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−µc j D c,n + c E × H + E D × H 0 +〈ρvn − j D n 〉∆µ − c E 0 2 × H D 2 · n · n , (124) where 2 denotes the dielectric, ie n points into the dielectric. The entropy production is now R sf = fn ∆T + 〈ρc vn − j D c,n〉∆µc + 〈vn gj〉−Π D jn ∆vt,j +〈ρvn − j D n 〉 ∆µ eff + n × E D + E · c ∆H 0 t + H D 2 + v2/c × D2 · c E 0 2 × n . (125) This Rsf yields 9 boundary conditions. Ignoring cross terms, we have especially n × E D + E = βsc ∆H 0 t, βs > 0 . (126) Here we have altogether 18 boundary conditions. They suffice to determine the 7 collective modes of the conductor and the 9 collective modes of the dielectric, to fix the normal component of B + B D and to determine the lab velocity of the interface. 5 Two Illustrative Examples The hydrodynamics is not simply a coarse-grained theory; it has a scaleinvariance built in: Varying the resolution, the hydrodynamics remains valid. This feature makes the theory especially general and widely applicable. The same applies to the boundary conditions derived above, which are valid where ever the bulk theory is. Below are two examples for the vacuum-conductor interface, where we shall calculate some coefficients from two simple models. In both cases, we shall employ a higher resolution hydrodynamic theory to obtain the coefficient of the coarser-grained one. More specifically, we shall in the first example calculate the coefficient in the boundary condition Eq(126), and understand the discontinuity of the coarsegrained magnetic field H as given by a highly conducting surface layer (and surface current) that is not explicitly accounted for in the given resolution. This is of course a trivial case, but it does serve to draw one scenario in which the discontinuity of the magnetic field is not surprising. The crucial point here is that the coarse-grained boundary conditions as presented above are applicable also for situations where a higher resolution theory is not easily 20
available, such as for a turbulent boundary layer. If we check our ambitions with respect to resolving the turbulence in details, this layer and its turbulence enhanced magnetic diffusivity should be well accounted for by the same boundary condition. The second example is less straight forward. Here, we employ the hydrodynamic theory for a dielectric as the higher resolution description of the conductor. This sounds somewhat surprising at first and should be explained. The dielectric theory (including a finite conductivity) contains a stationary, collective mode (the sq-mode) that is given by an exponential decay of the electromagnetic field from the system’s surface [4]. Because the decay length shrinks with the conductivity, the hydrodynamic theory for conductors does not contain this mode, (though this may be amended by including higher order gradient terms). Nevertheless, the derived boundary conditions are such that the long ranged effects of the sq-mode are well accounted for, albeit on a grain size on which the sq-mode is no longer resolved. This statement is explicitly proven by comparing the dielectric with the conductor theory. 5.1 The Highly Conducting Surface Layer We consider an infinitely long and conducting wire (region 1), of r0, which is located in a vacuum (region 2), and subjected to a parallel, constant and homogeneous electrical field E = E0 ez. coarse-grained theory Within the wire we have E D = const ez = E0 ez, (127) and from Eq(76) ∇×H = 1 βc ED , (128) in vacuum we have ∇×H =0. (129) The connecting condition, Eq(126), is 21
Page 1 and 2: Boundary Conditions of the Hydrodyn
Page 3 and 4: fields are involved. This is the re
Page 5 and 6: ∂νΠ µν = 0 (7) and is symmetr
Page 7 and 8: Putting all this together, the Π 0
Page 9 and 10: (E, B) µν ⎛ ⎞ ⎜ 0 −Ex −
Page 11 and 12: 0= ˙ D + j D el + ρel v − c ∇
Page 13 and 14: The hydrodynamic equations are with
Page 15 and 16: vt = v − (v · n) n , (82) vn =(v
Page 17 and 18: ⎛ ⎜ ⎝ −ΠD,eff t,i HD × n
Page 19: ∆(D D n + Dn) =−σsf , (118)
Page 23 and 24: To obtain a finite βs, we add a th
Page 25 and 26: with H D z = αc A e−x/λ λ A =

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