Hydrodynamic Scaling at ZBL

Parameters selected for 1kJ shot into 2X Xe cell

Conditions for validity of MHD

Initial conditions

Parameters for hydrodynamic scaling are selected from the region close to the physical process of interest (Ryutov 1999).  Values here are immediate postshock parameters selected from analytic blast wave model at the time when the blast wave reaches scale height.  Fluid velocity is approximated as shock velocity at high compression ratio.  All units are CGS except where specified - in particular, T is given in eV.

As of this writing the Sedov model is used due to uncertainties in inputs for the Liang/Kielty model.

h=0.635

A=131

γ=1.06

T=20.95

Z=0.55(Sqrt[T])

p=5.2863 (10^8)

ρ=0.0034

v=2.33343 (10^6)

nA=6.022141 (10^23)

ni=ρ ((1/A)) nA

ne=Z ni

B=092300

Λ=1.45

α=1

Collisionality

The localization condition necessary for hydrodynamic treatment of the plasma can be met through collisions or magnetic fields.  The smaller of the following parameters must be much less than one:

IonLocalization:=(10^(-4))(v/(B h))

IonLocalization

Collision:=8 (10^(-12))((v^4)/(Λ h ni))

Collision

Heat Conduction

The hydrodynamic equations require that convective thermal transport dominates thermal conduction.  This is expressed by requiring the Peclet number to be large:

Pe:=((h v)/χ)

The thermal diffusivity varies depending upon whether the electrons are magnetized.  The effective thermal diffusivity can be taken to be the minimum of the following values.  For unmagnetized electrons,

χ:=3.3 (10^(-3)) ((A (T^(5/2)))/(Λ Z(Z+1)ρ))

χ

For magnetized electrons,

χmag:= 8.6 (10^9)α(( (Sqrt[A]))/Z)(T/B)

χmag

Pe

PeMag:=((h v)/(χmag))

PeMag

χ/χmag

Radiation Flux

Ordinary hydrodynamics requires energy losses due to radiation to be negligible.  This is expressed analogously to heat conduction by requiring the radiation Peclet number to be large.  For radiative shock experiments, however, the radiation must be dynamically important.  The effect of radiation is conventionally taken into account in the lowered effective γ in the postshock gas.  Additionally, the modification of the self-similar Sedov solution due to Liang & Kielty reporduces the slowing of the shock trajectory due to constant energy loss at the shock front.  Cf. analytic blast wave model for detail & references.

Viscosity

Ideal hydrodynamics require that viscosity is negligible, with the condition that the Reynolds number is large:

Rey:=((h v)/ν)

The kinematic viscosity is given by ν.  All sources of viscosity must be added.  The photon viscosity is given by

νrad:=3 (10^(-9))(A/Z)((T^4)/(ρ^2))

νrad

The particle viscosity depends on magnetization, and the smaller of the following two values should be taken.  In the collisional limit,

νi:=3.3 (10^(-5))(((Sqrt[A])(T^(5/2)))/(Λ (Z^4) ρ))

νi

In the magnetized case,

νimag:=2 (10^8)((α T)/(Z B))

N[νimag]

ν:=νi+νrad

Rey

Magnetic Viscosity

Ideal MHD requires ohmic dissipation to be negligible.  This is expressed by requiring the magnetic Reynolds number to be large.  In practical units (Ryutov 1999) (kT in eV):

ReM:=0.08(Sqrt[((Z+1)/((Z^2)A))])h (T^2)

ReM

Hydrodynamic scaling parameters

Provided the above conditions are met, it can be shown (Ryutov 1999, 2000) that two systems which obey the ideal Euler or ideal MHD equations will show similar time evolution if their respective Euler number and β are similar.  The Euler number, as defined by Ryutov, is given by

Eu:=v(Sqrt[(ρ/p)])

Eu

β is given by

β:=((8 π p)/(B^2))

β

Mach Numbers

ρ0=0.0003
p0=55661.8

Cs=(((((γ p0)/(ρ0))))^(1/2))

M=(v/Cs)

vA=(B/(((4π ρ0))^(1/2)))

M_A=(v/vA)

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