Flow of grounded abyssal ocean currents along zonally-varying topography on a rotating sphere

Swaters, Gordon E.

doi:10.1080/03091929.2012.751381

Cited by 6 publications

(26 citation statements)

References 23 publications

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“…To see this, observe that it follows from that

h false(\overset{a}{true} (y), y false) = 0 ⟹ h_{0} false(τ (\tilde{a} false(y false), y) false) = 0 ⟹ τ false(\overset{a}{true} (y), y false) = a,

that is, a grounding must correspond, of course, to a streamline, which when substituted into implies that

h_{B} false(\overset{a}{true} (y), y false) = h_{B 0} false(a false),

i.e., the grounding is located along the isobath

h_{B 0} false(a false)

. As a corollary, should the bottom topography be independent of y the cross‐slope location of the grounding is constant with respect to y (see also ).…”

Section: Nonlinear Steady Solutionsmentioning

confidence: 99%

“…Formally, Eq. is a small Rossby number limit of the shallow water equations for a differentially rotating fluid that permits finite‐amplitude dynamical deflections in the thickness, or height, of the “grounded” current located immediately above the underlying variable topography, which is overlain by an infinitely deep but dynamically passive fluid layer , i.e., the reduced gravity approximation .…”

Section: The Model Equationmentioning

confidence: 99%

“…The steady or time‐independent solutions of satisfy the quasi‐linear hyperbolic partial differential equation

false(1 + β y false) false(\partial_{x} h_{B} false) h_{y} - false[β h + (1 + β y) (\partial_{y} h_{B}) false] h_{x} = β false(\partial_{x} h_{B} false) h,

which can be solved exactly for completely arbitrary bottom topography

h_{B} false(x, y false)

using the method of characteristics (for details see ). If we suppose that along

y = y_{0}

that

h_{B} false(x, y_{0} false) = h_{B 0} false(x false)

and that

h false(x, y_{0} false) = h_{0} false(x false)

, then the nonlinear solution to can be written in the form

h false(x, y false) = \frac{1 + β y}{1 + β y_{0}} h_{0} false(τ false),

h_{B} false(x, y false) = \frac{β (y_{0} - y)}{1 + β y_{0}} h_{0} false(τ false) + h_{B 0} false(τ false),

where

τ = x

when

y = y_{0}

.…”

Section: Nonlinear Steady Solutionsmentioning

confidence: 99%

“…The shock will form at the first y ‐value equatorward of y 0 for which

false| h_{x} false| \to \infty

(but h remains bounded). However, Swaters has shown that if the meridional velocity

v (x, y_{0})

associated with the solutions and along the inflow poleward boundary, given by,

v false(x, y_{0} false) = \frac{h_{0}^{'} false(x false) + h_{B 0}^{'} false(x false)}{1 + β y_{0}},

is equatorward for all x along y 0 within the abyssal current, then no shock forms in the solution and in the β‐plane region equatorward of

y = y_{0}

. Of course, this applies only to a mid‐latitude β‐plane or on a sphere that does not extend to the equator.…”

Section: Nonlinear Steady Solutionsmentioning

confidence: 99%

“…In addition, because

h_{B} = h_{B} false(x false)

it necessarily follows that the location of the groundings are invariant with respect to y , that is, their cross‐slope location is fixed by the location of the groundings along

y = y_{0}

. Graphical depictions of these properties can be seen in the many images contained in .…”

Section: An Example With Upslope and Down Slope Groundingsmentioning

confidence: 99%

See 4 more Smart Citations

Hamiltonian Structure and a Variational Principle for Grounded Abyssal Flow on a Sloping Bottom in a Mid‐Latitude β‐Plane

Swaters

2018

Stud Appl Math

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Observations, numerical simulations, and theoretical scaling arguments suggest that in mid‐latitudes, away from the high‐latitude source regions and the equator, the meridional transport of abyssal water masses along a continental slope correspond to geostrophic flows that are gravity or density driven and topographically steered. These dynamics are examined using a nonlinear reduced‐gravity geostrophic model that describes grounded abyssal meridional flow over sloping topography that crosses the planetary vorticity gradient. It is shown that this model possesses a noncanonical Hamiltonian formulation. General nonlinear steady solutions to the model can be obtained for arbitrary bottom topography. These solutions correspond to nonparallel shear flows that flow across the planetary vorticity gradient. If the in‐flow current along the poleward boundary is strictly equatorward, then no shock can form in the solution in the mid‐latitude domain. It is also shown that the steady solutions satisfy the first‐order necessary conditions for an extremal to a suitably constrained potential energy functional. Sufficient conditions for the definiteness of the second variation of the constrained energy functional are examined. The theory is illustrated with a nonlinear steady solution corresponding to an abyssal flow with upslope and down slope groundings in the height field.

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“…To see this, observe that it follows from that

h false(\overset{a}{true} (y), y false) = 0 ⟹ h_{0} false(τ (\tilde{a} false(y false), y) false) = 0 ⟹ τ false(\overset{a}{true} (y), y false) = a,

that is, a grounding must correspond, of course, to a streamline, which when substituted into implies that

h_{B} false(\overset{a}{true} (y), y false) = h_{B 0} false(a false),

i.e., the grounding is located along the isobath

h_{B 0} false(a false)

. As a corollary, should the bottom topography be independent of y the cross‐slope location of the grounding is constant with respect to y (see also ).…”

Section: Nonlinear Steady Solutionsmentioning

confidence: 99%

Section: The Model Equationmentioning

confidence: 99%

“…The steady or time‐independent solutions of satisfy the quasi‐linear hyperbolic partial differential equation

false(1 + β y false) false(\partial_{x} h_{B} false) h_{y} - false[β h + (1 + β y) (\partial_{y} h_{B}) false] h_{x} = β false(\partial_{x} h_{B} false) h,

which can be solved exactly for completely arbitrary bottom topography

h_{B} false(x, y false)

using the method of characteristics (for details see ). If we suppose that along

y = y_{0}

that

h_{B} false(x, y_{0} false) = h_{B 0} false(x false)

and that

h false(x, y_{0} false) = h_{0} false(x false)

, then the nonlinear solution to can be written in the form

h false(x, y false) = \frac{1 + β y}{1 + β y_{0}} h_{0} false(τ false),

h_{B} false(x, y false) = \frac{β (y_{0} - y)}{1 + β y_{0}} h_{0} false(τ false) + h_{B 0} false(τ false),

where

τ = x

when

y = y_{0}

.…”

Section: Nonlinear Steady Solutionsmentioning

confidence: 99%

“…The shock will form at the first y ‐value equatorward of y 0 for which

false| h_{x} false| \to \infty

(but h remains bounded). However, Swaters has shown that if the meridional velocity

v (x, y_{0})

associated with the solutions and along the inflow poleward boundary, given by,

v false(x, y_{0} false) = \frac{h_{0}^{'} false(x false) + h_{B 0}^{'} false(x false)}{1 + β y_{0}},

is equatorward for all x along y 0 within the abyssal current, then no shock forms in the solution and in the β‐plane region equatorward of

y = y_{0}

. Of course, this applies only to a mid‐latitude β‐plane or on a sphere that does not extend to the equator.…”

Section: Nonlinear Steady Solutionsmentioning

confidence: 99%

“…In addition, because

h_{B} = h_{B} false(x false)

it necessarily follows that the location of the groundings are invariant with respect to y , that is, their cross‐slope location is fixed by the location of the groundings along

y = y_{0}

. Graphical depictions of these properties can be seen in the many images contained in .…”

Section: An Example With Upslope and Down Slope Groundingsmentioning

confidence: 99%

See 3 more Smart Citations

Hamiltonian Structure and a Variational Principle for Grounded Abyssal Flow on a Sloping Bottom in a Mid‐Latitude β‐Plane

Swaters

2018

Stud Appl Math

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Meridional dynamics of grounded abyssal water masses on a sloping bottom in a mid-latitude -plane

Swaters¹

2018

J. Fluid Mech.

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Observations, numerical simulations and theoretical scaling arguments suggest that in mid-latitudes, away from the source regions and the equator, the meridional transport of abyssal water masses along a continental slope corresponds to planetary geostrophic flows that are gravity- or density-driven and topographically steered. We investigate these dynamics using a nonlinear reduced-gravity model that can describe grounded abyssal meridional flow over sloping topography that crosses the planetary vorticity gradient. Exact nonlinear steady and time-dependent solutions are obtained. The general steady theory is illustrated for a non-parallel equatorward flow that possesses a single along-slope grounding along the upslope flank of the current (complementing previous work). Four specific nonlinear time-dependent solutions are described. Two initial-value problems are solved exactly. The first initial configuration corresponds to an equatorward abyssal flow that has no cross-slope shear in the along-slope velocity and possesses a single grounding along the upslope flank of the current. The nonlinear time-dependent evolution of this initial current into a non-parallel shear flow is described. The second initial condition corresponds to an isolated radially symmetric grounded abyssal pool or dome. The nonlinear time-dependent evolution of this abyssal dome, which propagates equatorward with unsteady along- and cross-slope velocities while deforming into an elliptically shaped abyssal dome with $\unicode[STIX]{x1D6FD}$-induced diminishing height, is described. Finally, the nonlinear time-dependent boundary-value problem can be solved exactly in which the in-flow boundary condition on the poleward boundary of the mid-latitude domain corresponds to a time-dependent abyssal current with both an upslope and downslope grounding. Two specific time-dependent boundary conditions are examined. The first corresponds to a time-limited surge in the equatorward volume transport in the abyssal current along the poleward boundary. The second configuration corresponds to the nonlinear evolution of a finite-amplitude downslope plume or loop that forms in the abyssal current that is reminiscent of those seen in baroclinic instability simulations.

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Cross-equatorial flow of grounded abyssal ocean currents

Kim

Swaters

Sutherland

2014

Geophysical & Astrophysical Fluid Dynamics

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Flow of grounded abyssal ocean currents along zonally-varying topography on a rotating sphere

Cited by 6 publications

References 23 publications

Hamiltonian Structure and a Variational Principle for Grounded Abyssal Flow on a Sloping Bottom in a Mid‐Latitude β‐Plane

Hamiltonian Structure and a Variational Principle for Grounded Abyssal Flow on a Sloping Bottom in a Mid‐Latitude β‐Plane

Meridional dynamics of grounded abyssal water masses on a sloping bottom in a mid-latitude -plane

Cross-equatorial flow of grounded abyssal ocean currents

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