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source: palm/trunk/SOURCE/basic_constants_and_equations_mod.f90 @ 4505

Last change on this file since 4505 was 4502, checked in by schwenkel, 4 years ago
Implementation of ice microphysics
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1	!> @file basic_constants_and_equations_mod.f90
2	!------------------------------------------------------------------------------!
3	! This file is part of the PALM model system.
4	!
5	! PALM is free software: you can redistribute it and/or modify it under the
6	! terms of the GNU General Public License as published by the Free Software
7	! Foundation, either version 3 of the License, or (at your option) any later
8	! version.
9	!
10	! PALM is distributed in the hope that it will be useful, but WITHOUT ANY
11	! WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
12	! A PARTICULAR PURPOSE. See the GNU General Public License for more details.
13	!
14	! You should have received a copy of the GNU General Public License along with
15	! PALM. If not, see <http://www.gnu.org/licenses/>.
16	!
17	! Copyright 1997-2020 Leibniz Universitaet Hannover
18	!------------------------------------------------------------------------------!
19	!
20	! Current revisions:
21	! -----------------
22	!
23	!
24	! Former revisions:
25	! -----------------
26	! $Id: basic_constants_and_equations_mod.f90 4502 2020-04-17 16:14:16Z schwenkel $
27	! Implementation of ice microphysics
28	!
29	! 4400 2020-02-10 20:32:41Z suehring
30	! Move routine to transform coordinates from netcdf_interface_mod to
31	! basic_constants_and_equations_mod
32	!
33	! 4360 2020-01-07 11:25:50Z suehring
34	! Corrected "Former revisions" section
35	!
36	! 4088 2019-07-11 13:57:56Z Giersch
37	! Comment of barometric formula improved, function for ideal gas law revised
38	!
39	! 4084 2019-07-10 17:09:11Z knoop
40	! Changed precomputed fractions to be variable based
41	!
42	! 4055 2019-06-27 09:47:29Z suehring
43	! Added rgas_univ (universal gas constant) (E.C. Chan)
44	!
45	!
46	! 3655 2019-01-07 16:51:22Z knoop
47	! OpenACC port for SPEC
48	! 3361 2018-10-16 20:39:37Z knoop
49	! New module (introduced with modularization of bulk cloud physics model)
50	!
51	!
52	!
53	!
54	! Description:
55	! ------------
56	!> This module contains all basic (physical) constants
57	!> and
58	!> functions for the calculation of diagnostic quantities.
59	!------------------------------------------------------------------------------!
60	MODULE basic_constants_and_equations_mod
61
62
63	USE kinds
64
65	IMPLICIT NONE
66
67	REAL(wp), PARAMETER :: c_p = 1005.0_wp !< heat capacity of dry air (J kg-1 K-1)
68	REAL(wp), PARAMETER :: degc_to_k = 273.15_wp !< temperature (in K) of 0 deg C (K)
69	REAL(wp), PARAMETER :: g = 9.81_wp !< gravitational acceleration (m s-2)
70	REAL(wp), PARAMETER :: kappa = 0.4_wp !< von Karman constant
71	REAL(wp), PARAMETER :: l_m = 0.33E+06_wp !< latent heat of water melting (J kg-1)
72	REAL(wp), PARAMETER :: l_v = 2.5E+06_wp !< latent heat of water vaporization (J kg-1)
73	REAL(wp), PARAMETER :: l_s = l_m + l_v !< latent heat of water sublimation (J kg-1)
74	REAL(wp), PARAMETER :: molecular_weight_of_nacl = 0.05844_wp !< mol. m. NaCl (kg mol-1)
75	REAL(wp), PARAMETER :: molecular_weight_of_c3h4o4 = 0.10406_wp !< mol. m. malonic acid (kg mol-1)
76	REAL(wp), PARAMETER :: molecular_weight_of_nh4no3 = 0.08004_wp !< mol. m. ammonium sulfate (kg mol-1)
77	REAL(wp), PARAMETER :: molecular_weight_of_water = 0.01801528_wp !< mol. m. H2O (kg mol-1)
78	REAL(wp), PARAMETER :: pi = 3.141592654_wp !< PI
79	!$ACC DECLARE COPYIN(pi)
80	REAL(wp), PARAMETER :: rgas_univ = 8.31446261815324_wp !< universal gas constant (J K-1 mol-1)
81	REAL(wp), PARAMETER :: rho_l = 1.0E3_wp !< density of water (kg m-3)
82	REAL(wp), PARAMETER :: rho_nacl = 2165.0_wp !< density of NaCl (kg m-3)
83	REAL(wp), PARAMETER :: rho_c3h4o4 = 1600.0_wp !< density of malonic acid (kg m-3)
84	REAL(wp), PARAMETER :: rho_nh4no3 = 1720.0_wp !< density of ammonium sulfate (kg m-3)
85	REAL(wp), PARAMETER :: r_d = 287.0_wp !< sp. gas const. dry air (J kg-1 K-1)
86	REAL(wp), PARAMETER :: r_v = 461.51_wp !< sp. gas const. water vapor (J kg-1 K-1)
87	REAL(wp), PARAMETER :: sigma_sb = 5.67037E-08_wp !< Stefan-Boltzmann constant
88	REAL(wp), PARAMETER :: solar_constant = 1368.0_wp !< solar constant at top of atmosphere
89	REAL(wp), PARAMETER :: vanthoff_nacl = 2.0_wp !< van't Hoff factor for NaCl
90	REAL(wp), PARAMETER :: vanthoff_c3h4o4 = 1.37_wp !< van't Hoff factor for malonic acid
91	REAL(wp), PARAMETER :: vanthoff_nh4no3 = 2.31_wp !< van't Hoff factor for ammonium sulfate
92
93	REAL(wp), PARAMETER :: p_0 = 100000.0_wp !< standard pressure reference state
94
95	REAL(wp), PARAMETER :: g_d_cp = g / c_p !< precomputed g / c_p
96	REAL(wp), PARAMETER :: lv_d_cp = l_v / c_p !< precomputed l_v / c_p
97	REAL(wp), PARAMETER :: ls_d_cp = l_s / c_p !< precomputed l_s / c_p
98	REAL(wp), PARAMETER :: lv_d_rd = l_v / r_d !< precomputed l_v / r_d
99	REAL(wp), PARAMETER :: rd_d_rv = r_d / r_v !< precomputed r_d / r_v
100	REAL(wp), PARAMETER :: rd_d_cp = r_d / c_p !< precomputed r_d / c_p
101	REAL(wp), PARAMETER :: cp_d_rd = c_p / r_d !< precomputed c_p / r_d
102
103	REAL(wp) :: molecular_weight_of_solute = molecular_weight_of_nacl !< mol. m. NaCl (kg mol-1)
104	REAL(wp) :: rho_s = rho_nacl !< density of NaCl (kg m-3)
105	REAL(wp) :: vanthoff = vanthoff_nacl !< van't Hoff factor for NaCl
106
107
108	SAVE
109
110	PRIVATE magnus_0d, &
111	magnus_1d, &
112	magnus_tl_0d, &
113	magnus_tl_1d, &
114	magnus_0d_ice, &
115	magnus_1d_ice, &
116	ideal_gas_law_rho_0d, &
117	ideal_gas_law_rho_1d, &
118	ideal_gas_law_rho_pt_0d, &
119	ideal_gas_law_rho_pt_1d, &
120	exner_function_0d, &
121	exner_function_1d, &
122	exner_function_invers_0d, &
123	exner_function_invers_1d, &
124	barometric_formula_0d, &
125	barometric_formula_1d
126
127
128	INTERFACE convert_utm_to_geographic
129	MODULE PROCEDURE convert_utm_to_geographic
130	END INTERFACE convert_utm_to_geographic
131
132	INTERFACE magnus
133	MODULE PROCEDURE magnus_0d
134	MODULE PROCEDURE magnus_1d
135	END INTERFACE magnus
136
137	INTERFACE magnus_tl
138	MODULE PROCEDURE magnus_tl_0d
139	MODULE PROCEDURE magnus_tl_1d
140	END INTERFACE magnus_tl
141
142	INTERFACE magnus_ice
143	MODULE PROCEDURE magnus_0d_ice
144	MODULE PROCEDURE magnus_1d_ice
145	END INTERFACE magnus_ice
146
147	INTERFACE ideal_gas_law_rho
148	MODULE PROCEDURE ideal_gas_law_rho_0d
149	MODULE PROCEDURE ideal_gas_law_rho_1d
150	END INTERFACE ideal_gas_law_rho
151
152	INTERFACE ideal_gas_law_rho_pt
153	MODULE PROCEDURE ideal_gas_law_rho_pt_0d
154	MODULE PROCEDURE ideal_gas_law_rho_pt_1d
155	END INTERFACE ideal_gas_law_rho_pt
156
157	INTERFACE exner_function
158	MODULE PROCEDURE exner_function_0d
159	MODULE PROCEDURE exner_function_1d
160	END INTERFACE exner_function
161
162	INTERFACE exner_function_invers
163	MODULE PROCEDURE exner_function_invers_0d
164	MODULE PROCEDURE exner_function_invers_1d
165	END INTERFACE exner_function_invers
166
167	INTERFACE barometric_formula
168	MODULE PROCEDURE barometric_formula_0d
169	MODULE PROCEDURE barometric_formula_1d
170	END INTERFACE barometric_formula
171	!
172	!-- Public routines
173	PUBLIC convert_utm_to_geographic
174
175	CONTAINS
176
177
178	!------------------------------------------------------------------------------!
179	! Description:
180	! ------------
181	!> Convert UTM coordinates into geographic latitude and longitude. Conversion
182	!> is based on the work of KrÃŒger (1912) DOI: 10.2312/GFZ.b103-krueger28
183	!> and Karney (2013) DOI: 10.1007/s00190-012-0578-z
184	!> Based on a JavaScript of the geodesy function library written by chrisveness
185	!> https://github.com/chrisveness/geodesy
186	!------------------------------------------------------------------------------!
187	SUBROUTINE convert_utm_to_geographic( crs, eutm, nutm, lon, lat )
188
189	INTEGER(iwp) :: j !< loop index
190
191	REAL(wp), INTENT(in) :: eutm !< easting (UTM)
192	REAL(wp), INTENT(out) :: lat !< geographic latitude in degree
193	REAL(wp), INTENT(out) :: lon !< geographic longitude in degree
194	REAL(wp), INTENT(in) :: nutm !< northing (UTM)
195
196	REAL(wp) :: a !< 2pia is the circumference of a meridian
197	REAL(wp) :: cos_eta_s !< cos(eta_s)
198	REAL(wp) :: delta_i !<
199	REAL(wp) :: delta_tau_i !<
200	REAL(wp) :: e !< eccentricity
201	REAL(wp) :: eta !<
202	REAL(wp) :: eta_s !<
203	REAL(wp) :: n !< 3rd flattening
204	REAL(wp) :: n2 !< n^2
205	REAL(wp) :: n3 !< n^3
206	REAL(wp) :: n4 !< n^4
207	REAL(wp) :: n5 !< n^5
208	REAL(wp) :: n6 !< n^6
209	REAL(wp) :: nu !<
210	REAL(wp) :: nu_s !<
211	REAL(wp) :: sin_eta_s !< sin(eta_s)
212	REAL(wp) :: sinh_nu_s !< sinush(nu_s)
213	REAL(wp) :: tau_i !<
214	REAL(wp) :: tau_i_s !<
215	REAL(wp) :: tau_s !<
216	REAL(wp) :: x !< adjusted easting
217	REAL(wp) :: y !< adjusted northing
218
219	REAL(wp), DIMENSION(6) :: beta !< 6th order KrÃŒger expressions
220
221	REAL(wp), DIMENSION(8), INTENT(in) :: crs !< coordinate reference system, consists of
222	!< (/semi_major_axis,
223	!< inverse_flattening,
224	!< longitude_of_prime_meridian,
225	!< longitude_of_central_meridian,
226	!< scale_factor_at_central_meridian,
227	!< latitude_of_projection_origin,
228	!< false_easting,
229	!< false_northing /)
230
231	x = eutm - crs(7) ! remove false easting
232	y = nutm - crs(8) ! remove false northing
233	!
234	!-- from Karney 2011 Eq 15-22, 36:
235	e = SQRT( 1.0_wp / crs(2) * ( 2.0_wp - 1.0_wp / crs(2) ) )
236	n = 1.0_wp / crs(2) / ( 2.0_wp - 1.0_wp / crs(2) )
237	n2 = n * n
238	n3 = n * n2
239	n4 = n * n3
240	n5 = n * n4
241	n6 = n * n5
242
243	a = crs(1) / ( 1.0_wp + n ) * ( 1.0_wp + 0.25_wp * n2 &
244	+ 0.015625_wp * n4 &
245	+ 3.90625E-3_wp * n6 )
246
247	nu = x / ( crs(5) * a )
248	eta = y / ( crs(5) * a )
249
250	!-- According to KrÃŒger (1912), eq. 26*
251	beta(1) = 0.5_wp * n &
252	- 2.0_wp / 3.0_wp * n2 &
253	+ 37.0_wp / 96.0_wp * n3 &
254	- 1.0_wp / 360.0_wp * n4 &
255	- 81.0_wp / 512.0_wp * n5 &
256	+ 96199.0_wp / 604800.0_wp * n6
257
258	beta(2) = 1.0_wp / 48.0_wp * n2 &
259	+ 1.0_wp / 15.0_wp * n3 &
260	- 437.0_wp / 1440.0_wp * n4 &
261	+ 46.0_wp / 105.0_wp * n5 &
262	- 1118711.0_wp / 3870720.0_wp * n6
263
264	beta(3) = 17.0_wp / 480.0_wp * n3 &
265	- 37.0_wp / 840.0_wp * n4 &
266	- 209.0_wp / 4480.0_wp * n5 &
267	+ 5569.0_wp / 90720.0_wp * n6
268
269	beta(4) = 4397.0_wp / 161280.0_wp * n4 &
270	- 11.0_wp / 504.0_wp * n5 &
271	- 830251.0_wp / 7257600.0_wp * n6
272
273	beta(5) = 4583.0_wp / 161280.0_wp * n5 &
274	- 108847.0_wp / 3991680.0_wp * n6
275
276	beta(6) = 20648693.0_wp / 638668800.0_wp * n6
277
278	eta_s = eta
279	nu_s = nu
280	DO j = 1, 6
281	eta_s = eta_s - beta(j) * SIN(2.0_wp * j * eta) * COSH(2.0_wp * j * nu)
282	nu_s = nu_s - beta(j) * COS(2.0_wp * j * eta) * SINH(2.0_wp * j * nu)
283	ENDDO
284
285	sinh_nu_s = SINH( nu_s )
286	sin_eta_s = SIN( eta_s )
287	cos_eta_s = COS( eta_s )
288
289	tau_s = sin_eta_s / SQRT( sinh_nu_s2 + cos_eta_s2 )
290
291	tau_i = tau_s
292	delta_tau_i = 1.0_wp
293
294	DO WHILE ( ABS( delta_tau_i ) > 1.0E-12_wp )
295
296	delta_i = SINH( e * ATANH( e * tau_i / SQRT( 1.0_wp + tau_i**2 ) ) )
297
298	tau_i_s = tau_i * SQRT( 1.0_wp + delta_i**2 ) &
299	- delta_i * SQRT( 1.0_wp + tau_i**2 )
300
301	delta_tau_i = ( tau_s - tau_i_s ) / SQRT( 1.0_wp + tau_i_s**2 ) &
302	* ( 1.0_wp + ( 1.0_wp - e*2 ) tau_i**2 ) &
303	/ ( ( 1.0_wp - e*2 ) SQRT( 1.0_wp + tau_i**2 ) )
304
305	tau_i = tau_i + delta_tau_i
306
307	ENDDO
308
309	lat = ATAN( tau_i ) / pi * 180.0_wp
310	lon = ATAN2( sinh_nu_s, cos_eta_s ) / pi * 180.0_wp + crs(4)
311
312	END SUBROUTINE convert_utm_to_geographic
313
314	!------------------------------------------------------------------------------!
315	! Description:
316	! ------------
317	!> This function computes the magnus formula (Press et al., 1992).
318	!> The magnus formula is needed to calculate the saturation vapor pressure
319	!------------------------------------------------------------------------------!
320	FUNCTION magnus_0d( t )
321
322	IMPLICIT NONE
323
324	REAL(wp), INTENT(IN) :: t !< temperature (K)
325
326	REAL(wp) :: magnus_0d
327	!
328	!-- Saturation vapor pressure for a specific temperature:
329	magnus_0d = 611.2_wp * EXP( 17.62_wp * ( t - degc_to_k ) / &
330	( t - 29.65_wp ) )
331
332	END FUNCTION magnus_0d
333
334	!------------------------------------------------------------------------------!
335	! Description:
336	! ------------
337	!> This function computes the magnus formula (Press et al., 1992).
338	!> The magnus formula is needed to calculate the saturation vapor pressure
339	!------------------------------------------------------------------------------!
340	FUNCTION magnus_1d( t )
341
342	IMPLICIT NONE
343
344	REAL(wp), INTENT(IN), DIMENSION(:) :: t !< temperature (K)
345
346	REAL(wp), DIMENSION(size(t)) :: magnus_1d
347	!
348	!-- Saturation vapor pressure for a specific temperature:
349	magnus_1d = 611.2_wp * EXP( 17.62_wp * ( t - degc_to_k ) / &
350	( t - 29.65_wp ) )
351
352	END FUNCTION magnus_1d
353
354	!------------------------------------------------------------------------------!
355	! Description:
356	! ------------
357	!> This function computes the magnus formula (Press et al., 1992) using the
358	!> (ice-) liquid water potential temperature.
359	!> The magnus formula is needed to calculate the saturation vapor pressure over
360	!> a plane liquid water surface
361	!------------------------------------------------------------------------------!
362	FUNCTION magnus_tl_0d( t_l )
363
364	IMPLICIT NONE
365
366	REAL(wp), INTENT(IN) :: t_l !< liquid water temperature (K)
367
368	REAL(wp) :: magnus_tl_0d
369	!
370	!-- Saturation vapor pressure for a specific temperature:
371	magnus_tl_0d = 610.78_wp * EXP( 17.269_wp * ( t_l - 273.16_wp ) / &
372	( t_l - 35.86_wp ) )
373
374	END FUNCTION magnus_tl_0d
375
376	!------------------------------------------------------------------------------!
377	! Description:
378	! ------------
379	!> This function computes the magnus formula (Press et al., 1992) using the
380	!> (ice-) liquid water potential temperature.
381	!> The magnus formula is needed to calculate the saturation vapor pressure over
382	!> a plane liquid water surface
383	!------------------------------------------------------------------------------!
384	FUNCTION magnus_tl_1d( t_l )
385
386	IMPLICIT NONE
387
388	REAL(wp), INTENT(IN), DIMENSION(:) :: t_l !< liquid water temperature (K)
389
390	REAL(wp), DIMENSION(size(t_l)) :: magnus_tl_1d
391	!
392	!-- Saturation vapor pressure for a specific temperature:
393	magnus_tl_1d = 610.78_wp * EXP( 17.269_wp * ( t_l - 273.16_wp ) / &
394	( t_l - 35.86_wp ) )
395
396	END FUNCTION magnus_tl_1d
397
398	!------------------------------------------------------------------------------!
399	! Description:
400	! ------------
401	!> This function computes the magnus formula (Press et al., 1992).
402	!> The magnus formula is needed to calculate the saturation vapor pressure over
403	!> a plane ice surface
404	!------------------------------------------------------------------------------!
405	FUNCTION magnus_0d_ice( t )
406
407	IMPLICIT NONE
408
409	REAL(wp), INTENT(IN) :: t !< temperature (K)
410
411	REAL(wp) :: magnus_0d_ice
412	!
413	!-- Saturation vapor pressure for a specific temperature:
414	magnus_0d_ice = 611.2_wp * EXP( 22.46_wp * ( t - degc_to_k ) / &
415	( t - 0.53_wp ) )
416
417	END FUNCTION magnus_0d_ice
418
419	!------------------------------------------------------------------------------!
420	! Description:
421	! ------------
422	!> This function computes the magnus formula (Press et al., 1992).
423	!> The magnus formula is needed to calculate the saturation vapor pressure over
424	!> a plane ice surface
425	!------------------------------------------------------------------------------!
426	FUNCTION magnus_1d_ice( t )
427
428	IMPLICIT NONE
429
430	REAL(wp), INTENT(IN), DIMENSION(:) :: t !< temperature (K)
431
432	REAL(wp), DIMENSION(size(t)) :: magnus_1d_ice
433	!
434	!-- Saturation vapor pressure for a specific temperature:
435	magnus_1d_ice = 611.2_wp * EXP( 22.46_wp * ( t - degc_to_k ) / &
436	( t - 0.53_wp ) )
437
438	END FUNCTION magnus_1d_ice
439
440	!------------------------------------------------------------------------------!
441	! Description:
442	! ------------
443	!> Compute the ideal gas law for scalar arguments.
444	!------------------------------------------------------------------------------!
445	FUNCTION ideal_gas_law_rho_0d( p, t )
446
447	IMPLICIT NONE
448
449	REAL(wp), INTENT(IN) :: p !< pressure (Pa)
450	REAL(wp), INTENT(IN) :: t !< temperature (K)
451
452	REAL(wp) :: ideal_gas_law_rho_0d
453	!
454	!-- compute density according to ideal gas law:
455	ideal_gas_law_rho_0d = p / (r_d * t)
456
457	END FUNCTION ideal_gas_law_rho_0d
458
459	!------------------------------------------------------------------------------!
460	! Description:
461	! ------------
462	!> Compute the ideal gas law for 1-D array arguments.
463	!------------------------------------------------------------------------------!
464	FUNCTION ideal_gas_law_rho_1d( p, t )
465
466	IMPLICIT NONE
467
468	REAL(wp), INTENT(IN), DIMENSION(:) :: p !< pressure (Pa)
469	REAL(wp), INTENT(IN), DIMENSION(:) :: t !< temperature (K)
470
471	REAL(wp), DIMENSION(size(p)) :: ideal_gas_law_rho_1d
472	!
473	!-- compute density according to ideal gas law:
474	ideal_gas_law_rho_1d = p / (r_d * t)
475
476	END FUNCTION ideal_gas_law_rho_1d
477
478	!------------------------------------------------------------------------------!
479	! Description:
480	! ------------
481	!> Compute the ideal gas law for scalar arguments.
482	!------------------------------------------------------------------------------!
483	FUNCTION ideal_gas_law_rho_pt_0d( p, t )
484
485	IMPLICIT NONE
486
487	REAL(wp), INTENT(IN) :: p !< pressure (Pa)
488	REAL(wp), INTENT(IN) :: t !< temperature (K)
489
490	REAL(wp) :: ideal_gas_law_rho_pt_0d
491	!
492	!-- compute density according to ideal gas law:
493	ideal_gas_law_rho_pt_0d = p / (r_d * exner_function(p) * t)
494
495	END FUNCTION ideal_gas_law_rho_pt_0d
496
497	!------------------------------------------------------------------------------!
498	! Description:
499	! ------------
500	!> Compute the ideal gas law for 1-D array arguments.
501	!------------------------------------------------------------------------------!
502	FUNCTION ideal_gas_law_rho_pt_1d( p, t )
503
504	IMPLICIT NONE
505
506	REAL(wp), INTENT(IN), DIMENSION(:) :: p !< pressure (Pa)
507	REAL(wp), INTENT(IN), DIMENSION(:) :: t !< temperature (K)
508
509	REAL(wp), DIMENSION(size(p)) :: ideal_gas_law_rho_pt_1d
510	!
511	!-- compute density according to ideal gas law:
512	ideal_gas_law_rho_pt_1d = p / (r_d * exner_function(p) * t)
513
514	END FUNCTION ideal_gas_law_rho_pt_1d
515
516	!------------------------------------------------------------------------------!
517	! Description:
518	! ------------
519	!> Compute the exner function for scalar arguments.
520	!------------------------------------------------------------------------------!
521	FUNCTION exner_function_0d( p )
522
523	IMPLICIT NONE
524
525	REAL(wp), INTENT(IN) :: p !< pressure (Pa)
526
527	REAL(wp) :: exner_function_0d
528	!
529	!-- compute exner function:
530	exner_function_0d = ( p / p_0 )**( rd_d_cp )
531
532	END FUNCTION exner_function_0d
533
534	!------------------------------------------------------------------------------!
535	! Description:
536	! ------------
537	!> Compute the exner function for 1-D array arguments.
538	!------------------------------------------------------------------------------!
539	FUNCTION exner_function_1d( p )
540
541	IMPLICIT NONE
542
543	REAL(wp), INTENT(IN), DIMENSION(:) :: p !< pressure (Pa)
544
545	REAL(wp), DIMENSION(size(p)) :: exner_function_1d
546	!
547	!-- compute exner function:
548	exner_function_1d = ( p / p_0 )**( rd_d_cp )
549
550	END FUNCTION exner_function_1d
551
552	!------------------------------------------------------------------------------!
553	! Description:
554	! ------------
555	!> Compute the exner function for scalar arguments.
556	!------------------------------------------------------------------------------!
557	FUNCTION exner_function_invers_0d( p )
558
559	IMPLICIT NONE
560
561	REAL(wp), INTENT(IN) :: p !< pressure (Pa)
562
563	REAL(wp) :: exner_function_invers_0d
564	!
565	!-- compute exner function:
566	exner_function_invers_0d = ( p_0 / p )**( rd_d_cp )
567
568	END FUNCTION exner_function_invers_0d
569
570	!------------------------------------------------------------------------------!
571	! Description:
572	! ------------
573	!> Compute the exner function for 1-D array arguments.
574	!------------------------------------------------------------------------------!
575	FUNCTION exner_function_invers_1d( p )
576
577	IMPLICIT NONE
578
579	REAL(wp), INTENT(IN), DIMENSION(:) :: p !< pressure (Pa)
580
581	REAL(wp), DIMENSION(size(p)) :: exner_function_invers_1d
582	!
583	!-- compute exner function:
584	exner_function_invers_1d = ( p_0 / p )**( rd_d_cp )
585
586	END FUNCTION exner_function_invers_1d
587
588	!------------------------------------------------------------------------------!
589	! Description:
590	! ------------
591	!> Compute the barometric formula for scalar arguments. The calculation is
592	!> based on the assumption of a polytropic atmosphere and neutral
593	!> stratification, where the temperature lapse rate is g/cp.
594	!------------------------------------------------------------------------------!
595	FUNCTION barometric_formula_0d( z, t_0, p_0)
596
597	IMPLICIT NONE
598
599	REAL(wp), INTENT(IN) :: z !< height (m)
600	REAL(wp), INTENT(IN) :: t_0 !< temperature reference state (K)
601	REAL(wp), INTENT(IN) :: p_0 !< surface pressure (Pa)
602
603	REAL(wp) :: barometric_formula_0d
604	!
605	!-- compute barometric formula:
606	barometric_formula_0d = p_0 * ( (t_0 - g_d_cp * z) / t_0 )**( cp_d_rd )
607
608	END FUNCTION barometric_formula_0d
609
610	!------------------------------------------------------------------------------!
611	! Description:
612	! ------------
613	!> Compute the barometric formula for 1-D array arguments. The calculation is
614	!> based on the assumption of a polytropic atmosphere and neutral
615	!> stratification, where the temperature lapse rate is g/cp.
616	!------------------------------------------------------------------------------!
617	FUNCTION barometric_formula_1d( z, t_0, p_0)
618
619	IMPLICIT NONE
620
621	REAL(wp), INTENT(IN), DIMENSION(:) :: z !< height (m)
622	REAL(wp), INTENT(IN) :: t_0 !< temperature reference state (K)
623	REAL(wp), INTENT(IN) :: p_0 !< surface pressure (Pa)
624
625	REAL(wp), DIMENSION(size(z)) :: barometric_formula_1d
626	!
627	!-- compute barometric formula:
628	barometric_formula_1d = p_0 * ( (t_0 - g_d_cp * z) / t_0 )**( cp_d_rd )
629
630	END FUNCTION barometric_formula_1d
631
632	END MODULE basic_constants_and_equations_mod

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