Approximate Dispersion Relations for Waves on Arbitrary Shear Flows

Ellingsen, Simen Å.; Li, Yan

doi:10.1002/2017jc012994

Cited by 46 publications

(67 citation statements)

References 38 publications

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“…The most used of these was first presented by Skop () generalizing Stewart and Joy () and was developed to second‐order accuracy by Kirby and Chen (). This relation (generalized to the 3‐D case of a turning U ( z )) we call the 3DKC and to leading order may be written (see; Ellingsen & Li, )

\overset{c}{true} false(boldk false) \approx c_{0} false(1 - δ false); δ = \int_{- h}^{0} normald z \frac{boldk \cdot U^{'} false(z false) \sinh 2 k false(z + h false)}{k c_{0} \sinh 2 k h} .

…”

Section: Existing Approaches For Constant Hmentioning

confidence: 99%

“…Here and for later reference, we define

{boldU}_{0} = U (0), U_{0}^{'} = {boldU}^{'} (0), \tilde{c} (k) = c (k) - k \cdot {boldU}_{0} / k, Δ U = U - {boldU}_{0}, w_{0} = w (k, 0)

and

c_{0} = \sqrt{false(g false/ k + T k false/ ρ false) \tanh k h}

. We recently proposed an alternative approximation

\overset{c}{true} false(boldk false) \approx c_{0} false(\sqrt{δ^{2} + 1} - δ false),

which has certain advantages (Ellingsen & Li, ). Both approximations come with a second‐order accurate extension providing excellent accuracy at far greater cost.…”

Section: Existing Approaches For Constant Hmentioning

confidence: 99%

“…Here u and v are amplitudes of the horizontal velocities, p ( z ) is amplitude of the dynamic pressure, and ζ is the surface elevation, all in k space. The boundary value problem (3) may then be written following Ellingsen and Li (),

{\overset{w}{true}}^{'^{'}} - k^{2} \overset{w}{true} = \frac{boldk \cdot U^{''}}{boldk \cdot normalΔ boldU - k \overset{c}{true}} \overset{w}{true}; - h < z < η^{false(0 false)}; \overset{w}{true} false(- h false) = 0; \overset{w}{true} false(η^{false(0 false)} false) = 1 .

D_{R} false(boldk, \overset{c}{true} false) = 0,

where

D_{R} false(\overset{c}{true} false) \equiv {\overset{c}{true}}^{2} + \overset{c}{true} I_{c} false(\overset{c}{true} false) - c_{0 \overset{h}{true}}^{2},

I_{c} false(\overset{c}{true} false) = \frac{boldk \cdot U_{0}^{'} \tanh k \overset{h}{true}}{k^{2}} + \overset{c}{true} true \int_{- h}^{η^{(0)}} normald z \frac{boldk \cdot U^{''} \overset{w}{true} false(boldk, z false) \sinh k false(z + h false)}{k false(boldk \cdot normalΔ boldU - k \overset{c}{true} false) \cosh k \overset{h}{true}},

c_{0 <...>}

…”

Section: Dimmentioning

confidence: 99%

“…In principle, this returns the next order approximation of

{\overset{c}{true}}_{\approx}

\overset{c}{true}

and the accurate

\overset{w}{true}

of the input

{\overset{c}{true}}_{\approx} false(boldk false)

. Compared to the second‐order corrections to (Kirby & Chen, ) and (Ellingsen & Li, ), offers much faster computation.…”

Section: Dimmentioning

confidence: 99%

“…Using the resulting integral, the real part

\overset{c}{true}

, pertaining to wave propagation, is kept in the limit ϵ →0 + . (The imaginary part has a bearing on the stability of the critical layer; see, e.g., discussion in Ellingsen & Li, ; Shrira, ; Velthuizen & van Wijngaarden, ).…”

Section: Dimmentioning

confidence: 99%

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A Framework for Modeling Linear Surface Waves on Shear Currents in Slowly Varying Waters

Ellingsen

2019

JGR Oceans

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We present a theoretical and numerical framework—which we dub the direct integration method (DIM)—for simple, efficient, and accurate evaluation of surface wave models allowing presence of a current of arbitrary depth dependence and where bathymetry and ambient currents may vary slowly in horizontal directions. On horizontally constant water depth and shear current the DIM numerically evaluates the dispersion relation of linear surface waves to arbitrary accuracy, and we argue that for this purpose it is superior to two existing numerical procedures: the piecewise‐linear approximation and a method due to Dong and Kirby (2012, https://doi.org/10.9753/icce.v33.waves.2). The DIM moreover yields the full linearized flow field at little extra cost. We implement the DIM numerically with iterations of standard numerical methods. The wide applicability of the DIM in an oceanographic setting in four aspects is shown. First, we show how the DIM allows practical implementation of the wave action conservation equation recently derived by Quinn et al. (2017, https://doi.org/10.1016/j.ocemod.2017.03.003). Second, we demonstrate how the DIM handles with ease cases where existing methods struggle, that is, velocity profiles U(z) changing direction with vertical coordinate z and strongly sheared profiles. Third, we use the DIM to calculate and analyze the full linear flow field beneath a 2‐D ring wave upon a near‐surface wind‐driven exponential shear current, revealing striking qualitative differences compared to no shear. Finally we demonstrate that the DIM can be a real competitor to analytical dispersion relation approximations such as that of Kirby and Chen (1989, https://doi.org/10.1029/JC094iC01p01013) even for wave/ocean modeling.

show abstract

\overset{c}{true} false(boldk false) \approx c_{0} false(1 - δ false); δ = \int_{- h}^{0} normald z \frac{boldk \cdot U^{'} false(z false) \sinh 2 k false(z + h false)}{k c_{0} \sinh 2 k h} .

…”

Section: Existing Approaches For Constant Hmentioning

confidence: 99%

“…Here and for later reference, we define

{boldU}_{0} = U (0), U_{0}^{'} = {boldU}^{'} (0), \tilde{c} (k) = c (k) - k \cdot {boldU}_{0} / k, Δ U = U - {boldU}_{0}, w_{0} = w (k, 0)

and

c_{0} = \sqrt{false(g false/ k + T k false/ ρ false) \tanh k h}

. We recently proposed an alternative approximation

\overset{c}{true} false(boldk false) \approx c_{0} false(\sqrt{δ^{2} + 1} - δ false),

which has certain advantages (Ellingsen & Li, ). Both approximations come with a second‐order accurate extension providing excellent accuracy at far greater cost.…”

Section: Existing Approaches For Constant Hmentioning

confidence: 99%

{\overset{w}{true}}^{'^{'}} - k^{2} \overset{w}{true} = \frac{boldk \cdot U^{''}}{boldk \cdot normalΔ boldU - k \overset{c}{true}} \overset{w}{true}; - h < z < η^{false(0 false)}; \overset{w}{true} false(- h false) = 0; \overset{w}{true} false(η^{false(0 false)} false) = 1 .

D_{R} false(boldk, \overset{c}{true} false) = 0,

where

D_{R} false(\overset{c}{true} false) \equiv {\overset{c}{true}}^{2} + \overset{c}{true} I_{c} false(\overset{c}{true} false) - c_{0 \overset{h}{true}}^{2},

I_{c} false(\overset{c}{true} false) = \frac{boldk \cdot U_{0}^{'} \tanh k \overset{h}{true}}{k^{2}} + \overset{c}{true} true \int_{- h}^{η^{(0)}} normald z \frac{boldk \cdot U^{''} \overset{w}{true} false(boldk, z false) \sinh k false(z + h false)}{k false(boldk \cdot normalΔ boldU - k \overset{c}{true} false) \cosh k \overset{h}{true}},

c_{0 <...>}

…”

Section: Dimmentioning

confidence: 99%

“…In principle, this returns the next order approximation of

{\overset{c}{true}}_{\approx}

\overset{c}{true}

and the accurate

\overset{w}{true}

of the input

{\overset{c}{true}}_{\approx} false(boldk false)

. Compared to the second‐order corrections to (Kirby & Chen, ) and (Ellingsen & Li, ), offers much faster computation.…”

Section: Dimmentioning

confidence: 99%

“…Using the resulting integral, the real part

\overset{c}{true}

Section: Dimmentioning

confidence: 99%

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A Framework for Modeling Linear Surface Waves on Shear Currents in Slowly Varying Waters

Ellingsen

2019

JGR Oceans

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A novel method for water waves propagating in the presence of vortical mean flows over variable bathymetry

Touboul

Belibassakis

2019

J. Ocean Eng. Mar. Energy

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The Error in Predicted Phase Velocity of Surface Waves atop a Shear Current with Uncertainty

2019

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The effect of a depth-dependent shear current U (z) on surface wave dispersion is conventionally calculated by assuming U (z) to be an exactly known function, from which the resulting phase velocity c(k) is determined. This, however, is not the situation in reality. Field measurements of the current profile are performed at a finite number of discrete depths and with nonzero experimental uncertainty. Here we analyse how imperfect knowledge of U (z) affects estimates of c(k). We performed a numerical experiment simulating a large number of "measurements" of three different shear currents: an exponential profile, a 1/7-law profile, and a profile measured in the Columbia River delta. A number of measurement points were specified, the topmost of which at z = −h s (permitting simulation of measurement points which do not fully extend to the surface at z = 0), and measurements taken from a normal distribution with standard deviation ΔU. Four different methods of reconstructing a continuous U (z) from the measurements are compared with respect to mean value and variance of c(k). We find that an ordinary least-squares polynomial fit seems robust against mispredicting mean values at the expense of relatively high variance. Its performance is similar for all profiles, whereas a fit to an exponential form is excellent in one case and poor in another. A clear conclusion is the need for a measurement of the surface velocity U (0) when there is significant shear near the surface. For the exponential and Columbia profiles alike, errors due to extrapolation of U from z = −h s to 0 dominate the resulting error of c, especially for shorter wavelengths. In contrast, the error in c(k) decreases slowly with a higher density of measurement points, indicating that better, not more, velocity measurements should be invested in. A pseudospectral analysis of the linear operator corresponding to the three velocity profiles was performed. In all cases, the pseudospectrum shows strong asymmetry around the eigenvalue for c, indicating that a perturbation in the underlying current is more likely to push c to higher, not lower, values. This is in tentative agreement with our observation that for sufficiently large ΔU , c is found to have predominantly positive skewness, although the direct relationship between the two is not altogether obvious.

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Approximate Dispersion Relations for Waves on Arbitrary Shear Flows

Cited by 46 publications

References 38 publications

A Framework for Modeling Linear Surface Waves on Shear Currents in Slowly Varying Waters

A Framework for Modeling Linear Surface Waves on Shear Currents in Slowly Varying Waters

A novel method for water waves propagating in the presence of vortical mean flows over variable bathymetry

The Error in Predicted Phase Velocity of Surface Waves atop a Shear Current with Uncertainty

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