MHD Rankine Vortex: an exact solution
and implementation for solar chromosphere
Alexey V. Byalko
Landau Institute of Theoretical Physics RAS,
Chernogolovka, Moscow oblast, 142432 Russia
& journal “Priroda” RAS
email: alexey@byalko.ru
www.byalko.com/alexey
Rankine vortex in general hydrodynamic
Rankine W.J.M. Manual of Applied Mechanics, London, 1858.
Byalko A.V. Underwater Gas Tornado and Its Possible Occurrence in Nature, Procedia IUTAM 8 , 51 – 57, 2013.
𝑟
𝑣𝜑 = 𝑣0 𝑓
; 𝑓Rankine 𝑥 =
𝑟0
𝑥 at 𝑥 < 1;
𝑥 −1 at 𝑥 > 1.
The viscosity term for Rankine vortex in NavierStokes equations is equal to zero!
∆𝑣𝜑 −
𝑣𝜑
𝜕𝑣𝜑
𝑣𝜑
𝜈 𝜕
0 if 𝑟 < 𝑟0
≡
𝑟
−
𝜈
=
0 if 𝑟 > 𝑟0
𝑟2
𝑟 𝜕𝑟
𝜕𝑟
𝑟2
Stable 2-dimentional vortex
Jason L. et al. Universal Profile of the Vortex Condensate in Two-Dimensional Turbulence PRL
113, 254503 (2014)
An inverse turbulent cascade in a restricted
two-dimensional periodic domain creates a
condensate—a pair of coherent system-size
vortices.
The vortices have a universal internal structure
independent of the type of small-scale
dissipation, small-scale forcing, and boundary
conditions.
Radial profile of the mean polar velocity
U in log-lin coordinates.
Burgers—Rott vortex is an exact solution of Navier-Stokes equations
𝑟
2𝜈
𝑣𝑟 = −𝛼𝑟; 𝑣𝑧 = 2𝛼 𝑧 − 𝑧0 ; 𝑣𝜑 = 𝑣0 𝑓Burges−Rott ( ); 𝑟02 =
.
𝑟0
𝛼
where 𝛼 > 0; 𝑣0 , 𝑧0 are arbitrary constants and
𝑥 at 𝑥 ≪ 1;
𝑥 −1 at 𝑥 ≫ 1.
2
The viscosity term for Burges—Rott vortex is proportional to 𝜈 1 − 𝑒 −𝑥 (1 + 𝑥 2 − 𝑥 4 ) 𝑥 −3
(dashed curve), it is located near the vortex core x=1.
𝑓Burgers−Rott 𝑥 = 1 − 𝑒 −𝑥
2
𝑥 −1 =
Energy per unit height is distributed over all vortex nearly
uniformly:
𝑅 𝜌 𝑧 𝑣2
𝑅
𝜑
𝜀 𝑧 = 2𝜋
𝑟𝑑𝑟 = 𝜋𝑎2 𝜌 𝑧 ln 𝑞 .
2
𝑟0
0
𝑞Rankine = 𝑒 = 1.65; 𝑞B−R = 1.335; 𝑞2D ≈ 𝑒 = 2.72.
Fig.: Vortexes: Rankine (blue), Burges—Rott (red), 2dimentional (green). Viscosity term for Burges—Rott
(dashed)
MHD equations in cylindrical coordinates
An example: Bisnovatyi-Kogan G. S. Mon.Not.R. Astron. Soc. 376, 457–464, 2007.
1) All derivatives 𝜕/𝜕𝜑 → 0; 2) 𝐵𝑟 = 0. Then from equation div𝑩 = 0 we have 𝜕𝐵𝑧 /𝜕𝑧 = 0.
Two remained magnetic field equation are:
𝜕𝐵𝜑 𝜕 𝑣𝜑 𝐵𝑧
𝐵𝜑 𝜕𝑣𝑟
=
−
;
𝜕𝑡
𝜕𝑧
𝜕𝑟
𝜕𝐵𝑧
𝜕𝐵𝑧
= −𝑣𝑟
;
𝜕𝑡
𝜕𝑟
We will consider 𝐵𝑧 = const.
3) Currents are proportional to
rot𝑩 = −
𝜕𝐵𝜑
𝜕𝐵𝑧
1 𝜕(𝑟𝐵𝜑 )
,−
= 0,
.
𝜕𝑧
𝜕𝑟
𝑟 𝜕𝑟
To minimize currents in the core let us take
𝑟
;
𝑟0
𝑟
𝐵𝜑 𝑡, 𝑧, 𝑟 = 𝐵𝑧 𝑏 𝑡, 𝑧 𝑓
.
𝑟0
𝑣𝜑 𝑡, 𝑧, 𝑟 = 𝑎 𝑡, 𝑧 𝑓
Then at 𝑟 > 𝑟0 𝜕(𝑟𝐻𝜑 )/𝜕𝑟 = 0. It means that vertical current is in the core only. But radial
current flows everywhere.
Velocity MHD-equations linear on 𝑣𝑟 , 𝑣𝑧
𝑩 = 0, 𝐵𝜑 , 𝐵𝑧 ; 𝒗 = 𝑣𝑟 , 𝑣𝜑 , 𝑣𝑧 ; 𝑣𝑟 , 𝑣𝑧 ≪ 𝑣𝜑 ≪ 𝑢.
𝜕 ln 𝜌
𝜕 ln 𝜌
𝜕 ln 𝜌
𝜕𝑣𝑟 𝑣𝑟 𝜕𝑣𝑧
+ 𝑣𝑟
+ 𝑣𝑧
=−
− −
;
𝜕𝑡
𝜕𝑟
𝜕𝑧
𝜕𝑟
𝑟
𝜕𝑧
𝜕𝑣𝑟 𝑣𝜑2
𝜕 ln 𝜌
−
= −𝑢2
;
𝜕𝑡
𝑟
𝜕𝑟
𝜕𝑣𝜑
𝜕𝑣𝜑
𝐵𝑧 𝜕𝐵𝜑
+ 𝑣𝑟
=
;
𝜕𝑡
𝜕𝑟
4π𝜌 𝜕𝑧
𝐵𝜑 𝜕𝐵𝜑
𝜕𝑣𝑧
𝜕𝑝
=𝑔−
+
→ 𝜌 = 𝜌(𝑧).
𝜕𝑡
𝜌𝜕𝑧 4π𝜌 𝜕𝑧
Here 𝑢2 = 𝜕𝑝/𝜕𝜌 𝑠 is the sound velocity and density far from vortex core is a known function
of height 𝜌 = 𝜌∞ (𝑧).
Radial and vertical velocities we consider as small comparing with 𝑣𝜑 . In the first approximation
let us take 𝑣𝑟 = 0 and 𝑣𝑧 = 0. Then for functions a and b we have:
𝜕𝑏
𝜕𝑎 𝜕𝑎
𝐵𝑧 𝜕𝑏
= 𝐵𝑧
;
=
.
𝜕𝑡
𝜕𝑧 𝜕𝑡 4π𝜌(𝑧) 𝜕𝑧
If 𝜌 = const then a simple solution is the wave with Alfven velocity
𝑢𝐴 (𝑧) = 𝐵𝑧 / 4π𝜌(𝑧).
But we found an exact (quaziclassical) solution in media with changeable density.
Quazi-classical approximation
Let us search solution in media with changing density in a form
𝑟
𝑟
𝑣𝜑 𝑡, 𝑧, 𝑟 = 𝑎 𝑡, 𝑧 𝑓
; 𝐵𝜑 𝑡, 𝑧, 𝑟 = 𝐵𝑧 𝑏 𝑡, 𝑧 𝑓
.
𝑟0
𝑟0
𝑖𝜔𝑏 = 𝑎′ ; 𝑖𝜔𝑎 = 𝑢2 𝑏′.
𝑧
𝑧
𝑑𝑥
𝐶
𝑑𝑥
𝑎 = 𝐶 exp 𝑖𝜔𝑡 − 𝑖𝜔
+ 𝐷 𝑧 ;𝑏 = −
exp 𝑖𝜔𝑡 − 𝑖𝜔
+𝐷 𝑧 .
𝑢
𝑥
𝑢(𝑧)
𝑢
𝑥
𝑧0
𝑧0
𝑖𝜔
𝑢′ 𝑖𝜔
′
2
2
′
𝑎 = 𝑎 𝐷′ −
= 𝑖𝜔𝑏; 𝑢 𝑏′ = 𝑢 𝑏 𝐷 − −
= 𝑖𝜔𝑎.
𝑢
𝑢
𝑢
𝑢𝐷′ 2 − 𝑢𝐷′ 𝑢′ + 2𝑖𝜔 + 𝑖𝜔𝑢′ = 0.
𝑧
𝜔
𝑖𝜔𝑢′
𝑢′
1
𝑑𝑥
′
If ′ ≫ 1 then 𝑢𝐷 = ′
→ + 2𝑖𝜔; 𝐷 = ln 𝐶𝑢 + 2𝑖𝜔
;
𝑢
𝑢 + 2𝑖𝜔
2
2
𝑢
𝑥
𝑧0
𝑧
𝜔
𝜔2
𝑑𝑥
′
′
If ′ ≪ 1 then 𝑢𝐷 = 𝑢 + 2𝑖𝜔 + 𝑂
; 𝐷 = ln 𝐶𝑢 + 2𝑖𝜔
;
𝑢
𝑢′
𝑢
𝑥
𝑧0
𝑧
𝑧
𝑑𝑥
𝑑𝑥
𝑎 = 𝐶𝑢(𝑧) exp 𝑖𝜔𝑡 + 𝑖𝜔
; 𝑏 = −𝐶𝐵𝑧 exp 𝑖𝜔𝑡 + 𝑖𝜔
.
𝑢
𝑥
𝑢
𝑥
𝑧0
𝑧0
Ergo: starting with upward solution we receive the downward one with different predexponent.
An exact (quazi-classical) MHD solution: a torsion Alfven waves
The upward wave:
𝑧
𝑟
𝑑𝑥
𝑣𝜑 𝑡, 𝑧, 𝑟 = 𝑣𝜑0 𝑓
exp 𝑖𝜔𝑡 − 𝑖𝜔
;
𝑟0
𝑢
𝑥
𝑧0
𝑧
𝑣𝜑0
𝑟
𝑑𝑥
𝐵𝜑 𝑡, 𝑧, 𝑟 = 𝐵𝑧
𝑓
exp 𝑖𝜔𝑡 − 𝑖𝜔
.
𝑢(𝑧) 𝑟0
𝑢
𝑥
𝑧0
The downward wave:
𝑧
𝑢(𝑧)
𝑟
𝑑𝑥
𝑣𝜑 𝑡, 𝑧, 𝑟 = 𝑣𝜑0
𝑓
exp 𝑖𝜔𝑡 + 𝑖𝜔
;
𝑢(𝑧0 ) 𝑟0
𝑢
𝑥
𝑧0
𝑧
𝑟
𝑑𝑥
𝐵𝜑 𝑡, 𝑧, 𝑟 = 𝐵𝑧 𝑓
exp 𝑖𝜔𝑡 + 𝑖𝜔
.
𝑟0
𝑢
𝑥
𝑧0
Here 𝑢𝐴 (𝑧) = 𝐵𝑧 / 4𝜋𝜌(𝑧) is the Alfven velocity increasing with height
z. There are 2 integration constants: velocity amplitude 𝑣𝜑0 (a function
of frequency w): and height 𝑧0 .
Magnetic field amplitude is proportion to velocity amplitude but
decreasing.
This solution has strong dispersion: the wave length is increasing with
height – for the solar atmosphere exponentially!
Solar atmosphere velocities: sound (blue), turbulent (green), Alfven (red) and its
exponential upproximation (dashed). Pay attention that sound, Alfven and
turbulent velocities becomes of the same order at transition region (z = 1850 km).
Graphical presentation of torsion
Alfven upward wave
The torsion Alfven upward wave in dependence on height
z and time t for a frequency with a period T = 10 min.
Radial dependence of vortex density:
𝜕𝑣𝑟
𝜕𝑡
−
2
𝑣𝜑
𝑟
= −𝑢2
𝜕 ln 𝜌
.
𝜕𝑟
Neglecting small radial velocities we can solve its time-averaged equation. There is no
magnetic field in it; the radial variable could be separated from z and t. Thus we can
integrate it to obtain the density dependence:
2
𝑣𝜑2 (𝑟, 𝑡, 𝑧) 𝑡 𝑣𝜑0
𝜕 ln 𝜌
𝑟 1
2
=
= 2 𝑓
.
𝜕𝑟
𝑢2 𝑟
𝑟
𝑟
𝑢𝑠
0
2 ∞
𝑣𝜑0
𝑑𝑥
2
𝜌 𝑟 = 𝜌∞ exp −
𝑓 𝑥
.
𝑢𝑠
𝑥
𝑟
The question is can we treat angular velocities
as not small comparing with solar sound
velocity 𝑢𝑠 = 7 ÷ 8 km/s. The full answer is
not clear. I will suppose – yes, 𝑣𝜑 can reach
sound velocity at least by the order of
magnitude. It does not contradict to conditions
of Euler equations, since there is no density
contrast.
Since density near the core are lower than
outside the vortex the Alfven velocity is higher
in the core. It could lead to self-focusing of
torsion MHD waves.
Radial dependences of density for the
Burgers—Rott vortex in cases:
𝑣𝜑0 /𝑢𝑠 = 0.75 (blue line);
𝑣𝜑0 /𝑢𝑠 = 1.0 (yellow line);
𝑣𝜑0 /𝑢𝑠 = 1.25 (red line).
Radial velocities (km/s)
Wave energy transfer
For energy flux we study the upward wave only. Then the upward energy transfer from a
point (r, z) is equal
𝑧
1
1 𝑣02 𝐵𝑧2 2 𝑟
2
𝑞𝑧 =
𝑣 𝐵 𝐵 =
𝑓
cos 𝜔𝑡 − 𝜔 𝑑𝑥/𝑢(𝑥) .
4π 𝜑 𝜑 𝑧 4π 𝑢𝐴 𝑧
𝑟0
𝑧0
The upward energy flux density 𝑞𝑧 𝑟, 𝑧 is always positive, it pulsates with double frequency
2𝜔. Let us average it over time changing cosin squared => ½.
1 𝑣02 𝐵𝑧2 2 𝑟
𝑞𝑧 𝑡 =
𝑓
.
8π 𝑢𝐴 𝑧
𝑟0
Then we integrate the flux over vortex area and over the all frequency spectrum. The
resulting energy flux 𝑄𝑧 is equal to
𝑅
𝑄𝑧 =
0
𝑣02 𝑟02 𝐵𝑧2
𝑦𝑅
𝑅
𝑞𝑧 𝑡 2π 𝑟𝑑𝑟 =
ln
= 𝜋 𝑣02 𝑟02 𝜌 𝑧 𝑢𝐴 𝑧 ln 𝑦 .
2𝜌 𝑧 𝑢𝐴 𝑧
𝑟0
𝑟0
Here
𝑦Rankine = 𝑒 = 1.65; 𝑦B−R = 1.33; 𝑦2D = 𝑒 = 2.72.
Comparing it with the energy density of the vortex itself
𝑅
𝑅
𝑟
𝑦𝑅
2
2
𝜀 𝑧 =𝜋
𝜌 𝑧 𝑣𝜑 𝑟𝑑𝑟 = 𝜋𝜌 𝑧 𝑣0
𝑓2
𝑟𝑑𝑟 = 𝜋𝜌 𝑧 𝑣02 𝑟02 ln
.
𝑟
𝑟
0
0
0
0
we have
𝑄𝑧 𝑧 = 𝜀 𝑧 𝑢𝐴 𝑧 .
It means a remarkable relation:
The wave energy flux from a vortex is equal to its self energy density times
Alfven velocity.
It is correct at any height.
Net radiative cooling rates (NRCR), or net radiative losses is sum over all lines and continua.
In a steady state, the total net radiation cooling rate must be balanced by the non-radiative
(mechanical or magnetic) heating needed to produce the given atmosphere. Negative total rate
in the temperature minimum region occurs mainly due to ion H–. (E. Avrett et al.).
The problem is to find a way to transfer energy flux F2 from photosphere to chromosphere.
Photosphere
Chromosphere
Transition region
Solar flux: FS = 6.3153 107 Wm-2;
Integrated fluxes:
F1 = 1.28 108 Wm-2 = 2FS;
F2 = – 1.4 105 Wm-2 = – 2 10-3 FS;
F3 = 5.6 103 Wm-2 ~ 0.01% FS.
Net radiative cooling rates (NRCR), or net radiative losses is sum over all lines and continua.
In a steady state, the total net radiation cooling rate must be balanced by the non-radiative
(mechanical or magnetic) heating needed to produce the given atmosphere. Negative total rate
in the temperature minimum region occurs mainly due to ion H–. (E. Avrett et al.).
The problem is to find a way to transfer energy flux F2 from photosphere to chromosphere.
Photosphere
Chromosphere
Transition region
Solar flux: FS = 6.3153 107 Wm-2;
Integrated fluxes:
F1 = 1.28 108 Wm-2 = 2FS;
F2 = – 1.4 105 Wm-2 = – 2 10-3 FS;
F3 = 5.6 103 Wm-2 ~ 0.01% FS.
Results and discussion
1. A solution for MHD equations in ideal plasma was obtained. It is a
torsion Alfven wave with dispersion based on Rankine type vortex.
2. The upward mode of this wave transfer energy up in a pulsating way.
This upward energy transfer occurs in all vortex region.
3. The vortex wave energy flux is equal to the vortex energy times
Alfven velocity.
4. Energy dissipation occurs mainly in the vortex core due to viscosity
and all over the vortex due to electric conductivity.
5. Sound waves generated by pulsating torsion wave do not disturb the
main solution.
6. Density in the vortex core is lower than in its outside region. It opens
a possibility for a self-focusing of torsion wave and formation of
solitons.
7. An attempt to explain the negative region in the Net Radiative
Cooling Rate (NRCR) occur to be unsuccessful.
Wave energy dissipation
Energy dissipation due to viscosity is limited in the core of Rankine type vortex. Energy
dissipation due to conductivity is distributed over the all vortex uniformly:
2
𝑧
𝜕𝐵𝜑
𝑗2
𝑐2
𝑐 2 𝜌2 2 2 2 𝑟
𝑑𝑧1
2 𝜔𝑡 − 𝜔
𝑤 𝑟, 𝑧 = =
=
𝜔
𝑎
𝑓
sin
.
𝜔
𝜎 16𝜋 2 𝜎 𝜕𝑧
𝜎 𝐵𝑧2
𝑟0
𝑢
𝑧
𝑧0 𝐴 1
Its dimension is W/m3. Averaging it over time (sin^21/2) and integrating the dissipation term
over vortex area we receive:
2 2 2
𝑐 2 𝜔2 𝑣02 𝜌2 2 𝑟
𝑐
𝜔 𝑣0 𝜌 2 𝑟
𝑤 =
𝑓
=
𝑓
.
2𝜎𝐵𝑧2
𝑟0
𝑢𝐴
8𝜋𝜎
𝑟0
𝑅
𝑊=
0
𝑐
2𝜋𝑟𝑑𝑟 𝑤 =
𝑢𝐴
2 2 2 2
𝑟0 𝜔 𝑣0 𝜌
4𝜎
ln
𝑦𝑅
.
𝑟0