# Adiabatic formulation of the ECMWF model

## Presentation on theme: "Adiabatic formulation of the ECMWF model"— Presentation transcript:

Adiabatic formulation of the ECMWF model
Agathe Untch (office 11) Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Introduction Step by step guide through the decisions to be taken / choices to be made when designing the adiabatic formulation of a global Numerical Weather Prediction (NWP) model. In the process we are “constructing” the dynamical core of the ECMWF operational NWP Model. Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Introduction (cont.) A numerical model has to be: stable accurate efficient No compromise possible on stability! The relative importance given to accuracy versus efficiency depends on what the model is intended for. For example: an operational NWP model has to be very efficient to allow the running of all applications (data-assimilation, forecasts, ensemble prediction system) in a tight daily schedule. a research model might not have to be so efficient but can’t compromise on accuracy. Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Introduction (cont.) Essential to the performance of any NWP or climate-prediction model are a.) the form of the continuous governing equations (approximated or full Euler equations?) b.) boundary conditions imposed (conservation properties depend on these). c.) the numerical schemes chosen to discretize and integrate the governing equations. Adiabatic formulation of the ECMWF model

Euler Equations for a moist atmosphere on a rotating sphere
3D momentum equation Continuity equation Thermodynamic equation Humidity equation Transport equations of various physical/chemical species Equation of state Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Notations: total time derivative specific volume q specific humidity L latent heat Xi mass mixing ratios of physical or chemical species (e.g. aerosols, ozone) g gravity = gravitation g* + centrifugal force Spherical geopotential approximation is made: neglect Earth’s oblateness (~0.3%). => spherical geometry assumed! Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Euler Equations in spherical coordinates Momentum equations in spherical coordinates : x With Adiabatic formulation of the ECMWF model

Continuity equation in spherical coordinates
With Thermodynamic equation in spherical coordinates With Adiabatic formulation of the ECMWF model

Shallow atmosphere approximation
1. Replace r by the mean radius of the earth a and by where z is height above mean sea level. , 2. Neglect vertical and horizontal variations in g. 3. Neglect all metric terms not involving 4. Neglect the Coriolis terms containing (resulting from the horizontal component of ). a z a Adiabatic formulation of the ECMWF model

Euler Equations in shallow atmosphere approximation
is horizontal velocity n is the tracer for the hydrostatic approximation: n=0 => vertical momentum equation = hydrostatic eq. For n=1 we refer to these equations as Non-hydrostatic equations in Shallow atmosphere Approximation (NH-SA) Adiabatic formulation of the ECMWF model

=> Choice of predicted variables Predicted variables: or ?
(Allows enforcement of mass conservation) (Thermodyn. Computations are simpler) Combine continuity, thermodynamic & gas equation to obtain a prognostic equation for p. Form of NH-SA equations more commonly used in meteorology: => Adiabatic formulation of the ECMWF model

Hydrostatic Approximation (n=0)
Benefits from hydrostatic approximation Vertical momentum equation becomes a diagnostic relation (=> one prognostic variable (w) less!) Vertically propagating acoustic waves are eliminated (these are the fastest waves in the atmosphere, causing the biggest stability problems in numerical integrations!) Drawbacks of hydrostatic approximation Not valid for short horizontal scales (for mesoscale phenomena) Short gravity waves are distorted in the hydrostatic pressure field. Operational version of the ECMWF model is a hydrostatic model. operational horizontal resolution ~25km (T799), so hydrostatic approximation is (still) OK. Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Hydrostatic shallow atmosphere equations (Hydrostatic Primitive Equations (HPE)) p is monotonic function of z and can be used as vertical coordinate. (Eliassen (1949)) Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Choice of vertical coordinate Most commonly used vertical coordinates: Height above mean sea level z: - most natural vertical coordinate Pressure p (isobaric coordinate): - has advantages for thermodynamic calculations - makes the continuity equation a diagnostic relation in the hydrostatic system - can be extended for use in non-hydrostatic models Potential temperature (isentropic coordinate): good coordinate where atmosphere is stably stratified (potential temperature increases monotonic with z). adiabatic flow stays on isentropic surfaces (2D flow) good coord. in stratosphere, not very good in troposphere Adiabatic formulation of the ECMWF model

Choice of vertical coordinate (cont.)
Generalized vertical coordinate s: (Kasahara (1974), Staniforth & Wood (2003)) Any variable s which is a monotonic single-valued function of height z can be used as a vertical coordinate. Coordinate transformation rules (from z to any vertical coordinate s): Adiabatic formulation of the ECMWF model

Pressure p as vertical coordinate in the hydrostatic system
Hydrostatic relation between p and z: Coordinate transformation rules (from z to p): => with geopotential Adiabatic formulation of the ECMWF model

Hydrostatic Primitive Equations with pressure as vertical coordinate
Pressure gradient replaced by geopotential gradient (at constant pressure). pressure vertical velocity Continuity eq. is a diagnostic eq. in p-coordinates. Number of prognostic variables reduced to 3 (horizontal winds & T)! Geopotential computed from hydrostatic equation. Adiabatic formulation of the ECMWF model

Hydrostatic pressure as vertical coordinate
for a non-hydrostatic shallow atmosphere model Introduced by Laprise (1991) In the hydrostatic system with pressure as vertical coordinate the continuity equation is a diagnostic equation. The idea is to find a vertical coordinate for the NH system which makes the continuity equation a diagnostic equation. Continuity equation in generalized vertical coordinate s: (Kasahara(1974)) For every s for which => continuity is diagnostic eq.! Adiabatic formulation of the ECMWF model

Hydrostatic pressure as vertical coordinate
for a non-hydrostatic shallow atmosphere model (cont.) Choose and denote with the coordinate s for which i.e. For is the weight of a column of air (of unit area) above a point at height z, i.e. hydrostatic pressure. Adiabatic formulation of the ECMWF model

Hydrostatic pressure as vertical coordinate
for a non-hydrostatic shallow atmosphere model (cont.) in Z => D3 Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Boundary conditions Governing equations have to be solved subject to boundary conditions. The lower boundary of the atmosphere (surface of the earth) is a material boundary (air parcel cannot cross it!) velocity component perpendicular to surface has to vanish (e.g. at a flat and rigid surface vertical velocity w = 0) Unfortunately, the topography of the earth is far from flat, making it quite tricky to apply the lower boundary condition. Solution: Use a terrain-following vertical coordinate. For example: traditional sigma-coordinate (Phillips, 1957) Adiabatic formulation of the ECMWF model

Terrain-following vertical coordinate
We are looking for a vertical coordinate “s” which makes it easy to apply the condition of zero velocity normal to the boundary, even for very complex boundaries like the earth’s topography. The easiest case is a flat and rigid boundary where the boundary condition simply is: at the boundary. Therefore, we look to create a vertical coordinate which makes the upper and lower boundaries “flat”. That is, s is constant following the shape of the boundary (i.e. the boundary is a coordinate surface). at the top at the bottom & e.g. Adiabatic formulation of the ECMWF model

Terrain-following vertical coordinate (cont.)
The ECMWF model uses a terrain-following vertical coordinate based on hydrostatic pressure. The principle will be explained based on hydrostatic pressure : is a monotonic single-valued function of hydrostatic pressure and also depends on surface pressure in such a way that is the pressure at the top boundary.) (Where A simple function that fulfils these conditions is For this is the traditional sigma-coordinate of Phillips (1957) Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Sigma-coordinate (First introduced by Phillips (1957)) Drawback: Influence of topography is felt even in the upper levels far away from the surface. Remedy: Use hybrid sigma-pressure coordinates x Adiabatic formulation of the ECMWF model

Hybrid vertical coordinate
First introduced by Simmons and Burridge (1981). The difference to the sigma coordinate is in the way the monotonic relation between the new coordinate and hydrostatic pressure is defined: The functions A and B can be quite general and allow to design a hybrid sigma-pressure coordinate where the coordinate surfaces are sigma surfaces near the ground, gradually become more horizontal with increasing distance from the surface and turn into pure pressure surfaces in the stratosphere (B=0). In order that the top and bottom boundaries are coordinate surfaces (=> easy application of boundary condition), A and B have to fulfil: at the surface at the top Adiabatic formulation of the ECMWF model

sigma-coordinates & hybrid η-coordinates
Comparison of sigma-coordinates & hybrid η-coordinates Coordinate surfaces over a hill for sigma-coordinate η-coordinate Adiabatic formulation of the ECMWF model

Non-hydrostatic equations in hybrid vertical coordinate
prognostic continuity eq. Adiabatic formulation of the ECMWF model

Hydrostatic Primitive Equations in hybrid η vertical coordinate
In addition to the geopotential gradient term, a pressure gradient term again! Continuity equation is prognostic again because the (hydrostatic) pressure is not the vertical coordinate anymore. Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Hydrostatic equations of the ECMWF operational model (incorporating moisture) Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Notations: virtual temperature q specific humidity gas constant of dry air, gas constant of water vapour specific heat of dry air at constant pressure specific heat of water vapour at constant pressure p-coordinate vertical velocity X mass mixing ratio of physical or chemical species (e.g. aerosols, ozone) contributions from physical parametrizations horizontal diffusion terms Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
From the continuity equation with boundary conditions: we can derive (by vertical integration) the following equations: Needed for the energy-conversion term in the thermodynamic equation Needed for the semi-Lagrangian advection Prognostic equations for surface pressure B from def. of vert. coord. Adiabatic formulation of the ECMWF model

of the ECMWF hydrostatic model
Prognostic equations of the ECMWF hydrostatic model These equations are discretized and integrated in the ECMWF model. Adiabatic formulation of the ECMWF model

Discretisation in the ECMWF Model
Space discretisation In the horizontal: spectral transform method In the vertical: cubic finite-elements Time discretisation Semi-implicit semi-Lagrangian two-time-level scheme Adiabatic formulation of the ECMWF model

Horizontal discretisation
Options for discretisation are: in grid-point space only (grid-point model) finite-difference, finite volume methods (in spectral space only) in both grid-point and spectral space and transform back and forth between the two spaces (spectral transform method, spectral model) Gives the best of both worlds: Non-local operations (e.g. derivatives) are computed in spectral space (analytically) Local operations (e.g. products of terms) are computed in grid-point space The price to pay is in the cost of the transformations between the two spaces in finite-element space (basis functions with finite support) Adiabatic formulation of the ECMWF model

Horizontal discretisation (cont.)
ECMWF model uses the spectral transform method Representation in spectral space in terms of spherical harmonics: Ideally suited set of basis functions for spherical geometry (eigenfunctions of the Laplace operator). m: zonal wavenumber n: total wavenumber λ= longitude μ= sin(θ) θ: latitude Pnm: Associated Legendre functions of the first kind Adiabatic formulation of the ECMWF model

The horizontal spectral representation
Triangular truncation (isotropic) Spherical harmonics Fourier functions associated Legendre polynomials Legendre transform by Gaussian quadrature using NL  (2N+1)/2 “Gaussian” latitudes (linear grid) ((3N+1)/2 if quadratic grid) No “fast” algorithm available FFT (fast Fourier transform) using NF  2N+1 points (linear grid) (3N+1 if quadratic grid) Triangular truncation: n N m m = -N m = N Adiabatic formulation of the ECMWF model

Horizontal discretisation (cont.)
Representation in grid-point space is on the reduced Gaussian grid: Gaussian grid: grid of Guassian quadrature points (to facilitate accurate numerical computation of the integrals involved in the Fourier and Legendre transforms) - Gauss-Legendre quadrature in latitude: Grid-points in latitude are the zeros of the Legendre polynomial of order NG Gaussian latitudes NG  (2N+1)/2 for the linear grid. NG  (3N+1)/2 for the quadratic grid. - Gauss-Fourier quadrature in longitude: Grid-points in longitude are equidistantly spaced (Fourier) points 2N+1 for linear grid 3N+1 for quadratic grid Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
The Gaussian grid Reduced grid Full grid About 30% reduction in number of points • Associated Legendre functions are very small near the poles for large m Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
T T1279 25 km grid-spacing ( 843,490 grid-points) Current operational resolution 16 km grid-spacing (2,140,704 grid-points) Future operational resolution (from end 2009) Adiabatic formulation of the ECMWF model

Spectral transform method
Grid-point space -semi-Lagrangian advection -physical parametrizations FFT Inverse FFT Fourier Space Fourier Space Spectral space -horizontal gradients -semi-implicit calculations -horizontal diffusion LT Inverse LT FFT: Fast Fourier Transform, LT: Legendre Transform Adiabatic formulation of the ECMWF model

Horizontal discretisation (cont.)
Advantages of the spectral representation: a.) Horizontal derivatives are computed analytically => pressure-gradient terms are very accurate => no need to stagger variables on the grid b.) Spherical harmonics are eigenfunctions of the the Laplace operator => Solving the Helmholtz equation (arising from the semi-implicit method) is straightforward. => Applying high-order diffusion is very easy. Disadvantage: Computational cost of the Legendre transforms is high and grows faster with increasing horizontal resolution than the cost of the rest of the model. Adiabatic formulation of the ECMWF model

Comparison of cost profiles at different horizontal resolutions
Adiabatic formulation of the ECMWF model

Cost of Legendre transforms
Adiabatic formulation of the ECMWF model

Profile for T2047 on IBM p690+ (768 CPUs)
Legendre Transforms ~17% of total cost of model Physics ~36% of total cost Adiabatic formulation of the ECMWF model

Vertical discretisation
Variables are discretized on terrain-following pressure based hybrid η-levels. L91 L60 Vertical resolution of the operational ECMWF model: 91 hybrid η-levels resolving the atmosphere up to 0.01hPa (~80km) (upper mesosphere) Adiabatic formulation of the ECMWF model

Vertical discretisation (cont.)
Choices: - finite difference methods - finite element methods Operational version of the ECMWF model uses a cubic finite-element (FE) scheme based on cubic B-splines. No staggering of variables, i.e. all variables are held on the same vertical levels. (Good for semi-Lagrangian advection scheme.) Inspection of the governing equations shows that there are only vertical integrals (no derivatives) to be computed (if advection is done with semi-Lagrangian scheme). Adiabatic formulation of the ECMWF model

of the ECMWF hydrostatic model
Prognostic equations of the ECMWF hydrostatic model Reminder: slide 32 These equations are discretized and integrated in the ECMWF model. Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
From the continuity equation Reminder: slide 31 with boundary conditions: we can derive (by vertical integration) the following equations: Needed for the energy-conversion term in the thermodynamic equation Needed for the semi-Lagrangian advection Prognostic equations for surface pressure B from def. of vert. coord. Adiabatic formulation of the ECMWF model

Vertical integration in finite elements
can be approximated as Basis sets Applying the Galerkin method with test functions tj => Aji Bji Adiabatic formulation of the ECMWF model

Vertical integration in finite elements
Including the transformation from grid-point (GP) representation to finite-element representation (FE) and the projection of the result from FE to GP representation one obtains Matrix J depends only on the choice of the basis functions and the level spacing. It does not change during the integration of the model, so it needs to be computed only once during the initialisation phase of the model and stored. Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Cubic B-splines for regular spacing of levels (Prenter (1975)) Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Cubic B-splines as basis elements Basis elements for the represen- tation of the function to be integrated (integrand) f Basis elements for the representation of the integral F Adiabatic formulation of the ECMWF model

Benefits from using the finite-element scheme in the vertical
High order accuracy (8th order for cubic elements) Very accurate computation of the pressure-gradient term in conjunction with the spectral computation of horizontal derivatives More accurate vertical velocity for the semi-Lagrangian trajectory computation Improved ozone conservation Reduced vertical noise in the stratosphere No staggering of variables required in the vertical: good for semi-Lagrangian scheme because winds and advected variables are represented on the same vertical levels. Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Discretisation in time Decisions to be taken: Discretize on three time-levels (e.g. leapfrog scheme) - produce a computational mode (time-filtering needed) two time-levels - more efficient than three-time-level schemes - (less stable) How to discretize the right-hand sides of the equations in time: - explicitly - implicitly - semi-implicitly How to treat the advection: - in Eulerian way - in semi-Lagrangian way Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Discretization in time (cont.) Operational version of the ECMWF model uses Two-time-level scheme Semi-implicit treatment of the right-hand sides Semi-Lagrangian advection Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Time discretisation of the right-hand sides Discretisation of the right-hand side (RHS) of the equations: RHS taken at the centre of the time interval: explicit (second order) discretisation. Stability is subject to CFL-like criterion - RHS average of its value at initial time and at final time: implicit discretisation (generally stable) => leads to a difficult system to solve (iterative solvers) treat only some linearized terms of RHS implicitly (semi-implicit discretisation) Adiabatic formulation of the ECMWF model

Semi-implicit time integration
For compact notation define: “implicit correction term” Notations: X : advected variable RHS: right-hand side of the equation L: part of RHS treated implicitly Superscripts: “0” indicates value for explicit discretiz, “-” indicates value at start of time step “+” indicates value at end of time step => L=RHS => implicit scheme L= part of RHS => semi-implicit Benefit: slowing-down of the waves too fast for the explicit CFL cond. Drawback: overhead of having to solve an elliptic boundary-value prob. Adiabatic formulation of the ECMWF model

Semi-implicit time integration (cont.)
Choice of which terms in RHS to treat implicitly is guided by the knowledge of which waves cause instability because they are too fast (violate the CFL condition) and need to be slowed down with an implicit treatment. In a hydrostatic model, fastest waves are horizontally propagating external gravity waves (long surface gravity waves), Lamb waves (acoustic wave not filtered out by the hydrostatic approximation) and long internal gravity waves. => implicit treatment of the adjustment terms. L= linearization of part of RHS (i.e. terms supporting the fast modes) => good chance of obtaining a system of equations in the variables at “+” that can be solved almost analytically in a spectral model. Adiabatic formulation of the ECMWF model

Semi-implicit time integration (cont.)
semi-implicit corrections semi-implicit equations Adiabatic formulation of the ECMWF model

Semi-implicit time integration (cont.)
equations Reference state for linearization: ref. temperature ref. surf. pressure Where: => lin. geopotential for X=T => lin. energy conv. term for X=D Adiabatic formulation of the ECMWF model

Semi-implicit time integration (cont.)
equations Reference state for linearization: ref. temperature ref. surf. pressure By eliminating in above system all but one of the unknowns (D+) => operator acting only on the vertical unity operator Adiabatic formulation of the ECMWF model

Semi-implicit time integration (cont.)
Vertically coupled set of Helmholtz equations. Coupling through Uncouple by transforming to the eigenspace of this matrix gamma (i.e. diagonalize gamma). Unity matrix “I” stays diagonal. => One equation for each In spectral space (spherical harmonics space): because Once D+ has been computed, it is easy to compute the other variables at “+”. Adiabatic formulation of the ECMWF model

Semi-Lagrangian (SL) schemes are more efficient & more stable than Eulerian advection schemes. Coupling SL advection with semi-implicit treatment of the fast modes results in a very stable scheme where the timestep can be chosen on the basis of accuracy rather than for stability. Disadvantage: Lack of conservation of mass and tracer concentrations. (More difficult to enforce conservation than in Eulerian schemes) Adiabatic formulation of the ECMWF model

All equations are of this (Lagrangian) form: (See slide 32) Three time-level scheme: x x x x A D * x M Centred second order accurate scheme Ingredients of semi-Lagrangian advection are: 1.) Computation of the departure point (tajectory computation) 2.) Interpolation of the advected fields at the departure location Adiabatic formulation of the ECMWF model

of the ECMWF hydrostatic model
Prognostic equations of the ECMWF hydrostatic model Reminder: slide 32 These equations are discretized and integrated in the ECMWF model. Adiabatic formulation of the ECMWF model

Two-time-level second order accurate schemes : with Extrapolation in time to middle of time interval Unstable! => noisy forecasts Forecast of temperature at 200 hPa (from 1997/01/04) Adiabatic formulation of the ECMWF model

Stable extrapolating two-time-level semi-Lagrangian
(SETTLS): Taylor expansion to second order With and Forecast 200 hPa T from 1997/01/04 using SETTLS Adiabatic formulation of the ECMWF model

Interpolation in the semi-Lagrangian scheme
ECMWF model uses quasi-monotone quasi-cubic Lagrange interpolation Cubic Lagrange interpolation: with the weights x y Number of 1D cubic interpolations in two dimensions is 5, in three dimensions 21! To save on computations: cubic interpolation only for nearest neighbour rows, linear interpolation for rest => “quasi-cubic interpolation” => 7 cubic + 10 linear in 3 dimensions. Adiabatic formulation of the ECMWF model

Interpolation in the semi-Lagrangian scheme (cont)
Quasi-monotone interpolation: x interpolated cubically x: interpolation point quasi-monotone procedure: x x: grid points Quasi-monotone interpolation is used in the horizontal for all variables and also in the vertical for humidity and all “tracers” (e.g. ozone, aerosols). Has a detrimental effect on conservation, but prevents unphysical negative concentrations. Adiabatic formulation of the ECMWF model

Modified continuity & thermodynamic equations
Accuracy of cubic interpolation is much reduced when the field to be interpolated is rough (e.g. surface pressure over orography) Idea by Ritchie & Tanguay (1996): Subtract a time-independent term from the surface pressure which “contains” a large part of the orographic influence on surface pressure, advect the rest (smoother term) and treat the advection of the “rough term” with the right-hand side of the continuity equation [RHS]. Continuity equation => Reduced mass loss/gain during a forecast. Adiabatic formulation of the ECMWF model

Modified continuity & thermodynamic equations (cont)
Similar idea for thermodynamic equation: (Hortal &Temperton (2001)) Thermodynamic equation with Approximation to the change of T with height in the standard atmosphere. Reduces noise levels over orography in all fields, but in particular in vertical velocity. Adiabatic formulation of the ECMWF model

Momentum eq. is discretized in vector form (because a vector is continuous across the poles, components are not!) Interpolations at departure point are done for components u & v of the velocity vec- tor relative to the system of reference local at D. Interpolated values are to be used at A, so the change of reference system from D to A needs to be taken into account. Trajectories are arcs of great circles if constant (angular) velocity is assumed for the duration of a time step. X Y Z A V x D M i j A D Tangent plane projection Trajectory calculation Adiabatic formulation of the ECMWF model

Treatment of the Coriolis term
In three-time-level semi-Lagrangian: • treated explicitly with the rest of the RHS In two-time-level semi-Lagrangian: Extrapolation in time to the middle of the trajectory leads to instability (Temperton (1997)) Two stable options: • Advective treatment: (Vector R here is the position vector.) • Implicit treatment : Helmholtz eqs partially coupled for individual spectral components => tri-diagonal system to be solved. Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Summary of the adiabatic formulation of the operational ECMWF atmospheric model Hydrostatic shallow-atmosphere equations with pressure-based hybrid vertical coordinate Two-time-level semi-Lagrangian advection SETTLS (Stable Extrapolation Two-Time-Level Scheme) Quasi-monotone quasi-cubic Lagrange interpol. at departure point Linear interpolation for trajectory computations and RHS terms Modified continuity & thermodynamic equations to advect smoother fields (net of the orographic roughness) Semi-implicit treatment of linearized adjustment terms & Coriolis terms Cubic finite elements for the vertical integrals Spectral horizontal Helmholtz solver (and derivative computations) Uses the linear reduced Gaussian grid Adiabatic formulation of the ECMWF model

Adiabatic formulation of the ECMWF model
Thank you very much for your attention Adiabatic formulation of the ECMWF model