This paper is Part II of a two-part paper that documents the CM4X (Climate Model version 4X) hierarchy of coupled climate models developed at the Geophysical Fluid Dynamics Laboratory (GFDL). Part I of this paper is presented in \citeA{CM4X_partI}. Here we present a suite of case studies that examine ocean and sea ice features that are targeted for further research, which include sea level, eastern boundary upwelling, Arctic and Southern Ocean sea ice, Southern Ocean circulation, and North Atlantic circulation. The case studies are based on experiments that follow the protocol of version 6 from the Coupled Model Intercomparison Project (CMIP6). The analysis reveals a systematic improvement in the simulation fidelity of CM4X relative to its CM4.0 predecessor, as well as an improvement when refining the ocean/sea ice horizontal grid spacing from the $0.25^{\circ}$ of CM4X-p25 to the $0.125^{\circ}$ of CM4X-p125. Even so, there remain many outstanding biases, thus pointing to the need for further grid refinements, enhancements to numerical methods, and/or advances in parameterizations, each of which target long-standing model biases and limitations.

We propose a metric for measuring internal and forced variability in ensemble atmosphere, ocean, or climate models using information theory: Shannon entropy and mutual information. This metric differs from the standard ensemble-variance approaches. Information entropy quantifies variability by the size of the visited probability distribution, as opposed to variance that measures only its second moment. Shannon entropy and mutual information manage correlated fields, apply to any data, and are insensitive to outliers as well as a change of units or scale. Finally, we use an example featuring a highly skewed probability distribution (Arctic sea surface temperature) to show that the new metric is robust even with a sharp nonlinear cutoff (the freezing point). We apply these two metrics to quantify internal vs forced variability in (1) idealized Gaussian data, (2) an initial condition ensemble of a realistic coastal ocean model, (3) the Community Earth System Model large ensemble. Each case illustrates the advantages of the proposed metric over variance-based metrics. Furthermore, in the coastal ocean model, the new metric is adapted to further quantify the impact of different boundary forcing choices to aid in prioritizing model improvements–i.e., comparing different choices of extrinsic forcing. The metric can be applied to any ensemble of models where intrinsic and extrinsic factors compete to control variability and can be applied regardless of if the ensemble spread is Gaussian.

We present the GFDL-CM4X (Geophysical Fluid Dynamics Laboratory Climate Model version 4X) coupled climate model hierarchy. The primary application for CM4X is to investigate ocean and sea ice physics as part of a realistic coupled Earth climate model. CM4X utilizes an updated MOM6 (Modular Ocean Model version 6) ocean physics package relative to CM4.0, and there are two members of the hierarchy: one that uses a horizontal grid spacing of $0.25^{\circ}$ (referred to as CM4X-p25) and the other that uses a $0.125^{\circ}$ grid (CM4X-p125). CM4X also refines its atmospheric grid from the nominally 100~km (cubed sphere C96) of CM4.0 to 50~km (C192). Finally, CM4X simplifies the land model to allow for a more focused study of the role of ocean changes to global mean climate.   CM4X-p125 reaches a global ocean area mean heat flux imbalance of $-0.02~\mbox{W}~\mbox{m}^{-2}$ within $\mathcal{O}(150)$ years in a pre-industrial simulation, and retains that thermally equilibrated state over the subsequent centuries. This 1850 thermal equilibrium is characterized by roughly $400~\mbox{ZJ}$ less ocean heat than present-day, which corresponds to estimates for anthropogenic ocean heat uptake between 1850 and present-day. CM4X-p25 approaches its thermal equilibrium only after more than 1000 years, at which time its ocean has roughly $1100~\mbox{ZJ}$ {\it more} heat than its early 21st century ocean initial state. Furthermore, the root-mean-square sea surface temperature bias for historical simulations is roughly 20\% smaller in CM4X-p125 relative to CM4X-p25 (and CM4.0). We offer the {\it mesoscale dominance hypothesis} for why CM4X-p125 shows such favorable thermal equilibration properties.