Definition of a Generalized Diabatic Circulation Based on a Variational Approach

ISSN 000-4338, Izvestiya, Atmospheric and Oceanic Physics, 2007, Vol. 43, No. 4, pp. 436 44. Pleiades Publishing, Ltd., 2007. Original Russian Text A.S. Medvedev, 2007, published in Izvestiya AN. Fizika Atmosfery i Okeana, 2007, Vol. 43, No. 4, pp. 476 48.. Definition of a Generalized Diabatic Circulation Based on a Variational Approach A. S. Medvedev Max Planck Institute for Solar System Research, Max-Planck-Strasse 2, Katlenburg-Lindau, 379 Germany e-mail: medvedev@mps.mpg.de Received October 20, 2006; in final form, February 4, 2007 Abstract Diabatic-circulation diagnostics with the use of the distributions of heating rates and potential temperature requires that, in each particular case, a special and ambiguously defined correction to the stream function be introduced to turn a globally averaged vertical velocity to zero at any isobaric level. Up to now, the physical nature of this correction has been little explained and it has been usually written in a form that has not been substantiated to a sufficient extent. In this paper, this correction and its uncertainty are related to the eddy term, which is usually neglected in the concept of diabatic velocities. The decomposition of wave fluxes into advective and diffusion components is not unique. As a result, one can formulate a variational problem of minimizing the diffusion component of the wave flux and, thus, the problem of finding advective velocities, which involve the maximum of eddy-induced advection. A unique solution of this problem is obtained, and the relation of the solution to the standard diabatic circulation is studied. It is shown that, in the approximation of quasi-horizontal isentropes, the generalized diabatic stream function is identical with the standard stream function. This result partially justifies the correction that is commonly used in calculations of the diabatic circulation. DOI: 0.34/S000433807040056 This equation defines the vertical diabatic velocity w*, where θ and Q are the zonally averaged potential tem- INTRODUCTION The diabatic circulation denotes the advective transport on a meridional (latitudinal altitudinal) plane that is completely controlled by diabatic heating. The concept of the diabatic circulation is widely used in diagnostic studies of the atmosphere [ 4]. It is believed that diabatic velocities adequately describe the meridional transport of zonally averaged tracers and zonal mean potential temperature in particular [5]. The basic idea of the definition of the diabatic circulation is the assumption that the effect of disturbances in the thermodynamic equation that arises from zonal or any other averaging can be completely represented as the advection of potential temperature. In this case, the diabatic velocity ( v *, w* ) is the sum of the Eulerian mean velocity ( v, w) and the eddyinduced velocity, which completely involves eddy fluxes. In the simplified case where one can disregard the transience ( / t = 0) and neglect the meridional advection on the basis of scaling arguments ( v wθ z ), the averaged thermodynamic equation written in log-isobaric coordinates is reduced to w*θ z = Q. () perature and diabatic foreing, respectively, and the letter subscript denotes the partial derivative in the corresponding direction. The meridional component of the diabatic velocity v * can be obtained from the continuity equation for ( v *, w* ) by introducing the suitable stream function: v * = ψ z, w* = ψ y. The global integral of the function w* weighted by the domain s area is bound to be zero, because no global mass transport occurs through any isobaric level. In practice, this condition is usually not fulfilled for given distributions of Q and θ ; therefore, the introduction of a correction to w* or, in turn, to Q is always required. The author of [6] assumed that this correction appears in practice because of the possible inaccuracies in determining Q. There is an infinitely large number of latitude-dependent functions that make the global mean vertical velocity w* vanish. The sensitivity of the diabatic velocity to different procedures of correction were studied in [6]. However, only the latitude-independent correction to the vertical velocity is usually applied in practice. In this paper, the diabatic circulation obtained with the aid of such a correction will be referred to as the standard circulation. By the present time, no rigorous substantiation has been proposed for this method of reaching the global mass balance. It seems likely that this 436

DEFINITION OF A GENERALIZED DIABATIC CIRCULATIONS BASED ON 437 method is used in studies primarily because of its simplicity. In order to understand this problem, it is useful to consider the relationship between the diabatic and the residual circulation. The residual circulation on a meridional plane represents the advective transport defined as the sum of the Eulerian mean velocity and the eddy-induced velocity [7]. Unlike the diabatic circulation, in the residual circulation, the diffusive component of transport is not necessarily equal exactly to zero. The decomposition of the eddy term into advective and diffusion components is ambiguous. Since this term appears in the mean thermodynamic equation only in the form of the divergence of the v'θ' flux, where the primes denote deviations from the means, any nondivergent (solenoidal) flux can be added to the expression under the gradient sign ( ). This rotational flux, or gauge, contributes simultaneously to both the advective and diffusion components; therefore, the two components are generally other than zero [8]. In specific applications, additional constraints are required to separate the advective and diffusion eddy contributions. Thus, in general, the wave transport described by eddy fluxes cannot be taken into account completely through advection. For example, the eddy-induced diffusion in the context of the transformed Eulerian mean (TEM) formalism vanishes only in the quasi-geostrophic approximation for steady and adiabatic disturbances [7]. In a strict sense, such disturbances do not interact with the mean flow and do not produce the net transport of the mean potential temperature. The diabatic circulation in the form as it is defined in the first sentence of this paper need not occur in a general case. Nevertheless, it is possible to reformulate the diabatic circulation in a useful way. This paper proposes the representation of the eddy flux in the form that minimizes the diffusive part of the transport, thus making the advective part maximal. The velocity ( v *, w* ) defined in this manner will incorporate the maximum of the eddy-induced advective transport, whereas the latitudinal correction discussed above will be uniquely related to the least diffusive contribution of disturbances. This paper is organized as follows. Section presents the basic equations. The variational problem is formulated and solved in Section 2. The resulting diabatic circulation is compared to the standard circulation in Section 3.. BASIC EQUATIONS The thermodynamic equation for the mean potential temperature Θ is written in log-isobaric coordinates as Θ t + v Θ y + wθ z = Q ρ 0 ρ 0 v'θ', (2) where the overbar denotes zonal averaging, Q is the rate of diabatic heating or cooling, v' = (v', w') is the two-dimensional (on a meridional plane) eddy velocity, ρ 0 = ρ s exp( z/h) is the background density, H is the atmospheric scale height, and ρ s is the density at the level z = 0. The scalar-product operator ( ) is written in spherical coordinates as ( ) = (cosφ) (cosφ)/ y + / z, where φ is latitude, y = a φ, and a is the radius of the planet. In order to parameterize the eddy flux in terms of the mean potential temperature, the gradientflux relation v'θ' = K Θ. (3) is commonly used. In (3), K denotes the tensor whose components must be found and θ ( y, z ) θ is the gradient written in spherical coordinates. The tensor K can be uniquely decomposed into its symmetric and asymmetric components [9], which are associated with wave-induced advection (skew diffusion) and usual gradient diffusion, respectively. Thus, the divergence of the eddy flux on the right-hand side of (2) may be written as ρ 0 ρ 0 v'θ' = ṽ w θ z + ρ 0 ρ 0 K θ, (4) where the symmetric component of the tensor K is approximated by the local isotropic diffusion with the single parameter K(y, z) and the wave-induced velocities ( ṽ, w ) are determined by the asymmetric component of K. Then, thermodynamic equation (2) may be written as θ t + v * + w*θ z = Q + ρ 0 ρ 0 K θ, (5) where the velocities induced by disturbances are included into the residual velocity introduced above: v * = v + ṽ, w* = w + w. In the following, we will disregard the time dependence of θ t because it can always be restored by formally adding θ t to Q. The Eulerian mean velocity v ( v, w) satisfies the equation of mass conservation ρ 0 v = 0, (6) and it will be assumed that ( ṽ, w ) possesses the same conservation property. Therefore, the residual velocity ( v *, w* ) is pure solenoidal as well. Introducing the mass-weighted stream function Ψ v * = ( ρ 0 cosφ) Ψ z, w* = ( ρ 0 cosφ) Ψ y, (7) and substituting it into (5), we obtain Ψ z + Ψ y θ z = cosφρ ( 0 Q + ρ 0 K θ). (8) The lateral boundary condition for Ψ can be found from (7). First, the global integral of the mass- and area-weighted velocity w* must vanish at any logisobaric level, cosφw* dφ. It follows from NP the ρ SP 0

438 MEDVEDEV second equation in (7) that Ψ must be a periodic function at the South and North poles, Ψ(SP) = Ψ(NP). Second, no zonally averaged mass flux occurs through the poles, ρ 0 v * (SP) = ρ 0 v * (NP) = 0. Thus, it follows from the first equation in (7) that Ψ z vanishes at the poles. Consequently, the stream function Ψ must be constant at the two poles at all altitudes. An arbitrary constant can always be added to the function Ψ in (7) to satisfy the lateral boundary condition, the velocities ( v *, w* ) being unchanged. Therefore, the exact value of Ψ(SP) = Ψ(NP) = const is of no importance here. Next, it is convenient to introduce the unit vectors nˆ and ŝ so that nˆ is locally normal to the mean isentropes θ = const, i.e., directed along the local gradient of the mean potential temperature θ. The vector ŝ is orthogonal to nˆ and locally tangent to the isentropes. If ŷ = ( ŷ, 0) and ẑ = (0, ẑ ) are the unit vectors in the meridional and vertical directions, respectively, their relations to the new coordinates ŝ and nˆ can be formulated as follows: θ z ŝ = ---------ŷ --------- ẑ, nˆ = ---------ŷ + --------- ẑ. (9) θ θ θ θ With the use of (9), relation (8) may be rewritten in the (n, s)-coordinate system in a substantially simplified form: Ψ ------ s = (0) where θ n = θ / n = θ. This one-dimensional equation for Ψ must be solved with the periodic boundary conditions at the South and North poles Ψ(SP) = Ψ(NP) = const to ensure the closure of streamlines in an isolated domain, as was discussed somewhat earlier. For simplicity, we will assume that Ψ(SP) = Ψ(NP) = 0, although the subsequent derivation can be repeated with any nonzero constant. As a result, the following integral condition must be imposed on (0): NP Ψ( NP) = cosφθ n [ ρ 0 Q + ( ρ 0 KΘ n ) ] d s = 0, () n SP where the integral is taken along an isentrope θ = const from the South to the North Pole. Since the stream function Ψ is determined by Eq. (0) on isentropes, it is useful to compare it to the diabatic circulation written in isentropic coordinates. Eddy fluxes and temperature advection do not appear in the context of this formulation, and the diabatic heating rate divided by the parameter of static stability turns out to be exactly equal to the vertical diabatic velocity. However, a limitation of this approach lies in the fact that the velocities in an isentropic coordinate ----------- cosφ θ ρ Q + 0 ρ0 Kθ n, n n system are Lagrangian; i.e., their divergence is other than zero. In a strict sense, the stream function for such velocities cannot be introduced and, correspondingly, the meridional diabatic transport cannot be found from the continuity equation. Only under the additional assumptions that make it possible to negleet the divergent component of fluxes can one introduce such a stream function. In this case, the stream function can be found from an equation similar to Eq. (0) but without the diffusion term on the righthand side (see, for example, [0], Appendix C). It should be noted that the vertical velocity introduced in this manner is globally unbalanced, as is described in the Introduction, and the problem of choosing the correction arises as previously. The stream function Ψ and the corresponding transport velocities could be obtained from Eq. (0) if the diffusion coefficient K(s, n) were known. However, relation () imposes only a restriction on the function K(s, n) for the given Q but does not determine the dependence of K on the variables s and n. This uncertainty leads to the ambiguity of the circulation found exclusively from the distribution of Q, i.e., the diabatic circulation. In order to ensure closure, it is necessary to introduce additional assumptions. This study proposes to find the function K that minimizes the diffusion component of the eddy flux in (5). In turn, this would provide the maximum of the eddy-induced advective component, which is included into the velocity ( v *, w* ). This condition seems to be physically plausible because the diabatic velocities determined in this manner would maximally completely describe the advective transport in a meridional plane. This is precisely the ultimate goal of introducing the formalism of diabatic circulation. The mathematical development of this idea is presented in the next section. 2. SOLUTION OF THE VARIATIONAL PROBLEM In order to minimize diffusion fluxes in the domain, we will solve the conditional variational problem of finding the function K(s, n) that minimizes the integral NP SP cosφ( ρ 0 Kθ n ) 2 dsdn = min (2) with the constraint given by (). The cosine in (2) appears as a scaling factor in spherical coordinates. To simplify the notation, we will use the integral sign without limits when integration with respect to s from pole to pole (from SP to NP) is meant; otherwise, the limits will be indicated explicitly. Integration with respect to the variable n is always performed over the entire vertical extent of the domain of interest if no other limits are indicated explicitly. Since ρ 0 and are single-val- θ n

DEFINITION OF A GENERALIZED DIABATIC CIRCULATIONS BASED ON 439 ued and positive definite functions of s and n, it is convenient to introduce the variable Y, Y( s, n) ρ 0 Kθ n. (3) Thus, the problem is reduced to the finding of the function Y minimizing the integral cosφy 2 dsdn = min, (4) subject to the integral condition (see ()) φθ cos n Y n ds = ρ 0 cosφθ n Qds. (5) In order to solve this conditional variational problem, it is necessary to solve the corresponding Euler Lagrange equation L ----- L ------- L ------- = 0, (6) Y s Y s n Y n where Y n Y/ n, Y s Y/ s, and the function L is specified as L φ Y 2 = cos ( + 2λθ n Y n ). (7) In Eq. (7), the second term in parentheses follows from condition (5) and 2λ(n) is the constant Lagrange multiplier that is to be determined from (5). As is seen from (7), L does not depend explicitly on Y s. Therefore, the second term on the left-hand side of (6) is zero and (6) reduces to Y = λθ ( n ) n = λ n θ n + λθ ( n ) nn, (8) everywhere except at the poles. Differentiation of (8) with respect to n and substitution of Y n into (5) yield the following ordinary differential equation for λ: an ( )λ nn + 2bn ( )λ n + cn ( )λ = f( n), (9) where the coefficients can be determined because λ(n) can be factored outside the integral with respect to s: 2 an ( ) = cosφθ n ds, bn ( ) = cosφθ n ( θ n ) n ds, cn ( ) = cosφθ n ( θ n ) nn ds, (20) f( n) = ρ 0 cosφθ n Qds. Equation (9) can be solved both numerically and analytically if the initial conditions for λ and λ n are specified at a certain reference level in the domain, for example, at a lower level n = 0. However, λ and λ n determine the diffusion flux Y, which is a physically meaningful quantity. Therefore, λ(0) and λ n (0) can be chosen sufficiently arbitrarily except for the restriction that they satisfy relation (8) for the given distribution Y 0 = Y(s, 0). For example, [λ(0) = Y 0 / ( θ n ) nn, λ n (0) = 0] or [λ(0) = 0, λ n (0) = Y 0 θ n ] can equally be used as the boundary conditions for the solution of (9). Thus, the solution to the variational problem is obtained. A practical algorithm for numerical applications lies in the following. First, at each vertical level, the coefficients a(n), b(n), c(n), and f(n) are calculated in accordance with (20). After that, the Lagrange multiplier λ is found as a solution of ordinary differential equation (9). For this purpose, suitable initial conditions λ(0) and λ n (0) are to be chosen from a specified distribution of the diffusion flux at the lower boundary Y 0 in accordance with (8). Next, from relation (8) differentiated with respect to n, the solution for Y n can be found at all altitudes. Finally, the circulation stream function Ψ is calculated at all vertical levels from (0) by means of integrating with respect to s. 3. RELATION TO THE STANDARD DIABATIC CIRCULATION An immediate question that arises in connection with the circulation just introduced is associated with the relation between this circulation and the standard diabatic circulation, which is conventionally used in most applications. In a general case, it seems likely that these two types of circulation cannot be easily compared analytically. For simplicity, in this section, we restrict ourselves to the case where θ z. In this approximation, contours of the potential temperature θ become horizontal, i.e., coincide with log-isobaric levels, and θ z is no longer a function of φ. This approximation is well applicable in the stratosphere, where at least monthly mean isentropes have no meridional slope. Note that the diabatic circulation corresponding to this case was discussed in the Introduction in association with Eq. (). In accordance with the standard method, the vertical velocity in Eq. () is to be corrected uniformly at all latitudes to eliminate a nonzero globally averaged mass flux: w* ( st) Qyz (, ) = ---------------- + w 0 ( z), θ z (2) where the correction w 0 to the vertical velocity is obtained by means of integrating the two sides of Eq. (2) multiplied by cosφ and equating the left-hand side of the equation to zero: w 0 ( z) = -- 2 π/2 π/2 Qcosφ --------------- dφ. θ z (22) By analogy with (7), w ( st) can be expressed in terms of the standard diabatic stream function Ψ (st) as w( * st ) = ( cosφ) ( st) Ψ y. Substituting this expression into (22) ρ 0

440 MEDVEDEV and using the relation dy = adφ, we obtain an equation similar to (0): Ψ ( st) ------------- (23) φ aρ φ Q = cos ---- 0 θ + w 0. z Now, we turn to the diabatic circulation introduced in the previous section. In accordance with the approximation adopted here, the formulas of this section can be transformed by means of replacing the derivatives with respect to n to the vertical derivatives and the functions of (s, n) to the functions of (y, z). Since θ z and θ zz are no longer functions of φ, the integrals in (20), which are taken from pole to pole, are simplified. In particular, az ( ) 2aθ 2 = z, bz ( ) = 2aθ z ( θ z ) z, (24) cz ( ) = 2aθ z ( θ z ) zz, f( z) = 2aρ 0 w 0. The solution for the Lagrange multiplier λ(z) can be found from (9). However, since our purpose is to find Y z (z) rather than λ itself, we omit this step. Instead, we notice that Y and Y z (z) depend on z alone and (9) can be represented as λθ z ( ) zz = Y z ( z) = ρ 0 w 0 θ z. (25) The function Y z (z) is the desired solution to the variational problem. It follows from (25) that Y z is proportional to w 0. Thus, the generalized diabatic stream function is obtained from the equation identical to (23) and this stream function is exactly equal to the standard stream function in this approximation. Note that, in the case of interest, the diffusion flux required to reach a global mass equilibrium, Y( z) = ρ 0 Kθ z = ρ 0 w 0 ( z' )θ z dz', is independent of the latitude φ. z 0 (26) 4. CONCLUSIONS A nonzero global mean mass flux or, equivalently, radiation imbalance arises always when the velocities of diabatic circulation are calculated from specified distributions of heating rates and potential temperature. Therefore, the diagnostics of the diabatic circulation requires that a correction be introduced to ensure the absence of a global mass flux through any isobaric level. Although an infinite number of functions are able to provide the required balance, only a latitude-independent correction to the vertical velocity has been commonly used in studies to date. No physically plausible explanation for such a violation of the balance except for inaccuracies in determining heating rates and no rigorous rationalization of the form of this correction function have been proposed. In this paper, a generalized scheme of the diabatic circulation has been introduced. It is based on the fact that, in a general case, eddy heat fluxes cannot be completely described in the form of advection and, therefore, the wave-induced heating rates must necessarily be taken into account in the thermodynamic equation. However, the decomposition of the eddy fluxes into advective and diffusion components is not unique. Consequently, the values of the divergence of the diffusive eddy fluxes, i.e., of the wave heating, which must be added to the diabatic heating, are also ambiguous. In the context of the proposed approach, diabatic velocities are redefined so as to include the maximum of the eddy-induced advective transport, i.e., to describe advection on a meridional plane as fully as possible. As a result, the heating rate, which represents the convergence of diffusive eddy fluxes and is absent in the context of the conventional diabatic circulation, turns out to be related to the correction function that ensures the global mass balance. In this paper, the conditional variational problem of finding the separation that provides a minimum for the diffusive fluxes and, consequently, a maximum for the advection-related fluxes has been formulated and solved. A formula for the stream function has been derived that is more general than the formula commonly employed in calculations of the diabatic circulation. This formula can be used in practical applications. In the approximation of quasi-horizontal mean isentropes, which is a fairly close approximation for the stratosphere, this formula coincides with the standard one. The result obtained provides a partial substantiation for using the standard diabatic circulation with a latitude-independent correction to the vertical velocity. In the cases where the meridional slope of mean isentropes cannot be neglected, the generalized diabatic circulation differs from the standard diabatic circulation. The generalized diabatic velocities can be substituted into the mean momentum equation, and an expression similar to the divergence of the Elliassen Palm fluxes can be obtained. Thus, the generalized diabatic velocities represent one more formulation of the residual (transport) circulation. However, unlike the other approaches to the description of the residual circulation (TEM [8, ]), the transport velocities in this formulation are expressed as functions of the mean diabatic heating rather than in terms of eddy quantities. REFERENCES. J. E. Rosenfield, M. R. Schoeberl, and M. A. Geller, A Computation of the Stratospheric Diabatic Circulation Using an Accurate Radiative Transfer Model, J. Atmos. Sci. 36, 859 876 (987).

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