Boundary layers occur often in geophysical problems. In oceanography, two famous boundary layers are the Stommel and Munk layers, the latter forming the western-boundary currents in most OGCM solutions. Here, we look at the Munk boundary layers that develop in a solution forced by a τx wind patch, illustrating how their width changes as a function of the mixing coefficient, ν_{h}.

Domain: 60ºE–100ºE, 10ºN–50ºN Resolution: 0.1º β-plane:f = f_{0} + β(y – y_{0}), y_{0} = 30º Characteristic speed:c_{1} = 264 cm/s Forcing: switched-on, τx wind patch Mixing:ν_{h} = 10^{6} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
The wind has the form τ^{x} = τ_{o}X(x)Y(y)T(t), where

X(x) = 0.5{1+cos[2π(x–80º)/20º]}, 70º ≤ x ≤ 90º (D1a)

Y(y) = 0.5{1+cos[2π(y–30º)/20º]}, 20º ≤ y ≤ 40º (D1b)

T(t) = θ(t) (D1c)

θ is a step function, τ_{o} = 1.5 dyn/cm^{2}, and X(x) and Y(y) are both zero outside their designated ranges. There is a Munk layer in steady state, with a width of 0.35°, close to that predicted by theory (see discussion of Figure D).

Domain: 60ºE–100ºE, 10ºN–50ºN Resolution: 0.1º β-plane:f = f_{0} + β(y – y_{0}), y_{0} = 30º Characteristic speed:c_{1} = 264 cm/s Forcing: switched-on, τx wind patch Mixing:ν_{h} = 10^{6} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
As in Experiment D1, but with ν_{h} = 5e7 cm2/s. As expected, the Munk layer broadens by a factor of 10⅓ = 2.15 to 7°, consistent with theory (see discussion of Figure D).

Domain: 60ºE–100ºE, 10ºN–50ºN Resolution: 0.1º β-plane:f = f_{0} + β(y – y_{0}), y_{0} = 30º Characteristic speed:c_{1} = 264 cm/s Forcing: switched-on, τx wind patch Mixing:ν_{h} = 10^{6} cm^{2}/s, A = 0.00013 cm^{2}/s^{3} Movie time step: daily Description:
As in Experiment D1, but with ν_{h} = 5e5 cm2/s. As expected, the Munk layer thins by a factor of 2.15 to 0.15°, consistent with theory (see discussion of Figure D).

Figure D
The movies for Experiments D1−D3 show the spin-up of solutions for 3 different values of the horizontal-viscosity coefficient, ν_{h}. To illustrate the structure of the western-boundary currents more clearly, Figure D plots maps of v very near the western boundary. Theoretically, the distance offshore x_{m} where v attains its maximum value is

x_{m} = 2πr_{m}/(3√3) , (D1)

where r_{m} = (ν_{h}/β)⅓. For ν_{h} = 5e5, 5e6, and 5e7 cm^{2}/s, (D1) predicts x_{m} = 0.15°, 31°, and 0.67°, respectively, in excellent agreement with the values of 0.15°, 0.35°, and 0.7° estimated from Figure D.

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