Confusion about the horizontal pressure gradient force being equal to the gradient of the geopotential in pressure coordinates

Question

I see many sources in atmospheric dynamics express the following:

$\frac{1}{\rho}(\nabla p) = \nabla_p \phi$

For example this source equation 10.

The reasoning given is that $\frac{1}{\rho}(\frac{\partial p}{\partial x})_z = (\frac{\partial \phi}{\partial x})_p$ and $\frac{1}{\rho}(\frac{\partial p}{\partial y})_z = (\frac{\partial \phi}{\partial y})_p$. (see equation 9 in my linked pdf).

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While I agree with the above, shouldn't $\nabla_p \phi$ lie in a different plane that the $\nabla p$? The first should lie in the plane $p=const$ and the second should lie in the plane $z=const$.

Informally the $i$ and $j$ unit vectors should be different for these two gradients since we are in different coordinate systems. While I agree that the magnitudes attached to these gradients is indeed the same, the would be a bit tilted from each other?

I'm probably just missing something, and would appreciate some clarity, thank you.

Answer 1

The first should lie in the plane 𝑝=𝑐𝑜𝑛𝑠𝑡 and the second should lie in the plane 𝑧=𝑐𝑜𝑛𝑠𝑡

Yes … the gradient of pressure is calculated on height planes while the gradient of Geopotential height is calculated on pressure plane.

Informally the 𝑖 and 𝑗 unit vectors should be different for these two gradients since we are in different coordinate systems.

No both are sharing the same horizontal coordinate system …