On geostrophic wind. How can the Coriolis force cancel out the pressure gradient force when they are mutually perpendicular?

Question

If we are in Northern Hemisphere and there is a point where there is a high pressure, denote it by H, and there is a point where there is low pressure, denote it by L, (for simplicity assume them to lie latitudinally, that is H is on west and L on east).

The pressure gradient force will drive the wind from H to L, and since we are in the Northern Hemisphere, the wind gets deflected to the right by the Coriolis force.

The wind began with a velocity $\vec{V}$ initially directed towards L (that is in the direction of pressure gradient force P), but due to Coriolis force C, at right angles to the velocity, it turns rightwards and we get a geostrophic wind, that is a wind blowing parallel to isobars. But in all this, when did C get cancelled with P?

Answer 1

Consider the scenario just after the picture, the wind is blowing southwards, but it still have tendency to blow towards L because of Pressure gradient force, but now the Coriolis force will act rightwards, that is towards west and if |C| = |P|, the two opposite forces will cancel each other out and the wind will continue to flow in parallel fashion to isobars. The situation after a few moments will be:

$\vec{V'}$ is the velocity just after the situation shown in question diagram. Thus, that is how C and P will cancel each other out.

Answer 2

If a pressure gradient force poofed and appeared out of nothing, then it would be as you suggest:

• The wind would start out moving from high to low.
• The wind would continue to accelerate due to PGF, and the wind would shift direction (to the right in the NH) due to Coriolis.
• At each point during the turn, more of the Coriolis turn (to the right in the NH) would be becoming in opposition to the PGF rather than perpendicular to it... so gradually the speed acceleration decreases (more of the forces are perpendicular to motion rather than in line with the motion)... and the directional acceleration (after initially increasing due to increasing velocity $\implies$ increasing Coriolis) also gradually decreases (as the wind turns [more south in your picture], the PGF starts to counteract more of the coriolis).

In reality, at frictionless levels like 500 mb, because pressure development is generally gradual, the wind steadily adjusts as the changes gradually occur, and so the result is that winds are basically geostrophic always. I'm pretty sure it's still not absolutely truly perfectly 100% geostrophic, or a moving parcel of air at that level would never change direction, but the component of ageostrophic wind is so small as to basically allow it to be idealized as perfectly geostrophic.

I think your struggle is that you draw $\overrightarrow{v}$ showing it changing (having it curved), whereas you draw the forces as instantaneous snapshots. Instead, try to consider each instant as the $\overrightarrow{v}$ vector turns... your $\overrightarrow{c}$ vector is also changing direction to match.

So initially $\overrightarrow{c}$ and $\overrightarrow{P}$ don't cancel at all, but as the wind turns, more and more of them cancels until it reaches a steady state when $\overrightarrow{c}$ and $\overrightarrow{P}$ are completely equal and opposite and so the velocity vector is unchanging (in magnitude and direction) as well.