View source for Kelvin-Helmholtz Instability and Roll-up

<strong> Kelvin-Helmholtz Instability </strong> (KHI) ([[#helmholtz|Helmoltz 1868]], [[#kelvin1|Kelvin 1871]]) is a hydrodynamic instability in which immiscible, incompressible, and inviscid fluids are in relative and irrotational motion. In KHI, the velocity and density profiles are uniform in each fluid layer, but they are discontinuous at the (plane) interface between the two fluids. This discontinuity in the (tangential) velocity, i.e., the shear flow, induces vorticity at the interface; as a result, the interface becomes an unstable vortex sheet that rolls up into a spiral. KHI is observed in a wide rage of nature. Ocean mixing in meteorology ([[#smyth|Smyth and Moum 2012]]), motion of interstellar clouds ([[#murray|Murray et al. 1993]], [[#vietri|Vietri et al. 1997]]) and clumping in supernova remnants ([[#wang|Wang and Chevalier 2001]]) in astrophysics, production of unstable shear layer and vortices in high energy density plasmas ([[#harding|Harding et al. 2009]], [[#hurricane|Hurricane et al. 2009]]), and quantized vortices in quantum fluids ([[#blaauwgeers|Blaauwgeers et al. 2002]], [[#takeuchi|Takeuchi et al. 2010]]), etc. are given as such examples. 


The basic mechanism of KHI development is in the existence of a uniform velocity shear and that does not need gravity or density difference. Therefore, these effects, as well as the surface tension and viscosity, are neglected in most cases in order to simplify the theoretical and numerical calculations, in which KHI is treated as a fluid instability that a lighter fluid is superposed on a heavier fluid ([[#chandra|Chandrasekhar 1981]],  [[#drazin1|Drazin 1970]], [[#nayfeh|Nayfeh and Saric 1972]], [[#weissman|Weissman 1979]]). When a heavier fluid is superposed on a lighter fluid, the instability is distinguished as Rayleigh-Taylor Instability (RTI) ([[#chandra|Chandrasekhar 1981]], [[#rti|Andrew and David 2009]]). RTI is an instability caused by gravity, in which the uniform shear flow is not considered. There is another inviscid fluid instability with density stratification, which is referred to as Richtmyer-Meshkov Instability (RMI) ([[#richtmyer|Richtmyer 1960]], [[#meshkov|Meshkov 1969]]). RMI is a shock-induced interfacial instability and is important in various areas such as supernova explosion, supersonic combustion, non-uniform turbulences, and Inertial Confinement Fusion (ICF) ([[#rmi|Schilling and Jacobs 2008]], [[#nishihara|Nishihara et al. 2010]]). KHI, RTI, and RMI each possess different linear growth rates (KHI and RTI have exponential growth, while the growth rate of RMI is proportional to time <math>t</math>); therefore, they are distinguished at the linear stage. However, when they attain the fully nonlinear stage and complicated interfacial motion with roll-up appears, these three instabilities are not distinguished and frequently discussed as KHI.




==Linear stability theory==


[[Image:KHI_setting.jpg|thumb|368px|right|KHI|Schematic diagram of the interfacial instability.]]

The physical mechanism underlying KHI is shown in  <figref>KHI_setting.jpg</figref>. The interface is regarded as a vortex sheet in the two-dimensional plane <math>(x,y)\ ,</math> at which a horizontal velocity discontinuity exists. In  <figref>KHI_setting.jpg</figref>, the physical quantities for <math>y < 0</math> and <math>y > 0</math> are represented by the subscripts 1 and 2, respectively.
The uniform velocity distribution in each fluid is known as basic flow, which is given by <math>{\mathbf{ U}_1} = (U_1,0)</math> for <math>y < 0</math> and <math>{\mathbf{ U}_2} = (U_2,0)</math> for <math>y > 0</math> in the figure.
The system is governed by the normal velocity continuity condition (the kinematic boundary condition) and pressure continuous condition at the interface.
The former condition implies that the normal component of the interfacial velocity <math>{\mathbf{u}}_{int}</math>is equivalent to the normal velocity of each fluid; i.e.,
:<math>\label{normal}
{\mathbf{n}} \cdot {\mathbf{u}}_{int} = {\mathbf{n}}\cdot {\mathbf{u}}_1 ={\mathbf{n}} \cdot {\mathbf{u}}_2,
</math>

where
:<math>
 {\mathbf{n}} = \frac{1}{\sqrt{1+\left(\frac{\partial \eta}{\partial x}\right)^2}}\left(-\frac{\partial \eta}{\partial x},1 \right)
</math>
is the unit normal to the interface and the fluid velocity <math>{\mathbf{u}}_i</math> in each fluid <math>i</math> <math>(i=1,2)</math> is related to the velocity potential <math>\phi_i</math> as <math> {\mathbf{u}}_i =  {\mathbf{U}}_i+\nabla \phi_i\ .</math> The deviation of the interface <math>y = \eta(x,t)\ ,</math> where <math>t</math> denotes time, is measured from <math>y=0\ .</math>
Then, \eqref{normal} can be rewritten as
:<math>
\frac{\partial \phi_1}{\partial y} =\frac{\partial \eta}{\partial t} + \frac{\partial \phi_1}{\partial x}\frac{\partial \eta}{\partial x}, 
</math>
:<math>\label{kbc}
\frac{\partial \phi_2}{\partial y} = \frac{\partial \eta}{\partial t} + \frac{\partial \phi_2}{\partial x}\frac{\partial \eta}{\partial x}. 
</math>

The condition that the pressure <math>p_i</math> in fluid <math>i</math> <math>(i=1,2)</math> is continuous at the interface, i.e.; <math>p_1 = p_2</math> at <math>y=\eta(x,t)\ ,</math> gives
:<math>\label{bernoulli}
\rho_1\left\{\frac{\partial \phi_1}{\partial t} + \frac{1}{2}\left[\left(\frac{\partial \phi_1}{\partial x}\right)^2 + \left(\frac{\partial \phi_1}{\partial y}\right)^2 \right]\right\} + (\rho_1 - \rho_2)g\eta
=\left\{\rho_2\frac{\partial \phi_2}{\partial t} + \frac{1}{2}\left[\left(\frac{\partial \phi_2}{\partial x}\right)^2 + \left(\frac{\partial \phi_2}{\partial y}\right)^2 \right]\right\} + \frac{\sigma\left(\frac{\partial^2 \eta}{\partial x^2}\right)}{\left[1+\left(\frac{\partial \eta}{\partial x}\right)^2\right]^{3/2}} , 
</math>

where <math>\rho_i</math> is the density in fluid <math>i</math> <math>(i=1,2)\ ,</math> <math>g</math> is gravity, and <math>\sigma</math> is surface tension coefficient at the interface. In the case of KHI, <math>\rho_1 \geq \rho_2</math> is assumed. Equation \eqref{bernoulli} is known as the Bernoulli equation. Equations \eqref{kbc} and  \eqref{bernoulli} are the governing equations of KHI.



Substituting the basic flow <math>{\mathbf{U}}_i</math> <math>(i=1,2)</math> into \eqref{kbc} and  \eqref{bernoulli}, and taking up to the first order of the small disturbance, one obtains
:<math>
\frac{\partial \eta}{\partial t} - \frac{\partial \phi_1}{\partial y} = -U_1 \frac{\partial \eta}{\partial x},
</math>
:<math>\label{kbc1}
\frac{\partial \eta}{\partial t} - \frac{\partial \phi_2}{\partial y} = -U_2 \frac{\partial \eta}{\partial x}, 
</math>

and
:<math>\label{bernoulli1}
\rho_1\left(
\frac{\partial \phi_1}{\partial t} + U_1 \frac{\partial \phi_1}{\partial x}\right) + (\rho_1 - \rho_2)g\eta 
= \rho_2\left(\frac{\partial \phi_2}{\partial t} + U_2\frac{\partial \phi_2}{\partial x}\right) + \sigma\frac{\partial^2 \eta}{\partial x^2}, 
</math>

where all quantities with respect to <math>\phi_i</math> are taken at <math>y=0\ .</math>
From the incompressibility condition, Laplace equation, i.e., <math>\triangle \phi_i = 0 </math> holds in each fluid <math>i\ .</math> Thus, one gets solutions of \eqref{kbc1} and  \eqref{bernoulli1} as
:<math>
\eta(x,t) = {\rm Re}\left[A_0{\rm e}^{i (k x - \omega t)}\right],
</math>
:<math>
\phi_1(x,t) = {\rm Re}\left[iA_0\left(U_1 - \frac{\omega}{k}\right){\rm e}^{i(kx-\omega t)}\right]{\rm e}^{ky} \quad (y < 0),
</math>
:<math>\label{linear_sol}
\phi_2(x,t) ={\rm Re}\left[-iA_0\left(U_2 - \frac{\omega}{k}\right){\rm e}^{i(kx-\omega t)}\right]{\rm e}^{-ky} \quad (y > 0), 
</math>

where <math>k</math> is the wave number, <math>{\rm Re}</math> denotes the real part, <math>A_0</math> is a constant that corresponds to the initial amplitude of the interface, and
:<math>\label{growth}
\omega = \frac{\rho_1 U_1 + \rho_2 U_2}{\rho_1 + \rho_2}k \pm \sqrt{
\frac{-\rho_1\rho_2(U_1 - U_2)^2}{(\rho_1 + \rho_2)^2}k^2 + \frac{\rho_1 - \rho_2}{\rho_1 + \rho_2}gk + \frac{\sigma}{\rho_1 + \rho_2}k^3}. 
</math>

Here, the condition that the disturbance vanishes at <math>y=\pm \infty</math> is imposed. The system grows exponentially under linear approximation when
:<math>\label{imaginary}
F(k) \equiv -\frac{\rho_1\rho_2(U_1 - U_2)^2}{(\rho_1 + \rho_2)^2} + g(k) < 0,
</math>
where <math>g(k)</math> is defined by 
:<math>\label{g}
g(k) = \frac{\rho_1 - \rho_2}{\rho_1 + \rho_2}\frac{g}{k} + \frac{\sigma}{\rho_1 + \rho_2}k.
</math>


When <math>\rho_1 < \rho_2</math> and <math>U_1 \ne U_2 =0</math>, <math>F(k)</math> is always negative; <math>F(k) < 0</math>; therefore, the system is unstable and this instability is referred to as KHI. When <math>\rho_1 < \rho_2</math> and <math>U_1 = U_2 =0</math>, the system becomes unstable for the value of <math>g(k)</math> that satisfies the inequality <math>g(k) < 0</math>; i.e., for the wavenumber <math>k</math> that is in the range <math>0 < k < \sqrt{(\rho_2 - \rho_1)g/\sigma}</math>. This instability is referred to as (classical) RTI. 


When <math>\rho_1 > \rho_2</math>, there exists a minimum value <math>g(k) = g_m \equiv 2\sqrt{g\sigma(\rho_1 - \rho_2)}/(\rho_1 + \rho_2)</math> in <math>g(k)</math> for the wavenumber <math>k = k_c \equiv \sqrt{g(\rho_1 - \rho_2)/\sigma}</math> and the system is always stable (<math>F(k) > 0</math>) when <math>U_1 = U_2 =0</math>. Even though <math>U_1 \ne U_2 =0</math>, the system can become stable (including the marginally stable) for the case that the condition <math>|U_1 - U_2| \leq (\rho_1 + \rho_2)\sqrt{g_m/(\rho_1 \rho_2)}</math> is satisfied. These stable cases for <math>\rho_1 > \rho_2</math> are referred to as capillary-gravity waves. On the other hand, when <math>|U_1 - U_2| > (\rho_1 + \rho_2)\sqrt{g_m/(\rho_1 \rho_2)}</math>, there exists a range in <math>k</math> such that <math>F(k)</math> satisfies the inequality <math>F(k) < 0</math>. In this case, the system is unstable even if <math>\rho_1 > \rho_2</math> (refer to <figref>KHI_A01_s01.gif</figref>). 


The detailed calculations of the linear stability theory for KHI have been provided by Batchelor  ([[#batchelor|Batchelor 2000]])  (for the case that <math>\rho_1 = \rho_2</math> and <math>\sigma=0</math>), Chandrasekhar ([[#chandra|Chandrasekhar 1981]]), and Drazin and Reid ([[#dr|Drazin and Reid 2004]]). 

==Nonlinear theory==
===Birkhoff-Rott equation===

The nonlinear dynamics of the interface is described in terms of the dynamics of a vortex sheet. The temporal evolution of a sheet is expressed by an integro-differential equation known as the Birkhoff-Rott equation, which is derived from the Biot-Savart law ([[#birkhoff|Birkhoff 1962]], [[#rott|Rott 1956]]). For simplicity, we set <math>\rho_1 = \rho_2\ ,</math> <math>g=0\ ,</math> and <math>\sigma=0\ .</math> By denoting the interface as the complex form: <math>z(\theta,t) = x(\theta,t) + iy(\theta,t)\ ,</math> the Birkhoff-Rott equation is expressed as
:<math>\label{eqn_BR}
\frac{\partial z^*(\theta, t)}{\partial t} = \frac{1}{2\pi i}{\rm {P.V.}}\int_{-\infty}^\infty \frac{\gamma (\theta')}{z(\theta, t)-z(\theta', t)}d\theta' , 
</math>

where <math>\theta</math> is a Lagrangian parameter that parameterizes the interface, <math>z^*</math> is the complex conjugate of <math>z\ ,</math> <math>{\rm {P.V.}}</math> denotes the principal value integral, and <math> \gamma</math> is the vortex sheet strength defined as <math> \gamma = d\Gamma/d\theta</math> through the circulation <math> \Gamma\ .</math>
Equation \eqref{eqn_BR} gives the velocity of the sheet induced by the vorticity at the interface, and the configuration of the interface is obtained from the temporal integration of  \eqref{eqn_BR}. When density stratification does not exist, i.e., when the fluid density in the system is uniform, the circulation <math>\Gamma</math> is conserved along Lagrangian particle paths (Kelvin's circulation theorem; see [[#batchelor|Batchelor 2000]], [[#kelvin2|Kelvin 1869]]) and <math> \gamma</math> does not depend on time. The Birkhoff-Rott equation is equivalent to the 2D Euler system expressed by \eqref{kbc} and  \eqref{bernoulli}  (in a sense of weak solutions) when the initial interface is sufficiently smooth ([[#lopes|Lopes Filho et al. 2007]]).

===Moore's curvature singularity===

Taking the Lagrangian parameter <math>\theta</math> as the circulation <math>\Gamma\ ,</math> i.e., <math>\theta = \Gamma</math> <math>(-\infty < \Gamma < \infty)\ ,</math> Moore investigated the analyticity of the Birkhoff-Rott equation \eqref{eqn_BR} [[#moore|(Moore 1979)]]. Setting the initial condition as
:<math>
z(\Gamma,0) = \Gamma + i\epsilon \sin \Gamma \quad (\epsilon \ll 1)
</math>
and assuming the form
:<math>\label{eqn_z}
z(\Gamma,t) = \Gamma + 2i\sum_{n=1}^\infty A_n(t)\sin n\Gamma,
</math>
 
he obtained the asymptotic expansion for <math>n \gg 1</math> and <math>t \gg 1</math> as
:<math>\label{eqn_sol_z}
\epsilon^n A_{n0} \sim t^{-1}(2\pi)^{-1/2}(1+i)n^{-5/2}\exp\left[n\left(1 + \frac{1}{2} + \ln \frac{1}{4}\epsilon t \right) \right],
</math>
 
where <math>A_n</math> is expanded as <math>A_n = \epsilon^n A_{n0} + \epsilon^{n+2} A_{n2} + \epsilon^{n+4} A_{n4} + \cdots\ .</math>
The asymptotic solution to the Birkhoff-Rott equation loses its analyticity at a finite time <math>t=t_c</math> such that the condition <math>1 + t_c/2 + \ln t_c = \ln (4/\epsilon)</math> is satisfied.
Then, the shape of the sheet is given by
:<math>\label{eqn_power_law}
z(\Gamma,t_c) \sim \Gamma + \sum_{n=- \infty}^\infty C n^{-5/2} {\rm e}^{in\Gamma}
</math>

and
:<math>
z(\Gamma,t) = \Gamma + \frac{2\sqrt{3}}{3t}(1+i)\left[(1-{\rm e}^{i\Gamma}\epsilon\Theta)^{3/2} - (1-{\rm e}^{-i\Gamma}\epsilon\Theta)^{3/2}\right]
</math> 
at <math>\Gamma = \Gamma_c =2n\pi</math> <math>(n=0, \pm 1, \pm2, \cdots)\ ,</math> where <math>C</math> is a constant and <math>\Theta = t/4 \exp\left(t/2+1 \right)\ .</math>
Since <math>\epsilon\Theta \to 1</math> as <math>t \to t_c\ ,</math> the curvature of the sheet diverges as <math>|\Gamma - \Gamma_c|^{-1/2}</math> in the neighborhood of the singular point <math>\Gamma = \Gamma_c\ .</math> The true sheet strength <math>\kappa\ ,</math> defined as <math>\kappa = {d \Gamma}/{d s} =  \gamma/s_\theta\ ,</math> has the cuspidal form <math>\kappa(\Gamma,t) \propto |\Gamma - \Gamma_c|^{1/2}\ ,</math> where <math>s_\theta = \sqrt{x_\theta^2 + y_\theta^2}\ ,</math> the subscript <math>\theta</math> denotes differentiation with respect to that variable, and <math>d</math> denotes differentiation along the fluid particles.



The power law <math>n^{-5/2}</math> for the Fourier coefficient in \eqref{eqn_power_law} has been verified by several researchers via numerical ([[#krasny1|Krasny 1986]], [[#shelley|Shelley 1992]]) and analytical ([[#caflisch1|Caflisch and Semmes 1990]], [[#cbt|Cowley et al. 1999]], [[#mbo|Meiron et al. 1982]]) calculations. Baker et al. ([[#bcs|Baker et al. 1993]]) extended Moore's asymptotic theory for curvature singularity to the case of Atwood number <math>A \ne 0</math> and found that a family of exact solutions for which singularities develop on the fluid interface, where <math>A</math> is defined by the fluid densities <math>\rho_1</math> and <math>\rho_2</math> as <math>A = (\rho_2 - \rho_1)/(\rho_1 + \rho_2)\ .</math> 
They showed theoretically and numerically that these singularities reach the real axis in the complex plane in finite time for <math>A \ne 1\ ,</math> and then, Moore's curvature singularity appears in the physical plane. They also predicted that these singularities never reach the real axis in finite time when <math>A=1\ ,</math> which suggests that the curvature singularity does not appear for <math>A = 1\ .</math> Although the rigorous mathematical proof does not exist, their prediction for finite Atwood numbers including <math>A=1</math> is supported theoretically ([[#tanveer|Tanveer 1993]]) and numerically ([[#mn|Matsuoka and Nishihara 2006]]). 



==Numerical method==

===Zero surface tension case===



[[Image:KHI_xx_curvature.jpg|thumb|600px|center|fig_singularity|Moore’s curvature singularity (a) interfacial profile and (b) curvature of (a), where the regularized parameter <math>\delta=0</math> and the alternate point quadrature method is used in the calculation. The curvature of the vortex core is extremely
large, even though its interfacial amplitude is small and smooth.]]


Suppose that the initial disturbance is <math>2\pi</math>-periodic in the <math>x</math>-direction; <math>z(\theta + 2\pi, t) = 2\pi + z(\theta,t)\ ,</math> <math>\gamma(\theta+2\pi) = \gamma(\theta)\ .</math> Then, the Birkhoff-Rott equation \eqref{eqn_BR} is rewritten as
:<math>\label{eqn_periodic_BR}
\frac{\partial z^*(\theta, t)}{\partial t} = \frac{1}{4\pi i}{\rm {P.V.}}\int_0^{2\pi}\gamma(\theta')\cot\frac{1}{2}\left[ z(\theta, t)-z(\theta', t) \right]d\theta' .
</math>
 
Highly accurate calculations are required to detect singularity formation. Shelley ([[#shelley|Shelley 1992]]) carried out the spatial integration expressed by \eqref{eqn_periodic_BR} with spectral accuracy by using the alternate point quadrature method ([[#sidi|Sidi and Israeli 1988]]), and numerically verified the <math>-5/2</math>-type power law and curvature singularity predicted by Moore. 
This method does not include the error by the spatial integration, i.e., it is exact within the machine accuracy; however, the accuracy is typically lost at the critical time. Although it is first order accurate, the point vortex method presented by Krasny ([[#krasny1|Krasny 1986]]) remains consistent at the critical time at which the Moore's curvature singularity appears. For the mathematical discussion on the relation between the point vortex approximation and the critical time, refer to the reference ([[#chl|Caflisch et al. 1999]]).



Curvature singularity is an unphysical phenomenon. In real systems, the interface is a layer with a finite thickness; in addition, viscosity or surface tension affects the interface. Owing to these effects, singularity formation is prevented in reality; instead, the roll-up of a sheet is observed.
In order to perform long-time calculations beyond curvature singularity, Krasny introduced a small parameter <math>\delta</math> and regularized the Cauchy integral in the Birkhoff-Rott equation \eqref{eqn_periodic_BR}  ([[#krasny2|Krasny 1987]]). For the periodic case, this regularization yields
:<math>\frac{\partial x}{\partial t}(\theta, t) = -\frac{1}{4 \pi}\int_{0}^{2\pi}\frac{\sinh\left(y(\theta, t) - y(\theta', t)\right)\gamma(\theta', t)}{\cosh\left(y(\theta, t) - y(\theta', t)\right)-\cos\left(x(\theta, t)-x(\theta', t)\right) + \delta^2}d\theta',
</math> 
:<math>\label{eqn_velocity}
\frac{\partial y}{\partial t}(\theta, t) = \frac{1}{4 \pi}\int_{0}^{2\pi}\frac{\sin\left(y(\theta, t) - y(\theta', t)\right)\gamma(\theta', t)}{\cosh\left(y(\theta, t) - y(\theta', t)\right)-\cos\left(x(\theta, t)-x(\theta', t)\right) + \delta^2}d\theta'.
</math>
 
[[Image:KHI_s0.gif|thumb|368px|center|fig_s0|Roll-up of a vortex sheet, where <math>\delta=0.1</math> and <math>\sigma=0\ .</math>]]

This approach is known as the vortex blob method. The parameter <math>\delta</math> plays an important role in cutting higher-order Fourier modes. When <math>\delta \ne 0\ ,</math> \eqref{eqn_velocity} does not coincide with the 2D Euler system ([[#holm|Holm et al. 2006]]); however, this equation effectively describes real interfacial motion such as roll-up. The spatial integration in \eqref{eqn_velocity} can be carried out using the conventional trapezoidal rule.



The curvature singularity in KHI and the roll-up of a sheet are shown in  <figref>KHI_xx_curvature.jpg</figref> and  <figref>KHI_s0.gif</figref>, respectively. Here, the initial condition is taken as <math>x(\theta,0) =\theta + 0.05\sin\theta</math> and <math>y(\theta,0) = -0.05\sin\theta</math> for both cases. For the regularized parameter <math>\delta = 0\ ,</math> roll-up does not occur. The point <math>\theta = \theta_c</math> (<math>\Gamma_c</math> in Moore's analysis) at which the curvature diverges is known as the vortex core. The <math>N</math>-th derivative (<math>N \geq 2</math>) with respect to <math>\theta</math> diverges at the vortex core when <math>\delta=0\ .</math>
When <math>\delta \ne 0\ ,</math> the roll-up occurs and the vortex core becomes the center of a spiral. The speed of the roll-up slows down for larger values of <math>\delta\ .</math> In addition to the vortex blob method, there are some alternative regularization approaches for describing the roll-up of a vortex sheet ([[#bp|Baker and Pham 2006]], [[#holm|Holm et al. 2006]]).


Singularity formation is also detected in 3D vortex motion ([[#hou1|Hou et al. 2003]], [[#ishihara1|Ishihara and Kaneda 1995]], [[#ishihara2|Ishihara and Kaneda 1996]], [[#nitsche|Nitsche 2001]], [[#sakajo|Sakajo 2002]]). In this case, the vortex-induced velocity corresponding to the Birkhoff-Rott equation \eqref{eqn_periodic_BR}  is expressed by the Biot-Savart law.

===Finite surface tension case===

When the surface tension is taken into account, the roll-up can occur even though the regularized parameter <math>\delta = 0\ .</math> Hou '' et al''. numerically verified this fact for the case of the Atwood number <math>A=0\ ,</math> i.e., <math>\rho_1 = \rho_2</math> ([[#hou2|Hou et al. 1994]], [[#hou3|Hou et al. 1997]]). Their method is generalized to the case of <math>A \ne 0</math>  ([[#ceni|Ceniceros and Hou 1998]], [[#cm|Matsuoka 2009]]). The roll-up for <math>\sigma \ne 0</math> and <math>A \ne 0</math> is shown in  <figref>KHI_A01_s01.gif</figref>, where the initial condition is selected as <math>x(\theta,0)= \theta</math> and <math>y(\theta,0) = 0.1\sin\theta\ .</math> 
The parameters used here (refer to the figure caption) corresponds to the unstable case [<math>F(k)< 0</math>]. 
For the roll-up with finite surface tension, a phenomenon called "pinching singularity" is observed instead of Moore's curvature singularity ([[#hou2|Hou et al. 1994]]). This pinching phenomenon disappears as the surface tension coefficient increases ([[#cm|Matsuoka 2009]]). When <math>\rho_1 > \rho_2</math> (<math>A < 0\ ;</math> the lighter fluid is superposed on the heavier fluid) and the relative shear velocity <math>|U_1 - U_2| = 0\ ,</math> the system is linearly stable as described in the section "Linear stability theory" and the roll-up does not occur.

[[Image:KHI_A01_s01.gif|thumb|368px|center|fig_A01_s01|Interfacial motion with surface tension, where <math>A=-0.1</math>, <math>g=1.0</math>, <math> \sigma/(\rho_1+\rho_2) = 0.05\ ,</math>
and <math> |U_1 -U_2|=2.0\ .</math>]]


==Effects of viscosity and roll-up==

The vortex sheet model is not a realistic representation of shear layers in nature. In real flows, the short-wave disturbances are suppressed by viscosity which acts to thicken the vortex sheet. Moore introduced an intrinsic coordinate system and extended a vortex sheet model to a thin layer model with uniform vorticity distribution ([[#moore2|Moore 1978]]). Applying the Moore's curvilinear coordinates to the vorticity equation derived from the Navier-Stokes equation, Dhanak proposed an equation of motion of a viscous vortex sheet using the method of matched asymptotic expansions ([[#dhanak|Dhanak 1994]]). Sohn performed numerical computations of this Dhanak's viscous sheet model and obtained a result that the roll-up was weakened by viscosity ([[#sohn2|Sohn 2013]]). Tryggvason et al. ([[#tryggvason|Tryggvason et al. 1991]]) numerically solved full viscous Navier-Stokes equations and concluded that the viscous solution of a thin vorticity layer with high Reynolds number almost becomes the same one obtained by a vortex sheet model with relatively small regularization parameter <math>\delta</math>. Forbes et al. presented accurate numerical scheme for solving viscous flow without using the vortex sheet model ([[#forbes1|Chen and Forbes 2011]]) and found the roll-up of a thin viscous layer in a weakly compressible system ([[#forbes2|Forbes 2011]]). These numerical results suggest that viscosity acts as a regularization effect on the roll-up of vortex sheets.  


==Interfacial instabilities in magnetohydrodynamic and electro-magnetohydrodynamics flows ==

Interfacial instabilities in magnetohydrodynamic (MHD) or electro-magnetohydrodynamics (EMHD) flows are important for laboratory plasmas, both of magnetic and inertial confinement fusions, space plasmas, and astrophysics. When the magnetic field exists, the fluid instabilities such as KHI, RTI, and RMI tend to be suppressed. The pioneering work for vortex sheet motion in MHD flows can be found in ([[#chandra2|Chandrasekhar 1952]]), in which he studied convective motion in a magnetic field and concluded that the convection is inhibited by the magnetic field. The linear stability theory for KHI and RTI in MHD flows is summarized in his book ([[#chandra|Chandrasekhar 1981]]). In MHD or EMHD flows, not only a velocity shear but also discontinuity of the magnetic field can exist across an interface. When the magnetic field changes either in direction or magnitude or both across the interface, a current is induced in a thin layer. This current-carrying layer is referred to as a current sheet ([[#priest|Priest 2000]]) and it is important in magnetic reconnection processes in space plasmas and laboratory fusion plasmas. In the case that a vortex sheet and a current sheet coexist, the interface is called a current-vortex sheet ([[#axford|Axford 1960]]). 



Taking into account the plasma sheath region, Arshukova et al. theoretically discussed the deformation of a current-vortex sheet in MHD KHI and obtained a result that the instability is stronger for smaller curvature radius of the interface ([[#arshukova|Arshukova et al. 2002]]). Hunter and Thoo derived an amplitude equation to describe the weakly nonlinear motion of a current-vortex sheet in MHD KHI ([[#hunter|Hunter and Thoo 2011]]). Del Sarto et al. showed that the long time evolution of current layers in MHD and EMHD flows can be strongly affected by the onset of a secondary KHI which leads to the formation of fluid vortices inside the magnetic islands ([[#delsarto1|Del Sarto et al. 2003]], [[#delsarto2|Del Sarto et al. 2005]]). Their studies for the secondary KHI are extended to the three-dimensional magnetic reconnection in fusion relevant plasmas by Grasso et al. ([[#grasso|Grasso et al. 2007]]), in which the roll-up at highly nonlinear stages is observed. 



The roll-up of the interface is also observed in MHD RMI. Motivated by the origin of a strong magnetic field associated with supernova remnants (SNR's) ([[#uchiyama|Uchiyama et al. 2007]]) in astrophysics , Sano et al. investigated the interfacial motion in MHD RMI and concluded that the stretch of a vortex sheet causes magnetic field amplification in an SNR ([[#sano|Sano et al. 2012]]). Wheatley et al. found an oscillation of the interface in MHD RMI and claimed that the nonlinear growth of RMI can be inhibited by a magnetic field, which might lead to the suppression of the hydrodynamic instability in ICF ([[#wheatley|Wheatley et al. 2014]]). Generally, the roll-up in MHD flows tends to become weaker compared with the one in pure hydrodynamic flows, which suggests that the existence of a magnetic field might enable to prevent the occurrence of hydrodynamic instabilities such as KHI, RTI, and RMI. 




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== Further reading ==

* A. J. Majda and A. L. Bertozzi "Vorticity and incompressible flow" Cambridge University Press, 2002. ISBN 0-521-63057-6.

* P. G. Saffman,  "Vortex Dynamics" Cambridge University Press, 1992. ISBN 0-521-47739-5. 

==Internal references==

<span id = "AD2009">
* W. Cook Andrew and Youngs David (2009) ''Rayleigh-Taylor instability and mixing'' Scholarpedia, '''4'''(2):6092. ([http://www.scholarpedia.org/article/Rayleigh-Taylor_instability_and_mixing online])
</span>
<span id = "SJ2008">
* Oleg Schilling and Jeffrey W. Jacobs (2008) ''Richtmyer-Meshkov instability and re-accelerated inhomogeneous flows'' Scholarpedia, '''3'''(7):6090. ([http://www.scholarpedia.org/article/Richtmyer-Meshkov_instability_and_re-accelerated_inhomogeneous_flows online])

 
[[Category:Fluid Dynamics]]