# A Lagrangian Model to Predict the Modification of Near-Surface Scalar Mixing Ratios and Air–Water Exchange Fluxes in Offshore Flow

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DOI: 10.1007/s10546-011-9598-0

- Cite this article as:
- Rowe, M.D., Perlinger, J.A. & Fairall, C.W. Boundary-Layer Meteorol (2011) 140: 87. doi:10.1007/s10546-011-9598-0

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## Abstract

A model was developed to predict the modification with fetch in offshore flow of mixing ratio, air–water exchange flux, and near-surface vertical gradients in mixing ratio of a scalar due to air–water exchange. The model was developed for planning and interpretation of air–water exchange flux measurements in the coastal zone. The Lagrangian model applies a mass balance over the internal boundary layer (IBL) using the integral depth scale approach, previously applied to development of the nocturnal boundary layer overland. Surface fluxes and vertical profiles in the surface layer were calculated using the NOAA COARE bulk algorithm and gas transfer model (e.g., Blomquist et al. 2006, Geophys Res Lett 33:1–4). IBL height was assumed proportional to the square root of fetch, and estimates of the IBL growth rate coefficient, *α*, were obtained by three methods: (1) calibration of the model to a large dataset of air temperature and humidity modification over Lake Ontario in 1973, (2) atmospheric soundings from the 2004 New England Air Quality Study and (3) solution of a simplified diffusion equation and an estimate of eddy diffusivity from Monin–Obukhov similarity theory (MOST). Reasonable agreement was obtained between the calibrated and MOST values of *α* for stable, neutral, and unstable conditions, and estimates of *α* agreed with previously published parametrizations that were valid for the stable IBL only. The parametrization of *α* provides estimates of IBL height, and the model estimates modification of scalar mixing ratio, fluxes, and near-surface gradients, under conditions of coastal offshore flow (0–50 km) over a wide range in stability.

### Keywords

Air–sea gas exchangeBulk Richardson numberInternal boundary layerOffshore flowStability### List of Symbols

- d
*T*_{a} Land–lake air temperature modification

- d
*T*_{d} Land–lake dewpoint temperature modification

*f*Fraction of

*h*that defines the top of the surface layer*F*Flux per unit area at the surface

*g*Acceleration due to gravity

*H*(*x*)Integral depth scale

*H*(*x*)_{u}Upper portion of the integral depth scale above the surface layer

*H*(*x*)_{l}Lower, surface-layer portion of the integral depth scale

*h*Height of the internal boundary layer

*K*Turbulent eddy diffusivity

*k*_{a}Atmospheric gas transfer velocity

*L*Obukhov length

*n*Exponent that determines the shape of the IBL mixing ratio profile

*P*Atmospheric pressure

*R*Gas constant

*Ri*_{b}Bulk Richardson number

*Ri*_{b10}Bulk Richardson number defined using upstream, overland meteorological Variables at 10-m reference height

*r*(*z*)Gas mixing ratio as a function of height

*r*_{l}Upstream, overland mixed layer gas mixing ratio

*r*_{s}Gas mixing ratio at the surface

*T*Absolute temperature

*U*Mean wind speed in the

*x*direction*U*_{10}Wind speed at 10-m height

- \({\overline{U}}\)
Wind speed averaged vertically over the IBL

*u*_{*}Friction velocity

*x*Horizontal dimension aligned with the mean wind

*X*Fetch: distance travelled by the air mass over water from the coast

*z*Vertical dimension, positive upward

*z*_{m}Profile matching height; border between the surface layer and the IBL

*z*_{o}Aerodynamic roughness length

*z*_{r}Reference height at which wind speed or scalar has a known value

*α*IBL growth rate coefficient

*γ*Lapse rate: deviation of temperature or mixing ratio vertical profile from well-mixed condition

*γ*_{d}Dry adiabatic lapse rate,

*γ*_{d}= −0.0098 K m^{−1}*γ*_{e}Environmental lapse rate

*θ*Potential temperature

*θ*_{v}Virtual potential temperature

*θ*_{vl}Upstream, overland mixed layer

*θ*_{v}*θ*_{vs}*θ*_{v}of air at equilibrium with the water surface*κ*von Kármán constant, assumed to have a value of 0.4

- \({\Phi_{\rm H}(z/L)}\)
MOST gradient profile function for potential temperature

- Ψ
_{M}(*z*/*L*) MOST integral profile function for wind speed