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Salmon, R.  2018.  Entropy budget and coherent structures associated with a spectral closure model of turbulence. Journal of Fluid Mechanics. 857:806-822.   10.1017/jfm.2018.778   AbstractWebsite

We 'derive' the eddy-damped quasi-normal Markovian model (EDQNM) by a method that replaces the exact equation for the Fourier phases with a solvable stochastic model, and we analyse the entropy budget of the EDQNM. We show that a quantity that appears in the probability distribution of the phases may be interpreted as the rate at which entropy is transferred from the Fourier phases to the Fourier amplitudes. In this interpretation, the decrease in phase entropy is associated with the formation of structures in the flow, and the increase of amplitude entropy is associated with the spreading of the energy spectrum in wavenumber space. We use Monte Carlo methods to sample the probability distribution of the phases predicted by our theory. This distribution contains a single adjustable parameter that corresponds to the triad correlation time in the EDQNM. Flow structures form as the triad correlation time becomes very large, but the structures take the form of vorticity quadrupoles that do not resemble the monopoles and dipoles that are actually observed.

Salmon, R.  2014.  Analogous formulation of electrodynamics and two-dimensional fluid dynamics. Journal of Fluid Mechanics. 761   10.1017/jfm.2014.642   AbstractWebsite

A single, simply stated approximation transforms the equations for a two-dimensional perfect fluid into a form that is closely analogous to Maxwell's equations in classical electrodynamics. All the fluid conservation laws are retained in some form. Waves in the fluid interact only with vorticity and not with themselves. The vorticity is analogous to electric charge density, and point vortices are the analogues of point charges. The dynamics is equivalent to an action principle in which a set of fields and the locations of the point vortices are varied independently. We recover classical, incompressible, point vortex dynamics as a limiting case. Our full formulation represents the generalization of point vortex dynamics to the case of compressible flow.

Salmon, R.  2013.  Coupled systems of two-dimensional turbulence. Journal of Fluid Mechanics. 732   10.1017/jfm.2013.422   AbstractWebsite

Ordinary two-dimensional turbulence corresponds to a Hamiltonian dynamics that conserves energy and the vorticity on fluid particles. This paper considers coupled systems of two-dimensional turbulence with three distinct governing dynamics. One is a Hamiltonian dynamics that conserves the vorticity on fluid particles and a quantity analogous to the energy that causes the system members to develop a strong correlation in velocity. The other two dynamics considered are non-Hamiltonian. One conserves the vorticity on particles but has no conservation law analogous to energy conservation; the other conserves energy and enstrophy but it does not conserve the vorticity on fluid particles. The coupled Hamiltonian system behaves like two-dimensional turbulence, even to the extent of forming isolated coherent vortices. The other two dynamics behave very differently, but the behaviours of all four dynamics are accurately predicted by the methods of equilibrium statistical mechanics.

Salmon, R.  2013.  An alternative view of generalized Lagrangian mean theory. Journal of Fluid Mechanics. 719:165-182.   10.1017/jfm.2012.638   AbstractWebsite

If the variables describing wave-mean flow interactions are chosen to include a set of fluid-particle labels corresponding to the mean flow, then the generalized Lagrangian mean (GLM) theory takes the form of an ordinary classical field theory. Its only truly distinctive features then arise from the distinctive feature of fluid dynamics as a field theory, namely, the particle-relabelling symmetry property, which corresponds by Noether's theorem to the many vorticity conservation laws of fluid mechanics. The key feature of the formulation is that all the dependent variables depend on a common set of space-time coordinates. This feature permits an easy and transparent derivation of the GLM equations by use of the energy-momentum tensor formalism. The particle-relabelling symmetry property leads to the GLM potential vorticity law in which pseudo-momentum is the only wave activity term present. Thus the particle-relabelling symmetry explains the prominent importance of pseudo-momentum in GLM theory.

Salmon, R.  2012.  Statistical mechanics and ocean circulation. Communications in Nonlinear Science and Numerical Simulation. 17:2144-2152.   10.1016/j.cnsns.2011.05.044   AbstractWebsite

We apply equilibrium statistical mechanics based upon the conservation of energy and potential enstrophy to the mass-density distribution within the ocean, using a Monte Carlo method that conserves the buoyancy of each fluid particle. The equilibrium state resembles the buoyancy structure actually observed. (C) 2011 Elsevier B.V. All rights reserved.

Salmon, R.  2010.  The shape of the main thermocline, revisited. Journal of Marine Research. 68:541-568.   10.1357/002224010794657182   AbstractWebsite

Using the Monte Carlo method of statistical physics, we compute the equilibrium statistical mechanics of the shallow water equations, considered as a reduced-gravity model of the ocean's upper layer in a square ocean that spans the equator. The ensemble-averaged flow comprises a westward drift at low latitudes, associated with the poleward deepening of the main thermocline, and a more intense compensating eastward flow near the latitudes at which the layer depth vanishes. Inviscid numerical simulations with a model that exactly conserves mass, energy, and potential enstrophy support the theoretical prediction.

Salmon, R.  2009.  A shallow water model conserving energy and potential enstrophy in the presence of boundaries. Journal of Marine Research. 67:779-814. AbstractWebsite

We extend a previously developed method for constructing shallow water models that conserve energy and potential enstrophy to the case of flow bounded by rigid walls. This allows the method to be applied to ocean models. Our procedure splits the dynamics into a set of prognostic equations for variables (vorticity, divergence, and depth) chosen for their relation to the Casimir invariants of mass, circulation and potential enstrophy, and a set of diagnostic equations for variables that are the functional derivatives of the Hamiltonian with respect to the chosen prognostic variables. The form of the energy determines the form of the diagnostic equations. Our emphasis on conservation laws produces a novel form of the boundary conditions, but numerical test cases demonstrate the accuracy of our model and its extreme robustness, even in the case of vanishing viscosity.

Salmon, R.  2009.  An Ocean Circulation Model Based on Operator-Splitting, Hamiltonian Brackets, and the Inclusion of Sound Waves. Journal of Physical Oceanography. 39:1615-1633.   10.1175/2009jpo4134.1   AbstractWebsite

This paper offers a simple, entirely prognostic, ocean circulation model based on the separation of the complete dynamics, including sound waves, into elementary Poisson brackets. For example, one bracket corresponds to the propagation of sound waves in a single direction. Other brackets correspond to the rotation of the velocity vector by individual components of the vorticity and to the action of buoyancy force. The dynamics is solved by Strang splitting of the brackets. Key features of the method are the assumption that the sound waves propagate exactly one grid distance in a time step and the use of Riemann invariants to solve the sound-wave dynamics exactly. In these features the method resembles the lattice Boltzmann method, but the flexibility of more conventional methods is retained. As in the lattice Boltzmann method, very short time steps are required to prevent unrealistically strong coupling between the sound waves and the slow hydrodynamic motions of primary interest. However, the disadvantage of small time steps is more than compensated by the model's extreme simplicity, even in the presence of very complicated boundaries, and by its massively parallel form. Numerical tests and examples illustrate the practicality of the method.

Salmon, R.  2007.  A general method for conserving energy and potential enstrophy in shallow-water models. Journal of the Atmospheric Sciences. 64:515-531.   10.1175/jas3837.1   AbstractWebsite

The shallow-water equations may be posed in the form dF/dt = {F, H, Z}, where H is the energy, Z is the potential enstrophy, and the Nambu bracket {F, H, Z} is completely antisymmetric in its three arguments. This makes it very easy to construct numerical models that conserve analogs of the energy and potential enstrophy; one need only discretize the Nambu bracket in such a way that the antisymmetry property is maintained. Using this strategy, this paper derives explicit finite-difference approximations to the shallow-water equations that conserve mass, circulation, energy, and potential enstrophy on a regular square grid and on an unstructured triangular mesh. The latter includes the regular hexagonal grid as a special case.

Salmon, R.  2005.  A general method for conserving quantities related to potential vorticity in numerical models. Nonlinearity. 18:R1-R16.   10.1088/0951-7715/18/5/r01   AbstractWebsite

Nambu proposed a generalization of Hamiltonian dynamics in the form dF/dt = {F, H, Z}, which conserves H and Z because the Nambu bracket {F, H, Z} is completely antisymmetric. The equations of fluid dynamics fit Nambu's form with H the energy and Z a quantity related to potential vorticity. This formulation makes it easy, in principle, to construct numerical fluid-models that conserve analogues of H and Z; one need only discretize the Nambu bracket in such a way that the antisymmetry property is preserved. In practice, the bracket may contain apparent singularities that are cancelled by the functional derivatives of Z. Then the discretization must be carried out in such a way that the cancellation is maintained. Following this strategy, we derive numerical models of the shallow-water equations and the equations for incompressible flow in two and three dimensions. The models conserve the energy and an arbitrary moment of the potential vorticity. The conservation of potential enstrophy - the second moment of potential vorticity - is thought to be especially important because it prevents the spurious cascade of energy into high wavenumbers.

Dellar, PJ, Salmon R.  2005.  Shallow water equations with a complete Coriolis force and topography. Physics of Fluids. 17   10.1063/1.21156747   AbstractWebsite

This paper derives a set of two-dimensional equations describing a thin inviscid fluid layer flowing over topography in a frame rotating about an arbitrary axis. These equations retain various terms involving the locally horizontal components of the angular velocity vector that are discarded in the usual shallow water equations. The obliquely rotating shallow water equations are derived both by averaging the three-dimensional equations and from an averaged Lagrangian describing columnar motion using Hamilton's principle. They share the same conservation properties as the usual shallow water equations, for the same energy and modified forms of the momentum and potential vorticity. They may also be expressed in noncanonical Hamiltonian form using the usual shallow water Hamiltonian and Poisson bracket. The conserved potential vorticity takes the standard shallow water form, but with the vertical component of the rotation vector replaced by the component locally normal to the surface midway between the upper and lower boundaries. (c) 2005 American Institute of Physics.

Salmon, R.  2004.  Poisson-Bracket approach to the construction of energy- and potential-enstrophy-conserving algorithms for the shallow-water equations. Journal of the Atmospheric Sciences. 61:2016-2036.   10.1175/1520-0469(2004)061<2016:pattco>;2   AbstractWebsite

Arakawa and Lamb discovered a finite-difference approximation to the shallow-water equations that exactly conserves finite-difference approximations to the energy and potential enstrophy of the fluid. The Arakawa Lamb (AL) algorithm is a stunning and important achievement-stunning, because in the shallow-water case, neither energy nor potential enstrophy is a simple quadratic, and important because the simultaneous conservation of energy and potential enstrophy is known to prevent the spurious cascade of energy to high wavenumbers. However, the method followed by AL is somewhat ad hoc, and it is difficult to see how it might be generalized to other systems. In this paper, the AL algorithm is rederived and greatly generalized in a way that should permit still further generalizations. Beginning with the Hamiltonian formulation of shallow-water dynamics, its two essential ingredients-the Hamiltonian functional and the Poisson-bracket operator-are replaced by finite-difference approximations that maintain the desired conservation laws. Energy conservation is maintained if the discrete Poisson bracket retains the antisymmetry property of the exact bracket, a trivial constraint. Potential enstrophy is conserved if a set of otherwise arbitrary coefficients is chosen in such a way that a very large quadratic form contains only diagonal terms. Using a symbolic manipulation program to satisfy the potential-enstrophy constraint, it is found that the energy- and potential-enstrophy-conserving schemes corresponding to a stencil of 25 grid points contain 22 free parameters. The AL scheme corresponds to the vanishing of all free parameters. No parameter setting can increase the overall accuracy of the schemes beyond second order, but 19 of the free parameters may be independently adjusted to yield a scheme with fourth-order accuracy in the vorticity equation.

Salmon, R.  2002.  Numerical solution of the two-layer shallow water equations with bottom topography. Journal of Marine Research. 60:605-638.   10.1357/002224002762324194   AbstractWebsite

We present a simple, robust numerical method for solving the two-layer shallow water equations with arbitrary bottom topography. Using the technique of operator splitting, we write the equations as a pair of hyperbolic systems with readily computed characteristics, and apply third-order-upwind differences to the resulting wave equations. To prevent the thickness of either layer from vanishing, we modify the dynamics, inserting an artificial form of potential energy that becomes very large as the layer becomes very thin. Compared to high-order Riemann schemes with flux or slope limiters, our method is formally more accurate, probably less dissipative, and certainly more efficient. However, because we do not exactly conserve momentum and mass, bores move at the wrong speed unless we add explicit, momentum-conserving viscosity. Numerical solutions demonstrate the accuracy and stability of the method. Solutions corresponding to two-layer, wind-driven ocean flow require no explicit viscosity or hyperviscosity of any kind; the implicit hyperdiffusion associated with third-order-upwind differencing effectively absorbs the enstrophy cascade to small scales.

Salmon, R.  1999.  Lattice Boltzmann solutions of the three-dimensional planetary geostrophic equations. Journal of Marine Research. 57:847-884.   10.1357/002224099321514079   AbstractWebsite

We use the lattice Boltzmann method as the basis for a three-dimensional, numerical ocean circulation model in a rectangular basin. The fundamental dynamical variables are the populations of mass- and buoyancy-particles with prescribed discrete velocities. The particles obey collision rules that correspond, on the macroscopic scale, to planetary geostrophic dynamics. The advantages of the model are simplicity, stability, and massively parallel construction. By the special nature of its construction, the lattice Boltzmann model resolves upwelling boundary layers and unsteady convection. Solutions of the model show many of the features predicted by ocean circulation theories.

Salmon, R.  1999.  The lattice Boltzmann method as a basis for ocean circulation modeling. Journal of Marine Research. 57:503-535.   10.1357/002224099764805174   AbstractWebsite

We construct a lattice Boltzmann model of a single-layer, "reduced gravity" ocean in a square basin, with shallow water or planetary geostrophic dynamics, and boundary conditions of no slip or no stress. When the volume of the moving upper layer is sufficiently small, the motionless lower layer outcrops over a broad area of the northern wind gyre, and the pattern of separated and isolated western boundary currents agrees with the theory of Veronis (1973). Because planetary geostrophic dynamics omit inertia, lattice Boltzmann solutions of the planetary geostrophic equations do not require a lattice with the high degree of symmetry needed to correctly represent the Reynolds stress. This property gives planetary geostrophic dynamics a significant computational advantage over the primitive equations, especially in three dimensions.

Salmon, R.  1998.  Linear ocean circulation theory with realistic bathymetry. Journal of Marine Research. 56:833-884.   10.1357/002224098321667396   AbstractWebsite

The linear equations governing stratified, wind-driven flow in an ocean of arbitrary shape may be combined into a single "advection-diffusion" equation for the pressure phi, in which the "flow" advecting phi includes a delta-function contribution at the ocean bottom in the sense of southward "advection" of phi along western continental slopes. This interpretation of the phi-equation helps to explain numerical solutions obtained with a finite-element model incorporating realistic North Atlantic bathymetry.

Salmon, R.  1998.  Lectures on geophysical fluid dynamics. , New York: Oxford University Press Abstract
Miles, J, Salmon R.  1997.  On the vorticity of long gravity waves in water of variable depth. Wave Motion. 25:273-274.   10.1016/s0165-2125(96)00039-x   AbstractWebsite

Yoon and Liu's analysis of long gravity waves in water of slowly varying depth is modified to allow for conservation of potential vorticity in place of their (incorrect) conservation at conventional vorticity.

Becker, JM, Salmon R.  1997.  Eddy formation on a continental slope. Journal of Marine Research. 55:181-200.   10.1357/0022240973224418   AbstractWebsite

We consider a uniform-density ocean in which the depth-independent horizontal velocity is driven by a two-gyre wind. Numerical solutions of the governing vorticity equation reveal that the solutions with a flat ocean bottom differ greatly from those in which a continental shelf and slope are present along the western boundary. In the ocean basin with a western continental shelf, steady inertial circulations readily lose stability to unsteady inertial circulations that spawn eddies. However, eddies do not form in the corresponding solutions for the flat-bottom ocean. Reducing the friction leads to hat-bottom solutions that shed eddies far offshore, but still differ significantly from the corresponding continental-slope solutions, where eddies pinch off near the western boundary.

Salmon, R.  1996.  Large-scale semigeostrophic equations for use in ocean circulation models. Journal of Fluid Mechanics. 318:85-105.   10.1017/s0022112096007045   AbstractWebsite

Hamiltonian approximation methods yields approximate dynamical equations that apply to nearly geostrophic flow at scales larger than the internal Rossby deformation radius. These equations incorporate fluid inertia with the same order of accuracy as the semi-geostrophic equations, but are nearly as simple (in appropriate coordinates) as the equations obtained by completely omitting the inertia.

Salmon, R, Ford R.  1995.  A Simple-Model of the Joint Effect of Baroclinicity and Relief on Ocean Circulation. Journal of Marine Research. 53:211-230.   10.1357/0022240953213250   AbstractWebsite

We offer a simple model for studying the joint effect of baroclinicity and relief (jebar) on large-scale ocean circulation, based upon the planetary geostrophic equations. Applying a Galerkin approximation to the buoyancy equation, and asssuming that the temperature diffusion and vertical stratification are weak, we obtain a simple relation between the ocean temperature and the streamfunction psi for the vertically-averaged horizontal transport. Substituting this relation back into the vertically-averaged vorticity equation yields a single, generally nonlinear equation for psi, in which jebar corresponds to a clockwise 'advection' of Jr along the continental slope (for the realistic case of temperature increasing with psi). Numerical solutions resemble those obtained by Salmon (1994) using a more accurate model, and provide a physically transparent explanation for the northward excursion of the Gulf Stream along the western continental slope observed in the previous study.

Salmon, R.  1994.  Generalized 2-Layer Models of Ocean Circulation. Journal of Marine Research. 52:865-908.   10.1357/0022240943076939   AbstractWebsite

The assumption that surfaces of constant temperature and potential vorticity coincide leads to an exact, time-dependent reduction of the ideal thermocline equations in an ocean basin of arbitrary shape. After modifications to include forcing, dissipation, and the presence of the equator, these reduced equations form the basis for numerical models that are both more realistic and easier to solve than the conventional two-layer model.

Salmon, R, Smith LM.  1994.  Hamiltonian Derivation of the Nonhydrostatic Pressure-Coordinate Model. Quarterly Journal of the Royal Meteorological Society. 120:1409-1413.   10.1256/smsqj.51913   AbstractWebsite
Bogden, PS, Davis RE, Salmon R.  1993.  The North-Atlantic Circulation - Combining Simplified Dynamics with Hydrographic Data. Journal of Marine Research. 51:1-52.   10.1357/0022240933223855   AbstractWebsite

We estimate the time-averaged velocity field in the North Atlantic from observations of density, wind stress and bottom topography. The flow is assumed geostrophic, with prescribed Ekman pumping at the surface, and no normal component at the bottom. These data and dynamics determine velocity to within an arbitrary function of (Coriolis parameter)/(ocean depth), which we call the ''dynamical free mode.'' The free mode is selected to minimize mixing of potential density at mid-depth. This tracer-conservation criterion serves as a relatively weak constraint on the calculation. Estimates of vertical velocity are particularly sensitive to variations in the free mode and to errors in density. In contrast, horizontal velocities are relatively robust. Below the thermocline, we predict a strong O(1 cm/sec) westward flow across the entire North Atlantic, in a narrow range of latitude between 25N and 32N. This feature supports the qualitative (and controversial) conjecture by Wust (1935) of flow along the ''Mediterranean Salt Tongue.'' Along continental margins and at the Mid-Atlantic Ridge, predicted bottom velocity points along isobaths, with shallow water to the right. These flows agree with many long-term current measurements and with notions of the circulation based on tracer distributions. The results conflict with previous oceanographic-inverse models, which predict mid-depth flows an order of magnitude smaller and often in opposite directions. These discrepancies may he attributable to our relatively strong enforcement of the bottom boundary condition. This involves the plausible, although tenuous, assertion that the flow ''feels'' only the large-scale features of the bottom topography. Our objective is to investigate the consequences of using this hypothesis to estimate the North Atlantic circulation.

Salmon, R.  1992.  A 2-Layer Gulf-Stream over a Continental-Slope. Journal of Marine Research. 50:341-365.   10.1357/002224092784797610   AbstractWebsite

We consider the two-layer form of the planetary geostrophic equations, in which a simple Rayleigh friction replaces the inertia, on a western continental slope. In the frictionless limit, these equations can be written as characteristic equations in which the potential vorticities of the top and bottom layers play the role of Riemann invariants. The general solution is of two types. In the first type, the characteristics can cross, and friction is required to resolve the resulting shocks. In the second type, one of the two Riemann invariants is uniform, the remaining characteristic is a line of constant f/H, and the solutions take a simple explicit form. A solution resembling the Gulf Stream can be formed by combining three solutions of the second type. Compared to the corresponding solution for homogeneous fluid, the Gulf Stream and its seaward countercurrent are stronger, and the latter is concentrated in a thin frictional layer on the eastern edge of the Stream.