Traveled light through the atmosphere affected by a number of phenomena such
as scattering, absorption and turbulence (Mohammadein and
Abu-Bakr, 2010; Momeni and Moslehi-Fard, 2008;
Lewandowski, 2003). Turbulence has been investigated
not only in applied sciences but also in basic science research, such as physics
and mathematics research. Turbulence is a complicated dynamical phenomenon which
based on strong nonlinearity. This phenomenon is far from an equilibrium state
and may be understand in context of vortices. However, classical description
of vortices are not well-defined. Therefore quantum turbulence (Vinen,
2006; Tsubota, 2008; Kobayashi
and Tsubota, 2005) will be more convenient. Comparing quantum turbulence
and classical turbulence reveals definite differences which demonstrates the
importance of studying quantum turbulence. Turbulence in a classical viscous
fluid admitted vortices, but these vortices are unstable. Moreover, in order
to have conserved circulation we need quantum turbulence.
Thus, quantum turbulence is an easier system to study than classical turbulence
and has a much simpler model of turbulence than classical turbulence (Vinen
and Niemela, 2002). A vortex in superfluid with circular quantization is
called a quantized vortex. Quantized vortices also appropriate to study any
rotational motion of a superfluid which is different from a classical vortex
in viscous fluid. Thermal counter flow of superfluid turbulence has been studied
experimentally, where the normal fluid and superfluid flow assumed in opposite
directions. By using an injected heat current one can obtain flow which suggests
that the superflow becomes dissipative if the relative velocity between the
two fluids exceeds a critical value (Gorter and Mellink,
Since, the dynamics of quantized vortices is nonlinear and non-local, one can
understand vortex dynamics observations quantitatively. Superfluid turbulence
is often called quantum turbulence, which indeed study quantized vortices. Turbulence
phenomenon also affect on the laser beam. The subject of turbulence is also
important in optics and laser researches. Some of the important effects of turbulence
on the laser beam are for example phase-front distortion, scintillation and
beam broadening. Information about the turbulence profile is crucial to assist
the tomographic process in wide field Adaptive Optics (AO) system. Also study
of the turbulent layers may be used to reduce the impact of the delays which
exists in AO systems (Poyneer et al., 2009).
The turbulent flow of a fluid is a phenomenon, widely extend in the nature.
The air circulation in the lungs and also gas movement in the interstellar medium
are examples of turbulent flows.
Here, we interest to quantum turbulence in superfluid which is microscopic
theory of superfluid 4He that will provide a proper description of its behavior
on the small scales. In that case we deals with an equation describing the static
and dynamic behavior of the condensate wave function which is the non-linear
Schrodinger equation, or Ginsburg-Pitaevski-Gross equation (Pitaevskii,
1961; Gross, 1963):
where, ψ is the condensate wave function of particle and φ is the
wave function of electron. Also V0 = 4πdħ2/m
and U0 = 2πlħ2/μ are measures of the repulsive
interatomic forces in the fluid, where, l is the boson-impurity scattering length
and d is the boson diameter. Moreover, μ is the chemical potential. We
assume U0|V0 and set ħ = 1 for simplicity. A single-quantum
rectilinear vortex along r = 0 in cylindrical polar coordinates is described
by the following function:
where, f(r) = 0 and f(∞) = f0. Therefore, non-linear differential
Eq. 1 reduced to the following equation:
where, prime denote derivative with respect to r. In this study, we would like
to solve the Eq. 3 for various famous exponential potentials.
First of all we consider constant potential. Then, we examine Morse and Wood-Saxon
potentials and also a general exponential form of potential.
In the simplest case we assume:
where, E is a constant. This situation is special form of exponential function,
EeT, when r|1. In that case the Eq. 3 has the following
where, c1 and c2 are integration constants. In the Fig.
1, we can see that the wave function is periodic.
After studying the harmonic oscillator as a representation of molecule vibration,
one notice that a diatomic molecule which was actually bound using a harmonic
potential would never dissociate. The Morse potential realistically leads to
dissociation, making it more useful than the Harmonic potential. The Morse potential
is the simplest representative of the potential between two particles where
dissociation is possible. The Morse potential may be written in the following
where, C and a are arbitrary constants.
||Wave function in terms of r with constant potential for E
= 4 (dotted line), E = 5 (solid line), E = 6 (dashed line) and E = 25 (dash-dotted
||Wave function in terms of r with Morse potential for a = 1
(dotted line), a= 5 (solid line) and a = 25 (dashed line)
Numerically, we find behavior of the wave function in the Fig.
2, which shows periodic feature.
SIMPLE EXPONENTIAL POTENTIAL
Here, we consider simple exponential function which may be serves as a toy
model for interatomic potentials. In that case we assume that:
where, A and γ are arbitrary constants. Numerically, we find behavior
of the wave function in the Fig. 3, which shows periodic feature.
We find that the value of constant γ should be negative.
We can see that the solution with potential Eq. 10 is similar
to solution with the Morse potential.
GENERALIZED WOODS-SAXON POTENTIAL
Woods and Saxon introduced a potential to study elastic scattering (Woods
and Saxon, 1954). The Woods-Saxon potential plays an important role in microscopic
physics, since it can be used to describe the interaction of a nucleon with
the heavy nucleus. This potential is utilized to represent the mean field which
is felt by valance electron in Helium model (Dudek et
al., 2004). Generalized Woods-Saxon potential may be written as the
where, v, τ and ε are arbitrary constants. We can see that the generalized
Woods-Saxon potential with τ = 0, v = A and eεro1 limit
yields to the potential (10). Numerically, we find behavior of the wave function
in the Fig. 4, which shows periodic feature.
||Wave function in terms of r with exponential potential for
γ = -1 (dotted line), γ = -5 (solid line) and γ = -25 (dashed
||Wave function in terms of r with various potentials. Constant
potential (dash-dotted line), Morse potential (dashed line), Exponential
potential (dotted line), Generalized Woods-Saxon potential (solid line)
We can see that the solution with potential (10) is similar to solution with
the Morse and generalized Woods-Saxon potentials. We compare all solution in
a single plot to find differences of various models (Fig. 4).
In this study, we considered quantum turbulence and calculated wave function
from non-linear Schrodinger equation which is known as Ginsburg-Pitaevski-Gross
equation with various exponential potentials such as Morse and generalized Woods-Saxon
potentials. We found that the wave function has periodic behavior for exponential
form of potentials. There are still many interesting potentials which may be
used in the non-linear Schrodinger equation such as Dirac-Morse, Rosen-Morse,
Dirac-Rosen-Morse, Dirac-Eckart and Dirac-Scarf potentials (Alhaidari,
2001, 2003, 2004a, b).