JCM's dynamical core is called Dinosaur and was developed for use in NeuralGCM . The Dinosaur dycore is spectral and uses sigma levels (surfaces at fixed fractions of the surface pressure) for the vertical coordinate. It integrates the hydrostatic equations with a semi-implicit gravity wave scheme in addition to specified physics “terms” (see Sect. ). The dycore's prognostic variables include horizontal wind (vorticity and divergence), temperature (anomaly from a reference temperature), and log surface pressure. Additional prognostic variables can be specified as tracers. In the default configuration of JCM, the only additional tracer is specific humidity. Dinosaur supports a range of horizontal grid resolutions; these are labeled by their spectral truncation, the maximum wavenumber of spherical harmonic they can resolve. Dinosaur's grids use a quadratic truncation, meaning there are 3 grid points per wavelength for the highest resolved wavenumber.

Supported spectral truncations range from T21 (a 64×32 grid) to T425 (a 1280×640 grid), with JCM's default being T31 (96×48, roughly 4°×4° grid cell size at the equator). There are no technical obstacles to adding support for resolutions beyond T425, though other features may be of higher priority for high-resolution studies, such as support for more vertical levels or for a non-spectral dynamical core. (Spectral integration schemes must simulate the entire globe, making them inefficient for region-focused experiments, and the spectral transforms required at each timestep eventually bottleneck performance as they scale worse than linearly with the number of grid points.) Dinosaur is capable of operating with many vertical levels. In JCM we use the 8σ levels expected by the default physics suite (described below). Sigma layer boundaries are at σ values of 0, 0.05, 0.14, 0.26, 0.42, 0.60, 0.77, 0.90 and 1. Prognostic variables other than log surface pressure are specified at the center of the sigma levels (σ=0.025, 0.095, 0.20, 0.34, 0.51, 0.685, 0.835, 0.95).

Time integration in JCM is performed with the IMEX (implicit-explicit) SIL3 (semi-implicit Lorenz three-cycle) Runge-Kutta scheme, which treats the linear terms of the primitive equations implicitly and the nonlinear advection and physics terms explicitly . IMEX SIL3 is second order accurate (discretization error scales with the square of the timestep) for the implicit terms and third order accurate for the explicit terms. The SPEEDY Fortran model uses a leapfrog scheme, but for simplicity we use IMEX SIL3, which does not require keeping track of state at multiple time-steps. The default time step is 30 min, although this can be easily varied at model initialization time. After computation of parameterized tendencies at each time step, horizontal diffusion of the prognostic variables is computed and the surface pressure field is corrected to maintain constant global mean surface pressure (mass conservation).

JCM uses the physical parameterizations from SPEEDY (Simplified Parameterizations, primitivE-Equation DYnamics) version 41, which have been translated from Fortran90 to JAX. SPEEDY is an intermediate-complexity general circulation model developed at the International Centre for Theoretical Physics (ICTP) . SPEEDY represents physical processes with simplified schemes that operate on 8 vertical levels. The top two layers represent the stratosphere, the bottom layer represents the planetary boundary layer, and the intermediate layers represent the free troposphere. SPEEDY has been used in a large number of studies, both on its own as well as in coupled contexts (with many different ocean models). Its lightweight design makes it an efficient sandbox for prototyping numerical schemes or data assimilation methods , looking at long-term climate interactions , or looking at the large-scale impacts of a wide range of climate forcings .

The SPEEDY parameterizations were developed by simplifying more complex physics schemes while retaining underlying physical principles. These simplifying assumptions are specifically intended to work for a model with few (order 10) vertical levels. Here we provide a brief overview of each of the SPEEDY physics parameterizations, and full details are available in the original model description paper . The primary difference between the SPEEDY parameterizations and JCM is the handling of horizontal diffusion and orographic corrections, which is described in detail below. Neither SPEEDY nor JCM v1.1 implements a daily cycle.

Convection. Convection is simulated with a simplified mass-flux scheme based on Tiedtke's 1989 paper . Convection is activated in areas meeting two simultaneous criteria for conditional instability: humidity in the planetary boundary layer (the lowest layer) exceeding a prescribed threshold, and saturation moist static energy decreasing with height between the PBL and upper troposphere (the highest non-stratospheric layer). The cloud-base mass flux (at the top of the PBL) is calculated to relax the PBL humidity to its threshold value on a time scale of 6 h. The air in the updrafts is assumed to be saturated. Entrainment occurs in the lower half of the troposphere, and detrainment occurs only at the cloud-top level (determined by the conditional instability criterion).

Both JCM and speedy.f90 simulations begin at isothermal rest and run for three years. The models are forced by a monthly climatology of sea-surface temperature, surface albedo, land surface temperature, soil moisture, sea ice concentration, and snow cover along with realistic orography. This monthly climatology is obtained from ERA-Interim over the period 1979–2008 . In addition to sharing initial and boundary conditions, both simulations are run at T31 resolution with a time step of 40 min. The first three months of the model runs are discarded as model “spin up”. Figures  through validate the mean state of JCM against the mean state of speedy.f90. Figure  shows surface temperature and specific humidity in both models. The root-mean-square (RMS) difference of the 3-year simulations is shown in the bottom row. As mentioned in Sect. , JCM does not yet have orographic corrections. This leads to slight differences between the two models in temperature and specific humidity over land. The average RMS difference at the surface is 1.8 °C and 0.68 g kg⁻¹.

Figure shows the average meridional (left) and zonal (right) wind in the surface layer. Again the RMS difference is shown in the bottom row. Large-scale patterns in meridional wind are the same between the two models with slight differences in strength, with JCM winds tending to be stronger than in speedy.f90. The zonal winds are generally stronger everywhere in JCM than in speedy.f90, and is most notable in the mid-latitudes. The discrepancy between the zonal wind in speedy.f90 and JCM is likely due to differences in the implementation of the time stepping and horizontal diffusion. The JCM parameterizations have been tested against speedy.f90 to verify that they produce the same output. This does not include horizontal diffusion, which in speedy.f90 is implemented implicitly in coordination with the leap frog time stepping. Although the general formulation of the spectral dynamical cores is the same, they are not identical and the additional choices to use Runge-Kutta and explicit horizontal diffusion in the JCM lead to differing results. Speedy.f90 also implements additional linear drag on the mean vorticity and divergence in the stratosphere (this drag is undocumented and thus currently excluded from the JCM). JCM v1.1 has been tuned to reduce these differences where possible. While we aim to match the qualitative climate of the Fortran, we are not prioritizing reproducing speedy.f90 exactly, and instead use it as a starting point for an intermediate-complexity model. These architectural choices are also described in Sect. . The average RMS difference of meridional and zonal winds at the surface are 2.1 and 3.5 m s⁻¹ respectively. Figure  shows a comparison of the winds at the top of the troposphere. Again, the large-scale patterns are qualitatively similar, but JCM winds are stronger than in speedy.f90. The average RMS difference of meridional winds at the top of the troposphere is similar to at the surface, but the average RMS difference in the zonal winds is roughly 9 m s⁻¹.

In addition to validating the prognostic JCM state against SPEEDY, we verify the spatial patterns of cloud cover and precipitation. Figure  shows average stratiform cloud cover fraction, precipitation due to convection, and precipitation due to large scale condensation. The panel (a) indicates that low clouds are most prevalent along the eastern boundaries of ocean basins in both JCM and speedy.f90, with average cloud cover near 40 %. This pattern agrees well with the climatology described by . Panels (b), (c), (e) and (f) show precipitation patterns associated with different physical processes. Convective precipitation (b and e) is highest in the tropics, and is particularly strong near the inter-tropical convergence zone (ITCZ), the Pacific warm pool, the Indian Ocean, and the equatorial Atlantic. While JCM and speedy.f90 show the same spatial patterns, precipitation due to convection is more concentrated in speedy.f90 and more diffuse in JCM, likely due to the larger spatial scales of the horizontal diffusion. Precipitation due to large-scale condensation (c and f) is highest in the mid-latitudes with hotspots over the North Pacific Ocean, the Gulf Stream, the Himalayas, and the Southern Ocean. Large-scale condensation appears to be slightly stronger in JCM than in speedy.f90. Spectral ringing in low-resolution models is a common phenomenon . Artifacts of this type of ringing can be seen in Fig.  in the results from both models. However, the ringing in JCM is stronger than in speedy.f90. We have investigated whether this difference affects model stability, and we have found no indication that it grows over time in these low-resolution configurations (T31-T85). This includes running the model for 100-year runs and taking gradients through those long integrations.

Automatic differentiation (AD, also called algorithmic differentiation) computes derivatives of functions by recursively applying the chain rule to the computational graph. When AD is not available, tangent linear and adjoint models must be manually derived and implemented. Given the complexity of state-of-the-art Fortran models, this manual generation of gradients is expensive to implement and maintain. Thus, in most cases, gradient information is inaccessible or presents a substantial burden to model developers.

JAX provides JCM with its AD capabilities . In JAX, gradients can be computed in either forward (tangent linear) or reverse (adjoint) mode. Each mode has multiple use cases and can be applied to a wide variety of science problems. We use F(x) to denote the tangent linear model of a function F(x) with input parameters x. The relationship of F(x) to the adjoint model FT(x), where both are linearized about x, is 1〈F(x)v,w〉=〈v,FT(x)w〉, where v is the tangent vector, w is the co-tangent vector, and 〈⋅〉 denotes the inner product. Forward-mode differentiation is applied through the Jacobian vector product (JVP) and is equivalent to F(x)v. This evaluates the multiplication of a user-specified tangent vector, v, with the tangent linear model, F(x). Reverse-mode differentiation evaluates a vector-Jacobian product (VJP), equivalent to FT(x)w. This multiplies a user-specified co-tangent vector, w, with the AD-derived adjoint, FT(x), of the function F(x). Both VJP and JVP avoid computing the entire Jacobian of the function to minimize computational cost, but they can be used to reconstruct it if necessary. Example applications are discussed in more detail in the following sections.

In large-scale geophysical models integrated over long time windows, JVP typically costs approximately 1–2 times the cost of a nonlinear forward model integration. In contrast, VJP often costs approximately 3–6 times the cost of the forward integration, depending on memory constraints, checkpointing strategy, and solver structure . Reverse-mode differentiation is more computationally efficient than forward mode for gradient evaluation when the output dimension of the function of interest is much smaller than the input dimension (e.g., a scalar loss function). In this case, a single VJP evaluation produces the full gradient, whereas forward mode would require n separate JVP evaluations, where n is the input dimension. Conversely, forward-mode differentiation is more efficient when the number of inputs is much smaller than the number of outputs.

Calibration of ESMs against observations is a problem of interest to the geoscience community and benefits from gradient information. The goal of ESM calibration is to estimate physics parameters that generate the best match between a model trajectory and observations. Without gradient information, this process involves ensemble methods and hand-tuning by domain experts. Hand-tuning is limited in the degree to which it utilizes observational information and leaves much of the parameter space unexplored. Ensemble methods can be effective, but we often lack comparisons of these approximations against gradient-based approaches. Such comparisons are a focus for future work with the JCM. With gradient information, we can take a more systematic approach to calibration using gradient-based optimization techniques. Calibration with gradient information has the potential to assess model behaviour across more of the parameter space.

A popular approach for ESM calibration is to estimate model parameters against climate statistics . The optimization problem is formulated to minimize a loss defined by the misfit between observed and modeled statistics. In order to illustrate how JCM can be calibrated using AD, we show an idealized example that simultaneously estimates parameter values for stratiform cloud albedo and relative humidity threshold. We generate synthetic observed statistics by running the JCM in the default aquaplanet configuration, which has values of stratiform cloud albedo =0.5 and relative humidity threshold =0.3. The calculated statistics are the spatial and temporal averages of each model output variable over a 5 d model run. We initialize the starting values of stratiform cloud albedo and relative humidity threshold to be 0.1 and 0.75, respectively, and then optimize over the parameter space to find values that most closely reproduce the synthetic statistics. The observations denoted by y in Eq. () correspond to the simulated statistics. This example can be adapted or extended to use any climate statistics and any set of parameters.

In this example case, the function F(⋅) is a 5 d JCM simulation with stratiform cloud albedo and relative humidity threshold = x. The loss function L(⋅) is then defined as 2L(x)=12R-12y-F(x)22, where ⋅2 defines the L-2 norm, and R is the observational covariance matrix. R is assumed to be a diagonal matrix with variances =1. We use a gradient descent optimizer to solve for the value of x that minimizes the loss. There are many other types of optimization algorithms, but given the low complexity and smooth loss landscape, we chose the simplest approach. The gradient descent update is as follows: 3xk+1=xk-α∂L∂xk, where xk denotes the vector of parameter values at iteration k, α denotes the step size, and the search direction is the negative gradient of the loss with respect to xk. Figure  shows the calculated loss at each step of the optimization (blue dots) and the true parameter values (grey star). It can be seen that the optimizer succeeds at identifying the values that generated the simulated observations. We calculate the loss across the possible range for both parameters and include a visualization of the 2-dimensional loss landscape, where light colors signify lower cost.

The gradient descent steps are plotted as blue dots to show the loss at each iteration. We obtain ∂L/∂xk using reverse mode automatic differentiation, providing the direction for the next parameter update. The gradient descent optimizer successfully identifies the value of stratiform cloud albedo used to generate the “true” statistics.

JVP facilitates sensitivity experiments with JCM. When using JVP, the tangent vector is effectively a perturbation that is propagated through the tangent linear model. If the function we differentiate is a JCM trajectory, then JVP gives us the evolution of an input perturbation to the model. This could be a perturbed parameter value, model forcing, or initial conditions. The model run is represented by F(x), where argument x is the model input we will perturb. f denotes the model outputs of interest. Using forward mode automatic differentiation, we can calculate the change in the outputs, Δf, given perturbation Δx, which is approximated by 4Δf≈F(x0)Δx, where x0 is the input to the model forward run and F(⋅) is the tangent linear model. JVP is used to evaluate the right-hand side of Eq. (). The user only needs to provide the function F(x) and the perturbation Δx (i.e., the tangent vector).

As an example, we show the influence of a sea surface temperature (SST) perturbation on the future state of a default JCM run. Here x is the SST forcing for the model run and f is the predicted meridional wind, zonal wind, and top-of-atmosphere outgoing longwave radiation. F(x) is a function that executes the forward run of JCM with the given SSTs and isolates the desired predictions. We apply an SST perturbation, Δx, to obtain corresponding changes in the predictions, Δf. JVP propagates the perturbation (tangent vector) through the tangent linear model of F(x). Figure a shows an example SST perturbation. Figure b shows the change in meridional wind, zonal wind, and TOA longwave radiation at 12, 24, and 36 h in the future. The localized warm SST anomaly centered near the equator drives a rapid, large-scale dynamical response in the upper-level winds, with both meridional and zonal wind perturbations propagating globally within 12–36 h. The alternating meridional wind response straddling the equator resembles Rossby wave dispersion from an equatorial heat source . The TOA outgoing longwave radiation response remains local to the SST forcing throughout the 36 h period, indicating that while the dynamical adjustment is fast and far-reaching, the thermodynamic/radiative response is still confined near the heating source on this timescale. JVP circumvents the need for large ensembles and finite differences to estimate model sensitivities and instead directly calculates those quantities.

As an alternative example, we analyze the sensitivity of the predicted top-of-atmosphere (TOA) outgoing longwave radiation with respect to the open-ocean surface albedo. Instead of perturbing a forcing field, we perturb a model parameter. In this scenario, our output f of the forward map F(⋅) is the predicted TOA outgoing longwave radiation and our input x will be the JCM physics parameters. In order to extract the sensitivity of TOA outgoing longwave radiation to open-ocean surface albedo, we set all values of the tangent vector Δx to 0, except for the value corresponding to our parameter of interest which is set to 1. This method effectively extracts individual elements of the Jacobian. The gradient is always computed at the input value x0, which in this case is the set of default parameter values from SPEEDY. The code block in Fig.  implements the described experiment, calculating the gradient of f (longwave_rad.ftop) with respect to the open-ocean surface albedo (Parameters.albsea) using JVP (jax.jvp). Figure  shows the output of this calculation, which is the sensitivity of TOA outgoing longwave radiation after one and two days of simulation to a unit perturbation in open-ocean albedo.