Nonhydrostatic RSM - equations in hydrostatic . coordinate

Nonhydrostatic RSM

- Basic equations and concepts- definition of the coordinate- equations in hydrostatic � coordinate- multi-options to the coordinate- energetic

- Numerical techniques- implement into the RSM- modification to techniques- treatment in model physics- initialization

- Uniqueness- multi-evolved coordinate- spectral perturbation nesting- Mesoscale Spectral Model (MSM)

Nonhydrostatic RSM (MSM)definition of the coordinate

Following Juang (1992), the definitionof hydrostatic relation, used for thecoordinate transformation from zcoordinate to � coordinate, is given by

(2.1)

where satisfies the equation of statein hydrostatic system as

(2.2)and g is gravitational constant, R is gasconstant for dry air, z is physical height,

, and are density, pressureand virtual temperature in hydrostaticsystem.


Next, the vertical coordinate is definedas

(2.3)

where is hydrostatic pressure atsurface. Then the hydrostatic relation ofEq. 2.1 can be rewritten as

(2.4)

for the coordinate transformation.Obviously, it is the same definition asdescribed in Phillips (1957) for thehydrostatic system in � coordinates, but itis used to define the coordinate for thenonhydrostatic system here.


Using the above definition for thecoordinate, the coordinatetransformations can be given as

(2.5)

for vertical transformation, and

(2.6)

for others, where s can be either x or y forspacial derivative, or t for temporalderivative (see Eq. 5 in Juang 1992), and

is geopotential.

Nonhydrostatic RSM (MSM)equations in hydrostatic � coordinate

Using Eqs. 2.5 and 2.6, the fullycompressible nonhydrostatic system in zcoordinate (see Eq. 1 in Juang, 1992) canbe directly transformed into the �coordinate without any assumption as

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)

(2.12)

Nonhydrostatic RSM (MSM)equations in hydrostatic � coordinate

The symbols used are following:

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

(2.19)

(

Nonhydrostatic RSM (MSM)multi-options to the coordinate

How to specify the coordinate? It canbe time-independent or time-dependent.For time-dependent, it can be internally-evolved or externally-evolved. Forinternally-evolved, we started from massconservation. The conservation of densitycan be written in z coordinate as

(2.21)

which is the same as Eq. 1-51 in Haltinerand Williams (1980) but hydrostaticdensity is used here, not total density.The above equation can be written in �coordinate as

(2.22)

by the same manipulation described inHaltiner and Williams (1980)


Eq. 2.22 can be written as, by integratingthe entire vertical domain,

(2.23)

The internally-evolved hydrostatic surfacepressure can be obtained by Eq. 2.23

Next, the hydrostatic temperature hasto be determined by thermodynamicconservation in nonhydrostatic system.Again, with the coordinate transform andthe equation of state, the thermodynamicequation for hydrostatic temperature canbe manipulated into

(2.24)

where total wind is used to derive theinternally-evolved hydrostatictemperature.


The equations introduced here can becoded into the model with options, so wecan switch easily among these conditions.They can be summarized as

(2.25)

(2.26)

for two prognostic equations, and anotherform of Eq. 2.22

(2.27)

is necessary for solving Eq. 2.26 here.


Basically, we can run model with �T=�Q =1, but for simplicity and time-saving

(1) in cases of idealized studies�T =0, �Q =0 => time-independent

Or externally-evolved

(2) in cases of real-data experiments�T =1, �Q =1 => internally-evolved

or�T =0, �Q =1 => internally-evolvedsurface pressure and externally-evolved temperature.

Nonhydrostatic RSM (MSM)energetic

The total energy equation can beformed by the summation of the followingequations, which are from the previoussections, without considering sub-gridforcing and map factor for simplicity:

(2.28)

(2.29)

(2.30)

(2.31)

(2.32)

where total energy is

(2.33)


After some manipulations andcancellations of some identical terms, thesummation of Eqs. 2.28 to 2.32 can bewritten as

(2.34)

which can be easily shown that the resultof right-hand-side is zero by using

in the following:

(2.35)


Thus, the Eq. 2.34 can be written as

which shows the total energy isconserved. Note that, all prognosticequations are used except the hydrostatictemperature equation. It implies thehydrostatic temperature, Eq. 2.24, maynot be critical for energy conservation,and it can be specified by any method,either internally-evolved one or externallyevolved one, however, the continuityequation of Eq. 2.23 is necessary, i.e., theinternally-evolved surface pressure isrequired to have energy conservation.

Nonhydrostatic RSM (MSM)implement into the RSM

Variable decomposition for spectralcomputation

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)and

(3.7)

(3.8)where subscript b shows variables fromthe coarse outer grid system, subscript sshows variables on the surface.

Nonhydrostatic RSM (MSM)implement into the RSM

The fully compressible nonhydrostaticsystem in the hydrostatic � coordinatecan be rewritten into a perturbation formas

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

- For the internally-evolved, �Q = �T = 1, and allequations are used.

- For internally-evolved hydrostatic pressure butexternally-evolved hydrostatic temperature,�Q = 1, �T = 0, , , andEq.3.13 is not necessary.

- For externally-evolved, �T = �Q = 0,, , ,

and Eqs. 3.12 and 3.13 arenot necessary.

Nonhydrostatic RSM (MSM)modification to techniques

Most of the numerical methods used inthe RSM to stabilize the model integrationare adopted. They are - perturbation method, - spectral computation, - fourth-order horizontal diffusion, - implicit lateral boundary relaxation, - time filter, and - semi-implicit adjustment.a larger coefficient of the time filter andforward-time-weighted semi-implicitadjustment have to be used for a stableintegration with a larger time step. througha simple numerical stability analysis(JHK97).

Nonhydrostatic RSM (MSM)treatment in model physics

All model physics used in Juang andKanamitsu (1994) with someimprovements in Juang et al (1997) areused, however, the part of the modelphysics having a hydrostatic assumptionor relationship has to be reworked.

The model physics codes aremodified to have options to run with bothhydrostatic and nonhydrostatic models.All the vertical integrations of the air massin the hydrostatic model with (ps��)/ghave been changed to ��z.

Nonhydrostatic RSM (MSM)treatment in model physics

In the hydrostatic, the changes of thetemperature due to diabatic effectcontribute to the changes of thehydrostatic thickness as

(3.21)

In the nonhydrostatic, the change ofthe temperature due to any diabaticheating effect give the changes ofpressure as

(3.22)

A forward-time-weighted semi-implicitadjustment is used to re-adjust thehydrostatic reference state and thenonhydrostatic system together after the

physics step.

Nonhydrostatic RSM (MSM)initialization

There is no initialization used in the casesdescribed in this paper, even though thereis a forward digital filter initialization builtinto the model (JK94) as an option. Forreal data cases, the initial condition for theMSM is obtained only by three-dimensional interpolation from the globalor regional spectral model analysis oroutputs to the MSM. In other word, thehydrostatic state is used as the initialcondition for the MSM.

When the initial condition is assumedto be hydrostatic, some manipulations toobtain the vertical velocity, w (=dz/dt), isrequired because w is not explicitlyexisted in the hydrostatic system.


In order to have the vertical velocity tosatisfy the hydrostatic motion initially,Eqs. 2.14 and 2.26 are used to determinevertical velocity with all variables inhydrostatic state for initial condition. Eq.2.14 can be rewritten as

(3.23)

for w, where in the first term of the righthand side can be obtained by Eqs. 2.22and 2.23, and

(3.24)

and(3.25)

where matrix A is defined in Juang 1992.


Eqs. 3.24 and 3.25 that the initialvertical motion, w, in this method isadiabatic and hydrostatic.

In the dynamic step, all computation isadiabatic. If the w is given initially asmentioned in the previous transparency,then will be hydrostatic and adiabaticinitially, as the same as that in thehydrostatic model. Thus it is considered tobe a correct specification for the initialvalue of w, which is a hydrostatic motionunder the hydrostatic initial condition.

Nonhydrostatic RSM (MSM)configurations

- In DCLSYSFor hydrostatic run, you do##NON = ‘C-NON’;##HYD = ‘ ’;For nonhydrostatic run, you do##NON = ‘ ’;##HYD = ‘C-HYD’;

For multi-options of coordinate, weintroduce

##ET = ‘ ’; to turn on for �T=1##ET = ‘C-ET’; to turn off for �T=0

and ##EQ = ‘ ’; to turn on for �Q=1##EQ = ‘C-EQ’; to turn off for �Q=0

for future version.

Nonhydrostatic RSM - equations in hydrostatic . coordinate

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