PV   inversion is applied in the frame of a shallow-water model with few degrees of freedom but supporting Rossby and gravity waves. Its complete formulation may be found in Lorenz (1980). Intermediate solutions between quasi-geostrophy and the fully non-linear formulation have been exhibited by Gent and McWilliams (1982) for the same model.

THE LORENZ MODEL

A homogeneous fluid evolves over an infinite uniformly rotating plane domain over a topography h. The height of the free surface is H+z. The flow is forced by a height — i.e. mass — source F and is damped diffusively. The model is severely truncated to nine independent components (three wave numbers times three variables). To do this three horizontal wave numbers k_1,2,3 are introduced. Three orthogonal functions f_i = cos (k_i . r / L) are also introduced where L is a horizontal length scale and r is a vector related to the horizontal position. The streamfunction ψ , the velocity potential χ and the height h are projected onto the three modes leading to nine independent components x_1,2,3, y_1,2,3 and z_1,2,3 where :

ψ =2L² f ( y₁ f₁ + y₂ f₂ + y₃ f₃)

χ =2L² f( x₁ f₁ + x₂ f₂ + x₃ f₃)

z=2L² f² g^-1 ( z₁ f₁ + z₂ f₂ + z₃ f₃)

h=2L² f² g^-1 ( h₁ f₁ + h₂ f₂ + h₃ f₃)

F=2L² f³ g^-1 ( F₁ f₁ + F₂ f₂ + F₃ f₃)

The parameters chosen in the numerical experiment are those of Lorenz : f^-1=3h is the unit of time, H=8km, L=1080 km, g=10 ms^-2 and the topography (h₁,h₂,h₃)=(-0.1,0,0) is only in the direction of f₁. The forcing (F₁,F₂,F₃)=(Ro,0,0) with Ro the Rossby number defines the flow regime of the experiment.

PV   INVERSION

The model is run over for 150 days so that the solution converges towards the attractor free from spurious gravity waves activity induced by any imbalance present in the initial conditions. Then we choose one particular solution defined by its nine components. Potential vorticity inversion is applied to rebuild the nine components of the model knowing only the three components of the potential vorticity. The figure below represents y₃ versus z₃ (say the component associated to the smallest scale represented in the model of the vorticity versus the same component of the height) in the vicinity of the chosen state of the atmosphere. By definition the state of the atmosphere lies within the subspace of possible solutions of the model (the black solid line) having in mind the fuzziness of the slow quasi-manifold concept (Ford et al., 2000). The dashed line quite parallel to the trajectory of the model represents the subspace of geostrophic solutions. The distance between the solid curve and the dashed one may be interpreted as the ageostrophy of the flow. The long-dashed curve represents the solutions defined by a given potential vorticity.

Then, in this context inverting the potential vorticity consists in seeking the intersection between a first curve defining the potential vorticity and a second one associated to the chosen balance assumption. Note that the exact solution is the black dot that corresponds to the intersection between the model solutions and the curve defining the potential vorticity.

The model variables may also be expressed in terms of gravity and Rossby contributions. The pure Rossby or gravity motions are along the two vectors displayed on the figure. For instance an evolution of the fluid along Rossby modes is geostrophic. Particular attention must be paid to the orientation of the vector associated to the gravity waves. The angle between the curve with constant potential vorticity and the axis associated to pure gravity dynamics is near zero. It means that the potential vorticity field is not sensitive to gravity wave emission. Moreover, this feature enables good convergence of the algorithm as we will see later on.

The geostrophic inversion of P is the intersection of the two dashed curves (i.e. M₁ on the figure). The figure clearly demonstrates that if the model is run with this initial condition (the grey solid curve) then gravity waves of large amplitude are emitted due to the imbalance of the initial state. Appying the method pof Arbogast et al. (2008) reduces the imbalance of the solution by making the solution closer to the black dot. After a first dynamical initialization (M₁¹, both panels) the solution is closer to the subspace of balanced solutions free of spurious gravity waves but its potential vorticity is now different from the potential vorticity to be inverted. With this method the next iterate M₂ may be seen as a quasi-orthogonal projection of the initialized solution onto the subspace with fixed potential vorticity. The forecast is run using M₂ as initial condition and is associated to the black oscillating curve that shows little gravity wave emission compared with the run started with the geostrophic solution.

The quasi-orthogonality of the curve of constant potential vorticity with the model trajectory and the geostrophic subspace allows a rapid convergence. For high Rossby numbers regimes the trajectory of the model can make an angle very different from π/2 with the two other curves. Then the separation between motions along the subspace of model solutions and spurious gravity wave emission by time filtering would be less efficient and so would become the algorithm.