Talk:Primitive equations

What are $(u,v,\omega ,T,\phi )$, respectively? -- The Anome 15:49, 30 Aug 2003 (UTC)

$u$ and $v$ are the zonal (east and west) and meridoinal (north and south) velocities (I think), $\omega$ is the vertical velocity, T is the temperature, and $\phi$ is the geopotential.

Also, f is the Coriolis parameter, which takes into account the fact that we're dealing with a non-intertial coordinate system.

I'll double check the relavent texts before adding the definitions. --Loren

Primitive equations using sigma coordinate system, polar stereographic projection

Is this version really necessary? It doesn't seem to illustrate anything more than the simpler version aside from water vapor transport, which can easily be incorporated into the simpler version as analogous to the continuity equation. -Loren 07:34, 7 May 2007 (UTC)

I've temporarily commented out this section. There are just too many questions regarding some of the equations that need to be verified. -Loren 07:45, 7 May 2007 (UTC)

Commented text

Primitive equations using sigma coordinate system, polar stereographic projection

• According to the National Weather Service Handbook No. 1 - Facsimile Products, the primitive equations can be simplified into the following equations:

• Temperature:

${\frac {\partial T}{\partial t}}=u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}+w{\frac {\partial T}{\partial z}}$

• Zonal wind:

${\frac {\partial u}{\partial t}}=\eta v-{\frac {\partial \Phi }{\partial x}}-c_{p}\theta {\frac {\partial \pi }{\partial x}}-z{\frac {\partial u}{\partial \sigma }}-{\frac {\partial ({\frac {u_{2}+y}{2}})}{\partial x}}$

• Meridional Wind:

${\frac {\partial v}{\partial t}}=-\eta {\frac {u}{v}}-{\frac {\partial \Phi }{\partial y}}-c_{p}\theta {\frac {\partial \pi }{\partial y}}-z{\frac {\partial v}{\partial \sigma }}-{\frac {\partial ({\frac {u_{2}+y}{2}})}{\partial y}}$

• Precipitable water:

${\frac {\partial W}{\partial t}}=u{\frac {\partial W}{\partial x}}+v{\frac {\partial W}{\partial y}}+w{\frac {\partial W}{\partial z}}$

• Pressure Thickness:

${\frac {\partial }{\partial t}}{\frac {\partial p}{\partial \sigma }}=u{\frac {\partial }{\partial x}}x{\frac {\partial p}{\partial \sigma }}+v{\frac {\partial }{\partial y}}y{\frac {\partial p}{\partial \sigma }}+w{\frac {\partial }{\partial z}}z{\frac {\partial p}{\partial \sigma }}$

• These simplifications make it much easier to understand what is happening in the model. Things like the temperature (potential temperature), precipitable water, and to an extent the pressure thickness simply move from one spot on the grid to another with the wind. The wind is forecasted slightly differently. It uses geopotential, specific heat, the exner function π, and change in sigma coordinate.

Equation formatting

I've converted a bunch of equations from pseudo-notation into TeX markup. But two of them contained the strange term g(r/r), which looks like a typo of some sort to me. Somebody who actually knows the relevant equations should check and fix if needed. (I'd guess the second r should be r_0, but I'd be guessing.) -dmmaus 00:50, 7 August 2007 (UTC)

Frictional Force

I'm not going to change it, but is anyone familiar enough with that notation to verify its accuracy..and for that matter explain its variables? I know it as: $F_{rx}=\nu [{\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}$

$F_{ry}=\nu [{\frac {\partial ^{2}v}{\partial x^{2}}}+{\frac {\partial ^{2}v}{\partial y^{2}}}+{\frac {\partial ^{2}v}{\partial z^{2}}}$

$F_{rz}=\nu [{\frac {\partial ^{2}w}{\partial x^{2}}}+{\frac {\partial ^{2}w}{\partial y^{2}}}+{\frac {\partial ^{2}w}{\partial z^{2}}}$

This is obviously not as succinct as in vector notation, it can be compacted but...I learned it through a matter of substitutions which aren't discussed here. For example $\nu ={\frac {\mu }{\rho }}$ seems particularly useful. However it's never explained. In fact a lot of this is never explained; Cv, r, or m...it seems this entire section if not the entire page have been ripped straight from text books without any comprehension by the transcriber...OR no efforts by the writer to imbue understanding onto the readers. I could re-write this section in a notation I'm more accustomed to, but unless someone can explain some of the variable notations I can't help it as is.

—Preceding unsigned comment added by 68.3.13.207 (talk) 21:44, 27 July 2010 (UTC)

not necessarily diff. eq.

simply because an equation shows a differential or the "simpler" partial differential doesn't mean it is a differential equation.

it's not a diff eq unless you must find the form of eq for y (meaning you have no eq for y) and diff are involved.

for y`=x dx we know y=x^2/2+C. eq. involving integration (esp if one already knows the answer) is not a diff eq. and the ` was really not necessary since it represents what we already know, x dx.

in diff eq. one solves to find a function. substituting values or just integrating (calculus) is not diff eq.

but for diff eq we might have y`+P(x)y=Q(x) and we do not know what function y (thus not y` either) is, we only know it's a function that satisfies all other parts and sol'n including diff. one applies rules to find a full eq. for y, much different than a rule of integration.

for @^2v/@x^2 + @^2v/@y^2=0 might be a pde if we don't know v. but if we look it up in the table of pde sol'n, long ago solved and published, no diff eq. were used or needed.

Please describe variables

Some sections are poorly written. For example, an equation is thrown up, yet some of the variables are not explained anywhere! So how is a reader supposed to understand what they represent? — Preceding unsigned comment added by 146.142.1.10 (talk) 17:03, 25 October 2016 (UTC)

Discrepancy between Section 1 and Section 2

The Definitions section is very welcome, but seems to be at odds with the "Forces that cause atmospheric motion" section: it is very confusing to first write that $f$ denotes the Coriolis force, and then in the very first equation in the next section it is used for something else. Likewise, a lot of notation in that section are not defined in the Definitions section. A pass of harmonization would be very helpful.Ventricule (talk) 18:11, 30 August 2024 (UTC)