Shallow-water potential vorticity ( $q = (f + ζ) / h$ )

is a saddle. At equality the two meet at the rim. This is the onset criterion; once closed contours form, the potential vorticity of trapped fluid is not determined solely by its present upstream conditions.

Zero-potential-vorticity rotating channel flow ( $u = u_{1} + f y, h = h_{1} - f (u_{1} y + f y^{2} /2) / g$ )

 0  0

For a steady, slowly varying rotating channel, negligible transverse velocity and zero shallow-water potential vorticity give

u_{y} = f

. Cross-channel geostrophic balance gives

g h_{y} = - f u

, producing the displayed profiles. The specific energy

E = h + u^{2} / (2 g)

is constant across the channel, and

E + H

is constant along it. Zero potential vorticity is not the value in an ordinary finite-depth reservoir at rest in the rotating frame; it requires vanishing absolute vorticity or a deep-reservoir approximation.

Hydraulic control of a zero-PV rotating channel ( $Q_{c} = \frac{b}{g} [g h_{1} + (f b)^{2} /16 - f b /4]^{3}$ )

 0  0

At a hydraulic control, the specific-energy function of a zero-potential-vorticity rotating channel flow is stationary with respect to wall depth at fixed discharge and width. Writing

v = u_{1} + f b /2

gives

v (v + f b /2) = g h_{1}

and

Q_{c} = b v^{3} / g

, hence the displayed discharge. The fully wetted positive-flow solution requires

f b < 2 g h_{1}

; at equality the other wall dries and the first wall velocity vanishes. Beyond that limit the assumed two-wet-wall profile must be replaced.

Linearized shallow-water potential-vorticity anomaly ( $q = ζ - f η / H$ )

 0  0

For a constant-depth resting layer on an f-plane, the depth-scaled linear potential vorticity anomaly is

q = ζ - \frac{f}{H} η, ζ = v_{x} - u_{y} .

(1)

It satisfies

q_{t} = 0

under the linearized shallow water equations. The physical perturbation of

(f + ζ) / (H + η)

q / H

. The signed scalar

q

, not its absolute value, is used in potential-vorticity inversion and balanced energy pairings.

Shallow-water quasi-geostrophic potential vorticity ( $Q = \nabla^{2} ψ - ψ / L_{D}^{2} + β y$ )

 0  0

For small Rossby number and small free-surface displacement, shallow-water potential vorticity reduces, up to a constant factor and additive constant, to relative vorticity plus background PV gradients and the stretching term

- ψ / L_{D}^{2}

Cosine-ridge shallow-water geostrophic response ( $ϕ^{''} - L_{R}^{- 2} ϕ = - f b / h_{0}$ )

 0  0

For a uniform upstream current whose free-surface and bottom slopes compensate, steady quasi-geostrophic approximation flow with uniform incoming potential vorticity obeys the displayed inversion equation. Let

b = ϵ cos (x / L)

∣ x ∣ < a = π L /2

and zero outside. With

C = (f ϵ / h_{0}) / (L^{- 2} + L_{R}^{- 2})

, the decaying even response is

ϕ = C [cos (x / L) + (L_{R} / L) e^{- a / L_{R}} cosh (x / L_{R})]

inside and

ϕ = C (L_{R} / L) e^{- a / L_{R}} cosh (a / L_{R}) e^{- (∣ x ∣ - a) / L_{R}}

outside. Continuity of

ϕ

and

ϕ^{'}

fixes the coefficients. Its streamlines satisfy

y = y_{\infty} + ϕ (x) / U

, so the response extends beyond the compact ridge through exponentially decaying balanced tails.

Uniform-current obstruction to constant shallow-water QG potential vorticity ( $\overline{Q} = f + U y / L_{R}^{2}$ )

 0  0

On a constant f-plane with finite Rossby deformation radius, a geostrophic uniform current

(U, 0)

has streamfunction

- U y

and a sloping free surface. The full shallow-water quasi-geostrophic potential vorticity is

Q = f + \nabla^{2} Ψ - Ψ/ L_{R}^{2} + f b / H

, so a flat-bottom background has the displayed nonconstant value. Omitting its gradient is not a harmless streamfunction gauge choice. A uniform-current, constant-PV inversion must explicitly supply a compensating background, external balance, or a rigid-lid limit.

Gaussian topographic response with compensated uniform potential vorticity ( $ψ^{'} = f ϵ a^{2} e^{- r^{2} / a^{2}} / H$ )

 0  0

In a shallow-water potential-vorticity inversion with the background gradient explicitly compensated, uniform PV gives

(\nabla^{2} - L_{R}^{- 2}) ψ^{'} = - f b / H

. The displayed Gaussian function solves it for

b = ϵ [4 (1 - r^{2} / a^{2}) + a^{2} / L_{R}^{2}] e^{- r^{2} / a^{2}}

. This is not, without that compensation, a steady solution of the standard finite-depth constant-f current problem: its residual is

U ψ_{x}^{'} / L_{R}^{2}

. Closed streamlines can occur while Rossby number and surface displacement remain small.

Closed-streamline threshold for a Gaussian disturbance in uniform flow ( $∣ A ∣/ (∣ U ∣ a) = e /2$ )

 0  0

For total streamfunction

Ψ = - U y + A e^{- (x^{2} + y^{2}) / a^{2}}

, the maximum opposing zonal disturbance speed is

2/ e ∣ A ∣/ a

. Below the displayed threshold the zonal velocity never changes sign and streamlines cannot close. Above it two stagnation points appear on the symmetry axis: one has definite Hessian and is a center, the other has indefinite Hessian and is a saddle. Closed contours surround the center. Equality is a degenerate onset, not a finite-area closed-cell regime.

Potential-vorticity mixing over a finite bottom slope ( $Δ Q = - S y 1_{{∣ y ∣ < a}}, S = f ϵ / H$ )

 0  0

Homogenizing the potential vorticity over a finite constant bottom slope creates the odd anomaly displayed here while leaving the exterior unchanged. Inverting

(\partial_{y}^{2} - L_{R}^{- 2}) ψ = Δ Q

with decay at both infinities and matching

ψ, ψ_{y}

gives a central return current and flanking jets. The deformation radius sets the smoothing length. For

L_{R} ≪ a

, the central surface follows the bottom slope to leading order, but edge corrections and decaying exterior flow are essential.

Potential-vorticity step edge wave ( $c = U_{0} - Δ Q / (2 k^{2} + L_{R}^{- 2})$ )

 0  0

A positive jump

Δ Q

of shallow-water quasi-geostrophic potential vorticity supports a current

U = U_{0} e^{- ∣ y ∣/ L_{R}}

with

U_{0} = L_{R} Δ Q /2

. A disturbance has structure

ϕ = A e^{- k^{2} + L_{R}^{- 2} ∣ y ∣}

. Integrating the linear potential-vorticity conservation equation across the interface gives

(U_{0} - c) [ϕ_{y}] + Δ Qϕ (0) = 0

, hence the displayed dispersion relation. The wave propagates intrinsically against the mean current because the displaced PV interface induces the velocity that advects it; no planetary beta is required.

Barotropic deformation radius ( $L_{D} = g H / f$ )

 0  0

The barotropic deformation radius is the horizontal scale at which free-surface gravity and rotation balance. It contributes

L_{D}^{- 2}

to the denominator of the one-layer Rossby-wave dispersion relation.

Shallow-water potential vorticity ( $q = (f + ζ) / h$ )

Circular topographic anticyclone ( $q = - f ϵ 1_{r < a}$ )

Closed-streamline threshold for a circular topographic anticyclone ( $f ϵ a / (2 U) = 1$ )

Zero-potential-vorticity rotating channel flow ( $u = u_{1} + f y, h = h_{1} - f (u_{1} y + f y^{2} /2) / g$ )

Hydraulic control of a zero-PV rotating channel ( $Q_{c} = \frac{b}{g} [g h_{1} + (f b)^{2} /16 - f b /4]^{3}$ )

Linearized shallow-water potential-vorticity anomaly ( $q = ζ - f η / H$ )

Shallow-water quasi-geostrophic potential vorticity ( $Q = \nabla^{2} ψ - ψ / L_{D}^{2} + β y$ )

Cosine-ridge shallow-water geostrophic response ( $ϕ^{''} - L_{R}^{- 2} ϕ = - f b / h_{0}$ )

Uniform-current obstruction to constant shallow-water QG potential vorticity ( $\overline{Q} = f + U y / L_{R}^{2}$ )

Gaussian topographic response with compensated uniform potential vorticity ( $ψ^{'} = f ϵ a^{2} e^{- r^{2} / a^{2}} / H$ )

Closed-streamline threshold for a Gaussian disturbance in uniform flow ( $∣ A ∣/ (∣ U ∣ a) = e /2$ )

Potential-vorticity mixing over a finite bottom slope ( $Δ Q = - S y 1_{{∣ y ∣ < a}}, S = f ϵ / H$ )

Potential-vorticity step edge wave ( $c = U_{0} - Δ Q / (2 k^{2} + L_{R}^{- 2})$ )

Barotropic deformation radius ( $L_{D} = g H / f$ )

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