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Currently I hear a series of tutorial videos on atmospheric dynamics as a first starting preparation for reading more detailed material, such as Holton. In particular I refer to this part:

https://www.youtube.com/watch?v=YjugCNkLD0k&list=PL_cuIb7hx5lg_zHfUVsUrw6I66U4jq8Dq&index=10

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I can follow more or less the derivation of what we see here, namely that by using p instead of z for the vertical coordinate the pressure force on an air parcel reduces to gradient of geopotential $\Phi$:

Still I have some troubles with regard to the mathematical formalism:

What does $\left(\frac{\partial \Phi}{\partial x}\right)_p$ mean mathematically? I would have assumed, that $\Phi$ is a function of $x, y, p$: $\Phi = \Phi(x,y,p)$ and, therefore $\frac{\partial \Phi}{\partial x}$ means to keep y,p as constant and consider only the change of $\Phi$ caused by a change in x (this is the definition of partial derivative). Why do we write $\left(\frac{\partial \Phi}{\partial x}\right)_p$ instead? Once we agree on a special set of coordinates we dont have to specify what other coordinates are held constant for the partial derivative. But if we decide to do it, why is it then not written as $\left(\frac{\partial \Phi}{\partial x}\right)_{y,p}$ ? Looks a bit confusing.

Because of this notational discrepancy, I'm afraid I haven't quite gotten the "magic behind it" yet, maybe missing the most important point of all, and therefore just think I got it.

Can somebody explain, where I have my missing point?

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    $\begingroup$ You're right, the indices should be $y,p$ instead of just $p$, when following standard notation. Here, they were presumably a) just a bit lazy or b) wanted to keep emphasizing what the other independent coordinate is (y is trivial, but p or z or r needs to be specified). Don't loose sleep over it. $\endgroup$ Dec 17, 2022 at 13:46
  • $\begingroup$ So I can regard $\Phi$ as a regular multivariable function on x,y,p? The author always talks about "gradient on a surface of constant pressure" and I wonder what this should be in contrast to regular gradient. I see no difference. $\endgroup$
    – MichaelW
    Dec 17, 2022 at 14:43
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    $\begingroup$ Think of your 'regular' gradient as "gradient on a surface of constant z". Every scalar variable in meteorology is 3-D multivariate, $f=f(x_i, x_j, x_k)$, but clever vertical coordinates can be used, like $x_k=p$ in order to simplify the equations of motion. $\endgroup$ Dec 17, 2022 at 14:52

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The subscript $p$ denotes that it on a pressure surface. It is more or less a formality for a derivation, so it is commonly dropped. It is rooted in the chain rule

Put another way, it eliminates the complexity of this conundrum $\Phi=\Phi(x,y,p[x,y])$. By specifically stating that you are working on an isobaric surface, you turn

$$\frac{\partial \Phi}{\partial x}= (\frac{\partial \Phi}{\partial x})_{p=constant} + \frac{ \partial \Phi}{\partial p} (\frac{\partial p}{\partial x})_{z=constant} $$ into $$\frac{\partial \Phi}{\partial x}= (\frac{\partial \Phi}{\partial x})_{p} $$.

If you take this concept of chain rule and make it abstract, you could expand it to working on all sorts of surfaces. You could turn the equations from pressure coordinates to PV or temperature coordinates, for example.

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