The dynamics of rotating stratified flows are often described using linear Rossby and inertia–gravity waves (hereinafter called gravity waves), which have well-known properties. The total generation of energy of the linear gravity wave mode in the first case scales to leading order as the square of the Rossby number in agreement with independent estimates from laboratory experiments, although a part of the linear gravity wave mode is slaved to the Rossby wave mode without wavelike temporal behavior. Since they are generated by nonresonant interactions only, the energy transfers toward the respective linear wave mode with vanishing energy are small in both cases. The results are validated by a corresponding ensemble of numerical model simulations supporting the validity of the weak-interaction assumption necessary to derive the kinetic equation. The kinetic equation in both cases is numerically evaluated, for which energy is conserved within numerical precision. Two examples are considered: initially vanishing linear gravity wave energy in the presence of a fully developed Rossby wave field and the reversed case of initially vanishing linear Rossby wave energy in the presence of a realistic gravity wave field. Mixed triad wave–wave interactions between Rossby and gravity waves are analytically derived using the kinetic equation for models of different complexity. Also shown is the energy gain related to the slaved mode in the model ensemble (solid line and stars), the energy gain related to the remaining waves (dashed line and stars), and, where E ± denotes the energy related to the remaining waves generated in a model ensemble initialized with. Integrated wave energy gain after t = 10 in the model ensemble (solid line and dots) and predicted by the kinetic equation (dashed line and dots) as a function of Ro 2. (b) Integrated cumulative total energy gain by the balanced flow (blue), and the same quantity for absolute energy transfers within the gravity wave field (red). (28) as function of time and horizontal wavenumber k and integrated over vertical wavenumber m. (a) Nonresonant energy transfer ∂ t E 0 given by Eq. The white lines denote ω/ f = 2, 3, 4, 5, 10, and 20, and = 2 × 10 −11 m −1 s −1, where denotes the change of the Coriolis parameter with latitude in dimensional form. Wavenumbers and energies are shown in their dimensional form, corresponding to Ro =0.1 and Fr = 2 × 10 −3. (a),(b) Content spectra of ∂ t E + and (c) ∂ t E 0 by resonant interactions for t → ∞ using a Garrett–Munk-like spectrum for E + and initially E 0 = 0 for the vertically resolved model as a function of horizontal and vertical wavenumber.
WATER NO GRAVITY LONDON DISPERSIO SERIES
(c) Exemplary time series of h at a single point from the inverse Fourier transform of the slaved mode (solid) and the total wave mode (dashed). Also shown as a dashed line is the integrated energy gain related to the slaved wave mode as defined in the text. (b) Integrated wave energy gain in the model ensemble.
(a) Energy transfer toward (if positive) wave mode E ± in the model ensemble with 127 2 grid points. (c) The integrated cumulative energy gain for the balanced mode E 0 (blue line) and the wave mode E ± (red line). (24) with 127 2 grid points.Įnergy transfer (a) toward (if positive) balanced mode E 0 and (b) toward wave mode E ± by the mixed components in the first sum of Eq. The dotted, dashed, and dash–dotted lines correspond to a model with 127 2, 255 2, and 1023 2 grid points, respectively, and the solid lines to an evaluation of Eq. The solid, dashed, dotted, and dash–dotted lines in (b) correspond to calculations using 127 2, 255 2, 511 2 grid points, respectively.īalanced mode energy transfer in a model ensemble with 1000 members at three different times initialized and diagnosed as described in the text, and predicted by the kinetic equation. The energy follows a power law of k −3 for large k. Here, k denotes wavenumber modulus and ϕ is wavenumber angle, with k = k(cos ϕ, sin ϕ). (a) Energy distribution of the balanced mode and (b) the corresponding energy transfer within the balanced mode given by Eq.
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