Some aspects of Vorticity fields in Relativistic and Quantum Plasmas - PowerPoint PPT Presentation

ICTP-IAEA College on Plasma Physics, 2016 Some aspects of Vorticity fields in Relativistic and Quantum Plasmas Felipe A. Asenjo 1 Universidad Adolfo Ib´ a˜ nez, Chile ◮ Part I: Non-relativistic and Special relativistic Plasmas ◮ Part II: General relativistic Plasmas ◮ Part III: Quantum and Quantum Relativistic Plasmas 1 felipe.asenjo@uai.cl

ICTP-IAEA College on Plasma Physics, 2016 Part I: VORTICITY IN NON-RELATIVISTIC AND SPECIAL RELATIVISTIC PLASMAS Felipe A. Asenjo

Today... ◮ We explore the concept of vorticity fields in electromagnetism ◮ We introduce the concept of vorticity fields in a plasmas ◮ We study the generation of vorticity ◮ We introduce the concept of helicity

Maxwell equations

Maxwell equations Dynamics ∂ B = −∇ × E ∂ t ∂ E ∂ t + J = ∇ × B Constraints ∇ · B = 0 ∇ · E = ρ E , B electric and magnetic fields ρ, J charge and current densities (sources)

Maxwell equations The dynamics is consistent with the constraints ∇· → ∂ B = −∇ × E = ⇒ ∇ · B = 0 ∂ t ∇ · ∂ E ⇒ ∂ ∂ t ∇ · E = ∂ρ ∂ t + J = ∇ × B = ∂ t = −∇ · J via the continuity equation And they produce the wave-like equations � ∂ 2 � ∂ t 2 − ∇ 2 B = ∇ × J � ∂ 2 � E = − ∂ J ∂ t 2 − ∇ 2 ∂ t − ∇ ρ

Electromagnetic fields and potentials ∇ · B = 0 = ⇒ B = ∇ × A � ∂ A � ∂ B ⇒ ∂ A ∂ t = −∇ × E = ⇒ ∇ × ∂ t + E = 0 = ∂ t + E = −∇ φ

Electromagnetic fields and potentials ∇ · B = 0 = ⇒ B = ∇ × A The magnetic field is the vorticity of the electromagnetic field ∂ B � ∂ A � ⇒ ∂ A ∂ t = −∇ × E = ⇒ ∇ × ∂ t + E = 0 = ∂ t + E = −∇ φ The no-sources Maxwell equations become indetically satisfied The sources Maxwell equations are written as � ∂ 2 � � ∂φ � ∂ t 2 − ∇ 2 A = J − ∇ ∂ t + ∇ · A � � ∇ φ − ∂ A ∇ · = ρ ∂ t If Lorentz gauge is used ∂ t φ + ∇ · A = 0, then � ∂ 2 � ∂ 2 � � ∂ t 2 − ∇ 2 ∂ t 2 − ∇ 2 A = J , φ = ρ

Vorticity The vorticity field is any psedovector that is the rotational (curl) of a vector field (potential).

Magnetic helicity The vorticity field has associated a quantity called helicity � A · B d 3 x h = such that � ∂ A ∂ h A · ∂ B � ∂ t · B d 3 x + ∂ t d 3 x = ∂ t � � ( − E − ∇ φ ) · B d 3 x − A · ∇ × E d 3 x = � � E · B d 3 x − ( φ B + E × A ) · d 2 x ≡ − 2 � E · B d 3 x ≡ − 2

Non-relativistic plasma

Non-relativistic plasma fluid Fluid equation � ∂ � v = q ( E + v × B ) − 1 ∂ t + v · ∇ n ∇ p m Maxwell equations ∂ B = −∇ × E ∂ t ∂ E ∂ t + J = ∇ × B And an equation of state

Non-relativistic plasma fluid We re-write the fluid equation as m ∂ v ∂ t − m v × ( ∇ × v ) = q ( E + v × B ) − 1 2 ∇ v 2 − 1 n ∇ p where we have used a × ( ∇ × b ) = ( ∇ b ) · a − ( a · ∇ ) b m ∂ v � � �� B + m − 1 2 ∇ v 2 − 1 ∂ t = q E + v × q ∇ × v n ∇ p It appears the interesting field Ω = B + m q ∇ × v = ∇ × P that will be a generalized vorticity with the potential [the canonical momentum] P = A + m q v

Generalized vorticity Taking the curl of the previous equation � 1 ∂ ∇ × v � m = ∇ × E + ∇ × ( v × Ω) − 1 2 q ∇ × ∇ v 2 − ∇ × qn ∇ p ∂ t q can be written as ∂ Ω 1 ∂ t − ∇ × ( v × Ω) = qn 2 ∇ n × ∇ p and ∂ P ∂ t − v × Ω = − 1 qn ∇ p − ∇ φ

Fluid helicity The helicity associated to the fluid is � P · Ω d 3 x h = which satisfies � ∂ P ∂ h P · ∂ Ω � ∂ t · Ω d 3 x + ∂ t d 3 x = ∂ t � � � v × Ω − 1 · Ω d 3 x = qn ∇ p − ∇ φ � � � ∇ × ( v × Ω) + 1 d 3 x + P · qn 2 ∇ n × ∇ p � � � 1 1 2 qn ∇ p · Ω d 3 x + qn 2 P · ∇ n × ∇ p d 3 x ≡ − ≡ − qn ∇ p · Ω the helicity is conserved if p = p ( n ) . 2 2 Mahajan & Yoshida, Phys. Plasmas 18 , 055701 (2011).

Sources for Generalized vorticity ∂ t Ω − ∇ × ( v × Ω) = 1 n 2 ∇ n × ∇ p If p = p ( n ) , then ∇ n × ∇ p = 0 ∂ Ω ∂ t − ∇ × ( v × Ω) = 0 Therefore, if initiallhy the vorticity is null, it remains null for all times If ∇ n × ∇ p � = 0, then the term 1 n 2 ∇ n × ∇ p is so-called Biermann battery. It can generate vorticity from plasma thermodynamical inhomogenities.

◮ The conservation of helicity establishes topological constraints. It can forbid the creation (destruction) of vorticity in plasmas. ◮ We can see that the generalized helicity remains unchanged in ideal dynamics. This conservation implies serious contraints on the origin and dynamics of magnetic fields. ◮ Otherwise, the nonideal effects can change the helicity. For example, if gradients of pressure and temperature have different directions [Biermann battery]. ◮ An anisotropic pressure tensor may also generate vorticity.

Special relativistic plasma

Special Relativistic plasma fluids For relativistic plasmas there exist also a generalized voticity and a fluid helicity. Now the relativistic plasma fluid is a little more complicated. We have to consider: ◮ the rest-frame density of the lfuid n . ◮ the energy density of the fluid ǫ . ◮ the pressure of the fluid p . ◮ the enthalpy density of the fluid h = ǫ + p . ◮ the relativistic velocity, through the Lorentz factor γ = ( 1 − v 2 ) − 1 / 2 . ◮ coupled to Maxwell equations via the current density n γ v .

Special Relativistic plasma fluids - covariant form The relativistic ideal plasma description can be obtained from the conservation of the ideal fluid energy-momentum tensor ∂ ν T µν = 0, with T µν = ( ǫ + p ) U µ U ν + p η µν with U µ U µ = − 1 η µν = ( − 1 , 1 , 1 , 1 ) such that in the rest-frame, where U µ = ( 1 , 0 , 0 , 0 ) , we find T 00 = ǫ T 0 i = 0 T ij = p δ ij The equation for the plasma fluid is U ν ∂ ν ( mfU µ ) = qF µν U ν − 1 n ∂ µ p with f = ǫ + p mn

Special Relativistic plasma fluids - covariant form Also we have the continuity equation ∂ µ ( nU µ ) = 0 and Maxwell equations ∂ ν F µν = qnU µ

Magnetofluid Unification 3 Instead of solving the previous equations, let us look the big picture. The covariant fluid equation can be cast in the form qU ν M µν = T ∂ µ σ where the magnetofluid tensor is M µν = F µν + m q S µν with S µν = ∂ µ ( fU ν ) − ∂ ν ( fU µ ) and the entropy density follows ∂ µ σ = 1 nT ( ∂ µ p − mn ∂ µ f ) For an ideal relativistic gas f = K 3 ( m / T ) / K 2 ( m / T ) 3 Mahajan PRL 90 , 035001 (2003); Mahajan & Yoshida, PoP 18 , 055701 (2011).

Magnetofluid tensor (why is important) M µµ ≡ 0 M 0 i → ξ = E − m q ∂ t ( f γ v ) − m q ∇ ( f γ ) M ij → Ω = B + m q ∇ × ( f γ v ) The magnetofluid tensor is the natural extension to the covariant form of the plasma vorticity.

Magnetofluid tensor (why is important) M µµ ≡ 0 M 0 i → ξ = E − m q ∂ t ( f γ v ) − m q ∇ ( f γ ) M ij → Ω = B + m q ∇ × ( f γ v ) The magnetofluid tensor is the natural extension to the covariant form of the plasma vorticity. Equation qU ν M µν = T ∂ µ σ is the covariant vorticity equation for the plasma. ∂σ ⇒ v · ξ = − T (For µ = 0) = q γ ∂ t ⇒ ξ + v × Ω = T (For µ = i ) = q γ ∇ σ

Defining the potential (generalized canonical momentum) P µ = A µ + m q fU µ = ( P 0 , P ) then M µν = ∂ µ P ν − ∂ ν P µ In this way ξ = − ∂ P ∂ t − ∇P 0 , Ω = ∇ × P ⇒ ∇ × ξ = − ∂ Ω ⇒ 1 2 ε αβµν ∂ β M µν = 0 = ∂ t ⇐

Defining the potential (generalized canonical momentum) P µ = A µ + m q fU µ = ( P 0 , P ) then M µν = ∂ µ P ν − ∂ ν P µ In this way ξ = − ∂ P ∂ t − ∇P 0 , Ω = ∇ × P ⇒ ∇ × ξ = − ∂ Ω ⇒ 1 2 ε αβµν ∂ β M µν = 0 = ∂ t ⇐ ∂σ ⇒ v · ξ = − T (For µ = 0) = q γ ∂ t ⇒ ∂ P ∂ t − v × Ω = − T q γ ∇ σ − ∇P 0 (For µ = i ) = This last equation is the potential equation for the vortical dynamics!

Generalized relativistic vorticity and its dynamics Ω = ∇ × P = B + m q ∇ × ( f γ v ) � T ∂ Ω � ∂ t − ∇ × ( v × Ω) = −∇ × ∇ σ q γ ◮ The Generalized voticity has both kinematical and thermal relativistic corrections [NR limit γ → 1, f → 1]. ◮ The vortical dynamics contains those corrections. It appears a more general battery.

Generalized relativistic helicity 4 K µ = 1 2 ε µναβ P ν M αβ 4 Mahajan PRL 90 , 035001 (2003)

Generalized relativistic helicity 4 K µ = 1 2 ε µναβ P ν M αβ 1 2 ε µναβ ∂ µ P ν M αβ + 1 ∂ µ K µ 2 ε µναβ P ν ∂ µ M αβ = ε µναβ ∂ µ P ν M αβ = the Generalized helicity � � � K 0 d 3 x = ε 0 ijk P i M jk d 3 x = P · Ω d 3 x h ≡ 4 Mahajan PRL 90 , 035001 (2003)