Linearization of radiative heating and cooling rates for the case of non-scattering planetary atmospheres

ARTICLE IN PRESS

Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 1098–1117 www.elsevier.com/locate/jqsrt

Linearization of radiative heating and cooling rates for the case of non-scattering planetary atmospheres Eugene A. Ustinov Jet Propulsion Laboratory, California Institute of Technology, Mail Stop 169-237, 4800 Oak Grove Drive, Pasadena, CA 91109, USA Received 21 October 2007; received in revised form 7 December 2007; accepted 10 December 2007

Abstract Rates of radiative heating and cooling of the non-scattering planetary atmosphere and scattering underlying surface are analytically linearized with respect to the atmospheric and surface parameters. Resulting expressions for sensitivities of radiative heating and cooling rates are used to formulate the linearized 1D radiative model of atmospheric dynamics which directly incorporates the relevant linearized atmospheric and surface parameters. Applications to more sophisticated models of atmospheric dynamics, as well as extension to the case of presence of atmospheric scattering, are briefly discussed. r 2007 Elsevier Ltd. All rights reserved. Keywords: Radiative transfer; Sensitivity analysis; Heating and cooling rates

1. Introduction Radiative processes play a crucial role in the dynamics of planetary atmospheres, as well as in their energy balance. Transfer of the shortwave solar radiation in the atmosphere defines the primary source of the solar energy for the atmosphere–surface system, which provides the driving force for the atmospheric dynamics. Transfer of the thermal radiation in the atmosphere defines the ultimate sink of thermal energy of the atmosphere–surface system into the space. Thus, the accuracy of evaluation of both heating by solar radiation and cooling by thermal radiation is of crucial importance for accurate modeling of the dynamics of planetary atmospheres. Radiative transfer (RT) in general, and radiative heating and cooling rates in particular, depend on the atmospheric and surface parameters, which may be varied by atmospheric dynamics. Their variations, in their turn, induce variations of the shortwave sources and longwave sinks of radiative energy which drives this dynamics. Thus, atmospheric radiation provides a feedback, from dynamically varied parameters of the atmosphere–surface system to the dynamics of this system, and the accuracy of evaluation of this feedback is not less important for accurate modeling of the dynamics of lower planetary atmospheres than the evaluation of radiative heating and cooling rates themselves. Detailed quantitative modeling of radiative heating and cooling rates descends from the classic monograph by Goody [1]. Shortly thereafter these results were applied to atmospheres of Mars [2,3] and Venus [4] and Tel.: +1 818 354 2048.

E-mail address: [email protected] 0022-4073/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2007.12.012

ARTICLE IN PRESS E.A. Ustinov / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 1098–1117

Nomenclature Variables A Bn ; B cp cs E F Hg I n; I K Na Rn ; R p; p0 ; p00 p0 Q rn ; r T tn ; t u X z; z0 d kn ; k m n ;  tn ; t O

surface albedo Planck function atmospheric heat capacity at constant pressure surface heat capacity illumination at normal incidence illumination of the horizontal area atmospheric scale height intensity of radiation various weighting functions and sensitivities of- and to atmospheric parameters aerosol number density monochromatic radiances at the top of the atmosphere (TOA) atmospheric pressure surface pressure volume heating and cooling rates intermediate radiances temperature monochromatic transmittance functions cosine of nadir angle, measured from nadir toward zenith non-specified geophysical parameter altitude small variation of a given parameter; Dirac’s delta function total atmospheric absorption coefficient cosine of zenith angle surface emissivity monochromatic optical depth measured from TOA solid angle

Subscripts s d d IR  "; # + 0

relevant relevant relevant relevant relevant relevant relevant relevant

to the surface to diffuse (solar) radiation to Dirac’s delta function to thermal radiation to solar radiation to, respectively, upwelling and downwelling radiation to direct (solar) radiation to a linearization of the given variable

Superscripts ðBÞ ðkÞ ðþÞ ðÞ

relevant relevant relevant relevant

to to to to

the atmospheric Planck function the total atmospheric absorption coefficient IR heating IR cooling

1099

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1100

corresponding 1D radiative and radiative–convective models of the atmospheres of these planets were built. The dynamics of the atmosphere of Mars is driven essentially by radiative heating and cooling (for the summary of corresponding results through early 90s, see, e.g., Part V of the collective monograph [5]). Further progress is stimulated, in particular, by available data volume on the thermal structure of the Martian atmosphere from the TES (Thermal Emission Spectrometer) instrument flown on Mars Global Surveyor which stimulates assimilation of these data in the same fashion as for the Earth atmosphere (see, e.g., [6]). And, variational assimilation of meteorological data using full-physics adjoint models (see, e.g., [7]) implies a need in linearization of radiative effects in the atmospheric models. Radiative heating and cooling of the lower planetary atmospheres occurs under the conditions of local thermodynamic equilibrium (LTE). In the upper atmospheres of planets, non-LTE processes become important, and the assumptions based on LTE are not valid anymore (see, e.g., [8]). This paper is based on the LTE assumption and the applicability of the results obtained is thus limited to the lower planetary atmospheres. Applications to the non-LTE conditions are outside the scope of this study. In this paper we perform direct linearization of the radiative heating and cooling rates of a plane-parallel non-scattering planetary atmosphere and an underlying surface with respect to the atmospheric and surface parameters. The intention of this linearization is twofold. First, the linearization provides an analytic framework for direct computations of the sensitivity of the rates to atmospheric parameters, thus providing a possibility to evaluate and analyze the feedback between the dynamics and parameters of the model via radiative processes. Second, it provides a possibility to directly linearize the radiative forcing terms of various models of atmospheric dynamics, starting from the simplest 1D radiative model used as an illustration in this paper all the way up to the detailed GCM models. Thus, linearization of the forcing terms with respect to the atmospheric and surface parameters allows for complete linearization of the models as a whole. The approach pursued in this paper is based on a close analogy between the radiative heating and cooling rates of the atmosphere and surface on one hand and radiances observed in remote sensing, e.g., at the top of the atmosphere, on the other. Like radiative heating and cooling rates, the observed radiances depend, in general, on the vertical profiles of the involved atmospheric parameters throughout the atmospheric column and on the surface parameters. Recently, the author has developed a general linearization approach to evaluate the weighing functions of remote sensing for all atmospheric parameters involved in the RT in blackbody planetary atmospheres [9]. In this paper we will apply this approach to solar heating rates of the atmosphere and surface and thermal IR cooling rates, again, of the atmosphere and surface. Thus both in the solar and thermal IR spectral regions, we need to evaluate four groups of sensitivities of atmospheric and surface heating rates, QðpÞ and Qs to atmospheric and surface parameters X ðpÞ and X s (see Appendix A for general information): 1: 2: 3: 4:

of of of of

the the the the

atmospheric rates to the atmospheric parameters, dQðpÞ=dX ðp0 Þ; atmospheric rates to the surface parameters, qQðpÞ=qX s ; surface rates to the atmospheric parameters, dQs =dX ðpÞ; surface rates to the surface parameters, qQs =qX s .

In the next section we present the summary of results of [9] introducing the general linearization approach to be used. In Section 3 we present the formulation for the radiative heating and cooling rates. In Sections 4 and 5 we obtain the above four groups of sensitivities for the solar heating and thermal IR cooling rates, respectively. In Section 6 we apply the obtained results to the linearization of the 1D radiative model of atmospheric dynamics. Section 7 contains the discussion and summary of obtained results. The derivations below are done for monochromatic radiation but it is understood that the results need to be integrated over the relevant spectral regions: solar and thermal IR. 2. Weighting functions for remote sensing of blackbody planetary atmospheres This section contains a brief summary of the general linearization approach used in [9]. The monochromatic radiances Rn ðmÞ observed at the top of a blackbody atmosphere at the frequency n and the nadir viewing angle cos1 m are related to the geophysical parameters of the atmosphere and underlying surface through a

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following set of expressions (dependence on n and m is implied where necessary): Z 0 R ¼ tðp0 ÞeBs þ BðpÞ dtðpÞ,

1101

(1)

p0

tðpÞ ¼ expðtðpÞ=mÞ, Z

(2)

p

kðp0 ÞH g ðp0 Þ d ln p0 .

tðpÞ ¼

(3)

0

Here p is atmospheric pressure at a given level in the atmosphere, BðpÞ  BðTðpÞÞ and Bs  BðT s Þ are, respectively, blackbody radiances at atmospheric temperature TðpÞ and surface temperature T s , p0 is atmospheric pressure at the surface, e is the surface emissivity. Further on, tðpÞ and tðpÞ are, respectively, atmospheric transmittance along the given line of sight and optical depth, kðpÞ is the total (i.e. a sum over all atmospheric constituents) volume absorption coefficient of the atmosphere, and H g ðpÞ is the atmospheric scale height. We also need an alternative expression for radiances R, which allows direct linearization with respect to the transmittance function tðpÞ: Z p0 R ¼ Bð0Þ þ tðpÞ dBðpÞ þ tðp0 ÞðeBs  Bðp0 ÞÞ. (4) 0

Dependence of radiances R on atmospheric parameters has a form of a nonlinear integral operator, i.e. the radiances depend on profiles of these parameters in the whole atmospheric column. On the other hand, the variables BðpÞ, kðpÞ, and H g ðpÞ depend on corresponding atmospheric parameters locally. The atmospheric scale height H g ðpÞ varies only with the mean molecular mass (the reasons why the dependence on temperature is cancelled out are discussed in [9]). Thus, linearization of radiances R with respect to BðpÞ and kðpÞ in the form Z p0 dB R ¼ K ðBÞ ðpÞdBðpÞ d ln p, (5) 0

Z

p0

dk R ¼

K ðkÞ ðpÞdkðpÞ d ln p,

(6)

0

where K ðBÞ and K ðkÞ are corresponding kernels, provides a direct way to evaluate the weighing functions of the atmospheric parameters involved in radiative transfer. The Planck function depends solely on temperature and we have: dBðpÞ ¼

qB dTðpÞ. qT

The total volume absorption coefficient depends on temperature and other atmospheric parameters: X qk qk dkðpÞ ¼ dTðpÞ þ dX k ðpÞ. qT qX k k

(7)

(8)

The partial derivatives in Eqs. (7), (8) are available analytically with the sole exception of the partial derivative qk=qT, which may be computed from values of k tabulated for an appropriate set of temperatures and pressures. Assuming all of these partial derivatives to be known, we have: Temperature weighting functions: K ðTÞ ðpÞ ¼ K ðBÞ ðpÞ

qB qk þ K ðkÞ ðpÞ . qT qT

(9)

Weighting functions with respect to other atmospheric parameters: K ðX Þ ðpÞ ¼ K ðkÞ ðpÞ

qk . qX

(10)

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Expressions for K ðBÞ ðpÞ and K ðkÞ ðpÞ have the form [9]: 1 H g ðpÞkðpÞtðpÞ, m

(11)

1 K ðkÞ ðpÞ ¼  H g ðpÞðrðpÞ  tðpÞBðpÞÞ, m

(12)

K ðBÞ ðpÞ ¼

where Z

p

rðpÞ ¼ tðp0 ÞeBs þ

Bðp0 Þ dtðp0 Þ

(13)

p0

are intermediate radiances obtained when the observed radiances are integrated from the surface upwards (see Eq. (1)). Both transmittances tðpÞ and intermediate radiances rðpÞ are computed along with computation of the radiances R, and no additional time-consuming integrations are necessary. 3. Radiative heating and cooling in non-scattering atmospheres In this section we obtain two sets of expressions for monochromatic fluxes of, respectively, solar and thermal radiation in the form similar to Eqs. (1)–(4). These expressions are further used to obtain the expressions for radiative heating and cooling rates, which are used in subsequent sections to evaluate their sensitivities to atmospheric and surface parameters. As in the previous section, dependence on frequency n in this and subsequent sections is implied. 3.1. Heating by shortwave solar radiation The atmosphere is heated by solar radiation coming directly from the Sun and scattered back from the underlying surface. The surface is heated by solar radiation transmitted through the atmosphere (atmospheric scattering is neglected throughout this paper). First, we consider heating of the atmosphere. Let E  be the normal incidence illumination by solar radiation at the top of the atmosphere. Then, at some altitude z in the atmosphere, this illumination becomes: EðzÞ ¼ E  t ðzÞ,

(14)

where t ðzÞ is the transmittance function, which is expressed through the optical depth tðzÞ [cf. Eq. (2)]: t ðzÞ ¼ expðtðzÞ=m Þ,

(15)

where m is the cosine of solar zenith angle. We also need to express tðzÞ through the absorption coefficient kðzÞ [cf. Eq. (3)]: Z pðzÞ Z 1 tðzÞ ¼ kðp0 ÞH g ðp0 Þ d ln p0 ¼ kðz0 Þ dz0 . (16) z

0

The illumination of the horizontal area at the level z is F ðzÞ ¼ m EðzÞ.

(17)

Its decrease of F ðzÞ between levels z and z  dz can be evaluated from Eqs. (14)–(17): dF ðzÞ ¼ m E  dt ðzÞ ¼ E  t ðzÞkðzÞ dz

(18)

and its ratio to the thickness of corresponding layer dz is interpreted as a volume sink rate of the energy of direct solar radiation, which is converted to heating of the atmosphere. Correspondingly, its opposite is the volume heating rate due to direct solar radiation at the level z: Q+ ðzÞ ¼ 

dF . dz

(19)

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From the last of equalities in Eq. (18) we have: Q+ ðzÞ ¼ EðzÞkðzÞ.

(20)

Using Eq. (14), we obtain: Q+ ðzÞ ¼ E  t ðzÞkðzÞ.

(21)

Thus, the heating by direct solar radiation is defined by the values of the normal incidence illumination and of the absorption coefficient at the level z. It is worth reminding that all the derivations above are referred to monochromatic radiation. To obtain the corresponding total heating rate, Eq. (21) needs to be integrated over the whole solar spectral region. Eq. (21) describes heating of the atmosphere by direct solar radiation. We use this result to evaluate the heating of the atmosphere Qd ðzÞ due to diffuse solar radiation scattered by the underlying Lambertian surface with albedo A. We replace the illumination E  due to (monodirectional) solar radiation by the illumination due to radiation Ið0; mÞ dO scattered within an elementary solid angle dO in the direction with zenith angle cos1 m. Also, we replace transmittance t ðzÞ between the top of the atmosphere and the level z by the transmittance tðz0 ; z; mÞ between the underlying surface at the level z0 and the level z [cf. Eq. (B.1)]. We have [cf. Eqs. (14), (20)]: dQd ðzÞ ¼ Iðz0 ; mÞ dO  tðz; z0 ; mÞkðzÞ.

(22)

The intensity of radiation scattered by the underlying Lambertian surface is A m E  t ðz0 Þ. (23) p  Substituting Eq. (23) into Eq. (22) and integrating the transmittance tðz0 ; zÞ over the lower hemisphere we obtain: Iðz0 ; mÞ ¼

Qd ðzÞ ¼ 2m AE  t ðz0 Þht" ðz0 ; zÞikðzÞ,

(24)

where ht" ðz0 ; zÞi is diffuse transmittance for the upwelling radiation (see Appendix B). Combining Eqs. (21) and (24) and replacing the vertical coordinate z by atmospheric pressure p we obtain the net volume heating rate: Q ðpÞ ¼ E  ðt ðpÞ þ 2m At ðp0 Þht" ðp0 ; pÞiÞkðpÞ.

(25)

The heating rate of the surface is directly defined by the solar illumination at the bottom of the atmosphere and the albedo A of the surface: Qs ¼ ð1  AÞm E  tðp0 Þ.

(26)

Together, Eqs. (25) and (26) define the heating of the atmosphere and surface by solar radiation. 3.2. Cooling by thermal radiation In contrast to the heating by solar radiation, the thermal radiation may contribute to both heating and cooling of the atmosphere and surface, depending on the budget of radiative fluxes. In derivations below, we obtain the heating and cooling rates separately and then combine them into a net cooling rate. To compute the heating by thermal radiation we obtain the upward and downward fluxes of radiation illuminating the atmosphere at a given level z, then evaluate their decreases while moving correspondingly downward and upward between levels z and z  dz; further on, we relate these decreases to dz, and finally, taking the opposite of the sum of resulting radiative sink rates we obtain the corresponding heating rate due to thermal radiation. To obtain the upward and downward fluxes of thermal radiation we need to get the solution of the equation of RT over the whole atmospheric column. In the absence of atmospheric scattering the RT equation and its boundary conditions have the form: u

dI þ Iðt; uÞ ¼ BðtÞ; dt

Ið0; uÞ ¼ 0; u40;

Iðt0 ; uÞ ¼ eBs ; uo0,

(27)

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where t is the optical depth, and u ¼ cos y is the cosine of the elevation angle measured from the direction of increase of t (nadir). In the absence of atmospheric scattering, and if, as in most practical cases, the scattering of downwelling radiation by the underlying surface can be neglected, the intensities of upwelling ðuo0Þ and downwelling ðu40Þ radiation can be computed separately. In particular, for the upwelling radiation at the top of the atmosphere after integration of the RT equation from the bottom of the atmosphere to the top we obtain:       Z 0 t0 t dt I " ð0; uÞ ¼ exp  exp  BðtÞ  , (28) eBs þ juj juj juj t0 and, after denoting tðt; uÞ ¼ expðt=jujÞ, Eq. (28) is easily reduced to the form of Eq. (1). In a similar way, after integration of the RT equation from the bottom of the atmosphere to the given level t we obtain the intensity of upwelling radiation at this level:    0    Z t t0  t t t dt0 0 I " ðt; uÞ ¼ exp  exp  eBs þ Bðt Þ  . (29) juj juj juj t0 The intensity of downwelling radiation ðu40Þ in the atmosphere is obtained by integration of the RT equation from the top of the atmosphere to the given level t:   Z t t  t0 dt0 . (30) I # ðt; uÞ ¼ exp  Bðt0 Þ u u 0 Eqs. (29) and (30) can be rewritten in a way analogous to Eq. (1) using the transmittance function tðt; t0 ; uÞ describing transmittance between levels t0 and t in the atmosphere (see Appendix B) and switching to the atmospheric pressure p as a vertical coordinate. We have: Z p I " ðp; uÞ ¼ tðp; p0 ; uÞeBs þ Bðp0 Þ dtðp; p0 ; uÞ, (31) p0

Z

p

I # ðp; uÞ ¼

Bðp0 Þ dtðp; p0 ; uÞ.

(32)

0

As in paper [9] we will need alternative expressions, which allow direct linearization with respect to transmittance functions, and, further on, with respect to the absorption coefficient. Performing integration by parts we obtain: Z p0 I " ðp; uÞ ¼ BðpÞ þ tðp; p0 ; uÞðeBs  Bðp0 ÞÞ þ tðp; p0 ; uÞ dBðp0 Þ, (33) p

Z

0

I # ðp; uÞ ¼ BðpÞ  tðp; 0; uÞBð0Þ þ

tðp; p0 ; uÞ dBðp0 Þ.

(34)

p

The directions of integration in Eqs. (31)–(34), and throughout this paper are chosen as follows. If the integration is performed over dBðp0 Þ then the direction of integration is from the given level p into the atmosphere. If the integration is performed over dtðp; p0 ; uÞ, then the direction of integration is chosen so that dtðp; p0 ; uÞ40. The upward and downward radiative fluxes are obtained by integration of corresponding intensities, Eqs. (31), (32) over respective hemispheres: Z F "# ðpÞ ¼ I "# ðp; uÞjuj dO. (35) 2p

Using the reasoning analogous to that preceding Eq. (20) we can directly express the heating rate due to the thermal radiation in the form: QðþÞ IR ðpÞ ¼ 2pðhI " ðpÞi þ hI # ðpÞiÞkðpÞ,

(36)

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where the angular brackets represent integration over corresponding hemispheres. Integrating Eqs. (31), (32) over respective hemispheres and using diffuse transmittances for the upwelling and downwelling radiation (see Appendix B) we have: Z p hI " ðpÞi ¼ ht" ðp; p0 ÞieBs þ Bðp0 Þ dht" ðp; p0 Þi, (37) p0

Z

p

Bðp0 Þ dht# ðp; p0 Þi.

hI # ðpÞi ¼

(38)

0

We can now point out the direct analogy between Eq. (1) and Eqs. (37), (38) and rewrite these equations in the alternative form: Z p0 hI " ðpÞi ¼ BðpÞ þ ht" ðp; p0 Þi dBðp0 Þ þ ht" ðp; p0 ÞiðeBs  Bðp0 ÞÞ, (39) p

Z

0

ht# ðp; p0 Þi dBðp0 Þ.

hI # ðpÞi ¼ BðpÞ  ht# ðp; 0ÞiBð0Þ þ

(40)

p

The description of cooling by thermal radiation is more straightforward. From the RT equation written for an infinitesimally thin layer dz in the atmosphere we have: dI "# ðz; uÞ ¼

1 1 BðzÞ dtðzÞ ¼ kðzÞBðzÞ dz. juj juj

(41)

The corresponding infinitesimal upward and downward fluxes leaving the layer dz are [cf. Eq. (35)]: Z dF "# ðzÞ ¼ dI "# ðz; uÞjuj dO.

(42)

2p

Substituting Eq. (41) into separate expressions for dF " ðzÞ and dF # ðzÞ and summing them we obtain the net flux of thermal radiation leaving the layer dz: dF ðzÞ ¼ 4pkðzÞBðzÞ dz.

(43)

Its ratio to the thickness of the layer dz taken with an opposite sign yields the volume cooling rate. Changing the variable from z to p we have: QðÞ IR ðpÞ ¼ 4pkðpÞBðpÞ.

(44)

The sum of Eqs. (44) and (36) taken with the opposite sign yields the net volume cooling rate QIR ðpÞ due to thermal radiation at the given level z in the atmosphere. Using Eqs. (37), (38) we obtain: ! Z p Z p 0 0 0 0 QIR ðpÞ ¼ 2pkðpÞ ht" ðp; p0 ÞieBs  2BðpÞ þ Bðp Þ dht" ðp; p Þi þ Bðp Þ dht# ðp; p Þi . (45) p0

0

On the other hand, substituting the alternative expressions Eqs. (39), (40) into Eq. (36) we see that the terms containing BðpÞ in these equations compensate the cooling rate (Eq. (44)). We have:  QIR ðpÞ ¼  2pkðpÞ ht" ðp; p0 ÞiðeBs  Bðp0 ÞÞ  ht# ðp; 0ÞiBð0Þ Z þ p

p0

ht" ðp; p0 Þi dBðp0 Þ þ

Z

0

 ht# ðp; p0 Þi dBðp0 Þ .

ð46Þ

p

In this form of expression for QIR ðpÞ, cooling is accounted for in the terms with Bð0Þ and Bðp0 Þ describing fluxes of radiation emanating from the atmosphere at its upper and lower boundaries. Now we consider the net IR cooling rate for the underlying surface. The surface is heated by the downwelling thermal radiation of the atmosphere and is cooled by its proper thermal radiation. The heating rate can be obtained by evaluating the illumination due to the downwelling radiation at p0 and accounting for

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the emissivity of the surface: QðþÞ sIR ¼ 2pehuI # ðp0 ; uÞi,

(47)

where the angular brackets denote the integration over the lower hemisphere. The cooling rate is defined by the flux of thermal radiation from the surface taken with the opposite sign: QðÞ sIR ¼ 2pehuBs i ¼ peBs .

(48)

The sum of Eqs. (47) and (48) taken with the opposite sign yields the net cooling rate for the underlying surface: QsIR ¼ peð2huI # ðp0 ; uÞi  Bs Þ.

(49)

Eq. (49) together with Eq. (45), or with its alternative, Eq. (46), define the heating/cooling of the atmosphere and surface by thermal IR radiation. Eq. (45) (or its alternative, Eq. (46)) together with Eq. (49) define the cooling of the atmosphere and surface by thermal IR radiation. 4. Sensitivity analysis of the solar heating rates In this section we analyze sensitivities of the solar heating rates of the atmosphere, Eq. (25), and surface, Eq. (26) to both atmospheric and surface parameters. 4.1. Linearization of the atmospheric heating rate with respect to the atmospheric absorption coefficient Linearization of Eq. (25) with respect to atmospheric properties, which are contained in the transmittance functions tðpÞ, htðp0 ; pÞi and absorption coefficient kðpÞ, gives: dQ ðpÞ ¼ E  fkðpÞ½dt ðpÞ þ 2m Aðdt ðp0 Þhtðp0 ; pÞi þ t ðp0 Þdhtðp0 ; pÞiÞ þ dkðpÞ½t ðpÞ þ 2 m At ðp0 Þhtðp0 ; pÞig.

ð50Þ

Neglecting variations due to variations of the mean molecular mass of the atmosphere (see [9] for corresponding discussion) we associate all the variations of the transmittance functions involved in Eq. (50) with variations of the absorption coefficient kðpÞ. From Eqs. (15), (16) we have: Z p Z p0 1 1 0 0 0 dt ðpÞ ¼  t ðpÞ dkðp ÞH g ðp Þ d ln p ¼  t ðpÞ yðp  p0 Þdkðp0 ÞH g ðp0 Þ d ln p0 , (51) m m 0 0 where yðxÞ is the Heavyside function which equals unity for positive values of its argument and zero for negative ones. For the transmittance of the whole atmospheric column we have: Z p0 1 dtðp0 Þ ¼  tðp0 Þ dkðp0 ÞH g ðp0 Þ d ln p0 . (52) m 0 In a similar fashion, using Eqs. (B.1), (B.2), and (B.6) we have:   Z p0   Z p0 1 1 dkðp0 ÞH g ðp0 Þ d ln p0 ¼  t" ðp0 ; pÞ yðp0  pÞdkðp0 ÞH g ðp0 Þ d ln p0 . dht" ðp0 ; pÞi ¼  t" ðp0 ; pÞ m m p 0 (53) Also, using the Dirac delta function dðxÞ with the normalization requirement Z p0 dðp  p0 ÞH g ðp0 Þ d ln p0 ¼ 1

(54)

0

we express dkðpÞ in the form: Z p0 dkðpÞ ¼ dðp  p0 Þdkðp0 ÞH g ðp0 Þ d ln p0 . 0

(55)

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Now substituting Eqs. (52), (53), and (55) into Eq. (50) we express the variation of the solar heating rate dQ ðpÞ in the form of Eq. (B.6) with the kernel:      1 1 0 0 0 K ðkÞ ðp; p Þ ¼ E H ðpÞ tðpÞ  yðp  p ÞkðpÞ þ dðp  p Þ þ 2m Atðp Þ kðpÞ  htðp0 ; pÞi  g   0 m m     1  tðp0 ; pÞ yðp0  pÞ þ htðp0 ; pÞidðp  p0 Þ , ð56Þ m which represents the desired sensitivity of the atmospheric solar heating rate Q ðpÞ to the atmospheric absorption coefficient kðpÞ. 4.2. Sensitivities of the atmospheric solar heating rate to the atmospheric and surface parameters Variation of the total absorption coefficient kðpÞ due to variation of any local atmospheric parameter X ðpÞ can be expressed as: qk dX ðpÞ. qX Thus, the corresponding sensitivity of Q ðpÞ can be expressed through the sensitivity to kðpÞ as



ðX Þ ðkÞ 0 0 qk

K  ðp; p Þ ¼ K  ðp; p Þ . qX p0 dkðpÞ ¼

(57)

(58)

For example, let aðpÞ be the number density of atmospheric aerosol. Variation daðpÞ will result in corresponding variation of the total absorption coefficient: and

dk ¼ da ¼ ad ln a

(59)

qk

¼ aðpÞ. q ln a p

(60)

Sensitivities of Q ðpÞ, Eq. (25), to the surface parameters have the form of corresponding partial derivatives (see Eq. (A.3)). For the surface albedo A, we immediately obtain: qQ ðpÞ ¼ 2m E  t ðp0 Þhtðp0 ; pÞikðpÞ. qA For the surface pressure p0 we have:   qQ ðpÞ qtðp0 Þ q ¼ 2m E  kðpÞ htðp0 ; pÞi þ tðp0 Þ htðp0 ; pÞi . q ln p0 q ln p0 q ln p0

(61)

(62)

Using the definition of the transmittance function (Eq. (15)) together with the definition of optical depth (Eq. (16)) we have: qtðp0 Þ 1 ¼ H g ðp0 Þkðp0 Þtðp0 Þ q ln p0 m

(63)

and using Eq. (B.12) we obtain:

   qQ ðpÞ 1 1 tðp ; pÞ . ¼ 2m E  kðpÞH g ðp0 Þkðp0 Þtðp0 Þ htðp0 ; pÞi þ m m 0 q ln p0

(64)

4.3. Sensitivities of the surface solar heating rate to the atmospheric and surface parameters Varying the expression for the surface solar heating rate, Eq. (26), where all atmospheric parameters are contained in the transmittance tðp0 Þ, and using the expression for dtðp0 Þ, Eq. (52), we can express dQs in the

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form of Eq. (A.6) with the kernel: K ðkÞ s ðpÞ ¼ ð1  AÞE  tðp0 Þ H g ðpÞ,

(65)

which has a meaning of sensitivity to the atmospheric absorption coefficient. Sensitivity to any atmospheric parameter X ðpÞ is expressed through K ðkÞ s ðpÞ in a way analogous to Eq. (58):



Þ ðkÞ 0 0 qk

. (66) K ðX s ðp; p Þ ¼ K s ðp; p Þ qX p0 Sensitivity to the albedo A is obtained by taking a corresponding partial derivative of Eq. (26) qQs ¼ m E  tðp0 Þ. qA Sensitivity to the surface pressure p0 is obtained from Eqs. (61) and (63): qQs ¼ ð1  AÞE  H g ðp0 Þkðp0 Þtðp0 Þ q ln p0

(67)

(68)

or, comparing with Eq. (65) we can express it through the value of the sensitivity of Qs to the atmospheric absorption coefficient at the surface: qQs ¼ kðp0 ÞK ðkÞ s ðp0 Þ. q ln p0

(69)

5. Sensitivity analysis of the thermal IR heating/cooling rates Most of this section (Sections 5.1–5.3) is devoted to the sensitivities of the atmospheric cooling rate to the atmospheric parameters. The sensitivities of the atmospheric cooling rate to the surface parameters are obtained in Section 5.4, and sensitivities of the surface cooling rate to surface parameters are obtained in Section 5.5. 5.1. Linearization of the atmospheric cooling rate with respect to the Planck function Varying Eq. (45) with respect to the Planck function BðpÞ, changing the sign and direction of integration in the integral over ½p0 ; p and rearranging terms we have:   Z p Z p0 dB QIR ðpÞ ¼ 2pkðpÞ 2dBðpÞ þ dBðp0 Þ dht# ðp; p0 Þi þ dBðp0 Þ dht" ðp; p0 Þi . (70) 0

p

Differentials of the transmittance functions are expressed through their derivatives in the form: dht"# ðp; p0 Þi ¼ 

qht"# ðp; p0 Þi d ln p0 . q ln p0

(71)

From Eq. (B.9) we have:

  qht"# ðp; p0 Þi 1 0 0 0 tðp; p ; uÞ . ¼ H g ðp Þkðp Þ  q ln p0 juj

(72)

Representing the variation dBðpÞ in Eq. (70) in the form analogous to Eq. (55) we can rewrite Eq. (70) in the form of Eq. (A.6) Z p0 0 0 0 dB QIR ðpÞ ¼ K ðBÞ (73) IR ðp; p ÞdBðp Þ d ln p 0

with the kernel:

    1 0 0 0 0 0 tðp; p ðp; p Þ ¼ 2pH ðp Þ kðp Þ ; uÞ  2dðp  p Þ . K ðBÞ g IR juj

(74)

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5.2. Linearization of the atmospheric cooling rate with respect to the absorption coefficient Linearizing the expression for QIR ðpÞ (Eq. (46)) we have:  dk QIR ðpÞ ¼ 2pkðpÞ dht" ðp; p0 ÞiðeBs  Bðp0 ÞÞ  dht# ðp; 0ÞiBð0Þ Z

p0

þ

dht" ðp; p0 Þi dBðp0 Þ þ

Z p

p

0

 Q ðpÞ dht# ðp; p0 Þi dBðp0 Þ þ IR dkðpÞ. kðpÞ

ð75Þ

To evaluate the terms with variations of the transmittance functions, we first linearize the transmittance function tðp; p0 ; uÞ and then perform corresponding angular integrations. Applying the definition (Eq. (B.1)) to the function tðp; p0 ; uÞ we can express its variation through that of the absorption coefficient in the form [cf. Eq. (3)]: Z p 1 dkðp00 ÞH g ðp00 Þ d ln p00 . (76) dtðp; p0 ; uÞ ¼  tðp; p0 ; uÞ u 0 p (Note that for both upwelling and downwelling radiation, the values of u and of d ln p00 have the same sign.) The intention of the lengthy derivations below is to express the variational terms corresponding to the upwelling and downwelling radiation in the brackets in the right side of Eq. (75) through the variation of the absorption coefficient kðpÞ and the intensities of, respectively, upwelling and downwelling radiation, Eqs. (31) and (30). Since these intensities are computed anyway, in order to obtain the IR cooling rates, this provides additional savings of computing time because, as a result, we gain a possibility to compute the sensitivities of heating/cooling rates to atmospheric parameters in parallel with these rates themselves. Using Eq. (76) for the integral terms in Eq. (75) due to the downwelling radiation (p0 op; u40) we have:  Z p Z 0 Z 0 1 0 0 0 0 dtðp; p ; uÞ dBðp Þ ¼ dBðp Þ  tðp; p ; uÞ d ln p00 H g ðp00 Þdkðp00 Þ. (77) u p p p0 Changing the direction of integration in both integrals, changing the order of integration over the triangular domain of p0 and p00 arguments, and exchanging the notations p0 2p00 we obtain:   Z 0 Z p Z p0 1 dtðp; p0 ; uÞ dBðp0 Þ ¼ d ln p0 dkðp0 ÞH g ðp0 Þ dBðp00 Þ  tðp; p00 ; uÞ . (78) u p 0 0 For the integral terms in Eq. (75) due to the upwelling radiation ðp0 4p; uo0Þ we have:  Z p Z p0 Z p0 1 0 0 0 0 dtðp; p ; uÞ dBðp Þ ¼ dBðp Þ  tðp; p ; uÞ d ln p00 H g ðp00 Þdkðp00 Þ. u p p p0 Performing the same manipulations as those preceding Eq. (78) we obtain:   Z p0 Z p0 Z p0 1 0 0 0 0 0 00 00 dtðp; p ; uÞ dBðp Þ ¼ d ln p dkðp ÞH g ðp Þ dBðp Þ  tðp; p ; uÞ . juj p p p0

(79)

(80)

Variations of the transmittance function for the off-integral terms in Eq. (75) can be rewritten in the form:   Z p 1 0 0 0 ðp0 op; u40Þ, dtðp; 0; uÞ ¼ d ln p dkðp ÞH g ðp Þ  tðp; 0; uÞ (81) u 0 Z dtðp; p0 ; uÞ ¼ p

p0

  1 d ln p dkðp ÞH g ðp Þ  tðp; p0 ; uÞ juj 0

0

0

ðp0 4p; uo0Þ.

(82)

Multiplying Eq. (81) by Bð0Þ, subtracting it from Eq. (77), performing integration by parts to replace the integration over Bðp00 Þ by that over tðp; p00 Þ and using the definition (Eq. (30)) of the intensity of downwelling

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radiation I # ðp; uÞ we obtain: Z 0 Z dtðp; 0; uÞBð0Þ þ dtðp; p0 ; uÞ dBðp0 Þ ¼ p

p 0

  1 d ln p0 dkðp0 ÞH g ðp0 Þ  tðp; p0 ; uÞðI # ðp; u0 Þ  Bðp0 ÞÞ . u

(83)

Multiplying Eq. (82) by eBs  Bðp0 Þ, adding it to Eq. (77), performing manipulations analogous to those preceding Eq. (78) and using the definition (Eq. (31)) of the intensity of upwelling radiation I " ðp; uÞ we obtain: Z p0 dtðp; p0 ; uÞðeBs  Bðp0 ÞÞ þ dtðp; p0 ; uÞ dBðp0 Þ p   Z p0 1 0 0 0 0 0 0 ¼ d ln p dkðp ÞH g ðp Þ  tðp; p ; uÞðI " ðp ; uÞ  Bðp ÞÞ . ð84Þ juj p Performing angular integrations of Eqs. (83), (84) we can express the variational terms corresponding to the upwelling and downwelling radiation in the brackets in the right side of Eq. (75) as follows:   Z 0 Z p 1 0 0 0 0 0 0 0 0 dht# ðp; 0ÞiBð0Þ þ dtðp; p ; uÞ dBðp Þ ¼ d ln p dkðp ÞH g ðp Þ  tðp; p ; uÞðI # ðp ; uÞ  Bðp ÞÞ , (85) u p 0 Z dht" ðp; p0 ÞiðeBs  Bðp0 ÞÞ þ

p0

dht" ðp; p0 Þi dBðp0 Þ

p

Z

p0

¼ p

  1 d ln p0 dkðp0 ÞH g ðp0 Þ  tðp; p0 ; uÞðI " ðp0 ; uÞ  Bðp0 ÞÞ . juj

ð86Þ

In conclusion, we need to express the variation dkðpÞ in Eq. (75) in the form of Eq. (55). After that, Eq. (75) can be rewritten in the form of Eq. (A.6): Z p0 0 0 0 dQIR ðpÞ ¼ K ðkÞ (87) IR ðp; p Þdkðp Þ d ln p 0

with the kernel 0 0 0 0 0 0 K ðkÞ IR ðp; p Þ ¼ yðp  p ÞK # ðp; p Þ þ yðp  pÞK " ðp; p Þ þ dðp  p ÞK d ðpÞ,

where kernels K " ðp; p0 Þ and K # ðp; p0 Þ have the form:   1 0 0 0 0 0 tðp; p ; uÞðBðp ÞÞ  I "# ðp ; uÞ K "# ðp; p Þ ¼ 2pkðpÞH g ðp Þ juj

(88)

(89)

and K d ðpÞ ¼ H g ðpÞ

QIR ðpÞ . kðpÞ

(90)

5.3. Sensitivities of the atmospheric cooling rate to atmospheric parameters Using results of previous subsections we can now obtain the sensitivities of atmospheric heating/cooling rates to the atmospheric parameters. At a given frequency, the Planck function BðpÞ depends only on the atmospheric temperature TðpÞ. Dependence on the remaining atmospheric parameters involved in radiative heating and cooling is contained in the absorption coefficient kðpÞ which also may depend on atmospheric temperature. Thus, in general, variation of atmospheric temperature results in corresponding variations dBðpÞ ¼ ðqB=qTÞdTðpÞ and dkðpÞ ¼ ðqk=qTÞdTðpÞ. Substituting these variations in Eqs. (73) and (87) and summing resulting expressions we obtain an expression for the variation of QIR ðpÞ due to atmospheric

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temperature TðpÞ in the form of Eq. (A.6) with the kernel:





ðTÞ ðBÞ ðkÞ 0 0 qB

0 qk

þ K IR ðp; p Þ . K IR ðp; p Þ ¼ K IR ðp; p Þ

qT qT 0 0 Tðp Þ

1111

(91)

p

ðkÞ 0 0 Expressions for intermediate kernels K ðBÞ IR ðp; p Þ and K IR ðp; p Þ in Eq. (91) are given by Eqs. (74) and (88). If any other atmospheric parameter X ðpÞ is varied, its variation manifests itself in a corresponding variation of the absorption coefficient dkðpÞ ¼ ðqk=qX ÞdX ðpÞ. Substituting this variation in Eq. (87) we obtain an expression for the variation of QIR ðpÞ due to an atmospheric parameter (other than temperature) X ðpÞ in the form of Eq. (A.6) with the kernel:



Þ ðkÞ 0 0 qk

K ðX ðp; p Þ ¼ K ðp; p Þ , (92) IR IR qX 0 p

0 where the intermediate kernel K ðkÞ IR ðp; p Þ is given by Eq. (88).

5.4. Sensitivities of the atmospheric cooling rate to surface parameters Here we evaluate sensitivities to surface temperature T s , surface emissivity e, and atmospheric pressure at the surface p0 . From Eq. (45) we directly obtain:

qQIR qB

¼ 2pkðpÞht" ðp0 ; pÞie

, (93) qT qT s

Ts

qQIR ¼ 2pkðpÞht" ðp0 ; pÞiBs . qe The sensitivity to the surface pressure p0 is obtained from Eq. (45): ! Z p qQIR q q 0 0 ¼ 2pkðpÞ ht" ðp0 ; pÞieBs þ Bðp Þ dht" ðp ; pÞi . q ln p0 q ln p0 p0 q ln p0 For the first term inside parentheses, applying Eq. (B.12) we have:   q 1 ht" ðp0 ; pÞi ¼ H g ðp0 Þkðp0 Þ t" ðp0 ; pÞ . q ln p0 u

(94)

(95)

(96)

The second term inside the parentheses needs to be transformed to extract the transmittance from the differential. Using Eq. (96), the differential can be rewritten in the form:   q 1 0 0 0 0 0 t dht" ðp0 ; pÞi ¼ ht ðp ; pÞid ln p ¼ H ðp Þkðp Þ ðp ; pÞ d ln p0 (97) " g " q ln p0 u and we can represent the second term in the form of the derivative of an integral with respect to its lower limit, ln p0 . This derivative equals the value of the function under the integral sign at p0 taken with an opposite sign, and we have:   Z p q 1 0 0 Bðp Þ dht" ðp ; pÞi ¼ Bðp0 Þ t" ðp0 ; pÞ H g ðp0 Þkðp0 Þ. (98) q ln p0 p0 u Substituting Eqs. (96), (98) in Eq. (95) we obtain:   qQIR 1 ¼ 2pkðpÞ t" ðp0 ; pÞ H g ðp0 Þkðp0 ÞðBðp0 Þ  eBs Þ. u q ln p0

(99)

5.5. Sensitivities of the surface cooling rate to atmospheric and surface parameters Sensitivities with respect to atmospheric parameters are evaluated using the same linearization approach as in Sections 5.1–5.3. We start from Eq. (49). Variation of any atmospheric parameter results in variation of

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huI # ðp0 ; uÞi only. We have: dQsIR ¼ 2pehudI # ðp0 ; uÞi.

(100)

Using Eq. (32) we linearize QsIR with respect to BðpÞ: Z p0 dB QsIR ¼ 2pe dBðpÞ dhut# ðp0 ; pÞi.

(101)

0

From Eq. (B.11) we have dhut# ðp0 ; pÞi ¼ H g ðpÞkðpÞht# ðp0 ; pÞi d ln p.

(102)

Substituting Eq. (102) into Eq. (101) we rewrite the variation dQsIR in the form: Z p0 H g ðpÞkðpÞht# ðp0 ; pÞi dBðpÞ d ln p. dB QsIR ¼ 2pe

(103)

0

Thus, variation dB QsIR is transformed to the form of Eq. (A.5) with the kernel: K ðBÞ sIR ðpÞ ¼ 2peH g ðpÞkðpÞht# ðp0 ; pÞi.

(104)

Now we need to linearize QsIR with respect to kðpÞ. Linearizing Eq. (34) with respect to ht# ðp0 ; pÞi we have: ! Z 0 dt QsIR ¼ 2pe Bð0Þdhut# ðp0 ; 0Þi þ dhut# ðp0 ; pÞi dBðpÞ . (105) p0

Multiplying Eq. (B.14) by u and integrating it over the lower hemisphere we express the variation dhut# ðp; p0 Þi through dkðpÞ: Z p dk hut# ðp; p0 Þi ¼ ht# ðp; p0 Þi dkðp00 ÞH g ðp00 Þ d ln p00 . (106) p0

Substituting into Eq. (105) gives the expression of the variation of QsIR with respect to the absorption coefficient: " # Z p0 Z 0 Z p0 0 0 0 dk QsIR ¼ 2pe ht# ðp0 ; pÞiBð0Þ H g ðpÞdkðpÞ d ln p  dBðpÞht# ðp0 ; pÞi d ln p dkðp ÞH g ðp Þ . p0

0

p

(107) Now we transform the second double integral term in square brackets of Eq. (107). Changing the direction of integration in the integral over dBðpÞ, changing the order of integration over the triangular domain of p and p0 , and exchanging the notations p2p0 we obtain: Z 0 Z p0 Z p0 Z 0 0 0 0 dBðpÞht# ðp0 ; pÞi d ln p dkðp ÞH g ðp Þ ¼ d ln p dkðpÞH g ðpÞ dBðp0 Þht# ðp0 ; p0 Þi. (108) p0

p

p

0

Combining terms in square brackets in Eq. (107) we obtain:   Z p0 Z 0 dk QsIR ¼ 2pe d ln p dkðpÞH g ðpÞ ht# ðp0 ; 0ÞiBð0Þ  ht# ðp0 ; p0 Þi dBðp0 Þ . 0

(109)

p

This result can be rewritten in a more compact form. Comparing the terms in square brackets with Eq. (40) we have: Z p0 H g ðpÞ½BðpÞ  hI # ðpÞidkðpÞ d ln p. (110) dk QsIR ¼ 2pe 0

Thus, variation dk QsIR is transformed to the form of Eq. (A.5) with the kernel: K ðkÞ sIR ðpÞ ¼ 2peH g ðpÞ½BðpÞ  hI # ðpÞi.

(111)

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Using Eqs. (104), (111) we can obtain sensitivities of the surface cooling rate QsIR to the atmospheric temperature TðpÞ and to any other atmospheric parameter X ðpÞ. In analogy with Eqs. (91), (92) we have:



qB

qk

ðTÞ ðBÞ ðkÞ þ K sIR ðpÞ , (112) K sIR ðpÞ ¼ K sIR ðpÞ

qT TðpÞ qT p Þ ðkÞ K ðX sIR ðpÞ ¼ K sIR ðpÞ

qk

. qX p

(113)

Sensitivities of the surface heating/cooling rates to the surface parameters are obtained as partial derivatives with respect to the corresponding parameters. From Eq. (49) we immediately obtain sensitivities to the surface temperature T s and surface emissivity e:

qQsIR qB

¼ pe , (114) qT T s qT s qQsIR ¼ pBs . qe For the surface pressure we have from Eq. (49): qQsIR q ¼ 2p huI # ðp0 ; uÞi. q ln p0 q ln p0

(115)

(116)

Substituting Eq. (32) for I # ðp0 ; uÞ, using Eq. (B.12) to represent ht# ðp0 ; pÞi and evaluating the derivative of the resulting integral with respect to its upper limit we obtain ðht# ðp0 ; p0 Þi  1Þ: qQsIR ¼ 2pH g ðp0 Þkðp0 ÞBðp0 Þ. q ln p0

(117)

6. Example of application: linearized radiative 1D model The linearized model of radiative heating and cooling rates developed above makes it possible to explicitly linearize the radiative 1D model. For demonstration purposes, we assume that the field variables of this model consist of atmospheric temperature TðpÞ and surface temperature T s . Restoring the subscript n where necessary, to label the corresponding monochromatic quantities, we formulate the initial nonlinear model in the form: Z dTðpÞ ¼ ðQ;n ðpÞ  QIR;n ðpÞÞ dn, (118) cp dt Z dT s ¼ ðQs;n  QsIR;n Þ dn. (119) cs dt Here cp and cs are heat capacities of the atmosphere and surface respectively, and integration is carried out over the relevant spectral range. Model parameters include a vertical profile of number density of nonscattering gray aerosol N a ðpÞ, longwave surface emissivity en , and shortwave surface albedo An . We assume that these parameters experience some perturbations N 0a ðpÞ, e0n , and A0n which result in perturbations of the heating and cooling rates Q0;n ðpÞ, Q0IR;n ðpÞ, Q0s;n , and Q0sIR;n both directly and through ensuing perturbations of atmospheric and surface temperatures T 0 ðpÞ and T 0s . Linearization of the nonlinear system of Eqs. (118), (119) with respect to the above model parameters gives: Z dT 0 ðpÞ ¼ ðQ0;n ðpÞ  Q0IR;n ðpÞÞ dn, cp (120) dt Z dT 0s ¼ ðQ0s;n  Q0sIR;n Þ dn, (121) cs dt

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Perturbations Q0;n ðpÞ and Q0IR;n ðpÞ of atmospheric rates in the system of Eqs. (120), (121) have the form (see Appendix B): Z p0 Z p0 Q0n ðpÞ ¼ K nðTÞ ðp; p0 ÞT 0 ðp0 Þ d ln p0 þ K nðN a Þ ðp; p0 ÞN 0a ðp0 Þ d ln p0 0

0

qQ ðpÞ qQn ðpÞ 0 qQn ðpÞ 0 þ n T 0s þ An þ e. qT s qAn qen n

ð122Þ

Similarly, perturbations Q0s;n and Q0sIR;n of surface rates in the system of Eqs. (120), (121) have the form: Z p0 Z p0 qQs;n 0 qQs;n 0 qQs;n 0 0 0 aÞ Q0s;n ¼ K ðTÞ ðpÞT ðpÞ d ln p þ K ðN T þ A þ e. (123) s;n s;n ðpÞN a ðpÞ d ln p þ qT s s qAn n qen n 0 0 Representing the right-hand terms of the system of Eqs. (120), (121) in the form of Eqs. (122) and (123), respectively, and transferring the terms with field variables into the left sides of resulting equations we obtain a system of linear inhomogeneous differential equations with respect to the linearized field variables TðpÞ0 and T s :  Z  Z p0 Z Z dT 0 ðpÞ 0 sÞ þ K nðTTÞ ðp; p0 Þ dn T 0 ðp0 Þ d ln p0 þ K ðTT ðpÞ dn T ¼ SðTÞ (124) cp n s n ðpÞ dn, dt 0  Z  Z p0  Z Z dT 0s ðT s TÞ 0 ðT s T s Þ sÞ Kn þ cs ðpÞ dn T ðpÞ d ln p þ Kn dn T 0s ¼ S ðT dn, (125) n dt 0 where ðTÞ 0 0 K ðTTÞ ðp; p0 Þ ¼ K ðTÞ n IR;n ðp; p Þ  K ;n ðp; p Þ, sÞ ðpÞ ¼ K ðTT n

qQIR;n ðpÞ qQ;n ðpÞ  , qT s qT s

ðTÞ s TÞ K ðT ðpÞ ¼ K ðTÞ n sIR;n ðpÞ  K s;n ðpÞ,

qQsIR;n qQs;n  , qT s qT s Z p0 qQ;n ðpÞ 0 qQ;n ðpÞ 0 ðN a Þ ðTÞ 0 0 0 0 0 aÞ Sn ðpÞ ¼ ðK ðN An þ en , ;n ðp; p Þ  K IR;n ðp; p ÞÞN a ðp Þ d ln p þ qAn qen 0 Z p0 qQs;n 0 qQs;n 0 ðN a Þ ðT s Þ 0 aÞ Sn ¼ ðK s;n ðpÞ  K ðN A þ e. sIR;n ðpÞÞN a ðpÞ d ln p þ qAn n qen n 0 sTsÞ ¼ K ðT n

(126) (127) (128) (129)

(130)

(131)

The system of Eqs. (124), (125) provides the desired formulation of the linearized 1D radiative model. The expressions for the monochromatic sensitivities of the radiative heating and cooling rates to the atmospheric temperature TðpÞ and surface temperature T s used in Eqs. (126)–(129) were obtained in previous sections. The right-hand terms in Eqs. (130), (131) are expressed through the perturbations of the atmospheric and surface parameters, and monochromatic sensitivities to these parameters were also obtained in previous sections. This concludes the formulation of the linearized 1D radiative model. 7. Discussion and conclusion The results of this paper can be summarized as follows. Availability of sensitivities of radiative heating and cooling rates with respect to atmospheric and surface parameters makes it possible to perform the linearization of the rates with respect to these parameters. The initial nonlinear expressions for the heating and cooling rates were obtained in Section 3. Expressions for atmospheric solar heating and thermal IR cooling rates are given by Eqs. (25) and (45) [(46)], respectively. (Numbers in square brackets label the alternative

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1115

expressions.) Expressions for the surface solar heating and thermal IR cooling rates are given by Eqs. (26) and (49), respectively. Using the above expressions, four groups of sensitivities discussed in the Introduction were obtained in Sections 4 and 5. Sensitivities of the atmospheric solar heating and thermal IR cooling rates to the atmospheric parameters (Group 1) are given by Eqs. (58) and (91), (92), respectively. Sensitivities of those rates to the surface parameters (Group 2) are given by Eqs. (61), (64) and (93), (94), (99), respectively. Sensitivities of the surface solar heating and thermal IR cooling rates to the atmospheric parameters (Group 3) are given by Eqs. (66) and (112), (113), respectively. Sensitivities of those rates to the surface parameters (Group 4) are given by Eqs. (67), (68) [or (69)] and (114), (115), (117), respectively. In Section 6, the above results were applied to the linearization of the simplest 1D radiative model. The terms describing radiative heating and cooling rates, as sources of diabatic heat, enter into the system of primitive equations of atmospheric dynamics in the same form, via the thermodynamic equation, and the system of these equations can be linearized with respect to the atmospheric and surface parameters in the same manner as the 1D radiative model above. If convective adjustment takes place, it needs to be addressed separately. Corresponding linearization of the radiative–convective model deserves a separate effort. In this paper we considered the case of non-scattering planetary atmospheres. Based on the analogy with evaluations of weighting functions for remote sensing addressed in the introduction, one can anticipate that the adjoint sensitivity analysis of radiative transfer used to compute these weighting functions in presence of atmospheric scattering (see, e.g., [10]) can also be applied to the linearization of radiative heating and cooling rates in scattering atmospheres. This topic also deserves yet another separate effort. Acknowledgments The author is grateful to Prof. Peter Gierasch (Cornell University) and Prof. Andrew Ingersoll (California Institute of Technology) for fruitful discussions which contributed to the understanding and formulation of the problem considered in this paper. The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology under a contract with the National Aeronautics and Space Administration. Appendix A. Sensitivity analysis for discrete and continuous model parameters Sensitivities of models provide a quantitative description of responses of the output parameters to the variations of the input parameters of models. For discrete input parameters, such as the surface albedo A, corresponding sensitivity has the form of a partial derivative with respect to this parameter. For the continuous input parameters, such as the vertical profile of temperature TðpÞ, corresponding sensitivity has the form of a variational derivative. Specific cases are below. Let some discrete input parameter Di experience a variation dDi . Then the response of any output parameter Po (a discrete or continuous one) is dPo ¼

qPo dDi , qDi

(A.1)

where qPo =qDi is the sensitivity of the output parameter Po to the discrete input parameter Di . In particular, for the discrete output parameter Do we have: dDo ¼

qDo dDi . qDi

(A.2)

For the continuous output parameter F o ðxÞ the response at each value of its argument x is dF o ðxÞ ¼

qF o ðxÞ dDi . qDi

(A.3)

Let some continuous input parameter F i ðxÞ experience a variation dF i ðxÞ. Then the response of any output parameter Po (a discrete or continuous one) depends, in general, on the values of this variation in the whole

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domain of the argument x 2 X : Z dPo ¼ K PðFoi Þ ðxÞdF i ðxÞ dx.

(A.4)

In particular, for the discrete output parameter Do we have: Z iÞ dDo ¼ K ðF Do ðxÞdF i ðxÞ dx.

(A.5)

For the continuous output parameter F o ðxÞ, its response at each value of its argument x is Z 0 0 0 iÞ dF o ðxÞ ¼ K ðF F o ðx; x ÞdF i ðx Þ dx .

(A.6)

X

X

X

The kernels of Eqs. (A.5), (A.6) are corresponding variational derivatives of output parameters with respect to the input parameters: iÞ K ðF Do ðxÞ ¼

dDo , dF i ðxÞ

(A.7)

dF o ðxÞ . dF i ðx0 Þ

0 iÞ K ðF F o ðx; x Þ ¼

(A.8)

Thus, the sensitivities to the discrete input parameters are represented by corresponding partial derivatives, Eqs. (A.2) and (A.3). Sensitivities to the continuous input parameters are represented by corresponding variational derivatives, Eqs. (A.7) and (A.8), respectively. Appendix B. Transmittance functions This section contains the reference information on the transmittance functions for monodirectional and diffuse radiation, which are repeatedly used in the main text. Monodirectional transmittance between two levels, z and z0 in the direction cos1 u measured from the nadir is defined from the difference of corresponding optical depths as:    Z z0  tðzÞ  tðz0 Þ 1 0 00 00 tðz; z ; uÞ ¼ exp  kðz Þ dz . ¼ exp  (B.1) u u z It is assumed that the radiation is propagating from level z0 to level z. Then the numerator and denominator under the sign of exponent function in Eq. (B.1) always have the same sign. Diffuse transmittances for the upwelling and downwelling radiation are defined as angular integrals of Eq. (B.1) over, respectively, the lower and upper hemispheres: Z 0 0 ht" ðz; z Þi ¼ tðz; z0 ; uÞ du, (B.2) 1

ht# ðz; z0 Þi ¼

Z

1

tðz; z0 ; uÞ du.

(B.3)

0

Similarly, corresponding cosine-weighted and inverse cosine-weighted diffuse transmittances are defined as: Z 0 tðz; z0 ; uÞu du, (B.4) hut" ðz; z0 Þi ¼ 1

hut# ðz; z0 Þi ¼

Z

1

tðz; z0 ; uÞu du,

(B.5)

0



1 t" ðz; z0 Þ u



Z

0

tðz; z0 ; uÞ

¼ 1

du , u

(B.6)

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 Z 1 1 du t# ðz; z0 Þ ¼ tðz; z0 ; uÞ . u u 0

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(B.7)

Note that the diffuse transmittances (Eqs. (B.2)–(B.7)) can be expressed through corresponding exponential integral functions: huk t"# ðz; z0 Þi ¼ E 2k ðjtðzÞ  tðz0 ÞjÞ.

(B.8)

In the applications in the main text, the derivatives of diffuse transmittances given by Eqs. (B.2)–(B.5) are needed with respect to z0 . Substituting dtðz0 Þ ¼ kðz0 Þ dz we obtain:   q 1 0 0 0 t ht ðz; z Þi ¼ kðz Þ ðz; z Þ , (B.9) "# "# qz0 u q hut"# ðz; z0 Þi ¼ kðz0 Þht"# ðz; z0 Þi. (B.10) qz0 [Note that uo0 for the upwelling radiation in Eqs. (B.9), (B.10).] Most derivations in the main text are done using atmospheric pressure p as a vertical coordinate. Eqs. (B.1)–(B.7) are converted by straightforward replacement z; z0 ! p; p0 . To convert Eqs. (B.9), (B.10) we use the differential hydrostatic equation in the form dz ¼ H g ðpÞ d ln p

(B.11)

to obtain:

  q 1 0 0 0 0 t ht ðp; p Þi ¼ H ðp Þkðp Þ ðp; p Þ , "# g "# q ln p0 u

(B.12)

q hut"# ðp; p0 Þi ¼ H g ðp0 Þkðp0 Þht"# ðp; p0 Þi. q ln p0

(B.13)

In applications in the main text we also need an expression for the variation of the transmittance function tðp; p0 ; uÞ. Using Eq. (B.1) and switching to the pressure coordinate we obtain: Z p 1 dkðp00 ÞH g ðp00 Þ d ln p00 . (B.14) dtðp; p0 ; uÞ ¼  tðp; p0 ; uÞ u 0 p

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