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Many astrophysical plasmas are well modeled as perfect conductors. Ideal MHD assumes this limit. As a result, there is no electric field in the fluid's rest frame. In other frames, we generally have $\vec{E} = -\vec{v} \times \vec{B}$, so there is an electric field. However, the perfect conductivity constraint means we don't have to model the electric field -...

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The lack of the electric field in modeling plasmas stems from the Lorentz force, $$\mathbf F=q\mathbf E+q\boldsymbol\beta\times\mathbf B$$ where $\boldsymbol\beta=\mathbf v/c$. For most astrophysical plasmas, the force is zero, so we have that $$\mathbf E=-\boldsymbol\beta\times\mathbf B$$ So any time we see an electric field, we can simply replace it ...

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The same thing that makes the surface rotate: the fact that it always rotated. Because angular momentum is (almost) conserved our planet has no other option than to always rotate. (Of course in reality this picture is complicated by interactions with rest of the universe, but those are just small corrections). You can ask then why did it rotate in the first ...

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Live on earth is protected from solar wind by the earth's magnetic field. Charged particles from the sun (mostly) penetrate the earth's atmosphere with great velocity. These particles can be trapped by a magnetic field to follow circular path's around the magnetic field lines, thereby losing their energy due to collisions or bremstrahlung. From first ...

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One must be very careful in making the step from $\nabla\cdot\mathbf{B}=0$ to a statement such as "magnetic field lines do not start or end". Consider the field in the region of an X-point type magnetic null (in two dimensions). Take a 'volume' (i.e. an area) centred on the null point, and look at the field lines through the bounding curve. No matter how ...

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We have conservation of energy to contend with. The magnetic energy being released in the upper layers of the suns atmosphere originated in turbulent convection lower down. So in effect the upper layers of the sun (upper here meaning roughly from the photosphere and downward) are acting as a heat engine putting some energy into the suns magnetic field. So ...

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The earth's rotation is slowing down, due primarily to tidal interaction with the moon (transfering angular momentum to the moon). This frictional loss occurs near the surface. On this basis, if the rotational coupling of the inner core to the mantle is imperfect, it would be expected to lag with respect to the slowdown. I think precessional changes (cycle ...

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I think the easiest way is to start off with the conservation of momentum in the conservation-law form: $$\frac{\partial\boldsymbol\pi}{\partial t}=\nabla\cdot\mathbb T=0$$ where $\boldsymbol\pi=\rho\mathbf u$ and $$\mathbb T=\rho\mathbf u\mathbf u+\left(p+\frac12B^2\right)\mathbb I - \mathbf B\mathbf B$$ is the stress tensor and, as discussed before, $\... 5 The Osher paper does define what a weak solution is. We seek a solution$w$of$x$and$t$such that $$\partial_t w + \partial_x f(w) = 0$$ for a known function$f$(the flux function), given initial conditions $$w(x,0) = w_0(x)$$ for known$w_0$, for$-\infty < x < \infty$and$0 < t < \infty$. A weak solution is a bounded measurable ... 4 Due to relativity (hence the effects of Lorentz transformations on the electromagnetic tensor), the magnetic field vanishes, but an "apparent" electric field appears. 3 Q: When we write that, do we suppose a collisionless or collisional nature of the fluids? A: It's the energy-momentum tensor for a perfect fluid Chapter 2.26 Q: If this description corresponds to collisional fluids, why cosmological simulations are N-body simulations (collisionless) and are not simply based on hydrodynamics? A: Cosmological simulations are ... 3 If you consider the case of ideal MHD (perfectly conducting fluid) we have the limiting case where the magnetic field is frozen into the fluid. Thus, manipulating the magnetic field yields manipulation of the fluid and visa-versa. The Richtmyer Meshkov (RM) instability is suppressed in this limiting case by the application of a longitudinal magnetic field (... 3 A charged particle moving in a magnetic field experiences a force f= q*VxB. This force is perpendicular to both the mag field and the direction of motion. If it is a uniform magnetic field charged particles move in circles, and the frequency is called the cyclotron frequency. You can essentially decompose the velocity into a component parrallel to the mag ... 3 Effervescent tablets release$CO_2$, carbon dioxide, into the air when the reaction takes place. For example, sodium-based tablets may contain$NaHCO_3$, the sodium bicarbonate, and it may react with an organic compound – in this case vinegar (acetic acid) like this: $$NaHCO_3 + HC_2 H_3O_2 \to H_2 O + CO_2 + NaC_2 H_3 O_2$$ The water$H_2O$in the final ... 3 Let us start by summarising the governing equations of MHD. We have the reduced form of the Maxwell equations $$\nabla \times \mathbf{B} = \mu \mathbf{J},$$ $$\nabla . \mathbf{J} = 0,$$ $$\nabla \times \mathbf{E} = - \partial \mathbf{B} / \partial t,$$ $$\nabla.\mathbf{B} = 0$$ and the auxillary equations $$\mathbf{J} = \sigma(\mathbf{E} + u \times \... 3 The simple answer is that, if the rod is really full of an incompressible fluid, the centrifugal force will be balanced by a pressure gradient and will not, on its own, create a fluid flow. If the magnetic field is parallel to the axis of rotation, then it will tend to drive one charged species toward the ends of the rod, while the other charged species is ... 3 \nabla\cdot\mathbf B=0 does indicate that there are no magnetic monopoles, so there isn't a "starting" or "ending" point for field lines is mostly correct. So this must mean that magnetic field lines either form closed loops extend to infinity intersect the domain boundary (wall, stellar surface, etc) So the "starting & ending points" issue ... 2 You assume a magnetic field as static? Magnetic fields may represent and even present at level 1X as standing wave functions, which has led classical physical sciences to regard them as static. However, basic researches have shown magnetic fields to be "consistently dynamic" and "coherently interactive" (distributable both algebraically and geometrically ... 2 On a planetary scale the earths surface is a good insulator and we do not lose significant net heat to space or gain net heat from the sun. So heat generated may build up over time. Radioactive decay heat and gravitationally produced friction would tend to melt the interior of a sufficiently large rocky planet. Heavy elements would sink toward the center ... 2 According to our current understanding of the formation of planets, they are created from dust which originates in a previous supernova (or other large) explosion. This is called a nebula. http://en.wikipedia.org/wiki/Nebular_hypothesis As this nebula contracts into planets, there are generally two possible cases: that the dust that is eventually going to ... 2 Try looking at the figures in a 2005 simulation by Takahashi et.al. in Science magazine, that at least show recurring reversals at http://www.sciencemag.org/cgi/content/full/309/5733/459?cookietest=yes. Given that the viscosity, structure, and heat generation of the core are all to some degree unknown, and that the process may depend upon parametric ... 2 Let's start with a few issues considering Amperè's law. Amperè's law describes the magnetic field generated by current. The current can be localized at a certain point in space, but its magnetic field is spread everywhere. Applying Amperè's law to a certain point where current is zero does not generate a magnetic field. However, it doesn't necessarily mean ... 2 There exists a basic asymmetry: there are no magnetic monopoles of the number and dimensions that the electric monopoles exist ( there are theories with magnetic monopoles and people are looking for them but we are talking of masses much larger than electrons and quarks) . Hand waving, (because I have not checked the math just extending the symmetry that ... 2 Probably because matter on a large scale is electrically neutral and therefore the electric effects cancel out .This asymmetry arises from the fact that atom as a whole is electrically neutral. 2 This has a three-part answer. The first part concerns large-scale, quasi-static electric fields. The nice thing about electric fields is that one can always do a simple galilean transformation to a frame where this quasi-static electric field does not exist. So that is the first part (not very satisfactory, but true and practical). The second part ... 2 Background In the absence of an electric field, a charged particle experiences a force that is perpendicular to the magnetic field and its velocity relative to that field, called the Lorentz force. This is given by:$$ \mathbf{F}_{s} = q_{s} \ \mathbf{v}_{s} \times \mathbf{B} \tag{1}$$where$q_{s}$is the charge of species$s$,$\mathbf{v}_{s}$is the ... 1 Joule heating is typically associated with increases in random kinetic energy (i.e., heat) due to$\mathbf{j} \cdot \mathbf{E}\$. Ohmic dissipation and resistive heating are similar in a sense to Joule heating, as all three result from fluctuating electric fields acting as an effective drag force on an otherwise free flowing charged particle. Ion drag is ...

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