### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Model and data
- 3. Cyclic bred vectors
- 4. Bred vector dimension
- 5. Results
- 6. Conclusions
- Acknowledgements
- References

A coupled ocean–atmosphere dynamical ensemble prediction system is used to study coupled initialization and bred cyclic modes in the case of Tropical Cyclone (TC) Yasi. Ocean initial perturbations are constructed to identify the fastest-growing nonlinear modes in the ocean response to the TC. The ensemble provides a characterization of how initial and evolving dynamical ocean perturbations influence the coupled system through surface fluxes under extreme conditions. Results show how sea-surface temperature perturbations project into atmospheric perturbations of pressure and moisture content within the storm environment. By calculating the local bred vector dimension for ocean-surface velocity, we show that a low-dimensional subspace forms along the track of TC Yasi. The iterative approach to coupled initialization used in this study generates cyclic modes that are embedded on to the dynamics of regions critical to the coupled ocean–atmosphere TC dynamics. The ensemble mean forecasted sea-surface temperature and sea-surface height associated with the ocean response is in better agreement with observations, despite the biases the coupled model inherits from its component models. Both model and observations reveal a twin cold core structure in the ocean wake of TC Yasi.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Model and data
- 3. Cyclic bred vectors
- 4. Bred vector dimension
- 5. Results
- 6. Conclusions
- Acknowledgements
- References

Deterministic forecasts from oceanic and atmospheric modelling systems have inherent error growth when compared to analyses and observations. Models contain implicit bias and error, which comes from uncertainties in initial and boundary (forcing) conditions, parametrizations, representations and dynamical approximations. These uncertainties evolve with the flow and are communicated amongst variables along the natural dynamical pathways of the system. For example, in an ocean–atmosphere coupled model, over an appropriate time-scale, an error in sea-surface temperature (SST) can produce an error in the heat and moisture flux, which can be transferred to a pressure perturbation and subsequently to an error in the wind. This sequence of events leads to errors in the surface wind stress and momentum flux, which in turn can degrade the representation of surface currents, vertical mixing and SST. Some error in SST may also manifest as error in the latent heat flux then in turn error in atmospheric moisture content and precipitation. Errors are subject to the transport and feedback processes that occur in the system and there will be local variation in the sensitivity of the system. Understanding these uncertainties and their pathways is critical for weather, seasonal and climate prediction systems.

We can develop ways of gaining insight into how uncertainty propagates in dynamical systems by employing ensemble prediction methodologies. Running an ensemble of forecasts with optimal perturbations can sample the error covariance subspace and provide error and bias estimates characteristic of the dynamical system. The information gained can be used to quantify the extent to which uncertainty in one component of the system can project onto another. Its also possible to filter the fast-growing, less predictable parts of the forecast by running an ensemble with slightly different perturbations from the initial conditions and then averaging the members (Toth and Kalnay, 1997). Toth and Kalnay (1993) showed that random perturbations with high degrees of freedom cannot be used to properly sample error growth but, rather, perturbations representative of errors in the analysis related to their growing modes are preferred. The bred vectors (BV) of a full nonlinear model provide perturbations that contain information on the natural fastest-growing instabilities in the dynamical system (Yang *et al.*, 2009; Frederiksen *et al.*, 2010; Toth and Kalnay, 1993). The use of BVs has been popular in atmospheric and coupled prediction systems for many years and has positively influenced forecast skill (Toth and Kalnay, 1997). Their use in ocean forecasting has also recently been explored. Yin and Oey (2007) applied BVs to the Loop Current in the Gulf of Mexico to gain a more accurate forecast and diagnose that baroclinic instabilities were responsible for the largest error growth. O'Kane *et al.*(2011) showed that BV growth patterns typically anti-correlate with forecast error in the regions of largest dynamical instability of the East Australian Current. Ensembles based on BVs are useful when considering operational prediction systems because, while it is impossible to accurately sample the probability distribution function, it is only required that we sample the subspace of the growing error modes to accurately estimate analysis error. Apart from potentially improving forecast skill and providing uncertainty guidance, BVs can be used in other applications such as adaptive sampling to aid design of observation networks (Oke and O'Kane, 2010) and estimating flow-dependent error covariances related to specific forecasts and events (Corazza *et al.*, 2003). This becomes increasingly important for prediction systems that target extreme events yet are significantly limited by observation density. The BV ensemble approach should be useful for estimating error growth in coupled tropical cyclone prediction systems where coupling in the boundary layers is relatively tight and at short time-scales. The natural system is dominated by rapid dynamical adjustments. A prediction system that attempts to resolve these adjustments must therefore possess rapid dynamical error growth inherent in the time tendencies of the dynamical system. Trends in prediction systems are to explicitly resolve more of the physics by increasing resolution, thereby reducing dependence on subgrid-scale parametrizations. This means that it will be increasingly important to provide information regarding dynamical uncertainty. At present it is not known how beneficial a BV ensemble approach would be to understanding this in relation to coupled tropical cyclone simulations.

This work provides the opportunity to examine the ocean response and error characteristics of a TC-coupled dynamical prediction system. For this we choose to examine forecasts of TC Yasi, within the time window 31 January 2011 to 2 February, 0000 UTC. This case is useful as satellite observations show a clearly defined ocean response, it represents a severe test of modelling and observation systems and should provide an approximate upper bound on short-timescale error growth in the ocean and atmosphere. Figure 1 shows the bathymetry of the subdomain of the study area and the temporal mean (AVG) and standard deviations (SDN) of mean sea-level pressure (MSLP) and SST in the coupled control 72 h forecast. Also shown is the covariance of SST and heat flux, which provides an example of air–sea interaction variability. MSLP SDN illustrates the storm track, its extent and time variability during the forecast. SST SDN shows variability in surface temperature induced by the passage of the TC based on the initial control simulation. The coupled forecast contains implicit error and bias as it inherits this from the component models and also communicates this between the models through the surface fluxes. From the outset, the atmospheric model in this case is biased to a less than observed intensity tropical cyclone and also develops error in track (<100 km) over the forecast period. The ocean analysis for initial conditions is biased with underrepresented atmospheric fluxes, meaning that actual surface conditions may be cooler than analysed. Although we provide simple cross-validation of our simulations with observations, much of the interpretation we provide from the ensemble experiments is framed in terms of ‘perfect model’ scenarios.

### 2. Model and data

- Top of page
- Abstract
- 1. Introduction
- 2. Model and data
- 3. Cyclic bred vectors
- 4. Bred vector dimension
- 5. Results
- 6. Conclusions
- Acknowledgements
- References

We employ the Bureau of Meteorology's coupled limited area model (CLAM), which is based on the UK Met Office Unified Model (UM) version 6.4 (Davies *et al.*, 2005), the Ocean Atmosphere Sea Ice Soil coupler version 4 (OASIS4) (Redler *et al.*, 2010) and MOM4p1 (Griffies, 2009). CLAM is integrated into operational national weather and ocean prediction systems run at the National Meteorological and Oceanographic Centre (NMOC). The atmospheric prediction system is known as ACCESS-TC, which incorporates both real and synthetic observations for vortex initialization using four-dimensional variational data assimilation (4DVAR), which was developed for the UM by Rawlins *et al.*(2007). The ocean prediction system is known as OceanMAPS (Brassington *et al.*, 2007), which uses an ensemble optimal interpolation scheme called BODAS (Bluelink Ocean Data Assimilation System) (Oke *et al.*, 2008). The ocean model system includes adaptive initialization (Sandery *et al.*, 2011) and the GOTM *κ*-*ε* vertical mixing scheme. The ocean initialization period is the first 12 h of the forecast, where surface fluxes are linearly increased from zero to actual values to minimize inertial oscillations. CLAM sea-surface height, temperature, salinity and currents are initialized to the OceanMAPS analysis, while adjustments to the fluxes are also permitted. Surface heat and momentum fluxes between the ocean and atmospheric models are also coupled during the initialization and throughout the simulations at relatively high frequency (180 s). In this work we focus on the same 72 h period from the forecast base date of 31 January 2011, 0000 UTC. This period essentially contains the evolution of a well-developed TC through to landfall and subsequent decay. The region of the limited-area model covers an approximate area of 9 × 10^{6} km^{2} at the surface, providing spatial extent that can influence the atmospheric large-scale environmental flow in the forecast.

The ocean analysis (OceanMAPS) has underrepresentation of the ocean response to the TC, due to it being forced by a global, relatively coarse resolution (∼80 km) atmospheric model (ACCESS-G). One of the improvements for ocean prediction offered in the CLAM system is a better representation of the surface fluxes provided by the higher resolution (∼12 km) ACCESS-TC, which also incorporates 4DVAR and vortex initialization. This typically leads to better agreement in the simulated ocean response to the TC (Sandery *et al.*, 2010). The present simulations are not an exception to this. Therefore, rather than comparing to the analysis, we choose to explore the differences between the ensemble method and the control forecast in direct comparison to observations of SSH and SST from BODAS. It should be noted that information from observations used in BODAS enter CLAM through the OceanMAPS initial and boundary conditions. These are a combination of ARGO profiles, XBTs from the ships of opportunity program (SOOP), AMSR-E SST and Jason-1, Jason-2 and ENVISAT altimetric sea-surface height. We also present three GHRSST SST daily analysis products for comparison, namely NASA JPL Our-Ocean 1 km resolution daily SST analysis (Chao *et al.*, 2009), the UK Met Office OSTIA 12 km daily SST analysis (Donlon *et al.*, 2012) and the Bureau of Meteorology Regional Multi Sensor SST 12 km Analysis (RAMSSA) (Beggs *et al.*, 2011). For a more detailed description and intercomparison of the GHRSST products used in this study the reader is referred to Dash *et al.*(2012).

### 3. Cyclic bred vectors

- Top of page
- Abstract
- 1. Introduction
- 2. Model and data
- 3. Cyclic bred vectors
- 4. Bred vector dimension
- 5. Results
- 6. Conclusions
- Acknowledgements
- References

The cyclic bred vector approach has been examined in a number of previous studies (e.g. Frederiksen, 1997). Wei and Frederiksen (2005) first applied this to ensemble initialization in an examination of error growth during Southern Hemisphere mid-latitude blocking. More recently, this approach has been used to generate initial perturbations in ensemble seasonal prediction forecasts of the 1997 El Niño–1998 La Niña regime transitions (Frederiksen *et al.*, 2010). They characterized these iterated bred vectors as ‘cyclic modes’. As bred vectors may be regarded as the nonlinear generalization of leading Lyapunov vectors, cyclic modes may be regarded as the nonlinear generalization of leading finite-time normal modes. Cyclic modes for initialization have also been investigated in the context of data assimilation in the ‘running in place’ method described by Kalnay and Yang (2010), which was developed to capture the underlying dynamics in a spin-up of an EnKF and to improve the Gaussianity of the initial ensemble spread.

Forecasts initializing with cyclic bred ocean perturbations are generated within the coupled model. The ensemble is constructed first by running a control forecast (*CONTROL*) and then an initial perturbation vector (*PV* ) formed from an uncorrelated random perturbation of ocean temperature at all depths, representing an injection of subgrid-scale turbulence (Figure 2). The initial amplitude of the random perturbation is determined empirically to represent analysis error. There is a 12 h initialization period for both *CONTROL* and *PV*, followed by 60 h of free adjustment. At the rescaling period (72 h) the differences in state between the *PV* and *CONTROL* are calculated, rescaled and initialized into the analysis at time zero. This sequence repeats a further *n* times, providing *n* bred cyclic members for the same forecast period of interest. In our case we run *n* = 8 iterations. The rescale factor is taken to be equal to the ratio of initial root mean square (RMS) to evolved RMS anomaly using an L2 Euclidean norm defined as the RMS of thermocline temperatures at 254 m depth (hereafter notated as T254m) as in O'Kane *et al.*(2011). Rescaling is applied to the ocean model state variables sea level (*η*), temperature (*T*), salinity (*S*) and the zonal and meridional components of the velocity field (*u,v*). The growth factor is calculated as the inverse of the rescale factor. A third diagnostic is also used—the characteristic Lyapunov exponent—which is a quantity that characterizes the rate of separation of infinitesimally close trajectories in a dynamical system. This is calculated as the log of the growth factor. BV growth structures have two main degrees of freedom: rescaling period and amplitude of initial random perturbation. The rescaling period captures modes either growing or saturating within the time-scale of that period. We choose to examine the rescale period of 72 h, which provides an example of error growth over a typical TC forecast period. To measure the affect of the choice of initial perturbation amplitudes, three different ensembles are run with initial perturbation amplitudes of 0.1, 0.25 and 0.5 K respectively. The 0.1 K ensemble, however, is the basis for our results. An adaptive initialization scheme (Sandery *et al.*, 2011) is used to introduce all perturbations into the model. This tends to maximize the memory of the perturbation in the system and manages dynamical stability.

### 4. Bred vector dimension

- Top of page
- Abstract
- 1. Introduction
- 2. Model and data
- 3. Cyclic bred vectors
- 4. Bred vector dimension
- 5. Results
- 6. Conclusions
- Acknowledgements
- References

The bred vector dimension **Ψ** relates directly to the predictability of the system (Patil *et al.*, 2001) and is defined as

- (1)

where *σ*_{l} are the singular values of the *N* ×*L* matrix constructed from *L* column vectors and *N*/2 the number of horizontal grid points corresponding to a suitably determined local region. The *u* and *v* velocities of the *l*th local bred vector at the nearest *N*/2 horizontal points from the elements of the *l*th column vector are used. Prior to calculating **Ψ** the *u* and *v* velocities of length *N*/2 are normalized in two ways. The *u* velocities are rescaled such that both *u* and *v* have the same mean square. These two vectors are then combined to form a vector of length *N*, which is normalized to unit length. This process is repeated for all *L* local bred vectors, which is eight in our case. To determine the effective dimensionality of the subspace spanned by the local bred vectors we first form an *N* ×*L* matrix *B* and find the *L* ×*L* covariance matrix *C* =*B*^{T} *B*. The *σ*_{l} are calculated as the singular values of *B*, where are the eigenvalues of *C*. As each column of *B* has unit length

- (2)

low values of **Ψ** determine regions where the local bred vectors *B* effectively span a subspace of significantly lower dimension than that of the full space.

In the coupled TC case there are relatively fast coupled modes in the atmosphere and ocean; the ocean response over short time-scales (hours to days) can influence the intensity of the storm. This is different from other coupled dynamical systems (Peña and Kalnay, 2004), where coupled modes can have different time-scales in the ocean and atmosphere. The ensemble, in our system, produces a coupled initialization because in each member the atmosphere is allowed to dynamically respond to the bred initial ocean conditions. Rescaling the ocean only in the coupled system is a coupled initialization technique; however, this approach will breed ocean initial conditions naturally biased to the atmospheric forcing. This method may be useful in the context of coupled tropical cyclone forecasting because, at various stages during the life cycle of the storm, the ocean initial condition should account for the evolving response to the storm.

### 5. Results

- Top of page
- Abstract
- 1. Introduction
- 2. Model and data
- 3. Cyclic bred vectors
- 4. Bred vector dimension
- 5. Results
- 6. Conclusions
- Acknowledgements
- References

We choose to analyse a subset (∼6° in from each boundary) of the horizontal model domain corresponding to where the TC and ocean variability are largest and it is noted that the solutions here, over relatively short time-scales, have no significant influence from the open boundary conditions. The results from the control forecast for SST, SSH and surface velocity, for the hour at the end of the initialization period (12 h) and for the last hour (72 h) of the forecast, are shown in Figure 3. SSH has the spatial mean subtracted. Although pre-storm SSTs are several degrees warmer than the 26°C threshold for support of TCs, negative SSH indicates that the underlying heat content is relatively low. After the passage of the TC, a significant cool wake develops that also exposes some of the pre-existing underlying cold cores at the surface. The passage of the TC appears to reorganize the low SST anomalies and intensify a cold-core eddy around 15°S, 155°E. In the simulation, surface velocities evolve into mesoscale structures, influenced by inertial oscillations (not shown) that comprise part of a larger surface-layer disturbance induced by the storm.

The character of the ensemble is illustrated with the ensemble mean state and variability presented in Figures 4 and 5, respectively. In these plots we show ensemble SST, SSH and surface velocity at 12 h and 72 h as in Figure 3. Ensemble SST at 12 h shows that the bred vectors embed information about future error growth into the initial conditions of the subsequent forecast. This is evident in the three anomalously cool areas in the wake region (< ∼0.5 K). Ensemble average SSH and surface velocity, compared to the control, in the initialization period have already adjusted to the future surface fluxes and areas of dynamical growth. As with the control forecast, the ensemble average SST shows the existence of at least two cold core areas in the wake, with a possible third emerging further west. The central (153°E, 16.5°*S*) and eastern (155°E, 15.75°*S*) are the largest and are not exactly collocated with the initial perturbations but appear to be connected to westward advection of the initial instabilities. As with the ocean response to most TCs in the Northern (Southern) Hemisphere, the maximum cooling occurs on the centre right (left) side of the track due to a combination of surface divergence and inertial resonance (Bender and Ginis, 2000). The relatively low SSH associated with this surface divergence and subsurface cooling appear to be geostrophically balanced. It can be seen that individual bred vector forecasts create relatively small yet structured perturbations to the variables shown. Later it will be shown how these relatively small perturbations project into the atmosphere.

In Figure 6 the diagnostic Lyapunov exponents, growth rates and rescale factors *R* are shown. The normalized rescale factors *R*_{NORM} are calculated relative to the ensemble and time averaged *R* according to

- (3)

where *R*_{i}(*t*_{o}) is the initial value at time *t*_{o}*,R*_{e} is the evolved value at each model time step, the sum is over perturbations (iterations) and the overline denotes the time mean. *L* is the number of ensemble members and *l* is the ensemble member. Equation (3) is applied to each of the diagnostics. In Figure 6 we consider the time evolution of the relative rescale factor (top row), growth factor (middle row) and characteristic Lyapunov exponent (bottom row) for each cyclic perturbation of the evolving bred vectors and for each choice of initial rescaling amplitude, 0.1 K (left), 0.25 K (middle) and 0.5 K (right) at T254m. The initial perturbation evolution from an isotropic initial disturbance displays an entirely positive rescaling and negative growth over the 3-day period. This corresponds to the organization of nascent error structures that the system at the same time actively tries to damp. In all subsequent cycles we see that once the coherent error structures are formed they are able to grow, after some transient period of less than 1 day, as evident in the increasingly positive values in all three diagnostics. In all diagnostics, but most prominently growth factor and characteristic Lyapunov exponent, we observe clear oscillations in the relative growth rates and Lyapunov exponents in the evolution of cyclic modes. Frederiksen and Branstator (2001) observed similar behaviour in single propagating finite time normal modes (FTNM) in the Northern Hemisphere 300 hPa stream function from large-scale barotropic modes on seasonal and intra-seasonal time-scales. They showed that this type of oscillatory behaviour occurs because the growing cyclic mode (or FTNM) disturbances have both eastward and westward propagating components that can interfere constructively and destructively. These interference effects manifest as bursts of relative growth and decay evident in Figure 6. These effects are strongest in the final cyclic modes, which correspond to the fastest-growing nonlinearly modified FTNMs. In the experiments, the spatial variation of the mean state and its variability in the ensemble perturbation vectors for SST display little variation when changing initial random temperature perturbation amplitudes to 0.1, 0.25 and 0.5 K.

In Figure 7 we calculate the spatial pattern of the local bred vector dimension *ψ* and see from the ocean surface velocity that a low-dimensional subspace forms along the track of TC Yasi. Thus our iterative approach to coupled initialization generates cyclic modes that are embedded onto the dynamics in the region identified by low values in *ψ* (Figure 7). The regions of low *ψ* correspond to the local low-dimensional behaviour in the coupled ocean–atmosphere TC dynamics.

The ensemble average and standard deviation SST bred vectors for the ensemble at the beginning (1 h) and end of the forecast are shown in Figure 8. Here we can see that the ocean cool wake in SST that resulted from the 72 h forecast is embedded into the initial ensemble mean state with a bias down to −0.5 K along the TC track. This is relatively small compared to the actual ocean SST response, which was a cooling of up to 3 K. The SST ensemble standard deviation at the initial time shows the greatest variability in the initial conditions occurs where the leading ocean dynamical disturbances dominate the growth of the cool wake; however, the spread of up to 0.2 K appears to be relatively small. The colder initial conditions in the ensemble members leads to a less intense TC, weaker winds and less vertical mixing, so by the end of the forecast average SSTs in the bred vectors are warmer than the control forecast by up to 0.75 K. We can see that the time mean of the ensemble mean SST cyclic bred vectors would appear to cancel out the extreme parts of the signal, leaving behind a residual in the wake area. The ensemble variability in SST tends to be weaker and dominated by ocean currents and inertial oscillations after the passage of the storm at the end of the forecast. Ensemble average and standard deviation bred perturbation temperature sections in Figure 8 show that bred perturbations in the temperature field are largest below the surface mixed layer and generally within the 100–500 m depth range. This result shows generally that the time-evolving bred perturbations, within the most unstable region of the domain, have spatial coherence and the direction of growth is the same for most of the instabilities. The ensemble average temperature bred vector in the ensemble shows a cool bias in the upper ∼50 m and a warm bias in a layer of about the same thickness between ∼50 and 100 m. This result describes a dynamically consistent inherent model bias. TCs tend to drive deep vertical mixing that cools the surface layer and warms the base of the mixed layer. The BVs shown in Figure 8 show that this process occurs in the model and that model errors grow strongly along these pathways of the system.

In our coupled ensemble experiments we find that ocean perturbations have an impact on the atmosphere. Ensemble variability in the atmospheric model is directly related to perturbations originating from the ocean and coupled mode variability. This makes ocean-only bred vectors a diagnostic capable of capturing the variability of the atmospheric state directly induced by ocean perturbations in SST. The integrated effect in the atmospheric model can be seen in the instantaneous ensemble means and standard deviations of MSLP at 6-hourly intervals shown in Figure 9. Ensemble MSLP perturbations closely resemble variations of storm structure rather than intensity. Some variability may be due to phase differences; however, some appears to be directly related to asymmetric localized features surrounding the eye-wall region. Specific humidity at 4500 m height above the surface is shown in Figure 10 as ensemble standard deviations in successive 12-hourly time averages. Ensemble variability is relatively low during the initialization period, where it appears that only the inner storm structure at this height above the surface is perturbed during the initialization. The ensemble variability grows throughout the forecast into a relatively detailed pattern reflecting the projection of ocean perturbations into mid-troposphere moisture content. Ocean perturbations influence both the large-scale environment, leading to phase differences in storm translation, and air–sea fluxes that seem to have an influence at smaller spatial scales, i.e. the intensity of individual fronts and convective systems. Strong air–sea interaction in the TC scenario means that relatively fast-coupled interacting modes are present. Due to this, the breeding of the ocean state over the initialization and forecast period can be thought of as providing a coupled initialization for the ocean in response to the atmosphere. The bred vector for MSLP (not shown) shows a weak positive bias along the track and is consistent with the cool bias in the SST perturbation vector.

At the short time-scale of the forecast, and under TC conditions where retrievals are difficult due to sensor issues, there is usually a limited amount of ocean observations to draw on to help understand the ocean response and the model performance in resolving this. Nonetheless, a reasonable amount of SST and SSH information is available for this case. In Figures 11–13 observations and model comparisons are presented. The first dataset of observations we focus on here was captured by the operational ocean state estimation from BODAS at the Bureau of Meteorology. Figure 11 shows super-observations from BODAS (observations up-scaled to reduce representativeness error) (Oke and Sakov, 2008) for SSH and SST for the 2 February 2011 obtained for the last day of the forecast. Also shown in Figure 11 are the corresponding *CONTROL* forecast and the ensemble time-averaged data for the same period and locations. Figure 12 shows scatter-plots of the same data for SST and SSH for 2 February 2011 collocated with the *CONTROL* and the ensemble averaged data. There is a slight decrease in error in sea-surface height from *CONTROL* to the ensemble. The decrease in error is gained due to an improved initialization of the ocean to the storm (see Figure 11 at approximately 13.5°*S*, 155°E). The overall statistics for SSH, however, indicate that error with respect to the observation system for this variable in this environment is close to saturation. The representativeness error occurring in this environment can be high as the ocean model rapidly adjusts to the TC with large barotropic and baroclinic responses. In these results, the ensemble provides a less biased, more accurate representation of SSTs compared to *CONTROL*. There is an overall reduction of the mean bias in SST from 0.43 K in *CONTROL* to 0.36 K. The cool wakes in both *CONTROL* and the ensemble are biased warm compared to observations, which is consistent with the model TC being biased towards weaker than observed intensity. Figure 13 compares a 6-day composite of AVHRR imagery and three GHRSST foundation SST daily analysis products, namely NASA JPL Our-Ocean 1 km resolution daily SST analysis, the UK Met Office OSTIA 0.05° daily SST analysis product and the Bureau of Meteorology Regional Multi Sensor SST 12 km Analysis (RAMSSA) for 3 February 2011 with model control and ensemble foundation SST averaged over the last 24 h of the forecast. The differences in these fields demonstrates that resolving the TC wake in an SST analysis framework is a severe test of observation systems and analysis methods. The OSTIA product is the smoothest analysis due to larger correlation length scales. The G1 product appears to be the noisiest. RAMSSA also appears somewhat noisy in this case. Nonetheless, some of this information suggests the ocean response was characterized by the development of twin cold cores, which is in agreement with features brought out clearly in *CONTROL* and the ensemble.

In practice, coupled initialization of an ocean model to a TC may be done over shorter time-scales in order to create initial conditions more appropriate for the forecast. We carried out an experiment running the same ensemble as shown in Figure 2 with a shortened cycle period of 24 h. The results for 12 h time mean surface currents in the *CONTROL* forecast and in the ensemble mean for this experiment are shown in Figure 14. Note that for both *CONTROL* and the cyclic members the TC fluxes are linearly increased from zero to full strength to minimize inertial oscillations during the 12 h initialization period. This result clearly shows that the cyclic bred vectors initialize the surface currents in the ocean model to the TC fluxes in a manner more consistent with the TC than the *CONTROL* forecast.

### 6. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Model and data
- 3. Cyclic bred vectors
- 4. Bred vector dimension
- 5. Results
- 6. Conclusions
- Acknowledgements
- References

The results thus far demonstrate that cyclic bred vectors can be used as a coupled initialization approach. An extreme weather event and a 72 h forecast period were used to illustrate this. In this article, we presented the application of ensemble-based cyclic bred vectors to coupled TC forecasting. Small injections of subgrid scale turbulence in the ocean thermal state, regardless of the initial random distribution, were found to initiate finite-time normal modes that invariably lead to cool bias in the SST perturbation vector initial conditions at the surface and warm biases at the base of the mixed layer. The cyclic bred vectors capture coupled modes through account of the air–sea interaction allowed for in the model. The cooler initial conditions by <0.5 K in the ensemble members fed back into a less intense TC of the order of +5 hPa minimum central pressure with weaker winds driving less vertical mixing. This then allowed for warmer SSTs to evolve in the wake of the order of 0.25–0.5 K with respect to the control by the end of the forecast and a small intensification in the atmospheric state BV where minimum surface pressure was lowered by several hPa. Results show how error growth within this type of forecast can be calculated and uncertainty measured with a relatively inexpensive ensemble based on cyclic bred vectors. We found that ensemble spread at the surface of the ocean was far more convergent over time than at depth in the model. In this environment, surface fluxes dominate the ocean response, yet baroclinic instabilities, resulting from perturbations of water masses, dominate the growth (tendencies) of the wake. Coupling exists between these dynamical regimes. Our ensemble calculates perturbations that are relatively small changes in SST (<0.5 K) yet larger at depths between approximately 100 and 500 m (<2.0 K). Ensemble spread in the bred vector temperature sections showed areas in the domain possessing the largest growth, i.e. the areas most sensitive to the initial and subsequent perturbations.

We have presented a method that identifies errors and biases of the forecast that could be used in a hybrid data assimilation system. The ensemble time mean from this approach represents a nonlinearly filtered ocean initial state that has memory of the atmospheric state. We calculated the local bred vector dimension for ocean surface velocity and surface winds to show that a low-dimensional subspace forms along the track of TC Yasi. The iterative approach to coupled initialization used in this study generates cyclic finite-time normal modes that are embedded onto the dynamics of regions where there is local low-dimensional behaviour in the coupled ocean–atmosphere TC dynamics. This method may be used to initialize coupled ocean models to TC fluxes at any time within the forecast. The ensemble mean SST and sea-surface height, associated with the ocean response, were in better agreement with observations than the control forecast, despite the biases the coupled model inherited from its component models. This work also shows that information about future error growth can be gained from the cyclic bred vector ensemble as they capture and embed information about the dominant fastest-growing error modes into the initial conditions. By perturbing the ocean only, we were also able to sample how analysis error in the ocean state projects into the atmosphere. The coherent SST perturbations were found to lead to relatively small atmospheric perturbations that also projected into the areas of fastest-growing atmospheric instabilities.

The information gained from this study provides a rich example of coupled model sensitivity and the details of it. Ensemble spread in the atmospheric model for TC track and intensity is small; however, details of individual convection cells within the TC environment differ. Spread in mid-level tropospheric humidity and mean sea-level pressure shows coherent structures and consistent growth over the forecast period. The differences in atmospheric states are driven by small SST perturbations within ±0.5 K that force small changes in storm translation speed, direction and intensity. This study provides examples of length, time and nonlinear error growth scales for the TC case that will be subject to variation from case to case.