Relative wind stress

Relative wind stress is a shear stress that is produced by wind blowing over the surface of the ocean, or another large body of water. Relative wind stress is related to wind stress but takes the difference between the surface ocean current velocity and wind velocity into account. The units are Newton per meter squared [ N m − 2 ] {\displaystyle [Nm^{-2}]} or Pascal [ P a ] {\displaystyle [Pa]} . Wind stress over the ocean is important as it is a major source of kinetic energy input to the ocean which in turn drives large scale ocean circulation. The use of relative wind stress instead of wind stress, where the ocean current is assumed to be stationary, reduces the stress felt over the ocean in models. This leads to a decrease in the calculation of power input into the ocean of 20–35% and thus, results in a different simulation of the large scale ocean circulation.

Mathematical formulation The wind stress ( τ ) {\displaystyle (\tau )} acting on the ocean surface is usually parameterized using the turbulent drag formula

τ = C d ρ a | u → a | u → a {\displaystyle \quad \tau =C_{d}\rho _{a}|{\vec {u}}_{a}|{\vec {u}}_{a}} . where C d {\displaystyle C_{d}} is the turbulent drag coefficient (usually determined empirically), ρ a {\displaystyle \rho _{a}} is the air density, and u → a {\displaystyle {\vec {u}}_{a}} is the wind velocity vector, usually taken at 10m above sea level. This parameterization is commonly referred to as resting ocean approximation. From now on we will refer to wind stress in resting ocean approximation as simply resting ocean wind stress. On the other hand, relative wind stress ( τ r e l ) {\displaystyle (\tau _{rel})} makes use of the velocity of the surface wind relative to the velocity at the ocean surface u → o {\displaystyle {\vec {u}}_{o}} , as follows,

τ r e l = C d ρ a | u → a − u → o | ( u → a − u → o ) {\displaystyle \quad \tau _{rel}=C_{d}\rho _{a}|{\vec {u}}_{a}-{\vec {u}}_{o}|({\vec {u}}_{a}-{\vec {u}}_{o})} . where u → o {\displaystyle {\vec {u}}_{o}} is the surface ocean velocity and thus, the terms with ( u → a − u → o ) {\displaystyle ({\vec {u}}_{a}-{\vec {u}}_{o})} represent the wind velocity relative to the surface ocean velocity. Therefore, the difference between wind stress and relative wind stress is that relative wind stress takes into account the relative motion of the wind with respect to the surface ocean current.

Work done by the wind on the ocean The work wind does on the ocean can be computed by

P = τ ⋅ u → o {\displaystyle \qquad P=\tau \cdot {\vec {u}}_{o}}

where τ {\displaystyle \tau } is the chosen parameterization for the wind stress. Thus, in resting ocean approximation, the work done on the ocean by the wind is

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