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Past studies have suggested that regional trends in anthropogenic aerosols can influence the Pacific decadal oscillation (PDO) through modulation of the Aleutian low. However, the robustness of this connection is debated. This study analyzes changes to the Aleutian low in an ensemble of climate models forced with large, idealized global and regional black carbon (BC) and sulfate aerosol perturbations. To isolate the role of ocean feedbacks, the experiments are performed with an interactive ocean and with prescribed sea surface temperatures. The results show a robust weakening of the Aleutian low forced by a global tenfold increase in BC in both experiment configurations. A linearized steady-state primitive equation model is forced with diabatic heating anomalies to investigate the mechanisms through which heating from BC emissions influences the Aleutian low. The heating from BC absorption over India and East Asia generates Rossby wave trains that propagate into the North Pacific sector, forming an upper-tropospheric ridge. Sources of BC outside of East Asia enhance the weakening of the Aleutian low. The responses to a global fivefold and regional tenfold increase in sulfate aerosols over Asia show poor consistency across climate models, with a multimodel mean response that does not project strongly onto the Aleutian low. These findings for a large, idealized step increase in regional sulfate aerosol differ from previous studies that suggest the transient increase in sulfate aerosols over Asia during the early twenty-first century weakened the Aleutian low and induced a transition to a negative PDO phase.

We develop a convex integration scheme for constructing nonunique weak solutions to the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics) in both two and three dimensions. We also develop such a scheme for the construction of nonunique weak solutions to the three-dimensional viscous primitive equations, as well as the two-dimensional Prandtl equations. While in Boutros et al. (Calc Var Partial Differ Equ 62(8):219, 2023) the classical notion of weak solution to the hydrostatic Euler equations was generalised, we introduce here a further generalisation. For such generalised weak solutions, we show the existence and nonuniqueness for a large class of initial data. Moreover, we construct infinitely many examples of generalised weak solutions which do not conserve energy. The barotropic and baroclinic modes of solutions to the hydrostatic Euler equations (which are the average and the fluctuation of the horizontal velocity in the z -coordinate, respectively) that are constructed have different regularities.

The detailed dynamical mechanisms of the upper-tropospheric circulation response to the Madden–Julian oscillation (MJO) convection are examined by integrating a primitive equation model. A series of initial-value calculations with the climatological boreal winter background flow forced by the MJO-like thermal forcing successfully capture the key aspects of the observed circulation response to the MJO convection. This suggests that a large fraction of MJO-related circulation anomalies are direct responses to tropical heating in both the tropics and extratropics and can be largely explained by linear dynamics. It is found that MJO-like dipole heatings not only intensify tropical upper-tropospheric anomalies but also weaken them at certain regions because of the interaction between equatorial Kelvin and Rossby waves. The Rossby wave train primarily excited by horizontal divergence of upper-level perturbation flow propagates northeastward and then heads back to the equator. In this way, Rossby wave activity once generated over the subtropical Indian Ocean tends to enhance the equatorial upper-tropospheric anomalies over the tropical Atlantic and West Africa that have already been created by the zonally propagating equatorial Rossby and Kelvin waves. A ray path tracing reveals that a successive downstream development of Rossby wave train mostly results from the large-scale waves with zonal wavenumbers 2–3 in the Northern Hemisphere and 3–5 in the Southern Hemisphere. The sensitivity tests show that the overall results are quite robust. It is found, however, that the detailed circulation response to the MJO-like forcing is somewhat sensitive to the background flow. This suggests that MJO-related circulation anomalies may have nonnegligible long-term variability and change as background flow varies.

We study the effect of the fast rotation and vertical viscosity on the lifespan of solutions to the three-dimensional primitive equations (also known as the hydrostatic Navier-Stokes equations) with impermeable and stress-free boundary conditions. Firstly, for a short time interval, independent of the rate of rotation | Ω | , we establish the local well-posedness of solutions with initial data that is analytic in the horizontal variables and only L 2 in the vertical variable. Moreover, it is shown that the solutions immediately become analytic in all the variables with increasing-in-time (at least linearly) radius of analyticity in the vertical variable for as long as the solutions exist. On the other hand, the radius of analyticity in the horizontal variables might decrease with time, but as long as it remains positive the solution exists. Secondly, with fast rotation, i.e., large | Ω | , we show that the existence time of the solution can be prolonged, with “well-prepared” initial data. Finally, in the case of two spatial dimensions with Ω = 0 , we establish the global well-posedness provided that the initial data is small enough. The smallness condition on the initial data depends on the vertical viscosity and the initial radius of analyticity in the horizontal variables.

A hydrostatic, coupled-mode, shallow-water model (CSW) is described and used to diagnose and simulate tidal dynamics in the greater Mid-Atlantic Bight region. The reduced-physics model incorporates realistic stratification and topography, internal tide forcing from a priori estimates of the surface tide, and advection terms that describe first-order interactions of internal tides with slowly varying mean flow and mean buoyancy fields and their respective shear. The model is validated via comparisons with semianalytic models and nonlinear primitive equation models in several idealized and realistic simulations that include internal tide interactions with topography and mean flows. Then, 24 simulations of internal tide generation and propagation in the greater Mid-Atlantic Bight region are used to diagnose significant internal tide interactions with the Gulf Stream. The simulations indicate that locally generated mode-one internal tides refract and/or reflect at the Gulf Stream. The redirected internal tides often reappear at the shelf break, where their onshore energy fluxes are intermittent (i.e., noncoherent with surface tide) because meanders in the Gulf Stream alter their precise location, phase, and amplitude. These results provide an explanation for anomalous onshore energy fluxes that were previously observed at the New Jersey shelf break and linked to the irregular generation of nonlinear internal waves.

In this work, we derive the hydrostatic approximation by taking the small aspect ratio limit to the Navier–Stokes equations. The aspect ratio (the ratio of the depth to horizontal width) is a geometrical constraint in the general large scale geophysical motions meaning that the vertical scale is significantly smaller than horizontal. We derive the versatile relative entropy inequality . Applying the versatile relative entropy inequality we gave the rigorous derivation of the limit from the compressible Navier–Stokes equations to the compressible Primitive Equations. This is the first work where the relative entropy inequality was used for proving hydrostatic approximation - the compressible Primitive Equations.

In this paper, we establish the existence and uniqueness of the local strong solution to the Cauchy problem of the inhomogeneous incompressible primitive equations in a periodic domain. Inspired by the work (J. Li in J. Differ. Equ. 263:6512–6536, 2017), we remove the compatibility condition required in (Q. Jiu and F. Wang in Acta Math. Sci. Ser. B Engl. Ed. 40:1316–1334, 2020), and permit the initial density with vacuum. Meanwhile, we prove the existence of the eigenfunctions to the hydrostatic Stokes operator.

In this paper, we consider the initial-boundary value problem for the two-dimensional primitive equations of the large-scale oceanic dynamics. These models are often used to predict weather and climate change. Using the differential inequality technique, rigorous a priori bounds of solutions and the continuous dependence on the heat source are established. We show the application of symmetry in mathematical inequalities in practice.

First, a new sufficient condition for uniqueness of weak solutions is proved for the system of 2D viscous Primitive Equations. Second, global existence and uniqueness are established for several classes of weak solutions with partial initial regularity, including but not limited to those weak solutions with initial horizontal regularity, rather than vertical regularity. Moreover, new and improved regularity properties of weak solutions and strong solutions to the system of 2D viscous Primitive Equations are obtained. Our results and analyses for the problem with physical boundary conditions can be extended to those with other typical boundary conditions. Most of the results were not available before, even for the periodic case.

The two-dimensional inviscid hydrostatic primitive equations of the atmosphere with humidity and saturation are considered in the presence of topography. The model studied here describes the dynamics of the air or water in order to approximate global atmospheric flows. The heart of the paper is to derive a new set of transformed inviscid primitive equations using a version of the terrain-following coordinate systems and to develop an accurate numerical scheme to the equations. In this regard, a fully discrete numerical algorithm based on a Godunov-type finite volume method is proposed and its convergence tested. We then use this algorithm to simulate the flows above a mountain using the terrain-following coordinate system with a dynamic bottom pressure.