Geopotential Height

Geopotential Height

In practice, most measurements of the atmosphere are made of pressure not elevation due to the variation in gravity at different elevations and  latitudes. The pressure contours of the atmosphere are defined by a geopotential height (i.e. a “gravity-corrected elevation”). To understand this mathematically, consider the equation for hydrostatic equilibrium: We can replace by invoking the Ideal Gas Law which can rearranged as This is a solvable ordinary differential equation, which has the measured height as a function of pressure. Here, defines the…

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Parabolic Free Surface

Parabolic Free Surface

Imagine a still pond at the edge of a forest. Make it a beautiful autumn day because that’s a nice thought. The surface is like smooth glass despite the rugged shape of the ground beneath it. This is because the surface of a fluid defines a zero potential height. Essentially the only forces acting on it are gravity and gravity is acting perpendicular to the surface. But what if you could include another force on the pond? How would the…

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Adventures in Your Kitchen Sink: Stream Thinning

Adventures in Your Kitchen Sink: Stream Thinning

Turn on your faucet. Go on, I’ll wait. Make sure you keep the speed of the water low enough that the water still appears clear (i.e. laminar). Now consider the diameter of the stream near the faucet and then closer to the basin. You should notice that the stream thins as it flows down into the sink. How strange! Stream thinning is not just a quirk of your sink! It is, in fact, a consequence of the very fundamental law…

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What does “Frozen-In” Mean?

What does “Frozen-In” Mean?

For highly conductive plasma (conductivity (σ) → ∞), we often say that the plasma is ‘frozen-in’ the magnetic field lines, but what does this mean physically? Let’s begin by examining the mathematical (forgive me) behavior of the flow. From Ohm’s Law and Ampere’s Law1  Solve for E Take the curl of both sides By Faraday’s Law and the vector identify for Recall the Maxwell equation that says that there can be no magnetic monopoles and distribute the negative sign where…

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Parker Spirals

Parker Spirals

If the sun didn’t rotate, then the frozen-in magnetic field would be carried out radially with the solar wind (v = 440 km/s on average). Instead, the sun’s rotation (ω = 2.87 x 10-6 rad/s) drags the magnetic field configuration into a Parker Spiral. The angle between the radial flow and the spiral can be found with simple trigonometry. where the radial component of the magnetic field at some distance from the sun’s center can be found through the conservation…

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