text:
\section{Introduction}
Small--scale atmospheric phenomena are typically characterized by relatively
slow dynamics, that is, low Mach number flows for which the fast acoustic modes
are physically irrelevant.  Thus, numerical modeling of these flows
does not typically require explicitly resolving fast--propagating sound waves.
Two different approaches have been widely used to remove the time step
constraint that would result from resolving fast acoustic modes.
The first and more common approach solves the fully compressible
equations of motion
but limits the impact of acoustic modes, for instance, by advancing the acoustic
signal in time with an implicit time discretization or with multiple smaller time steps,
as originally considered for cloud models in \cite{Tapp1976}
and \cite{Klemp1978}.  A second alternative consists
of analytically filtering acoustic modes from the original compressible
equations, thus deriving a new set of equations, often called
sound--proof equations.  Within this category are
anelastic \cite{Bat53,OguPhi62,DutFic69,Gou69}
and pseudo--incompressible \cite{durran:1989} models.

Several anelastic formulations (see, e.g., \cite{Clark1979,LipHem82,Grabowski2002}),
and recently a pseudo--incompressible formalism
\cite{ONeill2013}, have been developed for moist flows.
In this paper we derive a low Mach number model for moist
atmospheric flows
with a general equation of state
using the low Mach number formalism in \cite{ABNZ:III}
as a starting point.
Here we use the term ``low Mach number model'' to refer
to a model in which the equations are valid approximations to the
fully compressible equations when the Mach number is small.
In atmospheric modeling, the equations which follow from
assuming the Mach number is small, thus the variation of the
pressure from the background pressure is small,
are also referred to as pseudo--incompressible equations
following \cite{durran:1989}.
The anelastic equations require additional assumptions on the
smallness of density and temperature variations to be valid.
(For a complete discussion on sound-proof equations
for atmospheric flows,
and their differences, see,
for instance, \cite{Klein2009} and references therein.)
We note that the low Mach number equations presented here do
not guarantee that a flow that initially satisfies the low Mach
number assumption will continue to satisfy it for all time.
The buoyancy forcing from a large density perturbation in a
domain with large vertical extent could accelerate
the flow into a regime where the Mach number is no longer small.
Once the flow reached this regime, the low Mach number equations
would no longer be valid approximations to the fully compressible
equations.  However, until that point, the equations are valid.

Analogous to
the moist pseudo--incompressible model of
\cite{ONeill2013} we consider only reversible
processes given by water phase changes, using here an exact
Clausius--Clapeyron formula for moist thermodynamics.
In contrast to \cite{ONeill2013}, however,
we derive the equations of motion in terms of conserved variables
(like \cite{Ooyama1990} in a compressible framework),
that is, using appropriate invariant variables such that terms
resulting from phase change are eliminated from the time evolution equations
\cite{Betts1973,Tripoli1981,Hauf1987}.
We also include the effects of the specific heats of both water
vapor and liquid water,
and consider an isentropic expansion factor ($\gamma$) that accounts
for variations in the water composition of moist air.
Although the low Mach number formulation holds for any moist
equation of state, for the purposes of numerical comparison with
compressible solutions we consider the special case where both
dry air and water vapor are assumed to be ideal gases.

While the larger time step allowed by the low Mach number formulation
can lead to greater computational efficiency than a purely compressible
formulation, it may also introduce
larger errors in the dynamics of moist flows as investigated in \cite{CASTROmoist}.
In addition to the latent heat release accompanying phase changes, thermodynamic properties
such as the specific heat of moist air, as well as thermodynamic variables
such as temperature, depend on the composition of the moist air,
thus change over the time step.  This motivates our use of invariant
variables as prognostic variables, namely total water content
and a specific enthalpy of moist air that
accounts for both sensible and latent heats.
In models where source terms related to phase transition appear explicitly
in the evolution equations, they are typically first neglected or lagged
in time and then corrected or estimated during a given time step;
the use of invariant variables removes the need of accounting
for such terms.
Nevertheless, the diabatic contribution of the latent heat release
must appear in the source term for the low Mach divergence constraint
on the velocity field.  In practice the latter involves computing the
rate of evaporation (or condensation).
Since no analytical expression exists for this rate,
one of the most common ways to estimate it deduces it from
the change in water vapor content necessary to ensure
that there is no supersaturated water vapor
at the end of the time step (cf. \cite{Soong1973}).
This variation is measured with respect to an initial estimate
of water vapor that does not necessarily respect the saturation
requirements either because it was initially advected without
accounting for phase changes
(see, e.g., \cite{Klemp1978,Grabowski1990,Bryan2002,Satoh2003,ONeill2013})
or because it considers a lagged
evaporation rate from the previous time step
(see, e.g., \cite{Grabowski2002}).
In our model, because water vapor is not used as a prognostic variable,
we cannot compute this variation of water vapor.
Instead we adopt a different approach, similar to \cite{LipHem82},
that estimates the evaporation rate based on the fact that
whenever a parcel is saturated, a Clausius--Clapeyron formula
relates the local values of water content to the thermodynamics
within the parcel.
The conservation equation for saturated water vapor then becomes a
time--varying constraint that guarantees thermodynamic equilibrium
from which the evaporation rate can be estimated.
The latter is thus computed as required during a
time step using the current values of
water content and thermodynamic variables,
diagnostically recovered from the invariant prognostic variables.

This paper is organized as follows.
We introduce the new low Mach number model for moist
atmospheric flows in Section~\ref{sec:gov_eqs},
and describe the moist thermodynamics in Section~\ref{sec:moist_thermo}.
In Section~\ref{sec:numer} we discuss the numerical implementation.
Finally, in Section~\ref{sec:simul}, we present
several numerical comparisons and discuss our findings.

\section{Low Mach number formulation}\label{sec:gov_eqs}
We begin by writing the fully compressible equations of motion
expressing conservation of mass, momentum, and enthalpy
in a constant gravitational field
in which we neglect Coriolis forces and viscous terms:
\begin{eqnarray}
\frac{\partial \rho}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho \boldsymbol{U} \right)
&=& 0, \label{eqn:cont} \\
\frac{\partial \left( \rho \boldsymbol{U}\right) }{\partial t} +
\boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho \boldsymbol{U} \boldsymbol{U} \right ) + \boldsymbol{\nabla} p & = & -\rho g\boldsymbol{\hat{e}}_z,
\label{eqn:momentum}\\
\frac{\partial ( \rho  \widehat{h} ) }{\partial t} +
\boldsymbol{\nabla} \boldsymbol{\cdot} ( \rho \widehat{h} \boldsymbol{U} ) - \frac{D p}{D t}
& =  &
\rho \mathcal{H},\label{eqn:enthalpy}
\end{eqnarray}
where the material derivative
is defined as
$D/D t = \partial /\partial t + \boldsymbol{U} \boldsymbol{\cdot} \boldsymbol{\nabla}$.
Here $\rho$ is the total density, $\boldsymbol{U}$ is the velocity,
and $\widehat{h}$ is the specific enthalpy of moist air.
The pressure, $p$, is defined by an equation of state (EOS).
The term $\mathcal{H}$ represents a source of heat (per unit mass and time)
to the system, such as thermal conduction or radiation.
We include gravitational acceleration given by
$\boldsymbol{g} = -g \boldsymbol{\hat{e}}_z$, where $\boldsymbol{\hat{e}}_z$ is the unit vector in the vertical
direction.

We formulate the moist atmospheric processes as in \cite{Romps2008},
with the additional simplification that at any grid point all phases
have the same velocity and temperature.
We treat moist air as an ideal mixture of dry air, water vapor,
and liquid water,  with the water phases in thermodynamic equilibrium,
so that only reversible processes are taken into account;
ice--phase microphysics, precipitation fallout, and subgrid--scale turbulence
are ignored.  Denoting by $q_a$, $q_v$, and $q_l$
the mass fraction of dry air, water vapor,
and liquid water, respectively, we note that $q_a+q_v+q_l = 1$
and write
\begin{eqnarray}
 \frac{\partial \left( \rho q_a \right)}{\partial t}
+ \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho q_a \boldsymbol{U} \right) & = & 0, \label{eqn:qa}\\
\frac{\partial \left( \rho q_v \right)}{\partial t}
+ \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho q_v \boldsymbol{U} \right) & =  & e_v, \label{eqn:qv}\\
\frac{\partial \left( \rho q_l \right)}{\partial t}
+ \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho q_l \boldsymbol{U} \right) & = & -e_v,\label{eqn:ql}
\end{eqnarray}
where $\rho$ is the total density.
Positive values of $e_v$ correspond to evaporation;
negative values correspond to condensation.
If we define the mass fraction of total water, $q_w =q_v +q_l$,
then we can replace (\ref{eqn:qv})--(\ref{eqn:ql}) by
\begin{equation}\label{eqn:qw}
\frac{\partial \left( \rho q_w \right)}{\partial t}
+ \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho q_w \boldsymbol{U} \right) = 0.
\end{equation}
(We note that the system including both (\ref{eqn:cont}) and (\ref{eqn:qa})--(\ref{eqn:ql})
or (\ref{eqn:cont}), (\ref{eqn:qa}), and (\ref{eqn:qw}) is over--specified;
in practice (\ref{eqn:cont}) need not be solved separately.)

We define the internal energy of
moist air, $\widehat{e},$ as in \cite{Romps2008},
\begin{equation}\label{eqn:ehat}
\widehat{e} = c_{vm} \left(T - T_{\mathrm{trip}}  \right) + q_v E_{0v},
\end{equation}
where the constant--volume specific heat of moist air is
given by
$c_{vm} = q_a c_{va} + q_v c_{vv} + q_l c_{vl},$
with the specific heats at constant volume:
$c_{va}$, $c_{vv}$, and $c_{vl}$, for
dry air, water vapor, and liquid water, respectively.
Here $T_{\mathrm{trip}}$ is the triple--point temperature
and $E_{0v}$ is the specific internal energy of water vapor
at the triple point,
while the enthalpy of moist air is given by
\begin{equation}\label{eqn:hhat}
\widehat{h} = \widehat{e} + \frac{p}{\rho}.
\end{equation}
Note that an evolution equation for the internal energy
\begin{equation}\label{eqn:int_energy}
\frac{\partial \left( \rho  \widehat{e} \right) }{\partial t} +
\boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho \widehat{e} \boldsymbol{U} \right)
+ p \left(\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U}\right) =
\rho \mathcal{H},
\end{equation}
can be used instead of (\ref{eqn:enthalpy}).
In either case, there are no source terms related to phase change
in (\ref{eqn:enthalpy}) or (\ref{eqn:int_energy}),
as observed in \cite{Ooyama1990} and \cite{CASTROmoist}.
Finally, to close the system
(\ref{eqn:cont})--(\ref{eqn:qa}) with (\ref{eqn:qw}),
we consider a general equation of state for moist air written as
$p=p(\rho,T,q_a,q_v,q_l)$.  (Because
$q_a+q_v+q_l = 1$ we could remove one of these three arguments from
the equation of state; for clarity of exposition, however, we leave
all three.)

In the low Mach number model, we write the pressure, $p(\boldsymbol{x},t)$,
with $\boldsymbol{x}=(x,y,z)$, as the sum of a base state pressure,
$p_0(z,t)$, and a perturbational, or dynamic, pressure,
$p'(\boldsymbol{x},t)$, such that
$|p'|/p_0 = O(M^2)$.
In contrast to anelastic models,
the perturbations in density and temperature themselves do not need to
be small for the equations to remain valid;
as long as those perturbations do not result in the flow violating the
assumption of a low Mach number, hence small $|p'|/p_0$, the equations
remain valid.
The base state is assumed to be in hydrostatic equilibrium,
i.e., $\boldsymbol{\nabla}p_0=-\rho_0 g \boldsymbol{\hat{e}}_z$, where $\rho_0=\rho_0(z,t)$ is the
base state density.
The fully compressible momentum equation (\ref{eqn:momentum})
could be rewritten as
\begin{equation*}\label{eqn:momentumLM_orig}
\frac{\partial \left( \rho \boldsymbol{U}\right) }{\partial t} +
\boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho \boldsymbol{U} \boldsymbol{U} \right ) + \boldsymbol{\nabla} p' =
-(\rho-\rho_0)g\boldsymbol{\hat{e}}_z,
\end{equation*}
with no approximation.  The low Mach number momentum equation
has an additional contribution to the buoyancy term,
\begin{equation}\label{eqn:momentumLM}
\frac{\partial \left( \rho \boldsymbol{U}\right) }{\partial t} +
\boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho \boldsymbol{U} \boldsymbol{U} \right ) + \boldsymbol{\nabla} p' =
-(\rho-\rho_0)g\boldsymbol{\hat{e}}_z -
\left( \frac{\rho_0}{\rho}
\left.\frac{\partial \rho}{\partial p_0}\right \vert_s p' \right)g\boldsymbol{\hat{e}}_z,
\end{equation}
as introduced by \cite{Klein2012}.  Here the derivative of $\rho$
with respect to $p_0$ is taken at constant entropy, $s,$
and is based on the low Mach number form of the equation of state, i.e.
$p_0(z,t) = p(\rho,T,q_a,q_v,q_l)$.  As pointed out in \cite{VLBWZ:2013},
this form of the momentum equation is analytically equivalent to the
momentum equation in the pseudo--incompressible equation set, i.e.,
\begin{equation*}\label{eqn:momentum_pseudo}
\frac{\partial \boldsymbol{U}}{\partial t} +
\boldsymbol{U} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{U} + c_{pa} \theta \boldsymbol{\nabla} \pi^\prime =
\frac{\theta - \theta_0}{\theta_0} g \boldsymbol{\hat{e}}_z
\end{equation*}
with Exner pressure, $\pi = \left( p /  p_{00} \right)^{R_a/c_{pa}}$
($p_{00}$ is a reference pressure,
while $R_a$ and $c_{pa}$ stand for the specific gas constant
and constant--pressure specific heat of dry air, respectively),
potential temperature, $\theta = T / \pi,$
and the base state potential temperature, $\theta_0$,
if we define
a linearized relationship between the perturbational pressures,
$p^\prime$ and $\pi^\prime.$

To complete the low Mach number model for moist atmospheric flows
we first replace $p$ by $p_0$ in the equation of state
and we then
differentiate $p_0 = p(\rho,T,q_a,q_v,q_l)$ along particle paths.
Following the derivation in \cite{ABNZ:III},
with details as given in Appendix \ref{app:LM_constraint},
we derive the following divergence constraint
for the low Mach number model:
\begin{equation}\label{eqn:divu_alpha}
\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U}
+ \alpha \frac{D p_0}{D t} = S,
\end{equation}
with
\begin{equation}\label{eqn:S}
\alpha = \frac{1}{\Gamma_1 p_0},
\qquad
S =
\left[
\frac{1}{\rho p_{\rho}} \left(p_{q_v} - p_{q_l} \right)
- \sigma \left(\widehat{h}_{q_v} - \widehat{h}_{q_l} \right)
\right]
\frac{e_v}{\rho}
+ \sigma \mathcal{H},
\end{equation}
where
$p_{\rho} = \partial p/\partial \rho \vert_{T,q_i}$,
$p_{T} = \partial p/\partial T \vert_{\rho,q_i}$,
$p_{q_i} =  \partial p/\partial q_i\vert_{\rho,T,(q_j,j\neq i)}$,
$\widehat{h}_p = \partial \widehat{h}/\partial p \vert_{T,q_i}$,
$\widehat{h}_{q_i} = \partial \widehat{h} / \partial q_i \vert_{T,p,(q_j,j\neq i)}$,
$\Gamma_1 = {\partial (\log p)}/{\partial (\log \rho)}\vert_s$, and
$\sigma = p_{T}/\left(\rho c_{pm} p_{\rho} \right)$,
where $c_{pm} = \partial \widehat{h}/\partial T \vert_{p,q_i}$
is the specific heat of moist air at constant pressure.
Here, as in (\ref{eqn:momentumLM}), $\vert_s$ refers to the derivative at
constant entropy.
Notice that both $\alpha$ and $S$ in the divergence constraint
(\ref{eqn:divu_alpha}) depend on the water composition of
moist air, $q_v$ and $q_l$.
Most importantly the constraint on the divergence of the velocity field retains
compressibility effects from stratification as well as latent heat release
and compositional changes.

Summarizing the low Mach number equation set
for moist atmospheric flows in the form we will use, we have
\begin{eqnarray}
\frac{\partial (\rho q_a)}{\partial t}  &=& - \boldsymbol{\nabla} \boldsymbol{\cdot} (\rho q_a \boldsymbol{U})
\label{eq:rhoqaupd} \\
\frac{\partial (\rho q_w)}{\partial t}  &=& - \boldsymbol{\nabla} \boldsymbol{\cdot} (\rho q_w \boldsymbol{U})
\label{eq:rhoqwupd} \\
\frac{\partial (\rho \widehat{h})}{\partial t}  &=& - \boldsymbol{\nabla} \boldsymbol{\cdot} (\rho \widehat{h} \boldsymbol{U})
+ \frac{Dp_0}{Dt} + \rho \mathcal{H}, \label{eq:rhohhatupd} \\
\frac{\partial\boldsymbol{U}}{\partial t} &=& -\boldsymbol{U} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{U}
 -\frac{1}{\rho} \boldsymbol{\nabla} p' - \frac{(\rho - \rho_0)}{\rho} g \boldsymbol{\hat{e}}_z
- \left( \frac{\rho_0}{\rho^2}
\left.\frac{\partial \rho}{\partial p_0}\right \vert_s p' \right)g\boldsymbol{\hat{e}}_z,
\label{eq:velupd} \\
 \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U} & = & -
\alpha \frac{D p_0}{D t} + S, \label{eq:constr}
\end{eqnarray}
where the total mass density is defined as
\begin{equation}
\rho =  \rho q_a + \rho q_w.
\end{equation}
Contrary to the original compressible equation set
(\ref{eqn:cont})--(\ref{eqn:enthalpy}) together
with (\ref{eqn:qa}) and (\ref{eqn:qw}) and a given equation
of state,
the total pressure field, $p$, is now decoupled
from the density and the other state variables
in the enthalpy and momentum equations
(\ref{eq:rhohhatupd}) and (\ref{eq:velupd})
(to be compared with (\ref{eqn:enthalpy}) and (\ref{eqn:momentum})).
Instead of having $\boldsymbol{\nabla} p$ in the momentum equation (\ref{eq:velupd}),
we now have $\boldsymbol{\nabla} p'$, where $p'$ is a perturbational pressure on the
background pressure $p_0$, that is no longer related to the other
state variables
through the equation of state.
That is, pressure is no longer advanced in time through system
(\ref{eq:rhoqaupd})--(\ref{eq:constr}),
and hence acoustic modes are filtered out from the original
compressible governing equations.
In practice, $p'$ is computed in such a way that
the divergence constraint (\ref{eq:constr}) is satisfied at a given time,
analogous to what is done for incompressible flows.
While the equation of state might at first glance appear to be missing
from the low Mach number equation set, it is in fact encapsulated in
the constraint, (\ref{eq:constr}), which was derived by
differentiation of the equation of state along particle paths.
This differs substantially from the anelastic approximation which results
from
replacing the density
by the background density in the continuity equation, which thus
contains no information about the equation of state.

An underlying assumption in the low Mach number approximation is that the
pressure remains close to the background pressure.
Heat release and large--scale convective motions
in a convectively unstable background
can both cause the background state to evolve in time.
As discussed in \cite{almgren:2000} and demonstrated numerically in
\cite{ABRZ:II} for an externally specified heating profile,
if the base state does not evolve in response to heating,
the low Mach number method quickly becomes invalid.
For the small--scale motions of interest here, the base state can
effectively be viewed as independent of time; however, for the sake
of completeness we retain the full time dependence of the base state
in the development of the methodology.

\section{Moist thermodynamics}\label{sec:moist_thermo}
Phase changes and thus variations in
water composition of moist air
are introduced in the flow dynamics
through the divergence constraint (\ref{eq:constr}),
specifically through $\alpha$ and $S$.
These parameters are evaluated at a given time
accounting for the current water composition
in terms of liquid and vapor, and thus accounting for
phase transitions and the current saturation requirements,
given the prognostic state variables.
To define $\alpha$ and $S$ according to (\ref{eqn:S})
in the divergence constraint (\ref{eq:constr}),
we here consider dry air and water vapor to be ideal gases
(see, e.g., \cite{Ooyama1990,Satoh2003,Klemp2007}), and note
that while the low Mach number formulation allows a more general
equation of state, this is a standard assumption in atmospheric modeling.

The partial pressures of dry air and water vapor are then given by
$p_a = \rho q_a R_a T$ and
$p_v = \rho q_v R_v T,$
where $R_a = R/M_a$ and $R_v = R/M_v$
are the specific gas constants for dry air and water vapor, respectively,
with $R$ the universal gas constant for ideal gases,
and $M_a$ and $M_v$ the molar masses of dry air and water, respectively.
The sum of the partial pressures defines the total pressure of a parcel,
\begin{equation}\label{eqn:pres}
p = p_a + p_v = \rho R_m T,
\end{equation}
where the specific gas constant of moist air is defined as
\begin{equation*}\label{eqn:R_m}
R_m = q_aR_a + q_vR_v = \left(\frac{q_a}{M_a} + \frac{q_v}{M_v}\right) R.
\end{equation*}
Additionally, the specific heat capacities at constant
pressure can be defined as
\begin{equation*}\label{eqn:cpm}
c_{pa} = c_{va} + R_a, \quad
c_{pv} = c_{vv} + R_v, \quad
c_{pm} = c_{vm} + R_m,
\end{equation*}
for dry air, water vapor, and moist air, respectively.
A common approximation in cloud models
is to neglect the specific heats of water vapor and liquid water
(see, e.g., \cite{Bryan2002} for a study and discussion on this topic).
This was also assumed in the moist pseudo--incompressible model
of \cite{ONeill2013}.
Here we consider specific heats for all three phases.

With this choice for the equation of state for moist air
(eq.~(\ref{eqn:pres})),
we have
\begin{equation}\label{eqn:alphapres}
\alpha = \frac{1}{\gamma_m p_0},
\qquad
S =
\left[
\frac{1}{(\epsilon q_a +  q_v)}
- \frac{L_e}{c_{pm} T}
\right]
\frac{e_v}{\rho}
+ \left[ \frac{1}{c_{pm} T}\right] \mathcal{H},
\end{equation}
where $\gamma_m= c_{pm} / c_{vm}$ is the
isentropic expansion factor of moist air,
$\epsilon = R_a/R_v=M_v/M_a$,
and the latent heat of vaporization, $L_e$,
is defined as
\begin{equation}\label{eqn:Le}
L_e = E_{0v} + R_v T + (c_{vv} - c_{vl})(T - T_{\mathrm{trip}}).
\end{equation}
Note that both $\alpha$ and $S$ depend
on the composition of the moist mixture even
when there is no phase transition, that is,
when $e_v =0$. We can now replace
$\Gamma_1$ as used in (\ref{eqn:divu_alpha})--(\ref{eqn:S})
by $\gamma_m$ for moist atmospheric flows.

The saturation vapor pressure
with respect to liquid water, $p_v^*$, is defined
by the following Clausius--Clapeyron relation:
\begin{equation}\label{eqn:pvstar}
p_v^*(T) = p_{\mathrm{trip}} \left(\frac{T}{T_{\mathrm{trip}}}\right)^{\alpha_v}
\exp \left [ \beta_v
\left( \frac{1}{T_{\mathrm{trip}}}-\frac{1}{T} \right) \right ],
\end{equation}
with constants $\alpha_v$ and $\beta_v$ given, for instance, by
\begin{equation}\label{eqn:constant_pvstar}
\alpha_v = \frac{c_{pv}-c_{vl}}{R_v}, \qquad
\beta_v = \frac{E_{0v} -(c_{vv}-c_{vl})T_{\mathrm{trip}}}{R_v},
\end{equation}
as in \cite{Romps2008}.
The saturated mass fraction of water vapor,
$q_v^*$, can be then computed from the EOS,
given in this case by
\begin{equation}\label{eqn:qvstar}
q_v^*(\rho,T) = \frac{p_v^*}{\rho R_v T},
\end{equation}
or equivalently by
\begin{equation}\label{eqn:qvstar_2}
q_v^*(q_a,p,T) = \frac{\epsilon q_a p_v^*}{p - p_v^*}.
\end{equation}
Following \cite{Ooyama1990,Satoh2003}, we assume
that air parcels cannot be supersaturated,
and thus $q_v$ cannot exceed its saturated value, $q_v^*$.
The water mass fractions,
$q_v$ and $q_l$, as well as the temperature
$T$, which are not explicitly computed in
(\ref{eq:rhoqaupd})--(\ref{eq:constr}),
are obtained by solving the following
nonlinear system of equations:
\begin{equation}\label{eqn:solveT}
\left.
\begin{array}{l}
\widehat{h} =
c_{pm}(q_a,q_v,q_l)  \left(T - T_{\mathrm{trip}}  \right) +
R_m(q_a,q_v) T_{\mathrm{trip}} +
q_v E_{0v}, \\[1.5ex]
q_v = \min\left[q_v^*(\rho,T),q_w\right],  \\[1.5ex]
q_l = q_w - q_v,
\end{array}
\right\}
\end{equation}
which satisfies the Clausius--Clapeyron relation
and the saturation requirements, given $(\rho,\widehat{h},q_a,q_w)$.

No approximation has been introduced so far to evaluate
the water composition and therefore the thermodynamic
properties of the moist fluid.
We now need to estimate the evaporation rate, $e_v$,
that is required to quantify the latent heat release
in the divergence constraint.
Considering that there is no analytical expression
for such a rate and that we cannot derive it from an approximate
value of vapor water as previously discussed in the Introduction,
we introduce the following approach.
Taking into account that
the evaporation rate
$e_v$ is different from zero
only when a change of phase is taking place, that is, when
$q_v = q_v^*$,
we can rewrite the conservation equation for $q_v$ (eq.~(\ref{eqn:qv})) as
\begin{equation}\label{eqn:ev_qvstar}
e_v =
\frac{\partial \left( \rho q_v^* \right)}{\partial t}
+ \boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho q_v^* \boldsymbol{U} \right) =
\rho \frac{D q_v^*}{D t}.
\end{equation}
We show in Appendix \ref{app:evp_rate}
that if $q_v = q_v^*$, then
\begin{equation}\label{eqn:ev_hhat_rho_a}
e_v = {\mathcal{A}}_{e} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U} +
{\mathcal{B}}_{e} \frac{D p_0}{D t} + {\mathcal{C}}_{e} \rho \mathcal{H},
\end{equation}
where
parameters
${\mathcal{A}}_{e}$, ${\mathcal{B}}_{e}$, and ${\mathcal{C}}_{e}$
are in general functions of $(\rho,T,q_a,q_v,q_l)$.
Otherwise, $e_v=0$ whenever $q_v < q_v^*$.
In \cite{LipHem82}, the term
$\boldsymbol{\nabla} \boldsymbol{\cdot} ( \rho q_v^* \boldsymbol{U})$ in (\ref{eqn:ev_qvstar})
is replaced with
$\boldsymbol{\nabla} \boldsymbol{\cdot} ( \rho q_{va} \boldsymbol{U} )$,
where $( \rho q_{va} )$ stands for the
water vapor content advected without considering
the evaporation rate in (\ref{eqn:qv}):
$\partial ( \rho q_{va}) / \partial t +
\boldsymbol{\nabla} \boldsymbol{\cdot} \left( \rho q_{va} \boldsymbol{U} \right) = 0$.

Finally, using (\ref{eqn:ev_qvstar})
to estimate $e_v$
in $S$ (eq. (\ref{eqn:alphapres}))
involves having a low Mach divergence constraint
where the source term depends on
the pressure and the velocity field, i.e.,
\begin{equation}\label{eqn:divu_alpha_dependency}
\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U}
+ \alpha \frac{D p_0}{D t} =
S\left(\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U} , \frac{D p_0}{D t} \right).
\end{equation}
A simpler expression can be obtained
by rearranging the terms in (\ref{eqn:divu_alpha_dependency}),
as shown in Appendix \ref{app:mod_cons},
which leads to
a modified divergence constraint:
\begin{equation}\label{eqn:divu_ev}
\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U}
+ \widetilde{\alpha} \frac{D p_0}{D t} = \widetilde{S},
\end{equation}
similar to the general low Mach divergence constraint
(\ref{eqn:divu_alpha}),
with
\begin{equation}\label{eqn:alphatilde}
\widetilde{\alpha} = \frac{1}{\widetilde{\gamma}_m p_0},
\qquad \widetilde{S} = \widetilde{\sigma} \mathcal{H},
\end{equation}
both depending on
$(\rho,T,q_a,q_v,q_l)$.
In particular
if no heat source is considered,
we have that $\widetilde{S} = 0$ and
all the information related to phase change
is included in $\widetilde{\alpha}$.
Now
$\Gamma_1$
in the general low Mach constraint
(\ref{eqn:divu_alpha})--(\ref{eqn:S})
is given by
a modified isentropic expansion factor of moist air:
$\widetilde{\gamma}_m$
(eq. (\ref{eqn:gammatilde1}) in Appendix \ref{app:mod_cons}).
Both divergence constraints, (\ref{eqn:divu_alpha_dependency})
and (\ref{eqn:divu_ev}), are analytically equivalent.

\section{Numerical methodology}\label{sec:numer}
In order to solve the low Mach number equation set
(\ref{eq:rhoqaupd})--(\ref{eq:constr}) we begin with the MAESTRO code,
which was originally designed to simulate
low Mach number stratified, reacting flows in astrophysical settings
\cite{ABRZ:I,ABRZ:II,ABNZ:III,multilevel}.

We recall that for the moist equation of state considered here
we replace $\Gamma_1$ in the original
MAESTRO notation by $\gamma_m$. Since $\gamma_m$ in general varies in
space and time, the solution procedure in the original
MAESTRO algorithm replaces
$\gamma_m(\boldsymbol{x})$ by  $\overline{\gamma}_m(z)$,
that is,
$\alpha (z) = 1/(\overline{\gamma}_m p_0)$
in the divergence constraint
(\ref{eqn:divu_alpha_dependency})
where
$\overline{\gamma}_m$
is the lateral average
of
$\gamma_m$,
\begin{equation}\label{eq:lateral_average}
\overline{\gamma}_m(z) = \frac{1}{A(\Omega_H)} \int_{\Omega_H} \gamma_m(\boldsymbol{x}) \; d\Omega,
\end{equation}
where ${\mathrm A}(\Omega_H) = \int_{\Omega_H} d\Omega,$ $\Omega_H$ is
a region at constant height for the plane--parallel atmosphere,
and $d\Omega$ represents an area measure.
The same follows for
$\widetilde{\gamma}_m$ in the modified divergence constraint
(\ref{eqn:divu_ev}):
$\widetilde{\alpha} (z)= 1/(\overline{\widetilde{\gamma}}_m p_0)$.
The introduction of an averaged
$\gamma_m$ (or $\widetilde{\gamma}_m$)
allows us to rewrite
the constraint
(\ref{eqn:divu_alpha_dependency})
(or (\ref{eqn:divu_ev})) as
\begin{equation}\label{constraint_eq}
\boldsymbol{\nabla} \boldsymbol{\cdot} (\beta_0 \boldsymbol{U}) = \beta_0 \left (S - \alpha \frac{\partial p_0}{\partial t}\right),
\end{equation}
as shown in \cite{ABRZ:I} (Appendix B),
with
\begin{equation*}\label{eq:beta_0}
\beta_0(z,t) = \beta(0,t) \exp \left ({\int_0^z
\alpha (z)
               \frac{\partial p_0}{\partial z^\prime} \, dz^\prime} \right ),
\end{equation*}
(with $\widetilde{S}$ and $\widetilde{\alpha}$ instead of $S$ and $\alpha$
if the modified divergence (\ref{eqn:divu_ev}) is considered).
Moreover,
the momentum equation (\ref{eq:velupd}), including the
correction term introduced by \cite{Klein2012},
can be equivalently recast as
\begin{equation}\label{eqn:momentumLM_beta}
\frac{\partial\boldsymbol{U}}{\partial t} = -\boldsymbol{U} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{U}
 -\frac{\beta_0}{\rho} \boldsymbol{\nabla} \left(\frac{p'}{\beta_0} \right)
- \frac{(\rho - \rho_0)}{\rho} g \boldsymbol{\hat{e}}_z,
\end{equation}
as derived in \cite{VLBWZ:2013}, and discussed in \cite{ABNZ_energy}.

MAESTRO thus solves the low Mach equation set
(\ref{eq:rhoqaupd})--(\ref{eq:rhohhatupd})
with the momentum equation (\ref{eqn:momentumLM_beta})
and the divergence constraint (\ref{constraint_eq}).
A predictor--corrector formalism is implemented
to solve the flow dynamics, as detailed in \cite{multilevel}.
In the predictor step
an estimate of the expansion of the base state is first computed,
and then an estimate of the flow variables at the new time level.
In the corrector step
results of the previous state update are used to compute a new base
state update as well as the full state update.

Since we are considering the time evolution of total water
(\ref{eq:rhoqwupd})
and the definition of enthalpy of moist air (\ref{eqn:hhat})
involves a conservation equation (\ref{eq:rhohhatupd})
without source terms related to phase change,
all the information related to variations in the moist
composition and latent heat release
is contained in the divergence constraint
(\ref{constraint_eq}).
Here
$q_v$ and $q_l$, as well
as the local temperature $T$, must be computed point--wise
in order to define $\alpha$, $S$, and $\beta_0$
(or $\widetilde{\alpha}$, $\widetilde{S}$, and $\beta_0$).
Point--wise values of $(q_v,q_l,T)$ are thus diagnostically recovered
by solving the nonlinear system (\ref{eqn:solveT}),
given the values of $(\rho,\widehat{h},q_a,q_w)$ at a given time;
we use the Newton solver described in \cite{CASTROmoist}.

We consider two approaches for handling phase transitions
depending on which divergence constraint is used
to define (\ref{constraint_eq})
in the numerical implementation:
(\ref{eqn:divu_alpha_dependency})
or (\ref{eqn:divu_ev}).
In the first case,
which uses (\ref{eqn:divu_alpha_dependency}),
the evaporation rate
is evaluated according to (\ref{eqn:ev_hhat_rho_a})
and introduced in the source term $S$ of the constraint.
As previously remarked, there is a dependence
of $S$ on the velocity field and the base state pressure.
Consequently, approximate or lagged values of $\boldsymbol{U}$ and $p_0$ are used to
estimate $e_v$ during the prediction step, which
is later recomputed with the updated values of
$\boldsymbol{U}$ and $p_0$ during the correction step.
Notice, however, that during both steps
the moist composition and hence phase transitions are
diagnostically recovered based on the current values
of $(\rho,\widehat{h},q_a,q_w)$.
The
second approach
consists of considering
the modified divergence constraint (\ref{eqn:divu_ev}),
in which case there is no need to estimate the evaporation rate
(\ref{eqn:ev_hhat_rho_a}).
Parameters $\widetilde{\alpha}$, $\widetilde{S}$, and $\beta_0$ in
(\ref{constraint_eq}) are computed with
$(\rho,T,q_a,q_v,q_l)$ given the current values
of $(\rho,\widehat{h},q_a,q_w)$ throughout the predictor--corrector
scheme for the flow dynamics.

Notice that considering the
lateral average of $\gamma_m(\boldsymbol{x})$ through
(\ref{eq:lateral_average}) is the only approximation
introduced in the present model in terms of thermodynamic
properties of the moist fluid.
The replacement of $\Gamma_1(\boldsymbol{x},t)$ by its
lateral average
$\overline{\Gamma}_1 (z,t)$ was demonstrated
in \cite{ABNZ:III} to have little effect on the astrophysical
flows studied there.
In the case of moist atmospheric flows,
$\Gamma_1$ is given by $\gamma_m$
which varies according to the local
moist air composition at a given time and position.
If the modified divergence constraint (\ref{eqn:divu_ev})
is considered,
$\Gamma_1$ is given by $\widetilde{\gamma}_m$ which varies
not only according to the moist air composition
but also the local values of $(\rho,T)$.
We can thus
consider the effects of localized variations in
$\Gamma_1$
following \cite{ABNZ:III},
by writing $\Gamma_1(\boldsymbol{x},t) = \overline{\Gamma}_1(z,t) + \delta \Gamma_1(\boldsymbol{x},t)$,
and hence,
\begin{equation*}
\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U}
 = - \frac{1}{(\overline{\Gamma}_1 + \delta \Gamma_1) p_0}
  \frac{Dp_0}{Dt} +
S,
\end{equation*}
instead of (\ref{eq:constr}).
Assuming that $\delta \Gamma_1 \ll \overline{\Gamma}_1$, we then have
\begin{equation}\label{eq:gammafull}
\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U} =
- \frac{1}{\overline{\Gamma}_1 p_0} \frac{Dp_0}{Dt}
+ S  +
\frac{\delta \Gamma_1}{\overline{\Gamma}_1^2 p_0}
\frac{Dp_0}{Dt},
\end{equation}
which leads to
\begin{equation}\label{constraint_eq_deltagamma1}
\boldsymbol{\nabla} \boldsymbol{\cdot} (\beta_0 \boldsymbol{U}) =
\beta_0 \left (S - \alpha \frac{\partial p_0}{\partial t}
+
\frac{\delta \Gamma_1}{\overline{\Gamma}_1^2 p_0}
\frac{Dp_0}{Dt}
\right).
\end{equation}
We will refer to the $\delta \Gamma_1$--correction whenever
(\ref{constraint_eq_deltagamma1}) is considered instead of
(\ref{constraint_eq}).
Notice that
$Dp_0/D t = \partial p_0/\partial t + \boldsymbol{U} \boldsymbol{\cdot} \boldsymbol{\nabla}p_0$,
and therefore, to solve (\ref{constraint_eq_deltagamma1})
a lagged $\boldsymbol{U}$ is used in evaluating the right--hand side,
as described in \cite{ABNZ:III}.

\section{Numerical simulations}\label{sec:simul}
To validate the low Mach number method we compare simulations
using the low Mach number method to simulations using a
fully compressible approach.
The first problem we consider is a benchmark problem
presented in \cite{Bryan2002} for moist flows
in an isentropically stratified background.
We investigate both the first and second approach to implementing phase
transitions, and find very good agreement between the
low Mach number and compressible simulations.
We then consider a problem described in \cite{GrabowskiClark1991}
for non--isentropic background states and both saturated and only
partially saturated media, which was also
studied in \cite{CASTROmoist}.
Finally, we show a comparison of three dimensional simulations.

\subsection{Isentropic background state}

Two--dimensional simulations of a benchmark test case are investigated in
\cite{Bryan2002}.
The computational domain is $10\,$km high and $20\,$km wide;
the background atmosphere is isentropically stratified,
at a uniform wet equivalent potential temperature, $\theta_{e0}=320\,$K,
and is saturated, that is, $q_v = q_v^*$ and $q_l >0$
everywhere in the domain.  A warm perturbation in potential temperature
is introduced in the domain, which leads to a rising thermal.
Here we use the configuration and parameters as given in \cite{CASTROmoist}.
The normal velocity is set equal to zero on the side and bottom
boundaries; at the top boundary the normal velocity is set equal to the
velocity corresponding to the base state evolution.  (See \cite{ABRZ:II}
for further detail about the base state evolution.)
Homogeneous Neumann boundary conditions are used in solving for the
perturbational pressure.
For the thermodynamic variables, we impose
homogeneous Neumann boundary conditions on the horizontal sides;
the background state is reconstructed by extrapolation at
vertical boundaries
and the full state is set equal to the base one there,
in order to determine the corresponding fluxes.
The time step for the low Mach number computations
is dictated by the advective CFL number
which is based on the velocity but not the sound speed;
we set this to $0.9$.

In the first approach to account for phase transitions
the divergence constraint is given by
(\ref{eqn:divu_alpha_dependency})
with $\alpha$ and $S$ defined by (\ref{eqn:alphapres}),
and the evaporation rate  (\ref{eqn:ev_hhat_rho_a}).
For this particular problem,
$\mathcal{H}=0$ in $S$, and hence it is not necessary
to compute ${\mathcal{C}}_{e}$ in (\ref{eqn:ev_hhat_rho_a}).
For a $256\times 128$ grid
of similar spatial resolution to that considered
in \cite{Bryan2002},
the maximum and minimum values for
the perturbational wet equivalent potential temperature
($\theta'_e = \theta_e - \theta_{e0}$) are
$4.05402\,$K and $-0.28931\,$K, respectively,
compared with the original
$4.09521\,$K and $-0.305695\,$K in \cite{Bryan2002}.
Additionally, our computation yields
$15.8199\,$m s$^{-1}$ and $-9.45586\,$m s$^{-1}$,
for the maximum and minimum vertical velocities, respectively,
to be compared with  $15.7130\,$m s$^{-1}$ and $-9.92698\,$m s$^{-1}$
in \cite{Bryan2002}.  The low Mach number code takes roughly
a factor of six less computational time than the compressible
simulation at the same resolution run with the
code described in \cite{CASTROmoist} at an acoustic CFL number
of $0.9$.

The second approach does not explicitly estimate $e_v$,
but rather considers a
modified divergence constraint (eq.~(\ref{eqn:divu_ev}))
with $\widetilde{\alpha}$ and $\widetilde{S}$ defined by (\ref{eqn:alphatilde}).
Since for this particular problem, $\mathcal{H}=0$, we have $\widetilde{S}=0$.
For a $256\times 128$ grid, the maximum and minimum values for
the perturbational wet equivalent potential temperature $\theta'_e$
are now given by $4.12911\,$K and $-0.30190\,$K, respectively.
The time steps taken with the second approach are
slightly larger than with the first approach, and
the overall computational time is roughly $10$ to $15\,$\%
less with this approach than with the first approach.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.49\textwidth]{Bryant_moist_comp_ny512256evp}
\includegraphics[width=0.49\textwidth]{Bryant_moist_comp_ny512256gamma}
\includegraphics[width=0.49\textwidth]{Bryant_moist_comp_ny512256correvp}
\includegraphics[width=0.49\textwidth]{Bryant_moist_comp_ny512256corrgamma}
\caption{Comparison with the compressible solution for the moist thermal simulations
at $1000\,$s.
Perturbational potential temperature is shown with contours every $1\,$K:
first (left) and second approach (right)
The simulations in the bottom row include the $\delta \Gamma_1$--correction.
}
\label{fig:compressible_comp}
\end{figure}

Figure~\ref{fig:compressible_comp} shows results from computations
of the problem described above;  in this figure $\theta'_e$ from the low
Mach number simulations is overlaid on the reference compressible solution,
which was computed with the fully coupled compressible solver
described in \cite{CASTROmoist} and validated against results from
\cite{Bryan2002}.
The low Mach number simulations were carried out on a uniform grid
of $512\times 256$;  the compressible solution was performed on a
finer grid of $1024\times 512$ in order to get a more accurate
reference solution.   In the top row of Figure~\ref{fig:compressible_comp}
we see that the low Mach number computation using the
first approach agrees well with the compressible solution. The
computation with the second approach, however, shows a significant
height difference of the top of the thermal.
In the bottom row of Figure~\ref{fig:compressible_comp} we
see that  accounting for the local variation of $\gamma_m(\boldsymbol{x})$
and $\widetilde{\gamma}_m (\boldsymbol{x})$ through the the $\delta \Gamma_1$--correction described
in \S\ref{sec:numer} improves the fidelity of both approaches.
With the first approach we see the improvement mostly in the tips;
with the second approach the height of the thermal is noticeably improved.
The fact that including the $\delta \Gamma_1$--correction
impacts the second approach more dramatically than the first is consistent
with the observation that replacing $\gamma_m(\boldsymbol{x})$ by $\overline{\gamma}_m$
in the first approach meant neglecting local variations in $\gamma_m(\boldsymbol{x})$
from $-8\times 10^{-5}$ to $6.5\times 10^{-4}$, while replacing
$\widetilde{\gamma}_m  (\boldsymbol{x})$ by $\overline{\widetilde{\gamma}}_m$
in the second approach meant neglecting local variations in $\widetilde{\gamma}_m  (\boldsymbol{x})$
at least an order of magnitude larger, i.e. from $-6.7\times 10^{-3}$ to $1.5\times 10^{-3}.$
This behavior follows naturally from the fact that in the modified
divergence constraint (eq.~(\ref{eqn:divu_ev})),
$\widetilde{\gamma}_m (\boldsymbol{x})$ accounts not only for the varying water composition but
also for all the latent heat released during
phase transitions (see Appendix \ref{app:mod_cons});
therefore, local variations in $\widetilde{\gamma}_m (\boldsymbol{x})$
are expected to be more important.

The formalism adopted for the present low Mach model considers
straightforwardly the effects of the specific heats of liquid water
and water vapor in the evaluation of the thermodynamic
properties of the moist fluid, and in particular in the definition of the
internal energy and specific enthalpy of moist air,
(\ref{eqn:ehat}) and (\ref{eqn:hhat}).
The latter is not possible within the moist pseudo-incompressible
model introduced in \cite{ONeill2013}, which relies on a
$\theta - \pi$
(potential temperature -- Exner pressure) formalism defined
with the specific heat of dry air.
Neglecting the specific heats of liquid water
and water vapor in our model amounts to consider the
Equation set A investigated in \cite{Bryan2002} for the
same isentropic background configuration.
This is also the benchmark problem considered in
\cite{ONeill2013} to validate their model.
Figure~\ref{fig:pseudo_comp} shows the low Mach number
solution when the specific heats of water are neglected
and contrasts it to the general case where all specific
heats are taken into account, like in Figure~\ref{fig:compressible_comp}.
In \cite{Bryan2002},
the maximum and minimum values for
$\theta'_e$ are
$2.13647\,$K and $-1.39627\,$K, respectively.
As discussed in \cite{Bryan2002}, the impact of approximating
thermodynamic properties of moist air can lead to important
variations for certain configurations, as illustrated in this example.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.49\textwidth]{PI_bubble}
\includegraphics[width=0.49\textwidth]{CompLM_PI_ny128_zoomout}
\caption{Solution to the isentropic background problem neglecting
the specific heats of liquid water and vapor water.
Left: perturbational potential temperature shown with contours every $0.5\,$K;
negative contours are dashed.
Contrast to Fig.~6(a) in \cite{Bryan2002}.
Right: comparison with the low Mach number solution shown in
Figure \ref{fig:compressible_comp} (bottom left);
perturbational potential temperature is shown with contours every $1.5\,$K.
}
\label{fig:pseudo_comp}
\end{figure}

\subsection{Non--isentropic background state}\label{sec:non-isen}
We next consider a non--isentropic background state
given by the hydrostatically balanced profiles
in \cite{Clark1984} (eq.~2).  For the following computations
we define a computational domain $4\,$km high and wide, with periodic horizontal
boundary conditions.
The boundary conditions at the top and bottom boundaries are as described
for the isentropic case,
except for the thermodynamic variables which
are extrapolated to determine the corresponding fluxes.
Again we use the configuration and parameters as given in \cite{CASTROmoist}.
All simulations with the low Mach formulation were performed on a uniform
grid of $256\times 256$.  As before, the reference compressible solutions
were computed on a finer grid, here $1024\times 1024$.

The initial distributions of water vapor and liquid water
in the atmosphere are set by the
relative humidity in the atmosphere, RH,
measured in percentage
and defined as RH$~= (p_v/p_v^*)\times 100$.
In particular if RH$_0 <100\,$\%,
then no liquid water should be initially present in the
atmosphere in order to guarantee the thermodynamic
equilibrium of the initial state.  Following \cite{CASTROmoist},
we consider in this study two cases:
first, a saturated medium with RH$_0 =100\,$\%
and $q_l>0$ everywhere in the domain as in the
isentropic background problem;
and a second configuration with RH$_0 =20\,$\%,
and hence, no liquid water in the initial background state.

Let us consider the first configuration
with an initially saturated environment.
As described in \cite{CASTROmoist},
we initially introduce a warm perturbation of temperature.
Figure~\ref{fig:Grabowski_satmoist}
shows solutions obtained with the low Mach formulation
using the first and second approach for the divergence constraint,
as well as the compressible reference solution.
Solutions are very similar in all three cases
even though the low Mach approximations yield thermals
rising slightly faster.  In contrast to the previous
example, introducing the $\delta \Gamma_1$--correction
does not visibly change the results.
For a $256\times 256$ grid and a CFL factor of $0.9$,
the time steps for the low Mach approximations are about
$1.4\,$s compared to $0.04\,$s with the compressible formulation.
The low Mach number simulation takes roughly a
factor of $13$ less computational time than the compressible simulation.
\begin{figure}[!ht]
\noindent
\includegraphics[width=0.32\textwidth]{Grabowski_satmoist_ny1024_CAS_ql}
\includegraphics[width=0.32\textwidth]{Grabowski_satmoist_ny256_case1evp_ql}
\includegraphics[width=0.32\textwidth]{Grabowski_satmoist_ny256_case1gamma_ql}\\
\includegraphics[width=0.32\textwidth]{Grabowski_satmoist_ny1024_CAS_vy}
\includegraphics[width=0.32\textwidth]{Grabowski_satmoist_ny256_case1evp_vy}
\includegraphics[width=0.32\textwidth]{Grabowski_satmoist_ny256_case1gamma_vy}
\caption{Initially saturated, non--isentropic background state.
Liquid water mass fraction (top) and vertical velocity (bottom) are shown
for the reference compressible (left), and low Mach number solution at $300\,$s
using the first (center) and second (right) approach.
Contours are every $10^{-4}$ (top) and $1\,$m s$^{-1}$ (bottom);
negative contours are dashed.}
\label{fig:Grabowski_satmoist}
\end{figure}

For the second configuration with RH$_0 =20\,$\%,
we consider the same previous temperature perturbation
and an additional circular perturbation
in the relative humidity, which is set to $100\,$\%,
with a transition layer, as detailed in \cite{CASTROmoist}.
Initially there is no liquid water in the domain.
Like Figure~\ref{fig:compressible_comp}, Figure~\ref{fig:Grabowski_nnsatmoist}
compares the two low Mach number solutions to the reference compressible solution;
the top row shows the results using laterally averaged $\gamma_m$ and $\widetilde{\gamma}_m,$ while
the bottom row shows the results using the $\delta \Gamma_1$--correction.
Here the deviation of $\gamma_m(\boldsymbol{x})$ from $\overline{\gamma}_m (z)$ in the first approach
ranges from $-4.2\times 10^{-3}$ to $6.9\times 10^{-4},$
which is considerably smaller than
the deviation of $\widetilde{\gamma}_m (\boldsymbol{x})$ from $\overline{\widetilde{\gamma}}_m (z)$ in the second
approach, which ranges from $-0.17$ to $3.3\times 10^{-2}$.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.49\textwidth]{Grabowski_nnsatmoist_comp_ny1024256evp}
\includegraphics[width=0.49\textwidth]{Grabowski_nnsatmoist_comp_ny1024256gamma}
\includegraphics[width=0.49\textwidth]{Grabowski_nnsatmoist_comp_ny1024256correvp}
\includegraphics[width=0.49\textwidth]{Grabowski_nnsatmoist_comp_ny1024256corrgamma}
\caption{Non--isentropic background state with a saturated perturbation.
Moist specific entropy is shown with contours every $50\,$J kg$^{-1}$ K$^{-1}$.
The low Mach number solution at $300\,$s (red)
using the first (left) and second (right) approach overlays the reference compressible solution (black).
On the bottom are simulations that use the $\delta \Gamma_1$--correction.
}
\label{fig:Grabowski_nnsatmoist}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.49\textwidth]{Grabowski_nnsatmoist_comp_ny1024256corrgamma_2ndorder}
\caption{Non--isentropic background state with a saturated perturbation.
Moist specific entropy is shown with contours every $50\,$J kg$^{-1}$ K$^{-1}$.
The low Mach solution at $300\,$s (red)
using the $\delta \Gamma_1$--correction given by (\ref{eq:gammafull_2ndorder})
overlays the reference compressible solution (black).}
\label{fig:Grabowski_nnsatmoist_2ndorder}
\end{figure}

Better agreement can be seen in Figure~\ref{fig:Grabowski_nnsatmoist_2ndorder},
where a term of order $\delta \Gamma_1^2$
is also considered in (\ref{eq:gammafull}) for the
$\delta \Gamma_1$--correction:
\begin{equation}\label{eq:gammafull_2ndorder}
\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U} =
- \frac{1}{\overline{\Gamma}_1 p_0} \frac{Dp_0}{Dt}
+ S  +
\frac{\delta \Gamma_1}{\overline{\Gamma}_1^2 p_0}
\frac{Dp_0}{Dt}
- \frac{\delta \Gamma_1^2}{\overline{\Gamma}_1^3 p_0}
\frac{Dp_0}{Dt}.
\end{equation}
The time steps used in the low Mach computations are
about $2.1\,$s, compared to $0.04\,$s with the
compressible formulation.
The low Mach number simulation takes roughly a
factor of $15$ less computational time than the compressible simulation.

\subsection{Three--dimensional simulation}
Finally, we consider two interacting thermals
rising in a three--dimensional, non--isentropically
stratified background in a domain $10\,$km on a side
and $15\,$km high.
The background state is defined as in
\S\ref{sec:non-isen},
with a relative humidity of RH$_0 =20\,$\%
and no liquid water in the initial configuration.
The following temperature perturbation is then introduced:
\begin{equation*}\label{eqn:temp_pert3D}
T' = 6 \cos ^2 \left( \frac{\pi L_1}{2} \right)
   + 6 \cos ^2 \left( \frac{\pi L_2}{2} \right),
\end{equation*}
where
$L_1 = \min (1, r_1), \;\;  L_2 = \min (1, r_2),$
and
\begin{equation*}\label{eq:pert_L3D_1}
r_1 = \frac{1}{4} \sqrt{\left({x-x_1} \right)^2
+\left({y-y_1} \right)^2 + \left({z-z_1} \right)^2 } \; ,
\end{equation*}
\begin{equation*}\label{eq:pert_L3D_2}
r_2 = \frac{1}{3}\sqrt{\left({x-x_2} \right)^2
+\left({y-y_2} \right)^2 + \left({z-z_2} \right)^2 },
\end{equation*}
with
$x_1 = y_1 = 5\,$km, $z_1 = 7.5\,$km
and
$x_2 = y_2 = 7\,$km,
$z_2 = 7.5\,$km.
Within the regions where the temperature is perturbed, we also
perturb the relative humidity by setting
RH equal to $50\,$\% for
$r_1 < 3\,$km and $r_2 < 2\,$km,
for each initial thermal,
with corresponding transition layers:
\begin{equation*}\label{eqn:rh_pert3D_1}
 {\rm RH} = {\rm RH}_0 + (50 - {\rm RH}_0)
\cos ^2 \left( \frac{\pi }{2}
\left[r_1-3\right] \right),
\
3 \leq r_1 \leq 4,
\end{equation*}
\begin{equation*}\label{eqn:rh_pert3D_2}
 {\rm RH} = {\rm RH}_0 + (50 - {\rm RH}_0)
\cos ^2 \left( \frac{\pi }{2}
\left[r_2-2\right] \right),
\
2 \leq r_2 \leq 3.
\end{equation*}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.49\textwidth]{CASt500crop}
\includegraphics[width=0.49\textwidth]{MAEt500crop}
\includegraphics[width=0.49\textwidth]{CASt1000crop}
\includegraphics[width=0.49\textwidth]{MAEt1000crop}
\caption{Two interacting thermals in a three--dimensional
non--isentropic background state.
Isosurfaces of liquid water for the reference compressible solutions
(left) and the low Mach number ones (right)
at times $500\,$s
($q_l=3.073\times 10^{-5}$)
(top) and $1000\,$s (bottom)
($q_l=3.214\times 10^{-5}$).
The low Mach number solver uses the modified divergence constraint
(\ref{eqn:divu_ev}) (second approach)
with the $\delta \Gamma_1$--correction given by (\ref{eq:gammafull_2ndorder}).}
\label{fig:Grabowski_3D}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.49\textwidth]{Image_HorizAvg_ql}
\includegraphics[width=0.49\textwidth]{Image_HorizAvg_ql_zoom}
\caption{Horizontal budgets of liquid water, $\overline{q_l}(z)$,
computed with (\ref{eq:lateral_average}) at $t = 200\,$s, $500\,$s and $1000\,$s.
Reference compressible and low Mach number
solutions are represented with solid and dashed lines, respectively.
More details can be appreciated in the zoomed region (right).}
\label{fig:budget_ql}
\end{figure}

We consider the low Mach number formalism using the second
approach (modified divergence constraint (\ref{eqn:divu_ev}))
to numerically implement phase transitions,
with the $\delta \Gamma_1$--correction given by (\ref{eq:gammafull_2ndorder}).
As before, the low Mach number solutions are contrasted
to reference compressible solutions.  For a uniform grid of
size $256 \times 256 \times 384$, the time step in the low Mach number simulation
is approximately $3\,$s, compared to $0.1\,$s in the compressible simulation.
For this particular problem, the total run time of the
low Mach number simulation is roughly a factor of $5$ less
than that of the compressible simulation.
Figure \ref{fig:Grabowski_3D} illustrates the formation
of liquid water as computed with both formulations.
The relative difference between the maximum values of $q_l$ in
the compressible and the low Mach number solutions is roughly $8\,$\%
at $t = 500\,$s and $4\,$\% at $t=1000\,$s, the two times shown in
Figure \ref{fig:Grabowski_3D}.
Figure \ref{fig:budget_ql} shows the horizontal budgets of liquid water,
$\overline{q_l}(z)$, computed using formula (\ref{eq:lateral_average}),
at $t = 200$s and at the two simulation times
shown in Figure \ref{fig:Grabowski_3D}.
For $t = 200\,$s practically the same solution is recovered
with both formulations, with a
relative difference of $0.2\,$\% between
the maximum values of $q_l$.
Recalling that $q_l$ is diagnostically recovered in both
the compressible and the low Mach number approach,
we can track the formation of liquid water by computing it
after each time step for comparison purposes.
Local values of $q_l$ larger than $10^{-10}$ appear
after $111\,$s and $112.5\,$s for the compressible
and low Mach number simulations, respectively.
The Mach number $M$ associated with this particular problem
remains lower than $0.05$ during the entire numerical simulation.

\section{Summary}
We have presented a new low Mach number model for moist
atmospheric flows with a general equation of state,
based on the low Mach number model for stratified reacting flows
introduced in \cite{ABNZ:III}.
In our model we consider only reversible processes,
namely water phase changes as in \cite{ONeill2013},
using an exact Clausius--Clapeyron formula for moist thermodynamics
and considering the effects of the specific heats of water
and the temperature dependency of the latent heat.
A set of invariant variables
was used as prognostic variables in the equations of motion,
including in particular
the total water content and a specific enthalpy of moist air
that accounts for the contribution of both sensible and latent heats.
The evolution equations
can thus be solved without needing
to estimate or neglect source terms related to phase change during
the time integration.
The mass fractions of water vapor and liquid water
are diagnostically recovered as required during a time step
by imposing the saturation requirements of an atmosphere at thermodynamic
equilibrium.  The latter is an important property since
updating
the solution while
ignoring the varying water composition,
may  negatively
impact the accuracy of the moist flow dynamics,
as investigated in \cite{CASTROmoist}.

We then considered a moist thermodynamic model that treats
dry air and water vapor as ideal gases to define the
equation of state for moist air.
In order to account for the latent heat release
in the low Mach divergence constraint for the velocity field,
the evaporation rate is estimated from the time variation
of saturated water vapor within a parcel.
The amount of saturated water vapor within a parcel
is determined by the
Clausius--Clapeyron formula as a function of the
local thermodynamical state;
the evolution of the state depends on the local advected motions.
An analytical expression for the evaporation rate was thus
derived that depends on local parameters given by
the temperature and pressure, the water composition,
and the velocity field.  Two approaches were then considered.
In the first, the rate of phase change can be computed to
evaluate the latent heat release; in the second,
a modified divergence constraint can be analytically
deduced by introducing the derived expression for the evaporation rate
in the original divergence constraint.
Both approaches are analytically equivalent and together
with the low Mach number equation set allow us to
characterize moist atmospheric flows.

The MAESTRO code\footnote{Available at
http://bender.astro.sunysb.edu/Maestro/download/}
\cite{multilevel}, originally designed to simulate
stratified reacting flows arising in astrophysical settings, was
adapted to model moist atmospheric flows.
A series of test problems was investigated
with both isentropic and non--isentropic background states,
as well as saturated and partially saturated regions in the
atmosphere.  Results were contrasted to
reference solutions obtained with a fully compressible
formulation.  Very good agreement with the reference
moist dynamics was shown using both the first and second approach
(with the $\delta \Gamma_1$--correction),
thus demonstrating that low Mach number models can serve as a reasonably
accurate and computationally efficient alternative to compressible
codes for small-scale moist atmospheric applications.

\section*{Acknowledgments}
This material is based upon work supported by the
U.S. Department of Energy, Office of Science,
Office of Advanced Scientific Computing Research,
Applied Mathematics program under contract number DE-AC02005CH11231.

redpajama_set_name: RedPajamaArXiv


Taking a page from its parent brand Facebook, Instagram has finally introduced its long-awaited business profiles to capitalize on the more than $3 billion market that its content creators have cultivated for the platform.
The business profile adds a very important functionality to Instagram — analytics. When you use a personal profile, you have no way to analyze the demographics of the people visiting your page. This all changes with the business profile. Analytics give you the information to target your audience precisely and hone in on the most profitable potential customers.
Business profiles also allow you to promote posts directly from Instagram instead of requiring a promotional effort through another platform. This reduces the amount of time from ad to customer drastically, and it also gives content creators who enjoy using Instagram more power to leverage the platform.
Creating an Instagram business profile also opens up your marketing options to the future innovations that will undoubtedly come to the business side of Instagram first. Among other things, the platform has promised even more robust analytics, partnerships with other major social media platforms, and a continually growing audience with more opportunities for direct revenue from the platform.
Facebook has created a page for people who are looking to convert their personal Instagram profiles into a business profile here.
In terms of social media marketing, Instagram is currently the king of the internet. Make sure that you bolster your ability to market your wares and analyze your campaign with an Instagram business profile. 802Social is here for all of your social media needs — call us when you are ready to take your online marketing campaign to the next level.
We've explored how to identify which social media platform is right for your business; now, we'll begin to explore the complexity and benefits of one of those platforms: Instagram. Instagram is an online mobile photo-sharing, video-sharing and social networking service. It's used as a tool for conveying a message in an attractive, pictorial way. The social platform is used for like sharing photos and :15-second videos to those that you are connected with and can be seen when utilizing hashtags.
It's expected Instagram usership will climb up to nearly 107 million by 2018...so the light bulb may be going off in your head: why am I not on Instagram for my business?
Over the course of the next few months, join us in exploring what Instagram can do for business, how to utilize it the best way and how to gain fans (along with other FAQs). We encourage you to come along on this journey with us to learn more about how to use Instagram for your business and decide for yourself if it's the right platform for you, your business and your customers.
The biggest piece of advice we have when using Instagram is to be thoughtful about each post you make. It's not about the numbers here, it's about quality, provocative content that can catch someone's eye, it is about presenting your content in a unique, clever and thoughtful way and trusting the rest will fall into place. It is certainly worth investing some money in professional photography and videography to get the kind of content you need, otherwise, iPhone photos are completely acceptable as long as they're not poor quality or uninteresting.
Another piece of advice we have is to use Instagram as a relationship tool with your customers. Show your customers and fans the "insider eyes" of what it's like to create your product, or what it's like in the day-in-the-life of an employee. Get creative with out you present your narrative.
Lastly, interact. Participate in the conversations, react to comments, and start your own conversations on other like-minded accounts photos. This will help you to build a community. Share (or #regram) other photos that you like that have to do with what you're promoting or marketing. Along the same point, encouraging your usership to provide you with content. This can be done with a simple call-to-action or you can run Instagram Contests (blog post coming soon).
As always, if it all sounds great, but a bit too overwhelming to manage, 802Social is here to help.


The Grand Theatre of Lemmings.
The theme of the street theatre event was Then And Now. It was a commemoration of WW1, with Mandy Rose and Helen Abbey singing Old Time Music Hall, the young Lemmings running circus skills for children and a poppy making workshop. There were letters from the trenches read by Charlie Peacock and my workshop was to collect words, thoughts, ideas, and memories from people passing by and to write them on postcards to be added to a time capsule. I wrote the poem below from those words and memories and it was read back to the people of Clacton at the end of the day.
Never has so much been owed to so few.
I still think of you every day, Walter.
You hated it when people wasted food.
which we sold for sweets when we were evacuated to Wales?
who gave their lives, and limbs for freedom.
A word, a sound, a bomb whistles by, or perhaps a sigh?
Missiles still fly, innocent people still die.


When properly installed, the LoadDox-Guard Expanding Aluminum Barrier has been designed to conform to OSHA Fall Protection Systems criteria and practices 1926.502 as defined in Federal Regulation Standards 29 CFR. The expansion unit utilizes heavy duty industrial strength rivets (shear spec.: 1,725 lbs., tensile spec.: 1,400 lbs., clamp spec.: 925 lbs.) to provide retention and a ½" round steel drop pin in the center to limit horizontal deflection. The hinge and lock mounting posts or brackets must be secured with locally obtained fasteners to ensure the LoadDox-Guard is capable of withstanding a minimum force of 200 lbs. in any direction to conform to OSHA recommendations. The user is solely responsible for determining the appropriate installation positioning for hazard protection to ensure correct operation of the LoadDox-Guard barrier. North American Safety Products, Inc. assumes no liability for accident or injury due to incorrectly or unsafe positioning or attachment.
Quickly folds to a compact size. No tools required.
Reduces maintenance time and minimizes set-up.


The 100 CLUB is the great way to support our village hall and at the same time have an inexpensive, twelve months' long flutter. For only £2 per month per number you get four chances to win a CASH PRIZE in every one of our monthly draws plus a special draw in December. All these exciting chances are yours just £24 a year and of course, you can have more than one number. A call to 01689 858583 will get you started at the first opportunity.


text:
Q: Is it possible to open a pop up in an iframe Is it possible to force a pop up to open in an iframe where the action to open the pop up has taken place?
I have a VueJs Application that displays a website inside an iframe and don't want the application to open new windows. But it is also not acceptable to just block all the popups with sand boxing, as some contain critical functionality.
Thus i am looking for any way to "force" target="_blank" onto the link on the webpage.
I know that there is no standard way to do so, but would be open for any suggestion on how to alternatively solve this.
Maybe there is a way to disable the browsers ability to open new windows and hijack the calls.
Looking forward to your answers.

A: That capability is all or nothing, mostly. Since it's a different site in the iframe, you likely don't have access to the iframe content and so can't control the links in it.
iframe has a sandbox attribute to which you could add the allow-popups option

*

*allow-popups: Allows popups (such as window.open(), target="_blank", or showModalDialog()). If this keyword is not used, the popup will silently fail to open.

As in:
<iframe ... sandbox="allow-popups"></iframe>

There's also:

*

*allow-popups-to-escape-sandbox: Lets the sandboxed document open new windows without those windows inheriting the sandboxing. For example, this can safely sandbox an advertisement without forcing the same restrictions upon the page the ad links to.

*allow-modals: Lets the resource open modal windows.

redpajama_set_name: RedPajamaStackExchange


I haven't blogged in over a month. I've been busy. But everyone says that, right? While each one of us has the same amount of time every day, it's how we spend that time that can change our lives, make or break a relationship, or even alter our doors of opportunity.
It's not about the best model or the most efficient To-Do List. It's not about being the early bird or the procrastinator. It's not even about clarifying priorities.
It's about self-discipline, mindset, and pace.
I believe the following strategies can apply to all consumers of time, but especially to the do-it-all survivors, recovering perfectionists, and abstract OCD-leaning fighters.
Perhaps it's not lacking the will to say No to new opportunities. It's not misunderstanding the amount of time available or how to weigh the importance of each priority. Believe it or not, some people have to wrestle with taking on less.
Maybe this stems from the obligation to be all-things-educator to save the world. Maybe it's the iconic educator lifestyle of complete sacrifice and martyrdom that's constantly praised on social media and in real life. Maybe the reason why some educators take on so much is rooted in relentless passion to become better, wanting to diversify portfolio experiences, or even for the love of the profession itself.
Teaching is a passion so sacred, it's easy to give. And give. And give. Until you mathematically run out of time to give–and to perform well in giving all along the way. When you factor in the #WholeEducator or #WholeTeacher, family, friends, and health peak priority.
For these kindred souls, self-discipline is needed to simply do less. To put less on the plate. To write down less items on the To-Do List. To move more things to Stephen Covey's Quadrant IV: Not Urgent–Not Important. It sounds so simple, yet, for many, it's a daily battle.
Binary thinking–that's an all-or-none mindset–is gripping. It stifles progress. It handcuffs good and productive people. It destroys confidence. And it can challenge relationships.
When you can't begin a one-hour project because you only have 30 minutes available, you lose those 30 minutes. When you need 4-6 hour blocks of time to make a meaningful impact in current studies and research, the 2-3 hour time slots go by the wayside. When you have just 15 minutes To-Do 3 things on the List, none of them get done.
In his post 5 Rules to Simplify Your Life, AJ Juliani has really helped me in Rule 3: Use a Checklist Instead of a To-Do List. Instead of saying that you have to get this done, you say that you're going to work on this for a specific amount of time.
Abstract Obsessive Compulsive Disorder can limit one from moving on. Stuck in a continual spin cycle, some ideas and details never come into fruition due to unrelenting thought. This means that both starting and stopping processes can be hard. Transitions are intense, because they require hard, definitive breaks, moving from thinking to doing.
With time being so finite, pacing guides can help those arrested in detailed analyses.
And maybe managing time and productivity are not as much about progressing at a consistent, steady pace over time. Maybe it's more simple than that. Maybe it's just about the will to keep moving forward.


The food pantry is available for people residing in Manchester who are experiencing financial hardship.
The pantry is open for food distribution on Mondays, Wednesdays and Fridays from 12:00 p.m. through 1:00 p.m. as well as Thursdays from 6:00 p.m. – 7:00 p.m.
Families and individuals can utilize the pantry one time per month and receive pre-packed groceries based on the number of individuals residing in the household. Choices of fresh fruits, vegetables, meats and bakery items are also available based on family size.
The bread line is open following the pantry distribution times. Clients using the pantry and others can come and get bakery and perishable items. Pantry clients may access the bread line as frequently as necessary.
Those accessing the food pantry need to bring current photo identification, proof of their residence in Manchester, proof of income (pay stub, earnings statement or bank statement), and birth certificates or health insurance cards for any children residing in the household.
In Manchester, affordable housing is extremely limited. According to the American Friends Service Committee, rents in the southern half of New Hampshire have increased as much as 37% over the last 5 years. Coupled with high utility rates, gasoline prices and the lack of living wage jobs; families and individuals can become "food insecure" very quickly. Food insecurity is the condition where a household has little or no money left over after paying other living expenses, to purchase food. For children living in these households it means a great disadvantage in learning. Children with poor nutrition have a more difficult time concentrating in school and are at risk for weakened immune systems. Poverty and hunger are real concerns in Manchester.
57% of students in the Manchester School District receive free or reduced-price school lunches during the school year. Some schools are in the high 80% and low 90% eligibility. Student eligibility for free or reduced-price lunch is a widely accepted alternative measure for the poverty rate since it captures the population that is economically disadvantaged.


text:
Android News / Master & Dynamic Launch MW50 Bluetooth Headphones
Master & Dynamic Launch MW50 Bluetooth Headphones
By Tom Dawson
The headphone market is an undeniably-large market to be part of these days, and as smartphones are evolving, with the humble headphone jack starting to be phased out by some, it's no wonder that consumers are starting to look ahead to a world of wireless headphones, using Bluetooth technology. This was the first year that wireless headphone sales started to fight back in a meaningful way against wired headphone sales, and while there are some inexpensive options out there, users are often left unimpressed by the clarity and overall build quality of these cheaper options. New York's Master & Dynamic is a luxury headphone brand that is trying very hard to change all of this, and first did so with last year's MW60, now they're back with the new MW50.
At $449, these won't be winning any awards for budget headphone of the year, but where they might win awards for is in their build quality, excellent clarity and long Bluetooth range. Similarly to the excellent MW60, the MW50 are made from genuine leather and all-aluminum parts for the outer-casing. These are a set of on-ear headphones that are designed to last the listener decades, and the fit and finish on these is where much of the money goes when purchasing a pair. Excellent looks and long-lasting build aside, master & Dynamic have a house sound that's warm, clear and resonant, appealing to modern listeners without doing your favorite artists injustice in that endeavor. To achieve that with the MW50, the firm is using a custom 40mm beryllium driver.
Where the Bluetooth side of things are concerned – these work just fine with a 3.5mm cable, too – the MW50 lasts for 16 hours on a single charge, which is recharged via the Type-C port on the ear cups, which is also where the play/pause, connect and skip buttons can be found. 30 percent smaller than the MW60, the MW50 are a more portable pair of wireless headphones, and with a range of 100 feet away from your phone or computer, they're more than happy to join you wherever you're headed. The MW50 are available from Master & Dynamic's website right now in either Black Leather or Brown Leather, and those interested in a closer look can take a look at the gallery below.
Master Dynamic MW50 Press 10
Master Dynamic MW50 Press 1
Top 10 Best Password Manager Android Apps – 2021
At this point in time, you've probably registered to a ton of sites / services.…
Top Five Alternatives To Samsung's New Galaxy S21 Series
Samsung's Galaxy S21 series has arrived and it's undeniably the best smartphone the company has…
By Daniel Golightly · January 20, 2021
You Can't Buy The USB-C Pixel Buds From Google Anymore
Once upon a time, the USB-C Pixel Buds felt like a thing of legend. A…
Tom Dawson
Former Editor-in-Chief
For years now I've had a heavy interest in technology, growing up with 8-bit computers and gaming consoles has fed into an addiction to everything that beeps. Android saved me from the boredom of iOS years ago and I love watching the platform grow. As an avid reader and writer nothing pleases me more than to write about the exciting world of Android, Google and mobile technology as a whole.

redpajama_set_name: RedPajamaCommonCrawl


February 1911. Bayou La Batre, Alabama. "Little Julia tending the baby at home. All the older ones are at the factory. She shucks [oysters] also. Alabama Canning Co." Photograph by Lewis Wickes Hine. View full size.
Submitted by clusk1020 on Fri, 02/28/2014 - 12:43pm.
This is such a sweet photo. I hope these little angels had wonderfully happy and long lives.
Submitted by Mr Mel on Sat, 02/08/2014 - 1:27pm.
Hines outdid himself with this one. The little girl's eyes are just amazing.


These types of reviews are always the hardest for me; the ones where it seems as though the rest of the world loves it and I'm in the super small minority that didn't. I believe that for me the insane about of anticipation and promotion leading up to the release of this book tainted my reading experience. There was months and months of build up making it nearly impossible not to hear something about this book. I have never seen so many ARCS given out to promote a book meaning that before it was even officially published there were dozens of reviews available. All over social media people were posting pictures of their tear stained faces and raving about how this book is the absolute best book that they've ever read and how it has changed their life, so no pressure right?
Colleen kept stressing that she thought that when starting this book you should go in blind and I think that for the most part I did. I questioned reading this book so soon after its release because I knew that with the amount of build up going in to it it would be hard for it to live up to the hype. The synopsis is wholly misguiding because it makes it sound like a love triangle is involved when that is not the case.
Lily, in the beginning, seems a bit lost. A couple chapters in she seems to have hit her stride: opening her own business, having her mom close by, made some friends. We learn more about her past, and her past relationship, when she starts to read an old journal of hers and this is where we meet Atlas, which is a great name don't you think? While this is happening she seems to fall in to a sort of easy relationship with Ryle, someone that she run-in with months prior and now seems endlessly tangled with her life. Her life seems to be coming together perfectly until a chance encounter with Atlas. As her relationship with Ryle develops she starts to question some of his actions. I think overall Lily was a very strong character doing the best she could in an impossible situation. My only problem is that I had trouble connecting with her. While I had no problems with her decisions there was just something about her that made it impossible to connect; that allowed me to see her situation as though I was in her shoes.
Atlas knows what it is like to need a hand which is why as an adult he is fiercely protective. Even though we learn a lot about him through Lily's journal Atlas the adult is a whole different animal. After being homeless Atlas is now an award winning chef with a bit of a chip on his shoulder. His love for Lily has never gone away and his infinite patience with her allows her to blossom, even if they are not together. I like the layers that make up Atlas: his soft side, his defensive side, his protective side. All put together they make one hell of a package.
I know when I have read a good book when I'm so moved by it that I'm simply unable to move on and unfortunately I didn't have that problem here. It was a nice story but didn't quite hit the mark for me.


text:
2012 in Review: Blackout Protests Against Blacklist Bills
By Parker Higgins
As the year draws to a close, EFF looks back at the major trends influencing digital rights in 2012 and discussing where we are in the fight for free expression, innovation, fair use, and privacy. Click here to read other blog posts in this series.
Coming into 2012, the Internet community was looking down the barrel of very dangerous legislation that would have created legal structures to silence legitimate speech in the name of curbing online "piracy." A House bill called the Stop Online Piracy Act (SOPA) and its Senate counterpart, the PROTECT IP Act (PIPA), had been debated, amended, and looked to be on the fast track for legislative approval.
That all changed on January 18. An historic "Blackout Day" protest, loosely coordinated by a coalition of public interest groups, startups, tech companies, and thousands of different websites, resulted in millions of emails and tens of thousands of calls to legislators. As sites across the web turned out their virtual lights at once, an important but otherwise arcane copyright bill became front-page news—and impossible for the content lobby and their favorite legislators to sneak past the public.
There were plenty of reasons to be concerned. After all, Congress has passed 15 laws aimed at stopping "piracy" over the last 30 years—an impressive record given the general absence of actual facts about the problem or the effectiveness of the proposed solutions. And even after an earlier protest, held in November 2011, resulted in nearly 90,000 calls to Congress in one day, SOPA's author (and Judiciary Committee chair) Lamar Smith seemed intent on pushing the bill through, dismissing the complaints as not being "legitimate."
But support for the bills began to crumble when met with the magnitude of the blackout protests. Over the course of January 18, people sent nearly a million emails to their legislators through the EFF's action center alone, and many million more through other sites. Some of the most popular sites on the Internet, like Google, Reddit, and Wikipedia, were among over 100,000 pages dark in protest.
It worked. Angry supporters of the bill may have derided the grassroots action—MPAA chief Chris Dodd called it "dangerous" and a "gimmick," while RIAA CEO Cary Sherman said it was based on "misinformation" and, strangely, a "misuse of power"—but legislators got the message, and began jumping ship from SOPA and PIPA almost immediately. Within a month, the bills were effectively dead.
The defeat of SOPA and PIPA marked an important victory in an ongoing struggle for copyright policy that's based on actual facts and real evidence. The January 18 protests, too, served as a template for a new era of online activism. In 2012, we've already seen similar battles play out all over the world; in 2013, we're sure to see even more.
SOPA/PIPA: Internet Blacklist Legislation
Deeplinks Blog by Alex Moss | May 21, 2019
Reddit Commenter's Fight for Anonymity Is a Win for Free Speech and Fair Use
A fight over unmasking an anonymous Reddit commenter has turned into a significant win for online speech and fair use. A federal court has affirmed the right to share copyrighted material for criticism and commentary, and shot down arguments that Internet users from outside the United States can't...
Press Release | May 14, 2019
YouTube User Fights Unfair Takedown Campaign from UFC
San Francisco – The creator of popular post-fight commentary videos on YouTube is demanding an end to the Ultimate Fighting Championship (UFC)'s unfair practice of sending takedown notices based on bogus copyright claims. The creator, John MacKay, is represented by the Electronic Frontier Foundation (EFF). MacKay operates the "Boxing...
Press Release | May 2, 2019
Media Alert: Court Hearing Monday for Redditor Fighting to Stay Anonymous
San Francisco – On Monday, May 6 at 11am, the Electronic Frontier Foundation (EFF) will argue that a San Francisco court should quash a subpoena from the Watch Tower Bible and Tract Society aimed at getting the identity of an anonymous Reddit commenter. Watch Tower is the supervising body...
Deeplinks Blog by Katharine Trendacosta | April 14, 2019
EFF's Tweet About an Overzealous DMCA Takedown Is Now Subject to an Overzealous Takedown
Get ready for a tale as good as anything you'd see on television. Here's the sequence of events: the website TorrentFreak publishes an article about a leak of TV episodes, including shows from the network Starz. TorrentFreak tweets its article, Starz sends a copyright takedown notice. TorrentFreak writes about the...
Deeplinks Blog by Danny O'Brien | March 26, 2019
EU's Parliament Signs Off on Disastrous Internet Law: What Happens Next?
In a stunning rejection of the will of five million online petitioners, and over 100,000 protestors this weekend, the European Parliament has abandoned common-sense and the advice of academics, technologists, and UN human rights experts, and approved the Copyright in the Digital Single Market Directive in its entirety...
Deeplinks Blog by Cory Doctorow | March 22, 2019
This Could Be It: Key Polish Political Party Comes Out Against Article 13
With only days to go before the final EU debate and vote on the new Copyright Directive (we're told the debate will be at 0900h CET on Tuesday, 26 March, and the vote will happen at 1200h CET), things could not be more urgent and fraught. That's why...
Deeplinks Blog by Jason Kelley | March 13, 2019
If It Really Wants To Restore Debate, Facebook Should Update Its Ad Policy
Last week, Facebook CEO Mark Zuckerberg announced a new "privacy-focused" direction for the company that, while sounding great in theory, also set off several alarm bells—including concerns about competition as the company moves to make its messaging properties indistinguishable from one another. As usual for Zuckerberg, it's all...
Deeplinks Blog by Elliot Harmon | March 1, 2019
Don't Sacrifice Fair Use to the Bots
Three years ago, we warned of a string of dangerous new policy proposals on the horizon. Under these proposals, platforms would be forced to implement copyright bots that sniffed all of the media that users uploaded to them, deleting your uploads with no human review. It's happening. The European...
Deeplinks Blog by Michael Barclay | February 25, 2019
EFF Asks the Supreme Court to Clean Up the Oracle v. Google Mess
EFF has just filed an amicus brief in support of Google's petition asking the U.S. Supreme Court to review the long-running case of Oracle v. Google. The case asks whether functional aspects of computer programs are copyrightable, and...


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Worlds Unbound: The Art of teamLab
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W034689 | $40.00
Laura Lee. Intellect, Bristol, 2021.
388 pp. Well illustrated (all col.). 27 x 22 cm. In English. Hardcover.
Worlds Unbound introduces and contextualizes the artistic output of the Tokyo-based digital art collective teamLab, known for its electrifying installations that transcend boundaries between gallery, public space, and popular entertainment. In the twenty years since its founding in 2001, teamLab has grown into a global multidisciplinary collaboration with more than six hundred participants, including engineers, computer graphics animators, mathematicians, graphic designers, architects, artists, and computer programmers. With lavish illustrations throughout, this book provides a comprehensive overview of teamLab's artistic vision and achievements, illuminating the remarkable scope of the group's groundbreaking art and its fundamental contributions to the pivotal field of new media art. Unpacking teamLab's digital environments, from framed works animated on a loop to immersive environments that unfold in real time and respond to viewer behavior, Worlds Unbound illustrates how the collective uses the concept of interdependency to interrelate the natural world, participatory culture, and the digital art experience.
Subject Headings: Asian Art (Western Style) ; Non-Western in a Western Style ; Western Art -- Japan -- Post-1945 ; Post-1970 ; Post-1990 ; Post-2000 -- Installation Art --
Artist(s): teamLab (group)


Bass Guitarist / Vocalist currently looking to join a busy function band, with great gear and transport.
Theatre credits include musicals "Oklahoma" (chorus member), "Guys & Dolls" (Lieutenant Brannigan), "Copacabana" (Maitre D & Pirate King), "42nd Street" (Mac) with the Wolverhampton Musical Comedy Company and plays Alan Ayckbourn's "Way Upstream" (Alistair) and Michael Frayn's "Alphabetical Order" (Wally) with Codsall Dramatic Society plus numerous summer and Christmas variety shows and voice overs for the "Lunar Men" project at Wolverhampton Art Gallery.
Nick France began playing bass guitar just after leaving school in 1980 and went on to sing and play in a series of local bands throughout the 80's. The first shot at major stardom was with the ill fated Birmingham based "Dreddin' The Weddin' " working with top Midlands drummer Pete Isherwood and breathing the same air as The Who and The Style Council for a short time.
For most of the 1990's he was bassist / vocalist with the acclaimed Birmingham based folk/rock outfit "Dewberry Firkin". The Firkin regularly supported circuit favourites including "The Steve Gibbons Band", "John Otway", "Quill", "Soul Survivors", Kerfuffle" and even played a Monster Raving Looney Rally with "Screaming Lord Sutch". More recently Nick was lead vocalist and bassist with Staffordshire based pop/rock band "The Special Stuff" and has had spells with folk / rock / ceilidh band "The Fox Hat", "The Rub" rhythm & blues band, Coventry folk/rock band "Bellatryx", Birmingham rock/pop bands "Jewels", "Classix", "Fools Life" and "Thirsty Work".
2013 saw a fresh challenge having now joined Disco/Soul Sensation "Dance Floor Deluxe" playing 70's Disco / Soul / Motown / Funk / Northern Soul classics.


I ask this because, on the telly this evening (Remembrance Sunday) there was a broadcast of the Queen at a Remembrance Day Service at Westminster Abbey.
And guess who was there to greet the Queen at the door? Sir bleedin' Roy Strong. When I knew him back in the 70s he wasn't a Sir - he was the somewhat dusty but quirky head honcho at the V&A museum. Now he's looking perky all dollied up in medieval robes on telly shaking hands with the Queen and chatting to Prince Charles like they're old mates. Cripes! Was I gobsmacked!
And then the BBC commentator says in his hushed tones "And here's Sir Roy Strong, the High Bailiff".
Never heard of the High Bailiff. What does the High Bailiff actually do for us?
How much does he get paid? Or is it just a comfy totally 'non-job' for ageing arty farty old farts like Roy Strong?
What does the High Bailiff do to earn his dosh? Does he actually do anything important? If so, what?
PS Here below is a photo of Sir Roy Strong pictured in his nightie in his back garden before he grew his new daft Catweasel handle bar tash and pointy Hogwarts's beard.
I couldn't find any pics of him in his daffy High Bailiff's togs. In fact, I couldn't even find any pics of the High Bailiff's ceremonial robes at all, with or without Sir Roy Strong inside them. I must be doing something wrong. Help!


Her argument: to avoid feelings of false guilt, we need to clearly distinguish friends from acquaintances.
Anxiety: managing it from information overload. Lifehacker says "Don't Voluntarily Take on Other People's Anxiety if You Can Avoid It," but Paul said "bear one another's burdens" (Gal 6:2). I think both are right, but Lifehacker goes too far. We love and help others, but we realize our limitation and trust in God's sovereignty FOR them. After all, Jesus said "Therefore I tell you, do not worry about your life, what you will eat or drink; or about your body, what you will wear. Is not life more than food, and the body more than clothes?" (Matt 6:25).
- how to recognize your selfishness (from personal experience this week): are you consistently showing up late? Do your conversations mainly revolve around you or the person you are talking to? Do you more often adjust to others schedules or have them adjust to your own?


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Another addition to our movies collection series. Gamora full costume as shown in Guardian of the galaxy 2. Full set of costume including accessories and specially made boots. Suitable for Adults during Parties, Halloween and Special occasions. Material: Clothing in cotton and denim, Accessories in canvas and Pleather.


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Agrostopoa es un género de plantas de la familia de las poáceas.

Taxonomía
El género fue descrito por Davidse, Soreng & P.M.Peterson y publicado en Novon 19(1): 33. 2009. La especie tipo es:  Agrostopoa wallisii (Mez) P.M.Peterson, Davidse & Soreng

Especies
  Agrostopoa barclayae Davidse, Soreng & P.M.Peterson
  Agrostopoa wallisii (Mez) P.M.Peterson, Davidse & Soreng
  Agrostopoa woodii Soreng, P.M.Peterson & Davidse

Referencias

Enlaces externos

Poinae

redpajama_set_name: RedPajamaWikipedia


The inaugural speech was delivered by vice chancellor Prof Dr. J. Mahapatra. Ms Oopali Operajita a senior advisor to India's parliamentary leaders and world leaders on public policy and international affairs was the Chief Guest of the evening which was attended by dignitaries, and guests, and students. Ms Operajita elaborated on needs of public health schools and education in India.


Several years ago I stopped making New Year's resolutions because within the first month (sometimes the first week!), I had broken them. You know, things like, "lose 30 pounds, have daily devotional time with God first thing every morning, travel internationally, have a hot marriage, be the world's best mom." Then I felt like a failure when they didn't happen–even remotely. Do you relate?
60% of people abandon them within six months.
Most people make the same New Year's Resolution 10 times without success.
I decided that instead of resolutions (or even "goals"), I'd pray to God for a Scripture and a "one word" and/or theme for the coming new year. This has worked out well for me. And I no longer feel like #wifefail or #momfail or whatever fail.
Last year's "one word" was intentional.
Each year, I've discovered how accurate the "one word" was for that year with both its blessings and challenges to overcome. Throughout 2017, I had to make very intentional choices.
For 2018, my "one word" is JOY. Whenever I begin to see or hear a word or a Scripture two, three, or more times, I know that God is speaking to my heart and wants to teach me something.
I've begun to see this word JOY a lot and recently found a Christmas decoration with two red cardinals (cardinals are very special to me), with the word JOY on it. I felt that was the final confirmation that I needed to use this as my one word for the coming year.
I look forward to this, as 2017 was hell, one of the worst years in my life (and I've had some bad ones before!).
On the Blue Letter Bible website, that word "joy" from Nehemiah 8:10 in the original language is chedvah. It simply means joy or gladness.
You can have God's joy even in the midst of painful circumstances, because our hope as believers is always in our faith in Jesus Christ and His second coming, not in what is happening around us. We fix our eyes on Him, not the storm.
1: the emotion evoked by well-being, success, or good fortune or by the prospect of possessing what one desires: delight.
2: a state of happiness or felicity: bliss.
3: a source or cause of delight.
Here's a virtual toast to you and me for a much better, joy-filled-to-overflowing year in 2018!
Do you make New Year's Resolutions or have a "one word," theme, and/or Scripture for the new year? Please share what you do and what yours is in the comments below.