text:
\section{Introduction \label{sec:intro}}
The interaction between surface water waves and underlying currents affects the wave properties significantly \citep{longuet1961changes}, as also confirmed by experiments, e.g., \cite{swan2001experimental}. The interaction is hence of practical interest for design of offshore structures, see recent comparisons of structural responses with and without this interaction considered \citep{chen2018fatigue,chen2018wave} where linear wave-current interaction models were used though; for coastal engineering applications \citep{kirby1983propagation, dong2012theoretical} and
in geophysical fluid dynamics \citep{kirby1984note, kirby1989surface}. Of most interest is the problem of two-dimensional (2D) periodic water waves propagating on steady current. A large number of theoretical studies and numerical investigations have been dedicated to this problem. However, it is still far from being fully understood. A comprehensive introduction to nonlinear wave-current interactions can be found in \cite{constantin2011nonlinear}.

In fact, it is only recently that the existence of the large-amplitude wave solutions for rotational flows with general vorticity (thus allowing arbitrary current profiles) in finite water depth has been proved by \cite{constantin2004exact}. Prior to this, only waves of small-amplitude were known to exist in the presence of vorticity (\cite{dubreil1934sur}). \cite{hur2006global} further extended the proof of \cite{constantin2004exact} to the case of infinite-depth with the additional restriction that the vorticity distribution is positive and monotone. \cite{hur2011stokeswaves} later completed the proof of existence of large-amplitude waves with infinite depth removing the mentioned restrictions on vorticity. Moreover, \cite{constantin2011periodic} and \cite{strauss2012vorticity} addressed the existence of periodic traveling waves of small and large amplitude in a flow with an arbitrary bounded but discontinuous vorticity.

Following the work by \cite{constantin2004exact}, rotational flows with constant and non-constant vorticity have received considerable attention. Notably, many numerical investigations, on both irrotational and rotational flows, appeared much earlier. For example, for irrotational flow in finite water depth, \cite{chappelear1961direct} expanded the velocity field and the wave profile in Fourier series and the Fourier coefficients were numerically determined from the Bernoulli equation by using the least squares method. The direct numerical simulation results were compared with the fifth-order Stokes theory. \cite{dalrymple1974finite} presented a numerical model for steady water waves propagating over a linear shear current (constant vorticity) where stream function was expressed as the superposition of a linearly varying current profile and an irrotational wave field. The results thereof showed that the vorticity has a significant impact on the wave height. \cite{simmen1985steady} and \cite{da1988steep} studied periodic steady water waves with constant vorticity for infinite and finite water depth, respectively. The numerical methods for irrotational flows and rotational flows with constant vorticity are generally based on the boundary integral method or the Fourier expansion. Recently, \cite{clamond2018accurate} developed a novel conformal mapping method for irrotational flows, where the governing equation is rendered of Babenko kind which is then solved using the Petviashvili method. The model is very efficient and is able to solve for large amplitude waves with quite large steepnesses over arbitrary depth. \cite{dyachenko2019stokes} further modified the Babenko equation to permit constant vorticity and applied Newton-GMRES method following the use of the Fourier expansion. There are also other methods that have been developed. For instance, \cite{ablowitz2006new} proposed a non-local formulation which was further applied to consider non-flat bottom and moving non-flat bottom boundaries \citep{fokas2012water,francius2017two}. \cite{wahlen2009steady} and \cite{ehrnstrom2011steady} devised a novel transform for small-amplitude waves with constant and linear vorticity, i.e., a vertical scaling transformation.

For waves on rotational flows with non-constant vorticity, early numerical investigations date back to \cite{dalrymple1973water,dalrymple1977numerical} where the author considered the effects of various current profiles on the finite-amplitude steady waves over finite water depth. The method was based on the Dubreil-Jacotin (DJ) transformation and finite difference discretization of the transformed rectangular domain. For a given boundary condition, the solution to internal points is improved by a number of sweeps; the error on the free surface boundary condition is then evaluated; subsequently, the solution on the free boundary is updated and the procedure is repeated until convergence. Using this method, \cite{dalrymple1976symmetric} showed the influence of the vorticity on wavelength and crest elevation of the wave, considering that the currents vary as trigonometric and hyperbolic sine and cosine functions of the depth. \cite{thomas1990wave} performed experimental studies for a finite-amplitude wave train interacting with a steady current containing an arbitrary distribution of vorticity, and moderate agreement was found between the measurements and the results obtained using the model of \cite{dalrymple1977numerical}.

\cite{ko2008large} presented a numerical continuation method to find large-amplitude 2D periodic steady water waves with arbitrary vorticity, which was used for discussing the relationship between amplitude, hydraulic head, depth and mass flux for variable vorticity cases \citep{ko2008effect}. This method neither uses truncated Fourier expansion nor makes any assumption on the water depth or wave amplitude. Further numerical analyses of water waves on rotational flows with discontinuous vorticity can be found in \cite{strauss2010steady}, \cite{constantin2011periodic}, and \cite{constantin2012dispersion}. The ethos of the approaches towards wave computation is that the local bifurcation from a laminar flow to a small-amplitude wave solution is predicted by a dispersion relation, the algorithm beginning with the linearized solution. The wave solution is then continued with increasing wave height until the solution becomes close to a wave in a flow with stagnation point (global bifurcation). The dispersion relation has been derived for small-amplitude periodic water waves traveling on rotational flows for various vorticity cases, e.g., see \cite{constantin2011periodic}; and \cite{constantin2012dispersion} and \cite{martin2015dispersion} for flows with two layers. \cite{karageorgis2012dispersion} discussed dispersion relations for several types of non-constant vorticities. High-order approximations for the constant vorticity case were discussed in \cite{constantin2015approximations}. \cite{amann2018numerical} expanded upon the method of \cite{ko2008large}, by allowing nonuniform grid points and enabling the continuation with vorticity as a bifurcation parameter. Using the latter scheme they successfully discovered a new part of the bifurcation curve for some critical constant vorticity values (see also \cite{kalimeris2018analytical}). Furthermore, \cite{constantin2015penalization} proposed a penalization method to solve for large-amplitude waves. Note that using the method of \cite{ko2008large}, the mean water depth is recovered from the solution, which leads to a small  variation in the value of mean water depth along the bifurcation curve \citep{ko2008effect}. The mean-depth can be held constant by following the reformulation proposed by  \cite{henry2013large,henry2013steady}.

Considerable understanding of water waves on irrotational flows and flows with constant vorticity has been established recently. For irrotational flows, \cite{constantin2006trajectories} proved analytically that the particle trajectories are not closed on Stokes waves and \cite{henry2006trajectories} extended the proof to deep water. \cite{constantin2008particle} and \cite{matioc2010particle} showed that there are no closed particle orbits in linear periodic water waves. \cite{borluk2012particle} reconstructed the velocity fields and the particle paths of long-crested waves on irrotational flows using the KdV approximation. These particle paths were found to be consistent with the Stokes drift but with increasing amplitude. \cite{carter2020particle} studied the particle paths in the nonlinear Schr{\"o}dinger models. \cite{grue2014velocity} experimentally studied the velocity field below large-amplitude periodic water waves and the results are generally consistent with nonlinear theories. \cite{grue2017experimental} further investigated the particle paths in steep waves at finite water depth (without background current) experimentally, focusing on the Stokes drift. It was shown that for steep waves on irrotational flow, two closed particle paths occur at two vertical positions, in contrast to the theoretical results \citep{constantin2006trajectories}.  The lower closed particle paths occur at a position quite close to the bed due to the boundary layer effect, whereas the theoretical study by \cite{constantin2006trajectories} assumes inviscid water. It is also clear from \cite{grue2017experimental} that the thickness of the boundary layer is about 5 percent of the water depth, which is significantly affected by viscosity effect. \cite{paprota2018particle} developed a theoretical model for studying the particle paths of nonlinear waves generated in a closed flume and the Stokes drift velocity obtained from experimental data \citep{paprota2016experimental} were found to be quite consistent with theories. For a summary of the present analytical knowledge on the properties of irrotational flows, including the velocity field, pressure and particle trajectories, see \cite{constantin2015flow}. For waves on a linear-shear flow, \cite{ehrnstrom2008streamlines} studied the pattern of streamlines and particle trajectories and pointed out that for non-positive vorticity the particles display a forward drift. Using an asymptotic expansion for the stream function, \cite{ali2013reconstruction} constructed the pressure beneath steady long gravity water waves traveling on flows with constant background vorticity. \cite{curtis2018particle} investigated the effect of linear background shear on the properties of wavetrains, particularly on particle paths, in deep water using nonlinear Schrödinger models. Numerically, for understanding the particle paths or trajectories, \cite{nachbin2014boundary} proposed a method for particle trajectories in Stokes waves in the boundary integral formulation for irrotational and constant vorticity cases. A summary of the flow structure beneath rotational flows with constant vorticity and the corresponding numerical method can be found in \cite{nachbin2017capturing}. Lately, \cite{dyachenko2019stokesII} recovered various characteristics of wave profiles in flows with constant vorticity.

For rotational flow with general vorticity but subject to certain constraints, \cite{constantin2007rotational} studied the points of maximal horizontal velocity of waves near stagnation. Further understanding on the velocity field has been established by \cite{varvaruca2008some} and \cite{basu2019some}.  \cite{vasan2014pressure} discussed the pressure beneath rotational flows with constant vorticity pointing out interesting applications in wave profile recovery
\citep{oliveras2012recovering,basu2017wave,basu2017estimation}.

For surface waves on rotational flows, there are still a number of open problems yet to be solved \citep{strauss2012vorticity}. However, it becomes extremely difficult to capture the properties of the surface wave and the flow structure analytically for complex cases. As manifested in the literature, numerical investigations can provide us with important insights for
such cases. To deal with the otherwise intractable problem of rotational flows with arbitrary voriticity distribution, contrary to the widely assumed linear shear flows, we focus on the shear flow with piece-wise constant vorticity which is the simplest case deviating from constant vorticity. This is a valid approximation for waves traveling on depth-varying currents \citep{dalrymple1974water,swan2001experimental}. In particular, we are concerned with shear flows with two layers. It is known that wind typically induces vorticity in the layer near the surface, while near the ocean shore, vorticity can be generated at the bottom, e.g., by tidal action \citep{ko2008effect}. In this study, we report on a new branch of the bifurcation curve for waves on rotational flows with discontinuous vorticity and investigate the pressure beneath the waves for flows with nearly approaching stagnation points. Further, we develop a method for computing particle paths based on the exact velocity field solved numerically. The particle paths are of increasing interest since they are related to the Lagrangian transport and the Stokes drift in ocean  \citep{bremer2016lagrangian,van2017stokes}.

The rest of this paper is structured as follows. In Section \ref{sec:prob} and Section \ref{sec:method}, we briefly introduce the problem and the numerical methods respectively. The numerical results are then presented in Section \ref{sec:result}. We close the study by a brief summary in Section \ref{sec:con}.

\section{Statement of the problem}\label{sec:prob}
We consider two-dimensional steady, periodic, rotational gravity traveling surface waves, of period $2L$ ($L>0$ being a constant) propagating over water flows bounded below by the flat bed $y=-d$, where $d>0$ is a constant. The water is assumed to be inviscid. In the numerical investigation, we shall place our focus on the case with discontinuous vorticity of piece-wise constant type.

\subsection{Governing equations}
The time-dependent water wave problem is presented in the Cartesian $(X,Y)$-coordinate system with the origin on the mean water level, having the $X$-axis pointing in the horizontal direction and the $Y$-axis pointing upwards. The steady character of the problem we consider allows us to repeal the time-dependence by using the change of coordinates $(X-ct,Y)\mapsto (x,y)$, where $c$ is the constant speed of the traveling waves and $t$ denotes time. Hereafter, we will be working in the moving frame $(x,y)$ with the origin placed under the crest, except for the computation of particle paths. In this setting, the motion of the water flow is governed by the Euler (or momentum) equations, the equation of mass conservation supplemented by the kinematic boundary conditions on the two boundaries of the flow and the dynamic boundary condition on the free surface. The latter condition at the free surface decouples the motion of the water from the motion of the air above, while the former kinematic conditions convey the fact that the bed and the free surface are impermeable boundaries, cf. \cite{constantin2011nonlinear}.

Aiming at a simplification of the analysis we introduce the stream function, denoted by $\psi$, and defined (up to a constant) by
\begin{equation}\label{eq:psi}
\psi_y=u-c\quad \mbox{and}\quad \psi_x=-v \ .
\end{equation}
Here, ($u,v$) is the velocity field in the $(x,y)$-coordinates.
The mass flux relative to the uniform flow at speed $c$ (normalized with respect to density) is
\begin{equation}\label{p0}
p_0=\int_{-d}^{\eta(x)}[u(x,y)-c]\mathrm{d} y,
\end{equation}
further referred to as relative mass flux. It turns out from mass conservation equation and from the kinematic boundary condition on the surface that $p_0$ is a constant.

From \eqref{eq:psi} and the kinematic condition on the free surface we infer that $\psi$ is constant on the free surface.
Therefore, we choose $\psi=0$ on the surface and thus, by \eqref{p0}, we have
 $\psi=-p_0$ at the bottom $y=-d$. From the first relation in \eqref{eq:psi}, we have
\begin{equation}
\psi(x,y
=-p_0+\int_{-d}^y\left[u(x,s)-c\right]\mathrm{d} s.
\end{equation}
Since we assume that $u<c$ throughout the fluid \citep{constantin2004exact}, $p_0$ is always negative.  Following \cite{constantin2004exact},
the fluid vorticity is defined as
\begin{equation}
\omega:=u_y-v_x.
\end{equation}
Moreover, the assumption of no-stagnation (since $u<c$) implies, cf. \cite{constantin2011nonlinear}, that there is a function of one variable, which we denote by $\gamma$, such that $\omega(x,y)=\gamma(\psi(x,y))$ for all $(x,y)$ in the water domain. Also from formula \eqref{eq:psi}, we have that $\omega=\Delta\psi$. Hence, the fluid vorticity satisfies
\begin{equation}\label{eq:vort-def}
\Delta\psi=u_y-v_x=\gamma(\psi).
\end{equation}
Here, $\gamma$ is the function already found by the previous argument involving the no-stagnation condition, as presented in the book by \cite{constantin2011nonlinear}.

The density normalized hydraulic head $E$ is expressed as
\begin{equation}\label{eq:head}
E=\frac{1}{2}\left[(u-c)^2+v^2\right]+P+\mathrm{g} y-\int_0^{\psi}\gamma(s)\mathrm{d} s
\end{equation}
which is a constant owing to conservation of energy. In the expression, $P$ is the pressure (normalized with respect to density) in the fluid and at the surface $P=P_{atm}$, the atmospheric pressure which also normalized with respect to density. Therefore, for a given water depth, we can introduce another constant $Q=2(E-P_{atm}+\mathrm{g} d)$, and the boundary condition on the surface is given by
\begin{equation}\label{eq:surf-eq}
\psi_x^2+\psi_y^2+2\mathrm{g}(y+d)=Q\quad\mbox{on}\quad \psi=0.
\end{equation}

Eventually, we obtain the fixed domain boundary-value problem from the free-surface wave problem by applying the following DJ transformation, i.e.
\begin{equation}
q=x\quad\mbox{and}\quad p=-\psi(x,y)
\end{equation}
to the relations \eqref{eq:vort-def} and \eqref{eq:surf-eq}, and including the boundary condition at $p=p_0$. The equations are
\begin{equation}\label{eq:eom}
\left.\begin{array}{lll}
(1+h_p^2)h_{pp}-2h_ph_qh_{qp}+h_{p}^2h_{qq}-\gamma(-p)h_p^3=0 &\mbox{in}& R,\\
1+h_q^2+(2\mathrm{g} h-Q)h_p^2=0 &\mbox{on}& p=0,\\
h=0 &\mbox{on}& p=p_0,
\end{array}\right\}
\end{equation}
where $h(q,p)=y+d$ is the height function and $R$ is the rectangle defined by $R=\left\{(q,p):-L<q<L,p_0<p<0\right\}$.

\subsection{Recovery of wave properties}
We will solve the problem \eqref{eq:eom} for the height function using a numerical method that will be introduced in the next section. For characterizing the waves, we shall recall the recovery of wave properties from the height function. Since in this formulation, the mean water depth $d$ is not fixed, it needs to be recovered first from $h$ by
\begin{equation}
d=\frac{1}{2L}\int_{-L}^{L}h(q,0)\mathrm{d} q \ .
\end{equation}
Afterwards, the water profile is determined by
\begin{equation}
y=h-d.
\end{equation}
The wave height $a$ is obtained as
\begin{equation}
a=h(0,0)-h(L,0)
\end{equation}
as we are concerned with waves of one crest and one trough per period. Further, the velocity field is recovered by
\begin{equation}\label{eq:uv}
(c-u,v)=\left(\frac{1}{h_p},-\frac{h_q}{h_p}\right),
\end{equation}
recalling the DJ transformation (see \cite{constantin2011nonlinear} for details). The pressure in the fluid domain can be obtained from the relation \eqref{eq:head} and here we define a relative pressure, $\tilde P=P-P_{atm}$, for the following discussions as
\begin{equation}\label{eq:P}
\tilde P=-\frac{1+h_q^2}{2h^2_p}-\mathrm{g} h+\frac{Q}{2}-\int_{0}^{p}\gamma(-s)\mathrm{d} s = -\frac{\left[(u-c)^2+v^2\right]}{2}-\mathrm{g} (y+d)+\frac{Q}{2}-\int_0^{p}\gamma(-s)\mathrm{d} s.
\end{equation}

For computing water particle paths or trajectories, the traveling speed of the wave is recovered \citep{constantin2011nonlinear} by using
\begin{equation}\label{eq:speed}
c= \kappa-\frac{1}{L}\int_0^L \left[u(x,-d)-c\right]\mathrm{d} x,
\end{equation}
where $\kappa$ is the average horizontal current strength on the bed. We then need to solve a set of differential equations \citep{constantin2010pressure,nachbin2014boundary}
\begin{equation}\label{eq:pathode}
\left.\begin{array}{l}
X^\prime=u(X-ct,Y),\\
Y^\prime=v(X-ct,Y),\\
(X(0),Y(0))=(X_0,Y_0),
\end{array}
\right\}
\end{equation}
where the prime denotes the derivative with respect to time. Here, we use a numerical integration method to compute the particle paths once the velocity field is solved, as will be detailed in the next section.

\section{Numerical methods}\label{sec:method}
In this section, we briefly introduce the numerical methods to solve \eqref{eq:eom}, for performing numerical continuation to arrive at the solution of waves in flows with closely
approaching stagnation points, and for computing the particle trajectories from the velocity field in the Eulerian setting. The dispersion relations and the linearized solutions for local bifurcation from the laminar flow solutions to small-amplitude wave solutions are also reviewed for the sake of completeness.

\subsection{Discretization and implementation}
Following the methods in \cite{ko2008large} and \cite{amann2018numerical}, we use a finite difference scheme to discretize the computational domain. Taking advantage of the symmetry of $h$ function with respect to $q=0$ \citep{constantin2007symmetry}, we set the computational domain as a fixed rectangle $[0,L]\times[p_0,0]$ to improve the computation efficiency. We first discretize the domain using uniformly distributed grid points and then refine the grid size near the surface and the bottom under the crest ($q=0$). In the case of discontinuous vorticity, we additionally refine the grid size near the location where the vorticity changes abruptly. We apply the central difference scheme at interior grid points, and at left and right boundary points by mirroring $h$ along $q=0$ and $q=L$, considering the periodicity and symmetry properties of the waves in absence of stagnation points. At the boundaries $p=0$ and $p=p_0$, we apply the backward and forward difference schemes, respectively. Note that the refinement is achieved by reducing the grid size smoothly so that the central difference still has a second-order accuracy, approximately.

For implementation, we use the open-source library \emph{Eigen} \citep{eigenweb} for solving the nonlinear algebraic equations subsequent to the finite difference discretization. We use the predictor-corrector method detailed in \cite{amann2018numerical} for numerical continuation and write our own code on the basis of \emph{Eigen}. In the following numerical analysis, we discretize the domain first by an equidistant $250\times750$ grid with additional refinement at locations where the stagnation point may occur.  Following the refinement, the smallest grid size achieved is ${1/8}^{\rm{th}}$ of the initial size prior to the mentioned refinement.

\begin{figure}
	\centering
   \includegraphics[width=0.6\textwidth]{fig1-setting.pdf}
   \caption{\raggedleft The setting of nonlinear periodic waves traveling on rotational flow with discontinuous vorticity. In the figure, $\gamma_1$ and $\gamma_2$ are positive. }
   \label{fig1:setting}
\end{figure}

\subsection{Dispersion relations and linearization solutions for local bifurcation}
As previously mentioned, we focus on the discontinuous vorticity. The vorticity function is expressed as
\begin{equation}
\gamma=\left\{\begin{array}{ll}
\gamma_1,&p\in(p_1,0),\\
\gamma_2,&p\in(p_0,p_1).
\end{array}\right.
\end{equation}
where $p_1$ corresponds to the value of the streamline at the interface $y=y_1$, and $\gamma_1$ and $\gamma_2$ are the constant vorticity values of the surface layer and the layer adjacent to the bed. The setting of the problem is illustrated in Figure \ref{fig1:setting}. For the case of $\gamma_1=\gamma_2=0$, the problem reduces to irrotational flow, and the dispersion relation and the linearized solutions are available in \cite{constantin2011nonlinear}. For the situation of a rotational flow over an irrotational and vice versa, the dispersion relations are explicitly derived in  \cite{constantin2011periodic} and \cite{constantin2012dispersion}, respectively. For steady water waves on two rotational layers, see \cite{martin2015dispersion}.

The dispersion relations are nonlinear equations. Here, we solve them using the Newton method. For large vorticity, we apply a simple continuation scheme by using the solution for smaller vorticity for initialization \citep{seydel2009practical}.

\subsection{Computation of the particle trajectories}
We numerically compute the particle paths in the physical domain, i.e. in the fixed $(X,Y)$-coordinate. For a particle located at $(X(t),Y(t))$ at time $t$, its velocity is interpolated from the velocities at the neighboring four grid points (following appropriate translation) using a similar interpolation method as in \cite{umeyama2012eulerian}. The interpolation scheme is illustrated in figure \ref{fig2:interpolation}. Note that in our case, the cell is of trapezoidal shape in the physical domain.  Subsequently, we use numerical integration to solve the first-order differential equations \eqref{eq:pathode}. This simple interpolation scheme is found to be sufficiently accurate.

\begin{figure}
	\centering
   \includegraphics[width=0.7\textwidth]{fig2-interpolation.pdf}
   \caption{Interpolation strategy for instantaneous particle velocity. In the figure, $\mathbf{u}=[u\ \ v]^\top$ is the velocity vector; $l_1, l_2, l_3$ and $l_4$ are respective distances of the particle to the four neighboring grid points.}
   \label{fig2:interpolation}
\end{figure}

\section{Results}\label{sec:result}
Following \cite{ko2008large,ko2008effect} and \cite{amann2018numerical}, we assign $\mathrm{g}=9.8$, $2L=2\upi$ and $p_0=-2$. We focus on the new part of the bifurcation curves, pressure distribution, and particle paths under flows with nearly approaching stagnation points. Note that in the present system all the variables are expressed in SI unit and a physical system of a different wavelength can be scaled to the $2\pi$-periodic system analyzed in this paper by scaling $\mathrm{g}$ and $\gamma$ using the wave number \citep{constantin2011nonlinear}. However, the scaling has no influence on the laminar flow solution. In the following, as will be seen, the current velocity at the crest (under zero-amplitude waves) in most of the cases is less than 2.5 m/s, which is physically realistic in oceans.

\subsection{Critical vorticity and the new part of the branch of bifurcation diagram}
For the case of constant vorticity, it is known that when the vorticity is below a critical value $\gamma_{crit}$, the stagnation point first appears at the bottom and at that stage numerical continuation breaks down \citep{ko2008large}. However, \cite{amann2018numerical} and \cite{kalimeris2018analytical} found that when $\gamma<\gamma_{crit}$ but is very close to
$\gamma_{crit}$, the branch of the bifurcation diagram ($a-Q$ plot) has another part which is bounded below and above by two values of the bifurcation parameter $Q$. These bounds
correspond to two different waves in flows with stagnation point at the bottom and at the surface. On the new part of the branch, the wave height is further increased, and it is thus referred to as the upper part. To reach this upper part of the solutions, we need to overcome the gap of the bifurcation curve corresponding to waves in flows with stagnation points. For this purpose, \cite{amann2018numerical} resorted to the $\gamma-$continuation scheme where a solution on the bifurcation diagram corresponding to $\gamma_0$ ($\gamma_0>\gamma_{crit}>\gamma$) is used to initialize local bifurcation to arrive at one solution on the upper part corresponding to $\gamma$. Once one solution is obtained, the continuation in the solution set corresponding to $\gamma$ can be performed again with $Q$ or the wave height as the bifurcation parameter. In the following, we apply this scheme when needed.

\begin{figure}
   \centering$\begin{array}{cc}
   \includegraphics[trim={0.4cm 0.8cm 0 0},clip,height=3.8cm]{fig3a-a-Q.pdf} &
   \includegraphics[trim={0.6cm 0.8cm 0 0},clip,height=3.8cm]{fig3b-a-Q.pdf} \\
   \end{array}$
   \caption{Bifurcation diagrams for cases with $p_1=-0.5, \gamma_1=0$ (at the top layer) and various different values of $\gamma_2$ (at the bottom layer): (a) full diagrams, and (b) a close-up view of the upper parts.}
   \label{fig3:bifurcation}
\end{figure}

\begin{figure}
   \centering$\begin{array}{cc}
   \includegraphics[trim={0.6cm 0.6cm 0 0},clip,height=3.8cm]{fig4a-streamlines.pdf} &
   \includegraphics[trim={0.6cm 0.6cm 0 0},clip,height=3.8cm]{fig4b-streamlines.pdf} \\
   \includegraphics[trim={0.6cm 0.6cm 0 0},clip,height=3.8cm]{fig4c-streamlines.pdf} &
   \includegraphics[trim={0.6cm 0.6cm 0 0},clip,height=3.8cm]{fig4d-streamlines.pdf} \\
   \end{array}$
   \caption{Streamlines of flows at the endpoints of each of the two parts of the bifurcation curve corresponding to $p_1=-0.5, \gamma_1=0$ (at the top layer), and $\gamma_2=-3.23$ (at the bottom layer): (a) laminar flow, (b) wave at the upper bound of the lower part, (c) wave at the lower bound of the upper part, and (d) wave at the upper bound of the upper part. In the figures, the insets show the background current profile when no wave motion.}
   \label{fig4:streamlines}
\end{figure}

\begin{figure}
   \centering$\begin{array}{cc}
   \includegraphics[trim={0.8cm 0.8cm 0 0},clip,height=3.8cm]{fig5a-velocity.pdf} &
   \includegraphics[trim={0.7cm 0.8cm 0 0},clip,height=3.8cm]{fig5b-velocity.pdf} \\
   \end{array}$
   \caption{Horizontal velocity, $c-u$, under the crest along the bifurcation curve for $\gamma_1=0$ (at the top layer), $p_1=-0.5$, and various different values of $\gamma_2$ (at the bottom layer): (a) at the bottom and (b) at the surface. In the figures, the insets are closeup view of the curves when approaching the largest wave height.}
   \label{fig5:velocity}
\end{figure}

Proceeding from the observations of \cite{amann2018numerical}, we investigated the case of discontinuous vorticity and found that the upper part also exists. Furthermore, for the case of discontinuous vorticity, the upper bound on this part always corresponds to waves in flows with the stagnation point at the surface, while its lower bound corresponds to waves in flows with stagnation point either at the bottom or at an internal point where vorticity jump occurs. We first show an example of the former situation. An irrotational flow ($\gamma_1=0, p_1=-0.5$) was considered on a rotational flow with vorticity $\gamma_2$. We varied $\gamma_2$ to find out the critical value $\gamma_{2,crit}$ below which the stagnation point first occurs at the bottom. It was found that $\gamma_{2,crit}$ is slightly less than $-3.22$. Note that with the vorticity values $\gamma_1=0$ and $\gamma_2=-3.23$ and hence a shearing laminar flow with profile shown in figure 4a over a total water depth of 0.787 m, we have the surface velocity, $u=-2.07$ m/s (the current flow direction is opposite to the wave propagation direction). Eq. (2.5) with a bifurcation value of $Q = 26.85$
(see figure 3a) leads to the relative surface velocity with respect to the wave speed as $c-u=3.38$ m/s, since $u<c$. The wave speed $c$ is $1.31$ m/s as also verified by the dispersion relation for this case. In the case of positive vorticity in the bottom layer, e.g., $\gamma_1=0$ and $\gamma_2=3$, the water depth is 0.74 m, the current velocity at the surface is around 1.53 m/s, and at the local bifurcation the wave speed is $c=3.69$ m/s. These values of the physical parameters indicate that the setting considered in this paper are realistic.

We then further decreased $\gamma_2$ and found the upper part, as shown in figure \ref{fig3:bifurcation}. For the case of $\gamma_2=-3.23$, the streamlines of wave solutions at the four endpoints (marked with cross in figure \ref{fig3:bifurcation}) of the two parts of the bifurcation curve are shown in figure \ref{fig4:streamlines}. Figure \ref{fig4:streamlines} clearly shows that the stagnation point first occurs at the bottom and then at the crest. Figure \ref{fig5:velocity} shows the horizontal velocities $c-u$ at the bottom and at the surface under the crest along the bifurcation curves. Note that both waves in figure \ref{fig4:streamlines}b and figure \ref{fig4:streamlines}c correspond to flows close to having a stagnation point at the bottom, and the horizontal velocities $c-u$ at the bottom in these two cases are comparable, as in figure \ref{fig5:velocity}a, while the wave in figure  \ref{fig4:streamlines}c has a larger height, a sharper crest and smaller horizontal velocity $c-u$ at the surface (see figure \ref{fig5:velocity}b). It is seen from figure \ref{fig5:velocity} that the velocity $c-u$ at the bottom is first decreasing and then increasing along the bifurcation curve, with the presence of a gap for $\gamma_2<\gamma_{2,crit}$, while the velocity $c-u$ at the crest is decreasing monotonically. Indeed, at the uppermost end of the bifurcation diagram in all those three cases, the velocities $c-u$ at the surface are still larger than those at the bottom, the stagnation point is expected to occur at the crest because the velocity at the surface is decreasing while at the bottom is increasing with respect to the increasing wave height.

\begin{figure}
   \centering$\begin{array}{cc}
   \includegraphics[trim={0.75cm 0.8cm 0 0},clip,height=3.8cm]{fig6a-a-Q.pdf} &
   \includegraphics[trim={0.75cm 0.8cm 0 0},clip,height=3.8cm]{fig6b-a-Q.pdf} \\
   \end{array}$
   \caption{Bifurcation diagrams for cases with $p_1=-0.7, \gamma_2=0$ (at the bottom layer) and various different values of $\gamma_1$ (at the top layer): (a) full diagrams, and (b) a close-up view of the upper parts.}
   \label{fig6:bifurcation}
\end{figure}

\begin{figure}
   \centering$\begin{array}{cc}
   \includegraphics[trim={0.6cm 0.6cm 0 0},clip,height=3.8cm]{fig7a-streamlines.pdf} &
   \includegraphics[trim={0.6cm 0.6cm 0 0},clip,height=3.8cm]{fig7b-streamlines.pdf} \\
   \includegraphics[trim={0.6cm 0.6cm 0 0},clip,height=3.8cm]{fig7c-streamlines.pdf} &
   \includegraphics[trim={0.6cm 0.6cm 0 0},clip,height=3.8cm]{fig7d-streamlines.pdf} \\
   \end{array}$
   \caption{Streamlines of flows at the endpoints of each of the two parts of the bifurcation curve corresponding to $p_1=-0.7$, $\gamma_1=-10.42$ (at the top layer), and $\gamma_2=0$ (at the bottom layer): (a) laminar flow, (b) wave at the upper bound of the lower part, (c) wave at the lower bound of the upper part, and (d) wave at the upper bound of the upper part. In the figures, the insets show the background current profile when no wave motion.}
   \label{fig7:streamlines}
\end{figure}

\begin{figure}
   \centering$\begin{array}{cc}
   \includegraphics[trim={0.6cm 0.8cm 0 0},clip,height=3.8cm]{fig8a-velocity.pdf} &
   \includegraphics[trim={0.6cm 0.8cm 0 0},clip,height=3.8cm]{fig8b-velocity.pdf} \\
   \end{array}$
   \caption{Horizontal velocity, $c-u$, under the crest along the bifurcation curve for $\gamma_2=0$ (at the bottom layer), $p_1=-0.7$, and varying $\gamma_1$ (at the top layer): (a) at the position where vorticity jumps ($y=y_1$ corresponds to $p=p_1$) and (b) at the surface. In the figures, the insets are closeup view of the curves when approaching the largest wave height.}
   \label{fig8:velocity}
\end{figure}

We further consider a layer ($p_1=-0.7$) of rotational flow with vorticity $\gamma_1$ on an irrotational flow ($\gamma_2=0$), following the setting in \cite{ko2008effect} and \cite{constantin2011periodic}. We found that the critical vorticity $\gamma_{1,crit}$ is slightly below $-10.41$ (a value refined from $-10$ in \cite{ko2008effect}). By using $\gamma_1$ as the bifurcation parameter, we also found the upper part of the branch of the bifurcation diagram when $\gamma_1$ is less but very close to $\gamma_{1,crit}$. Figure \ref{fig6:bifurcation} shows the bifurcation diagrams for $\gamma_1=-10.41, -10.42$, and $-10.45$. For $\gamma_1=-10.42$, the streamlines of the waves corresponding to the four endpoints of the bifurcation curve are shown in figure \ref{fig7:streamlines}, which clearly illustrates that the stagnation point first occurs internally and later occurs at the crest with further increased wave height. Figure \ref{fig8:velocity} shows the variation of the velocities $c-u$ at $p=p_1$ and at the surface along the bifurcation curves. The velocity $c-u$ at the position where vorticity jumps, is first decreasing and then increasing with respect to the wave height. Notably, the velocity decreases slightly again when close to the wave in flows with stagnation point at the surface. Again, the velocity $c-u$ on the crest is decreasing monotonically along with the increasing wave height.

To summarize, in this subsection, we have observed new features of the bifurcation curves with fixed mass flux, for large amplitude waves on flows with discontinuous vorticity. Specifically, we show that along the bifurcation curve the stagnation point can first occur at the bottom or internally and then occur again at the surface for some particular vorticity distributions.

\subsection{Pressure distribution}
We now present the pressure distribution for waves with stagnation points at the bottom, at an internal position and at the surface, for the case with discontinuous vorticity.

\begin{figure}
   \centering
   \includegraphics[trim={0.2cm 0.2cm 0 0},clip,height=4cm]{fig9-pressure.pdf}
   \caption{Pressure distribution for waves approaching waves with stagnation point at the crest for $p_1=-0.5$, $\gamma_1=0$ (at the top layer), and: (a) $\gamma_2=3$, (b) $\gamma_2=-2$, and (c) $\gamma_2=-3.23$. In the figure, the insets show the corresponding background shear flow when no wave motion.}
   \label{fig9:pressure}
\end{figure}

\begin{figure}
   \centering$\begin{array}{cc}
   \includegraphics[trim={0.5cm 0.2cm 0 0},clip,height=3.8cm]{fig10a-pressure.pdf} &
   \includegraphics[trim={0.5cm 0.2cm 0 0},clip,height=3.8cm]{fig10b-pressure.pdf} \\
   \end{array}$
   \caption{Pressure distribution in flows with an approaching stagnation point at the crest for $p_1=-0.5$, $\gamma_1=0$ (at the top layer) and various different values of $\gamma_2$ along (a) the bottom, and (b) the crest line.}
   \label{fig10:pressure}
\end{figure}

\begin{figure}
   \centering
   \includegraphics[trim={0.2cm 0.2cm 0 0},clip,height=4cm]{fig11-pressure.pdf}
   \caption{Pressure distribution  in flows with an approaching stagnation point at the bottom for $p_1=-0.5$, $\gamma_1=0$ (at the top layer), and: (a) $\gamma_2=-3.23$ (below gap), (b) $\gamma_2=-3.23$ (above gap), and (c) $\gamma_2=-5$. In the figure, the insets show the corresponding background shear flow when no wave motion.}
   \label{fig11:pressure}
\end{figure}

\begin{figure}
   \centering$\begin{array}{cc}
   \includegraphics[trim={0.5cm 0.2cm 0 0},clip,height=3.8cm]{fig12a-pressure.pdf} &
   \includegraphics[trim={0.5cm 0.2cm 0 0},clip,height=3.8cm]{fig12b-pressure.pdf} \\
   \end{array}$
   \caption{Pressure distribution  in flows with an approaching stagnation point  at the bottom for $p_1=-0.5$, $\gamma_1=0$ (at the top layer) and various different values of $\gamma_2$ along (a) the bottom, and (b) the crest line.}
   \label{fig12:pressure}
\end{figure}

\begin{figure}
   \centering
   \includegraphics[trim={0.2cm 0.2cm 0 0},clip,height=4cm]{fig13-pressure.pdf}
   \caption{Pressure distribution  in flows with an approaching stagnation point  at the crest for $p_1=-0.7$, $\gamma_2=0$ (at the bottom layer), and: (a) $\gamma_1=5$, (b) $\gamma_1=-5$, and (c) $\gamma_1=-10.42$. In the figure, the insets show the corresponding background shear flow when no wave motion.}
   \label{fig13:pressure}
\end{figure}

\begin{figure}
   \centering$\begin{array}{cc}
   \includegraphics[trim={0.5cm 0.2cm 0 0},clip,height=3.8cm]{fig14a-pressure.pdf} &
   \includegraphics[trim={0.5cm 0.2cm 0 0},clip,height=3.8cm]{fig14b-pressure.pdf} \\
   \end{array}$
   \caption{Pressure distribution in flows with an approaching stagnation point  at the crest for $p_1=-0.7$, $\gamma_2=0$ (at the bottom layer) and various different values of $\gamma_1$ along (a) the stream line $\psi=-p_1$, and (b) the crest line.}
   \label{fig14:pressure}
\end{figure}

\begin{figure}
   \centering
   \includegraphics[trim={0.2cm 0.2cm 0 0},clip,height=4cm]{fig15-pressure.pdf}
   \caption{Pressure distribution in flows with an approaching stagnation point  at a position where vorticity jump occurs, for $p_1=-0.7$, $\gamma_2=0$ (at the bottom layer), and: (a) $\gamma_1=-10.42$ (below gap), (b) $\gamma_1=-10.42$ (above gap), and (c) $\gamma_1=-12$. In the figures, the insets show the background current profile when no wave motion.}
   \label{fig15:pressure}
\end{figure}

\begin{figure}
   \centering$\begin{array}{cc}
   \includegraphics[trim={0.5cm 0.2cm 0 0},clip,height=3.8cm]{fig16a-pressure.pdf} &
   \includegraphics[trim={0.5cm 0.2cm 0 0},clip,height=3.8cm]{fig16b-pressure.pdf} \\
   \end{array}$
   \caption{Pressure distribution in flows with an approaching stagnation point  at a position where vorticity jump occurs for $p_1=-0.7$, $\gamma_2=0$ and various different values of $\gamma_1$ along (a) the stream line $\psi=-p_1$, and (b) the crest line.}
   \label{fig16:pressure}
\end{figure}

For an irrotational flow on a rotational flow with $p_1=-0.5$ and $\gamma_1=0$, we consider the cases corresponding to $\gamma_2=3, -2$ and $-5$. For waves close to the case of flows
having a stagnation point at the crest, the pressure distributions are shown in figure \ref{fig9:pressure}, and figure \ref{fig10:pressure} compares the pressure along the bottom and along the crest line. For waves in flows close to having a stagnation point at the bottom, the pressure distribution demonstrated in figure \ref{fig11:pressure} and figure \ref{fig12:pressure} shows the pressure along the bottom and the crest line. Two features of the pressure distributions are consistent with the pressure beneath a Stokes wave as proved by \cite{constantin2010pressure} and those can be stated as \lq{}\lq{the pressure in the fluid strictly decreases horizontally away from a crest line toward its neighboring trough lines, and the pressure strictly increases with depth}\rq{}\rq{}. Even though, when $\gamma_2$ is close to the critical value, the pressure along the bottom displays a \lq\lq{plateau}\rq\rq{} behavior, as previously pointed out by \cite{amann2018numerical} for the case of constant vorticity, the pressure is still decreasing away from the crest line, slowly.

\begin{figure}
    \centering$\begin{array}{cc}
   \includegraphics[trim={0.5cm 0.2cm 0 0},clip,height=3.8cm]{fig17a-velocity.pdf} &
   \includegraphics[trim={0.5cm 0.2cm 0 0},clip,height=3.8cm]{fig17b-velocity.pdf} \\
   \end{array}$
   \caption{Variation of velocities, $c-u$ and $v$, with  $p_1=-0.7$, $\gamma_2=0$ (at the bottom layer), and various different values of $\gamma_1$: (a) $c-u$ and $v$ along the streamline $\psi=-p_1$, and (b) $c-u$ along the crest line ($v$ is zero along this line and hence not shown).}
   \label{fig17:velocity}
\end{figure}

For a rotational flow on an irrotational flow, we analyze the cases for $\gamma_1=5, -5$, and $-10.42$. Figure \ref{fig13:pressure} shows the pressure distribution in flows close to having a stagnation point at the crest, and variations of the pressure along the bottom and along the crest line are illustrated in figure \ref{fig14:pressure}. The pressure distribution in flows when internal stagnation points are almost developed is illustrated in figure \ref{fig15:pressure}. Correspondingly, figure \ref{fig16:pressure} shows the pressure along the streamline $\psi=-p_1$ and along the crest line. From these figures, it is seen that none of the facts regarding the pressure distribution for an irrotational Stokes wave \citep{constantin2010pressure} hold, as also observed by \cite{ribeiro2017flow} for rotational flows with constant vorticity. When the vorticity $\gamma_1$ is below a critical value, the pressure along the streamline $\psi=-p_1$ achieves a local minimum under the crest, as opposed to the maximum there when $\gamma_1$ is large. For smaller $\gamma_1$, the relative pressure $\tilde P$ at this position becomes negative (i.e. sub-atmospheric), which is the minimum along the streamline. The pressure attaining a value less than the atmospheric pressure has also been observed by \cite{ali2013reconstruction} for the case of constant vorticity using a model based on an asymptotic expansion for the stream functions. When $\gamma_1$ is below a critical value, the pressure below but close to the layer where the vorticity jump occurs also exhibits the \lq{}\lq{plateau}\rq{}\rq{} behavior, internally, see figure \ref{fig15:pressure}. Figures \ref{fig14:pressure}-\ref{fig16:pressure} show that when $\gamma_1$ is much larger than the critical value, the pressure is increasing with depth along the crest line. Interestingly, when $\gamma_1$ is quite close to $\gamma_{1,crit}$, the pressure $\tilde P$ along the crest line first increases slightly, and then decreases, and increases again when close to the position where vorticity jump occurs; the overall pressure in the surface layer is close to the atmospheric pressure, see figure \ref{fig14:pressure}b for $\gamma_1=-10.42$. With a further decrease in the value of $\gamma_1$ below the critical value, the relative pressure in the surface layer becomes negative, first decreasing from the crest and then increasing near the position where the vorticity jump takes place. To understand the reason for the development of a negative relative pressure $\tilde P$ in the surface layer near the crest line, we recall the expression \eqref{eq:P}.  For the cases illustrated in figure \ref{fig14:pressure}, the term $\int_0^p\gamma(-s)\mathrm{d} s$ in \eqref{eq:P} is positive in the surface layer and increases rapidly with depth since $|\gamma_1|$ is large, the term $\mathrm{g}(y+d)$ is also large considering the large wave height, and the velocity $c-u$ is also not small, as shown in figure \ref{fig17:velocity}; hence $\tilde P$ in the surface layer becomes negative. Further decreasing $\gamma_1$ (with increasing $|\gamma_1|$), it seems that the minimal pressure along the crest line will keep decreasing.

\subsection{Particle trajectories}
We proceed to compute the particle trajectories under the waves corresponding to flows with nearly developed stagnation points, for the previously mentioned cases. To recover the wave speed using expression \eqref{eq:speed}, we assume that the current strength at the bed is zero, i.e., $\kappa=0$. Note that the choice of a different setting (adding an arbitrary uniform current) may lead to variations of the resulting particle paths. For example, when studying particle paths under waves on linear shear flow, \cite{ribeiro2017flow} set the current strength at the mid-depth to be zero (to ensure a flow with no net mass flux in the case of zero-amplitude waves). However, we here focus on the relative horizontal displacements of the particles along the depth, which are not affected by the presence of a uniform background current.

\begin{figure}
   \centering
   \includegraphics[trim={0.2cm 0.2cm 0 0},clip,height=4cm]{fig18-path.pdf}
   \caption{(a) Particle trajectories under the waves in flows close to having a stagnation point at the crest with $p_1=-0.5$, $\gamma_1=0$, and $\gamma_2=-2$ in the moving frame; (b) the background current profile when no wave motion.}
   \label{fig18:path}
\end{figure}

In the following computation and illustration of the particle trajectories, we set that: (i) the initial positions of the particles are under a zero-crossing point of the wave profile \citep{paprota2018particle}; (ii) the particles trajectories are plotted for the waves traveling rightwards for two wave periods, i.e., for a time of $4L/c=4\pi/c$, and the final position of
the particles under zero-amplitude waves are also indicated in the graphs for comparison. In the graphs, six particle trajectories are plotted. Three of those particles are located at the surface, at the interface, and at a position very close to the bottom but not exactly at the bottom. The other three particles include one located in the top layer and two located in the bottom layer. The initial positions are denoted by dots and the final positions are indicated by square markers.

\begin{figure}
   \centering
   $\begin{array}{c}
   \includegraphics[trim={0.4cm 0.2cm 0 0},clip,height=4cm]{fig19ab-path.pdf}\\
   \includegraphics[trim={0.4cm 0.2cm 0 0},clip,height=4cm]{fig19cd-path.pdf}\\
   \includegraphics[trim={0.4cm 0.2cm 0 0},clip,height=4cm]{fig19ef-path.pdf}\\
   \end{array}$
   \caption{Particle trajectories under the waves in flows close to having a stagnation point near the crest, for $p_1=-0.5$ and $\gamma_1=0$: (a) $\gamma_2=3$; (c) $\gamma_2=-2$; (e) $\gamma_2=-3.23$. The background current profiles corresponding to zero-amplitude waves are shown in (b), (d), and (f), respectively. In the plot of the particle paths, the gray lines indicate the final position of the particles if under zero-amplitude waves after the time $4L/c$.}
   \label{fig19:paths}
\end{figure}

\begin{figure}
   \centering
   $\begin{array}{c}
   \includegraphics[trim={0.4cm 0.2cm 0 0},clip,height=4cm]{fig20ab-path.pdf}\\
   \includegraphics[trim={0.4cm 0.2cm 0 0},clip,height=4cm]{fig20cd-path.pdf}\\
   \end{array}$
   \caption{Particle trajectories under the waves in flows close to having a stagnation point at the bottom, for $p_1=-0.5$ and $\gamma_1=0$: (a) $\gamma_2=-3.23$ (below the gap); (c) $\gamma_2=-3.25$ (below the gap). The background current profiles corresponding to zero-amplitude waves are shown in (b) and (d), respectively. In the plot of the particle paths, the gray lines indicate the final position of the particles if under zero-amplitude waves after the time $4L/c$.}
   \label{fig20:paths}
\end{figure}

\begin{figure}
   \centering
   $\begin{array}{c}
   \includegraphics[trim={0.4cm 0.2cm 0 0},clip,height=4cm]{fig21ab-path.pdf}\\
   \includegraphics[trim={0.4cm 0.2cm 0 0},clip,height=4cm]{fig21cd-path.pdf}\\
   \includegraphics[trim={0.4cm 0.2cm 0 0},clip,height=4cm]{fig21ef-path.pdf}\\
   \end{array}$
   \caption{Particle trajectories under the waves in flows close to having a stagnation point near the crest, for $p_1=-0.7$ and $\gamma_2=0$: (a) $\gamma_1=5$; (c) $\gamma_1=-5$; (e) $\gamma_1=-10.42$. The background current profiles corresponding to zero-amplitude waves are shown in (b,d,f), respectively. In the plot of the particle paths, the gray lines indicate the final position of the particles if under zero-amplitude waves after the time $4L/c$.}
   \label{fig21:paths}
\end{figure}

\begin{figure}
   \centering
   $\begin{array}{c}
   \includegraphics[trim={0.4cm 0.2cm 0 0},clip,height=4cm]{fig22ab-path.pdf}\\
   \includegraphics[trim={0.4cm 0.2cm 0 0},clip,height=4cm]{fig22cd-path.pdf}\\
   \end{array}$
   \caption{Particle trajectories under the waves in flows close to having a stagnation point at the bottom, for $p_1=-0.7$ and $\gamma_2=0$: (a) $\gamma_2=-10.42$ (below the gap); (c) $\gamma_2=-12$. The background current profiles corresponding to zero-amplitude waves are shown in (b) and (d), respectively. In the plot of the particle paths, the gray lines indicate the final position of the particles if under zero-amplitude waves after the time $4L/c$.}
   \label{fig22:paths}
\end{figure}

First of all, we plot the computed particle trajectories in the moving frame, as illustrated in figure \ref{fig18:path}, which theoretically are the corresponding streamlines. The streamlines passing through the initial particle locations are plotted for comparison, and a good agreement is seen, suggesting the accuracy of the method for computing particle trajectories.

For an irrotational layer on a rotational layer, the particle trajectories in flows close to having a stagnation point at the crest are plotted in figure \ref{fig19:paths} and those in flows close to having a stagnation point at the bottom are plotted in figure \ref{fig20:paths}. We show that as theoretically predicted by \cite{constantin2006trajectories} when a particle reaches
the crest of the wave in flows having a stagnation point at the crest, it does not pause there but moves further (figure \ref{fig19:paths}a). However, for waves in flows close to having
an internal stagnation point or a stagnation point at the bottom, a particle reaching the stagnation point tends to pause at that point in the moving frame, similar to the case shown in \cite{ribeiro2017flow} for flows
with a bottom stagnation point. Some observations on the features of the particle trajectories are listed below:
\begin{enumerate}
\item In both figure \ref{fig19:paths} and figure \ref{fig20:paths}, the horizontal position of the particles are on the right of the gray lines (which are the position of the particles with no waves), indicating that the waves induce a positive drift of the particles in the direction of the propagating waves. This is consistent with what was proved by \cite{constantin2012particle} and the experimental results \citep{paprota2018particle} for irrotational flows.
\item In the top irrotational layer, the horizontal displacement (with respect to the background current induced displacement) or dirft is larger when the particle is closer to the surface than when it is near the interface of the two layers, which is consistent with experimental and approximate theoretical results \citep{paprota2018particle,curtis2018particle}. In the bottom rotational layer, when the vorticity is positive, the wave induced drift is larger when the particle is close to the interface as compared to the drift induced when the particle is close to the bottom
surface. For negative (adverse) vorticity in the bottom layer, the largest drift tends to occur at the bottom surface (i.e. the bottom flat bed) when the flow is close to having a stagnation point at the bottom, as also observed in \cite{ribeiro2017flow} for linear shear flow.
\item For flows close to having a stagnation point at the crest, as shown in figure \ref{fig19:paths}, the negative vorticity in the bottom layer clearly increases the drift of the particles in the top irrotational layer relative to the background current, particularly for particles at the surface.
\item For flows close to having a stagnation point at the bottom, as in figure \ref{fig20:paths}, a stronger negative vorticity in the bottom layer tends to decrease the relative drifts of the particles at different vertical positions in the top layer, as observed by comparing figure \ref{fig19:paths}c and figure \ref{fig20:paths}c.
\end{enumerate}

With a shear layer on an irrotational layer, figure \ref{fig21:paths} shows the particle trajectories for flows close to having a stagnation point at the crest and figure \ref{fig22:paths} demonstrates the particle trajectories for flows close to having a stagnation point at the location when the vorticity jump occurs. Again, we can see from both figures that
the wave-induced drifts are positive in the direction of the propagating waves. Some conspicuous features are observed as described below:
\begin{enumerate}
\item In the bottom irrotational layer, the drift is always larger when the particle is close to the interface, consistent with particle paths in irrotional flow \citep{paprota2018particle}.
\item In the top rotational layer, when the vorticity is positive, the wave-induced drift is larger when the particle is near the surface as comapred to the drift when the particle
is close to the interface of the two layers; while with negative vorticity, for flows close to having a stagnation point at the interface, the particle with the largest drift is at the interface.
\item A negative vorticity in the top layer tends to increase the wave-induced drifts of particles in both layers.
\end{enumerate}

\section{Conclusions}\label{sec:con}
In this paper, we investigate the behavior of flow along a new bifurcation curve, and the pressure and particle trajectories under
waves traveling on a rotational flow with discontinuous vorticity (close to having a stagnation point). For this purpose, we have developed a method for computing the particle paths using the velocity field obtained from a numerical continuation method based on DJ transform of the Euler equations for periodic waves on rotational flow. The continuation method generates a group of waves of fixed relative mass flux until a point where a flow with a stagnation point is reached.

By using the previously mentioned methods, we have recovered new branches of the bifurcation curves not connected to the trivial (laminar) solution, for waves on rotational flow with discontinuous vorticity. Particularly, we show that along the bifurcation curves, the stagnation point can first occur at the position where the vorticity jumps and then again occur at the surface. On the pressure distribution, we show that for the case of shear layer on an irrotational layer, the pressure in the top layer under the crest could be less than the atmosphere pressure and the pressure in the top layer near the interface displays a \lq\lq{plateau}\rq\rq{} behavior when the vorticity is large with a negative sign (i.e., adverse). This \lq\lq{plateau}\rq\rq{} behavior of the pressure has only been observed along the bottom for linear shear flow. We further show the particle trajectories in flows close to having a stagnation point for various cases and qualitatively discuss the influence of vorticity of the rotational layer on the drifts of the particles in both layers.

\section*{Declaration of Interests}
The authors report no conflict of interest.

\section*{Acknowledgments}
The authors are grateful to the anonymous reviewers for their comments and suggestions which helped us improve the paper. L. Chen acknowledges the partial support of the National Natural Science Foundation of China (grant number: 51978506). C. I. Martin acknowledges the support of the Austrian Science Fund (FWF) through research grant P 30878-N32.

\bibliographystyle{jfm}

redpajama_set_name: RedPajamaArXiv


text:
Australian, född 1858, död , var ett engelskt fullblod, mest känd för att ha exporterats till USA där han blev en mycket framgångsrik och inflytelserik avelshingst.

Bakgrund
Australian var en fuxhingst efter West Australian och under Emilia (efter Young Emilius). Han föddes upp av William E. Duncombe och ägdes av Alexander Keene Richards och Robert A. Alexander. Han fick från början namnet Millington.

Australian kom från den tredje kullen föl av West Australian som segrade i 2000 Guineas, Epsom Derby, St Leger och Ascot Gold Cup 1853, och har i efterhand erkänts som den första Triple Crown-vinnaren. West Australian betraktades av samtida experter som en av de bästa brittiska hästarna på artonhundratalet. Efter tävlingskarriären hade han måttliga framgångar som avelshingst till segrare i England och Frankrike, och var genom sin son Solon till stor del ansvarig för överlevnaden av Godolphin Arabians blodslinje i Europa.

Efter slutet av tävlingskarriären blev Australian avelshingst och tillbringade större delen av sin avelskarriär på Woodburn Stud. Han blev far till hästar som vann 410 löp, och blev slagen av Lexington för titeln ledande avelshingst i Nordamerika vid sex tillfällen mellan 1871 och 1877.

Referenser

Noter

Brittiska galopphästar
Hästar födda 1858
Hästar avlidna 1879
Godolphin Arabians faderslinje
Fullblodsfamilj 11

redpajama_set_name: RedPajamaWikipedia


text:
package com.restfb.types;

import java.io.Serializable;

import com.restfb.Facebook;
import com.restfb.util.ReflectionUtils;

/**
 * Represents a location (address and latitude/longitude).
 *

 * This is used by several Graph API types, e.g. <tt>{@link Post}</tt> and <tt>{@link Page}</tt>.
 *
 * @author Mark Allen
 * @since 1.6.8
 */
public class Location implements Serializable {
  @Facebook
  private String street;

  @Facebook
  private String city;

  @Facebook
  private String state;

  @Facebook
  private String country;

  @Facebook
  private String zip;

  @Facebook
  private Double latitude;

  @Facebook
  private Double longitude;

  private static final long serialVersionUID = 1L;

  /**
   * @see java.lang.Object#hashCode()
   */
  @Override
  public int hashCode() {
    return ReflectionUtils.hashCode(this);
  }

  /**
   * @see java.lang.Object#equals(java.lang.Object)
   */
  @Override
  public boolean equals(Object that) {
    return ReflectionUtils.equals(this, that);
  }

  /**
   * @see java.lang.Object#toString()
   */
  @Override
  public String toString() {
    return ReflectionUtils.toString(this);
  }

  /**
   * The street address of this location.
   *
   * @return The street address of this location.
   */
  public String getStreet() {
    return street;
  }

  /**
   * The city name of this location.
   *
   * @return The city name of this location.
   */
  public String getCity() {
    return city;
  }

  /**
   * The state name of this location.
   *
   * @return The state name of this location.
   */
  public String getState() {
    return state;
  }

  /**
   * The country name of this location.
   *
   * @return The country name of this location.
   */
  public String getCountry() {
    return country;
  }

  /**
   * The postal code of this location.
   *
   * @return The postal code of this location.
   */
  public String getZip() {
    return zip;
  }

  /**
   * The latitude of this location.
   *
   * @return The latitude of this location.
   */
  public Double getLatitude() {
    return latitude;
  }

  /**
   * The longitude of this location.
   *
   * @return The longitude of this location.
   */
  public Double getLongitude() {
    return longitude;
  }
}

redpajama_set_name: RedPajamaGithub


6 Money Saving Mama Tips - From a Mom of 3!
Fry Sauce and Grits is a blog written by 3 sisters who have some pretty big things going on right now, Caitlin got engaged and Melanie had her first baby- congrats!!
I love Ikea and their sleek and small-home-friendly design. Thanks for sharing, Jessica!


text:
IT-Repairs
Mac Specialists
Date of Statement: 28th October 2020
This website is provided and operated by Xytron Ltd a company incorporated and registered in England and Wales with company number 05199954 whose registered office is at:
449 Twyford Road, Rotherwas Industrial Estate, Hereford, Herefordshire, HR2 6JR
These terms and conditions tell you the rules for using our website www.it-repairs.co.uk and the services provided by the Company. By using our website you accept these terms.
IT-Repairs is part of Xytron Ltd a Group of companies in the technology sector.
Here you will find our Terms & Conditions which we kindly ask you read before employing our services.
Diagnosis Cost:
IT-Repairs charge a diagnosis fee of £60 including VAT per computer. This covers the expenses incurred in diagnosing the Customer computer and takes 24HRs to complete. The results will then be emailed or SMS message to the Customer with the costings to repair the machine to the standard requested.
All repairs are carried out to a high standard. We provide you with a warranty on all parts and serviced rendered by our Company. The warranty period ranges from 90 days for iPhone screen replacement to 1 year for MacBook Pro battery replacement. Please ask a member of staff for details and warranty length when you're requesting details on our services.
All successfully completed repairs (unless stated otherwise via direct contact with a customer) come with a 12 month (365 calendar day) guarantee. In the rare event a part we have fitted has failed, we will replace it free of charge as long as it falls within the warranty period.
We cannot warranty a device if it has suffered any physical or liquid damage, which has occurred after the initial repair.
We cannot warranty the whole device, and can warranty only the parts we have used to repair the issue. We cannot be held liable for other issues unrelated to the repair. For example, if we replace the screen, we cannot be held responsible if the device develops a logic board fault in the warranty period.
Abandoned Devices:
Devices that are diagnosed as non-repairable or not cost effective for repair, will be disposed of if abandoned for over 30 days at our sole discretion. Any devices abandoned over 180 days without prior arrangements will be disposed of at our discretion. It is the customers responsibility to verify that we have the correct contact information on file which is provided to reception staff when the equipment is delivered to our address.
Data Privacy:
IT-Repairs recommends that all users ensure a full backup of important data held on devices is carried out, prior to engaging the services of the Company. If you are unable to carry out a full backup, please inform a member of our staff, who will employ best efforts to ensure data is backed up.
We pride ourselves on delivering a very secure & trustworthy service. This means employing AES encryption on all communications, namely SSL on our website, encrypted backup servers and fully compliant with GDPR which came into UK Law in May 2018.
If you have any questions or concerns regarding data security and how we handle your data, please ask to speak with our Data Protection Officer Mr Richard Cuthbertson.
The Company or "Us":
(Xytron Limited)
449 Twyford Road
Rotherwas Industrial Estate
HR2 6JR
www.it-repairs.co.uk
IT-Repairs is VAT registered: GB 889578726 – we charge 20% VAT on all UK sales
Within the website, links to external sites and resources may be provided and these are third parties. The links to such sites are provided purely for information only and aren't an endorsement by IT-Repairs. It is the responsibility of the third party organization to ensure the content on its website(s) complies with all laws and regulations. IT-Repairs have no control over the contents of third party sites or resources.
We may make changes to these terms
From time to time we may amend these terms. Every time you visit our website or engage our services, please check and familiarize yourself with our terms of engagement to ensure you fully understand.
Hereford HR2 6JR
09.00 – 16.00 (15:00 Friday)
Company No
© Copyright 2020 IT-Repairs Hereford
© Copyright 2004 - 2021 IT-Repairs Hereford
Business Address:Twyford Road,Rotherwas Industrial Estate,Hereford,HR2 6JR,GB |Tel: 01432 842624 |Email: [email protected].
Company No. 05199954 |VAT No.889 5787 26 |Business hours are 09.00 a.m. to 16.00 p.m. Monday to Friday IT-Repairs Hereford Rated 4.9 / 5 based on 157 reviews. | Review Us

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: with biber, how to declare that all citations in a given file are of a given category? I am using biblatex with Biber. I am using categories to separate citations that belong to different types and to produce different bibliographies accordingly.
Let's say I have a.bib file that contains bib records (let's say with labels a1, a2, ...) that all belong to a category c1, and a b.bib file that contains bib records (labels b1, b2, ...) that all belong to category c2.
Currently, I write something like this:
\usepackage[citestyle=numeric-comp,backend=biber,defernumbers=true]{biblatex}
\DeclareBibliographyCategory{c1}
\DeclareBibliographyCategory{c2}
\addtocategory{c1}{a1,a2,...}
\addtocategory{c2}{b1,b2,...}
\addbibresource{a.bib}
\addbibresource{b.bib}

Is there an easy way to say: "all bib records in a.bib belong to c1, without listing them all?"

A: biblatex offers two main solutions to the task of splitting/categorising entries into arbitrary subsets

*

*bibliography categories and

*keywords.

Bibliography categories work on the fly on the document side. You declare categories with \DeclareBibliographyCategory and add entries to categories with \addtocategory in your document. Categories are useful for ad-hoc classification and grouping that may vary from document to document. One great example of categories is How to split bibliography into "works cited" and "works not cited"?, where categories are used to split the bibliography in cited works and those added with \nocite.
Keywords on the other hand are added to the .bib file in the keywords field. They require no further set-up in the document and can be used directly to filter bibliographies with keyword=<keyword> or for general purpose tests with \ifkeyword{<keyword>}{<true>}{<false>}. Since keywords come from the .bib file, they are usually tied to a specific entry and not to a document.
For the most part, categories and keywords can be used in a similar fashion once they are known/defined.

When Biber/BibTeX has processed your .bib file and passes the data on to biblatex in the .bbl there is no information about the data source of each entry. That makes it extremely hard to generate categories from .bib files, since the relevant data is not available any more.
There is, however, a trick to let Biber add different keywords for each .bib file as shown in biblatex: multiple bibliographies categorised by different .bib files.
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{csquotes}
\usepackage[backend=biber,style=authoryear]{biblatex}

\DeclareSourcemap{
  \maps[datatype=bibtex]{
    \map{
      \perdatasource{\jobname-primary.bib}
      \step[fieldset=keywords, fieldvalue={, primary}, append]
    }
    \map{
      \perdatasource{\jobname-secondary.bib}
      \step[fieldset=keywords, fieldvalue={, secondary}, append]
    }
  }
}

\usepackage{filecontents}
\begin{filecontents}{\jobname-primary.bib}
@BOOK{hectic,
  AUTHOR    = {Henry Hectic},
  TITLE     = {How Horticulturalists Howl},
  PUBLISHER = {Honorary Books: Henage},
  YEAR      = {2000}
}
\end{filecontents}
\begin{filecontents}{\jobname-secondary.bib}
@BOOK{flutter,
  AUTHOR    = {Frederick Flutter},
  TITLE     = {Fraternising with Flowers},
  PUBLISHER = {Frippery Pamphlets: Folkestone},
  YEAR      = {1995}
}
\end{filecontents}

\addbibresource{\jobname-primary.bib}
\addbibresource{\jobname-secondary.bib}

\begin{document}
Some citations: \cite{hectic}, \cite{flutter}.
\printbibliography[title=Primary Sources, keyword=primary]
\printbibliography[title=Secondary Sources, keyword=secondary]
\end{document}

If you insist on categories, this can be used as a roundabout way to produce categories in the end as follows.
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{csquotes}
\usepackage[backend=biber,style=authoryear]{biblatex}

\DeclareSourcemap{
  \maps[datatype=bibtex]{
    \map{
      \perdatasource{\jobname-primary.bib}
      \step[fieldset=keywords, fieldvalue={, primary}, append]
    }
    \map{
      \perdatasource{\jobname-secondary.bib}
      \step[fieldset=keywords, fieldvalue={, secondary}, append]
    }
  }
}

\DeclareBibliographyCategory{primary}
\DeclareBibliographyCategory{secondary}

\AtDataInput{%
  \ifkeyword{primary}
    {\addtocategory{primary}{\thefield{entrykey}}}
    {}%
  \ifkeyword{secondary}
    {\addtocategory{secondary}{\thefield{entrykey}}}
    {}%
}

\usepackage{filecontents}
\begin{filecontents}{\jobname-primary.bib}
@BOOK{hectic,
  AUTHOR    = {Henry Hectic},
  TITLE     = {How Horticulturalists Howl},
  PUBLISHER = {Honorary Books: Henage},
  YEAR      = {2000}
}
\end{filecontents}
\begin{filecontents}{\jobname-secondary.bib}
@BOOK{flutter,
  AUTHOR    = {Frederick Flutter},
  TITLE     = {Fraternising with Flowers},
  PUBLISHER = {Frippery Pamphlets: Folkestone},
  YEAR      = {1995}
}
\end{filecontents}

\addbibresource{\jobname-primary.bib}
\addbibresource{\jobname-secondary.bib}

\begin{document}
Some citations: \cite{hectic}, \cite{flutter}.
\printbibliography[title=Primary Sources, category=primary]
\printbibliography[title=Secondary Sources, category=secondary]
\end{document}

But I doubt that that is going to provide a great advantage.

redpajama_set_name: RedPajamaStackExchange


text:
NEW ENGINE, ENHANCED SERVICEABILITY HIGHLIGHT ADVANCEMENTS FOR THE 2010 TEREX® PR950B HALF/FULL LANE MILLING MACHINE
Equipment World Staff (EQW)
OKLAHOMA CITY, February 19, 2010 – For the 2010 paving season, the Terex® PR950B half/full lane milling machine features a new, quieter C27 Tier II diesel engine and enhanced cooling system. Developing 947 hp (706 kW) at 1,800 RPM, the PR950B offers plenty of power to make up to 15-in (381-mm) deep plunge cuts with its standard 86 in (2,184 mm) half-lane cutter. Operating at a 14 percent reduced engine RPM, the PR950B emits lower noise levels, improving the operating environment around the machine.
The new PR950B design also enhances machine serviceability. Working from a platform, the service technician now has unobstructed access to the engine, filters and hydraulic pumps for servicing. A new service panel offers quick troubleshooting of the hydraulic system to quickly diagnose issues in the field.
A new 20 CFM air compressor available for the PR950B allows service personnel to quickly and efficiently change cutter teeth. In contrast to previous designs that required the machine engine to run for changing teeth, the new stand-alone air compressor is powered by its own 10 hp (7 kW) diesel engine that is much quieter and consumes considerably less fuel. The engine draws fuel from the main diesel tank.
The PR950B milling machine's quick-mount cutter housing design offers fast removal for simple maintenance and easy cutter change-out. Four cutter widths are now available – standard 86-in (2,184 mm); 103 in (2,612 mm); 135 in (3,429 mm); and full lane 150 in (3,810 mm) – increasing machine versatility to tackle a variety of applications. The exclusive tapered cutter housing carries substantially less material, reducing clean-up. The machine's high pressure water system with 1,220 gallon (4,618 liter) tank capacity effectively cools the cutter teeth for optimal service life. When tooth changes are required, the hydraulically operated rear cutter door can be raised up to 32 in (813 mm) for unimpeded access to the rotor.
The Terex® PR950B features wide, 42-in (1,067 mm) upper and lower conveyors to quickly channel material away from the cutter, optimizing production capabilities. The upper conveyor swings 60 degrees to either left or right of center, allowing material to be quickly discharged directly into a truck on either side of the machine.
Featuring a hydrostatic drive system, four machine work speeds match cutter productivity to the material being milled. Speeds range from a low of 0 – 74 fpm (0 – 23 mpm) to a high of 175 fpm (0 – 53 mpm) for high production mainline milling.
Available in either a three- or four-track configuration, the PR950B includes all-track drive for maximum traction. All-track steering delivers exceptional on-site maneuverability and a class-leading 8-ft (2,438-mm) turning radius for both track designs.
About Terex Roadbuilding
Part of Terex Corporation (NYSE: TEX), Terex Roadbuilding is a diversified manufacturer of equipment for use in the construction, infrastructure and road building industries. The equipment offered includes over 150 products, including automatic roller and bridge pavers, under-bridge aerial access platforms, mixer trucks, slipform pavers, curb & barrier machines, ready mix plants, asphalt pavers, material transfer devices and vehicles, asphalt plants, warm mix asphalt systems, reclaimer/stabilizers and milling machines. In addition to its global Oklahoma City office, Terex Roadbuilding operates manufacturing facilities in Canton, S.D., Fort Wayne, Ind., Rock Hill, S.C., and Rio Grande de Sul, Brazil. More company information can be found at www.terexrb.com and www.terex.com.
Commercial class pavers: Smaller and lighter but more sophisticated than ever
Introducing the AggreScreed by TerraTec Industries
Are You Sitting Down? Ditch Witch 420sx Released.


It perfectly fits my remotes and a fun candle. The best part is that it doesn't overwhelm the space and complements it perfectly.
So there's a quick, easy, and inexpensive way to corral those remotes and have a little bit of whimsy added to your living space!
What a cool idea. And useful. My four remotes float all over the place.


text:
<?xml version="1.0" encoding="UTF-8" ?>
<!--

       Copyright 2009-2012 The MyBatis Team

       Licensed under the Apache License, Version 2.0 (the "License");
       you may not use this file except in compliance with the License.
       You may obtain a copy of the License at

          http://www.apache.org/licenses/LICENSE-2.0

       Unless required by applicable law or agreed to in writing, software
       distributed under the License is distributed on an "AS IS" BASIS,
       WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
       See the License for the specific language governing permissions and
       limitations under the License.

-->

<!DOCTYPE sqlMap PUBLIC "-//ibatis.apache.org//DTD SQL Map 2.0//EN"
    "http://ibatis.apache.org/dtd/sql-map-2.dtd">

<sqlMap namespace="Item">

  <typeAlias alias="item" type="com.ibatis.jpetstore.domain.Item"/>

  <cacheModel id="itemCache" type="LRU">
    <flushInterval hours="24"/>
    <property name="size" value="100"/>
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  <cacheModel id="quantityCache" type="LRU">
    <flushInterval hours="24"/>
    <flushOnExecute statement="updateInventoryQuantity"/>
    <property name="size" value="100"/>
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  <select id="getItemListByProduct" resultClass="item" parameterClass="string" cacheModel="itemCache">
    SELECT
    I.ITEMID,
    LISTPRICE,
    UNITCOST,
    SUPPLIER AS supplierId,
    I.PRODUCTID AS "product.productId",
    NAME AS "product.name",
    DESCN AS "product.description",
    CATEGORY AS "product.categoryId",
    STATUS,
    ATTR1 AS attribute1,
    ATTR2 AS attribute2,
    ATTR3 AS attribute3,
    ATTR4 AS attribute4,
    ATTR5 AS attribute5
    FROM ITEM I, PRODUCT P
    WHERE P.PRODUCTID = I.PRODUCTID
    AND I.PRODUCTID = #value#
  </select>

  <select id="getItem" resultClass="item" parameterClass="string" cacheModel="quantityCache">
    select
    I.ITEMID,
    LISTPRICE,
    UNITCOST,
    SUPPLIER AS supplierId,
    I.PRODUCTID AS "product.productId",
    NAME AS "product.name",
    DESCN AS "product.description",
    CATEGORY AS "product.categoryId",
    STATUS,
    ATTR1 AS attribute1,
    ATTR2 AS attribute2,
    ATTR3 AS attribute3,
    ATTR4 AS attribute4,
    ATTR5 AS attribute5,
    QTY AS quantity
    from ITEM I, INVENTORY V, PRODUCT P
    where P.PRODUCTID = I.PRODUCTID
    and I.ITEMID = V.ITEMID
    and I.ITEMID = #value#
  </select>

  <select id="getInventoryQuantity" resultClass="int" parameterClass="string">
    SELECT QTY AS value
    FROM INVENTORY
    WHERE ITEMID = #itemId#
  </select>

  <update id="updateInventoryQuantity" parameterClass="map">
    UPDATE INVENTORY SET
    QTY = QTY - #increment#
    WHERE ITEMID = #itemId#
  </update>

</sqlMap>


Low maintenance open spray jets which excel in performance and reliability. Deflector snaps on to frame to temporarily restricts spray diameter for young trees.
– Open frame and spray plate.
– Wide range of spray patterns and trajectories.
– Colour coded nozzles for flow rate identification.
– 'Quick' thread fo easy insertion into tubing.
Specifically designed for orchards, vineyards and horticultural crops.


Madame Ranevskaya (Rampling) is a spoiled aging aristocratic lady, who returns from a trip to Paris to face the loss of her magnificent Cherry Orchard estate after a default on the mortgage... See full � the brand gap pdf download In The Cherry Orchard, an improverished landowning family is unable to face the fact that their estate is about to be auctioned off. Lopakhin, a local merchant, presents numerous options to save the estate including cutting down their prized cherry orchard but stricken by denial the family leave the estate to the sound of axes.
The earth is great and beautiful; it is full of wonderful places. [A pause] Think, Anya, your grandfather, your great-grandfather and all your ancestors were serf-owners, owners of living souls.
Key plays, such as The Cherry Orchard and The Seagull, receive dedicated chapters while lesser-known works and genres are also brought to light. The volume concludes with appendices of primary sources, lists of works, and a select bibliography.


text:
Chapter 494: Misery and Suffering
On an island in the South Sea, the world of immortals.
The South Sea Gourmet, who resembled a meatball more than a human, carefully turned a freshly killed beast on a spit over a bonfire. Suddenly, a shudder ran down his spine.
"Dammit!" He turned to the Azure Province in shock. "The body of the eyes escaped and ate the body of the ears! The crack leading to the outer world has been opened!
"This won't do, I must go to Crimson Province. If the eyes eat my body and open up the crack there, the world of immortals will be doomed!"
The South Sea Gourmet abandoned his meal and rushed to Crimson Province in Nephrite Major.
At the same time, the two Wayfarers in the world looked up in unison toward the direction of Azure Province.
"Where did the demonic nature of the body come from? I only wanted to devour the old man and take his place, but he… he wants to destroy every living thing!" Art Saint Wayfarer shuddered. Next to him, Zhao Changkong observed the situation with knitted brows.
"Attention, Nephrite Imperial Army!" commanded Zhao Changkong after three breaths of hesitation.
"Wait!" the Art Saint hurriedly stopped him. "The yin spirits from outside the realm are not to be trifled with. No amount of low-level immortals will be able to stop them! Numbers alone mean nothing.
"Allow me!" Lifting the veil shrouding him, Art Saint Wayfarer threw out the portrait of a beautiful woman and made his way to Azure Province. "If I don't return, Your Majesty, find the South Sea Gourmet in Crimson Province. He will protect you for a century in my stead." With that, he disappeared into the distance.
Letting out a deep sigh, Zhao Changkong brooded. Gravely, he bowed to the air. "Please take the Ingress Path to Azure Province, royal uncle."
Amid the fading hum, the pristine white path disappeared from Xiankan.
Following Wayfarer, Wanfeng arrived at Azure Province as well, with her emerald bamboo stick. A hazy green light had enveloped the province, keeping the swarming yin spirits firmly confined within. Those who hadn't been able to make it to the immortal cities or larger settlements had all been devoured by the terrifying spirits.
Unlike other large provinces, however, Azure Province was highly experienced in dealing with swarms of enemies like these. Immortals and cultivators alike moved to their appropriate places in an orderly manner and activated all of the great formations in the cities and settlements, made sturdy by the Azure Province Seal.
The spirits weren't breaking into the cities any time soon. With their continuously increasing numbers, though, it was only a matter of time before the defenses of the cities failed.
The governor of the Azure Province was a peerless immortal. He could enter heaven and earth to search for aether dao fruit and become a dao immortal at any time. However, he currently felt like he was being suffocated to death.
Terrifying waves of yin spirits came with great ferocity, and a battle formation of a million golden immortals and their bearers were completely destroyed after only a little more than ten breaths. The Azure Governor felt a stabbing pain in his chest.
"Die!!!" he snarled. The seal in his hand shook violently as he drew upon the power of Azure Province to blast the spirits back. Every one of his attacks disintegrated a great number of enemies.
"The power of heaven and earth!" Inspiration struck Lu Yun. "The vast power of the world is these spirits' greatest bane!"
He made a move following his observation of the governor's actions, retracting his combat arts and tapping into the world's power with his nascent spirit. Great energy of the land rippled out like water, creating immense waves in the air that swept over many of the yin spirits and rendering them into ashes.
But there were simply too many of them, and an endless supply of yin spirits continued to emerge from the ancient tomb. Although Lu Yun had killed tens of millions of spirits with a single attack, their casualties were quickly replenished.
"There's no end to this. The key to it all is the ancient tomb... I must seal it!" Hellfire burned around him as he changed course and struggled to make his way to the tomb.
Although the yin spirits were afraid of hellfire, their sheer numbers prevented Lu Yun from moving about freely. It felt as if he was wading his way through a marsh, each step slow and ponderous.
Other than the immortal cities, ordinary settlements like mortal towns and cultivator cities were falling, losing countless residents to the invasion of the yin spirits. In fact, even the larger immortal cities were struggling as well.
A slight tremor passed through Azure Province, followed by the descent of a white path. Its great heavenly power destroyed countless yin spirits with a casual shake, and a wizened peak arcane dao immortal stood upon it.
"Governor of Azure Province, why did you not request help from the heavenly court when the province was struck by such a calamity? You are an official of the imperial court! Is this meant as a challenge to the court's authority?!"
The nine-fruit arcane dao immortal destroyed yin spirits as easily as flipping his hand. The path under his feet landed within the province and extended in thousands of different directions like an enormous dragon, protecting the immortal cities teetering on the edge of annihilation.
The arrival of the Ingress Path lifted a great amount of pressure from Azure Province, and many more elites had traveled upon it to stem the tide of evil.
The flood of spirits was innumerable, but most of their strength fell between peerless immortals and normal cultivators. With the newcomers' concerted efforts, half of them were immediately decimated.
"Ancestor!" the Azure governor suddenly wailed in sorrow.
A skeleton lay crouched over the crack in the ancient tomb at the center of the province, its flesh and essence all devoured. It was the origin dao immortal serving the province—or rather, his remains after failing to seal the tomb with his own body and being eaten by the endless tide.
"What happened here?!" Zhao Changkong's royal uncle gazed gravely at the skeleton. "He's an origin dao immortal. Even with his dao injuries, he shouldn't have been killed and eaten by mere yin spirits… No, watch out!!"
Scarlet shadows emerged from nowhere and lunged at the elites coming to the province's rescue. A dao immortal was consumed before he could even react, falling helplessly to the ground as a withered, dried husk.
"Waughh!!" Piercing screams rang from the tomb, and from the crack swarmed a horde of scarlet shadows. A pungent, metallic smell permeated the air.


text:
'use strict';

var ApiDeployService = require('../service/api-deploy/api-deploy-service');
var ApiDeployEvent = require('../service/api-deploy/api-deploy-event');

var pub = {};

pub.getParameters = function (event) {
    return ApiDeployEvent.getParameters(event);
};

pub.createResource = function createResource(event, context, eventParams, callback) {
    ApiDeployService.deployApi(eventParams, function (error) {
        if (error) {
            return callback(error);
        }
        getForResponse(event, context, eventParams.params.stageName, eventParams.params.restApiId, callback);
    });
};

pub.deleteResource = function deleteResource(_event, _context, _eventParams, callback) {
    callback();
};

pub.updateResource = function updateResource(event, context, eventParams, callback) {
    pub.createResource(event, context, eventParams, callback);
};

module.exports = pub;

/* eslint max-params: 0 */
function getForResponse(_event, _context, stageName, restApiId, callback) {
    ApiDeployService.getForResponse(stageName, restApiId, function (error, apiStage) {
        if (error) {
            return callback(error);
        }
        apiStage.physicalResourceId = restApiId + '/' + stageName;
        return callback(null, apiStage);
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Below are some resources to help you learn about the Nakba.
You can also read IMEU's Quick Facts: The Palestinian Nakba.
Relive the journey of Nakba refugees using the interactive tool on the Middle East Monitor website.
Read about the Nakba on the Electronic Intifada.
Visualizing Palestine has created a visual map and list of the status of the 536 Palestinian villages depopulated by Israel to illustrate that return is possible. You can also visit their interactive map showing 143 years of colonization and 69 years of the Nakba.
Read the article "It's Time For Our Prime Minister to Commeorate Nakba Day" by Tyler Levitan of Independent Jewish Voices.
Read the Palestinian BDS National Committee (BNC) statement on the Nakba.
The US Campaign for Palestinian rights have announced a political education campaign, Together We Rise, to mark Nakba Day. You can sign up for the series and preview the curriculum.
Jewish Voice for Peace (JVP) has created an educational project, Facing the Nakba.
The Nakba did not end in 1948 and continues until today, in the form of Israel's ongoing theft of Palestinian land for settlements and for Jewish communities inside Israel, its destruction of Palestinian homes and agricultural land, revocation of residency rights , deportations, periodic brutal military assaults that result in mass civilian casualties such as the one that took place in Gaza in the summer of 2014, and the denial of the internationally-recognized legal right of return of millions of stateless Palestinian refugees.


Weber works closely with utilities to solve problems in underground networks. Connection of large cables at suitable working heights for field personnel and reliability are but two ongoing issues raised on a regular basis. The range of composite pillars and Langmatz small glass-reinforced polycarbonate enclosures help provide some of the answers.
The other side of the solution is the wide range of reliable switchgear at Weber's disposal. Tested to Australian conditions the reliability of the switchgear and the connecting bus work is unquestioned. Factory procedures are continually updated as processes improve. Making high current connections that last for extended periods is what is needed in an electricity network – that's what Weber does well.
See the download for a list of standard designs or email your enquiry.


Our dining room is on a first come, first served basis. However, if you have a large party of up to 50 people, or are planning an event, or need catering services, drop us a line and we'll hook you up.
We are a family owned and operated small business, therefore your opinion and loyalty means much more to us.
As the brand continues to grow with the addition of my families involvement, we have chosen to rebrand ourselves and target a healthier food segment as well as continuing our great reputation as the Premier Steakhouse and Banquet facility. After much consideration we opted to change our name to the Texas Border Grill to better reflect the diversity of menu selection, and product development.
After teaming up with Chef Michael Boggs in 2009, the two perfected the menu and experienced great success with Texas Border Grill. Now, the pair is looking to open another location in the Middle Georgia area. Look out for it in August 2016!
We are conveniently located between Macon and Warner Robins, Just 2 miles off I-75. Exit at the Hartley Bridge Road Exit # 155, Turn East and then at the end of Hartley Bridge Turn Right at the light and we are just 1/4 mile on the right.


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Sustainability Advocate
Ruth Ann Barrett of EarthSayers.tv, Voices of Sustainability
Tag Archives: Yale Program on Climate Change Communication
Climate Justice and Sustainability Advocacy
The chief curator (me) at EarthSayers.tv, Voices of Sustainability, has created a new channel addressing Climate Justice. The impetus for doing so arose out of a study by the Yale Program for Climate Change Communication. They conduct scientific research on public climate change knowledge, attitudes, policy preferences, and behavior, and the underlying psychological, cultural, and political factors that influence them. Of particular interest was their audience research, Global Warming's Six Americas. A must read.
For climate change communicators I assume that this study formed the foundation for your present programs and campaigns. However, for those of us sustainability advocates with an environmental, social, and economic bent the Yale Program research may have been missed given the information overload that climatographer Mark Trexler addressed in his whitepaper, The Problem of Infinite Information in Corporate Climate Change Decision-Making.
Initially, these are the voices we are advancing on the topic of Climate Justice with more to come. Start with Linda Haydock of the Inter-community Peace & Justice Center's What is Climate Justice? Continue on with Mary Robinson, former President of Ireland and former United Nations High Commissioner for Human Rights on "Climate Change as a Human Rights Issue" and Julia Olson of Our Children's Trust on "Securing the Legal Right to a Stable Climate." If you parent or work with the more youthful among us, especially millennials, you might want to listen to the conversation between Dr. James Hansen and his grand daughter, Sophie Kivlehan on " Young Peoples Burden."
There are other voices to include Pope Francis, M.E. Tucker, David Korten, Anthony Leiserowitz, Tim Brennan and HH Dalai Lama.
"Most Americans say global warming is personally important to them, but don't talk or hear about it much." Yale Program Climate Note of September 29, 2016
I can't do much about the talking part, but I can improve on the number of people, like you, hearing about it. It's up to you to re-frame the conversation around climate change to climate justice and talk with your family, friends, and work colleagues.
"More than half of those who are interested in global warming or think the issue is important "rarely" or "never" talk about it with family and friends (57% and 54% respectively)."
Ruth Ann Barrett, Sustainability Advocate, October 7, 2016, Portland, Oregon.
P.S. What sparked my interest was the Yale Program report, Faith, Morality, and the Environment: Portraits of Global Warming's Six America's which led to an EarthSayers' initiative, Faith and Climate Justice. More on Faith and Climate Justice in my next blog post.
This entry was posted in Climate Change, sustainability and tagged climate change, climate justice, EarthSayers.tv, global warming, Julia Olson, linda Haydock, Mark Trexler, Mary Robinson, Our Children's Trust, ruth ann barrett, Yale Program on Climate Change Communication on October 7, 2016 by Ruth Ann.
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A blog published by EarthSayers.tv, the Voices of Sustainability, featuring Ruth Ann Barrett, Curator of EarthSayers.tv, a specialized search engine to the visionaries, leaders, teachers – people from all walks of life – speaking on behalf of Mother Earth and her children. www.EarthSayers.org.
Contact is Ruth Ann Barrett, ruthann@earthsayers.tv, 415-377-1835.
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The CP DB1 transformer discharge box facilitates fast discharging of power transformers after a test of the winding resistance.
A transformer must be discharged whenever the measurement of one phase has been completed. Thus customers can save a lot of time by measuring with the CP DB1.


Let's rock the dance floor of the legendary and unique Crash Club! The disco is located in the heart of Munich Schwabing and is famous for its rock music and 80s sound since over 50 years! Wear your coolest pair of jeans, slip on that leather jacket, and join us for a trip back in time to some of the best years of music history! You will spend a special night with the very best of 80s rock & pop hits and hit the dance floor with other global minds.
Come early if you would like to grab a free tattoo sticker or meet & greet old and new friends before the DJ turns up the music! The first 50 attendees will also get a free welcome drink!
The guest list will be closed around 17:00 the day of the event. Please make sure to sign up to enter for free (Albatross Members) or at a reduced entrance fee (Basic Members)!
As always, do not hesitate to invite your international friends and other global minds. If you capture some special moments during the evening on your camera, please upload them here afterwards so we can all re-live the event!
We are looking forward to rocking out with you!


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AGL drops Vocus acquisition talks
By Brendon Foye on Jun 17, 2019 9:58AM
Energy retailer AGL has already pulled out of talks to acquire Vocus for more than $3 billion after less than a week.
Last Tuesday, AGL put forward a non-binding, indicative offer to buy all of Vocus's shares for $4.85 each, valuing the company at $3.017 billion, while it completed due diligence on Vocus. That deal has been scrapped after six days as AGL is no longer confident it would create sufficient value for its shareholders, according to chief executive Brett Redman.
"We believe there will be material opportunities for AGL as energy and data value streams continue to converge and the traditional energy sector accelerates its transformation. The approach to Vocus reflected our view that the Vocus asset base has attributes that could support the execution of this strategy and benefit our customers," said Redman.
"However, we are no longer confident that an acquisition of Vocus at the proposed terms would represent sufficient certainty of creating value for AGL shareholders."
AGL is the second suitor in as many months to cancel acquisition talks with Vocus. Global investment firm EQT Infrastructure offered $3.26 billion to buy Vocus in May, but binned talks after a week of "accelerated" due diligence.
Vocus's shares were trading at $4.36 each at the time of writing.
Update: Vocus chief executive Kevin Russell issued the following comment:
"As we have repeatedly said, this is a three year turnaround. We have great confidence that our strategy and ability to execute our business plan will deliver significant value to our shareholders in the medium to long term.
"There is growing demand for our strategically valuable network assets and we have a substantial opportunity for Vocus Networks to gain market share. This is the core of our business. The Vocus management team will now be able to focus all of their attention on realising the opportunity that we have ahead of us."
agl telco vocus
By Brendon Foye
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