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ATmospheric, Meteorological, and Environmental Technologies RAMS Technical Overview

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ATmospheric, Meteorological, and Environmental Technologies Technical Description of RAMS Basic equations Radiation Grid structure and nesting Microphysics Time differencing Soil/vegetation Advection Turbulence Convective parameterization Data analysis Boundary conditions Nudging Surface layer

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ATmospheric, Meteorological, and Environmental Technologies RAMS General Equations Start from Navier-Stokes and mass continuity Reynolds average (assuming averaging operator) Quasi-Boussinesq approximation Fickian diffusion assumption for variance and covariance terms Simplification of mass continuity equation

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ATmospheric, Meteorological, and Environmental Technologies RAMS Basic Equations

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ATmospheric, Meteorological, and Environmental Technologies Ice-Liquid Potential Temperature il Main prognostic thermodynamic variable in RAMS Conservative in a parcel in dry and moist adiabatic processes Changes in a parcel only when precipitation added or removed

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ATmospheric, Meteorological, and Environmental Technologies RAMS Grid - Stagger Standard C grid (Mesinger and Arakawa, 1976). All thermodynamic and moisture variables are defined in “center” of the box Velocity components, u, v, and w staggered ½ x, ½ y, and ½ z x y v u v uT x z w u w uT

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ATmospheric, Meteorological, and Environmental Technologies RAMS Grid - Projection Horizontal grid uses a rotated polar- stereographic projection Pole of the projection is rotated to an area near the center of the domain Minimizes distortion of the projection in the main area of interest Appropriate “map factors” are used in all horizontal derivative terms. EQ

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ATmospheric, Meteorological, and Environmental Technologies RAMS Grid - z Derivative terms (Clark, 1977), in tensor notation: H - height of the top of the grid z g - local topography height Vertical structure of the RAMS grid z terrain-following coordinate Gal-Chen and Somerville, 1975 Clark, 1977 Tripoli and Cotton, 1982 Top of the model domain is flat Bottom follows the terrain Coordinates:

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ATmospheric, Meteorological, and Environmental Technologies RAMS Grid - z winds Transformation can be viewed as a simple mapping of the horizontal velocity components to the terrain-following system since they remain horizontal in Cartesian space Eliminates the complication of dealing with non-orthogonal velocity components Vertical component, w *, has an imposed value of 0 at z*, which implies no mass flux through the ground surface. Relationship between Cartesian wind components and the z components:

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ATmospheric, Meteorological, and Environmental Technologies RAMS Grid Nesting Adaptive nesting scheme Unlimited number of nests User-specified space/time ratios Horizontal and vertical nesting Mass conservative, reversible Movable 40 km 10 km 2 km z 8 km

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ATmospheric, Meteorological, and Environmental Technologies RAMS Grid Nesting Two way interaction Coarse grid run first for timestep, then interpolate to nested grid boundaries Tendency at nest boundary computed Tendency applied for each of nest timesteps At end of nest timesteps, nested grid averaged and coarse grid values replaced

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ATmospheric, Meteorological, and Environmental Technologies RAMS Time Differencing Hybrid Scheme U,V,W,P scalars tt time

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ATmospheric, Meteorological, and Environmental Technologies RAMS - Time Differencing Time-split scheme for acoustic modes For simplified two- dimensional, dry, inviscid, quasi-Boussinesq equation set Vertical and horizontal coordinate transformations have been removed for clarity. c = speed of sound

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ATmospheric, Meteorological, and Environmental Technologies RAMS - Time Differencing (continued) Other considerations Inclusion of a Crank-Nicholson scheme in the matrix solution RAMS contains a reduction of c, the speed of sound Reduction of c to 40% or higher has no significant effect on model solution Time-split scheme for acoustic modes Vertical implicit scheme (Tripoli and Cotton, 1982; Durran, 1983) allows a longer small time step by solving the vertical pressure gradient and vertical divergence terms. Solved with a tri-diagonal matrix technique.

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ATmospheric, Meteorological, and Environmental Technologies RAMS Advection Two types of advection schemes for the hybrid time differencing: standard leapfrog-type schemes for the velocity components forward-upstream schemes for scalar variables

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ATmospheric, Meteorological, and Environmental Technologies Flux Form Advection Advective schemes are configured in flux form in order to conserve mass and momentum. Considering the x-direction, the advective terms can be generically written assuming constant grid spacing and omitting topographical and spherical transformations:

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ATmospheric, Meteorological, and Environmental Technologies Leapfrog Advective Fluxes Second-order leapfrog fluxes: Fourth-order leapfrog fluxes:

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ATmospheric, Meteorological, and Environmental Technologies Forward-in-Time Advection Forward advection fluxes computed with the second-order or the sixth- order forward upstream advection schemes Tremback et al. (1987). Same family of schemes as the classical first-order forward upstream scheme and Crowley (1968) second-order scheme. Two different forms of the flux scheme First fits a polynomial to the field being advected then integrates the function. For schemes of order three+, this does not reduce to the advective form for constant grid spacing and advecting velocity. Second form makes this requirement. More accurate than the first but does require that the grid spacing be constant. First form (integrated flux form) used in the vertical where the grid spacing is stretched. Second form is used in the horizontal

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ATmospheric, Meteorological, and Environmental Technologies Forward Advective Fluxes Second-order fluxes: Sixth-order integrated fluxes:

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ATmospheric, Meteorological, and Environmental Technologies Other advection issues Positive definite (v4.3 and higher) Non-oscillatory (too diffusive) Parallelism (only 2 nd -order schemes implemented at this time)

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ATmospheric, Meteorological, and Environmental Technologies RAMS had several options (v3b and earlier): Orlanski (1976) computes c from: Klemp and Lilly (1978) scheme averages the Orlanski phase velocities in the vertical, then applies the averaged velocities to the entire vertical column. Klemp and Wilhelmson (1978) scheme simply specifies a constant value as a typical gravity wave phase velocity ( m/s) Because of parallelism and experimentation, only specified phase velocity scheme remains. Radiative boundary condition is only applied to the normal velocity components. Due to the grid stagger, other variables are defined 1/2 x outside of the normal velocity components. Options exist for zero gradient conditions, constant inflow and/or outflow conditions, or radiative outflow conditions. Also, cyclic boundary conditions for certain idealized simulations. RAMS Lateral Boundaries The general form on the C-grid stagger that is used in RAMS is the basic radiative condition : u - wind component normal to the boundary c – phase velocity

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ATmospheric, Meteorological, and Environmental Technologies Wall (w=0) Rayleigh friction layer where an extra term has been added to the basic prognostic equations: - prognostic variables of u, v, w, and 0 - initial value of these variables - timescale which is defined as linear function of height, varying from a value of infinity at the bottom of the absorbing layer to a maximum value at the top of the model domain which is usually set between 60 and 300 seconds. RAMS Top Boundary

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ATmospheric, Meteorological, and Environmental Technologies RAMS Convective Parameterization A simplified Kuo convective parameterization modification of the generalized form of the Kuo (1974) parameterization described by Molinari (1985) standard Kuo-type scheme is an equilibrium scheme; convection acts to consume the convective instability that is supplied by the larger scales. terms in the thermodynamic and moisture equations due to moist convection are written as: I - rate at which the resolvable scale is supplying moisture to a column; parameterized (Molinari and Corsetti (1985) ) as the resolved vertical flux of water vapor through the lifting condensation level (LCL). b - defined by Kuo (1974) as moisture partitioning parameter; determines what fraction of I is used to increase the moisture of the column. (1-b)I is precipitated. (1-b) (precipitation efficiency) is computed according to the empirical function given by Fritsch and Chappell (1980).

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ATmospheric, Meteorological, and Environmental Technologies RAMS Convective Parameterization Q 1 and Q 2 - vertical profiles of the convective heating and moistening Q 1 is computed from the difference between the environmental and a convective profile. Convective profile is a weighted average between the updraft and downdraft profiles. Updraft: profile of the moist adiabat of the source level air lifted to its LCL. Downdrafts: begin at the level of the E minimum of the sounding at the same temperature as the environment. at cloud base, they are 2 K colder than the environment at the surface, they are 5 K colder. Other levels interpolated linearly in height Weighting function for the downdraft relative to the updraft is also somewhat arbitrarily defined to be 1% of the updraft at their beginning level, 10% at the LCL, 20% at the level of maximum downdraft mass flux (about 800 m as described by Knupp, 1985) and 100% at the surface. Weighting function is used to define the convective profile. Below cloud base, the environmental temperature is used instead of the updraft.

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ATmospheric, Meteorological, and Environmental Technologies RAMS Convective Parameterization For Q 2, two actual regions of moisture tendency are defined. Region below cloud base is dried at the rate I. Profile of this drying is the total water mixing ratio which forces the vertical profile of the mixing ratio toward a constant value below cloud base. Anvil region is moistened by the rate bI with the moistening profile constant from 2/3 of the height between the convective source level and the cloud top (consistent with the profile of English, 1973.). No requirement built into the scheme to reach the moist adiabat in the limit. Therefore, a check is made on the potential temperature profile with the convective tendencies added to see if any level will exceed the moist adiabatic value after the total convective tendencies are applied. The moisture supply rate, I, is reduced if this is the case and all convective tendencies are recomputed with this new value to ensure mass and energy conservation.

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ATmospheric, Meteorological, and Environmental Technologies RAMS Surface Layer Surface layer fluxes of heat, momentum, and water vapor into the atmosphere are computed with the scheme of Louis (1979). Approximates the profile functions of Businger et al. (1971) (which need to be solved iteratively) with non-iterative analytic expressions. Computed fluxes serve as the lower boundary for the sub-grid diffusion scheme for the atmosphere. The expressions for the surface layer fluxes can be written as:

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ATmospheric, Meteorological, and Environmental Technologies RAMS Radiation Three schemes Mahrer/Pielke (1977) - simple scheme with several approximations - does not treat clouds in any way Chen/Cotton (1983) - more complicated – emissivity- based scheme - longwave and shortwave cloud effects Harrington (1999) - most complicated - two-stream scheme - designed for detailed treat of ice habits, particles, etc.

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ATmospheric, Meteorological, and Environmental Technologies New Radiation Parameterization developed at CSU by several students, finished by Jerry Harrington (now at Penn State) two-stream scheme computational time proportional to NZ, not NZ 2 more accurate than Chen-Cotton (somewhat slower) uses McClatchey soundings for levels above model top interacts with hydrometeor species based on computed size spectrum and shape future work: couple radiation closer to microphysics, radiation part of microphysics thermodynamic budget

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ATmospheric, Meteorological, and Environmental Technologies RAMS Microphysics - Physical Processes Cloud droplet nucleation Ice nucleation Vapor diffusional growth Evaporation/sublimation Heat diffusion Freezing/melting Shedding Sedimentation Collisions between hydrometeors Secondary ice production

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ATmospheric, Meteorological, and Environmental Technologies RAMS Microphysics - Hydrometeor Types 1.Cloud droplets 2.Rain 3.Pristine ice (crystals) 4.Snow 5.Aggregates 6.Graupel 7.Hail H G S A C P R

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ATmospheric, Meteorological, and Environmental Technologies RAMS Surface Parameterization - LEAF-2 Extension of work at CSU: McCumber and Pielke, 1981; Tremback and Kessler, 1985; Lee, 1992; Walko et al., 1996 Resistance form of equations with concept of canopy air Sub-grid scale “patches” Multi-layer snow cover model and frozen soil Interaction with optional models, such as TOPMODEL for runoff and sub-surface horizontal water transport and CENTURY for dynamic plant growth

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ATmospheric, Meteorological, and Environmental Technologies RAMS/LEAF-2 Canopy air Old method: TATA RARA TGTG New method: TATA RDRD TGTG RARA TCTC

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ATmospheric, Meteorological, and Environmental Technologies RAMS/LEAF-2 Patches Patch 1Patch 2Patch 3 Fraction =.25 LAI = 1 Veg fraction = 1 Fraction =.5 LAI = 2 Veg fraction = 1 Fraction =.25 LAI = 2 Veg fraction =.5

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ATmospheric, Meteorological, and Environmental Technologies Reynolds averaging of the prognostic differential equations for momentum and conservative scalars is performed to partition advective transport into resolved and unresolved components. Unresolved flux components may be expressed in terms of covariances of the form (momentum):scalars: where subscripts i and j - spatial directions [1,2,3] u i - transporting velocity component u j - transported velocity component - transported scalar An overbar represents the Reynolds average, and a prime the deviation from that average. Contribution to the tendency of the resolved variables due to turbulent transport is given by the convergence of the turbulent fluxes. RAMS Turbulence

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ATmospheric, Meteorological, and Environmental Technologies RAMS parameterizes the unresolved transport using K-theory, in which the covariances are evaluated as the product of an eddy mixing coefficient and the gradient of the transported quantity. For scalars, this parameterization takes the form: where K hi is the eddy mixing coefficient for scalars which applies to the i-direction. K hi is never negative, which restricts the parameterized eddy fluxes to always be down- gradient. Four standard options for computing K mi and K hi 1)Basic Smagorinsky (1963) scheme which relates the mixing coefficients to the fluid strain or deformation rate, 2)Basic Smagorinsky scheme with corrections for the influence of Brunt-Vaisala frequency (Hill, 1974) and Richardson number (Lilly, 1962). 3)Prognostic TKE-based scheme (Mellor-Yamada) 4)Prognostic TKE-based scheme (Deardorf) 5)E- and E-l- (implemented in custom v4.3 by Silvia Trini Castelli, CNR/Torino) RAMS Turbulence

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ATmospheric, Meteorological, and Environmental Technologies “RAMS format” data files (RALPH) “First guess” gridded field Upper air observations Surface observations Data preparation (reformatting) RAMS/ISAN ISentropic ANalysis Hybrid isentropic/ z Gridded analysis fields for initial and boundary conditions / 4DDA (varfiles) RAMS Data Analysis

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ATmospheric, Meteorological, and Environmental Technologies RAMS / ISAN ISentropic ANalysis package Data analysis on a hybrid isentropic/ z vertical coordinate Prepares data analyses for RAMS initial fields, lateral boundary conditions, and four-dimensional data assimilation Two steps: Isentropic stage - prepares data analysis for isentropic coordinate, z coordinate, and surface Varfile (variable initialization file) stage - combines isentropic, z, and surface analysis into integrated z analysis

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ATmospheric, Meteorological, and Environmental Technologies RAMS - Nudging “Nudging” type of four-dimensional data assimilation (4DDA) scheme Model fields can be nudged toward observational data as a simulation progresses. Lateral boundary condition of Davies (1978) and the nudging top boundary condition are also nudging where represents the prognostic variables of u, v, il, , and r T. - timescale which controls the strength of the nudging term and varies in three dimensions Configuration of the timescale structure can be broken into three parts: 1)lateral boundary 2) top boundary 3) domain interior Final timescale defined for any grid point is the minimum of the three computed timescales (the one that will provide the maximum nudging strength).

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ATmospheric, Meteorological, and Environmental Technologies Lateral nudging ground Top nudging Interior nudging Lateral nudging RAMS Nudging Regions

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ATmospheric, Meteorological, and Environmental Technologies RAMS - Lateral Boundary Nudging Lateral boundary nudging is an implementation of the Davies (1978) scheme Number of grid points in a boundary region (of only the coarsest resolution grid in a nested grid run) are nudged toward the data analysis. Two purposes: 1)introducing time-varying information into the model domain 2)damping information propagating from the model interior toward the lateral boundary. In the x (east-west) direction, the form of the timescale structure is a parabolic function defined as: where lat - nudging timescale for the lateral boundary regions, B - timescale specified for the actual boundary point, x B - x coordinate of the boundary point, x I - x coordinate of the interior point where the lateral boundary timescale goes to infinity. This equation is applied only between x B and x I.

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ATmospheric, Meteorological, and Environmental Technologies Patterned after the Rayleigh friction absorbing layer where a layer below the model top is forced back to an initial horizontally-homogeneous state in order to damp vertically- propagating gravity waves, thus reducing wave reflections from the model top. When the model is run with observational data initialization, the absorbing layer is nudged to a inhomogeneous observed state. In the nudging region near the model top, the nudging timescale is defined as a simple linear function of height: where top - nudging timescale for the top boundary region, T - timescale specified for the actual top boundary point, z T - height of the top boundary point, and z I - eight of the base of the nudging layer where the timescale goes to infinity. This equation is applied only between z T and z I. RAMS - Top Boundary Nudging

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ATmospheric, Meteorological, and Environmental Technologies Analysis nudging for 4DDA scheme where the observational data is first objectively analyzed to the model grid, then the model field is nudged to the gridded analysis. Contrasts with observational nudging where the model fields are nudged to the observational data only at those grid points which are in the vicinity of the observations. Both techniques have their advantages and disadvantages: analysis nudging generally being somewhat more efficient and easier to apply. observational nudging has the advantage of only performing the nudging where the observations warrant RAMS – Interior (4DDA) Nudging

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