Resolving conflicts

CHANGES

@@ -1,4 +1,17 @@
-#23.11
+# 23.12
+    -- AMReX submodule set to 23.12 release hash (9b733ec)
+
+    -- Inclusion of Kessler microphysics option (#1324)
+
+    -- Fix for particles with redistribute call (#1314)
+
+    -- Inclusion of radiation (#1311)
+
+    -- Inclusion of refluxing for two-way coupling (#1289)
+
+    -- Metgrid initialization with SST and LandMask (#1280)
+
+# 23.11
 
   -- AMReX submodule set to 23.11 release hash (ae7b64b)
 

Docs/sphinx_doc/index.rst

@@ -49,6 +49,7 @@ In addition to this documentation, there is API documentation for ERF generated
 
    theory/DryEquations.rst
    theory/WetEquations.rst
+   theory/Initialization.rst
    theory/Buoyancy.rst
    theory/Microphysics.rst
    theory/DNSvsLES.rst

Docs/sphinx_doc/theory/Buoyancy.rst

@@ -112,13 +112,18 @@ if we neglect :math:`\frac{p^\prime}{\bar{p_0}}`.
 
 Type 4
 ------
-.. math:: 
+This expression for buoyancy is from `khairoutdinov2003cloud`_ and `bryan2002benchmark`_.
+
+.. _`khairoutdinov2003cloud`: https://journals.ametsoc.org/view/journals/atsc/60/4/1520-0469_2003_060_0607_crmota_2.0.co_2.xml
+.. _`bryan2002benchmark`: https://journals.ametsoc.org/view/journals/mwre/130/12/1520-0493_2002_130_2917_absfmn_2.0.co_2.xml
+
+.. math::
 
     \begin{equation}
     \mathbf{B} = \rho'\mathbf{g} \approx -\rho\Bigg(\frac{T'}{T} + 0.61 q_v' - q_c - q_p - \frac{p'}{p}\Bigg),
     \end{equation}
 
-The derivation follows. The total density is given by :math:`\rho = \rho_d(1 + q_v + q_c + q_p)`, which can be written as 
+The derivation follows. The total density is given by :math:`\rho = \rho_d(1 + q_v + q_c + q_p)`, which can be written as
 
 .. math::
 
@@ -155,9 +160,9 @@ Using :math:`- 0.61 q_v + q_c + q_p \ll 1`, we have
 
 Since the background values of cloud water and precipitate mass mixing ratios -- :math:`q_c` and :math:`q_p` are zero, we have :math:`q_c' = q_c` and :math:`q_p' = q_p`. Hence, we have
 
-.. math:: 
+.. math::
 
-	\begin{equation}
-	\rho'\approx -\rho\Bigg(\frac{T'}{T} + 0.61 q_v' - q_c - q_p - \frac{p'}{p}\Bigg),
-	\end{equation}
+    \begin{equation}
+    \rho'\approx -\rho\Bigg(\frac{T'}{T} + 0.61 q_v' - q_c - q_p - \frac{p'}{p}\Bigg),
+    \end{equation}
 

Docs/sphinx_doc/theory/Initialization.rst

@@ -0,0 +1,149 @@
+
+ .. role:: cpp(code)
+    :language: c++
+
+ .. role:: f(code)
+    :language: fortran
+
+.. _Initialization:
+
+Initialization
+==================
+
+To initialize a background (base) state for a simulation with a hydrostatic atmosphere, the hydrostatic equation balancing the pressure gradient
+and gravity must be satisfied. This section details the procedure for the initialization of the background state. The procedure is similar for the
+cases with and without moisture, the only difference being that the density for the cases with moisture has to be the total density
+:math:`\rho = \rho_d(1 + q_t)`, where :math:`\rho_d` is the dry density, and :math:`q_t` is the total mass mixing ratio -- water vapor and liquid water, instead
+of the dry density :math:`\rho_d` for cases without moisture.
+
+Computation of the dry density
+-------------------------------
+We express the total pressure :math:`p` as:
+
+.. math::
+   p = \rho_d R_d T_v.
+
+By definition, we have:
+
+.. math::
+   T_v = \theta_m\left(\frac{p}{p_0}\right)^{R_d/C_p},
+
+.. math::
+   T = \theta_d\left(\frac{p}{p_0}\right)^{R_d/C_p}.
+
+This gives:
+
+.. math::
+   p = \rho_d R_d \theta_m\left(\frac{p}{p_0}\right)^{R_d/C_p},
+
+which, on rearranging, yields:
+
+.. math::
+   p = p_0\left(\frac{\rho_d R_d \theta_m}{p_0}\right)^\gamma.
+
+To obtain :math:`\theta_m`, consider the density of dry air:
+
+.. math::
+   \rho_d = \frac{p - p_v}{R_d T}.
+
+Substituting for :math:`\rho_d` from the above equation, we get:
+
+.. math::
+   \frac{p}{T_v} = \frac{p - p_v}{T},
+
+which implies:
+
+.. math::
+   \frac{T_v}{T} = \frac{p}{p - p_v} = \frac{1}{1-\left(\cfrac{p_v}{p}\right)}.
+
+We also have:
+
+.. math::
+   q_v = \frac{\rho_v}{\rho_d} = \frac{p_v}{R_v T}\frac{R_d T}{p-p_v} = \frac{r p_v}{p - p_v},
+
+where :math:`p_v` is the partial pressure of water vapor, :math:`r = R_d/R_v \approx 0.622`, and :math:`q_v` is the vapor mass mixing ratio—the ratio of the
+mass of vapor to the mass of dry air. Rearranging and using :math:`q_v \ll r`, we get:
+
+.. math::
+   \frac{p_v}{p} = \frac{1}{1 + \left(\cfrac{r}{q_v}\right)} \approx \frac{q_v}{r},
+
+which, on substitution in the equation for :math:`\frac{T_v}{T}`, gives:
+
+.. math::
+   \frac{T_v}{T} = \frac{1}{1 - \left(\cfrac{q_v}{r}\right)}.
+
+As :math:`q_v \ll 1`, a binomial expansion, ignoring higher-order terms, gives:
+
+.. math::
+   T_v = T\left(1 + \frac{R_v}{R_d}q_v\right).
+
+Hence, the density of dry air is given by:
+
+.. math::
+   \rho_d = \frac{p}{R_d T_v} = \frac{p}{R_d T\left(1 + \cfrac{R_v}{R_d}q_v\right)}.
+
+
+Initialization with a second-order integration of the hydrostatic equation
+----------------------------------------------------------------------------
+
+We have the hydrostatic equation given by
+
+.. math::
+
+    \frac{\partial p}{\partial z} = -\rho g,
+
+where :math:`\rho = \rho_d(1 + q_t)`, :math:`\rho_d` is the dry density, and :math:`q_t` is the total mass mixing ratio -- water vapor and liquid water. Using an average value of :math:`\rho` for the integration from :math:`z = z(k-1)` to :math:`z(k)`, we get
+
+.. math::
+
+    p(k) = p(k-1) - \frac{(\rho(k-1) + \rho(k))}{2} g\Delta z.
+
+The density at a point is a function of the pressure, potential temperature, and relative humidity. The latter two quantities are computed using user-specified profiles, and hence, for simplicity, we write :math:`\rho(k) = f(p(k))`. Hence
+
+.. math::
+
+    p(k) = p(k-1) - \frac{\rho(k-1)}{2}g\Delta z - \frac{f(p(k))}{2}g\Delta z.
+
+Now, we define
+
+.. math::
+
+    F(p(k)) \equiv p(k) - p(k-1) + \frac{\rho(k-1)}{2}g\Delta z + \frac{f(p(k))}{2}g\Delta z = 0.
+
+This is a non-linear equation in :math:`p(k)`. Consider a Newton-Raphson iteration (where :math:`n` denotes the iteration number) procedure
+
+.. math::
+
+    F(p+\delta p) \approx F(p) + \delta p \frac{\partial F}{\partial p} = 0,
+
+which implies
+
+.. math::
+
+    \delta p = -\frac{F}{F'},
+
+with the gradient being evaluated as
+
+.. math::
+
+    F' = \frac{F(p+\epsilon) - F(p)}{\epsilon},
+
+and the iteration update is given by
+
+.. math::
+
+    p^{n+1} = p^n + \delta p.
+
+For the first cell (:math:`k=0`), which is at a height of :math:`z = \frac{\Delta z}{2}` from the base, we have
+
+.. math::
+
+    p(0) = p_0 - \rho(0)g\frac{\Delta z}{2},
+
+where :math:`p_0 = 1e5 \, \text{N/m}^2` is the pressure at the base. Hence, we define
+
+.. math::
+
+    F(p(0)) \equiv p(0) - p_0 + \rho(0)g\frac{\Delta z}{2},
+
+and the Newton-Raphson procedure is the same.

Docs/sphinx_doc/theory/Microphysics.rst

@@ -7,8 +7,12 @@
 
 .. _Microphysics:
 
+Microphysics model
+====================
+
 Kessler Microphysics model
-===========================
+---------------------------
+The Kessler microphysics model is a simple version of cloud microphysics which has precipitation only in the form of rain. Hence :math:`q_p = q_r`.
 Governing equations for the microphysical quantities for Kessler microphysics from `gabervsek2012dry`_ are
 
 .. math::
@@ -24,14 +28,78 @@ Governing equations for the microphysical quantities for Kessler microphysics fr
 where :math:`C_c` is the rate of condensation of water vapor to cloud water, :math:`E_c` is the rate of evaporation of cloud water to water vapor,
 :math:`A_c` is the autoconversion of cloud water to rain, :math:`K_c` is the accretion of cloud water to rain drops, :math:`E_r` is the evaporation of
 rain to water vapor and :math:`F_r` is the sedimentation of rain. The parametrization used is given in `klemp1978simulation`_, and is given
-below. Note that in all the equations, :math:`p` is specified in millibars and :math:`\overline{\rho}` is specified in g cm :math:`^{-3}`.
+below. Note that in all the equations, :math:`p` is specified in millibars and :math:`\overline{\rho}` is specified in g cm :math:`^{-3}`. The parametrization
+of the source terms are given below.
 
 .. _`gabervsek2012dry`: https://journals.ametsoc.org/view/journals/mwre/140/10/mwr-d-11-00144.1.xml
+
+.. raw:: latex
+
+   \newgeometry{left=2cm,right=2cm,top=2cm,bottom=2cm}
+
+Rate of condensation of water vapor/evaporation of cloud water
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+From `klemp1978simulation`_, we have the following expressions.
+
 .. _`klemp1978simulation`: https://journals.ametsoc.org/view/journals/atsc/35/6/1520-0469_1978_035_1070_tsotdc_2_0_co_2.xml
 
+If the air is not saturated, i.e. :math:`q_v > q_{vs}`
+
+.. math::
+    C_c = \frac{q_v - q_{vs}}{1 + \cfrac{q_{vs}^*4093L}{C_p(T-36)^2}}
+
+If the air is not saturated, i.e. :math:`q_v < q_{vs}`, then cloud water evaporates to water vapor at the rate
+
+.. math::
+    E_c = \frac{q_{vs} - q_v}{1 + \cfrac{q_{vs}^*4093L}{C_p(T-36)^2}}
+
+Rate of autoconversion of cloud water into rain
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The rate of autoconversion of cloud water into rain is given by
+
+.. math::
+    A_c = k_1(q_c - a)
+
+where :math:`k_1 = 0.001` s\ :sup:`-1`, :math:`a = 0.001` kg kg\ :sup:`-1`.
+
+Rate of accretion of cloud water into rain water drops
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The rate of accretion of cloud water into rain water drops is given by
+
+.. math::
+    K_c = k_2q_cq_r^{0.875}
+
+where :math:`k_2= 2.2` s\ :sup:`-1`.
+
+The rate of evaporation of rain into water vapor
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The rate of evaporation of rain into water vapor is given by
+
+.. math::
+    E_r = \cfrac{1}{\overline{\rho}}\cfrac{(1- q_v/q_s)C(\overline{\rho}q_r)^{0.525}}{5.4\times10^5 + 2.55\times10^6/(\overline{p}q_s)},
+
+where the ventilation factor :math:`C = 1.6 + 124.9(\overline{\rho}q_r)^{0.2046}`.
+
+Terminal fall velocity of rain
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The terminal fall velocity of rain is given by
+
+.. math::
+    V = 3634(\overline{\rho}q_r)^{0.1346}\Bigg(\cfrac{\overline{\rho}}{\rho_0}\Bigg)^{-\frac{1}{2}}~\text{[cm/s]}
+
+.. raw:: latex
+
+   \restoregeometry
+
+
 
 Single Moment Microphysics Model
-===================================
+----------------------------------
 The conversion rates among the moist hydrometeors are parameterized assuming that
 
 .. math::
@@ -54,7 +122,7 @@ Marat F. Khairoutdinov and David A. Randall's (J. Atm Sciences, 607, 1983).
 The implementation of microphysics model in ERF is similar to the that in the SAM code (http://rossby.msrc.sunysb.edu/~marat/SAM.html)
 
 Accretion
-------------------
+~~~~~~~~~~~~
 There are several different type of accretional growth mechanisms that need to be included; these describe
 the interaction of water vapor and cloud water with rain water.
 
@@ -69,21 +137,21 @@ The accretion of raindrops by accretion of cloud water is
    Q_{racw} = \frac{\pi E_{RW}n_{0R}\alpha q_{c}\Gamma(3+b)}{4\lambda_{R}^{3+b}}(\frac{\rho_{0}}{\rho})^{1/2}
 
 The bergeron Process
-------------------------
+~~~~~~~~~~~~~~~~~~~~~~
 The cloud water transform to snow by deposition and rimming can be written as
 
 .. math::
    Q_{sfw} = N_{150}\left(\alpha_{1}m_{150}^{\alpha_{2}}+\pi E_{iw}\rho q_{c}R_{150}^{2}U_{150}\right)
 
 Autoconversion
-------------------------
+~~~~~~~~~~~~~~~~~~~~~~
 The collision and coalescence of cloud water to from raindrops is parameterized as following
 
 .. math::
    Q_{raut} = \rho\left(q_{c}-q_{c0}\right)^{2}\left[1.2 \times 10^{-4}+{1.569 \times 10^{-12}N_{1}/[D_{0}(q_{c}-q_{c0})]}\right]^{-1}
 
 Evaporation
-------------------------
+~~~~~~~~~~~~~~~~~~~~~~
 The evaporation rate of rain is
 
 .. math::