1. Stochastic processes and random fields 2. Continuous semimartingales and stochastic integrals 3. Semimartingales with spatial parameter and stochastic integrals 4. Stochastic flows 5. Convergence of stochastic flows 6. Stochastic partial differential equations.

Stochastic flows and stochastic differential equations

If model parameterizations of unresolved physics, such as the variety of upper ocean mixing processes, are to hold over the large range of time and space scales of importance to climate, they must be strongly physically based. Observations, theories, and models of oceanic vertical mixing are surveyed. Two distinct regimes are identified: ocean mixing in the boundary layer near the surface under a variety of surface forcing conditions (stabilizing, destabilizing, and wind driven), and mixing in the ocean interior due to internal waves, shear instability, and double diffusion (arising from the different molecular diffusion rates of heat and salt). Mixing schemes commonly applied to the upper ocean are shown not to contain some potentially important boundary layer physics. Therefore a new parameterization of oceanic boundary layer mixing is developed to accommodate some of this physics. It includes a scheme for determining the boundary layer depth h, where the turbulent contribution to the vertical shear of a bulk Richardson number is parameterized. Expressions for diffusivity and nonlocal transport throughout the boundary layer are given. The diffusivity is formulated to agree with similarity theory of turbulence in the surface layer and is subject to the conditions that both it and its vertical gradient match the interior values at h. This nonlocal “K profile parameterization” (KPP) is then verified and compared to alternatives, including its atmospheric counterparts. Its most important feature is shown to be the capability of the boundary layer to penetrate well into a stable thermocline in both convective and wind-driven situations. The diffusivities of the aforementioned three interior mixing processes are modeled as constants, functions of a gradient Richardson number (a measure of the relative importance of stratification to destabilizing shear), and functions of the double-diffusion density ratio, Rρ. Oceanic simulations of convective penetration, wind deepening, and diurnal cycling are used to determine appropriate values for various model parameters as weak functions of vertical resolution. Annual cycle simulations at ocean weather station Papa for 1961 and 1969–1974 are used to test the complete suite of parameterizations. Model and observed temperatures at all depths are shown to agree very well into September, after which systematic advective cooling in the ocean produces expected differences. It is argued that this cooling and a steady salt advection into the model are needed to balance the net annual surface heating and freshwater input. With these advections, good multiyear simulations of temperature and salinity can be achieved. These results and KPP simulations of the diurnal cycle at the Long-Term Upper Ocean Study (LOTUS) site are compared with the results of other models. It is demonstrated that the KPP model exchanges properties between the mixed layer and thermocline in a manner consistent with observations, and at least as well or better than alternatives.

Oceanic vertical mixing: a review and a model with a nonlocal boundary layer parameterization

Abstract In this study, a mechanism is demonstrated whereby a large reduction in the Atlantic thermohaline circulation (THC) can induce global-scale changes in the Tropics that are consistent with paleoevidence of the global synchronization of millennial-scale abrupt climate change. Using GFDL’s newly developed global coupled ocean–atmosphere model (CM2.0), the global response to a sustained addition of freshwater to the model’s North Atlantic is simulated. This freshwater forcing substantially weakens the Atlantic THC, resulting in a southward shift of the intertropical convergence zone over the Atlantic and Pacific, an El Nino–like pattern in the southeastern tropical Pacific, and weakened Indian and Asian summer monsoons through air–sea interactions.

Simulated Global Response to a Substantial Weakening of the Atlantic Thermohaline Circulation

第四紀研究 = The Quaternary research

[1] Analyses of global climate from measurements dating back to the nineteenth century show an ‘Atlantic Multidecadal Oscillation’ (AMO) as a leading large-scale pattern of multidecadal variability in surface temperature Yet it is not possible to determine whether these fluctuations are genuinely oscillatory from the relatively short observational record alone Using a 1400 year climate model calculation, we are able to simulate the observed pattern and amplitude of the AMO The results imply the AMO is a genuine quasi-periodic cycle of internal climate variability persisting for many centuries, and is related to variability in the oceanic thermohaline circulation (THC) This relationship suggests we can attempt to reconstruct past THC changes, and we infer an increase in THC strength over the last 25 years Potential predictability associated with the mode implies natural THC and AMO decreases over the next few decades independent of anthropogenic climate change

A Signature of Persistent Natural Thermohaline Circulation Cycles in Observed Climate

In this work, we examine the solution properties of the Burgers’ equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine–Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.

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The Burgers' equation with stochastic transport: shock formation, local and global existence of smooth solutions

We extend the Ito–Wentzell formula for the evolution of a time-dependent stochastic field along a semimartingale to k-form-valued stochastic processes. The result is the Kunita–Ito–Wentzell (KIW) formula for k-forms. We also establish a correspondence between the KIW formula for k-forms derived here and a certain class of stochastic fluid dynamics models which preserve the geometric structure of deterministic ideal fluid dynamics. This geometric structure includes Eulerian and Lagrangian variational principles, Lie–Poisson Hamiltonian formulations and natural analogues of the Kelvin circulation theorem, all derived in the stochastic setting.

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Implications of Kunita–Itô–Wentzell Formula for k -Forms in Stochastic Fluid Dynamics

We extend the Ito-Wentzell formula for the evolution of a time-dependent stochastic field along a semimartingale to $k$-form-valued stochastic processes. The result is the Kunita-Ito-Wentzell (KIW) formula for $k$-forms. We also establish a correspondence between the KIW formula for $k$-forms derived here and a certain class of stochastic fluid dynamics models which preserve the geometric structure of deterministic ideal fluid dynamics. This geometric structure includes Eulerian and Lagrangian variational principles, Lie--Poisson Hamiltonian formulations and natural analogues of the Kelvin circulation theorem, all derived in the stochastic setting.

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Implications of Kunita-It\^o-Wentzell formula for $k$-forms in stochastic fluid dynamics

The prediction of climate change and its impact on extreme weather events is one of the great societal and intellectual challenges of our time. The first part of the problem is to make the distinction between weather and climate. The second part is to understand the dynamics of the fluctuations of the physical variables. The third part is to predict how the variances of the fluctuations are affected by statistical correlations in their fluctuating dynamics. This paper investigates a framework called LA SALT which can meet all three parts of the challenge for the problem of climate change. As a tractable example of this framework, we consider the Euler–Boussinesq (EB) equations for an incompressible stratified fluid flowing under gravity in a vertical plane with no other external forcing. All three parts of the problem are solved for this case. In fact, for this problem, the framework also delivers global well-posedness of the dynamics of the physical variables and closed dynamical equations for the moments of their fluctuations. Thus, in a well-posed mathematical setting, the framework developed in this paper shows that the mean field dynamics combines with an intricate array of correlations in the fluctuation dynamics to drive the evolution of the mean statistics. The results of the framework for 2D EB model analysis define its climate, as well as climate change, weather dynamics, and change of weather statistics, all in the context of a model system of SPDEs with unique global strong solutions.

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Modelling the Climate and Weather of a 2D Lagrangian-Averaged Euler-Boussinesq Equation with Transport Noise

Atmospheric reanalyses are commonly used to force numerical ocean models, but despite large discrepancies reported between different products, the impact of reanalysis uncertainty on the si...

/pdf/impacts-of-atmospheric-reanalysis-uncertainty-on-atlantic-16fcd0jux2.pdf

So Takao

Papers

The Burgers' equation with stochastic transport: shock formation, local and global existence of smooth solutions

Implications of Kunita–Itô–Wentzell Formula for k -Forms in Stochastic Fluid Dynamics

Implications of Kunita-It\^o-Wentzell formula for $k$-forms in stochastic fluid dynamics

Modelling the Climate and Weather of a 2D Lagrangian-Averaged Euler-Boussinesq Equation with Transport Noise

Impacts of Atmospheric Reanalysis Uncertainty on Atlantic Overturning Estimates at 25°N