White House issues new dire climate report: Scientists: Extreme weather will worsen if pollutants aren't curbed

[Ban Hammer 13,352]

Time to go whack some weeds before it gets too hot and the yellow jackets start waking up.

have you killed off the nests around there?

[scandal 951]

Holy smokes brother scandal... I was lost for a moment but I followed the logic behind the math and presto-... my fluid dynamics circuits understood. I am glad to have another scientist... have you a computational background as it may, in OT!! Now I know whom to bother the next time I have flow rate issues with my lab HPLC rigs...

Now I retire for the evening. 12:30 AM, South Side of Chicago, and I can hear tropospheric precipitation colliding with the planet's crust along with a slight increase in air pressure outside the window I am located at.

Bill... even with those upper atmospheric variations in air speed and density... keep in mind that since CO2 has mass, gravity will keep it more condensed at lower altitudes, and hence the molecular motion points I stated earlier back. Good reading from you and Scandal-ous!

I was thinking about that last night on the way to pick up the wife from work. The heavier gases should concentrate at lower altitudes. As far as why a liquid might enable simplication over gases, the changes in the system as observed in the Ideal Gas Law could be ignored when dealing with liquids. I also considered the difference in liquid currents at sea level versus air currents at altitude.

I am so far away from that stuff. The math is still recognizable, but the theory behind the equations is a blur. I looked at my old physics books a few weeks ago, and I was lost trying to understand Newton's Second Law.

Yeah I get you... even though CO2 isn't as heavy as some other pollutant gases, its still sufficient mass to collect due to gravity.

Regarding PV=nRT... although the atmosphere is a closed system, its still large enough for conditions to not be constant enough and I think this gas law is inapplicable to both kinds of fluid systems due to its principal use in high temperature/low pressure dynamics. Although we could visualize it being more relevant as atmospheric pressure becomes more static and temperature increases... but I think the formula is far beyond even those limits. The formula also tends to ignore the interaction between molecules and the energy game between them... I guess this is almost negligible for tiny gases like H2 in a large atmosphere... but CO2 has a little more beef to it. Yes I had to look in my chem text.

Anyway... back to biology for the next couple of hours.

Hal - the weather in Chicago was pretty amazing yesterday, wasn't it? It was almost pitch darkness at 11AM yesterday, looked like the end of the world!

Regarding the change in density of atmosphere (not just C02, but all the atmospheric gases) at lower altitudes as compared to higher ones -this is accommodated by a single continuous model with one fluid regime. The mass and momentum continuity expressions allow for a non-constant fluid density. As I said before, it's the divergence which must be zero in an incompressible flow. There can be (and is) a non-zero convection term.

Bill, you are quite right at higher and higher elevations when you get to a rarified atmosphere at some point the fluid regime cannot be modeled as an Eulerian flow and needs to be tracked as particulates using statistical mechanics.

Like any model, N-S makes assumptions about where it is being employed. If those assumptions no longer hold true, the model is not appropriate. What I presume people do when looking at problems that span the lower atmosphere to higher regions is to do an asymptotic matching between N-S based solutions below and statistical mechanics ones above.

Since there appears to be some genuine interest here regarding N-S, here's a brief laymans overview of its significance and meaning.

There are 2 key assumptions behind it: (1) a "fluid" consists of particles. The particle sizes and the inter-particle distances are much smaller than the spatial dimensions of the solution space. Hence, molecules of H2O or CO2 or N2 are much smaller than vessels containing them or over which they flow (beer bottles, centrifuge tubes, bathtubs, airfoils, turbines, oceans, atmospheres).

(2) The basic conservation principles of physics: conservation of mass, of momentum, of angular momentum (for rotating flows), of energy, and of enthalpy apply.

Due to (1), we assume an averaging approach is valid. We can take a volume of space (imagine a cube 1cmX1cmX1cm) and consider all the fluid within that cube, and flowing through the surfaces of the cube. If you now take the limit (mathematically) and shrink that cube down to an infinitesimal size, that is the idealized fluid "particle". And the model will hold true as long as the fluid properties are smoothly continuous at the scale of that particle.

Due to (2), we can now write down balance equations.

Conservation of mass is a simple statement that you can't make something out of nothing. Hence if you have a closed system, no matter how the fluid moves about the total mass must be constant. Or if it's not a closed system, then the total mass at a given moment must equal the mass that was there an instant ago + the mas that was transferred into the bounding surface of the system - the mass that transferred out through the bounding surface. Let's apply that very principle to each of the infinitesimal elements we defined in (1). Each little tiny cube will have its own mass transfer balance. And then we'll sum up (i.e. integrate) over all the little tiny cubes.

So let's express mathematically the rate of change of fluid momentum to the convective flows across the surface:

First term is time-rate of change of "stuff" (density) inside the fluid volume. Second term is the convection - the rate of "drift" of "stuff" over the membrane of the surface. It's literally the rate of change (vector spatial derivative) of the velocity vector of the fluid (v) as that "current" drags stuff in and out of our little box. Clearly, the sum of the two must be identically zero - that's conservation of mass.

We can do a similar thing for momentum conservation. Newton's second law tells us that applied force is related to the accelaration, F=ma. But 'a' (acceleration) is the change in velocity, dv/dt. And in fact (m * dv/dt) is really more generally d(mv)/dt (since the mass density is not always constant), which is dp/dt (where p= momentum = mv). So the more general statement of Newton's second law is F= dp/dt , the applied force is equal to the change in momentum.

When you "plug in" F=dp/dt into the convective model for an infinitesimal volume, you get:

The first term is the time rate of change of momentum (the mass 'm' is now normalized as the density per unit volume rho).

The second term is the convective term, just as before in the mass balance, where we have the fluid velocity v 'dragging' momentum in and out over the bounding surface. The 'vv' expression is a tensor product - since these are both vectors. Finally, 'Q' is the applied external force, the famous 'F' of Newton's F=ma. It is both a generalized external body force (e.g. gravity), and any inter-particle forces (e.g. frictional viscosity) between neighboring fluid particles.

Since p is a vector quantity, so in 3 dimensions we have 3 separate balance equations, one in each dimension. (The momentum in a closed system along each of x,y,z axes must be independently conserved). Thus the equation above is really 3 separate scalar equations all expressed as a single vector equation.

We now have 4 independent variables to be solved for - the density rho, and the velocity components in each dimension (for Cartesian coordinates, typically expressed as u,v,w). These variables are in fact field functions, they vary over the continuum. I.w u is u(x,y,z,t).

And we have 4 equations (one mass balance, 3 momentum balance).

Consequently we should be able to integrate and solve for the 4 field functions, assuming suitable initial and boundary values. Note that until here the system is first-order (we have only first derivatives in time and space, no second derivatives).

However -- the missing step is 'Q'. We need to account for fluid viscosity. In 'simple' Netownian fluids, we have a linear stress/strain relationship. Consequently Q becomes a term that associates a constant viscosity and a second order derivative of the velocity field.

But in other settings things get more complex, and we cannot solve mass/momentum balance on their own, but rather need to introduce the energy equation to be able to get a complete solution for all field variables.

As I said, I'm recalling this from 20 years ago, plus quick Google hits. I may be off on some statements, but generally it's the outline.

N-S is a system of coupled non-linear PDEs (Partial differential equations) in space and time. The equations are solved (usually numerically, in some rare cases closed form solutions exist for simple geometries) to provide the key field functions of interest - density and velocity fields.

The equations Bill provided at the top are N-S expressed in a particular spherical coordinate system suitable for climate studies of Earth. That's why you get the trig expressions appearing, it's due to using (r, theta, phi) coordinates rather than Cartesian (x,y,z).

N-S is fascinating because of the non-linearity and tight coupling of the equations.

Other famous PDE systems such as Schrodinger's Equation (in Quantum Mechanics) or Black-Scholes (in option pricing theory) and even Maxwell's Equations (in Electromagnetism) also have their own inherent beauty. But in my (very limited) knowledge of all of these, Navier-Stokes are by far the biggest ###### to solve numerically.

Enough!

[I AM NOT THAT GUY 5,715]

Time to go whack some weeds before it gets too hot and the yellow jackets start waking up.

have you killed off the nests around there?

Yep. A few years ago. But, I am prepared in case a queen snuck in over the winter. They are small this time of year, and hard to recognize until it is too late.

[one...two...tree 2,762]

Here's my contribution to the current conversation...

[tmma 467]

Navier-Stokes ....I love the complexities and nuances of the ([iMO] imperfect) Navier Stokes derived equation...

Have a good afternoon all....interesting thought proking reading!! I'm on my way out the door, ......I'll be back

[HAL 9000 96]

OMG OMG OMG OMG OMG!!!

This is becoming the First VJ Nerd Symposium.

And yes... Scandal... I'm not sure what parts you call home... but around here I actually needed coffee by 1PM to remind myself not to fall asleep due to lack of light. Today its another story though- although all the excess rain closed the beaches- can you believe that!!

I wish I were more versed in the mathematics of atmospheric dynamics. Lots of my friends work on biological computational models so the transformations are not all alien to me, but given the correct assumptions we're talking about quite a few parallel models. Naturally... things have their limits.

[one...two...tree 2,762]

I just want to know if you nerds can build a better beer bong? Cuz that would so awesome, dude.

[I AM NOT THAT GUY 5,715]

I just want to know if you nerds can build a better beer bong? Cuz that would so awesome, dude.

Looks like she is having sex tonight, and regetting it in the morning.

[HAL 9000 96]

Well we know who's never been to a nerd parté.

All this nerd knows is that lab work today has been accomplished without any regrets.

Have a good rest-of-weekend folks.

[scandal 951]

OMG OMG OMG OMG OMG!!!

This is becoming the First VJ Nerd Symposium.

And yes... Scandal... I'm not sure what parts you call home... but around here I actually needed coffee by 1PM to remind myself not to fall asleep due to lack of light. Today its another story though- although all the excess rain closed the beaches- can you believe that!!

I wish I were more versed in the mathematics of atmospheric dynamics. Lots of my friends work on biological computational models so the transformations are not all alien to me, but given the correct assumptions we're talking about quite a few parallel models. Naturally... things have their limits.

I live in the near-western burbs. I work downtown in the Loop. I commute on the Metra UP-West line. You're on the south side, I gather?

My son is entering his high school senior year, and is brilliant. He's doing AP level classes in calculus and physics, and finds them boring. He wants to do either physics or math in college, which I'm encouraging him to do. Down the line as he thinks about a career, I'm trying to get him to see the light -- that the best way I know of to combine a love for mathematical models and numerical solutions, with a very lucrative career and comfortable lifestyle, is to be a quant. Had I known 20 years ago, what I know today, I would have aspired to be a quant on Wall St. As it is, I rub shoulders with the guys in our Research division who build our trading models. But I am but a lowly peon Project Manager, who spends his time building Gantt charts and defining schedules and priorities.

I figure anyone who can grok N-S can certainly build binomial trees for Black Scholes models. Option theory is comparatively easy. The tricky part is not the model or the math I think, it's the need to constantly recalculate solutions quickly in real-time to adjust theoretical values as market conditions change, and underlying price and volatility change. To do that you need fast, parallel numerical techniques, great hardware multicore processors, DMA NICs, and low-latency market data systems snd execution venues. I like working at a trading firm, you get to play with lots of neat toys and really really smart people. And they're all focused on making $$$$$!!

[scandal 951]

This thread needs a few more pictures to illustrate the very deep points being made.. don't you think?

[Ban Hammer 13,352]

This thread needs a few more pictures to illustrate the very deep points being made.. don't you think?

the posts are about the only "deep points" made

[one...two...tree 2,762]

I just want to know if you nerds can build a better beer bong? Cuz that would so awesome, dude.

Looks like she is having sex tonight, and regetting it in the morning.

[one...two...tree 2,762]

This thread needs a few more pictures to illustrate the very deep points being made.. don't you think?

What element on the Periodic Chart is she bending over for, for $500 Alex.

[Hilarious Clinton 452]

I just want to know if you nerds can build a better beer bong? Cuz that would so awesome, dude.

ummm why are the guys looking the other way?