This is the fourth of a series of papers on topological phase transitions (TPTs), including

- quantum phase transitions,
- formation of galactic spiral structure,
- sunspots and solar eruptions, and
- interior separation of fluid flows.

**1.** A TPT refers to the change in the topological structure in the physical space of the solutions of the governing partial differential equation (PDE) models of the underlying physical problem.

Boundary-layer separation phenomenon is one of the most important processes in fluid flows, and there is a long history of studies which go back at least, if not earlier, to the pioneering work of L. Prandtl in 1904. Basically, in the boundary-layer, the shear flow can detach/separate from the boundary, generating vortices and leading to more complicated turbulent behavior. The fundamental level understanding of this challenge problem boils down to TPTs of fluid flows. Mathematically, the velocity field of the fluid satisfies the Navier-Stokes equations (NSEs) or their variations, and defines its own topological structure in the physical space , where is the physical domain that the fluid occupies. Then the TPT associated with the boundary-layer or interior separations studies transition of the topological structure of as the system control parameter varies.

**2.** It is clear then the geometric theory of incompressible flows developed by the authors plays a crucial role for the study of TPTs of fluids; and the complete account of this geometric theory is given in the authors’ research monograph

One component of this geometric theory is the necessary and sufficient conditions for structural stability of divergence-free vector fields. Another component of the theory crucial for the study in this paper is the theorems on structural bifurcations. These theorems form the kinematic theory for understanding the topological phase transitions associated with fluid flows.

**3.** The most difficult and important aspect of TPTs associated with fluid flows is to make connections between the solutions of the NSEs and their structure in the physical space. The first such connection is the *separation equation* [MW05, Theorem 5.4.1] for the NSEs with the rigid boundary condition:

In this paper, we derive the following separation equation for the NSE with the free-slip boundary condition:

where and are the divergence-free parts of the external forcing and the nonlinear term as defined:

The separation equations (1) and (2) provide necessary and sufficient conditions for the flow separation at a boundary point:

*the complete information for boundary-layer separation is encoded in the separation equations (1) and (2), which are therefore crucial for all applications; and**the separation equations (1) and (2) link precisely the separation point , the external forcing and the initial velocity field .*

By exploring the leading order terms of the forcing and the initial velocity field (Taylor expansions), more detailed condition, called *predicable* condition, are derived in [3, 14] for the Dirichlet boundary condition case, and in this paper for the free boundary condition case.

The separation equations (1) and (2), as well as the predicable conditions determine precisely when, where, and how a boundary-layer separation occurs.

**4.** For example, using the separation equations (1) and (2), we derive

*precise criteria on critical curvature for generating vortices from boundary tip points,**critical velocity formula for surface turbulence; and**the mechanism of the formation of the subpolar gyre and the formation of the small scale wind-driven vortex oceanic flows, in the north Atlantic ocean.*

In particular, for the wind-driven north Atlantic circulations, with careful analysis using the separation equations (1) and (2), we derive the following conclusions:

*If the mid-latitude seasonal wind strength exceeds certain threshold ,vortices near the north Atlantic coast will form. Moreover, the scale (radius) of the vortices is an increasing function of ;**the condition for the initial formation of the subpolar gyre is that the curvature of at the tip point on the east coast of Canada is sufficiently large, and the combined effect of the convexity of the tangential component of the Gulf stream shear flow and the strength of the tangential friction force is positive; and**the vortex separated from the boundary tip point is then amplified and maintained by the wind stress, the strong Gulf stream current and the Coriolis effect, leading to the big subpolar gyre that we observe.*

*Tian Ma, Shouhong Wang*

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