cBottle: Generative Climate Emulator

• cBottle is a conditional generative climate emulator that uses a two-stage HEALPix-based cascaded diffusion architecture to produce realistic atmospheric states.
• It employs a coarse global generator and a super-resolution stage to enable tasks like downscaling, bias correction, channel in-filling, and sensitivity analysis.
• The framework supports probabilistic analysis of extreme events and differentiable sensitivity studies by integrating SST and calendar conditioning with efficient gradient computations.

Searching arXiv for the specified cBottle papers and closely related work to ground the article. Climate in a Bottle (cBottle) is a generative diffusion-based framework for emulating global atmospheric states from climate simulation and reanalysis data, with an explicit emphasis on conditioning on sea surface temperatures (SSTs) and solar-time information. It is presented as a route toward a generative foundation model for the kilometer-scale global atmosphere, combining a coarse global generator with a super-resolution stage, operating on the equal-area HEALPix sphere, and supporting downstream tasks including downscaling, bias correction, channel in-filling, sensitivity analysis, and probabilistic analysis of extreme events (Brenowitz et al., 10 May 2025).

1. Definition and conceptual scope

In its primary formulation, cBottle is a conditionally generative climate emulator rather than an auto-regressive forecast model. The framework is designed to synthesize atmospheric states that are compatible with prescribed boundary conditions, especially monthly average SSTs, day of year, and hour of day, while remaining flexible across data modalities such as reanalysis and high-resolution simulation (Brenowitz et al., 10 May 2025). This places cBottle in a class of climate emulators aimed at compressing and interacting with petabyte-scale climate datasets through learned generative structure rather than through storage of explicit trajectories.

The name also aligns with a broader compression-and-emulation philosophy. A statistical precursor appears in “Compression and Conditional Emulation of Climate Model Output” (Guinness et al., 2016), which represents a large climate field through stored summary statistics $S(Y)$ together with a conditional model $f(Y \mid S(Y))$. In that earlier framework, decompression is performed through either the conditional expectation $E(Y \mid S(Y))$ or a conditional draw from $f(Y \mid S(Y))$, with uncertainty quantification arising from conditional simulation rather than exact recovery (Guinness et al., 2016). This suggests that cBottle can be understood not only as a specific diffusion architecture, but also as part of a longer methodological trajectory in which climate data are represented compactly and regenerated probabilistically.

A concise chronology of the cBottle literature covered here is as follows.

Paper	Main role	arXiv id
“Climate in a Bottle: Towards a Generative Foundation Model for the Kilometer-Scale Global Atmosphere”	Core cBottle framework	(Brenowitz et al., 10 May 2025)
“Sensitivity Analysis for Climate Science with Generative Flow Models”	Adjoint gradients for SST sensitivity using cBottle	(Dobra et al., 1 Nov 2025)
“Towards accurate extreme event likelihoods from diffusion model climate emulators”	PF-ODE likelihoods, guidance, and importance sampling with cBottle	(Manshausen et al., 5 May 2026)
“Compression and Conditional Emulation of Climate Model Output”	Statistical antecedent for summary-plus-emulator compression	(Guinness et al., 2016)

Within the cBottle literature itself, the framework is not limited to unconditional generation. It is explicitly used as an emulator of ERA5-like atmospheric states, as a coarse-to-fine generator of kilometer-scale global fields, as a differentiable model for sensitivity analysis, and as a probabilistic model whose learned density can be interrogated through the probability-flow ODE (Brenowitz et al., 10 May 2025).

2. Architectural design on the sphere

The core cBottle system uses a two-stage cascaded diffusion design. The first stage is a globally trained coarse-resolution image generator at HPX64, corresponding to roughly 100 km resolution and about 49,152 pixels on the sphere. The second stage is a locally trained 16× super-resolution model that upsamples from HPX64 to HPX1024, corresponding to about 5–6 km resolution and 12,582,912 pixels (Brenowitz et al., 10 May 2025). The two stages can be used independently: the coarse model can generate global climate fields directly, the super-resolution model can refine existing coarse inputs, and the full cascade can generate high-resolution atmospheric samples end to end.

A major technical choice is the use of HEALPix as the spherical discretization. HEALPix provides equal-area pixels, iso-latitude structure, and hierarchical multi-resolution indexing, avoiding the distortions of a regular latitude-longitude grid in global modeling (Brenowitz et al., 10 May 2025). For coarse global modeling, cBottle processes the 12 HEALPix faces with face-wise convolutions and explicit padding:

$$x = \left( x_1, x_2, \cdots x_{12} \right), \quad x_f \in \mathbb{R}^{c_{in} \times 2^l\times 2^l}$$

$$x^p = \mathcal{P}\left(x; k\right) = \left( x^p_1, x^p_2, \cdots x^p_{12} \right), \quad x^p_f \in \mathbb{R}^{c_{in} \times (2^l+2k)\times (2^l+2k)}$$

$$y_f = \mathcal{K}_{conv} \ast x^p_f, \quad y_f \in \mathbb{R}^{c_{out} \times 2^l\times 2^l}$$

$$y = \left( y_1, y_2, \cdots y_{12} \right), \quad y \in \mathbb{R}^{c_{out} \times 12 \times 2^l\times 2^l}.$$

The diffusion backbone is a Song-UNet-style denoising architecture modified for HEALPix-aware convolutions and padding, learned position embeddings, and calendar embeddings; the video variant additionally introduces temporal self-attention (Brenowitz et al., 10 May 2025). The model variants are cBottle-3d, cBottle-video, and cBottle-SR. The coarse and video models are trained globally on HPX64, while the super-resolution model is trained on 128×128 patches extracted from HPX1024 data (Brenowitz et al., 10 May 2025).

The super-resolution stage addresses a specific scalability constraint. A naive full global HPX1024 diffusion model would require over 2,000 GB of GPU memory. cBottle instead uses overlapping patch-based multi-diffusion, defining the global denoiser as an overlap-averaged combination of local denoisers:

$$\mathcal{D}_\theta(x_t) = \frac{ \sum_i \mathcal{F}_i^{-1}\left( \Phi_\theta \left( \mathcal{F}_i(x_t) \,\middle|\, \mathcal{F}_i(y), \sigma \right) \right) }{ \sum_i \mathcal{F}_i^{-1}(J_N) }.$$

The HEALPix version employs padded patch extraction,

$$\mathcal{G}_i(x_t) = \mathcal{F}_i(\mathcal{P}(x_t; N/2)),$$

with overlap used to suppress patch-boundary seams (Brenowitz et al., 10 May 2025). This is a practical systems contribution as much as a modeling one, because it makes global kilometer-scale generative super-resolution computationally feasible.

3. Training data, variables, and conditioning structure

cBottle is trained on two large global datasets: ERA5 reanalysis and ICON Cycle 3 km-scale simulation (Brenowitz et al., 10 May 2025). ERA5 contributes observationally constrained realism and is used as hourly data from 1980–2018, with training on 1980–2017 and validation on 2018 after regridding from latitude-longitude to HEALPix. ICON contributes storm-resolving simulated structure, with roughly five years of data, using 3-hourly 3D fields and 30-minute 2D fields, training on 2020-01-20 to 2024-03-06 and validation on 2024-03-06 to 2025-07-22 (Brenowitz et al., 10 May 2025).

The framework is multimodal. The main paper lists 2D surface and radiative variables such as precipitation, 2 m temperature, winds, sea ice concentration, cloud water/ice, outgoing longwave radiation, outgoing shortwave radiation, and downwelling shortwave radiation, together with 3D pressure-level variables $f(Y \mid S(Y))$0 at multiple pressure levels (Brenowitz et al., 10 May 2025). In the sensitivity-analysis paper, cBottle is described as generating 45 atmospheric variables at about 100 km resolution, including profile variables such as temperature, zonal and meridional wind, and geopotential height; column variables such as cloud water/ice and outgoing longwave/shortwave radiation; and surface variables such as precipitation, sea level pressure, sea ice concentration, SST, 2 m temperature, and near-surface winds (Dobra et al., 1 Nov 2025). In the extreme-event likelihood study, the coarse generator configuration is described as producing full atmospheric states including winds, temperature, humidity, and geopotential on eight pressure levels at HPX64 resolution (Manshausen et al., 5 May 2026).

Conditioning is central to the scientific design. The coarse generator is conditioned on day of year, hour of day, monthly-averaged SST, and a dataset label indicating ERA5 or ICON (Brenowitz et al., 10 May 2025). The time embeddings are periodic:

$f(Y \mid S(Y))$1

$f(Y \mid S(Y))$2

where $f(Y \mid S(Y))$3 is local solar time in radians (Brenowitz et al., 10 May 2025). In the sensitivity-analysis use case, the conditioning variables are written as day of year $f(Y \mid S(Y))$4, second of day $f(Y \mid S(Y))$5, and SST forcing $f(Y \mid S(Y))$6, with experiments typically varying $f(Y \mid S(Y))$7 while keeping $f(Y \mid S(Y))$8, $f(Y \mid S(Y))$9, and the latent noise $E(Y \mid S(Y))$0 fixed (Dobra et al., 1 Nov 2025).

The diffusion training objective is the standard denoising score-matching loss:

$E(Y \mid S(Y))$1

with the denoiser approximating the score by

$E(Y \mid S(Y))$2

Sampling uses the Elucidated Diffusion Model sampler with $E(Y \mid S(Y))$3 (Brenowitz et al., 10 May 2025).

A distinctive methodological point is the use of a much wider noise range than is typical in natural-image diffusion models. The paper identifies a failure mode termed signal leak bias: if the maximum noise level is too small, the model can exploit residual signal in the noisy input rather than learning the intended conditioning relationship. To mitigate this, cBottle uses a log-uniform noise schedule,

$E(Y \mid S(Y))$4

with settings such as $E(Y \mid S(Y))$5 for the coarse model, $E(Y \mid S(Y))$6 for the video model, and a different log-normal schedule for super-resolution with inference $E(Y \mid S(Y))$7 (Brenowitz et al., 10 May 2025).

4. Scientific behavior and downstream uses

The principal cBottle paper evaluates the model across a wide battery of climate diagnostics. These include diurnal-to-seasonal variability, large-scale modes of variability, tropical cyclone statistics, trends of climate change, weather extremes, heavy-tailed precipitation distributions, and high-wavenumber power spectra (Brenowitz et al., 10 May 2025). The reported tests include seasonal sea ice concentration, seasonal cycles of temperature, precipitation, and radiation, the diurnal cycle of surface precipitation and temperature, the Northern Annular Mode / Arctic Oscillation, ENSO / Nino3.4-related precipitation patterns, and spatio-temporal coherence in video mode through lagged correlation maps and regression slope maps of mean sea level pressure (Brenowitz et al., 10 May 2025).

These evaluations are designed to test different aspects of generative climate fidelity. Diurnal and seasonal diagnostics test whether the calendar conditioning is used correctly. EOF structure for the Northern Annular Mode and teleconnections tied to Nino3.4 test planetary-scale organization rather than local texture alone. Tropical-cyclone diagnostics test whether the generator places organized vortices in plausible regions with plausible occurrence rates. Trend and exceedance diagnostics probe whether the model reproduces secular or tail-related signals rather than only mean climatology (Brenowitz et al., 10 May 2025).

The framework is also presented as a foundation-model-like system because it supports tasks beyond forward generation. Zero-shot bias correction is used to translate between datasets without training a separate pairwise model for each source-target pair. Zero-shot downscaling uses the super-resolution stage to refine coarse ERA5 fields into finer-scale structure resembling ICON-like variability. Channel in-filling treats missing variables as a conditional generation problem; the paper specifically demonstrates in-filling of outgoing shortwave radiation, outgoing longwave radiation, and surface downwelling shortwave radiation, including mitigation of stripe artifacts in ERA5 radiation fields (Brenowitz et al., 10 May 2025).

The channel-filling formulation is expressed through a conditional score approximation,

$E(Y \mid S(Y))$8

where observed and unobserved channels are separated by slicing operators (Brenowitz et al., 10 May 2025). This is one of the clearest indications that cBottle is intended as a general conditional model over atmospheric state spaces rather than only a sampler of complete fields.

A recurring scientific theme is that cBottle is conditioned on boundary information rather than on the full resolved dynamical history. This makes it suitable for “what-if” generation under alternative SSTs and solar conditions, but it also implies that the generated states are best interpreted as samples from a learned conditional climate distribution rather than as deterministic forecasts (Brenowitz et al., 10 May 2025).

5. Differentiable sensitivity analysis with SST forcing

A major extension of cBottle is its use for sensitivity analysis in climate science. “Sensitivity Analysis for Climate Science with Generative Flow Models” formulates cBottle as a flow model in the probability-flow ODE form of an EDM-type diffusion model:

$E(Y \mid S(Y))$9

with generation defined by integrating backward from $f(Y \mid S(Y))$0 to $f(Y \mid S(Y))$1 to obtain $f(Y \mid S(Y))$2 (Dobra et al., 1 Nov 2025). In this setting, cBottle acts as an emulator of ERA5-like atmospheric states conditioned on day of year $f(Y \mid S(Y))$3, second of day $f(Y \mid S(Y))$4, and SST forcing $f(Y \mid S(Y))$5.

The scientific objective is to compute gradients of a scalar quantity of interest $f(Y \mid S(Y))$6 with respect to SST forcing:

$f(Y \mid S(Y))$7

Examples of $f(Y \mid S(Y))$8 include globally integrated outgoing radiation, precipitation near the coast of Peru, wind shear in the tropical Pacific, and other derived atmospheric diagnostics (Dobra et al., 1 Nov 2025). These quantities are used to study storm intensity and predictability, radiative feedbacks, the influence of observations, and large-scale responses to warming patterns.

The technical contribution is the use of the adjoint state method to avoid naive backpropagation through all ODE solver steps. The adjoint state is defined as

$f(Y \mid S(Y))$9

and satisfies

$$x = \left( x_1, x_2, \cdots x_{12} \right), \quad x_f \in \mathbb{R}^{c_{in} \times 2^l\times 2^l}$$0

Defining

$$x = \left( x_1, x_2, \cdots x_{12} \right), \quad x_f \in \mathbb{R}^{c_{in} \times 2^l\times 2^l}$$1

the desired gradient is recovered at terminal time:

$$x = \left( x_1, x_2, \cdots x_{12} \right), \quad x_f \in \mathbb{R}^{c_{in} \times 2^l\times 2^l}$$2

The paper summarizes the computation through the coupled system

$$x = \left( x_1, x_2, \cdots x_{12} \right), \quad x_f \in \mathbb{R}^{c_{in} \times 2^l\times 2^l}$$3

integrated from $$x = \left( x_1, x_2, \cdots x_{12} \right), \quad x_f \in \mathbb{R}^{c_{in} \times 2^l\times 2^l}$$4 to $$x = \left( x_1, x_2, \cdots x_{12} \right), \quad x_f \in \mathbb{R}^{c_{in} \times 2^l\times 2^l}$$5, with result $$x = \left( x_1, x_2, \cdots x_{12} \right), \quad x_f \in \mathbb{R}^{c_{in} \times 2^l\times 2^l}$$6 (Dobra et al., 1 Nov 2025).

A central concern is whether these gradients are meaningful rather than merely numerically available. To address this, the paper proposes a gradient self-consistency check. For a small perturbation $$x = \left( x_1, x_2, \cdots x_{12} \right), \quad x_f \in \mathbb{R}^{c_{in} \times 2^l\times 2^l}$$7,

$$x = \left( x_1, x_2, \cdots x_{12} \right), \quad x_f \in \mathbb{R}^{c_{in} \times 2^l\times 2^l}$$8

For full fields, the paper introduces a directional propagation ODE and defines the grad-check relative error

$$x = \left( x_1, x_2, \cdots x_{12} \right), \quad x_f \in \mathbb{R}^{c_{in} \times 2^l\times 2^l}$$9

Using 32 SST pairs separated by one day, the study computes this metric for all 45 variables and reports that qualitative and quantitative checks suggest that cBottle’s gradients are often sensible and self-consistent, with variables higher in the atmosphere often showing lower relative error (Dobra et al., 1 Nov 2025).

The practical significance is the reported reduction in computational cost from weeks on a supercomputer with physical models to hours on a GPU with the generative emulator (Dobra et al., 1 Nov 2025). The paper is explicit, however, that these results provide initial evidence rather than proof of physical correctness, especially for out-of-distribution SST perturbations relevant to climate change. A further subtlety is that cBottle’s SST conditioning is tied to day of year through monthly SST climatology interpolation, so varying $$x^p = \mathcal{P}\left(x; k\right) = \left( x^p_1, x^p_2, \cdots x^p_{12} \right), \quad x^p_f \in \mathbb{R}^{c_{in} \times (2^l+2k)\times (2^l+2k)}$$0 while fixing $$x^p = \mathcal{P}\left(x; k\right) = \left( x^p_1, x^p_2, \cdots x^p_{12} \right), \quad x^p_f \in \mathbb{R}^{c_{in} \times (2^l+2k)\times (2^l+2k)}$$1 can create an inconsistent input scenario; for seasonal analyses, the paper therefore derives a full derivative

$$x^p = \mathcal{P}\left(x; k\right) = \left( x^p_1, x^p_2, \cdots x^p_{12} \right), \quad x^p_f \in \mathbb{R}^{c_{in} \times (2^l+2k)\times (2^l+2k)}$$2

with

$$x^p = \mathcal{P}\left(x; k\right) = \left( x^p_1, x^p_2, \cdots x^p_{12} \right), \quad x^p_f \in \mathbb{R}^{c_{in} \times (2^l+2k)\times (2^l+2k)}$$3

and

$$x^p = \mathcal{P}\left(x; k\right) = \left( x^p_1, x^p_2, \cdots x^p_{12} \right), \quad x^p_f \in \mathbb{R}^{c_{in} \times (2^l+2k)\times (2^l+2k)}$$4

This coupling is a technical reminder that differentiability in a generative emulator does not remove the need for careful conditioning semantics (Dobra et al., 1 Nov 2025).

6. Probability densities, extreme events, and methodological limits

A second major extension treats cBottle not merely as a sampler but as a probabilistic model whose approximate likelihood can be extracted from the probability-flow ODE. In the extreme-event study, cBottle is used in its coarse-generator form at HPX64, conditioned on SSTs and solar position, and equipped with a coarse TC-probability map at HPX8 that enables guidance toward tropical-cyclone states (Manshausen et al., 5 May 2026).

The study reviews the diffusion-model likelihood relation

$$x^p = \mathcal{P}\left(x; k\right) = \left( x^p_1, x^p_2, \cdots x^p_{12} \right), \quad x^p_f \in \mathbb{R}^{c_{in} \times (2^l+2k)\times (2^l+2k)}$$5

where $$x^p = \mathcal{P}\left(x; k\right) = \left( x^p_1, x^p_2, \cdots x^p_{12} \right), \quad x^p_f \in \mathbb{R}^{c_{in} \times (2^l+2k)\times (2^l+2k)}$$6 is the ODE velocity field (Manshausen et al., 5 May 2026). The divergence is estimated with the Skilling–Hutchinson trace estimator,

$$x^p = \mathcal{P}\left(x; k\right) = \left( x^p_1, x^p_2, \cdots x^p_{12} \right), \quad x^p_f \in \mathbb{R}^{c_{in} \times (2^l+2k)\times (2^l+2k)}$$7

The paper stresses that the resulting quantity is the probability under the learned model rather than necessarily the true climate distribution (Manshausen et al., 5 May 2026).

Guided tropical-cyclone generation is written as

$$x^p = \mathcal{P}\left(x; k\right) = \left( x^p_1, x^p_2, \cdots x^p_{12} \right), \quad x^p_f \in \mathbb{R}^{c_{in} \times (2^l+2k)\times (2^l+2k)}$$8

where $$x^p = \mathcal{P}\left(x; k\right) = \left( x^p_1, x^p_2, \cdots x^p_{12} \right), \quad x^p_f \in \mathbb{R}^{c_{in} \times (2^l+2k)\times (2^l+2k)}$$9 is a classifier predicting TC probability, $$y_f = \mathcal{K}_{conv} \ast x^p_f, \quad y_f \in \mathbb{R}^{c_{out} \times 2^l\times 2^l}$$0 is the binary cross-entropy loss, $$y_f = \mathcal{K}_{conv} \ast x^p_f, \quad y_f \in \mathbb{R}^{c_{out} \times 2^l\times 2^l}$$1 is the target condition, and $$y_f = \mathcal{K}_{conv} \ast x^p_f, \quad y_f \in \mathbb{R}^{c_{out} \times 2^l\times 2^l}$$2 is a guidance strength (Manshausen et al., 5 May 2026). To quantify how much more likely a sample becomes under guidance, the paper defines an odds ratio

$$y_f = \mathcal{K}_{conv} \ast x^p_f, \quad y_f \in \mathbb{R}^{c_{out} \times 2^l\times 2^l}$$3

with a log-decomposition

$$y_f = \mathcal{K}_{conv} \ast x^p_f, \quad y_f \in \mathbb{R}^{c_{out} \times 2^l\times 2^l}$$4

This is then used for importance sampling of rare events:

$$y_f = \mathcal{K}_{conv} \ast x^p_f, \quad y_f \in \mathbb{R}^{c_{out} \times 2^l\times 2^l}$$5

Empirically, using 4797 September samples from 1980–2018 boundary conditions, the paper reports that guidance raises TC frequency by about an order of magnitude, that importance sampling brings the estimate back in line with the unguided control, and that for the most extreme threshold the importance-sampled estimate has lower uncertainty than the unguided Monte Carlo estimate (Manshausen et al., 5 May 2026). The paper therefore positions cBottle as useful not only for generating extremes but also for estimating their likelihood under the learned model.

At the same time, the limitations are emphasized clearly. cBottle’s TC probabilities can be biased low relative to ERA5 in some settings; odds ratios can have high variance, especially under strong guidance; the full guided plus odds-ratio procedure is about 33× slower than a single unguided sample in the prototype; classifier guidance is cumbersome because probability estimation requires double differentiation through the classifier; and counterfactual attribution would require additional models trained on pre-industrial SSTs, future SSTs, or masked-region settings for spatially localized analyses (Manshausen et al., 5 May 2026). The Antarctic heatwave case study further shows that cBottle-assigned log probability is not strongly correlated with anomaly magnitude, with probabilities dominated more by seasonal variability than by extremeness per se (Manshausen et al., 5 May 2026). This directly cautions against equating model likelihood with physical extremeness or attribution relevance.

Taken together, these results define the current status of cBottle. It is a global generative climate emulator trained on ERA5 and ICON, built as a HEALPix-based cascaded diffusion model, capable of generating physically plausible atmospheric states conditioned on SST and solar forcing, and extensible to differentiable sensitivity analysis and PF-ODE likelihood calculations (Brenowitz et al., 10 May 2025). The published results indicate promise across climatology, extremes, and downstream conditional tasks, while also delimiting unresolved issues of gradient validity under distribution shift, likelihood calibration for extremes, computational cost, and the interpretation of learned probabilities in physically counterfactual settings (Dobra et al., 1 Nov 2025).