drag_suite_run()

subroutine drag_suite::drag_suite_run	(	integer, intent(in)	IM,
		integer, intent(in)	KM,
		real(kind=kind_phys), dimension(:,:), intent(inout)	dvdt,
		real(kind=kind_phys), dimension(:,:), intent(inout)	dudt,
		real(kind=kind_phys), dimension(:,:), intent(inout)	dtdt,
		real(kind=kind_phys), dimension(:,:), intent(in)	U1,
		real(kind=kind_phys), dimension(:,:), intent(in)	V1,
		real(kind=kind_phys), dimension(:,:), intent(in)	T1,
		real(kind=kind_phys), dimension(:,:), intent(in)	Q1,
		integer, dimension(:), intent(in)	KPBL,
		real(kind=kind_phys), dimension(:,:), intent(in)	PRSI,
		real(kind=kind_phys), dimension(:,:), intent(in)	DEL,
		real(kind=kind_phys), dimension(:,:), intent(in)	PRSL,
		real(kind=kind_phys), dimension(:,:), intent(in)	PRSLK,
		real(kind=kind_phys), dimension(:,:), intent(in)	PHII,
		real(kind=kind_phys), dimension(:,:), intent(in)	PHIL,
		real(kind=kind_phys), intent(in)	DELTIM,
		integer, intent(in)	KDT,
		real(kind=kind_phys), dimension(:), intent(in)	var,
		real(kind=kind_phys), dimension(:), intent(in)	oc1,
		real(kind=kind_phys), dimension(:,:), intent(in)	oa4,
		real(kind=kind_phys), dimension(:,:), intent(in)	ol4,
		real(kind=kind_phys), dimension(:), intent(in)	varss,
		real(kind=kind_phys), dimension(:), intent(in)	oc1ss,
		real(kind=kind_phys), dimension(:,:), intent(in)	oa4ss,
		real(kind=kind_phys), dimension(:,:), intent(in)	ol4ss,
		real(kind=kind_phys), dimension(:), intent(in)	THETA,
		real(kind=kind_phys), dimension(:), intent(in)	SIGMA,
		real(kind=kind_phys), dimension(:), intent(in)	GAMMA,
		real(kind=kind_phys), dimension(:), intent(in)	ELVMAX,
		real(kind=kind_phys), dimension(:,:), intent(inout)	dtaux2d_ms,
		real(kind=kind_phys), dimension(:,:), intent(inout)	dtauy2d_ms,
		real(kind=kind_phys), dimension(:,:), intent(inout)	dtaux2d_bl,
		real(kind=kind_phys), dimension(:,:), intent(inout)	dtauy2d_bl,
		real(kind=kind_phys), dimension(:,:), intent(inout)	dtaux2d_ss,
		real(kind=kind_phys), dimension(:,:), intent(inout)	dtauy2d_ss,
		real(kind=kind_phys), dimension(:,:), intent(inout)	dtaux2d_fd,
		real(kind=kind_phys), dimension(:,:), intent(inout)	dtauy2d_fd,
		real(kind=kind_phys), dimension(:), intent(inout)	dusfc,
		real(kind=kind_phys), dimension(:), intent(inout)	dvsfc,
		real(kind=kind_phys), dimension(:), intent(inout)	dusfc_ms,
		real(kind=kind_phys), dimension(:), intent(inout)	dvsfc_ms,
		real(kind=kind_phys), dimension(:), intent(inout)	dusfc_bl,
		real(kind=kind_phys), dimension(:), intent(inout)	dvsfc_bl,
		real(kind=kind_phys), dimension(:), intent(inout)	dusfc_ss,
		real(kind=kind_phys), dimension(:), intent(inout)	dvsfc_ss,
		real(kind=kind_phys), dimension(:), intent(inout)	dusfc_fd,
		real(kind=kind_phys), dimension(:), intent(inout)	dvsfc_fd,
		real(kind=kind_phys), dimension(:), intent(in)	slmsk,
		real(kind=kind_phys), dimension(:), intent(in)	br1,
		real(kind=kind_phys), dimension(:), intent(in)	hpbl,
		real(kind=kind_phys), intent(in)	g,
		real(kind=kind_phys), intent(in)	cp,
		real(kind=kind_phys), intent(in)	rd,
		real(kind=kind_phys), intent(in)	rv,
		real(kind=kind_phys), intent(in)	fv,
		real(kind=kind_phys), intent(in)	pi,
		integer, intent(in)	imx,
		real(kind=kind_phys), dimension(:), intent(in)	cdmbgwd,
		integer, intent(in)	me,
		integer, intent(in)	master,
		logical, intent(in)	lprnt,
		integer, intent(in)	ipr,
		real(kind=kind_phys), dimension(:), intent(inout)	rdxzb,
		real(kind=kind_phys), dimension(:), intent(in)	dx,
		integer, intent(in)	gwd_opt,
		logical, intent(in)	do_gsl_drag_ls_bl,
		logical, intent(in)	do_gsl_drag_ss,
		logical, intent(in)	do_gsl_drag_tofd,
		real(kind=kind_phys), dimension(:,:,:), intent(inout)	dtend,
		integer, dimension(:,:), intent(in)	dtidx,
		integer, intent(in)	index_of_process_orographic_gwd,
		integer, intent(in)	index_of_temperature,
		integer, intent(in)	index_of_x_wind,
		integer, intent(in)	index_of_y_wind,
		logical, intent(in)	ldiag3d,
		logical, intent(in)	ldiag_ugwp,
		logical, intent(in)	ugwp_seq_update,
		real(kind=kind_phys), dimension(:,:), intent(in)	spp_wts_gwd,
		integer, intent(in)	spp_gwd,
		character(len=*), intent(out)	errmsg,
		integer, intent(out)	errflg
	)

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	type	dimensions	kind	intent
im	horizontal_loop_extent	horizontal loop extent	count	integer	()		in
km	vertical_layer_dimension	number of vertical layers	count	integer	()		in
dvdt	process_split_cumulative_tendency_of_y_wind	meridional wind tendency due to model physics	m s-2	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	inout
dudt	process_split_cumulative_tendency_of_x_wind	zonal wind tendency due to model physics	m s-2	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	inout
dtdt	process_split_cumulative_tendency_of_air_temperature	air temperature tendency due to model physics	K s-1	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	inout
u1	x_wind	zonal wind	m s-1	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	in
v1	y_wind	meridional wind	m s-1	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	in
t1	air_temperature	mid-layer temperature	K	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	in
q1	specific_humidity	mid-layer specific humidity of water vapor	kg kg-1	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	in
kpbl	vertical_index_at_top_of_atmosphere_boundary_layer	vertical index at top atmospheric boundary layer	index	integer	(ccpp_constant_one:horizontal_loop_extent)		in
prsi	air_pressure_at_interface	interface pressure	Pa	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_interface_dimension)	kind_phys	in
del	air_pressure_difference_between_midlayers	difference between mid-layer pressures	Pa	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	in
prsl	air_pressure	mid-layer pressure	Pa	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	in
prslk	dimensionless_exner_function	mid-layer Exner function	none	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	in
phii	geopotential_at_interface	interface geopotential	m2 s-2	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_interface_dimension)	kind_phys	in
phil	geopotential	mid-layer geopotential	m2 s-2	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	in
deltim	timestep_for_physics	physics time step	s	real	()	kind_phys	in
kdt	index_of_timestep	current time step index	index	integer	()		in
var	standard_deviation_of_subgrid_orography	standard deviation of subgrid height_above_mean_sea_level	m	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	in
oc1	convexity_of_subgrid_orography	convexity of subgrid height_above_mean_sea_level	none	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	in
oa4	asymmetry_of_subgrid_orography	asymmetry of subgrid height_above_mean_sea_level	none	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:4)	kind_phys	in
ol4	fraction_of_grid_box_with_subgrid_orography_higher_than_critical_height	horizontal fraction of grid box covered by subgrid height_above_mean_sea_level higher than critical height	frac	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:4)	kind_phys	in
varss	standard_deviation_of_subgrid_orography_small_scale	standard deviation of subgrid height_above_mean_sea_level small scale	m	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	in
oc1ss	convexity_of_subgrid_orography_small_scale	convexity of subgrid height_above_mean_sea_level small scale	none	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	in
oa4ss	asymmetry_of_subgrid_orography_small_scale	asymmetry of subgrid height_above_mean_sea_level small scale	none	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:4)	kind_phys	in
ol4ss	fraction_of_grid_box_with_subgrid_orography_higher_than_critical_height_small_scale	horizontal fraction of grid box covered by subgrid height_above_mean_sea_level higher than critical height small scale	frac	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:4)	kind_phys	in
theta	angle_from_east_of_maximum_subgrid_orographic_variations	angle with respect to east of maximum subgrid orographic variations	degree	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	in
sigma	slope_of_subgrid_orography	slope of subgrid height_above_mean_sea_level	none	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	in
gamma	anisotropy_of_subgrid_orography	anisotropy of subgrid height_above_mean_sea_level	none	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	in
elvmax	maximum_subgrid_orography	maximum of subgrid height_above_mean_sea_level	m	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	in
dtaux2d_ms	tendency_of_x_wind_due_to_mesoscale_orographic_gravity_wave_drag	x wind tendency from mesoscale gwd	m s-2	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	inout
dtauy2d_ms	tendency_of_y_wind_due_to_mesoscale_orographic_gravity_wave_drag	y wind tendency from mesoscale gwd	m s-2	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	inout
dtaux2d_bl	tendency_of_x_wind_due_to_blocking_drag	x wind tendency from blocking drag	m s-2	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	inout
dtauy2d_bl	tendency_of_y_wind_due_to_blocking_drag	y wind tendency from blocking drag	m s-2	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	inout
dtaux2d_ss	tendency_of_x_wind_due_to_small_scale_gravity_wave_drag	x wind tendency from small scale gwd	m s-2	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	inout
dtauy2d_ss	tendency_of_y_wind_due_to_small_scale_gravity_wave_drag	y wind tendency from small scale gwd	m s-2	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	inout
dtaux2d_fd	tendency_of_x_wind_due_to_form_drag	x wind tendency from form drag	m s-2	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	inout
dtauy2d_fd	tendency_of_y_wind_due_to_form_drag	y wind tendency from form drag	m s-2	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	inout
dusfc	instantaneous_x_stress_due_to_gravity_wave_drag	zonal surface stress due to orographic gravity wave drag	Pa	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	inout
dvsfc	instantaneous_y_stress_due_to_gravity_wave_drag	meridional surface stress due to orographic gravity wave drag	Pa	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	inout
dusfc_ms	vertically_integrated_x_momentum_flux_due_to_mesoscale_orographic_gravity_wave_drag	integrated x momentum flux from mesoscale gwd	Pa	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	inout
dvsfc_ms	vertically_integrated_y_momentum_flux_due_to_mesoscale_orographic_gravity_wave_drag	integrated y momentum flux from mesoscale gwd	Pa	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	inout
dusfc_bl	vertically_integrated_x_momentum_flux_due_to_blocking_drag	integrated x momentum flux from blocking drag	Pa	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	inout
dvsfc_bl	vertically_integrated_y_momentum_flux_due_to_blocking_drag	integrated y momentum flux from blocking drag	Pa	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	inout
dusfc_ss	vertically_integrated_x_momentum_flux_due_to_small_scale_gravity_wave_drag	integrated x momentum flux from small scale gwd	Pa	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	inout
dvsfc_ss	vertically_integrated_y_momentum_flux_due_to_small_scale_gravity_wave_drag	integrated y momentum flux from small scale gwd	Pa	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	inout
dusfc_fd	vertically_integrated_x_momentum_flux_due_to_form_drag	integrated x momentum flux from form drag	Pa	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	inout
dvsfc_fd	vertically_integrated_y_momentum_flux_due_to_form_drag	integrated y momentum flux from form drag	Pa	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	inout
slmsk	area_type	landmask: sea/land/ice=0/1/2	flag	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	in
br1	bulk_richardson_number_at_lowest_model_level	bulk Richardson number at the surface	none	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	in
hpbl	atmosphere_boundary_layer_thickness	PBL thickness	m	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	in
g	gravitational_acceleration	gravitational acceleration	m s-2	real	()	kind_phys	in
cp	specific_heat_of_dry_air_at_constant_pressure	specific heat of dry air at constant pressure	J kg-1 K-1	real	()	kind_phys	in
rd	gas_constant_of_dry_air	ideal gas constant for dry air	J kg-1 K-1	real	()	kind_phys	in
rv	gas_constant_water_vapor	ideal gas constant for water vapor	J kg-1 K-1	real	()	kind_phys	in
fv	ratio_of_vapor_to_dry_air_gas_constants_minus_one	(rv/rd) - 1 (rv = ideal gas constant for water vapor)	none	real	()	kind_phys	in
pi	pi	ratio of a circle's circumference to its diameter	none	real	()	kind_phys	in
imx	number_of_equatorial_longitude_points	number of longitude points along the equator	count	integer	()		in
cdmbgwd	multiplicative_tunable_parameters_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	real	(ccpp_constant_one:4)	kind_phys	in
me	mpi_rank	rank of the current MPI task	index	integer	()		in
master	mpi_root	master MPI-rank	index	integer	()		in
lprnt	flag_print	flag for debugging printouts	flag	logical	()		in
ipr	horizontal_index_of_printed_column	horizontal index of column used in debugging printouts	index	integer	()		in
rdxzb	level_of_dividing_streamline	level of the dividing streamline	none	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	inout
dx	characteristic_grid_lengthscale	size of the grid cell	m	real	(ccpp_constant_one:horizontal_loop_extent)	kind_phys	in
gwd_opt	control_for_drag_suite_gravity_wave_drag	flag to choose gwd scheme	flag	integer	()		in
do_gsl_drag_ls_bl	do_gsl_drag_suite_mesoscale_orographic_and_blocking_drag	flag to activate GSL drag suite - mesoscale GWD and blocking	flag	logical	()		in
do_gsl_drag_ss	do_gsl_drag_suite_small_scale_orographic_drag	flag to activate GSL drag suite - small-scale GWD	flag	logical	()		in
do_gsl_drag_tofd	do_gsl_drag_suite_turbulent_orographic_form_drag	flag to activate GSL drag suite - turb orog form drag	flag	logical	()		in
dtend	cumulative_change_of_state_variables	diagnostic tendencies for state variables	mixed	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension, ccpp_constant_one:cumulative_change_of_state_variables_outer_index_max)	kind_phys	inout
dtidx	cumulative_change_of_state_variables_outer_index	index of state-variable and process in last dimension of diagnostic tendencies array AKA cumulative_change_index	index	integer	(ccpp_constant_one:number_of_tracers_plus_one_hundred, ccpp_constant_one:number_of_cumulative_change_processes)		in
index_of_process_orographic_gwd	index_of_orographic_gravity_wave_drag_process_in_cumulative_change_index	index of orographic gravity wave drag process in second dimension of array cumulative change index	index	integer	()		in
index_of_temperature	index_of_temperature_in_cumulative_change_index	index of temperature in first dimension of array cumulative change index	index	integer	()		in
index_of_x_wind	index_of_x_wind_in_cumulative_change_index	index of x-wind in first dimension of array cumulative change index	index	integer	()		in
index_of_y_wind	index_of_y_wind_in_cumulative_change_index	index of x-wind in first dimension of array cumulative change index	index	integer	()		in
ldiag3d	flag_for_diagnostics_3d	flag for 3d diagnostic fields	flag	logical	()		in
ldiag_ugwp	flag_for_unified_gravity_wave_physics_diagnostics	flag for CIRES UGWP Diagnostics	flag	logical	()		in
ugwp_seq_update	do_ugwp_sequential_update	flag for ugwp sequential update	flag	logical	()		in
spp_wts_gwd	spp_weights_for_gravity_wave_drag_scheme	spp weights for gravity wave drag scheme	1	real	(ccpp_constant_one:horizontal_loop_extent, ccpp_constant_one:vertical_layer_dimension)	kind_phys	in
spp_gwd	control_for_gravity_wave_drag_spp_perturbations	control for gravity wave drag spp perturbations	count	integer	()		in
errmsg	ccpp_error_message	error message for error handling in CCPP	none	character	()	len=*	out
errflg	ccpp_error_code	error code for error handling in CCPP	1	integer	()		out

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

1. Orographic Gravity Wave Drag Section

Referenced by unified_ugwp::unified_ugwp_run().

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