Primitive Equation Research Articles

Ocean kinetic energy backscatter parametrizations on unstructured grids: Impact on mesoscale turbulence in a channel

We present a new energy backscatter parametrization for primitive equation ocean models at eddy-permitting resolution, specifically for unstructured grids. Traditional eddy parametrizations in terms of viscosity closures lead to excessive dissipation of kinetic energy when used with eddy-permitting meshes. Implemented into the FESOM2 ocean model, the backscatter parametrization leads to a more realistic total dissipation of kinetic energy. It maintains a reservoir of dissipated energy and reinjects this subgrid energy at larger scales at a controlled rate. The separation between dissipation and backscatter scales is achieved by using different-order differential operators and/or spatial smoothing. This ensures numerical model stability.We perform sensitivity studies with different choices of parameter settings and viscosity schemes in a configuration with a baroclinically unstable flow in a zonally reentrant channel with a horizontally uniform mesh. The best backscatter setup substantially improves eddy-permitting simulations at 1/4° and 1/6° resolution, bringing them close to a 1/12° eddy-resolving reference. Improvements are largest for levels of kinetic energy and variability in temperature and vertical velocity. A selected optimal default scheme is then tested in a mixed resolution setup – a channel with narrow transitions between an eddy-permitting and an eddy-resolving subdomain. The backscatter scheme is able to adapt dynamically to the different resolutions and moves the diagnostics closer to the high resolution reference throughout the domain.Our study is a first step toward using backscatter in global variable-mesh ocean models and suggests potential for substantial improvements of ocean mean state and variability at reduced computational cost.

Comments on “A Wave-Resolving Simulation of Langmuir Circulations with a Nonhydrostatic Free-Surface Model: Comparison with Craik–Leibovich Theory and an Alternative Eulerian View of the Driving Mechanism”

AbstractThe results of the subject paper are reviewed wherein credible Langmuir cells are produced by a numerical solution of the primitive fluid dynamic equations with a free surface. Whereas it is a major achievement, the claim that the same results support the general application of the so-called vortex force equations is challenged.

The generation and maintenance of hollow PV towers in a forced primitive equation model

The generation and maintenance of hollow PV towers in a forced primitive equation model

Sharp well-posedness and ill-posedness of the three-dimensional primitive equations of geophysics in Fourier–Besov spaces

We study the well-posedness and ill-posedness for the Cauchy problem of the three-dimensional primitive equations describing the large-scale oceanic and atmospheric circulations. By using the Littlewood–Paley analysis technique, we prove that the Cauchy problem of the three-dimensional primitive equations with the Prandtl number P=1 is locally well-posed in the Fourier–Besov spaces FḂp,r2−3p(R3) for 1<p≤∞,1≤r<∞ and FḂ1,r−1(R3) for 1≤r≤2, and is globally well-posed in these spaces when the initial data are small. We also verify that such problem is ill-posed in FḂ1,r−1(R3) for 2<r≤∞, which implies that our work completes a dichotomy of well-posedness and ill-posedness for the three-dimensional primitive equations in the Fourier–Besov space framework.

Boundary layers for the subcritical modes of the 3D primitive equations in a cube

In this article we study the boundary layers for the subcritical modes of the viscous Linearized Primitive Equations (LPEs) in a cube at small viscosity. The boundary layers include the parabolic boundary layers, ordinary boundary layers, and their interaction-corner layers. The boundary layer correctors are determined by a phenomenological study reminiscent of the Prandtl corrector approach and then a rigorous convergence result is proved which a posteriori justifies the phenomenological study.

On Material Transport by Shelfbreak Eddies

AbstractThe Lagrangian motion in the eddy field produced from an unstable retrograde jet along the shelf break is studied from idealized numerical experiments with a primitive equation model. The jet is initially in thermal wind balance with a cross-isobath density gradient and is not subjected to any atmospheric forcing. Over the course of the model integration, the jet becomes unstable and produces a quasi-stationary eddy field over a 2-month period. During this period, the cross-slope flow at the shelf break is characterized by along-slope correlation scales of O(10) km and temporal correlation scales of a few days. The relative dispersion of parcels across isobaths is found to increase with time as tb, where 1 &lt; b &lt; 2. This mixed diffusive–ballistic regime appears to reflect the combined effects of (i) the short length scales of velocity correlation at the shelf break and (ii) the seaward excursion of monopolar and dipolar vortices. Cross-slope dispersion is greater offshore of the front than inshore of the front, as offshore parcels are both subducted onshore below density surfaces and translated offshore with eddies. Nonetheless, the exchange of parcels across the jet remains very limited on the monthly time scale. Particles originating from the bottom experience upward displacements of a few tens of meters and seaward displacements of O(100) km, suggesting that the eddy activity engendered by an unstable along-slope jet provides another mechanism for bottom boundary layer detachment near the shelf edge.

The Limits of the Primitive Equations of Dynamics for Warm, Slowly Rotating Small Neptunes and Super Earths

Abstract We present significant differences in the simulated atmospheric flow for warm, tidally locked small Neptunes and super Earths (based on a nominal GJ 1214b) when solving the simplified, and commonly used, primitive dynamical equations or the full Navier–Stokes equations. The dominant prograde, superrotating zonal jet is markedly different between the simulations, which are performed using practically identical numerical setups, within the same model. The differences arise due to the breakdown of the so-called “shallow-fluid” and traditional approximations, which worsens when rotation rates are slowed, and day–night temperature contrasts are increased. The changes in the zonal advection between simulations solving the full and simplified equations, give rise to significant differences in the atmospheric redistribution of heat, altering the position of the hottest part of the atmosphere and temperature contrast between the daysides and nightsides. The implications for the atmospheric chemistry, and therefore, observations need to be studied with a model including a more detailed treatment of the radiative transfer and chemistry. Small Neptunes and super Earths are extremely abundant and important, potentially bridging the structural properties (mass, radius, and composition) of terrestrial and gas giant planets. Our results indicate care is required when interpreting the output of models solving the primitive equations of motion for such planets.

EXISTENCE OF TIME PERIODIC SOLUTIONS FOR THE 3-D VISCOUS PRIMITIVE EQUATIONS OF LARGE-SCALE DRY ATMOSPHERE

In this paper, we consider the existence of time periodic solutions of the 3-D viscous primitive equations of large-scale dry atmosphere. We used the Galerkin method. Firstly, by Leray-Schauder fixed point theorem, we prove the existence of approximate solutions of the primitive equations, then we show the convergence of the approximate solutions, and we also get the uniqueness to the primitive equations.

Global Existence of Weak Solutions to the Compressible Primitive Equations of Atmospheric Dynamics with Degenerate Viscosities

We show the existence of global weak solutions to the three-dimensional compressible primitive equations of atmospheric dynamics with degenerate viscosities. In analogy with the case of the compressible Navier-Stokes equations, the weak solutions satisfy the basic energy inequality, the Bresh-Desjardins entropy inequality and the Mellet-Vasseur estimate. These estimates play an important role in establishing the compactness of the vertical velocity of the approximating solutions, and therefore are essential to recover the vertical velocity in the weak solutions.

Laplace transform integration of a baroclinic model

A time integration scheme based on the Laplace transform (LT) has been implemented in a baroclinic primitive equation model. The LT scheme provides an attractive alternative to the popular semi‐implicit (SI) scheme. Analysis shows that it is more accurate than SI for both linear and nonlinear terms of the equations. Numerical experiments confirm the superior performance of the LT scheme. The algorithmic complexity of the LT scheme is comparable with SI, with just an additional transformation to vertical eigenmodes each time step, and it provides the possibility of improving weather forecasts at comparable computational cost.

Applications of generalized trigonometric functions with two parameters II

Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the $p$-Laplacian, which is known as a typical nonlinear differential operator. Compared to GTFs with one parameter, there are few applications of GTFs with two parameters to differential equations. We will apply GTFs with two parameters to studies on the inviscid primitive equations of oceanic and atmospheric dynamics, new formulas of Gaussian hypergeometric functions, and the $L^q$-Lyapunov inequality for the one-dimensional $p$-Laplacian.

Random attractor for the 3D viscous primitive equations driven by fractional noises

We develop a new and general method to prove the existence of the random attractor (strong attractor) for the primitive equations (PEs) of large-scale ocean and atmosphere dynamics under non-periodic boundary conditions and driven by infinite-dimensional additive fractional Wiener processes. In contrast to our new method, the common method, compact Sobolev embedding theorem, is to obtain the time-uniform a priori estimates in some Sobolev space whose regularity is higher than the solution space. But this method can not be applied to the 3D stochastic PEs with the non-periodic boundary conditions. Therefore, the existence of universal attractor (weak attractor) was established in previous works (see [15,16]). The main idea of our method is that we first derive that P-almost surely the solution operator of stochastic PEs is compact. Then we construct a compact absorbing set by virtue of the compact property of the solution operator and the existence of a absorbing set. We should point out that our method has some advantages over the common method of using compact Sobolev embedding theorem, i.e., using our method we only need to obtain time-uniform a priori estimates in the solution space to prove the existence of random attractor for the corresponding stochastic system, while the common method need to establish time-uniform a priori estimates in a more regular functional space than the solution space. Take the stochastic PEs for example, as the unique strong solution to the stochastic PEs belongs to C([0,T];(H1(℧))3), in view of our method, we only need to obtain the time-uniform a priori estimates in the solution space (H1(℧))3 to prove the existence of random attractor for this stochastic system, while the common method need to establish time-uniform a priori estimates for the solution in the functional space (H2(℧))3. However, time-uniform a priori estimates in (H2(℧))3 for the solution to stochastic PEs are too difficult to be established. The present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations driven by Wiener noises, fractional noises and even jump noises. In a forth coming paper, using this new method we [46] prove the existence of random attractor for the stochastic nematic liquid crystals equations. This is the first result about the long-time behavior of stochastic nematic liquid crystals equations.

Supercritical heat transfer characteristics of couple stress convection flow from a vertical cylinder using an equation of state approach

The present work describes numerical simulations of the supercritical heat transfer characteristics of couple stress fluid flow from a vertical cylinder using the equation of state approach. Redlich-Kwong (RK-EOS) and Van der Waals (VW-EOS) equations of state (EOS) are deployed to derive the equation for the thermal expansion coefficient (β). The β values calculated based on RK-EOS are sufficiently close to the experimental values when compared with those based on VW-EOS. Due to the presence of couple stresses in the fluid, highly nonlinear coupled partial differential equations are generated. These primitive equations are reduced to dimensionless form by using suitable non-dimensional quantities. An unconditionally stable finite difference technique is used to solve the normalized conservation equations under physically viable boundary conditions, in order to describe the natural convection heat transfer characteristics of couple stress fluid external to a vertical cylinder in the supercritical region. For different values of the emerging physical parameters, numerical data for couple stress fluid for the case of Nitrogen is generated and presented in the form of graphs and tables. The present computations indicate that the velocity field is suppressed close to the cylinder whereas it is enhanced away from the cylinder with increasing values of reduced temperature in the supercritical fluid region. The reverse behavior is observed for reduced pressure. The present results are compared with the previous results and found to be in good agreement. Applications of the model include super-critical coating dynamics in the aerospace and medical industries.

Estimating Subseasonal Variability and Trends in Global Atmosphere Using Reanalysis Data.

A new measure of subseasonal variability is introduced that provides a scale‐dependent estimation of vertically and meridionally integrated atmospheric variability in terms of the normal modes of linearized primitive equations. Applied to the ERA‐Interim data, the new measure shows that subseasonal variability decreases for larger zonal wave numbers. Most of variability is due to balanced (Rossby mode) dynamics but the portion associated with the inertio‐gravity (IG) modes increases as the scale reduces. Time series of globally integrated variability anomalies in ERA‐Interim show an increase in variability after year 2000. In recent years the anomalies have been about 2% above the 1981–2010 average. The relative increase in variability projecting on the IG modes is larger and more persistent than for the Rossby modes. Although the IG part is a small component of the subseasonal variability, it is an important effect likely reflecting the observed increase in the tropical precipitation variability.

Transient contributions to the forcing of the atmospheric annual cycle

The forcing of the global circulation is examined using a primitive equation model and a 38-year reanalysis dataset. One-timestep integrations are initialised with selected sets of initial conditions, and the forcing budget for the mean annual cycle is deduced. This budget consists of sources and sinks of momentum, temperature and humidity which are balanced by dynamical terms. The associated timescale interactions are examined in detail. The time-mean forcing is balanced by time-mean fluxes, annual cycle interactions and transient fluxes. The annual cycle of the forcing is balanced by the interaction of annual cycle anomalies with the time-mean flow and with themselves (this latter cycle-cycle interaction term is found to be important for the moisture supply over West Africa). Transient interactions on other timescales also contribute to the forcing of the annual cycle, but the interaction term between the annual cycle and other timescales is small, as is the storage term associated with seasonal tendencies. This objectively derived empirical forcing is then used to drive the dynamical model. The resulting simple GCM is called DREAM (Dynamical Research Empirical Atmospheric Model). This is the first time this approach has been used with an annual cycle. The systematic errors of DREAM compared to the reanalysis chiefly concern the momentum balance in the southern hemisphere jet. Perpetual season simulations are similar to individual seasons from the annual cycle run, consistent with the small seasonal tendency term in the forcing.

Local Finite-Amplitude Wave Activity as a Diagnostic for Rossby Wave Packets

AbstractUpper-tropospheric Rossby wave packets (RWPs) are important dynamical features, because they are often associated with weather systems and sometimes act as precursors to high-impact weather. The present work introduces a novel diagnostic to identify RWPs and to quantify their amplitude. It is based on the local finite-amplitude wave activity (LWA) of Huang and Nakamura, which is generalized to the primitive equations in isentropic coordinates. The new diagnostic is applied to a specific episode containing large-amplitude RWPs and compared with a more traditional diagnostic based on the envelope of the meridional wind. In this case, LWA provides a more coherent picture of the RWPs and their zonal propagation. This difference in performance is demonstrated more explicitly in the framework of an idealized barotropic model simulation, where LWA is able to follow an RWP into its fully nonlinear stage, including cutoff formation and wave breaking, while the envelope diagnostic yields reduced amplitudes in such situations.

Stochastic 2D primitive equations: Central limit theorem and moderate deviation principle

In this paper, we establish a central limit theorem and a moderate deviation for two-dimensional stochastic primitive equations driven by multiplicative noise. This is the first result about the limit theorem and the moderate deviations for stochastic primitive equations. The proof of the results relies on the weak convergence method and some delicate and careful apriori estimates.

Jet formation in salt-finger convection: a modified Rayleigh–Bénard problem

Large-scale coherent structures such as jets in Rayleigh–Bénard convection and related systems are receiving increasing attention. This paper studies, both numerically and theoretically, the process of jet formation in two-dimensional salt-finger convection. The approach utilizes an asymptotically derived system of equations referred to as the modified Rayleigh–Bénard convection (MRBC) model, valid in the geophysically and astrophysically relevant limit in which the solute diffuses much more slowly than heat. In these equations, convection is driven by a destabilizing salinity gradient while the effects of the stabilizing temperature gradient manifest themselves as an additional anisotropic dissipation acting on large scales. The MRBC system is specified by two external parameters: the Schmidt number $\mathit{Sc}$ (ratio of viscosity to solutal diffusivity) and the Rayleigh ratio $\mathit{Ra}$ (ratio between the Rayleigh numbers of the destabilizing solutal stratification and the stabilizing thermal stratification). Two distinct $\mathit{Ra}$ regimes are explored for fixed $\mathit{Sc}=1$. In all cases studied the system develops a horizontal jet structure that is maintained self-consistently by turbulent fluctuations, but coarsens over time. For intermediate Rayleigh ratios (e.g. $\mathit{Ra}=6$), the MRBC model captures the relaxation oscillations superposed on the jet structure observed at similar parameter values in direct numerical simulations of the primitive equations. For smaller Rayleigh ratios (e.g. $\mathit{Ra}=2$), a regime for which direct numerical simulation of the primitive equations is difficult because of the presence of fast gravity waves, the MRBC model reveals the existence of statistically steady jets whose properties are studied in detail. Three hierarchical models, the MRBC and further reductions in the form of quasilinear and single-mode approximations, are used to confirm that jets form and are sustained as a result of the interaction between fluctuations (salt fingers) and large-scale horizontally averaged horizontal flows (jets). Even though the small-scale structures exhibited by the three models exhibit clear differences, all three produce the same power-law spectrum of the mean fields at large vertical scales: in all, the spectrum of the mean streamfunction scales as $m^{-3}$ and the mean salinity field scales as $m^{-1}$, with $m$ the vertical wavenumber. A theoretical explanation of these observations based on the dominant balances in the mean and fluctuation equations is provided. As a consequence, the jets have a zigzag profile, a conclusion that is consistent with numerical simulations. Based on numerical observations, a three-component phenomenological model consisting of a linearly growing mode, a linearly damped mode and a mean mode is proposed to explain the observed transition from statistically steady jet structure to jets with superposed oscillations that takes place with increasing Rayleigh ratio.

On the backward uniqueness of the stochastic primitive equations with additive noise

The previous works focus on the uniqueness for the initial-value problems of stochastic primitive equations. Uniqueness for the initial-value problems means that if the two initial conditions are the same, then the two solutions coincide with each other. However there is no work to answer what will happen to the solutions if the two initial conditions are different. This problem for the stochastic three dimensional primitive equations is addressed by the backward uniqueness established in this article. The backward uniqueness means that if two solutions intersect at time $t>0, $ then they are equal everywhere on the interval $(0, t).$ In other words, given two different initial-value conditions, the corresponding two solutions will never cross in the future. Hence this article can be viewed as a further study of the dependence of the solutions on the initial data.

Diagnosing, modeling, and testing a multiplicative stochastic Gent-McWilliams parameterization

A depth-independent isotropic Gent-McWilliams (GM) transport parameter κ is diagnosed from an idealized eddy-resolving primitive equation simulation. The optimal depth-independent isotropic GM parameterization is only able to model less than 50% of the diagnosed total tendency of temperature induced by unresolved mesoscale eddies. A spatio-temporal stochastic model of the GM parameter is developed based on the diagnosed values; the graphical lasso is used to estimate the spatial correlation structure. The stochastic model is used as a stochastic parameterization in low-resolution model simulations. The low-resolution stochastic simulation does a poor job of reproducing the temporal mean of large-scale temperature. Deterministic GM parameterizations and multiplicative stochastic GM parameterizations with unrealistic structure result in significantly more-accurate large-scale temperature in the low-resolution simulations. These results suggest that either the depth-independence or the isotropy of the GM parameterization are unrealistic as models of the eddy tracer transport, or that a stochastic GM parameterization should include an additive component.