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Register a new userModelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model
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The stochastic variational approach for geophysical fluid dynamics was introduced by Holm (Proc Roy Soc A, 2015) as a framework for deriving stochastic parameterisations for unresolved scales. This paper applies the variational stochastic parameterisation in a two-layer quasi-geostrophic model for a beta-plane channel flow configuration. We present a new method for estimating the stochastic forcing (used in the parameterisation) to approximate unresolved components using data from the high resolution deterministic simulation, and describe a procedure for computing physically-consistent initial conditions for the stochastic model. We also quantify uncertainty of coarse grid simulations relative to the fine grid ones in homogeneous (teamed with small-scale vortices) and heterogeneous (featuring horizontally elongated large-scale jets) flows, and analyse how the spread of stochastic solutions depends on different parameters of the model. The parameterisation is tested by comparing it with the true eddy-resolving solution that has reached some statistical equilibrium and the deterministic solution modelled on a low-resolution grid. The results show that the proposed parameterisation significantly depends on the resolution of the stochastic model and gives good ensemble performance for both homogeneous and heterogeneous flows, and the parameterisation lays solid foundations for data assimilation.
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We present high-precision experimental and numerical studies of the Nusselt number $Nu$ as functions of the Rayleigh number $Ra$ in geostrophic rotating convection with domain aspect ratio ${Gamma}$ varying from 0.4 to 3.8 and the Ekman number Ek from $2.0{times}10^{-7}$ to $2.7{times}10^{-5}$. The heat-transport data $Nu(Ra)$ reveal a gradual transition from buoyancy-dominated to geostrophic convection at large $Ek$, whereas the transition becomes sharp with decreasing $Ek$. We determine the power-law scaling of $Nu{sim}Ra^{gamma}$, and show that the boundary flows give rise to pronounced enhancement of $Nu$ in a broad range of the geostrophic regime, leading to reduction of the scaling exponent ${gamma}$ in small ${Gamma}$ cells. The present work provides new insight into the heat-transport scaling in geostrophic convection and may explain the discrepancies observed in previous studies.
Wave--current interaction (WCI) dynamics energizes and mixes the ocean thermocline by producing a combination of Langmuir circulation, internal waves and turbulent shear flows, which interact over a wide range of time scales. Two complementary approaches exist for approximating different aspects of WCI dynamics. These are the Generalized Lagrangian Mean (GLM) approach and the Gent--McWilliams (GM) approach. Their complementarity is evident in their Kelvin circulation theorems. GLM introduces a wave pseudomomentum per unit mass into its Kelvin circulation integrand, while GM introduces a an additional `bolus velocity to transport its Kelvin circulation loop. The GLM approach models Eulerian momentum, while the GM approach models Lagrangian transport. In principle, both GLM and GM are based on the Euler--Boussinesq (EB) equations for an incompressible, stratified, rotating flow. The differences in their Kelvin theorems arise from differences in how they model the flow map in the Lagrangian for the Hamilton variational principle underlying the EB equations. A recently developed approach for uncertainty quantification in fluid dynamics constrains fluid variational principles to require that Lagrangian trajectories undergo Stochastic Advection by Lie Transport (SALT). Here we introduce stochastic closure strategies for quantifying uncertainty in WCI by adapting the SALT approach to both the GLM and GM approximations of the EB variational principle. In the GLM framework, we introduce a stochastic group velocity for transport of wave properties, relative to the frame of motion of the Lagrangian mean flow velocity and a stochastic pressure contribution from the fluctuating kinetic energy. In the GM framework we introduce a stochastic bolus velocity in addition to the mean drift velocity by imposing the SALT constraint in the GM variational principle.
For the generalized surface quasi-geostrophic equation $$left{ begin{aligned} & partial_t theta+ucdot abla theta=0, quad text{in } mathbb{R}^2 times (0,T), & u= abla^perp psi, quad psi = (-Delta)^{-s}theta quad text{in } mathbb{R}^2 times (0,T) , end{aligned} right. $$ $0<s<1$, we consider for $kge1$ the problem of finding a family of $k$-vortex solutions $theta_varepsilon(x,t)$ such that as $varepsilonto 0$ $$ theta_varepsilon(x,t) rightharpoonup sum_{j=1}^k m_jdelta(x-xi_j(t)) $$ for suitable trajectories for the vortices $x=xi_j(t)$. We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem $$(-Delta)^sW = (W-1)^gamma_+, quad text{in } mathbb{R}^2, quad 1<gamma < frac{1+s}{1-s}$$ whose existence and uniqueness have recently been proven in cite{chan_uniqueness_2020}.
We consider the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing in the whole space. When the dispersive amplitude parameter is large enough, we prove the global well-posedness of strong solution to the equation with large initial data. We also show the strong convergence result as the amplitude parameter goes to $infty$. Both results rely on the Strichartz-type estimates for the corresponding linear equation.
We examine a two dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface. The two media are separated by a free common interface. The gravity driven surface and internal water waves (at the common interface between the media) in the presence of a depth-dependent current are studied under certain physical assumptions. Both media are considered incompressible and with prescribed vorticities. Using the Hamiltonian approach the Hamiltonian of the system is constructed in terms of wave variables and the equations of motion are calculated. The resultant equations of motion are then analysed to show that wave-current interaction is influenced only by the current profile in the strips adjacent to the surface and the interface. Small amplitude and long-wave approximations are also presented.
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