GFS gwdps Main

This subroutine includes orographic gravity wave drag and mountain blocking. More...

Detailed Description

The time tendencies of zonal and meridional wind are altered to include the effect of mountain induced gravity wave drag from subgrid scale orography including convective breaking, shear breaking and the presence of critical levels.

Argument Table

local_name	standard_name	long_name	units	rank	type	kind	intent	optional
im	horizontal_loop_extent	horizontal loop extent	count	0	integer		in	F
ix	horizontal_dimension	horizontal dimension	count	0	integer		in	F
km	vertical_dimension	number of vertical layers	count	0	integer		in	F
A	tendency_of_y_wind_due_to_model_physics	meridional wind tendency due to model physics	m s-2	2	real	kind_phys	inout	F
B	tendency_of_x_wind_due_to_model_physics	zonal wind tendency due to model physics	m s-2	2	real	kind_phys	inout	F
C	tendency_of_air_temperature_due_to_model_physics	air temperature tendency due to model physics	K s-1	2	real	kind_phys	inout	F
u1	x_wind	zonal wind	m s-1	2	real	kind_phys	in	F
v1	y_wind	meridional wind	m s-1	2	real	kind_phys	in	F
t1	air_temperature	mid-layer temperature	K	2	real	kind_phys	in	F
q1	water_vapor_specific_humidity	mid-layer specific humidity of water vapor	kg kg-1	2	real	kind_phys	in	F
kpbl	vertical_index_at_top_of_atmosphere_boundary_layer	vertical index at top atmospheric boundary layer	index	1	integer		in	F
prsi	air_pressure_at_interface	interface pressure	Pa	2	real	kind_phys	in	F
del	air_pressure_difference_between_midlayers	difference between mid-layer pressures	Pa	2	real	kind_phys	in	F
prsl	air_pressure	mid-layer pressure	Pa	2	real	kind_phys	in	F
prslk	dimensionless_exner_function_at_model_layers	mid-layer Exner function	none	2	real	kind_phys	in	F
phii	geopotential_at_interface	interface geopotential	m2 s-2	2	real	kind_phys	in	F
phil	geopotential	mid-layer geopotential	m2 s-2	2	real	kind_phys	in	F
deltim	time_step_for_physics	physics time step	s	0	real	kind_phys	in	F
kdt	index_of_time_step	current time step index	index	0	integer		in	F
hprime	standard_deviation_of_subgrid_orography	standard deviation of subgrid orography	m	1	real	kind_phys	in	F
oc	convexity_of_subgrid_orography	convexity of subgrid orography	none	1	real	kind_phys	in	F
oa4	asymmetry_of_subgrid_orography	asymmetry of subgrid orography	none	2	real	kind_phys	in	F
clx4	fraction_of_grid_box_with_subgrid_orography_higher_than_critical_height	horizontal fraction of grid box covered by subgrid orography higher than critical height	frac	2	real	kind_phys	in	F
theta	angle_from_east_of_maximum_subgrid_orographic_variations	angle with respect to east of maximum subgrid orographic variations	degrees	1	real	kind_phys	in	F
sigma	slope_of_subgrid_orography	slope of subgrid orography	none	1	real	kind_phys	in	F
gamma	anisotropy_of_subgrid_orography	anisotropy of subgrid orography	none	1	real	kind_phys	in	F
elvmax	maximum_subgrid_orography	maximum of subgrid orography	m	1	real	kind_phys	inout	F
dusfc	instantaneous_x_stress_due_to_gravity_wave_drag	zonal surface stress due to orographic gravity wave drag	Pa	1	real	kind_phys	out	F
dvsfc	instantaneous_y_stress_due_to_gravity_wave_drag	meridional surface stress due to orographic gravity wave drag	Pa	1	real	kind_phys	out	F
g	gravitational_acceleration	gravitational acceleration	m s-2	0	real	kind_phys	in	F
cp	specific_heat_of_dry_air_at_constant_pressure	specific heat of dry air at constant pressure	J kg-1 K-1	0	real	kind_phys	in	F
rd	gas_constant_dry_air	ideal gas constant for dry air	J kg-1 K-1	0	real	kind_phys	in	F
rv	gas_constant_water_vapor	ideal gas constant for water vapor	J kg-1 K-1	0	real	kind_phys	in	F
imx	number_of_equatorial_longitude_points	number of longitude points along the equator	count	0	integer		in	F
nmtvr	number_of_statistical_measures_of_subgrid_orography	number of statistical measures of subgrid orography	count	0	integer		in	F
cdmbgwd	multiplication_factors_for_mountain_blocking_and_orographic_gravity_wave_drag	multiplic. factors for (1) mountain blocking drag coeff. and (2) ref. level orographic gravity wave drag	none	1	real	kind_phys	in	F
me	mpi_rank	rank of the current MPI task	index	0	integer		in	F
lprnt	flag_print	flag for debugging printouts	flag	0	logical		in	F
ipr	horizontal_index_of_printed_column	horizontal index of column used in debugging printouts	index	0	integer		in	F
rdxzb	level_of_dividing_streamline	level of the dividing streamline	none	1	real	kind_phys	out	F
errmsg	error_message	error message for error handling in CCPP	none	0	character	len=*	out	F
errflg	error_flag	error flag for error handling in CCPP	flag	0	integer		out	F

GFS Orographic GWD Scheme General Algorithm

1. Calculate subgrid mountain blocking
2. Calculate orographic wave drag

The NWP model gravity wave drag (GWD) scheme in the GFS has two main components: how the surface stress is computed, and then how that stress is distributed over a vertical column where it may interact with the models momentum. Each of these depends on the large scale environmental atmospheric state and assumptions about the sub-grid scale processes. In Alpert GWD (1987) based on linear, two-dimensional non-rotating, stably stratified flow over a mountain ridge, sub-grid scale gravity wave motions are assumed which propagate away from the mountain. Described in Alpert (1987), the flux measured over a "low level" vertically averaged layer, in the atmosphere defines a base level flux. "Low level" was taken to be the first 1/3 of the troposphere in the 1987 implementation. This choice was meant to encompass a thick low layer for vertical averages of the environmental (large scale) flow quantities. The vertical momentum flux or gravity wave stress in a grid box due to a single mountain is given as in Pierrehumbert, (1987) (PH):

\( \tau = \frac {\rho \: U^{3}\: G(F_{r})} {\Delta X \; N } \)

emetic \( \Delta X \) is a grid increment, N is the Brunt Viasala frequency

\( N(\sigma) = \frac{-g \: \sigma \: \frac{\partial\Theta}{\partial\sigma}}{\Theta \:R \:T} \)

The environmental variables are calculated from a mass weighted vertical average over a base layer. G(Fr) is a monotonically increasing function of Froude number,

\( F_{r} = \frac{N h^{'}}{U} \)

where U is the wind speed calculated as a mass weighted vertical average in the base layer, and h', is the vertical displacement caused by the orography variance. An effective mountain length for the gravity wave processes,

\( l^{*} = \frac{\Delta X}{m} \)

where m is the number of mountains in a grid box, can then be defined to obtain the form of the base level stress

\( \tau = \frac {\rho \: U^{3} \: G(F_{r})} {N \;l^{*}} \)

giving the stress induced from the surface in a model grid box. PH gives the form for the function G(Fr) as

\( G(F_{r}) = \bar{G}\frac{F^{2}_{r}}{F^{2}_{r}\: + \:a^{2}} \)

Where \( \bar{G} \) is an order unity non-dimensional saturation flux set to 1 and 'a' is a function of the mountain aspect ratio also set to 1 in the 1987 implementation of the GFS GWD. Typical values of U=10m/s, N=0.01 1/s, l*=100km, and a=1, gives a flux of 1 Pascal and if this flux is made to go to zero linearly with height then the decelerations would be about 10/m/s/day which is consistent with observations in PH.

In Kim, Moorthi, Alpert's (1998, 2001) GWD currently in GFS operations, the GWD scheme has the same physical basis as in Alpert (1987) with the addition of enhancement factors for the amplitude, G, and mountain shape details in G(Fr) to account for effects from the mountain blocking. A factor, E m’, is an enhancement factor on the stress in the Alpert '87 scheme. The E ranges from no enhancement to an upper limit of 3, E=E(OA)[1-3], and is a function of OA, the Orographic Asymmetry defined in KA (1995) as

Orographic Asymmetry (OA) = \( \frac{ \bar{x} \; - \; \sum\limits_{j=1}^{N_{b}} x_{j} \; n_{j} }{\sigma_{x}} \)

where Nb is the total number of bottom blocks in the mountain barrier, \( \sigma_{x} \) is the standard deviation of the horizontal distance defined by

\( \sigma_{x} = \sqrt{ \frac{\sum\limits_{j=1}^{N_{b}} \; (x_{j} \; - \; \bar{x} )^2}{N_{x}} } \)

where Nx is the number of grid intervals for the large scale domain being considered. So the term, E(OA)m’/ \( \Delta X \) in Kim's scheme represents a multiplier on G shown in Alpert's eq (1), where m’ is the number of mountains in a sub-grid scale box. Kim increased the complexity of m’ making it a function of the fractional area of the sub-grid mountain and the asymmetry and convexity statistics which are found from running a gravity wave model for a large number of cases:

\( m^{'} = C_{m} \Delta X \left[ \frac{1 \; + \; \sum\limits_{x} L_{h} }{\Delta X} \right]^{OA+1} \)

Where, according to Kim, \( \sum \frac{L_{h}}{\Delta X} \) is the fractional area covered by the subgrid-scale orography higher than a critical height \( h_{c} = Fr_{c} U_{0}/N_{0} \) , over the "low level" vertically averaged layer, for a grid box with the interval \( \Delta X \). Each \( L_{n}\) is the width of a segment of orography intersection at the critical height:

\( Fr_{0} = \frac{N_{0} \; h^{'}}{U_{0}} \)

\( G^{'}(OC,Fr_{0}) = \frac{Fr_{0}^{2}}{Fr_{0}^{2} \; + \; a^{2}} \)

\( a^{2} = \frac{C_{G}}{OC} \)

\( E(OA, Fr_{0}) = (OA \; + \; 2)^{\delta} \) and \( \delta \; = \; \frac{C_{E} \; Fr_{0}}{Fr_{c}} \) where \( Fr_{c} \) is as in Alpert.

This represents a closed scheme, somewhat empirical adjustments to the original scheme to calculate the surface stress.

Momentum is deposited by the sub-grid scale gravity waves break due to the presence of convective mixing assumed to occur when the minimum Richardson number:

Orographic Convexity (OC) = \( \frac{ \sum\limits_{j=1}^{N_{x}} \; (h_{j} \; - \; \bar{h})^4 }{N_{x} \;\sigma_{h}^4} \) , and where \( \sigma_{h} = \sqrt{ \frac{\sum\limits_{j=1}^{N_{x}} \; (h_{j} \; - \; \bar{h} )^2}{N_{x}} } \)

This represents a closed scheme, somewhat empirical adjustments to the original scheme to calculate the surface stress.

Momentum is deposited by the sub-grid scale gravity waves break due to the presence of convective mixing assumed to occur when the minimum Richardson number:

\( Ri_{m} = \frac{Ri(1 \; - \; Fr)}{(1 \; + \; \sqrt{Ri}Fr)^2} \)

Is less than 1/4 Or if critical layers are encountered in a layer the the momentum flux will vanish. The critical layer is defined when the base layer wind becomes perpendicular to the environmental wind. Otherwise, wave breaking occurs at a level where the amplification of the wave causes the local Froude number or similarly a truncated (first term of the) Scorer parameter, to be reduced below a critical value by the saturation hypothesis (Lindzen,). This is done through eq 1 which can be written as

\( \tau = \rho U N k h^{'2} \)

For small Froude number this is discretized in the vertical so at each level the stress is reduced by ratio of the Froude or truncated Scorer parameter, \( \frac{U^{2}}{N^{2}} = \frac{N \tau_{l-1}}{\rho U^{3} k} \) , where the stress is from the layer below beginning with that found near the surface. The respective change in momentum is applied in that layer building up from below.

An amplitude factor is part of the calibration of this scheme which is a function of the model resolution and the vertical diffusion. This is because the vertical diffusion and the GWD account encompass similar physical processes. Thus, one needs to run the model over and over for various amplitude factors for GWD and vertical diffusion.

In addition, there is also mountain blocking from lift and frictional forces. Improved integration between how the GWD is calculated and the mountain blocking of wind flow around sub-grid scale orography is underway at NCEP. The GFS already has convectively forced GWD an independent process. The next step is to test

GFS Orographic GWD Scheme Detailed Algorithm

Functions/Subroutines
subroutine	gwdps::gwdps_run ( IM, IX, KM, A, B, C, U1, V1, T1, Q1, KPBL, PRSI, DEL, PRSL, PRSLK, PHII, PHIL, DELTIM, KDT, HPRIME, OC, OA4, CLX4, THETA, SIGMA, GAMMA, ELVMAX, DUSFC, DVSFC, G, CP, RD, RV, IMX, nmtvr, cdmbgwd, me, lprnt, ipr, rdxzb, errmsg, errflg)