SOUND EMISSION AND IRREVERSIBLE DYNAMICS DURING VORTEX - - PowerPoint PPT Presentation
arXiv:2005.02047, arXiv.2005.02048 SOUND EMISSION AND IRREVERSIBLE DYNAMICS DURING VORTEX RECONNECTIONS IN QUANTUM FLUIDS DAVIDE PROMENT, UNIVERSITY OF EAST ANGLIA (UK) Joint work with: Alberto Villois and Giorgio Krstulovic arXiv:2005.02047,
Superfluid liquid helium [Public Domain, Wikipedia] Neutron stars [Robert Schulze, Wikipedia] Bose-Einstein condensates [top: JILA group, bottom: Ketterle et al.]
Instability and reconnect
ion in
the head-on collision of two vortex rings
Department of Mechanical & Manufacturing Engineering,
University of Melbourne, Parkville 3052, Australia
ONE mechanism by which fluid flows increase their complexity is through the instability of vortex filaments. When an instability brings vortex filaments of opposite circulation together, the filaments may break and rejoin in a process known as reconnection. This process of instability and reconnection leads to some funda- mental changes in the topology of flows. Here we present experi- mental observations of a special type of instability in which two colliding vortex rings become unstable and reconnect to form a series of smaller rings. Although this phenomenon was briefly noted more than a decade agot, no detailed observations were made, and little is known about the mechanisms involved. We have used coloured dyes to reveal the detailed structure of the small rings and many other features, including a short-wavelength insta- bility around the circumference of the colliding rings. At high Reynolds number, collision leads to a turbulent cloud, with the
The experiment was conducted in water in a glass tank (1.22 m long, 0.36 m wide and 0.47 m deep) in which were immersed two horizontally opposed nozzles spaced 220 mm apart. The position of one nozzle could be finely adjusted to make the rings collide exactly head-on. Both nozzles were connected to a piston, which ejected short, equal pulses of water from both nozzles simultaneously to produce two identical vortex rings travelling towards each other. Accurate, repeatable results were achieved
sequence of photographs showing different stages
The initial Reynolds number of each ring is roughly 1,000. The first photograph in the sequence, a, has been arbitrarily
assigned as t = 0.00 s. The elapsed time for the subsequent stages of the collision is shown in each photograph, with different time intervals adopted to illustrate the main features
rolled-up spiral dye sheets which become squashed and flattened during the collision. Each turn of the sheet forms a fold which becomes one of the concentric ribs on the
not believe that the membrane contains much vorticity.
NATURE· VOL 357 . 21 MAY 1992
LETTERS TO NATURE
by driving the piston with an electronically controlled stepping motor: this meant that the circulation and Reynolds number of the rings could also be determined. The vortex rings were made visible by releasing neutrally buoyant dyes around the circumfer- ence of each nozzle; the resulting flow patterns were recorded using a video recorder. Figure 1 shows different stages of a head-on collision for a Reynolds number (Re) of -1,000. (The initial Reynolds number
each ring is defined by UD / IJ, where U is the initial translation velocity, D is the diameter of the ring and IJ is the kinematic viscosity.) Figure 1
b shows that when the two vortex rings are
close to one another, the velocity induced by one ring on the
reasonably well, but when each ring has increased in size to about four times its initial diameter, a symmetrical instability in the form of azimuthal waviness begins to develop. As time progresses, the waves on the rings grow until they touch at the locations of maximum inward displacement. At the points of contact, the segments of the two vortex filaments eventually become interconnected to form small rings, a process commonly referred to as 'vortex reconnection'. As can be seen in Fig. 1
e,
the observation that each small ring is made up of both red and blue dye indicates that it consists of segments from both of the
Throughout the process of reconnection, the original vortex rings continue to grow in diameter, albeit at a slower rate. This growth is associated with stretching of the contact regions between the waves, and seems to be related to the reconnection
rings cease to exist, and the small rings then convect away radially from the central axis at slightly different speeds. The azimuthal waves that occur during the collision do not always form a regular pattern around the rings: the wavelength of the instability varies along the circumference and from run to run.
225
ion in
Department of Mechanical & Manufacturing Engineering,
University of Melbourne, Parkville 3052, Australia
ONE mechanism by which fluid flows increase their complexity is through the instability of vortex filaments. When an instability brings vortex filaments of opposite circulation together, the filaments may break and rejoin in a process known as reconnection. This process of instability and reconnection leads to some funda- mental changes in the topology of flows. Here we present experi- mental observations of a special type of instability in which two colliding vortex rings become unstable and reconnect to form a series of smaller rings. Although this phenomenon was briefly noted more than a decade agot, no detailed observations were made, and little is known about the mechanisms involved. We have used coloured dyes to reveal the detailed structure of the small rings and many other features, including a short-wavelength insta- bility around the circumference of the colliding rings. At high Reynolds number, collision leads to a turbulent cloud, with the
The experiment was conducted in water in a glass tank (1.22 m long, 0.36 m wide and 0.47 m deep) in which were immersed two horizontally opposed nozzles spaced 220 mm apart. The position of one nozzle could be finely adjusted to make the rings collide exactly head-on. Both nozzles were connected to a piston, which ejected short, equal pulses of water from both nozzles simultaneously to produce two identical vortex rings travelling towards each other. Accurate, repeatable results were achieved
sequence of photographs showing different stages
The initial Reynolds number of each ring is roughly 1,000. The first photograph in the sequence, a, has been arbitrarily
assigned as t = 0.00 s. The elapsed time for the subsequent stages of the collision is shown in each photograph, with different time intervals adopted to illustrate the main features
rolled-up spiral dye sheets which become squashed and
flattened during the collision. Each turn of the sheet forms a fold which becomes one of the concentric ribs on the
not believe that the membrane contains much vorticity.
NATURE· VOL 357 . 21 MAY 1992
LETTERS TO NATURE
by driving the piston with an electronically controlled stepping motor: this meant that the circulation and Reynolds number of the rings could also be determined. The vortex rings were made visible by releasing neutrally buoyant dyes around the circumfer- ence of each nozzle; the resulting flow patterns were recorded using a video recorder. Figure 1 shows different stages of a head-on collision for a Reynolds number (Re) of -1,000. (The initial Reynolds number
each ring is defined by UD / IJ, where U is the initial translation velocity, D is the diameter of the ring and IJ is the kinematic viscosity.) Figure 1
b shows that when the two vortex rings are
close to one another, the velocity induced by one ring on the
reasonably well, but when each ring has increased in size to about four times its initial diameter, a symmetrical instability in the form of azimuthal waviness begins to develop. As time progresses, the waves on the rings grow until they touch at the locations of maximum inward displacement. At the points of contact, the segments of the two vortex filaments eventually become interconnected to form small rings, a process commonly referred to as 'vortex reconnection'. As can be seen in Fig. 1
e,
the observation that each small ring is made up of both red and blue dye indicates that it consists of segments from both of the
Throughout the process of reconnection, the original vortex rings continue to grow in diameter, albeit at a slower rate. This growth is associated with stretching of the contact regions between the waves, and seems to be related to the reconnection
rings cease to exist, and the small rings then convect away radially from the central axis at slightly different speeds. The azimuthal waves that occur during the collision do not always form a regular pattern around the rings: the wavelength of the instability varies along the circumference and from run to run.
225
Instability and reconnect
ion in
the head-on collision of two vortex rings
Department of Mechanical & Manufacturing Engineering,
University of Melbourne, Parkville 3052, Australia
ONE mechanism by which fluid flows increase their complexity is through the instability of vortex filaments. When an instability brings vortex filaments of opposite circulation together, the filaments may break and rejoin in a process known as reconnection. This process of instability and reconnection leads to some funda- mental changes in the topology of flows. Here we present experi- mental observations of a special type of instability in which two colliding vortex rings become unstable and reconnect to form a series of smaller rings. Although this phenomenon was briefly noted more than a decade agot, no detailed observations were made, and little is known about the mechanisms involved. We have used coloured dyes to reveal the detailed structure of the small rings and many other features, including a short-wavelength insta- bility around the circumference of the colliding rings. At high Reynolds number, collision leads to a turbulent cloud, with the
The experiment was conducted in water in a glass tank (1.22 m long, 0.36 m wide and 0.47 m deep) in which were immersed two horizontally opposed nozzles spaced 220 mm apart. The position of one nozzle could be finely adjusted to make the rings collide exactly head-on. Both nozzles were connected to a piston, which ejected short, equal pulses of water from both nozzles simultaneously to produce two identical vortex rings travelling towards each other. Accurate, repeatable results were achieved
sequence of photographs showing different stages
The initial Reynolds number of each ring is roughly 1,000. The first photograph in the sequence, a, has been arbitrarily
assigned as t = 0.00 s. The elapsed time for the subsequent stages of the collision is shown in each photograph, with different time intervals adopted to illustrate the main features
rolled-up spiral dye sheets which become squashed and flattened during the collision. Each turn of the sheet forms a fold which becomes one of the concentric ribs on the
not believe that the membrane contains much vorticity.
NATURE· VOL 357 . 21 MAY 1992
LETTERS TO NATURE
by driving the piston with an electronically controlled stepping motor: this meant that the circulation and Reynolds number of the rings could also be determined. The vortex rings were made visible by releasing neutrally buoyant dyes around the circumfer- ence of each nozzle; the resulting flow patterns were recorded using a video recorder. Figure 1 shows different stages of a head-on collision for a Reynolds number (Re) of -1,000. (The initial Reynolds number
each ring is defined by UD / IJ, where U is the initial translation velocity, D is the diameter of the ring and IJ is the kinematic viscosity.) Figure 1
b shows that when the two vortex rings are
close to one another, the velocity induced by one ring on the
reasonably well, but when each ring has increased in size to about four times its initial diameter, a symmetrical instability in the form of azimuthal waviness begins to develop. As time progresses, the waves on the rings grow until they touch at the locations of maximum inward displacement. At the points of contact, the segments of the two vortex filaments eventually become interconnected to form small rings, a process commonly referred to as 'vortex reconnection'. As can be seen in Fig. 1
e,
the observation that each small ring is made up of both red and blue dye indicates that it consists of segments from both of the
Throughout the process of reconnection, the original vortex rings continue to grow in diameter, albeit at a slower rate. This growth is associated with stretching of the contact regions between the waves, and seems to be related to the reconnection
rings cease to exist, and the small rings then convect away radially from the central axis at slightly different speeds. The azimuthal waves that occur during the collision do not always form a regular pattern around the rings: the wavelength of the instability varies along the circumference and from run to run.
225
[Kleckner & Irvine, Nature 2013]
Trefoil decaying in classical viscous fluids [Kurstulovic, private communication] Trefoil decaying in classical quantum fluids [Proment et al., PRE 2012]
[Serafini et al., PRL 2015] [Paoletti et al., PNAS 2008]
δ−(t) δ+(t)
Vortex reconnections in superfluid liquid helium (top) and BEC of cold gases (bottom)
vfil(x, t) = − Γ 4π ∫ℒ [x − R(ℓ, t)] × dR(ℓ, t) x − R(ℓ, t)
3
Local induction approximation (LIA)
· R(t) = Γ 4π ln ( L0 a0 ) + 풪(1) κ ̂ b
[Saffman, Vortex Dynamics ; Pismen, Vortices in Nonlinear Fields]
fit
iℏ ∂ψ ∂t + ℏ2 2m ∇2ψ − g|ψ|2ψ = 0
linear momentum and energy, that is
N = ∫ |ψ|2dV , P = iℏ 2 ∫ (ψ∇ψ* − ψ*∇ψ) dV and H = ∫ ℏ2 2m |∇ψ|2 + g 2 |ψ|4dV
[Pitaevskii & Stringari, 2003]
iℏ ∂ψ ∂t + ℏ2 2m ∇2ψ − g|ψ|2ψ = 0
i ∂ψ ∂t = c 2ξ (−ξ2∇2ψ + m ρ0 |ψ|2ψ)
is the only inherent length scale of the system
|ψ0| = ρ0/m ξ = ℏ2/(2mgρ0) c = gρ0/m2
[Pitaevskii & Stringari, 2003]
∂ρ ∂t + ∇ ⋅ (ρv) = 0 ∂v ∂t + (v ⋅ ∇)v = − c2 ρ0 ∇ρ + c2ξ2∇ ∇2 ρ ρ
i ∂ψ ∂t = c 2ξ (−ξ2∇2ψ + m ρ0 |ψ|2ψ)
[Nore et al., Phys. Fluids 1997]
[Pitaevskii, JETP 1961]
x
y 0.2 0.4 0.6 0.8 1
i ∂ψ ∂t = c 2ξ (−ξ2∇2ψ + m ρ0 |ψ|2ψ)
[Koplik & Levine, PRL 1993]
[Villois et al., JPhysA 2016]
Example of the evolution of the density field of an Hopf link realisation
ρ
δ±(t) = A± Γ|t − tr| ϕ+ = 2arcot(Ar) , where Ar = A+/A−
[Nazarenko & West, JLTP 2003]
δ±(t) ≤ ξ ⟹ i ∂ψ ∂t = c 2ξ (−ξ2∇2ψ + m ρ0 |ψ|2ψ)
before and after, only the pre-factors change
a plane and their projections are branches of an hyperbola
δ ∝ t1/2 A±
[Villois et al., PRFluids 2017]
x
y 0.2 0.4 0.6 0.8 1
0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5
Red points data of this work, other symbols are from [Villois et al., PRFluids 2018] and [Galantucci et al., PNAS 2019]
δ±(t) = A± Γ|t − tr| ϕ+ = 2arcot(Ar) given Ar = A+/A−
Biot-Savart model Schrödinger equation
input
δ−(t) ξ
<latexit sha1_base64="Ya0DY5y/uo686nVfTbTb0EXnpo=">ACEnicbVDLTsJAFJ36RHxVXLqZSExwIWnRJdENy4xkUdCkUynA0yYdpqZWwNp+AvXbvUb3Bm3/oCf4F84BRYKnuQmJ+fcm3v8WPBNTjOl7Wyura+sZnbym/v7O7t2weFhpaJoqxOpZCq5RPNBI9YHTgI1oVI6EvWNMf3mR+85EpzWV0D+OYdULSj3iPUwJG6toFL2ACyMNZCU69fh97I961i07ZmQIvE3dOimiOWtf+9gJk5BFQAXRu06MXRSoBTwSZ5L9EsJnRI+qxtaERCpjvp9PYJPjFKgHtSmYoAT9XfEykJtR6HvukMCQz0opeJ/3qZAlIKvXA9K46KY/iBFhEZ/t7icAgcZYPDrhiFMTYEIVNy9gOiCKUDAp5k027mISy6RKbvn5crdRbF6PU8ph47QMSohF12iKrpFNVRHFI3QM3pBr9aT9Wa9Wx+z1hVrPnOI/sD6/AHSH518</latexit>δ+(t) ξ
<latexit sha1_base64="vqgSZr8nhlK16ec2D1Xx6GjNX4=">ACEnicbVDLSsNAFJ34rPUV69LNYBEqQkmqoMuiG5cV7AOaWiaTaTt0kgkzN9IS+heu3eo3uBO3/oCf4F84abvQ1gMXDufcy73+LHgGhzny1pZXVvf2Mxt5bd3dvf27YNCQ8tEUVanUkjV8olmgkesDhwEa8WKkdAXrOkPbzK/+ciU5jK6h3HMOiHpR7zHKQEjde2CFzAB5OGsBKdev4+9Ee/aRafsTIGXiTsnRTRHrWt/e4GkScgioIJo3XadGDopUcCpYJO8l2gWEzokfdY2NCIh0510evsEnxglwD2pTEWAp+rviZSEWo9D3SGBAZ60cvEf71MASmFXjgAeledlEdxAiyis/29RGCQOMsHB1wxCmJsCKGKmxcwHRBFKJgU8yYbdzGJZdKolN3zcuXuoli9nqeUQ0foGJWQiy5RFd2iGqojikboGb2gV+vJerPerY9Z64o1nzlEf2B9/gDO1Z16</latexit>δ±(t) ⌧ ξ
<latexit sha1_base64="x3GdmI1tmJLmLHqbEY/35CXFgI=">ACFHicbVDLTgJBEJzF+Jrfdy8TCQmeCG7aKJHohePmMgjYZHMDgNMmNnZzPQakfAbnr3qN3gzXr37Cf6Fs8BwUo6qVR1p7srjAU34HlfTmZpeWV1Lbue29jc2t5xd/dqRiWasipVQulGSAwTPGJV4CBYI9aMyFCweji4Sv36PdOGq+gWhjFrSdKLeJdTAlZquwdBhwkgd0EsC3ASCIGDB952817RmwAvEn9G8miGStv9DjqKJpJFQAUxpul7MbRGRAOngo1zQWJYTOiA9FjT0ohIZlqjyfVjfGyVDu4qbSsCPF/T4yINGYoQ9spCfTNvJeK/3qpAkoJM3cAdC9aIx7FCbCITvd3E4FB4TQh3OGaURBDSwjV3L6AaZ9oQsHmLPZ+PNJLJaqeifFks3Z/ny5SylLDpER6iAfHSOyugaVAVUfSIntELenWenDfn3fmYtmac2cw+gPn8wfsXp6m</latexit>Biot-Savart model
R−
1,2
<latexit sha1_base64="8fNreUZBNgXcmMFVCQLOMbLg4vI=">ACD3icbVDLSsNAFJ34rPXRqEs3g0VwoSWpgi6LblxWsQ9oY5hMJ+3QyUyYmQgl5CNcu9VvcCdu/Q/wb9w0mahrQcuHM65l3vCWJGlXacL2tpeWV1b20Ud7c2t6p2Lt7bSUSiUkLCyZkN0CKMpJS1PNSDeWBEUBI51gfJ37nUciFRX8Xk9i4kVoyGlIMdJG8u1K2g9CeJc9nPqpe1LPfLvq1Jwp4CJxC1IFBZq+/d0fCJxEhGvMkFI914m1lyKpKWYkK/cTRWKEx2hIeoZyFBHlpdPDM3hklAEMhTFNZyqvydSFCk1iQLTGSE9UvNeLv7r5YoWgqm5A3R46aWUx4kmHM/2hwmDWsA8HDigkmDNJoYgLKl5AeIRkghrE2HZOPOJ7FI2vWae1ar35XG1dFSiVwA7BMXDBWiAG9AELYBAp7BC3i1nqw36936mLUuWcXMPvgD6/MHUtScIA=</latexit>R+
1,2
<latexit sha1_base64="HI63mBs/BbivdJEnIYA94mzeKGw=">ACD3icbVDLSsNAFJ34rPXRqEs3g0UQlJUQZdFNy6r2Ae0MUymk3boZCbMTIQS8hGu3eo3uBO3foKf4F84abPQ1gMXDufcy73BDGjSjvOl7W0vLK6tl7aKG9ube9U7N29thKJxKSFBROyGyBFGOWkpalmpBtLgqKAkU4wvs79ziORigp+rycx8SI05DSkGkj+XYl7QchvMseTvzUPa1nvl1as4UcJG4BamCAk3f/u4PBE4iwjVmSKme68TaS5HUFDOSlfuJIjHCYzQkPUM5iojy0unhGTwygCGQpriGk7V3xMpipSaRIHpjJAeqXkvF/1ckULwdTcATq89FLK40QTjmf7w4RBLWAeDhxQSbBmE0MQltS8APEISYS1ibBsnHnk1gk7XrNPavVb8+rjasipRI4AIfgGLjgAjTADWiCFsAgAc/gBbxaT9ab9W59zFqXrGJmH/yB9fkDT5KcHg=</latexit> i ∂ψ ∂t = − Γ 4π ∇2ψ
ψr(x, y, z) = 1 ζ5/2 p [z − A(x cos θ + y sin θ)2 + B(−x cos θ + y sin θ)2 2ζ ] + i [z − Cx2 + Dy2 2ζ ]
A general second-order polynomial solution at the reconnection time having two nodal-lines (vortices) is given by
tr = 0
(a)
p = ± 1 , (A, B, C, D) ∈ ℝ , θ ∈ [0,π] , ζ > 0 is a generic length scale
intersection of and
as this is a quadratic form
Re(ψr) = 0 Im(ψr) = 0 θ = 0
i ∂ψ ∂t = − Γ 4π ∇2ψ ⟹ ψ(x, y, z, t) = (1 + it Γ 4π ∇2 ) ψr(x, y, z)
− C − A B − D x2 + y2 = A + B + C + D 2(B − D)pπ Γt
Once evolved in time, the wave-function reads
φ+ φ+ x/ζ y/ζ (a) t < 0, R−
1,2
t > 0, R+
1,2
z = BC − AD 2(B − D)ζ x2 + D(C + D) + B(A + B) 4(B − D)pπζ Γt
concavity parameter Λ x/ζ z/ζ (b) t < 0, R−
1,2
t > 0, R+
1,2
projections onto the y = 0 plane projections onto the z = 0 plane
2Λ < [tan2 ( ϕ+ 2 ) − 1](B + D) ∩ [(p = − 1 ∩ D > B) ∪ (p = 1 ∩ D < B)] − C − A B − D x2 + y2 = A + B + C + D 2(B − D)pπ Γt
φ+ φ+ x/ζ y/ζ (a) t < 0, R−
1,2
t > 0, R+
1,2
z = BC − AD 2(B − D)ζ x2 + D(C + D) + B(A + B) 4(B − D)pπζ Γt
concavity parameter Λ x/ζ z/ζ (b) t < 0, R−
1,2
t > 0, R+
1,2
projections onto the y = 0 plane projections onto the z = 0 plane
All reconnection angles and concavity parameter are possible!
ϕ+ Λ
Biot-Savart model Schrödinger equation
input
δ−(t) ξ
<latexit sha1_base64="Ya0DY5y/uo686nVfTbTb0EXnpo=">ACEnicbVDLTsJAFJ36RHxVXLqZSExwIWnRJdENy4xkUdCkUynA0yYdpqZWwNp+AvXbvUb3Bm3/oCf4F84BRYKnuQmJ+fcm3v8WPBNTjOl7Wyura+sZnbym/v7O7t2weFhpaJoqxOpZCq5RPNBI9YHTgI1oVI6EvWNMf3mR+85EpzWV0D+OYdULSj3iPUwJG6toFL2ACyMNZCU69fh97I961i07ZmQIvE3dOimiOWtf+9gJk5BFQAXRu06MXRSoBTwSZ5L9EsJnRI+qxtaERCpjvp9PYJPjFKgHtSmYoAT9XfEykJtR6HvukMCQz0opeJ/3qZAlIKvXA9K46KY/iBFhEZ/t7icAgcZYPDrhiFMTYEIVNy9gOiCKUDAp5k027mISy6RKbvn5crdRbF6PU8ph47QMSohF12iKrpFNVRHFI3QM3pBr9aT9Wa9Wx+z1hVrPnOI/sD6/AHSH518</latexit>δ+(t) ξ
<latexit sha1_base64="vqgSZr8nhlK16ec2D1Xx6GjNX4=">ACEnicbVDLSsNAFJ34rPUV69LNYBEqQkmqoMuiG5cV7AOaWiaTaTt0kgkzN9IS+heu3eo3uBO3/oCf4F84abvQ1gMXDufcy73+LHgGhzny1pZXVvf2Mxt5bd3dvf27YNCQ8tEUVanUkjV8olmgkesDhwEa8WKkdAXrOkPbzK/+ciU5jK6h3HMOiHpR7zHKQEjde2CFzAB5OGsBKdev4+9Ee/aRafsTIGXiTsnRTRHrWt/e4GkScgioIJo3XadGDopUcCpYJO8l2gWEzokfdY2NCIh0510evsEnxglwD2pTEWAp+rviZSEWo9D3SGBAZ60cvEf71MASmFXjgAeledlEdxAiyis/29RGCQOMsHB1wxCmJsCKGKmxcwHRBFKJgU8yYbdzGJZdKolN3zcuXuoli9nqeUQ0foGJWQiy5RFd2iGqojikboGb2gV+vJerPerY9Z64o1nzlEf2B9/gDO1Z16</latexit>δ±(t) ⌧ ξ
<latexit sha1_base64="x3GdmI1tmJLmLHqbEY/35CXFgI=">ACFHicbVDLTgJBEJzF+Jrfdy8TCQmeCG7aKJHohePmMgjYZHMDgNMmNnZzPQakfAbnr3qN3gzXr37Cf6Fs8BwUo6qVR1p7srjAU34HlfTmZpeWV1Lbue29jc2t5xd/dqRiWasipVQulGSAwTPGJV4CBYI9aMyFCweji4Sv36PdOGq+gWhjFrSdKLeJdTAlZquwdBhwkgd0EsC3ASCIGDB952817RmwAvEn9G8miGStv9DjqKJpJFQAUxpul7MbRGRAOngo1zQWJYTOiA9FjT0ohIZlqjyfVjfGyVDu4qbSsCPF/T4yINGYoQ9spCfTNvJeK/3qpAkoJM3cAdC9aIx7FCbCITvd3E4FB4TQh3OGaURBDSwjV3L6AaZ9oQsHmLPZ+PNJLJaqeifFks3Z/ny5SylLDpER6iAfHSOyugaVAVUfSIntELenWenDfn3fmYtmac2cw+gPn8wfsXp6m</latexit>Biot-Savart model
R−
1,2
<latexit sha1_base64="8fNreUZBNgXcmMFVCQLOMbLg4vI=">ACD3icbVDLSsNAFJ34rPXRqEs3g0VwoSWpgi6LblxWsQ9oY5hMJ+3QyUyYmQgl5CNcu9VvcCdu/Q/wb9w0mahrQcuHM65l3vCWJGlXacL2tpeWV1b20Ud7c2t6p2Lt7bSUSiUkLCyZkN0CKMpJS1PNSDeWBEUBI51gfJ37nUciFRX8Xk9i4kVoyGlIMdJG8u1K2g9CeJc9nPqpe1LPfLvq1Jwp4CJxC1IFBZq+/d0fCJxEhGvMkFI914m1lyKpKWYkK/cTRWKEx2hIeoZyFBHlpdPDM3hklAEMhTFNZyqvydSFCk1iQLTGSE9UvNeLv7r5YoWgqm5A3R46aWUx4kmHM/2hwmDWsA8HDigkmDNJoYgLKl5AeIRkghrE2HZOPOJ7FI2vWae1ar35XG1dFSiVwA7BMXDBWiAG9AELYBAp7BC3i1nqw36936mLUuWcXMPvgD6/MHUtScIA=</latexit>R+
1,2
<latexit sha1_base64="HI63mBs/BbivdJEnIYA94mzeKGw=">ACD3icbVDLSsNAFJ34rPXRqEs3g0UQlJUQZdFNy6r2Ae0MUymk3boZCbMTIQS8hGu3eo3uBO3foKf4F84abPQ1gMXDufcy73BDGjSjvOl7W0vLK6tl7aKG9ube9U7N29thKJxKSFBROyGyBFGOWkpalmpBtLgqKAkU4wvs79ziORigp+rycx8SI05DSkGkj+XYl7QchvMseTvzUPa1nvl1as4UcJG4BamCAk3f/u4PBE4iwjVmSKme68TaS5HUFDOSlfuJIjHCYzQkPUM5iojy0unhGTwygCGQpriGk7V3xMpipSaRIHpjJAeqXkvF/1ckULwdTcATq89FLK40QTjmf7w4RBLWAeDhxQSbBmE0MQltS8APEISYS1ibBsnHnk1gk7XrNPavVb8+rjasipRI4AIfgGLjgAjTADWiCFsAgAc/gBbxaT9ab9W59zFqXrGJmH/yB9fkDT5KcHg=</latexit>[Pismen, 1999]
momentum: P±
fil = κ
2 ∫ℒ R± × dR± energy: E±
LIA ∝ ∫ℒ
|dR±| ⟹ ΔPfil = P+
fil − P− fil
ΔELIA = E+
LIA − E− LIA
δ(t) = δlin
[Pismen, 1999]
and , so that they satisfy the shape found in the linear theory is
ϕ+ Λ
R−
1(ℓ, t) = {− δ−(t)
2 cot ( ϕ+ 2 ) sinh(ℓ), δ−(t) 2 cosh(ℓ), z−(ℓ, t)} R−
2(ℓ, t) = = {
δ−(t) 2 cot ( ϕ+ 2 ) sinh(ℓ), − δ−(t) 2 cosh(ℓ), z−(ℓ, t)} R+
1(ℓ, t) = {− δ+(t)
2 cosh(ℓ), δ+(t) 2 tan ( ϕ+ 2 ) sinh(ℓ), z+(ℓ, t)} R+
2(ℓ, t) = {
δ+(t) 2 cosh(ℓ), − δ+(t) 2 tan ( ϕ+ 2 ) sinh(ℓ), z+(ℓ, t)} , where l ∈ ℝ
momentum: P±
fil = κ
2 ∫ℒ R± × dR± energy: E±
LIA ∝ ∫ℒ
|dR±| ⟹ ΔPfil = P+
fil − P− fil
ΔELIA = E+
LIA − E− LIA
δ(t) = δlin
As the filaments are branches
infinite length. We compute their integrals in a finite cylinder parallel to the z-axis, centred at the reconnection point (the origin) and of radius R > δlin
δlin R (a) t < 0 δlin R (b) t > 0
L−(R/δlin) = 1 2 ln 8(R/δlin)2 + (A2
r − 1) + 2
[4 (R/δlin)
2 − 1] [4 (R/δlin) 2 + A2 r ]
A2
r + 1
L+(R/δlin) = 1 2 ln 8A2
r (R/δlin)2 + (1 − A2 r ) + 2Ar
[4 (R/δlin)
2 − 1] [4A2 r (R/δlin) 2 + 1]
A2
r + 1
The limits of integration, in the parametrisation of the filaments, are given by
ΔPfil ∝ (0,0,1 + A2
r
Ar ) = (0,0, − 2 csc ϕ+) ΔELIA(Ar, Λ/ζ, δlin, R/δlin) ∝ ∫
L+(R/δlin) −L+(R/δlin)
∂R+
1
∂ℓ + ∂R+
2
∂ℓ dℓ − ∫
L−(R/δlin) −L−(R/δlin)
∂R−
1
∂ℓ + ∂R−
2
∂ℓ dℓ
1 2 3 4
0.5 1.0 1.5
1 2 3 4
0.5 1.0 1.5
Λ = 0 Λ ↔ − Λ R/δlin |Λ| → ∞
momentum: P±
fil = κ
2 ∫ℒ R± × dR± energy: E±
LIA ∝ ∫ℒ
|dR±| ⟹ ΔPfil = P+
fil − P− fil
ΔELIA = E+
LIA − E− LIA
Ppulse = − ΔPfil ∝ (0, 0, 1 + A2
r
Ar ) ⟹ ΔPwav,z > 0
Example of sound pulse emission propagating along the positive z-axis
c
20 40 0.975 0.98 0.985 0.99 0.995 1 1.005
20 0.975 0.98 0.985 0.99 0.995 1 1.005
distribution of the rates of approach and separation is asymmetric, evidence
an irreversible dynamics explained by the emission
A− A+
reconnections in quantum fluids (GP model)
0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5
1.04
b) c) d)
sound pulse only propagates towards the positive z-axis
linear theory and BS (and LIA)
time
Biot-Savart model Schrödinger equation
input
δ−(t) ξ
<latexit sha1_base64="Ya0DY5y/uo686nVfTbTb0EXnpo=">ACEnicbVDLTsJAFJ36RHxVXLqZSExwIWnRJdENy4xkUdCkUynA0yYdpqZWwNp+AvXbvUb3Bm3/oCf4F84BRYKnuQmJ+fcm3v8WPBNTjOl7Wyura+sZnbym/v7O7t2weFhpaJoqxOpZCq5RPNBI9YHTgI1oVI6EvWNMf3mR+85EpzWV0D+OYdULSj3iPUwJG6toFL2ACyMNZCU69fh97I961i07ZmQIvE3dOimiOWtf+9gJk5BFQAXRu06MXRSoBTwSZ5L9EsJnRI+qxtaERCpjvp9PYJPjFKgHtSmYoAT9XfEykJtR6HvukMCQz0opeJ/3qZAlIKvXA9K46KY/iBFhEZ/t7icAgcZYPDrhiFMTYEIVNy9gOiCKUDAp5k027mISy6RKbvn5crdRbF6PU8ph47QMSohF12iKrpFNVRHFI3QM3pBr9aT9Wa9Wx+z1hVrPnOI/sD6/AHSH518</latexit>δ+(t) ξ
<latexit sha1_base64="vqgSZr8nhlK16ec2D1Xx6GjNX4=">ACEnicbVDLSsNAFJ34rPUV69LNYBEqQkmqoMuiG5cV7AOaWiaTaTt0kgkzN9IS+heu3eo3uBO3/oCf4F84abvQ1gMXDufcy73+LHgGhzny1pZXVvf2Mxt5bd3dvf27YNCQ8tEUVanUkjV8olmgkesDhwEa8WKkdAXrOkPbzK/+ciU5jK6h3HMOiHpR7zHKQEjde2CFzAB5OGsBKdev4+9Ee/aRafsTIGXiTsnRTRHrWt/e4GkScgioIJo3XadGDopUcCpYJO8l2gWEzokfdY2NCIh0510evsEnxglwD2pTEWAp+rviZSEWo9D3SGBAZ60cvEf71MASmFXjgAeledlEdxAiyis/29RGCQOMsHB1wxCmJsCKGKmxcwHRBFKJgU8yYbdzGJZdKolN3zcuXuoli9nqeUQ0foGJWQiy5RFd2iGqojikboGb2gV+vJerPerY9Z64o1nzlEf2B9/gDO1Z16</latexit>δ±(t) ⌧ ξ
<latexit sha1_base64="x3GdmI1tmJLmLHqbEY/35CXFgI=">ACFHicbVDLTgJBEJzF+Jrfdy8TCQmeCG7aKJHohePmMgjYZHMDgNMmNnZzPQakfAbnr3qN3gzXr37Cf6Fs8BwUo6qVR1p7srjAU34HlfTmZpeWV1Lbue29jc2t5xd/dqRiWasipVQulGSAwTPGJV4CBYI9aMyFCweji4Sv36PdOGq+gWhjFrSdKLeJdTAlZquwdBhwkgd0EsC3ASCIGDB952817RmwAvEn9G8miGStv9DjqKJpJFQAUxpul7MbRGRAOngo1zQWJYTOiA9FjT0ohIZlqjyfVjfGyVDu4qbSsCPF/T4yINGYoQ9spCfTNvJeK/3qpAkoJM3cAdC9aIx7FCbCITvd3E4FB4TQh3OGaURBDSwjV3L6AaZ9oQsHmLPZ+PNJLJaqeifFks3Z/ny5SylLDpER6iAfHSOyugaVAVUfSIntELenWenDfn3fmYtmac2cw+gPn8wfsXp6m</latexit>Biot-Savart model R−
1,2
<latexit sha1_base64="8fNreUZBNgXcmMFVCQLOMbLg4vI=">ACD3icbVDLSsNAFJ34rPXRqEs3g0VwoSWpgi6LblxWsQ9oY5hMJ+3QyUyYmQgl5CNcu9VvcCdu/Q/wb9w0mahrQcuHM65l3vCWJGlXacL2tpeWV1b20Ud7c2t6p2Lt7bSUSiUkLCyZkN0CKMpJS1PNSDeWBEUBI51gfJ37nUciFRX8Xk9i4kVoyGlIMdJG8u1K2g9CeJc9nPqpe1LPfLvq1Jwp4CJxC1IFBZq+/d0fCJxEhGvMkFI914m1lyKpKWYkK/cTRWKEx2hIeoZyFBHlpdPDM3hklAEMhTFNZyqvydSFCk1iQLTGSE9UvNeLv7r5YoWgqm5A3R46aWUx4kmHM/2hwmDWsA8HDigkmDNJoYgLKl5AeIRkghrE2HZOPOJ7FI2vWae1ar35XG1dFSiVwA7BMXDBWiAG9AELYBAp7BC3i1nqw36936mLUuWcXMPvgD6/MHUtScIA=</latexit>R+
1,2
<latexit sha1_base64="HI63mBs/BbivdJEnIYA94mzeKGw=">ACD3icbVDLSsNAFJ34rPXRqEs3g0UQlJUQZdFNy6r2Ae0MUymk3boZCbMTIQS8hGu3eo3uBO3foKf4F84abPQ1gMXDufcy73BDGjSjvOl7W0vLK6tl7aKG9ube9U7N29thKJxKSFBROyGyBFGOWkpalmpBtLgqKAkU4wvs79ziORigp+rycx8SI05DSkGkj+XYl7QchvMseTvzUPa1nvl1as4UcJG4BamCAk3f/u4PBE4iwjVmSKme68TaS5HUFDOSlfuJIjHCYzQkPUM5iojy0unhGTwygCGQpriGk7V3xMpipSaRIHpjJAeqXkvF/1ckULwdTcATq89FLK40QTjmf7w4RBLWAeDhxQSbBmE0MQltS8APEISYS1ibBsnHnk1gk7XrNPavVb8+rjasipRI4AIfgGLjgAjTADWiCFsAgAc/gBbxaT9ab9W59zFqXrGJmH/yB9fkDT5KcHg=</latexit>20 40 0.975 0.98 0.985 0.99 0.995 1 1.005
the origin of the irreversible dynamics by showing that the energy of the sound pulse is only positive when that is for
A+ > A− 0 ≤ ϕ+ ≤ π/2
a “superposition” of (quasi-)linear waves, or a full nonlinear structure
asymptotic matching theory
time
Biot-Savart model Schrödinger equation
input
δ−(t) ξ
<latexit sha1_base64="Ya0DY5y/uo686nVfTbTb0EXnpo=">ACEnicbVDLTsJAFJ36RHxVXLqZSExwIWnRJdENy4xkUdCkUynA0yYdpqZWwNp+AvXbvUb3Bm3/oCf4F84BRYKnuQmJ+fcm3v8WPBNTjOl7Wyura+sZnbym/v7O7t2weFhpaJoqxOpZCq5RPNBI9YHTgI1oVI6EvWNMf3mR+85EpzWV0D+OYdULSj3iPUwJG6toFL2ACyMNZCU69fh97I961i07ZmQIvE3dOimiOWtf+9gJk5BFQAXRu06MXRSoBTwSZ5L9EsJnRI+qxtaERCpjvp9PYJPjFKgHtSmYoAT9XfEykJtR6HvukMCQz0opeJ/3qZAlIKvXA9K46KY/iBFhEZ/t7icAgcZYPDrhiFMTYEIVNy9gOiCKUDAp5k027mISy6RKbvn5crdRbF6PU8ph47QMSohF12iKrpFNVRHFI3QM3pBr9aT9Wa9Wx+z1hVrPnOI/sD6/AHSH518</latexit>δ+(t) ξ
<latexit sha1_base64="vqgSZr8nhlK16ec2D1Xx6GjNX4=">ACEnicbVDLSsNAFJ34rPUV69LNYBEqQkmqoMuiG5cV7AOaWiaTaTt0kgkzN9IS+heu3eo3uBO3/oCf4F84abvQ1gMXDufcy73+LHgGhzny1pZXVvf2Mxt5bd3dvf27YNCQ8tEUVanUkjV8olmgkesDhwEa8WKkdAXrOkPbzK/+ciU5jK6h3HMOiHpR7zHKQEjde2CFzAB5OGsBKdev4+9Ee/aRafsTIGXiTsnRTRHrWt/e4GkScgioIJo3XadGDopUcCpYJO8l2gWEzokfdY2NCIh0510evsEnxglwD2pTEWAp+rviZSEWo9D3SGBAZ60cvEf71MASmFXjgAeledlEdxAiyis/29RGCQOMsHB1wxCmJsCKGKmxcwHRBFKJgU8yYbdzGJZdKolN3zcuXuoli9nqeUQ0foGJWQiy5RFd2iGqojikboGb2gV+vJerPerY9Z64o1nzlEf2B9/gDO1Z16</latexit>δ±(t) ⌧ ξ
<latexit sha1_base64="x3GdmI1tmJLmLHqbEY/35CXFgI=">ACFHicbVDLTgJBEJzF+Jrfdy8TCQmeCG7aKJHohePmMgjYZHMDgNMmNnZzPQakfAbnr3qN3gzXr37Cf6Fs8BwUo6qVR1p7srjAU34HlfTmZpeWV1Lbue29jc2t5xd/dqRiWasipVQulGSAwTPGJV4CBYI9aMyFCweji4Sv36PdOGq+gWhjFrSdKLeJdTAlZquwdBhwkgd0EsC3ASCIGDB952817RmwAvEn9G8miGStv9DjqKJpJFQAUxpul7MbRGRAOngo1zQWJYTOiA9FjT0ohIZlqjyfVjfGyVDu4qbSsCPF/T4yINGYoQ9spCfTNvJeK/3qpAkoJM3cAdC9aIx7FCbCITvd3E4FB4TQh3OGaURBDSwjV3L6AaZ9oQsHmLPZ+PNJLJaqeifFks3Z/ny5SylLDpER6iAfHSOyugaVAVUfSIntELenWenDfn3fmYtmac2cw+gPn8wfsXp6m</latexit>Biot-Savart model R−
1,2
<latexit sha1_base64="8fNreUZBNgXcmMFVCQLOMbLg4vI=">ACD3icbVDLSsNAFJ34rPXRqEs3g0VwoSWpgi6LblxWsQ9oY5hMJ+3QyUyYmQgl5CNcu9VvcCdu/Q/wb9w0mahrQcuHM65l3vCWJGlXacL2tpeWV1b20Ud7c2t6p2Lt7bSUSiUkLCyZkN0CKMpJS1PNSDeWBEUBI51gfJ37nUciFRX8Xk9i4kVoyGlIMdJG8u1K2g9CeJc9nPqpe1LPfLvq1Jwp4CJxC1IFBZq+/d0fCJxEhGvMkFI914m1lyKpKWYkK/cTRWKEx2hIeoZyFBHlpdPDM3hklAEMhTFNZyqvydSFCk1iQLTGSE9UvNeLv7r5YoWgqm5A3R46aWUx4kmHM/2hwmDWsA8HDigkmDNJoYgLKl5AeIRkghrE2HZOPOJ7FI2vWae1ar35XG1dFSiVwA7BMXDBWiAG9AELYBAp7BC3i1nqw36936mLUuWcXMPvgD6/MHUtScIA=</latexit>R+
1,2
<latexit sha1_base64="HI63mBs/BbivdJEnIYA94mzeKGw=">ACD3icbVDLSsNAFJ34rPXRqEs3g0UQlJUQZdFNy6r2Ae0MUymk3boZCbMTIQS8hGu3eo3uBO3foKf4F84abPQ1gMXDufcy73BDGjSjvOl7W0vLK6tl7aKG9ube9U7N29thKJxKSFBROyGyBFGOWkpalmpBtLgqKAkU4wvs79ziORigp+rycx8SI05DSkGkj+XYl7QchvMseTvzUPa1nvl1as4UcJG4BamCAk3f/u4PBE4iwjVmSKme68TaS5HUFDOSlfuJIjHCYzQkPUM5iojy0unhGTwygCGQpriGk7V3xMpipSaRIHpjJAeqXkvF/1ckULwdTcATq89FLK40QTjmf7w4RBLWAeDhxQSbBmE0MQltS8APEISYS1ibBsnHnk1gk7XrNPavVb8+rjasipRI4AIfgGLjgAjTADWiCFsAgAc/gBbxaT9ab9W59zFqXrGJmH/yB9fkDT5KcHg=</latexit>20 40 0.975 0.98 0.985 0.99 0.995 1 1.005
maths problem) by letting different regularisation scales (viscosity in classical fluid, dispersion in quantum fluids) tends to zero
the rates varies, for experimental applications in quantum fluids where thermal excitations are always present (statistical mechanics problem)
A±
Acknowledgments G.K., D.P . and A.V. were supported by the cost-share Royal Society International Exchanges Scheme (IE150527) in conjunction with CNRS. A.V. and D.P . were supported by the EPSRC First Grant scheme (EP/P023770/1). G.K. and D.P . acknowledge the Federation Doeblin for supporting D.P . during his sojourn in Nice. G.K. was also supported by the ANR JCJC GIANTE ANR-18-CE30-0020-0.1 and by the EU Horizon 2020 research and innovation programme under the grant agreement No 823937 in the framework of Marie Skodowska-Curie HALT project. Part of this work has been presented at the workshop “Irreversibility and Turbulence” hosted by Fondation Les Treilles in September 2017. G.K. and D.P . acknowledge Fondation Les Treilles and all participants of the workshop for the frightful scientific discussions and support.