Using Jacobian sensitivities to assess a linearization of the relaxed Arakawa Schubert convection scheme

Transcription

1 Quarterly Journalof the Royal Meteorological Society Q. J. R. Meteorol. Soc. 1: , April 214 B DOI:1.12/qj.221 Using Jacobian sensitivities to assess a linearization of the relaxed Arakawa Schubert convection scheme D. Holdaway a,b * and R. Errico a,c a Global Modeling and Assimilation Office, NASA Goddard Space Flight Center, MD, USA b Goddard Earth Sciences Technology and Research, Universities Space Research Association, MD, USA c Goddard Earth Sciences Technology and Research, Morgan State University, MD, USA *Correspondence to: D. Holdaway, Code 61.1, Goddard Space Flight Center, Greenbelt, MD 2771, USA. dan.holdaway@nasa.gov The inclusion of linearized moist physics can increase the accuracy of 4D-Var data assimilation and adjoint-based sensitivity analysis. Moist processes such as convection can exhibit nonlinear behaviour. As a result, representation of these processes in a linear way requires much care; a straightforward linearization may yield a poor approximation to the behaviour of perturbations of interest and could contain numerical instability. Here, an extensive numerical study of the Jacobian of the relaxed Arakawa Schubert (RAS) convection scheme is shown. A Jacobian based on perturbations at individual model levels can be used to understand the physical behaviour of the RAS scheme, predict how sensitive that behaviour is to the prognostic variables and determine the stability of a linearization of the scheme. The linearity of the scheme is also considered by making structured perturbations, constructed from the principle components of the model variables. Based on the behaviour of the Jacobian operator and the results when using structured perturbations, a suitable method for linearizing the RAS scheme is determined. For deep, strong convection, the structures of the RAS Jacobian are reasonably simple, the rate at which finite-amplitude estimates of the structures change with respect to input perturbations is small and the eigenmodes of the Jacobian are not prohibitively unstable. For deep convection, an exact linearization is therefore suitable. For shallow convection, the RAS scheme can be more sensitive to the input prognostic variables due to the faster time-scales and proximity to switches. Linearization of the RAS therefore requires some simplifications to smooth the behaviour for shallow convection. It is noted that the physical understanding of the scheme gained from examining the Jacobian provides a useful tool to the developers of nonlinear physical parametrizations. Key Words: adjoint; tangent linear; convection; 4D-Var; sensitivity; principle components; parametrization Received 29 January 213; Revised 29 April 213; Accepted 5 June 213; Published online in Wiley Online Library 1 August Introduction Over the past decade, weather centres have been directing significant effort at implementing 4D-variational data assimilation (4D-Var); see Rabier (25) for a review. Many centres already have working versions of 4D-Var and at NASA s Global Modeling and Assimilation Office (GMAO) a 4D-Var of the Goddard Earth Observing System (GEOS) cubed-sphere dynamical core is nearing completion. In 4D-Var, the cost function giving the optimal analysis is based on the forecast trajectory over an analysis window, allowing for flow-dependent covariances. When iterating over the analysis window, the algorithm requires the tangent linear and adjoint versions of the forward model operator (Courtier et al., 1994). The tangent linear and adjoint, described in detail by Errico (1997), are a linearization of the model and give a first-order approximation of perturbations of the model trajectory and the gradient of a cost function, respectively. A large portion of the development of 4D-Var includes the assembly of the tangent linear and adjoint models for the dynamics and physics routines. For large-scale features of the atmosphere, the behaviour of perturbations is fairly linear, at least on the time-scale of the analysis window (usually around 6 hours). As a result, the general approach to constructing the tangent linear and adjoint of the dynamics is often to linearize the model exactly. In contrast, perturbations for smaller scale physics can exhibit strong nonlinearities over the analysis window. If strong nonlinearity exists then an exact linearization of the model may not accurately represent the perturbation trajectory and could even result in numerical instability. Moist processes in the atmosphere can be very nonlinear, resulting from mechanisms such as re-evaporation, deposition, c 213 Royal Meteorological Society

2 132 D. Holdaway and R. Errico latent heating and sudden fast updraughts and downdraughts. Further, numerical schemes can include discontinuous modelling and artificial processes that increase the nonlinearity, such as the trigger function for the onset of convection, removal of supersaturation and removal of negative specific humidity. Careful consideration is required when trying to represent the moist schemes and these underlying nonlinear processes in a linear way. Retaining enough of the physical behaviour to give improvement to the desired application whilst obtaining enough linearity to give sensible results is a delicate balance. The inclusion of linearized moist physics in 4D-Var has been shown to be very beneficial (Errico and Raeder, 1999; Janisková et al., 1999b; Mahfouf and Rabier, 2; Tompkins and Janiskova, 24; Lopez and Moreau, 25; Errico et al., 27; Amerault et al., 28; Stiller, 29; Stiller and Ballard, 29). Despite the difficulties encountered in linearizing moist physics parametrizations and the effort directed towards this problem in recent years, it remains an important and active area of research. In addition to the 4D-Var application, the linear model is also used in the observation operator when assimilating observations. This technique is employed in 4D-Var, 3D-Var and ensemble methods that avoid covariance localizations in observation space. The general improvement of the assimilation of moist observations is a considerable current challenge in numerical weather prediction (Errico et al., 27; Lopez, 27; Bauer et al., 211). Assimilating an observation that is sensitive to clouds and precipitation requires a linear representation of precipitation rates and cloudy and rainy satellite radiances in model space. This is an area where much improvement is still needed and, with the expected launch of the Global Precipitation Measurement satellite in 214 (Hou et al., 28), remains a pressing issue. Another important objective in numerical weather prediction is to develop a better general understanding of the behaviour of moist processes in the atmosphere and how to properly represent them in model parametrizations. One way to achieve this is through adjoint-based sensitivity and adjoint-based observationimpact analysis that includes moist physics (Errico et al., 23; Lopez, 23; Mahfouf and Bilodeau, 27). In adjoint-based sensitivity analysis, the adjoint is used to propagate a sensitivity (gradient) backwards in time and so determine where errors or improvements to model initial conditions can have the biggest effect (e.g. Langland et al., 1996). In adjoint-based observation impacts, an adjoint is used to estimate the change in a forecasterror metric due to arbitrary sets of observations that were used to improve initial conditions (Errico, 27; Gelaro et al., 27; Langland and Baker, 24). Using these methods can indicate regions where improvement would be beneficial in the model and in the observations. As new model parametrizations are developed, examining their behaviour in a linear sense by looking at sensitivities to initial conditions can provide much insight into their function and alert one to any spurious behaviour. Global circulation models generally divide moist processes into convection and large-scale condensation and schemes specifically tailored to each process are used. In the GEOS model, the relaxed Arakawa Schubert (RAS) scheme is used for the convection (Arakawa and Schubert, 1974; Moorthi and Suarez, 1992) and for large-scale condensation the scheme developed by Bacmeister et al. (26) is used. Both schemes are designed to act on a single column of the atmosphere. Inputs and outputs are onedimensional vectors. The objective of this article is to determine the best way to linearize convection in the GEOS data assimilation and adjoint model. Doing so effectively will make improvements to the forthcoming 4D-Var system and to the adjoint tools that can be used in the observation operator, for sensitivity analysis and for observation-impact analysis. An older, more simple, version of the RAS scheme has been used successfully in adjoint models using an approximation of the Jacobian (Errico et al., 1994), so it is an interesting scheme to examine in more detail. Any attempt to model the atmosphere in a linear way requires care. A linearization must follow the perturbation trajectory of the nonlinear model as closely as possible and be numerically stable and computationally efficient. Depending on the linearity and stability of the original scheme, three general routes to linearization can be envisioned: (1) develop a new linear and stable parametrization; (2) simplify or smooth the existing nonlinear scheme to improve linearity or stability; or (3) linearize the existing nonlinear scheme exactly. All of these options come with advantages and disadvantages. Developing a completely new scheme provides much flexibility but requires a very detailed understanding of the processes being represented and the workings of the nonlinear scheme. Unless great care is used, the approach would result in solutions far from the actual perturbation trajectory. Simplifying an existing scheme would seem easier than starting from scratch. However, simplifying in a sensible way can be difficult if the scheme being considered is not well documented or communication with the original architect of the scheme is not possible. Linearizing a scheme exactly may seem desirable for closeness to the original perturbation trajectory, but if the scheme supports strong nonlinearity or instability or is inefficient it would cause problems for the linearization. Efforts to include linearized physics in the European Centre for Medium-Range Weather Forecasts (ECMWF) and Météo-France assimilation systems have been based on both development of simplified linear schemes and smoothing of current schemes (Mahfouf, 1999; Janisková et al., 1999a,b, 22). Given that any approach to tangent linear and adjoint development comes with difficulties, it is useful to make a preliminary study of the system and choose the most appropriate route. Without this understanding, months could be spent developing an exact adjoint, only to discover that the perturbation trajectory that it gives diverges from the nonlinear perturbation trajectory within one time step or has a very strong numerical instability. Similarly, a simple linear and stable replacement scheme could be developed when a linearization of the original scheme would produce results closer to the actual perturbation trajectory. Finite-difference estimates of a Jacobian of a scheme provide a very useful tool for examining the linearity and choosing an appropriate approach to linearization (Errico, 1996; Mahfouf et al., 1996; Fillion and Belair, 24). In general terms, a Jacobian is a vector or matrix of first-order partial derivatives of a function with respect to a set of its arguments. When applied to a scheme, the Jacobian can be used to describe the rate at which the outputs change with respect to the inputs. Finite-difference estimates of Jacobian elements are generated by comparing the outputs of the scheme when the inputs are unperturbed and perturbed. If a scheme is highly nonlinear, then different small perturbations to the inputs could result in outputs with large differences. The Jacobian elements need to be examined for infinitesimal and realistic, finite-sized, perturbations. Realistic perturbations are required to understand how the scheme will perform in the presence of the kinds of errors encountered in realistic situations. Infinitesimal perturbations are small enough that behaviour is close to the continuous behaviour but not so small that roundoff error becomes problematic. If the elements obtained with a particular input perturbation size are very different from those obtained using a different-sized input perturbation, then nonlinearity is implied. Functions that describe the tendency produced by the scheme are considered. The sensitivity of these functions with respect to each input variable makes up the Jacobian elements. To begin with, the Jacobian is constructed by considering the sensitivity to perturbations made at each model level independently. Doing so gives an approximation of the operator of the tangent linear model and allows for a straightforward check on both linearity and stability. When examining the Jacobian to estimate nonlinearity, however, such perturbations may be unrepresentative of the c 213 Royal Meteorological Society Q. J. R. Meteorol. Soc. 1: (214)

3 Jacobian Sensitivities of RAS Convection Scheme 1321 structures of perturbations encountered in practical problems. For example, since errors are correlated in the vertical, a temperature perturbation of 1 K at a single model level and K at all other levels would almost never be encountered in practice. Estimating nonlinearity from such a perturbation may yield skewed conclusions. The robustness of the results can be improved by considering perturbations that have a realistic vertical structure. A set of perturbation structures is obtained from the principal components of the input variable covariances. The sensitivity of the scheme is computed with respect to the set of principle-component perturbation structures. In 4D-Var, as well as sensitivity and observation-impact experiments, the algorithm involves products of the tangent linear and adjoint operators. If an instability occurs in the linear scheme then it could cause the solution to diverge quickly, especially given the simplification of the other physics that would otherwise be constrained in the nonlinear model. The stability of the linear scheme can be determined by an eigendecomposition of the Jacobian computed from single-level perturbations. If the Jacobian supports some very rapidly growing modes, then the linearized model will likely perform very poorly. If the study reveals a scheme to be very nonlinear or very unstable then an exact linearization of that scheme would not be useful and the first option listed above should be taken. If the scheme has some linear properties or some stability then it may be possible to linearize based on some approximation or smoothing of the scheme, as in the second option. If the scheme is fairly linear and stable then an exact linearization of the scheme would be possible, as in the third option. Studying the Jacobian sensitivities is a very useful technique, not just for determining linearity but for examining the nonlinear model itself. Since it describes where the response to input perturbations occurs, it provides insight into the physical behaviour of the nonlinear scheme (Fillion and Mahfouf, 23). The results can be used to highlight where non-physical or spurious behaviour occurs or where sensitivities should actually be larger. The technique offers a very useful tool in the development of the physical parametrizations themselves. The methodology outlined in this article is very versatile and can be applied to any scheme. The article is arranged as follows: Section 2 describes the generation of an approximation of the tangent linear model operator, presents an examination of the behaviour of the operator and shows a check of the stability of the RAS scheme. Section 3 describes the process of using principle components to test the linearity based on structured perturbations. Section 4 offers some concluding remarks. 2. Jacobian approximation of the tangent linear model operator In this section the behaviour, linearity and stability of the RAS scheme is examined. Jacobian sensitivities are formulated so as to replicate the structure of the tangent linear model operator matrix. This is achieved by making perturbations for each input variable at individual model levels. A range of sizes of perturbations is applied to gauge the sensitivity of the scheme with respect to the inputs. In making infinitesimal-sized perturbations, an approximation of the operator of the tangent linear model is obtained, allowing for a check on the stability of a potential linearization of the scheme Generating the Jacobian The GEOS prognostic variables that are passed to the RAS convection scheme are horizontal velocities u and v, potential temperature θ, specific humidity q and surface pressure p s. The scheme updates u, v, θ and q and produces a number of rainfall, cloud-fraction and mass-flux diagnostics. Most of these diagnostics are passed through the large-scale condensation scheme, where they are combined with other effects. The sensitivity of the scheme with respect to θ and q is considered in the description of the Jacobian, since these are the primary fields that have an impact. The analysis can be extended to account for all other input variables. The effect of the convection, or other moist physics, scheme can be represented as θ = H(θ, q), (1) t q = Q(θ, q), (2) t where H is called the heating rate and Q the moistening rate. In practice, the RAS scheme produces increments θ new and q new as opposed to a tendency. The heating and moistening rates can be inferred from these increments: H = (θ new θ) / t, (3) Q = ( q new θ ) / t, (4) where t is the time step of the scheme. The RAS scheme considers different cloud depths sequentially; see Moorthi and Suarez (1992) for a general description of the algorithm. The heating and moistening rates of a nonlinear scheme could also be written in vector notation, y = M (x). (5) The vector y contains the tendencies at each model level, vector x contains the inputs at each model level. Model M is the forward operator of the nonlinear scheme. Variables are linearized, using the Taylor series, into reference and perturbation parts, e.g. θ = θ (r) + θ ; superscript (r) is used to denote the reference part and superscript is used to denote perturbation part. Linearizing Eqs (1) and (2) gives θ = H (r) t θ θ + H (r) q q, (6) q = Q (r) t θ θ + Q (r) q q. (7) Writteninmatrixform,Eqs(6)and(7)are t ( θ q H ) = θ Q θ Adopting vector notation, Eq. (8) is H (r) ( ) q θ Q. (8) q q y = Mx. (9) Equation (9) is the linearized version of Eq. (5) and is the formal tangent linear model. Matrix M is referred to as the forward operator of the tangent linear model. In the discretized model, the vectors x and y contain the perturbations of θ and q at each model level. For this two-variable example, each vector has dimension 2n, wheren is the number of vertical levels in the model. The matrix M has dimension 2n 2n; each component within M, for example H/ θ, is a matrix with dimension n n. The transpose, ) T ( ) T Q M T = ( H θ ( H q is the operator of the adjoint model, J x θ ) T ( Q q ) T, (1) = MT J y. (11) c 213 Royal Meteorological Society Q. J. R. Meteorol. Soc. 1: (214)

4 1322 D. Holdaway and R. Errico The adjoint model only makes sense within the context of a function J = J(y). In 4D-Var, J is the cost function that is minimised. In the tangent linear model, perturbed variables are propagated forward in time. The output of the adjoint is a sensitivity with respect to original input x, so it propagates the sensitivity backwards in time. The behaviours of both the tangent linear and adjoint models are represented in the behaviour of M. Inpractice,M is not explicitly calculated and operated on as it would be prohibitively expensive to do so. However, since the behaviour of the linearized system is represented in M, studying its behaviour gives insight into how useful solutions of a linearization would be for approximating perturbation behaviour in the nonlinear context. Rather than undertaking the potentially arduous task of deriving the formal tangent linear model in order to examine M, it can be approximated using a perturbation method applied to the nonlinear model. Given a set of inputs, the heating and moistening rates can be determined, as in Eqs (3) and (4). The finite-difference approximation of the gradient of the scheme with respect to the inputs is found by adding a perturbation to each input; for example, θ + θ. The resulting change to the heating and moistening rates is then computed as, H = H = H(θ + θ, q) H(θ, q). The sensitivity is then obtained by dividing the difference in the heating and moistening rates (outputs) by the difference in the inputs, e.g. θ = θ.the Jacobian of the sensitivities is J = H H θ q Q Q θ q. (12) Matrix J is the finite-difference approximation of the tangent linear operator M. Perturbations are added to each input a single model level at a time. Matrix J has columns corresponding to each individual perturbation. The leftmost column of J would be obtained by perturbing θ at the first model level, which is the highest model level. A perturbation at a single level gives new heating and moistening rates that can be different from the original ones at any model level; hence, each perturbation produces a whole column of J. Perturbing the inputs in this way means that for small enough perturbations J M. The above Jacobian matrix considers the heating and moistening rates. In the RAS scheme, the general structure of the convection depends on the profiles of θ, q and the surface pressure p s. The model uses an η coordinate to specify pressure throughout the column (Mesinger, 1984). Pressure is not updated by the RAS scheme. For the u and v part of the system, acceleration rates A u and A v are defined. The input profiles of horizontal velocities u and v are not important in determining the convective behaviour, so sensitivities with respect to u and v are zero except in their own equation. The convection can affect the structure of u and v through cumulus friction, so that sensitivity is considered in the full analysis. The full Jacobian matrix that is examined is J = A u u A v v A u θ A v θ A u q A u p s A v p s A v q H H H θ q p s Q Q Q θ q p s. (13) Each component in J in Eq. (13) has dimension n n, except for the last column of components, which is a vector since p s is a single value. The current GEOS model has n = model levels, so the full J has 288 rows and 289 columns. Having one fewer row is due to the RAS scheme not updating surface pressure p s.each column of J multiplies a model variable at a model level. The first column would multiply u at level 1, the second column u at level 2 and so on. Model levels are indexed descending in height, so level 1 is the model lid. In order to generate the entire Jacobian matrix for one global grid point, the RAS scheme must be run 289 times; this can be reduced by making perturbations only in the vertical region where convection is active. Any reference to partial derivatives henceforth in the text refers to the finite-difference approximation of the derivative. In order to produce realistic perturbations, standard deviations σ for the model inputs at each level are used. The standard deviation data are obtained from an estimate of backgrounderror variances, obtained from an observing system simulation experiment (Errico et al., 212). The perturbation to the inputs is χ k =±δ ( σ χ )k, (14) where k is the vertical level being perturbed, χ is each model input variable u, v, θ, q and p s and ( σ χ is the )k standard deviation. A range of perturbations are obtained using δ = 1 4,1 3,1 2,1 1,.5, 1. and 2.. When δ = 1 4 the perturbation is infinitesimal; perturbations that are realistic are obtained when δ O(1). Assuming sufficient numerical accuracy is available, generating the Jacobian matrix using δ = 1 4 versus δ = 1 4 will produce negligible difference in the structure, unless a switch is altered in the numerics. If negligible difference is found then that structure can be assumed to be the structure of the tangent linear model s forward operator. If that structure remains consistent for larger δ then it suggests linear behaviour in the model Jacobian structure and behaviour Generating the Jacobian matrix requires input profiles for u, v, θ, q and p s. These are obtained from runs of an older version of the GEOS weather forecast model, run using a latitude longitude grid. Two model runs are performed for the same time period with two different resolutions: low resolution 2.5 and high resolution.5. After an initial spin up, the input profiles for the Jacobian matrix are chosen at 12 UTC from 2 July 211. This time and date provides a range of summer and winter conditions across the globe and provides daytime over both land and ocean. The RAS scheme uses buoyancy as the driving force of convection. A limitation of the scheme in comparison with other convection schemes is that it does not consider the vertical velocity w, so dynamically driven convection is not modelled directly. In order to understand the behaviour of the scheme, the Jacobian matrix is examined by eye. The study is focused by selecting a subset of profiles, chosen to cover a range of convective depths, topographic conditions, latitudes, time of day and meteorological phenomena. In all, hundreds of Jacobian matrices were examined by eye. Here two select cases are shown, chosen to highlight general behaviour and linearity properties of the scheme Linear deep convection case The first case shown (Figure 1) is for a very deep convective profile that exhibits a large degree of linearity. The location of this profile on the globe is approximately 3 N, 17 E, which is over land. Other profiles that have strong deep convection were examined over ocean and land, over a range of latitudes, covering different dynamical features and for both resolution cases. All such profiles that were examined had behaviour similar to the profile examined here. Figure 1 shows the characteristics of the profile and demonstrates the effect of the scheme on the atmosphere at this location. The top panels give the input profiles for u, v, θ and q; surface pressure is p s = hpa. The middle panels of the figure show the acceleration, heating and moistening c 213 Royal Meteorological Society Q. J. R. Meteorol. Soc. 1: (214)

5 Jacobian Sensitivities of RAS Convection Scheme 1323 Pressure (hpa) Pressure (hpa) Pressure (hpa) (a) 2 8 u (r), v (r) (ms 1 ) (d) A u (r), Av (r) (ms 2 ) u (r) v (r) 1 1 x 1 4 (g) Precip Source (kgkg 1 s 1 ) 2 4 x 1 8 A (r) u A (r) v (b) θ (r) (K) 3 (e) H (r) (Ks 1 ) 1 x 1 4 (h) Cumulative Mass Flux (kgm 2 s 1 ).5.1 (c) q (r) (kgkg 1 ).1.2 (f) Q (r) (kgkg 1 s 1 ) 2 1 x 1 8 (i) Updraught Areal Fraction.5 Figure 1. The upper panels show the input profiles of (a) horizontal velocity, (b) potential temperature and (c) specific humidity for a typical deep convection case. The middle panels show (d) acceleration, (e) heating and (f) moistening rates as produced by the nonlinear RAS scheme for the input profiles in the upper panels. The lowest panels show (g) precipitation flux, (h) upward mass flux and (i) updraught areal fraction. rates. The lowest panels provide some of the RAS diagnostics, precipitation flux, upward cumulative mass flux across each level and the updraught areal fraction. In practice, the precipitation flux and updraught terms are modified to take into account largescale condensation and thus produce more complete diagnostics. However, it is useful to understand how these properties of the scheme behave. For this profile, convection is active from around 12 hpa down to around 9 hpa, or vertical levels For the profile shown in Figure 1, water vapour is condensing in the convective layer, causing an increase in potential temperature and a reduction in specific humidity. The instability and condensing moisture result in a precipitation flux throughout the convective layer. Upwelling is seen in the mass flux and updraught. Figure 2 shows the Jacobian matrix for the deep convective profile shown in Figure 1. The Jacobian matrix is generated using an infinitesimal perturbation, δ = 1 4. In the figure, the elements for the heating and moistening rate sensitivities are shown; the gradients of the acceleration rates A u and A v with respect to the inputs are shown in Figure 3. The x-axisineach panel shows the index of the model level at which the perturbation is made and the y-axis shows the index at which the response occurs. Each panel of the Jacobian matrix is scaled individually, since each has different units and multiplies inputs with differing magnitudes. Jacobian matrix elements could be normalized with respect to the individual standard deviations to approximate which parts of the Jacobian matrix are dominant. However, different parts of the Jacobian matrix may dominate for different Vertical level Vertical level Vertical level global profiles, so linearity and behaviour for the whole Jacobian matrix is considered. The first aspect to note from Figure 2 is the relatively simple Jacobian structure for this deep convection profile. There are clear dominant features that can be understood physically and often responses occur at the same model level as the input perturbation, or a model level adjacent to it, evident in the dominant diagonal features. Figure 2(a) and (b) (upper panels) show H/ θ and H/ q. A positive Jacobian element at a given location signifies that a positive perturbation results in the heating rate being increased at that location; a negative perturbation would cause a decrease. Locations where the Jacobian elements are negative means positive perturbations reduce the heating rate and negative perturbations increase it. There are three dominant features in Figure 2(a). Firstly, there exists a diagonal feature consisting of a positive element below the level of the perturbation and a negative element at the level above. Secondly, there is a response throughout the convective layer to a perturbation in temperature at level ; for this profile, level is the level just above the cloud base. Thirdly, there is a response throughout the convective layer to a perturbation at level. The dominant feature in Figure 2(b) is due to perturbations made to specific humidity in the sub-convective layer. The most dominant feature in Figure 2(a) is the diagonal structure. The structure indicates that a positive perturbation (temperature increase) at a given level would result in reducing the heating rate at the level above the perturbation and increasing the heating rate at the level below. The heating rate at the level of the perturbation is approximately unchanged. This positive negative response describes the change to the vertical transport of heat and can be seen clearly in eqs (A3) (A33) in Moorthi and Suarez (1992). A positive perturbation at a given level reduces the transport of dry and moist static energy at the level above in the tendency calculation and increases it at the level below. The change cancels with itself in the calculation at the level where the perturbation lies. Physically, the positive negative response demonstrates how a perturbation affects the upward transport of heat locally. Positively perturbing temperature at a given level stabilizes (relative to air coming upwards) and reduces the amount of moist convective air passing through that level, increasing latent heating below and decreasing it above. The perturbation is effectively directed downwards. The elements in column are positive, indicating that positively perturbing temperature at level increases the heating rate at levels throughout most of the convective region; negatively perturbing it would reduce heating. By increasing temperature near the cloud base, the energy available for convection is increased. As a result more moist air is transported upwards, warming the atmosphere aloft. The response is felt most strongly where the heating rate itself is largest, which occurs around levels, 51 and. This is where the action of the convection is dominant, likely due to some property of the underlying wind, temperature and moisture profiles. This effect is similar to when a temperature inversion or wind shear blocks deep convection and the transported moisture spreads out, forming anvil-type cloud. The response to the perturbation at level produces negative elements in column, indicating that a positive perturbation to temperature here would reduce the heating rate at other levels. Unlike when making a positive perturbation low down, at level, making a positive perturbation at level inhibits convection by stabilizing the atmosphere relative to a rising particle. This inhibition of convection by a positive temperature perturbation reduces the heating rate beneath. The sensitivity is largest at level : this is the level where the heating rate is largest and also where the mass flux reaches its maximum, seen in Figure 1(e) and (h). Smaller sensitivity is seen for perturbations at other high levels but it is not contoured in the plot. The heating-rate sensitivity with respect to q is shown in Figure 2(b). The response to making a perturbation within c 213 Royal Meteorological Society Q. J. R. Meteorol. Soc. 1: (214)

6 1324 D. Holdaway and R. Errico (a) H / θ (s 1 ) (H) x 1 5 (b) H / q (Ks 1 kg 1 kg) (H).5.5 (c) Q / θ (kgkg 1 s 1 K 1 ) (Q) x 1 8 (d) Q / q (s 1 ) (Q) x Figure 2. The Jacobian matrix for a typical deep convection case, found using infinitesimal perturbations, 1 4 σ χ. Each panel shows the structure of a quadrant of the Jacobian matrix, effectively a plot of the matrix J, Eq. (5). The region where convection is occurring is shown. Clockwise from top left, the panels show (a) H/ θ,(b) H/ q,(d) Q/ q and (c) Q/ θ. (a) A u / u (s 1 ) x 1 5 (b) A u / θ (ms 2 K 1 ) 2 x 1 5 (c) A / q (ms 2 kgkg 1 ) u (A u ) Perturbation Index (u) (A u ) (A u ) (d) A v / v (s 1 ) (A v ) Perturbation Index (v) x 1 5 (e) A v / θ (ms 2 K 1 ) (A v ) x 1 5 (f) A / q (ms 2 kgkg 1 ) v (A v ) Figure 3. As for Figure 2 but showing the A u (upper panels) and A v (lower panels) parts of the Jacobian matrix. the sub-convective layer region is the dominant feature. A perturbation here alters the heating rate throughout the convective layer. A positive perturbation in the specific humidity would result in an increase in the heating-rate magnitude above. A negative perturbation would result in reducing the heating rate. The observed effect is similar to the response seen when perturbing temperature at level in Figure 2(a). Positively perturbing specific humidity in the sub-convective layer increases the amount of moisture that will be transported upward by the scheme and thus the amount of latent heating of the atmosphere above. The RAS scheme makes a computation of the energy in the sub-convective layer using a sum over that region, hence why perturbations in that layer all result in approximately the same response. It is in this sum that the moisture can have the most c 213 Royal Meteorological Society Q. J. R. Meteorol. Soc. 1: (214)

7 Jacobian Sensitivities of RAS Convection Scheme 1325 impact on the strength of convection, which is why the sensitivity is largest here. Again the dominant response occurs at the levels where the heating rate produced by the scheme is the largest, seen by comparing Figure 2(b) with Figure 1(e). This shows that, at least for small perturbations, the magnitude of the heating rate changes slightly while its vertical structure remains relatively unaltered. Specific humidity decreases approximately exponentially with height, so columns to the left of the panel, which are already smaller, would generally multiply small perturbations and be eclipsed even further. Figure 2(c) and (d) (the lower two panels) show the effect of making perturbations on the moistening rate. Examining the features in these two panels, it is clear that the dominant response to perturbations occurs at level. From Figure 1(c), it is clear that level is the location of a double turning point and inflexion in the specific humidity profile, where the specific humidity briefly increases upwards. As moist air rises convectively it is replaced, through subsidence, by drier air. The drying of the atmosphere for this profile occurs predominately at the same location, seen in Figure 1(f). Likewise, with the anvil cloud example, the properties of the atmospheric profile cause the action of the convection to be concentrated in a specific region. Similarly to the heating-rate sensitivities, the effect of any perturbations is felt predominantly where the action is strongest. The moistening rate is negative (drying), so a positive Jacobian element implies that a positive perturbation reduces the magnitude of the moistening rate and reduces the amount of drying. Negative Jacobian elements imply that positive perturbations increase the drying. Figure 2(d) shows Q/ q. As in Figure 2(a), the diagonal features are evident, showing how perturbations effect the dry and moist static energy transport calculations locally. A positive perturbation to the specific humidity at a given level results in a decrease in the moistening rate (increase in magnitude) at the level above and an increase in the moistening rate (reduction in magnitude) at the level below. Again, a positive perturbation reduces the upward transport of moist air across that layer, moistening the level below and drying the level above. As seen in Figure 2(b), the dominant sensitivity is to perturbations of specific humidity in the sub-convective layer and the largest response occurs at level. That the response is seen at level can be understood through the two step process described above. Firstly, perturbing moisture in the sub-convective layer changes the potential for convection and thus the amount of moist air being convected upwards. Secondly, this changes the amount of subsidence of dry air from above, which is focused through level. Increasing the specific humidity in the sub-convective layer will, through convection and subsidence, decrease the negative moistening rate (increase the amount of drying). Figure 2(c) shows Q/ θ. Again the sensitivity is felt most at level, where the turning points and maximum drying occur. As for H/ θ, the dominant sensitivity is to perturbations of potential temperature at levels and, though the sign of the elements is now reversed due to the negative moistening rate. The physical reasoning behind the sensitivity to these levels remains the same. However, now the response is felt only at level rather than throughout the column. Increasing temperature at level, for example, increases convection, warming the atmosphere aloft. Seen in Figure 2(a), this results in increased subsidence and increased drying at level, represented by the negative Jacobian elements at this location in this column. It is clear from Figure 2(b) and (d) that the RAS scheme has little sensitivity to the specific humidity above the sub-cloud layer. This is considered a weakness in convection schemes. Bechtold et al. (28) show an improvement in atmospheric variability in the ECMWF model when a dependence on humidity in this region is accounted for in the convection scheme. The results here demonstrate how the Jacobian can be used to understand what the scheme is sensitive to, or in this case what it is not sensitive to. This can aid in pinpointing where development and improvement in the schemes will be most beneficial. This is especially helpful if one is not particularly familiar with the scheme in question. Figure 3 shows the acceleration-rate sensitivities with respect to velocity, potential temperature and specific humidity. The horizontal wind components are not used for computing the strength of convection, but wind speed can be affected by cumulus friction. For the sensitivity with respect to the velocities, the dominant structure is the diagonal features seen previously, representing the change in upward transport of momentum. For the sensitivity with respect to the potential temperature, the dominant sensitivity is due to perturbations at the top of the convective layer, where heating and acceleration rates are simultaneously large. There also exists the sensitivity to perturbations in temperature at the level just above the cloud base, level. For the structure at the top of the convective layer the response lies at the same level as the perturbation, showing that changes in temperature locally impact the cumulus friction. For the sensitivity with respect to specific humidity the dominant feature is again due to perturbations in the sub-convective layer and the response is largest where the acceleration rates are largest. Plots of the sensitivity with respect to the surface pressure p s are omitted. Here and elsewhere, the findings are that the sensitivity with respect to p s is very small for the heating, moistening and acceleration rates. This implies that it would be possible to neglect linearized pressure terms in a tangent linear and adjoint version of the RAS scheme. From examining the Jacobian for this and other profiles for infinitesimal perturbations, we obtain an understanding of some of the general properties of the RAS scheme. These can be summarized as follows. (1) The RAS scheme is sensitive to perturbations of temperature at the level above the cloud base and the levels where the heating rate is largest. (2) For moisture, the scheme is most sensitive to perturbations in the sub-convective layer. The RAS scheme performs an integration over this layer and uses it to determine the strength of convection. A perturbation at any level in the sub-convective region results in the same response but proportional to the layer thickness. (3) There is a clear relationship between the matrix structures and the heating, moistening and acceleration rates. The response to perturbations is felt most strongly where the heating and moistening rates are strongest. These are the levels where the action of the scheme is dominant due to some underlying features in the model variable profiles. (4) The calculation of vertical transport of dry and moist static energies results in a diagonal Jacobian feature. Perturbations result in an increase in heating and moistening below and a decrease above. (5) The scheme is not particularly sensitive to perturbations in specific humidity above the cloud base layer. A great deal of insight into the properties of the scheme is provided by looking at the Jacobian matrix for an infinitesimal perturbation. However, this alone provides little insight into the linearity of the scheme. One way to explore the linearity in the scheme is to make different sized perturbations to the inputs and re-examine the Jacobian matrix. Of particular interest is whether the structure of the Jacobian matrix remains consistent even when realistically sized perturbations are made. Figure 4 shows the effect of changing δ, the factor multiplying the perturbation. The figure shows δ as 1 1 and 2., a subset of the δ values considered. Structures shown are representative of results for the whole range of δ. The figure shows (a,e) H/ θ, (b,f) H/ q, (c,g) Q/ θ and (d,h) Q/ q. Results are similar for all other parts of the Jacobian matrix. For all sizes of factor used to multiply the perturbation, there is negligible change to the structure of the Jacobian matrix, implying that the scheme is very linear for this profile. A number of deep convective profiles were examined, all of which had linear or close to linear properties. In some cases the magnitude of the structure would change for the c 213 Royal Meteorological Society Q. J. R. Meteorol. Soc. 1: (214)

8 1326 D. Holdaway and R. Errico (a) H / θ (b) H / q (c) Q / θ (d) Q / q δ =.1 (e) (f) (g) (h) δ = x x x 1 5 Figure 4. From left to right, this figure shows (a, e) H/ θ,(b,f) H/ q, (c,g) Q/ θ and (d,h) Q/ q. The top row shows the structure of the Jacobian matrix for δ =.1 and the bottom row shows the structure of the Jacobian matrix for δ = 2.. larger perturbations but the overall behaviour would not differ; responses occur at the same levels and are of the same sign. Given that for the deepest convection the scheme appears to be linear or close to linear, an exact linearization of the RAS scheme should produce reasonable results for the deepest convection, provided the linearized scheme produces stable results Switches and nonlinearity A scheme can be nonlinear in the sense that the underlying functions have complex structure, such as many turning and inflexion points. A gradient approximation, such as in a firstorder Taylor series, will be less accurate in regions where this kind of behaviour exists. Another way in which a scheme can be nonlinear is through discontinuities in the functions, represented by conditional switches in the numerics. An example would be a numerical routine that rains only when supersaturation is detected. Unfortunately, many moist physics schemes involve such sometimes artificial switches. Although switches are not used excessively in the RAS scheme, they can and will affect the linearity. The central algorithm in the RAS scheme is based on considering an ensemble of possible clouds within each vertical column. Each cloud in the ensemble can have different properties, such as different detrainment levels. This follows a realistic understanding of atmospheric convection for the grid scale being used, which could indeed have clouds detraining at different levels. The number of clouds in the ensemble is decided using conditional statements that rely on the input variables. If the input variables are perturbed, then these switches can change and the overall behaviour of the scheme can change discretely. The effect of this would be to have structures of the Jacobian matrix that look very different, possibly even with very small perturbations. Instead of the simple behaviour seen above, the location of the dominant features could be hard to predict. Whether a linearization of the scheme will be useful depends on how nonlinear the underlying functions are and also how often switches are altered. In this Jacobian matrix study, the influence of switches could be controlled by constraining based on the unperturbed inputs, however this could place the scheme artificially in a non-physical regime and so produce misleading results. There are some places in the RAS algorithm where values must be chosen based on fixed values. For example, the entrainment parameter (eq. (A18) in Moorthi and Suarez, 1992) depends on a value that is determined from a resolutiondependent constant divided by the square root of the difference between the surface temperature multiplied by 1 and its nearest lower integer. If the constant were 2, corresponding to 1 resolution, and the surface temperature were K then the value would be 2/ ( ) = Adding an infinitesimal perturbation of 1 4 to the temperature would give 239. instead, altering the eventual entrainment parameter by a significant amount. Having such a change would likely give misleading Jacobian matrix elements and invalidate the stability testing results. When a linearized version of a scheme is developed, the effect of these kinds of nonlinear switches must be considered and often controlled in some way. Figure 5 shows the Jacobian matrix for the same magnitudes of δ used in Figure 4. The profile examined here, taken from a grid box centred at 18 N, 42 W, exhibits shallow convection. The convection occurs from model levels up to, approximately 9 83 hpa. From Figure 5, it is clear that for δ = 1 1 and δ = 2. the structure of the Jacobian matrix is different. Upon further investigation, it is evident that for this particular profile when going from δ = 1 1 to δ = 1 a switch has been crossed. The number of cloud soundings is increased and the depth of c 213 Royal Meteorological Society Q. J. R. Meteorol. Soc. 1: (214)

9 Jacobian Sensitivities of RAS Convection Scheme 1327 (a) H / θ (b) H / q (c) Q / θ (d) Q / q δ =.1 (e) (f) (g) (h) δ = x x x 1 4 Figure 5. As for Figure 4, but for a shallow convective profile for which the crossing of a switch has been observed. convection is increased; whereas for δ = 1 1 the convection goes as high as level, for δ = 2. itonlygoesuptolevel. These findings show that the strength of the convection is not calculated in a smooth linear way but through switches that can alter, depending on features in the input variables. The case shown here is chosen in order to highlight a potentially problematic numerical switch. The majority of cases for which the structure of the Jacobian matrix has been examined do not suffer these large discrete changes as δ is altered. Often when nonlinearity is encountered it is only once δ = 2.. However, a linearization will be inaccurate when profiles such as that seen above are encountered. The question going forward is whether the presence of such profiles, however few, will invalidate the linear approximation. The RAS scheme does not explicitly model deep and shallow convection separately. It is the same algorithm for all cloud depths. Differences in depths result from the scheme simply not finding clouds that detrain at higher levels. That profiles supporting shallower convection appear more likely to exhibit nonlinearity is due to the switches that decide the number of clouds in the ensemble being more sensitive to perturbations. The conditions for convection reaching a certain discrete model level are closer to the threshold. Further, for shallow convection the highest model levels where the clouds are detraining are much closer together. Perturbing specific humidity in the sub-cloud layer would only have to increase the highest point of detrainment by a relatively small amount to change the number of clouds considered in the ensemble discretely. Changing the number of clouds in the ensemble for deep convection would require a perturbation that changes the highest point of detrainment much more Stability check There are two issues regarding the stability of the linearized model. The first is in regards to the presence of growing solutions due to the underlying behaviour in the model. The second is in regards to choosing a numerically stable time-stepping scheme. Unstable growing behaviour is not uncommon. Indeed, the modelling of certain atmospheric phenomena requires the representation of instabilities of this nature. When instabilities occur in the atmosphere they are eventually disrupted or diffused away through nonlinear dynamical and physical mechanisms. When a linearization is implemented, these mechanisms cannot be represented or may be highly simplified, rendering the linearization incapable of properly modelling the restoration of stability. In 4D-Var, a product of the linear operators is required in order to propagate the sensitivity across the analysis window, 18 time steps (six hours) in the current GEOS configuration. If large unchecked growth occurs in the scheme, it could quickly lead to unrealistic structures in the perturbation trajectory fields. Care is also required when matching the time-stepping scheme to the properties of the physics schemes (Durran, 1999). There are situations that can arise in which the particular choice of time stepping would be stable for the nonlinear scheme but not necessarily stable for the linearized scheme. An example of this would be if, say, the linearization resulted in a term containing division by a very small value. The same time step and scheme used for the nonlinear model may not yield stable results for the linearized model. Laroche et al. (22) present a study of stability in a linearization of a simplified planetary boundary-layer scheme. The stability can be analyzed using an eigendecomposition. The equations in the RAS scheme consider one column of the atmosphere at a time. Solutions of the linearized system take the form e.g. θ exp (λt), whereλ is complex. The imaginary part describes the frequency of the solution and the real part describes the growth rate of the solution. Seeking solutions in this form in Eq. (9) gives λy = Mx. (15) c 213 Royal Meteorological Society Q. J. R. Meteorol. Soc. 1: (214)