Test models for filtering and prediction of moisture-coupled tropical waves

Authors

  • John Harlim,

    Corresponding author
    1. Department of Mathematics, North Carolina State University, Raleigh, NC, USA
    • Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695, USA.
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  • Andrew J. Majda

    1. Department of Mathematics and Center for Atmospheric and Ocean Science, Courant Institute of Mathematical Sciences, New York University, NY, USA
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Abstract

The filtering/data assimilation and prediction of moisture-coupled tropical waves is a contemporary topic with significant implications for extended-range forecasting. The development of efficient algorithms to capture such waves is limited by the unstable multiscale features of tropical convection which can organize large-scale circulations and the sparse observations of the moisture-coupled wave in both the horizontal and vertical. The approach proposed here is to address these difficult issues of data assimilation and prediction through a suite of analogue models which, despite their simplicity, capture key features of the observational record and physical processes in moisture-coupled tropical waves. The analogue models emphasized here involve the multicloud convective parametrization based on three cloud types (congestus, deep, and stratiform) above the boundary layer. Two test examples involving an MJO-like turbulent travelling wave and the initiation of a convectively coupled wave train are introduced to illustrate the approach. A suite of reduced filters with judicious model errors for data assimilation of sparse observations of tropical waves, based on linear stochastic models in a moisture-coupled eigenmode basis is developed here and applied to the two test problems. Both the reduced filter and 3D-Var with a full moist background covariance matrix can recover the unobserved troposphere humidity and precipitation rate; on the other hand, 3D-Var with a dry background covariance matrix fails to recover these unobserved variables. The skill of the reduced filtering methods in recovering the unobserved precipitation, congestus, and stratiform heating rates as well as the front-to-rear tilt of the convectively coupled waves exhibits a subtle dependence on the sparse observation network and the observation time. Copyright © 2012 Royal Meteorological Society

1. Introduction

Observational data indicate that, through the complex interaction of heating and moist convection, tropical atmosphere flows are organized on a hierarchy of scales (Nakazawa, 1988) ranging from cumulus clouds of a few kilometres to mesoscale convective systems (Houze, 2004) to equatorial synoptic-scale convectively coupled Kelvin waves and two-day waves (Kiladis, et al., 2009) to planetary-scale intraseasonal organized circulations such as the Madden–Julian Oscillation (MJO; Zhang, 2005). These moisture- coupled tropical waves like the MJO exert a substantial influence on intraseasonal prediction in the Tropics, Subtropics, and midlatitudes (Moncrieff, et al., 2007). Despite the continued research efforts by the climate community, the present coarse-resolution general circulation models (GCMs), used for prediction of weather and climate, poorly represent variability associated with tropical convection (Lau and Waliser, 2005; Zhang, 2005; Lin, et al., 2006). Given the importance of moisture-coupled tropical waves for short-term climate and medium- to long-range weather prediction, new strategies for the filtering or data assimilation and prediction of moisture-coupled tropical waves are needed and this is the topic of the present article.

The approach proposed here is to address the issues of data assimilation and prediction through a suite of analogue models which, despite their simplicity, capture key features of the observational record and physical processes in moisture-coupled tropical waves. This approach is analogous to the use of various versions of the Lorenz-96 model (Lorenz, 1996; Wilks, 2005; Majda, et al., 2005; Abramov and Majda, 2007; Crommelin and Vanden-Eijnden, 2008; Harlim and Majda, 2008a, 2010a; Majda and Harlim, 2012, and references therein) to gain insight into basic issues for midlatitude filtering, prediction, and parametrization. The viability of this approach for moisture-coupled tropical waves rests on recent advances in simplified modelling of convectively coupled tropical waves and the MJO which predict key physical features of these waves such as their phase speed, dispersion relation, front-to-rear tilt (Kiladis, et al., 2005, 2009), and circulation in qualitative agreement with observations (Khouider and Majda, 2006a,b, 2007, 2008a,b; Majda, et al., 2007; Majda and Stechmann, 2009a,b, 2011) through simplified moisture-coupled models. The analogue models emphasized here involve the multicloud convective parametrization based on three cloud types (congestus, deep, and stratiform) above the boundary layer (Khouider and Majda, 2006a,b, 2007, 2008a,b). The convective closure of the multicloud model takes into account the energy available for congestus and deep convection and uses a nonlinear moisture switch that allows for natural transitions between congestus and deep convection as well as for stratiform downdraughts which cool and dry the boundary layer. As a simplified model with two vertical baroclinic modes, the multicloud model is very successful in capturing most of the spectrum of convectively coupled waves (Kiladis, et al., 2009; Khouider and Majda, 2008b; Han and Khouider, 2010), as well as the nonlinear organization of large-scale envelopes mimicking across-scale interactions of the MJO and convectively coupled waves (Khouider and Majda, 2007, 2008a). Furthermore, the multicloud parametrization has been used in the next-generation GCM (High-Order Method Modeling Environment, HOMME) of the National Center for Atmospheric Research and is very successful in simulating the MJO and convectively coupled equatorial waves, at a coarse resolution of 170 km in the idealized case of a uniform sea-surface temperature (SST) aquaplanet setting (Khouider, et al., 2011). A stochastic version of the multicloud model has been utilized recently as a novel convective parametrization to improve the physical variability of deficient deterministic convective parametrizations (Frenkel, et al., 2012; Khouider, et al., 2010).

The filtering skill for the recovery of troposphere moisture, heating profiles, precipitation, and vertical tilts in circulation and temperature from sparse noisy partial observations is studied here for a turbulent MJO-like travelling wave (Majda, et al., 2007) and for the temporal development of a convectively coupled wave train. A suite of filters with judicious model errors, based on linear stochastic models (Harlim and Majda, 2008a, 2010a; Majda and Harlim, 2012) on a moisture-coupled eigenmode basis is developed here and applied to the two test problems as well as related 3D-Var algorithms with a full moist background covariance matrix or a dry background covariance (Žagar, et al., 2004b; Žagar, et al., 2004a). These results are the first demonstration of the utility of the analogue multicloud models for gaining insight for data assimilation and prediction of moisture-coupled tropical waves.

The plan for the remainder of the article is as follows. In section 2, the suite of simplified tropical models for filtering and prediction is reviewed; section 3 illustrates two simplified cases, an MJO analogue wave (Majda, et al., 2007) and the temporal development of a convectively coupled tropical wave train which illustrate phenomena in the models and also serve as examples for filtering in subsequent sections of the article. The suite of filters with judicious model errors for moisture-coupled tropical waves is introduced in section 4. Filtering skill for these algorithms applied to the MJO analogue wave and the development of a convectively coupled wave train is reported in section 5. Section 6 is a concluding discussion and summary.

2. Test models with moisture-coupled tropical waves

The test models proposed here begin with two coupled shallow-water systems: a direct heating mode forced by a bulk precipitation rate from deep penetrative clouds (Neelin and Zeng, 2000) and a second vertical baroclinic mode forced by the upper-level heating (cooling) and lower-level cooling (heating) of stratiform and congestus clouds, respectively (Khouider and Majda, 2006a). Below, for simplicity in exposition, we present these equations without explicit nonlinear advection effects and coupling to barotropic winds. This allows us to emphasize moisture-coupled tropical waves here, but we comment later in this section about how nonlinear advection and barotropic winds enrich the dynamics of the test models. Thus, the test models begin with two equatorial shallow-water equations

equation image(1)

for j = 1,2. The equations in (1) are obtained by a Galerkin projection of the hydrostatic primitive equations with constant buoyancy frequency onto the first two baroclinic modes. More details of their derivation may be found in (Neelin and Zeng, 2000; Frierson, et al., 2004; Stechmann and Majda, 2009). In (1), vj = (uj,vj)j=1,2 represent the first and second baroclinic velocities assuming equation image and equation image vertical profiles, respectively, while θj,j = 1,2 are the corresponding potential temperature components with the vertical profiles equation image and equation image, respectively. Therefore, the total velocity field is approximated by

equation image

where V is the horizontal velocity and w is the vertical velocity. The total potential temperature is given approximately by

equation image

Here HT ≈ 16 km is the height of the tropical troposphere with 0 ≤ zHT and equation image while equation image is the incompressible barotropic wind which is set to zero hereafter, for the sake of simplicity. In (1), P ≥ 0 models the heating from deep convection while Hs, Hc are the stratiform and congestus heating rates. Conceptually, the direct heating mode has a positive component and serves to heat the whole troposphere and is associated with a vertical shear flow. The second baroclinic mode is heated by the congestus clouds, Hc, from below and by the stratiform clouds, Hs, from above and therefore cooled by Hc from above and by Hs from below. It is associated with a jet shear flow in the middle troposphere (Khouider and Majda, 2006a, 2007, 2008a,, b). The terms S1 and S2 are the radiative cooling rates associated with the first and second baroclinic modes respectively.

The system of equations in (1) is augmented by an equation for the boundary-layer equivalent potential temperature, θeb, and another for the vertically integrated moisture content, q.

equation image(2)

In (2), hb ≈ 500 m is the height of the moist boundary layer while equation image, and equation image are parameters associated with a prescribed moisture background and perturbation vertical profiles. According to the first equation in (2), θeb changes in response to the downdraughts, D, and the sea surface evaporation E. A detailed pedagogical derivation of the moisture equation starting from the equations of bulk cloud microphysics is presented in Khouider and Majda (2006b). The approximate numerical values of equation image and equation image, follow directly from the derivation, while the coefficient equation image arises from the background moisture gradient. We use the standard value equation image (Neelin and Zeng, 2000; Frierson, et al., 2004).

In full generality, the parametrizations in (1) and (2) automatically have conservation of an approximation to vertically integrated moist static energy. Notice that, the precipitation rate in (2), balances the vertical average of the total convective heating rate in (1), therefore leading to the conservation of the vertical average of the equivalent potential temperature

equation image

when the external forces, namely, the radiative cooling rates, S1, S2, and the evaporative heating, E, are set to zero. Also note that the sensible heating flux has been ignored in (1) for simplicity since this is a relatively small contribution in the Tropics. Here and elsewhere in the text

equation image

The equations in (1) and (2) for the prognostic variables q,θebj,vj,j = 1,2, are written in non-dimensional units where the equatorial Rossby deformation radius, Le ≈ 1500 km is the length-scale, the first baroclinic dry gravity wave speed, c ≈ 50 m s−1, is the velocity scale, T = Le/c ≈ 8 h is the associated time-scale, and the dry-static stratification equation image K is the temperature unit scale. For convenience, the basic bulk parameters of the model are listed in Table 1.

Table 1. Bulk constants in the two-layer mode model.
SymbolValueDefinition
Cd0.001Boundary-layer turbulent momentum friction
hb500 mBoundary-layer height
HT16 kmHeight of the tropical troposphere.
Le≈ 1500 kmEquatorial deformation radius, length-scale
N 0.01 s−1Brunt–Väisälä buoyancy frequency
equation image0.9Moisture stratification factor
T = Le/c ≈ 8 hTime-scale
equation image0.1Baroclinic contribution to the moisture
  (nonlinear) convergence associated with the
  moisture anomalies
equation image≈ 15 KDry static stratification, temperature scale
α20.1Relative contribution of θ2 to the middle
  troposphere θe
θ0300 KReference temperature
equation image0.8Baroclinic contribution to the moisture
  convergence associated with the moisture
  background
τR50 daysNewtonian cooling relaxation time
τw75 daysRayleigh-wind friction relaxation time

2.1. The convective parametrization

The surface evaporative heating, E, in (2) obeys an adjustment equation toward the boundary-layer saturation equivalent potential temperature, equation image,

equation image(3)

with τe is the evaporative time-scale. The middle tropospheric equivalent potential temperature anomaly is defined approximately by

equation image(4)

Notice that the coefficient equation image in (4) results from the vertical average of the first baroclinic potential temperature, equation image, while the small value for α2 adds a non-zero contribution from θ2 to θem to include its contribution from the lower middle troposphere although its vertical average is zero. The multicloud model closure is based on a moisture switch parameter Λ (Khouider and Majda, 2006a; 2008a,b), which serves as a measure for the moistness and dryness of the middle troposphere. When the discrepancy between the boundary layer and the middle troposphere equivalent potential temperature is above some fixed threshold, θ+, the atmosphere is defined as dry. Moist parcels rising from the boundary layer will have their moisture quickly diluted by entrainment of dry air, hence losing buoyancy and ceasing to convect. In this case, we set Λ = 1 which automatically inhibits deep convection in the model (see below). When this discrepancy is below some lower value, θ, we have a relatively moist atmosphere and we set Λ = Λ < 1. The function Λ is then interpolated (linearly) between these two values. More precisely, we set

equation image(5)

The value of θ represents a threshold below which the free troposphere is locally moist and ‘accepts’ only deep convection while the value of θ+ defines complete dryness.

Therefore, the precipitation, P, and the downdraughts, D, obey

equation image(6)

while the stratiform and congestus heating rate, Hs and Hc, solve the relaxation-type equations

equation image(7)

and

equation image(8)

respectively. The dynamical equations in (1), (2), (7), and (8) define the multicloud model. Notice that, as anticipated above, when the middle troposphere is dry, Λ = 1, deep convection is completely inhibited, even if P0, i.e. convective available potential energy (CAPE) is positive, whereas congestus heating is favoured. Other variants of the equation in (8) for Hc can be utilized where changes in Hc respond to low-level CAPE (Khouider and Majda, 2008a,b).

The quantities P0 and D0 represent respectively the maximum allowable deep convective heating/precipitation and downdraughts, independent of the value of the switch function Λ. Notice that conceptually the model is not bound to any type of convective parametrization. A Betts–Miller relaxation type parametrization as well as a CAPE parametrization can be used to set up a closure for P0. Here we let

equation image(9)

where f+ = max(f,0) and equation image is a threshold constant value measuring a significant fraction of the tropospheric saturation and τconv, a1, a2, a0 are parameters specified below. In particular, the coefficient a0 is related to the inverse buoyancy relaxation time of Fuchs.

The downdraughts are closed by

equation image(10)

where m0 is a scaling of the downdraught mass flux and equation image is a prescribed precipitation/deep convective heating at radiative convective equilibrium. Here μ2 is a parameter allowing for stratiform and congestus mass flux anomalies (Majda and Shefter, 2001; Majda, et al., 2004). Finally the radiative cooling rates, S1, S2 in (1) are given by a simple Newtonian cooling model

equation image(11)

where equation image are the radiative cooling rates at radiative convective equilibrium (RCE). This is a spatially homogeneous steady-state solution where the convective heating is balanced by the radiative cooling. The basic constants in the model convective parametrization and the typical values utilized here are given in Table 2. The physical features incorporated in the multi-cloud model are discussed in detail in (Khouider and Majda, 2006a, 2007, 2008a,b).

Table 2. Parameters in the convective parametrization. The parameters in the middle panel will be chosen differently for the MJO-analogue case in section 3.1 and the temporal development of a convectively coupled wave train in section 3.2. The parameters in the lower panel are determined at the RCE state.
SymbolValueDefinition
A,B  Linear fitting constant interpolating the switch
  function Λ
equation image1 K day−1Second baroclinic radiative cooling rate
αs0.25Stratiform heating adjustment coefficient
γ20.1Relative contribution of θ2 to convective
  parametrization
equation image10 KDiscrepancy between boundary layer θe at its
  saturated value and at the RCE state
θ±10, 20 KTemperature threshold used to define the switch
  function Λ
a0 Inverse buoyancy time-scale of convective
  parametrization
a1 Relative contribution of θeb to convective
  parametrization
a2 Relative contribution of q to convective
  parametrization
αc Congestus heating adjustment coefficient
equation image Discrepancy between boundary and middle
  troposphere potential temperature at RCE value
Λ Lower threshold of the switch function Λ
μ2 Relative contribution of stratiform and congestus
  mass flux anomalies to the downdraughts
τc Congestus heating adjustment time
τconv Deep convection adjustment time
τs Stratiform heating adjustment time
m0 Scaling of downdraught mass flux
equation image Threshold beyond which condensation takes place
  in the Betts–Miller scheme
equation image Second baroclinic radiative cooling rate
τe≈ 8 hEvaporation time-scale in the boundary layer

2.2. Moisture-coupled phenomena in the test models

As already noted in the introduction, the dynamic multicloud models in (1), (2), (7), (8) capture a number of observational features of equatorial convectively coupled waves and the MJO. These phenomena occur in multi-wave dynamical models with strong moisture coupling through (2), nonlinear on–off switches like (5), (9), (10) and nonlinear saturation of moisture-coupled instabilities (Khouider and Majda, 2006a, 2007, 2008a,b; Khouider, et al. 2011). All of these features present major challenges for contemporary data assimilation and prediction strategies. Two detailed analogue examples are presented in section 3.

As described in detail in Khouider and Majda (2006b), the multicloud models in a limiting regime also include the quasi-equilibrium models (Neelin and Zeng, 2000; Frierson, et al., 2004; Pauluis, et al., 2008) which mimic the Betts–Miller and Arakawa–Schubert parametrizations of GCMs. Such models arise formally by keeping the first baroclinic mode in (1), retaining the moisture equation in (2) with D = 0, setting Λ = 1 in (6), and using P0 in (9) with a1 = 0 while ignoring all remaining boundary-layer and cloud equations. There are many interesting exact solutions of the nonlinear dynamics with moisture switches in this quasi-equilibrium regime, e.g. large-scale precipitation fronts, which serve as interesting test problems for filtering with nonlinear switches and moisture-coupled waves (Frierson, et al., 2004; Pauluis, et al., 2008; Stechmann and Majda, 2006); the behaviour of finite-ensemble Kalman filters (Evensen, 1994; Anderson, 2001; Bishop, et al., 2001; Hunt, et al., 2007) and particle filters (van Leeuwen, 2010; Anderson, 2010) are particularly interesting in this context with moisture-coupled switches and exact solutions. Žagar (2012) give other interesting uses of similar models as tests for tropical data assimilation. However, rigorous mathematical theory establishes that these quasi-equilibrium models have no instabilities or positive Lyapunov exponents (Majda and Souganidis, 2010), unlike realistic tropical convection and the full multicloud models. More realism in the quasi-equilibrium tropical models can be achieved by allowing active barotropic dynamics and coupled nonlinear advection which allows for tropical–extratropical wave interactions (Lin, et al., 2000; Majda and Biello, 2003; Biello and Majda, 2004). Examples with these features are developed by Khouider and Majda (2005a,b).

3. Examples of moisture-coupled tropical waves in the test model

In this section, we describe two concrete examples with solutions which will be used as the truth for generating synthetic observations (as we will describe in section 4). The two specific examples include an MJO-like travelling wave (Majda, et al., 2007) and the initiation of a convectively coupled wave train that mimics the solutions of explicit simulations with a cloud-resolving model (Grabowski). Following the basic set-up in Khouider and Majda (2006a), Khouider and Majda (2007), we consider the multicloud model in (1), (2), (7), (8) on a periodic equatorial ring without rotation, β = 0, without barotropic wind, equation image, and with a uniform background SST given by constant equation image. With this set-up, the wind velocity in (1), (2) has only the zonal wind component, vj = uj, resolved at every 40 km on an equatorial belt of 40 000 km.

3.1. An MJO-like turbulent travelling wave

In our first example, we consider the parameter regime for an intraseasonal MJO-like turbulent travelling wave. Following Majda, et al. (2007), we set the bulk parameters in Table 1, equation image, equation image, equation image, equation image, equation image and the convective parameters in Table 2, equation image, a0 = 12, a1 = 0.1, a2 = 0.9, μ2 = 0.5, αc = 0.5, Λ = 0.2. The intraseasonal time-scale is generated through τconv = 12 h which is consistent with the current observational estimates for large-scale consumption of CAPE, and τs = τc = 7 days which is also consistent with the current observational record for low-level moistening and congestus cloud development in the MJO.

The linear stability analysis for this parameter regime has been studied in detail in Majda, et al. (2007). Here, we summarize for convenience some of the important features for eastward propagating waves: the unstable wavenumbers 2 and 3 have growth rates of roughly (30 days)−1 and phase speed of 6.9 and 5.8 m s−1, respectively. These unstable modes have westward-tilted vertical structure for heating, velocity, and temperature, with clear first and second baroclinic mode contributions and low-level higher potential temperature leading and within the deep convection (Figure 9 below). In Figure 1, we show the contour plot of the precipitation P (which is exactly the deep heating rate for this model) at the statistical steady state from a numerical simulation between 5000 and 5200 days. The main feature here is an eastward moving wavenumber 2 MJO-like wave with phase speed 6.1 m s−1. Within the envelope of this wave are intense westward moving small-scale fluctuations. These fluctuations occur irregularly and there are often long breaks between intense deep convective events. All of these features are observed in the MJO (Zhang, 2005).

Figure 1.

Contour plot of the deep convective heating P(x,t) from a numerical simulation of the multicloud model with parameter values in section 3.1, Tables 1 and 2. Heating values > 2 K day−1 are shaded in grey and > 10 K day−1 in black.

3.2. Initiation of a convectively coupled wave train

In this second example, we consider the three-cloud model with enhanced congestus heating (Khouider and Majda, 2008a) with slightly different parametrization from the above. In particular, the total precipitation, P, is different from the deep convection heating rate, Hd, and is defined as

equation image(12)

allowing for stratiform and congestus rain. The key feature in this new parametrization is attributed to the asymmetric heating rate contribution in the upper- and lower-level atmosphere with non-zero ξs and ξc, respectively. This new feature replaces the first baroclinic heating equation in (1) with

equation image(13)

The moisture equation in (2) remains unchanged, except that now we remove the scale factor equation image in front of P since it is already included in (12).

The new congestus parametrization uses exactly the same switch function Λ in (5) with the middle-troposphere equivalent potential temperature approximation in (4). The precipitation, P, in (6) is replaced by

equation image(14)

with bulk energy available for deep convection given by

equation image(15)

In (15), parameter equation image is the bulk convective heating determined at the RCE state. The downdraught in (6) is also replaced by

equation image(16)

Compared to (6), this new parametrization assigns Λ = 0 for the moisture switch minimum threshold and ignores the factor Λ in the original downdraught equation. The corresponding dynamical equations for the stratiform and congestus heating are

equation image(17)
equation image(18)

where

equation image(19)

denotes a ‘bulk energy’ for congestus heating.

In our numerical experiment, we use the same parameter values as in Khouider and Majda (2008a). The bulk constants in Table 1 are not changed. The convective parameters in Table 2 are used with Λ = 0, μ2 = 0.25, αc = 0.1, τs = 3 h, τc = 1 h, a0 = 5, a1 = a2 = 0.5, τconv = 2 h, and equation image K. The additional new parameters for the enhanced congestus parametrization include: the coefficients representing contributions of stratiform and congestus clouds to the first baroclinic heating, ξs = 0.5 and ξc = 1.25, respectively; the inverse convective buoyancy time-scale associated with congestus clouds, a0′ = 2; and the bulk convective heating equation image that is determined at RCE. Interested readers should consult Khouider and Majda (2008a) for the details of the linear stability analysis.

Here, we are interested in the initiation of a convectively coupled wave train to mimic the high-resolution two-dimensional explicit cloud-resolving model solutions in Grabowski. In particular, we integrate the model with a localized piece of a single unstable linear wave of small amplitude centred at 20 000 km as the initial condition. (Figure 2 shows a space–time plot of the first two baroclinic velocities, potential temperatures, congestus and deep heating rates, moisture, and precipitation.) Note that this set-up is exactly the regime analyzed in Frenkel, et al. (2011), in which they focused on understanding the effect of the diurnal cycle, and we neglect the diurnal cycle here. Notice there are fast-moving waves (q, Hd, and P in Figure 2) during the first 2 days moving away from the 20 000 km mark. After about 8–10 days, additional waves appear; this wave initiation is partly due to the convectively coupled wave interactions with faster-moving gravity waves. After about 100 days, these waves mature to a wave train of six individual eastward moving waves with a wave speed of approximately 14.5 m s−1 (Figure 3). Such wave structure and wave train organization resembles the structure found in the explicit simulations with the cloud-resolving model of Grabowski. Moreover, the mature waves have a total convective heating pattern (with backward and upward tilt in the wind and temperature fields, upper-tropospheric positive temperature anomalies slightly leading the region of the upward motion, which is in phase with the heating anomalies, with low-level convergence) which is very similar to convectively coupled Kelvin waves observed in nature (Wheeler and Kiladis, 1999; Wheeler, et al., 2000; Straub and Kiladis, 2002).

Figure 2.

Initiation of a convectively coupled wave train. The space–time plot here is constructed with coarse spatial and temporal resolutions at every 2000 km and 24 h. (This coarse dataset is sampled from solutions with higher resolutions at every 40 km and 3 h.) The contour intervals are 0.25 m s−1 for the zonal wind, 0.025 K for the potential temperature and humidity, and 0.05 K day−1 for the heating rates and precipitation. For u1,u212,q, solid black (dashed grey) contours denote positive (negative) values. For Hc,Hd,P, solid black (dashed grey) contours denote heating rates greater (smaller) than 1 K day−1.

Figure 3.

Moving average of the vertical structure in a reference frame of 14.5 m s−1 from a time period 500–1000 days. The contour intervals are 0.07 K for the potential temperature, 0.54 K day−1 for the total convective heating, and 0.35 m s−1 for the horizontal velocity. Solid (dashed) contours denote positive (negative) values.

4. Algorithms for filtering moisture-coupled waves from sparse observations

In this section, we first describe the sparse observation networks and then discuss in detail the reduced stochastic filtering algorithms.

4.1. Sparse observation networks

In the present article, we consider horizontally sparse observations at every 2 000 km. This means we only have M = 20 observations at xj = jh, h = 2π/40000 km in a non-dimensionalized unit assuming that the equatorial belt circumference is 40 000 km. For compact notation, we define Ψj,m = (u1,u212eb,q,Hs,Hc)T; we use subscripts j and m to specify that each component in Ψ is evaluated at grid point xj and discrete time tm, respectively. We define a general observation model

equation image(20)

where G is an observation operator that maps the model state to the observation state space and σj,m are eight-dimensional independent Gaussian white noises with mean zero and diagonal covariance matrix Ro. Vertically, we consider four observation networks with specific G:

SO (Surface observations). Here, we consider observing the wind, potential temperature at a surface height zs = 100 m, and the equivalent boundary-layer potential temperature θeb. The corresponding observation operator is a 3×8 matrix G with non-zero components

equation image(21)

where G,G′ are the vertical baroclinic profiles defined in section 2.

SO+MT (Surface observations + middle troposphere temperature). This observation network includes temperature at middle-troposphere height zm = 8 km in addition to SO. The corresponding observation operator is a 4×8 matrix G with non-zero components

equation image(22)

in addition to (21).

SO+MTV (Surface observations + middle troposphere temperature and velocity). This observation network includes velocity at middle-troposphere height zm = 8 km in addition to SO+MT. The corresponding observation operator is a 5×8 matrix G with non-zero components

equation image(23)

in addition to (21) and (22).

CO (Complete observations). This vertically complete observation network is defined with G = I for diagnostic purposes.

4.2. Filtering algorithms

In this article, we consider the simplest version of our reduced stochastic filters, the Mean Stochastic Model (MSM; Harlim and Majda, 2008a, 2010a,b; Majda and Harlim, 2012). The new feature in the present context is that we have multiple variables Ψj as opposed to a scalar field, and therefore we need to design the MSM in an appropriate coordinate expansion to avoid parametrizing various coupling terms.

As in Harlim and Majda (2008a), our design of the filter prior model is based on the standard approach for modelling turbulent fluctuations (Majda, et al., 1999, 2008; Majda and Timofeyev, 2004; DelSole, 2004), i.e. we introduce model errors through linearizing the nonlinear models about a frozen constant state and replacing the truncated nonlinearity with a dissipation and spatially correlated noise (white in time) to mimic rapid energy transfer between different scales. In the present context, we consider the linearized multicloud model about the RCE,

equation image(24)

where Ψ′ denotes the perturbation field about the RCE and equation image denotes the linearized differential operator of the multicloud model at RCE. A comprehensive study of the linear stability analysis of (24) involves solving eigenvalues of an 8 × 8 dispersion matrix, ω(k), and was reported in Majda, et al. (2007) for the MJO-like wave and in Khouider and Majda (2008a) and Frenkel, et al. (2011) for the multicloud model with enhanced congestus heating.

Consider a numerical discretization for (24) with spatial mesh size of Δx = 2000 km such that the model state space is essentially similar to the observation state space. With this approximation, the PDE in (24) becomes

equation image(25)

where equation image are the discrete Fourier components of {Ψj}j=1,…,M. Now consider an eigenvalue decomposition, iω(k)Zk = ZkΛk, where Λk is a diagonal matrix of the eigenvalues and Zk is a matrix whose columns are the corresponding eigenvectors. Then we can write (25) as a diagonal system,

equation image(26)

with the transformation

equation image(27)

4.2.1. The MSM filter

The MSM is defined through the stochastic differential system,

equation image(28)

for |k| ≤ M/2. In (28), Γk,Ωk, and σk are diagonal matrices with diagonal components obtained through regression fitting to the climatological statistics, while the forcing term is proportional to the climatological mean field, equation image; here, the angle bracket 〈·〉 denotes an average. Notice that the realizability of this stochastic model (referred to as MSM-1; Majda, et al., 2010; Harlim and Majda, 2010b; Majda and Harlim, 2012) is guaranteed since Γk is always positive definite as opposed to the alternative approach which sets Ωk = −iΛk (Penland, 1989; DelSole, 2000). Throughout this article, the climatological statistics are computed from solutions of the full multicloud model resolved at 40 km grid points with different temporal resolutions for the two cases: the MJO-like travelling wave and the initiation of a convectively coupled wave train (sections 5.1 and 5.2).

The discrete-time Kalman filtering problem with the MSM as the prior model is defined for each horizontal wavenumber k as

equation image(29)
equation image(30)

where the observation model in (30) is the discrete Fourier component of the canonical observation model in (20) with Gaussian noises, equation image. The discrete filter model in (29) has coefficients

equation image(31)
equation image(32)

and unbiased Gaussian noises ηk,m with covariance matrix

equation image(33)

These coefficients are obtained by evaluating the analytical solutions of the stochastic differential system in (28) at observation time interval Δt = tm+1tm and applying the transformation in (27).

The MSM filter in (29)–(30) is computationally very cheap since it only involves M/2 + 1 independent 8 × 8 Kalman filtering problems, ignoring cross-correlations between different horizontal wavenumbers. Such a diagonal approximation may seem to be counterintuitive since it generates severe model errors, but we have shown that it provides high filtering skill beyond the perfect model simulations in various contexts including the regularly spaced sparse observations (Harlim and Majda, 2008b), irregularly spaced sparse observations (Harlim, 2011), strongly chaotic nonlinear dynamical systems (Harlim and Majda, 2008a, 2010a), and midlatitude baroclinic wave dynamics (Harlim and Majda, 2010b).

Applying the Kalman filter formula on each wavenumber in (29)–(30) provides the following background (or prior) mean and error covariance estimates,

equation image(34)
equation image(35)

and analysis (or posterior) mean and error covariance estimates

equation image(36)

where Kk.m is the Kalman gain matrix.

4.2.2. The complete 3D-Var

For diagnostic purposes, we also consider a 3D-Var version in the MSM framework above. That is, we simply set the background-error covariance matrix to be independent of time,

equation image(37)

and repeat the mean prior and posterior updates in (34), (36) with a constant Kalman gain matrix,

equation image

We called this approach the complete 3D-Var because the forward model parameters in (31), (32), (33) and the background covariance matrix in (37) are determined from complete solutions of the multicloud model in (1), including the moisture and heating variables from (2), (7), (8). This formulation is significantly different from an earlier approach with variational techniques (Žagar, et al. 2004b; Žagar, et al., 2004a) in which the background covariance matrix is parametrized in an eigenmode basis constructed from the dry equatorial waveguide.

4.2.3. The ‘dry and cold’ 3D-Var

To mimic the approach in Žagar, et al. (2004b), Žagar, et al. (2004a), we consider only using the wind and temperature data, u1,u212eb, to construct the ‘dry and cold’ eigenmode basis and background covariance matrix Bk. Technically, we still use the MSM model in (28), but replace the transformation in (27) by

equation image(38)

In this sense, the parameters Γk,Ωk, and σk in (28) are fitted to climatological statistics of equation image based on only the wind and temperature variables. Repeating the 3D-Var algorithm described above in this set-up provides an honest ‘dry and cold’ version analogous to the earlier approach in Žagar, et al. (2004b), Žagar, et al. (2004a).

Besides the eigenmode basis difference, we should note that the ‘dry and cold’ 3D-Var here is computationally much cheaper than that in Žagar, et al. (2004b), Žagar, et al. (2004a) since we perform both the prior and posterior updates in the diagonalized Fourier basis with reduced stochastic filters through (34)–(36) as opposed to their approach that propagates the nonlinear dry shallow-water equations in physical space and applies the analysis step in the spectral diagonal basis. On each data assimilation step, their approach requires back-and-forth transformations in between the physical and spectral spaces with a rotational transformation matrix that is quite often ill-conditioned, as reported in Žagar, et al. (2004b).

For diagnostic purposes, we will also consider the ‘moist and cold’ 3D-Var in the numerical simulations in section 5.1; this model is constructed exactly like the ‘dry and cold’ model described above with moisture q in addition to the wind and temperatures, u1,u212eb.

5. Filtering skill for moisture-coupled tropical waves

In this section, we report the numerical results of implementing the filtering algorithms in section 4.2 to assimilate the synthetic sparse observation networks defined in section 4.1 on the two examples discussed in section 3.

In the numerical simulations below, we consider the precise observations case with Ro = 0 and small observation noises with positive definite covariance matrix Ro > 0. In the non-zero noise case, we choose the observation noise variance to be 10% of the climatological variance of each variable. In this sense, the noise variances are less than both the peak of the energy spectrum and the smallest average signal amplitude.

5.1. MJO-like turbulent travelling wave

Our goal here is to check the filtering skill in recovering the structure of the MJO-like travelling wave (section 3.1) with the MSM forward model in (28) with parameters (31)–(33), which are specified from a time series of 8 000 days with temporal resolution of 6 h at the climatological state.

First, we compare the moving average of u1, u2, θ1, θ2, θeb, q, Hs, Hc, P obtained from the true solutions of the test model in section 3.1 and the posterior mean estimates in (36). The moving average is taken in a reference frame at 6.1 m s−1 from a time period of 750–1000 days. In Figures 4–8, we show the moving average from assimilations with observation time interval of 24 h for complete observations (CO) with Ro = 0, and for all observation networks discussed in section 4.1, CO, SO+MTV, SO+MT, SO with small observation noise covariance, Ro > 0. For observation network CO without observation errors, Ro = 0 (Figure 4), the three schemes (MSM filter, complete 3D-Var and ‘dry and cold’ 3D-Var) are identical and they perfectly recover the averaged MJO structure except for slight overestimation on the stratiform heating and precipitation.

Figure 4.

MJO-like waves with Δt = 24 h, Ro = 0 and complete observations (CO). The moving average is in a reference frame at 6.1 m s−1 of the model variables. True (grey dashes), posterior mean state of the complete 3D-Var (circles), MSM filter (squares), and the ‘dry and cold’ 3D-Var (diamonds).

Figure 5.

MJO-like waves with Δt = 24 h, Ro > 0 and complete observations (CO). The moving average is in a reference frame at 6.1 m s−1 of the model variables. True (grey dashes), posterior mean state of the complete 3D-Var (circles), MSM filter (squares), the ‘dry and cold’ 3D-Var (diamonds), and the ‘moist and cold’ 3D-Var (asterisks).

Figure 6.

As Figure 5, but for surface observations plus middle-troposphere potential temperature and velocity (SO+MTV).

Figure 7.

As Figure 5, but for surface observations plus middle-troposphere potential temperature (SO+MT).

Figure 8.

As Figure 5, but for surface observations (SO).

In the presence of observation noise, we include results with ‘moist and cold’ 3D-Var (detailed discussion appears at the end of section 4.2.3). We find that all the four schemes are able to recover u1 and θeb with any observation network. When middle-troposphere wind observation is absent (SO+MT and SO in Figures 7, 8), the estimate for u1 slightly degrades but is completely wrong for u2. The MSM filter overestimates θ2 by roughly 0.1 K, even with surface and middle-troposphere potential temperature observations; we find that this poor estimation is attributed to an inaccurate mean estimate (on the zeroth horizontal mode) of θ2. The MSM filter, the complete and ‘moist and cold’ 3D-Var are able to recover the oscillating structure of the moisture q with any observation network (with slight errors for the MSM filter with SO) reflecting the active and suppressed convective phases of the MJO-like wave. On the other hand, the ‘dry and cold’ 3D-Var cannot produce q accurately even with observation network CO, and simply predicts a dry atmosphere (with zero moisture profile) when the moisture is unobserved. All the four filters are not able to reproduce the stratiform and congestus heating profiles when they are not observed.

Except for the SO network, both the complete and ‘moist and cold’ 3D-Var are able to reasonably recover the precipitation rate, P, which in this model is exactly the deep convection heating rate; here, the ‘cold and dry’ 3D-Var precipitation estimate is very inaccurate (Figures 57). On the other hand, the MSM filter captures the peak of the precipitation on all the three observation networks (CO, SO+MTV and SO+MT), but overestimates the profile on the last two observation networks. This overestimation of the precipitation (as well as those observed when we assimilate only the SO network; Figure 8) can be explained as follows. From the precipitation budget in (9), it is obvious that the contributions of θeb, q, and θ2 to the convective parametrization are small (with scale factors a1 = 0.1,a2 = 0.5, a0γ2 = 1.2, respectively) relative to θ1 (with scale factor a0 = 12). Therefore, the wet filtered state (with large precipitation estimates as seen in Figure 8) is attributed to the slight underestimation of the first baroclinic potential temperature, θ1. The complete 3D-Var underestimates θ1 by as much as 0.5 K; this yields a spatially uniform precipitation rate of about 2.3 K day−1. The MSM filter underestimates θ1 by as much as 1.5 K and its corresponding precipitation estimate is about 20 K day−1.

In Figures 9–12, we show the detailed vertical structure of the total potential temperature Θ, the velocity vector field (V, w), the total convective heating, and horizontal velocity from the MJO-like wave in section 3.1 and the complete 3D-Var estimates with observation networks SO+MTV, SO+MT, and SO, respectively. In particular, the vertical tilted structure in the potential temperature is recovered with any of these three observation networks; similar recovery (not shown) is also obtained with the MSM filter and the ‘moist and cold’ 3D-Var; the ‘dry and cold’ 3D-Var also recovers this tilted structure except with observation network SO. On the other hand, the tilted structure in the horizontal velocity with low-level convergence that is in phase with the deep convective heating is not recovered whenever the middle-troposphere wind observation is absent. Notice also that the deep convective heating is recovered except with observation network SO; similar recovery (not shown) is also attained with the MSM filter and the ‘moist and cold’ 3D-Var but not with the ‘dry and cold’ 3D-Var.

Figure 9.

The true vertical profile of the MJO-like waves computed with moving average in a reference frame at 6.1 m s−1. The contour intervals are 0.07 K for the potential temperature, 0.29 K day−1 for the total convective heating, and 1 m s−1 for the horizontal velocity. Solid (dashed) contours denote positive (negative) values.

Figure 10.

The vertical profile from complete 3D-Var estimate with observation network SO+MTV and Ro > 0, and Δt = 24 h. Contours are as in Figure 9.

Figure 11.

As Figure 10, but for observation network SO+MT.

Figure 12.

As Figure 10, but for observation network SO.

We also find that both the complete and ‘moist and cold’ 3D-Var are able to reconstruct the detailed precipitation structure in Figure 1 except when assimilated with observation network SO (results are not shown). The MSM filter is able to capture the peak but overestimates the detail profile. The ‘moist and cold’ 3D-Var reproduces the eastward MJO-like signal, but fails to capture the westward intermittent moist fluctuations within the MJO envelope as shown in Figure 1.

We also repeated the numerical experiments above with different observation time intervals ranging from 6 h to 8 days with the complete 3D-Var and MSM filter (Figure 13 shows the average RMS errors on the MSM filter case). Particularly noteworthy is that the posterior estimates have roughly similar RMS errors for the observed variables independent of the observation times; for the unobserved variables, the RMS errors for the shorter observation times are larger than those for the longer observation times! This latter result can be understood as follows. The dynamical operator ℱk in (31) is essentially marginally stable (with largest eigenvalue 0.9899) for Δt = 6 h and is strictly stable (with largest eigenvalue 0.8836) for longer Δt = 72 h. The observability condition, which is a necessary condition for accurate filtered solutions when the dynamical operator is marginally stable (Anderson and Moore, 1979; Majda and Harlim, 2012), is practically violated here; our test with SO+MT observation network suggests that the observability matrix is ill-conditioned with equation image. This explains why the longer observation times produce more accurate filtered solutions. Thus, with the crude spatial observation network and the inefficient behaviour of MSM at short times, this simple filtering strategy necessarily cannot capture subgrid-scale features of the wave with high skill; by design this is also true for 3D-Var. We encounter similar behaviour of filtered solutions in the next example in section 5.2.

Figure 13.

Average RMS errors as functions of observation time interval (in days): observation error (thin dashed line), climatological errors (dash-dotted line), CO (bold solid line), SO+MTV (bold dashes), SO+MT (circles) and SO(squares).

In Figure 13, we include the climatological errors (dash-dotted line) and observation errors (thin dashes) for diagnostic purposes. Recall that the observation-error covariance Ro in our experiments is 10% of the climatological variances and the observation errors are only relevant for diagnostic purposes when the corresponding variable is observed. So, in real time, the MSM filter with sparse observation networks SO+MTV, SO+MT has reasonable skill as long as its RMS errors are below the climatological errors. In this sense, we observe that the MSM filter is very skilful for variables u1eb and q for any observation network as well as for θ1 for observation networks other than SO. Our conjecture is that, on these variables, the RMS errors will increase as the observation time interval is near its slowest decaying time (70 days for this model). For the other variables, the filtering skill is not better than the climatological variability and further improvement will be addressed in future work.

5.2. Initiation of a convectively coupled wave train

Here, our goal is to check the filtering skill in recovering the transient behaviour of initiation of a convectively coupled wave train (section 3.2) with the MSM forward model in (28) where parameters (31)–(33) are specified from a time series at the climatological state for the period of time 500–1000 days with temporal resolution of 3 h.

In Figures 14–17, we report the space–time plot of the filtered estimates at the initial period of time 0–50 days from the complete 3D-Var with observation time Δt = 24 h, observation noise variance Ro > 0, and observation networks SOMTV, SOMT, and SO. By eye, we can see that the emerging pattern in Figure 2 is recovered for all variables except for the deep convection heating rate with the complete observation network! This poor estimate is attributed to an overestimation of θ1 (which sets the available convective heating Qd in (15) to zero). In this case, the precipitation budget in the filtered solution is dominated by the stratiform and congestus heating rates. On the other hand, even if the pattern of Hd is always captured with networks SO+MTV, SO+MT, SO, its accuracy is questionable as we will see below.

Figure 14.

Space–time plot from the complete 3D-Var estimate with observation network CO and Ro > 0, and Δt = 24 h. The contour intervals are 0.25 m s−1 for the zonal wind, 0.025 K for the potential temperature and humidity, and 0.05 K day−1 for the heating rates and precipitation. For u1,u212,q, solid black (dashed grey) contours denote positive (negative) values, and for Hc,Hd,P solid black (dashed grey) contours denote heating rates greater (smaller) than 1 K day−1.

Figure 15.

As Figure 14, but for observation network SO+MTV, and with contour intervals 0.05 K day−1 for the congestus heating rate, and 0.25 K day−1 for the deep convective heating and precipitation.

Figure 16.

As Figure 15, but for observation network SO+MT.

Figure 17.

As Figure 15, but for observation network SO.

To be more precise, we quantify the filter skill with the average RMS error and pattern correlation (PC) between the posterior mean estimate and the truth at the initiation period of time 0–75 days before these waves lock into a wave train of six waves as shown in Figure 3. In Figures 18–25, we plot these two performance measures as functions of observation times for observation networks CO, SO+MTV, SO+MT, and SO. In each panel, we compare four numerical experiments including the MSM filter with Ro = 0 and Ro > 0, and the complete 3D-Var with Ro = 0 and Ro > 0.

Figure 18.

RMS errors as functions of observation time interval for observation network CO: MSM filter (grey), complete 3D-Var (black), Ro = 0 (solid line and dashes with no markers), Ro > 0 (dashes with square/circle markers).

Figure 19.

Pattern correlations as functions of observation time interval for observation network CO: MSM filter (grey), complete 3D-Var (black), Ro = 0 (solid line and dashes with no markers), Ro > 0 (dashes with square/circle markers).

Figure 20.

As Figure 18, but for observation network SO+MTV.

Figure 21.

As Figure 19, but for observation network SO+MTV.

Figure 22.

As Figure 18, but for observation network SO+MT.

Figure 23.

As Figure 19, but for observation network SO+MT.

Figure 24.

As Figure 18, but for observation network SO.

Figure 25.

As Figure 19, but for observation network SO.

From the average RMS errors (Figures 18, 20, 22, 24), we find that the filtering skill of the MSM filter and the complete 3D-Var are not different at all except for the wind variables when the middle-troposphere wind is not observed and Ro > 0; there, the RMS errors of the complete 3D-Var are smaller than those of the MSM filter (Figures 22, 24), but their PCs are identical (Figures 23, 25). When observations are complete (CO) and Ro = 0, both the MSM filter and complete 3D-Var are able to recover the truth (with negligible errors and PC one).

The average RMS errors for simulations with Ro > 0 (dashes with markers) for variables θ12,q,Hc decay as functions of observation time even with complete observation network (Figure 18). We find that the larger errors with shorter observation times here are attributed to the violation of practical controllability (Anderson and Moore, 1979; Harlim and Majda, 2008b; Majda and Harlim, 2012) which is also a necessary condition for optimal filtering when the system is marginally stable (here ℱk has a maximum eigenvalue close to 1). Additionally, we observe a similar decaying pattern for the error as function of observation time with SO+MTV, SO+MT, and SO for the unobserved variables q, Hc, Hd, and P even when the observed wind and potential temperatures have no errors (Ro = 0). Here, the larger errors in the unobserved variables for shorter observation times are attributed to the violation of practical observability as explained in section 5.1.

When Ro > 0, the RMS errors of the deep convection heating rate Hd are roughly 1 K day−1 (Figure 18) with observation network CO but the PCs are roughly zero (the PC curves are below 0.5 in Figure 19). The PC confirms the inability to recover Hd as shown in Figure 14. Both filtering schemes with the other observation networks (SO+MTV, SO+MT, SO) recover the structure of Hd (with PC of roughly 0.6 from Figures 21, 23, 25) but their errors are very large (as much as 10 K from Figures 20, 22, 24). The failure even to capture the deep convection pattern with CO is attributed to overestimation of θ1 as explained earlier by contrasting the detailed space–time structure of θ1 in Figures 2 and 14. The average RMS errors show such a tendency for failure with larger error with CO compared to those with the other networks, but they do not inform us whether the potential temperature estimates are above or below the truth, which is important for accurate precipitation estimation.

Finally, notice that with networks SO+MTV, SO+MT, and SO, the RMS errors of the unobserved variables for assimilation with Ro = 0 are larger than those with Ro > 0; for example, see the errors for variables θ1,q,Hc,Hd,P in Figure 24. We find that these larger errors with Ro = 0 are due to an ill-conditioned Kalman gain matrix in (36) with sparse observation networks with operator equation image.

6. Summary and concluding discussion

In this article, we use multicloud models (Khouider and Majda, 2006a, 2007, 2008a; Majda, et al., 2007) as the test models for filtering moist tropical convection. In particular, we aim to establish guidelines for the future design of filtering schemes in assimilating and predicting tropical atmospheric dynamics. We view the multicloud model, with convective parametrization which includes three cloud types (congestus, deep, and stratiform) above the boundary layer, as a candidate for the simplest toy model for moisture-coupled tropical waves (analogous to the Lorenz-96 for midlatitude weather dynamics) for the following reasons: it is very successful in capturing most of the spectrum of the convectively coupled waves (Khouider and Majda, 2008a,b; Kiladis, et al., 2009) as well as the nonlinear organization of large-scale envelopes mimicking across-scale interactions of the MJO (our first example in section 3.1) and convectively coupled waves (our second example in section 3.2). More importantly, this model also captures the vertical profile with front and rear tilting, the phase speed, and dispersion relations that match the observational record (Kiladis, et al., 2005; Kiladis, et al. 2009).

Here, we demonstrate the filtering skill with a suite of reduced stochastic filters with model errors, based on linear stochastic models (Harlim and Majda, 2008a; 2010a; Majda and Harlim, 2012) in capturing the intraseasonal MJO-like wave (Majda, et al., 2007) and the transient initiation of a convectively coupled wave train that resembles the results from simulations with a cloud-resolving model (Grabowski). From these numerical experiments, we find the following:

  • (i)The key factor for accurate precipitation estimates is an accurate estimation of the first baroclinic potential temperature. Our test problems suggests that slight overestimation in θ1 produces a dry atmosphere with no rain at all and slight underestimation in θ1 produces a wet atmosphere with unrealistic high precipitation rate;
  • (ii)Our simple reduced stochastic filters are able to recover moisture and precipitation field profiles (even when online observations of these variables are not available) provided that the filter forward prior model is designed in a moisture-coupled eigenmode basis. This result suggests that the future design of tropical data assimilation algorithms should account for a moisture-coupled eigenmode basis instead of dry eigenmode basis as in Žagar, et al. (2004b), Žagar, et al. (2004a);
  • (iii)A better estimate for the tropical convection wave patterns requires more than surface wind and potential temperature observations;
  • (iv)The skill of the reduced filtering methods with horizontally and vertically sparse observations suggests that more accurate filtered solutions are achieved with less frequent observation times. Such a counterintuitive finding is justified through an analysis of the classical observability and controllability conditions which are necessary for optimal filtering, especially when the observation time-scale is too short relative to the time-scale of the true signal.

We hope that the encouraging results in this article can convince researchers who are interested in tropical data assimilation to investigate:

  • (i)The potential of improving the estimate accuracy with more sophisticated filtering schemes through the test models here; this includes testing with the ensemble Kalman filters (Evensen, 1994; Anderson, 2001; Bishop, et al., 2001; Hunt, et al., 2007), particle filters with small ensemble sizes (Anderson, 2010; van Leeuwen, 2010), and other reduced stochastic filters (Gershgorin, et al., 2010a,b; Majda and Harlim, 2012) or the Gaussian closure filter (Branicki, et al., 2012) with stochastic parametrizations that account for model errors ‘on the fly’;
  • (ii)Exploration of the filtering skill for other parameter regimes in multicloud models with more realistic SST profile, active barotropic dynamics and coupled nonlinear advection which allows for tropical–extratropical wave interactions (Lin, et al., 2000; Majda and Biello, 2003; Biello and Majda, 2004);
  • (iii)Use of the test models to design appropriate filters to cope with various observation networks with irregularly spaced sparse observations (Harlim, 2011) and satellite measurements.

Acknowledgements

The authors thank Boualem Khouider, Samuel Stechmann, and Yevgeniy Frenkel for sharing the relevant multicloud model source codes. The research of JH is partially supported by the Office of Naval Research grant N00014-11-1-0310, the NC State start-up fund, and the NC State Faculty Research and Professional Development fund. The research of AJM is partially supported by the National Science Foundation grant DMS-0456713 and the Office of Naval Research grants ONR DRI N00014-10-1-0554 and N00014-11-1-0306.

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