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In most derivations of the Ekman spiral we begin by assuming a steady state in which only frictional (drag) and Coriolis forces are acting- and then we equate them to get the spiral. I have two issues with this: a) how are Coriolis force and friction ever going to cancel out- are they not perpendicular to each other by definition? b) how is this a steady state solution if there’s only friction but no energy input? Or rather, the friction term must both input and dissipate energy from the system, but how? I’m afraid I’ve misunderstood something quite seriously because every textbook I check simply equates friction and Coriolis without addressing the perpendicular thing, so it’s probably not an issue to begin with but I haven’t been able to figure out why not. Any help would be appreciated! Here’s a picture of the derivation in Stewart’s Introduction to Physical Oceanography: A section of Stewart’s Introduction to Physical Oceanography with a force diagram showing Coriolis and drag force to be perpendicular, then equating them below to derive the Ekman spiral

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  • $\begingroup$ 1st of all, the fluid needs to be already in motion. As there are no other external forcing only Earth's rotation (Coriolis) and friction are acting on the fluid. They are indeed perpendicular and they won't "cancel out." They will force the fluid to rotate. Being in steady state means they won't change in time unless the forcing changes. $\endgroup$
    – arkaia
    Commented Apr 11 at 8:22
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    $\begingroup$ Figured it out! The tilting of the velocity vectors with depth means that the frictional pull exerted by the layer above is in a different direction to the drag from the layer below, so the net is not in the same direction as the motion but ends up being perpendicular to it —> the constant wind stress keeps the whole system in steady state $\endgroup$ Commented Apr 11 at 21:00

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