An evaluation of the performance of vertical mixing parameterizations for tidal mixing in the Regional Ocean Modeling System (ROMS)

Robertson, Robin; Dong, Changming

doi:10.1186/s40562-019-0146-y

Geoscience Letters

Official Journal of the Asia Oceania Geosciences Society (AOGS)

Table 1 The dispersion relations and group speeds for a variety of barotropic and baroclinic waves

From: An evaluation of the performance of vertical mixing parameterizations for tidal mixing in the Regional Ocean Modeling System (ROMS)

Wave type	Dispersion	Horizontal group speed
Two-layer interfacial wave: ∞ over ∞	$\omega = \sqrt {\frac{{(\mathop \rho \nolimits_{2} - \mathop \rho \nolimits_{1} )}}{{(\mathop \rho \nolimits_{1} + \mathop \rho \nolimits_{2} )}}gk}$	$\mathop c\nolimits_{\text{g}} = \frac{1}{2}\sqrt {\frac{{(\mathop \rho \nolimits_{2} - \mathop \rho \nolimits_{1} )}}{{(\mathop \rho \nolimits_{1} + \mathop \rho \nolimits_{2} )}}\frac{g}{k}}$
Two-layer interfacial wave: finite over ∞: barotropic	$\omega = \sqrt {gk}$	$\mathop c\nolimits_{\text{g}} = \frac{1}{2}\sqrt {\frac{g}{k}}$
Two-layer interfacial wave: finite over ∞: baroclinic shallow water	$\omega = \sqrt {g^{\prime}H} k$ $g^{\prime} = g\left( {\frac{{\mathop \rho \nolimits_{2} - \mathop \rho \nolimits_{1} }}{{\mathop \rho \nolimits_{2} }}} \right)$	$\mathop c\nolimits_{\text{g}} = \sqrt {g^{\prime}H}$
Continuous stratification without rotation	$\omega = N\sqrt {\frac{{\mathop k\nolimits^{2} + \mathop l\nolimits^{2} }}{{\mathop k\nolimits^{2} + \mathop l\nolimits^{2} + \mathop m\nolimits^{2} }}} = N\cos \theta$	$\mathop c\nolimits_{{{\text{g}}x}} = N\mathop m\nolimits^{2} \mathop {\left( {\mathop k\nolimits^{2} + \mathop m\nolimits^{2} } \right)}\nolimits^{ - 3/2}$
Continuous stratification with rotation	$\mathop {\mathop \omega \nolimits_{j} }\nolimits^{2} = \frac{{\mathop f\nolimits^{2} \mathop {\mathop m\nolimits_{j} }\nolimits^{2} + \mathop N\nolimits_{{}}^{2} \mathop k\nolimits^{2} }}{{\mathop {\mathop k\nolimits^{2} + m}\nolimits_{j}^{2} }}$	$\mathop c\nolimits_{{{\text{g}}j}} = \frac{{\mathop {\mathop k\nolimits_{{}} \mathop m\nolimits_{j}^{2} \left( {\mathop N\nolimits^{2} - \mathop f\nolimits^{2} } \right)}\nolimits^{{}} }}{{\mathop {\left( {\mathop f\nolimits^{2} \mathop m\nolimits_{j}^{2} + \mathop N\nolimits^{2} \mathop k\nolimits_{{}}^{2} } \right)}\nolimits^{1/2} \mathop {\left( {\mathop k\nolimits_{{}}^{2} + \mathop m\nolimits_{j}^{2} } \right)}\nolimits^{3/2} }}$
Barotropic Kelvin wave	$\omega = \sqrt {gH} k$	$\mathop c\nolimits_{{{\text{g}}x}} = \sqrt {gH}$
Baroclinic Kelvin wave	$\omega = \sqrt {g^{\prime}H} k$	$\mathop c\nolimits_{{{\text{g}}x}} = \sqrt {g^{\prime}H}$
Barotropic Poincaré wave	$\omega = \sqrt {\mathop f\nolimits^{2} + gH\left( {\mathop k\nolimits^{2} + \mathop l\nolimits^{2} } \right)}$	$\mathop c\nolimits_{{{\text{g}}x}} = \frac{gHk}{{\sqrt {\mathop f\nolimits^{2} + gH\mathop k\nolimits^{2} } }}$
Baroclinic Poincaré wave	$\omega = \sqrt {\mathop f\nolimits^{2} + \mathop c\nolimits^{2} \left( {\mathop k\nolimits^{2} + \mathop l\nolimits^{2} } \right)}$ $c = \sqrt {g^{\prime}H}$ or $c = {\raise0.7ex\hbox{$N$} \!\mathord{\left/ {\vphantom {N m}}\right.\kern-0pt} \!\lower0.7ex\hbox{$m$}} = {\raise0.7ex\hbox{${NH}$} \!\mathord{\left/ {\vphantom {{NH} {j\pi }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${j\pi }$}}$	$\mathop c\nolimits_{{{\text{g}}x}} = \frac{{\mathop c\nolimits^{2} k}}{{\sqrt {\mathop f\nolimits^{2} + \mathop c\nolimits^{2} \mathop k\nolimits^{2} } }}$
Shallow water surface gravity wave	$\omega = \sqrt {gH} k$	$\mathop c\nolimits_{{{\text{g}}x}} = \sqrt {gH}$
Deep water surface gravity wave	$\omega = \sqrt {gk}$	$\mathop c\nolimits_{\text{g}} = \frac{1}{2}\sqrt {\frac{g}{k}}$

ω is the frequency, k and l are the horizontal wavenumbers (l = 0, wave propagation aligned in x-direction), m is the vertical wave number, and c_g the horizontal group speed. The water depth is H and the upper layer thickness is H_U. Gravity is g and the reduced gravity is gʹ. ρ₁ and ρ₂ are the densities of the upper and lower layers, respectively. N is the Brunt–Väisäla frequency and f is the inertial frequency for the latitude. The mode number is represented by j. The equations were taken from Kundu et al. (2015)

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Wave type	Dispersion	Horizontal group speed
Two-layer interfacial wave: ∞ over ∞	\(\omega = \sqrt {\frac{{(\mathop \rho \nolimits_{2} - \mathop \rho \nolimits_{1} )}}{{(\mathop \rho \nolimits_{1} + \mathop \rho \nolimits_{2} )}}gk}\)	\(\mathop c\nolimits_{\text{g}} = \frac{1}{2}\sqrt {\frac{{(\mathop \rho \nolimits_{2} - \mathop \rho \nolimits_{1} )}}{{(\mathop \rho \nolimits_{1} + \mathop \rho \nolimits_{2} )}}\frac{g}{k}}\)
Two-layer interfacial wave: finite over ∞: barotropic	\(\omega = \sqrt {gk}\)	\(\mathop c\nolimits_{\text{g}} = \frac{1}{2}\sqrt {\frac{g}{k}}\)
Two-layer interfacial wave: finite over ∞: baroclinic shallow water	\(\omega = \sqrt {g^{\prime}H} k\) \(g^{\prime} = g\left( {\frac{{\mathop \rho \nolimits_{2} - \mathop \rho \nolimits_{1} }}{{\mathop \rho \nolimits_{2} }}} \right)\)	\(\mathop c\nolimits_{\text{g}} = \sqrt {g^{\prime}H}\)
Continuous stratification without rotation	\(\omega = N\sqrt {\frac{{\mathop k\nolimits^{2} + \mathop l\nolimits^{2} }}{{\mathop k\nolimits^{2} + \mathop l\nolimits^{2} + \mathop m\nolimits^{2} }}} = N\cos \theta\)	\(\mathop c\nolimits_{{{\text{g}}x}} = N\mathop m\nolimits^{2} \mathop {\left( {\mathop k\nolimits^{2} + \mathop m\nolimits^{2} } \right)}\nolimits^{ - 3/2}\)
Continuous stratification with rotation	\(\mathop {\mathop \omega \nolimits_{j} }\nolimits^{2} = \frac{{\mathop f\nolimits^{2} \mathop {\mathop m\nolimits_{j} }\nolimits^{2} + \mathop N\nolimits_{{}}^{2} \mathop k\nolimits^{2} }}{{\mathop {\mathop k\nolimits^{2} + m}\nolimits_{j}^{2} }}\)	\(\mathop c\nolimits_{{{\text{g}}j}} = \frac{{\mathop {\mathop k\nolimits_{{}} \mathop m\nolimits_{j}^{2} \left( {\mathop N\nolimits^{2} - \mathop f\nolimits^{2} } \right)}\nolimits^{{}} }}{{\mathop {\left( {\mathop f\nolimits^{2} \mathop m\nolimits_{j}^{2} + \mathop N\nolimits^{2} \mathop k\nolimits_{{}}^{2} } \right)}\nolimits^{1/2} \mathop {\left( {\mathop k\nolimits_{{}}^{2} + \mathop m\nolimits_{j}^{2} } \right)}\nolimits^{3/2} }}\)
Barotropic Kelvin wave	\(\omega = \sqrt {gH} k\)	\(\mathop c\nolimits_{{{\text{g}}x}} = \sqrt {gH}\)
Baroclinic Kelvin wave	\(\omega = \sqrt {g^{\prime}H} k\)	\(\mathop c\nolimits_{{{\text{g}}x}} = \sqrt {g^{\prime}H}\)
Barotropic Poincaré wave	\(\omega = \sqrt {\mathop f\nolimits^{2} + gH\left( {\mathop k\nolimits^{2} + \mathop l\nolimits^{2} } \right)}\)	\(\mathop c\nolimits_{{{\text{g}}x}} = \frac{gHk}{{\sqrt {\mathop f\nolimits^{2} + gH\mathop k\nolimits^{2} } }}\)
Baroclinic Poincaré wave	\(\omega = \sqrt {\mathop f\nolimits^{2} + \mathop c\nolimits^{2} \left( {\mathop k\nolimits^{2} + \mathop l\nolimits^{2} } \right)}\) \(c = \sqrt {g^{\prime}H}\) or \(c = {\raise0.7ex\hbox{$N$} \!\mathord{\left/ {\vphantom {N m}}\right.\kern-0pt} \!\lower0.7ex\hbox{$m$}} = {\raise0.7ex\hbox{${NH}$} \!\mathord{\left/ {\vphantom {{NH} {j\pi }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${j\pi }$}}\)	\(\mathop c\nolimits_{{{\text{g}}x}} = \frac{{\mathop c\nolimits^{2} k}}{{\sqrt {\mathop f\nolimits^{2} + \mathop c\nolimits^{2} \mathop k\nolimits^{2} } }}\)
Shallow water surface gravity wave	\(\omega = \sqrt {gH} k\)	\(\mathop c\nolimits_{{{\text{g}}x}} = \sqrt {gH}\)
Deep water surface gravity wave	\(\omega = \sqrt {gk}\)	\(\mathop c\nolimits_{\text{g}} = \frac{1}{2}\sqrt {\frac{g}{k}}\)