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

Swaters, Gordon E.

doi:10.1017/jfm.2018.17

Cited by 1 publication

(16 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

“…In standard notation , the nondimensional model is the nonlinear hyperbolic partial differential equation

h_{t} + \partial (h + h_{B}, \frac{h}{1 + β y}) = 0,

where the Jacobian

\partial false(A, B false) \equiv A_{x} B_{y} - A_{y} B_{x}

(subscripts denote partial differentiation unless otherwise denoted),

(x, y)

are the eastward or zonal and northward or meridional Cartesian coordinates, respectively,

h (x, y, t) \geq 0

is the thickness or height of the abyssal current above the bottom topography

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

(see Fig. ), and the coefficient

1 + β y

is the linearly varying Coriolis parameter associated with the β‐plane approximation where for typical length scales

β ≃ 0.02

(but cannot be neglected over the meridional basin length scales of physical relevance). Formally, Eq.…”

Section: The Model Equationmentioning

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%

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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…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%

“…In standard notation , the nondimensional model is the nonlinear hyperbolic partial differential equation

h_{t} + \partial (h + h_{B}, \frac{h}{1 + β y}) = 0,

where the Jacobian

\partial false(A, B false) \equiv A_{x} B_{y} - A_{y} B_{x}

(subscripts denote partial differentiation unless otherwise denoted),

(x, y)

are the eastward or zonal and northward or meridional Cartesian coordinates, respectively,

h (x, y, t) \geq 0

is the thickness or height of the abyssal current above the bottom topography

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

(see Fig. ), and the coefficient

1 + β y

is the linearly varying Coriolis parameter associated with the β‐plane approximation where for typical length scales

β ≃ 0.02

(but cannot be neglected over the meridional basin length scales of physical relevance). Formally, Eq.…”

Section: The Model Equationmentioning

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%

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

Self Cite

View full text Add to dashboard Cite

show abstract

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

Cited by 1 publication

References 31 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

Contact Info

Product

Resources

About