Vertical alignment of stagnation points in pseudo-plane ideal flows

Recent studies of pseudo-plane ideal flow (PIF) reveal a ubiquitous presence of vortex alignment in both homogeneous and stratified fluids, and in both inertial and rotating reference frames as well. The exact solutions of a steady-state PIF model suggest that stagnation points tend to be vertically aligned and the concentric structure represents a fixed-point phenomenon of the Euler equations. Exception occurs in the rotating frame when a flow holds inertial period and skew center becomes possible. Properties of stagnation points based on Morse theory are obtained, leading to a topological explanation of vertical alignment via pressure Hessian. The study thus uncovers a new aspect of vortex behavior in ideal fluid that requires vortex center to align with the direction of gravity when vortex evolution reaches a laminar end state characterized by steady pseudo-plane velocities. Though the phenomenon arises from the constraint of the Euler equations, under specific conditions the topological theory is applicable to viscous fluid and explains the curvilinear tilting of von K\'arm\'an swirling vortex.


I. Introduction
Diagnoses of atmospheric and oceanic flows have revealed a basic state called Gravest Empirical Mode (GEM) in which a scalar property, such as temperature, has invariant vertical profile along a streamfunction contour Watts 2001, Sun 2005). The temperature data from a baroclinic current fluctuate around the basic GEM profile with small rms residues. The low-dimensional structure motivated Sun (2008) to develop a steady pseudo-plane ideal flow (PIF) model based on the notion that geostrophic flows are quasi-horizontal and stratified turbulence tends to collapse into a laminar end state with negligible vertical velocity. Such pseudo-plane flows have vertically varying horizontal velocities but no vertical velocity (Saccomandi 1994).  McWilliams et al. (1999). Colors indicate positive and negative values of potential vorticity. Reproduced with permission from J. Fluid Mech. 401, 1-26 (1999). Copyright 1999 Cambridge University Press. Meanwhile numerical experiments show that geostrophic vortices in a freely evolving turbulence have an intriguing tendency toward vertical alignment (Polvani 1991, Nof and Dewar 1994, Viera 1995, McWilliams et al. 1999. The phenomenon, as depicted in Figure 1, is different from the equivalent-barotropic structure of the GEM field and represents a weak form of vertical coherence with wider presence in geophysical fluids. Compared with the alignment of vorticity and strain rate in small-scale turbulence (Ashurst et al. 1987), the macroscale vortex alignment is less well-known and a theoretical explanation of its physical mechanism remains elusive.
Previous experiments reveal that evolution to an aligned state involves transient processes such as filamentation and radiation of gravity waves, but do not explain why the system shall evolve to that state. The question has to be addressed by a theoretical analysis of the end state. Recent studies of exact solutions to the PIF model shed new light on this aspect: except for inertial cases, the steady PIF solutions display universal concentric formulation in both rotating and non-rotating fluids. That is, the horizontal position of stagnation point is fixed and vertical alignment of stagnation points occurs in elliptic, hyperbolic and multipolar strain flows (Sun 2016(Sun , 2017. The concentric structure has obvious connection with the vortex-alignment phenomenon, because stagnation point is the common feature of steady vortices and pseudo-plane velocity characterizes the laminar end state of vortex evolution. We intend to develop a topological theory for the concentric structure of the PIF solutions, and thereby provide a theoretical explanation for the vortex-alignment phenomenon. As it turns out, the phenomenon stems from the Euler equations and has omnipresence beyond geophysical flows. Sun (2008) derived the PIF model from the stratified-turbulence equations of Riley et al. (1981)  (1)

II. Steady pseudo-plane ideal flow
Here p is the pressure perturbation divided by a mean density 0 ρ , ρ is the density perturbation scaled by 0 / g ρ , and f is the Coriolis parameter in a reference frame rotating with constant angular velocity ( 0 f = for the inertial reference frame). The continuity equation (4) for incompressible fluid leads to a streamfunction ( , , ) x The overdetermination of the PIF model and its symmetry implication have been examined by Sun (2008).
In differential topology, Morse theory enables us to analyze the topology of a manifold by studying differentiable functions on that manifold (Morse 1925). The Hessian matrix as a square matrix of second-order partial derivatives plays an important role in Morse theory.
For pseudo-plane flows, the Hessian matrix deals with function ( , ) x y ψ of two variables and its determinant, called the Hessian, is written as The Morse lemma states that nondegenerate critical points are isolated.
The lemma is a sufficient condition but not a necessary one, because a degenerate critical point may also be isolated. A nondegenerate critical point in non-divergent flows is either a center or a saddle, depending on the sign of its Hessian. But a degenerate critical point can be any topologic type, including cusp (Brøns and Hartnack 1999).
In contrast to previous studies of flow topology that examined node, focus and saddle (Tobak and Peake 1982, Perry and Chong 1987, Aref and Brøns 1998, Heil et al. 2017, this study addresses vertical alignment of stagnation points in pseudo-plane flows and is not concerned with their exact topological type, as long as they are isolated. Examination of vertical alignment is only meaningful for isolated critical points.

III. Examples of exact solutions
We first provide some examples of exact solutions to the PIF model and examine their stagnation points. As listed in Table I, four solutions are rotating fluid and the other five are non-rotating fluid. The derivation of these solutions can be found in Sun (2016Sun ( , 2017.
and for hyperbolic flow Hessian, the origin is an isolated stagnation point and a pressure center.
The jet direction rotates vertically. The non-isolated stagnation points at each depth form a critical line x zy = .
S8. Quartic circular vortex The origin is the only stagnation point at each depth, and is an isolated degenerate critical point for both streamfunction and pressure.  and Marston (1989). At each depth the origin is the only critical point for streamfunction and pressure, and therefore is degenerate isolated.  Table   III according to their density fields and pressure critical points.

IV. Properties of stagnation point
We restrict our attention to those pseudo-plane flows with isolated stagnation points. Because viscosity is absent in the PIF model, pseudo-plane flows at different depths can only interact via pressure. The key step of geometric analysis, therefore, is to find the critical-point relation between streamfunction and pressure. There are three lemmas on this regard.
which means a pressure critical point has the same Hessian degeneracy as the stagnation point. If 0 Hψ ≠ , equality (7) requires 0 Hp > , meaning nondegenerate stagnation point in the inertial frame must correspond to a pressure extremum. The above lemma is thus proved. rotating frame Together the three lemmas describe critical-point relation between different variables. The Hessian relation between streamfunction and pressure, as summarized in Table IV, enables us to study the vertical coherence of stagnation points. We begin with homogeneous fluid and use a phenomenological approach to prove the following theorem. and has a depth-independent pressure perturbation field.
Projection of the tilting line of pressure critical points on the depth-independent pressure field results in a horizontal critical line, violating the assumption of isolated pressure critical points. Therefore the line of stagnation points can not tilt, and the above theorem is proved.
A flow without pressure perturbation, such as S1, is not subject to Theorem 1 and a tilting line of stagnation points is allowed. If the pressure critical points are not isolated, such as in S2, a skew stagnation center could occur. Theorem 1 requires concentric structure in a homogeneous flow with isolated pressure critical points, such as S6 and S9 (belong to Type I and Type II of Table III).
In the following we conduct a critical-point analysis of pseudo-plane quadratic flows and obtain three lemmas with analytical proofs.

Lemma 4. A stagnation point in quadratic flows is isolated if and only if it is nondegenerate.
Nondegeneracy is a sufficient condition for isolated critical point in the The above lemma means the isolated stagnation point of a quadratic flow can not be degenerate (see quadratic solutions S1-S7 in Table II for verification) and we only need to consider the nondegenerate type hereafter.

Appendix A. Tilting circular vortex in viscous fluid
For a viscous incompressible fluid between two parallel disks rotating around different axes at the same angular velocity, Berker (1982) obtained a steady pseudo-plane solution without vertical circulation, which can be regarded as an equilibrium state of the well-known von x y is straight. In that case Theorem 1 becomes applicable and requires the spatial line to be vertical, which is contradictory.