currents in excess of 1 ms -1 and vast intertidal areas. The period of study (9 th February -9 th March 2008) coincides with a period of observations within the Dee Estuary (14 th February -9 th March 2008, Bolaños and Souza, 2010), which are supplemented with nearshore measurements from the Liverpool Bay Coastal Observatory (COBS: http://cobs.noc.ac.uk/, Howarth and Palmer, 2011) to investigate a 30 day period. The wind characteristics during this period are dominated by southwest to westerly winds (Fig. 1)

Model setup and observations
This study focuses on accurately simulating the coastal circulation, currents and surge elevation, within a hypertidal region or freshwater influence (Polton et al., 2011). We apply the Proudman Oceanographic Laboratory Coastal Ocean Modelling System (POLCOMS, Holt and James, 2001) to simulate the coastal circulation within Liverpool Bay. POLCOMS is coupled to the General Ocean Turbulence model (GOTM, Holt and Umlauf, 2008) and we use the k-ε scheme with the stability functions derived from Canuto et al. (2001). The Liverpool Bay computational domain has a horizontal resolution of ~180 m with 10 vertical sigma-levels within the water column and is nested, in one-way, within a 1.8 km Irish Sea model (Fig. 2).
Both model domains are forced by ~12 km hindcast atmospheric data from the UK Met Office numerical weather prediction model, the wind speed and air pressure is provided hourly, while the air temperature, specific humidity and cloud cover are provided 3 hourly. External To do this the model data is extracted at a depth of 10% below the surface (i.e., the -0.1 sigma-level). The surge elevation is obtained by applying harmonic tidal analysis to pressure sensor records and to the modelled total elevations at the same locations. The tidal analysis is performed using t-tide (Pawlowicz et al., 2002) with all the available shallow water constituents considered. The limited period (< 1 month of data) is likely to cause slight discrepancies between the analysed surge and that from the long-term tide gauge data (problems extracting the surge elevation are discussed further by Brown et al., 2012). For consistency, the same method has been applied to each location for both model and observation.
Wind observations are available from the COBS at Hilbre Island in the mouth of the Dee, a few hundred meters from the Hilbre Channel ADCP deployment. The long-term observed wind climate (Fig. 1a) shows local influence from the estuary causing a strong south-easterly wind contribution, in addition to more typical wind direction from the southwest through to north west (as observed offshore within the bay, see Wolf et al., 2010). For the period when observations were available for model validation the winds were dominated by south-westerly to westerly conditions (Fig. 1b)

Validation method
Intercomparison between the two numerical experiments requires a quantitative assessment in order to determine which modelling approach is best. To that end, we introduce a time-varying comparative accuracy metric which is based on the absolute error of each model experiment.
This metric is used to show the time variation in the comparison of the wind stress parameterisation performance. Using standard error metrics (RMS error and mean of the timevarying Bias) applied to the full study period allows the overall accuracy of each model experiment to be determined. For each mooring location we determine the time-varying differential accuracy (DA) as the difference between the absolute errors of each numerical experiment:

Time-varying accuracy
The time-varying results of the differential accuracy metric (described above) are shown in No correlation between the wind direction and the better performing wind stress parameterisation is identified for any of the properties studied. The time-averaged differential accuracy metric (

Discussion and concluding statements
Three locations within Liverpool Bay have been used to test the accuracy of two wind stress parameterisations within a coastal model: the COARE bulk formulae and the standard Charnock relation. The locations represent nearshore and estuarine environments. This study defines a time-varying differential accuracy metric, which is shown to be a simple, but affective method to compare model accuracy over time. Under these (very) fetch limited conditions, dominated by south-westerly to westerly winds, error analysis confirms both methods of wind stress parameterisation give valid results compared with ADCP and pressure sensor observations. A differential accuracy metric is applied to identify which method performs with highest accuracy in coastal seas. Over the duration of the study period either formula can perform more accurately at an instance, but on average over the longer term the standard Charnock relation performs with higher accuracy. A short 25-hour period of north-westerly to northerly winds is focused on ( Charnock coefficient. In this application the additional smooth surface roughness length (see Smith, 1988) is unlikely to have much influence in the COARE bulk formulae since wind speeds only occasionally drop below 3 m/s during the study. In Liverpool Bay the largest fetches are from west to northwest directions, but are still more limited than open ocean conditions. No correlation between the wind direction (fetch) and which model performs more accurately has been found. The standard Charnock coefficient has been set here to a value that gives good surge simulation across eastern Irish Sea region. The value of this parameter is thought to be related to the model grid resolution (0.0275 for ~ 12 km grid and 0.0185 for ~1.8 km grid, Brown and Wolf, 2009